Fix support for mathematical types

[tautschnig]

We were missing front-end support for reals and did not consistently support all of integers, naturals, rationals, reals but instead would only handle varying subsets in each place.

• Each commit message has a non-empty body, explaining why the change was made.
• Methods or procedures I have added are documented, following the guidelines provided in CODING_STANDARD.md.
• The feature or user visible behaviour I have added or modified has been documented in the User Guide in doc/cprover-manual/
• Regression or unit tests are included, or existing tests cover the modified code (in this case I have detailed which ones those are in the commit message).
• n/a My commit message includes data points confirming performance improvements (if claimed).
• My PR is restricted to a single feature or bugfix.
• n/a White-space or formatting changes outside the feature-related changed lines are in commits of their own.

[codecov]

Codecov Report

❌ Patch coverage is 63.53591% with 66 lines in your changes missing coverage. Please review.
✅ Project coverage is 80.48%. Comparing base (5f4c82d) to head (9d13355).

Files with missing lines	Patch %	Lines
src/ansi-c/c_typecast.cpp	57.69%	22 Missing ⚠️
src/solvers/smt2/smt2_conv.cpp	73.23%	19 Missing ⚠️
src/util/real.cpp	45.45%	6 Missing ⚠️
src/util/simplify_expr.cpp	28.57%	5 Missing ⚠️
src/solvers/flattening/boolbv_width.cpp	0.00%	4 Missing ⚠️
src/util/real.h	42.85%	4 Missing ⚠️
src/util/rational_tools.cpp	66.66%	3 Missing ⚠️
src/ansi-c/c_typecheck_type.cpp	75.00%	1 Missing ⚠️
src/util/arith_tools.cpp	50.00%	1 Missing ⚠️
src/util/pointer_offset_size.cpp	50.00%	1 Missing ⚠️

Additional details and impacted files

@@             Coverage Diff             @@
##           develop    #8324      +/-   ##
===========================================
+ Coverage    80.43%   80.48%   +0.05%     
===========================================
  Files         1695     1697       +2     
  Lines       208285   208385     +100     
  Branches        73       73              
===========================================
+ Hits        167524   167726     +202     
+ Misses       40761    40659     -102     

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[kroening]

Representing real numbers like floating-point numbers will at the very least irritate. Please add a comment why this is the right choice.

[tautschnig]

Hmm, which approach would you have chosen instead?

[kroening]

E.g., you could have represented the algebraic numbers via the root of a polynomial; I'd be curious what the various SMT solvers output as models.

[polgreen]

z3 and cvc5 seem to output Reals as fractions, unless the denominator is 1, when they just output a singular decimal.

Whilst they use decimals, I've not managed to make it produce a number that isn't an integer as the nominator or denominator, or as a singular decimal.

(set-logic ALL)
(declare-fun x () Real)
(declare-fun y () Real)
(declare-fun z () Real)
(assert (= y (/ 100 30)))
(assert (= x 0.111111))
(assert (= z 7))
(check-sat)
(get-model)

z3 says:

(
  (define-fun z () Real
    7.0)
  (define-fun y () Real
    (/ 10.0 3.0))
  (define-fun x () Real
    (/ 111111.0 1000000.0))
)

cvc5 says

(
(define-fun x () Real (/ 111111 1000000))
(define-fun y () Real (/ 10 3))
(define-fun z () Real 7.0)
)

For irrational numbers, they output the root of a polynomial.

(set-logic ALL)
(declare-fun y () Real)
(assert (= (* y y) 2 ))
(check-sat)
(get-model)

z3 says

(
  (define-fun y () Real
    (root-obj (+ (^ x 2) (- 2)) 1))
)

cvc5 segfaults...

[kroening]

I suspect we'll eventually want the algebraic numbers.

[tautschnig]

I'll try to address this (even when I'd argue that the current PR already addresses a number of CBMC invariant failures that the included regression tests cause when attempted without the code changes).

[kroening]

No need to implement this right now; my suggestion is to add a comment, then merge this PR.