(* *)
(**************************************************************************)
-include "basic_2/rt_computation/cpxs_tdeq.ma".
+include "basic_2/rt_computation/cpxs_teqx.ma".
include "basic_2/rt_computation/cpxs_cpxs.ma".
include "basic_2/rt_computation/csx_csx.ma".
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-(* Properties with uncounted context-sensitive rt-computation for terms *****)
+(* Properties with unbound context-sensitive rt-computation for terms *******)
(* Basic_1: was just: sn3_intro *)
-lemma csx_intro_cpxs: ∀h,o,G,L,T1.
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
+lemma csx_intro_cpxs: ∀h,G,L,T1.
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄) →
+ ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄.
/4 width=1 by cpx_cpxs, csx_intro/ qed-.
(* Basic_1: was just: sn3_pr3_trans *)
-lemma csx_cpxs_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
+lemma csx_cpxs_trans: ∀h,G,L,T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2
/2 width=3 by csx_cpx_trans/
qed-.
-(* Eliminators with uncounted context-sensitive rt-computation for terms ****)
+(* Eliminators with unbound context-sensitive rt-computation for terms ******)
-lemma csx_ind_cpxs_tdeq: ∀h,o,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → R T2) → R T1
+lemma csx_ind_cpxs_teqx: ∀h,G,L. ∀Q:predicate term.
+ (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀T0. ⦃G, L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛[h, o] T2 → R T2.
-#h #o #G #L #R #IH #T1 #H @(csx_ind … H) -T1
+ ∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T0. ⦃G,L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛ T2 → Q T2.
+#h #G #L #Q #IH #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02
-@IH -IH /3 width=3 by csx_cpxs_trans, csx_tdeq_trans/ -HT1 #V2 #HTV2 #HnTV2
-lapply (tdeq_tdneq_trans … HT02 … HnTV2) -HnTV2 #H
-elim (tdeq_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
-lapply (tdneq_tdeq_canc_dx … H … HV02) -H #HnTV0
-elim (tdeq_dec h o T1 T0) #H
-[ lapply (tdeq_tdneq_trans … H … HnTV0) -H -HnTV0 #Hn10
+@IH -IH /3 width=3 by csx_cpxs_trans, csx_teqx_trans/ -HT1 #V2 #HTV2 #HnTV2
+lapply (teqx_tneqx_trans … HT02 … HnTV2) -HnTV2 #H
+elim (teqx_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
+lapply (tneqx_teqx_canc_dx … H … HV02) -H #HnTV0
+elim (teqx_dec T1 T0) #H
+[ lapply (teqx_tneqx_trans … H … HnTV0) -H -HnTV0 #Hn10
lapply (cpxs_trans … HT10 … HTV0) -T0 #H10
- elim (cpxs_tdneq_fwd_step_sn … H10 … Hn10) -H10 -Hn10
- /3 width=8 by tdeq_trans/
-| elim (cpxs_tdneq_fwd_step_sn … HT10 … H) -HT10 -H #T #V #HT1 #HnT1 #HTV #HVT0
- elim (tdeq_cpxs_trans … HVT0 … HTV0) -T0
- /3 width=8 by cpxs_trans, tdeq_trans/
+ elim (cpxs_tneqx_fwd_step_sn … H10 … Hn10) -H10 -Hn10
+ /3 width=8 by teqx_trans/
+| elim (cpxs_tneqx_fwd_step_sn … HT10 … H) -HT10 -H #T #V #HT1 #HnT1 #HTV #HVT0
+ elim (teqx_cpxs_trans … HVT0 … HTV0) -T0
+ /3 width=8 by cpxs_trans, teqx_trans/
]
qed-.
(* Basic_2A1: was: csx_ind_alt *)
-lemma csx_ind_cpxs: ∀h,o,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → R T2) → R T1
+lemma csx_ind_cpxs: ∀h,G,L. ∀Q:predicate term.
+ (∀T1. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛ T2 → ⊥) → Q T2) → Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → R T.
-#h #o #G #L #R #IH #T #HT
-@(csx_ind_cpxs_tdeq … IH … HT) -IH -HT // (**) (* full auto fails *)
+ ∀T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q T.
+#h #G #L #Q #IH #T #HT
+@(csx_ind_cpxs_teqx … IH … HT) -IH -HT // (**) (* full auto fails *)
qed-.