X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpr.ma;h=978c42b7cb91d2920a50002bb23b9941f32a75a6;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=6bcfa173e8ec96dfc351c475a5820c71c14026ed;hpb=984856dbab870ddc3156040df69b1f1846cc3aaf;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
index 6bcfa173e..978c42b7c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
@@ -12,6 +12,10 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/xoa/ex_6_6.ma".
+include "ground_2/xoa/ex_7_7.ma".
+include "ground_2/xoa/or_4.ma".
+include "ground_2/insert_eq/insert_eq_0.ma".
 include "basic_2/rt_transition/cpm.ma".
 
 (* CONTEXT-SENSITIVE PARALLEL R-TRANSITION FOR TERMS ************************)
@@ -21,77 +25,77 @@ include "basic_2/rt_transition/cpm.ma".
 (* Note: cpr_flat: does not hold in basic_1 *)
 (* Basic_1: includes: pr2_thin_dx *)
 lemma cpr_flat: ∀h,I,G,L,V1,V2,T1,T2.
-                ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[h] T2 →
-                ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡[h] ⓕ{I}V2.T2.
+                ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L❫ ⊢ T1 ➡[h] T2 →
+                ❪G,L❫ ⊢ ⓕ[I]V1.T1 ➡[h] ⓕ[I]V2.T2.
 #h * /2 width=1 by cpm_cast, cpm_appl/
-qed. 
+qed.
 
 (* Basic_1: was: pr2_head_1 *)
-lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
-                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T.
+lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
+                   ∀T. ❪G,L❫ ⊢ ②[I]V1.T ➡[h] ②[I]V2.T.
 #h * /2 width=1 by cpm_bind, cpr_flat/
 qed.
 
 (* Basic inversion properties ***********************************************)
 
-lemma cpr_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h] T2 →
-                     ∨∨ T2 = ⓪{J}
-                      | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≡ T2 &
+lemma cpr_inv_atom1: ∀h,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ➡[h] T2 →
+                     ∨∨ T2 = ⓪[J]
+                      | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h] V2 & ⇧*[1] V2 ≘ T2 &
                                    L = K.ⓓV1 & J = LRef 0
-                      | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≡ T2 &
-                                     L = K.ⓑ{I}V & J = LRef (⫯i).
+                      | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ➡[h] T & ⇧*[1] T ≘ T2 &
+                                   L = K.ⓘ[I] & J = LRef (↑i).
 #h #J #G #L #T2 #H elim (cpm_inv_atom1 … H) -H *
-/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_3_intro/
+[2,4:|*: /3 width=8 by or3_intro0, or3_intro1, or3_intro2, ex4_4_intro, ex4_3_intro/ ]
 [ #n #_ #_ #H destruct
 | #n #K #V1 #V2 #_ #_ #_ #_ #H destruct
 ]
 qed-.
 
 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
-lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
-#h #G #L #T2 #s #H elim (cpm_inv_sort1 … H) -H * // #_ #H destruct
+lemma cpr_inv_sort1: ∀h,G,L,T2,s. ❪G,L❫ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
+#h #G #L #T2 #s #H elim (cpm_inv_sort1 … H) -H //
 qed-.
 
-lemma cpr_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[h] T2 →
-                     T2 = #0 ∨
-                     ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 & ⬆*[1] V2 ≡ T2 &
-                                L = K.ⓓV1.
+lemma cpr_inv_zero1: ∀h,G,L,T2. ❪G,L❫ ⊢ #0 ➡[h] T2 →
+                     ∨∨ T2 = #0
+                      | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h] V2 & ⇧*[1] V2 ≘ T2 &
+                                   L = K.ⓓV1.
 #h #G #L #T2 #H elim (cpm_inv_zero1 … H) -H *
 /3 width=6 by ex3_3_intro, or_introl, or_intror/
 #n #K #V1 #V2 #_ #_ #_ #H destruct
 qed-.
 
-lemma cpr_inv_lref1: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ➡[h] T2 →
-                     T2 = #(⫯i) ∨
-                     ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ➡[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
+lemma cpr_inv_lref1: ∀h,G,L,T2,i. ❪G,L❫ ⊢ #↑i ➡[h] T2 →
+                     ∨∨ T2 = #(↑i)
+                      | ∃∃I,K,T. ❪G,K❫ ⊢ #i ➡[h] T & ⇧*[1] T ≘ T2 & L = K.ⓘ[I].
 #h #G #L #T2 #i #H elim (cpm_inv_lref1 … H) -H *
-/3 width=7 by ex3_4_intro, or_introl, or_intror/
+/3 width=6 by ex3_3_intro, or_introl, or_intror/
 qed-.
 
-lemma cpr_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[h] T2 → T2 = §l.
+lemma cpr_inv_gref1: ∀h,G,L,T2,l. ❪G,L❫ ⊢ §l ➡[h] T2 → T2 = §l.
 #h #G #L #T2 #l #H elim (cpm_inv_gref1 … H) -H //
 qed-.
 
 (* Basic_1: includes: pr0_gen_cast pr2_gen_cast *)
-lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝ V1. U1 ➡[h] U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
-                              U2 = ⓝV2.T2
-                     ) ∨ ⦃G, L⦄ ⊢ U1 ➡[h] U2.
+lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓝ V1.U1 ➡[h] U2 →
+                     ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L❫ ⊢ U1 ➡[h] T2 &
+                                 U2 = ⓝV2.T2
+                      | ❪G,L❫ ⊢ U1 ➡[h] U2.
 #h #G #L #V1 #U1 #U2 #H elim (cpm_inv_cast1 … H) -H
 /2 width=1 by or_introl, or_intror/ * #n #_ #H destruct
 qed-.
 
-lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[h] T2 &
-                                 U2 = ⓕ{I}V2.T2
-                      | (⦃G, L⦄ ⊢ U1 ➡[h] U2 ∧ I = Cast)
-                      | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
-                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h] T2 & U1 = ⓛ{p}W1.T1 &
-                                            U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
-                      | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≡ V2 &
-                                              ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h] T2 &
-                                              U1 = ⓓ{p}W1.T1 &
-                                              U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
+lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓕ[I]V1.U1 ➡[h] U2 →
+                     ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L❫ ⊢ U1 ➡[h] T2 &
+                                 U2 = ⓕ[I]V2.T2
+                      | (❪G,L❫ ⊢ U1 ➡[h] U2 ∧ I = Cast)
+                      | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 & ❪G,L❫ ⊢ W1 ➡[h] W2 &
+                                            ❪G,L.ⓛW1❫ ⊢ T1 ➡[h] T2 & U1 = ⓛ[p]W1.T1 &
+                                            U2 = ⓓ[p]ⓝW2.V2.T2 & I = Appl
+                      | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V & ⇧*[1] V ≘ V2 &
+                                              ❪G,L❫ ⊢ W1 ➡[h] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ➡[h] T2 &
+                                              U1 = ⓓ[p]W1.T1 &
+                                              U2 = ⓓ[p]W2.ⓐV2.T2 & I = Appl.
 #h * #G #L #V1 #U1 #U2 #H
 [ elim (cpm_inv_appl1 … H) -H *
   /3 width=13 by or4_intro0, or4_intro2, or4_intro3, ex7_7_intro, ex6_6_intro, ex3_2_intro/
@@ -100,6 +104,39 @@ lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 
 ]
 qed-.
 
+(* Basic eliminators ********************************************************)
+
+lemma cpr_ind (h): ∀Q:relation4 genv lenv term term.
+                   (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
+                   (∀G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h] V2 → Q G K V1 V2 →
+                     ⇧*[1] V2 ≘ W2 → Q G (K.ⓓV1) (#0) W2
+                   ) → (∀I,G,K,T,U,i. ❪G,K❫ ⊢ #i ➡[h] T → Q G K (#i) T →
+                     ⇧*[1] T ≘ U → Q G (K.ⓘ[I]) (#↑i) (U)
+                   ) → (∀p,I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h] T2 →
+                     Q G L V1 V2 → Q G (L.ⓑ[I]V1) T1 T2 → Q G L (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
+                   ) → (∀I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L❫ ⊢ T1 ➡[h] T2 →
+                     Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ[I]V1.T1) (ⓕ[I]V2.T2)
+                   ) → (∀G,L,V,T1,T,T2. ⇧*[1] T ≘ T1 → ❪G,L❫ ⊢ T ➡[h] T2 →
+                     Q G L T T2 → Q G L (+ⓓV.T1) T2
+                   ) → (∀G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h] T2 → Q G L T1 T2 →
+                     Q G L (ⓝV.T1) T2
+                   ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V2 → ❪G,L❫ ⊢ W1 ➡[h] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h] T2 →
+                     Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
+                     Q G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
+                   ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h] V → ❪G,L❫ ⊢ W1 ➡[h] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ➡[h] T2 →
+                     Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
+                     ⇧*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2)
+                   ) →
+                   ∀G,L,T1,T2. ❪G,L❫ ⊢ T1 ➡[h] T2 → Q G L T1 T2.
+#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2
+@(insert_eq_0 … 0) #n #H
+@(cpm_ind … H) -G -L -T1 -T2 -n [2,4,11:|*: /3 width=4 by/ ]
+[ #G #L #s #H destruct
+| #n #G #K #V1 #V2 #W2 #_ #_ #_ #H destruct
+| #n #G #L #U1 #U2 #T #_ #_ #H destruct
+]
+qed-.
+
 (* Basic_1: removed theorems 12:
             pr0_subst0_back pr0_subst0_fwd pr0_subst0
             pr0_delta1