(* Advanced properties ******************************************************)
-lemma lpxs_pair: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → ∀V1,V2. ⦃h, L1⦄ ⊢ V1 ➡*[h, g] V2 →
- ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
+lemma lpxs_pair: ∀h,o,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
+ ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡*[h, o] V2 →
+ ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, o] L2.ⓑ{I}V2.
/2 width=1 by TC_lpx_sn_pair/ qed.
(* Advanced inversion lemmas ************************************************)
-lemma lpxs_inv_pair1: ∀h,g,I,K1,L2,V1. ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2 →
- ∃∃K2,V2. ⦃h, K1⦄ ⊢ ➡*[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[h, g] V2 & L2 = K2.ⓑ{I}V2.
+lemma lpxs_inv_pair1: ∀h,o,I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, o] L2 →
+ ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡*[h, o] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, o] V2 & L2 = K2.ⓑ{I}V2.
/3 width=3 by TC_lpx_sn_inv_pair1, lpx_cpxs_trans/ qed-.
-lemma lpxs_inv_pair2: ∀h,g,I,L1,K2,V2. ⦃h, L1⦄ ⊢ ➡*[h, g] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃h, K1⦄ ⊢ ➡*[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[h, g] V2 & L1 = K1.ⓑ{I}V1.
+lemma lpxs_inv_pair2: ∀h,o,I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡*[h, o] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡*[h, o] K2 & ⦃G, K1⦄ ⊢ V1 ➡*[h, o] V2 & L1 = K1.ⓑ{I}V1.
/3 width=3 by TC_lpx_sn_inv_pair2, lpx_cpxs_trans/ qed-.
+(* Advanced eliminators *****************************************************)
+
+lemma lpxs_ind_alt: ∀h,o,G. ∀R:relation lenv.
+ R (⋆) (⋆) → (
+ ∀I,K1,K2,V1,V2.
+ ⦃G, K1⦄ ⊢ ➡*[h, o] K2 → ⦃G, K1⦄ ⊢ V1 ➡*[h, o] V2 →
+ R K1 K2 → R (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpx_cpxs_trans/ qed-.
+
(* Properties on context-sensitive extended parallel computation for terms **)
-lemma lpxs_cpx_trans: ∀h,g. s_r_trans … (cpx h g) (lpxs h g).
-/3 width=5 by s_r_trans_TC2, lpx_cpxs_trans/ qed-.
+lemma lpxs_cpx_trans: ∀h,o,G. c_r_transitive … (cpx h o G) (λ_.lpxs h o G).
+/3 width=5 by c_r_trans_LTC2, lpx_cpxs_trans/ qed-.
-lemma lpxs_cpxs_trans: ∀h,g. s_rs_trans … (cpx h g) (lpxs h g).
-/3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ qed-.
+(* Note: alternative proof: /3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ *)
+lemma lpxs_cpxs_trans: ∀h,o,G. c_rs_transitive … (cpx h o G) (λ_.lpxs h o G).
+#h #o #G @c_r_to_c_rs_trans @c_r_trans_LTC2
+@c_rs_trans_TC1 /2 width=3 by lpx_cpxs_trans/ (**) (* full auto too slow *)
+qed-.
-lemma cpxs_bind2: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
- ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
-#h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
-lapply (lpxs_cpxs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
-qed.
+lemma cpxs_bind2: ∀h,o,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, o] T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, o] ⓑ{a,I}V2.T2.
+/4 width=5 by lpxs_cpxs_trans, lpxs_pair, cpxs_bind/ qed.
(* Inversion lemmas on context-sensitive ext parallel computation for terms *)
-lemma cpxs_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
+lemma cpxs_inv_abst1: ∀h,o,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, o] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡*[h, o] T2 &
U2 = ⓛ{a}V2.T2.
-#h #g #a #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5/
+#h #o #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
-lapply (lpxs_cpx_trans … HT02 (L.ⓛV1) ?) /2 width=1/ -HT02 #HT02
-lapply (cpxs_strap1 … HV10 … HV02) -V0
-lapply (cpxs_trans … HT10 … HT02) -T0 /2 width=5/
+lapply (lpxs_cpx_trans … HT02 (L.ⓛV1) ?)
+/3 width=5 by lpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro/
qed-.
-lemma cpxs_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
+lemma cpxs_inv_abbr1: ∀h,o,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, o] U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, o] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, o] T2 &
U2 = ⓓ{a}V2.T2
) ∨
- ∃∃T2. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
-#h #g #a #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
+ ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡*[h, o] T2 & ⬆[0, 1] U2 ≡ T2 & a = true.
+#h #o #a #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
#U0 #U2 #_ #HU02 * *
[ #V0 #T0 #HV10 #HT10 #H destruct
elim (cpx_inv_abbr1 … HU02) -HU02 *
[ #V2 #T2 #HV02 #HT02 #H destruct
- lapply (lpxs_cpx_trans … HT02 (L.ⓓV1) ?) /2 width=1/ -HT02 #HT02
- lapply (cpxs_strap1 … HV10 … HV02) -V0
- lapply (cpxs_trans … HT10 … HT02) -T0 /3 width=5/
+ lapply (lpxs_cpx_trans … HT02 (L.ⓓV1) ?)
+ /4 width=5 by lpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
| #T2 #HT02 #HUT2
- lapply (lpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02 /2 width=1/ -V0 #HT02
- lapply (cpxs_trans … HT10 … HT02) -T0 /3 width=3/
+ lapply (lpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
+ /4 width=3 by lpxs_pair, cpxs_trans, ex3_intro, or_intror/
]
| #U1 #HTU1 #HU01
elim (lift_total U2 0 1) #U #HU2
- lapply (cpx_lift … HU02 (L.ⓓV1) … HU01 … HU2) -U0 /2 width=1/ /4 width=3/
+ /6 width=12 by cpxs_strap1, cpx_lift, drop_drop, ex3_intro, or_intror/
]
qed-.
(* More advanced properties *************************************************)
-lemma lpxs_pair2: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 →
- ∀V1,V2. ⦃h, L2⦄ ⊢ V1 ➡*[h, g] V2 → ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
+lemma lpxs_pair2: ∀h,o,I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 →
+ ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡*[h, o] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, o] L2.ⓑ{I}V2.
/3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.
+
+(* Properties on supclosure *************************************************)
+
+lemma lpx_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #K1 #HKL1 elim (lpx_fqu_trans … H12 … HKL1) -L1
+ /3 width=5 by cpx_cpxs, fqu_fqup, ex3_2_intro/
+| #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+ #L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fqu_trans … H2 … HL0) -L
+ #L #T3 #HT3 #HT32 #HL2 elim (fqup_cpx_trans … HT0 … HT3) -T
+ /3 width=7 by cpxs_strap1, fqup_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma lpx_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡[h, o] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 [ /2 width=5 by ex3_2_intro/ ]
+#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+#L0 #T0 #HT10 #HT0 #HL0 elim (lpx_fquq_trans … H2 … HL0) -L
+#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpx_trans … HT0 … HT3) -T
+/3 width=7 by cpxs_strap1, fqus_strap1, ex3_2_intro/
+qed-.
+
+lemma lpxs_fquq_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐⸮ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
+[ /2 width=5 by ex3_2_intro/
+| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
+ lapply (lpx_cpxs_trans … HT1 … HK1) -HT1
+ elim (lpx_fquq_trans … HT2 … HK1) -K
+ /3 width=7 by lpxs_strap2, cpxs_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma lpxs_fqup_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐+ ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #HT12 #K1 #H @(lpxs_ind_dx … H) -K1
+[ /2 width=5 by ex3_2_intro/
+| #K1 #K #HK1 #_ * #L #T #HT1 #HT2 #HL2 -HT12
+ lapply (lpx_cpxs_trans … HT1 … HK1) -HT1
+ elim (lpx_fqup_trans … HT2 … HK1) -K
+ /3 width=7 by lpxs_strap2, cpxs_trans, ex3_2_intro/
+]
+qed-.
+
+lemma lpxs_fqus_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀K1. ⦃G1, K1⦄ ⊢ ➡*[h, o] L1 →
+ ∃∃K2,T. ⦃G1, K1⦄ ⊢ T1 ➡*[h, o] T & ⦃G1, K1, T⦄ ⊐* ⦃G2, K2, T2⦄ & ⦃G2, K2⦄ ⊢ ➡*[h, o] L2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 /2 width=5 by ex3_2_intro/
+#G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #K1 #HLK1 elim (IH1 … HLK1) -L1
+#L0 #T0 #HT10 #HT0 #HL0 elim (lpxs_fquq_trans … H2 … HL0) -L
+#L #T3 #HT3 #HT32 #HL2 elim (fqus_cpxs_trans … HT3 … HT0) -T
+/3 width=7 by cpxs_trans, fqus_strap1, ex3_2_intro/
+qed-.
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