- the relation for pointwise extensions now takes a binder as argument

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / computation / lpxs_cpxs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
index b22ee7c897f3ebaa4001bd02b4a36b4706d86452..d6e614042fbc14df90c0a4458e378086a3d44997 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_cpxs.ma
@@ -34,13 +34,27 @@ lemma lpxs_inv_pair2: ∀h,g,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h, g] K2.ⓑ{I}V
                       ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h, g] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 & L1 = K1.ⓑ{I}V1.
 /3 width=3 by TC_lpx_sn_inv_pair2, lpx_cpxs_trans/ qed-.
 
+(* Advanced eliminators *****************************************************)
+
+lemma lpxs_ind_alt: ∀h,g,G. ∀R:relation lenv.
+                    R (⋆) (⋆) → (
+                       ∀I,K1,K2,V1,V2.
+                       ⦃G, K1⦄ ⊢ ➡*[h, g] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h, g] V2 →
+                       R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+                    ) →
+                    ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → R L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpx_cpxs_trans/ qed-.
+
 (* Properties on context-sensitive extended parallel computation for terms **)
 
-lemma lpxs_cpx_trans: ∀h,g,G. s_r_trans … (cpx h g G) (lpxs h g G).
-/3 width=5 by s_r_trans_TC2, lpx_cpxs_trans/ qed-.
+lemma lpxs_cpx_trans: ∀h,g,G. s_r_transitive … (cpx h g G) (λ_.lpxs h g G).
+/3 width=5 by s_r_trans_LTC2, lpx_cpxs_trans/ qed-.
 
-lemma lpxs_cpxs_trans: ∀h,g,G. s_rs_trans … (cpx h g G) (lpxs h g G).
-/3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ qed-.
+(* Note: alternative proof: /3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ *)
+lemma lpxs_cpxs_trans: ∀h,g,G. s_rs_transitive … (cpx h g G) (λ_.lpxs h g G).
+#h #g #G @s_r_to_s_rs_trans @s_r_trans_LTC2
+@s_rs_trans_TC1 /2 width=3 by lpx_cpxs_trans/ (**) (* full auto too slow *)
+qed-.
 
 lemma cpxs_bind2: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
                   ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
@@ -89,9 +103,32 @@ lemma lpxs_pair2: ∀h,g,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 →
 
 (* Properties on supclosure *************************************************)
 
-lemma lpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+lemma lpx_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
+  /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
+| #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+  #L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fqu_trans … H2 … HL0) -L
+  #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
+  /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma lpx_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
+                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 [ /2 width=5 by ex3_2_intro/ ]
+#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+#L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fquq_trans … H2 … HL0) -L
+#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
+/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
+qed-.
+
+lemma lpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
                        ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
-                       â\88\83â\88\83K2,T. â¦\83G1, K1â¦\84 â\8a¢ T1 â\9e¡*[h, g] T & â¦\83G1, K1, Tâ¦\84 â\8a\83⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
+                       â\88\83â\88\83K2,T. â¦\83G1, K1â¦\84 â\8a¢ T1 â\9e¡*[h, g] T & â¦\83G1, K1, Tâ¦\84 â\8a\90⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
 [ /2 width=5 by ex3_2_intro/
 | #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
@@ -101,22 +138,24 @@ lemma lpxs_fquq_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2,
 ]
 qed-.
 
-lemma lpxs_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+lemma lpxs_fqup_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+                       ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
+                       ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
+[ /2 width=5 by ex3_2_intro/
+| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
+  lapply (lpx_cpxs_trans … HT1 … HK1) -HT1
+  elim (lpx_fqup_trans … HT2 … HK1) -K
+  /3 width=7 by lpxs_strap2, cpxs_trans, ex3_2_intro/
+]
+qed-.
+
+lemma lpxs_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
                        ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, g] L1 →
-                       â\88\83â\88\83K2,T. â¦\83G1, K1â¦\84 â\8a¢ T1 â\9e¡*[h, g] T & â¦\83G1, K1, Tâ¦\84 â\8a\83* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
+                       â\88\83â\88\83K2,T. â¦\83G1, K1â¦\84 â\8a¢ T1 â\9e¡*[h, g] T & â¦\83G1, K1, Tâ¦\84 â\8a\90* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, g] L2.
 #h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
 #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
 #L0 #T0 #HT10 #HT0 #HL0 elim (lpxs_fquq_trans … H2 … HL0) -L
 #L #T3 #HT3 #HT32 #HL2 elim (fqus_cpxs_trans … HT3 … HT0) -T
 /3 width=7 by cpxs_trans, fqus_strap1, ex3_2_intro/
 qed-.
-
-lemma lpx_fqus_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
-                      ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, g] L1 →
-                      ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, g] T & ⦃G1, K1, T⦄ ⊃* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, g] L2.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
-#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
-#L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fquq_trans … H2 … HL0) -L
-#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
-/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
-qed-.