typing_ctxt_elims [elim!]:
"Γ, Δ ⊨c: "(nat ==> (nat ==>)\Rightarrow ctxt \Rightarrow type ==> bool"
"\, Δtt (E t) : T <==
typing_ctxt_correct1:
shows "Γ, Δ ⊨!]:"\<Gamma,>\^>x\^subt ♢
by(induction E arbitrary: Γ Δ T r; force)
typing_ctxt_correct2:
shows "Γ, Δ ⊨ctxt E : T <== U ==> Γ, Δ ⊨T r : U ==> Γ, Δ ⊨T (ctxt_subst E r) : T"
by(induction arbitrary: r rule: typ [intro!]: "\lbrakk🚫
ctxt_subst_basecase:
"∀n. c[n = n ♢]C = c"
"∀n. t[n = n ♢]T = t"
by(induct c and t) (auto)
ctxt_subst_caseApp:
"∀n E s. (c[n=n (liftM_ctxt E n)]C)[n=n (♢\∙ (liftM_trm s n))]C = c[n=n ((liftM_ctxt E n) \∙ (liftM_trm s n))]C"
"∀n E s. (t[n=n (liftM_ctxt E n)]T)[n=n (♢\∙ (liftM_trm s n))]T = t[n=n ((liftM_ctxt E n) \∙ (liftM_trm s n))]T"
by (induction c and t) (auto simp add: liftLM_comm_ctxt liftLM_comm liftMM_comm_ctxt liftMM_comm liftM_ctxt_struct_subst)
ctxt_subst:
assumes "Γ, Δ ⊨ctxt E : U <== T"
shows "(ctxt_subst E (μ T : c)) <----* μ U : (c[0 = 0 (liftM_ctxt E 0)]C)"
assms proof(induct E arbitrary: U T Γ Δ c)
case ContextEmpty
have ctxtEmpty_inv: "Γ, Δ ⊨ctxt♢ : U <== T ==> U = T"
by(cases Γ Δ "♢" rule: typing_ctxt.cases, fastforce, simp)
show ?case
using ContextEmpty by (clarsimp dest!: ctxtEmpty_inv simp: ctxt_subst_basecase)
case (ContextApp E x)
then show ?case
by clarsimp (rule rtc_term_trans, rule rtc_appL, assumption, rule step_term, force, clarsimp simp add: ctxt_subst_caseApp(1))
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