+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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-.
-
-lemma fsle_inv_frees_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 #f2 #Hf2
-elim (fsle_frees_trans_eq … H2L … Hf2) // -L2 -T2
-/3 width=6 by frees_mono, sle_eq_repl_back1/
-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_trans_rc: ∀L1,L,T1,T. |L1| = |L| → ⦃L1, T1⦄ ⊆ ⦃L, T⦄ →
- ∀L2,T2. |L| = |L2| → ⦃L, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
-#L1 #L #T1 #T #HL1
-* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
-#L2 #T2 #HL2
-* #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
-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/
-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-.
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