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 *********************************************************)
/2 width=2 by cpg_ess, ex_intro/ qed.
lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 →
- â¬\86*[1] V2 â\89¡ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
+ â¬\86*[1] V2 â\89\98 W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
#h * #G #K #V1 #V2 #W2 *
/3 width=4 by cpg_delta, cpg_ell, ex_intro/
qed.
-lemma cpx_lref: ∀h,I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T →
- â¬\86*[1] T â\89¡ U â\86\92 â¦\83G, K.â\93\91{I}Vâ¦\84 â\8a¢ #⫯i ⬈[h] U.
-#h #I #G #K #V #T #U #i *
+lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T →
+ â¬\86*[1] T â\89\98 U â\86\92 â¦\83G, K.â\93\98{I}â¦\84 â\8a¢ #â\86\91i ⬈[h] U.
+#h #I #G #K #T #U #i *
/3 width=4 by cpg_lref, ex_intro/
qed.
/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.
qed.
lemma cpx_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
- â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V â\86\92 â¬\86*[1] V â\89¡ V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
+ â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V â\86\92 â¬\86*[1] V â\89\98 V2 → ⦃G, L⦄ ⊢ W1 ⬈[h] W2 →
⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 →
⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
#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 →
∨∨ T2 = ⓪{J}
| ∃∃s. T2 = ⋆(next h s) & J = Sort s
- | â\88\83â\88\83I,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¬\86*[1] V2 â\89¡ T2 &
+ | â\88\83â\88\83I,K,V1,V2. â¦\83G, Kâ¦\84 â\8a¢ V1 â¬\88[h] V2 & â¬\86*[1] V2 â\89\98 T2 &
L = K.ⓑ{I}V1 & J = LRef 0
- | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2 &
- L = K.ⓑ{I}V & J = LRef (⫯i).
+ | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 &
+ L = K.ⓘ{I} & J = LRef (↑i).
#h #J #G #L #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
-/4 width=9 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_5_intro, ex4_4_intro, ex2_intro, ex_intro/
+/4 width=8 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_4_intro, ex2_intro, ex_intro/
qed-.
lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] T2 →
- T2 = ⋆s ∨ T2 = ⋆(next h s).
+ ∨∨ T2 = ⋆s | T2 = ⋆(next h s).
#h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H *
/2 width=1 by or_introl, or_intror/
qed-.
lemma cpx_inv_zero1: ∀h,G,L,T2. ⦃G, L⦄ ⊢ #0 ⬈[h] T2 →
- T2 = #0 ∨
- ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≡ T2 &
- L = K.ⓑ{I}V1.
+ ∨∨ T2 = #0
+ | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
+ L = K.ⓑ{I}V1.
#h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
qed-.
-lemma cpx_inv_lref1: â\88\80h,G,L,T2,i. â¦\83G, Lâ¦\84 â\8a¢ #⫯i ⬈[h] T2 →
- T2 = #(⫯i) ∨
- ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
+lemma cpx_inv_lref1: â\88\80h,G,L,T2,i. â¦\83G, Lâ¦\84 â\8a¢ #â\86\91i ⬈[h] T2 →
+ ∨∨ T2 = #(↑i)
+ | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{I}.
#h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
-/4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
+/4 width=6 by ex3_3_intro, ex_intro, or_introl, or_intror/
qed-.
lemma cpx_inv_gref1: ∀h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ⬈[h] T2 → T2 = §l.
#h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
qed-.
-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
- ) ∨
- ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T & ⬆*[1] U2 ≡ T &
- p = true & I = Abbr.
+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
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[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/
qed-.
-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
- ) ∨
- ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[h] T & ⬆*[1] U2 ≡ T & p = true.
+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
+ | ∃∃T. ⬆*[1] T ≘ T1 & ⦃G, L⦄ ⊢ T ⬈[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-.
| ∃∃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
- | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V & â¬\86*[1] V â\89¡ V2 &
+ | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V & â¬\86*[1] V â\89\98 V2 &
⦃G, L⦄ ⊢ W1 ⬈[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
(* 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_bind: ∀h,I,G,K,T2,i. ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] T2 →
+ ∨∨ T2 = #(↑i)
+ | ∃∃T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2.
+#h #I #G #L #T2 #i * #c #H elim (cpg_inv_lref1_bind … 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.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
U1 = ⓛ{p}W1.T1 &
U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
- | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V & â¬\86*[1] V â\89¡ V2 &
+ | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â¬\88[h] V & â¬\86*[1] V â\89\98 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 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
/3 width=4 by ex2_2_intro, ex_intro/
qed-.
+
+(* Basic eliminators ********************************************************)
+
+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 →
+ 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 →
+ Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+ ) → (∀G,L,V,T1,T,T2. ⬆*[1] T ≘ T1 → ⦃G, L⦄ ⊢ T ⬈[h] T2 → 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 →
+ 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 →
+ 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 → 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-.
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