Created
February 26, 2014 00:04
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Trouble with heterogeneous equality
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Set Implicit Arguments. | |
Require Import Arith.Arith_base List Omega. | |
Require Import Wellfounded.Lexicographic_Product. | |
Require Import Relation_Operators. | |
Require Import JMeq. | |
(* Notation for heterogeneous equality *) | |
Infix "==" := JMeq (at level 70, no associativity). | |
(* kind definitions *) | |
Inductive Kind : Set := | |
| Star : Kind | |
| KFun : Kind -> Kind -> Kind. | |
Notation "*" := Star. | |
Notation "k '==>' k'" := (KFun k k') (at level 60, right associativity). | |
Definition VarCtx := list Kind. | |
Definition ConCtx := list Kind. | |
(* type definition *) | |
Inductive Ty (C : ConCtx) (V : VarCtx) : Kind -> Set := | |
| var : forall k, In k V -> Ty C V k | |
| con : forall k, In k C -> Ty C V k | |
| app : forall k k', Ty C V (k ==> k') -> Ty C V k -> Ty C V k'. | |
Definition eq_ty_dec : forall {C V k}(t t' : Ty C V k), {t == t'} + {~ (t == t')}. | |
intros C V k t ; | |
induction t ; intro t' ; destruct t' ; | |
try repeat (match goal with | |
| [V : VarCtx |- context[var _ _ _ _ == var _ _ _ _]] => left | |
| [|- context[con _ _ _ _ == con _ _ _ _]] => left | |
| [|- context[app _ _ == app _ _]] => left | |
| [|- _ ] => let x := fresh "H" in right ; intro x | |
end). |
Wow... This is really a much better solution! Thanks for your time, Jason!
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What you want is not doable without proof irrelevance, because
In
is not unique; if the element appears multiple times, then there are many proofs. However, if you're willing to use a decidable version ofIn
, then it works: