(* 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.
+ λG,L,T1,T2. ∃c. ⦃G,L⦄ ⊢ T1 ⬈[eq_f,c,h] T2.
interpretation
"unbound context-sensitive parallel rt-transition (term)"
(* Basic properties *********************************************************)
(* Basic_2A1: was: cpx_st *)
-lemma cpx_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ⬈[h] ⋆(next h s).
+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_delta: ∀h,I,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 →
- ⬆*[1] V2 ≘ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈[h] W2.
+lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 →
+ ⬆*[1] V2 ≘ 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,T,U,i. ⦃G, K⦄ ⊢ #i ⬈[h] T →
- ⬆*[1] T ≘ U → ⦃G, K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U.
+lemma cpx_lref: ∀h,I,G,K,T,U,i. ⦃G,K⦄ ⊢ #i ⬈[h] T →
+ ⬆*[1] T ≘ U → ⦃G,K.ⓘ{I}⦄ ⊢ #↑i ⬈[h] U.
#h #I #G #K #T #U #i *
/3 width=4 by cpg_lref, ex_intro/
qed.
lemma cpx_bind: ∀h,p,I,G,L,V1,V2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
- ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L.ⓑ{I}V1⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2.
#h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
/3 width=2 by cpg_bind, 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.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ T1 ⬈[h] T2 →
+ ⦃G,L⦄ ⊢ ⓕ{I}V1.T1 ⬈[h] ⓕ{I}V2.T2.
#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):
- ∀T1,T. ⬆*[1] T ≘ T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬈[h] T2 →
- ∀V. ⦃G, L⦄ ⊢ +ⓓV.T1 ⬈[h] T2.
+ ∀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.
-lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ⬈[h] T2.
+lemma cpx_eps: ∀h,G,L,V,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[h] T2 → ⦃G,L⦄ ⊢ ⓝV.T1 ⬈[h] T2.
#h #G #L #V #T1 #T2 *
/3 width=2 by cpg_eps, ex_intro/
qed.
(* Basic_2A1: was: cpx_ct *)
-lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ⬈[h] V2.
+lemma cpx_ee: ∀h,G,L,V1,V2,T. ⦃G,L⦄ ⊢ V1 ⬈[h] V2 → ⦃G,L⦄ ⊢ ⓝV1.T ⬈[h] V2.
#h #G #L #V1 #V2 #T *
/3 width=2 by cpg_ee, ex_intro/
qed.
lemma cpx_beta: ∀h,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 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ⬈[h] ⓓ{p}ⓝW2.V2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] 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 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
/3 width=2 by cpg_beta, ex_intro/
qed.
lemma cpx_theta: ∀h,p,G,L,V1,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 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ⬈[h] ⓓ{p}W2.ⓐV2.T2.
+ ⦃G,L⦄ ⊢ V1 ⬈[h] V → ⬆*[1] V ≘ 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 *
/3 width=4 by cpg_theta, 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.
+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 * /2 width=2 by cpx_flat, cpx_bind/
qed.
(* Basic inversion lemmas ***************************************************)
-lemma cpx_inv_atom1: ∀h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ⬈[h] T2 →
+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
- | ∃∃I,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
+ | ∃∃I,K,V1,V2. ⦃G,K⦄ ⊢ V1 ⬈[h] V2 & ⬆*[1] V2 ≘ T2 &
L = K.ⓑ{I}V1 & J = LRef 0
- | ∃∃I,K,T,i. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 &
+ | ∃∃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=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 →
+lemma cpx_inv_sort1: ∀h,G,L,T2,s. ⦃G,L⦄ ⊢ ⋆s ⬈[h] T2 →
∨∨ 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 →
+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 &
+ | ∃∃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: ∀h,G,L,T2,i. ⦃G, L⦄ ⊢ #↑i ⬈[h] T2 →
+lemma cpx_inv_lref1: ∀h,G,L,T2,i. ⦃G,L⦄ ⊢ #↑i ⬈[h] T2 →
∨∨ T2 = #(↑i)
- | ∃∃I,K,T. ⦃G, K⦄ ⊢ #i ⬈[h] T & ⬆*[1] T ≘ T2 & L = K.ⓘ{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=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.
+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 &
+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 &
+ | ∃∃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 &
+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.
+ | ∃∃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-.
-lemma cpx_inv_abst1: ∀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 &
+lemma cpx_inv_abst1: ∀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.
#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
/3 width=5 by ex3_2_intro, ex_intro/
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 &
+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
- | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[h] W2 &
- ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[h] T2 &
+ | ∃∃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
- | ∃∃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 &
+ | ∃∃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.
#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
/4 width=13 by or3_intro0, or3_intro1, or3_intro2, ex6_7_intro, ex5_6_intro, ex3_2_intro, ex_intro/
qed-.
-lemma cpx_inv_cast1: ∀h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[h] T2 &
+lemma cpx_inv_cast1: ∀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
- | ⦃G, L⦄ ⊢ U1 ⬈[h] U2
- | ⦃G, L⦄ ⊢ V1 ⬈[h] U2.
+ | ⦃G,L⦄ ⊢ U1 ⬈[h] U2
+ | ⦃G,L⦄ ⊢ V1 ⬈[h] U2.
#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
/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 →
+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.
+ | ∃∃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 →
+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.
+ | ∃∃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 &
+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 &
+ | (⦃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 &
+ | ∃∃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
(* Basic forward lemmas *****************************************************)
-lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
- ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 &
+lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ⦃G,L⦄ ⊢ -ⓑ{I}V1.T1 ⬈[h] T → ∀p.
+ ∃∃V2,T2. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈[h] ⓑ{p,I}V2.T2 &
T = -ⓑ{I}V2.T2.
#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/
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 →
+ (∀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 →
+ ) → (∀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 →
+ ) → (∀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 →
+ ) → (∀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 →
+ ) → (∀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 →
+ ) → (∀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 →
+ ) → (∀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 →
+ ) → (∀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 →
+ ) → (∀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.
+ ∀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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