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Author SHA1 Message Date
Leonardo de Moura
ce10bfeaf6 perf(kernel/metavar): improve performance of has_assigned_metavar by avoiding for_each+exception 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 15:19:59 -07:00
Leonardo de Moura
57d9d23bd4 feat(kernel/for_each): allow function F to interrupt for_each search 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 14:58:02 -07:00
Leonardo de Moura
2dd44bdf1a perf(kernel/for_each): delay initialization of visited set 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 14:52:08 -07:00
Leonardo de Moura
c3c66b6c90 feat(make): add THREAD_SAFE build option 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 11:50:35 -07:00
Leonardo de Moura
471bbd4040 refactor(kernel/metavar): combine several splay_trees (at metavar_env) into a single one 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 11:02:19 -07:00
Leonardo de Moura
5e34f410b3 refactor(splay_map): modify splay_find signature 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 10:47:15 -07:00
Leonardo de Moura
5812dfcf44 perf(kernel/justification): remove cache from depends_on 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 10:26:05 -07:00
Leonardo de Moura
2b5c951de3 perf(name): add quick_cmp for hierarchical names 
It first compare names using hash codes.

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 09:58:06 -07:00
Leonardo de Moura
66f4834dbc perf(kernel/metavar): add quick test that catches many cases 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 09:24:01 -07:00
Leonardo de Moura
f38178311c perf(kernel/expr_eq): delay hashtable initialization 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 09:21:47 -07:00
Leonardo de Moura
412bc792c9 fix(style): missing include 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 08:10:28 -07:00
Leonardo de Moura
0c21f45292 fix(kernel/unification_constraint): memory leak 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-25 08:06:21 -07:00
Leonardo de Moura
a5c3829d1b feat(kernel): add unexpected_metavar_occurrence exception 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-24 19:56:44 -07:00
Leonardo de Moura
ca6a6d71e5 fix(kernel/printer): bug when printing let expression 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-24 18:05:23 -07:00
Leonardo de Moura
872b698bc3 feat(elaborator): add option m_assume_injectivity for getting more concise solutions 
We may miss solutions, but the solutions found are much more readable.
For example, without this option, for elaboration problem

Theorem Example4 (a b c d e : N) (H: (a = b ∧ b = e ∧ b = c) ∨ (a = d ∧ d = c)) : (h a c) = (h c a) :=
    DisjCases H
              (fun H1 : _,
                   let AeqC := Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1))
                   in CongrH AeqC (Symm AeqC))
              (fun H1 : _,
                   let AeqC := Trans (Conjunct1 H1) (Conjunct2 H1)
                   in CongrH AeqC (Symm AeqC))

the elaborator generates

Theorem Example4 (a b c d e : N) (H : a = b ∧ b = e ∧ b = c ∨ a = d ∧ d = c) : (h a c) = (h c a) :=
    DisjCases
        H
        (λ H1 : if
               Bool
               (if Bool (a = b) (if Bool (if Bool (if Bool (b = e) (if Bool (b = c) ⊥ ⊤) ⊤) ⊥ ⊤) ⊥ ⊤) ⊤)
               ⊥
               ⊤,
           let AeqC := Trans (Conjunct1 H1) (Conjunct2 (Conjunct2 H1)) in CongrH AeqC (Symm AeqC))
        (λ H1 : if Bool (if Bool (a = d) (if Bool (d = c) ⊥ ⊤) ⊤) ⊥ ⊤,
           let AeqC := Trans (Conjunct1 H1) (Conjunct2 H1) in CongrH AeqC (Symm AeqC))

The solution is correct, but it is not very readable. The problem is that the elaborator expands the definitions of \/ and /\.

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-24 16:47:50 -07:00
Leonardo de Moura
bbc265ded4 feat(frontends/lean): hook new elaborator in the default frontend 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-24 15:14:29 -07:00
Leonardo de Moura
434c33f225 feat(metavar): automatically apply beta-reduction when instantiating metavariable applications (i.e., expressions of the form (?m a)), when the metavariable is a lambda 
This feature is useful for problems that require higher-order matching and/or unification.

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-23 16:35:50 -07:00
Leonardo de Moura
873e56844a refactor(beta_reduction): add function apply_beta 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-23 16:03:52 -07:00
Leonardo de Moura
f1e0d6ec29 refactor(beta_reduction): move beta reduction functions to the kernel, delete reduce.cpp file and tests 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-23 15:44:26 -07:00
Leonardo de Moura
c1e451151a feat(replace_visitor): add an abstract class for applying transformations on expressions 
I also removed replace_using_ctx since it is subsumed by the new class.

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-23 15:01:37 -07:00
Leonardo de Moura
13531b7d3e refactor(kernel): rename trace to justification 
Motivations:

- We have been writing several comments of the form "... trace/justification..." and "this trace object justify ...".
- Avoid confusion with util/trace.h

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-23 13:42:17 -07:00
Leonardo de Moura
17b48010b7 fix(unification_constraint): fix printer for max constraints 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-23 12:00:29 -07:00
Leonardo de Moura
274b11530f feat(metavar): improve apply_local_context 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-23 11:59:36 -07:00
Leonardo de Moura
b57f492e2d fix(kernel/printer): improve printer for Type expressions 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-23 11:58:49 -07:00
Leonardo de Moura
891d22b3de feat(kernel/context): add method for remove context entries at positions [s, s+n). 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 15:52:24 -07:00
Leonardo de Moura
874f67c605 feat(normalizer): remove normalization rule t == t ==> true 
This normalization rule is not really a computational rule.
It is essentially encoding the reflexivity axiom as computation.
It can also be abaused. For example, with this rule,
the following definition is valid:

Theorem Th : a = a := Refl b

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 14:02:48 -07:00
Leonardo de Moura
c3e87f106f fix(kernel/trace): fix typo in depends_on 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:37 -07:00
Leonardo de Moura
183f5a1ccf feat(elaborator): solve unification constraints 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
a1710aeeb9 feat(elaborator): add trace objects for elaborator 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
dc0e7a4472 feat(pos_info_provider): add position information provider for expressions 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
8663ac550f feat(kernel/trace): add function depends_on for trace objects 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
b1b49e86e7 test(elaborator): add simple test 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
1548ffabb1 feat(elaborator): add new elaborator interface 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
827c65b5e9 feat(kernel): add static_assert for update_metavar 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
61ccaf741c fix(frontend/lean): minor modification to be able to execute lean frontend while refactoring elaborator 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
7cf83800c0 refactor(metavar): implement metavar_env, and use unification_constraint and trace objects in the type_checker, light_checker 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
1f0eab7a14 test(type_checker): add new tests for type_checker trace objects 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
fc288929a2 feat(type_checker): add trace objects to justify constraints created by the type checker 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
5b1b03bafd refactor(is_convertible): move from normalizer to type_checker class 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
85bfa45e6a refactor(kernel_exception): delete kernel_exception_formatter, and implement kernel_exception pretty printer as a virtual method 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
d843d432d3 refactor(kernel): move printer and formatter objects to the kernel 
The printer and formatter objects are not trusted code.
We moved them to the kernel to be able to provide them as an argument to the trace objects.
Another motivation is to eliminate the kernel_exception_formatter hack.
With the formatter in the kernel, we can implement the pretty printer for kernel exceptions as a virtual method.

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
ddb90d3038 feat(kernel): add unification_constraint and trace objects to the kernel 
Trace objects will be used to justify steps performed by engines such as the elaborator. We use them to implement non-chronological backtracking in the elaborator. They are also use to justify to the user why something did not work.

The unification constraints are in the kernel because the type checker may create them when type checking a term containing metavariables.

Remark: a minimalistic kernel does not need to include metavariables, unification constraints, nor trace objects. We include these objects in our kernel to minimize code duplication.

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
59914a36f3 refactor(metavar): reorganize and simplify metavariables 
- Use hierarchical names instead of unsigned integers to identify metavariables.
- Associate type with metavariable.
- Replace metavar_env with substitution.
- Rename meta_ctx --> local_ctx
- Rename meta_entry --> local_entry
- Disable old elaborator
- Rename unification_problems to unification_constraints
- Add metavar_generator
- Fix metavar unit tests
- Modify type checker to use metavar_generator
- Fix placeholder module

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-22 08:15:36 -07:00
Leonardo de Moura
0783805671 feat(kernel): add weight to kernel definitions 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-15 14:50:08 -07:00
Leonardo de Moura
aa5be3262f fix(type_checker): fix inconsistent cache bug in type_checker, add tests that expose the problem 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-10-01 19:23:55 -07:00
Soonho Kong
e3b762e909 feat(kernel): add static_assert to expr,expr_eq,replace 2013-10-01 16:47:36 -07:00
Soonho Kong
a832173f5f feat(kernel/expr): add expr::operator() which takes 8 args 2013-10-01 00:19:30 -07:00
Leonardo de Moura
e5d312dc18 fix(kernel): the hash code of expressions was not being used to compare them 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-26 08:41:20 -07:00
Leonardo de Moura
98b4e09063 refactor(kernel): simplify expr_eq 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-25 22:09:08 -07:00
Leonardo de Moura
6477708d78 refactor(debug): improve lean_unreachable(), now we can avoid 'fake' return statements 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-25 21:27:20 -07:00
Leonardo de Moura
9f0dab1add fix(kernel): add declarations for operator<<(std::ostream&, expr const&) and operator<<(std::ostream&, context const&) in the kernel 
The actual implementation of these two operators is outside of the
kernel. They are implemented in the file 'library/printer.cpp'.
We declare them in the kernel to prevent the following problem.
Suppose there is a file 'foo.cpp' that does not include
'library/printer.h', but contains

    expr a;
    ...
    std::cout << a << "\n";
    ...

The compiler does not generate an error message. It silently uses the
operator bool() to coerce the expression into a Boolean. This produces
counter-intuitive behavior, and may confuse developers.
 2013-09-25 17:45:54 -07:00
Leonardo de Moura
ba0528c298 Implement total order on expressions 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-24 12:16:32 -07:00
Leonardo de Moura
46d6c41835 Fix bug in the type checker (when type checking terms with meta-variables). 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-22 19:12:19 -07:00
Leonardo de Moura
1647e44510 Fix memory corruption bug 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-22 18:53:58 -07:00
Leonardo de Moura
16a6a54df1 Fix abuse of operator-> overload 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-22 16:41:51 -07:00
Leonardo de Moura
c847d27763 Improve higher order unification 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-21 00:41:49 -07:00
Leonardo de Moura
651c5d6751 Fix warnings and bugs related to unused variables. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-19 22:41:07 -07:00
Soonho Kong
ab6ca82e6f Update to suppress unused-parameter warnings 2013-09-19 22:40:34 -07:00
Leonardo de Moura
80a1b96925 Remove duplicate solutions in the higher order matching module. Simplify solutions when higher-order matching is used, and we don't have a residue. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-19 18:59:28 -07:00
Leonardo de Moura
2f29ff70d7 Implement higher-order unification 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-18 20:46:00 -07:00
Leonardo de Moura
ad901ce087 Use consistent naming conventions 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-17 14:43:07 -07:00
Leonardo de Moura
3df6149daa Add support for metavariables in the type checker. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-17 11:09:59 -07:00
Leonardo de Moura
9f9dcf9a29 Fix bug in the creation of children environments. Remove unnecessary function. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-17 07:37:33 -07:00
Leonardo de Moura
30b19c314a Add basic support for metavariables at is_convertible. Swap is_convertible arguments to make it more intuitive. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-17 07:15:47 -07:00
Leonardo de Moura
a26c7d47f2 Add simplification to add_lift 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-17 03:14:02 -07:00
Leonardo de Moura
99e8d2feae Add simplification rule to add_inst 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-17 02:57:28 -07:00
Leonardo de Moura
21c7a45f67 Relax definition of identity_stack. Fix printer for metavariable contexts. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-17 02:09:45 -07:00
Leonardo de Moura
d3bce584f4 Add support for new metavariable representation in the normalizer. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-17 01:57:21 -07:00
Leonardo de Moura
da09e7217a Cleanup meta_entry code 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-16 19:32:28 -07:00
Leonardo de Moura
99a163f11d Simplify metavariable context. Now, we have only 'lift' and 'inst' instead of 'subst', 'lift' and 'lower' 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-16 19:32:28 -07:00
Leonardo de Moura
cad562a448 Add support for metavariables in the normalizer. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-16 19:32:28 -07:00
Leonardo de Moura
1be2a30c8c Fix bug in normalizer. We must create a scope whenever we extend the value stack. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-16 19:32:28 -07:00
Leonardo de Moura
f79f046294 Add partial support for metavariables in the normalizer 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-15 21:23:50 -07:00
Leonardo de Moura
2800292947 Add timestamp to metavar_env 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-15 19:50:48 -07:00
Leonardo de Moura
5a4bc331d2 Make unification_problems a virtual class. Associate a 'standard' context with each metavariable in metavar_env 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-15 19:38:36 -07:00
Leonardo de Moura
63e102055e Move metavariables to the kernel. This is the first step for implementing the new elaborator. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-15 12:09:01 -07:00
Soonho Kong
bc60b47295 Apply coding style 2013-09-13 18:48:09 -07:00
Leonardo de Moura
bcc3827a99 Modify Doxygen file to extract all elements even the undocumented ones. Disable warnings for undocumented entities. Add extra comments. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-13 13:46:22 -07:00
Leonardo de Moura
d54834279e Use consistent coding style for if-then-else 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-13 12:57:40 -07:00
Leonardo de Moura
8c735f1daa Use consistent coding style for spaces after ',' 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-13 12:49:03 -07:00
Leonardo de Moura
2c68117adf Tag TODOs 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-13 12:25:21 -07:00
Leonardo de Moura
0c09e4524a Use consistent names for import functions, and library files. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-13 08:58:34 -07:00
Soonho Kong
5c3866cd71 Use fullpath in #include directives, add missing STL headers 2013-09-13 03:35:29 -07:00
Leonardo de Moura
26097475fd Use fullpath in #include directives. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-12 20:04:10 -07:00
Leonardo de Moura
572c7ced2a Add #include<atomic> to expr.h. We need it when #define LEAN_THREAD_UNSAFE_REF_COUNT is used 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-11 19:48:55 -07:00
Leonardo de Moura
33c4b44b2b Encapsulate context implementation. The current implementantion based on lists may be a performance problem in the future, and we should be able to change it without affecting the whole code base. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-07 11:15:11 -07:00
Leonardo de Moura
bab11b57ad Move Symm and Trans back to basic_thms.cpp 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-06 23:49:35 -07:00
Leonardo de Moura
c674bb3790 Add castlib as an independent library 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-06 23:40:47 -07:00
Leonardo de Moura
7a9d53d0d7 Refactor arith libraries 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-06 23:19:47 -07:00
Leonardo de Moura
b92bbeb83b Add casting propagation and normalization 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-06 20:45:26 -07:00
Leonardo de Moura
c0c2f52087 Add Cast, DomInj and RanInj. Improve operator << for lean_frontend objects. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-06 18:32:15 -07:00
Leonardo de Moura
8840b37258 Fix type checker and elaborator for let expressions. Fix get_coercions (we need to pass the context). Fix pretty printer for def_type_mismatch_exception. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-06 11:02:00 -07:00
Leonardo de Moura
2459c4ae7c Add (optional) type to let declarations 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-06 10:06:26 -07:00
Leonardo de Moura
87d3961158 Improve elaborator error messages 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-04 16:36:58 -07:00
Leonardo de Moura
be7fa0932a Add unicode name for the types: Nat, Int and Real 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-04 09:03:41 -07:00
Leonardo de Moura
d41160f8a5 Modify environment. Now, when a builtin value is declared, if it has a unicode alternative representation, then we add it as a definition. Now, everything that occurs in the environment has been 'declared'. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-04 08:53:00 -07:00
Leonardo de Moura
e955c054ca Modify type checker. Now, it only accepts builtin values that have been declared in the environment. The idea is to be able to track which classes of builtin values have been used in a given environment. We want to be able to quantify the size of the trusted code base for a particular development. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-04 08:30:04 -07:00
Leonardo de Moura
887f696f66 Factor duplicate code. Add more comments 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-03 23:27:58 -07:00
Leonardo de Moura
fc9e395818 Define absolute value function and notation for it. Add new example. 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-03 20:39:54 -07:00
Leonardo de Moura
3992c4b8f9 Define divides, and add examples 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2013-09-03 20:18:20 -07:00