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 "ground_2/lib/star.ma".
16 include "basic_2/notation/relations/suptermplus_6.ma".
17 include "basic_2/notation/relations/suptermplus_7.ma".
18 include "basic_2/s_transition/fqu.ma".
20 (* PLUS-ITERATED SUPCLOSURE *************************************************)
22 definition fqup: bool → tri_relation genv lenv term ≝
25 interpretation "extended plus-iterated structural successor (closure)"
26 'SupTermPlus b G1 L1 T1 G2 L2 T2 = (fqup b G1 L1 T1 G2 L2 T2).
28 interpretation "plus-iterated structural successor (closure)"
29 'SupTermPlus G1 L1 T1 G2 L2 T2 = (fqup true G1 L1 T1 G2 L2 T2).
31 (* Basic properties *********************************************************)
33 lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
34 ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
35 /2 width=1 by tri_inj/ qed.
37 lemma fqup_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
38 ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
39 ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
40 /2 width=5 by tri_step/ qed.
42 lemma fqup_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
43 ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
44 ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
45 /2 width=5 by tri_TC_strap/ qed.
47 lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+[b] ⦃G, L, V⦄.
48 /2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
50 lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G, L.ⓑ{I}V, T⦄.
51 /2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
53 lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G, L.ⓧ, T⦄.
54 /3 width=1 by fqu_clear, fqu_fqup/ qed.
56 lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+[b] ⦃G, L, T⦄.
57 /2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
59 lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G, L, V2⦄.
60 /2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
62 lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I1}V1, T⦄.
63 /2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
65 lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I2}V2, T⦄.
66 /2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
68 (* Basic eliminators ********************************************************)
70 lemma fqup_ind: ∀b,G1,L1,T1. ∀R:relation3 ….
71 (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
72 (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
73 ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → R G2 L2 T2.
74 #b #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
75 @(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
78 lemma fqup_ind_dx: ∀b,G2,L2,T2. ∀R:relation3 ….
79 (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → R G1 L1 T1) →
80 (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
81 ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → R G1 L1 T1.
82 #b #G2 #L2 #T2 #R #IH1 #IH2 #G1 #L1 #T1 #H
83 @(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
86 (* Basic_2A1: removed theorems 1: fqup_drop *)