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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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15 include "basic_2/rt_computation/cpxs_tdeq.ma".
16 include "basic_2/rt_computation/cpxs_cpxs.ma".
17 include "basic_2/rt_computation/csx_csx.ma".
19 (* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
21 (* Properties with unbound context-sensitive rt-computation for terms *******)
23 (* Basic_1: was just: sn3_intro *)
24 lemma csx_intro_cpxs: ∀h,o,G,L,T1.
25 (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
26 ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
27 /4 width=1 by cpx_cpxs, csx_intro/ qed-.
29 (* Basic_1: was just: sn3_pr3_trans *)
30 lemma csx_cpxs_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
31 ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
32 #h #o #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
33 /2 width=3 by csx_cpx_trans/
36 (* Eliminators with unbound context-sensitive rt-computation for terms ******)
38 lemma csx_ind_cpxs_tdeq: ∀h,o,G,L. ∀Q:predicate term.
39 (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
40 (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) → Q T1
42 ∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
43 ∀T0. ⦃G, L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛[h, o] T2 → Q T2.
44 #h #o #G #L #Q #IH #T1 #H @(csx_ind … H) -T1
45 #T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02
46 @IH -IH /3 width=3 by csx_cpxs_trans, csx_tdeq_trans/ -HT1 #V2 #HTV2 #HnTV2
47 lapply (tdeq_tdneq_trans … HT02 … HnTV2) -HnTV2 #H
48 elim (tdeq_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
49 lapply (tdneq_tdeq_canc_dx … H … HV02) -H #HnTV0
50 elim (tdeq_dec h o T1 T0) #H
51 [ lapply (tdeq_tdneq_trans … H … HnTV0) -H -HnTV0 #Hn10
52 lapply (cpxs_trans … HT10 … HTV0) -T0 #H10
53 elim (cpxs_tdneq_fwd_step_sn … H10 … Hn10) -H10 -Hn10
54 /3 width=8 by tdeq_trans/
55 | elim (cpxs_tdneq_fwd_step_sn … HT10 … H) -HT10 -H #T #V #HT1 #HnT1 #HTV #HVT0
56 elim (tdeq_cpxs_trans … HVT0 … HTV0) -T0
57 /3 width=8 by cpxs_trans, tdeq_trans/
61 (* Basic_2A1: was: csx_ind_alt *)
62 lemma csx_ind_cpxs: ∀h,o,G,L. ∀Q:predicate term.
63 (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
64 (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) → Q T1
66 ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → Q T.
67 #h #o #G #L #Q #IH #T #HT
68 @(csx_ind_cpxs_tdeq … IH … HT) -IH -HT // (**) (* full auto fails *)