1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "static_2/syntax/lveq_lveq.ma".
16 include "static_2/static/fsle_fqup.ma".
18 (* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
20 (* Advanced inversion lemmas ************************************************)
22 lemma fsle_frees_trans:
23 ∀L1,L2,T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
24 ∀f2. L2 ⊢ 𝐅+⦃T2⦄ ≘ f2 →
25 ∃∃n1,n2,f1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 & L1 ≋ⓧ*[n1,n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
26 #L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
27 lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
28 lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
29 lapply (sle_eq_repl_back2 … Hn … Hgf2) -g2
30 /2 width=6 by ex3_3_intro/
33 lemma fsle_frees_trans_eq:
35 ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ → ∀f2. L2 ⊢ 𝐅+⦃T2⦄ ≘ f2 →
36 ∃∃f1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 & f1 ⊆ f2.
37 #L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
38 elim (fsle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
39 elim (lveq_inj_length … H2L) // -L2 #H1 #H2 destruct
40 /2 width=3 by ex2_intro/
43 lemma fsle_inv_frees_eq:
45 ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
46 ∀f1. L1 ⊢ 𝐅+⦃T1⦄ ≘ f1 → ∀f2. L2 ⊢ 𝐅+⦃T2⦄ ≘ f2 →
48 #L1 #L2 #H1L #T1 #T2 #H2L #f1 #Hf1 #f2 #Hf2
49 elim (fsle_frees_trans_eq … H2L … Hf2) // -L2 -T2
50 /3 width=6 by frees_mono, sle_eq_repl_back1/
53 (* Main properties **********************************************************)
55 theorem fsle_trans_sn:
56 ∀L1,L2,T1,T. ⦃L1,T1⦄ ⊆ ⦃L2,T⦄ →
57 ∀T2. ⦃L2,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
59 * #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
61 * #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
62 lapply (frees_mono … Hf0 … Hg0) -Hf0 -Hg0 #Hfg0
63 elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
64 lapply (sle_eq_repl_back1 … Hf … Hfg0) -f0
65 /4 width=10 by sle_tls, sle_trans, ex4_4_intro/
68 theorem fsle_trans_dx:
69 ∀L1,T1,T. ⦃L1,T1⦄ ⊆ ⦃L1,T⦄ →
70 ∀L2,T2. ⦃L1,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
72 * #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
74 * #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
75 lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
76 elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
77 lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
78 /4 width=10 by sle_tls, sle_trans, ex4_4_intro/
81 theorem fsle_trans_rc:
82 ∀L1,L,T1,T. |L1| = |L| → ⦃L1,T1⦄ ⊆ ⦃L,T⦄ →
83 ∀L2,T2. |L| = |L2| → ⦃L,T⦄ ⊆ ⦃L2,T2⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄.
85 * #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
87 * #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
88 lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
89 elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
90 elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
91 lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
92 /3 width=10 by lveq_length_eq, sle_trans, ex4_4_intro/
95 theorem fsle_bind_sn_ge:
97 ∀V1,T1,T. ⦃L1,V1⦄ ⊆ ⦃L2,T⦄ → ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2,T⦄ →
98 ∀p,I. ⦃L1,ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2,T⦄.
99 #L1 #L2 #HL #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
100 elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
101 elim (lveq_inj_void_sn_ge … H1n1 … H1n2) -H1n2 // #H1 #H2 #H3 destruct
102 elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
103 <tls_xn in H2n2; #H2n2
104 /4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
107 theorem fsle_flat_sn:
108 ∀L1,L2,V1,T1,T. ⦃L1,V1⦄ ⊆ ⦃L2,T⦄ → ⦃L1,T1⦄ ⊆ ⦃L2,T⦄ →
109 ∀I. ⦃L1,ⓕ{I}V1.T1⦄ ⊆ ⦃L2,T⦄.
110 #L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #I
111 elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
112 elim (lveq_inj … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
113 elim (sor_isfin_ex f1 f2) /2 width=3 by frees_fwd_isfin/ #f #Hf #_
114 /4 width=12 by frees_flat, sor_inv_sle, sor_tls, ex4_4_intro/
117 theorem fsle_bind_eq:
118 ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
119 ∀I2,T1,T2. ⦃L1.ⓧ,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
120 ∀p,I1. ⦃L1,ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.T2⦄.
122 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I2 #T1 #T2
123 * #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p #I1
124 elim (lveq_inj_length … H1L) // #H1 #H2 destruct
125 elim (lveq_inj_length … H2L) // -HL -H2L #H1 #H2 destruct
126 elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
127 elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
128 /4 width=15 by frees_bind_void, frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
132 ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
133 ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1,T1⦄ ⊆ ⦃L2.ⓑ{I2}V2,T2⦄ →
134 ∀p. ⦃L1,ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2,ⓑ{p,I2}V2.T2⦄.
136 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I1 #I2 #T1 #T2
137 * #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
138 elim (lveq_inv_pair_pair … H2L) -H2L #H2L #H1 #H2 destruct
139 elim (lveq_inj … H2L … H1L) -H1L #H1 #H2 destruct
140 elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
141 elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
142 /4 width=15 by frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
146 ∀L1,L2,V1,V2. ⦃L1,V1⦄ ⊆ ⦃L2,V2⦄ →
147 ∀T1,T2. ⦃L1,T1⦄ ⊆ ⦃L2,T2⦄ →
148 ∀I1,I2. ⦃L1,ⓕ{I1}V1.T1⦄ ⊆ ⦃L2,ⓕ{I2}V2.T2⦄.
149 /3 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn/ qed-.