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199 commits

Author SHA1 Message Date
Leonardo de Moura
0c1674ab70 feat(builtin): quotient types 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-03-12 00:20:46 -07:00
Leonardo de Moura
e0eacd1f9f feat(builtin): simpler encoding of sum types 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-03-03 18:30:45 -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
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
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
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
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
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
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
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
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
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
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
Leonardo de Moura
354d5607af chore(builtin/sum): cleanup 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 21:25:57 -08:00
Leonardo de Moura
4b4b5e3345 fix(frontends/lean): import explicit versions when using the command 'using' 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 17:33:58 -08:00
Leonardo de Moura
a51139e63b feat(frontends/lean): position information in error messages 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 17:26:38 -08:00
Leonardo de Moura
ef321e730f feat(builtin/tactic): add the 'skip' (bogus) tactic for ignoring a proof hole in a big proof 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 12:47:23 -08:00
Leonardo de Moura
581ae0c83b chore(build): fix build dependencies 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 09:30:33 -08:00
Leonardo de Moura
db3bcdba55 refactor(builtin): move pair definition and theorems to pair.lean 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 09:20:52 -08:00
Leonardo de Moura
daf7075ce4 refactor(builtin/sum): use new 'have' expression to formalize optional-types 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 09:15:12 -08:00
Leonardo de Moura
5d698a60a7 refactor(builtin/sum): use new 'have' expression to formalize sum-types 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 08:59:05 -08:00
Leonardo de Moura
ba9a8f9d98 feat(frontends/lean): add 'show' expression syntax sugar 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 07:50:22 -08:00
Leonardo de Moura
90f5a4f813 feat(builtin/sum): cleanup, and avoid unicode character that is not available in some platforms 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-06 00:20:49 -08:00
Leonardo de Moura
c01f82aeb7 feat(builtin): add sum types 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-05 23:04:44 -08:00
Leonardo de Moura
87da23649b feat(builtin/optional): prove dichotomy and induction theorems for optional types 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-05 19:49:50 -08:00
Leonardo de Moura
30570c843f feat(builtin): add optional type 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-05 17:33:06 -08:00
Leonardo de Moura
f4ec874c6e refactor(builtin): remove dead module heq 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-04 14:42:28 -08:00
Leonardo de Moura
0283887ee9 refactor(builtin/kernel): move the heq axioms into kernel.lean 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
 2014-02-04 14:17:34 -08:00