products.json

{
    "content": [
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct",
            "prop": "(nproduct::'a set ⇒ ('a ⇒ nat) ⇒ nat) = iterate times",
            "ty": "def",
            "type": "'a set ⇒ ('a ⇒ nat) ⇒ nat"
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "neutral_nat_mul",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "theI, {P: λx::nat. ∀y. x * y = y ∧ y * x = y, a: (1::nat)}",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ (∀y::nat. 1 * y = y ∧ y * 1 = y) ⟶ (∀x::nat. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1) ⟶ (∀y::nat. (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) * y = y ∧ y * (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) = y)"
                },
                {
                    "args": "∀y::nat. 1 * y = y ∧ y * 1 = y",
                    "id": "1",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ∀y::nat. 1 * y = y ∧ y * 1 = y"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [
                        "0",
                        "1"
                    ],
                    "rule": "apply_fact",
                    "th": "⊢ (∀x::nat. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1) ⟶ (∀y::nat. (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) * y = y ∧ y * (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) = y)"
                },
                {
                    "args": "∀x::nat. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1",
                    "id": "3",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ∀x::nat. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "2",
                        "3"
                    ],
                    "rule": "apply_fact",
                    "th": "⊢ ∀y::nat. (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) * y = y ∧ y * (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) = y"
                },
                {
                    "args": "THE x::nat. ∀y. x * y = y ∧ y * x = y",
                    "id": "5",
                    "prevs": [
                        "3",
                        "4"
                    ],
                    "rule": "apply_fact_for",
                    "th": "⊢ (THE x::nat. ∀y. x * y = y ∧ y * x = y) = 1"
                },
                {
                    "args": "neutral_def, neutral times = (1::nat)",
                    "id": "6",
                    "prevs": [
                        "5"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ neutral times = (1::nat)"
                },
                {
                    "args": "",
                    "id": "7",
                    "prevs": [
                        "6"
                    ],
                    "rule": "intros",
                    "th": "⊢ neutral times = (1::nat)"
                }
            ],
            "prop": "neutral times = (1::nat)",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_forward_step",
                    "param_P": "(%x::nat. ∀y. x * y = y ∧ y * x = y)",
                    "param_a": "(1::nat)",
                    "theorem": "theI"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal": "(∀y::nat. 1 * y = y ∧ y * 1 = y)",
                    "goal_id": "1",
                    "method_name": "cut"
                },
                {
                    "goal_id": "1",
                    "method_name": "z3"
                },
                {
                    "fact_ids": [
                        "0",
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "apply_fact"
                },
                {
                    "fact_ids": [
                        "2"
                    ],
                    "goal": "(∀x::nat. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1)",
                    "goal_id": "3",
                    "method_name": "cut"
                },
                {
                    "goal_id": "3",
                    "method_name": "z3"
                },
                {
                    "fact_ids": [
                        "2",
                        "3"
                    ],
                    "goal_id": "4",
                    "method_name": "apply_fact"
                },
                {
                    "fact_ids": [
                        "3"
                    ],
                    "goal_id": "5",
                    "method_name": "apply_prev"
                }
            ],
            "ty": "thm",
            "vars": {}
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "monoidal_nat_mul",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "",
                    "id": "0",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ ∀x::nat. ∀y. x * y = y * x"
                },
                {
                    "args": "x, nat",
                    "id": "0.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "y, nat",
                    "id": "0.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR y"
                },
                {
                    "args": "mult_comm, {x: x, y: y}",
                    "id": "0.2",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ x * y = y * x"
                },
                {
                    "args": "",
                    "id": "0.3",
                    "prevs": [
                        "0.0",
                        "0.1",
                        "0.2"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀x::nat. ∀y. x * y = y * x"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ ∀x::nat. ∀y. ∀z. x * (y * z) = x * y * z"
                },
                {
                    "args": "x, nat",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "y, nat",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR y"
                },
                {
                    "args": "z, nat",
                    "id": "1.2",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR z"
                },
                {
                    "args": "mult_assoc, x * (y * z) = x * y * z",
                    "id": "1.3",
                    "prevs": [],
                    "rule": "rewrite_goal",
                    "th": "⊢ x * (y * z) = x * y * z"
                },
                {
                    "args": "",
                    "id": "1.4",
                    "prevs": [
                        "1.0",
                        "1.1",
                        "1.2",
                        "1.3"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀x::nat. ∀y. ∀z. x * (y * z) = x * y * z"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ ∀x::nat. neutral times * x = x"
                },
                {
                    "args": "x, nat",
                    "id": "2.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "mult_1_left, 1 * x = x",
                    "id": "2.1",
                    "prevs": [],
                    "rule": "rewrite_goal",
                    "th": "⊢ 1 * x = x"
                },
                {
                    "args": "neutral_nat_mul, neutral times * x = x",
                    "id": "2.2",
                    "prevs": [
                        "2.1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ neutral times * x = x"
                },
                {
                    "args": "",
                    "id": "2.3",
                    "prevs": [
                        "2.0",
                        "2.2"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀x::nat. neutral times * x = x"
                },
                {
                    "args": "conjI",
                    "id": "3",
                    "prevs": [
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "⊢ (∀x::nat. ∀y. ∀z. x * (y * z) = x * y * z) ∧ (∀x::nat. neutral times * x = x)"
                },
                {
                    "args": "conjI",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "apply_theorem",
                    "th": "⊢ (∀x::nat. ∀y. x * y = y * x) ∧ (∀x::nat. ∀y. ∀z. x * (y * z) = x * y * z) ∧ (∀x::nat. neutral times * x = x)"
                },
                {
                    "args": "monoidal_def, monoidal (times::nat ⇒ nat ⇒ nat)",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "",
                    "id": "6",
                    "prevs": [
                        "5"
                    ],
                    "rule": "intros",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                }
            ],
            "prop": "monoidal (times::nat ⇒ nat ⇒ nat)",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "monoidal_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "conjI"
                },
                {
                    "goal_id": "0",
                    "method_name": "introduction",
                    "names": "x,y"
                },
                {
                    "goal_id": "0.2",
                    "method_name": "apply_backward_step",
                    "theorem": "mult_comm"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "conjI"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction",
                    "names": "x,y,z"
                },
                {
                    "goal_id": "1.3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "mult_assoc"
                },
                {
                    "goal_id": "2",
                    "method_name": "introduction",
                    "names": "x"
                },
                {
                    "goal_id": "2.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_nat_mul"
                },
                {
                    "goal_id": "2.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "mult_1_left"
                }
            ],
            "ty": "thm",
            "vars": {}
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_empty",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "monoidal_nat_mul",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_empty, {f: f, op: (times::nat ⇒ nat ⇒ nat)}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ iterate times ∅ f = neutral times"
                },
                {
                    "args": "neutral_nat_mul, iterate times ∅ f = 1",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal_sym",
                    "th": "⊢ iterate times ∅ f = 1"
                },
                {
                    "args": "nproduct_def, nproduct ∅ f = 1",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ∅ f = 1"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ∅ f = 1"
                }
            ],
            "prop": "nproduct ∅ f = 1",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "true",
                    "theorem": "neutral_nat_mul"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_empty"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_insert",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_insert, {f: f, op: (times::nat ⇒ nat ⇒ nat), s: s, x: x}",
                    "id": "2",
                    "prevs": [
                        "1",
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ iterate times (insert x s) f = (if x ∈ s then iterate times s f else f x * iterate times s f)"
                },
                {
                    "args": "nproduct_def, nproduct (insert x s) f = (if x ∈ s then nproduct s f else f x * nproduct s f)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s ⊢ nproduct (insert x s) f = (if x ∈ s then nproduct s f else f x * nproduct s f)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ nproduct (insert x s) f = (if x ∈ s then nproduct s f else f x * nproduct s f)"
                }
            ],
            "prop": [
                "finite s ",
                "⟶ nproduct (insert x s) f = (if x ∈ s then nproduct s f else f x * nproduct s f)"
            ],
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_insert"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "s": "'a set",
                "x": "'a"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_support",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "iterate_support, iterate times (support times f s) f = iterate times s f",
                    "id": "0",
                    "prevs": [],
                    "rule": "rewrite_goal",
                    "th": "⊢ iterate times (support times f s) f = iterate times s f"
                },
                {
                    "args": "nproduct_def, nproduct (support times f s) f = nproduct s f",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct (support times f s) f = nproduct s f"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct (support times f s) f = nproduct s f"
                }
            ],
            "prop": "nproduct (support times f s) f = nproduct s f",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "iterate_support"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_union",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "finite t",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite t ⊢ finite t"
                },
                {
                    "args": "disjoint s t",
                    "id": "2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "disjoint s t ⊢ disjoint s t"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "3",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_union, {f: f, op: (times::nat ⇒ nat ⇒ nat), s: s, t: t}",
                    "id": "4",
                    "prevs": [
                        "3",
                        "0",
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s, finite t, disjoint s t ⊢ iterate times (s ∪ t) f = iterate times s f * iterate times t f"
                },
                {
                    "args": "nproduct_def, nproduct (s ∪ t) f = nproduct s f * nproduct t f",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, finite t, disjoint s t ⊢ nproduct (s ∪ t) f = nproduct s f * nproduct t f"
                },
                {
                    "args": "",
                    "id": "6",
                    "prevs": [
                        "0",
                        "1",
                        "2",
                        "5"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ finite t ⟶ disjoint s t ⟶ nproduct (s ∪ t) f = nproduct s f * nproduct t f"
                }
            ],
            "prop": [
                "finite s ",
                "⟶ finite t ⟶ disjoint s t ⟶ nproduct (s ∪ t) f = nproduct s f * nproduct t f"
            ],
            "steps": [
                {
                    "goal_id": "3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_union"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "s": "'a set",
                "t": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_image",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y ⊢ ∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_image, {f: f, g: g, op: (times::nat ⇒ nat ⇒ nat), s: s}",
                    "id": "2",
                    "prevs": [
                        "1",
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y ⊢ iterate times (image f s) g = iterate times s (g ∘ f)"
                },
                {
                    "args": "nproduct_def, nproduct (image f s) g = nproduct s (g ∘ f)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y ⊢ nproduct (image f s) g = nproduct s (g ∘ f)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y) ⟶ nproduct (image f s) g = nproduct s (g ∘ f)"
                }
            ],
            "prop": [
                "(∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y) ",
                "⟶ nproduct (image f s) g = nproduct s (g ∘ f)"
            ],
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_image"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ 'b",
                "g": "'b ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_injection",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "∀x. x ∈ s ⟶ p x ∈ s",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ s ⟶ p x ∈ s ⊢ ∀x. x ∈ s ⟶ p x ∈ s"
                },
                {
                    "args": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y",
                    "id": "2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y ⊢ ∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "3",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_injection, {f: f, op: (times::nat ⇒ nat ⇒ nat), p: p, s: s}",
                    "id": "4",
                    "prevs": [
                        "3",
                        "0",
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s, ∀x. x ∈ s ⟶ p x ∈ s, ∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y ⊢ iterate times s (f ∘ p) = iterate times s f"
                },
                {
                    "args": "nproduct_def, nproduct s (f ∘ p) = nproduct s f",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ∀x. x ∈ s ⟶ p x ∈ s, ∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y ⊢ nproduct s (f ∘ p) = nproduct s f"
                },
                {
                    "args": "",
                    "id": "6",
                    "prevs": [
                        "0",
                        "1",
                        "2",
                        "5"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ (∀x. x ∈ s ⟶ p x ∈ s) ⟶ (∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y) ⟶ nproduct s (f ∘ p) = nproduct s f"
                }
            ],
            "prop": [
                "finite s ",
                "⟶ (∀x. x ∈ s ⟶ p x ∈ s) ",
                "   ⟶ (∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y) ⟶ nproduct s (f ∘ p) = nproduct s f"
            ],
            "steps": [
                {
                    "goal_id": "3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_injection"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "p": "'a ⇒ 'a",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_add_split",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "m ≤ n + 1",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "m ≤ n + 1 ⊢ m ≤ n + 1"
                },
                {
                    "args": "natseg_add_split, {p: p}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "m ≤ n + 1 ⊢ {m..n + p} = {m..n} ∪ {n + 1..n + p}"
                },
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "2",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "finite_natseg, {m: n + 1, n: n + p}",
                    "id": "3",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({n + 1..n + p})"
                },
                {
                    "args": "n < n + 1",
                    "id": "4",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ n < n + 1"
                },
                {
                    "args": "disjI1, {A: n < n + 1, B: n + p < m ∨ n < m ∨ n + p < n + 1}",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "m ≤ n + 1 ⊢ n < n + 1 ∨ n + p < m ∨ n < m ∨ n + p < n + 1"
                },
                {
                    "args": "disjoint_natseg, disjoint ({m..n}) ({n + 1..n + p})",
                    "id": "6",
                    "prevs": [
                        "5"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ n + 1 ⊢ disjoint ({m..n}) ({n + 1..n + p})"
                },
                {
                    "args": "nproduct_union, {f: f, s: {m..n}, t: {n + 1..n + p}}",
                    "id": "7",
                    "prevs": [
                        "2",
                        "3",
                        "6"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "m ≤ n + 1 ⊢ nproduct ({m..n} ∪ {n + 1..n + p}) f = nproduct ({m..n}) f * nproduct ({n + 1..n + p}) f"
                },
                {
                    "args": "nproduct ({m..n + p}) f = nproduct ({m..n}) f * nproduct ({n + 1..n + p}) f",
                    "id": "8",
                    "prevs": [
                        "1",
                        "7"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "m ≤ n + 1 ⊢ nproduct ({m..n + p}) f = nproduct ({m..n}) f * nproduct ({n + 1..n + p}) f"
                },
                {
                    "args": "",
                    "id": "9",
                    "prevs": [
                        "0",
                        "8"
                    ],
                    "rule": "intros",
                    "th": "⊢ m ≤ n + 1 ⟶ nproduct ({m..n + p}) f = nproduct ({m..n}) f * nproduct ({n + 1..n + p}) f"
                }
            ],
            "prop": [
                "m ≤ n + 1 ",
                "⟶ nproduct ({m..n + p}) f = nproduct ({m..n}) f * nproduct ({n + 1..n + p}) f"
            ],
            "steps": [
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "apply_forward_step",
                    "param_p": "p",
                    "theorem": "natseg_add_split"
                },
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_union"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "goal_id": "4",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "disjoint_natseg"
                },
                {
                    "goal_id": "4",
                    "method_name": "apply_backward_step",
                    "theorem": "disjI1"
                },
                {
                    "goal_id": "4",
                    "method_name": "z3"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat",
                "n": "nat",
                "p": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_pos_lt",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ (∀x. x ∈ ∅ ⟶ 0 < f x) ⟶ 0 < nproduct ∅ f"
                },
                {
                    "args": "∀x. x ∈ ∅ ⟶ 0 < f x",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ ∅ ⟶ 0 < f x ⊢ ∀x. x ∈ ∅ ⟶ 0 < f x"
                },
                {
                    "args": "(0::nat) < 1",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ (0::nat) < 1"
                },
                {
                    "args": "nproduct_empty, 0 < nproduct ∅ f",
                    "id": "1.2",
                    "prevs": [
                        "1.1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ∀x. x ∈ ∅ ⟶ 0 < f x ⊢ 0 < nproduct ∅ f"
                },
                {
                    "args": "",
                    "id": "1.3",
                    "prevs": [
                        "1.0",
                        "1.2"
                    ],
                    "rule": "intros",
                    "th": "finite s ⊢ (∀x. x ∈ ∅ ⟶ 0 < f x) ⟶ 0 < nproduct ∅ f"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ ∀x. ∀s1. ((∀x1. x1 ∈ s1 ⟶ 0 < f x1) ⟶ 0 < nproduct s1 f) ⟶ ¬(x ∈ s1) ⟶ finite s1 ⟶ (∀x1. x1 ∈ insert x s1 ⟶ 0 < f x1) ⟶ 0 < nproduct (insert x s1) f"
                },
                {
                    "args": "x, 'a",
                    "id": "2.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "s, 'a set",
                    "id": "2.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR s"
                },
                {
                    "args": "(∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f",
                    "id": "2.2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "(∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f"
                },
                {
                    "args": "¬(x ∈ s)",
                    "id": "2.3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(x ∈ s) ⊢ ¬(x ∈ s)"
                },
                {
                    "args": "∀x1. x1 ∈ insert x s ⟶ 0 < f x1",
                    "id": "2.4",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x1. x1 ∈ insert x s ⟶ 0 < f x1 ⊢ ∀x1. x1 ∈ insert x s ⟶ 0 < f x1"
                },
                {
                    "args": "nproduct_insert, {f: f, x: x}",
                    "id": "2.5",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ nproduct (insert x s) f = (if x ∈ s then nproduct s f else f x * nproduct s f)"
                },
                {
                    "args": "if_not_P",
                    "id": "2.6",
                    "prevs": [
                        "2.5",
                        "2.3"
                    ],
                    "rule": "rewrite_fact",
                    "th": "finite s, ¬(x ∈ s) ⊢ nproduct (insert x s) f = f x * nproduct s f"
                },
                {
                    "args": "eq_refl, {x: x}",
                    "id": "2.7",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ x = x"
                },
                {
                    "args": "disjI1, {A: x = x, B: x ∈ s}",
                    "id": "2.8",
                    "prevs": [
                        "2.7"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ x = x ∨ x ∈ s"
                },
                {
                    "args": "member_insert, x ∈ insert x s",
                    "id": "2.9",
                    "prevs": [
                        "2.8"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ x ∈ insert x s"
                },
                {
                    "args": "",
                    "id": "2.10",
                    "prevs": [
                        "2.4",
                        "2.9"
                    ],
                    "rule": "apply_fact",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ 0 < f x"
                },
                {
                    "args": "",
                    "id": "2.11",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ ∀x. x ∈ s ⟶ 0 < f x"
                },
                {
                    "args": "x1, 'a",
                    "id": "2.11.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x1"
                },
                {
                    "args": "x1 ∈ s",
                    "id": "2.11.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "x1 ∈ s ⊢ x1 ∈ s"
                },
                {
                    "args": "disjI2, {A: x1 = x, B: x1 ∈ s}",
                    "id": "2.11.2",
                    "prevs": [
                        "2.11.1"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s, x1 ∈ s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ x1 = x ∨ x1 ∈ s"
                },
                {
                    "args": "member_insert, x1 ∈ insert x s",
                    "id": "2.11.3",
                    "prevs": [
                        "2.11.2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, x1 ∈ s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ x1 ∈ insert x s"
                },
                {
                    "args": "",
                    "id": "2.11.4",
                    "prevs": [
                        "2.4",
                        "2.11.3"
                    ],
                    "rule": "apply_fact",
                    "th": "finite s, x1 ∈ s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ 0 < f x1"
                },
                {
                    "args": "",
                    "id": "2.11.5",
                    "prevs": [
                        "2.11.0",
                        "2.11.1",
                        "2.11.4"
                    ],
                    "rule": "intros",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ ∀x1. x1 ∈ s ⟶ 0 < f x1"
                },
                {
                    "args": "",
                    "id": "2.12",
                    "prevs": [
                        "2.2",
                        "2.11"
                    ],
                    "rule": "apply_fact",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ 0 < nproduct s f"
                },
                {
                    "args": "conjI",
                    "id": "2.13",
                    "prevs": [
                        "2.10",
                        "2.12"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ 0 < f x ∧ 0 < nproduct s f"
                },
                {
                    "args": "lt_mult, 0 < f x * nproduct s f",
                    "id": "2.14",
                    "prevs": [
                        "2.13"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ 0 < f x * nproduct s f"
                },
                {
                    "args": "0 < nproduct (insert x s) f",
                    "id": "2.15",
                    "prevs": [
                        "2.6",
                        "2.14"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ 0 < f x1, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f ⊢ 0 < nproduct (insert x s) f"
                },
                {
                    "args": "",
                    "id": "2.16",
                    "prevs": [
                        "2.0",
                        "2.1",
                        "2.2",
                        "2.3",
                        "0",
                        "2.4",
                        "2.15"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀x. ∀s. ((∀x1. x1 ∈ s ⟶ 0 < f x1) ⟶ 0 < nproduct s f) ⟶ ¬(x ∈ s) ⟶ finite s ⟶ (∀x1. x1 ∈ insert x s ⟶ 0 < f x1) ⟶ 0 < nproduct (insert x s) f"
                },
                {
                    "args": "finite_induct_strong",
                    "id": "3",
                    "prevs": [
                        "0",
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s ⊢ (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f"
                }
            ],
            "prop": "finite s ⟶ (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < nproduct s f",
            "steps": [
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "revert_intro"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_induct_strong"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction"
                },
                {
                    "goal_id": "1.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_empty"
                },
                {
                    "goal_id": "1.1",
                    "method_name": "z3"
                },
                {
                    "goal_id": "2",
                    "method_name": "introduction",
                    "names": "x,s"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "2.5",
                    "method_name": "apply_forward_step",
                    "param_f": "f",
                    "param_x": "x",
                    "theorem": "nproduct_insert"
                },
                {
                    "fact_ids": [
                        "2.5",
                        "2.3"
                    ],
                    "goal_id": "2.6",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "2.6"
                    ],
                    "goal_id": "2.7",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "2.7",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "lt_mult"
                },
                {
                    "goal_id": "2.7",
                    "method_name": "apply_backward_step",
                    "theorem": "conjI"
                },
                {
                    "fact_ids": [
                        "2.4"
                    ],
                    "goal_id": "2.7",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2.7",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "member_insert"
                },
                {
                    "goal_id": "2.7",
                    "method_name": "apply_backward_step",
                    "theorem": "disjI1"
                },
                {
                    "goal_id": "2.7",
                    "method_name": "apply_backward_step",
                    "theorem": "eq_refl"
                },
                {
                    "fact_ids": [
                        "2.2"
                    ],
                    "goal_id": "2.11",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2.11",
                    "method_name": "introduction",
                    "names": "x1"
                },
                {
                    "fact_ids": [
                        "2.4"
                    ],
                    "goal_id": "2.11.2",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2.11.2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "member_insert"
                },
                {
                    "goal_id": "2.11.2",
                    "method_name": "apply_backward_step",
                    "theorem": "disjI2"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "p": "'a ⇒ 'a",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_pos_lt_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x ⊢ ∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x"
                },
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x ⊢ ∀x. x ∈ {m..n} ⟶ 0 < f x"
                },
                {
                    "args": "x, nat",
                    "id": "2.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "x ∈ {m..n}",
                    "id": "2.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "x ∈ {m..n} ⊢ x ∈ {m..n}"
                },
                {
                    "args": "member_natseg",
                    "id": "2.2",
                    "prevs": [
                        "2.1"
                    ],
                    "rule": "rewrite_fact",
                    "th": "x ∈ {m..n} ⊢ m ≤ x ∧ x ≤ n"
                },
                {
                    "args": "conjD1",
                    "id": "2.3",
                    "prevs": [
                        "2.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "x ∈ {m..n} ⊢ m ≤ x"
                },
                {
                    "args": "conjD2",
                    "id": "2.4",
                    "prevs": [
                        "2.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "x ∈ {m..n} ⊢ x ≤ n"
                },
                {
                    "args": "x",
                    "id": "2.5",
                    "prevs": [
                        "0",
                        "2.3",
                        "2.4"
                    ],
                    "rule": "apply_fact_for",
                    "th": "x ∈ {m..n}, ∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x ⊢ 0 < f x"
                },
                {
                    "args": "",
                    "id": "2.6",
                    "prevs": [
                        "2.0",
                        "2.1",
                        "2.5"
                    ],
                    "rule": "intros",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x ⊢ ∀x. x ∈ {m..n} ⟶ 0 < f x"
                },
                {
                    "args": "nproduct_pos_lt",
                    "id": "3",
                    "prevs": [
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x ⊢ 0 < nproduct ({m..n}) f"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x) ⟶ 0 < nproduct ({m..n}) f"
                }
            ],
            "prop": "(∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x) ⟶ 0 < nproduct ({m..n}) f",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_pos_lt"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "goal_id": "2",
                    "method_name": "introduction",
                    "names": "x"
                },
                {
                    "fact_ids": [
                        "2.1"
                    ],
                    "goal_id": "2.2",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "member_natseg"
                },
                {
                    "fact_ids": [
                        "2.2"
                    ],
                    "goal_id": "2.3",
                    "method_name": "apply_forward_step",
                    "theorem": "conjD1"
                },
                {
                    "fact_ids": [
                        "2.2"
                    ],
                    "goal_id": "2.4",
                    "method_name": "apply_forward_step",
                    "theorem": "conjD2"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "2.5",
                    "method_name": "apply_prev"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_offset",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "nproduct_image, {f: λi. i + p, g: f, s: {m..n}}",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ (∀x. ∀y. x ∈ {m..n} ⟶ y ∈ {m..n} ⟶ x + p = y + p ⟶ x = y) ⟶ nproduct (image (λi. i + p) ({m..n})) f = nproduct ({m..n}) (f ∘ (λi. i + p))"
                },
                {
                    "args": "comp_fun_def",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_fact",
                    "th": "⊢ (∀x. ∀y. x ∈ {m..n} ⟶ y ∈ {m..n} ⟶ x + p = y + p ⟶ x = y) ⟶ nproduct (image (λi. i + p) ({m..n})) f = nproduct ({m..n}) (λx. f (x + p))"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ ∀x. ∀y. x ∈ {m..n} ⟶ y ∈ {m..n} ⟶ x + p = y + p ⟶ x = y"
                },
                {
                    "args": "x, nat",
                    "id": "2.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "y, nat",
                    "id": "2.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR y"
                },
                {
                    "args": "x ∈ {m..n}",
                    "id": "2.2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "x ∈ {m..n} ⊢ x ∈ {m..n}"
                },
                {
                    "args": "y ∈ {m..n}",
                    "id": "2.3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "y ∈ {m..n} ⊢ y ∈ {m..n}"
                },
                {
                    "args": "x + p = y + p",
                    "id": "2.4",
                    "prevs": [],
                    "rule": "assume",
                    "th": "x + p = y + p ⊢ x + p = y + p"
                },
                {
                    "args": "add_cancel_right",
                    "id": "2.5",
                    "prevs": [
                        "2.4"
                    ],
                    "rule": "apply_theorem",
                    "th": "x + p = y + p ⊢ x = y"
                },
                {
                    "args": "",
                    "id": "2.6",
                    "prevs": [
                        "2.0",
                        "2.1",
                        "2.2",
                        "2.3",
                        "2.4",
                        "2.5"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀x. ∀y. x ∈ {m..n} ⟶ y ∈ {m..n} ⟶ x + p = y + p ⟶ x = y"
                },
                {
                    "args": "",
                    "id": "3",
                    "prevs": [
                        "1",
                        "2"
                    ],
                    "rule": "apply_fact",
                    "th": "⊢ nproduct (image (λi. i + p) ({m..n})) f = nproduct ({m..n}) (λx. f (x + p))"
                },
                {
                    "args": "natseg_offset_image, nproduct ({m + p..n + p}) f = nproduct ({m..n}) (λi. f (i + p))",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({m + p..n + p}) f = nproduct ({m..n}) (λi. f (i + p))"
                },
                {
                    "args": "",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({m + p..n + p}) f = nproduct ({m..n}) (λi. f (i + p))"
                }
            ],
            "prop": "nproduct ({m + p..n + p}) f = nproduct ({m..n}) (λi. f (i + p))",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "natseg_offset_image"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_forward_step",
                    "param_f": "(λi. i + p)",
                    "param_g": "f",
                    "param_s": "{m..n}",
                    "theorem": "nproduct_image"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "comp_fun_def"
                },
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2",
                    "method_name": "introduction",
                    "names": "x,y"
                },
                {
                    "fact_ids": [
                        "2.4"
                    ],
                    "goal_id": "2.5",
                    "method_name": "apply_backward_step",
                    "theorem": "add_cancel_right"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat",
                "n": "nat",
                "p": "nat"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_sing",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "nproduct_insert, {f: f, s: (∅::'a set), x: x}",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite (∅::'a set) ⟶ nproduct ({x}) f = (if x ∈ ∅ then nproduct ∅ f else f x * nproduct ∅ f)"
                },
                {
                    "args": "finite_empty, {}",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite (∅::'a set)"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [
                        "0",
                        "1"
                    ],
                    "rule": "apply_fact",
                    "th": "⊢ nproduct ({x}) f = (if x ∈ ∅ then nproduct ∅ f else f x * nproduct ∅ f)"
                },
                {
                    "args": "member_empty, {x: x}",
                    "id": "3",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ ¬(x ∈ ∅)"
                },
                {
                    "args": "if_not_P",
                    "id": "4",
                    "prevs": [
                        "2",
                        "3"
                    ],
                    "rule": "rewrite_fact",
                    "th": "⊢ nproduct ({x}) f = f x * nproduct ∅ f"
                },
                {
                    "args": "nproduct_empty",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "rewrite_fact",
                    "th": "⊢ nproduct ({x}) f = f x * 1"
                },
                {
                    "args": "mult_1_right",
                    "id": "6",
                    "prevs": [
                        "5"
                    ],
                    "rule": "rewrite_fact",
                    "th": "⊢ nproduct ({x}) f = f x"
                },
                {
                    "args": "",
                    "id": "7",
                    "prevs": [
                        "6"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({x}) f = f x"
                }
            ],
            "prop": "nproduct ({x}) f = f x",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "apply_forward_step",
                    "param_f": "f",
                    "param_s": "({}::'a set)",
                    "param_x": "x",
                    "theorem": "nproduct_insert"
                },
                {
                    "goal": "finite (∅::'a set)",
                    "goal_id": "1",
                    "method_name": "cut"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_empty"
                },
                {
                    "fact_ids": [
                        "0",
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "apply_fact"
                },
                {
                    "goal": "~(x ∈ ∅)",
                    "goal_id": "3",
                    "method_name": "cut"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "member_empty"
                },
                {
                    "fact_ids": [
                        "2",
                        "3"
                    ],
                    "goal_id": "4",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "4"
                    ],
                    "goal_id": "5",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "nproduct_empty"
                },
                {
                    "fact_ids": [
                        "5"
                    ],
                    "goal_id": "6",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "mult_1_right"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "x": "'a"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_sing_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "nproduct_sing, nproduct ({n}) f = f n",
                    "id": "0",
                    "prevs": [],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({n}) f = f n"
                },
                {
                    "args": "natseg_sing, nproduct ({n..n}) f = f n",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({n..n}) f = f n"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({n..n}) f = f n"
                }
            ],
            "prop": "nproduct ({n..n}) f = f n",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "natseg_sing"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_sing"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_0_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "",
                    "id": "0",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ m = 0 ⟶ nproduct (if m = 0 then {0} else ∅) f = (if m = 0 then f 0 else 1)"
                },
                {
                    "args": "m = 0",
                    "id": "0.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "m = 0 ⊢ m = 0"
                },
                {
                    "args": "nproduct_sing, nproduct ({0}) f = f 0",
                    "id": "0.1",
                    "prevs": [],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({0}) f = f 0"
                },
                {
                    "args": "if_P, nproduct (if m = 0 then {0} else ∅) f = (if m = 0 then f 0 else 1)",
                    "id": "0.2",
                    "prevs": [
                        "0.1",
                        "0.0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m = 0 ⊢ nproduct (if m = 0 then {0} else ∅) f = (if m = 0 then f 0 else 1)"
                },
                {
                    "args": "",
                    "id": "0.3",
                    "prevs": [
                        "0.0",
                        "0.2"
                    ],
                    "rule": "intros",
                    "th": "⊢ m = 0 ⟶ nproduct (if m = 0 then {0} else ∅) f = (if m = 0 then f 0 else 1)"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ ¬(m = 0) ⟶ nproduct (if m = 0 then {0} else ∅) f = (if m = 0 then f 0 else 1)"
                },
                {
                    "args": "¬(m = 0)",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(m = 0) ⊢ ¬(m = 0)"
                },
                {
                    "args": "nproduct_empty, nproduct ∅ f = 1",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ∅ f = 1"
                },
                {
                    "args": "if_not_P, nproduct (if m = 0 then {0} else ∅) f = (if m = 0 then f 0 else 1)",
                    "id": "1.2",
                    "prevs": [
                        "1.1",
                        "1.0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "¬(m = 0) ⊢ nproduct (if m = 0 then {0} else ∅) f = (if m = 0 then f 0 else 1)"
                },
                {
                    "args": "",
                    "id": "1.3",
                    "prevs": [
                        "1.0",
                        "1.2"
                    ],
                    "rule": "intros",
                    "th": "⊢ ¬(m = 0) ⟶ nproduct (if m = 0 then {0} else ∅) f = (if m = 0 then f 0 else 1)"
                },
                {
                    "args": "classical_cases",
                    "id": "2",
                    "prevs": [
                        "0",
                        "1"
                    ],
                    "rule": "apply_theorem",
                    "th": "⊢ nproduct (if m = 0 then {0} else ∅) f = (if m = 0 then f 0 else 1)"
                },
                {
                    "args": "natseg_0, nproduct ({m..0}) f = (if m = 0 then f 0 else 1)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({m..0}) f = (if m = 0 then f 0 else 1)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({m..0}) f = (if m = 0 then f 0 else 1)"
                }
            ],
            "prop": "nproduct ({m..0}) f = (if m = 0 then f 0 else 1)",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "natseg_0"
                },
                {
                    "case": "m = 0 ",
                    "goal_id": "0",
                    "method_name": "cases"
                },
                {
                    "goal_id": "0",
                    "method_name": "introduction"
                },
                {
                    "fact_ids": [
                        "0.0"
                    ],
                    "goal_id": "0.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "if_P"
                },
                {
                    "goal_id": "0.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_sing"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction"
                },
                {
                    "fact_ids": [
                        "1.0"
                    ],
                    "goal_id": "1.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "goal_id": "1.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_empty"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_Suc_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "",
                    "id": "0",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ m ≤ Suc n ⟶ nproduct (if m ≤ Suc n then insert (Suc n) ({m..n}) else {m..n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
                },
                {
                    "args": "m ≤ Suc n",
                    "id": "0.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "m ≤ Suc n ⊢ m ≤ Suc n"
                },
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "0.1",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "nproduct_insert, {f: f, x: Suc n}",
                    "id": "0.2",
                    "prevs": [
                        "0.1"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ nproduct (insert (Suc n) ({m..n})) f = (if Suc n ∈ {m..n} then nproduct ({m..n}) f else f (Suc n) * nproduct ({m..n}) f)"
                },
                {
                    "args": "less_eq_refl, {n: n}",
                    "id": "0.3",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ n ≤ n"
                },
                {
                    "args": "not_lt, ¬(n < n)",
                    "id": "0.4",
                    "prevs": [
                        "0.3"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ Suc n ⊢ ¬(n < n)"
                },
                {
                    "args": "disjI2, {A: ¬(m ≤ Suc n), B: ¬(n < n)}",
                    "id": "0.5",
                    "prevs": [
                        "0.4"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "m ≤ Suc n ⊢ ¬(m ≤ Suc n) ∨ ¬(n < n)"
                },
                {
                    "args": "de_morgan_thm1, ¬(m ≤ Suc n ∧ n < n)",
                    "id": "0.6",
                    "prevs": [
                        "0.5"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ Suc n ⊢ ¬(m ≤ Suc n ∧ n < n)"
                },
                {
                    "args": "lesseq_Suc_less, ¬(m ≤ Suc n ∧ Suc n ≤ n)",
                    "id": "0.7",
                    "prevs": [
                        "0.6"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ Suc n ⊢ ¬(m ≤ Suc n ∧ Suc n ≤ n)"
                },
                {
                    "args": "member_natseg, ¬(Suc n ∈ {m..n})",
                    "id": "0.8",
                    "prevs": [
                        "0.7"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ Suc n ⊢ ¬(Suc n ∈ {m..n})"
                },
                {
                    "args": "if_not_P",
                    "id": "0.9",
                    "prevs": [
                        "0.2",
                        "0.8"
                    ],
                    "rule": "rewrite_fact",
                    "th": "m ≤ Suc n ⊢ nproduct (insert (Suc n) ({m..n})) f = f (Suc n) * nproduct ({m..n}) f"
                },
                {
                    "args": "mult_comm, {x: f (Suc n), y: nproduct ({m..n}) f}",
                    "id": "0.10",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ f (Suc n) * nproduct ({m..n}) f = nproduct ({m..n}) f * f (Suc n)"
                },
                {
                    "args": "nproduct (insert (Suc n) ({m..n})) f = nproduct ({m..n}) f * f (Suc n)",
                    "id": "0.11",
                    "prevs": [
                        "0.9",
                        "0.10"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "m ≤ Suc n ⊢ nproduct (insert (Suc n) ({m..n})) f = nproduct ({m..n}) f * f (Suc n)"
                },
                {
                    "args": "if_P, nproduct (if m ≤ Suc n then insert (Suc n) ({m..n}) else {m..n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)",
                    "id": "0.12",
                    "prevs": [
                        "0.11",
                        "0.0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ Suc n ⊢ nproduct (if m ≤ Suc n then insert (Suc n) ({m..n}) else {m..n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
                },
                {
                    "args": "",
                    "id": "0.13",
                    "prevs": [
                        "0.0",
                        "0.12"
                    ],
                    "rule": "intros",
                    "th": "⊢ m ≤ Suc n ⟶ nproduct (if m ≤ Suc n then insert (Suc n) ({m..n}) else {m..n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ ¬(m ≤ Suc n) ⟶ nproduct (if m ≤ Suc n then insert (Suc n) ({m..n}) else {m..n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
                },
                {
                    "args": "¬(m ≤ Suc n)",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(m ≤ Suc n) ⊢ ¬(m ≤ Suc n)"
                },
                {
                    "args": "if_not_P, nproduct (if m ≤ Suc n then insert (Suc n) ({m..n}) else {m..n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)",
                    "id": "1.1",
                    "prevs": [
                        "1.0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "¬(m ≤ Suc n) ⊢ nproduct (if m ≤ Suc n then insert (Suc n) ({m..n}) else {m..n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
                },
                {
                    "args": "",
                    "id": "1.2",
                    "prevs": [
                        "1.0",
                        "1.1"
                    ],
                    "rule": "intros",
                    "th": "⊢ ¬(m ≤ Suc n) ⟶ nproduct (if m ≤ Suc n then insert (Suc n) ({m..n}) else {m..n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
                },
                {
                    "args": "classical_cases",
                    "id": "2",
                    "prevs": [
                        "0",
                        "1"
                    ],
                    "rule": "apply_theorem",
                    "th": "⊢ nproduct (if m ≤ Suc n then insert (Suc n) ({m..n}) else {m..n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
                },
                {
                    "args": "natseg_Suc, nproduct ({m..Suc n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({m..Suc n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({m..Suc n}) f = (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
                }
            ],
            "prop": [
                "nproduct ({m..Suc n}) f = ",
                "(if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f)"
            ],
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "natseg_Suc"
                },
                {
                    "case": " m ≤ Suc n",
                    "goal_id": "0",
                    "method_name": "cases"
                },
                {
                    "goal_id": "0",
                    "method_name": "introduction"
                },
                {
                    "fact_ids": [
                        "0.0"
                    ],
                    "goal_id": "0.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "if_P"
                },
                {
                    "goal": "finite {m..n}",
                    "goal_id": "0.1",
                    "method_name": "cut"
                },
                {
                    "goal_id": "0.1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "fact_ids": [
                        "0.1"
                    ],
                    "goal_id": "0.2",
                    "method_name": "apply_forward_step",
                    "param_f": "f",
                    "param_x": "Suc n",
                    "theorem": "nproduct_insert"
                },
                {
                    "goal": "~(Suc n ∈ {m..n})",
                    "goal_id": "0.3",
                    "method_name": "cut"
                },
                {
                    "goal_id": "0.3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "member_natseg"
                },
                {
                    "goal_id": "0.3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "lesseq_Suc_less"
                },
                {
                    "goal_id": "0.3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "de_morgan_thm1"
                },
                {
                    "goal_id": "0.3",
                    "method_name": "apply_backward_step",
                    "theorem": "disjI2"
                },
                {
                    "goal_id": "0.3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "not_lt"
                },
                {
                    "goal_id": "0.3",
                    "method_name": "apply_backward_step",
                    "theorem": "less_eq_refl"
                },
                {
                    "fact_ids": [
                        "0.2",
                        "0.8"
                    ],
                    "goal_id": "0.9",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "0.9"
                    ],
                    "goal_id": "0.10",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "0.10",
                    "method_name": "apply_backward_step",
                    "theorem": "mult_comm"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction"
                },
                {
                    "fact_ids": [
                        "1.0"
                    ],
                    "goal_id": "1.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "if_not_P"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_eq",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀x. x ∈ s ⟶ f x = g x",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ s ⟶ f x = g x ⊢ ∀x. x ∈ s ⟶ f x = g x"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_eq",
                    "id": "2",
                    "prevs": [
                        "1",
                        "0"
                    ],
                    "rule": "apply_theorem",
                    "th": "∀x. x ∈ s ⟶ f x = g x ⊢ iterate times s f = iterate times s g"
                },
                {
                    "args": "nproduct_def, nproduct s f = nproduct s g",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "∀x. x ∈ s ⟶ f x = g x ⊢ nproduct s f = nproduct s g"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀x. x ∈ s ⟶ f x = g x) ⟶ nproduct s f = nproduct s g"
                }
            ],
            "prop": "(∀x. x ∈ s ⟶ f x = g x) ⟶ nproduct s f = nproduct s g",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_eq"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "g": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_eq_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = g i",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = g i ⊢ ∀i. m ≤ i ⟶ i ≤ n ⟶ f i = g i"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = g i ⊢ ∀x. x ∈ {m..n} ⟶ f x = g x"
                },
                {
                    "args": "i, nat",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR i"
                },
                {
                    "args": "i ∈ {m..n}",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "i ∈ {m..n} ⊢ i ∈ {m..n}"
                },
                {
                    "args": "member_natseg",
                    "id": "1.2",
                    "prevs": [
                        "1.1"
                    ],
                    "rule": "rewrite_fact",
                    "th": "i ∈ {m..n} ⊢ m ≤ i ∧ i ≤ n"
                },
                {
                    "args": "conjD1",
                    "id": "1.3",
                    "prevs": [
                        "1.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "i ∈ {m..n} ⊢ m ≤ i"
                },
                {
                    "args": "conjD2",
                    "id": "1.4",
                    "prevs": [
                        "1.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "i ∈ {m..n} ⊢ i ≤ n"
                },
                {
                    "args": "i",
                    "id": "1.5",
                    "prevs": [
                        "0",
                        "1.3",
                        "1.4"
                    ],
                    "rule": "apply_fact_for",
                    "th": "i ∈ {m..n}, ∀i. m ≤ i ⟶ i ≤ n ⟶ f i = g i ⊢ f i = g i"
                },
                {
                    "args": "",
                    "id": "1.6",
                    "prevs": [
                        "1.0",
                        "1.1",
                        "1.5"
                    ],
                    "rule": "intros",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = g i ⊢ ∀i. i ∈ {m..n} ⟶ f i = g i"
                },
                {
                    "args": "nproduct_eq",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "apply_theorem",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = g i ⊢ nproduct ({m..n}) f = nproduct ({m..n}) g"
                },
                {
                    "args": "",
                    "id": "3",
                    "prevs": [
                        "0",
                        "2"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀i. m ≤ i ⟶ i ≤ n ⟶ f i = g i) ⟶ nproduct ({m..n}) f = nproduct ({m..n}) g"
                }
            ],
            "prop": "(∀i. m ≤ i ⟶ i ≤ n ⟶ f i = g i) ⟶ nproduct ({m..n}) f = nproduct ({m..n}) g",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_eq"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction",
                    "names": "i"
                },
                {
                    "fact_ids": [
                        "1.1"
                    ],
                    "goal_id": "1.2",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "member_natseg"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1.3",
                    "method_name": "apply_prev"
                },
                {
                    "fact_ids": [
                        "1.2"
                    ],
                    "goal_id": "1.3",
                    "method_name": "apply_backward_step",
                    "theorem": "conjD1"
                },
                {
                    "fact_ids": [
                        "1.2"
                    ],
                    "goal_id": "1.4",
                    "method_name": "apply_backward_step",
                    "theorem": "conjD2"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "g": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_eq_0",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ (1::nat) = 0 ⟶ (∃x. x ∈ ∅ ∧ f x = 0)"
                },
                {
                    "args": "(1::nat) = 0",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "(1::nat) = 0 ⊢ (1::nat) = 0"
                },
                {
                    "args": "∃x. x ∈ ∅ ∧ f x = 0",
                    "id": "1.1",
                    "prevs": [
                        "1.0"
                    ],
                    "rule": "z3",
                    "th": "(1::nat) = 0 ⊢ ∃x. x ∈ ∅ ∧ f x = 0"
                },
                {
                    "args": "",
                    "id": "1.2",
                    "prevs": [
                        "1.0",
                        "1.1"
                    ],
                    "rule": "intros",
                    "th": "finite s ⊢ (1::nat) = 0 ⟶ (∃x. x ∈ ∅ ∧ f x = 0)"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ (∃x. x ∈ ∅ ∧ f x = 0) ⟶ (1::nat) = 0"
                },
                {
                    "args": "∃x. x ∈ ∅ ∧ f x = 0",
                    "id": "2.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∃x. x ∈ ∅ ∧ f x = 0 ⊢ ∃x. x ∈ ∅ ∧ f x = 0"
                },
                {
                    "args": "x, 'a",
                    "id": "2.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "x ∈ ∅ ∧ f x = 0",
                    "id": "2.2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "x ∈ ∅ ∧ f x = 0 ⊢ x ∈ ∅ ∧ f x = 0"
                },
                {
                    "args": "conjD1",
                    "id": "2.3",
                    "prevs": [
                        "2.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "x ∈ ∅ ∧ f x = 0 ⊢ x ∈ ∅"
                },
                {
                    "args": "member_empty, (1::nat) = 0",
                    "id": "2.4",
                    "prevs": [
                        "2.3"
                    ],
                    "rule": "resolve_theorem",
                    "th": "x ∈ ∅ ∧ f x = 0 ⊢ (1::nat) = 0"
                },
                {
                    "args": "∃x. x ∈ ∅ ∧ f x = 0",
                    "id": "2.5",
                    "prevs": [
                        "2.0",
                        "2.0",
                        "2.1",
                        "2.2",
                        "2.4"
                    ],
                    "rule": "intros",
                    "th": "finite s ⊢ (∃x. x ∈ ∅ ∧ f x = 0) ⟶ (1::nat) = 0"
                },
                {
                    "args": "iffI",
                    "id": "3",
                    "prevs": [
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s ⊢ (1::nat) = 0 ⟷ (∃x. x ∈ ∅ ∧ f x = 0)"
                },
                {
                    "args": "nproduct_empty, nproduct ∅ f = 0 ⟷ (∃x. x ∈ ∅ ∧ f x = 0)",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s ⊢ nproduct ∅ f = 0 ⟷ (∃x. x ∈ ∅ ∧ f x = 0)"
                },
                {
                    "args": "",
                    "id": "5",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ ∀x. ∀s1. nproduct s1 f = 0 ⟷ (∃x1. x1 ∈ s1 ∧ f x1 = 0) ⟶ ¬(x ∈ s1) ⟶ finite s1 ⟶ nproduct (insert x s1) f = 0 ⟷ (∃x1. x1 ∈ insert x s1 ∧ f x1 = 0)"
                },
                {
                    "args": "x, 'a",
                    "id": "5.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "s, 'a set",
                    "id": "5.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR s"
                },
                {
                    "args": "nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0)",
                    "id": "5.2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0) ⊢ nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0)"
                },
                {
                    "args": "¬(x ∈ s)",
                    "id": "5.3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(x ∈ s) ⊢ ¬(x ∈ s)"
                },
                {
                    "args": "nproduct_insert, {f: f, x: x}",
                    "id": "5.4",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ nproduct (insert x s) f = (if x ∈ s then nproduct s f else f x * nproduct s f)"
                },
                {
                    "args": "if_not_P",
                    "id": "5.5",
                    "prevs": [
                        "5.4",
                        "5.3"
                    ],
                    "rule": "rewrite_fact",
                    "th": "finite s, ¬(x ∈ s) ⊢ nproduct (insert x s) f = f x * nproduct s f"
                },
                {
                    "args": "f x = 0 ∨ (∃x1. x1 ∈ s ∧ f x1 = 0) ⟷ (∃x1. (x1 = x ∨ x1 ∈ s) ∧ f x1 = 0)",
                    "id": "5.6",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ f x = 0 ∨ (∃x1. x1 ∈ s ∧ f x1 = 0) ⟷ (∃x1. (x1 = x ∨ x1 ∈ s) ∧ f x1 = 0)"
                },
                {
                    "args": "member_insert, f x = 0 ∨ (∃x1. x1 ∈ s ∧ f x1 = 0) ⟷ (∃x1. x1 ∈ insert x s ∧ f x1 = 0)",
                    "id": "5.7",
                    "prevs": [
                        "5.6"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ¬(x ∈ s), nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0) ⊢ f x = 0 ∨ (∃x1. x1 ∈ s ∧ f x1 = 0) ⟷ (∃x1. x1 ∈ insert x s ∧ f x1 = 0)"
                },
                {
                    "args": "f x = 0 ∨ nproduct s f = 0 ⟷ (∃x1. x1 ∈ insert x s ∧ f x1 = 0)",
                    "id": "5.8",
                    "prevs": [
                        "5.2",
                        "5.7"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "finite s, ¬(x ∈ s), nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0) ⊢ f x = 0 ∨ nproduct s f = 0 ⟷ (∃x1. x1 ∈ insert x s ∧ f x1 = 0)"
                },
                {
                    "args": "mult_eq_0, f x * nproduct s f = 0 ⟷ (∃x1. x1 ∈ insert x s ∧ f x1 = 0)",
                    "id": "5.9",
                    "prevs": [
                        "5.8"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ¬(x ∈ s), nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0) ⊢ f x * nproduct s f = 0 ⟷ (∃x1. x1 ∈ insert x s ∧ f x1 = 0)"
                },
                {
                    "args": "nproduct (insert x s) f = 0 ⟷ (∃x1. x1 ∈ insert x s ∧ f x1 = 0)",
                    "id": "5.10",
                    "prevs": [
                        "5.5",
                        "5.9"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "finite s, ¬(x ∈ s), nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0) ⊢ nproduct (insert x s) f = 0 ⟷ (∃x1. x1 ∈ insert x s ∧ f x1 = 0)"
                },
                {
                    "args": "",
                    "id": "5.11",
                    "prevs": [
                        "5.0",
                        "5.1",
                        "5.2",
                        "5.3",
                        "0",
                        "5.10"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀x. ∀s. nproduct s f = 0 ⟷ (∃x1. x1 ∈ s ∧ f x1 = 0) ⟶ ¬(x ∈ s) ⟶ finite s ⟶ nproduct (insert x s) f = 0 ⟷ (∃x1. x1 ∈ insert x s ∧ f x1 = 0)"
                },
                {
                    "args": "finite_induct_strong",
                    "id": "6",
                    "prevs": [
                        "0",
                        "4",
                        "5"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s ⊢ nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0)"
                },
                {
                    "args": "",
                    "id": "7",
                    "prevs": [
                        "0",
                        "6"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0)"
                }
            ],
            "prop": "finite s ⟶ nproduct s f = 0 ⟷ (∃x. x ∈ s ∧ f x = 0)",
            "steps": [
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_induct_strong"
                },
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_empty"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "iffI"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction"
                },
                {
                    "fact_ids": [
                        "1.0"
                    ],
                    "goal_id": "1.1",
                    "method_name": "z3"
                },
                {
                    "goal_id": "2",
                    "method_name": "introduction"
                },
                {
                    "fact_ids": [
                        "2.0"
                    ],
                    "goal_id": "2.1",
                    "method_name": "exists_elim",
                    "names": "x"
                },
                {
                    "fact_ids": [
                        "2.2"
                    ],
                    "goal_id": "2.3",
                    "method_name": "apply_forward_step",
                    "theorem": "conjD1"
                },
                {
                    "fact_ids": [
                        "2.3"
                    ],
                    "goal_id": "2.4",
                    "method_name": "apply_resolve_step",
                    "theorem": "member_empty"
                },
                {
                    "goal_id": "5",
                    "method_name": "introduction",
                    "names": "x,s"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "5.4",
                    "method_name": "apply_forward_step",
                    "param_f": "f",
                    "param_x": "x",
                    "theorem": "nproduct_insert"
                },
                {
                    "fact_ids": [
                        "5.4",
                        "5.3"
                    ],
                    "goal_id": "5.5",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "5.5"
                    ],
                    "goal_id": "5.6",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "5.6",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "mult_eq_0"
                },
                {
                    "fact_ids": [
                        "5.2"
                    ],
                    "goal_id": "5.6",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "5.6",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "member_insert"
                },
                {
                    "goal_id": "5.6",
                    "method_name": "z3"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_eq_0_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "nproduct_eq_0, {f: f, s: {m..n}}",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n}) ⟶ nproduct ({m..n}) f = 0 ⟷ (∃x. x ∈ {m..n} ∧ f x = 0)"
                },
                {
                    "args": "member_natseg",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_fact",
                    "th": "⊢ finite ({m..n}) ⟶ nproduct ({m..n}) f = 0 ⟷ (∃x. (m ≤ x ∧ x ≤ n) ∧ f x = 0)"
                },
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "2",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "",
                    "id": "3",
                    "prevs": [
                        "1",
                        "2"
                    ],
                    "rule": "apply_fact",
                    "th": "⊢ nproduct ({m..n}) f = 0 ⟷ (∃x. (m ≤ x ∧ x ≤ n) ∧ f x = 0)"
                },
                {
                    "args": "conj_assoc, nproduct ({m..n}) f = 0 ⟷ (∃x. m ≤ x ∧ x ≤ n ∧ f x = 0)",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({m..n}) f = 0 ⟷ (∃x. m ≤ x ∧ x ≤ n ∧ f x = 0)"
                },
                {
                    "args": "",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({m..n}) f = 0 ⟷ (∃x. m ≤ x ∧ x ≤ n ∧ f x = 0)"
                }
            ],
            "prop": "nproduct ({m..n}) f = 0 ⟷ (∃x. m ≤ x ∧ x ≤ n ∧ f x = 0)",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "apply_forward_step",
                    "param_f": "f",
                    "param_s": "{m..n}",
                    "theorem": "nproduct_eq_0"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "member_natseg"
                },
                {
                    "goal_id": "2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "conj_assoc"
                },
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_restrict",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ ∀x. x ∈ s ⟶ (if x ∈ s then f x else 1) = f x"
                },
                {
                    "args": "x, 'a",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "x ∈ s",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "x ∈ s ⊢ x ∈ s"
                },
                {
                    "args": "if_P, (if x ∈ s then f x else 1) = f x",
                    "id": "1.2",
                    "prevs": [
                        "1.1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "x ∈ s ⊢ (if x ∈ s then f x else 1) = f x"
                },
                {
                    "args": "",
                    "id": "1.3",
                    "prevs": [
                        "1.0",
                        "1.1",
                        "1.2"
                    ],
                    "rule": "intros",
                    "th": "finite s ⊢ ∀x. x ∈ s ⟶ (if x ∈ s then f x else 1) = f x"
                },
                {
                    "args": "nproduct_eq",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s ⊢ nproduct s (λx. if x ∈ s then f x else 1) = nproduct s f"
                },
                {
                    "args": "",
                    "id": "3",
                    "prevs": [
                        "0",
                        "2"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ nproduct s (λi. if i ∈ s then f i else 1) = nproduct s f"
                }
            ],
            "prop": "finite s ⟶ nproduct s (λi. if i ∈ s then f i else 1) = nproduct s f",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_eq"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction",
                    "names": "x"
                },
                {
                    "fact_ids": [
                        "1.1"
                    ],
                    "goal_id": "1.2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "if_P"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_restrict_set",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "monoidal_nat_mul",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_restrict_set, {P: P, f: f, op: (times::nat ⇒ nat ⇒ nat), s: s}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ iterate times ({x. x ∈ s ∧ P x}) f = iterate times s (λx. if P x then f x else neutral times)"
                },
                {
                    "args": "neutral_nat_mul, iterate times ({i. i ∈ s ∧ P i}) f = iterate times s (λi. if P i then f i else 1)",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal_sym",
                    "th": "⊢ iterate times ({i. i ∈ s ∧ P i}) f = iterate times s (λi. if P i then f i else 1)"
                },
                {
                    "args": "nproduct_def, nproduct ({i. i ∈ s ∧ P i}) f = nproduct s (λi. if P i then f i else 1)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({i. i ∈ s ∧ P i}) f = nproduct s (λi. if P i then f i else 1)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({i. i ∈ s ∧ P i}) f = nproduct s (λi. if P i then f i else 1)"
                }
            ],
            "prop": "nproduct ({i. i ∈ s ∧ P i}) f = nproduct s (λi. if P i then f i else 1)",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "true",
                    "theorem": "neutral_nat_mul"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_restrict_set"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "P": "'a ⇒ bool",
                "f": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_le",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ (∀x. x ∈ ∅ ⟶ f x ≤ g x) ⟶ nproduct ∅ f ≤ nproduct ∅ g"
                },
                {
                    "args": "∀x. x ∈ ∅ ⟶ f x ≤ g x",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ ∅ ⟶ f x ≤ g x ⊢ ∀x. x ∈ ∅ ⟶ f x ≤ g x"
                },
                {
                    "args": "less_eq_refl, {n: (1::nat)}",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ (1::nat) ≤ 1"
                },
                {
                    "args": "nproduct_empty, nproduct ∅ f ≤ nproduct ∅ g",
                    "id": "1.2",
                    "prevs": [
                        "1.1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ∀x. x ∈ ∅ ⟶ f x ≤ g x ⊢ nproduct ∅ f ≤ nproduct ∅ g"
                },
                {
                    "args": "",
                    "id": "1.3",
                    "prevs": [
                        "1.0",
                        "1.2"
                    ],
                    "rule": "intros",
                    "th": "finite s ⊢ (∀x. x ∈ ∅ ⟶ f x ≤ g x) ⟶ nproduct ∅ f ≤ nproduct ∅ g"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ ∀x. ∀s1. ((∀x1. x1 ∈ s1 ⟶ f x1 ≤ g x1) ⟶ nproduct s1 f ≤ nproduct s1 g) ⟶ ¬(x ∈ s1) ⟶ finite s1 ⟶ (∀x1. x1 ∈ insert x s1 ⟶ f x1 ≤ g x1) ⟶ nproduct (insert x s1) f ≤ nproduct (insert x s1) g"
                },
                {
                    "args": "x, 'a",
                    "id": "2.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "s, 'a set",
                    "id": "2.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR s"
                },
                {
                    "args": "(∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g",
                    "id": "2.2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "(∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g"
                },
                {
                    "args": "¬(x ∈ s)",
                    "id": "2.3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(x ∈ s) ⊢ ¬(x ∈ s)"
                },
                {
                    "args": "∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1",
                    "id": "2.4",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1 ⊢ ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1"
                },
                {
                    "args": "nproduct_insert, {f: f, x: x}",
                    "id": "2.5",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ nproduct (insert x s) f = (if x ∈ s then nproduct s f else f x * nproduct s f)"
                },
                {
                    "args": "if_not_P",
                    "id": "2.6",
                    "prevs": [
                        "2.5",
                        "2.3"
                    ],
                    "rule": "rewrite_fact",
                    "th": "finite s, ¬(x ∈ s) ⊢ nproduct (insert x s) f = f x * nproduct s f"
                },
                {
                    "args": "nproduct_insert, {f: g, x: x}",
                    "id": "2.7",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ nproduct (insert x s) g = (if x ∈ s then nproduct s g else g x * nproduct s g)"
                },
                {
                    "args": "if_not_P",
                    "id": "2.8",
                    "prevs": [
                        "2.7",
                        "2.3"
                    ],
                    "rule": "rewrite_fact",
                    "th": "finite s, ¬(x ∈ s) ⊢ nproduct (insert x s) g = g x * nproduct s g"
                },
                {
                    "args": "member_insert",
                    "id": "2.9",
                    "prevs": [
                        "2.4"
                    ],
                    "rule": "rewrite_fact",
                    "th": "∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1 ⊢ ∀x1. x1 = x ∨ x1 ∈ s ⟶ f x1 ≤ g x1"
                },
                {
                    "args": "eq_refl, {x: x}",
                    "id": "2.10",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ x = x"
                },
                {
                    "args": "disjI1, {A: x = x, B: x ∈ s}",
                    "id": "2.11",
                    "prevs": [
                        "2.10"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ x = x ∨ x ∈ s"
                },
                {
                    "args": "x",
                    "id": "2.12",
                    "prevs": [
                        "2.9",
                        "2.11"
                    ],
                    "rule": "apply_fact_for",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ f x ≤ g x"
                },
                {
                    "args": "",
                    "id": "2.13",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ ∀x. x ∈ s ⟶ f x ≤ g x"
                },
                {
                    "args": "x1, 'a",
                    "id": "2.13.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x1"
                },
                {
                    "args": "x1 ∈ s",
                    "id": "2.13.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "x1 ∈ s ⊢ x1 ∈ s"
                },
                {
                    "args": "disjI2, {A: x1 = x, B: x1 ∈ s}",
                    "id": "2.13.2",
                    "prevs": [
                        "2.13.1"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s, x1 ∈ s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ x1 = x ∨ x1 ∈ s"
                },
                {
                    "args": "x1",
                    "id": "2.13.3",
                    "prevs": [
                        "2.9",
                        "2.13.2"
                    ],
                    "rule": "apply_fact_for",
                    "th": "finite s, x1 ∈ s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ f x1 ≤ g x1"
                },
                {
                    "args": "",
                    "id": "2.13.4",
                    "prevs": [
                        "2.13.0",
                        "2.13.1",
                        "2.13.3"
                    ],
                    "rule": "intros",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ ∀x1. x1 ∈ s ⟶ f x1 ≤ g x1"
                },
                {
                    "args": "",
                    "id": "2.14",
                    "prevs": [
                        "2.2",
                        "2.13"
                    ],
                    "rule": "apply_fact",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ nproduct s f ≤ nproduct s g"
                },
                {
                    "args": "le_mult2",
                    "id": "2.15",
                    "prevs": [
                        "2.12",
                        "2.14"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ f x * nproduct s f ≤ g x * nproduct s g"
                },
                {
                    "args": "nproduct (insert x s) f ≤ g x * nproduct s g",
                    "id": "2.16",
                    "prevs": [
                        "2.6",
                        "2.15"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ nproduct (insert x s) f ≤ g x * nproduct s g"
                },
                {
                    "args": "nproduct (insert x s) f ≤ nproduct (insert x s) g",
                    "id": "2.17",
                    "prevs": [
                        "2.8",
                        "2.16"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "finite s, ¬(x ∈ s), ∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1, (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g ⊢ nproduct (insert x s) f ≤ nproduct (insert x s) g"
                },
                {
                    "args": "",
                    "id": "2.18",
                    "prevs": [
                        "2.0",
                        "2.1",
                        "2.2",
                        "2.3",
                        "0",
                        "2.4",
                        "2.17"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀x. ∀s. ((∀x1. x1 ∈ s ⟶ f x1 ≤ g x1) ⟶ nproduct s f ≤ nproduct s g) ⟶ ¬(x ∈ s) ⟶ finite s ⟶ (∀x1. x1 ∈ insert x s ⟶ f x1 ≤ g x1) ⟶ nproduct (insert x s) f ≤ nproduct (insert x s) g"
                },
                {
                    "args": "finite_induct_strong",
                    "id": "3",
                    "prevs": [
                        "0",
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s ⊢ (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g"
                }
            ],
            "prop": "finite s ⟶ (∀x. x ∈ s ⟶ f x ≤ g x) ⟶ nproduct s f ≤ nproduct s g",
            "steps": [
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "revert_intro"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_induct_strong"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction"
                },
                {
                    "goal_id": "1.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_empty"
                },
                {
                    "goal_id": "1.1",
                    "method_name": "apply_backward_step",
                    "theorem": "less_eq_refl"
                },
                {
                    "goal_id": "2",
                    "method_name": "introduction",
                    "names": "x,s"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "2.5",
                    "method_name": "apply_forward_step",
                    "param_f": "f",
                    "param_x": "x",
                    "theorem": "nproduct_insert"
                },
                {
                    "fact_ids": [
                        "2.5",
                        "2.3"
                    ],
                    "goal_id": "2.6",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "2.7",
                    "method_name": "apply_forward_step",
                    "param_f": "g",
                    "param_x": "x",
                    "theorem": "nproduct_insert"
                },
                {
                    "fact_ids": [
                        "2.7",
                        "2.3"
                    ],
                    "goal_id": "2.8",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "2.8"
                    ],
                    "goal_id": "2.9",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "fact_ids": [
                        "2.6"
                    ],
                    "goal_id": "2.9",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "2.9",
                    "method_name": "apply_backward_step",
                    "theorem": "le_mult2"
                },
                {
                    "fact_ids": [
                        "2.4"
                    ],
                    "goal_id": "2.9",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "member_insert"
                },
                {
                    "fact_ids": [
                        "2.9"
                    ],
                    "goal_id": "2.10",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2.10",
                    "method_name": "apply_backward_step",
                    "theorem": "disjI1"
                },
                {
                    "goal_id": "2.10",
                    "method_name": "apply_backward_step",
                    "theorem": "eq_refl"
                },
                {
                    "fact_ids": [
                        "2.2"
                    ],
                    "goal_id": "2.13",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2.13",
                    "method_name": "introduction",
                    "names": "x1"
                },
                {
                    "fact_ids": [
                        "2.9"
                    ],
                    "goal_id": "2.13.2",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2.13.2",
                    "method_name": "apply_backward_step",
                    "theorem": "disjI2"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "g": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_le_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i ≤ g i",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i ≤ g i ⊢ ∀i. m ≤ i ⟶ i ≤ n ⟶ f i ≤ g i"
                },
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i ≤ g i ⊢ ∀x. x ∈ {m..n} ⟶ f x ≤ g x"
                },
                {
                    "args": "i, nat",
                    "id": "2.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR i"
                },
                {
                    "args": "i ∈ {m..n}",
                    "id": "2.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "i ∈ {m..n} ⊢ i ∈ {m..n}"
                },
                {
                    "args": "member_natseg",
                    "id": "2.2",
                    "prevs": [
                        "2.1"
                    ],
                    "rule": "rewrite_fact",
                    "th": "i ∈ {m..n} ⊢ m ≤ i ∧ i ≤ n"
                },
                {
                    "args": "conjD1",
                    "id": "2.3",
                    "prevs": [
                        "2.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "i ∈ {m..n} ⊢ m ≤ i"
                },
                {
                    "args": "conjD2",
                    "id": "2.4",
                    "prevs": [
                        "2.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "i ∈ {m..n} ⊢ i ≤ n"
                },
                {
                    "args": "i",
                    "id": "2.5",
                    "prevs": [
                        "0",
                        "2.3",
                        "2.4"
                    ],
                    "rule": "apply_fact_for",
                    "th": "i ∈ {m..n}, ∀i. m ≤ i ⟶ i ≤ n ⟶ f i ≤ g i ⊢ f i ≤ g i"
                },
                {
                    "args": "",
                    "id": "2.6",
                    "prevs": [
                        "2.0",
                        "2.1",
                        "2.5"
                    ],
                    "rule": "intros",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i ≤ g i ⊢ ∀i. i ∈ {m..n} ⟶ f i ≤ g i"
                },
                {
                    "args": "nproduct_le",
                    "id": "3",
                    "prevs": [
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i ≤ g i ⊢ nproduct ({m..n}) f ≤ nproduct ({m..n}) g"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀i. m ≤ i ⟶ i ≤ n ⟶ f i ≤ g i) ⟶ nproduct ({m..n}) f ≤ nproduct ({m..n}) g"
                }
            ],
            "prop": "(∀i. m ≤ i ⟶ i ≤ n ⟶ f i ≤ g i) ⟶ nproduct ({m..n}) f ≤ nproduct ({m..n}) g",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_le"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "goal_id": "2",
                    "method_name": "introduction",
                    "names": "i"
                },
                {
                    "fact_ids": [
                        "2.1"
                    ],
                    "goal_id": "2.2",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "member_natseg"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "2.3",
                    "method_name": "apply_prev"
                },
                {
                    "fact_ids": [
                        "2.2"
                    ],
                    "goal_id": "2.3",
                    "method_name": "apply_backward_step",
                    "theorem": "conjD1"
                },
                {
                    "fact_ids": [
                        "2.2"
                    ],
                    "goal_id": "2.4",
                    "method_name": "apply_backward_step",
                    "theorem": "conjD2"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "g": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_eq_1",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀x. x ∈ s ⟶ f x = 1",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ s ⟶ f x = 1 ⊢ ∀x. x ∈ s ⟶ f x = 1"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "neutral_nat_mul, ∀x. x ∈ s ⟶ f x = neutral times",
                    "id": "2",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "∀x. x ∈ s ⟶ f x = 1 ⊢ ∀x. x ∈ s ⟶ f x = neutral times"
                },
                {
                    "args": "iterate_eq_neutral",
                    "id": "3",
                    "prevs": [
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "∀x. x ∈ s ⟶ f x = 1 ⊢ iterate times s f = neutral times"
                },
                {
                    "args": "neutral_nat_mul, iterate times s f = 1",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "rewrite_goal_sym",
                    "th": "∀x. x ∈ s ⟶ f x = 1 ⊢ iterate times s f = 1"
                },
                {
                    "args": "nproduct_def, nproduct s f = 1",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "∀x. x ∈ s ⟶ f x = 1 ⊢ nproduct s f = 1"
                },
                {
                    "args": "",
                    "id": "6",
                    "prevs": [
                        "0",
                        "5"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀x. x ∈ s ⟶ f x = 1) ⟶ nproduct s f = 1"
                }
            ],
            "prop": "(∀x. x ∈ s ⟶ f x = 1) ⟶ nproduct s f = 1",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "true",
                    "theorem": "neutral_nat_mul"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_eq_neutral"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                },
                {
                    "goal_id": "2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_eq_1_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = 1",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = 1 ⊢ ∀i. m ≤ i ⟶ i ≤ n ⟶ f i = 1"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = 1 ⊢ ∀x. x ∈ {m..n} ⟶ f x = 1"
                },
                {
                    "args": "i, nat",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR i"
                },
                {
                    "args": "i ∈ {m..n}",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "i ∈ {m..n} ⊢ i ∈ {m..n}"
                },
                {
                    "args": "member_natseg",
                    "id": "1.2",
                    "prevs": [
                        "1.1"
                    ],
                    "rule": "rewrite_fact",
                    "th": "i ∈ {m..n} ⊢ m ≤ i ∧ i ≤ n"
                },
                {
                    "args": "conjD1",
                    "id": "1.3",
                    "prevs": [
                        "1.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "i ∈ {m..n} ⊢ m ≤ i"
                },
                {
                    "args": "conjD2",
                    "id": "1.4",
                    "prevs": [
                        "1.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "i ∈ {m..n} ⊢ i ≤ n"
                },
                {
                    "args": "i",
                    "id": "1.5",
                    "prevs": [
                        "0",
                        "1.3",
                        "1.4"
                    ],
                    "rule": "apply_fact_for",
                    "th": "i ∈ {m..n}, ∀i. m ≤ i ⟶ i ≤ n ⟶ f i = 1 ⊢ f i = 1"
                },
                {
                    "args": "",
                    "id": "1.6",
                    "prevs": [
                        "1.0",
                        "1.1",
                        "1.5"
                    ],
                    "rule": "intros",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = 1 ⊢ ∀i. i ∈ {m..n} ⟶ f i = 1"
                },
                {
                    "args": "nproduct_eq_1",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "apply_theorem",
                    "th": "∀i. m ≤ i ⟶ i ≤ n ⟶ f i = 1 ⊢ nproduct ({m..n}) f = 1"
                },
                {
                    "args": "",
                    "id": "3",
                    "prevs": [
                        "0",
                        "2"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀i. m ≤ i ⟶ i ≤ n ⟶ f i = 1) ⟶ nproduct ({m..n}) f = 1"
                }
            ],
            "prop": "(∀i. m ≤ i ⟶ i ≤ n ⟶ f i = 1) ⟶ nproduct ({m..n}) f = 1",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_eq_1"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction",
                    "names": "i"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1.2",
                    "method_name": "apply_prev"
                },
                {
                    "fact_ids": [
                        "1.1"
                    ],
                    "goal_id": "1.2",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "member_natseg"
                },
                {
                    "fact_ids": [
                        "1.2"
                    ],
                    "goal_id": "1.3",
                    "method_name": "apply_backward_step",
                    "theorem": "conjD1"
                },
                {
                    "fact_ids": [
                        "1.2"
                    ],
                    "goal_id": "1.4",
                    "method_name": "apply_backward_step",
                    "theorem": "conjD2"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_mul_gen",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite ({x. x ∈ s ∧ ¬(f x = 1)})",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite ({x. x ∈ s ∧ ¬(f x = 1)}) ⊢ finite ({x. x ∈ s ∧ ¬(f x = 1)})"
                },
                {
                    "args": "finite ({x. x ∈ s ∧ ¬(g x = 1)})",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite ({x. x ∈ s ∧ ¬(g x = 1)}) ⊢ finite ({x. x ∈ s ∧ ¬(g x = 1)})"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "2",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "neutral_nat_mul, finite ({x. x ∈ s ∧ ¬(f x = neutral times)})",
                    "id": "3",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite ({x. x ∈ s ∧ ¬(f x = 1)}), finite ({x. x ∈ s ∧ ¬(g x = 1)}) ⊢ finite ({x. x ∈ s ∧ ¬(f x = neutral times)})"
                },
                {
                    "args": "support_def, finite (support times f s)",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite ({x. x ∈ s ∧ ¬(f x = 1)}), finite ({x. x ∈ s ∧ ¬(g x = 1)}) ⊢ finite (support times f s)"
                },
                {
                    "args": "neutral_nat_mul, finite ({x. x ∈ s ∧ ¬(g x = neutral times)})",
                    "id": "5",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite ({x. x ∈ s ∧ ¬(f x = 1)}), finite ({x. x ∈ s ∧ ¬(g x = 1)}) ⊢ finite ({x. x ∈ s ∧ ¬(g x = neutral times)})"
                },
                {
                    "args": "support_def, finite (support times g s)",
                    "id": "6",
                    "prevs": [
                        "5"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite ({x. x ∈ s ∧ ¬(f x = 1)}), finite ({x. x ∈ s ∧ ¬(g x = 1)}) ⊢ finite (support times g s)"
                },
                {
                    "args": "iterate_op_gen",
                    "id": "7",
                    "prevs": [
                        "2",
                        "4",
                        "6"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite ({x. x ∈ s ∧ ¬(f x = 1)}), finite ({x. x ∈ s ∧ ¬(g x = 1)}) ⊢ iterate times s (λx. f x * g x) = iterate times s f * iterate times s g"
                },
                {
                    "args": "nproduct_def, nproduct s (λx. f x * g x) = nproduct s f * nproduct s g",
                    "id": "8",
                    "prevs": [
                        "7"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite ({x. x ∈ s ∧ ¬(f x = 1)}), finite ({x. x ∈ s ∧ ¬(g x = 1)}) ⊢ nproduct s (λx. f x * g x) = nproduct s f * nproduct s g"
                },
                {
                    "args": "",
                    "id": "9",
                    "prevs": [
                        "0",
                        "1",
                        "8"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite ({x. x ∈ s ∧ ¬(f x = 1)}) ⟶ finite ({x. x ∈ s ∧ ¬(g x = 1)}) ⟶ nproduct s (λx. f x * g x) = nproduct s f * nproduct s g"
                }
            ],
            "prop": [
                "finite ({x. x ∈ s ∧ ¬(f x = 1)}) ",
                "⟶ finite ({x. x ∈ s ∧ ¬(g x = 1)}) ",
                "   ⟶ nproduct s (λx. f x * g x) = nproduct s f * nproduct s g"
            ],
            "steps": [
                {
                    "goal_id": "2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_op_gen"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                },
                {
                    "goal_id": "3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "support_def"
                },
                {
                    "goal_id": "3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_nat_mul"
                },
                {
                    "goal_id": "5",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "support_def"
                },
                {
                    "goal_id": "5",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "g": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_mul",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "finite_restrict, {P: λx. ¬(f x = 1), s: s}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ finite ({x. x ∈ s ∧ ¬(f x = 1)})"
                },
                {
                    "args": "finite_restrict, {P: λx. ¬(g x = 1), s: s}",
                    "id": "2",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ finite ({x. x ∈ s ∧ ¬(g x = 1)})"
                },
                {
                    "args": "nproduct_mul_gen",
                    "id": "3",
                    "prevs": [
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s ⊢ nproduct s (λx. f x * g x) = nproduct s f * nproduct s g"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ nproduct s (λx. f x * g x) = nproduct s f * nproduct s g"
                }
            ],
            "prop": "finite s ⟶ nproduct s (λx. f x * g x) = nproduct s f * nproduct s g",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_mul_gen"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_restrict"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_restrict"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "g": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_mul_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "nproduct_mul, {f: f, g: g, s: {m..n}}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ nproduct ({m..n}) (λx. f x * g x) = nproduct ({m..n}) f * nproduct ({m..n}) g"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({m..n}) (λx. f x * g x) = nproduct ({m..n}) f * nproduct ({m..n}) g"
                }
            ],
            "prop": "nproduct ({m..n}) (λx. f x * g x) = nproduct ({m..n}) f * nproduct ({m..n}) g",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_mul"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "g": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_const",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "nat_nat_power_def_1, 1 = c ^ (0::nat)",
                    "id": "1",
                    "prevs": [],
                    "rule": "rewrite_goal",
                    "th": "⊢ 1 = c ^ (0::nat)"
                },
                {
                    "args": "card_empty, 1 = c ^ card (∅::'a set)",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s ⊢ 1 = c ^ card (∅::'a set)"
                },
                {
                    "args": "nproduct_empty, nproduct ∅ (λx::'a. c) = c ^ card (∅::'a set)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s ⊢ nproduct ∅ (λx::'a. c) = c ^ card (∅::'a set)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ ∀x::'a. ∀s1. nproduct s1 (λx1. c) = c ^ card s1 ⟶ ¬(x ∈ s1) ⟶ finite s1 ⟶ nproduct (insert x s1) (λx1. c) = c ^ card (insert x s1)"
                },
                {
                    "args": "x, 'a",
                    "id": "4.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "s, 'a set",
                    "id": "4.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR s"
                },
                {
                    "args": "nproduct s (λx. c) = c ^ card s",
                    "id": "4.2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "nproduct s (λx. c) = c ^ card s ⊢ nproduct s (λx. c) = c ^ card s"
                },
                {
                    "args": "¬(x ∈ s)",
                    "id": "4.3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(x ∈ s) ⊢ ¬(x ∈ s)"
                },
                {
                    "args": "nproduct_insert, {f: λx1::'a. c, x: x}",
                    "id": "4.4",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ nproduct (insert x s) (λx1. c) = (if x ∈ s then nproduct s (λx1. c) else c * nproduct s (λx1. c))"
                },
                {
                    "args": "if_not_P",
                    "id": "4.5",
                    "prevs": [
                        "4.4",
                        "4.3"
                    ],
                    "rule": "rewrite_fact",
                    "th": "finite s, ¬(x ∈ s) ⊢ nproduct (insert x s) (λx1. c) = c * nproduct s (λx1. c)"
                },
                {
                    "args": "card_insert, {x: x}",
                    "id": "4.6",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ card (insert x s) = (if x ∈ s then card s else Suc (card s))"
                },
                {
                    "args": "if_not_P",
                    "id": "4.7",
                    "prevs": [
                        "4.6",
                        "4.3"
                    ],
                    "rule": "rewrite_fact",
                    "th": "finite s, ¬(x ∈ s) ⊢ card (insert x s) = Suc (card s)"
                },
                {
                    "args": "",
                    "id": "4.8",
                    "prevs": [
                        "4.2",
                        "4.5"
                    ],
                    "rule": "rewrite_fact_with_prev",
                    "th": "finite s, ¬(x ∈ s), nproduct s (λx. c) = c ^ card s ⊢ nproduct (insert x s) (λx1. c) = c * c ^ card s"
                },
                {
                    "args": "nat_nat_power_def_2, nproduct (insert x s) (λx1. c) = c ^ Suc (card s)",
                    "id": "4.9",
                    "prevs": [
                        "4.8"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ¬(x ∈ s), nproduct s (λx. c) = c ^ card s ⊢ nproduct (insert x s) (λx1. c) = c ^ Suc (card s)"
                },
                {
                    "args": "nproduct (insert x s) (λx1. c) = c ^ card (insert x s)",
                    "id": "4.10",
                    "prevs": [
                        "4.7",
                        "4.9"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "finite s, ¬(x ∈ s), nproduct s (λx. c) = c ^ card s ⊢ nproduct (insert x s) (λx1. c) = c ^ card (insert x s)"
                },
                {
                    "args": "",
                    "id": "4.11",
                    "prevs": [
                        "4.0",
                        "4.1",
                        "4.2",
                        "4.3",
                        "0",
                        "4.10"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀x::'a. ∀s. nproduct s (λx1. c) = c ^ card s ⟶ ¬(x ∈ s) ⟶ finite s ⟶ nproduct (insert x s) (λx1. c) = c ^ card (insert x s)"
                },
                {
                    "args": "finite_induct_strong",
                    "id": "5",
                    "prevs": [
                        "0",
                        "3",
                        "4"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s ⊢ nproduct s (λx. c) = c ^ card s"
                },
                {
                    "args": "",
                    "id": "6",
                    "prevs": [
                        "0",
                        "5"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ nproduct s (λx. c) = c ^ card s"
                }
            ],
            "prop": "finite s ⟶ nproduct s (λx. c) = c ^ card s",
            "steps": [
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_induct_strong"
                },
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_empty"
                },
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "card_empty"
                },
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nat_nat_power_def_1"
                },
                {
                    "goal_id": "4",
                    "method_name": "introduction",
                    "names": "x,s"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "4.4",
                    "method_name": "apply_forward_step",
                    "param_f": "(λx1::'a. c)",
                    "param_x": "x",
                    "theorem": "nproduct_insert"
                },
                {
                    "fact_ids": [
                        "4.4",
                        "4.3"
                    ],
                    "goal_id": "4.5",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "4.6",
                    "method_name": "apply_forward_step",
                    "param_x": "x",
                    "theorem": "card_insert"
                },
                {
                    "fact_ids": [
                        "4.6",
                        "4.3"
                    ],
                    "goal_id": "4.7",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "4.7"
                    ],
                    "goal_id": "4.8",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "4.8",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nat_nat_power_def_2"
                },
                {
                    "fact_ids": [
                        "4.2",
                        "4.5"
                    ],
                    "goal_id": "4.8",
                    "method_name": "rewrite_fact_with_prev"
                }
            ],
            "ty": "thm",
            "vars": {
                "c": "nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_const_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "nproduct_const, {c: c, s: {m..n}}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ nproduct ({m..n}) (λx. c) = c ^ card ({m..n})"
                },
                {
                    "args": "card_natseg, nproduct ({m..n}) (λx. c) = c ^ (n + 1 - m)",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal_sym",
                    "th": "⊢ nproduct ({m..n}) (λx. c) = c ^ (n + 1 - m)"
                },
                {
                    "args": "",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({m..n}) (λx. c) = c ^ (n + 1 - m)"
                }
            ],
            "prop": "nproduct ({m..n}) (λx. c) = c ^ (n + 1 - m)",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "true",
                    "theorem": "card_natseg"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_const"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                }
            ],
            "ty": "thm",
            "vars": {
                "c": "nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_const_natseg_1",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "nat_plus_minus, c ^ (n + 1 - 1) = c ^ n",
                    "id": "0",
                    "prevs": [],
                    "rule": "rewrite_goal",
                    "th": "⊢ c ^ (n + 1 - 1) = c ^ n"
                },
                {
                    "args": "nproduct_const_natseg, nproduct ({1..n}) (λx. c) = c ^ n",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({1..n}) (λx. c) = c ^ n"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({1..n}) (λx. c) = c ^ n"
                }
            ],
            "prop": "nproduct ({1..n}) (λx. c) = c ^ n",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_const_natseg"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nat_plus_minus"
                }
            ],
            "ty": "thm",
            "vars": {
                "c": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_one",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀x. x ∈ s ⟶ (1::nat) = 1",
                    "id": "0",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ∀x. x ∈ s ⟶ (1::nat) = 1"
                },
                {
                    "args": "nproduct_eq_1",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem",
                    "th": "⊢ nproduct s (λx. 1) = 1"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct s (λn. 1) = 1"
                }
            ],
            "prop": "nproduct s (λn. 1) = 1",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_eq_1"
                },
                {
                    "goal_id": "0",
                    "method_name": "z3"
                }
            ],
            "ty": "thm",
            "vars": {
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_closed",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "P 1",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "P 1 ⊢ P 1"
                },
                {
                    "args": "∀x. ∀y. P x ⟶ P y ⟶ P (x * y)",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ ∀x. ∀y. P x ⟶ P y ⟶ P (x * y)"
                },
                {
                    "args": "∀a. a ∈ s ⟶ P (f a)",
                    "id": "2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀a. a ∈ s ⟶ P (f a) ⊢ ∀a. a ∈ s ⟶ P (f a)"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "3",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "neutral_nat_mul, P (neutral times)",
                    "id": "4",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "P 1, ∀a. a ∈ s ⟶ P (f a), ∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ P (neutral times)"
                },
                {
                    "args": "",
                    "id": "5",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "P 1, ∀a. a ∈ s ⟶ P (f a), ∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ ∀x. x ∈ s ∧ ¬(f x = neutral times) ⟶ P (f x)"
                },
                {
                    "args": "x, 'a",
                    "id": "5.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "x ∈ s ∧ ¬(f x = neutral times)",
                    "id": "5.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "x ∈ s ∧ ¬(f x = neutral times) ⊢ x ∈ s ∧ ¬(f x = neutral times)"
                },
                {
                    "args": "conjD1",
                    "id": "5.2",
                    "prevs": [
                        "5.1"
                    ],
                    "rule": "apply_theorem",
                    "th": "x ∈ s ∧ ¬(f x = neutral times) ⊢ x ∈ s"
                },
                {
                    "args": "x",
                    "id": "5.3",
                    "prevs": [
                        "2",
                        "5.2"
                    ],
                    "rule": "apply_fact_for",
                    "th": "P 1, ∀a. a ∈ s ⟶ P (f a), x ∈ s ∧ ¬(f x = neutral times), ∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ P (f x)"
                },
                {
                    "args": "",
                    "id": "5.4",
                    "prevs": [
                        "5.0",
                        "5.1",
                        "5.3"
                    ],
                    "rule": "intros",
                    "th": "P 1, ∀a. a ∈ s ⟶ P (f a), ∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ ∀x. x ∈ s ∧ ¬(f x = neutral times) ⟶ P (f x)"
                },
                {
                    "args": "iterate_closed",
                    "id": "6",
                    "prevs": [
                        "3",
                        "4",
                        "1",
                        "5"
                    ],
                    "rule": "apply_theorem",
                    "th": "P 1, ∀a. a ∈ s ⟶ P (f a), ∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ P (iterate times s f)"
                },
                {
                    "args": "nproduct_def, P (nproduct s f)",
                    "id": "7",
                    "prevs": [
                        "6"
                    ],
                    "rule": "rewrite_goal",
                    "th": "P 1, ∀a. a ∈ s ⟶ P (f a), ∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ P (nproduct s f)"
                },
                {
                    "args": "",
                    "id": "8",
                    "prevs": [
                        "0",
                        "1",
                        "2",
                        "7"
                    ],
                    "rule": "intros",
                    "th": "⊢ P 1 ⟶ (∀x. ∀y. P x ⟶ P y ⟶ P (x * y)) ⟶ (∀a. a ∈ s ⟶ P (f a)) ⟶ P (nproduct s f)"
                }
            ],
            "prop": "P 1 ⟶ (∀x. ∀y. P x ⟶ P y ⟶ P (x * y)) ⟶ (∀a. a ∈ s ⟶ P (f a)) ⟶ P (nproduct s f)",
            "steps": [
                {
                    "goal_id": "3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_closed"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                },
                {
                    "goal_id": "4",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_nat_mul"
                },
                {
                    "goal_id": "5",
                    "method_name": "introduction",
                    "names": "x"
                },
                {
                    "fact_ids": [
                        "2"
                    ],
                    "goal_id": "5.2",
                    "method_name": "apply_prev"
                },
                {
                    "fact_ids": [
                        "5.1"
                    ],
                    "goal_id": "5.2",
                    "method_name": "apply_backward_step",
                    "theorem": "conjD1"
                }
            ],
            "ty": "thm",
            "vars": {
                "P": "nat ⇒ bool",
                "f": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_related",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "R 1 1",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "R 1 1 ⊢ R 1 1"
                },
                {
                    "args": "∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1)",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1)"
                },
                {
                    "args": "finite s",
                    "id": "2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "∀i. i ∈ s ⟶ R (f i) (g i)",
                    "id": "3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀i. i ∈ s ⟶ R (f i) (g i) ⊢ ∀i. i ∈ s ⟶ R (f i) (g i)"
                },
                {
                    "args": "iterate_related, {R: R, f: f, g: g, op: (times::nat ⇒ nat ⇒ nat), s: s}",
                    "id": "4",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat) ⟶ R (neutral times) (neutral times) ⟶ (∀x1. ∀y1. ∀x2. ∀y2. R x1 x2 ⟶ R y1 y2 ⟶ R (x1 * y1) (x2 * y2)) ⟶ finite s ⟶ (∀x. x ∈ s ⟶ R (f x) (g x)) ⟶ R (iterate times s f) (iterate times s g)"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "5",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "neutral_nat_mul, R (neutral times) (neutral times)",
                    "id": "6",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, R 1 1, ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ R (neutral times) (neutral times)"
                },
                {
                    "args": "",
                    "id": "7",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s, R 1 1, ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ ∀x1. ∀y1. ∀x2. ∀y2. R x1 x2 ⟶ R y1 y2 ⟶ R (x1 * y1) (x2 * y2)"
                },
                {
                    "args": "m, nat",
                    "id": "7.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR m"
                },
                {
                    "args": "n, nat",
                    "id": "7.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR n"
                },
                {
                    "args": "m1, nat",
                    "id": "7.2",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR m1"
                },
                {
                    "args": "n1, nat",
                    "id": "7.3",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR n1"
                },
                {
                    "args": "R m m1",
                    "id": "7.4",
                    "prevs": [],
                    "rule": "assume",
                    "th": "R m m1 ⊢ R m m1"
                },
                {
                    "args": "R n n1",
                    "id": "7.5",
                    "prevs": [],
                    "rule": "assume",
                    "th": "R n n1 ⊢ R n n1"
                },
                {
                    "args": "m, m1, n, n1",
                    "id": "7.6",
                    "prevs": [
                        "1",
                        "7.4",
                        "7.5"
                    ],
                    "rule": "apply_fact_for",
                    "th": "finite s, R m m1, R n n1, R 1 1, ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ R (m * n) (m1 * n1)"
                },
                {
                    "args": "",
                    "id": "7.7",
                    "prevs": [
                        "7.0",
                        "7.1",
                        "7.2",
                        "7.3",
                        "7.4",
                        "7.5",
                        "7.6"
                    ],
                    "rule": "intros",
                    "th": "finite s, R 1 1, ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ ∀m. ∀n. ∀m1. ∀n1. R m m1 ⟶ R n n1 ⟶ R (m * n) (m1 * n1)"
                },
                {
                    "args": "",
                    "id": "8",
                    "prevs": [
                        "4",
                        "5",
                        "6",
                        "7",
                        "2",
                        "3"
                    ],
                    "rule": "apply_fact",
                    "th": "finite s, R 1 1, ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ R (iterate times s f) (iterate times s g)"
                },
                {
                    "args": "nproduct_def, R (nproduct s f) (nproduct s g)",
                    "id": "9",
                    "prevs": [
                        "8"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, R 1 1, ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ R (nproduct s f) (nproduct s g)"
                },
                {
                    "args": "",
                    "id": "10",
                    "prevs": [
                        "0",
                        "1",
                        "2",
                        "3",
                        "9"
                    ],
                    "rule": "intros",
                    "th": "⊢ R 1 1 ⟶ (∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1)) ⟶ finite s ⟶ (∀i. i ∈ s ⟶ R (f i) (g i)) ⟶ R (nproduct s f) (nproduct s g)"
                }
            ],
            "prop": [
                "R 1 1 ",
                "⟶ (∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1)) ",
                "   ⟶ finite s ⟶ (∀i. i ∈ s ⟶ R (f i) (g i)) ⟶ R (nproduct s f) (nproduct s g)"
            ],
            "steps": [
                {
                    "goal_id": "4",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "4",
                    "method_name": "apply_forward_step",
                    "param_R": "R",
                    "param_f": "f",
                    "param_g": "g",
                    "param_op": "(times::nat=>nat=>nat)",
                    "param_s": "s",
                    "theorem": "iterate_related"
                },
                {
                    "fact_ids": [
                        "4"
                    ],
                    "goal_id": "5",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "5",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                },
                {
                    "goal_id": "6",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_nat_mul"
                },
                {
                    "goal_id": "7",
                    "method_name": "introduction",
                    "names": "m,n,m1,n1"
                },
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "7.6",
                    "method_name": "apply_prev"
                }
            ],
            "ty": "thm",
            "vars": {
                "R": "nat ⇒ nat ⇒ bool",
                "f": "'a ⇒ nat",
                "g": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_closed_nonempty",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "¬(s = ∅)",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(s = ∅) ⊢ ¬(s = ∅)"
                },
                {
                    "args": "∀x. ∀y. P x ⟶ P y ⟶ P (x * y)",
                    "id": "2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ ∀x. ∀y. P x ⟶ P y ⟶ P (x * y)"
                },
                {
                    "args": "∀a. a ∈ s ⟶ P (f a)",
                    "id": "3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀a. a ∈ s ⟶ P (f a) ⊢ ∀a. a ∈ s ⟶ P (f a)"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "4",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_closed_nonempty",
                    "id": "5",
                    "prevs": [
                        "4",
                        "2",
                        "0",
                        "1",
                        "3"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s, ¬(s = ∅), ∀a. a ∈ s ⟶ P (f a), ∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ P (iterate times s f)"
                },
                {
                    "args": "nproduct_def, P (nproduct s f)",
                    "id": "6",
                    "prevs": [
                        "5"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ¬(s = ∅), ∀a. a ∈ s ⟶ P (f a), ∀x. ∀y. P x ⟶ P y ⟶ P (x * y) ⊢ P (nproduct s f)"
                },
                {
                    "args": "",
                    "id": "7",
                    "prevs": [
                        "0",
                        "1",
                        "2",
                        "3",
                        "6"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ ¬(s = ∅) ⟶ (∀x. ∀y. P x ⟶ P y ⟶ P (x * y)) ⟶ (∀a. a ∈ s ⟶ P (f a)) ⟶ P (nproduct s f)"
                }
            ],
            "prop": [
                "finite s ",
                "⟶ ¬(s = ∅) ",
                "   ⟶ (∀x. ∀y. P x ⟶ P y ⟶ P (x * y)) ⟶ (∀a. a ∈ s ⟶ P (f a)) ⟶ P (nproduct s f)"
            ],
            "steps": [
                {
                    "goal_id": "4",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "4",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_closed_nonempty"
                },
                {
                    "goal_id": "4",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "P": "nat ⇒ bool",
                "f": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_related_nonempty",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1)",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1)"
                },
                {
                    "args": "finite s",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "¬(s = ∅)",
                    "id": "2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(s = ∅) ⊢ ¬(s = ∅)"
                },
                {
                    "args": "∀i. i ∈ s ⟶ R (f i) (g i)",
                    "id": "3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀i. i ∈ s ⟶ R (f i) (g i) ⊢ ∀i. i ∈ s ⟶ R (f i) (g i)"
                },
                {
                    "args": "iterate_related_nonempty, {R: R, f: f, g: g, op: (times::nat ⇒ nat ⇒ nat), s: s}",
                    "id": "4",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat) ⟶ (∀x1. ∀y1. ∀x2. ∀y2. R x1 x2 ⟶ R y1 y2 ⟶ R (x1 * y1) (x2 * y2)) ⟶ finite s ⟶ ¬(s = ∅) ⟶ (∀x. x ∈ s ⟶ R (f x) (g x)) ⟶ R (iterate times s f) (iterate times s g)"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "5",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "",
                    "id": "6",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s, ¬(s = ∅), ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ ∀x1. ∀y1. ∀x2. ∀y2. R x1 x2 ⟶ R y1 y2 ⟶ R (x1 * y1) (x2 * y2)"
                },
                {
                    "args": "m, nat",
                    "id": "6.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR m"
                },
                {
                    "args": "n, nat",
                    "id": "6.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR n"
                },
                {
                    "args": "m1, nat",
                    "id": "6.2",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR m1"
                },
                {
                    "args": "n1, nat",
                    "id": "6.3",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR n1"
                },
                {
                    "args": "R m m1",
                    "id": "6.4",
                    "prevs": [],
                    "rule": "assume",
                    "th": "R m m1 ⊢ R m m1"
                },
                {
                    "args": "R n n1",
                    "id": "6.5",
                    "prevs": [],
                    "rule": "assume",
                    "th": "R n n1 ⊢ R n n1"
                },
                {
                    "args": "m, m1, n, n1",
                    "id": "6.6",
                    "prevs": [
                        "0",
                        "6.4",
                        "6.5"
                    ],
                    "rule": "apply_fact_for",
                    "th": "finite s, R m m1, R n n1, ¬(s = ∅), ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ R (m * n) (m1 * n1)"
                },
                {
                    "args": "",
                    "id": "6.7",
                    "prevs": [
                        "6.0",
                        "6.1",
                        "6.2",
                        "6.3",
                        "6.4",
                        "6.5",
                        "6.6"
                    ],
                    "rule": "intros",
                    "th": "finite s, ¬(s = ∅), ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ ∀m. ∀n. ∀m1. ∀n1. R m m1 ⟶ R n n1 ⟶ R (m * n) (m1 * n1)"
                },
                {
                    "args": "",
                    "id": "7",
                    "prevs": [
                        "4",
                        "5",
                        "6",
                        "1",
                        "2",
                        "3"
                    ],
                    "rule": "apply_fact",
                    "th": "finite s, ¬(s = ∅), ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ R (iterate times s f) (iterate times s g)"
                },
                {
                    "args": "nproduct_def, R (nproduct s f) (nproduct s g)",
                    "id": "8",
                    "prevs": [
                        "7"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ¬(s = ∅), ∀i. i ∈ s ⟶ R (f i) (g i), ∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1) ⊢ R (nproduct s f) (nproduct s g)"
                },
                {
                    "args": "",
                    "id": "9",
                    "prevs": [
                        "0",
                        "1",
                        "2",
                        "3",
                        "8"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1)) ⟶ finite s ⟶ ¬(s = ∅) ⟶ (∀i. i ∈ s ⟶ R (f i) (g i)) ⟶ R (nproduct s f) (nproduct s g)"
                }
            ],
            "prop": [
                "(∀m. ∀n. ∀m1. ∀n1. R m n ⟶ R m1 n1 ⟶ R (m * m1) (n * n1)) ",
                "⟶ finite s ",
                "   ⟶ ¬(s = ∅) ⟶ (∀i. i ∈ s ⟶ R (f i) (g i)) ⟶ R (nproduct s f) (nproduct s g)"
            ],
            "steps": [
                {
                    "goal_id": "4",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "4",
                    "method_name": "apply_forward_step",
                    "param_R": "R",
                    "param_f": "f",
                    "param_g": "g",
                    "param_op": "(times::nat=>nat=>nat)",
                    "param_s": "s",
                    "theorem": "iterate_related_nonempty"
                },
                {
                    "fact_ids": [
                        "4"
                    ],
                    "goal_id": "5",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "5",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                },
                {
                    "goal_id": "6",
                    "method_name": "introduction",
                    "names": "m,n,m1,n1"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "6.6",
                    "method_name": "apply_prev"
                }
            ],
            "ty": "thm",
            "vars": {
                "R": "nat ⇒ nat ⇒ bool",
                "f": "'a ⇒ nat",
                "g": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_clauses_left",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "m ≤ n",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "m ≤ n ⊢ m ≤ n"
                },
                {
                    "args": "natseg_lrec",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem",
                    "th": "m ≤ n ⊢ {m..n} = insert m ({m + 1..n})"
                },
                {
                    "args": "nproduct_insert, {f: f, s: {m + 1..n}, x: m}",
                    "id": "2",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m + 1..n}) ⟶ nproduct (insert m ({m + 1..n})) f = (if m ∈ {m + 1..n} then nproduct ({m + 1..n}) f else f m * nproduct ({m + 1..n}) f)"
                },
                {
                    "args": "finite_natseg, {m: m + 1, n: n}",
                    "id": "3",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m + 1..n})"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "2",
                        "3"
                    ],
                    "rule": "apply_fact",
                    "th": "⊢ nproduct (insert m ({m + 1..n})) f = (if m ∈ {m + 1..n} then nproduct ({m + 1..n}) f else f m * nproduct ({m + 1..n}) f)"
                },
                {
                    "args": "¬(m + 1 ≤ m)",
                    "id": "5",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ¬(m + 1 ≤ m)"
                },
                {
                    "args": "disjI1, {A: ¬(m + 1 ≤ m), B: ¬(m ≤ n)}",
                    "id": "6",
                    "prevs": [
                        "5"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "m ≤ n ⊢ ¬(m + 1 ≤ m) ∨ ¬(m ≤ n)"
                },
                {
                    "args": "de_morgan_thm1, ¬(m + 1 ≤ m ∧ m ≤ n)",
                    "id": "7",
                    "prevs": [
                        "6"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ n ⊢ ¬(m + 1 ≤ m ∧ m ≤ n)"
                },
                {
                    "args": "member_natseg, ¬(m ∈ {m + 1..n})",
                    "id": "8",
                    "prevs": [
                        "7"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ n ⊢ ¬(m ∈ {m + 1..n})"
                },
                {
                    "args": "if_not_P",
                    "id": "9",
                    "prevs": [
                        "4",
                        "8"
                    ],
                    "rule": "rewrite_fact",
                    "th": "m ≤ n ⊢ nproduct (insert m ({m + 1..n})) f = f m * nproduct ({m + 1..n}) f"
                },
                {
                    "args": "nproduct ({m..n}) f = f m * nproduct ({m + 1..n}) f",
                    "id": "10",
                    "prevs": [
                        "1",
                        "9"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "m ≤ n ⊢ nproduct ({m..n}) f = f m * nproduct ({m + 1..n}) f"
                },
                {
                    "args": "",
                    "id": "11",
                    "prevs": [
                        "0",
                        "10"
                    ],
                    "rule": "intros",
                    "th": "⊢ m ≤ n ⟶ nproduct ({m..n}) f = f m * nproduct ({m + 1..n}) f"
                }
            ],
            "prop": "m ≤ n ⟶ nproduct ({m..n}) f = f m * nproduct ({m + 1..n}) f",
            "steps": [
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "apply_forward_step",
                    "theorem": "natseg_lrec"
                },
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_forward_step",
                    "param_f": "f",
                    "param_s": "{m + 1..n}",
                    "param_x": "m",
                    "theorem": "nproduct_insert"
                },
                {
                    "goal": "finite ({m + 1..n})",
                    "goal_id": "3",
                    "method_name": "cut"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "fact_ids": [
                        "2",
                        "3"
                    ],
                    "goal_id": "4",
                    "method_name": "apply_fact"
                },
                {
                    "goal": "~(m ∈ {m + 1..n})",
                    "goal_id": "5",
                    "method_name": "cut"
                },
                {
                    "goal_id": "5",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "member_natseg"
                },
                {
                    "goal_id": "5",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "de_morgan_thm1"
                },
                {
                    "goal_id": "5",
                    "method_name": "apply_backward_step",
                    "theorem": "disjI1"
                },
                {
                    "goal_id": "5",
                    "method_name": "z3"
                },
                {
                    "fact_ids": [
                        "4",
                        "8"
                    ],
                    "goal_id": "9",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_clauses_right",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "",
                    "id": "0",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ (0::nat) < 0 ⟶ m ≤ 0 ⟶ nproduct ({m..0}) f = nproduct ({m..0 - 1}) f * f 0"
                },
                {
                    "args": "(0::nat) < 0",
                    "id": "0.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "(0::nat) < 0 ⊢ (0::nat) < 0"
                },
                {
                    "args": "m ≤ 0",
                    "id": "0.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "m ≤ 0 ⊢ m ≤ 0"
                },
                {
                    "args": "nproduct ({m..0}) f = nproduct ({m..0 - 1}) f * f 0",
                    "id": "0.2",
                    "prevs": [
                        "0.0"
                    ],
                    "rule": "z3",
                    "th": "(0::nat) < 0 ⊢ nproduct ({m..0}) f = nproduct ({m..0 - 1}) f * f 0"
                },
                {
                    "args": "",
                    "id": "0.3",
                    "prevs": [
                        "0.0",
                        "0.1",
                        "0.2"
                    ],
                    "rule": "intros",
                    "th": "⊢ (0::nat) < 0 ⟶ m ≤ 0 ⟶ nproduct ({m..0}) f = nproduct ({m..0 - 1}) f * f 0"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ ∀n. (0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n) ⟶ 0 < Suc n ⟶ m ≤ Suc n ⟶ nproduct ({m..Suc n}) f = nproduct ({m..Suc n - 1}) f * f (Suc n)"
                },
                {
                    "args": "n, nat",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR n"
                },
                {
                    "args": "0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n ⊢ 0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n"
                },
                {
                    "args": "0 < Suc n",
                    "id": "1.2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "0 < Suc n ⊢ 0 < Suc n"
                },
                {
                    "args": "m ≤ Suc n",
                    "id": "1.3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "m ≤ Suc n ⊢ m ≤ Suc n"
                },
                {
                    "args": "if_P, (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f) = nproduct ({m..n}) f * f (Suc n)",
                    "id": "1.4",
                    "prevs": [
                        "1.3"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ Suc n ⊢ (if m ≤ Suc n then nproduct ({m..n}) f * f (Suc n) else nproduct ({m..n}) f) = nproduct ({m..n}) f * f (Suc n)"
                },
                {
                    "args": "nproduct_Suc_natseg, nproduct ({m..Suc n}) f = nproduct ({m..n}) f * f (Suc n)",
                    "id": "1.5",
                    "prevs": [
                        "1.4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "0 < Suc n, m ≤ Suc n, 0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n ⊢ nproduct ({m..Suc n}) f = nproduct ({m..n}) f * f (Suc n)"
                },
                {
                    "args": "suc_sub1, nproduct ({m..Suc n}) f = nproduct ({m..Suc n - 1}) f * f (Suc n)",
                    "id": "1.6",
                    "prevs": [
                        "1.5"
                    ],
                    "rule": "rewrite_goal",
                    "th": "0 < Suc n, m ≤ Suc n, 0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n ⊢ nproduct ({m..Suc n}) f = nproduct ({m..Suc n - 1}) f * f (Suc n)"
                },
                {
                    "args": "",
                    "id": "1.7",
                    "prevs": [
                        "1.0",
                        "1.1",
                        "1.2",
                        "1.3",
                        "1.6"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀n. (0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n) ⟶ 0 < Suc n ⟶ m ≤ Suc n ⟶ nproduct ({m..Suc n}) f = nproduct ({m..Suc n - 1}) f * f (Suc n)"
                },
                {
                    "args": "nat_induct, {P: λn. 0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n, x: n}",
                    "id": "2",
                    "prevs": [
                        "0",
                        "1"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ 0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n"
                },
                {
                    "args": "",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "intros",
                    "th": "⊢ 0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n"
                }
            ],
            "prop": "0 < n ⟶ m ≤ n ⟶ nproduct ({m..n}) f = nproduct ({m..n - 1}) f * f n",
            "steps": [
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "revert_intro"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "revert_intro"
                },
                {
                    "goal_id": "0",
                    "method_name": "induction",
                    "theorem": "nat_induct",
                    "var": "n"
                },
                {
                    "goal_id": "0",
                    "method_name": "introduction"
                },
                {
                    "fact_ids": [
                        "0.0"
                    ],
                    "goal_id": "0.2",
                    "method_name": "z3"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction",
                    "names": "n"
                },
                {
                    "goal_id": "1.4",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "suc_sub1"
                },
                {
                    "goal_id": "1.4",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_Suc_natseg"
                },
                {
                    "fact_ids": [
                        "1.3"
                    ],
                    "goal_id": "1.4",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "if_P"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_superset",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "u ⊆ v",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "u ⊆ v ⊢ u ⊆ v"
                },
                {
                    "args": "∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = 1",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = 1 ⊢ ∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = 1"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "2",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "neutral_nat_mul, ∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = neutral times",
                    "id": "3",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "u ⊆ v, ∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = 1 ⊢ ∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = neutral times"
                },
                {
                    "args": "iterate_superset",
                    "id": "4",
                    "prevs": [
                        "2",
                        "0",
                        "3"
                    ],
                    "rule": "apply_theorem",
                    "th": "u ⊆ v, ∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = 1 ⊢ iterate times v f = iterate times u f"
                },
                {
                    "args": "nproduct_def, nproduct v f = nproduct u f",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "u ⊆ v, ∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = 1 ⊢ nproduct v f = nproduct u f"
                },
                {
                    "args": "",
                    "id": "6",
                    "prevs": [
                        "0",
                        "1",
                        "5"
                    ],
                    "rule": "intros",
                    "th": "⊢ u ⊆ v ⟶ (∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = 1) ⟶ nproduct v f = nproduct u f"
                }
            ],
            "prop": "u ⊆ v ⟶ (∀x. x ∈ v ⟶ ¬(x ∈ u) ⟶ f x = 1) ⟶ nproduct v f = nproduct u f",
            "steps": [
                {
                    "goal_id": "2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_superset"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                },
                {
                    "goal_id": "3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "u": "'a set",
                "v": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "nproduct_univ",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "support times f univ ⊆ s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "support times f univ ⊆ s ⊢ support times f univ ⊆ s"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_univ",
                    "id": "2",
                    "prevs": [
                        "1",
                        "0"
                    ],
                    "rule": "apply_theorem",
                    "th": "support times f univ ⊆ s ⊢ iterate times s f = iterate times univ f"
                },
                {
                    "args": "nproduct_def, nproduct s f = nproduct univ f",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "support times f univ ⊆ s ⊢ nproduct s f = nproduct univ f"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ support times f univ ⊆ s ⟶ nproduct s f = nproduct univ f"
                }
            ],
            "prop": "support times f univ ⊆ s ⟶ nproduct s f = nproduct univ f",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_univ"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_pair",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "monoidal_nat_mul",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_pair, {f: f, m: m, n: n, op: (times::nat ⇒ nat ⇒ nat)}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ iterate times ({2 * m..2 * n + 1}) f = iterate times ({m..n}) (λi. f (2 * i) * f (2 * i + 1))"
                },
                {
                    "args": "nproduct_def, nproduct ({2 * m..2 * n + 1}) f = nproduct ({m..n}) (λi. f (2 * i) * f (2 * i + 1))",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({2 * m..2 * n + 1}) f = nproduct ({m..n}) (λi. f (2 * i) * f (2 * i + 1))"
                },
                {
                    "args": "",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({2 * m..2 * n + 1}) f = nproduct ({m..n}) (λi. f (2 * i) * f (2 * i + 1))"
                }
            ],
            "prop": [
                "nproduct ({2 * m..2 * n + 1}) f = ",
                "nproduct ({m..n}) (λi. f (2 * i) * f (2 * i + 1))"
            ],
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_pair"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ nat",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_reflect",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "monoidal_nat_mul",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_reflect, {m: m, n: n, op: (times::nat ⇒ nat ⇒ nat), x: x}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ iterate times ({m..n}) x = (if n < m then neutral times else iterate times ({0..n - m}) (λi. x (n - i)))"
                },
                {
                    "args": "neutral_nat_mul, iterate times ({m..n}) x = (if n < m then 1 else iterate times ({0..n - m}) (λi. x (n - i)))",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal_sym",
                    "th": "⊢ iterate times ({m..n}) x = (if n < m then 1 else iterate times ({0..n - m}) (λi. x (n - i)))"
                },
                {
                    "args": "nproduct_def, nproduct ({m..n}) x = (if n < m then 1 else nproduct ({0..n - m}) (λi. x (n - i)))",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({m..n}) x = (if n < m then 1 else nproduct ({0..n - m}) (λi. x (n - i)))"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({m..n}) x = (if n < m then 1 else nproduct ({0..n - m}) (λi. x (n - i)))"
                }
            ],
            "prop": [
                "nproduct ({m..n}) x = ",
                "(if n < m then 1 else nproduct ({0..n - m}) (λi. x (n - i)))"
            ],
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "true",
                    "theorem": "neutral_nat_mul"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_reflect"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "m": "nat",
                "n": "nat",
                "x": "nat ⇒ nat"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "nproduct_delete",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "a ∈ s",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "a ∈ s ⊢ a ∈ s"
                },
                {
                    "args": "monoidal_nat_mul",
                    "id": "2",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_delete, {a: a, f: f, op: (times::nat ⇒ nat ⇒ nat), s: s}",
                    "id": "3",
                    "prevs": [
                        "2",
                        "0",
                        "1"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s, a ∈ s ⊢ f a * iterate times (delete s a) f = iterate times s f"
                },
                {
                    "args": "nproduct_def, f a * nproduct (delete s a) f = nproduct s f",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, a ∈ s ⊢ f a * nproduct (delete s a) f = nproduct s f"
                },
                {
                    "args": "",
                    "id": "5",
                    "prevs": [
                        "0",
                        "1",
                        "4"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ a ∈ s ⟶ f a * nproduct (delete s a) f = nproduct s f"
                }
            ],
            "prop": "finite s ⟶ a ∈ s ⟶ f a * nproduct (delete s a) f = nproduct s f",
            "steps": [
                {
                    "goal_id": "2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_delete"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "a": "'a",
                "f": "'a ⇒ nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_fact",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀i::nat. 1 ≤ i ⟶ i ≤ 0 ⟶ i = 1",
                    "id": "0",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ∀i::nat. 1 ≤ i ⟶ i ≤ 0 ⟶ i = 1"
                },
                {
                    "args": "nproduct_eq_1_natseg",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem",
                    "th": "⊢ nproduct ({1..0}) (λi. i) = 1"
                },
                {
                    "args": "fact_def_1, nproduct ({1..0}) (λm. m) = fact 0",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct ({1..0}) (λm. m) = fact 0"
                },
                {
                    "args": "",
                    "id": "3",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "⊢ ∀n. nproduct ({1..n}) (λm. m) = fact n ⟶ nproduct ({1..Suc n}) (λm. m) = fact (Suc n)"
                },
                {
                    "args": "n, nat",
                    "id": "3.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR n"
                },
                {
                    "args": "nproduct ({1..n}) (λm. m) = fact n",
                    "id": "3.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "nproduct ({1..n}) (λm. m) = fact n ⊢ nproduct ({1..n}) (λm. m) = fact n"
                },
                {
                    "args": "nat_zero_Suc_neq, {n: n}",
                    "id": "3.2",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ ¬(0 = Suc n)"
                },
                {
                    "args": "ineq_sym",
                    "id": "3.3",
                    "prevs": [
                        "3.2"
                    ],
                    "rule": "apply_theorem",
                    "th": "nproduct ({1..n}) (λm. m) = fact n ⊢ ¬(Suc n = 0)"
                },
                {
                    "args": "le_1_2",
                    "id": "3.4",
                    "prevs": [
                        "3.3"
                    ],
                    "rule": "apply_theorem",
                    "th": "nproduct ({1..n}) (λm. m) = fact n ⊢ 1 ≤ Suc n"
                },
                {
                    "args": "mult_comm, {x: fact n, y: Suc n}",
                    "id": "3.5",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ fact n * Suc n = Suc n * fact n"
                },
                {
                    "args": "fact_def_2, fact n * Suc n = fact (Suc n)",
                    "id": "3.6",
                    "prevs": [
                        "3.5"
                    ],
                    "rule": "rewrite_goal",
                    "th": "nproduct ({1..n}) (λm. m) = fact n ⊢ fact n * Suc n = fact (Suc n)"
                },
                {
                    "args": "nproduct ({1..n}) (λm. m) * Suc n = fact (Suc n)",
                    "id": "3.7",
                    "prevs": [
                        "3.1",
                        "3.6"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "nproduct ({1..n}) (λm. m) = fact n ⊢ nproduct ({1..n}) (λm. m) * Suc n = fact (Suc n)"
                },
                {
                    "args": "if_P, (if 1 ≤ Suc n then nproduct ({1..n}) (λm. m) * Suc n else nproduct ({1..n}) (λm. m)) = fact (Suc n)",
                    "id": "3.8",
                    "prevs": [
                        "3.7",
                        "3.4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "nproduct ({1..n}) (λm. m) = fact n ⊢ (if 1 ≤ Suc n then nproduct ({1..n}) (λm. m) * Suc n else nproduct ({1..n}) (λm. m)) = fact (Suc n)"
                },
                {
                    "args": "nproduct_Suc_natseg, nproduct ({1..Suc n}) (λm. m) = fact (Suc n)",
                    "id": "3.9",
                    "prevs": [
                        "3.8"
                    ],
                    "rule": "rewrite_goal",
                    "th": "nproduct ({1..n}) (λm. m) = fact n ⊢ nproduct ({1..Suc n}) (λm. m) = fact (Suc n)"
                },
                {
                    "args": "",
                    "id": "3.10",
                    "prevs": [
                        "3.0",
                        "3.1",
                        "3.9"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀n. nproduct ({1..n}) (λm. m) = fact n ⟶ nproduct ({1..Suc n}) (λm. m) = fact (Suc n)"
                },
                {
                    "args": "nat_induct, {P: λn. nproduct ({1..n}) (λm. m) = fact n, x: n}",
                    "id": "4",
                    "prevs": [
                        "2",
                        "3"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ nproduct ({1..n}) (λm. m) = fact n"
                },
                {
                    "args": "",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct ({1..n}) (λm. m) = fact n"
                }
            ],
            "prop": "nproduct ({1..n}) (λm. m) = fact n",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "induction",
                    "theorem": "nat_induct",
                    "var": "n"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "fact_def_1"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "nproduct_eq_1_natseg"
                },
                {
                    "goal_id": "0",
                    "method_name": "z3"
                },
                {
                    "goal_id": "3",
                    "method_name": "introduction",
                    "names": "n"
                },
                {
                    "goal_id": "3.2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_Suc_natseg"
                },
                {
                    "goal": "1 ≤ Suc n",
                    "goal_id": "3.2",
                    "method_name": "cut"
                },
                {
                    "goal_id": "3.2",
                    "method_name": "apply_backward_step",
                    "theorem": "le_1_2"
                },
                {
                    "goal_id": "3.2",
                    "method_name": "apply_backward_step",
                    "theorem": "ineq_sym"
                },
                {
                    "goal_id": "3.2",
                    "method_name": "apply_backward_step",
                    "theorem": "nat_zero_Suc_neq"
                },
                {
                    "fact_ids": [
                        "3.4"
                    ],
                    "goal_id": "3.5",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "if_P"
                },
                {
                    "fact_ids": [
                        "3.1"
                    ],
                    "goal_id": "3.5",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "3.5",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "fact_def_2"
                },
                {
                    "goal_id": "3.5",
                    "method_name": "apply_backward_step",
                    "theorem": "mult_comm"
                }
            ],
            "ty": "thm",
            "vars": {
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "nproduct_delta",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "monoidal_nat_mul",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::nat ⇒ nat ⇒ nat)"
                },
                {
                    "args": "iterate_delta, {a: a, f: λx::'a. b, op: (times::nat ⇒ nat ⇒ nat), s: s}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ iterate times s (λx. if x = a then b else neutral times) = (if a ∈ s then b else neutral times)"
                },
                {
                    "args": "neutral_nat_mul, iterate times s (λx. if x = a then b else 1) = (if a ∈ s then b else 1)",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal_sym",
                    "th": "⊢ iterate times s (λx. if x = a then b else 1) = (if a ∈ s then b else 1)"
                },
                {
                    "args": "nproduct_def, nproduct s (λx. if x = a then b else 1) = (if a ∈ s then b else 1)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ nproduct s (λx. if x = a then b else 1) = (if a ∈ s then b else 1)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ nproduct s (λx. if x = a then b else 1) = (if a ∈ s then b else 1)"
                }
            ],
            "prop": "nproduct s (λx. if x = a then b else 1) = (if a ∈ s then b else 1)",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "nproduct_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "true",
                    "theorem": "neutral_nat_mul"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_delta"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_nat_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "a": "'a",
                "b": "nat",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "iproduct",
            "prop": "(iproduct::'a set ⇒ ('a ⇒ int) ⇒ int) = iterate times",
            "ty": "def",
            "type": "'a set ⇒ ('a ⇒ int) ⇒ int"
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "neutral_int_mul",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "theI, {P: λx::int. ∀y. x * y = y ∧ y * x = y, a: (1::int)}",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ (∀y::int. 1 * y = y ∧ y * 1 = y) ⟶ (∀x::int. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1) ⟶ (∀y::int. (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) * y = y ∧ y * (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) = y)"
                },
                {
                    "args": "∀y::int. 1 * y = y ∧ y * 1 = y",
                    "id": "1",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ∀y::int. 1 * y = y ∧ y * 1 = y"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [
                        "0",
                        "1"
                    ],
                    "rule": "apply_fact",
                    "th": "⊢ (∀x::int. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1) ⟶ (∀y::int. (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) * y = y ∧ y * (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) = y)"
                },
                {
                    "args": "∀x::int. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1",
                    "id": "3",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ∀x::int. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "2",
                        "3"
                    ],
                    "rule": "apply_fact",
                    "th": "⊢ ∀y::int. (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) * y = y ∧ y * (THE x. ∀y1. x * y1 = y1 ∧ y1 * x = y1) = y"
                },
                {
                    "args": "THE x::int. ∀y. x * y = y ∧ y * x = y",
                    "id": "5",
                    "prevs": [
                        "3",
                        "4"
                    ],
                    "rule": "apply_fact_for",
                    "th": "⊢ (THE x::int. ∀y. x * y = y ∧ y * x = y) = 1"
                },
                {
                    "args": "neutral_def, neutral times = (1::int)",
                    "id": "6",
                    "prevs": [
                        "5"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ neutral times = (1::int)"
                },
                {
                    "args": "",
                    "id": "7",
                    "prevs": [
                        "6"
                    ],
                    "rule": "intros",
                    "th": "⊢ neutral times = (1::int)"
                }
            ],
            "prop": "neutral times = (1::int)",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_forward_step",
                    "param_P": "(%x::int. ∀y. x * y = y ∧ y * x = y)",
                    "param_a": "(1::int)",
                    "theorem": "theI"
                },
                {
                    "goal": "(∀y::int. 1 * y = y ∧ y * 1 = y)",
                    "goal_id": "1",
                    "method_name": "cut"
                },
                {
                    "goal_id": "1",
                    "method_name": "z3"
                },
                {
                    "fact_ids": [
                        "0",
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "apply_fact"
                },
                {
                    "goal": "(∀x::int. (∀y. x * y = y ∧ y * x = y) ⟶ x = 1)",
                    "goal_id": "3",
                    "method_name": "cut"
                },
                {
                    "goal_id": "3",
                    "method_name": "z3"
                },
                {
                    "fact_ids": [
                        "2",
                        "3"
                    ],
                    "goal_id": "4",
                    "method_name": "apply_fact"
                },
                {
                    "fact_ids": [
                        "3"
                    ],
                    "goal_id": "5",
                    "method_name": "apply_prev"
                }
            ],
            "ty": "thm",
            "vars": {}
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "monoidal_int_mul",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀x::int. ∀y. x * y = y * x",
                    "id": "0",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ∀x::int. ∀y. x * y = y * x"
                },
                {
                    "args": "∀x::int. ∀y. ∀z. x * (y * z) = x * y * z",
                    "id": "1",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ∀x::int. ∀y. ∀z. x * (y * z) = x * y * z"
                },
                {
                    "args": "∀x::int. 1 * x = x",
                    "id": "2",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ ∀x::int. 1 * x = x"
                },
                {
                    "args": "neutral_int_mul, ∀x::int. neutral times * x = x",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ ∀x::int. neutral times * x = x"
                },
                {
                    "args": "conjI",
                    "id": "4",
                    "prevs": [
                        "1",
                        "3"
                    ],
                    "rule": "apply_theorem",
                    "th": "⊢ (∀x::int. ∀y. ∀z. x * (y * z) = x * y * z) ∧ (∀x::int. neutral times * x = x)"
                },
                {
                    "args": "conjI",
                    "id": "5",
                    "prevs": [
                        "0",
                        "4"
                    ],
                    "rule": "apply_theorem",
                    "th": "⊢ (∀x::int. ∀y. x * y = y * x) ∧ (∀x::int. ∀y. ∀z. x * (y * z) = x * y * z) ∧ (∀x::int. neutral times * x = x)"
                },
                {
                    "args": "monoidal_def, monoidal (times::int ⇒ int ⇒ int)",
                    "id": "6",
                    "prevs": [
                        "5"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ monoidal (times::int ⇒ int ⇒ int)"
                },
                {
                    "args": "",
                    "id": "7",
                    "prevs": [
                        "6"
                    ],
                    "rule": "intros",
                    "th": "⊢ monoidal (times::int ⇒ int ⇒ int)"
                }
            ],
            "prop": "monoidal (times::int ⇒ int ⇒ int)",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "monoidal_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "conjI"
                },
                {
                    "goal_id": "0",
                    "method_name": "z3"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "conjI"
                },
                {
                    "goal_id": "1",
                    "method_name": "z3"
                },
                {
                    "goal_id": "2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "neutral_int_mul"
                },
                {
                    "goal_id": "2",
                    "method_name": "z3"
                }
            ],
            "ty": "thm",
            "vars": {}
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "iproduct_empty",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "monoidal_int_mul",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::int ⇒ int ⇒ int)"
                },
                {
                    "args": "iterate_empty, {f: f, op: (times::int ⇒ int ⇒ int)}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ iterate times ∅ f = neutral times"
                },
                {
                    "args": "neutral_int_mul, iterate times ∅ f = 1",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal_sym",
                    "th": "⊢ iterate times ∅ f = 1"
                },
                {
                    "args": "iproduct_def, iproduct ∅ f = 1",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ iproduct ∅ f = 1"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ iproduct ∅ f = 1"
                }
            ],
            "prop": "iproduct ∅ f = 1",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "iproduct_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "true",
                    "theorem": "neutral_int_mul"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_empty"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_int_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ int"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "iproduct_insert",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "monoidal_int_mul",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::int ⇒ int ⇒ int)"
                },
                {
                    "args": "iterate_insert, {f: f, op: (times::int ⇒ int ⇒ int), s: s, x: x}",
                    "id": "2",
                    "prevs": [
                        "1",
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ iterate times (insert x s) f = (if x ∈ s then iterate times s f else f x * iterate times s f)"
                },
                {
                    "args": "iproduct_def, iproduct (insert x s) f = (if x ∈ s then iproduct s f else f x * iproduct s f)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s ⊢ iproduct (insert x s) f = (if x ∈ s then iproduct s f else f x * iproduct s f)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ iproduct (insert x s) f = (if x ∈ s then iproduct s f else f x * iproduct s f)"
                }
            ],
            "prop": [
                "finite s ",
                "⟶ iproduct (insert x s) f = (if x ∈ s then iproduct s f else f x * iproduct s f)"
            ],
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "iproduct_def"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_insert"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_int_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ int",
                "s": "'a set",
                "x": "'a"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "product",
            "prop": "(product::'a set ⇒ ('a ⇒ real) ⇒ real) = iterate times",
            "ty": "def",
            "type": "'a set ⇒ ('a ⇒ real) ⇒ real"
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "product_empty",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "monoidal_real_mul",
                    "id": "0",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::real ⇒ real ⇒ real)"
                },
                {
                    "args": "iterate_empty, {f: f, op: (times::real ⇒ real ⇒ real)}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "⊢ iterate times ∅ f = neutral times"
                },
                {
                    "args": "neutral_real_mul, iterate times ∅ f = 1",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "rewrite_goal_sym",
                    "th": "⊢ iterate times ∅ f = 1"
                },
                {
                    "args": "product_def, product ∅ f = 1",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ product ∅ f = 1"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ product ∅ f = 1"
                }
            ],
            "prop": "product ∅ f = 1",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "product_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "true",
                    "theorem": "neutral_real_mul"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_empty"
                },
                {
                    "goal_id": "0",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_real_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ real"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "product_insert",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "monoidal_real_mul",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::real ⇒ real ⇒ real)"
                },
                {
                    "args": "iterate_insert, {f: f, op: (times::real ⇒ real ⇒ real), s: s, x: x}",
                    "id": "2",
                    "prevs": [
                        "1",
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ iterate times (insert x s) f = (if x ∈ s then iterate times s f else f x * iterate times s f)"
                },
                {
                    "args": "product_def, product (insert x s) f = (if x ∈ s then product s f else f x * product s f)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s ⊢ product (insert x s) f = (if x ∈ s then product s f else f x * product s f)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ product (insert x s) f = (if x ∈ s then product s f else f x * product s f)"
                }
            ],
            "prop": [
                "finite s ",
                "⟶ product (insert x s) f = (if x ∈ s then product s f else f x * product s f)"
            ],
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "product_def"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_insert"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_real_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ real",
                "s": "'a set",
                "x": "'a"
            }
        },
        {
            "attributes": [
                "hint_rewrite"
            ],
            "name": "product_support",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "iterate_support, iterate times (support times f s) f = iterate times s f",
                    "id": "0",
                    "prevs": [],
                    "rule": "rewrite_goal",
                    "th": "⊢ iterate times (support times f s) f = iterate times s f"
                },
                {
                    "args": "product_def, product (support times f s) f = product s f",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "rewrite_goal",
                    "th": "⊢ product (support times f s) f = product s f"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [
                        "1"
                    ],
                    "rule": "intros",
                    "th": "⊢ product (support times f s) f = product s f"
                }
            ],
            "prop": "product (support times f s) f = product s f",
            "steps": [
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "product_def"
                },
                {
                    "goal_id": "0",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "iterate_support"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ real",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "product_union",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "finite t",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite t ⊢ finite t"
                },
                {
                    "args": "disjoint s t",
                    "id": "2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "disjoint s t ⊢ disjoint s t"
                },
                {
                    "args": "monoidal_real_mul",
                    "id": "3",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::real ⇒ real ⇒ real)"
                },
                {
                    "args": "iterate_union, {f: f, op: (times::real ⇒ real ⇒ real), s: s, t: t}",
                    "id": "4",
                    "prevs": [
                        "3",
                        "0",
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s, finite t, disjoint s t ⊢ iterate times (s ∪ t) f = iterate times s f * iterate times t f"
                },
                {
                    "args": "product_def, product (s ∪ t) f = product s f * product t f",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, finite t, disjoint s t ⊢ product (s ∪ t) f = product s f * product t f"
                },
                {
                    "args": "",
                    "id": "6",
                    "prevs": [
                        "0",
                        "1",
                        "2",
                        "5"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ finite t ⟶ disjoint s t ⟶ product (s ∪ t) f = product s f * product t f"
                }
            ],
            "prop": [
                "finite s ",
                "⟶ finite t ⟶ disjoint s t ⟶ product (s ∪ t) f = product s f * product t f"
            ],
            "steps": [
                {
                    "goal_id": "3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "product_def"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_union"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_real_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ real",
                "s": "'a set",
                "t": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "product_image",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y ⊢ ∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y"
                },
                {
                    "args": "monoidal_real_mul",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::real ⇒ real ⇒ real)"
                },
                {
                    "args": "iterate_image, {f: f, g: g, op: (times::real ⇒ real ⇒ real), s: s}",
                    "id": "2",
                    "prevs": [
                        "1",
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y ⊢ iterate times (image f s) g = iterate times s (g ∘ f)"
                },
                {
                    "args": "product_def, product (image f s) g = product s (g ∘ f)",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y ⊢ product (image f s) g = product s (g ∘ f)"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y) ⟶ product (image f s) g = product s (g ∘ f)"
                }
            ],
            "prop": [
                "(∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ f x = f y ⟶ x = y) ",
                "⟶ product (image f s) g = product s (g ∘ f)"
            ],
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "product_def"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_image"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_real_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ 'b",
                "g": "'b ⇒ real",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "product_injection",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "∀x. x ∈ s ⟶ p x ∈ s",
                    "id": "1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ s ⟶ p x ∈ s ⊢ ∀x. x ∈ s ⟶ p x ∈ s"
                },
                {
                    "args": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y",
                    "id": "2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y ⊢ ∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y"
                },
                {
                    "args": "monoidal_real_mul",
                    "id": "3",
                    "prevs": [],
                    "rule": "apply_theorem",
                    "th": "⊢ monoidal (times::real ⇒ real ⇒ real)"
                },
                {
                    "args": "iterate_injection, {f: f, op: (times::real ⇒ real ⇒ real), p: p, s: s}",
                    "id": "4",
                    "prevs": [
                        "3",
                        "0",
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s, ∀x. x ∈ s ⟶ p x ∈ s, ∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y ⊢ iterate times s (f ∘ p) = iterate times s f"
                },
                {
                    "args": "product_def, product s (f ∘ p) = product s f",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ∀x. x ∈ s ⟶ p x ∈ s, ∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y ⊢ product s (f ∘ p) = product s f"
                },
                {
                    "args": "",
                    "id": "6",
                    "prevs": [
                        "0",
                        "1",
                        "2",
                        "5"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ (∀x. x ∈ s ⟶ p x ∈ s) ⟶ (∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y) ⟶ product s (f ∘ p) = product s f"
                }
            ],
            "prop": [
                "finite s ",
                "⟶ (∀x. x ∈ s ⟶ p x ∈ s) ",
                "   ⟶ (∀x. ∀y. x ∈ s ⟶ y ∈ s ⟶ p x = p y ⟶ x = y) ⟶ product s (f ∘ p) = product s f"
            ],
            "steps": [
                {
                    "goal_id": "3",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "product_def"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "iterate_injection"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "monoidal_real_mul"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ real",
                "p": "'a ⇒ 'a",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "product_add_split",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "m ≤ n + 1",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "m ≤ n + 1 ⊢ m ≤ n + 1"
                },
                {
                    "args": "natseg_add_split, {p: p}",
                    "id": "1",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "m ≤ n + 1 ⊢ {m..n + p} = {m..n} ∪ {n + 1..n + p}"
                },
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "2",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "finite_natseg, {m: n + 1, n: n + p}",
                    "id": "3",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({n + 1..n + p})"
                },
                {
                    "args": "n < n + 1 ∨ n + p < m ∨ n < m ∨ n + p < n + 1",
                    "id": "4",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ n < n + 1 ∨ n + p < m ∨ n < m ∨ n + p < n + 1"
                },
                {
                    "args": "disjoint_natseg, disjoint ({m..n}) ({n + 1..n + p})",
                    "id": "5",
                    "prevs": [
                        "4"
                    ],
                    "rule": "rewrite_goal",
                    "th": "m ≤ n + 1 ⊢ disjoint ({m..n}) ({n + 1..n + p})"
                },
                {
                    "args": "product_union, {f: f, s: {m..n}, t: {n + 1..n + p}}",
                    "id": "6",
                    "prevs": [
                        "2",
                        "3",
                        "5"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "m ≤ n + 1 ⊢ product ({m..n} ∪ {n + 1..n + p}) f = product ({m..n}) f * product ({n + 1..n + p}) f"
                },
                {
                    "args": "product ({m..n + p}) f = product ({m..n}) f * product ({n + 1..n + p}) f",
                    "id": "7",
                    "prevs": [
                        "1",
                        "6"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "m ≤ n + 1 ⊢ product ({m..n + p}) f = product ({m..n}) f * product ({n + 1..n + p}) f"
                },
                {
                    "args": "",
                    "id": "8",
                    "prevs": [
                        "0",
                        "7"
                    ],
                    "rule": "intros",
                    "th": "⊢ m ≤ n + 1 ⟶ product ({m..n + p}) f = product ({m..n}) f * product ({n + 1..n + p}) f"
                }
            ],
            "prop": [
                "m ≤ n + 1 ",
                "⟶ product ({m..n + p}) f = product ({m..n}) f * product ({n + 1..n + p}) f"
            ],
            "steps": [
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "apply_forward_step",
                    "param_p": "p",
                    "theorem": "natseg_add_split"
                },
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "product_union"
                },
                {
                    "goal_id": "2",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "goal_id": "3",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "goal_id": "4",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "disjoint_natseg"
                },
                {
                    "goal_id": "4",
                    "method_name": "z3"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ real",
                "m": "nat",
                "n": "nat",
                "p": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "product_pos_le",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ (∀x. x ∈ ∅ ⟶ 0 ≤ f x) ⟶ 0 ≤ product ∅ f"
                },
                {
                    "args": "∀x. x ∈ ∅ ⟶ 0 ≤ f x",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ ∅ ⟶ 0 ≤ f x ⊢ ∀x. x ∈ ∅ ⟶ 0 ≤ f x"
                },
                {
                    "args": "(0::real) ≤ 1",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ (0::real) ≤ 1"
                },
                {
                    "args": "product_empty, 0 ≤ product ∅ f",
                    "id": "1.2",
                    "prevs": [
                        "1.1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ∀x. x ∈ ∅ ⟶ 0 ≤ f x ⊢ 0 ≤ product ∅ f"
                },
                {
                    "args": "",
                    "id": "1.3",
                    "prevs": [
                        "1.0",
                        "1.2"
                    ],
                    "rule": "intros",
                    "th": "finite s ⊢ (∀x. x ∈ ∅ ⟶ 0 ≤ f x) ⟶ 0 ≤ product ∅ f"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ ∀x. ∀s1. ((∀x1. x1 ∈ s1 ⟶ 0 ≤ f x1) ⟶ 0 ≤ product s1 f) ⟶ ¬(x ∈ s1) ⟶ finite s1 ⟶ (∀x1. x1 ∈ insert x s1 ⟶ 0 ≤ f x1) ⟶ 0 ≤ product (insert x s1) f"
                },
                {
                    "args": "a, 'a",
                    "id": "2.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR a"
                },
                {
                    "args": "s, 'a set",
                    "id": "2.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR s"
                },
                {
                    "args": "(∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f",
                    "id": "2.2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "(∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f ⊢ (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f"
                },
                {
                    "args": "¬(a ∈ s)",
                    "id": "2.3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(a ∈ s) ⊢ ¬(a ∈ s)"
                },
                {
                    "args": "∀x. x ∈ insert a s ⟶ 0 ≤ f x",
                    "id": "2.4",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ insert a s ⟶ 0 ≤ f x ⊢ ∀x. x ∈ insert a s ⟶ 0 ≤ f x"
                },
                {
                    "args": "product_insert, {f: f, x: a}",
                    "id": "2.5",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ product (insert a s) f = (if a ∈ s then product s f else f a * product s f)"
                },
                {
                    "args": "if_not_P",
                    "id": "2.6",
                    "prevs": [
                        "2.5",
                        "2.3"
                    ],
                    "rule": "rewrite_fact",
                    "th": "finite s, ¬(a ∈ s) ⊢ product (insert a s) f = f a * product s f"
                },
                {
                    "args": "a ∈ insert a s",
                    "id": "2.7",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ a ∈ insert a s"
                },
                {
                    "args": "a",
                    "id": "2.8",
                    "prevs": [
                        "2.4",
                        "2.7"
                    ],
                    "rule": "apply_fact_for",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 ≤ f x, (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f ⊢ 0 ≤ f a"
                },
                {
                    "args": "",
                    "id": "2.9",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 ≤ f x, (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f ⊢ ∀x. x ∈ s ⟶ 0 ≤ f x"
                },
                {
                    "args": "x, 'a",
                    "id": "2.9.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR x"
                },
                {
                    "args": "x ∈ s",
                    "id": "2.9.1",
                    "prevs": [],
                    "rule": "assume",
                    "th": "x ∈ s ⊢ x ∈ s"
                },
                {
                    "args": "x ∈ insert a s",
                    "id": "2.9.2",
                    "prevs": [
                        "2.9.1"
                    ],
                    "rule": "z3",
                    "th": "x ∈ s ⊢ x ∈ insert a s"
                },
                {
                    "args": "x",
                    "id": "2.9.3",
                    "prevs": [
                        "2.4",
                        "2.9.2"
                    ],
                    "rule": "apply_fact_for",
                    "th": "finite s, x ∈ s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 ≤ f x, (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f ⊢ 0 ≤ f x"
                },
                {
                    "args": "",
                    "id": "2.9.4",
                    "prevs": [
                        "2.9.0",
                        "2.9.1",
                        "2.9.3"
                    ],
                    "rule": "intros",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 ≤ f x, (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f ⊢ ∀x. x ∈ s ⟶ 0 ≤ f x"
                },
                {
                    "args": "",
                    "id": "2.10",
                    "prevs": [
                        "2.2",
                        "2.9"
                    ],
                    "rule": "apply_fact",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 ≤ f x, (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f ⊢ 0 ≤ product s f"
                },
                {
                    "args": "real_le_mul",
                    "id": "2.11",
                    "prevs": [
                        "2.8",
                        "2.10"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 ≤ f x, (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f ⊢ 0 ≤ f a * product s f"
                },
                {
                    "args": "0 ≤ product (insert a s) f",
                    "id": "2.12",
                    "prevs": [
                        "2.6",
                        "2.11"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 ≤ f x, (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f ⊢ 0 ≤ product (insert a s) f"
                },
                {
                    "args": "",
                    "id": "2.13",
                    "prevs": [
                        "2.0",
                        "2.1",
                        "2.2",
                        "2.3",
                        "0",
                        "2.4",
                        "2.12"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀a. ∀s. ((∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f) ⟶ ¬(a ∈ s) ⟶ finite s ⟶ (∀x. x ∈ insert a s ⟶ 0 ≤ f x) ⟶ 0 ≤ product (insert a s) f"
                },
                {
                    "args": "finite_induct_strong",
                    "id": "3",
                    "prevs": [
                        "0",
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s ⊢ (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f"
                }
            ],
            "prop": "finite s ⟶ (∀x. x ∈ s ⟶ 0 ≤ f x) ⟶ 0 ≤ product s f",
            "steps": [
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "revert_intro"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_induct_strong"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction"
                },
                {
                    "goal_id": "1.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "product_empty"
                },
                {
                    "goal_id": "1.1",
                    "method_name": "z3"
                },
                {
                    "goal_id": "2",
                    "method_name": "introduction",
                    "names": "a,s"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "2.5",
                    "method_name": "apply_forward_step",
                    "param_f": "f",
                    "param_x": "a",
                    "theorem": "product_insert"
                },
                {
                    "fact_ids": [
                        "2.5",
                        "2.3"
                    ],
                    "goal_id": "2.6",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "2.6"
                    ],
                    "goal_id": "2.7",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "2.7",
                    "method_name": "apply_backward_step",
                    "theorem": "real_le_mul"
                },
                {
                    "fact_ids": [
                        "2.4"
                    ],
                    "goal_id": "2.7",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2.7",
                    "method_name": "z3"
                },
                {
                    "fact_ids": [
                        "2.2"
                    ],
                    "goal_id": "2.9",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2.9",
                    "method_name": "introduction",
                    "names": "x"
                },
                {
                    "fact_ids": [
                        "2.4"
                    ],
                    "goal_id": "2.9.2",
                    "method_name": "apply_prev"
                },
                {
                    "fact_ids": [
                        "2.9.1"
                    ],
                    "goal_id": "2.9.2",
                    "method_name": "z3"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ real",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "product_pos_le_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 ≤ f x",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 ≤ f x ⊢ ∀x. m ≤ x ⟶ x ≤ n ⟶ 0 ≤ f x"
                },
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "∀x. m ≤ x ∧ x ≤ n ⟶ 0 ≤ f x",
                    "id": "2",
                    "prevs": [
                        "0"
                    ],
                    "rule": "z3",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 ≤ f x ⊢ ∀x. m ≤ x ∧ x ≤ n ⟶ 0 ≤ f x"
                },
                {
                    "args": "member_natseg, ∀x. x ∈ {m..n} ⟶ 0 ≤ f x",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 ≤ f x ⊢ ∀x. x ∈ {m..n} ⟶ 0 ≤ f x"
                },
                {
                    "args": "product_pos_le",
                    "id": "4",
                    "prevs": [
                        "1",
                        "3"
                    ],
                    "rule": "apply_theorem",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 ≤ f x ⊢ 0 ≤ product ({m..n}) f"
                },
                {
                    "args": "",
                    "id": "5",
                    "prevs": [
                        "0",
                        "4"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀x. m ≤ x ⟶ x ≤ n ⟶ 0 ≤ f x) ⟶ 0 ≤ product ({m..n}) f"
                }
            ],
            "prop": "(∀x. m ≤ x ⟶ x ≤ n ⟶ 0 ≤ f x) ⟶ 0 ≤ product ({m..n}) f",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "product_pos_le"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "goal_id": "2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "member_natseg"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "2",
                    "method_name": "z3"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ real",
                "m": "nat",
                "n": "nat"
            }
        },
        {
            "attributes": [
                "hint_backward",
                "hint_backward1"
            ],
            "name": "product_pos_lt",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "finite s",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "finite s ⊢ finite s"
                },
                {
                    "args": "",
                    "id": "1",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ (∀x. x ∈ ∅ ⟶ 0 < f x) ⟶ 0 < product ∅ f"
                },
                {
                    "args": "∀x. x ∈ ∅ ⟶ 0 < f x",
                    "id": "1.0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ ∅ ⟶ 0 < f x ⊢ ∀x. x ∈ ∅ ⟶ 0 < f x"
                },
                {
                    "args": "(0::real) < 1",
                    "id": "1.1",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ (0::real) < 1"
                },
                {
                    "args": "product_empty, 0 < product ∅ f",
                    "id": "1.2",
                    "prevs": [
                        "1.1"
                    ],
                    "rule": "rewrite_goal",
                    "th": "finite s, ∀x. x ∈ ∅ ⟶ 0 < f x ⊢ 0 < product ∅ f"
                },
                {
                    "args": "",
                    "id": "1.3",
                    "prevs": [
                        "1.0",
                        "1.2"
                    ],
                    "rule": "intros",
                    "th": "finite s ⊢ (∀x. x ∈ ∅ ⟶ 0 < f x) ⟶ 0 < product ∅ f"
                },
                {
                    "args": "",
                    "id": "2",
                    "prevs": [],
                    "rule": "subproof",
                    "th": "finite s ⊢ ∀x. ∀s1. ((∀x1. x1 ∈ s1 ⟶ 0 < f x1) ⟶ 0 < product s1 f) ⟶ ¬(x ∈ s1) ⟶ finite s1 ⟶ (∀x1. x1 ∈ insert x s1 ⟶ 0 < f x1) ⟶ 0 < product (insert x s1) f"
                },
                {
                    "args": "a, 'a",
                    "id": "2.0",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR a"
                },
                {
                    "args": "s, 'a set",
                    "id": "2.1",
                    "prevs": [],
                    "rule": "variable",
                    "th": "⊢ _VAR s"
                },
                {
                    "args": "(∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f",
                    "id": "2.2",
                    "prevs": [],
                    "rule": "assume",
                    "th": "(∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f ⊢ (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f"
                },
                {
                    "args": "¬(a ∈ s)",
                    "id": "2.3",
                    "prevs": [],
                    "rule": "assume",
                    "th": "¬(a ∈ s) ⊢ ¬(a ∈ s)"
                },
                {
                    "args": "∀x. x ∈ insert a s ⟶ 0 < f x",
                    "id": "2.4",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. x ∈ insert a s ⟶ 0 < f x ⊢ ∀x. x ∈ insert a s ⟶ 0 < f x"
                },
                {
                    "args": "product_insert, {f: f, x: a}",
                    "id": "2.5",
                    "prevs": [
                        "0"
                    ],
                    "rule": "apply_theorem_for",
                    "th": "finite s ⊢ product (insert a s) f = (if a ∈ s then product s f else f a * product s f)"
                },
                {
                    "args": "if_not_P",
                    "id": "2.6",
                    "prevs": [
                        "2.5",
                        "2.3"
                    ],
                    "rule": "rewrite_fact",
                    "th": "finite s, ¬(a ∈ s) ⊢ product (insert a s) f = f a * product s f"
                },
                {
                    "args": "a ∈ insert a s",
                    "id": "2.7",
                    "prevs": [],
                    "rule": "z3",
                    "th": "⊢ a ∈ insert a s"
                },
                {
                    "args": "a",
                    "id": "2.8",
                    "prevs": [
                        "2.4",
                        "2.7"
                    ],
                    "rule": "apply_fact_for",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 < f x, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f ⊢ 0 < f a"
                },
                {
                    "args": "∀x. x ∈ s ⟶ 0 < f x",
                    "id": "2.9",
                    "prevs": [
                        "2.4"
                    ],
                    "rule": "z3",
                    "th": "∀x. x ∈ insert a s ⟶ 0 < f x ⊢ ∀x. x ∈ s ⟶ 0 < f x"
                },
                {
                    "args": "",
                    "id": "2.10",
                    "prevs": [
                        "2.2",
                        "2.9"
                    ],
                    "rule": "apply_fact",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 < f x, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f ⊢ 0 < product s f"
                },
                {
                    "args": "real_lt_mul",
                    "id": "2.11",
                    "prevs": [
                        "2.8",
                        "2.10"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 < f x, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f ⊢ 0 < f a * product s f"
                },
                {
                    "args": "0 < product (insert a s) f",
                    "id": "2.12",
                    "prevs": [
                        "2.6",
                        "2.11"
                    ],
                    "rule": "rewrite_goal_with_prev",
                    "th": "finite s, ¬(a ∈ s), ∀x. x ∈ insert a s ⟶ 0 < f x, (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f ⊢ 0 < product (insert a s) f"
                },
                {
                    "args": "",
                    "id": "2.13",
                    "prevs": [
                        "2.0",
                        "2.1",
                        "2.2",
                        "2.3",
                        "0",
                        "2.4",
                        "2.12"
                    ],
                    "rule": "intros",
                    "th": "⊢ ∀a. ∀s. ((∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f) ⟶ ¬(a ∈ s) ⟶ finite s ⟶ (∀x. x ∈ insert a s ⟶ 0 < f x) ⟶ 0 < product (insert a s) f"
                },
                {
                    "args": "finite_induct_strong",
                    "id": "3",
                    "prevs": [
                        "0",
                        "1",
                        "2"
                    ],
                    "rule": "apply_theorem",
                    "th": "finite s ⊢ (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f"
                },
                {
                    "args": "",
                    "id": "4",
                    "prevs": [
                        "0",
                        "3"
                    ],
                    "rule": "intros",
                    "th": "⊢ finite s ⟶ (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f"
                }
            ],
            "prop": "finite s ⟶ (∀x. x ∈ s ⟶ 0 < f x) ⟶ 0 < product s f",
            "steps": [
                {
                    "fact_ids": [
                        "1"
                    ],
                    "goal_id": "2",
                    "method_name": "revert_intro"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_induct_strong"
                },
                {
                    "goal_id": "1",
                    "method_name": "introduction"
                },
                {
                    "goal_id": "1.1",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "product_empty"
                },
                {
                    "goal_id": "1.1",
                    "method_name": "z3"
                },
                {
                    "goal_id": "2",
                    "method_name": "introduction",
                    "names": "a,s"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "2.5",
                    "method_name": "apply_forward_step",
                    "param_f": "f",
                    "param_x": "a",
                    "theorem": "product_insert"
                },
                {
                    "fact_ids": [
                        "2.5",
                        "2.3"
                    ],
                    "goal_id": "2.6",
                    "method_name": "rewrite_fact",
                    "sym": "false",
                    "theorem": "if_not_P"
                },
                {
                    "fact_ids": [
                        "2.6"
                    ],
                    "goal_id": "2.7",
                    "method_name": "rewrite_goal_with_prev"
                },
                {
                    "goal_id": "2.7",
                    "method_name": "apply_backward_step",
                    "theorem": "real_lt_mul"
                },
                {
                    "fact_ids": [
                        "2.4"
                    ],
                    "goal_id": "2.7",
                    "method_name": "apply_prev"
                },
                {
                    "goal_id": "2.7",
                    "method_name": "z3"
                },
                {
                    "fact_ids": [
                        "2.2"
                    ],
                    "goal_id": "2.9",
                    "method_name": "apply_prev"
                },
                {
                    "fact_ids": [
                        "2.4"
                    ],
                    "goal_id": "2.9",
                    "method_name": "z3"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "'a ⇒ real",
                "s": "'a set"
            }
        },
        {
            "attributes": [
                "hint_backward"
            ],
            "name": "product_pos_lt_natseg",
            "num_gaps": 0,
            "proof": [
                {
                    "args": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x",
                    "id": "0",
                    "prevs": [],
                    "rule": "assume",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x ⊢ ∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x"
                },
                {
                    "args": "finite_natseg, {m: m, n: n}",
                    "id": "1",
                    "prevs": [],
                    "rule": "apply_theorem_for",
                    "th": "⊢ finite ({m..n})"
                },
                {
                    "args": "∀x. m ≤ x ∧ x ≤ n ⟶ 0 < f x",
                    "id": "2",
                    "prevs": [
                        "0"
                    ],
                    "rule": "z3",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x ⊢ ∀x. m ≤ x ∧ x ≤ n ⟶ 0 < f x"
                },
                {
                    "args": "member_natseg, ∀x. x ∈ {m..n} ⟶ 0 < f x",
                    "id": "3",
                    "prevs": [
                        "2"
                    ],
                    "rule": "rewrite_goal",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x ⊢ ∀x. x ∈ {m..n} ⟶ 0 < f x"
                },
                {
                    "args": "product_pos_lt",
                    "id": "4",
                    "prevs": [
                        "1",
                        "3"
                    ],
                    "rule": "apply_theorem",
                    "th": "∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x ⊢ 0 < product ({m..n}) f"
                },
                {
                    "args": "",
                    "id": "5",
                    "prevs": [
                        "0",
                        "4"
                    ],
                    "rule": "intros",
                    "th": "⊢ (∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x) ⟶ 0 < product ({m..n}) f"
                }
            ],
            "prop": "(∀x. m ≤ x ⟶ x ≤ n ⟶ 0 < f x) ⟶ 0 < product ({m..n}) f",
            "steps": [
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "product_pos_lt"
                },
                {
                    "goal_id": "1",
                    "method_name": "apply_backward_step",
                    "theorem": "finite_natseg"
                },
                {
                    "goal_id": "2",
                    "method_name": "rewrite_goal",
                    "sym": "false",
                    "theorem": "member_natseg"
                },
                {
                    "fact_ids": [
                        "0"
                    ],
                    "goal_id": "2",
                    "method_name": "z3"
                }
            ],
            "ty": "thm",
            "vars": {
                "f": "nat ⇒ real",
                "m": "nat",
                "n": "nat"
            }
        }
    ],
    "description": "Products of natural numbers and real numbers.",
    "imports": [
        "sums"
    ],
    "name": "products"
}