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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/notation/relations/pconvstar_4.ma".
16 include "basic_2/conversion/cpc.ma".
18 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
20 definition cpcs: relation4 genv lenv term term ≝
23 interpretation "context-sensitive parallel equivalence (term)"
24 'PConvStar G L T1 T2 = (cpcs G L T1 T2).
26 (* Basic eliminators ********************************************************)
28 lemma cpcs_ind: ∀G,L,T1. ∀R:predicate term. R T1 →
29 (∀T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → R T → R T2) →
30 ∀T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 → R T2.
31 normalize /3 width=6 by TC_star_ind/
34 lemma cpcs_ind_dx: ∀G,L,T2. ∀R:predicate term. R T2 →
35 (∀T1,T. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → R T → R T1) →
36 ∀T1. ⦃G, L⦄ ⊢ T1 ⬌* T2 → R T1.
37 normalize /3 width=6 by TC_star_ind_dx/
40 (* Basic properties *********************************************************)
42 (* Basic_1: was: pc3_refl *)
43 lemma cpcs_refl: ∀G,L. reflexive … (cpcs G L).
44 /2 width=1 by inj/ qed.
46 (* Basic_1: was: pc3_s *)
47 lemma cpcs_sym: ∀G,L. symmetric … (cpcs G L).
48 normalize /3 width=1 by cpc_sym, TC_symmetric/
51 lemma cpc_cpcs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
52 /2 width=1 by inj/ qed.
54 lemma cpcs_strap1: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
55 normalize /2 width=3 by step/
58 lemma cpcs_strap2: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
59 normalize /2 width=3 by TC_strap/
62 (* Basic_1: was: pc3_pr2_r *)
63 lemma cpr_cpcs_dx: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
64 /3 width=1 by cpc_cpcs, or_introl/ qed.
66 (* Basic_1: was: pc3_pr2_x *)
67 lemma cpr_cpcs_sn: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡ T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
68 /3 width=1 by cpc_cpcs, or_intror/ qed.
70 lemma cpcs_cpr_strap1: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
71 /3 width=3 by cpcs_strap1, or_introl/ qed-.
73 (* Basic_1: was: pc3_pr2_u *)
74 lemma cpcs_cpr_strap2: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
75 /3 width=3 by cpcs_strap2, or_introl/ qed-.
77 lemma cpcs_cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
78 /3 width=3 by cpcs_strap1, or_intror/ qed-.
80 lemma cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
81 /3 width=3 by cpr_cpcs_dx, cpcs_strap1, or_intror/ qed-.
83 (* Basic_1: was: pc3_pr2_u2 *)
84 lemma cpcs_cpr_conf: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
85 /3 width=3 by cpcs_strap2, or_intror/ qed-.
87 (* Basic_1: removed theorems 9:
88 clear_pc3_trans pc3_ind_left
89 pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
90 pc3_pr2_fqubst0 pc3_pr2_fqubst0_back pc3_fqubst0
91 pc3_gen_abst pc3_gen_abst_shift
93 (* Basic_1: removed local theorems 6:
94 pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left