+++ /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/unfold/ltpsss.ma".
-include "basic_2/static/aaa_ltpss.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties about iterated parallel unfold ********************************)
-
-lemma aaa_ltpsss_conf: ∀L1,T,A. L1 ⊢ T ÷ A →
- ∀L2,d,e. L1 [d, e] ▶** L2 → L2 ⊢ T ÷ A.
-#L1 #T #A #HT #L2 #d #e #HL12
-@(TC_Conf3 … (λL,A. L ⊢ T ÷ A) … HT ? HL12) /2 width=5/
-qed.
-
-lemma aaa_ltpsss_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ÷ A → ∀L2,d,e. L1 [d, e] ▶** L2 →
- ∀T2. L2 ⊢ T1 [d, e] ▶* T2 → L2 ⊢ T2 ÷ A.
-/3 width=5/ qed.
-
-lemma aaa_ltpsss_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ÷ A → ∀L2,d,e. L1 [d, e] ▶** L2 →
- ∀T2. L2 ⊢ T1 [d, e] ▶ T2 → L2 ⊢ T2 ÷ A.
-/3 width=5/ qed.
lemma lpx_inv_TC: ∀R. reflexive ? R →
∀L1,L2. lpx (TC … R) L1 L2 → TC … (lpx R) L1 L2.
-#R #HR #L1 #L2 #H elim H -L1 -L2 /2 width=1/ /3 width=3/
+#R #HR #L1 #L2 #H elim H -L1 -L2 /3 width=1/ /3 width=3/
qed.
lemma lpx_append: ∀R,K1,K2. lpx R K1 K2 → ∀L1,L2. lpx R 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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_ldrop.ma".
-include "basic_2/static/aaa_lift.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties about parallel unfold *****************************************)
-
-(* Note: lemma 500 *)
-lemma aaa_ltpss_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
-#L1 #T1 #A #H elim H -L1 -T1 -A
-[ #L1 #k #L2 #d #e #_ #T2 #H
- >(tpss_inv_sort1 … H) -H //
-| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H
- [ #H destruct
- elim (lt_or_ge i d) #Hdi
- [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct
- /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
- | elim (lt_or_ge i (d + e)) #Hide
- [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct
- /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
- | -Hdi
- lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide
- /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
- elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=7/
- ]
-| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
-| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
-| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
-| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
-]
-qed.
-
-lemma aaa_ltpss_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
-/3 width=7/ qed.
-
-lemma aaa_ltpss_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
- ∀L2,d,e. L1 ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
-/2 width=7/ qed.
-
-lemma aaa_tpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T2 ⁝ A.
-/2 width=7/ qed.
-
-lemma aaa_tps_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T2 ⁝ A.
-/2 width=7/ 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/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_dx_ldrop.ma".
+include "basic_2/static/aaa_lift.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+(* Note: lemma 500 *)
+lemma aaa_ltpss_dx_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
+#L1 #T1 #A #H elim H -L1 -T1 -A
+[ #L1 #k #L2 #d #e #_ #T2 #H
+ >(tpss_inv_sort1 … H) -H //
+| #I #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ -Hdi #K2 #V2 #HK12 #HV12 #H destruct
+ /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
+ | elim (lt_or_ge i (d + e)) #Hide
+ [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K2 #V2 #HK12 #HV12 #H destruct
+ /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
+ | -Hdi
+ lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide
+ /3 width=8 by aaa_lref/ (**) (* too slow without trace *)
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 /3 width=7/
+ ]
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
+| #a #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /4 width=4/
+| #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
+| #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct /3 width=4/
+]
+qed.
+
+lemma aaa_ltpss_dx_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
+/3 width=7/ qed.
+
+lemma aaa_ltpss_dx_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
+ ∀L2,d,e. L1 ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
+/2 width=7/ qed.
+
+lemma aaa_tpss_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 → L ⊢ T2 ⁝ A.
+/2 width=7/ qed.
+
+lemma aaa_tps_conf: ∀L,T1,A. L ⊢ T1 ⁝ A →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 → L ⊢ T2 ⁝ A.
+/2 width=7/ 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/unfold/ltpss_sn_alt.ma".
+include "basic_2/static/aaa_ltpss_dx.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma aaa_ltpss_sn_conf: ∀L1,T,A. L1 ⊢ T ⁝ A →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 → L2 ⊢ T ⁝ A.
+#L1 #T #A #HT #L2 #d #e #HL12
+lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
+@(TC_Conf3 … (λL,A. L ⊢ T ⁝ A) … HT ? HL12) /2 width=5/
+qed.
+
+lemma aaa_ltpss_sn_tpss_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 → L2 ⊢ T2 ⁝ A.
+/3 width=5/ qed.
+
+lemma aaa_ltpss_sn_tps_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 → L2 ⊢ T2 ⁝ A.
+/3 width=5/ 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/unfold/tpss_tpss.ma".
-include "basic_2/unfold/ltpss_tpss.ma".
-include "basic_2/static/ssta_lift.ma".
-
-(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
-
-(* Properties about parallel unfold *****************************************)
-
-(* Note: apparently this was missing in basic_1 *)
-lemma ssta_ltpss_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 &
- L2 ⊢ U1 ▶* [d, e] U2.
-#h #g #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
-[ #L1 #k1 #l1 #Hkl1 #L2 #d #e #_ #T2 #H
- >(tpss_inv_sort1 … H) -H /3 width=3/
-| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H [ | -HVW1 ]
- [ #H destruct
- elim (lt_or_ge i d) #Hdi [ -HVW1 | ]
- [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK2) #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1
- >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
- | elim (lt_or_ge i (d + e)) #Hide [ -HVW1 | -Hdi -IHVW1 ]
- [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
- elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
- lapply (ldrop_fwd_ldrop2 … HLK2) #H
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
- | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
- elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
- lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
- elim (IHVW1 … HK12 … HV1W2) -IHVW1 -HK12 -HV1W2 #V2 #HWV2 #HW1V2
- elim (lift_total V2 0 (i+1)) #U2 #HVU2
- lapply (ssta_lift … HWV2 … HLK2 … HWT2 … HVU2) -HWV2 -HWT2 #HTU2
- lapply (tpss_lift_ge … HW1V2 … HLK2 … HWU1 … HVU2) // -HW1V2 -HLK2 -HWU1 -HVU2 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /2 width=3/
- ]
-| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
- elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
- [ #H destruct
- elim (lt_or_ge i d) #Hdi [ -HWV1 ]
- [ elim (ltpss_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss11 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1
- >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
- | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -Hdi ]
- [ elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
- elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
- lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
- elim (lift_total W2 0 (i+1)) #U2 #HWU2
- lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1 >minus_plus #H
- lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
- | lapply (ltpss_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
- ]
- ]
- | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
- elim (ltpss_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
- elim (ltpss_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
- lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
- ]
-| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
-| #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
-| #L1 #V1 #W1 #T1 #U1 #l #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H
- elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
- elim (IHVW1 … HV12) -IHVW1 -HV12 //
- elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
-]
-qed.
-
-lemma ssta_ltpss_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/3 width=5/ qed.
-
-lemma ssta_ltpss_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g, l] U1 →
- ∀L2,d,e. L1 ▶* [d, e] L2 →
- ∃∃U2. ⦃h, L2⦄ ⊢ T •[g, l] U2 & L2 ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma ssta_tpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g, l] U1 →
- ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g, l] U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ qed.
-
-lemma ssta_tps_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g, l] U1 →
- ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
- ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g, l] U2 & L ⊢ U1 ▶* [d, e] U2.
-/2 width=5/ 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/unfold/tpss_tpss.ma".
+include "basic_2/unfold/ltpss_dx_tpss.ma".
+include "basic_2/static/ssta_lift.ma".
+
+(* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
+
+(* Properties about dx parallel unfold **************************************)
+
+(* Note: apparently this was missing in basic_1 *)
+lemma ssta_ltpss_dx_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 &
+ L2 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U #l #H elim H -L1 -T1 -U -l
+[ #L1 #k1 #l1 #Hkl1 #L2 #d #e #_ #T2 #H
+ >(tpss_inv_sort1 … H) -H /3 width=3/
+| #L1 #K1 #V1 #W1 #U1 #i #l #HLK1 #HVW1 #HWU1 #IHVW1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H [ | -HVW1 ]
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ -HVW1 | ]
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1
+ >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
+ | elim (lt_or_ge i (d + e)) #Hide [ -HVW1 | -Hdi -IHVW1 ]
+ [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #V2 #HK12 #HV12 #H destruct
+ elim (IHVW1 … HK12 … HV12) -IHVW1 -HK12 -HV12 #W2 #HVW2 #HW12
+ lapply (ldrop_fwd_ldrop2 … HLK2) #H
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … H … HWU1 … HWU2) // -HW12 -H -HWU1 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #HVW2 #HWT2
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K0 #V0 #HK12 #HV12 #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK2) -HLK2 #HLK2
+ lapply (tpss_trans_eq … HV12 HVW2) -V2 #HV1W2
+ elim (IHVW1 … HK12 … HV1W2) -IHVW1 -HK12 -HV1W2 #V2 #HWV2 #HW1V2
+ elim (lift_total V2 0 (i+1)) #U2 #HVU2
+ lapply (ssta_lift … HWV2 … HLK2 … HWT2 … HVU2) -HWV2 -HWT2 #HTU2
+ lapply (tpss_lift_ge … HW1V2 … HLK2 … HWU1 … HVU2) // -HW1V2 -HLK2 -HWU1 -HVU2 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /2 width=3/
+ ]
+| #L1 #K1 #W1 #V1 #U1 #i #l #HLK1 #HWV1 #HWU1 #IHWV1 #L2 #d #e #HL12 #T2 #H
+ elim (tpss_inv_lref1 … H) -H [ | -HWV1 -HWU1 -IHWV1 ]
+ [ #H destruct
+ elim (lt_or_ge i d) #Hdi [ -HWV1 ]
+ [ elim (ltpss_dx_ldrop_conf_le … HL12 … HLK1 ?) -L1 /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss11 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1
+ >minus_plus <plus_minus_m_m // -Hdi /3 width=6/
+ | elim (lt_or_ge i (d + e)) #Hide [ -HWV1 | -IHWV1 -Hdi ]
+ [ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HLK2
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ #K2 #W2 #HK12 #HW12 #H destruct
+ elim (IHWV1 … HK12 … HW12) -IHWV1 -HK12 #V2 #HWV2 #_
+ lapply (ldrop_fwd_ldrop2 … HLK2) #HLK
+ elim (lift_total W2 0 (i+1)) #U2 #HWU2
+ lapply (tpss_lift_ge … HW12 … HLK … HWU1 … HWU2) // -HW12 -HLK -HWU1 >minus_plus #H
+ lapply (tpss_weak … H d e ? ?) [1,2: normalize [ >commutative_plus <plus_minus_m_m // | /2 width=1/ ]] -Hdi -Hide /3 width=6/
+ | lapply (ltpss_dx_ldrop_conf_ge … HL12 … HLK1 ?) -L1 // -Hide /3 width=6/
+ ]
+ ]
+ | * #K2 #V2 #W2 #Hdi #Hide #HLK2 #_ #_
+ elim (ltpss_dx_ldrop_conf_be … HL12 … HLK1 ? ?) -L1 // /2 width=2/ #X #H #HL2K0
+ elim (ltpss_dx_inv_tpss21 … H ?) -H /2 width=1/ -Hdi -Hide #K0 #V0 #_ #_ #H destruct
+ lapply (ldrop_mono … HL2K0 … HLK2) -HL2K0 -HLK2 #H destruct
+ ]
+| #a #I #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_bind1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 /2 width=1/ -HL12 /3 width=5/
+| #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
+| #L1 #V1 #W1 #T1 #U1 #l #_ #_ #IHVW1 #IHTU1 #L2 #d #e #HL12 #X #H
+ elim (tpss_inv_flat1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
+ elim (IHVW1 … HV12) -IHVW1 -HV12 //
+ elim (IHTU1 … HT12) -IHTU1 -HT12 // -HL12 /3 width=5/
+]
+qed.
+
+lemma ssta_ltpss_dx_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 &
+ L2 ⊢ U1 ▶* [d, e] U2.
+/3 width=5/ qed.
+
+lemma ssta_ltpss_dx_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g, l] U1 →
+ ∀L2,d,e. L1 ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •[g, l] U2 & L2 ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma ssta_tpss_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g, l] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g, l] U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ qed.
+
+lemma ssta_tps_conf: ∀h,g,L,T1,U1,l. ⦃h, L⦄ ⊢ T1 •[g, l] U1 →
+ ∀T2,d,e. L ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L⦄ ⊢ T2 •[g, l] U2 & L ⊢ U1 ▶* [d, e] U2.
+/2 width=5/ 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/unfold/ltpss_sn_alt.ma".
+include "basic_2/static/ssta_ltpss_dx.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties about sn parallel unfold **************************************)
+
+lemma ssta_ltpss_sn_conf: ∀h,g,L1,T,U1,l. ⦃h, L1⦄ ⊢ T •[g, l] U1 →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T •[g, l] U2 & L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T #U1 #l #HTU1 #L2 #d #e #HL12
+lapply (ltpss_sn_ltpssa … HL12) -HL12 #HL12
+@(ltpssa_ind … HL12) -L2 [ /2 width=3/ ] -HTU1
+#L #L2 #HL1 #HL2 * #U #HTU #HU1
+lapply (ltpssa_ltpss_sn … HL1) -HL1 #HL1
+elim (ssta_ltpss_dx_conf … HTU … HL2) -HTU #U2 #HTU2 #HU2
+lapply (ltpss_dx_tpss_trans_eq … HU2 … HL2) -HU2 -HL2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL1) -HU2 -HL1 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma ssta_ltpss_sn_tpss_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶* [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 &
+ L1 ⊢ U1 ▶* [d, e] U2.
+#h #g #L1 #T1 #U1 #l #HTU1 #L2 #d #e #HL12 #T2 #HT12
+elim (ssta_ltpss_sn_conf … HTU1 … HL12) -HTU1 #U #HT1U #HU1
+elim (ssta_tpss_conf … HT1U … HT12) -T1 #U2 #HTU2 #HU2
+lapply (ltpss_sn_tpss_trans_eq … HU2 … HL12) -HU2 -HL12 #HU2
+lapply (tpss_trans_eq … HU1 HU2) -U /2 width=3/
+qed.
+
+lemma ssta_ltpss_sn_tps_conf: ∀h,g,L1,T1,U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
+ ∀L2,d,e. L1 ⊢ ▶* [d, e] L2 →
+ ∀T2. L2 ⊢ T1 ▶ [d, e] T2 →
+ ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 &
+ L1 ⊢ U1 ▶* [d, e] U2.
+/3 width=3/ qed.
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