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[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / equivalence / cpcs_cpcs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
index 44cbd7491ac504e20e946156af9753ed06336748..125cc70feaa77829b63db3bafc5a5a2695093da0 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
@@ -20,8 +20,8 @@ include "basic_2/equivalence/cpcs_cprs.ma".
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpcs_inv_cprs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 →
-                     ∃∃T. L ⊢ T1 ➡* T & L ⊢ T2 ➡* T.
+lemma cpcs_inv_cprs: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                     ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
 #L #T1 #T2 #H @(cpcs_ind … H) -T2
 [ /3 width=3/
 | #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
@@ -33,27 +33,27 @@ lemma cpcs_inv_cprs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 →
 qed-.
 
 (* Basic_1: was: pc3_gen_sort *)
-lemma cpcs_inv_sort: ∀L,k1,k2. L ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
+lemma cpcs_inv_sort: ∀L,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
 #L #k1 #k2 #H
 elim (cpcs_inv_cprs … H) -H #T #H1
 >(cprs_inv_sort1 … H1) -T #H2
 lapply (cprs_inv_sort1 … H2) -L #H destruct //
 qed-.
 
-lemma cpcs_inv_abst1: ∀a,L,W1,T1,T. L ⊢ ⓛ{a}W1.T1 ⬌* T →
-                      ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+lemma cpcs_inv_abst1: ∀a,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
 #a #L #W1 #T1 #T #H
 elim (cpcs_inv_cprs … H) -H #X #H1 #H2
 elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
 @(ex2_2_intro … H2) -H2 /2 width=2/ (**) (* explicit constructor, /3 width=6/ is slow *)
 qed-.
 
-lemma cpcs_inv_abst2: ∀a,L,W1,T1,T. L ⊢ T ⬌* ⓛ{a}W1.T1 →
-                      ∃∃W2,T2. L ⊢ T ➡* ⓛ{a}W2.T2 & L ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
+lemma cpcs_inv_abst2: ∀a,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
 /3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
 
 (* Basic_1: was: pc3_gen_sort_abst *)
-lemma cpcs_inv_sort_abst: ∀a,L,W,T,k. L ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
+lemma cpcs_inv_sort_abst: ∀a,L,W,T,k. ⦃G, L⦄ ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
 #a #L #W #T #k #H
 elim (cpcs_inv_cprs … H) -H #X #H1
 >(cprs_inv_sort1 … H1) -X #H2
@@ -63,7 +63,7 @@ qed-.
 (* Basic_1: was: pc3_gen_lift *)
 lemma cpcs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
                      ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
-                     L ⊢ U1 ⬌* U2 → K ⊢ T1 ⬌* T2.
+                     ⦃G, L⦄ ⊢ U1 ⬌* U2 → K ⊢ T1 ⬌* T2.
 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
 elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
 elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
@@ -87,17 +87,17 @@ lapply (lprs_cprs_trans … HT1 … HL12) -HT1
 lapply (lprs_cprs_trans … HT2 … HL12) -L2 /2 width=3/
 qed-.
 
-lemma cpr_cprs_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
+lemma cpr_cprs_conf_cpcs: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_strip … HT1 … HT2) /2 width=3 by cpr_cprs_div/
 qed-.
 
-lemma cprs_cpr_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡ T2 → L ⊢ T2 ⬌* T1.
+lemma cprs_cpr_conf_cpcs: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_strip … HT1 … HT2) /2 width=3 by cprs_cpr_div/
 qed-.
 
-lemma cprs_conf_cpcs: ∀L,T,T1,T2. L ⊢ T ➡* T1 → L ⊢ T ➡* T2 → L ⊢ T1 ⬌* T2.
+lemma cprs_conf_cpcs: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T #T1 #T2 #HT1 #HT2
 elim (cprs_conf … HT1 … HT2) /2 width=3/
 qed-.
@@ -118,24 +118,24 @@ lemma lpr_cpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 → L
 /3 width=5 by lpr_cprs_conf, cpr_cprs/ qed-.
 
 (* Basic_1: was only: pc3_thin_dx *)
-lemma cpcs_flat: ∀L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L ⊢ T1 ⬌* T2 →
-                 ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
+lemma cpcs_flat: ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
 #L #V1 #V2 #HV12 #T1 #T2 #HT12 #I
 elim (cpcs_inv_cprs … HV12) -HV12 #V #HV1 #HV2
 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ (**) (* /3 width=5/ is too slow *)
 qed.
 
-lemma cpcs_flat_dx_cpr_rev: ∀L,V1,V2. L ⊢ V2 ➡ V1 → ∀T1,T2. L ⊢ T1 ⬌* T2 →
-                            ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
+lemma cpcs_flat_dx_cpr_rev: ∀L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                            ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
 /3 width=1/ qed.
 
 lemma cpcs_bind_dx: ∀a,I,L,V,T1,T2. L.ⓑ{I}V ⊢ T1 ⬌* T2 →
-                    L ⊢ ⓑ{a,I}V. T1 ⬌* ⓑ{a,I}V. T2.
+                    ⦃G, L⦄ ⊢ ⓑ{a,I}V. T1 ⬌* ⓑ{a,I}V. T2.
 #a #I #L #V #T1 #T2 #HT12
 elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_div, cprs_bind/ (**) (* /3 width=5/ is a bit slow *)
 qed.
 
-lemma cpcs_bind_sn: ∀a,I,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
+lemma cpcs_bind_sn: ∀a,I,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
 #a #I #L #V1 #V2 #T #HV12
 elim (cpcs_inv_cprs … HV12) -HV12 /3 width=5 by cprs_div, cprs_bind/ (**) (* /3 width=5/ is a bit slow *)
 qed.
@@ -150,7 +150,7 @@ qed-.
 (* Basic_1: was: pc3_lift *)
 lemma cpcs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
                  ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
-                 K ⊢ T1 ⬌* T2 → L ⊢ U1 ⬌* U2.
+                 K ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
 #L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
 elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
 elim (lift_total T d e) #U #HTU
@@ -158,14 +158,14 @@ lapply (cprs_lift … HT1 … HLK … HTU1 … HTU) -T1 #HU1
 lapply (cprs_lift … HT2 … HLK … HTU2 … HTU) -K -T2 -T -d -e /2 width=3/
 qed.
 
-lemma cpcs_strip: ∀L,T1,T. L ⊢ T ⬌* T1 → ∀T2. L ⊢ T ⬌ T2 →
-                  ∃∃T0. L ⊢ T1 ⬌ T0 & L ⊢ T2 ⬌* T0.
+lemma cpcs_strip: ∀L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
+                  ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
 #L #T1 #T @TC_strip1 /2 width=3/ qed-.
 
 (* More inversion lemmas ****************************************************)
 
-lemma cpcs_inv_abst_sn: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
-                        ∧∧ L ⊢ W1 ⬌* W2 & L.ⓛW1 ⊢ T1 ⬌* T2 & a1 = a2.
+lemma cpcs_inv_abst_sn: ∀a1,a2,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & L.ⓛW1 ⊢ T1 ⬌* T2 & a1 = a2.
 #a1 #a2 #L #W1 #W2 #T1 #T2 #H
 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
 elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
@@ -175,8 +175,8 @@ lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1/ -HT2 #HT2
 /4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
 qed-.
 
-lemma cpcs_inv_abst_dx: ∀a1,a2,L,W1,W2,T1,T2. L ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
-                        ∧∧ L ⊢ W1 ⬌* W2 & L. ⓛW2 ⊢ T1 ⬌* T2 & a1 = a2.
+lemma cpcs_inv_abst_dx: ∀a1,a2,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & L. ⓛW2 ⊢ T1 ⬌* T2 & a1 = a2.
 #a1 #a2 #L #W1 #W2 #T1 #T2 #HT12
 lapply (cpcs_sym … HT12) -HT12 #HT12
 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1/
@@ -185,23 +185,23 @@ qed-.
 (* Main properties **********************************************************)
 
 (* Basic_1: was pc3_t *)
-theorem cpcs_trans: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+theorem cpcs_trans: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
 
-theorem cpcs_canc_sn: ∀L,T,T1,T2. L ⊢ T ⬌* T1 → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+theorem cpcs_canc_sn: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3 by cpcs_trans, cpcs_sym/ qed-. (**) (* /3 width=3/ is too slow *)
 
-theorem cpcs_canc_dx: ∀L,T,T1,T2. L ⊢ T1 ⬌* T → L ⊢ T2 ⬌* T → L ⊢ T1 ⬌* T2.
+theorem cpcs_canc_dx: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
 /3 width=3 by cpcs_trans, cpcs_sym/ qed-. (**) (* /3 width=3/ is too slow *)
 
-lemma cpcs_bind1: ∀a,I,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V1 ⊢ T1 ⬌* T2 →
-                  L ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+lemma cpcs_bind1: ∀a,I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V1 ⊢ T1 ⬌* T2 →
+                  ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
 #a #I #L #V1 #V2 #HV12 #T1 #T2 #HT12
 @(cpcs_trans … (ⓑ{a,I}V1.T2)) /2 width=1/
 qed.
 
-lemma cpcs_bind2: ∀a,I,L,V1,V2. L ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V2 ⊢ T1 ⬌* T2 →
-                  L ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
+lemma cpcs_bind2: ∀a,I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V2 ⊢ T1 ⬌* T2 →
+                  ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
 #a #I #L #V1 #V2 #HV12 #T1 #T2 #HT12
 @(cpcs_trans … (ⓑ{a,I}V2.T1)) /2 width=1/
 qed.