update in basic_2 and static_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_computation / cpms.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
index 8c73990c836a36645b04e75c0b0c57967993cc69..454ee83127c6499287d18df4e8decbff5ce1ae03 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
@@ -37,14 +37,14 @@ lemma cpms_ind_sn (h) (G) (L) (T2) (Q:relation2 …):
                   Q 0 T2 →
                   (∀n1,n2,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T → ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → Q n2 T → Q (n1+n2) T1) →
                   ∀n,T1. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T1.
-#h #G #L #T2 #R @ltc_ind_sn_refl //
+#h #G #L #T2 #Q @ltc_ind_sn_refl //
 qed-.
 
 lemma cpms_ind_dx (h) (G) (L) (T1) (Q:relation2 …):
                   Q 0 T1 →
                   (∀n1,n2,T,T2. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T → Q n1 T → ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → Q (n1+n2) T2) →
                   ∀n,T2. ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2 → Q n T2.
-#h #G #L #T1 #R @ltc_ind_dx_refl //
+#h #G #L #T1 #Q @ltc_ind_dx_refl //
 qed-.
 
 (* Basic properties *********************************************************)
@@ -79,10 +79,17 @@ lemma cpms_appl_dx (n) (h) (G) (L):
 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
 qed.
 
-(* Basic_2A1: uses: cprs_zeta *)
 lemma cpms_zeta (n) (h) (G) (L):
-                ∀T2,T. ⬆*[1] T2 ≘ T →
-                ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+                ∀T1,T. ⬆*[1] T ≘ T1 →
+                ∀V,T2. ⦃G, L⦄ ⊢ T ➡*[n, h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+#n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
+/3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/
+qed.
+
+(* Basic_2A1: uses: cprs_zeta *)
+lemma cpms_zeta_dx (n) (h) (G) (L):
+                   ∀T2,T. ⬆*[1] T2 ≘ T →
+                   ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
 #n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/
 qed.
@@ -95,6 +102,16 @@ lemma cpms_eps (n) (h) (G) (L):
 /3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/
 qed.
 
+lemma cpms_ee (n) (h) (G) (L):
+              ∀U1,U2. ⦃G, L⦄ ⊢ U1 ➡*[n, h] U2 →
+              ∀T. ⦃G, L⦄ ⊢ ⓝU1.T ➡*[↑n, h] U2.
+#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
+[ /3 width=1 by cpm_cpms, cpm_ee/
+| #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1
+  /3 width=3 by cpms_step_sn, cpm_ee/
+]
+qed.
+
 (* Basic_2A1: uses: cprs_beta_dx *)
 lemma cpms_beta_dx (n) (h) (G) (L):
                    ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
@@ -116,21 +133,53 @@ lemma cpms_theta_dx (n) (h) (G) (L):
 /4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/
 qed.
 
+(* Basic properties with r-transition ***************************************)
+
+(* Basic_1: was: pr3_refl *)
+lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0).
+/2 width=1 by cpm_cpms/ qed.
+
+(* Advanced properties ******************************************************)
+
+lemma cpms_sort (h) (G) (L) (n):
+                ∀s. ⦃G,L⦄ ⊢ ⋆s ➡*[n,h] ⋆((next h)^n s).
+#h #G #L #n elim n -n [ // ]
+#n #IH #s <plus_SO
+/3 width=3 by cpms_step_dx, cpm_sort/
+qed.
+
 (* Basic inversion lemmas ***************************************************)
 
 lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[n, h] X2 → X2 = ⋆(((next h)^n) s).
 #n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
 #n1 #n2 #X #X2 #_ #IH #HX2 destruct
-elim (cpm_inv_sort1 … HX2) -HX2 * // #H1 #H2 destruct
-/2 width=3 by refl, trans_eq/
+elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct //
 qed-.
 
-(* Basic properties with r-transition ***************************************)
-
-(* Basic_1: was: pr3_refl *)
-lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0).
-/2 width=1 by cpm_cpms/ qed.
+lemma cpms_inv_cast1 (h) (n) (G) (L):
+      ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
+      ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
+       | ⦃G, L⦄ ⊢ T1 ➡*[n,h] X2
+       | ∃∃m. ⦃G, L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
+#h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2
+[ /3 width=5 by or3_intro0, ex3_2_intro/
+| #n1 #n2 #X #X2 #_ * [ * || * ]
+  [ #W #T #HW1 #HT1 #H #HX2 destruct
+    elim (cpm_inv_cast1 … HX2) -HX2 [ * || * ]
+    [ #W2 #T2 #HW2 #HT2 #H destruct
+      /4 width=5 by cpms_step_dx, ex3_2_intro, or3_intro0/
+    | #HX2 /3 width=3 by cpms_step_dx, or3_intro1/
+    | #m #HX2 #H destruct <plus_n_Sm
+      /4 width=3 by cpms_step_dx, ex2_intro, or3_intro2/
+    ]
+  | #HX #HX2 /3 width=3 by cpms_step_dx, or3_intro1/
+  | #m #HX #H #HX2 destruct
+    /4 width=3 by cpms_step_dx, ex2_intro, or3_intro2/
+  ]
+]
+qed-.
 
-(* Basic_2A1: removed theorems 4:
+(* Basic_2A1: removed theorems 5:
               sta_cprs_scpds lstas_scpds scpds_strap1 scpds_fwd_cprs
+              scpds_inv_lstas_eq
 *)