112 lines
3 KiB
Coq
112 lines
3 KiB
Coq
Require Import Autosubst2.fintype Autosubst2.syntax Autosubst2.core ssreflect.
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From Ltac2 Require Ltac2.
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Import Ltac2.Notations.
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Import Ltac2.Control.
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From Hammer Require Import Tactics.
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Definition renaming_ok {n m} (Γ : fin n -> PTm n) (Δ : fin m -> PTm m) (ξ : fin m -> fin n) :=
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forall (i : fin m), ren_PTm ξ (Δ i) = Γ (ξ i).
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Lemma up_injective n m (ξ : fin n -> fin m) :
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(forall i j, ξ i = ξ j -> i = j) ->
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forall i j, (upRen_PTm_PTm ξ) i = (upRen_PTm_PTm ξ) j -> i = j.
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Proof.
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sblast inv:option.
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Qed.
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Local Ltac2 rec solve_anti_ren () :=
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let x := Fresh.in_goal (Option.get (Ident.of_string "x")) in
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intro $x;
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lazy_match! Constr.type (Control.hyp x) with
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| fin _ -> _ _ => (ltac1:(case;hauto lq:on rew:off use:up_injective))
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| _ => solve_anti_ren ()
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end.
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Local Ltac solve_anti_ren := ltac2:(Control.enter solve_anti_ren).
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Lemma ren_injective n m (a b : PTm n) (ξ : fin n -> fin m) :
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(forall i j, ξ i = ξ j -> i = j) ->
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ren_PTm ξ a = ren_PTm ξ b ->
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a = b.
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Proof.
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move : m ξ b. elim : n / a => //; try solve_anti_ren.
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Qed.
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Inductive HF : Set :=
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| H_Pair | H_Abs | H_Univ | H_Bind (p : BTag) | H_Nat | H_Suc | H_Zero | H_Bot.
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Definition ishf {n} (a : PTm n) :=
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match a with
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| PPair _ _ => true
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| PAbs _ => true
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| PUniv _ => true
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| PBind _ _ _ => true
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| PNat => true
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| PSuc _ => true
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| PZero => true
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| _ => false
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end.
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Definition toHF {n} (a : PTm n) :=
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match a with
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| PPair _ _ => H_Pair
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| PAbs _ => H_Abs
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| PUniv _ => H_Univ
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| PBind p _ _ => H_Bind p
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| PNat => H_Nat
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| PSuc _ => H_Suc
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| PZero => H_Zero
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| _ => H_Bot
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end.
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Fixpoint ishne {n} (a : PTm n) :=
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match a with
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| VarPTm _ => true
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| PApp a _ => ishne a
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| PProj _ a => ishne a
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| PBot => true
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| PInd _ n _ _ => ishne n
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| _ => false
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end.
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Definition isbind {n} (a : PTm n) := if a is PBind _ _ _ then true else false.
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Definition isuniv {n} (a : PTm n) := if a is PUniv _ then true else false.
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Definition ispair {n} (a : PTm n) :=
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match a with
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| PPair _ _ => true
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| _ => false
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end.
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Definition isnat {n} (a : PTm n) := if a is PNat then true else false.
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Definition iszero {n} (a : PTm n) := if a is PZero then true else false.
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Definition issuc {n} (a : PTm n) := if a is PSuc _ then true else false.
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Definition isabs {n} (a : PTm n) :=
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match a with
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| PAbs _ => true
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| _ => false
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end.
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Definition ishf_ren n m (a : PTm n) (ξ : fin n -> fin m) :
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ishf (ren_PTm ξ a) = ishf a.
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Proof. case : a => //=. Qed.
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Definition isabs_ren n m (a : PTm n) (ξ : fin n -> fin m) :
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isabs (ren_PTm ξ a) = isabs a.
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Proof. case : a => //=. Qed.
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Definition ispair_ren n m (a : PTm n) (ξ : fin n -> fin m) :
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ispair (ren_PTm ξ a) = ispair a.
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Proof. case : a => //=. Qed.
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Definition ishne_ren n m (a : PTm n) (ξ : fin n -> fin m) :
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ishne (ren_PTm ξ a) = ishne a.
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Proof. move : m ξ. elim : n / a => //=. Qed.
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Lemma renaming_shift n m Γ (ρ : fin n -> PTm m) A :
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renaming_ok (funcomp (ren_PTm shift) (scons (subst_PTm ρ A) Γ)) Γ shift.
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Proof. sfirstorder. Qed.
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