update in basic_2 and apps_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_transition / cpr.ma

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 2d0608f9237369077ceb79b172fed79fbbc71e0c..b99663e05ee843c231c29a44359e3be07a8e61b6 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpr.ma
@@ -25,24 +25,24 @@ 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,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,0] T2 →
+                ❪G,L❫ ⊢ ⓕ[I]V1.T1 ➡[h,0] ⓕ[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.
+lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 →
+                   ∀T. ❪G,L❫ ⊢ ②[I]V1.T ➡[h,0] 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 →
+lemma cpr_inv_atom1: ∀h,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ➡[h,0] T2 →
                      ∨∨ T2 = ⓪[J]
-                      | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h] V2 & ⇧[1] V2 ≘ T2 &
+                      | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,0] V2 & ⇧[1] V2 ≘ T2 &
                                    L = K.ⓓV1 & J = LRef 0
-                      | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ➡[h] T & ⇧[1] T ≘ T2 &
+                      | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ➡[h,0] T & ⇧[1] T ≘ T2 &
                                    L = K.ⓘ[I] & J = LRef (↑i).
 #h #J #G #L #T2 #H elim (cpm_inv_atom1 … H) -H *
 [2,4:|*: /3 width=8 by or3_intro0, or3_intro1, or3_intro2, ex4_4_intro, ex4_3_intro/ ]
@@ -52,48 +52,48 @@ lemma cpr_inv_atom1: ∀h,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ➡[h] T2 →
 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.
+lemma cpr_inv_sort1: ∀h,G,L,T2,s. ❪G,L❫ ⊢ ⋆s ➡[h,0] 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 →
+lemma cpr_inv_zero1: ∀h,G,L,T2. ❪G,L❫ ⊢ #0 ➡[h,0] T2 →
                      ∨∨ T2 = #0
-                      | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h] V2 & ⇧[1] V2 ≘ T2 &
+                      | ∃∃K,V1,V2. ❪G,K❫ ⊢ V1 ➡[h,0] 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 →
+lemma cpr_inv_lref1: ∀h,G,L,T2,i. ❪G,L❫ ⊢ #↑i ➡[h,0] T2 →
                      ∨∨ T2 = #(↑i)
-                      | ∃∃I,K,T. ❪G,K❫ ⊢ #i ➡[h] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
+                      | ∃∃I,K,T. ❪G,K❫ ⊢ #i ➡[h,0] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
 #h #G #L #T2 #i #H elim (cpm_inv_lref1 … H) -H *
 /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,0] 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 &
+lemma cpr_inv_cast1: ∀h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓝ V1.U1 ➡[h,0] U2 →
+                     ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L❫ ⊢ U1 ➡[h,0] T2 &
                                  U2 = ⓝV2.T2
-                      | ❪G,L❫ ⊢ U1 ➡[h] U2.
+                      | ❪G,L❫ ⊢ U1 ➡[h,0] 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 &
+lemma cpr_inv_flat1: ∀h,I,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓕ[I]V1.U1 ➡[h,0] U2 →
+                     ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L❫ ⊢ U1 ➡[h,0] 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 &
+                      | (❪G,L❫ ⊢ U1 ➡[h,0] U2 ∧ I = Cast)
+                      | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 & ❪G,L❫ ⊢ W1 ➡[h,0] W2 &
+                                            ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,0] 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 &
+                      | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V & ⇧[1] V ≘ V2 &
+                                              ❪G,L❫ ⊢ W1 ➡[h,0] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,0] T2 &
                                               U1 = ⓓ[p]W1.T1 &
                                               U2 = ⓓ[p]W2.ⓐV2.T2 & I = Appl.
 #h * #G #L #V1 #U1 #U2 #H
@@ -108,26 +108,26 @@ qed-.
 
 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 →
+                   (∀G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ➡[h,0] 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 →
+                   ) → (∀I,G,K,T,U,i. ❪G,K❫ ⊢ #i ➡[h,0] 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 →
+                   ) → (∀p,I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡[h,0] 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 →
+                   ) → (∀I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ T1 ➡[h,0] 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 →
+                   ) → (∀G,L,V,T1,T,T2. ⇧[1] T ≘ T1 → ❪G,L❫ ⊢ T ➡[h,0] 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 →
+                   ) → (∀G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,0] 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 →
+                   ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V2 → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ➡[h,0] 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 →
+                   ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ➡[h,0] V → ❪G,L❫ ⊢ W1 ➡[h,0] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ➡[h,0] 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.
+                   ∀G,L,T1,T2. ❪G,L❫ ⊢ T1 ➡[h,0] 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/ ]