(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
definition cpx (h): relation4 genv lenv term term ≝
- λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[c, h] T2.
+ λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[eq_f, c, h] T2.
interpretation
"uncounted context-sensitive parallel rt-transition (term)"
(* Basic properties *********************************************************)
-lemma cpx_atom: ∀h,I,G,L. ⦃G, L⦄ ⊢ ⓪{I} ⬈[h] ⓪{I}.
-/2 width=2 by cpg_atom, ex_intro/ qed.
-
(* Basic_2A1: was: cpx_st *)
lemma cpx_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s).
/2 width=2 by cpg_ess, ex_intro/ qed.
lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2.
⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
-#h #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
-/3 width=2 by cpg_flat, ex_intro/
+#h * #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
+/3 width=5 by cpg_appl, cpg_cast, ex_intro/
qed.
lemma cpx_zeta: ∀h,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ⬈[h] T →
/3 width=4 by cpg_theta, ex_intro/
qed.
+(* Basic_2A1: includes: cpx_atom *)
lemma cpx_refl: ∀h,G,L. reflexive … (cpx h G L).
-/2 width=2 by ex_intro/ qed.
+/3 width=2 by cpg_refl, ex_intro/ qed.
+
+(* Advanced properties ******************************************************)
lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ⬈[h] ②{I}V2.T.
-#h #I #G #L #V1 #V2 *
-/3 width=2 by cpg_pair_sn, ex_intro/
+#h * /2 width=2 by cpx_flat, cpx_bind/
qed.
(* Basic inversion lemmas ***************************************************)
/3 width=5 by ex3_2_intro, ex_intro/
qed-.
-lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
- U2 = ⓕ{I}V2.T2
- | (⦃G, L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
- | (⦃G, L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
- U1 = ⓛ{p}W1.T1 &
- U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
- | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≡ V2 &
- ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
- U1 = ⓓ{p}W1.T1 &
- U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
-#h #I #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_flat1 … H) -H *
-/4 width=14 by or5_intro0, or5_intro1, or5_intro2, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro, ex_intro, conj/
-qed-.
-
lemma cpx_inv_appl1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ⬈[h] U2 →
∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
U2 = ⓐV2.T2
/4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/
qed-.
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] T2 →
+ T2 = #0 ∨
+ ∃∃V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2.
+#h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
+/4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/
+qed-.
+
+lemma cpx_inv_lref1_pair: ∀h,I,G,K,V,T2,i. ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ⬈[h] T2 →
+ T2 = #(⫯i) ∨
+ ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2.
+#h #I #G #L #V #T2 #i * #c #H elim (cpg_inv_lref1_pair … H) -H *
+/4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/
+qed-.
+
+lemma cpx_inv_flat1: ∀h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[h] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
+ U2 = ⓕ{I}V2.T2
+ | (⦃G, L⦄ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
+ | (⦃G, L⦄ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
+ | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
+ ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
+ U1 = ⓛ{p}W1.T1 &
+ U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
+ | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V & ⬆*[1] V ≡ V2 &
+ ⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
+ U1 = ⓓ{p}W1.T1 &
+ U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
+#h * #G #L #V1 #U1 #U2 #H
+[ elim (cpx_inv_appl1 … H) -H *
+ /3 width=14 by or5_intro0, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro/
+| elim (cpx_inv_cast1 … H) -H [ * ]
+ /3 width=14 by or5_intro0, or5_intro1, or5_intro2, ex3_2_intro, conj/
+]
+qed-.
+
(* Basic forward lemmas *****************************************************)
lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.