X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_computation%2Fcpms.ma;h=64215875127b70271ae305ae841247cabf71aba2;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=cbfd28726da4b217c36efd0377120dc984680995;hpb=f129bbbfda0e65a5f92ec086246f6e288376d4f9;p=helm.git

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 cbfd28726..642158751 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpms.ma
@@ -31,40 +31,185 @@ interpretation
    "context-sensitive parallel r-computation (term)"
    'PRedStar h G L T1 T2 = (cpms h G L O T1 T2).
 
+(* Basic eliminators ********************************************************)
+
+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 #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 #Q @ltc_ind_dx_refl //
+qed-.
+
 (* Basic properties *********************************************************)
 
-lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n, h] T2.
+(* Basic_1: includes: pr1_pr0 *)
+(* Basic_1: uses: pr3_pr2 *)
+(* Basic_2A1: includes: cpr_cprs *)
+lemma cpm_cpms (h) (G) (L): ∀n,T1,T2. ❪G,L❫ ⊢ T1 ➡[n,h] T2 → ❪G,L❫ ⊢ T1 ➡*[n,h] T2.
 /2 width=1 by ltc_rc/ qed.
 
-lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T →
-                                ∀n2,T2. ⦃G, L⦄ ⊢ T ➡*[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
+lemma cpms_step_sn (h) (G) (L): ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡[n1,h] T →
+                                ∀n2,T2. ❪G,L❫ ⊢ T ➡*[n2,h] T2 → ❪G,L❫ ⊢ T1 ➡*[n1+n2,h] T2.
 /2 width=3 by ltc_sn/ qed-.
 
-lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[n1, h] T →
-                                ∀n2,T2. ⦃G, L⦄ ⊢ T ➡[n2, h] T2 → ⦃G, L⦄ ⊢ T1 ➡*[n1+n2, h] T2.
+lemma cpms_step_dx (h) (G) (L): ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡*[n1,h] T →
+                                ∀n2,T2. ❪G,L❫ ⊢ T ➡[n2,h] T2 → ❪G,L❫ ⊢ T1 ➡*[n1+n2,h] T2.
 /2 width=3 by ltc_dx/ qed-.
 
+(* Basic_2A1: uses: cprs_bind_dx *)
+lemma cpms_bind_dx (n) (h) (G) (L):
+                   ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
+                   ∀I,T1,T2. ❪G,L.ⓑ[I]V1❫ ⊢ T1 ➡*[n,h] T2 →
+                   ∀p. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ➡*[n,h] ⓑ[p,I]V2.T2.
+#n #h #G #L #V1 #V2 #HV12 #I #T1 #T2 #H #a @(cpms_ind_sn … H) -T1
+/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind/ qed.
+
+lemma cpms_appl_dx (n) (h) (G) (L):
+                   ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
+                   ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
+                   ❪G,L❫ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
+#n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
+/3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
+qed.
+
+lemma cpms_zeta (n) (h) (G) (L):
+                ∀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.
+
+(* Basic_2A1: uses: cprs_eps *)
+lemma cpms_eps (n) (h) (G) (L):
+               ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
+               ∀V. ❪G,L❫ ⊢ ⓝV.T1 ➡*[n,h] T2.
+#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
+/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 →
+                   ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h] W2 →
+                   ∀T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[n,h] T2 →
+                   ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[n,h] ⓓ[p]ⓝW2.V2.T2.
+#n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
+/4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/
+qed.
+
+(* Basic_2A1: uses: cprs_theta_dx *)
+lemma cpms_theta_dx (n) (h) (G) (L):
+                    ∀V1,V. ❪G,L❫ ⊢ V1 ➡[h] V →
+                    ∀V2. ⇧*[1] V ≘ V2 →
+                    ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h] W2 →
+                    ∀T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[n,h] T2 →
+                    ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[n,h] ⓓ[p]W2.ⓐV2.T2.
+#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
+/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.
 
-(* Basic eliminators ********************************************************)
+(* Advanced properties ******************************************************)
 
-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 //
+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_dx
+/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 #H #_ destruct //
 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 //
+lemma cpms_inv_lref1_ctop (n) (h) (G):
+      ∀X2,i. ❪G,⋆❫ ⊢ #i ➡*[n,h] X2 → ∧∧ X2 = #i & n = 0.
+#n #h #G #X2 #i #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+  elim (cpm_inv_lref1_ctop … HX2) -HX2 #H1 #H2 destruct
+  /2 width=1 by conj/
+]
+qed-.
+
+lemma cpms_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 @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+  elim (cpm_inv_zero1_unit … HX2) -HX2 #H1 #H2 destruct
+  /2 width=1 by conj/
+]
+qed-.
+
+lemma cpms_inv_gref1 (n) (h) (G) (L):
+      ∀X2,l. ❪G,L❫ ⊢ §l ➡*[n,h] X2 → ∧∧ X2 = §l & n = 0.
+#n #h #G #L #X2 #l #H @(cpms_ind_dx … H) -X2
+[ /2 width=1 by conj/
+| #n1 #n2 #X #X2 #_ * #HX #Hn1 #HX2 destruct
+  elim (cpm_inv_gref1 … HX2) -HX2 #H1 #H2 destruct
+  /2 width=1 by conj/
+]
+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
 *)