(**************************************************************************)
include "basic_2/substitution/ldrop.ma".
-include "basic_2/unfold/frsups.ma".
include "basic_2/static/sd.ma".
(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
.
interpretation "stratified static type assignment (term)"
- 'StaticType h g l L T U = (ssta h g l L T U).
+ 'StaticType h g L T U l = (ssta h g l L T U).
definition ssta_step: ∀h. sd h → lenv → relation term ≝ λh,g,L,T,U.
- ∃l. ⦃h, L⦄ ⊢ T •[g, l+1] U.
+ ∃l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄.
(* Basic inversion lemmas ************************************************)
-fact ssta_inv_sort1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀k0. T = ⋆k0 →
+fact ssta_inv_sort1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀k0. T = ⋆k0 →
deg h g k0 l ∧ U = ⋆(next h k0).
#h #g #L #T #U #l * -L -T -U -l
[ #L #k #l #Hkl #k0 #H destruct /2 width=1/
qed.
(* Basic_1: was just: sty0_gen_sort *)
-lemma ssta_inv_sort1: ∀h,g,L,U,k,l. ⦃h, L⦄ ⊢ ⋆k •[g, l] U →
+lemma ssta_inv_sort1: ∀h,g,L,U,k,l. ⦃h, L⦄ ⊢ ⋆k •[g] ⦃l, U⦄ →
deg h g k l ∧ U = ⋆(next h k).
/2 width=4/ qed-.
-fact ssta_inv_lref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀j. T = #j →
- (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g, l] W &
+fact ssta_inv_lref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀j. T = #j →
+ (∃∃K,V,W. ⇩[0, j] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
⇧[0, j + 1] W ≡ U
) ∨
- (∃∃K,W,V,l0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g, l0] V &
+ (∃∃K,W,V,l0. ⇩[0, j] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
⇧[0, j + 1] W ≡ U & l = l0 + 1
).
#h #g #L #T #U #l * -L -T -U -l
qed.
(* Basic_1: was just: sty0_gen_lref *)
-lemma ssta_inv_lref1: ∀h,g,L,U,i,l. ⦃h, L⦄ ⊢ #i •[g, l] U →
- (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g, l] W &
+lemma ssta_inv_lref1: ∀h,g,L,U,i,l. ⦃h, L⦄ ⊢ #i •[g] ⦃l, U⦄ →
+ (∃∃K,V,W. ⇩[0, i] L ≡ K. ⓓV & ⦃h, K⦄ ⊢ V •[g] ⦃l, W⦄ &
⇧[0, i + 1] W ≡ U
) ∨
- (∃∃K,W,V,l0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g, l0] V &
+ (∃∃K,W,V,l0. ⇩[0, i] L ≡ K. ⓛW & ⦃h, K⦄ ⊢ W •[g] ⦃l0, V⦄ &
⇧[0, i + 1] W ≡ U & l = l0 + 1
).
/2 width=3/ qed-.
-fact ssta_inv_gref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀p0. T = §p0 → ⊥.
+fact ssta_inv_gref1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀p0. T = §p0 → ⊥.
#h #g #L #T #U #l * -L -T -U -l
[ #L #k #l #_ #p0 #H destruct
| #L #K #V #W #U #i #l #_ #_ #_ #p0 #H destruct
| #L #W #T #U #l #_ #p0 #H destruct
qed.
-lemma ssta_inv_gref1: ∀h,g,L,U,p,l. ⦃h, L⦄ ⊢ §p •[g, l] U → ⊥.
+lemma ssta_inv_gref1: ∀h,g,L,U,p,l. ⦃h, L⦄ ⊢ §p •[g] ⦃l, U⦄ → ⊥.
/2 width=9/ qed-.
-fact ssta_inv_bind1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U →
+fact ssta_inv_bind1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
∀a,I,X,Y. T = ⓑ{a,I}Y.X →
- ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g, l] Z & U = ⓑ{a,I}Y.Z.
+ ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
#h #g #L #T #U #l * -L -T -U -l
[ #L #k #l #_ #a #I #X #Y #H destruct
| #L #K #V #W #U #i #l #_ #_ #_ #a #I #X #Y #H destruct
qed.
(* Basic_1: was just: sty0_gen_bind *)
-lemma ssta_inv_bind1: ∀h,g,a,I,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •[g, l] U →
- ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g, l] Z & U = ⓑ{a,I}Y.Z.
+lemma ssta_inv_bind1: ∀h,g,a,I,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓑ{a,I}Y.X •[g] ⦃l, U⦄ →
+ ∃∃Z. ⦃h, L.ⓑ{I}Y⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓑ{a,I}Y.Z.
/2 width=3/ qed-.
-fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ∀X,Y. T = ⓐY.X →
- ∃∃Z. ⦃h, L⦄ ⊢ X •[g, l] Z & U = ⓐY.Z.
+fact ssta_inv_appl1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ∀X,Y. T = ⓐY.X →
+ ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
#h #g #L #T #U #l * -L -T -U -l
[ #L #k #l #_ #X #Y #H destruct
| #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
qed.
(* Basic_1: was just: sty0_gen_appl *)
-lemma ssta_inv_appl1: ∀h,g,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓐY.X •[g, l] U →
- ∃∃Z. ⦃h, L⦄ ⊢ X •[g, l] Z & U = ⓐY.Z.
+lemma ssta_inv_appl1: ∀h,g,L,Y,X,U,l. ⦃h, L⦄ ⊢ ⓐY.X •[g] ⦃l, U⦄ →
+ ∃∃Z. ⦃h, L⦄ ⊢ X •[g] ⦃l, Z⦄ & U = ⓐY.Z.
/2 width=3/ qed-.
-fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U →
- ∀X,Y. T = ⓝY.X → ⦃h, L⦄ ⊢ X •[g, l] U.
+fact ssta_inv_cast1_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ →
+ ∀X,Y. T = ⓝY.X → ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.
#h #g #L #T #U #l * -L -T -U -l
[ #L #k #l #_ #X #Y #H destruct
| #L #K #V #W #U #l #i #_ #_ #_ #X #Y #H destruct
qed.
(* Basic_1: was just: sty0_gen_cast *)
-lemma ssta_inv_cast1: ∀h,g,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X •[g, l] U →
- ⦃h, L⦄ ⊢ X •[g, l] U.
+lemma ssta_inv_cast1: ∀h,g,L,X,Y,U,l. ⦃h, L⦄ ⊢ ⓝY.X •[g] ⦃l, U⦄ →
+ ⦃h, L⦄ ⊢ X •[g] ⦃l, U⦄.
/2 width=4/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma ssta_inv_frsupp: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ⦃L, U⦄ ⧁+ ⦃L, T⦄ → ⊥.
-#h #g #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #l #_ #H
- elim (frsupp_inv_atom1_frsups … H)
-| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H
- elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
- elim (lift_inv_lref2_be … H ? ?) -H //
-| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H
- elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
- elim (lift_inv_lref2_be … H ? ?) -H //
-| #a #I #L #V #T #U #l #_ #IHTU #H
- elim (frsupp_inv_bind1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
- lapply (frsups_fwd_fw … H) -H normalize
- <associative_plus <associative_plus #H
- elim (le_plus_xySz_x_false … H)
-| #L #V #T #U #l #_ #IHTU #H
- elim (frsupp_inv_flat1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
- lapply (frsups_fwd_fw … H) -H normalize
- <associative_plus <associative_plus #H
- elim (le_plus_xySz_x_false … H)
-| /3 width=4/
-]
-qed-.
-
-fact ssta_inv_refl_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → T = U → ⊥.
-#h #g #L #T #U #l #H elim H -L -T -U -l
-[ #L #k #l #_ #H
- lapply (next_lt h k) destruct -H -e0 (**) (* destruct: these premises are not erased *)
- <e1 -e1 #H elim (lt_refl_false … H)
-| #L #K #V #W #U #i #l #_ #_ #HWU #_ #H destruct
- elim (lift_inv_lref2_be … HWU ? ?) -HWU //
-| #L #K #W #V #U #i #l #_ #_ #HWU #_ #H destruct
- elim (lift_inv_lref2_be … HWU ? ?) -HWU //
-| #a #I #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
-| #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
-| #L #W #T #U #l #HTU #_ #H destruct
- elim (ssta_inv_frsupp … HTU ?) -HTU /2 width=1/
-]
-qed-.
-
-lemma ssta_inv_refl: ∀h,g,T,L,l. ⦃h, L⦄ ⊢ T •[g, l] T → ⊥.
-/2 width=8 by ssta_inv_refl_aux/ qed-.
-
-lemma ssta_inv_frsups: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g, l] U → ⦃L, U⦄ ⧁* ⦃L, T⦄ → ⊥.
-#h #g #L #T #U #L #HTU #H elim (frsups_inv_all … H) -H
-[ * #_ #H destruct /2 width=6 by ssta_inv_refl/
-| /2 width=8 by ssta_inv_frsupp/
-]
-qed-.