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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/reduction/cir.ma".
16 include "basic_2/reduction/crx.ma".
18 (* CONTEXT-SENSITIVE EXTENDED IRREDUCIBLE TERMS *****************************)
20 definition cix: ∀h. sd h → lenv → predicate term ≝ λh,g,L,T. ⦃h, L⦄ ⊢ 𝐑[g]⦃T⦄ → ⊥.
22 interpretation "context-sensitive extended irreducibility (term)"
23 'NotReducible h g L T = (cix h g L T).
25 (* Basic inversion lemmas ***************************************************)
27 lemma cix_inv_sort: ∀h,g,L,k,l. deg h g k (l+1) → ⦃h, L⦄ ⊢ 𝐈[g]⦃⋆k⦄ → ⊥.
30 lemma cix_inv_delta: ∀h,g,I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃h, L⦄ ⊢ 𝐈[g]⦃#i⦄ → ⊥.
33 lemma cix_inv_ri2: ∀h,g,I,L,V,T. ri2 I → ⦃h, L⦄ ⊢ 𝐈[g]⦃②{I}V.T⦄ → ⊥.
36 lemma cix_inv_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓑ{a,I}V.T⦄ →
37 ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[g]⦃T⦄.
40 lemma cix_inv_bind: ∀h,g,a,I,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓑ{a,I}V.T⦄ →
41 ∧∧ ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ & ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[g]⦃T⦄ & ib2 a I.
42 #h #g #a * [ elim a -a ]
43 [ #L #V #T #H elim H -H /3 width=1/
44 |*: #L #V #T #H elim (cix_inv_ib2 … H) -H /2 width=1/ /3 width=1/
48 lemma cix_inv_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓐV.T⦄ →
49 ∧∧ ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ & ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ & 𝐒⦃T⦄.
50 #h #g #L #V #T #HVT @and3_intro /3 width=1/
51 generalize in match HVT; -HVT elim T -T //
52 * // #a * #U #T #_ #_ #H elim H -H /2 width=1/
55 lemma cix_inv_flat: ∀h,g,I,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓕ{I}V.T⦄ →
56 ∧∧ ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ & ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ & 𝐒⦃T⦄ & I = Appl.
58 [ elim (cix_inv_appl … H) -H /2 width=1/
59 | elim (cix_inv_ri2 … H) -H /2 width=1/
63 (* Basic forward lemmas *****************************************************)
65 lemma cix_inv_cir: ∀h,g,L,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ → L ⊢ 𝐈⦃T⦄.
68 (* Basic properties *********************************************************)
70 lemma cix_sort: ∀h,g,L,k. deg h g k 0 → ⦃h, L⦄ ⊢ 𝐈[g]⦃⋆k⦄.
71 #h #g #L #k #Hk #H elim (crx_inv_sort … H) -L #l #Hkl
72 lapply (deg_mono … Hk Hkl) -h -k <plus_n_Sm #H destruct
75 lemma tix_lref: ∀h,g,i. ⦃h, ⋆⦄ ⊢ 𝐈[g]⦃#i⦄.
76 #h #g #i #H elim (trx_inv_atom … H) -H #k #l #_ #H destruct
79 lemma cix_gref: ∀h,g,L,p. ⦃h, L⦄ ⊢ 𝐈[g]⦃§p⦄.
80 #h #g #L #p #H elim (crx_inv_gref … H)
83 lemma cix_ib2: ∀h,g,a,I,L,V,T. ib2 a I → ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ → ⦃h, L.ⓑ{I}V⦄ ⊢ 𝐈[g]⦃T⦄ →
84 ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓑ{a,I}V.T⦄.
85 #h #g #a #I #L #V #T #HI #HV #HT #H
86 elim (crx_inv_ib2 … HI H) -HI -H /2 width=1/
89 lemma cix_appl: ∀h,g,L,V,T. ⦃h, L⦄ ⊢ 𝐈[g]⦃V⦄ → ⦃h, L⦄ ⊢ 𝐈[g]⦃T⦄ → 𝐒⦃T⦄ → ⦃h, L⦄ ⊢ 𝐈[g]⦃ⓐV.T⦄.
90 #h #g #L #V #T #HV #HT #H1 #H2
91 elim (crx_inv_appl … H2) -H2 /2 width=1/