+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/exclaim_5.ma".
-include "basic_2/rt_computation/cpms.ma".
-
-(* NATIVE VALIDITY FOR TERMS ************************************************)
-
-(* activate genv *)
-(* Basic_2A1: uses: snv *)
-inductive nv (a) (h): relation3 genv lenv term ≝
-| nv_sort: ∀G,L,s. nv a h G L (⋆s)
-| nv_zero: ∀I,G,K,V. nv a h G K V → nv a h G (K.ⓑ{I}V) (#0)
-| nv_lref: ∀I,G,K,i. nv a h G K (#i) → nv a h G (K.ⓘ{I}) (#↑i)
-| nv_bind: ∀p,I,G,L,V,T. nv a h G L V → nv a h G (L.ⓑ{I}V) T → nv a h G L (ⓑ{p,I}V.T)
-| nv_appl: ∀n,p,G,L,V,W0,T,U0. (a = Ⓣ → n = 1) → nv a h G L V → nv a h G L T →
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 → ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0 → nv a h G L (ⓐV.T)
-| nv_cast: ∀G,L,U,T,U0. nv a h G L U → nv a h G L T →
- ⦃G, L⦄ ⊢ U ➡*[h] U0 → ⦃G, L⦄ ⊢ T ➡*[1, h] U0 → nv a h G L (ⓝU.T)
-.
-
-interpretation "native validity (term)"
- 'Exclaim a h G L T = (nv a h G L T).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact nv_inv_zero_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → X = #0 →
- ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
-#a #h #G #L #X * -G -L -X
-[ #G #L #s #H destruct
-| #I #G #K #V #HV #_ /2 width=5 by ex2_3_intro/
-| #I #G #K #i #_ #H destruct
-| #p #I #G #L #V #T #_ #_ #H destruct
-| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #H destruct
-| #G #L #U #T #U0 #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma nv_inv_zero (a) (h): ∀G,L. ⦃G, L⦄ ⊢ #0 ![a, h] →
- ∃∃I,K,V. ⦃G, K⦄ ⊢ V ![a, h] & L = K.ⓑ{I}V.
-/2 width=3 by nv_inv_zero_aux/ qed-.
-
-fact nv_inv_lref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀i. X = #(↑i) →
- ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
-#a #h #G #L #X * -G -L -X
-[ #G #L #s #j #H destruct
-| #I #G #K #V #_ #j #H destruct
-| #I #G #L #i #Hi #j #H destruct /2 width=4 by ex2_2_intro/
-| #p #I #G #L #V #T #_ #_ #j #H destruct
-| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #j #H destruct
-| #G #L #U #T #U0 #_ #_ #_ #_ #j #H destruct
-]
-qed-.
-
-lemma nv_inv_lref (a) (h): ∀G,L,i. ⦃G, L⦄ ⊢ #↑i ![a, h] →
- ∃∃I,K. ⦃G, K⦄ ⊢ #i ![a, h] & L = K.ⓘ{I}.
-/2 width=3 by nv_inv_lref_aux/ qed-.
-
-fact nv_inv_gref_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀l. X = §l → ⊥.
-#a #h #G #L #X * -G -L -X
-[ #G #L #s #l #H destruct
-| #I #G #K #V #_ #l #H destruct
-| #I #G #K #i #_ #l #H destruct
-| #p #I #G #L #V #T #_ #_ #l #H destruct
-| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #l #H destruct
-| #G #L #U #T #U0 #_ #_ #_ #_ #l #H destruct
-]
-qed-.
-
-(* Basic_2A1: uses: snv_inv_gref *)
-lemma nv_inv_gref (a) (h): ∀G,L,l. ⦃G, L⦄ ⊢ §l ![a, h] → ⊥.
-/2 width=8 by nv_inv_gref_aux/ qed-.
-
-fact nv_inv_bind_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] →
- ∀p,I,V,T. X = ⓑ{p,I}V.T →
- ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
- & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
-#a #h #G #L #X * -G -L -X
-[ #G #L #s #q #Z #X1 #X2 #H destruct
-| #I #G #K #V #_ #q #Z #X1 #X2 #H destruct
-| #I #G #K #i #_ #q #Z #X1 #X2 #H destruct
-| #p #I #G #L #V #T #HV #HT #q #Z #X1 #X2 #H destruct /2 width=1 by conj/
-| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #q #Z #X1 #X2 #H destruct
-| #G #L #U #T #U0 #_ #_ #_ #_ #q #Z #X1 #X2 #H destruct
-]
-qed-.
-
-(* Basic_2A1: uses: snv_inv_bind *)
-lemma nv_inv_bind (a) (h): ∀p,I,G,L,V,T. ⦃G, L⦄ ⊢ ⓑ{p,I}V.T ![a, h] →
- ∧∧ ⦃G, L⦄ ⊢ V ![a, h]
- & ⦃G, L.ⓑ{I}V⦄ ⊢ T ![a, h].
-/2 width=4 by nv_inv_bind_aux/ qed-.
-
-fact nv_inv_appl_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀V,T. X = ⓐV.T →
- ∃∃n,p,W0,U0. a = Ⓣ → n = 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
-#a #h #G #L #X * -L -X
-[ #G #L #s #X1 #X2 #H destruct
-| #I #G #K #V #_ #X1 #X2 #H destruct
-| #I #G #K #i #_ #X1 #X2 #H destruct
-| #p #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
-| #n #p #G #L #V #W0 #T #U0 #Ha #HV #HT #HVW0 #HTU0 #X1 #X2 #H destruct /3 width=7 by ex5_4_intro/
-| #G #L #U #T #U0 #_ #_ #_ #_ #X1 #X2 #H destruct
-]
-qed-.
-
-(* Basic_2A1: uses: snv_inv_appl *)
-lemma nv_inv_appl (a) (h): ∀G,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ![a, h] →
- ∃∃n,p,W0,U0. a = Ⓣ → n = 1 & ⦃G, L⦄ ⊢ V ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ V ➡*[1, h] W0 & ⦃G, L⦄ ⊢ T ➡*[n, h] ⓛ{p}W0.U0.
-/2 width=3 by nv_inv_appl_aux/ qed-.
-
-fact nv_inv_cast_aux (a) (h): ∀G,L,X. ⦃G, L⦄ ⊢ X ![a, h] → ∀U,T. X = ⓝU.T →
- ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
-#a #h #G #L #X * -G -L -X
-[ #G #L #s #X1 #X2 #H destruct
-| #I #G #K #V #_ #X1 #X2 #H destruct
-| #I #G #K #i #_ #X1 #X2 #H destruct
-| #p #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
-| #n #p #G #L #V #W0 #T #U0 #_ #_ #_ #_ #_ #X1 #X2 #H destruct
-| #G #L #U #T #U0 #HV #HT #HU0 #HTU0 #X1 #X2 #H destruct /2 width=3 by ex4_intro/
-]
-qed-.
-
-(* Basic_2A1: uses: snv_inv_appl *)
-lemma nv_inv_cast (a) (h): ∀G,L,U,T. ⦃G, L⦄ ⊢ ⓝU.T ![a, h] →
- ∃∃U0. ⦃G, L⦄ ⊢ U ![a, h] & ⦃G, L⦄ ⊢ T ![a, h] &
- ⦃G, L⦄ ⊢ U ➡*[h] U0 & ⦃G, L⦄ ⊢ T ➡*[1, h] U0.
-/2 width=3 by nv_inv_cast_aux/ qed-.
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