Simplify by using builtin list and a custom induction principle
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quoting.v
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quoting.v
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@ -1,8 +1,5 @@
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Require Import Coq.Lists.List.
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Require Import Coq.Program.Equality.
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Require Import Omega.
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Require Import String.
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Require Import Bool.
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Import ListNotations.
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Open Scope list_scope.
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Open Scope string_scope.
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@ -16,151 +13,116 @@ Inductive label : Type := (* efficiency hack *)
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Inductive value : Type :=
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| vBool : bool -> value
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| vSymbol : label -> value
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| vRecord: value -> values -> value
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| vSeq : values -> value
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with values : Type :=
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| vsNil : values
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| vsCons : value -> values -> values.
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| vRecord: value -> list value -> value
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| vSeq : list value -> value.
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Inductive pat : Type :=
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| pDiscard : pat
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| pCapture : pat -> pat
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| pBool : bool -> pat
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| pSymbol : label -> pat
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| pRecord: pat -> pats -> pat
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| pSeq : pats -> pat
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with pats : Type :=
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| psNil : pats
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| psCons : pat -> pats -> pats.
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| pRecord: pat -> list pat -> pat
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| pSeq : list pat -> pat.
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Definition vs1 v : values := vsCons v vsNil.
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Definition ps1 p : pats := psCons p psNil.
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Definition vQuote v : value := vRecord (vSymbol lQuote) [v].
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Definition vQuote v : value := vRecord (vSymbol lQuote) (vs1 v).
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Lemma value_dec : forall x y : value, { x = y } + { x <> y }
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with values_dec : forall xs ys : values, { xs = ys } + { xs <> ys }.
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repeat decide equality.
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Fixpoint value_dec (x y : value) : { x = y } + { x <> y }.
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repeat decide equality.
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Defined.
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Lemma pat_dec : forall x y : pat, { x = y } + { x <> y }
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with pats_dec : forall xs ys : pats, { xs = ys } + { xs <> ys }.
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repeat decide equality.
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Fixpoint pat_dec (x y : pat) : { x = y } + { x <> y }.
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repeat decide equality.
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Defined.
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Fixpoint qp p : value :=
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match p with
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| pDiscard => vRecord (vSymbol lDiscard) vsNil
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| pCapture p' => vRecord (vSymbol lCapture) (vs1 (qp p'))
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| pDiscard => vRecord (vSymbol lDiscard) []
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| pCapture p' => vRecord (vSymbol lCapture) [qp p']
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| pBool b => vBool b
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| pSymbol s => vSymbol s
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| pRecord (pSymbol lDiscard) psNil => vRecord (vQuote (vSymbol lDiscard)) vsNil
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| pRecord (pSymbol lCapture) (psCons p' psNil) => vRecord (vQuote (vSymbol lCapture)) (vs1 (qp p'))
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| pRecord (pSymbol lQuote) (psCons p' psNil) => vRecord (vQuote (vSymbol lQuote)) (vs1 (qp p'))
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| pRecord l ps => vRecord (qp l) (qps ps)
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| pSeq ps => vSeq (qps ps)
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end
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with qps ps : values :=
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match ps with
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| psNil => vsNil
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| psCons p' ps' => vsCons (qp p') (qps ps')
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end.
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| pRecord (pSymbol lDiscard) [] => vRecord (vQuote (vSymbol lDiscard)) []
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| pRecord (pSymbol lCapture) [p'] => vRecord (vQuote (vSymbol lCapture)) [qp p']
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| pRecord (pSymbol lQuote) [p'] => vRecord (vQuote (vSymbol lQuote)) [qp p']
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| pRecord l ps => vRecord (qp l) (map qp ps)
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| pSeq ps => vSeq (map qp ps)
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end.
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Fixpoint raw_uqp v : pat :=
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match v with
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| vBool b => pBool b
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| vSymbol s => pSymbol s
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| vRecord l fs => pRecord (raw_uqp l) (raw_uqps fs)
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| vSeq vs => pSeq (raw_uqps vs)
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end
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with raw_uqps vs : pats :=
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match vs with
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| vsNil => psNil
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| vsCons v vs' => psCons (raw_uqp v) (raw_uqps vs')
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end.
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| vRecord l fs => pRecord (raw_uqp l) (map raw_uqp fs)
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| vSeq vs => pSeq (map raw_uqp vs)
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end.
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Fixpoint uqp v : pat :=
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match v with
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| vRecord (vSymbol lDiscard) vsNil => pDiscard
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| vRecord (vSymbol lCapture) (vsCons v' vsNil) => pCapture (uqp v')
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| vRecord (vSymbol lQuote) (vsCons v' vsNil) => raw_uqp v'
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| vRecord (vSymbol lDiscard) [] => pDiscard
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| vRecord (vSymbol lCapture) [v'] => pCapture (uqp v')
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| vRecord (vSymbol lQuote) [v'] => raw_uqp v'
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| vBool b => pBool b
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| vSymbol s => pSymbol s
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| vRecord l fs => pRecord (uqp l) (uqps fs)
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| vSeq vs => pSeq (uqps vs)
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end
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with uqps vs : pats :=
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match vs with
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| vsNil => psNil
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| vsCons v vs' => psCons (uqp v) (uqps vs')
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end.
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| vRecord l fs => pRecord (uqp l) (map uqp fs)
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| vSeq vs => pSeq (map uqp vs)
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end.
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Lemma quoting_for_record_sensible :
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forall p ps,
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p <> pSymbol lDiscard ->
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p <> pSymbol lCapture ->
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p <> pSymbol lQuote ->
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uqp (qp (pRecord p ps)) = pRecord (uqp (qp p)) (uqps (qps ps)).
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Fixpoint pat_ind'
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(P : pat -> Prop)
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(PDiscard : P pDiscard)
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(PCapture : forall p, P p -> P (pCapture p))
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(PBool : forall b, P (pBool b))
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(PSymbol : forall l, P (pSymbol l))
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(PRecord : forall p ps, P p -> Forall P ps -> P (pRecord p ps))
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(PSeq : forall ps, Forall P ps -> P (pSeq ps))
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p
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: P p.
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Proof.
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destruct p; try reflexivity.
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destruct l; try congruence; reflexivity.
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intros; destruct p; try reflexivity.
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destruct l; try reflexivity; destruct p0; try reflexivity.
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destruct p0; try reflexivity.
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destruct p0; try reflexivity.
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induction p; auto.
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apply PRecord.
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apply IHp; try assumption.
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induction l; [ apply Forall_nil | ].
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apply Forall_cons; [ apply pat_ind'; try assumption | ].
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apply IHl; apply IHp.
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apply PSeq.
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induction l; [ apply Forall_nil | ].
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apply Forall_cons; [ apply pat_ind'; try assumption | ].
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apply IHl.
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Defined.
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Lemma Forall_map_map_id :
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forall ps,
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Forall (fun p : pat => uqp (qp p) = p) ps ->
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(map uqp (map qp ps)) = ps.
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Proof.
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intros; rewrite map_map; induction H; [ reflexivity | ].
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simpl; rewrite IHForall; rewrite H; reflexivity.
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Qed.
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Theorem quoting_sensible : forall p, uqp (qp p) = p
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with quoting_list_sensible : forall ps, uqps (qps ps) = ps.
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Ltac solve_map_map_id := repeat f_equal; apply Forall_map_map_id; assumption.
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Theorem quoting_sensible : forall p, uqp (qp p) = p.
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Proof.
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clear quoting_sensible.
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induction p; try reflexivity.
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remember (qp (pCapture p)) as p'; simpl in Heqp'; rewrite Heqp'; simpl; rewrite IHp; reflexivity.
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induction p using pat_ind';
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try solve [ reflexivity
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| rewrite <- IHp at 2; reflexivity
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| simpl; solve_map_map_id ].
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remember (pat_dec p (pSymbol lDiscard)) as HisDiscard; inversion HisDiscard.
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remember (pats_dec p0 psNil) as HisEmpty; inversion HisEmpty.
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rewrite H, H0; reflexivity.
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rewrite H; remember p0 as p0'; destruct p0'.
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congruence.
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simpl.
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rewrite Heqp0'.
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rewrite <- quoting_list_sensible with (ps := p0).
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rewrite <- Heqp0'.
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reflexivity.
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destruct p; simpl; try rewrite <- IHp; try solve [ solve_map_map_id ].
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remember (pat_dec p (pSymbol lCapture)) as HisCapture; inversion HisCapture;
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[ rewrite H0; clear H0 | ].
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remember p0 as p0'; destruct p0'; [ reflexivity | ].
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rewrite Heqp0'.
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destruct p0';
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[ simpl | unfold qp];
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rewrite <- quoting_list_sensible with (ps := p0) at 2;
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rewrite <- Heqp0';
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reflexivity.
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(* label pSymbol *)
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destruct l;
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[ destruct ps; [ reflexivity | inversion H; simpl; rewrite H2; solve_map_map_id ]
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| | | simpl; solve_map_map_id ];
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try solve [ destruct ps; [ reflexivity | ];
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destruct ps; [ inversion H; simpl; rewrite H2; reflexivity | ];
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inversion H; simpl; rewrite H2;
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inversion H3; simpl; rewrite H6;
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solve_map_map_id ].
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remember (pat_dec p (pSymbol lQuote)) as HisQuote; inversion HisQuote;
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[ rewrite H1; clear H1 | ].
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remember p0 as p0'; destruct p0'; [ reflexivity | ].
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rewrite Heqp0'.
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destruct p0';
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[ simpl | unfold qp];
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rewrite <- quoting_list_sensible with (ps := p0) at 2;
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rewrite <- Heqp0';
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reflexivity.
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rewrite quoting_for_record_sensible; try assumption.
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rewrite <- IHp at 2.
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rewrite <- quoting_list_sensible with (ps := p0) at 2.
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reflexivity.
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rewrite <- quoting_list_sensible with (ps := p) at 2; reflexivity.
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clear quoting_list_sensible.
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induction ps; try reflexivity.
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simpl.
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rewrite quoting_sensible.
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f_equal.
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apply IHps.
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(* label pRecord *)
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destruct p; try solve_map_map_id;
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destruct l0; destruct l; try solve_map_map_id;
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destruct l; solve_map_map_id.
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Qed.
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