update in ground_2 and basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / static / fsle_fsle.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/fsle_fsle.ma b/matita/matita/contribs/lambdadelta/basic_2/static/fsle_fsle.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/syntax/lveq_lveq.ma".
+include "basic_2/static/fsle_fqup.ma".
+
+(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fsle_frees_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+                        ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
+                        ∃∃n1,n2,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 &
+                                    L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[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
+/2 width=6 by ex3_3_intro/
+qed-.
+
+lemma fsle_frees_trans_eq: ∀L1,L2. |L1| = |L2| →
+                           ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 →
+                           ∃∃f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ 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
+/2 width=3 by ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem fsle_trans_sn: ∀L1,L2,T1,T. ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
+                       ∀T2. ⦃L2, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+#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/
+qed-.
+
+theorem fsle_trans_dx: ∀L1,T1,T. ⦃L1, T1⦄ ⊆ ⦃L1, T⦄ →
+                       ∀L2,T2. ⦃L1, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+#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/
+qed-.
+
+theorem fsle_bind_sn_ge: ∀L1,L2. |L2| ≤ |L1| →
+                         ∀V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, T⦄ →
+                         ∀p,I. ⦃L1, ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+#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/
+qed.
+
+theorem fsle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
+                      ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+#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/
+qed.
+
+theorem fsle_bind_eq: ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+                      ∀I2,T1,T2. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
+                      ∀p,I1. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+#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/
+qed.
+
+theorem fsle_bind: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+                   ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
+                   ∀p. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+#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/
+qed.
+
+theorem fsle_flat: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+                   ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+                   ∀I1,I2. ⦃L1, ⓕ{I1}V1.T1⦄ ⊆ ⦃L2, ⓕ{I2}V2.T2⦄.
+/3 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn/ qed-.