(* activate genv *)
inductive lstas (h): nat → relation4 genv lenv term term ≝
-| lstas_sort: ∀G,L,l,k. lstas h l G L (⋆k) (⋆((next h)^l k))
-| lstas_ldef: ∀G,L,K,V,W,U,i,l. ⬇[i] L ≡ K.ⓓV → lstas h l G K V W →
- ⬆[0, i+1] W ≡ U → lstas h l G L (#i) U
+| lstas_sort: ∀G,L,d,k. lstas h d G L (⋆k) (⋆((next h)^d k))
+| lstas_ldef: ∀G,L,K,V,W,U,i,d. ⬇[i] L ≡ K.ⓓV → lstas h d G K V W →
+ ⬆[0, i+1] W ≡ U → lstas h d G L (#i) U
| lstas_zero: ∀G,L,K,W,V,i. ⬇[i] L ≡ K.ⓛW → lstas h 0 G K W V →
lstas h 0 G L (#i) (#i)
-| lstas_succ: ∀G,L,K,W,V,U,i,l. ⬇[i] L ≡ K.ⓛW → lstas h l G K W V →
- ⬆[0, i+1] V ≡ U → lstas h (l+1) G L (#i) U
-| lstas_bind: ∀a,I,G,L,V,T,U,l. lstas h l G (L.ⓑ{I}V) T U →
- lstas h l G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
-| lstas_appl: ∀G,L,V,T,U,l. lstas h l G L T U → lstas h l G L (ⓐV.T) (ⓐV.U)
-| lstas_cast: ∀G,L,W,T,U,l. lstas h l G L T U → lstas h l G L (ⓝW.T) U
+| lstas_succ: ∀G,L,K,W,V,U,i,d. ⬇[i] L ≡ K.ⓛW → lstas h d G K W V →
+ ⬆[0, i+1] V ≡ U → lstas h (d+1) G L (#i) U
+| lstas_bind: ∀a,I,G,L,V,T,U,d. lstas h d G (L.ⓑ{I}V) T U →
+ lstas h d G L (ⓑ{a,I}V.T) (ⓑ{a,I}V.U)
+| lstas_appl: ∀G,L,V,T,U,d. lstas h d G L T U → lstas h d G L (ⓐV.T) (ⓐV.U)
+| lstas_cast: ∀G,L,W,T,U,d. lstas h d G L T U → lstas h d G L (ⓝW.T) U
.
interpretation "nat-iterated static type assignment (term)"
- 'StaticTypeStar h G L l T U = (lstas h l G L T U).
+ 'StaticTypeStar h G L d T U = (lstas h d G L T U).
(* Basic inversion lemmas ***************************************************)
-fact lstas_inv_sort1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀k0. T = ⋆k0 →
- U = ⋆((next h)^l k0).
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #k0 #H destruct //
-| #G #L #K #V #W #U #i #l #_ #_ #_ #k0 #H destruct
+fact lstas_inv_sort1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀k0. T = ⋆k0 →
+ U = ⋆((next h)^d k0).
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #k0 #H destruct //
+| #G #L #K #V #W #U #i #d #_ #_ #_ #k0 #H destruct
| #G #L #K #W #V #i #_ #_ #k0 #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #k0 #H destruct
-| #a #I #G #L #V #T #U #l #_ #k0 #H destruct
-| #G #L #V #T #U #l #_ #k0 #H destruct
-| #G #L #W #T #U #l #_ #k0 #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #k0 #H destruct
+| #a #I #G #L #V #T #U #d #_ #k0 #H destruct
+| #G #L #V #T #U #d #_ #k0 #H destruct
+| #G #L #W #T #U #d #_ #k0 #H destruct
qed-.
(* Basic_1: was just: sty0_gen_sort *)
-lemma lstas_inv_sort1: ∀h,G,L,X,k,l. ⦃G, L⦄ ⊢ ⋆k •*[h, l] X → X = ⋆((next h)^l k).
+lemma lstas_inv_sort1: ∀h,G,L,X,k,d. ⦃G, L⦄ ⊢ ⋆k •*[h, d] X → X = ⋆((next h)^d k).
/2 width=5 by lstas_inv_sort1_aux/
qed-.
-fact lstas_inv_lref1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀j. T = #j → ∨∨
- (∃∃K,V,W. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l] W &
+fact lstas_inv_lref1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀j. T = #j → ∨∨
+ (∃∃K,V,W. ⬇[j] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d] W &
⬆[0, j+1] W ≡ U
) |
(∃∃K,W,V. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V &
- U = #j & l = 0
+ U = #j & d = 0
) |
- (∃∃K,W,V,l0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, l0] V &
- ⬆[0, j+1] V ≡ U & l = l0+1
+ (∃∃K,W,V,d0. ⬇[j] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d0] V &
+ ⬆[0, j+1] V ≡ U & d = d0+1
).
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #j #H destruct
-| #G #L #K #V #W #U #i #l #HLK #HVW #HWU #j #H destruct /3 width=6 by or3_intro0, ex3_3_intro/
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #j #H destruct
+| #G #L #K #V #W #U #i #d #HLK #HVW #HWU #j #H destruct /3 width=6 by or3_intro0, ex3_3_intro/
| #G #L #K #W #V #i #HLK #HWV #j #H destruct /3 width=5 by or3_intro1, ex4_3_intro/
-| #G #L #K #W #V #U #i #l #HLK #HWV #HWU #j #H destruct /3 width=8 by or3_intro2, ex4_4_intro/
-| #a #I #G #L #V #T #U #l #_ #j #H destruct
-| #G #L #V #T #U #l #_ #j #H destruct
-| #G #L #W #T #U #l #_ #j #H destruct
+| #G #L #K #W #V #U #i #d #HLK #HWV #HWU #j #H destruct /3 width=8 by or3_intro2, ex4_4_intro/
+| #a #I #G #L #V #T #U #d #_ #j #H destruct
+| #G #L #V #T #U #d #_ #j #H destruct
+| #G #L #W #T #U #d #_ #j #H destruct
]
qed-.
-lemma lstas_inv_lref1: ∀h,G,L,X,i,l. ⦃G, L⦄ ⊢ #i •*[h, l] X → ∨∨
- (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l] W &
+lemma lstas_inv_lref1: ∀h,G,L,X,i,d. ⦃G, L⦄ ⊢ #i •*[h, d] X → ∨∨
+ (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d] W &
⬆[0, i+1] W ≡ X
) |
(∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, 0] V &
- X = #i & l = 0
+ X = #i & d = 0
) |
- (∃∃K,W,V,l0. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, l0] V &
- ⬆[0, i+1] V ≡ X & l = l0+1
+ (∃∃K,W,V,d0. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d0] V &
+ ⬆[0, i+1] V ≡ X & d = d0+1
).
/2 width=3 by lstas_inv_lref1_aux/
qed-.
X = #i
).
#h #G #L #X #i #H elim (lstas_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
-#K #W #V #
l #_ #_ #_ <plus_n_Sm #H destruct
+#K #W #V #
d #_ #_ #_ <plus_n_Sm #H destruct
qed-.
(* Basic_1: was just: sty0_gen_lref *)
-lemma lstas_inv_lref1_S: ∀h,G,L,X,i,l. ⦃G, L⦄ ⊢ #i •*[h, l+1] X →
- (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, l+1] W &
+lemma lstas_inv_lref1_S: ∀h,G,L,X,i,d. ⦃G, L⦄ ⊢ #i •*[h, d+1] X →
+ (∃∃K,V,W. ⬇[i] L ≡ K.ⓓV & ⦃G, K⦄ ⊢ V •*[h, d+1] W &
⬆[0, i+1] W ≡ X
) ∨
- (∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, l] V &
+ (∃∃K,W,V. ⬇[i] L ≡ K.ⓛW & ⦃G, K⦄ ⊢ W •*[h, d] V &
⬆[0, i+1] V ≡ X
).
-#h #G #L #X #i #l #H elim (lstas_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
+#h #G #L #X #i #d #H elim (lstas_inv_lref1 … H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
#K #W #V #_ #_ #_ <plus_n_Sm #H destruct
qed-.
-fact lstas_inv_gref1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀p0. T = §p0 → ⊥.
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #p0 #H destruct
-| #G #L #K #V #W #U #i #l #_ #_ #_ #p0 #H destruct
+fact lstas_inv_gref1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀p0. T = §p0 → ⊥.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #p0 #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #p0 #H destruct
| #G #L #K #W #V #i #_ #_ #p0 #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #p0 #H destruct
-| #a #I #G #L #V #T #U #l #_ #p0 #H destruct
-| #G #L #V #T #U #l #_ #p0 #H destruct
-| #G #L #W #T #U #l #_ #p0 #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #p0 #H destruct
+| #a #I #G #L #V #T #U #d #_ #p0 #H destruct
+| #G #L #V #T #U #d #_ #p0 #H destruct
+| #G #L #W #T #U #d #_ #p0 #H destruct
qed-.
-lemma lstas_inv_gref1: ∀h,G,L,X,p,l. ⦃G, L⦄ ⊢ §p •*[h, l] X → ⊥.
+lemma lstas_inv_gref1: ∀h,G,L,X,p,d. ⦃G, L⦄ ⊢ §p •*[h, d] X → ⊥.
/2 width=9 by lstas_inv_gref1_aux/
qed-.
-fact lstas_inv_bind1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀b,J,X,Y. T = ⓑ{b,J}Y.X →
- ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •*[h, l] Z & U = ⓑ{b,J}Y.Z.
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #b #J #X #Y #H destruct
-| #G #L #K #V #W #U #i #l #_ #_ #_ #b #J #X #Y #H destruct
+fact lstas_inv_bind1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀b,J,X,Y. T = ⓑ{b,J}Y.X →
+ ∃∃Z. ⦃G, L.ⓑ{J}Y⦄ ⊢ X •*[h, d] Z & U = ⓑ{b,J}Y.Z.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #b #J #X #Y #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #b #J #X #Y #H destruct
| #G #L #K #W #V #i #_ #_ #b #J #X #Y #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #b #J #X #Y #H destruct
-| #a #I #G #L #V #T #U #l #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
-| #G #L #V #T #U #l #_ #b #J #X #Y #H destruct
-| #G #L #W #T #U #l #_ #b #J #X #Y #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #b #J #X #Y #H destruct
+| #a #I #G #L #V #T #U #d #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #V #T #U #d #_ #b #J #X #Y #H destruct
+| #G #L #W #T #U #d #_ #b #J #X #Y #H destruct
]
qed-.
(* Basic_1: was just: sty0_gen_bind *)
-lemma lstas_inv_bind1: ∀h,a,I,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, l] X →
- ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, l] U & X = ⓑ{a,I}V.U.
+lemma lstas_inv_bind1: ∀h,a,I,G,L,V,T,X,d. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T •*[h, d] X →
+ ∃∃U. ⦃G, L.ⓑ{I}V⦄ ⊢ T •*[h, d] U & X = ⓑ{a,I}V.U.
/2 width=3 by lstas_inv_bind1_aux/
qed-.
-fact lstas_inv_appl1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀X,Y. T = ⓐY.X →
- ∃∃Z. ⦃G, L⦄ ⊢ X •*[h, l] Z & U = ⓐY.Z.
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #X #Y #H destruct
-| #G #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
+fact lstas_inv_appl1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀X,Y. T = ⓐY.X →
+ ∃∃Z. ⦃G, L⦄ ⊢ X •*[h, d] Z & U = ⓐY.Z.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #X #Y #H destruct
| #G #L #K #W #V #i #_ #_ #X #Y #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
-| #a #I #G #L #V #T #U #l #_ #X #Y #H destruct
-| #G #L #V #T #U #l #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
-| #G #L #W #T #U #l #_ #X #Y #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #d #_ #X #Y #H destruct
+| #G #L #V #T #U #d #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #W #T #U #d #_ #X #Y #H destruct
]
qed-.
(* Basic_1: was just: sty0_gen_appl *)
-lemma lstas_inv_appl1: ∀h,G,L,V,T,X,l. ⦃G, L⦄ ⊢ ⓐV.T •*[h, l] X →
- ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l] U & X = ⓐV.U.
+lemma lstas_inv_appl1: ∀h,G,L,V,T,X,d. ⦃G, L⦄ ⊢ ⓐV.T •*[h, d] X →
+ ∃∃U. ⦃G, L⦄ ⊢ T •*[h, d] U & X = ⓐV.U.
/2 width=3 by lstas_inv_appl1_aux/
qed-.
-fact lstas_inv_cast1_aux: ∀h,G,L,T,U,l. ⦃G, L⦄ ⊢ T •*[h, l] U → ∀X,Y. T = ⓝY.X →
- ⦃G, L⦄ ⊢ X •*[h, l] U.
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #X #Y #H destruct
-| #G #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
+fact lstas_inv_cast1_aux: ∀h,G,L,T,U,d. ⦃G, L⦄ ⊢ T •*[h, d] U → ∀X,Y. T = ⓝY.X →
+ ⦃G, L⦄ ⊢ X •*[h, d] U.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #X #Y #H destruct
| #G #L #K #W #V #i #_ #_ #X #Y #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
-| #a #I #G #L #V #T #U #l #_ #X #Y #H destruct
-| #G #L #V #T #U #l #_ #X #Y #H destruct
-| #G #L #W #T #U #l #HTU #X #Y #H destruct //
+| #G #L #K #W #V #U #i #d #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #d #_ #X #Y #H destruct
+| #G #L #V #T #U #d #_ #X #Y #H destruct
+| #G #L #W #T #U #d #HTU #X #Y #H destruct //
]
qed-.
(* Basic_1: was just: sty0_gen_cast *)
-lemma lstas_inv_cast1: ∀h,G,L,W,T,U,l. ⦃G, L⦄ ⊢ ⓝW.T •*[h, l] U → ⦃G, L⦄ ⊢ T •*[h, l] U.
+lemma lstas_inv_cast1: ∀h,G,L,W,T,U,d. ⦃G, L⦄ ⊢ ⓝW.T •*[h, d] U → ⦃G, L⦄ ⊢ T •*[h, d] U.
/2 width=4 by lstas_inv_cast1_aux/
qed-.
projects / helm.git / blobdiff