Update the syntactic reduction lemmas
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Yiyun Liu
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1 changed files with 107 additions and 13 deletions
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@ -2022,6 +2022,30 @@ Module EReds.
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hauto lq:on rew:off ctrs:rtc, ERed.R inv:ERed.R, rtc.
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Qed.
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Lemma suc_inv n (a : PTm n) C :
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rtc ERed.R (PSuc a) C ->
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exists b, rtc ERed.R a b /\ C = PSuc b.
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Proof.
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move E : (PSuc a) => u hu.
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move : a E.
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elim : u C / hu=>//=.
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- hauto l:on.
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- hauto lq:on rew:off ctrs:rtc, ERed.R inv:ERed.R, rtc.
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Qed.
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Lemma ind_inv n P (a : PTm n) b c C :
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rtc ERed.R (PInd P a b c) C ->
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exists P0 a0 b0 c0, rtc ERed.R P P0 /\ rtc ERed.R a a0 /\
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rtc ERed.R b b0 /\ rtc ERed.R c c0 /\
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C = PInd P0 a0 b0 c0.
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Proof.
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move E : (PInd P a b c) => u hu.
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move : P a b c E.
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elim : u C / hu.
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- hauto lq:on ctrs:rtc.
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- hauto lq:on rew:off ctrs:rtc, ERed.R inv:ERed.R, rtc.
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Qed.
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End EReds.
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#[export]Hint Constructors ERed.R RRed.R EPar.R : red.
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@ -2050,6 +2074,18 @@ Module EJoin.
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hauto lq:on rew:off use:EReds.bind_inv.
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Qed.
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Lemma suc_inj n (A0 A1 : PTm n) :
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R (PSuc A0) (PSuc A1) ->
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R A0 A1.
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Proof.
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hauto lq:on rew:off use:EReds.suc_inv.
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Qed.
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Lemma ind_inj n P0 P1 (a0 a1 : PTm n) b0 b1 c0 c1 :
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R (PInd P0 a0 b0 c0) (PInd P1 a1 b1 c1) ->
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R P0 P1 /\ R a0 a1 /\ R b0 b1 /\ R c0 c1.
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Proof. hauto q:on use:EReds.ind_inv. Qed.
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End EJoin.
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Module RERed.
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@ -2361,6 +2397,17 @@ Module REReds.
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induction h; hauto lq:on rew:off ctrs:rtc use:RERed.ToBetaEta, RReds.nf_refl, @rtc_once, ERed.nf_preservation.
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Qed.
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Lemma suc_inv n (a : PTm n) C :
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rtc RERed.R (PSuc a) C ->
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exists b, rtc RERed.R a b /\ C = PSuc b.
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Proof.
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move E : (PSuc a) => u hu.
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move : a E.
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elim : u C / hu=>//=.
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- hauto l:on.
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- hauto lq:on rew:off ctrs:rtc, RERed.R inv:RERed.R, rtc.
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Qed.
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End REReds.
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Module LoRed.
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@ -2699,12 +2746,37 @@ Proof.
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move : ihP hP' => /[apply]. move => [P4][hP0]hP1.
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eauto using EReds.IndCong, rtc_refl.
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+ move => P2 a0 a1 b0 c0 + [*]. subst.
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move {ihP}.
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move => h. exists (PInd P1 a1 b c).
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eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
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+ move => P2 a0 b0 b1 c0 hb [*] {ihP}. subst.
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Admitted.
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eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
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+ move => P2 a0 b0 c0 c1 h [*] {ihP}. subst.
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eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
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- move => P a0 a1 b c ha iha u.
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elim /ERed.inv => //=_;
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try solve [move => P0 P1 a2 b0 c0 hP[*]; subst;
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eauto 20 using rtc_refl, EReds.IndCong, rtc_l].
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move => P0 P1 a2 b0 c0 hP[*]. subst.
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move : iha hP => /[apply].
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move => [? [*]].
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eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
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- move => P a b0 b1 c hb ihb u.
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elim /ERed.inv => //=_;
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try solve [
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move => P0 P1 a0 b2 c0 hP [*]; subst;
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eauto 20 using rtc_refl, EReds.IndCong, rtc_l].
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move => P0 a0 b2 b3 c0 h [*]. subst.
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move : ihb h => /[apply]. move => [b2 [*]].
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eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
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- move => P a b c0 c1 hc ihc u.
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elim /ERed.inv => //=_;
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try solve [
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move => P0 P1 a0 b0 c hP [*]; subst;
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eauto 20 using rtc_refl, EReds.IndCong, rtc_l].
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move => P0 a0 b0 c2 c3 h [*]. subst.
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move : ihc h => /[apply]. move => [c2][*].
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eauto 20 using rtc_refl, EReds.IndCong, rtc_l.
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- qauto l:on inv:ERed.R ctrs:rtc use:EReds.SucCong.
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Qed.
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Lemma ered_confluence n (a b c : PTm n) :
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rtc ERed.R a b ->
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@ -2820,16 +2892,16 @@ Module BJoin.
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exists v. sfirstorder use:@relations.rtc_transitive.
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Qed.
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Lemma AbsCong n (a b : PTm (S n)) :
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R a b ->
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R (PAbs a) (PAbs b).
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Proof. hauto lq:on use:RReds.AbsCong unfold:R. Qed.
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(* Lemma AbsCong n (a b : PTm (S n)) : *)
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(* R a b -> *)
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(* R (PAbs a) (PAbs b). *)
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(* Proof. hauto lq:on use:RReds.AbsCong unfold:R. Qed. *)
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Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
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R a0 a1 ->
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R b0 b1 ->
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R (PApp a0 b0) (PApp a1 b1).
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Proof. hauto lq:on use:RReds.AppCong unfold:R. Qed.
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(* Lemma AppCong n (a0 a1 b0 b1 : PTm n) : *)
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(* R a0 a1 -> *)
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(* R b0 b1 -> *)
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(* R (PApp a0 b0) (PApp a1 b1). *)
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(* Proof. hauto lq:on use:RReds.AppCong unfold:R. Qed. *)
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End BJoin.
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Module DJoin.
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@ -2957,6 +3029,13 @@ Module DJoin.
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sauto lq:on rew:off use:REReds.univ_inv.
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Qed.
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Lemma suc_inj n (A0 A1 : PTm n) :
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R (PSuc A0) (PSuc A1) ->
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R A0 A1.
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Proof.
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hauto lq:on rew:off use:REReds.suc_inv.
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Qed.
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Lemma hne_app_inj n (a0 b0 a1 b1 : PTm n) :
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R (PApp a0 b0) (PApp a1 b1) ->
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ishne a0 ->
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@ -3251,6 +3330,13 @@ Module ESub.
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sauto lq:on rew:off inv:Sub1.R.
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Qed.
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Lemma suc_inj n (a b : PTm n) :
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R (PSuc a) (PSuc b) ->
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R a b.
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Proof.
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sauto lq:on use:EReds.suc_inv inv:Sub1.R.
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Qed.
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End ESub.
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Module Sub.
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@ -3400,6 +3486,14 @@ Module Sub.
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sauto lq:on rew:off use:REReds.univ_inv.
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Qed.
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Lemma suc_inj n (A0 A1 : PTm n) :
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R (PSuc A0) (PSuc A1) ->
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R A0 A1.
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Proof.
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sauto q:on use:REReds.suc_inv.
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Qed.
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Lemma cong n m (a b : PTm (S n)) c d (ρ : fin n -> PTm m) :
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R a b -> DJoin.R c d -> R (subst_PTm (scons c ρ) a) (subst_PTm (scons d ρ) b).
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Proof.
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