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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 "basic_2/rt_computation/csx_aaa.ma".
16 include "basic_2/rt_computation/fpbs_aaa.ma".
17 include "basic_2/rt_computation/fpbs_fpb.ma".
18 include "basic_2/rt_computation/fsb_csx.ma".
20 (* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
22 (* Main properties with atomic arity assignment for terms *******************)
24 (* Note: this is the "big tree" theorem *)
25 theorem aaa_fsb: ∀h,o,G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ≥[h, o] 𝐒⦃G, L, T⦄.
26 /3 width=2 by aaa_csx, csx_fsb/ qed.
28 (* Advanced eliminators with atomic arity assignment for terms **************)
30 fact aaa_ind_fpb_aux: ∀h,o. ∀Q:relation3 ….
31 (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
32 (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
35 ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
36 #h #o #R #IH #G #L #T #H @(csx_ind_fpb … H) -G -L -T
37 #G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
38 #G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h o … G2 … L2 … T2 … HTA1) -A1
39 /2 width=2 by fpb_fpbs/
42 lemma aaa_ind_fpb: ∀h,o. ∀Q:relation3 ….
43 (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
44 (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
47 ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
48 /4 width=4 by aaa_ind_fpb_aux, aaa_csx/ qed-.
50 fact aaa_ind_fpbg_aux: ∀h,o. ∀Q:relation3 ….
51 (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
52 (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
55 ∀G,L,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ∀A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
56 #h #o #Q #IH #G #L #T #H @(csx_ind_fpbg … H) -G -L -T
57 #G1 #L1 #T1 #H1 #IH1 #A1 #HTA1 @IH -IH //
58 #G2 #L2 #T2 #H12 elim (fpbs_aaa_conf h o … G2 … L2 … T2 … HTA1) -A1
59 /2 width=2 by fpbg_fwd_fpbs/
62 lemma aaa_ind_fpbg: ∀h,o. ∀Q:relation3 ….
63 (∀G1,L1,T1,A. ⦃G1, L1⦄ ⊢ T1 ⁝ A →
64 (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
67 ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → Q G L T.
68 /4 width=4 by aaa_ind_fpbg_aux, aaa_csx/ qed-.