X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpr.ma;h=6411f701ce1ddd4233da43869d4361803beef9c2;hb=e880d6eab5e1700f4a625ddcd7d0fa8f0cce2dcc;hp=382bb119be09cc145ba829527584b88eadd43ffe;hpb=b6e1db4f1b0f1d5121f2b214562f96c5b0fa544e;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 382bb119b..6411f701c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
@@ -16,19 +16,33 @@ include "basic_2/rt_transition/cpm.ma".
 
 (* CONTEXT-SENSITIVE PARALLEL R-TRANSITION FOR TERMS ************************)
 
+(* Basic properties *********************************************************)
+
+(* 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.
+#h * /2 width=1 by cpm_cast, cpm_appl/
+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.
+#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 &
+                      | ∃∃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/
-[ #n #_ #_ #H destruct
-| #n #K #V1 #V2 #_ #_ #_ #_ #H destruct
-]
+/3 width=8 by tri_lt, or3_intro0, or3_intro1, or3_intro2, ex4_4_intro, ex4_3_intro/
+#n #_ #_ #H destruct
 qed-.
 
 (* Basic_1: includes: pr0_gen_sort pr2_gen_sort *)
@@ -37,25 +51,34 @@ lemma cpr_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h] T2 → T2 = ⋆s.
 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.
+                     ∨∨ 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.
 #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.
+#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
@@ -63,22 +86,76 @@ lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h] U2 
                       | ∃∃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 &
+                      | ∃∃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 #I #G #L #V1 #U1 #U2 #H elim (cpm_inv_flat1 … H) -H *
-/3 width=13 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex7_7_intro, ex6_6_intro, ex3_2_intro, conj/
-#n #_ #_ #H destruct
+#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/
+| elim (cpr_inv_cast1 … H) -H [ * ]
+  /3 width=5 by or4_intro0, or4_intro1, ex3_2_intro, conj/
+]
 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.
-#h #G #L #V1 #U1 #U2 #H elim (cpm_inv_cast1 … H) -H
-/2 width=1 by or_introl, or_intror/ * #n #_ #H destruct
+(* Basic eliminators ********************************************************)
+
+lemma cpr_ind (h): ∀R:relation4 genv lenv term term.
+                   (∀I,G,L. R G L (⓪{I}) (⓪{I})) →
+                   (∀G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[h] V2 → R G K V1 V2 →
+                     ⬆*[1] V2 ≘ W2 → R G (K.ⓓV1) (#0) W2
+                   ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[h] T → R G K (#i) T →
+                     ⬆*[1] T ≘ U → R 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 →
+                     R G L V1 V2 → R G (L.ⓑ{I}V1) T1 T2 → R 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 →
+                     R G L V1 V2 → R G L T1 T2 → R G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+                   ) → (∀G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[h] T → R G (L.ⓓV) T1 T →
+                     ⬆*[1] T2 ≘ T → R G L (+ⓓV.T1) T2
+                   ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → R G L T1 T2 →
+                     R 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 →
+                     R G L V1 V2 → R G L W1 W2 → R G (L.ⓛW1) T1 T2 →
+                     R 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 →
+                     R G L V1 V → R G L W1 W2 → R G (L.ⓓW1) T1 T2 →
+                     ⬆*[1] V ≘ V2 → R G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+                   ) →
+                   ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h] T2 → R G L T1 T2.
+#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #G #L #T1 #T2
+* #c #HC #H generalize in match HC; -HC
+elim H -c -G -L -T1 -T2
+[ /2 width=3 by ex2_intro/
+| #G #L #s #H
+  lapply (isrt_inv_01 … H) -H #H destruct
+| /3 width=4 by ex2_intro/
+| #c #G #L #V1 #V2 #W2 #_ #_ #_ #H
+  elim (isrt_inv_plus_SO_dx … H) -H // #n #_ #H destruct
+| /3 width=4 by ex2_intro/
+| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #H
+  elim (isrt_O_inv_max … H) -H #HcV #HcT
+  /4 width=3 by isr_inv_shift, ex2_intro/
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #H
+  elim (isrt_O_inv_max … H) -H #HcV #HcT
+  /4 width=3 by isr_inv_shift, ex2_intro/
+| #cU #cT #G #L #U1 #U2 #T1 #T2 #HUT #HU12 #HT12 #IHU #IHT #H
+  elim (isrt_O_inv_max … H) -H #HcV #HcT
+  /3 width=3 by ex2_intro/
+| /4 width=4 by isrt_inv_plus_O_dx, ex2_intro/
+| /4 width=4 by isrt_inv_plus_O_dx, ex2_intro/
+| #c #G #L #U1 #U2 #T #_ #_ #H
+  elim (isrt_inv_plus_SO_dx … H) -H // #n #_ #H destruct
+| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #IHV #IHW #IHT #H
+  lapply (isrt_inv_plus_O_dx … H ?) -H // #H
+  elim (isrt_O_inv_max … H) -H #H #HcT
+  elim (isrt_O_inv_max … H) -H #HcV #HcW
+  /4 width=3 by isr_inv_shift, ex2_intro/
+| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #IHV #IHW #IHT #H
+  lapply (isrt_inv_plus_O_dx … H ?) -H // #H
+  elim (isrt_O_inv_max … H) -H #H #HcT
+  elim (isrt_O_inv_max … H) -H #HcV #HcW
+  /4 width=4 by isr_inv_shift, ex2_intro/
+]
 qed-.
 
 (* Basic_1: removed theorems 12:
@@ -88,7 +165,3 @@ qed-.
             pr2_gen_csort pr2_gen_cflat pr2_gen_cbind
             pr2_gen_ctail pr2_ctail
 *)
-(* Basic_1: removed local theorems 4:
-            pr0_delta_eps pr0_cong_delta
-            pr2_free_free pr2_free_delta
-*)