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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 *)
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15 include "basic_2/notation/relations/suptermstar_6.ma".
16 include "basic_2/s_transition/fquq.ma".
18 (* STAR-ITERATED SUPCLOSURE *************************************************)
20 definition fqus: tri_relation genv lenv term ≝ tri_TC … fquq.
22 interpretation "star-iterated structural successor (closure)"
23 'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus G1 L1 T1 G2 L2 T2).
25 (* Basic eliminators ********************************************************)
27 lemma fqus_ind: ∀G1,L1,T1. ∀R:relation3 …. R G1 L1 T1 →
28 (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
29 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G2 L2 T2.
30 #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
31 @(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
34 lemma fqus_ind_dx: ∀G2,L2,T2. ∀R:relation3 …. R G2 L2 T2 →
35 (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
36 ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ → R G1 L1 T1.
37 #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
38 @(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
41 (* Basic properties *********************************************************)
43 lemma fqus_refl: tri_reflexive … fqus.
44 /2 width=1 by tri_inj/ qed.
46 lemma fquq_fqus: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
47 /2 width=1 by tri_inj/ qed.
49 lemma fqus_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
50 ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
51 /2 width=5 by tri_step/ qed-.
53 lemma fqus_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄ →
54 ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄.
55 /2 width=5 by tri_TC_strap/ qed-.
57 (* Basic inversion lemmas ***************************************************)
59 lemma fqus_inv_fqu_sn: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
60 (∧∧ G1 = G2 & L1 = L2 & T1 = T2) ∨
61 ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐ ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐* ⦃G2, L2, T2⦄.
62 #G1 #G2 #L1 #L2 #T1 #T2 #H12 @(fqus_ind_dx … H12) -G1 -L1 -T1 /3 width=1 by and3_intro, or_introl/
63 #G1 #G #L1 #L #T1 #T * /3 width=5 by ex2_3_intro, or_intror/
64 * #HG #HL #HT #_ destruct //
67 lemma fqus_inv_sort1: ∀G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄ →
68 (∧∧ G1 = G2 & L1 = L2 & ⋆s = T2) ∨
69 ∃∃J,L,V. ⦃G1, L, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V.
70 #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
71 #G #L #T #H elim (fqu_inv_sort1 … H) -H /3 width=5 by ex2_3_intro, or_intror/
74 lemma fqus_inv_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐* ⦃G2, L2, T2⦄ →
75 ∨∨ ∧∧ G1 = G2 & L1 = L2 & #i = T2
76 | ∃∃J,L,V. ⦃G1, L, V⦄ ⊐* ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = 0
77 | ∃∃J,L,V,j. ⦃G1, L, #j⦄ ⊐* ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = ⫯j.
78 #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
79 #G #L #T #H elim (fqu_inv_lref1 … H) -H * /3 width=7 by or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro/
82 lemma fqus_inv_gref1: ∀G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐* ⦃G2, L2, T2⦄ →
83 (∧∧ G1 = G2 & L1 = L2 & §l = T2) ∨
84 ∃∃J,L,V. ⦃G1, L, §l⦄ ⊐* ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V.
85 #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
86 #G #L #T #H elim (fqu_inv_gref1 … H) -H /3 width=5 by ex2_3_intro, or_intror/
89 lemma fqus_inv_bind1: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
90 ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
91 | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
92 | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄
93 | ∃∃J,L,V,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≡ ⓑ{p,I}V1.T1 & L1 = L.ⓑ{J}V.
94 #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/
95 #G #L #T #H elim (fqu_inv_bind1 … H) -H *
96 [3: #J #V ] #H1 #H2 #H3 #H destruct
97 /3 width=7 by or4_intro1, or4_intro2, or4_intro3, ex3_4_intro/
100 lemma fqus_inv_flat1: ∀I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
101 ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2
102 | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
103 | ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄
104 | ∃∃J,L,V,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≡ ⓕ{I}V1.T1 & L1 = L.ⓑ{J}V.
105 #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/
106 #G #L #T #H elim (fqu_inv_flat1 … H) -H *
107 [3: #J #V ] #H1 #H2 #H3 #H destruct
108 /3 width=7 by or4_intro1, or4_intro2, or4_intro3, ex3_4_intro/
111 (* Advanced inversion lemmas ************************************************)
113 lemma fqus_inv_atom1: ∀I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐* ⦃G2, L2, T2⦄ →
114 ∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2.
115 #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/
116 #G #L #T #H elim (fqu_inv_atom1 … H)
119 lemma fqus_inv_sort1_pair: ∀I,G1,G2,L1,L2,V1,T2,s. ⦃G1, L1.ⓑ{I}V1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄ →
120 (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐* ⦃G2, L2, T2⦄.
121 #I #G1 #G2 #L1 #L2 #V #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
122 #G #L #T #H elim (fqu_inv_sort1_pair … H) -H
123 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
126 lemma fqus_inv_zero1_pair: ∀I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐* ⦃G2, L2, T2⦄ →
127 (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄.
128 #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
129 #G #L #T #H elim (fqu_inv_zero1_pair … H) -H
130 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
133 lemma fqus_inv_lref1_pair: ∀I,G1,G2,L1,L2,V1,T2,i. ⦃G1, L1.ⓑ{I}V1, #⫯i⦄ ⊐* ⦃G2, L2, T2⦄ →
134 (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #(⫯i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐* ⦃G2, L2, T2⦄.
135 #I #G1 #G2 #L1 #L2 #V #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
136 #G #L #T #H elim (fqu_inv_lref1_pair … H) -H
137 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
140 lemma fqus_inv_gref1_pair: ∀I,G1,G2,L1,L2,V1,T2,l. ⦃G1, L1.ⓑ{I}V1, §l⦄ ⊐* ⦃G2, L2, T2⦄ →
141 (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐* ⦃G2, L2, T2⦄.
142 #I #G1 #G2 #L1 #L2 #V #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
143 #G #L #T #H elim (fqu_inv_gref1_pair … H) -H
144 #H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
147 (* Basic_2A1: removed theorems 1: fqus_drop *)