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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 *)
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15 include "static_2/s_computation/fqup_weight.ma".
16 include "static_2/static/lsubf_lsubr.ma".
18 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
20 (* Advanced properties ******************************************************)
22 (* Note: this replaces lemma 1400 concluding the "big tree" theorem *)
23 lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅*⦃T⦄ ≘ f.
24 #L #T @(fqup_wf_ind_eq (Ⓣ) … (⋆) L T) -L -T
25 #G0 #L0 #T0 #IH #G #L * *
26 [ /3 width=2 by frees_sort, ex_intro/
27 | cases L -L /3 width=2 by frees_atom, ex_intro/
29 [ cases I -I #I [2: #V ] #HG #HL #HT destruct
31 /3 width=2 by frees_pair, fqu_fqup, fqu_lref_O, ex_intro/
32 | -IH /3 width=2 by frees_unit, ex_intro/
34 | #i #HG #HL #HT destruct
35 elim (IH G L (#i)) -IH
36 /3 width=2 by frees_lref, fqu_fqup, fqu_drop, ex_intro/
38 | /3 width=2 by frees_gref, ex_intro/
39 | #p #I #V #T #HG #HL #HT destruct
40 elim (IH G L V) // #f1 #HV
41 elim (IH G (L.ⓑ{I}V) T) -IH // #f2 #HT
42 elim (sor_isfin_ex f1 (⫱f2))
43 /3 width=6 by frees_fwd_isfin, frees_bind, isfin_tl, ex_intro/
44 | #I #V #T #HG #HL #HT destruct
45 elim (IH G L V) // #f1 #HV
46 elim (IH G L T) -IH // #f2 #HT
47 elim (sor_isfin_ex f1 f2)
48 /3 width=6 by frees_fwd_isfin, frees_flat, ex_intro/
52 (* Advanced main properties *************************************************)
54 theorem frees_bind_void: ∀f1,L,V. L ⊢ 𝐅*⦃V⦄ ≘ f1 → ∀f2,T. L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 →
55 ∀f. f1 ⋓ ⫱f2 ≘ f → ∀p,I. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f.
56 #f1 #L #V #Hf1 #f2 #T #Hf2 #f #Hf #p #I
57 elim (frees_total (L.ⓑ{I}V) T) #f0 #Hf0
58 lapply (lsubr_lsubf … Hf2 … Hf0) -Hf2 /2 width=5 by lsubr_unit/ #H02
59 elim (pn_split f2) * #g2 #H destruct
60 [ elim (lsubf_inv_push2 … H02) -H02 #g0 #Z #Y #H02 #H0 #H destruct
61 lapply (lsubf_inv_refl … H02) -H02 #H02
62 lapply (sor_eq_repl_fwd2 … Hf … H02) -g2 #Hf
63 /2 width=5 by frees_bind/
64 | elim (lsubf_inv_unit2 … H02) -H02 * [ #g0 #Y #_ #_ #H destruct ]
65 #z1 #g0 #z #Z #Y #X #H02 #Hz1 #Hz #H0 #H destruct
66 lapply (lsubf_inv_refl … H02) -H02 #H02
67 lapply (frees_mono … Hz1 … Hf1) -Hz1 #H1
68 lapply (sor_eq_repl_back1 … Hz … H02) -g0 #Hz
69 lapply (sor_eq_repl_back2 … Hz … H1) -z1 #Hz
70 lapply (sor_comm … Hz) -Hz #Hz
71 lapply (sor_mono … f Hz ?) // -Hz #H
72 lapply (sor_inv_sle_sn … Hf) -Hf #Hf
73 lapply (frees_eq_repl_back … Hf0 (↑f) ?) /2 width=5 by eq_next/ -z #Hf0
74 @(frees_bind … Hf1 Hf0) -Hf1 -Hf0 (**) (* constructor needed *)
75 /2 width=1 by sor_sle_dx/
79 (* Advanced inversion lemmas ************************************************)
81 lemma frees_inv_bind_void: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
82 ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
84 elim (frees_inv_bind … H) -H #f1 #f2 #Hf1 #Hf2 #Hf
85 elim (frees_total (L.ⓧ) T) #f0 #Hf0
86 lapply (lsubr_lsubf … Hf0 … Hf2) -Hf2 /2 width=5 by lsubr_unit/ #H20
87 elim (pn_split f0) * #g0 #H destruct
88 [ elim (lsubf_inv_push2 … H20) -H20 #g2 #I #Y #H20 #H2 #H destruct
89 lapply (lsubf_inv_refl … H20) -H20 #H20
90 lapply (sor_eq_repl_back2 … Hf … H20) -g2 #Hf
91 /2 width=5 by ex3_2_intro/
92 | elim (lsubf_inv_unit2 … H20) -H20 * [ #g2 #Y #_ #_ #H destruct ]
93 #z1 #z0 #g2 #Z #Y #X #H20 #Hz1 #Hg2 #H2 #H destruct
94 lapply (lsubf_inv_refl … H20) -H20 #H0
95 lapply (frees_mono … Hz1 … Hf1) -Hz1 #H1
96 lapply (sor_eq_repl_back1 … Hg2 … H0) -z0 #Hg2
97 lapply (sor_eq_repl_back2 … Hg2 … H1) -z1 #Hg2
98 @(ex3_2_intro … Hf1 Hf0) -Hf1 -Hf0 (**) (* constructor needed *)
99 /2 width=3 by sor_comm_23_idem/
103 lemma frees_ind_void: ∀Q:relation3 ….
105 ∀f,L,s. 𝐈⦃f⦄ → Q L (⋆s) f
107 ∀f,i. 𝐈⦃f⦄ → Q (⋆) (#i) (⫯*[i]↑f)
110 L ⊢ 𝐅*⦃V⦄ ≘ f → Q L V f→ Q (L.ⓑ{I}V) (#O) (↑f)
112 ∀f,I,L. 𝐈⦃f⦄ → Q (L.ⓤ{I}) (#O) (↑f)
115 L ⊢ 𝐅*⦃#i⦄ ≘ f → Q L (#i) f → Q (L.ⓘ{I}) (#(↑i)) (⫯f)
117 ∀f,L,l. 𝐈⦃f⦄ → Q L (§l) f
120 L ⊢ 𝐅*⦃V⦄ ≘ f1 → L.ⓧ ⊢𝐅*⦃T⦄≘ f2 → f1 ⋓ ⫱f2 ≘ f →
121 Q L V f1 → Q (L.ⓧ) T f2 → Q L (ⓑ{p,I}V.T) f
124 L ⊢ 𝐅*⦃V⦄ ≘ f1 → L ⊢𝐅*⦃T⦄ ≘ f2 → f1 ⋓ f2 ≘ f →
125 Q L V f1 → Q L T f2 → Q L (ⓕ{I}V.T) f
127 ∀L,T,f. L ⊢ 𝐅*⦃T⦄ ≘ f → Q L T f.
128 #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #L #T
129 @(fqup_wf_ind_eq (Ⓕ) … (⋆) L T) -L -T #G0 #L0 #T0 #IH #G #L * *
130 [ #s #HG #HL #HT #f #H destruct -IH
131 lapply (frees_inv_sort … H) -H /2 width=1 by/
133 [ #i #HG #HL #HT #f #H destruct -IH
134 elim (frees_inv_atom … H) -H #g #Hg #H destruct /2 width=1 by/
135 | #L #I * [ cases I -I #I [ | #V ] | #i ] #HG #HL #HT #f #H destruct
136 [ elim (frees_inv_unit … H) -H #g #Hg #H destruct /2 width=1 by/
137 | elim (frees_inv_pair … H) -H #g #Hg #H destruct
138 /4 width=2 by fqu_fqup, fqu_lref_O/
139 | elim (frees_inv_lref … H) -H #g #Hg #H destruct
140 /4 width=2 by fqu_fqup/
143 | #l #HG #HL #HT #f #H destruct -IH
144 lapply (frees_inv_gref … H) -H /2 width=1 by/
145 | #p #I #V #T #HG #HL #HT #f #H destruct
146 elim (frees_inv_bind_void … H) -H /3 width=7 by/
147 | #I #V #T #HG #HL #HT #f #H destruct
148 elim (frees_inv_flat … H) -H /3 width=7 by/