- notational change for cpg and cpx

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma
index 037113f24082ad5bb7c583564e6161e8365e7a20..d21a9042710095a8e18dc330a738be00764eef94 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/cpxs.ma
@@ -27,45 +27,45 @@ interpretation "extended context-sensitive parallel computation (term)"
 (* Basic eliminators ********************************************************)
 
 lemma cpxs_ind: ∀h,o,G,L,T1. ∀R:predicate term. R T1 →
-                (â\88\80T,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[h, o] T2 → R T → R T2) →
-                â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 → R T2.
+                (â\88\80T,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â¬\88[h, o] T2 → R T → R T2) →
+                â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2 → R T2.
 #h #o #L #G #T1 #R #HT1 #IHT1 #T2 #HT12
 @(TC_star_ind … HT1 IHT1 … HT12) //
 qed-.
 
 lemma cpxs_ind_dx: ∀h,o,G,L,T2. ∀R:predicate term. R T2 →
-                   (â\88\80T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, o] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[h, o] T2 → R T → R T1) →
-                   â\88\80T1. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 → R T1.
+                   (â\88\80T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h, o] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h, o] T2 → R T → R T1) →
+                   â\88\80T1. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2 → R T1.
 #h #o #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
 qed-.
 
 (* Basic properties *********************************************************)
 
-lemma cpxs_refl: â\88\80h,o,G,L,T. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[h, o] T.
+lemma cpxs_refl: â\88\80h,o,G,L,T. â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h, o] T.
 /2 width=1 by inj/ qed.
 
-lemma cpx_cpxs: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2.
+lemma cpx_cpxs: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2.
 /2 width=1 by inj/ qed.
 
-lemma cpxs_strap1: â\88\80h,o,G,L,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T →
-                   â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2.
+lemma cpxs_strap1: â\88\80h,o,G,L,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T →
+                   â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T â¬\88[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2.
 normalize /2 width=3 by step/ qed.
 
-lemma cpxs_strap2: â\88\80h,o,G,L,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, o] T →
-                   â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2.
+lemma cpxs_strap2: â\88\80h,o,G,L,T1,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h, o] T →
+                   â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2.
 normalize /2 width=3 by TC_strap/ qed.
 
 lemma lsubr_cpxs_trans: ∀h,o,G. lsub_trans … (cpxs h o G) lsubr.
 /3 width=5 by lsubr_cpx_trans, LTC_lsub_trans/
 qed-.
 
-lemma cprs_cpxs: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡* T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2.
+lemma cprs_cpxs: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88* T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2.
 #h #o #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /3 width=3 by cpxs_strap1, cpr_cpx/
 qed.
 
 lemma cpxs_sort: ∀h,o,G,L,s,d1. deg h o s d1 →
-                 â\88\80d2. d2 â\89¤ d1 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â\9e¡*[h, o] ⋆((next h)^d2 s).
+                 â\88\80d2. d2 â\89¤ d1 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â¬\88*[h, o] ⋆((next h)^d2 s).
 #h #o #G #L #s #d1 #Hkd1 #d2 @(nat_ind_plus … d2) -d2 /2 width=1 by cpx_cpxs/
 #d2 #IHd2 #Hd21 >iter_SO
 @(cpxs_strap1 … (⋆(iter d2 ℕ (next h) s)))
@@ -75,68 +75,68 @@ lemma cpxs_sort: ∀h,o,G,L,s,d1. deg h o s d1 →
 ]
 qed.
 
-lemma cpxs_bind_dx: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, o] V2 →
-                    â\88\80I,T1,T2. â¦\83G, L. â\93\91{I}V1â¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 →
-                    â\88\80a. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{a,I}V1.T1 â\9e¡*[h, o] ⓑ{a,I}V2.T2.
+lemma cpxs_bind_dx: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h, o] V2 →
+                    â\88\80I,T1,T2. â¦\83G, L. â\93\91{I}V1â¦\84 â\8a¢ T1 â¬\88*[h, o] T2 →
+                    â\88\80a. â¦\83G, Lâ¦\84 â\8a¢ â\93\91{a,I}V1.T1 â¬\88*[h, o] ⓑ{a,I}V2.T2.
 #h #o #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: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, o] V2 →
-                    â\88\80T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 →
-                    â\88\80I. â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â\9e¡*[h, o] ⓕ{I}V2.T2.
+lemma cpxs_flat_dx: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h, o] V2 →
+                    â\88\80T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2 →
+                    â\88\80I. â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â¬\88*[h, o] ⓕ{I}V2.T2.
 #h #o #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_sn: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, o] T2 →
-                    â\88\80V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, o] V2 →
-                    â\88\80I. â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â\9e¡*[h, o] ⓕ{I}V2.T2.
+lemma cpxs_flat_sn: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h, o] T2 →
+                    â\88\80V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88*[h, o] V2 →
+                    â\88\80I. â¦\83G, Lâ¦\84 â\8a¢ â\93\95{I}V1.T1 â¬\88*[h, o] ⓕ{I}V2.T2.
 #h #o #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_pair_sn: â\88\80h,o,I,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, o] V2 →
-                    â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â\91¡{I}V1.T â\9e¡*[h, o] ②{I}V2.T.
+lemma cpxs_pair_sn: â\88\80h,o,I,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88*[h, o] V2 →
+                    â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â\91¡{I}V1.T â¬\88*[h, o] ②{I}V2.T.
 #h #o #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
 /3 width=3 by cpxs_strap1, cpx_pair_sn/
 qed.
 
 lemma cpxs_zeta: ∀h,o,G,L,V,T1,T,T2. ⬆[0, 1] T2 ≡ T →
-                 â¦\83G, L.â\93\93Vâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ +â\93\93V.T1 â\9e¡*[h, o] T2.
+                 â¦\83G, L.â\93\93Vâ¦\84 â\8a¢ T1 â¬\88*[h, o] T â\86\92 â¦\83G, Lâ¦\84 â\8a¢ +â\93\93V.T1 â¬\88*[h, o] T2.
 #h #o #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_eps: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 →
-                â\88\80V. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dV.T1 â\9e¡*[h, o] T2.
+lemma cpxs_eps: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2 →
+                â\88\80V. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dV.T1 â¬\88*[h, o] T2.
 #h #o #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
 /3 width=3 by cpxs_strap1, cpx_cpxs, cpx_eps/
 qed.
 
-lemma cpxs_ct: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡*[h, o] V2 →
-               â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dV1.T â\9e¡*[h, o] V2.
+lemma cpxs_ct: â\88\80h,o,G,L,V1,V2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88*[h, o] V2 →
+               â\88\80T. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dV1.T â¬\88*[h, o] V2.
 #h #o #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,o,a,G,L,V1,V2,W1,W2,T1,T2.
-                    â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, o] V2 â\86\92 â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡[h, o] W2 →
-                    â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{a}W1.T1 â\9e¡*[h, o] ⓓ{a}ⓝW2.V2.T2.
+                    â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h, o] V2 â\86\92 â¦\83G, L.â\93\9bW1â¦\84 â\8a¢ T1 â¬\88*[h, o] T2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88[h, o] W2 →
+                    â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\9b{a}W1.T1 â¬\88*[h, o] ⓓ{a}ⓝW2.V2.T2.
 #h #o #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,o,a,G,L,V1,V,V2,W1,W2,T1,T2.
-                     â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, o] V â\86\92 â¬\86[0, 1] V â\89¡ V2 â\86\92 â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 →
-                     â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡[h, o] W2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{a}W1.T1 â\9e¡*[h, o] ⓓ{a}W2.ⓐV2.T2.
+                     â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h, o] V â\86\92 â¬\86[0, 1] V â\89¡ V2 â\86\92 â¦\83G, L.â\93\93W1â¦\84 â\8a¢ T1 â¬\88*[h, o] T2 →
+                     â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88[h, o] W2 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\93\90V1.â\93\93{a}W1.T1 â¬\88*[h, o] ⓓ{a}W2.ⓐV2.T2.
 #h #o #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: â\88\80h,o,G,L,U2,s. â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â\9e¡*[h, o] U2 →
+lemma cpxs_inv_sort1: â\88\80h,o,G,L,U2,s. â¦\83G, Lâ¦\84 â\8a¢ â\8b\86s â¬\88*[h, o] U2 →
                       ∃∃n,d. deg h o s (n+d) & U2 = ⋆((next h)^n s).
 #h #o #G #L #U2 #s #H @(cpxs_ind … H) -U2
 [ elim (deg_total h o s) #d #Hkd
@@ -150,10 +150,10 @@ lemma cpxs_inv_sort1: ∀h,o,G,L,U2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[h, o] U2 →
 ]
 qed-.
 
-lemma cpxs_inv_cast1: â\88\80h,o,G,L,W1,T1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dW1.T1 â\9e¡*[h, o] U2 →
-                      â\88¨â\88¨ â\88\83â\88\83W2,T2. â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡*[h, o] W2 & â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 & U2 = ⓝW2.T2
-                       | â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] U2
-                       | â¦\83G, Lâ¦\84 â\8a¢ W1 â\9e¡*[h, o] U2.
+lemma cpxs_inv_cast1: â\88\80h,o,G,L,W1,T1,U2. â¦\83G, Lâ¦\84 â\8a¢ â\93\9dW1.T1 â¬\88*[h, o] U2 →
+                      â\88¨â\88¨ â\88\83â\88\83W2,T2. â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88*[h, o] W2 & â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2 & U2 = ⓝW2.T2
+                       | â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] U2
+                       | â¦\83G, Lâ¦\84 â\8a¢ W1 â¬\88*[h, o] U2.
 #h #o #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
@@ -163,14 +163,14 @@ lapply (cpxs_strap1 … HW1 … HW2) -W
 lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5 by or3_intro0, ex3_2_intro/
 qed-.
 
-lemma cpxs_inv_cnx1: â\88\80h,o,G,L,T,U. â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[h, o] U â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â\9e¡[h, o] 𝐍⦃T⦄ → T = U.
+lemma cpxs_inv_cnx1: â\88\80h,o,G,L,T,U. â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h, o] U â\86\92 â¦\83G, Lâ¦\84 â\8a¢ â¬\88[h, o] 𝐍⦃T⦄ → T = U.
 #h #o #G #L #T #U #H @(cpxs_ind_dx … H) -T //
 #T0 #T #H1T0 #_ #IHT #H2T0
 lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
 qed-.
 
-lemma cpxs_neq_inv_step_sn: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡*[h, o] T2 → (T1 = T2 → ⊥) →
-                            â\88\83â\88\83T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, o] T & T1 = T â\86\92 â\8a¥ & â¦\83G, Lâ¦\84 â\8a¢ T â\9e¡*[h, o] T2.
+lemma cpxs_neq_inv_step_sn: â\88\80h,o,G,L,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h, o] T2 → (T1 = T2 → ⊥) →
+                            â\88\83â\88\83T. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88[h, o] T & T1 = T â\86\92 â\8a¥ & â¦\83G, Lâ¦\84 â\8a¢ T â¬\88*[h, o] T2.
 #h #o #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