Finish renaming
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Yiyun Liu
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1 changed files with 48 additions and 87 deletions
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@ -2133,26 +2133,16 @@ Lemma pars_univ_inv n i (c : PTm n) :
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rtc Par.R (PUniv i) c ->
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extract c = PUniv i.
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Proof.
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have : prov (PUniv i) (Univ i : PTm n) by sfirstorder.
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have : prov (PUniv i) (PUniv i : PTm n) by sfirstorder.
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move : prov_pars. repeat move/[apply].
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apply prov_extract.
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Qed.
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Lemma pars_const_inv n i (c : PTm n) :
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rtc Par.R (Const i) c ->
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extract c = Const i.
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rtc Par.R (PConst i) c ->
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extract c = PConst i.
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Proof.
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have : prov (Const i) (Const i : PTm n) by sfirstorder.
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move : prov_pars. repeat move/[apply].
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apply prov_extract.
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Qed.
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Lemma pars_pi_inv n p (A : PTm n) B C :
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rtc Par.R (TBind p A B) C ->
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exists A0 B0, extract C = TBind p A0 B0 /\
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rtc Par.R A A0 /\ rtc Par.R B B0.
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Proof.
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have : prov (TBind p A B) (TBind p A B) by hauto lq:on ctrs:prov, rtc.
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have : prov (PConst i) (PConst i : PTm n) by sfirstorder.
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move : prov_pars. repeat move/[apply].
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apply prov_extract.
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Qed.
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@ -2167,32 +2157,21 @@ Proof.
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Qed.
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Lemma pars_univ_inj n i j (C : PTm n) :
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rtc Par.R (Univ i) C ->
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rtc Par.R (Univ j) C ->
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rtc Par.R (PUniv i) C ->
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rtc Par.R (PUniv j) C ->
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i = j.
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Proof.
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sauto l:on use:pars_univ_inv.
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Qed.
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Lemma pars_const_inj n i j (C : PTm n) :
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rtc Par.R (Const i) C ->
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rtc Par.R (Const j) C ->
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rtc Par.R (PConst i) C ->
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rtc Par.R (PConst j) C ->
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i = j.
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Proof.
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sauto l:on use:pars_const_inv.
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Qed.
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Lemma pars_pi_inj n p0 p1 (A0 A1 : PTm n) B0 B1 C :
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rtc Par.R (TBind p0 A0 B0) C ->
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rtc Par.R (TBind p1 A1 B1) C ->
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exists A2 B2, p1 = p0 /\ rtc Par.R A0 A2 /\ rtc Par.R A1 A2 /\
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rtc Par.R B0 B2 /\ rtc Par.R B1 B2.
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Proof.
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move /pars_pi_inv => [A2 [B2 [? [h0 h1]]]].
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move /pars_pi_inv => [A3 [B3 [? [h2 h3]]]].
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exists A2, B2. hauto l:on.
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Qed.
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Definition join {n} (a b : PTm n) :=
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exists c, rtc Par.R a c /\ rtc Par.R b c.
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@ -2214,26 +2193,17 @@ Lemma join_refl n (a : PTm n) : join a a.
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Proof. hauto lq:on ctrs:rtc unfold:join. Qed.
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Lemma join_univ_inj n i j :
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join (Univ i : PTm n) (Univ j) -> i = j.
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join (PUniv i : PTm n) (PUniv j) -> i = j.
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Proof.
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sfirstorder use:pars_univ_inj.
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Qed.
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Lemma join_const_inj n i j :
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join (Const i : PTm n) (Const j) -> i = j.
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join (PConst i : PTm n) (PConst j) -> i = j.
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Proof.
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sfirstorder use:pars_const_inj.
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Qed.
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Lemma join_pi_inj n p0 p1 (A0 A1 : PTm n) B0 B1 :
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join (TBind p0 A0 B0) (TBind p1 A1 B1) ->
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p0 = p1 /\ join A0 A1 /\ join B0 B1.
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Proof.
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move => [c []].
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move : pars_pi_inj; repeat move/[apply].
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sfirstorder unfold:join.
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Qed.
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Lemma join_substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
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join a b ->
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join (subst_PTm ρ a) (subst_PTm ρ b).
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@ -2242,26 +2212,24 @@ Proof. hauto lq:on unfold:join use:Pars.substing. Qed.
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Fixpoint ne {n} (a : PTm n) :=
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match a with
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| VarPTm i => true
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| TBind _ A B => false
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| App a b => ne a && nf b
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| Abs a => false
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| Univ _ => false
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| Proj _ a => ne a
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| Pair _ _ => false
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| Const _ => false
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| Bot => true
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| PApp a b => ne a && nf b
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| PAbs a => false
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| PUniv _ => false
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| PProj _ a => ne a
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| PPair _ _ => false
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| PConst _ => false
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| PBot => true
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end
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with nf {n} (a : PTm n) :=
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match a with
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| VarPTm i => true
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| TBind _ A B => nf A && nf B
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| App a b => ne a && nf b
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| Abs a => nf a
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| Univ _ => true
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| Proj _ a => ne a
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| Pair a b => nf a && nf b
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| Const _ => true
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| Bot => true
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| PApp a b => ne a && nf b
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| PAbs a => nf a
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| PUniv _ => true
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| PProj _ a => ne a
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| PPair a b => nf a && nf b
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| PConst _ => true
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| PBot => true
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end.
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Lemma ne_nf n a : @ne n a -> nf a.
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@ -2290,37 +2258,30 @@ Proof.
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Qed.
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Lemma wne_app n (a b : PTm n) :
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wne a -> wn b -> wne (App a b).
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wne a -> wn b -> wne (PApp a b).
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Proof.
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move => [a0 [? ?]] [b0 [? ?]].
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exists (App a0 b0). hauto b:on drew:off use:RPars'.AppCong.
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exists (PApp a0 b0). hauto b:on drew:off use:RPars'.AppCong.
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Qed.
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Lemma wn_abs n a (h : wn a) : @wn n (Abs a).
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Lemma wn_abs n a (h : wn a) : @wn n (PAbs a).
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Proof.
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move : h => [v [? ?]].
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exists (Abs v).
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exists (PAbs v).
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eauto using RPars'.AbsCong.
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Qed.
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Lemma wn_bind n p A B : wn A -> wn B -> wn (@TBind n p A B).
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Proof.
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move => [A0 [? ?]] [B0 [? ?]].
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exists (TBind p A0 B0).
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hauto lqb:on use:RPars'.BindCong.
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Qed.
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Lemma wn_pair n (a b : PTm n) : wn a -> wn b -> wn (Pair a b).
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Lemma wn_pair n (a b : PTm n) : wn a -> wn b -> wn (PPair a b).
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Proof.
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move => [a0 [? ?]] [b0 [? ?]].
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exists (Pair a0 b0).
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exists (PPair a0 b0).
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hauto lqb:on use:RPars'.PairCong.
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Qed.
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Lemma wne_proj n p (a : PTm n) : wne a -> wne (Proj p a).
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Lemma wne_proj n p (a : PTm n) : wne a -> wne (PProj p a).
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Proof.
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move => [a0 [? ?]].
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exists (Proj p a0). hauto lqb:on use:RPars'.ProjCong.
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exists (PProj p a0). hauto lqb:on use:RPars'.ProjCong.
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Qed.
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Create HintDb nfne.
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@ -2348,10 +2309,10 @@ Proof.
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Qed.
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Lemma ext_wn n (a : PTm n) :
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wn (App a Bot) ->
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wn (PApp a PBot) ->
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wn a.
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Proof.
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move E : (App a (Bot)) => a0 [v [hr hv]].
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move E : (PApp a (PBot)) => a0 [v [hr hv]].
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move : a E.
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move : hv.
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elim : a0 v / hr.
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@ -2362,9 +2323,9 @@ Proof.
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case : a0 hr0=>// => b0 b1.
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elim /RPar'.inv=>// _.
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+ move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
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have ? : b3 = (Bot) by hauto lq:on inv:RPar'.R. subst.
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suff : wn (Abs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
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have : wn (subst_PTm (scons (Bot) VarPTm) a3) by sfirstorder.
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have ? : b3 = (PBot) by hauto lq:on inv:RPar'.R. subst.
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suff : wn (PAbs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
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have : wn (subst_PTm (scons (PBot) VarPTm) a3) by sfirstorder.
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move => h. apply wn_abs.
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move : h. apply wn_antirenaming.
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hauto lq:on rew:off inv:option.
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@ -2374,24 +2335,24 @@ Qed.
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Module Join.
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Lemma ProjCong p n (a0 a1 : PTm n) :
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join a0 a1 ->
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join (Proj p a0) (Proj p a1).
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join (PProj p a0) (PProj p a1).
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Proof. hauto lq:on use:Pars.ProjCong unfold:join. Qed.
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Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
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join a0 a1 ->
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join b0 b1 ->
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join (Pair a0 b0) (Pair a1 b1).
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join (PPair a0 b0) (PPair a1 b1).
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Proof. hauto lq:on use:Pars.PairCong unfold:join. Qed.
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Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
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join a0 a1 ->
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join b0 b1 ->
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join (App a0 b0) (App a1 b1).
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join (PApp a0 b0) (PApp a1 b1).
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Proof. hauto lq:on use:Pars.AppCong. Qed.
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Lemma AbsCong n (a b : PTm (S n)) :
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join a b ->
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join (Abs a) (Abs b).
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join (PAbs a) (PAbs b).
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Proof. hauto lq:on use:Pars.AbsCong. Qed.
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Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
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@ -2415,11 +2376,11 @@ Module Join.
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End Join.
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Lemma abs_eq n a (b : PTm n) :
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join (Abs a) b <-> join a (App (ren_PTm shift b) (VarPTm var_zero)).
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join (PAbs a) b <-> join a (PApp (ren_PTm shift b) (VarPTm var_zero)).
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Proof.
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split.
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- move => /Join.weakening h.
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have {h} : join (App (ren_PTm shift (Abs a)) (VarPTm var_zero)) (App (ren_PTm shift b) (VarPTm var_zero))
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have {h} : join (PApp (ren_PTm shift (PAbs a)) (VarPTm var_zero)) (PApp (ren_PTm shift b) (VarPTm var_zero))
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by hauto l:on use:Join.AppCong, join_refl.
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simpl.
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move => ?. apply : join_transitive; eauto.
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@ -2431,12 +2392,12 @@ Proof.
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Qed.
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Lemma pair_eq n (a0 a1 b : PTm n) :
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join (Pair a0 a1) b <-> join a0 (Proj PL b) /\ join a1 (Proj PR b).
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join (PPair a0 a1) b <-> join a0 (PProj PL b) /\ join a1 (PProj PR b).
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Proof.
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split.
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- move => h.
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have /Join.ProjCong {}h := h.
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have h0 : forall p, join (if p is PL then a0 else a1) (Proj p (Pair a0 a1))
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have h0 : forall p, join (if p is PL then a0 else a1) (PProj p (PPair a0 a1))
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by hauto lq:on use:join_symmetric, Join.FromPar, Par.ProjPair', Par.refl.
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hauto lq:on rew:off use:join_transitive, join_symmetric.
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- move => [h0 h1].
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@ -2447,11 +2408,11 @@ Proof.
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Qed.
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Lemma join_pair_inj n (a0 a1 b0 b1 : PTm n) :
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join (Pair a0 a1) (Pair b0 b1) <-> join a0 b0 /\ join a1 b1.
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join (PPair a0 a1) (PPair b0 b1) <-> join a0 b0 /\ join a1 b1.
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Proof.
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split; last by hauto lq:on use:Join.PairCong.
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move /pair_eq => [h0 h1].
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have : join (Proj PL (Pair b0 b1)) b0 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
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have : join (Proj PR (Pair b0 b1)) b1 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
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have : join (PProj PL (PPair b0 b1)) b0 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
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have : join (PProj PR (PPair b0 b1)) b1 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
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eauto using join_transitive.
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Qed.
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