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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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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 #G #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
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 #G #L #T2 #R #HT2 #IHT2 #T1 #HT12
38 @(TC_star_ind_dx … HT2 IHT2 … HT12) //
41 (* Basic properties *********************************************************)
43 (* Basic_1: was: pc3_refl *)
44 lemma cpcs_refl: ∀G,L. reflexive … (cpcs G L).
47 (* Basic_1: was: pc3_s *)
48 lemma cpcs_sym: ∀G,L. symmetric … (cpcs G L).
49 #G #L @TC_symmetric // qed.
51 lemma cpc_cpcs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T2.
54 lemma cpcs_strap1: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T ⬌ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
57 lemma cpcs_strap2: ∀G,L,T1,T,T2. ⦃G, L⦄ ⊢ T1 ⬌ T → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
60 (* Basic_1: was: pc3_pr2_r *)
61 lemma cpr_cpcs_dx: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
64 (* Basic_1: was: pc3_pr2_x *)
65 lemma cpr_cpcs_sn: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T2 ➡ T1 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
68 lemma cpcs_cpr_strap1: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
71 (* Basic_1: was: pc3_pr2_u *)
72 lemma cpcs_cpr_strap2: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
75 lemma cpcs_cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
78 lemma cpr_div: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡ T → ∀T2. ⦃G, L⦄ ⊢ T2 ➡ T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
81 (* Basic_1: was: pc3_pr2_u2 *)
82 lemma cpcs_cpr_conf: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ➡ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
85 (* Basic_1: removed theorems 9:
86 clear_pc3_trans pc3_ind_left
87 pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
88 pc3_pr2_fqubst0 pc3_pr2_fqubst0_back pc3_fqubst0
89 pc3_gen_abst pc3_gen_abst_shift
91 (* Basic_1: removed local theorems 6:
92 pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left