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Author SHA1 Message Date
Leonardo de Moura
aa8240985a test(examples/lean): small version of algebraic hierarchy (proof of concept) 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-24 20:51:19 -08:00
Leonardo de Moura
309e7ba880 fix(library/elaborator): temporary fix for bug reported by Jeremy 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-24 15:26:21 -08:00
Leonardo de Moura
16844fff73 feat(builtin): simulate binary encoding 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-20 20:44:05 -08:00
Leonardo de Moura
d79e9af210 fix(frontends/lean): help msg 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-18 09:31:30 -08:00
Leonardo de Moura
f781ad823c doc(builtin): Diaconescu’s theorem 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-18 09:12:09 -08:00
Leonardo de Moura
e9dada5e14 refactor(builtin/kernel): use standard definition for 'or' and 'and' 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-17 12:05:34 -08:00
Leonardo de Moura
4692e04d70 feat(builtin/proof_irrel): prove proof irrelevance 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-16 18:06:10 -08:00
Leonardo de Moura
c526e5ec00 feat(builtin/kernel): prove false_elim without using case 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-15 14:26:13 -08:00
Leonardo de Moura
1739b5c153 fix(kernel/type_checker): caching bug 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-12 10:43:01 -08:00
Leonardo de Moura
c740d9d799 fix(builtin/num): bug in the factorial definition 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-12 10:04:32 -08:00
Leonardo de Moura
45a0dbcc34 feat(builtin/num): define fact and exp 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-12 09:51:49 -08:00
Leonardo de Moura
368fcb5ff9 refactor(builtin/kernel): rename refute to by_contradiction 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-12 08:49:19 -08:00
Leonardo de Moura
69bccb6014 feat(builtin): define list, cons, nil and prove basic theorems 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-11 21:32:05 -08:00
Leonardo de Moura
bfe64a7031 fix(library/elaborator): hack for fixing a bug due to pairs/projs, this is temporary fix until we build a new elaborator 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-11 21:30:26 -08:00
Leonardo de Moura
0878b44fc7 feat(frontends/lean): allow user to import several theories using a single import 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-11 17:15:12 -08:00
Leonardo de Moura
fddaa7bc3a doc(examples/lean): add theorem sent by Jeremy Avigad 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-11 14:10:29 -08:00
Leonardo de Moura
1796c2b992 doc(examples/lean): proof of concept 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-11 07:59:49 -08:00
Leonardo de Moura
56ad1c1728 doc(examples/lean/abelian): proof of concept 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-10 18:17:15 -08:00
Leonardo de Moura
65076816fa doc(examples/lean/group): prove basic theorems 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-10 15:49:24 -08:00
Leonardo de Moura
acd04d3c2b doc(examples/lean): setoid and group examples 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-10 14:41:29 -08:00
Leonardo de Moura
11a2b3016f fix(builtin/num): remove hacks for making the elaborator happy 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-10 14:05:51 -08:00
Leonardo de Moura
b7b868de85 fix(library/elaborator): bug reported by Jeremy Avigad 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-10 14:01:22 -08:00
Leonardo de Moura
a2d2e36f04 refactor(frontends/lean): remove notation for creating tuples 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-10 09:03:42 -08:00
Leonardo de Moura
4e08a3233e fix(builtin): build dependency issue 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-10 08:33:07 -08:00
Leonardo de Moura
273f78d1cb feat(builtin/num): prove strong induction and other theorems for num 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 21:09:50 -08:00
Leonardo de Moura
b119c11473 feat(builtin/tactic): add simp_no_assump tactic 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 20:28:30 -08:00
Leonardo de Moura
4317f67bd2 fix(builtin/tactic): more meaningful error message when skip tactic is used in a full proof 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 18:57:33 -08:00
Leonardo de Moura
57982135d9 fix(library/simplifier): bug using congr1 theorem 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 16:24:29 -08:00
Leonardo de Moura
c45c1748d8 refactor(builtin/kernel): reorder congr1 arguments 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 16:15:44 -08:00
Leonardo de Moura
fd6f8b1945 refactor(builtin/num): simplify proofs using 'by simp' 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 15:03:45 -08:00
Leonardo de Moura
2d70e2f4f2 fix(library/tactic/goal): bug in the proof builder 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 15:02:36 -08:00
Leonardo de Moura
4c76f6abb9 chore(builtin/num): remove leftover 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 11:39:26 -08:00
Leonardo de Moura
8df7c7b02d feat(kernel/type_checker): remove fallback that expands opaque definitions in the type checker 
We should not rely on this feature. It can be quite expensive.
We invoke is_convertible in several places, in particular, if we are using overloading. For example, the frontend uses is_convertible to check which overload should be used. Thus, it will make several calls such as

   is_convertible(num, Nat)

If is_convertible starts unfolding opaque definitions, we would keep expanding num.

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 11:31:24 -08:00
Leonardo de Moura
4c4c8b3e0d feat(builtin/num): prove basic theorems using simplifier 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 11:24:53 -08:00
Leonardo de Moura
633ed6bb69 fix(frontends/lean/parser): bug in add_rewrite 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 09:46:56 -08:00
Leonardo de Moura
b24c085cb0 feat(frontends/lean): avoid warning message 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-09 09:26:51 -08:00
Leonardo de Moura
d6167eae32 feat(builtin/num): define add and mul 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-08 22:28:15 -08:00
Leonardo de Moura
cc4148a98d feat(builtin/num): primitive recursion theorem 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-08 22:05:57 -08:00
Leonardo de Moura
f28c56b188 feat(builtin/num): add auxiliary definitions and theorems for proving the primitive recursion theorem 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-08 19:36:17 -08:00
Leonardo de Moura
fa4b60963b feat(builtin/num): define lt predicate, and prove basic theorems 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-08 10:57:17 -08:00
Leonardo de Moura
0760b5b53d doc(todo): update todo list 2014-02-08 09:23:56 -08:00
Leonardo de Moura
ded72f94b2 doc(todo): update todo list 2014-02-08 09:23:13 -08:00
Leonardo de Moura
1f3e0f7a38 chore(builtin/num): update object file 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-08 09:16:52 -08:00
Leonardo de Moura
aeaa803f9a feat(builtin): add num type (the base type that will be used to build nat, int, real) 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-08 09:12:53 -08:00
Leonardo de Moura
1c43020fc9 fix(library/tactic/goal): bug creating main proof builder 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-08 09:11:07 -08:00
Leonardo de Moura
24528ff685 fix(library/elaborator): fix glitches in the elaborator that were forcing us to provide parameters explicitly 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-07 18:02:08 -08:00
Leonardo de Moura
1ec01f5757 refactor(builtin): merge pair.lean with kernel.lean, and add basic theorems 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-07 16:04:44 -08:00
Leonardo de Moura
5454e2af32 doc(todo): remove item from todo list 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-07 15:04:10 -08:00
Leonardo de Moura
ad7b13104f feat(*): add support for heterogeneous equality in the parser, elaborator and simplifier, adjusts unit test to reflect changes 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-07 15:03:16 -08:00
Leonardo de Moura
6d7ec9d7b6 refactor(kernel): add heterogeneous equality back to expr 
The main motivation is that we will be able to move equalities between universes.

For example, suppose we have
    A : (Type i)
    B : (Type i)
    H : @eq (Type j) A B
where j > i

We didn't find any trick for deducing (@eq (Type i) A B) from H.
Before this commit, heterogeneous equality as a constant with type

   heq : {A B : (Type U)} : A -> B -> Bool

So, from H, we would only be able to deduce

   (@heq (Type j) (Type j) A B)

Not being able to move the equality back to a smaller universe is
problematic in several cases. I list some instances in the end of the commit message.

With this commit, Heterogeneous equality is a special kind of expression.
It is not a constant anymore. From H, we can deduce

   H1 : A == B

That is, we are essentially "erasing" the universes when we move to heterogeneous equality.
Now, since A and B have (Type i), we can deduce (@eq (Type i) A B) from H1. The proof term is

  (to_eq (Type i) A B (to_heq (Type j) A B H))  :  (@eq (Type i) A B)

So, it remains to explain why we need this feature.

For example, suppose we want to state the Pi extensionality axiom.

axiom hpiext {A A' : (Type U)} {B : A → (Type U)} {B' : A' → (Type U)} :
      A = A' → (∀ x x', x == x' → B x == B' x') → (∀ x, B x) == (∀ x, B' x)

This axiom produces an "inflated" equality at (Type U) when we treat heterogeneous
equality as a constant. The conclusion

     (∀ x, B x) == (∀ x, B' x)

is syntax sugar for

   (@heq (Type U) (Type U) (∀ x : A, B x) (∀ x : A', B' x))

Even if A, A', B, B' live in a much smaller universe.

As I described above, it doesn't seem to be a way to move this equality back to a smaller universe.

So, if we wanted to keep the heterogeneous equality as a constant, it seems we would
have to support axiom schemas. That is, hpiext would be parametrized by the universes where
A, A', B and B'. Another possibility would be to have universe polymorphism like Agda.
None of the solutions seem attractive.

So, we decided to have heterogeneous equality as a special kind of expression.
And use the trick above to move equalities back to the right universe.

BTW, the parser is not creating the new heterogeneous equalities yet.
Moreover, kernel.lean still contains a constant name heq2 that is the heterogeneous
equality as a constant.

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-07 10:28:10 -08:00