Finish renaming

theories/fp_red.v

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