(* *)
(**************************************************************************)
-include "basic_2/notation/relations/lrsubeqv_4.ma".
+include "basic_2/notation/relations/lrsubeqv_5.ma".
include "basic_2/dynamic/snv.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
(* Note: this is not transitive *)
-inductive lsubsv (h:sh) (g:sd h): relation lenv ≝
-| lsubsv_atom: lsubsv h g (⋆) (⋆)
-| lsubsv_pair: ∀I,L1,L2,V. lsubsv h g L1 L2 →
- lsubsv h g (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubsv_abbr: ∀L1,L2,W,V,W1,V2,l. ⦃h, L1⦄ ⊢ ⓝW.V ¡[h, g] → ⦃h, L2⦄ ⊢ W ¡[h, g] →
- ⦃h, L1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ → ⦃h, L2⦄ ⊢ W •[h, g] ⦃l, V2⦄ →
- lsubsv h g L1 L2 → lsubsv h g (L1.ⓓⓝW.V) (L2.ⓛW)
+inductive lsubsv (h) (g) (G): relation lenv ≝
+| lsubsv_atom: lsubsv h g G (⋆) (⋆)
+| lsubsv_pair: ∀I,L1,L2,V. lsubsv h g G L1 L2 →
+ lsubsv h g G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubsv_abbr: ∀L1,L2,W,V,W1,V2,l. ⦃G, L1⦄ ⊢ ⓝW.V ¡[h, g] → ⦃G, L2⦄ ⊢ W ¡[h, g] →
+ ⦃G, L1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ → ⦃G, L2⦄ ⊢ W •[h, g] ⦃l, V2⦄ →
+ lsubsv h g G L1 L2 → lsubsv h g G (L1.ⓓⓝW.V) (L2.ⓛW)
.
interpretation
"local environment refinement (stratified native validity)"
- 'LRSubEqV h g L1 L2 = (lsubsv h g L1 L2).
+ 'LRSubEqV h g G L1 L2 = (lsubsv h g G L1 L2).
(* Basic inversion lemmas ***************************************************)
-fact lsubsv_inv_atom1_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 → L1 = ⋆ → L2 = ⋆.
-#h #g #L1 #L2 * -L1 -L2
+fact lsubsv_inv_atom1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 → L1 = ⋆ → L2 = ⋆.
+#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
| #L1 #L2 #W #V #V1 #V2 #l #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom1: ∀h,g,L2. h ⊢ ⋆ ¡⊑[h, g] L2 → L2 = ⋆.
-/2 width=5 by lsubsv_inv_atom1_aux/ qed-.
+lemma lsubsv_inv_atom1: ∀h,g,G,L2. G ⊢ ⋆ ¡⊑[h, g] L2 → L2 = ⋆.
+/2 width=6 by lsubsv_inv_atom1_aux/ qed-.
-fact lsubsv_inv_pair1_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
+fact lsubsv_inv_pair1_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
∀I,K1,X. L1 = K1.ⓑ{I}X →
- (∃∃K2. h ⊢ K1 ¡⊑[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[h, g] & ⦃h, K2⦄ ⊢ W ¡[h, g] &
- ⦃h, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
- h ⊢ K1 ¡⊑[h, g] K2 &
+ (∃∃K2. G ⊢ K1 ¡⊑[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,W1,V2,l. ⦃G, K1⦄ ⊢ X ¡[h, g] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+ ⦃G, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃G, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
+ G ⊢ K1 ¡⊑[h, g] K2 &
I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
-#h #g #L1 #L2 * -L1 -L2
+#h #g #G #L1 #L2 * -L1 -L2
[ #J #K1 #X #H destruct
| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3/
| #L1 #L2 #W #V #W1 #V2 #l #HV #HW #HW1 #HV2 #HL12 #J #K1 #X #H destruct /3 width=12/
]
qed-.
-lemma lsubsv_inv_pair1: ∀h,g,I,K1,L2,X. h ⊢ K1.ⓑ{I}X ¡⊑[h, g] L2 →
- (∃∃K2. h ⊢ K1 ¡⊑[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,W,V,W1,V2,l. ⦃h, K1⦄ ⊢ X ¡[h, g] & ⦃h, K2⦄ ⊢ W ¡[h, g] &
- ⦃h, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
- h ⊢ K1 ¡⊑[h, g] K2 &
+lemma lsubsv_inv_pair1: ∀h,g,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ¡⊑[h, g] L2 →
+ (∃∃K2. G ⊢ K1 ¡⊑[h, g] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,W,V,W1,V2,l. ⦃G, K1⦄ ⊢ X ¡[h, g] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+ ⦃G, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃G, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
+ G ⊢ K1 ¡⊑[h, g] K2 &
I = Abbr & L2 = K2.ⓛW & X = ⓝW.V.
/2 width=3 by lsubsv_inv_pair1_aux/ qed-.
-fact lsubsv_inv_atom2_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 → L2 = ⋆ → L1 = ⋆.
-#h #g #L1 #L2 * -L1 -L2
+fact lsubsv_inv_atom2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 → L2 = ⋆ → L1 = ⋆.
+#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
| #L1 #L2 #W #V #V1 #V2 #l #_ #_ #_ #_ #_ #H destruct
]
qed-.
-lemma lsubsv_inv_atom2: ∀h,g,L1. h ⊢ L1 ¡⊑[h, g] ⋆ → L1 = ⋆.
-/2 width=5 by lsubsv_inv_atom2_aux/ qed-.
+lemma lsubsv_inv_atom2: ∀h,g,G,L1. G ⊢ L1 ¡⊑[h, g] ⋆ → L1 = ⋆.
+/2 width=6 by lsubsv_inv_atom2_aux/ qed-.
-fact lsubsv_inv_pair2_aux: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
+fact lsubsv_inv_pair2_aux: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
∀I,K2,W. L2 = K2.ⓑ{I}W →
- (∃∃K1. h ⊢ K1 ¡⊑[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[h, g] & ⦃h, K2⦄ ⊢ W ¡[h, g] &
- ⦃h, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
- h ⊢ K1 ¡⊑[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
-#h #g #L1 #L2 * -L1 -L2
+ (∃∃K1. G ⊢ K1 ¡⊑[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,W1,V2,l. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+ ⦃G, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃G, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
+ G ⊢ K1 ¡⊑[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+#h #g #G #L1 #L2 * -L1 -L2
[ #J #K2 #U #H destruct
| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3/
| #L1 #L2 #W #V #W1 #V2 #l #HV #HW #HW1 #HV2 #HL12 #J #K2 #U #H destruct /3 width=10/
]
qed-.
-lemma lsubsv_inv_pair2: ∀h,g,I,L1,K2,W. h ⊢ L1 ¡⊑[h, g] K2.ⓑ{I}W →
- (∃∃K1. h ⊢ K1 ¡⊑[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
- ∃∃K1,V,W1,V2,l. ⦃h, K1⦄ ⊢ ⓝW.V ¡[h, g] & ⦃h, K2⦄ ⊢ W ¡[h, g] &
- ⦃h, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃h, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
- h ⊢ K1 ¡⊑[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
+lemma lsubsv_inv_pair2: ∀h,g,I,G,L1,K2,W. G ⊢ L1 ¡⊑[h, g] K2.ⓑ{I}W →
+ (∃∃K1. G ⊢ K1 ¡⊑[h, g] K2 & L1 = K1.ⓑ{I}W) ∨
+ ∃∃K1,V,W1,V2,l. ⦃G, K1⦄ ⊢ ⓝW.V ¡[h, g] & ⦃G, K2⦄ ⊢ W ¡[h, g] &
+ ⦃G, K1⦄ ⊢ V •[h, g] ⦃l+1, W1⦄ & ⦃G, K2⦄ ⊢ W •[h, g] ⦃l, V2⦄ &
+ G ⊢ K1 ¡⊑[h, g] K2 & I = Abst & L1 = K1. ⓓⓝW.V.
/2 width=3 by lsubsv_inv_pair2_aux/ qed-.
(* Basic_forward lemmas *****************************************************)
-lemma lsubsv_fwd_lsubr: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 → L1 ⊑ L2.
-#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+lemma lsubsv_fwd_lsubr: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 → L1 ⊑ L2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
qed-.
(* Basic properties *********************************************************)
-lemma lsubsv_refl: ∀h,g,L. h ⊢ L ¡⊑[h, g] L.
-#h #g #L elim L -L // /2 width=1/
+lemma lsubsv_refl: ∀h,g,G,L. G ⊢ L ¡⊑[h, g] L.
+#h #g #G #L elim L -L // /2 width=1/
qed.
-lemma lsubsv_cprs_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
- ∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* T2.
-/3 width=5 by lsubsv_fwd_lsubr, lsubr_cprs_trans/
+lemma lsubsv_cprs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
+ ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ➡* T2 → ⦃G, L1⦄ ⊢ T1 ➡* T2.
+/3 width=6 by lsubsv_fwd_lsubr, lsubr_cprs_trans/
qed-.
(* Properties on context-sensitive parallel equivalence for terms ***********)
-lemma lsubsv_cpcs_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
- ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
-/3 width=5 by lsubsv_fwd_lsubr, lsubr_cpcs_trans/
+lemma lsubsv_cpcs_trans: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
+ ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+/3 width=6 by lsubsv_fwd_lsubr, lsubr_cpcs_trans/
qed-.
(* Properties for the preservation results **********************************)
-fact lsubsv_sstas_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
- ∀L2,T. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L2, T⦄ → ⦃h, L2⦄ ⊢ T ¡[h, g] →
- ∀L1. h ⊢ L1 ¡⊑[h, g] L2 → ∀U2. ⦃h, L2⦄ ⊢ T •*[h, g] U2 →
- ∃∃U1. ⦃h, L1⦄ ⊢ T •*[h, g] U1 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 #T #HLT0 #HT #L1 #HL12 #U2 #H @(sstas_ind … H) -U2 [ /2 width=3/ ]
+fact lsubsv_sstas_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsubsv h g G1 L1 T1) →
+ ∀G,L2,T. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L2, T⦄ → ⦃G, L2⦄ ⊢ T ¡[h, g] →
+ ∀L1. G ⊢ L1 ¡⊑[h, g] L2 → ∀U2. ⦃G, L2⦄ ⊢ T •*[h, g] U2 →
+ ∃∃U1. ⦃G, L1⦄ ⊢ T •*[h, g] U1 & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L2 #T #HLT0 #HT #L1 #HL12 #U2 #H @(sstas_ind … H) -U2 [ /2 width=3/ ]
#U2 #W #l #HTU2 #HU2W * #U1 #HTU1 #HU12
lapply (IH1 … HT … HL12) // #H
-lapply (snv_sstas_aux … IH2 … HTU1) // /3 width=4 by ygt_yprs_trans, lsubsv_yprs/ -H #HU1
+lapply (snv_sstas_aux … IH2 … HTU1) // /3 width=5 by ygt_yprs_trans, lsubsv_yprs/ -H #HU1
lapply (snv_sstas_aux … IH2 … HTU2) // #H
-lapply (IH1 … H … HL12) [ /3 width=4 by ygt_yprs_trans, sstas_yprs/ ] -H #HU2
+lapply (IH1 … H … HL12) [ /3 width=5 by ygt_yprs_trans, sstas_yprs/ ] -H #HU2
elim (snv_fwd_ssta … HU1) #l1 #W1 #HUW1
elim (lsubsv_ssta_trans … HU2W … HL12) -HU2W #W2 #HUW2 #HW2
elim (ssta_cpcs_lpr_aux … IH4 IH3 … HU1 HU2 … HUW1 … HUW2 … HU12 L1) -HU1 -HU2 -HUW2 -HU12 //
-[2,3: /4 width=4 by ygt_yprs_trans, sstas_yprs, lsubsv_yprs/ ] -L2 -L0 -T0 -U2 #H #HW12 destruct
+[2,3: /4 width=5 by ygt_yprs_trans, sstas_yprs, lsubsv_yprs/ ] -G0 -L2 -L0 -T0 -U2 #H #HW12 destruct
lapply (cpcs_trans … HW12 … HW2) -W2 /3 width=4/
qed-.
-fact lsubsv_cpds_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
- ∀L2,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L2, T1⦄ → ⦃h, L2⦄ ⊢ T1 ¡[h, g] →
- ∀L1. h ⊢ L1 ¡⊑[h, g] L2 → ∀T2. ⦃h, L2⦄ ⊢ T1 •*➡*[h, g] T2 →
- ∃∃T. ⦃h, L1⦄ ⊢ T1 •*➡*[h, g] T & L1 ⊢ T2 ➡* T.
-#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 #T1 #HLT0 #HT1 #L1 #HL12 #T2 * #T #HT1T #HTT2
+fact lsubsv_cpds_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsubsv h g G1 L1 T1) →
+ ∀G,L2,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G,L2, T1⦄ → ⦃G, L2⦄ ⊢ T1 ¡[h, g] →
+ ∀L1. G ⊢ L1 ¡⊑[h, g] L2 → ∀T2. ⦃G, L2⦄ ⊢ T1 •*➡*[h, g] T2 →
+ ∃∃T. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g] T & ⦃G, L1⦄ ⊢ T2 ➡* T.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L2 #T1 #HLT0 #HT1 #L1 #HL12 #T2 * #T #HT1T #HTT2
lapply (lsubsv_cprs_trans … HL12 … HTT2) -HTT2 #HTT2
-elim (lsubsv_sstas_aux … IH4 IH3 IH2 IH1 … HLT0 … HL12 … HT1T) // -L2 -L0 -T0 #T0 #HT10 #HT0
+elim (lsubsv_sstas_aux … IH4 IH3 IH2 IH1 … HLT0 … HL12 … HT1T) // -G0 -L2 -L0 -T0 #T0 #HT10 #HT0
lapply (cpcs_cprs_strap1 … HT0 … HTT2) -T #HT02
elim (cpcs_inv_cprs … HT02) -HT02 /3 width=3/
qed-.
(* Properties concerning basic local environment slicing ********************)
(* Note: the constant 0 cannot be generalized *)
-lemma lsubsv_ldrop_O1_conf: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
+lemma lsubsv_ldrop_O1_conf: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
∀K1,e. ⇩[0, e] L1 ≡ K1 →
- ∃∃K2. h ⊢ K1 ¡⊑[h, g] K2 & ⇩[0, e] L2 ≡ K2.
-#h #g #L1 #L2 #H elim H -L1 -L2
+ ∃∃K2. G ⊢ K1 ¡⊑[h, g] K2 & ⇩[0, e] L2 ≡ K2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3/
| #I #L1 #L2 #V #_ #IHL12 #K1 #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK1
qed-.
(* Note: the constant 0 cannot be generalized *)
-lemma lsubsv_ldrop_O1_trans: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 →
+lemma lsubsv_ldrop_O1_trans: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 →
∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. h ⊢ K1 ¡⊑[h, g] K2 & ⇩[0, e] L1 ≡ K1.
-#h #g #L1 #L2 #H elim H -L1 -L2
+ ∃∃K1. G ⊢ K1 ¡⊑[h, g] K2 & ⇩[0, e] L1 ≡ K1.
+#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3/
| #I #L1 #L2 #V #_ #IHL12 #K2 #e #H
elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK2
(* Forward lemmas on lenv refinement for atomic arity assignment ************)
-lemma lsubsv_fwd_lsuba: ∀h,g,L1,L2. h ⊢ L1 ¡⊑[h, g] L2 → L1 ⁝⊑ L2.
-#h #g #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
+lemma lsubsv_fwd_lsuba: ∀h,g,G,L1,L2. G ⊢ L1 ¡⊑[h, g] L2 → G ⊢ L1 ⁝⊑ L2.
+#h #g #G #L1 #L2 #H elim H -L1 -L2 // /2 width=1/
#L1 #L2 #W #V #W1 #V2 #l #HV #HW #_ #_ #_ #IHL12 -W1 -V2
elim (snv_fwd_aaa … HV) -HV #A #HV
elim (snv_fwd_aaa … HW) -HW #B #HW
(* Properties concerning stratified native validity *************************)
-fact snv_lsubsv_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
- ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_lsubsv h g L1 T1.
-#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 * * [||||*] //
-[ #i #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
+fact snv_lsubsv_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsubsv h g G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_lsubsv h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L2 * * [||||*] //
+[ #i #HG0 #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_lref … H) -H #I2 #K2 #W2 #HLK2 #HW2
elim (lsubsv_ldrop_O1_trans … HL12 … HLK2) -HL12 #X #H #HLK1
elim (lsubsv_inv_pair2 … H) -H * #K1
/5 width=8 by snv_lref, fsupp_ygt, fsupp_lref/ (**) (* auto too slow without trace *)
| #V #W1 #V2 #l #HV #_ #_ #_ #_ #_ #H destruct /2 width=5/
]
-| #p #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2 -IH1
+| #p #HG0 #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2 -IH1
elim (snv_inv_gref … H)
-| #a #I #V #T #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
+| #a #I #V #T #HG0 #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_bind … H) -H /4 width=4/
-| #V #T #HL0 #HT0 #H #L1 #HL12 destruct
+| #V #T #HG0 #HL0 #HT0 #H #L1 #HL12 destruct
elim (snv_inv_appl … H) -H #a #W #W0 #U #l #HV #HT #HVW #HW0 #HTU
lapply (lsubsv_cprs_trans … HL12 … HW0) -HW0 #HW0
elim (lsubsv_ssta_trans … HVW … HL12) -HVW #W1 #HVW1 #HW1
elim (cpcs_inv_cprs … H) -H #W0 #HW10 #HW0
lapply (cpds_cprs_trans … (ⓛ{a}W0.U2) HTU ?) [ /2 width=1/ ] -HTU -HW0
/4 width=8 by snv_appl, fsupp_ygt/ (**) (* auto too slow without trace *)
-| #W #T #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
+| #W #T #HG0 #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_cast … H) -H #U #l #HW #HT #HTU #HUW
lapply (lsubsv_cpcs_trans … HL12 … HUW) -HUW #HUW
elim (lsubsv_ssta_trans … HTU … HL12) -HTU #U0 #HTU0 #HU0
(* Properties on stratified static type assignment **************************)
-lemma lsubsv_ssta_trans: ∀h,g,L2,T,U2,l. ⦃h, L2⦄ ⊢ T •[h, g] ⦃l, U2⦄ →
- ∀L1. h ⊢ L1 ¡⊑[h, g] L2 →
- ∃∃U1. ⦃h, L1⦄ ⊢ T •[h, g] ⦃l, U1⦄ & L1 ⊢ U1 ⬌* U2.
-#h #g #L2 #T #U #l #H elim H -L2 -T -U -l
+lemma lsubsv_ssta_trans: ∀h,g,G,L2,T,U2,l. ⦃G, L2⦄ ⊢ T •[h, g] ⦃l, U2⦄ →
+ ∀L1. G ⊢ L1 ¡⊑[h, g] L2 →
+ ∃∃U1. ⦃G, L1⦄ ⊢ T •[h, g] ⦃l, U1⦄ & ⦃G, L1⦄ ⊢ U1 ⬌* U2.
+#h #g #G #L2 #T #U #l #H elim H -G -L2 -T -U -l
[ /3 width=3/
-| #L2 #K2 #X #Y #U #i #l #HLK2 #_ #HYU #IHXY #L1 #HL12
+| #G #L2 #K2 #X #Y #U #i #l #HLK2 #_ #HYU #IHXY #L1 #HL12
elim (lsubsv_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
elim (lsubsv_inv_pair2 … H) -H * #K1 [ | -HYU -IHXY -HLK1 ]
[ #HK12 #H destruct
elim (lift_total T 0 (i+1)) /3 width=11/
| #V #W1 #V2 #l0 #_ #_ #_ #_ #_ #H destruct
]
-| #L2 #K2 #Y #X #U #i #l #HLK2 #HYX #HYU #IHYX #L1 #HL12
+| #G #L2 #K2 #Y #X #U #i #l #HLK2 #HYX #HYU #IHYX #L1 #HL12
elim (lsubsv_ldrop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
elim (lsubsv_inv_pair2 … H) -H * #K1 [ -HYX | -IHYX ]
[ #HK12 #H destruct
elim (lift_total W 0 (i+1))
/4 width=11 by cpcs_lift, ex2_intro, ssta_ldef, ssta_cast/
]
-| #a #I #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #HL12
+| #a #I #G #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #HL12
elim (IHTU2 (L1.ⓑ{I}V2) …) [2: /2 width=1/ ] -L2 /3 width=3/
-| #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #HL12
+| #G #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #HL12
elim (IHTU2 … HL12) -L2 /3 width=5/
-| #L2 #W2 #T2 #U2 #l #_ #IHTU2 #L1 #HL12
+| #G #L2 #W2 #T2 #U2 #l #_ #IHTU2 #L1 #HL12
elim (IHTU2 … HL12) -L2 /3 width=3/
]
qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/nativevalid_4.ma".
+include "basic_2/notation/relations/nativevalid_5.ma".
include "basic_2/computation/cpds.ma".
include "basic_2/equivalence/cpcs.ma".
(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
-inductive snv (h:sh) (g:sd h): lenv → predicate term ≝
-| snv_sort: ∀L,k. snv h g L (⋆k)
-| snv_lref: ∀I,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → snv h g K V → snv h g L (#i)
-| snv_bind: ∀a,I,L,V,T. snv h g L V → snv h g (L.ⓑ{I}V) T → snv h g L (ⓑ{a,I}V.T)
-| snv_appl: ∀a,L,V,W,W0,T,U,l. snv h g L V → snv h g L T →
+(* activate genv *)
+inductive snv (h:sh) (g:sd h): relation3 genv lenv term ≝
+| snv_sort: ∀G,L,k. snv h g G L (⋆k)
+| snv_lref: ∀I,G,L,K,V,i. ⇩[0, i] L ≡ K.ⓑ{I}V → snv h g G K V → snv h g G L (#i)
+| snv_bind: ∀a,I,G,L,V,T. snv h g G L V → snv h g G (L.ⓑ{I}V) T → snv h g G L (ⓑ{a,I}V.T)
+| snv_appl: ∀a,G,L,V,W,W0,T,U,l. snv h g G L V → snv h g G L T →
⦃G, L⦄ ⊢ V •[h, g] ⦃l+1, W⦄ → ⦃G, L⦄ ⊢ W ➡* W0 →
- ⦃G, L⦄ ⊢ T •*➡*[h, g] ⓛ{a}W0.U → snv h g L (ⓐV.T)
-| snv_cast: ∀L,W,T,U,l. snv h g L W → snv h g L T →
- ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ → ⦃G, L⦄ ⊢ U ⬌* W → snv h g L (ⓝW.T)
+ ⦃G, L⦄ ⊢ T •*➡*[h, g] ⓛ{a}W0.U → snv h g G L (ⓐV.T)
+| snv_cast: ∀G,L,W,T,U,l. snv h g G L W → snv h g G L T →
+ ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ → ⦃G, L⦄ ⊢ U ⬌* W → snv h g G L (ⓝW.T)
.
interpretation "stratified native validity (term)"
- 'NativeValid h g L T = (snv h g L T).
+ 'NativeValid h g G L T = (snv h g G L T).
(* Basic inversion lemmas ***************************************************)
-fact snv_inv_lref_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀i. X = #i →
- ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ¡[h, g].
-#h #g #L #X * -L -X
-[ #L #k #i #H destruct
-| #I #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5/
-| #a #I #L #V #T #_ #_ #i #H destruct
-| #a #L #V #W #W0 #T #U #l #_ #_ #_ #_ #_ #i #H destruct
-| #L #W #T #U #l #_ #_ #_ #_ #i #H destruct
+fact snv_inv_lref_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀i. X = #i →
+ ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ¡[h, g].
+#h #g #G #L #X * -G -L -X
+[ #G #L #k #i #H destruct
+| #I #G #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5/
+| #a #I #G #L #V #T #_ #_ #i #H destruct
+| #a #G #L #V #W #W0 #T #U #l #_ #_ #_ #_ #_ #i #H destruct
+| #G #L #W #T #U #l #_ #_ #_ #_ #i #H destruct
]
-qed.
-
-lemma snv_inv_lref: ∀h,g,L,i. ⦃G, L⦄ ⊢ #i ¡[h, g] →
- ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ¡[h, g].
-/2 width=3/ qed-.
-
-fact snv_inv_gref_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀p. X = §p → ⊥.
-#h #g #L #X * -L -X
-[ #L #k #p #H destruct
-| #I #L #K #V #i #_ #_ #p #H destruct
-| #a #I #L #V #T #_ #_ #p #H destruct
-| #a #L #V #W #W0 #T #U #l #_ #_ #_ #_ #_ #p #H destruct
-| #L #W #T #U #l #_ #_ #_ #_ #p #H destruct
+qed-.
+
+lemma snv_inv_lref: ∀h,g,G,L,i. ⦃G, L⦄ ⊢ #i ¡[h, g] →
+ ∃∃I,K,V. ⇩[0, i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ¡[h, g].
+/2 width=3 by snv_inv_lref_aux/ qed-.
+
+fact snv_inv_gref_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀p. X = §p → ⊥.
+#h #g #G #L #X * -G -L -X
+[ #G #L #k #p #H destruct
+| #I #G #L #K #V #i #_ #_ #p #H destruct
+| #a #I #G #L #V #T #_ #_ #p #H destruct
+| #a #G #L #V #W #W0 #T #U #l #_ #_ #_ #_ #_ #p #H destruct
+| #G #L #W #T #U #l #_ #_ #_ #_ #p #H destruct
]
-qed.
-
-lemma snv_inv_gref: ∀h,g,L,p. ⦃G, L⦄ ⊢ §p ¡[h, g] → ⊥.
-/2 width=7/ qed-.
-
-fact snv_inv_bind_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀a,I,V,T. X = ⓑ{a,I}V.T →
- ⦃G, L⦄ ⊢ V ¡[h, g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ T ¡[h, g].
-#h #g #L #X * -L -X
-[ #L #k #a #I #V #T #H destruct
-| #I0 #L #K #V0 #i #_ #_ #a #I #V #T #H destruct
-| #b #I0 #L #V0 #T0 #HV0 #HT0 #a #I #V #T #H destruct /2 width=1/
-| #b #L #V0 #W0 #W00 #T0 #U0 #l #_ #_ #_ #_ #_ #a #I #V #T #H destruct
-| #L #W0 #T0 #U0 #l #_ #_ #_ #_ #a #I #V #T #H destruct
+qed-.
+
+lemma snv_inv_gref: ∀h,g,G,L,p. ⦃G, L⦄ ⊢ §p ¡[h, g] → ⊥.
+/2 width=8 by snv_inv_gref_aux/ qed-.
+
+fact snv_inv_bind_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀a,I,V,T. X = ⓑ{a,I}V.T →
+ ⦃G, L⦄ ⊢ V ¡[h, g] ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ T ¡[h, g].
+#h #g #G #L #X * -G -L -X
+[ #G #L #k #a #I #V #T #H destruct
+| #I0 #G #L #K #V0 #i #_ #_ #a #I #V #T #H destruct
+| #b #I0 #G #L #V0 #T0 #HV0 #HT0 #a #I #V #T #H destruct /2 width=1/
+| #b #G #L #V0 #W0 #W00 #T0 #U0 #l #_ #_ #_ #_ #_ #a #I #V #T #H destruct
+| #G #L #W0 #T0 #U0 #l #_ #_ #_ #_ #a #I #V #T #H destruct
]
-qed.
+qed-.
-lemma snv_inv_bind: ∀h,g,a,I,L,V,T. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T ¡[h, g] →
- ⦃G, L⦄ ⊢ V ¡[h, g] ∧ ⦃h, L.ⓑ{I}V⦄ ⊢ T ¡[h, g].
-/2 width=4/ qed-.
+lemma snv_inv_bind: ∀h,g,a,I,G,L,V,T. ⦃G, L⦄ ⊢ ⓑ{a,I}V.T ¡[h, g] →
+ ⦃G, L⦄ ⊢ V ¡[h, g] ∧ ⦃G, L.ⓑ{I}V⦄ ⊢ T ¡[h, g].
+/2 width=4 by snv_inv_bind_aux/ qed-.
-fact snv_inv_appl_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀V,T. X = ⓐV.T →
+fact snv_inv_appl_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀V,T. X = ⓐV.T →
∃∃a,W,W0,U,l. ⦃G, L⦄ ⊢ V ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
⦃G, L⦄ ⊢ V •[h, g] ⦃l+1, W⦄ & ⦃G, L⦄ ⊢ W ➡* W0 &
⦃G, L⦄ ⊢ T •*➡*[h, g] ⓛ{a}W0.U.
-#h #g #L #X * -L -X
-[ #L #k #V #T #H destruct
-| #I #L #K #V0 #i #_ #_ #V #T #H destruct
-| #a #I #L #V0 #T0 #_ #_ #V #T #H destruct
-| #a #L #V0 #W0 #W00 #T0 #U0 #l #HV0 #HT0 #HVW0 #HW00 #HTU0 #V #T #H destruct /2 width=8/
-| #L #W0 #T0 #U0 #l #_ #_ #_ #_ #V #T #H destruct
+#h #g #G #L #X * -L -X
+[ #G #L #k #V #T #H destruct
+| #I #G #L #K #V0 #i #_ #_ #V #T #H destruct
+| #a #I #G #L #V0 #T0 #_ #_ #V #T #H destruct
+| #a #G #L #V0 #W0 #W00 #T0 #U0 #l #HV0 #HT0 #HVW0 #HW00 #HTU0 #V #T #H destruct /2 width=8/
+| #G #L #W0 #T0 #U0 #l #_ #_ #_ #_ #V #T #H destruct
]
-qed.
+qed-.
-lemma snv_inv_appl: ∀h,g,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ¡[h, g] →
+lemma snv_inv_appl: ∀h,g,G,L,V,T. ⦃G, L⦄ ⊢ ⓐV.T ¡[h, g] →
∃∃a,W,W0,U,l. ⦃G, L⦄ ⊢ V ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
⦃G, L⦄ ⊢ V •[h, g] ⦃l+1, W⦄ & ⦃G, L⦄ ⊢ W ➡* W0 &
⦃G, L⦄ ⊢ T •*➡*[h, g] ⓛ{a}W0.U.
-/2 width=3/ qed-.
+/2 width=3 by snv_inv_appl_aux/ qed-.
-fact snv_inv_cast_aux: ∀h,g,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀W,T. X = ⓝW.T →
+fact snv_inv_cast_aux: ∀h,g,G,L,X. ⦃G, L⦄ ⊢ X ¡[h, g] → ∀W,T. X = ⓝW.T →
∃∃U,l. ⦃G, L⦄ ⊢ W ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ & ⦃G, L⦄ ⊢ U ⬌* W.
-#h #g #L #X * -L -X
-[ #L #k #W #T #H destruct
-| #I #L #K #V #i #_ #_ #W #T #H destruct
-| #a #I #L #V #T0 #_ #_ #W #T #H destruct
-| #a #L #V #W0 #W00 #T0 #U #l #_ #_ #_ #_ #_ #W #T #H destruct
-| #L #W0 #T0 #U0 #l #HW0 #HT0 #HTU0 #HUW0 #W #T #H destruct /2 width=4/
+#h #g #G #L #X * -G -L -X
+[ #G #L #k #W #T #H destruct
+| #I #G #L #K #V #i #_ #_ #W #T #H destruct
+| #a #I #G #L #V #T0 #_ #_ #W #T #H destruct
+| #a #G #L #V #W0 #W00 #T0 #U #l #_ #_ #_ #_ #_ #W #T #H destruct
+| #G #L #W0 #T0 #U0 #l #HW0 #HT0 #HTU0 #HUW0 #W #T #H destruct /2 width=4/
]
-qed.
+qed-.
-lemma snv_inv_cast: ∀h,g,L,W,T. ⦃G, L⦄ ⊢ ⓝW.T ¡[h, g] →
+lemma snv_inv_cast: ∀h,g,G,L,W,T. ⦃G, L⦄ ⊢ ⓝW.T ¡[h, g] →
∃∃U,l. ⦃G, L⦄ ⊢ W ¡[h, g] & ⦃G, L⦄ ⊢ T ¡[h, g] &
⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ & ⦃G, L⦄ ⊢ U ⬌* W.
-/2 width=3/ qed-.
+/2 width=3 by snv_inv_cast_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma snv_fwd_ssta: ∀h,g,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃∃l,U. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄.
-#h #g #L #T #H elim H -L -T
-[ #L #k elim (deg_total h g k) /3 width=3/
-| * #L #K #V #i #HLK #_ * #l0 #W #HVW
+lemma snv_fwd_ssta: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃∃l,U. ⦃G, L⦄ ⊢ T •[h, g] ⦃l, U⦄.
+#h #g #G #L #T #H elim H -G -L -T
+[ #G #L #k elim (deg_total h g k) /3 width=3/
+| * #G #L #K #V #i #HLK #_ * #l0 #W #HVW
[ elim (lift_total W 0 (i+1)) /3 width=8/
| elim (lift_total V 0 (i+1)) /3 width=8/
]
-| #a #I #L #V #T #_ #_ #_ * /3 width=3/
-| #a #L #V #W #W1 #T0 #T1 #l #_ #_ #_ #_ #_ #_ * /3 width=3/
-| #L #W #T #U #l #_ #_ #HTU #_ #_ #_ /3 width=3/ (**) (* auto fails without the last #_ *)
+| #a #I #G #L #V #T #_ #_ #_ * /3 width=3/
+| #a #G #L #V #W #W1 #T0 #T1 #l #_ #_ #_ #_ #_ #_ * /3 width=3/
+| #G #L #W #T #U #l #_ #_ #HTU #_ #_ #_ /3 width=3/ (**) (* auto fails without the last #_ *)
]
qed-.
(* Forward lemmas on atomic arity assignment for terms **********************)
-lemma snv_fwd_aaa: ∀h,g,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃A. ⦃G, L⦄ ⊢ T ⁝ A.
-#h #g #L #T #H elim H -L -T
+lemma snv_fwd_aaa: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ∃A. ⦃G, L⦄ ⊢ T ⁝ A.
+#h #g #G #L #T #H elim H -G -L -T
[ /2 width=2/
-| #I #L #K #V #i #HLK #_ * /3 width=6/
-| #a * #L #V #T #_ #_ * #B #HV * #A #HA /3 width=2/
-| #a #L #V #W #W0 #T #U #l #_ #_ #HVW #HW0 #HTU * #B #HV * #X #HT
+| #I #G #L #K #V #i #HLK #_ * /3 width=6/
+| #a * #G #L #V #T #_ #_ * #B #HV * #A #HA /3 width=2/
+| #a #G #L #V #W #W0 #T #U #l #_ #_ #HVW #HW0 #HTU * #B #HV * #X #HT
lapply (cpds_aaa h g … HV W0 ?) [ -HTU /3 width=4/ ] -W #HW0 (**) (* auto fail without -HTU *)
lapply (cpds_aaa … HT … HTU) -HTU #H
elim (aaa_inv_abst … H) -H #B0 #A #H1 #HU #H2 destruct
lapply (aaa_mono … H1 … HW0) -W0 #H destruct /3 width=4/
-| #L #W #T #U #l #_ #_ #HTU #HUW * #B #HW * #A #HT
+| #G #L #W #T #U #l #_ #_ #HTU #HUW * #B #HW * #A #HT
lapply (aaa_cpcs_mono … HUW A … HW) -HUW /2 width=7/ -HTU #H destruct /3 width=3/
]
qed-.
-lemma snv_fwd_csn: ∀h,g,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2/
+lemma snv_fwd_csn: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
+#h #g #G #L #T #H elim (snv_fwd_aaa … H) -H /2 width=2/
qed-.
(* Inductive premises for the preservation results **************************)
-definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation2 lenv term ≝
- λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[h, g].
-
-definition IH_ssta_cpr_lpr: ∀h:sh. sd h → relation2 lenv term ≝
- λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ →
- ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
-
-definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
- λh,g,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
+definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+
+definition IH_ssta_cpr_lpr: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L1,T1. ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀U1,l. ⦃G, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄ & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+
+definition IH_snv_ssta: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L,T. ⦃G, L⦄ ⊢ T ¡[h, g] →
∀U,l. ⦃G, L⦄ ⊢ T •[h, g] ⦃l+1, U⦄ → ⦃G, L⦄ ⊢ U ¡[h, g].
-definition IH_snv_lsubsv: ∀h:sh. sd h → relation2 lenv term ≝
- λh,g,L2,T. ⦃h, L2⦄ ⊢ T ¡[h, g] →
- ∀L1. h ⊢ L1 ¡⊑[h, g] L2 → ⦃h, L1⦄ ⊢ T ¡[h, g].
+definition IH_snv_lsubsv: ∀h:sh. sd h → relation3 genv lenv term ≝
+ λh,g,G,L2,T. ⦃G, L2⦄ ⊢ T ¡[h, g] →
+ ∀L1. G ⊢ L1 ¡⊑[h, g] L2 → ⦃G, L1⦄ ⊢ T ¡[h, g].
(* Properties for the preservation results **********************************)
-fact snv_cprs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[h, g].
-#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #T2 #H
+fact snv_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G0 #L0 #T0 #IH #G #L1 #T1 #HLT0 #HT1 #T2 #H
elim H -T2 [ /2 width=6/ ] -HT1
/4 width=6 by ygt_yprs_trans, cprs_yprs/
qed-.
-fact ssta_cprs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ →
- ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #T2 #H
+fact ssta_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀U1,l. ⦃G, L1⦄ ⊢ T1 •[h, g] ⦃l, U1⦄ →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •[h, g] ⦃l, U2⦄ & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #T2 #H
elim H -T2 [ /2 width=7/ ]
#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
elim (IHT1 L1) // -IHT1 #U #HTU #HU1
elim (IH1 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
-[2: /3 width=9 by snv_cprs_lpr_aux/
+[2: /3 width=10 by snv_cprs_lpr_aux/
|3: /5 width=6 by ygt_yprs_trans, cprs_yprs/
-] -L0 -T0 -T1 -T #U2 #HTU2 #HU2
+] -G0 -L0 -T0 -T1 -T #U2 #HTU2 #HU2
lapply (lpr_cpcs_conf … HL12 … HU1) -L1 #HU1
lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
qed-.
-fact ssta_cpcs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1,T2. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T2⦄ →
- ⦃h, L1⦄ ⊢ T1 ¡[h, g] → ⦃h, L1⦄ ⊢ T2 ¡[h, g] →
- ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l1, U1⦄ →
- ∀U2,l2. ⦃h, L1⦄ ⊢ T2 •[h, g] ⦃l2, U2⦄ →
- L1 ⊢ T1 ⬌* T2 → ∀L2. L1 ⊢ ➡ L2 →
- l1 = l2 ∧ L2 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #T2 #H01 #H02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H #L2 #HL12
+fact ssta_cpcs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1,T2. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L1, T1⦄ → ⦃G0, L0, T0⦄ >[h, g] ⦃G, L1, T2⦄ →
+ ⦃G, L1⦄ ⊢ T1 ¡[h, g] → ⦃G, L1⦄ ⊢ T2 ¡[h, g] →
+ ∀U1,l1. ⦃G, L1⦄ ⊢ T1 •[h, g] ⦃l1, U1⦄ →
+ ∀U2,l2. ⦃G, L1⦄ ⊢ T2 •[h, g] ⦃l2, U2⦄ →
+ ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ l1 = l2 ∧ ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #T2 #H01 #H02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H #L2 #HL12
elim (cpcs_inv_cprs … H) -H #T #H1 #H2
elim (ssta_cprs_lpr_aux … H01 HT1 … HTU1 … H1 … HL12) -T1 /2 width=1/ #W1 #H1 #HUW1
elim (ssta_cprs_lpr_aux … H02 HT2 … HTU2 … H2 … HL12) -T2 /2 width=1/ #W2 #H2 #HUW2 -L0 -T0
lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
qed-.
-fact snv_sstas_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- ∀L,T. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
+fact snv_sstas_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ ∀G,L,T. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L, T⦄ → ⦃G, L⦄ ⊢ T ¡[h, g] →
∀U. ⦃G, L⦄ ⊢ T •*[h, g] U → ⦃G, L⦄ ⊢ U ¡[h, g].
-#h #g #L0 #T0 #IH #L #T #H01 #HT #U #H
+#h #g #G0 #L0 #T0 #IH #G #L #T #H01 #HT #U #H
@(sstas_ind … H) -U // -HT /4 width=5 by ygt_yprs_trans, sstas_yprs/
qed-.
-fact snv_sstas_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- ∀L1,T. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T⦄ → ⦃h, L1⦄ ⊢ T ¡[h, g] →
- ∀U. ⦃h, L1⦄ ⊢ T •*[h, g] U → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ U ¡[h, g].
-/4 width=7 by snv_sstas_aux, ygt_yprs_trans, sstas_yprs/
+fact snv_sstas_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ ∀G,L1,T. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L1, T⦄ → ⦃G, L1⦄ ⊢ T ¡[h, g] →
+ ∀U. ⦃G, L1⦄ ⊢ T •*[h, g] U → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ U ¡[h, g].
+/4 width=8 by snv_sstas_aux, ygt_yprs_trans, sstas_yprs/
qed-.
-fact sstas_cprs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1. ⦃h, L1⦄ ⊢ T1 •*[h, g] U1 → ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[h, g] U2 & L2 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #H
+fact sstas_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*[h, g] U1 → ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*[h, g] U2 & ⦃G, L2⦄ ⊢ U1 ⬌* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 #H
@(sstas_ind … H) -U1 [ /3 width=5 by lpr_cprs_conf, ex2_intro/ ]
#U1 #W1 #l1 #HTU1 #HUW1 #IHTU1 #T2 #HT12 #L2 #HL12
elim (IHTU1 … HT12 … HL12) -IHTU1 #U2 #HTU2 #HU12
lapply (snv_cprs_lpr_aux … IH2 … HT1 … HT12 … HL12) // #HT2
elim (snv_sstas_fwd_correct … HTU2) // #W2 #l2 #HUW2
elim (IH1 … HUW1 U1 … HL12) -HUW1 //
-[2: /3 width=7 by snv_sstas_aux/
-|3: /3 width=4 by ygt_yprs_trans, sstas_yprs/
+[2: /3 width=8 by snv_sstas_aux/
+|3: /3 width=5 by ygt_yprs_trans, sstas_yprs/
] #W #HU1W #HW1
elim (ssta_cpcs_lpr_aux … IH2 IH1 … HU1W … HUW2 … HU12 L2) // -IH1 -HU1W -HU12
-[2: /4 width=8 by snv_sstas_aux, ygt_yprs_trans, cprs_lpr_yprs/
-|3: /3 width=10 by snv_sstas_lpr_aux/
+[2: /4 width=9 by snv_sstas_aux, ygt_yprs_trans, cprs_lpr_yprs/
+|3: /3 width=11 by snv_sstas_lpr_aux/
|4,5: /4 width=5 by ygt_yprs_trans, cprs_lpr_yprs, sstas_yprs/
-] -L0 -T0 -L1 -T1 -HT2 #H #HW12 destruct
+] -G0 -L0 -T0 -L1 -T1 -HT2 #H #HW12 destruct
lapply (cpcs_trans … HW1 … HW12) -W /3 width=4/
qed-.
-fact cpds_cprs_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[h, g] →
- ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
- ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[h, g] U2 & L2 ⊢ U1 ➡* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+fact cpds_cprs_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ ∀G,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L1, T1⦄ → ⦃G, L1⦄ ⊢ T1 ¡[h, g] →
+ ∀U1. ⦃G, L1⦄ ⊢ T1 •*➡*[h, g] U1 →
+ ∀T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+ ∃∃U2. ⦃G, L2⦄ ⊢ T2 •*➡*[h, g] U2 & ⦃G, L2⦄ ⊢ U1 ➡* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
elim (sstas_cprs_lpr_aux … IH3 IH2 IH1 … H01 … HTW1 … HT12 … HL12) // -L0 -T0 -T1 #W2 #HTW2 #HW12
lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
elim (cpcs_inv_cprs … H) -H /3 width=3/
qed-.
-fact ssta_cpds_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
+fact ssta_cpds_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ ∀G,L,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G, L, T1⦄ → ⦃G, L⦄ ⊢ T1 ¡[h, g] →
∀l,U1. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l+1, U1⦄ → ∀T2. ⦃G, L⦄ ⊢ T1 •*➡*[h, g] T2 →
∃∃U,U2. ⦃G, L⦄ ⊢ U1 •*[h, g] U & ⦃G, L⦄ ⊢ T2 •*[h, g] U2 & ⦃G, L⦄ ⊢ U ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
elim (sstas_strip … HT1T … HTU1) #HU1T destruct [ -HT1T | -L0 -T0 -T1 ]
-[ elim (ssta_cprs_lpr_aux … IH2 IH1 … HTU1 … HTT2 L) // -L0 -T0 -T /3 width=5/
+[ elim (ssta_cprs_lpr_aux … IH2 IH1 … HTU1 … HTT2 L) // -G0 -L0 -T0 -T /3 width=5/
| @(ex3_2_intro …T2 HU1T) // /2 width=1/
]
qed-.
(* Relocation properties ****************************************************)
-lemma snv_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ T ¡[h, g] → ∀L,d,e. ⇩[d, e] L ≡ K →
+lemma snv_lift: ∀h,g,G,K,T. ⦃G, K⦄ ⊢ T ¡[h, g] → ∀L,d,e. ⇩[d, e] L ≡ K →
∀U. ⇧[d, e] T ≡ U → ⦃G, L⦄ ⊢ U ¡[h, g].
-#h #g #K #T #H elim H -K -T
-[ #K #k #L #d #e #_ #X #H
+#h #g #G #K #T #H elim H -G -K -T
+[ #G #K #k #L #d #e #_ #X #H
>(lift_inv_sort1 … H) -X -K -d -e //
-| #I #K #K0 #V #i #HK0 #_ #IHV #L #d #e #HLK #X #H
+| #I #G #K #K0 #V #i #HK0 #_ #IHV #L #d #e #HLK #X #H
elim (lift_inv_lref1 … H) * #Hid #H destruct
[ elim (ldrop_trans_le … HLK … HK0) -K /2 width=2/ #X #HL0 #H
elim (ldrop_inv_skip2 … H) -H /2 width=1/ -Hid #L0 #W #HLK0 #HVW #H destruct
/3 width=8/
| lapply (ldrop_trans_ge … HLK … HK0 ?) -K // -Hid /3 width=8/
]
-| #a #I #K #V #T #_ #_ #IHV #IHT #L #d #e #HLK #X #H
+| #a #I #G #K #V #T #_ #_ #IHV #IHT #L #d #e #HLK #X #H
elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct
/4 width=4/
-| #a #K #V #V0 #V1 #T #T1 #l #_ #_ #HV0 #HV01 #HT1 #IHV #IHT #L #d #e #HLK #X #H
+| #a #G #K #V #V0 #V1 #T #T1 #l #_ #_ #HV0 #HV01 #HT1 #IHV #IHT #L #d #e #HLK #X #H
elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
elim (lift_total V0 d e) #W0 #HVW0
elim (lift_total V1 d e) #W1 #HVW1
elim (lift_total T1 (d+1) e) #U1 #HTU1
@(snv_appl … a … W0 … W1 … U1 l)
[ /2 width=4/ | /2 width=4/ | /2 width=9/ | /2 width=9/ ]
- @(cpds_lift … HLK … HTU … HT1) /2 width=1/
-| #K #V0 #T #V #l #_ #_ #HTV #HV0 #IHV0 #IHT #L #d #e #HLK #X #H
+ @(cpds_lift … HT1 … HLK … HTU) /2 width=1/
+| #G #K #V0 #T #V #l #_ #_ #HTV #HV0 #IHV0 #IHT #L #d #e #HLK #X #H
elim (lift_inv_flat1 … H) -H #W0 #U #HVW0 #HTU #H destruct
elim (lift_total V d e) #W #HVW
@(snv_cast … W l) [ /2 width=4/ | /2 width=4/ | /2 width=9/ | /2 width=9/ ]
]
qed.
-lemma snv_inv_lift: ∀h,g,L,U. ⦃G, L⦄ ⊢ U ¡[h, g] → ∀K,d,e. ⇩[d, e] L ≡ K →
- ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊢ T ¡[h, g].
-#h #g #L #U #H elim H -L -U
-[ #L #k #K #d #e #_ #X #H
+lemma snv_inv_lift: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ U ¡[h, g] → ∀K,d,e. ⇩[d, e] L ≡ K →
+ ∀T. ⇧[d, e] T ≡ U → ⦃G, K⦄ ⊢ T ¡[h, g].
+#h #g #G #L #U #H elim H -G -L -U
+[ #G #L #k #K #d #e #_ #X #H
>(lift_inv_sort2 … H) -X -L -d -e //
-| #I #L #L0 #W #i #HL0 #_ #IHW #K #d #e #HLK #X #H
+| #I #G #L #L0 #W #i #HL0 #_ #IHW #K #d #e #HLK #X #H
elim (lift_inv_lref2 … H) * #Hid #H destruct
[ elim (ldrop_conf_le … HLK … HL0) -L /2 width=2/ #X #HK0 #H
elim (ldrop_inv_skip1 … H) -H /2 width=1/ -Hid #K0 #V #HLK0 #HVW #H destruct
/3 width=8/
| lapply (ldrop_conf_ge … HLK … HL0 ?) -L // -Hid /3 width=8/
]
-| #a #I #L #W #U #_ #_ #IHW #IHU #K #d #e #HLK #X #H
+| #a #I #G #L #W #U #_ #_ #IHW #IHU #K #d #e #HLK #X #H
elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=4/
-| #a #L #W #W0 #W1 #U #U1 #l #_ #_ #HW0 #HW01 #HU1 #IHW #IHU #K #d #e #HLK #X #H
+| #a #G #L #W #W0 #W1 #U #U1 #l #_ #_ #HW0 #HW01 #HU1 #IHW #IHU #K #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
elim (ssta_inv_lift1 … HW0 … HLK … HVW) -HW0 #V0 #HV0 #HVW0
elim (cprs_inv_lift1 … HW01 … HLK … HVW0) -W0 #V1 #HVW1 #HV01
- elim (cpds_inv_lift1 … HLK … HTU … HU1) -HU1 #X #H #HTU
+ elim (cpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #H #HTU
elim (lift_inv_bind2 … H) -H #Y #T1 #HY #HTU1 #H destruct
lapply (lift_inj … HY … HVW1) -HY #H destruct /3 width=8/
-| #L #W0 #U #W #l #_ #_ #HUW #HW0 #IHW0 #IHU #K #d #e #HLK #X #H
+| #G #L #W0 #U #W #l #_ #_ #HUW #HW0 #IHW0 #IHU #K #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #V0 #T #HVW0 #HTU #H destruct
elim (ssta_inv_lift1 … HUW … HLK … HTU) -HUW #V #HTV #HVW
- lapply (cpcs_inv_lift … HLK … HVW … HVW0 ?) // -W /3 width=4/
+ lapply (cpcs_inv_lift G … HLK … HVW … HVW0 ?) // -W /3 width=4/
]
qed-.
(* Advanced properties ******************************************************)
-lemma snv_fsup_conf: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ →
- ⦃h, L1⦄ ⊢ T1 ¡[h, g] → ⦃h, L2⦄ ⊢ T2 ¡[h, g].
-#h #g #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2
-[ #I1 #L1 #V1 #H
+lemma snv_fsup_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I1 #G1 #L1 #V1 #H
elim (snv_inv_lref … H) -H #I2 #L2 #V2 #H #HV2
lapply (ldrop_inv_O2 … H) -H #H destruct //
|2: *
|5,6: /3 width=7 by snv_inv_lift/
]
-[1,3: #a #I #L1 #V1 #T1 #H elim (snv_inv_bind … H) -H //
-|2,4: * #L1 #V1 #T1 #H
+[1,3: #a #I #G1 #L1 #V1 #T1 #H elim (snv_inv_bind … H) -H //
+|2,4: * #G1 #L1 #V1 #T1 #H
[1,3: elim (snv_inv_appl … H) -H //
|2,4: elim (snv_inv_cast … H) -H //
]
(* Properties on context-free parallel reduction for local environments *****)
-fact snv_cpr_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h g L1 T1.
-#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L1 * * [||||*]
-[ #k #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
+fact snv_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsubsv h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_cpr_lpr h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [||||*]
+[ #k #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #_ destruct -IH4 -IH3 -IH2 -IH1 -H1
>(cpr_inv_sort1 … H2) -X //
-| #i #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
+| #i #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
- lapply (fsupp_lref … HLK1) #HKL
+ lapply (fsupp_lref … G1 … HLK1) #HKL
elim (cpr_inv_lref1 … H2) -H2
[ #H destruct -HLK1
lapply (IH1 … HV12 … HK12) -IH1 -HV12 -HK12 // -HV1 [ /2 width=1/ ] -HKL /2 width=5/
lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
lapply (IH1 … HVW0 … HK12) -IH1 -HVW0 -HK12 // -HV1 [ /2 width=1/ ] -HKL /3 width=7/
]
-| #p #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
+| #p #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2 -IH1
elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_bind … H1) -H1 #HV1 #HT1
elim (cpr_inv_bind1 … H2) -H2 *
[ #V2 #T2 #HV12 #HT12 #H destruct
lapply (IH1 … HT1 … HT12 (L2.ⓓV1) ?) -IH1 [1,2: /2 width=1/ ] -L1 -T1 #HT2
lapply (snv_inv_lift … HT2 L2 … HXT2) -T2 // /2 width=1/
]
-| #V1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
elim (snv_inv_appl … H1) -H1 #a #W10 #W1 #U1 #l #HV1 #HT1 #HVW1 #HW10 #HTU1
elim (cpr_inv_appl1 … H2) -H2 *
[ #V2 #T2 #HV12 #HT12 #H destruct -IH4
lapply (IH1 … HW202 … HL12) // [ /2 width=1/ ] -HW20 #HW2
lapply (IH1 … HT20 … HT202 … (L2.ⓛW2) ?) [1,2: /2 width=1/ ] -HT20 #HT2
lapply (IH2 … HV2W3) //
- [ @(ygt_yprs_trans … L1 L1 … V1) (**) (* auto /4 width=5/ is a bit slow even with trace *)
+ [ @(ygt_yprs_trans … G1 G1 … L1 L1 … V1) (**) (* auto /4 width=5/ is a bit slow even with trace *)
[ /2 width=1 by fsupp_ygt/
| /3 width=1 by cprs_lpr_yprs, cpr_cprs/
]
elim (ssta_fwd_correct … HV2W3) <minus_plus_m_m #U3 #HWU3
elim (ssta_cpcs_lpr_aux … IH1 IH3 … HWU3 … HWU2 … HW32 … L2) // -IH3
[2: /4 width=5 by ygt_yprs_trans, fsupp_ygt, cprs_lpr_yprs, cpr_cprs/
- |3: @(ygt_yprs_trans … L1 L2 … V2) (**) (* auto not tried *)
- [ @(ygt_yprs_trans … L1 L1 … V1)
+ |3: @(ygt_yprs_trans … G1 G1 L1 L2 … V2) (**) (* auto not tried *)
+ [ @(ygt_yprs_trans … G1 G1 L1 L1 … V1)
[ /2 width=1 by fsupp_ygt/
| /3 width=1 by cprs_lpr_yprs, cpr_cprs/
]
] #H #_ destruct -IH2 -U3
lapply (IH4 … HT2 (L2.ⓓⓝW2.V2) ?)
[ /3 width=5/
- | @(ygt_yprs_trans … (L1.ⓛW20) … T2) (**) (* auto /5 width=5/ is too slow even with trace timeout=35 *)
- [ /4 width=4 by ygt_yprs_trans, fsupp_ygt, ypr_yprs, ypr_cpr/
+ | @(ygt_yprs_trans … G1 G1 … (L1.ⓛW20) … T2) (**) (* auto /5 width=5/ is too slow even with trace timeout=35 *)
+ [ /4 width=5 by ygt_yprs_trans, fsupp_ygt, ypr_yprs, ypr_cpr/
| /4 width=1 by ypr_yprs, ypr_lpr, lpr_pair/
]
] -L1 -V1 -T20 -U2 /3 width=4/
lapply (cpcs_canc_sn … H2 HW10) -W10 #H2
elim (lift_total X 0 1) #W20 #H3
lapply (ssta_lift … H1 (L2.ⓓW2) … HV02 … H3) /2 width=1/ -H1 #HVW20
- lapply (cpcs_lift (L2.ⓓW2) … H3 … HW13 H2) /2 width=1/ -HW13 -H3 -H2 #HW320
+ lapply (cpcs_lift … (L2.ⓓW2) … H3 … HW13 H2) /2 width=1/ -HW13 -H3 -H2 #HW320
lapply (cpcs_cprs_strap1 … HW320 … HW1) -W3 #HW20
elim (cpcs_inv_cprs … HW20) -HW20 #W3 #HW203 #HW3
lapply (cpds_cprs_trans … (ⓛ{a}W3.U2) HTU2 ?) [ /2 width=1/ ] -HW3 -HTU2 #HTU2
lapply (IH1 … HT02 (L2.ⓓW2) ?) // [1,2: /2 width=1/ ] -L1 #HT2
lapply (snv_lift … HV0 (L2.ⓓW2) … HV02) /2 width=1/ -HV0 -HV02 /3 width=8/
]
-| #W1 #T1 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH2
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct -IH4 -IH2
elim (snv_inv_cast … H1) -H1 #U1 #l #HW1 #HT1 #HTU1 #HUW1
elim (cpr_inv_cast1 … H2) -H2
[ * #W2 #T2 #HW12 #HT12 #H destruct
(* Properties on stratified static type assignment for terms ****************)
-fact snv_ssta_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_ssta h g L1 T1.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [||||*]
-[ #k #HL0 #HT0 #_ #X #l #H2 destruct -IH3 -IH2 -IH1
+fact snv_ssta_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_ssta h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [||||*]
+[ #k #HG0 #HL0 #HT0 #_ #X #l #H2 destruct -IH3 -IH2 -IH1
elim (ssta_inv_sort1 … H2) -H2 #_ #H destruct //
-| #i #HL0 #HT0 #H1 #X #l #H2 destruct -IH3 -IH2
+| #i #HG0 #HL0 #HT0 #H1 #X #l #H2 destruct -IH3 -IH2
elim (snv_inv_lref … H1) -H1 #I #K1 #V1 #HLK1 #HV1
elim (ssta_inv_lref1 … H2) -H2 * #K0 #V0 #W1 [| #l ] #H #HVW1 #HX [| #_ ]
lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (fsupp_lref … HLK1) #H
+ lapply (fsupp_lref … G1 … HLK1) #H
lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /4 width=7/
-| #p #HL0 #HT0 #H1 #X #l #H2 destruct -IH3 -IH2 -IH1
+| #p #HG0 #HL0 #HT0 #H1 #X #l #H2 destruct -IH3 -IH2 -IH1
elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HL0 #HT0 #H1 #X #l #H2 destruct -IH3 -IH2
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #l #H2 destruct -IH3 -IH2
elim (snv_inv_bind … H1) -H1 #HV1 #HT1
elim (ssta_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct /4 width=5/
-| #V1 #T1 #HL0 #HT0 #H1 #X #l #H2 destruct
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #l #H2 destruct
elim (snv_inv_appl … H1) -H1 #a #W1 #W0 #T0 #l0 #HV1 #HT1 #HVW1 #HW10 #HT10
elim (ssta_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
lapply (IH1 … HT1 … HTU1) -IH1 /2 width=1/ #HU1
elim (cpcs_inv_abst2 … HU0) -HU0 #W2 #U2 #HU2 #HU02
elim (cprs_inv_abst … HU02) -HU02 #HW02 #_
lapply (cprs_trans … HW10 … HW02) -W0 /3 width=10 by snv_appl, ex2_intro/ (**) (* auto is too slow without trace *)
-| #W1 #T1 #HL0 #HT0 #H1 #X #l #H2 destruct -IH3 -IH2
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #X #l #H2 destruct -IH3 -IH2
elim (snv_inv_cast … H1) -H1 #U1 #l0 #HW1 #HT1 #HTU1 #HUW1
lapply (ssta_inv_cast1 … H2) -H2 /3 width=5/
]
(* Properties on sn parallel reduction for local environments ***************)
-fact ssta_cpr_lpr_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[h, g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- ∀L1,T1. L0 = L1 → T0 = T1 → IH_ssta_cpr_lpr h g L1 T1.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 * * [|||| *]
-[ #k #_ #_ #_ #X2 #l #H2 #X3 #H3 #L2 #HL12 -IH3 -IH2 -IH1
+fact ssta_cpr_lpr_aux: ∀h,g,G0,L0,T0.
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_ssta h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_ssta_cpr_lpr h g G1 L1 T1) →
+ ∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_ssta_cpr_lpr h g G1 L1 T1.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| *]
+[ #k #_ #_ #_ #_ #X2 #l #H2 #X3 #H3 #L2 #HL12 -IH3 -IH2 -IH1
elim (ssta_inv_sort1 … H2) -H2 #Hkl #H destruct
>(cpr_inv_sort1 … H3) -X3 /4 width=6/
-| #i #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+| #i #HG0 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
elim (snv_inv_lref … H1) -H1 #I0 #K0 #V0 #H #HV1
elim (ssta_inv_lref1 … H2) -H2 * #K1
#V1 #W1 [| #l0 ] #HLK1 #HVW1 #HWU1 [| #H destruct ]
lapply (ldrop_mono … H … HLK1) -H #H destruct
- lapply (fsupp_lref … HLK1) #HKV1
+ lapply (fsupp_lref … G1 … HLK1) #HKV1
elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
elim (lpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
lapply (ldrop_fwd_ldrop2 … HLK2) #H2
lapply (ssta_lift … HVW2 … H2 … HW0 … HWU2) -HVW2 -HW0
lapply (cpcs_lift … H2 … HWU1 … HWU2 HW12) -H2 -W1 /3 width=6/
]
-| #p #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
+| #p #_ #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
elim (snv_inv_gref … H1)
-| #a #I #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
elim (snv_inv_bind … H1) -H1 #_ #HT1
elim (ssta_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
elim (cpr_inv_bind1 … H3) -H3 *
lapply (cpcs_bind1 true … V1 V1 … HU12) // -HU12 #HU12
lapply (cpcs_cpr_strap1 … HU12 U ?) -HU12 /2 width=3/
]
-| #V1 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct
elim (snv_inv_appl … H1) -H1 #a #W1 #W10 #U10 #l0 #HV1 #HT1 #HVW1 #HW10 #HTU10
elim (ssta_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
elim (cpr_inv_appl1 … H3) -H3 *
lapply (IH3 … HVW1) -IH3 // [ /2 width=1/ ] #HW1
elim (ssta_cpcs_lpr_aux … IH2 IH1 … HWX1 … HWV22 … L1) -HWX1 //
[2: /2 width=1/
- |3: /4 width=4 by ygt_yprs_trans, fsupp_ygt, sstas_yprs, ssta_sstas/
+ |3: /4 width=5 by ygt_yprs_trans, fsupp_ygt, sstas_yprs, ssta_sstas/
] #H #_ destruct -X1
lapply (IH2 … HV1 … HV12 … HL12) [ /2 width=1/ ] #HV2
lapply (IH2 … HW2 … HW20 … HL12) [ /2 width=1/ ] -IH2 #H2W20
lapply (cpcs_flat … HV10 … HU02 Appl) -HV10 -HU02 #HU02
lapply (cpcs_cpr_strap1 … HU02 (ⓓ{b}W2.ⓐV2.U2) ?) [ /2 width=3/ ] -V0 /4 width=3/
]
-| #U0 #T1 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+| #U0 #T1 #HG0 #HL0 #HT0 #H1 #X2 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
elim (snv_inv_cast … H1) -H1 #T0 #l0 #_ #HT1 #HT10 #_
lapply (ssta_inv_cast1 … H2) -H2 #HTU1
elim (ssta_mono … HT10 … HTU1) -HT10 #H1 #H2 destruct
(* Forward_lemmas on iterated stratified static type assignment for terms ***)
-lemma snv_sstas_fwd_correct: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ¡[h, g] → ⦃G, L⦄ ⊢ T1 •* [h, g] T2 →
+lemma snv_sstas_fwd_correct: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ¡[h, g] → ⦃G, L⦄ ⊢ T1 •* [h, g] T2 →
∃∃U2,l. ⦃G, L⦄ ⊢ T2 •[h, g] ⦃l, U2⦄.
-#h #g #L #T1 #T2 #HT1 #HT12
+#h #g #G #L #T1 #T2 #HT1 #HT12
elim (snv_fwd_ssta … HT1) -HT1 /2 width=5 by sstas_fwd_correct/
qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/btpredstarproper_6.ma".
+include "basic_2/notation/relations/btpredstarproper_8.ma".
include "basic_2/dynamic/ysc.ma".
include "basic_2/dynamic/yprs.ma".
(* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
-inductive ygt (h) (g) (L1) (T1): relation2 lenv term ≝
-| ygt_inj : ∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≻[h, g] ⦃L2, T2⦄ →
- ygt h g L1 T1 L2 T2
-| ygt_step: ∀L,L2,T. ygt h g L1 T1 L T → ⦃G, L⦄ ⊢ ➡ L2 → ygt h g L1 T1 L2 T
+inductive ygt (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| ygt_inj : ∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ygt h g G1 L1 T1 G2 L2 T2
+| ygt_step: ∀G,L,L2,T. ygt h g G1 L1 T1 G L T → ⦃G, L⦄ ⊢ ➡ L2 → ygt h g G1 L1 T1 G L2 T
.
interpretation "'big tree' proper parallel computation (closure)"
- 'BTPRedStarProper h g L1 T1 L2 T2 = (ygt h g L1 T1 L2 T2).
+ 'BTPRedStarProper h g G1 L1 T1 G2 L2 T2 = (ygt h g G1 L1 T1 G2 L2 T2).
(* Basic forvard lemmas *****************************************************)
-lemma ygt_fwd_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
-#h #g #L1 #L2 #T1 #T2 #H elim H -L2 -T2
-/3 width=4 by yprs_strap1, ysc_ypr, ypr_lpr/
+lemma ygt_fwd_yprs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
+/3 width=5 by yprs_strap1, ysc_ypr, ypr_lpr/
qed-.
(* Basic properties *********************************************************)
-lemma ysc_ygt: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[h, g] ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
-/3 width=4/ qed.
+lemma ysc_ygt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+/3 width=5/ qed.
-lemma ygt_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ ≽[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
-#h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2
+lemma ygt_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
lapply (ygt_fwd_yprs … H1) #H0
-elim (ypr_inv_ysc … H2) -H2 [| * #HL2 #H destruct ] /2 width=4/
+elim (ypr_inv_ysc … H2) -H2 [| * #HG2 #HL2 #HT2 destruct ] /2 width=5/
qed-.
-lemma ygt_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ >[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
-#h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -L2 -T2
-[ /3 width=4 by ygt_inj, yprs_strap2/ | /2 width=3/ ]
+lemma ygt_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -G2 -L2 -T2
+[ /3 width=5 by ygt_inj, yprs_strap2/ | /2 width=3/ ]
qed-.
-lemma ygt_yprs_trans: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ ≥[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
-#h #g #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(yprs_ind … HT2) -L2 -T2 //
-/2 width=4 by ygt_strap1/
+lemma ygt_yprs_trans: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(yprs_ind … HT2) -G2 -L2 -T2 //
+/2 width=5 by ygt_strap1/
qed-.
-lemma yprs_ygt_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ →
- ∀L2,T2. h ⊢ ⦃L, T⦄ >[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
-#h #g #L1 #L #T1 #T #HT1 @(yprs_ind … HT1) -L -T //
-/3 width=4 by ygt_strap2/
+lemma yprs_ygt_trans: ∀h,g,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ >[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G #L1 #L #T1 #T #HT1 @(yprs_ind … HT1) -G -L -T //
+/3 width=5 by ygt_strap2/
qed-.
-lemma fsupp_ygt: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[h, g] ⦃L2, T2⦄.
-#h #g #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -L2 -T2 /3 width=1/ /3 width=4/
+lemma fsupp_ygt: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -G2 -L2 -T2 /3 width=1/ /3 width=5/
qed.
-lemma cprs_ygt: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
- h ⊢ ⦃L, T1⦄ >[h, g] ⦃L, T2⦄.
-#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2
+lemma cprs_ygt: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
+ ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2
[ #H elim H //
| #T #T2 #_ #HT2 #IHT1 #HT12
elim (term_eq_dec T1 T) #H destruct
]
qed.
-lemma sstas_ygt: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 → (T1 = T2 → ⊥) →
- h ⊢ ⦃L, T1⦄ >[h, g] ⦃L, T2⦄.
-#h #g #L #T1 #T2 #H @(sstas_ind … H) -T2
+lemma sstas_ygt: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 → (T1 = T2 → ⊥) →
+ ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H @(sstas_ind … H) -T2
[ #H elim H //
| #T #T2 #l #_ #HT2 #IHT1 #HT12 -HT12
elim (term_eq_dec T1 T) #H destruct
]
qed.
-lemma lsubsv_ygt: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[h, g] L1 → (L1 = L2 → ⊥) →
- h ⊢ ⦃L1, T⦄ >[h, g] ⦃L2, T⦄.
+lemma lsubsv_ygt: ∀h,g,G,L1,L2,T. G ⊢ L2 ¡⊑[h, g] L1 → (L1 = L2 → ⊥) →
+ ⦃G, L1, T⦄ >[h, g] ⦃G, L2, T⦄.
/4 width=1/ qed.
(* Main properties **********************************************************)
-theorem ygt_trans: ∀h,g. bi_transitive … (ygt h g).
-/3 width=4 by ygt_fwd_yprs, ygt_yprs_trans/ qed-.
+theorem ygt_trans: ∀h,g. tri_transitive … (ygt h g).
+/3 width=5 by ygt_fwd_yprs, ygt_yprs_trans/ qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/btpred_6.ma".
+include "basic_2/notation/relations/btpred_8.ma".
include "basic_2/relocation/fsup.ma".
include "basic_2/reduction/lpr.ma".
include "basic_2/dynamic/lsubsv.ma".
(* "BIG TREE" PARALLEL REDUCTION FOR CLOSURES *******************************)
-inductive ypr (h) (g) (L1) (T1): relation2 lenv term ≝
-| ypr_fsup : ∀L2,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ypr h g L1 T1 L2 T2
-| ypr_lpr : ∀L2. L1 ⊢ ➡ L2 → ypr h g L1 T1 L2 T1
-| ypr_cpr : ∀T2. L1 ⊢ T1 ➡ T2 → ypr h g L1 T1 L1 T2
-| ypr_ssta : ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l+1, T2⦄ → ypr h g L1 T1 L1 T2
-| ypr_lsubsv: ∀L2. h ⊢ L2 ¡⊑[h, g] L1 → ypr h g L1 T1 L2 T1
+inductive ypr (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| ypr_fsup : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → ypr h g G1 L1 T1 G2 L2 T2
+| ypr_lpr : ∀L2. ⦃G1, L1⦄ ⊢ ➡ L2 → ypr h g G1 L1 T1 G1 L2 T1
+| ypr_cpr : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡ T2 → ypr h g G1 L1 T1 G1 L1 T2
+| ypr_ssta : ∀T2,l. ⦃G1, L1⦄ ⊢ T1 •[h, g] ⦃l+1, T2⦄ → ypr h g G1 L1 T1 G1 L1 T2
+| ypr_lsubsv: ∀L2. G1 ⊢ L2 ¡⊑[h, g] L1 → ypr h g G1 L1 T1 G1 L2 T1
.
interpretation
"'big tree' parallel reduction (closure)"
- 'BTPRed h g L1 T1 L2 T2 = (ypr h g L1 T1 L2 T2).
+ 'BTPRed h g G1 L1 T1 G2 L2 T2 = (ypr h g G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma ypr_refl: ∀h,g. bi_reflexive … (ypr h g).
+lemma ypr_refl: ∀h,g. tri_reflexive … (ypr h g).
/2 width=1/ qed.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/btpredstar_6.ma".
+include "basic_2/notation/relations/btpredstar_8.ma".
include "basic_2/substitution/fsupp.ma".
include "basic_2/computation/lprs.ma".
include "basic_2/dynamic/ypr.ma".
(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
-definition yprs: ∀h. sd h → bi_relation lenv term ≝
- λh,g. bi_TC … (ypr h g).
+definition yprs: ∀h. sd h → tri_relation genv lenv term ≝
+ λh,g. tri_TC … (ypr h g).
interpretation "'big tree' parallel computation (closure)"
- 'BTPRedStar h g L1 T1 L2 T2 = (yprs h g L1 T1 L2 T2).
+ 'BTPRedStar h g G1 L1 T1 G2 L2 T2 = (yprs h g G1 L1 T1 G2 L2 T2).
(* Basic eliminators ********************************************************)
-lemma yprs_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term. R L1 T1 →
- (∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≽[h, g] ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄ → R L2 T2.
-/3 width=7 by bi_TC_star_ind/ qed-.
+lemma yprs_ind: ∀h,g,G1,L1,T1. ∀R:relation3 genv lenv term. R G1 L1 T1 →
+ (∀L,G2,G,L2,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → R G L T → R G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2.
+/3 width=8 by tri_TC_star_ind/ qed-.
-lemma yprs_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term. R L2 T2 →
- (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≥[h, g] ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄ → R L1 T1.
-/3 width=7 by bi_TC_star_ind_dx/ qed-.
+lemma yprs_ind_dx: ∀h,g,G2,L2,T2. ∀R:relation3 genv lenv term. R G2 L2 T2 →
+ (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ → ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G L T → R G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ → R G1 L1 T1.
+/3 width=8 by tri_TC_star_ind_dx/ qed-.
(* Basic properties *********************************************************)
-lemma yprs_refl: ∀h,g. bi_reflexive … (yprs h g).
+lemma yprs_refl: ∀h,g. tri_reflexive … (yprs h g).
/2 width=1/ qed.
-lemma ypr_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
+lemma ypr_yprs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
/2 width=1/ qed.
-lemma yprs_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ ≽[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
-/2 width=4/ qed-.
+lemma yprs_strap1: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≽[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/2 width=5/ qed-.
-lemma yprs_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ ≥[h, g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
-/2 width=4/ qed-.
+lemma yprs_strap2: ∀h,g,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/2 width=5/ qed-.
(* Note: this is a general property of bi_TC *)
-lemma fsupp_yprs: ∀h,g,L1,L2,T1,T2. ⦃L1, T1⦄ ⊃+ ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
-#h #g #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -L2 -T2 /3 width=1/ /3 width=4/
+lemma fsupp_yprs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fsupp_ind … L2 T2 H) -G2 -L2 -T2 /3 width=1/ /3 width=5/
qed.
-lemma cprs_yprs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → h ⊢ ⦃L, T1⦄ ≥[h, g] ⦃L, T2⦄.
-#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4 by ypr_cpr, yprs_strap1/
+lemma cprs_yprs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=5 by ypr_cpr, yprs_strap1/
qed.
-lemma lprs_yprs: ∀h,g,L1,L2,T. L1 ⊢ ➡* L2 → h ⊢ ⦃L1, T⦄ ≥[h, g] ⦃L2, T⦄.
-#h #g #L1 #L2 #T #H @(lprs_ind … H) -L2 // /3 width=4 by ypr_lpr, yprs_strap1/
+lemma lprs_yprs: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
+#h #g #G #L1 #L2 #T #H @(lprs_ind … H) -L2 // /3 width=5 by ypr_lpr, yprs_strap1/
qed.
-lemma sstas_yprs: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 →
- h ⊢ ⦃L, T1⦄ ≥[h, g] ⦃L, T2⦄.
-#h #g #L #T1 #T2 #H @(sstas_ind … H) -T2 // /3 width=4 by ypr_ssta, yprs_strap1/
+lemma sstas_yprs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 •*[h, g] T2 →
+ ⦃G, L, T1⦄ ≥[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #H @(sstas_ind … H) -T2 // /3 width=5 by ypr_ssta, yprs_strap1/
qed.
-lemma lsubsv_yprs: ∀h,g,L1,L2,T. h ⊢ L2 ¡⊑[h, g] L1 → h ⊢ ⦃L1, T⦄ ≥[h, g] ⦃L2, T⦄.
+lemma lsubsv_yprs: ∀h,g,G,L1,L2,T. G ⊢ L2 ¡⊑[h, g] L1 → ⦃G, L1, T⦄ ≥[h, g] ⦃G, L2, T⦄.
/3 width=1/ qed.
-lemma cprs_lpr_yprs: ∀h,g,L1,L2,T1,T2. L1 ⊢ T1 ➡* T2 → L1 ⊢ ➡ L2 →
- h ⊢ ⦃L1, T1⦄ ≥[h, g] ⦃L2, T2⦄.
-/3 width=4 by yprs_strap1, ypr_lpr, cprs_yprs/
+lemma cprs_lpr_yprs: ∀h,g,G,L1,L2,T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L1⦄ ⊢ ➡ L2 →
+ ⦃G, L1, T1⦄ ≥[h, g] ⦃G, L2, T2⦄.
+/3 width=5 by yprs_strap1, ypr_lpr, cprs_yprs/
qed.
(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
-theorem yprs_trans: ∀h,g. bi_transitive … (yprs h g).
-/2 width=4/ qed-.
+theorem yprs_trans: ∀h,g. tri_transitive … (yprs h g).
+/2 width=5/ qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/btpredproper_6.ma".
+include "basic_2/notation/relations/btpredproper_8.ma".
include "basic_2/dynamic/ypr.ma".
(* "BIG TREE" PROPER PARALLEL REDUCTION FOR CLOSURES ************************)
-inductive ysc (h) (g) (L1) (T1): relation2 lenv term ≝
-| ysc_fsup : ∀L2,T2. ⦃L1, T1⦄ ⊃ ⦃L2, T2⦄ → ysc h g L1 T1 L2 T2
-| ysc_cpr : ∀T2. L1 ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → ysc h g L1 T1 L1 T2
-| ysc_ssta : ∀T2,l. ⦃h, L1⦄ ⊢ T1 •[h, g] ⦃l+1, T2⦄ → ysc h g L1 T1 L1 T2
-| ysc_lsubsv: ∀L2. h ⊢ L2 ¡⊑[h, g] L1 → (L1 = L2 → ⊥) → ysc h g L1 T1 L2 T1
+inductive ysc (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| ysc_fsup : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → ysc h g G1 L1 T1 G2 L2 T2
+| ysc_cpr : ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) → ysc h g G1 L1 T1 G1 L1 T2
+| ysc_ssta : ∀T2,l. ⦃G1, L1⦄ ⊢ T1 •[h, g] ⦃l+1, T2⦄ → ysc h g G1 L1 T1 G1 L1 T2
+| ysc_lsubsv: ∀L2. G1 ⊢ L2 ¡⊑[h, g] L1 → (L1 = L2 → ⊥) → ysc h g G1 L1 T1 G1 L2 T1
.
interpretation
"'big tree' proper parallel reduction (closure)"
- 'BTPRedProper h g L1 T1 L2 T2 = (ysc h g L1 T1 L2 T2).
+ 'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (ysc h g G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma ysc_ypr: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[h, g] ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L2, T2⦄.
-#h #g #L1 #L2 #T1 #T2 * -L2 -T2 /2 width=1/ /2 width=2/
+lemma ysc_ypr: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1/ /2 width=2/
qed.
(* Inversion lemmas on "big tree" parallel reduction for closures ***********)
-lemma ypr_inv_ysc: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≽[h, g] ⦃L2, T2⦄ →
- h ⊢ ⦃L1, T1⦄ ≻[h, g] ⦃L2, T2⦄ ∨ (L1 ⊢ ➡ L2 ∧ T1 = T2).
-#h #g #L1 #L2 #T1 #T2 * -L2 -T2 /3 width=1/ /3 width=2/
+lemma ypr_inv_ysc: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ ∨
+ ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡ L2 & T1 = T2.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /3 width=1/ /3 width=2/
[ #T2 #HT12 elim (term_eq_dec T1 T2) #H destruct /3 width=1/ /4 width=1/
| #L2 #HL21 elim (lenv_eq_dec L1 L2) #H destruct /3 width=1/ /4 width=1/
]
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≽ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'BTPRed $h $g $L1 $T1 $L2 $T2 }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≽ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'BTPRed $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≻ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'BTPRedProper $h $g $L1 $T1 $L2 $T2 }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≻ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'BTPRedProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ ≥ break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'BTPRedStar $h $g $L1 $T1 $L2 $T2 }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≥ break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'BTPRedStar $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( h ⊢ break ⦃ term 46 L1, break term 46 T1 ⦄ > break [ term 46 g ] break ⦃ term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'BTPRedStarProper $h $g $L1 $T1 $L2 $T2 }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ > break [ term 46 h, break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'BTPRedStarProper $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( h ⊢ break term 46 L1 ¡ ⊑ break [ term 46 g ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LRSubEqV $h $g $L1 $L2 }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( G ⊢ break term 46 L1 ¡ ⊑ break [ term 46 h, break term 46 g ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEqV $h $g $G $L1 $L2 }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 h , break term 46 L ⦄ ⊢ break term 46 T ¡ break [ term 46 g ] )"
- non associative with precedence 45
- for @{ 'NativeValid $h $g $L $T }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G , break term 46 L ⦄ ⊢ break term 46 T ¡ break [ term 46 h, break term 46 g ] )"
+ non associative with precedence 45
+ for @{ 'NativeValid $h $g $G $L $T }.
Closure of context-sensitive extended computation
for native validity.
</news>
- <news date="In progress.">
+ <news date="2013 August 7.">
Passive support for global environments.
</news>
<news date="2013 July 27.">
projects / helm.git / commitdiff