ty3/arity_props ty3_predicative
ty3/arity_props ty3_repellent
ty3/arity_props ty3_acyclic
-ty3/arity_props ty3_sn3
ty3/dec ty3_inference
-ty3/fwd ty3_gen_appl
ty3/fwd tys3_gen_nil
ty3/fwd tys3_gen_cons
ty3/fwd_nf2 ty3_gen_appl_nf2
@(cprs_trans … HV1) -HV1 /2 width=1/
qed.
+lemma cprs_beta_dx: ∀L,V1,V2,W,T1,T2.
+ L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡* T2 → L ⊢ ⓐV1.ⓛW.T1 ➡* ⓓV2.T2.
+#L #V1 #V2 #W #T1 #T2 #HV12 #HT12 @(cprs_ind … HT12) -T2
+[ /3 width=1/
+| -HV12 #T #T2 #_ #HT2 #IHT1
+ lapply (cpr_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+ @(cprs_trans … IHT1) -V1 -W -T1 /3 width=1/
+]
+qed.
+
(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
lemma lcpr_cprs_trans: ∀L1,L2. L1 ⊢ ➡ L2 →
∀T1,T2. L2 ⊢ T1 ➡* T2 → L1 ⊢ T1 ➡* 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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/computation/csn_aaa.ma".
+include "basic_2/equivalence/lcpcs_aaa.ma".
+include "basic_2/dynamic/nta.ma".
+
+(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************)
+
+(* Forward lemmas on atomic arity assignment for terms **********************)
+
+lemma nta_fwd_aaa: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∃∃A. L ⊢ T ⁝ A & L ⊢ U ⁝ A.
+#h #L #T #U #H elim H -L -T -U
+[ /2 width=3/
+| #L #K #V #W #U #i #HLK #_ #HWU * #B #HV #HW
+ lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/
+| #L #K #W #V #U #i #HLK #_ #HWU * #B #HW #_ -V
+ lapply (ldrop_fwd_ldrop2 … HLK) /3 width=9/
+| * #L #V #W #T #U #_ #_ * #B #HV #HW * #A #HT #HU
+ [ /3 width=3/ | /3 width=5/ ]
+| #L #V #W #T #U #_ #_ * #B #HV #HW * #X #H1 #H2
+ elim (aaa_inv_abst … H1) -H1 #B1 #A1 #HW1 #HT #H destruct
+ elim (aaa_inv_abst … H2) -H2 #B2 #A #_ #HU #H destruct
+ lapply (aaa_mono … HW1 … HW) -HW1 #H destruct /4 width=5/
+| #L #V #W #T #U #_ #_ * #X #HT #HUX * #A #H #_ -W
+ elim (aaa_inv_appl … H) -H #B #HV #HUA
+ lapply (aaa_mono … HUA … HUX) -HUX #H destruct /3 width=5/
+| #L #T #U #_ * #A #HT #HU /3 width=3/
+| #L #T #U1 #U2 #V2 #_ #HU12 #_ * #X #HT #HU1 * #A #HU2 #_
+ lapply (aaa_cpcs_mono … HU12 … HU1 … HU2) -U1 #H destruct /2 width=3/
+]
+qed-.
+
+(* Note: this is the stong normalization property *)
+(* Basic_1: was only: ty3_sn3 *)
+theorem nta_fwd_csn: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → L ⊢ ⬇* T ∧ L ⊢ ⬇* U.
+#h #L #T #U #H elim (nta_fwd_aaa … H) -H /3 width=2/
+qed-.
(* Advanced forvard lemmas **************************************************)
-fact nta_fwd_appl1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∀X,Y. T = ⓐY.X →
+fact nta_fwd_pure1_aux: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∀X,Y. T = ⓐY.X →
∃∃V,W. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & L ⊢ ⓐY.V ⬌* U.
#h #L #T #U #H elim H -L -T -U
[ #L #k #X #Y #H destruct
]
qed.
-lemma nta_fwd_appl1: ∀h,L,X,Y,U. ⦃h, L⦄ ⊢ ⓐY.X : U →
+lemma nta_fwd_pure1: ∀h,L,X,Y,U. ⦃h, L⦄ ⊢ ⓐY.X : U →
∃∃V,W. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V & L ⊢ ⓐY.V ⬌* U.
/2 width=3/ qed-.
(* *)
(**************************************************************************)
+(* FRAGMENT 0
+include "basic_2/unfold/delift_lift.ma".
+
+
+include "basic_2/equivalence/cpcs_cpcs.ma".
+include "basic_2/unfold/delift_delift.ma".
+
+
+
+axiom pippo1: ∀T1,V. ∃∃T2. ⓓV.T1 ➡ T2 & ⋆.ⓓV ⊢ T1 ▼*[0, 1] ≡ T2.
+(*
+#T1 #V elim (pippo0 (⋆.ⓓV) 0 1 ? T1)
+[ #T2 #HT12 @(ex2_1_intro … HT12)
+ @tpr
+*)
+
+axiom pippo: ∀L,V,T1. ∃∃T2. L ⊢ ⓓV.T1 ➡ T2 & L.ⓓV ⊢ T1 ▼*[0, 1] ≡ T2.
+
+axiom cprs_inv_appl1_cpcs: ∀L,V1,T1,U2. L ⊢ ⓐV1. T1 ➡* U2 → (
+ ∃∃V2,T2. L ⊢ V1 ➡* V2 & L ⊢ T1 ➡* T2 &
+ L ⊢ U2 ➡* ⓐV2. T2
+ ) ∨
+ ∃∃V2,W,T. L ⊢ V1 ➡* V2 &
+ L ⊢ T1 ➡* ⓛW. T & L ⊢ ⓓV2. T ⬌* U2.
+#L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
+#U #U2 #_ #HU2 * *
+[ #V0 #T0 #HV10 #HT10 #HUT0
+ elim (cprs_strip … HUT0 … HU2) -U #U #H #HU2
+ elim (cpr_inv_appl1 … H) -H *
+ [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
+ | #V2 #W2 #T #T2 #HV02 #HT2 #H1 #H2 destruct
+ lapply (cprs_strap1 … HV10 HV02) -V0 #HV12
+ lapply (cprs_div ? (ⓓV2.T) ? ? ? HU2) -HU2 /2 width=1/ /3 width=6/
+ | #V #V2 #W0 #W2 #T #T2 #HV0 #HW02 #HT2 #HV2 #H1 #H2 destruct
+ lapply (cprs_strap1 … HV10 HV0) -V0 #HV1
+ lapply (cprs_trans … HT10 (ⓓW2.T2) ?) -HT10 /2 width=1/ -W0 -T #HT1
+ elim (pippo L W2 T2) #T3 #H1T3 #H2T3
+ elim (pippo L W2 (ⓐV2.T2)) #X #H1X #H
+ elim (delift_inv_flat1 … H) -H #V3 #Y #HV23 #HY #H destruct
+
+
+ @or_introl @(ex3_2_intro … HV1 HT1) -HV1 -HT1
+
+ ]
+| /4 width=8/
+]
+qed-.
+
+
+include "basic_2/computation/cprs_lcprs.ma".
+include "basic_2/dynamic/nta.ma".
+
+axiom cprs_inv_appl_abst: ∀L,V,T,W,U. L ⊢ ⓐV.T ➡* ⓛW.U →
+ ∃∃V0,W0,T0,W1,U1. ⇧[O, 1] W ≡ W1 &
+ ⇧[O + 1, 1] U ≡ U1 &
+ L ⊢ V ➡* V0 & L ⊢ T ➡* ⓛW0.T0 &
+ L.ⓓV0 ⊢ T0 ➡* ⓛW1.U1.
+#L #V #T #W #U #H
+elim (cprs_inv_appl1 … H) -H *
+[ #V0 #T0 #_ #_ #H destruct
+| #V0 #W0 #T0 #HV0 #HT0 #H
+ elim (cprs_inv_abbr1 … H) -H *
+ [ #V1 #T1 #_ #_ #H destruct
+ | #X #H #HT01
+ elim (lift_inv_bind1 … H) -H #W1 #U1 #HW1 #HU1 #H destruct /2 width=10/
+ ]
+| #V0 #V1 #V2 #T0 #HV0 #HV01 #HT0 #H
+ elim (cprs_inv_abbr1 … H) -H *
+ [ #V3 #T3 #_ #_ #H destruct
+ | #X #H #HT3
+ elim (lift_inv_bind1 … H) -H #W3 #U3 #HW3 #HU3 #H destruct /2 width=10/
+
+axiom pippo: ∀h,L,T,U. ⦃h, L⦄ ⊢ T : U → ∀X,Y. L ⊢ T ➡* ⓛX.Y →
+ ∃Z. L ⊢ U ➡* ⓛX.Z.
+#h #L #T #U #H elim H -L -T -U
+[
+|
+|
+|
+|
+| #L #V #W #T #U #HTU #HUW #IHTU #IHUW #X #Y #HTY
+ elim (cprs_inv_appl1 … HTY) -HTY *
+ [ #V0 #T0 #_ #_ #H destruct
+ | #V0 #W0 #T0 #HV0 #HT0 #HTY
+ elim (IHTU … HT0) -IHTU -HT0 #Z #HUZ
+ elim (cprs_inv_abbr1 … HTY) -HTY *
+ [ #V1 #T1 #_ #_ #H destruct #X0
+
+
include "basic_2/dynamic/nta_ltpss.ma".
include "basic_2/dynamic/nta_thin.ma".
include "basic_2/dynamic/lsubn_nta.ma".
-lemma cpr_beta: ∀L,V1,V2,W,T1,T2.
- V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2.
-#L #V1 #V2 #W #T1 #T2 #HV12 * #T #HT1 #HT2
-lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2
-lapply (tpss_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
-@(ex2_1_intro … (ⓓV2.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too skow *)
-qed.
-
-lemma cpr_beta2: ∀L,V1,V2,W,T1,T2.
- L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2.
-#L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2
-lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2
-lapply (tpss_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
-@(ex2_1_intro … (ⓓV.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too skow *)
-qed.
+include "basic_2/hod/ntas_lift.ma".
+
+ elim (nta_inv_pure1 … HUW) -HUW #V0 #U0 #U1 #HV0 #HU0 #HU0W #HU01
+ @(ex2_2_intro … HYW)
+ [2:
+
+
+axiom pippo_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀Y,X. Z = ⓐY.X →
+ ∀W,T. L ⊢ X ➡* ⓛW.T → ⦃h, L⦄ ⊢ Y : W.
+#h #L #Z #U #H elim H -L -Z -U
+[
+|
+|
+|
+|
+| #L #V #W #T #U #HTU #_ #_ #IHUW #Y #X #H #W0 #T0 #HX destruct
+ lapply (IHUW Y U ? ?) -IHUW -W // #T
+ @(ex2_2_intro … HYW)
+ [2:
+
+axiom pippo: ∀h,L,V,X,U. ⦃h, L⦄ ⊢ ⓐV.X : U →
+ ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ V : W.
+#h #L #V #X #Y #H
+
+*)
(* NATIVE TYPE ASSIGNMENT ON TERMS ******************************************)
(* Properties on context-free parallel reduction for local environments ******)
-
-lemma nta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 →
+(*
+axiom nta_ltpr_cprs_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 →
+ ∀T2. L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊢ T2 : U.
+#h #L1 #T1 #U #H @(nta_ind_alt … H) -L1 -T1 -U
+[ #L1 #k #L2 #_ #T2 #H
+ >(cprs_inv_sort1 … H) -H //
+|
+|
+|
+|
+| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #HL12 #T2 #H
+ elim (cprs_inv_appl1 … H) -H *
+ [ #V2 #T0 #HV12 #HT10 #H destruct
+ elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=2/ ] #U
+ @(nta_conv … (ⓐV2.U1)) (* /2 width=1/*) [ /4 width=2/] (**) (* explicit constructor, /5 width=5/ is too slow *)
+ | #V2 #W2 #T0 #HV12 #HT10 #HT02
+ lapply (IHTU1 … HL12 (ⓛW2.T0) ?) -IHTU1 /2 width=1/ -HT10 #H
+ elim (nta_inv_bind1 … H) -H #W #U0 #HW2 #HTU0 #HU01
+ elim (cpcs_inv_abst1 … HU01) -HU01 #W #U #HU1 #HU0
+ lapply (IHUW1 … HL12 (ⓐV2.ⓛW.U) ?) -IHUW1 -HL12 /2 width=1/ -HV12 #H
+
+
+
+ elim (nta_fwd_pure1 … H) -H #W0 #U2 #HVU2 #H #HW01
+ elim (nta_inv_bind1 … H) -H #W3 #U3 #HW3 #HU3 #H
+ elim (cpcs_inv_abst1 … H) -H #W4 #U4
+*)
+(*
+axiom nta_ltpr_tpr_conf: ∀h,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → ∀L2. L1 ➡ L2 →
∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U.
#h #L1 #T1 #U #H @(nta_ind_alt … H) -L1 -T1 -U
[ #L1 #k #L2 #_ #T2 #H
elim (nta_fwd_correct … H1) #T #H2
elim (nta_inv_bind1 … H1) -H1 #W #U2 #HW2 #HTU2 #H
elim (cpcs_inv_abst … H Abst W2) -H #_ #HU21
- elim (nta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0
+ elim (nta_inv_bind1 … H2) -H2 #W0 #U0 #_ #H #_ -T -W0
lapply (lsubn_nta_trans … HTU2 (L2.ⓓV2) ?) -HTU2 /2 width=1/ #HTU2
- @(nta_conv … (ⓓV2.U2))
- [2: /2 width=2/
- |3: -h -L1 -T0 -T2 -W -U0
- |4: /3 width=2/
- ]
+ @(nta_conv … (ⓓV2.U2)) /2 width=2/ /3 width=2/ (**) (* explicit constructor, /4 width=5/ is too slow *)
| #V0 #V2 #W0 #W2 #T0 #T2 #_ #_ #_ #_ #H destruct
]
-
-
-
-
-
-| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- lapply (cpr_tpss … HV12) #HV
- elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=4/ ] #U #HU
- @(nta_conv … (ⓐV2.U1)) // /3 width=1/ /4 width=5/ (**) (* explicit constructor, /5 width=5/ is too slow *)
+| #L1 #V1 #W1 #T1 #U1 #_ #_ #IHTU1 #IHUW1 #L2 #HL12 #X #H
+ elim (tpr_inv_appl1 … H) -H *
+ [ #V2 #T2 #HV12 #HT12 #H destruct
+ elim (nta_fwd_correct h L2 (ⓐV1.T1) (ⓐV1.U1) ?) [2: /3 width=2/ ] #U
+ @(nta_conv … (ⓐV2.U1)) /2 width=1/ /4 width=2/ (**) (* explicit constructor, /5 width=5/ is too slow *)
+ | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
+ lapply (IHTU1 … HL12 (ⓛW2.T2) ?) -IHTU1 /2 width=1/ -T0 #H
+ elim (nta_inv_bind1 … H) -H #W #U2 #HW2 #HTU2 #HU21
+ lapply (IHUW1 … HL12 (ⓐV2.U1) ?) -IHUW1 -HL12 /2 width=1/ #H
+ elim (nta_inv_pure1 … H) -H #V0 #U0 #U #HV20 #HU10 #HU0W1 #HU0
+ @(nta_conv … (ⓓV2.U2))
+ [2: @nta_bind //
+ @(lsubn_nta_trans … HTU2) @lsubn_abbr //
+(*
+ lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB
+ lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0
+ lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/
+*)
+*)
+(*
+axiom pippo: ⦃h, L⦄ ⊢ ⓐV.X : Y →
+ ∃∃W,T. L ⊢ X ➡* ⓛW.T & ⦃h, L⦄ ⊢ ⓐV : W.
+
+*)
+(* SEGMENT 2
| #L1 #T1 #U1 #W1 #_ #_ #IHTU1 #IHUW1 #L2 #d #e #HL12 #X #H
elim (tpss_inv_flat1 … H) -H #U2 #T2 #HU12 #HT12 #H destruct
lapply (cpr_tpss … HU12) /4 width=4/
@(nta_conv … U11) /2 width=5/ (**) (* explicot constructor, /3 width=7/ is too slow *)
]
qed.
+*)
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
+(* SEGMENT 3
fact nta_ltpr_tpr_conf_aux: ∀h,L,T,L1,T1,U. ⦃h, L1⦄ ⊢ T1 : U → L = L1 → T = T1 →
∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊢ T2 : U.
-| #L1 #V1 #T1 #B #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct
- elim (tpr_inv_appl1 … H) -H *
- [ #V2 #T2 #HV12 #HT12 #H destruct
- lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB
- lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ /2 width=3/
- | #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
- elim (nta_inv_abst … HT1) -HT1 #B0 #A0 #HB0 #HA0 #H destruct
- lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB
- lapply (IH … HB0 … HL12 W2 ?) -HB0 /width=5/ #HB0
- lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 /width=5/ -T0 /2 width=1/ -L1 -V1 /4 width=7/
| #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
elim (nta_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0
lapply (IH … HW0 … HL12 … HW02) -HW0 /width=5/ #HW2
/2 width=9/ qed.
-lemma nta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A.
+axiom nta_ltpr_conf: ∀L1,T,A. L1 ⊢ T : A → ∀L2. L1 ➡ L2 → L2 ⊢ T : A.
/2 width=5/ qed.
-lemma nta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A.
+axiom nta_tpr_conf: ∀L,T1,A. L ⊢ T1 : A → ∀T2. T1 ➡ T2 → L ⊢ T2 : A.
/2 width=5/ qed.
+*)
elim (nta_inv_bind1 … H) -H #W2 #U2 #_ #HTU2 #H
@(cpcs_trans … H) -X /3 width=1/
| #L #V #W1 #T #U1 #_ #_ #_ #IHTU1 #X #H
- elim (nta_fwd_appl1 … H) -H #U2 #W2 #_ #HTU2 #H
+ elim (nta_fwd_pure1 … H) -H #U2 #W2 #_ #HTU2 #H
@(cpcs_trans … H) -X /3 width=1/
| #L #V #W1 #T #U1 #_ #_ #IHTU1 #_ #X #H
- elim (nta_fwd_appl1 … H) -H #U2 #W2 #_ #HTU2 #H
+ elim (nta_fwd_pure1 … H) -H #U2 #W2 #_ #HTU2 #H
@(cpcs_trans … H) -X /3 width=1/
| #L #T #U1 #_ #_ #X #H
elim (nta_inv_cast1 … H) -H //
(* *)
(**************************************************************************)
-include "basic_2/substitution/ldrop_sfr.ma".
include "basic_2/unfold/delift_lift.ma".
include "basic_2/unfold/thin_ldrop.ma".
include "basic_2/equivalence/cpcs_delift.ma".
lapply (cpcs_inv_abst … H Abst W) -H //
qed.
+lemma cpcs_inv_abst1: ∀L,W1,T1,T. L ⊢ ⓛW1.T1 ⬌* T →
+ ∃∃W2,T2. L ⊢ T ➡* ⓛW2.T2 & L ⊢ ⓛW1.T1 ➡* ⓛW2.T2.
+#L #W1 #T1 #T #H
+elim (cpcs_inv_cprs … H) -H #X #H1 #H2
+elim (cprs_inv_abst1 Abst W1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
+@(ex2_2_intro … H2) -H2 /2 width=2/ (**) (* explicit constructor, /3 width=6/ is slow *)
+qed-.
+
(* Basic_1: was: pc3_gen_lift *)
lemma cpcs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ (**) (* /3 width=5/ is too slow *)
qed.
+lemma cpcs_flat_dx_tpr_rev: ∀L,V1,V2. V2 ➡ V1 → ∀T1,T2. L ⊢ T1 ⬌* T2 →
+ ∀I. L ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
+/3 width=1/ qed.
+
lemma cpcs_abst: ∀L,V1,V2. L ⊢ V1 ⬌* V2 →
∀V,T1,T2. L.ⓛV ⊢ T1 ⬌* T2 → L ⊢ ⓛV1. T1 ⬌* ⓛV2. T2.
#L #V1 #V2 #HV12 #V #T1 #T2 #HT12
lemma cpcs_bind_sn: ∀I,L,V1,V2,T. L ⊢ V1 ⬌* V2 → L ⊢ ⓑ{I}V1. T ⬌* ⓑ{I}V2. T.
* /2 width=1/ /2 width=2/ qed.
+lemma cpcs_beta_dx: ∀L,V1,V2,W,T1,T2.
+ L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ⬌* T2 → L ⊢ ⓐV1.ⓛW.T1 ⬌* ⓓV2.T2.
+#L #V1 #V2 #W #T1 #T2 #HV12 #HT12
+elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
+lapply (cprs_beta_dx … HV12 HT1) -HV12 -HT1 #HT1
+lapply (cprs_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+@(cprs_div … HT1) /2 width=1/
+qed.
+
+lemma cpcs_beta_dx_tpr_rev: ∀L,V1,V2,W,T1,T2.
+ V1 ➡ V2 → L.ⓛW ⊢ T2 ⬌* T1 →
+ L ⊢ ⓓV2.T2 ⬌* ⓐV1.ⓛW.T1.
+/4 width=1/ qed.
+
(* Note: it does not hold replacing |L1| with |L2| *)
lemma cpcs_lsubs_trans: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
∀L2. L2 ≼ [0, |L1|] L1 → L2 ⊢ T1 ⬌* 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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/dynamic/nta.ma".
+
+(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************)
+
+definition ntas: sh → lenv → relation term ≝
+ λh,L. star … (nta h L).
+
+interpretation "higher order native type assignment (term)"
+ 'NativeTypeStar h L T U = (ntas h L T U).
+
+(* Basic eliminators ********************************************************)
+(*
+lemma cprs_ind: ∀L,T1. ∀R:predicate term. R T1 →
+ (∀T,T2. L ⊢ T1 ➡* T → L ⊢ T ➡ T2 → R T → R T2) →
+ ∀T2. L ⊢ T1 ➡* T2 → R T2.
+#L #T1 #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+*)
+axiom ntas_ind_dx: ∀h,L,T2. ∀R:predicate term. R T2 →
+ (∀T1,T. ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → R T → R T1) →
+ ∀T1. ⦃h, L⦄ ⊢ T1 :* T2 → R T1.
+(*
+#h #L #T2 #R #HT2 #IHT2 #T1 #HT12
+@(star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+*)
+(* Basic properties *********************************************************)
+
+lemma ntas_refl: ∀h,L,T. ⦃h, L⦄ ⊢ T :* T.
+// qed.
+
+lemma ntas_strap1: ∀h,L,T1,T,T2.
+ ⦃h, L⦄ ⊢ T1 :* T → ⦃h, L⦄ ⊢ T : T2 → ⦃h, L⦄ ⊢ T1 :* T2.
+/2 width=3/ qed.
+
+lemma ntas_strap2: ∀h,L,T1,T,T2.
+ ⦃h, L⦄ ⊢ T1 : T → ⦃h, L⦄ ⊢ T :* T2 → ⦃h, L⦄ ⊢ T1 :* T2.
+/2 width=3/ qed.
--- /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/dynamic/nta_lift.ma".
+include "basic_2/hod/ntas.ma".
+
+(* HIGHER ORDER NATIVE TYPE ASSIGNMENT ON TERMS *****************************)
+
+(* Advanced properties on native type assignment for terms ******************)
+
+lemma nta_pure_ntas: ∀h,L,U,W,Y. ⦃h, L⦄ ⊢ U :* ⓛW.Y → ∀T. ⦃h, L⦄ ⊢ T : U →
+ ∀V. ⦃h, L⦄ ⊢ V : W → ⦃h, L⦄ ⊢ ⓐV.T : ⓐV.U.
+#h #L #U #W #Y #H @(ntas_ind_dx … H) -U /2 width=1/ /3 width=2/
+qed.
+
+axiom pippo: ∀h,L,T,W,Y. ⦃h, L⦄ ⊢ T :* ⓛW.Y → ∀U. ⦃h, L⦄ ⊢ T : U →
+ ∃Z. ⦃h, L⦄ ⊢ U :* ⓛW.Z.
+(* REQUIRES SUBJECT CONVERSION
+#h #L #T #W #Y #H @(ntas_ind_dx … H) -T
+[ #U #HYU
+ elim (nta_fwd_correct … HYU) #U0 #HU0
+ elim (nta_inv_bind1 … HYU) #W0 #Y0 #HW0 #HY0 #HY0U
+*)
+
+(* Advanced inversion lemmas on native type assignment for terms ************)
+
+fact nta_inv_pure1_aux: ∀h,L,Z,U. ⦃h, L⦄ ⊢ Z : U → ∀X,Y. Z = ⓐY.X →
+ ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V &
+ L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T.
+#h #L #Z #U #H elim H -L -Z -U
+[ #L #k #X #Y #H destruct
+| #L #K #V #W #U #i #_ #_ #_ #_ #X #Y #H destruct
+| #L #K #W #V #U #i #_ #_ #_ #_ #X #Y #H destruct
+| #I #L #V #W #T #U #_ #_ #_ #_ #X #Y #H destruct
+| #L #V #W #Z #U #HVW #HZU #_ #_ #X #Y #H destruct /2 width=7/
+| #L #V #W #Z #U #HZU #_ #_ #IHUW #X #Y #H destruct
+ elim (IHUW U Y ?) -IHUW // /3 width=9/
+| #L #Z #U #_ #_ #X #Y #H destruct
+| #L #Z #U1 #U2 #V2 #_ #HU12 #_ #IHTU1 #_ #X #Y #H destruct
+ elim (IHTU1 ???) -IHTU1 [4: // |2,3: skip ] #W #V #T #HYW #HXV #HU1 #HVT
+ lapply (cpcs_trans … HU1 … HU12) -U1 /2 width=7/
+]
+qed.
+
+(* Basic_1: was only: ty3_gen_appl *)
+lemma nta_inv_pure1: ∀h,L,Y,X,U. ⦃h, L⦄ ⊢ ⓐY.X : U →
+ ∃∃W,V,T. ⦃h, L⦄ ⊢ Y : W & ⦃h, L⦄ ⊢ X : V &
+ L ⊢ ⓐY.V ⬌* U & ⦃h, L⦄ ⊢ V :* ⓛW.T.
+/2 width=3/ qed-.
+
+axiom nta_inv_appl1: ∀h,L,Z,Y,X,U. ⦃h, L⦄ ⊢ ⓐZ.ⓛY.X : U →
+ ∃∃W. ⦃h, L⦄ ⊢ Z : Y & ⦃h, L⦄ ⊢ ⓛY.X : ⓛY.W &
+ L ⊢ ⓐZ.ⓛY.W ⬌* U.
+(* REQUIRES SUBJECT REDUCTION
+#h #L #Z #Y #X #U #H
+elim (nta_inv_pure1 … H) -H #W #V #T #HZW #HXV #HVU #HVT
+elim (nta_inv_bind1 … HXV) -HXV #Y0 #X0 #HY0 #HX0 #HX0V
+lapply (cpcs_trans … (ⓐZ.ⓛY.X0) … HVU) -HVU /2 width=1/ -HX0V #HX0U
+@(ex3_1_intro … HX0U) /2 width=2/
+*)
notation "hvbox( h ⊢ break term 46 L1 : ⊑ break term 46 L2 )"
non associative with precedence 45
for @{ 'CrSubEqN $h $L1 $L2 }.
+
+(* Higher order dynamic typing **********************************************)
+
+notation "hvbox( ⦃ h , break L ⦄ ⊢ break term 46 T1 :* break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'NativeTypeStar $h $L $T1 $T2 }.
L.ⓛV ⊢ T1 ➡ T2 → L ⊢ ⓛV1. T1 ➡ ⓛV2. T2.
#L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02
lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02
-@(ex2_1_intro … (ⓛV0.T0)) /2 width=1/ -V1 -T1 (**) (* explicit constructors *)
-@tpss_bind // -V0
-@(tpss_lsubs_trans (L.ⓛV)) // -T0 -T2 /2 width=1/
+lapply (tpss_lsubs_trans … HT02 (L.ⓛV2) ?) -HT02 /2 width=1/ #HT02
+@(ex2_1_intro … (ⓛV0.T0)) /2 width=1/ (* explicit constructors *)
qed.
+lemma cpr_beta: ∀L,V1,V2,W,T1,T2.
+ L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2.
+#L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2
+lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2
+lapply (tpss_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+@(ex2_1_intro … (ⓓV.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too slow *)
+qed.
+
+lemma cpr_beta_dx: ∀L,V1,V2,W,T1,T2.
+ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2.
+/3 width=1/ qed.
+
(* Advanced inversion lemmas ************************************************)
(* Basic_1: was: pr2_gen_lref *)
(* *)
(**************************************************************************)
+include "basic_2/substitution/ldrop_sfr.ma".
include "basic_2/unfold/tpss_lift.ma".
include "basic_2/unfold/delift.ma".
lapply (lift_conf_be … HVU2 … HV2U ?) //
>commutative_plus in ⊢ (??%??→?); <minus_plus_m_m /3 width=6/
qed.
-
+
+fact sfr_delift_aux: ∀L,T,T1,d,e. d + e ≤ |L| → ≽ [d, e] L → T = T1 →
+ ∃T2. L ⊢ T1 ▼*[d, e] ≡ T2.
+#L #T @(cw_wf_ind … L T) -L -T #L #T #IH * * /2 width=2/
+[ #i #d #e #Hde #HL #H destruct
+ elim (lt_or_ge i d) #Hdi [ /3 width=2/ ]
+ elim (lt_or_ge i (d+e)) #Hide [2: /3 width=2/ ]
+ lapply (lt_to_le_to_lt … Hide Hde) #Hi
+ elim (ldrop_O1 … Hi) -Hi #I #K #V1 #HLK
+ lapply (sfr_inv_ldrop … HLK … HL ? ?) // #H destruct
+ lapply (ldrop_pair2_fwd_cw … HLK (#i)) #HKL
+ lapply (ldrop_fwd_ldrop2 … HLK) #HLK0
+ lapply (ldrop_fwd_O1_length … HLK0) #H
+ lapply (sfr_ldrop_trans_be_up … HLK0 … HL ? ?) -HLK0 -HL
+ [1,2: /2 width=1/ | <minus_n_O <minus_plus ] #HK
+ elim (IH … HKL … HK ?) -IH -HKL -HK
+ [3: // |2: skip |4: >H -H /2 width=1/ ] -Hde -H #V2 #V12 (**) (* H erased two times *)
+ elim (lift_total V2 0 d) /3 width=7/
+| #I #V1 #T1 #d #e #Hde #HL #H destruct
+ elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12
+ elim (IH (L.ⓑ{I}V1) T1 ? ? (d+1) e ? ? ?) -IH [3,6: // |2: skip |4,5: /2 width=1/ ] -Hde -HL #T2 #HT12
+ lapply (delift_lsubs_trans … HT12 (L.ⓑ{I}V2) ?) -HT12 /2 width=1/ /3 width=4/
+| #I #V1 #T1 #d #e #Hde #HL #H destruct
+ elim (IH … V1 … Hde HL ?) [2,4: // |3: skip ] #V2 #HV12
+ elim (IH … T1 … Hde HL ?) -IH -Hde -HL [3,4: // |2: skip ] /3 width=2/
+]
+qed.
+
+lemma sfr_delift: ∀L,T1,d,e. d + e ≤ |L| → ≽ [d, e] L →
+ ∃T2. L ⊢ T1 ▼*[d, e] ≡ T2.
+/2 width=2/ qed-.
+
(* Advanced inversion lemmas ************************************************)
lemma delift_inv_lref1_lt: ∀L,U2,i,d,e. L ⊢ #i ▼*[d, e] ≡ U2 → i < d → U2 = #i.
<key name="ex">5 2</key>
<key name="ex">5 3</key>
<key name="ex">5 4</key>
+ <key name="ex">5 5</key>
<key name="ex">6 4</key>
<key name="ex">6 6</key>
<key name="ex">7 6</key>
interpretation "multiple existental quantifier (5, 4)" 'Ex P0 P1 P2 P3 P4 = (ex5_4 ? ? ? ? P0 P1 P2 P3 P4).
+(* multiple existental quantifier (5, 5) *)
+
+inductive ex5_5 (A0,A1,A2,A3,A4:Type[0]) (P0,P1,P2,P3,P4:A0→A1→A2→A3→A4→Prop) : Prop ≝
+ | ex5_5_intro: ∀x0,x1,x2,x3,x4. P0 x0 x1 x2 x3 x4 → P1 x0 x1 x2 x3 x4 → P2 x0 x1 x2 x3 x4 → P3 x0 x1 x2 x3 x4 → P4 x0 x1 x2 x3 x4 → ex5_5 ? ? ? ? ? ? ? ? ? ?
+.
+
+interpretation "multiple existental quantifier (5, 5)" 'Ex P0 P1 P2 P3 P4 = (ex5_5 ? ? ? ? ? P0 P1 P2 P3 P4).
+
(* multiple existental quantifier (6, 4) *)
inductive ex6_4 (A0,A1,A2,A3:Type[0]) (P0,P1,P2,P3,P4,P5:A0→A1→A2→A3→Prop) : Prop ≝
non associative with precedence 20
for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.$P4) }.
+(* multiple existental quantifier (5, 5) *)
+
+notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P0) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P1) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P2) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P3) (λ${ident x0}.λ${ident x1}.λ${ident x2}.λ${ident x3}.λ${ident x4}.$P4) }.
+
+notation < "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 , ident x4 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4)"
+ non associative with precedence 20
+ for @{ 'Ex (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P0) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P1) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P2) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P3) (λ${ident x0}:$T0.λ${ident x1}:$T1.λ${ident x2}:$T2.λ${ident x3}:$T3.λ${ident x4}:$T4.$P4) }.
+
(* multiple existental quantifier (6, 4) *)
notation > "hvbox(∃∃ ident x0 , ident x1 , ident x2 , ident x3 break . term 19 P0 break & term 19 P1 break & term 19 P2 break & term 19 P3 break & term 19 P4 break & term 19 P5)"
projects / helm.git / commitdiff