X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fcomputation%2Fcpxs.ma;h=588208177e01d4b1ecb664f933aed770a523e4cc;hb=33f8507cadd3b36dc9afa227d8968dda66fe2034;hp=028798162fbfaecb0d47a9a2fa1971f7ec2e666d;hpb=f62eeb3c7824564ccbe4fff6e75ddee46ca39cc0;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
index 028798162..588208177 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
@@ -12,135 +12,170 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/notation/relations/predstar_6.ma".
 include "basic_2/reduction/cnx.ma".
 include "basic_2/computation/cprs.ma".
 
 (* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
 
-definition cpxs: ∀h. sd h → lenv → relation term ≝
-                 λh,g. LTC … (cpx h g).
+definition cpxs: ∀h. sd h → relation4 genv lenv term term ≝
+                 λh,g,G. LTC … (cpx h g G).
 
 interpretation "extended context-sensitive parallel computation (term)"
-   'PRedStar h g L T1 T2 = (cpxs h g L T1 T2).
+   'PRedStar h g G L T1 T2 = (cpxs h g G L T1 T2).
 
 (* Basic eliminators ********************************************************)
 
-lemma cpxs_ind: ∀h,g,L,T1. ∀R:predicate term. R T1 →
-                (∀T,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T → ⦃h, L⦄ ⊢ T ➡[g] T2 → R T → R T2) →
-                ∀T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → R T2.
-#h #g #L #T1 #R #HT1 #IHT1 #T2 #HT12
+lemma cpxs_ind: ∀h,g,G,L,T1. ∀R:predicate term. R T1 →
+                (∀T,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ T ➡[h, g] T2 → R T → R T2) →
+                ∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T2.
+#h #g #L #G #T1 #R #HT1 #IHT1 #T2 #HT12
 @(TC_star_ind … HT1 IHT1 … HT12) //
 qed-.
 
-lemma cpxs_ind_dx: ∀h,g,L,T2. ∀R:predicate term. R T2 →
-                   (∀T1,T. ⦃h, L⦄ ⊢ T1 ➡[g] T → ⦃h, L⦄ ⊢ T ➡*[g] T2 → R T → R T1) →
-                   ∀T1. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → R T1.
-#h #g #L #T2 #R #HT2 #IHT2 #T1 #HT12
+lemma cpxs_ind_dx: ∀h,g,G,L,T2. ∀R:predicate term. R T2 →
+                   (∀T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T → ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → R T → R T1) →
+                   ∀T1. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → R T1.
+#h #g #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cpxs_refl: ∀h,g,L,T. ⦃h, L⦄ ⊢ T ➡*[g] T.
-/2 width=1/ qed.
+lemma cpxs_refl: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T.
+/2 width=1 by inj/ qed.
 
-lemma cpx_cpxs: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
-/2 width=1/ qed.
+lemma cpx_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+/2 width=1 by inj/ qed.
 
-lemma cpxs_strap1: ∀h,g,L,T1,T. ⦃h, L⦄ ⊢ T1 ➡*[g] T →
-                   ∀T2. ⦃h, L⦄ ⊢ T ➡[g] T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
-normalize /2 width=3/ qed.
+lemma cpxs_strap1: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+normalize /2 width=3 by step/ qed.
 
-lemma cpxs_strap2: ∀h,g,L,T1,T. ⦃h, L⦄ ⊢ T1 ➡[g] T →
-                   ∀T2. ⦃h, L⦄ ⊢ T ➡*[g] T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
-normalize /2 width=3/ qed.
+lemma cpxs_strap2: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
+                   ∀T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+normalize /2 width=3 by TC_strap/ qed.
 
-lemma lsubx_cpxs_trans: ∀h,g. lsub_trans … (cpxs h g) lsubx.
-/3 width=5 by lsubx_cpx_trans, TC_lsub_trans/
+lemma lsubr_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) lsubr.
+/3 width=5 by lsubr_cpx_trans, LTC_lsub_trans/
 qed-.
 
-axiom cprs_cpxs: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → ⦃h, L⦄ ⊢ T1 ➡*[g] T2.
-(*
-#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/
+lemma cprs_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /3 width=3 by cpxs_strap1, cpr_cpx/
 qed.
-*)
-lemma cpxs_bind_dx: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
-                    ∀I,T1,T2. ⦃h, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[g] T2 →
-                    ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
-#h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
-/3 width=1/ /3 width=3/
+
+lemma cpxs_sort: ∀h,g,G,L,k,l1. deg h g k l1 →
+                 ∀l2. l2 ≤ l1 → ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] ⋆((next h)^l2 k).
+#h #g #G #L #k #l1 #Hkl1 #l2 @(nat_ind_plus … l2) -l2 /2 width=1 by cpx_cpxs/
+#l2 #IHl2 #Hl21 >iter_SO
+@(cpxs_strap1 … (⋆(iter l2 ℕ (next h) k)))
+[ /3 width=3 by lt_to_le/
+| @(cpx_st … (l1-l2-1)) <plus_minus_m_m
+  /2 width=1 by deg_iter, monotonic_le_minus_r/
+]
+qed.
+
+lemma cpxs_bind_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                    ∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
+                    ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
+#h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
+/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/
+qed.
+
+lemma cpxs_flat_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+                    ∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
+#h #g #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
+/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
 qed.
 
-lemma cpxs_flat_dx: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
-                    ∀T1,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 →
-                    ∀I. ⦃h, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[g] ⓕ{I} V2. T2.
-#h #g #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /3 width=1/ /3 width=5/
+lemma cpxs_flat_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
+                    ∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                    ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
+#h #g #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
 qed.
 
-lemma cpxs_flat_sn: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 →
-                    ∀V1,V2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
-                    ∀I. ⦃h, L⦄ ⊢ ⓕ{I} V1. T1 ➡*[g] ⓕ{I} V2. T2.
-#h #g #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=1/ /3 width=5/
+lemma cpxs_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡*[h, g] ②{I}V2.T.
+#h #g #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=3 by cpxs_strap1, cpx_pair_sn/
 qed.
 
-lemma cpxs_zeta: ∀h,g,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
-                 ⦃h, L.ⓓV⦄ ⊢ T1 ➡*[g] T → ⦃h, L⦄ ⊢ +ⓓV.T1 ➡*[g] T2.
-#h #g #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/
+lemma cpxs_zeta: ∀h,g,G,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
+                 ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2.
+#h #g #G #L #V #T1 #T #T2 #HT2 #H @(cpxs_ind_dx … H) -T1
+/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
 qed.
 
-lemma cpxs_tau: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡*[g] T2 → ∀V. ⦃h, L⦄ ⊢ ⓝV.T1 ➡*[g] T2.
-#h #g #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/
+lemma cpxs_eps: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
+                ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
 qed.
 
-lemma cpxs_ti: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 → ∀T. ⦃h, L⦄ ⊢ ⓝV1.T ➡*[g] V2.
-#h #g #L #V1 #V2 #H elim H -V2 /2 width=3/ /3 width=1/
+lemma cpxs_ct: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+               ∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2.
+#h #g #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ct/
 qed.
 
-lemma cpxs_beta_dx: ∀h,g,a,L,V1,V2,W1,W2,T1,T2.
-                    ⦃h, L⦄ ⊢ V1 ➡[g] V2 → ⦃h, L.ⓛW1⦄ ⊢ T1 ➡*[g] T2 → ⦃h, L⦄ ⊢ W1 ➡[g] W2 →
-                    ⦃h, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[g] ⓓ{a}ⓝW2.V2.T2.
-#h #g #a #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2 /3 width=1/
-/4 width=7 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/ (**) (* auto too slow without trace *)
+lemma cpxs_beta_dx: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
+                    ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 →
+                    ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
+#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
+/4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/
 qed.
 
-lemma cpxs_theta_dx: ∀h,g,a,L,V1,V,V2,W1,W2,T1,T2.
-                     ⦃h, L⦄ ⊢ V1 ➡[g] V → ⇧[0, 1] V ≡ V2 → ⦃h, L.ⓓW1⦄ ⊢ T1 ➡*[g] T2 →
-                     ⦃h, L⦄ ⊢ W1 ➡[g] W2 → ⦃h, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[g] ⓓ{a}W2.ⓐV2.T2.
-#h #g #a #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ]
-/4 width=9 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/ (**) (* auto too slow without trace *)
+lemma cpxs_theta_dx: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
+                     ⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 →
+                     ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
+#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 
+/4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cpxs_inv_sort1: ∀h,g,L,U2,k. ⦃h, L⦄ ⊢ ⋆k ➡*[g] U2 →
+lemma cpxs_inv_sort1: ∀h,g,G,L,U2,k. ⦃G, L⦄ ⊢ ⋆k ➡*[h, g] U2 →
                       ∃∃n,l. deg h g k (n+l) & U2 = ⋆((next h)^n k).
-#h #g #L #U2 #k #H @(cpxs_ind … H) -U2
+#h #g #G #L #U2 #k #H @(cpxs_ind … H) -U2
 [ elim (deg_total h g k) #l #Hkl
   @(ex2_2_intro … 0 … Hkl) -Hkl //
 | #U #U2 #_ #HU2 * #n #l #Hknl #H destruct
   elim (cpx_inv_sort1 … HU2) -HU2
-  [ #H destruct /2 width=4/
+  [ #H destruct /2 width=4 by ex2_2_intro/
   | * #l0 #Hkl0 #H destruct -l
     @(ex2_2_intro … (n+1) l0) /2 width=1 by deg_inv_prec/ >iter_SO //
   ]
 ]
 qed-.
 
-lemma cpxs_inv_cast1: ∀h,g,L,W1,T1,U2. ⦃h, L⦄ ⊢ ⓝW1.T1 ➡*[g] U2 →
-                      ∨∨ ∃∃W2,T2. ⦃h, L⦄ ⊢ W1 ➡*[g] W2 & ⦃h, L⦄ ⊢ T1 ➡*[g] T2 & U2 = ⓝW2.T2
-                       | ⦃h, L⦄ ⊢ T1 ➡*[g] U2
-                       | ⦃h, L⦄ ⊢ W1 ➡*[g] U2.
-#h #g #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
-#U2 #U #_ #HU2 * /3 width=3/ *
+lemma cpxs_inv_cast1: ∀h,g,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[h, g] U2 →
+                      ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 & U2 = ⓝW2.T2
+                       | ⦃G, L⦄ ⊢ T1 ➡*[h, g] U2
+                       | ⦃G, L⦄ ⊢ W1 ➡*[h, g] U2.
+#h #g #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
+#U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
 #W #T #HW1 #HT1 #H destruct
-elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3/ *
+elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
 #W2 #T2 #HW2 #HT2 #H destruct
 lapply (cpxs_strap1 … HW1 … HW2) -W
-lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5/
+lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5 by or3_intro0, ex3_2_intro/
 qed-.
 
-lemma cpxs_inv_cnx1: ∀h,g,L,T,U. ⦃h, L⦄ ⊢ T ➡*[g] U → ⦃h, L⦄ ⊢ 𝐍[g]⦃T⦄ → T = U.
-#h #g #L #T #U #H @(cpxs_ind_dx … H) -T //
+lemma cpxs_inv_cnx1: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ ➡[h, g] 𝐍⦃T⦄ → T = U.
+#h #g #G #L #T #U #H @(cpxs_ind_dx … H) -T //
 #T0 #T #H1T0 #_ #IHT #H2T0
-lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
+lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
+qed-.
+
+lemma cpxs_neq_inv_step_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) →
+                            ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T & T1 = T → ⊥ & ⦃G, L⦄ ⊢ T ➡*[h, g] T2.
+#h #g #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
+[ #H elim H -H //
+| #T1 #T #H1 #H2 #IH2 #H12 elim (eq_term_dec T1 T) #H destruct
+  [ -H1 -H2 /3 width=1 by/
+  | -IH2 /3 width=4 by ex3_intro/ (**) (* auto fails without clear *)
+  ]
+]
 qed-.