update in basic_2 and apps_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_equivalence / cpcs_cprs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cprs.ma
index 90fd0231372ba45ccfb34a2eae16335d4b5863f5..838b2d620f5ab62328f9d691208cc63a115a6f65 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_equivalence/cpcs_cprs.ma
@@ -21,7 +21,7 @@ include "basic_2/rt_equivalence/cpcs.ma".
 (* Inversion lemmas with context sensitive r-computation on terms ***********)
 
 lemma cpcs_inv_cprs (h) (G) (L): ∀T1,T2. ❪G,L❫ ⊢ T1 ⬌*[h] T2 →
-                                 ∃∃T. ❪G,L❫ ⊢ T1 ➡*[h] T & ❪G,L❫ ⊢ T2 ➡*[h] T.
+                                 ∃∃T. ❪G,L❫ ⊢ T1 ➡*[h,0] T & ❪G,L❫ ⊢ T2 ➡*[h,0] T.
 #h #G #L #T1 #T2 #H @(cpcs_ind_dx … H) -T2
 [ /3 width=3 by ex2_intro/
 | #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
@@ -44,7 +44,7 @@ qed-.
 (* Basic_2A1: was: cpcs_inv_abst1 *)
 lemma cpcs_inv_abst_sn (h) (G) (L):
                        ∀p,W1,T1,X. ❪G,L❫ ⊢ ⓛ[p]W1.T1 ⬌*[h] X →
-                       ∃∃W2,T2. ❪G,L❫ ⊢ X ➡*[h] ⓛ[p]W2.T2 & ❪G,L❫ ⊢ ⓛ[p]W1.T1 ➡*[h] ⓛ[p]W2.T2.
+                       ∃∃W2,T2. ❪G,L❫ ⊢ X ➡*[h,0] ⓛ[p]W2.T2 & ❪G,L❫ ⊢ ⓛ[p]W1.T1 ➡*[h,0] ⓛ[p]W2.T2.
 #h #G #L #p #W1 #T1 #T #H
 elim (cpcs_inv_cprs … H) -H #X #H1 #H2
 elim (cpms_inv_abst_sn … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
@@ -54,7 +54,7 @@ qed-.
 (* Basic_2A1: was: cpcs_inv_abst2 *)
 lemma cpcs_inv_abst_dx (h) (G) (L):
                        ∀p,W1,T1,X. ❪G,L❫ ⊢ X ⬌*[h] ⓛ[p]W1.T1 →
-                       ∃∃W2,T2. ❪G,L❫ ⊢ X ➡*[h] ⓛ[p]W2.T2 & ❪G,L❫ ⊢ ⓛ[p]W1.T1 ➡*[h] ⓛ[p]W2.T2.
+                       ∃∃W2,T2. ❪G,L❫ ⊢ X ➡*[h,0] ⓛ[p]W2.T2 & ❪G,L❫ ⊢ ⓛ[p]W1.T1 ➡*[h,0] ⓛ[p]W2.T2.
 /3 width=1 by cpcs_inv_abst_sn, cpcs_sym/ qed-.
 
 (* Basic_1: was: pc3_gen_sort_abst *)
@@ -69,70 +69,70 @@ qed-.
 (* Properties with context sensitive r-computation on terms *****************)
 
 (* Basic_1: was: pc3_pr3_r *)
-lemma cpcs_cprs_dx (h) (G) (L): ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_dx (h) (G) (L): ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[h,0] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 #h #G #L #T1 #T2 #H @(cprs_ind_dx … H) -T2
 /3 width=3 by cpcs_cpr_step_dx, cpcs_step_dx, cpc_cpcs/
 qed.
 
 (* Basic_1: was: pc3_pr3_x *)
-lemma cpcs_cprs_sn (h) (G) (L): ∀T1,T2. ❪G,L❫ ⊢ T2 ➡*[h] T1 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
+lemma cpcs_cprs_sn (h) (G) (L): ∀T1,T2. ❪G,L❫ ⊢ T2 ➡*[h,0] T1 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 #h #G #L #T1 #T2 #H @(cprs_ind_sn … H) -T2
 /3 width=3 by cpcs_cpr_div, cpcs_step_sn, cpcs_cprs_dx/
 qed.
 
 (* Basic_2A1: was: cpcs_cprs_strap1 *)
 lemma cpcs_cprs_step_dx (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ⬌*[h] T →
-                                     ∀T2. ❪G,L❫ ⊢ T ➡*[h] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
+                                     ∀T2. ❪G,L❫ ⊢ T ➡*[h,0] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 #h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_dx … H) -T2 /2 width=3 by cpcs_cpr_step_dx/
 qed-.
 
 (* Basic_2A1: was: cpcs_cprs_strap2 *)
-lemma cpcs_cprs_step_sn (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[h] T →
+lemma cpcs_cprs_step_sn (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[h,0] T →
                                      ∀T2. ❪G,L❫ ⊢ T ⬌*[h] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 #h #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_sn … H) -T1 /2 width=3 by cpcs_cpr_step_sn/
 qed-.
 
 lemma cpcs_cprs_div (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ⬌*[h] T →
-                                 ∀T2. ❪G,L❫ ⊢ T2 ➡*[h] T → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
+                                 ∀T2. ❪G,L❫ ⊢ T2 ➡*[h,0] T → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 #h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_sn … H) -T2 /2 width=3 by cpcs_cpr_div/
 qed-.
 
 (* Basic_1: was: pc3_pr3_conf *)
-lemma cpcs_cprs_conf (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T ➡*[h] T1 →
+lemma cpcs_cprs_conf (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T ➡*[h,0] T1 →
                                   ∀T2. ❪G,L❫ ⊢ T ⬌*[h] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 #h #G #L #T1 #T #H #T2 #HT2 @(cprs_ind_dx … H) -T1 /2 width=3 by cpcs_cpr_conf/
 qed-.
 
 (* Basic_1: was: pc3_pr3_t *)
 (* Basic_1: note: pc3_pr3_t should be renamed *)
-lemma cprs_div (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[h] T →
-                            ∀T2. ❪G,L❫ ⊢ T2 ➡*[h] T → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
+lemma cprs_div (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[h,0] T →
+                            ∀T2. ❪G,L❫ ⊢ T2 ➡*[h,0] T → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 #h #G #L #T1 #T #HT1 #T2 #H @(cprs_ind_sn … H) -T2
 /2 width=3 by cpcs_cpr_div, cpcs_cprs_dx/
 qed.
 
-lemma cprs_cpr_div (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[h] T →
-                                ∀T2. ❪G,L❫ ⊢ T2 ➡[h] T → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
+lemma cprs_cpr_div (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ➡*[h,0] T →
+                                ∀T2. ❪G,L❫ ⊢ T2 ➡[h,0] T → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 /3 width=5 by cpm_cpms, cprs_div/ qed-.
 
-lemma cpr_cprs_div (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ➡[h] T →
-                                ∀T2. ❪G,L❫ ⊢ T2 ➡*[h] T → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cprs_div (h) (G) (L): ∀T1,T. ❪G,L❫ ⊢ T1 ➡[h,0] T →
+                                ∀T2. ❪G,L❫ ⊢ T2 ➡*[h,0] T → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 /3 width=3 by cpm_cpms, cprs_div/ qed-.
 
-lemma cpr_cprs_conf_cpcs (h) (G) (L): ∀T,T1. ❪G,L❫ ⊢ T ➡*[h] T1 →
-                                      ∀T2. ❪G,L❫ ⊢ T ➡[h] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
+lemma cpr_cprs_conf_cpcs (h) (G) (L): ∀T,T1. ❪G,L❫ ⊢ T ➡*[h,0] T1 →
+                                      ∀T2. ❪G,L❫ ⊢ T ➡[h,0] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 #h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
 /2 width=3 by cpr_cprs_div/
 qed-.
 
-lemma cprs_cpr_conf_cpcs (h) (G) (L): ∀T,T1. ❪G,L❫ ⊢ T ➡*[h] T1 →
-                                      ∀T2. ❪G,L❫ ⊢ T ➡[h] T2 → ❪G,L❫ ⊢ T2 ⬌*[h] T1.
+lemma cprs_cpr_conf_cpcs (h) (G) (L): ∀T,T1. ❪G,L❫ ⊢ T ➡*[h,0] T1 →
+                                      ∀T2. ❪G,L❫ ⊢ T ➡[h,0] T2 → ❪G,L❫ ⊢ T2 ⬌*[h] T1.
 #h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
 /2 width=3 by cprs_cpr_div/
 qed-.
 
-lemma cprs_conf_cpcs (h) (G) (L): ∀T,T1. ❪G,L❫ ⊢ T ➡*[h] T1 →
-                                  ∀T2. ❪G,L❫ ⊢ T ➡*[h] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
+lemma cprs_conf_cpcs (h) (G) (L): ∀T,T1. ❪G,L❫ ⊢ T ➡*[h,0] T1 →
+                                  ∀T2. ❪G,L❫ ⊢ T ➡*[h,0] T2 → ❪G,L❫ ⊢ T1 ⬌*[h] T2.
 #h #G #L #T #T1 #HT1 #T2 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
 /2 width=3 by cprs_div/
 qed-.
@@ -147,7 +147,7 @@ elim (cpcs_inv_cprs … HT12) -HT12
 /3 width=5 by cprs_flat, cprs_div/
 qed.
 
-lemma cpcs_flat_dx_cpr_rev (h) (G) (L): ∀V1,V2. ❪G,L❫ ⊢ V2 ➡[h] V1 →
+lemma cpcs_flat_dx_cpr_rev (h) (G) (L): ∀V1,V2. ❪G,L❫ ⊢ V2 ➡[h,0] V1 →
                                         ∀T1,T2. ❪G,L❫ ⊢ T1 ⬌*[h] T2 →
                                         ∀I. ❪G,L❫ ⊢ ⓕ[I]V1.T1 ⬌*[h] ⓕ[I]V2.T2.
 /3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.