(* *)
(**************************************************************************)
-include "basic_2/computation/cprs_cprs.ma".
-include "basic_2/computation/lprs.ma".
+include "basic_2/rt_computation/cprs_cprs.ma".
+include "basic_2/rt_computation/lprs_cpms.ma".
-(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+(* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
(* Advanced properties ******************************************************)
-lemma lprs_pair: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
- ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
-/2 width=1 by TC_lpx_sn_pair/ qed.
+(* Basic_2A1: was: lprs_pair2 *)
+lemma lprs_pair_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
+ ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h] V2 →
+ ∀I. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h] L2.ⓑ{I}V2.
+/3 width=3 by lprs_pair, lprs_cpms_trans/ qed.
-(* Advanced inversion lemmas ************************************************)
+(* Properties on context-sensitive parallel r-computation for terms *********)
-lemma lprs_inv_pair1: ∀I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡* L2 →
- ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
- L2 = K2.ⓑ{I}V2.
-/3 width=3 by TC_lpx_sn_inv_pair1, lpr_cprs_trans/ qed-.
-
-lemma lprs_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡* K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
- L1 = K1.ⓑ{I}V1.
-/3 width=3 by TC_lpx_sn_inv_pair2, lpr_cprs_trans/ qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma lprs_ind_alt: ∀G. ∀R:relation lenv.
- R (⋆) (⋆) → (
- ∀I,K1,K2,V1,V2.
- ⦃G, K1⦄ ⊢ ➡* K2 → ⦃G, K1⦄ ⊢ V1 ➡* V2 →
- R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L1 L2.
-/3 width=4 by TC_lpx_sn_ind, lpr_cprs_trans/ qed-.
-
-(* Properties on context-sensitive parallel computation for terms ***********)
-
-lemma lprs_cpr_trans: ∀G. b_c_transitive … (cpr G) (λ_. lprs G).
-/3 width=5 by b_c_trans_CTC2, lpr_cprs_trans/ qed-.
-
-(* Basic_1: was just: pr3_pr3_pr3_t *)
-(* Note: alternative proof /3 width=5 by s_r_trans_CTC1, lprs_cpr_trans/ *)
-lemma lprs_cprs_trans: ∀G. b_rs_transitive … (cpr G) (λ_. lprs G).
-#G @b_c_to_b_rs_trans @b_c_trans_CTC2
-@b_rs_trans_TC1 /2 width=3 by lpr_cprs_trans/ (**) (* full auto too slow *)
-qed-.
-
-lemma lprs_cprs_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #H @(lprs_ind … H) -L1 /2 width=3 by ex2_intro/
+lemma lprs_cprs_conf_dx (h) (G): ∀L0.∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+#h #G #L0 #T0 #T1 #HT01 #L1 #H
+@(lprs_ind_dx … H) -L1 /2 width=3 by ex2_intro/
#L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
elim (cprs_lpr_conf_dx … HT1 … HL1) -HT1 #T2 #HT2
elim (cprs_lpr_conf_dx … HT0 … HL1) -L #T3 #HT3
/3 width=5 by cprs_trans, ex2_intro/
qed-.
-lemma lprs_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-/3 width=3 by lprs_cprs_conf_dx, cpr_cprs/ qed-.
+lemma lprs_cpr_conf_dx (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+/3 width=3 by lprs_cprs_conf_dx, cpm_cpms/ qed-.
-(* Note: this can be proved on its own using lprs_ind_dx *)
-lemma lprs_cprs_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01
+(* Note: this can be proved on its own using lprs_ind_sn *)
+lemma lprs_cprs_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡*[h] T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+#h #G #L0 #T0 #T1 #HT01 #L1 #HL01
elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01
-/3 width=3 by lprs_cprs_trans, ex2_intro/
+/3 width=3 by lprs_cpms_trans, ex2_intro/
qed-.
-lemma lprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-/3 width=3 by lprs_cprs_conf_sn, cpr_cprs/ qed-.
-
-lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=5 by lprs_cprs_trans, lprs_pair, cprs_bind/ qed.
-
-(* Inversion lemmas on context-sensitive parallel computation for terms *****)
-
-(* Basic_1: was: pr3_gen_abst *)
-lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
- ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
- U2 = ⓛ{a}W2.T2.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5 by ex3_2_intro/
-#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
-elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
-lapply (lprs_cpr_trans … HT02 (L.ⓛV1) ?)
-/3 width=5 by lprs_pair, cprs_trans, cprs_strap1, ex3_2_intro/
-qed-.
-
-lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
- ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
-#a #G #L #W1 #W2 #T1 #T2 #H elim (cprs_inv_abst1 … H) -H
-#W #T #HW1 #HT1 #H destruct /2 width=1 by conj/
-qed-.
-
-(* Basic_1: was pr3_gen_abbr *)
-lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
- U2 = ⓓ{a}V2.T2
- ) ∨
- ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⬆[0, 1] U2 ≘ T2 & a = true.
-#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
-#U0 #U2 #_ #HU02 * *
-[ #V0 #T0 #HV10 #HT10 #H destruct
- elim (cpr_inv_abbr1 … HU02) -HU02 *
- [ #V2 #T2 #HV02 #HT02 #H destruct
- lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?)
- /4 width=5 by lprs_pair, cprs_trans, cprs_strap1, ex3_2_intro, or_introl/
- | #T2 #HT02 #HUT2
- lapply (lprs_cpr_trans … HT02 (L.ⓓV1) ?) -HT02
- /4 width=3 by lprs_pair, cprs_trans, ex3_intro, or_intror/
- ]
-| #U1 #HTU1 #HU01 elim (lift_total U2 0 1)
- #U #HU2 lapply (cpr_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0
- /4 width=3 by cprs_strap1, drop_drop, ex3_intro, or_intror/
-]
-qed-.
-
-(* More advanced properties *************************************************)
-
-lemma lprs_pair2: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
- ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
-/3 width=3 by lprs_pair, lprs_cprs_trans/ qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_2A1: uses: scpds_inv_abst1 *)
-lemma cpms_inv_abst_sn (n) (h) (G) (L):
- ∀p,V1,T1,X2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡*[n, h] X2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[n, h] T2 &
- X2 = ⓛ{p}V2.T2.
-#n #h #G #L #p #V1 #T1 #X2 #H
-@(cpms_ind_dx … H) -X2 /2 width=5 by ex3_2_intro/
-#n1 #n2 #X #X2 #_ * #V #T #HV1 #HT1 #H1 #H2 destruct
-elim (cpm_inv_abst1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H2 destruct
-@ex3_2_intro [1,2,5: // ]
-[ /2 width=3 by cprs_step_dx/
-| @(cpms_trans … HT1) -T1 -V2 -n1
-
-/3 width=5 by cprs_step_dx, cpms_step_dx, ex3_2_intro/
-
-
-
-#d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
-lapply (da_inv_bind … Hd1) -Hd1 #Hd1
-elim (lstas_inv_bind1 … H1) -H1 #U #HTU1 #H destruct
-elim (cprs_inv_abst1 … H2) -H2 #V2 #T2 #HV12 #HUT2 #H destruct
-/3 width=6 by ex4_2_intro, ex3_2_intro/
-qed-.
-
-lemma scpds_inv_abbr_abst: ∀h,o,a1,a2,G,L,V1,W2,T1,T2,d. ⦃G, L⦄ ⊢ ⓓ{a1}V1.T1 •*➡*[h, o, d] ⓛ{a2}W2.T2 →
- ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 •*➡*[h, o, d] T & ⬆[0, 1] ⓛ{a2}W2.T2 ≘ T & a1 = true.
-#h #o #a1 #a2 #G #L #V1 #W2 #T1 #T2 #d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
-lapply (da_inv_bind … Hd1) -Hd1 #Hd1
-elim (lstas_inv_bind1 … H1) -H1 #U1 #HTU1 #H destruct
-elim (cprs_inv_abbr1 … H2) -H2 *
-[ #V2 #U2 #HV12 #HU12 #H destruct
-| /3 width=6 by ex4_2_intro, ex3_intro/
-]
-qed-.
-*)
+lemma lprs_cpr_conf_sn (h) (G): ∀L0. ∀T0,T1:term. ⦃G, L0⦄ ⊢ T0 ➡[h] T1 →
+ ∀L1. ⦃G, L0⦄ ⊢ ➡*[h] L1 →
+ ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡*[h] T & ⦃G, L1⦄ ⊢ T0 ➡*[h] T.
+/3 width=3 by lprs_cprs_conf_sn, cpm_cpms/ qed-.
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