From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Tue, 21 Apr 2020 12:35:09 +0000 (+0200)
Subject: update in basic_2
X-Git-Tag: make_still_working~181
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=b2cdc4abd9ac87e39bc51b0d9c38daea179adbd5;p=helm.git

update in basic_2

+ rpx_req: cpx_teqx_conf_rex replaced by cpx_teqx_repl_reqx
---

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_feqx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_feqx.ma
index 882543af4..8017c74ce 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_feqx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_feqx.ma
@@ -19,14 +19,15 @@ include "basic_2/rt_computation/cpxs_reqx.ma".
 
 (* Properties with sort-irrelevant equivalence for closures *****************)
 
+(* to be updated *)
 lemma feqx_cpxs_trans:
       ∀G1,G2,L1,L2,T1,T. ❪G1,L1,T1❫ ≛ ❪G2,L2,T❫ →
       ∀T2. ❪G2,L2❫ ⊢ T ⬈* T2 →
       ∃∃T0. ❪G1,L1❫ ⊢ T1 ⬈* T0 & ❪G1,L1,T0❫ ≛ ❪G2,L2,T2❫.
-#G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
+#G1 #G2 #L1 #L2 #T1 #T #H #T2 #H2T2
 elim (feqx_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
-elim (reqx_cpxs_trans … HT2 … HL12) #T0 #HT0 #HT02
-lapply (cpxs_reqx_conf_dx … HT2 … HL12) -HL12 #HL12
-elim (teqx_cpxs_trans … HT1 … HT0) -T #T #HT1 #HT0
-/4 width=5 by feqx_intro_dx, teqx_trans, ex2_intro/
+lapply (reqx_cpxs_trans … H2T2 … HL12) #H1T2
+lapply (cpxs_reqx_conf_dx … H2T2 … HL12) -HL12 #HL12
+lapply (teqx_cpxs_trans … HT1 … H1T2) -T #HT12
+/3 width=3 by feqx_intro_dx, ex2_intro/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_reqx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_reqx.ma
index b523f08b8..ee504e981 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_reqx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_reqx.ma
@@ -21,20 +21,17 @@ include "basic_2/rt_computation/cpxs_teqx.ma".
 
 (* Basic_2A1: was just: lleq_cpxs_trans *)
 lemma reqx_cpxs_trans (G):
-      ∀L0,T0,T1. ❪G,L0❫ ⊢ T0 ⬈* T1 → ∀L2. L2 ≛[T0] L0 →
-      ∃∃T. ❪G,L2❫ ⊢ T0 ⬈* T & T ≛ T1.
-#G #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 /2 width=3 by ex2_intro/
-#T0 #T #HT0 #_ #IH #L2 #HL2
-elim (reqx_cpx_trans … HL2 … HT0) #U1 #H1 #H2
-elim (IH L2) -IH /2 width=5 by cpx_reqx_conf_dx/ -L0 #U2 #H3 #H4
-elim (teqx_cpxs_trans … H2 … H3) -T #U0 #H2 #H3
-/3 width=5 by cpxs_strap2, teqx_trans, ex2_intro/
+      ∀L0,T0,T1. ❪G,L0❫ ⊢ T0 ⬈* T1 → ∀L2. L2 ≛[T0] L0 → ❪G,L2❫ ⊢ T0 ⬈* T1.
+#G #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 //
+#T0 #T #H0T0 #_ #IH #L2 #HL2
+lapply (reqx_cpx_trans … HL2 … H0T0) #H2T0
+lapply (IH L2 ?) -IH /2 width=5 by cpx_reqx_conf_dx/ -L0 #H2T1
+/2 width=3 by cpxs_strap2/
 qed-.
 
 (* Basic_2A1: was just: cpxs_lleq_conf *)
 lemma cpxs_reqx_conf (G):
-      ∀L0,T0,T1. ❪G,L0❫ ⊢ T0 ⬈* T1 → ∀L2. L0 ≛[T0] L2 →
-      ∃∃T. ❪G,L2❫ ⊢ T0 ⬈* T & T ≛ T1.
+      ∀L0,T0,T1. ❪G,L0❫ ⊢ T0 ⬈* T1 → ∀L2. L0 ≛[T0] L2 → ❪G,L2❫ ⊢ T0 ⬈* T1.
 /3 width=3 by reqx_cpxs_trans, reqx_sym/ qed-.
 
 (* Basic_2A1: was just: cpxs_lleq_conf_dx *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_teqx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_teqx.ma
index f3164e75e..0de7eb677 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_teqx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs_teqx.ma
@@ -19,29 +19,28 @@ include "basic_2/rt_computation/cpxs.ma".
 
 (* Properties with sort-irrelevant equivalence for terms ********************)
 
-lemma teqx_cpxs_trans (G) (L):
-      ∀U1,T1. U1 ≛ T1 → ∀T2. ❪G,L❫ ⊢ T1 ⬈* T2 →
-      ∃∃U2. ❪G,L❫ ⊢ U1 ⬈* U2 & U2 ≛ T2.
-#G #L #U1 #T1 #HUT1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /2 width=3 by ex2_intro/
-#T #T2 #_ #HT2 * #U #HU1 #HUT elim (teqx_cpx_trans … HUT … HT2) -T -T1
-/3 width=3 by ex2_intro, cpxs_strap1/
+lemma teqx_cpxs_trans (G) (L) (T):
+      ∀T1. T1 ≛ T → ∀T2. ❪G,L❫ ⊢ T ⬈* T2 → ❪G,L❫ ⊢ T1 ⬈* T2.
+#G #L #T #T1 #HT1 #T2 #HT2 @(cpxs_ind … HT2) -T2
+[ /3 width=1 by teqx_cpx, cpx_cpxs/
+| /2 width=3 by cpxs_strap1/
+]
 qed-.
 
 (* Note: this requires teqx to be symmetric *)
 (* Nasic_2A1: uses: cpxs_neq_inv_step_sn *)
 lemma cpxs_tneqx_fwd_step_sn (G) (L):
       ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈* T2 → (T1 ≛ T2 → ⊥) →
-      ∃∃T,T0. ❪G,L❫ ⊢ T1 ⬈ T & T1 ≛ T → ⊥ & ❪G,L❫ ⊢ T ⬈* T0 & T0 ≛ T2.
+      ∃∃T. ❪G,L❫ ⊢ T1 ⬈ T & T1 ≛ T → ⊥ & ❪G,L❫ ⊢ T ⬈* T2.
 #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
 [ #H elim H -H //
-| #T1 #T0 #HT10 #HT02 #IH #Hn12
-  elim (teqx_dec T1 T0) [ -HT10 -HT02 #H10 | -IH #Hn10 ]
-  [ elim IH -IH /3 width=3 by teqx_trans/ -Hn12
-    #T3 #T4 #HT03 #Hn03 #HT34 #H42
-    elim (teqx_cpx_trans … H10 … HT03) -HT03 #T5 #HT15 #H53
-    elim (teqx_cpxs_trans … H53 … HT34) -HT34 #T6 #HT56 #H64
-    /5 width=8 by teqx_canc_sn, (* 2x *) teqx_trans, ex4_2_intro/
-  | /3 width=6 by ex4_2_intro/
+| #T1 #T0 #HT10 #HT02 #IH #HnT12
+  elim (teqx_dec T1 T0) [ -HT10 -HT02 #HT10 | -IH #HnT10 ]
+  [ elim IH -IH /3 width=3 by teqx_trans/ -HnT12
+    #T #HT0 #HnT0 #HT2
+    lapply (teqx_cpx_trans … HT10 … HT0) -HT0 #HT1
+    /4 width=4 by teqx_canc_sn, ex3_intro/
+  | /3 width=4 by ex3_intro/
   ]
 ]
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.ma
index baba7a131..6995ad656 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_cpxs.ma
@@ -36,27 +36,23 @@ qed-.
 
 (* Eliminators with extended context-sensitive rt-computation for terms *****)
 
-lemma csx_ind_cpxs_teqx (G) (L):
+fact csx_ind_cpxs_aux (G) (L):
       ∀Q:predicate term.
       (∀T1. ❪G,L❫ ⊢ ⬈*𝐒 T1 →
         (∀T2. ❪G,L❫ ⊢ T1 ⬈* T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
       ) →
       ∀T1. ❪G,L❫ ⊢ ⬈*𝐒 T1 →
-      ∀T0. ❪G,L❫ ⊢ T1 ⬈* T0 → ∀T2. T0 ≛ T2 → Q T2.
+      ∀T2. ❪G,L❫ ⊢ T1 ⬈* T2 → Q T2.
 #G #L #Q #IH #T1 #H @(csx_ind … H) -T1
-#T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02
-@IH -IH /3 width=3 by csx_cpxs_trans, csx_teqx_trans/ -HT1 #V2 #HTV2 #HnTV2
-lapply (teqx_tneqx_trans … HT02 … HnTV2) -HnTV2 #H
-elim (teqx_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
-lapply (tneqx_teqx_canc_dx … H … HV02) -H #HnTV0
-elim (teqx_dec T1 T0) #H
-[ lapply (teqx_tneqx_trans … H … HnTV0) -H -HnTV0 #Hn10
-  lapply (cpxs_trans … HT10 … HTV0) -T0 #H10
-  elim (cpxs_tneqx_fwd_step_sn … H10 …  Hn10) -H10 -Hn10
-  /3 width=8 by teqx_trans/
-| elim (cpxs_tneqx_fwd_step_sn … HT10 … H) -HT10 -H #T #V #HT1 #HnT1 #HTV #HVT0
-  elim (teqx_cpxs_trans … HVT0 … HTV0) -T0
-  /3 width=8 by cpxs_trans, teqx_trans/
+#T1 #HT1 #IH1 #T2 #HT12
+@IH -IH /2 width=3 by csx_cpxs_trans/ -HT1 #V2 #HTV2 #HnTV2
+elim (teqx_dec T1 T2) #H
+[ lapply (teqx_tneqx_trans … H … HnTV2) -H -HnTV2 #Hn12
+  lapply (cpxs_trans … HT12 … HTV2) -T2 #H12
+  elim (cpxs_tneqx_fwd_step_sn … H12 …  Hn12) -H12 -Hn12
+  /3 width=4 by/
+| elim (cpxs_tneqx_fwd_step_sn … HT12 … H) -HT12 -H
+  /3 width=6 by cpxs_trans/
 ]
 qed-.
 
@@ -67,5 +63,5 @@ lemma csx_ind_cpxs (G) (L) (Q:predicate …):
       ) →
       ∀T. ❪G,L❫ ⊢ ⬈*𝐒 T → Q T.
 #G #L #Q #IH #T #HT
-@(csx_ind_cpxs_teqx … IH … HT) -IH -HT // (**) (* full auto fails *)
+@(csx_ind_cpxs_aux … IH … HT) -IH -HT // (**) (* full auto fails *)
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx.ma
index 86dc07b45..72d7a1f4b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_csx.ma
@@ -23,7 +23,8 @@ lemma csx_teqx_trans (G) (L):
       ∀T1. ❪G,L❫ ⊢ ⬈*𝐒 T1 →
       ∀T2. T1 ≛ T2 → ❪G,L❫ ⊢ ⬈*𝐒 T2.
 #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
-@csx_intro #T1 #HT21 #HnT21 elim (teqx_cpx_trans … HT2 … HT21) -HT21
+@csx_intro #T1 #HT21 #HnT21
+lapply (teqx_cpx_trans … HT2 … HT21) -HT21 #HT1
 /4 width=5 by teqx_repl/
 qed-.
 
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_reqx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_reqx.ma
index f6f8970e3..5d35690cc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_reqx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/csx_reqx.ma
@@ -26,8 +26,8 @@ lemma csx_reqx_conf (G) (L1):
 #G #L1 #T #H
 @(csx_ind … H) -T #T1 #_ #IH #L2 #HL12
 @csx_intro #T2 #HT12 #HnT12
-elim (reqx_cpx_trans … HL12 … HT12) -HT12
-/5 width=5 by cpx_reqx_conf_sn, csx_teqx_trans, teqx_trans/
+lapply (reqx_cpx_trans … HL12 … HT12) -HT12
+/3 width=4 by cpx_reqx_conf_sn/
 qed-.
 
 (* Basic_2A1: uses: csx_lleq_trans *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_csx.ma
index 1839df0f8..9991033e1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/fsb_csx.ma
@@ -50,7 +50,7 @@ generalize in match IHu; -IHu generalize in match H10; -H10
     elim (teqx_dec T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
     [ elim (teqx_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
       /4 width=7 by fsb_fpbs_trans, teqx_reqx_lpx_fpbs, fpbs_fqup_trans/
-    | elim (cpxs_tneqx_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
+    | elim (cpxs_tneqx_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #HT02 #HnT02 #HT24
       elim (fpbs_cpx_tneqx_trans … H10 … HT02 HnT02) -T0 #T0 #HT10 #HnT10 #H02
       /3 width=14 by fpbs_cpxs_teqx_fqup_lpx_trans/
     ]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_cnx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_cnx.ma
index 810fb85f3..ba56c516c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_cnx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnx_cnx.ma
@@ -22,6 +22,6 @@ include "basic_2/rt_transition/cnx.ma".
 lemma cnx_teqx_trans (G) (L):
       ∀T1. ❪G,L❫ ⊢ ⬈𝐍 T1 → ∀T2. T1 ≛ T2 → ❪G,L❫ ⊢ ⬈𝐍 T2.
 #G #L #T1 #HT1 #T2 #HT12 #T #HT2
-elim (teqx_cpx_trans … HT12 … HT2) -HT2 #T0 #HT10 #HT0
-lapply (HT1 … HT10) -HT1 -HT10 /2 width=5 by teqx_repl/ (**) (* full auto fails *)
+lapply (teqx_cpx_trans … HT12 … HT2) -HT2 #H
+lapply (HT1 … H) -HT1 -H /2 width=5 by teqx_canc_sn/ (**) (* full auto fails *)
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_feqx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_feqx.ma
index a0ec5d8f8..333a402d3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_feqx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_feqx.ma
@@ -20,14 +20,15 @@ include "basic_2/rt_transition/rpx_reqx.ma".
 
 (* Properties with sort-irrelevant equivalence for closures *****************)
 
+(**) (* to update *)
 lemma feqx_cpx_trans:
       ∀G1,G2,L1,L2,T1,T. ❪G1,L1,T1❫ ≛ ❪G2,L2,T❫ →
       ∀T2. ❪G2,L2❫ ⊢ T ⬈ T2 →
       ∃∃T0. ❪G1,L1❫ ⊢ T1 ⬈ T0 & ❪G1,L1,T0❫ ≛ ❪G2,L2,T2❫.
 #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
 elim (feqx_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
-elim (reqx_cpx_trans … HL12 … HT2) #T0 #HT0 #HT02
-lapply (cpx_reqx_conf_dx … HT2 … HL12) -HL12 #HL12
-elim (teqx_cpx_trans … HT1 … HT0) -T #T #HT1 #HT0
-/4 width=5 by feqx_intro_dx, teqx_trans, ex2_intro/
+lapply (reqx_cpx_trans … HL12 … HT2) #H
+lapply (cpx_reqx_conf_dx … HT2 … HL12) -HT2 -HL12 #HL12
+lapply (teqx_cpx_trans … HT1 … H) -T #HT12
+/3 width=3 by feqx_intro_dx, ex2_intro/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_reqx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_reqx.ma
index 9e75e8f96..19aeffac6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_reqx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/fpb_reqx.ma
@@ -30,7 +30,7 @@ lemma teqx_fpb_trans:
   elim (teqx_fqu_trans … H … HU21) -H
   /3 width=5 by fpb_fqu, ex3_2_intro/
 | #T1 #HUT1 #HnUT1
-  elim (teqx_cpx_trans … HU21 … HUT1) -HUT1
+  lapply (teqx_cpx_trans … HU21 … HUT1) -HUT1
   /6 width=5 by fpb_cpx, teqx_canc_sn, teqx_trans, ex3_2_intro/
 | /6 width=5 by fpb_lpx, rpx_teqx_div, teqx_reqx_conf, ex3_2_intro/
 ]
@@ -44,7 +44,7 @@ lemma reqx_fpb_trans:
 #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
 [ #G #L2 #U #H2 elim (reqx_fqu_trans … H2 … HT) -K2
   /3 width=5 by fpb_fqu, ex3_2_intro/
-| #U #HTU #HnTU elim (reqx_cpx_trans … HT … HTU) -HTU
+| #U #HTU #HnTU lapply (reqx_cpx_trans … HT … HTU) -HTU
   /5 width=11 by fpb_cpx, cpx_reqx_conf_sn, teqx_trans, teqx_reqx_conf, ex3_2_intro/ (**) (* time: 36s on dev *)
 | #L2 #HKL2 #HnKL2 elim (reqx_lpx_trans … HKL2 … HT) -HKL2
   /6 width=5 by fpb_lpx, (* 2x *) reqx_canc_sn, ex3_2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_reqx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_reqx.ma
index f75050ec6..73f134d10 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_reqx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/lpx_reqx.ma
@@ -20,13 +20,14 @@ include "basic_2/rt_transition/rpx_lpx.ma".
 
 (* Properties with sort-irrelevant equivalence for local environments *******)
 
+(**) (* to update as reqx_rpx_trans *)
 (* Basic_2A1: uses: lleq_lpx_trans *)
 lemma reqx_lpx_trans (G):
       ∀L2,K2. ❪G,L2❫ ⊢ ⬈ K2 → ∀L1. ∀T:term. L1 ≛[T] L2 →
       ∃∃K1. ❪G,L1❫ ⊢ ⬈ K1 & K1 ≛[T] K2.
 #G #L2 #K2 #HLK2 #L1 #T #HL12
 lapply (lpx_rpx … T HLK2) -HLK2 #HLK2
-elim (reqx_rpx_trans … HLK2 … HL12) -L2 #K #H #HK2
-elim (rpx_inv_lpx_req … H) -H #K1 #HLK1 #HK1
-/3 width=5 by req_reqx_trans, ex2_intro/
+lapply (reqx_rpx_trans … HL12 … HLK2) -L2 #H
+elim (rpx_inv_lpx_req … H) -H #K1 #HLK1 #HK12
+/3 width=3 by req_reqx, ex2_intro/
 qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_reqx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_reqx.ma
index 19fc79235..29964b29c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_reqx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/rpx_reqx.ma
@@ -48,127 +48,114 @@ lemma rpx_teqx_div (G):
       ∀T1,T2. T1 ≛ T2 → ∀L1,L2. ❪G,L1❫ ⊢ ⬈[T2] L2 → ❪G,L1❫ ⊢ ⬈[T1] L2.
 /2 width=5 by teqx_rex_div/ qed-.
 
-lemma cpx_teqx_conf_rex (G): R_confluent2_rex … (cpx G) cdeq (cpx G) cdeq.
-#G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
-[ #G #L0 #s0 #s1 #X0 #H0 #L1 #HL01 #L2 #HL02
-  elim (teqx_inv_sort1 … H0) -H0 #s2 #H destruct
-  /3 width=3 by teqx_sort, ex2_intro/
-| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
-  >(teqx_inv_lref1 … H0) -H0
-  elim (rpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
-  elim (reqx_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
-  elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
-  elim (teqx_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
-| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
-  >(teqx_inv_lref1 … H0) -H0
-  elim (rpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
-  elim (reqx_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
-  elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
-  elim (teqx_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
-| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
-  elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
-  elim (rpx_inv_bind … H1) -H1 #HL01 #H1
-  elim (reqx_inv_bind … H2) -H2 #HL02 #H2
-  lapply (reqx_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
-  elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
-  elim (IHT … HT02 … H1 … H2) -L0 -T0
-  /3 width=5 by cpx_bind, teqx_pair, ex2_intro/
-| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
-  elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
-  elim (rpx_inv_flat … H1) -H1 #HL01 #H1
-  elim (reqx_inv_flat … H2) -H2 #HL02 #H2
-  elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
-  elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
-  /3 width=5 by cpx_flat, teqx_pair, ex2_intro/
-| #G #L0 #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
-  elim (teqx_inv_pair1 … H0) -H0 #V2 #U2 #HV02 #HU02 #H destruct
-  elim (rpx_inv_bind … H1) -H1 #HL01 #H1
-  elim (reqx_inv_bind … H2) -H2 #HL02 #H2
-  lapply (rpx_inv_lifts_bi … H1 (Ⓣ) … HTU0) -H1 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H1
-  lapply (reqx_inv_lifts_bi … H2 (Ⓣ) … HTU0) -H2 [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #H2
+lemma cpx_teqx_repl_reqx (G) (L0) (T0):
+      ∀T1. ❪G,L0❫ ⊢ T0 ⬈ T1 → ∀T2. T0 ≛ T2 → ∀T3. T1 ≛ T3 →
+      ∀L2. L0 ≛[T0] L2 → ❪G,L2❫ ⊢ T2 ⬈ T3.
+#G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1
+[ * #x0 #G #L0 #X2 #HX2 #X3 #HX3 #L2 #_
+  [ elim (teqx_inv_sort1 … HX2) -HX2 #x2 #H destruct
+    elim (teqx_inv_sort1 … HX3) -HX3 #x3 #H destruct //
+  | lapply (teqx_inv_lref1 … HX2) -HX2 #H destruct
+    lapply (teqx_inv_lref1 … HX3) -HX3 #H destruct //
+  | lapply (teqx_inv_gref1 … HX2) -HX2 #H destruct
+    lapply (teqx_inv_gref1 … HX3) -HX3 #H destruct //
+  ]
+| #G #L0 #s0 #s1 #X2 #HX2 #X3 #HX3 #L2 #HL02
+  elim (teqx_inv_sort1 … HX2) -HX2 #s2 #H destruct
+  elim (teqx_inv_sort1 … HX3) -HX3 #s3 #H destruct //
+| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #X2 #HX2 #X3 #HX3 #L2 #HL2
+  lapply (teqx_inv_lref1 … HX2) -HX2 #H destruct
+  elim (reqx_inv_zero_pair_sn … HL2) -HL2 #K2 #V2 #HK02 #HV02 #H destruct
+  elim (teqx_inv_lifts_sn … HX3 … HVW1) -W1 #V3 #HVX3 #HV13
+  /3 width=3 by cpx_delta/
+| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #X2 #HX2 #X3 #HX3 #L2 #HL2
+  lapply (teqx_inv_lref1 … HX2) -HX2 #H destruct
+  elim (reqx_inv_lref_bind_sn … HL2) -HL2 #I2 #K2 #HK02 #H destruct
+  elim (teqx_inv_lifts_sn … HX3 … HVW1) -W1 #V3 #HVX3 #HV13
+  /3 width=3 by cpx_lref/
+| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X2 #HX2 #X3 #HX3 #L2 #HL02
+  elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #HV02 #HT02 #H destruct
+  elim (teqx_inv_pair1 … HX3) -HX3 #V3 #T3 #HV13 #HT13 #H destruct
+  elim (reqx_inv_bind … HL02) -HL02 #HV0 #HT0
+  lapply (reqx_bind_repl_dx … HT0 (BPair I V2) ?) -HT0
+  /2 width=1 by ext2_pair/ #HT0
+  /3 width=1 by cpx_bind/
+| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X2 #HX2 #X3 #HX3 #L2 #HL02
+  elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #HV02 #HT02 #H destruct
+  elim (teqx_inv_pair1 … HX3) -HX3 #V3 #T3 #HV13 #HT13 #H destruct
+  elim (reqx_inv_flat … HL02) -HL02 #HV0 #HT0
+  /3 width=1 by cpx_flat/
+| #G #L0 #V0 #U0 #T0 #T1 #HTU0 #_ #IH #X2 #HX2 #X3 #HX3 #L2 #HL02
+  elim (teqx_inv_pair1 … HX2) -HX2 #V2 #U2 #HV02 #HU02 #H destruct
+  elim (reqx_inv_bind … HL02) -HL02 #HV0 #HU0
+  lapply (reqx_inv_lifts_bi … HU0 (Ⓣ) … HTU0) -HU0
+  [6:|*: /3 width=2 by drops_refl, drops_drop/ ] #HT0
   elim (teqx_inv_lifts_sn … HU02 … HTU0) -U0 #T2 #HTU2 #HT02
-  elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
-  /3 width=5 by cpx_zeta, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
-  elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
-  elim (rpx_inv_flat … H1) -H1 #HL01 #H1
-  elim (reqx_inv_flat … H2) -H2 #HL02 #H2
-  elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
-  /3 width=3 by cpx_eps, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
-  elim (teqx_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
-  elim (rpx_inv_flat … H1) -H1 #HL01 #H1
-  elim (reqx_inv_flat … H2) -H2 #HL02 #H2
-  elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
-  /3 width=3 by cpx_ee, ex2_intro/
-| #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
-  elim (teqx_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
-  elim (teqx_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
-  elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
-  elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
-  elim (reqx_inv_flat … H2) -H2 #H2LV0 #H2
-  elim (reqx_inv_bind … H2) -H2 #H2LW0 #H2LT0
-  lapply (reqx_bind_repl_dx … H2LT0 (BPair Abst W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
-  elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
-  elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
-  elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
-  /4 width=7 by cpx_beta, teqx_pair, ex2_intro/ (* note: 2 teqx_pair *)
-| #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
-  elim (teqx_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
-  elim (teqx_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
-  elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
-  elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
-  elim (reqx_inv_flat … H2) -H2 #H2LV0 #H2
-  elim (reqx_inv_bind … H2) -H2 #H2LW0 #H2LT0
-  lapply (reqx_bind_repl_dx … H2LT0 (BPair Abbr W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
-  elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
-  elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
-  elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
-  elim (teqx_lifts_sn … HV1 … HVU1) -V1
-  /4 width=9 by cpx_theta, teqx_pair, ex2_intro/ (* note: 2 teqx_pair *)
+  /3 width=3 by cpx_zeta/
+| #G #L0 #V0 #T0 #T1 #_ #IH #X2 #HX2 #X3 #HX3 #L2 #HL02
+  elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #_ #HT02 #H destruct
+  elim (reqx_inv_flat … HL02) -HL02 #HV0 #HT0
+  /3 width=1 by cpx_eps/
+| #G #L0 #V0 #T0 #T1 #_ #IH #X2 #HX2 #X3 #HX3 #L2 #HL02
+  elim (teqx_inv_pair1 … HX2) -HX2 #V2 #T2 #HV02 #_ #H destruct
+  elim (reqx_inv_flat … HL02) -HL02 #HV0 #HT1
+  /3 width=1 by cpx_ee/
+| #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X2 #HX2 #X3 #HX3 #L2 #HL02
+  elim (teqx_inv_pair1 … HX2) -HX2 #V2 #X #HV02 #HX #H destruct
+  elim (teqx_inv_pair1 … HX) -HX #W2 #T2 #HW02 #HT02 #H destruct
+  elim (teqx_inv_pair1 … HX3) -HX3 #X #T3 #HX #HT13 #H destruct
+  elim (teqx_inv_pair1 … HX) -HX #W3 #V3 #HW13 #HV13 #H destruct
+  elim (reqx_inv_flat … HL02) -HL02 #HV0 #HL02
+  elim (reqx_inv_bind … HL02) -HL02 #HW0 #HT0
+  lapply (reqx_bind_repl_dx … HT0 (BPair Abst W2) ?) -HT0
+  /2 width=1 by ext2_pair/ #H2T0
+  /3 width=1 by cpx_beta/
+| #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X2 #HX2 #X3 #HX3 #L2 #HL02
+  elim (teqx_inv_pair1 … HX2) -HX2 #V2 #X #HV02 #HX #H destruct
+  elim (teqx_inv_pair1 … HX) -HX #W2 #T2 #HW02 #HT02 #H destruct
+  elim (teqx_inv_pair1 … HX3) -HX3 #W3 #X #HW13 #HX #H destruct
+  elim (teqx_inv_pair1 … HX) -HX #U3 #T3 #HU13 #HT13 #H destruct
+  elim (reqx_inv_flat … HL02) -HL02 #HV0 #HL02
+  elim (reqx_inv_bind … HL02) -HL02 #HW0 #HT0
+  lapply (reqx_bind_repl_dx … HT0 (BPair Abbr W2) ?) -HT0
+  /2 width=1 by ext2_pair/ #HT0
+  elim (teqx_inv_lifts_sn … HU13 … HVU1) -U1 #V3 #HVU3 #HV13
+  /3 width=3 by cpx_theta/
 ]
 qed-.
 
 lemma cpx_teqx_conf (G) (L):
-      ∀T0:term. ∀T1. ❪G,L❫ ⊢ T0 ⬈ T1 → ∀T2. T0 ≛ T2 →
-      ∃∃T. T1 ≛ T & ❪G,L❫ ⊢ T2 ⬈ T.
-#G #L #T0 #T1 #HT01 #T2 #HT02
-elim (cpx_teqx_conf_rex … HT01 … HT02 L … L) -HT01 -HT02
-/2 width=3 by rex_refl, ex2_intro/
-qed-.
+      ∀T0:term. ∀T1. ❪G,L❫ ⊢ T0 ⬈ T1 → ∀T2. T0 ≛ T2 → ❪G,L❫ ⊢ T2 ⬈ T1.
+/2 width=7 by cpx_teqx_repl_reqx/ qed-.
 
 lemma teqx_cpx_trans (G) (L):
-      ∀T2. ∀T0:term. T2 ≛ T0 → ∀T1. ❪G,L❫ ⊢ T0 ⬈ T1 →
-      ∃∃T. ❪G,L❫ ⊢ T2 ⬈ T & T ≛ T1.
-#G #L #T2 #T0 #HT20 #T1 #HT01
-elim (cpx_teqx_conf … HT01 T2) -HT01 /3 width=3 by teqx_sym, ex2_intro/
+      ∀T2. ∀T0:term. T2 ≛ T0 → ∀T1. ❪G,L❫ ⊢ T0 ⬈ T1 → ❪G,L❫ ⊢ T2 ⬈ T1.
+/3 width=3 by cpx_teqx_conf, teqx_sym/
 qed-.
 
+lemma teqx_cpx (G) (L):
+      ∀T1,T2:term. T1 ≛ T2 → ❪G,L❫ ⊢ T1 ⬈ T2.
+/2 width=3 by teqx_cpx_trans/ qed.
+
 (* Basic_2A1: uses: cpx_lleq_conf *)
 lemma cpx_reqx_conf (G):
-      ∀L0,T0,T1. ❪G,L0❫ ⊢ T0 ⬈ T1 → ∀L2. L0 ≛[T0] L2 →
-      ∃∃T. ❪G,L2❫ ⊢ T0 ⬈ T & T1 ≛ T.
-#G #L0 #T0 #T1 #HT01 #L2 #HL02
-elim (cpx_teqx_conf_rex … HT01 T0 … L0 … HL02) -HT01 -HL02
-/2 width=3 by rex_refl, ex2_intro/
-qed-.
+      ∀L0,T0,T1. ❪G,L0❫ ⊢ T0 ⬈ T1 → ∀L2. L0 ≛[T0] L2 → ❪G,L2❫ ⊢ T0 ⬈ T1.
+/2 width=7 by cpx_teqx_repl_reqx/ qed-.
 
 (* Basic_2A1: uses: lleq_cpx_trans *)
 lemma reqx_cpx_trans (G):
-      ∀L2,L0,T0. L2 ≛[T0] L0 → ∀T1. ❪G,L0❫ ⊢ T0 ⬈ T1 →
-      ∃∃T. ❪G,L2❫ ⊢ T0 ⬈ T & T ≛ T1.
-#G #L2 #L0 #T0 #HL20 #T1 #HT01
-elim (cpx_reqx_conf … HT01 L2) -HT01
-/3 width=3 by reqx_sym, teqx_sym, ex2_intro/
+      ∀L2,L0,T0. L2 ≛[T0] L0 → ∀T1. ❪G,L0❫ ⊢ T0 ⬈ T1 → ❪G,L2❫ ⊢ T0 ⬈ T1.
+/3 width=3 by cpx_reqx_conf, reqx_sym/
 qed-.
 
-lemma rpx_reqx_conf (G) (T): confluent2 … (rpx G T) (reqx T).
-/3 width=6 by rpx_fsge_comp, reqx_fsge_comp, cpx_teqx_conf_rex, rex_conf/ qed-.
+lemma rpx_reqx_conf (G) (T):
+      confluent1 … (rpx G T) (reqx T).
+/3 width=7 by reqx_fsge_comp, cpx_teqx_repl_reqx, rex_conf1/ qed-.
 
-lemma reqx_rpx_trans (G) (T):
-      ∀L2,K2. ❪G,L2❫ ⊢ ⬈[T] K2 → ∀L1. L1 ≛[T] L2 →
-      ∃∃K1. ❪G,L1❫ ⊢ ⬈[T] K1 & K1 ≛[T] K2.
-#G #T #L2 #K2 #HLK2 #L1 #HL12
-elim (rpx_reqx_conf … HLK2 L1)
-/3 width=3 by reqx_sym, ex2_intro/
-qed-.
+lemma reqx_rpx_trans (G) (T) (L):
+      ∀L1. L1 ≛[T] L → ∀L2. ❪G,L❫ ⊢ ⬈[T] L2 → ❪G,L1❫ ⊢ ⬈[T] L2.
+/3 width=3 by rpx_reqx_conf, reqx_sym/ qed-.
+
+lemma reqx_rpx (G) (T):
+      ∀L1,L2. L1 ≛[T] L2 → ❪G,L1❫ ⊢ ⬈[T] L2.
+/2 width=3 by reqx_rpx_trans/ qed.