update in ground_2, static_2, basic_2

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
index db5bdfe780fa125c758e186f1775c5744258a6fc..f161bf0615d97d7578c7de65b625d14b3a7f9827 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma
@@ -12,6 +12,10 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/xoa/ex_4_1.ma".
+include "ground_2/xoa/ex_4_3.ma".
+include "ground_2/xoa/ex_5_6.ma".
+include "ground_2/xoa/ex_6_7.ma".
 include "basic_2/notation/relations/pred_6.ma".
 include "basic_2/notation/relations/pred_5.ma".
 include "basic_2/rt_transition/cpg.ma".
@@ -36,19 +40,19 @@ lemma cpm_ess: ∀h,G,L,s. ⦃G,L⦄ ⊢ ⋆s ➡[1,h] ⋆(⫯[h]s).
 /2 width=3 by cpg_ess, ex2_intro/ qed.
 
 lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
-                 â¬\86*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ➡[n,h] W2.
+                 â\87§*[1] V2 ≘ W2 → ⦃G,K.ⓓV1⦄ ⊢ #0 ➡[n,h] W2.
 #n #h #G #K #V1 #V2 #W2 *
 /3 width=5 by cpg_delta, ex2_intro/
 qed.
 
 lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 →
-               â¬\86*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ➡[↑n,h] W2.
+               â\87§*[1] V2 ≘ W2 → ⦃G,K.ⓛV1⦄ ⊢ #0 ➡[↑n,h] W2.
 #n #h #G #K #V1 #V2 #W2 *
 /3 width=5 by cpg_ell, ex2_intro, isrt_succ/
 qed.
 
 lemma cpm_lref: ∀n,h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T →
-                â¬\86*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ➡[n,h] U.
+                â\87§*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ➡[n,h] U.
 #n #h #I #G #K #T #U #i *
 /3 width=5 by cpg_lref, ex2_intro/
 qed.
@@ -77,7 +81,7 @@ qed.
 
 (* Basic_2A1: includes: cpr_zeta *)
 lemma cpm_zeta (n) (h) (G) (L):
-               â\88\80T1,T. â¬\86*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
+               â\88\80T1,T. â\87§*[1] T ≘ T1 → ∀T2. ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
                ∀V. ⦃G,L⦄ ⊢ +ⓓV.T1 ➡[n,h] T2.
 #n #h #G #L #T1 #T #HT1 #T2 *
 /3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/
@@ -104,7 +108,7 @@ qed.
 
 (* Basic_2A1: includes: cpr_theta *)
 lemma cpm_theta: ∀n,h,p,G,L,V1,V,V2,W1,W2,T1,T2.
-                 â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V â\86\92 â¬\86*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
+                 â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V â\86\92 â\87§*[1] V ≘ V2 → ⦃G,L⦄ ⊢ W1 ➡[h] W2 →
                  ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 →
                  ⦃G,L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n,h] ⓓ{p}W2.ⓐV2.T2.
 #n #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 *
@@ -132,11 +136,11 @@ qed.
 lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G,L⦄ ⊢ ⓪{J} ➡[n,h] T2 →
                      ∨∨ T2 = ⓪{J} ∧ n = 0
                       | ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s & n = 1
-                      | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[n,h] V2 & â¬\86*[1] V2 ≘ T2 &
+                      | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[n,h] V2 & â\87§*[1] V2 ≘ T2 &
                                    L = K.ⓓV1 & J = LRef 0
-                      | â\88\83â\88\83m,K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[m,h] V2 & â¬\86*[1] V2 ≘ T2 &
+                      | â\88\83â\88\83m,K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[m,h] V2 & â\87§*[1] V2 ≘ T2 &
                                      L = K.ⓛV1 & J = LRef 0 & n = ↑m
-                      | â\88\83â\88\83I,K,T,i. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡[n,h] T & â¬\86*[1] T ≘ T2 &
+                      | â\88\83â\88\83I,K,T,i. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡[n,h] T & â\87§*[1] T ≘ T2 &
                                    L = K.ⓘ{I} & J = LRef (↑i).
 #n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or5_intro0, conj/
@@ -161,9 +165,9 @@ qed-.
 
 lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[n,h] T2 →
                      ∨∨ T2 = #0 ∧ n = 0
-                      | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[n,h] V2 & â¬\86*[1] V2 ≘ T2 &
+                      | â\88\83â\88\83K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[n,h] V2 & â\87§*[1] V2 ≘ T2 &
                                    L = K.ⓓV1
-                      | â\88\83â\88\83m,K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[m,h] V2 & â¬\86*[1] V2 ≘ T2 &
+                      | â\88\83â\88\83m,K,V1,V2. â¦\83G,Kâ¦\84 â\8a¢ V1 â\9e¡[m,h] V2 & â\87§*[1] V2 ≘ T2 &
                                      L = K.ⓛV1 & n = ↑m.
 #n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
@@ -175,9 +179,19 @@ lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G,L⦄ ⊢ #0 ➡[n,h] T2 →
 ]
 qed-.
 
+lemma cpm_inv_zero1_unit (n) (h) (I) (K) (G):
+      ∀X2. ⦃G,K.ⓤ{I}⦄ ⊢ #0 ➡[n,h] X2 → ∧∧ X2 = #0 & n = 0.
+#n #h #I #G #K #X2 #H
+elim (cpm_inv_zero1 … H) -H *
+[ #H1 #H2 destruct /2 width=1 by conj/
+| #Y #X1 #X2 #_ #_ #H destruct
+| #m #Y #X1 #X2 #_ #_ #H destruct
+]
+qed.
+
 lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[n,h] T2 →
                      ∨∨ T2 = #(↑i) ∧ n = 0
-                      | â\88\83â\88\83I,K,T. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡[n,h] T & â¬\86*[1] T ≘ T2 & L = K.ⓘ{I}.
+                      | â\88\83â\88\83I,K,T. â¦\83G,Kâ¦\84 â\8a¢ #i â\9e¡[n,h] T & â\87§*[1] T ≘ T2 & L = K.ⓘ{I}.
 #n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/
 | #I #K #V2 #HV2 #HVT2 #H destruct
@@ -185,16 +199,31 @@ lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ➡[n,h] T2 →
 ]
 qed-.
 
+lemma cpm_inv_lref1_ctop (n) (h) (G):
+      ∀X2,i. ⦃G,⋆⦄ ⊢ #i ➡[n,h] X2 → ∧∧ X2 = #i & n = 0.
+#n #h #G #X2 * [| #i ] #H
+[ elim (cpm_inv_zero1 … H) -H *
+  [ #H1 #H2 destruct /2 width=1 by conj/
+  | #Y #X1 #X2 #_ #_ #H destruct
+  | #m #Y #X1 #X2 #_ #_ #H destruct
+  ]
+| elim (cpm_inv_lref1 … H) -H *
+  [ #H1 #H2 destruct /2 width=1 by conj/
+  | #Z #Y #X0 #_ #_ #H destruct
+  ]
+]
+qed.
+
 lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G,L⦄ ⊢ §l ➡[n,h] T2 → T2 = §l ∧ n = 0.
 #n #h #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H
-#H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/ 
+#H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/
 qed-.
 
 (* Basic_2A1: includes: cpr_inv_bind1 *)
 lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n,h] U2 →
                      ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 &
                                  U2 = ⓑ{p,I}V2.T2
-                      | â\88\83â\88\83T. â¬\86*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡[n,h] U2 & 
+                      | â\88\83â\88\83T. â\87§*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡[n,h] U2 &
                              p = true & I = Abbr.
 #n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H *
 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
@@ -211,7 +240,7 @@ qed-.
 lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ➡[n,h] U2 →
                      ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ➡[n,h] T2 &
                                  U2 = ⓓ{p}V2.T2
-                      | â\88\83â\88\83T. â¬\86*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 & p = true.
+                      | â\88\83â\88\83T. â\87§*[1] T ≘ T1 & ⦃G,L⦄ ⊢ T ➡[n,h] U2 & p = true.
 #n #h #p #G #L #V1 #T1 #U2 #H
 elim (cpm_inv_bind1 … H) -H
 [ /3 width=1 by or_introl/
@@ -227,7 +256,7 @@ lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ➡[n,h] U
 #n #h #p #G #L #V1 #T1 #U2 #H
 elim (cpm_inv_bind1 … H) -H
 [ /3 width=1 by or_introl/
-| * #T #_ #_ #_ #H destruct  
+| * #T #_ #_ #_ #H destruct
 ]
 qed-.
 
@@ -235,7 +264,7 @@ lemma cpm_inv_abst_bi: ∀n,h,p1,p2,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ ⓛ{p1}V1.T1
                        ∧∧ ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ➡[n,h] T2 & p1 = p2.
 #n #h #p1 #p2 #G #L #V1 #V2 #T1 #T2 #H
 elim (cpm_inv_abst1 … H) -H #XV #XT #HV #HT #H destruct
-/2 width=1 by and3_intro/  
+/2 width=1 by and3_intro/
 qed-.
 
 (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
@@ -246,7 +275,7 @@ lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G,L⦄ ⊢ ⓐ V1.U1 ➡[n,h] U2 
                       | ∃∃p,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 & ⦃G,L⦄ ⊢ W1 ➡[h] W2 &
                                             ⦃G,L.ⓛW1⦄ ⊢ T1 ➡[n,h] T2 &
                                             U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
-                      | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V & â¬\86*[1] V ≘ V2 &
+                      | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â\9e¡[h] V & â\87§*[1] V ≘ V2 &
                                               ⦃G,L⦄ ⊢ W1 ➡[h] W2 & ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 &
                                               U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
 #n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
@@ -306,18 +335,18 @@ lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
                    (∀I,G,L. Q 0 G L (⓪{I}) (⓪{I})) →
                    (∀G,L,s. Q 1 G L (⋆s) (⋆(⫯[h]s))) →
                    (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
-                     â¬\86*[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
+                     â\87§*[1] V2 ≘ W2 → Q n G (K.ⓓV1) (#0) W2
                    ) → (∀n,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ➡[n,h] V2 → Q n G K V1 V2 →
-                     â¬\86*[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
+                     â\87§*[1] V2 ≘ W2 → Q (↑n) G (K.ⓛV1) (#0) W2
                    ) → (∀n,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ➡[n,h] T → Q n G K (#i) T →
-                     â¬\86*[1] T ≘ U → Q n G (K.ⓘ{I}) (#↑i) (U)
+                     â\87§*[1] T ≘ U → Q n G (K.ⓘ{I}) (#↑i) (U)
                    ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ➡[n,h] T2 →
                      Q 0 G L V1 V2 → Q n G (L.ⓑ{I}V1) T1 T2 → Q n G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
                    ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
                      Q 0 G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓐV1.T1) (ⓐV2.T2)
                    ) → (∀n,G,L,V1,V2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[n,h] V2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
                      Q n G L V1 V2 → Q n G L T1 T2 → Q n G L (ⓝV1.T1) (ⓝV2.T2)
-                   ) â\86\92 (â\88\80n,G,L,V,T1,T,T2. â¬\86*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
+                   ) â\86\92 (â\88\80n,G,L,V,T1,T,T2. â\87§*[1] T ≘ T1 → ⦃G,L⦄ ⊢ T ➡[n,h] T2 →
                      Q n G L T T2 → Q n G L (+ⓓV.T1) T2
                    ) → (∀n,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
                      Q n G L T1 T2 → Q n G L (ⓝV.T1) T2
@@ -328,7 +357,7 @@ lemma cpm_ind (h): ∀Q:relation5 nat genv lenv term term.
                      Q n G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
                    ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G,L⦄ ⊢ V1 ➡[h] V → ⦃G,L⦄ ⊢ W1 ➡[h] W2 → ⦃G,L.ⓓW1⦄ ⊢ T1 ➡[n,h] T2 →
                      Q 0 G L V1 V → Q 0 G L W1 W2 → Q n G (L.ⓓW1) T1 T2 →
-                     â¬\86*[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+                     â\87§*[1] V ≘ V2 → Q n G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
                    ) →
                    ∀n,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 → Q n G L T1 T2.
 #h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2