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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/notation/relations/suptermplus_6.ma".
16 include "basic_2/relocation/fqu.ma".
18 (* PLUS-ITERATED SUPCLOSURE *************************************************)
20 definition fqup: tri_relation genv lenv term ≝ tri_TC … fqu.
22 interpretation "plus-iterated structural successor (closure)"
23 'SupTermPlus G1 L1 T1 G2 L2 T2 = (fqup G1 L1 T1 G2 L2 T2).
25 (* Basic properties *********************************************************)
27 lemma fqu_fqup: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄.
28 /2 width=1 by tri_inj/ qed.
30 lemma fqup_strap1: ∀G1,G,G2,L1,L,L2,T1,T,T2.
31 ⦃G1, L1, T1⦄ ⊃+ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃ ⦃G2, L2, T2⦄ →
32 ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄.
33 /2 width=5 by tri_step/ qed.
35 lemma fqup_strap2: ∀G1,G,G2,L1,L,L2,T1,T,T2.
36 ⦃G1, L1, T1⦄ ⊃ ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊃+ ⦃G2, L2, T2⦄ →
37 ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄.
38 /2 width=5 by tri_TC_strap/ qed.
40 lemma fqup_ldrop: ∀G1,G2,L1,K1,K2,T1,T2,U1,e. ⇩[e] L1 ≡ K1 → ⇧[0, e] T1 ≡ U1 →
41 ⦃G1, K1, T1⦄ ⊃+ ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊃+ ⦃G2, K2, T2⦄.
42 #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #e #HLK1 #HTU1 #HT12 elim (eq_or_gt … e) #H destruct
43 [ >(ldrop_inv_O2 … HLK1) -L1 <(lift_inv_O2 … HTU1) -U1 //
44 | /3 width=5 by fqup_strap2, fqu_drop_lt/
48 lemma fqup_lref: ∀I,G,L,K,V,i. ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊃+ ⦃G, K, V⦄.
49 /3 width=6 by fqu_lref_O, fqu_fqup, lift_lref_ge, fqup_ldrop/ qed.
51 lemma fqup_pair_sn: ∀I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊃+ ⦃G, L, V⦄.
52 /2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
54 lemma fqup_bind_dx: ∀a,I,G,L,V,T. ⦃G, L, ⓑ{a,I}V.T⦄ ⊃+ ⦃G, L.ⓑ{I}V, T⦄.
55 /2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
57 lemma fqup_flat_dx: ∀I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊃+ ⦃G, L, T⦄.
58 /2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
60 lemma fqup_flat_dx_pair_sn: ∀I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊃+ ⦃G, L, V2⦄.
61 /2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
63 lemma fqup_bind_dx_flat_dx: ∀a,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{a,I1}V1.ⓕ{I2}V2.T⦄ ⊃+ ⦃G, L.ⓑ{I1}V1, T⦄.
64 /2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
66 lemma fqup_flat_dx_bind_dx: ∀a,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{a,I2}V2.T⦄ ⊃+ ⦃G, L.ⓑ{I2}V2, T⦄.
67 /2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
69 (* Basic eliminators ********************************************************)
71 lemma fqup_ind: ∀G1,L1,T1. ∀R:relation3 ….
72 (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
73 (∀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) →
74 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → R G2 L2 T2.
75 #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
76 @(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
79 lemma fqup_ind_dx: ∀G2,L2,T2. ∀R:relation3 ….
80 (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → R G1 L1 T1) →
81 (∀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) →
82 ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → R G1 L1 T1.
83 #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
84 @(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
87 (* Basic forward lemmas *****************************************************)
89 lemma fqup_fwd_fw: ∀G1,G2,L1,L2,T1,T2.
90 ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
91 #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
92 /3 width=3 by fqu_fwd_fw, transitive_lt/
95 (* Advanced eliminators *****************************************************)
97 lemma fqup_wf_ind: ∀R:relation3 …. (
98 ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
100 ) → ∀G1,L1,T1. R G1 L1 T1.
101 #R #HR @(f3_ind … fw) #n #IHn #G1 #L1 #T1 #H destruct /4 width=1 by fqup_fwd_fw/
104 lemma fqup_wf_ind_eq: ∀R:relation3 …. (
105 ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
106 ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → R G2 L2 T2
107 ) → ∀G1,L1,T1. R G1 L1 T1.
108 #R #HR @(f3_ind … fw) #n #IHn #G1 #L1 #T1 #H destruct /4 width=7 by fqup_fwd_fw/