include "basic_2/notation/relations/predty_5.ma".
include "basic_2/rt_transition/cpg.ma".
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
definition cpx (h): relation4 genv lenv term term ≝
λG,L,T1,T2. ∃c. ⦃G, L⦄ ⊢ T1 ⬈[eq_f, c, h] T2.
interpretation
- "uncounted context-sensitive parallel rt-transition (term)"
+ "unbound context-sensitive parallel rt-transition (term)"
'PRedTy h G L T1 T2 = (cpx h G L T1 T2).
(* Basic properties *********************************************************)
/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 →
- ⬆*[1] T2 ≘ T → ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
-#h #G #L #V #T1 #T #T2 *
+lemma cpx_zeta (h) (G) (L):
+ ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬈[h] T2 →
+ ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
+#h #G #L #T1 #T #HT1 #T2 *
/3 width=4 by cpg_zeta, ex_intro/
qed.
#h * /2 width=2 by cpx_flat, cpx_bind/
qed.
+lemma cpg_cpx (h) (Rt) (c) (G) (L):
+ ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[Rt,c,h] T2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2.
+#h #Rt #c #G #L #T1 #T2 #H elim H -c -G -L -T1 -T2
+/2 width=3 by cpx_theta, cpx_beta, cpx_ee, cpx_eps, cpx_zeta, cpx_flat, cpx_bind, cpx_lref, cpx_delta/
+qed.
+
(* Basic inversion lemmas ***************************************************)
lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[h] T2 →
lemma cpx_inv_bind1: ∀h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] U2 →
∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 &
U2 = ⓑ{p,I}V2.T2
- | â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â¬\88[h] T & â¬\86*[1] U2 â\89\98 T &
+ | â\88\83â\88\83T. â¬\86*[1] T â\89\98 T1 & â¦\83G, Lâ¦\84 â\8a¢ T â¬\88[h] U2 &
p = true & I = Abbr.
#h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
/4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/
lemma cpx_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[h] U2 →
∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T2 &
U2 = ⓓ{p}V2.T2
- | â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â¬\88[h] T & â¬\86*[1] U2 â\89\98 T & p = true.
+ | â\88\83â\88\83T. â¬\86*[1] T â\89\98 T1 & â¦\83G, Lâ¦\84 â\8a¢ T â¬\88[h] U2 & p = true.
#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
/4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/
qed-.
(* Basic eliminators ********************************************************)
-lemma cpx_ind: ∀h. ∀R:relation4 genv lenv term term.
- (∀I,G,L. R G L (⓪{I}) (⓪{I})) →
- (∀G,L,s. R G L (⋆s) (⋆(next h s))) →
- (∀I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → R G K V1 V2 →
- ⬆*[1] V2 ≘ W2 → R G (K.ⓑ{I}V1) (#0) W2
- ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → R G K (#i) T →
- ⬆*[1] T ≘ U → R G (K.ⓘ{I}) (#↑i) (U)
+lemma cpx_ind: ∀h. ∀Q:relation4 genv lenv term term.
+ (∀I,G,L. Q G L (⓪{I}) (⓪{I})) →
+ (∀G,L,s. Q G L (⋆s) (⋆(next h s))) →
+ (∀I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 → Q G K V1 V2 →
+ ⬆*[1] V2 ≘ W2 → Q G (K.ⓑ{I}V1) (#0) W2
+ ) → (∀I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T → Q G K (#i) T →
+ ⬆*[1] T ≘ U → Q G (K.ⓘ{I}) (#↑i) (U)
) → (∀p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
- R G L V1 V2 → R G (L.ⓑ{I}V1) T1 T2 → R G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
+ Q G L V1 V2 → Q G (L.ⓑ{I}V1) T1 T2 → Q G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
) → (∀I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ T1 ⬈[h] T2 →
- R G L V1 V2 → R G L T1 T2 → R G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
- ) â\86\92 (â\88\80G,L,V,T1,T,T2. â¦\83G, L.â\93\93Vâ¦\84 â\8a¢ T1 â¬\88[h] T â\86\92 R G (L.â\93\93V) T1 T →
- ⬆*[1] T2 ≘ T → R G L (+ⓓV.T1) T2
- ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → R G L T1 T2 →
- R G L (ⓝV.T1) T2
- ) → (∀G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → R G L V1 V2 →
- R G L (ⓝV1.T) V2
+ Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) â\86\92 (â\88\80G,L,V,T1,T,T2. â¬\86*[1] T â\89\98 T1 â\86\92 â¦\83G, Lâ¦\84 â\8a¢ T â¬\88[h] T2 â\86\92 Q G L T T2 →
+ Q G L (+ⓓV.T1) T2
+ ) → (∀G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
+ Q G L (ⓝV.T1) T2
+ ) → (∀G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
+ Q G L (ⓝV1.T) V2
) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 →
- R G L V1 V2 → R G L W1 W2 → R G (L.ⓛW1) T1 T2 →
- R G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
+ Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
+ Q G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
- R G L V1 V → R G L W1 W2 → R G (L.ⓓW1) T1 T2 →
- ⬆*[1] V ≘ V2 → R G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+ Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
+ ⬆*[1] V ≘ V2 → Q G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
) →
- ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → R G L T1 T2.
-#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
+ ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → Q G L T1 T2.
+#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
* #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/
qed-.