partial commit in static_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / static / fsle_fsle.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/fsle_fsle.ma b/matita/matita/contribs/lambdadelta/static_2/static/fsle_fsle.ma
index 588182317cdf7db2a2614bd0b2d76aac30b45061..7dafa423bfc2e97ccc46766c90ca3cffc06d9458 100644 (file)
--- a/matita/matita/contribs/lambdadelta/static_2/static/fsle_fsle.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/static/fsle_fsle.ma
@@ -19,21 +19,21 @@ include "static_2/static/fsle_fqup.ma".
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma fsle_frees_trans: 
-      â\88\80L1,L2,T1,T2. â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,T2â¦\84 →
-      â\88\80f2. L2 â\8a¢ ð\9d\90\85+â¦\83T2â¦\84 ≘ f2 →
-      â\88\83â\88\83n1,n2,f1. L1 â\8a¢ ð\9d\90\85+â¦\83T1â¦\84 â\89\98 f1 & L1 â\89\8bâ\93§*[n1,n2] L2 & ⫱*[n1]f1 â\8a\86 ⫱*[n2]f2.
+lemma fsle_frees_trans:
+      â\88\80L1,L2,T1,T2. â\9dªL1,T1â\9d« â\8a\86 â\9dªL2,T2â\9d« →
+      â\88\80f2. L2 â\8a¢ ð\9d\90\85+â\9dªT2â\9d« ≘ f2 →
+      â\88\83â\88\83n1,n2,f1. L1 â\8a¢ ð\9d\90\85+â\9dªT1â\9d« â\89\98 f1 & L1 â\89\8bâ\93§*[n1,n2] L2 & â«°*[n1]f1 â\8a\86 â«°*[n2]f2.
 #L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
 lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
-lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
-lapply (sle_eq_repl_back2 … Hn … Hgf2) -g2
+lapply (pr_tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
+lapply (pr_sle_eq_repl_back_dx … Hn … Hgf2) -g2
 /2 width=6 by ex3_3_intro/
 qed-.
 
 lemma fsle_frees_trans_eq:
       ∀L1,L2. |L1| = |L2| →
-      â\88\80T1,T2. â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,T2â¦\84 â\86\92 â\88\80f2. L2 â\8a¢ ð\9d\90\85+â¦\83T2â¦\84 ≘ f2 →
-      â\88\83â\88\83f1. L1 â\8a¢ ð\9d\90\85+â¦\83T1â¦\84 ≘ f1 & f1 ⊆ f2.
+      â\88\80T1,T2. â\9dªL1,T1â\9d« â\8a\86 â\9dªL2,T2â\9d« â\86\92 â\88\80f2. L2 â\8a¢ ð\9d\90\85+â\9dªT2â\9d« ≘ f2 →
+      â\88\83â\88\83f1. L1 â\8a¢ ð\9d\90\85+â\9dªT1â\9d« ≘ f1 & f1 ⊆ f2.
 #L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
 elim (fsle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
 elim (lveq_inj_length … H2L) // -L2 #H1 #H2 destruct
@@ -42,45 +42,66 @@ qed-.
 
 lemma fsle_inv_frees_eq:
       ∀L1,L2. |L1| = |L2| →
-      â\88\80T1,T2. â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,T2â¦\84 →
-      â\88\80f1. L1 â\8a¢ ð\9d\90\85+â¦\83T1â¦\84 â\89\98 f1 â\86\92 â\88\80f2. L2 â\8a¢ ð\9d\90\85+â¦\83T2â¦\84 ≘ f2 →
+      â\88\80T1,T2. â\9dªL1,T1â\9d« â\8a\86 â\9dªL2,T2â\9d« →
+      â\88\80f1. L1 â\8a¢ ð\9d\90\85+â\9dªT1â\9d« â\89\98 f1 â\86\92 â\88\80f2. L2 â\8a¢ ð\9d\90\85+â\9dªT2â\9d« ≘ f2 →
       f1 ⊆ f2.
 #L1 #L2 #H1L #T1 #T2 #H2L #f1 #Hf1 #f2 #Hf2
 elim (fsle_frees_trans_eq … H2L … Hf2) // -L2 -T2
-/3 width=6 by frees_mono, sle_eq_repl_back1/
+/3 width=6 by frees_mono, pr_sle_eq_repl_back_sn/
+qed-.
+
+lemma fsle_frees_conf:
+      ∀L1,L2,T1,T2. ❪L1,T1❫ ⊆ ❪L2,T2❫ →
+      ∀f1. L1 ⊢ 𝐅+❪T1❫ ≘ f1 →
+      ∃∃n1,n2,f2. L2 ⊢ 𝐅+❪T2❫ ≘ f2 & L1 ≋ⓧ*[n1,n2] L2 & ⫰*[n1]f1 ⊆ ⫰*[n2]f2.
+#L1 #L2 #T1 #T2 * #n1 #n2 #g1 #g2 #Hg1 #Hg2 #HL #Hn #f1 #Hf1
+lapply (frees_mono … Hg1 … Hf1) -Hg1 -Hf1 #Hgf1
+lapply (pr_tls_eq_repl n1 … Hgf1) -Hgf1 #Hgf1
+lapply (pr_sle_eq_repl_back_sn … Hn … Hgf1) -g1
+/2 width=6 by ex3_3_intro/
+qed-.
+
+lemma fsle_frees_conf_eq:
+      ∀L1,L2. |L1| = |L2| →
+      ∀T1,T2. ❪L1,T1❫ ⊆ ❪L2,T2❫ → ∀f1. L1 ⊢ 𝐅+❪T1❫ ≘ f1 →
+      ∃∃f2. L2 ⊢ 𝐅+❪T2❫ ≘ f2 & f1 ⊆ f2.
+#L1 #L2 #H1L #T1 #T2 #H2L #f1 #Hf1
+elim (fsle_frees_conf … H2L … Hf1) -T1 #n1 #n2 #f2 #Hf2 #H2L #Hf12
+elim (lveq_inj_length … H2L) // -L1 #H1 #H2 destruct
+/2 width=3 by ex2_intro/
 qed-.
 
 (* Main properties **********************************************************)
 
 theorem fsle_trans_sn:
-        â\88\80L1,L2,T1,T. â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,Tâ¦\84 →
-        â\88\80T2. â¦\83L2,Tâ¦\84 â\8a\86 â¦\83L2,T2â¦\84 â\86\92 â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,T2â¦\84.
+        â\88\80L1,L2,T1,T. â\9dªL1,T1â\9d« â\8a\86 â\9dªL2,Tâ\9d« →
+        â\88\80T2. â\9dªL2,Tâ\9d« â\8a\86 â\9dªL2,T2â\9d« â\86\92 â\9dªL1,T1â\9d« â\8a\86 â\9dªL2,T2â\9d«.
 #L1 #L2 #T1 #T
 * #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
 #T2
 * #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
 lapply (frees_mono … Hf0 … Hg0) -Hf0 -Hg0 #Hfg0
 elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
-lapply (sle_eq_repl_back1 … Hf … Hfg0) -f0
-/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
+lapply (pr_sle_eq_repl_back_sn … Hf … Hfg0) -f0
+/4 width=10 by pr_sle_tls, pr_sle_trans, ex4_4_intro/
 qed-.
 
 theorem fsle_trans_dx:
-        â\88\80L1,T1,T. â¦\83L1,T1â¦\84 â\8a\86 â¦\83L1,Tâ¦\84 →
-        â\88\80L2,T2. â¦\83L1,Tâ¦\84 â\8a\86 â¦\83L2,T2â¦\84 â\86\92 â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,T2â¦\84.
+        â\88\80L1,T1,T. â\9dªL1,T1â\9d« â\8a\86 â\9dªL1,Tâ\9d« →
+        â\88\80L2,T2. â\9dªL1,Tâ\9d« â\8a\86 â\9dªL2,T2â\9d« â\86\92 â\9dªL1,T1â\9d« â\8a\86 â\9dªL2,T2â\9d«.
 #L1 #T1 #T
 * #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
 #L2 #T2
 * #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
 lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
 elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
-lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
-/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
+lapply (pr_sle_eq_repl_back_dx … Hg … Hgf0) -g0
+/4 width=10 by pr_sle_tls, pr_sle_trans, ex4_4_intro/
 qed-.
 
 theorem fsle_trans_rc:
-        â\88\80L1,L,T1,T. |L1| = |L| â\86\92 â¦\83L1,T1â¦\84 â\8a\86 â¦\83L,Tâ¦\84 →
-        â\88\80L2,T2. |L| = |L2| â\86\92 â¦\83L,Tâ¦\84 â\8a\86 â¦\83L2,T2â¦\84 â\86\92 â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,T2â¦\84.
+        â\88\80L1,L,T1,T. |L1| = |L| â\86\92 â\9dªL1,T1â\9d« â\8a\86 â\9dªL,Tâ\9d« →
+        â\88\80L2,T2. |L| = |L2| â\86\92 â\9dªL,Tâ\9d« â\8a\86 â\9dªL2,T2â\9d« â\86\92 â\9dªL1,T1â\9d« â\8a\86 â\9dªL2,T2â\9d«.
 #L1 #L #T1 #T #HL1
 * #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
 #L2 #T2 #HL2
@@ -88,62 +109,62 @@ theorem fsle_trans_rc:
 lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
 elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
 elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
-lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
-/3 width=10 by lveq_length_eq, sle_trans, ex4_4_intro/
+lapply (pr_sle_eq_repl_back_dx … Hg … Hgf0) -g0
+/3 width=10 by lveq_length_eq, pr_sle_trans, ex4_4_intro/
 qed-.
 
 theorem fsle_bind_sn_ge:
         ∀L1,L2. |L2| ≤ |L1| →
-        â\88\80V1,T1,T. â¦\83L1,V1â¦\84 â\8a\86 â¦\83L2,Tâ¦\84 â\86\92 â¦\83L1.â\93§,T1â¦\84 â\8a\86 â¦\83L2,Tâ¦\84 →
-        â\88\80p,I. â¦\83L1,â\93\91{p,I}V1.T1â¦\84 â\8a\86 â¦\83L2,Tâ¦\84.
+        â\88\80V1,T1,T. â\9dªL1,V1â\9d« â\8a\86 â\9dªL2,Tâ\9d« â\86\92 â\9dªL1.â\93§,T1â\9d« â\8a\86 â\9dªL2,Tâ\9d« →
+        â\88\80p,I. â\9dªL1,â\93\91[p,I]V1.T1â\9d« â\8a\86 â\9dªL2,Tâ\9d«.
 #L1 #L2 #HL #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
 elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
 elim (lveq_inj_void_sn_ge … H1n1 … H1n2) -H1n2 // #H1 #H2 #H3 destruct
-elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
-<tls_xn in H2n2; #H2n2
-/4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
+elim (pr_sor_isf_bi f1 (⫰f2)) /3 width=3 by frees_fwd_isfin, pr_isf_tl/ #f #Hf #_
+<pr_tls_swap in H2n2; #H2n2
+/4 width=12 by frees_bind_void, pr_sor_inv_sle_bi, pr_sor_tls, ex4_4_intro/
 qed.
 
 theorem fsle_flat_sn:
-        â\88\80L1,L2,V1,T1,T. â¦\83L1,V1â¦\84 â\8a\86 â¦\83L2,Tâ¦\84 â\86\92 â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,Tâ¦\84 →
-        â\88\80I. â¦\83L1,â\93\95{I}V1.T1â¦\84 â\8a\86 â¦\83L2,Tâ¦\84.
+        â\88\80L1,L2,V1,T1,T. â\9dªL1,V1â\9d« â\8a\86 â\9dªL2,Tâ\9d« â\86\92 â\9dªL1,T1â\9d« â\8a\86 â\9dªL2,Tâ\9d« →
+        â\88\80I. â\9dªL1,â\93\95[I]V1.T1â\9d« â\8a\86 â\9dªL2,Tâ\9d«.
 #L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #I
 elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
 elim (lveq_inj … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
-elim (sor_isfin_ex f1 f2) /2 width=3 by frees_fwd_isfin/ #f #Hf #_
-/4 width=12 by frees_flat, sor_inv_sle, sor_tls, ex4_4_intro/
+elim (pr_sor_isf_bi f1 f2) /2 width=3 by frees_fwd_isfin/ #f #Hf #_
+/4 width=12 by frees_flat, pr_sor_inv_sle_bi, pr_sor_tls, ex4_4_intro/
 qed.
 
 theorem fsle_bind_eq:
-        â\88\80L1,L2. |L1| = |L2| â\86\92 â\88\80V1,V2. â¦\83L1,V1â¦\84 â\8a\86 â¦\83L2,V2â¦\84 →
-        â\88\80I2,T1,T2. â¦\83L1.â\93§,T1â¦\84 â\8a\86 â¦\83L2.â\93\91{I2}V2,T2â¦\84 →
-        â\88\80p,I1. â¦\83L1,â\93\91{p,I1}V1.T1â¦\84 â\8a\86 â¦\83L2,â\93\91{p,I2}V2.T2â¦\84.
+        â\88\80L1,L2. |L1| = |L2| â\86\92 â\88\80V1,V2. â\9dªL1,V1â\9d« â\8a\86 â\9dªL2,V2â\9d« →
+        â\88\80I2,T1,T2. â\9dªL1.â\93§,T1â\9d« â\8a\86 â\9dªL2.â\93\91[I2]V2,T2â\9d« →
+        â\88\80p,I1. â\9dªL1,â\93\91[p,I1]V1.T1â\9d« â\8a\86 â\9dªL2,â\93\91[p,I2]V2.T2â\9d«.
 #L1 #L2 #HL #V1 #V2
 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I2 #T1 #T2
 * #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p #I1
 elim (lveq_inj_length … H1L) // #H1 #H2 destruct
 elim (lveq_inj_length … H2L) // -HL -H2L #H1 #H2 destruct
-elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-/4 width=15 by frees_bind_void, frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
+elim (pr_sor_isf_bi f1 (⫰f2)) /3 width=3 by frees_fwd_isfin, pr_isf_tl/ #f #Hf #_
+elim (pr_sor_isf_bi g1 (⫰g2)) /3 width=3 by frees_fwd_isfin, pr_isf_tl/ #g #Hg #_
+/4 width=15 by frees_bind_void, frees_bind, pr_sor_monotonic_sle, pr_sle_tl, ex4_4_intro/
 qed.
 
 theorem fsle_bind:
-        â\88\80L1,L2,V1,V2. â¦\83L1,V1â¦\84 â\8a\86 â¦\83L2,V2â¦\84 →
-        â\88\80I1,I2,T1,T2. â¦\83L1.â\93\91{I1}V1,T1â¦\84 â\8a\86 â¦\83L2.â\93\91{I2}V2,T2â¦\84 →
-        â\88\80p. â¦\83L1,â\93\91{p,I1}V1.T1â¦\84 â\8a\86 â¦\83L2,â\93\91{p,I2}V2.T2â¦\84.
+        â\88\80L1,L2,V1,V2. â\9dªL1,V1â\9d« â\8a\86 â\9dªL2,V2â\9d« →
+        â\88\80I1,I2,T1,T2. â\9dªL1.â\93\91[I1]V1,T1â\9d« â\8a\86 â\9dªL2.â\93\91[I2]V2,T2â\9d« →
+        â\88\80p. â\9dªL1,â\93\91[p,I1]V1.T1â\9d« â\8a\86 â\9dªL2,â\93\91[p,I2]V2.T2â\9d«.
 #L1 #L2 #V1 #V2
 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I1 #I2 #T1 #T2
 * #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
 elim (lveq_inv_pair_pair … H2L) -H2L #H2L #H1 #H2 destruct
 elim (lveq_inj … H2L … H1L) -H1L #H1 #H2 destruct
-elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-/4 width=15 by frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
+elim (pr_sor_isf_bi f1 (⫰f2)) /3 width=3 by frees_fwd_isfin, pr_isf_tl/ #f #Hf #_
+elim (pr_sor_isf_bi g1 (⫰g2)) /3 width=3 by frees_fwd_isfin, pr_isf_tl/ #g #Hg #_
+/4 width=15 by frees_bind, pr_sor_monotonic_sle, pr_sle_tl, ex4_4_intro/
 qed.
 
 theorem fsle_flat:
-        â\88\80L1,L2,V1,V2. â¦\83L1,V1â¦\84 â\8a\86 â¦\83L2,V2â¦\84 →
-        â\88\80T1,T2. â¦\83L1,T1â¦\84 â\8a\86 â¦\83L2,T2â¦\84 →
-        â\88\80I1,I2. â¦\83L1,â\93\95{I1}V1.T1â¦\84 â\8a\86 â¦\83L2,â\93\95{I2}V2.T2â¦\84.
+        â\88\80L1,L2,V1,V2. â\9dªL1,V1â\9d« â\8a\86 â\9dªL2,V2â\9d« →
+        â\88\80T1,T2. â\9dªL1,T1â\9d« â\8a\86 â\9dªL2,T2â\9d« →
+        â\88\80I1,I2. â\9dªL1,â\93\95[I1]V1.T1â\9d« â\8a\86 â\9dªL2,â\93\95[I2]V2.T2â\9d«.
 /3 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn/ qed-.