--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/reduction/tpr.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
+
+definition cpr: lenv → term → term → Prop ≝
+ λL,T1,T2. ∃∃T. T1 ⇒ T & L ⊢ T [0, |L|] ≫ T2.
+
+interpretation
+ "context-sensitive parallel reduction (term)"
+ 'PRed L T1 T2 = (cpr L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cpr_pr: ∀T1,T2. T1 ⇒ T2 → ∀L. L ⊢ T1 ⇒ T2.
+/2/ qed.
+
+lemma cpr_tps: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 ⇒ T2.
+/3 width=5/ qed.
+
+lemma cpr_refl: ∀L,T. L ⊢ T ⇒ T.
+/2/ qed.
+
+(* NOTE: new property *)
+lemma cpr_flat: ∀I,L,V1,V2,T1,T2.
+ L ⊢ V1 ⇒ V2 → L ⊢ T1 ⇒ T2 → L ⊢ 𝕗{I} V1. T1 ⇒ 𝕗{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 * /3 width=5/
+qed.
+
+lemma cpr_delta: ∀L,K,V1,V2,V,i.
+ ↓[0, i] L ≡ K. 𝕓{Abbr} V1 → K ⊢ V1 [0, |L| - i - 1] ≫ V2 →
+ ↑[0, i + 1] V2 ≡ V → L ⊢ #i ⇒ V.
+#L #K #V1 #V2 #V #i #HLK #HV12 #HV2
+@ex2_1_intro [2: // | skip ] /3 width=8/ (**) (* /4/ is too slow *)
+qed.
+
+lemma cpr_cast: ∀L,V,T1,T2.
+ L ⊢ T1 ⇒ T2 → L ⊢ 𝕗{Cast} V. T1 ⇒ T2.
+#L #V #T1 #T2 * /3/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
--- /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 "lambda-delta/reduction/tpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
+
+inductive lpr: lenv → lenv → Prop ≝
+| lpr_sort: lpr (⋆) (⋆)
+| lpr_item: ∀K1,K2,I,V1,V2.
+ lpr K1 K2 → V1 ⇒ V2 → lpr (K1. 𝕓{I} V1) (K2. 𝕓{I} V2) (*𝕓*)
+.
+
+interpretation
+ "context-free parallel reduction (environment)"
+ 'PRed L1 L2 = (lpr L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpr_inv_item1_aux: ∀L1,L2. L1 ⇒ L2 → ∀K1,I,V1. L1 = K1. 𝕓{I} V1 →
+ ∃∃K2,V2. K1 ⇒ K2 & V1 ⇒ V2 & L2 = K2. 𝕓{I} V2.
+#L1 #L2 * -L1 L2
+[ #K1 #I #V1 #H destruct
+| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct - K1 I V1 /2 width=5/
+]
+qed.
+
+lemma lpr_inv_item1: ∀K1,I,V1,L2. K1. 𝕓{I} V1 ⇒ L2 →
+ ∃∃K2,V2. K1 ⇒ K2 & V1 ⇒ V2 & L2 = K2. 𝕓{I} V2.
+/2/ qed.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/tps.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+inductive tpr: term → term → Prop ≝
+| tpr_sort : ∀k. tpr (⋆k) (⋆k)
+| tpr_lref : ∀i. tpr (#i) (#i)
+| tpr_bind : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
+ tpr (𝕓{I} V1. T1) (𝕓{I} V2. T2)
+| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
+ tpr (𝕗{I} V1. T1) (𝕗{I} V2. T2)
+| tpr_beta : ∀V1,V2,W,T1,T2.
+ tpr V1 V2 → tpr T1 T2 →
+ tpr (𝕚{Appl} V1. 𝕚{Abst} W. T1) (𝕚{Abbr} V2. T2)
+| tpr_delta: ∀V1,V2,T1,T2,T.
+ tpr V1 V2 → tpr T1 T2 → ⋆. 𝕓{Abbr} V2 ⊢ T2 [0, 1] ≫ T →
+ tpr (𝕚{Abbr} V1. T1) (𝕚{Abbr} V2. T)
+| tpr_theta: ∀V,V1,V2,W1,W2,T1,T2.
+ tpr V1 V2 → ↑[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 →
+ tpr (𝕚{Appl} V1. 𝕚{Abbr} W1. T1) (𝕚{Abbr} W2. 𝕚{Appl} V. T2)
+| tpr_zeta : ∀V,T,T1,T2. ↑[0,1] T1 ≡ T → tpr T1 T2 →
+ tpr (𝕚{Abbr} V. T) T2
+| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (𝕚{Cast} V. T1) T2
+.
+
+interpretation
+ "context-free parallel reduction (term)"
+ 'PRed T1 T2 = (tpr T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma tpr_refl: ∀T. T ⇒ T.
+#T elim T -T //
+#I elim I -I /2/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tpr_inv_sort1_aux: ∀U1,U2. U1 ⇒ U2 → ∀k. U1 = ⋆k → U2 = ⋆k.
+#U1 #U2 * -U1 U2
+[ #k0 #k #H destruct -k0 //
+| #i #k #H destruct
+| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
+| #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct
+| #V1 #V2 #T1 #T2 #T #_ #_ #_ #k #H destruct
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct
+| #V #T #T1 #T2 #_ #_ #k #H destruct
+| #V #T1 #T2 #_ #k #H destruct
+]
+qed.
+
+lemma tpr_inv_sort1: ∀k,U2. ⋆k ⇒ U2 → U2 = ⋆k.
+/2/ qed.
+
+lemma tpr_inv_lref1_aux: ∀U1,U2. U1 ⇒ U2 → ∀i. U1 = #i → U2 = #i.
+#U1 #U2 * -U1 U2
+[ #k #i #H destruct
+| #j #i #H destruct -j //
+| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
+| #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
+| #V1 #V2 #T1 #T2 #T #_ #_ #_ #i #H destruct
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
+| #V #T #T1 #T2 #_ #_ #i #H destruct
+| #V #T1 #T2 #_ #i #H destruct
+]
+qed.
+
+lemma tpr_inv_lref1: ∀i,U2. #i ⇒ U2 → U2 = #i.
+/2/ qed.
+
+lemma tpr_inv_abbr1_aux: ∀U1,U2. U1 ⇒ U2 → ∀V1,T1. U1 = 𝕚{Abbr} V1. T1 →
+ ∨∨ ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Abbr} V2. T2
+ | ∃∃V2,T2,T. V1 ⇒ V2 & T1 ⇒ T2 &
+ ⋆. 𝕓{Abbr} V2 ⊢ T2 [0, 1] ≫ T &
+ U2 = 𝕚{Abbr} V2. T
+ | ∃∃T. ↑[0,1] T ≡ T1 & T ⇒ U2.
+#U1 #U2 * -U1 U2
+[ #k #V #T #H destruct
+| #i #V #T #H destruct
+| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #V #T #H destruct -I V1 T1 /3 width=5/
+| #I #V1 #V2 #T1 #T2 #_ #_ #V #T #H destruct
+| #V1 #V2 #W #T1 #T2 #_ #_ #V #T #H destruct
+| #V1 #V2 #T1 #T2 #T #HV12 #HT12 #HT2 #V0 #T0 #H destruct -V1 T1 /3 width=7/
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #V0 #T0 #H destruct
+| #V #T #T1 #T2 #HT1 #HT12 #V0 #T0 #H destruct -V T /3/
+| #V #T1 #T2 #_ #V0 #T0 #H destruct
+]
+qed.
+
+lemma tpr_inv_abbr1: ∀V1,T1,U2. 𝕚{Abbr} V1. T1 ⇒ U2 →
+ ∨∨ ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Abbr} V2. T2
+ | ∃∃V2,T2,T. V1 ⇒ V2 & T1 ⇒ T2 &
+ ⋆. 𝕓{Abbr} V2 ⊢ T2 [0, 1] ≫ T &
+ U2 = 𝕚{Abbr} V2. T
+ | ∃∃T. ↑[0,1] T ≡ T1 & tpr T U2.
+/2/ qed.
+
+lemma tpr_inv_abst1_aux: ∀U1,U2. U1 ⇒ U2 → ∀V1,T1. U1 = 𝕚{Abst} V1. T1 →
+ ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Abst} V2. T2.
+#U1 #U2 * -U1 U2
+[ #k #V #T #H destruct
+| #i #V #T #H destruct
+| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #V #T #H destruct -I V1 T1 /2 width=5/
+| #I #V1 #V2 #T1 #T2 #_ #_ #V #T #H destruct
+| #V1 #V2 #W #T1 #T2 #_ #_ #V #T #H destruct
+| #V1 #V2 #T1 #T2 #T #_ #_ #_ #V0 #T0 #H destruct
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #V0 #T0 #H destruct
+| #V #T #T1 #T2 #_ #_ #V0 #T0 #H destruct
+| #V #T1 #T2 #_ #V0 #T0 #H destruct
+]
+qed.
+
+lemma tpr_inv_abst1: ∀V1,T1,U2. 𝕚{Abst} V1. T1 ⇒ U2 →
+ ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Abst} V2. T2.
+/2/ qed.
+
+lemma tpr_inv_bind1: ∀V1,T1,U2,I. 𝕓{I} V1. T1 ⇒ U2 →
+ ∨∨ ∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕓{I} V2. T2
+ | ∃∃V2,T2,T. V1 ⇒ V2 & T1 ⇒ T2 &
+ ⋆. 𝕓{Abbr} V2 ⊢ T2 [0, 1] ≫ T &
+ U2 = 𝕚{Abbr} V2. T & I = Abbr
+ | ∃∃T. ↑[0,1] T ≡ T1 & tpr T U2 & I = Abbr.
+#V1 #T1 #U2 * #H
+[ elim (tpr_inv_abbr1 … H) -H * /3 width=7/
+| /3/
+]
+qed.
+
+lemma tpr_inv_appl1_aux: ∀U1,U2. U1 ⇒ U2 → ∀V1,U0. U1 = 𝕚{Appl} V1. U0 →
+ ∨∨ ∃∃V2,T2. V1 ⇒ V2 & U0 ⇒ T2 &
+ U2 = 𝕚{Appl} V2. T2
+ | ∃∃V2,W,T1,T2. V1 ⇒ V2 & T1 ⇒ T2 &
+ U0 = 𝕚{Abst} W. T1 &
+ U2 = 𝕓{Abbr} V2. T2
+ | ∃∃V2,V,W1,W2,T1,T2. V1 ⇒ V2 & W1 ⇒ W2 & T1 ⇒ T2 &
+ ↑[0,1] V2 ≡ V &
+ U0 = 𝕚{Abbr} W1. T1 &
+ U2 = 𝕚{Abbr} W2. 𝕚{Appl} V. T2.
+#U1 #U2 * -U1 U2
+[ #k #V #T #H destruct
+| #i #V #T #H destruct
+| #I #V1 #V2 #T1 #T2 #_ #_ #V #T #H destruct
+| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #V #T #H destruct -I V1 T1 /3 width=5/
+| #V1 #V2 #W #T1 #T2 #HV12 #HT12 #V #T #H destruct -V1 T /3 width=8/
+| #V1 #V2 #T1 #T2 #T #_ #_ #_ #V0 #T0 #H destruct
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #V0 #T0 #H
+ destruct -V1 T0 /3 width=12/
+| #V #T #T1 #T2 #_ #_ #V0 #T0 #H destruct
+| #V #T1 #T2 #_ #V0 #T0 #H destruct
+]
+qed.
+
+lemma tpr_inv_appl1: ∀V1,U0,U2. 𝕚{Appl} V1. U0 ⇒ U2 →
+ ∨∨ ∃∃V2,T2. V1 ⇒ V2 & U0 ⇒ T2 &
+ U2 = 𝕚{Appl} V2. T2
+ | ∃∃V2,W,T1,T2. V1 ⇒ V2 & T1 ⇒ T2 &
+ U0 = 𝕚{Abst} W. T1 &
+ U2 = 𝕓{Abbr} V2. T2
+ | ∃∃V2,V,W1,W2,T1,T2. V1 ⇒ V2 & W1 ⇒ W2 & T1 ⇒ T2 &
+ ↑[0,1] V2 ≡ V &
+ U0 = 𝕚{Abbr} W1. T1 &
+ U2 = 𝕚{Abbr} W2. 𝕚{Appl} V. T2.
+/2/ qed.
+
+lemma tpr_inv_cast1_aux: ∀U1,U2. U1 ⇒ U2 → ∀V1,T1. U1 = 𝕚{Cast} V1. T1 →
+ (∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Cast} V2. T2)
+ ∨ T1 ⇒ U2.
+#U1 #U2 * -U1 U2
+[ #k #V #T #H destruct
+| #i #V #T #H destruct
+| #I #V1 #V2 #T1 #T2 #_ #_ #V #T #H destruct
+| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #V #T #H destruct -I V1 T1 /3 width=5/
+| #V1 #V2 #W #T1 #T2 #_ #_ #V #T #H destruct
+| #V1 #V2 #T1 #T2 #T #_ #_ #_ #V0 #T0 #H destruct
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #V0 #T0 #H destruct
+| #V #T #T1 #T2 #_ #_ #V0 #T0 #H destruct
+| #V #T1 #T2 #HT12 #V0 #T0 #H destruct -V T1 /2/
+]
+qed.
+
+lemma tpr_inv_cast1: ∀V1,T1,U2. 𝕚{Cast} V1. T1 ⇒ U2 →
+ (∃∃V2,T2. V1 ⇒ V2 & T1 ⇒ T2 & U2 = 𝕚{Cast} V2. T2)
+ ∨ T1 ⇒ U2.
+/2/ qed.
+
+lemma tpr_inv_flat1: ∀V1,U0,U2,I. 𝕗{I} V1. U0 ⇒ U2 →
+ ∨∨ ∃∃V2,T2. V1 ⇒ V2 & U0 ⇒ T2 &
+ U2 = 𝕗{I} V2. T2
+ | ∃∃V2,W,T1,T2. V1 ⇒ V2 & T1 ⇒ T2 &
+ U0 = 𝕚{Abst} W. T1 &
+ U2 = 𝕓{Abbr} V2. T2 & I = Appl
+ | ∃∃V2,V,W1,W2,T1,T2. V1 ⇒ V2 & W1 ⇒ W2 & T1 ⇒ T2 &
+ ↑[0,1] V2 ≡ V &
+ U0 = 𝕚{Abbr} W1. T1 &
+ U2 = 𝕚{Abbr} W2. 𝕚{Appl} V. T2 &
+ I = Appl
+ | (U0 ⇒ U2 ∧ I = Cast).
+#V1 #U0 #U2 * #H
+[ elim (tpr_inv_appl1 … H) -H * /3 width=12/
+| elim (tpr_inv_cast1 … H) -H [1: *] /3 width=5/
+]
+qed.
+
+lemma tpr_inv_lref2_aux: ∀T1,T2. T1 ⇒ T2 → ∀i. T2 = #i →
+ ∨∨ T1 = #i
+ | ∃∃V,T,T0. ↑[O,1] T0 ≡ T & T0 ⇒ #i &
+ T1 = 𝕚{Abbr} V. T
+ | ∃∃V,T. T ⇒ #i & T1 = 𝕚{Cast} V. T.
+#T1 #T2 * -T1 T2
+[ #k #i #H destruct
+| #j #i /2/
+| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
+| #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
+| #V1 #V2 #T1 #T2 #T #_ #_ #_ #i #H destruct
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
+| #V #T #T1 #T2 #HT1 #HT12 #i #H destruct /3 width=6/
+| #V #T1 #T2 #HT12 #i #H destruct /3/
+]
+qed.
+
+lemma tpr_inv_lref2: ∀T1,i. T1 ⇒ #i →
+ ∨∨ T1 = #i
+ | ∃∃V,T,T0. ↑[O,1] T0 ≡ T & T0 ⇒ #i &
+ T1 = 𝕓{Abbr} V. T
+ | ∃∃V,T. T ⇒ #i & T1 = 𝕗{Cast} V. T.
+/2/ 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 "lambda-delta/substitution/tps_lift.ma".
+include "lambda-delta/reduction/tpr.ma".
+
+(* Relocation properties ****************************************************)
+
+lemma tpr_lift: ∀T1,T2. T1 ⇒ T2 →
+ ∀d,e,U1. ↑[d, e] T1 ≡ U1 → ∀U2. ↑[d, e] T2 ≡ U2 → U1 ⇒ U2.
+#T1 #T2 #H elim H -H T1 T2
+[ #k #d #e #U1 #HU1 #U2 #HU2
+ lapply (lift_mono … HU1 … HU2) -HU1 #H destruct -U1;
+ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct -U2 //
+| #i #d #e #U1 #HU1 #U2 #HU2
+ lapply (lift_mono … HU1 … HU2) -HU1 #H destruct -U1;
+ lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct -U2 //
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
+ elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
+ elim (lift_inv_bind1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct -X2 /3/
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
+ elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct -X2 /3/
+| #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct -X1;
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct -X;
+ elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct -X2 /3/
+| #V1 #V2 #T1 #T2 #T0 #HV12 #HT12 #HT2 #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
+ elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct -X1;
+ elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct -X2;
+ elim (lift_total T2 (d + 1) e) #U2 #HTU2
+ @tpr_delta
+ [4: @(tps_lift_le … HT2 … HTU2 HTU0 ?) /2/ |1: skip |2: /2/ |3: /2/ ] (**) (*/3. is too slow *)
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
+ elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct -X1;
+ elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct -X;
+ elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct -X2;
+ elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct -X;
+ elim (lift_trans_ge … HV2 … HV3 ?) -HV2 HV3 V // /3/
+| #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX #T0 #HT20
+ elim (lift_inv_bind1 … HX) -HX #V3 #T3 #_ #HT3 #HX destruct -X;
+ elim (lift_trans_ge … HT1 … HT3 ?) -HT1 HT3 T // /3 width=6/
+| #V #T1 #T2 #_ #IHT12 #d #e #X #HX #T #HT2
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct -X /3/
+]
+qed.
+
+lemma tpr_inv_lift: ∀T1,T2. T1 ⇒ T2 →
+ ∀d,e,U1. ↑[d, e] U1 ≡ T1 →
+ ∃∃U2. ↑[d, e] U2 ≡ T2 & U1 ⇒ U2.
+#T1 #T2 #H elim H -H T1 T2
+[ #k #d #e #U1 #HU1
+ lapply (lift_inv_sort2 … HU1) -HU1 #H destruct -U1 /2/
+| #i #d #e #U1 #HU1
+ lapply (lift_inv_lref2 … HU1) -HU1 * * #Hid #H destruct -U1 /3/
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct -X;
+ elim (IHV12 … HV01) -IHV12 HV01;
+ elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct -X;
+ elim (IHV12 … HV01) -IHV12 HV01;
+ elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/
+| #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X;
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y;
+ elim (IHV12 … HV01) -IHV12 HV01;
+ elim (IHT12 … HT01) -IHT12 HT01 /3 width=5/
+| #V1 #V2 #T1 #T2 #T0 #_ #_ #HT20 #IHV12 #IHT12 #d #e #X #HX
+ elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct -X;
+ elim (IHV12 … HWV1) -IHV12 HWV1 #W2 #HWV2 #HW12
+ elim (IHT12 … HUT1) -IHT12 HUT1 #U2 #HUT2 #HU12
+ elim (tps_inv_lift1_le … HT20 … HUT2 ?) -HT20 HUT2 // [3: /2 width=5/ |2: skip ] #U0 #HU20 #HUT0
+ @ex2_1_intro [2: /2/ |1: skip |3: /2/ ] (**) (* /3 width=5/ is slow *)
+| #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X;
+ elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y;
+ elim (IHV12 … HV01) -IHV12 HV01 #V3 #HV32 #HV03
+ elim (IHW12 … HW01) -IHW12 HW01 #W3 #HW32 #HW03
+ elim (IHT12 … HT01) -IHT12 HT01 #T3 #HT32 #HT03
+ elim (lift_trans_le … HV32 … HV2 ?) -HV32 HV2 V2 // #V2 #HV32 #HV2
+ @ex2_1_intro [2: /3/ |1: skip |3: /2/ ] (**) (* /4 width=5/ is slow *)
+| #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V0 #T0 #_ #HT0 #H destruct -X;
+ elim (lift_div_le … HT1 … HT0 ?) -HT1 HT0 T // #T #HT0 #HT1
+ elim (IHT12 … HT1) -IHT12 HT1 /3 width=5/
+| #V #T1 #T2 #_ #IHT12 #d #e #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct -X;
+ elim (IHT12 … HT01) -IHT12 HT01 /3/
+]
+qed.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/lift_weight.ma".
+include "lambda-delta/substitution/tps_tps.ma".
+include "lambda-delta/reduction/tpr_lift.ma".
+include "lambda-delta/reduction/tpr_tps.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+(* Confluence lemmas ********************************************************)
+
+lemma tpr_conf_sort_sort: ∀k. ∃∃X. ⋆k ⇒ X & ⋆k ⇒ X.
+/2/ qed.
+
+lemma tpr_conf_lref_lref: ∀i. ∃∃X. #i ⇒ X & #i ⇒ X.
+/2/ qed.
+
+lemma tpr_conf_bind_bind:
+ ∀I,V0,V1,T0,T1,V2,T2. (
+ ∀X0:term. #X0 < #V0 + #T0 + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
+ ∃∃X. 𝕓{I} V1. T1 ⇒ X & 𝕓{I} V2. T2 ⇒ X.
+#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
+elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 HT02 IH /3 width=5/
+qed.
+
+lemma tpr_conf_bind_delta:
+ ∀V0,V1,T0,T1,V2,T2,T. (
+ ∀X0:term. #X0 < #V0 + #T0 + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 →
+ T0 ⇒ T1 → T0 ⇒ T2 → ⋆. 𝕓{Abbr} V2 ⊢ T2 [O,1] ≫ T →
+ ∃∃X. 𝕓{Abbr} V1. T1 ⇒ X & 𝕓{Abbr} V2. T ⇒ X.
+#V0 #V1 #T0 #T1 #V2 #T2 #T #IH #HV01 #HV02 #HT01 #HT02 #HT2
+elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 HT02 IH // -V0 T0 #T0 #HT10 #HT20
+elim (tpr_tps_bind … HV2 HT20 … HT2) -HT20 HT2 /3 width=5/
+qed.
+
+lemma tpr_conf_bind_zeta:
+ ∀X2,V0,V1,T0,T1,T. (
+ ∀X0:term. #X0 < #V0 + #T0 +1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → T0 ⇒ T1 → T ⇒ X2 → ↑[O, 1] T ≡ T0 →
+ ∃∃X. 𝕓{Abbr} V1. T1 ⇒ X & X2 ⇒ X.
+#X2 #V0 #V1 #T0 #T1 #T #IH #HV01 #HT01 #HTX2 #HT0
+elim (tpr_inv_lift … HT01 … HT0) -HT01 #U #HUT1 #HTU
+lapply (tw_lift … HT0) -HT0 #HT0
+elim (IH … HTX2 … HTU) -HTX2 HTU IH /3/
+qed.
+
+lemma tpr_conf_flat_flat:
+ ∀I,V0,V1,T0,T1,V2,T2. (
+ ∀X0:term. #X0 < #V0 + #T0 + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
+ ∃∃T0. 𝕗{I} V1. T1 ⇒ T0 & 𝕗{I} V2. T2 ⇒ T0.
+#I #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
+elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 HT02 /3 width=5/
+qed.
+
+lemma tpr_conf_flat_beta:
+ ∀V0,V1,T1,V2,W0,U0,T2. (
+ ∀X0:term. #X0 < #V0 + (#W0 + #U0 + 1) + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 →
+ U0 ⇒ T2 → 𝕓{Abst} W0. U0 ⇒ T1 →
+ ∃∃X. 𝕗{Appl} V1. T1 ⇒ X & 𝕓{Abbr} V2. T2 ⇒ X.
+#V0 #V1 #T1 #V2 #W0 #U0 #T2 #IH #HV01 #HV02 #HT02 #H
+elim (tpr_inv_abst1 … H) -H #W1 #U1 #HW01 #HU01 #H destruct -T1;
+elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
+elim (IH … HT02 … HU01) -HT02 HU01 IH /3 width=5/
+qed.
+
+lemma tpr_conf_flat_theta:
+ ∀V0,V1,T1,V2,V,W0,W2,U0,U2. (
+ ∀X0:term. #X0 < #V0 + (#W0 + #U0 + 1) + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 → ↑[O,1] V2 ≡ V →
+ W0 ⇒ W2 → U0 ⇒ U2 → 𝕓{Abbr} W0. U0 ⇒ T1 →
+ ∃∃X. 𝕗{Appl} V1. T1 ⇒ X & 𝕓{Abbr} W2. 𝕗{Appl} V. U2 ⇒ X.
+#V0 #V1 #T1 #V2 #V #W0 #W2 #U0 #U2 #IH #HV01 #HV02 #HV2 #HW02 #HU02 #H
+elim (IH … HV01 … HV02) -HV01 HV02 // #VV #HVV1 #HVV2
+elim (lift_total VV 0 1) #VVV #HVV
+lapply (tpr_lift … HVV2 … HV2 … HVV) #HVVV
+elim (tpr_inv_abbr1 … H) -H *
+(* case 1: bind *)
+[ -HV2 HVV2 #WW #UU #HWW0 #HUU0 #H destruct -T1;
+ elim (IH … HW02 … HWW0) -HW02 HWW0 // #W #HW2 #HWW
+ elim (IH … HU02 … HUU0) -HU02 HUU0 IH // #U #HU2 #HUU
+ @ex2_1_intro [2: @tpr_theta |1:skip |3: @tpr_bind ] /2 width=7/ (**) (* /4 width=7/ is too slow *)
+(* case 2: delta *)
+| -HV2 HVV2 #WW2 #UU2 #UU #HWW2 #HUU02 #HUU2 #H destruct -T1;
+ elim (IH … HW02 … HWW2) -HW02 HWW2 // #W #HW02 #HWW2
+ elim (IH … HU02 … HUU02) -HU02 HUU02 IH // #U #HU2 #HUUU2
+ elim (tpr_tps_bind … HWW2 HUUU2 … HUU2) -HUU2 HUUU2 #UUU #HUUU2 #HUUU1
+ @ex2_1_intro
+ [2: @tpr_theta [6: @HVV |7: @HWW2 |8: @HUUU2 |1,2,3,4: skip | // ]
+ |1:skip
+ |3: @tpr_delta [3: @tpr_flat |1: skip ] /2 width=5/
+ ] (**) (* /5 width=14/ is too slow *)
+(* case 3: zeta *)
+| -HW02 HVV HVVV #UU1 #HUU10 #HUUT1
+ elim (tpr_inv_lift … HU02 … HUU10) -HU02 #UU #HUU2 #HUU1
+ lapply (tw_lift … HUU10) -HUU10 #HUU10
+ elim (IH … HUUT1 … HUU1) -HUUT1 HUU1 IH // -HUU10 #U #HU2 #HUUU2
+ @ex2_1_intro
+ [2: @tpr_flat
+ |1: skip
+ |3: @tpr_zeta [2: @lift_flat |1: skip |3: @tpr_flat ]
+ ] /2 width=5/ (**) (* /5 width=5/ is too slow *)
+]
+qed.
+
+lemma tpr_conf_flat_cast:
+ ∀X2,V0,V1,T0,T1. (
+ ∀X0:term. #X0 < #V0 + # T0 + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → T0 ⇒ T1 → T0 ⇒ X2 →
+ ∃∃X. 𝕗{Cast} V1. T1 ⇒ X & X2 ⇒ X.
+#X2 #V0 #V1 #T0 #T1 #IH #_ #HT01 #HT02
+elim (IH … HT01 … HT02) -HT01 HT02 IH /3/
+qed.
+
+lemma tpr_conf_beta_beta:
+ ∀W0:term. ∀V0,V1,T0,T1,V2,T2. (
+ ∀X0:term. #X0 < #V0 + (#W0 + #T0 + 1) + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
+ ∃∃X. 𝕓{Abbr} V1. T1 ⇒X & 𝕓{Abbr} V2. T2 ⇒ X.
+#W0 #V0 #V1 #T0 #T1 #V2 #T2 #IH #HV01 #HV02 #HT01 #HT02
+elim (IH … HV01 … HV02) -HV01 HV02 //
+elim (IH … HT01 … HT02) -HT01 HT02 IH /3 width=5/
+qed.
+
+lemma tpr_conf_delta_delta:
+ ∀V0,V1,T0,T1,TT1,V2,T2,TT2. (
+ ∀X0:term. #X0 < #V0 +#T0 + 1→
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 → T0 ⇒ T1 → T0 ⇒ T2 →
+ ⋆. 𝕓{Abbr} V1 ⊢ T1 [O, 1] ≫ TT1 →
+ ⋆. 𝕓{Abbr} V2 ⊢ T2 [O, 1] ≫ TT2 →
+ ∃∃X. 𝕓{Abbr} V1. TT1 ⇒ X & 𝕓{Abbr} V2. TT2 ⇒ X.
+#V0 #V1 #T0 #T1 #TT1 #V2 #T2 #TT2 #IH #HV01 #HV02 #HT01 #HT02 #HTT1 #HTT2
+elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
+elim (IH … HT01 … HT02) -HT01 HT02 IH // #T #HT1 #HT2
+elim (tpr_tps_bind … HV1 HT1 … HTT1) -HT1 HTT1 #U1 #TTU1 #HTU1
+elim (tpr_tps_bind … HV2 HT2 … HTT2) -HT2 HTT2 #U2 #TTU2 #HTU2
+elim (tps_conf … HTU1 … HTU2) -HTU1 HTU2 #U #HU1 #HU2
+@ex2_1_intro [2,3: @tpr_delta |1: skip ] /width=10/ (**) (* /3 width=10/ is too slow *)
+qed.
+
+lemma tpr_conf_delta_zeta:
+ ∀X2,V0,V1,T0,T1,TT1,T2. (
+ ∀X0:term. #X0 < #V0 +#T0 + 1→
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → T0 ⇒ T1 → ⋆. 𝕓{Abbr} V1 ⊢ T1 [O,1] ≫ TT1 →
+ T2 ⇒ X2 → ↑[O, 1] T2 ≡ T0 →
+ ∃∃X. 𝕓{Abbr} V1. TT1 ⇒ X & X2 ⇒ X.
+#X2 #V0 #V1 #T0 #T1 #TT1 #T2 #IH #_ #HT01 #HTT1 #HTX2 #HTT20
+elim (tpr_inv_lift … HT01 … HTT20) -HT01 #TT2 #HTT21 #HTT2
+lapply (tps_inv_lift1_eq … HTT1 … HTT21) -HTT1 #HTT1 destruct -T1;
+lapply (tw_lift … HTT20) -HTT20 #HTT20
+elim (IH … HTX2 … HTT2) -HTX2 HTT2 IH /3/
+qed.
+
+lemma tpr_conf_theta_theta:
+ ∀VV1,V0,V1,W0,W1,T0,T1,V2,VV2,W2,T2. (
+ ∀X0:term. #X0 < #V0 + (#W0 + #T0 + 1) + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ V0 ⇒ V1 → V0 ⇒ V2 → W0 ⇒ W1 → W0 ⇒ W2 → T0 ⇒ T1 → T0 ⇒ T2 →
+ ↑[O, 1] V1 ≡ VV1 → ↑[O, 1] V2 ≡ VV2 →
+ ∃∃X. 𝕓{Abbr} W1. 𝕗{Appl} VV1. T1 ⇒ X & 𝕓{Abbr} W2. 𝕗{Appl} VV2. T2 ⇒ X.
+#VV1 #V0 #V1 #W0 #W1 #T0 #T1 #V2 #VV2 #W2 #T2 #IH #HV01 #HV02 #HW01 #HW02 #HT01 #HT02 #HVV1 #HVV2
+elim (IH … HV01 … HV02) -HV01 HV02 // #V #HV1 #HV2
+elim (IH … HW01 … HW02) -HW01 HW02 // #W #HW1 #HW2
+elim (IH … HT01 … HT02) -HT01 HT02 IH // #T #HT1 #HT2
+elim (lift_total V 0 1) #VV #HVV
+lapply (tpr_lift … HV1 … HVV1 … HVV) -HV1 HVV1 #HVV1
+lapply (tpr_lift … HV2 … HVV2 … HVV) -HV2 HVV2 HVV #HVV2
+@ex2_1_intro [2,3: @tpr_bind |1:skip ] /2 width=5/ (**) (* /4 width=5/ is too slow *)
+qed.
+
+lemma tpr_conf_zeta_zeta:
+ ∀V0:term. ∀X2,TT0,T0,T1,T2. (
+ ∀X0:term. #X0 < #V0 + #TT0 + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ T0 ⇒ T1 → T2 ⇒ X2 →
+ ↑[O, 1] T0 ≡ TT0 → ↑[O, 1] T2 ≡ TT0 →
+ ∃∃X. T1 ⇒ X & X2 ⇒ X.
+#V0 #X2 #TT0 #T0 #T1 #T2 #IH #HT01 #HTX2 #HTT0 #HTT20
+lapply (lift_inj … HTT0 … HTT20) -HTT0 #H destruct -T0;
+lapply (tw_lift … HTT20) -HTT20 #HTT20
+elim (IH … HT01 … HTX2) -HT01 HTX2 IH /2/
+qed.
+
+lemma tpr_conf_tau_tau:
+ ∀V0,T0:term. ∀X2,T1. (
+ ∀X0:term. #X0 < #V0 + #T0 + 1 →
+ ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ T0 ⇒ T1 → T0 ⇒ X2 →
+ ∃∃X. T1 ⇒ X & X2 ⇒ X.
+#V0 #T0 #X2 #T1 #IH #HT01 #HT02
+elim (IH … HT01 … HT02) -HT01 HT02 IH /2/
+qed.
+
+(* Confluence ***************************************************************)
+
+lemma tpr_conf_aux:
+ ∀Y0:term. (
+ ∀X0:term. #X0 < #Y0 → ∀X1,X2. X0 ⇒ X1 → X0 ⇒ X2 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X
+ ) →
+ ∀X0,X1,X2. X0 ⇒ X1 → X0 ⇒ X2 → X0 = Y0 →
+ ∃∃X. X1 ⇒ X & X2 ⇒ X.
+#Y0 #IH #X0 #X1 #X2 * -X0 X1
+[ #k1 #H1 #H2 destruct -Y0;
+ lapply (tpr_inv_sort1 … H1) -H1
+(* case 1: sort, sort *)
+ #H1 destruct -X2 //
+| #i1 #H1 #H2 destruct -Y0;
+ lapply (tpr_inv_lref1 … H1) -H1
+(* case 2: lref, lref *)
+ #H1 destruct -X2 //
+| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_bind1 … H1) -H1 *
+(* case 3: bind, bind *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+ /3 width=7 by tpr_conf_bind_bind/ (**) (* /3 width=7/ is too slow *)
+(* case 4: bind, delta *)
+ | #V2 #T2 #T #HV02 #HT02 #HT2 #H1 #H2 destruct -X2 I
+ /3 width=9 by tpr_conf_bind_delta/ (**) (* /3 width=9/ is too slow *)
+(* case 5: bind, zeta *)
+ | #T #HT0 #HTX2 #H destruct -I
+ /3 width=8 by tpr_conf_bind_zeta/ (**) (* /3 width=8/ is too slow *)
+ ]
+| #I #V0 #V1 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_flat1 … H1) -H1 *
+(* case 6: flat, flat *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+ /3 width=7 by tpr_conf_flat_flat/ (**) (* /3 width=7/ is too slow *)
+(* case 7: flat, beta *)
+ | #V2 #W #U0 #T2 #HV02 #HT02 #H1 #H2 #H3 destruct -T0 X2 I
+ /3 width=8 by tpr_conf_flat_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 8: flat, theta *)
+ | #V2 #V #W0 #W2 #U0 #U2 #HV02 #HW02 #HT02 #HV2 #H1 #H2 #H3 destruct -T0 X2 I
+ /3 width=11 by tpr_conf_flat_theta/ (**) (* /3 width=11/ is too slow *)
+(* case 9: flat, tau *)
+ | #HT02 #H destruct -I
+ /3 width=6 by tpr_conf_flat_cast/ (**) (* /3 width=6/ is too slow *)
+ ]
+| #V0 #V1 #W0 #T0 #T1 #HV01 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_appl1 … H1) -H1 *
+(* case 10: beta, flat (repeated) *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+ @ex2_1_comm /3 width=8 by tpr_conf_flat_beta/
+(* case 11: beta, beta *)
+ | #V2 #WW0 #TT0 #T2 #HV02 #HT02 #H1 #H2 destruct -W0 T0 X2
+ /3 width=8 by tpr_conf_beta_beta/ (**) (* /3 width=8/ is too slow *)
+(* case 12, beta, theta (excluded) *)
+ | #V2 #VV2 #WW0 #W2 #TT0 #T2 #_ #_ #_ #_ #H destruct
+ ]
+| #V0 #V1 #T0 #T1 #TT1 #HV01 #HT01 #HTT1 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_abbr1 … H1) -H1 *
+(* case 13: delta, bind (repeated) *)
+ [ #V2 #T2 #HV02 #T02 #H destruct -X2
+ @ex2_1_comm /3 width=9 by tpr_conf_bind_delta/
+(* case 14: delta, delta *)
+ | #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2
+ /3 width=11 by tpr_conf_delta_delta/ (**) (* /3 width=11/ is too slow *)
+(* case 15: delta, zata *)
+ | #T2 #HT20 #HTX2
+ /3 width=10 by tpr_conf_delta_zeta/ (**) (* /3 width=10/ is too slow *)
+ ]
+| #VV1 #V0 #V1 #W0 #W1 #T0 #T1 #HV01 #HVV1 #HW01 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_appl1 … H1) -H1 *
+(* case 16: theta, flat (repeated) *)
+ [ #V2 #T2 #HV02 #HT02 #H destruct -X2
+ @ex2_1_comm /3 width=11 by tpr_conf_flat_theta/
+(* case 17: theta, beta (repeated) *)
+ | #V2 #WW0 #TT0 #T2 #_ #_ #H destruct
+(* case 18: theta, theta *)
+ | #V2 #VV2 #WW0 #W2 #TT0 #T2 #V02 #HW02 #HT02 #HVV2 #H1 #H2 destruct -W0 T0 X2
+ /3 width=14 by tpr_conf_theta_theta/ (**) (* /3 width=14/ is too slow *)
+ ]
+| #V0 #TT0 #T0 #T1 #HTT0 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_abbr1 … H1) -H1 *
+(* case 19: zeta, bind (repeated) *)
+ [ #V2 #T2 #HV02 #T02 #H destruct -X2
+ @ex2_1_comm /3 width=8 by tpr_conf_bind_zeta/
+(* case 20: zeta, delta (repeated) *)
+ | #V2 #T2 #TT2 #HV02 #HT02 #HTT2 #H destruct -X2
+ @ex2_1_comm /3 width=10 by tpr_conf_delta_zeta/
+(* case 21: zeta, zeta *)
+ | #T2 #HTT20 #HTX2
+ /3 width=9 by tpr_conf_zeta_zeta/ (**) (* /3 width=9/ is too slow *)
+ ]
+| #V0 #T0 #T1 #HT01 #H1 #H2 destruct -Y0;
+ elim (tpr_inv_cast1 … H1) -H1
+(* case 22: tau, flat (repeated) *)
+ [ * #V2 #T2 #HV02 #HT02 #H destruct -X2
+ @ex2_1_comm /3 width=6 by tpr_conf_flat_cast/
+(* case 23: tau, tau *)
+ | #HT02
+ /2 by tpr_conf_tau_tau/
+ ]
+]
+qed.
+
+theorem tpr_conf: ∀T0:term. ∀T1,T2. T0 ⇒ T1 → T0 ⇒ T2 →
+ ∃∃T. T1 ⇒ T & T2 ⇒ T.
+#T @(tw_wf_ind … T) -T /3 width=6 by tpr_conf_aux/
+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 "lambda-delta/reduction/lpr.ma".
+
+(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
+
+axiom tpr_tps_lpr: ∀L1,L2. L1 ⇒ L2 → ∀T1,T2. T1 ⇒ T2 →
+ ∀d,e,U1. L1 ⊢ T1 [d, e] ≫ U1 →
+ ∃∃U2. U1 ⇒ U2 & L2 ⊢ T2 [d, e] ≫ U2.
+
+lemma tpr_tps_bind: ∀I,V1,V2,T1,T2,U1. V1 ⇒ V2 → T1 ⇒ T2 →
+ ⋆. 𝕓{I} V1 ⊢ T1 [0, 1] ≫ U1 →
+ ∃∃U2. U1 ⇒ U2 & ⋆. 𝕓{I} V2 ⊢ T2 [0, 1] ≫ U2.
+/3 width=5/ qed.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/leq.ma".
+include "lambda-delta/substitution/lift.ma".
+
+(* DROPPING *****************************************************************)
+
+inductive drop: lenv → nat → nat → lenv → Prop ≝
+| drop_sort: ∀d,e. drop (⋆) d e (⋆)
+| drop_comp: ∀L1,L2,I,V. drop L1 0 0 L2 → drop (L1. 𝕓{I} V) 0 0 (L2. 𝕓{I} V)
+| drop_drop: ∀L1,L2,I,V,e. drop L1 0 e L2 → drop (L1. 𝕓{I} V) 0 (e + 1) L2
+| drop_skip: ∀L1,L2,I,V1,V2,d,e.
+ drop L1 d e L2 → ↑[d,e] V2 ≡ V1 →
+ drop (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
+.
+
+interpretation "dropping" 'RDrop L1 d e L2 = (drop L1 d e L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
+#d #e #L1 #L2 #H elim H -H d e L1 L2
+[ //
+| #L1 #L2 #I #V #_ #IHL12 #H1 #H2
+ >(IHL12 H1 H2) -IHL12 H1 H2 L1 //
+| #L1 #L2 #I #V #e #_ #_ #_ #H
+ elim (plus_S_eq_O_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H
+ elim (plus_S_eq_O_false … H)
+]
+qed.
+
+lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
+/2 width=5/ qed.
+
+lemma drop_inv_sort1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
+ L2 = ⋆.
+#d #e #L1 #L2 * -d e L1 L2
+[ //
+| #L1 #L2 #I #V #_ #H destruct
+| #L1 #L2 #I #V #e #_ #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
+]
+qed.
+
+lemma drop_inv_sort1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
+/2 width=5/ qed.
+
+lemma drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
+ ∀K,I,V. L1 = K. 𝕓{I} V →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↓[d, e - 1] K ≡ L2).
+#d #e #L1 #L2 * -d e L1 L2
+[ #d #e #_ #K #I #V #H destruct
+| #L1 #L2 #I #V #HL12 #H #K #J #W #HX destruct -L1 I V
+ >(drop_inv_refl … HL12) -HL12 K /3/
+| #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
+]
+qed.
+
+lemma drop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
+ (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+ (0 < e ∧ ↓[0, e - 1] K ≡ L2).
+/2/ qed.
+
+lemma drop_inv_drop1: ∀e,K,I,V,L2.
+ ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
+#e #K #I #V #L2 #H #He
+elim (drop_inv_O1 … H) -H * // #H destruct -e;
+elim (lt_refl_false … He)
+qed.
+
+lemma drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
+ ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L2 = K2. 𝕓{I} V2.
+#d #e #L1 #L2 * -d e L1 L2
+[ #d #e #_ #I #K #V #H destruct
+| #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
+| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z
+ /2 width=5/
+]
+qed.
+
+lemma drop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
+ ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L2 = K2. 𝕓{I} V2.
+/2/ qed.
+
+lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+ ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
+ ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L1 = K1. 𝕓{I} V1.
+#d #e #L1 #L2 * -d e L1 L2
+[ #d #e #_ #I #K #V #H destruct
+| #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
+| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
+| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z
+ /2 width=5/
+]
+qed.
+
+lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
+ ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
+ L1 = K1. 𝕓{I} V1.
+/2/ qed.
+
+(* Basic properties *********************************************************)
+
+lemma drop_refl: ∀L. ↓[0, 0] L ≡ L.
+#L elim L -L /2/
+qed.
+
+lemma drop_drop_lt: ∀L1,L2,I,V,e.
+ ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
+#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
+qed.
+
+lemma drop_leq_drop1: ∀L1,L2,d,e. L1 [d, e] ≈ L2 →
+ ∀I,K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{I} V →
+ d ≤ i → i < d + e →
+ ∃∃K2. K1 [0, d + e - i - 1] ≈ K2 &
+ ↓[0, i] L2 ≡ K2. 𝕓{I} V.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e
+[ #d #e #I #K1 #V #i #H
+ lapply (drop_inv_sort1 … H) -H #H destruct
+| #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #I #K1 #V #i #_ #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #V #e #HL12 #IHL12 #J #K1 #W #i #H #_ #Hie
+ elim (drop_inv_O1 … H) -H * #Hi #HLK1
+ [ -IHL12 Hie; destruct -i K1 J W;
+ <minus_n_O <minus_plus_m_m /2/
+ | -HL12;
+ elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
+ ]
+| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #I #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
+ lapply (plus_S_le_to_pos … Hdi) #Hi
+ lapply (drop_inv_drop1 … H ?) -H // #HLK1
+ elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
+]
+qed.
+
+(* Basic forvard lemmas *****************************************************)
+
+lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
+ ↓[O, e + 1] L1 ≡ K2.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+ elim (drop_inv_O1 … H) -H * #He #H
+ [ -IHL1; destruct -e K2 I2 V2 /2/
+ | @drop_drop >(plus_minus_m_m e 1) /2/
+ ]
+]
+qed.
+
+lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e.
+ ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
+#L1 elim L1 -L1
+[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
+ elim (drop_inv_O1 … H) -H * #He #H
+ [ -IHL1; destruct -e K2 I2 V2 //
+ | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
+ ]
+]
+qed.
+
+lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
+#L1 elim L1 -L1
+[ #L2 #e #H >(drop_inv_sort1 … H) -H //
+| #K1 #I1 #V1 #IHL1 #L2 #e #H
+ elim (drop_inv_O1 … H) -H * #He #H
+ [ -IHL1; destruct -e L2 //
+ | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize
+ >minus_le_minus_minus_comm //
+ ]
+]
+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 "lambda-delta/substitution/lift_lift.ma".
+include "lambda-delta/substitution/drop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Main properties **********************************************************)
+
+theorem drop_mono: ∀d,e,L,L1. ↓[d, e] L ≡ L1 →
+ ∀L2. ↓[d, e] L ≡ L2 → L1 = L2.
+#d #e #L #L1 #H elim H -H d e L L1
+[ #d #e #L2 #H
+ >(drop_inv_sort1 … H) -H L2 //
+| #K1 #K2 #I #V #HK12 #_ #L2 #HL12
+ <(drop_inv_refl … HK12) -HK12 K2
+ <(drop_inv_refl … HL12) -HL12 L2 //
+| #L #K #I #V #e #_ #IHLK #L2 #H
+ lapply (drop_inv_drop1 … H ?) -H /2/
+| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
+ elim (drop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct -X
+ >(lift_inj … HVT1 … HVT2) -HVT1 HVT2
+ >(IHLK1 … HLK2) -IHLK1 HLK2 //
+]
+qed.
+
+theorem drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
+ ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
+ ↓[0, e2 - e1] L1 ≡ L2.
+#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
+[ #d #e #e2 #L2 #H
+ >(drop_inv_sort1 … H) -H L2 //
+| #K1 #K2 #I #V #HK12 #_ #e2 #L2 #H #_ <minus_n_O
+ <(drop_inv_refl … HK12) -HK12 K2 //
+| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
+ lapply (drop_inv_drop1 … H ?) -H /2/ #HL2
+ <minus_plus_comm /3/
+| #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
+ lapply (transitive_le 1 … Hdee2) // #He2
+ lapply (drop_inv_drop1 … H ?) -H // -He2 #HL2
+ lapply (transitive_le (1 + e) … Hdee2) // #Hee2
+ @drop_drop_lt >minus_minus_comm /3/ (**) (* explicit constructor *)
+]
+qed.
+
+theorem drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
+ ∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 →
+ e2 < d1 → let d ≝ d1 - e2 - 1 in
+ ∃∃K1,V1. ↓[0, e2] L1 ≡ K1. 𝕓{I} V1 &
+ ↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2.
+#d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
+[ #d #e #e2 #K2 #I #V2 #H
+ lapply (drop_inv_sort1 … H) -H #H destruct
+| #L1 #L2 #I #V #_ #_ #e2 #K2 #J #V2 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #V #e #_ #_ #e2 #K2 #J #V2 #_ #H
+ elim (lt_zero_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #K2 #J #V #H #He2d
+ elim (drop_inv_O1 … H) -H *
+ [ -IHL12 He2d #H1 #H2 destruct -e2 K2 J V /2 width=5/
+ | -HL12 -HV12 #He #HLK
+ elim (IHL12 … HLK ?) -IHL12 HLK [ <minus_minus /3 width=5/ | /2/ ] (**) (* a bit slow *)
+ ]
+]
+qed.
+
+theorem drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
+ ∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 →
+ ∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2.
+#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
+[ #d #e #e2 #L2 #H
+ >(drop_inv_sort1 … H) -H L2 /2/
+| #K1 #K2 #I #V #HK12 #_ #e2 #L2 #HL2 #H
+ >(drop_inv_refl … HK12) -HK12 K1;
+ lapply (le_O_to_eq_O … H) -H #H destruct -e2 /2/
+| #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
+ lapply (le_O_to_eq_O … H) -H #H destruct -e2;
+ elim (IHL12 … HL2 ?) -IHL12 HL2 // #L0 #H #HL0
+ lapply (drop_inv_refl … H) -H #H destruct -L1 /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
+ elim (drop_inv_O1 … H) -H *
+ [ -He2d IHL12 #H1 #H2 destruct -e2 L /3 width=5/
+ | -HL12 HV12 #He2 #HL2
+ elim (IHL12 … HL2 ?) -IHL12 HL2 L2
+ [ >minus_le_minus_minus_comm // /3/ | /2/ ]
+ ]
+]
+qed.
+
+theorem drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
+ ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2.
+#d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
+[ #d #e #e2 #L2 #H
+ >(drop_inv_sort1 … H) -H L2 //
+| #K1 #K2 #I #V #HK12 #_ #e2 #L2 #H #_ normalize
+ >(drop_inv_refl … HK12) -HK12 K1 //
+| /3/
+| #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
+ lapply (drop_inv_drop1 … H ?) -H // #HL2
+ @drop_drop_lt // >le_plus_minus // @IHL12 /2/ (**) (* explicit constructor *)
+]
+qed.
+
+theorem drop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L.
+ ↓[d1, e1] L1 ≡ L → ↓[0, e2] L ≡ L2 → d1 ≤ e2 →
+ ↓[0, e2 + e1] L1 ≡ L2.
+#e1 #e1 #e2 >commutative_plus /2 width=5/
+qed.
+
+axiom drop_div: ∀e1,L1,L. ↓[0, e1] L1 ≡ L → ∀e2,L2. ↓[0, e2] L2 ≡ L →
+ ∃∃L0. ↓[0, e1] L0 ≡ L2 & ↓[e1, e2] L0 ≡ L1.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/syntax/length.ma".
+
+(* LOCAL ENVIRONMENT EQUALITY ***********************************************)
+
+inductive leq: lenv → nat → nat → lenv → Prop ≝
+| leq_sort: ∀d,e. leq (⋆) d e (⋆)
+| leq_comp: ∀L1,L2,I1,I2,V1,V2.
+ leq L1 0 0 L2 → leq (L1. 𝕓{I1} V1) 0 0 (L2. 𝕓{I2} V2)
+| leq_eq: ∀L1,L2,I,V,e. leq L1 0 e L2 → leq (L1. 𝕓{I} V) 0 (e + 1) (L2.𝕓{I} V)
+| leq_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
+ leq L1 d e L2 → leq (L1. 𝕓{I1} V1) (d + 1) e (L2. 𝕓{I2} V2)
+.
+
+interpretation "local environment equality" 'Eq L1 d e L2 = (leq L1 d e L2).
+
+(* Basic properties *********************************************************)
+
+lemma leq_refl: ∀d,e,L. L [d, e] ≈ L.
+#d elim d -d
+[ #e elim e -e [ #L elim L -L /2/ | #e #IHe #L elim L -L /2/ ]
+| #d #IHd #e #L elim L -L /2/
+]
+qed.
+
+lemma leq_sym: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L2 [d, e] ≈ L1.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e /2/
+qed.
+
+lemma leq_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≈ L2 → 0 < d →
+ ∀I1,I2,V1,V2. L1. 𝕓{I1} V1 [d, e] ≈ L2. 𝕓{I2} V2.
+
+#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2/
+qed.
+
+lemma leq_fwd_length: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e; normalize //
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma leq_inv_sort1_aux: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 #d #e #H elim H -H L1 L2 d e
+[ //
+| #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #H destruct
+| #L1 #L2 #I #V #e #_ #_ #H destruct
+| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #_ #H destruct
+qed.
+
+lemma leq_inv_sort1: ∀L2,d,e. ⋆ [d, e] ≈ L2 → L2 = ⋆.
+/2 width=5/ qed.
+
+lemma leq_inv_sort2: ∀L1,d,e. L1 [d, e] ≈ ⋆ → L1 = ⋆.
+/3/ qed.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/syntax/term.ma".
+
+(* RELOCATION ***************************************************************)
+
+inductive lift: term → nat → nat → term → Prop ≝
+| lift_sort : ∀k,d,e. lift (⋆k) d e (⋆k)
+| lift_lref_lt: ∀i,d,e. i < d → lift (#i) d e (#i)
+| lift_lref_ge: ∀i,d,e. d ≤ i → lift (#i) d e (#(i + e))
+| lift_bind : ∀I,V1,V2,T1,T2,d,e.
+ lift V1 d e V2 → lift T1 (d + 1) e T2 →
+ lift (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
+| lift_flat : ∀I,V1,V2,T1,T2,d,e.
+ lift V1 d e V2 → lift T1 d e T2 →
+ lift (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
+.
+
+interpretation "relocation" 'RLift T1 d e T2 = (lift T1 d e T2).
+
+(* Basic properties *********************************************************)
+
+lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i.
+#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/
+qed.
+
+lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T.
+#T elim T -T
+[ //
+| #i #d elim (lt_or_ge i d) /2/
+| #I elim I -I /2/
+]
+qed.
+
+lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
+#T1 elim T1 -T1
+[ /2/
+| #i #d #e elim (lt_or_ge i d) /3/
+| * #I #V1 #T1 #IHV1 #IHT1 #d #e
+ elim (IHV1 d e) -IHV1 #V2 #HV12
+ [ elim (IHT1 (d+1) e) -IHT1 /3/
+ | elim (IHT1 d e) -IHT1 /3/
+ ]
+]
+qed.
+
+lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
+ d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
+ ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2.
+#d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
+[ /3/
+| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
+| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
+ lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
+ <(arith_d1 i e2 e1) // /3/
+| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
+ elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
+ elim (IHT (d2+1) … ? ? He12) /3 width = 5/
+| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
+ elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
+ elim (IHT d2 … ? ? He12) /3 width = 5/
+]
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+#d #e #T1 #T2 #H elim H -H d e T1 T2 /3/
+qed.
+
+lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
+/2/ qed.
+
+lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
+#d #e #T1 #T2 * -d e T1 T2 //
+[ #i #d #e #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
+qed.
+
+lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
+/2 width=5/ qed.
+
+lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
+ (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
+#d #e #T1 #T2 * -d e T1 T2
+[ #k #d #e #i #H destruct
+| #j #d #e #Hj #i #Hi destruct /3/
+| #j #d #e #Hj #i #Hi destruct /3/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+]
+qed.
+
+lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
+ (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
+/2/ qed.
+
+lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i.
+#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (lt_refl_false … Hdd)
+qed.
+
+lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
+#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (lt_refl_false … Hdd)
+qed.
+
+lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+ T2 = 𝕓{I} V2. U2.
+#d #e #T1 #T2 * -d e T1 T2
+[ #k #d #e #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
+]
+qed.
+
+lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+ T2 = 𝕓{I} V2. U2.
+/2/ qed.
+
+lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T2 = 𝕗{I} V2. U2.
+#d #e #T1 #T2 * -d e T1 T2
+[ #k #d #e #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #i #d #e #_ #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5/
+]
+qed.
+
+lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T2 = 𝕗{I} V2. U2.
+/2/ qed.
+
+lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
+#d #e #T1 #T2 * -d e T1 T2 //
+[ #i #d #e #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+]
+qed.
+
+lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
+/2 width=5/ qed.
+
+lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
+ (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
+#d #e #T1 #T2 * -d e T1 T2
+[ #k #d #e #i #H destruct
+| #j #d #e #Hj #i #Hi destruct /3/
+| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+]
+qed.
+
+lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
+ (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
+/2/ qed.
+
+lemma lift_inv_lref2_lt: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → i < d → T1 = #i.
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (plus_lt_false … Hdd)
+qed.
+
+lemma lift_inv_lref2_ge: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (plus_lt_false … Hdd)
+qed.
+
+lemma lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
+ ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+ T1 = 𝕓{I} V1. U1.
+#d #e #T1 #T2 * -d e T1 T2
+[ #k #d #e #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5/
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
+]
+qed.
+
+lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕓{I} V2. U2 →
+ ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+ T1 = 𝕓{I} V1. U1.
+/2/ qed.
+
+lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
+ ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T1 = 𝕗{I} V1. U1.
+#d #e #T1 #T2 * -d e T1 T2
+[ #k #d #e #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #i #d #e #_ #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width = 5/
+]
+qed.
+
+lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕗{I} V2. U2 →
+ ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T1 = 𝕗{I} V1. U1.
+/2/ 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 "lambda-delta/substitution/lift.ma".
+
+(* RELOCATION ***************************************************************)
+
+(* Main properies ***********************************************************)
+
+theorem lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2.
+#d #e #T1 #U #H elim H -H d e T1 U
+[ #k #d #e #X #HX
+ lapply (lift_inv_sort2 … HX) -HX //
+| #i #d #e #Hid #X #HX
+ lapply (lift_inv_lref2_lt … HX ?) -HX //
+| #i #d #e #Hdi #X #HX
+ lapply (lift_inv_lref2_ge … HX ?) -HX /2/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+]
+qed.
+
+theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
+ d1 ≤ d2 →
+ ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1.
+#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+ lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
+ lapply (lift_inv_lref2_lt … Hi ?) -Hi /3/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
+ elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2
+ [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
+ | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
+ @(ex2_1_intro … #(i - e2))
+ [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
+ | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
+ ]
+ ]
+| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+ lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
+ elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
+ >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/
+| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+ lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
+ elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
+ elim (IHU … HU2 ?) /3 width = 5/
+]
+qed.
+
+theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
+#d #e #T #U1 #H elim H -H d e T U1
+[ #k #d #e #X #HX
+ lapply (lift_inv_sort1 … HX) -HX //
+| #i #d #e #Hid #X #HX
+ lapply (lift_inv_lref1_lt … HX ?) -HX //
+| #i #d #e #Hdi #X #HX
+ lapply (lift_inv_lref1_ge … HX ?) -HX //
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+]
+qed.
+
+theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
+ d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+ >(lift_inv_sort1 … HT2) -HT2 //
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
+ lapply (lift_inv_lref1_lt … HT2 Hid2) /2/
+| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
+ lapply (lift_inv_lref1_ge … HT2 ?) -HT2
+ [ @(transitive_le … Hd21 ?) -Hd21 /2/
+ | -Hd21 /2/
+ ]
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
+ lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) /2/
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
+ lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
+ lapply (IHT12 … HT20 ? ?) /2/
+]
+qed.
+
+theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
+ ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+[ #k #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_sort1 … HX) -HX /2/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
+ lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
+ elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
+ lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
+ lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
+ >plus_plus_comm_23 /4/
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
+ elim (IHV12 … HV20 ?) -IHV12 HV20 //
+ elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
+ elim (IHV12 … HV20 ?) -IHV12 HV20 //
+ elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+]
+qed.
+
+theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
+ ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+[ #k #d1 #e1 #d2 #e2 #X #HX #_
+ >(lift_inv_sort1 … HX) -HX /2/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
+ lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
+ lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e
+ lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2
+ lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
+ elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
+ [2: >plus_plus_comm_23] /4/
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+ elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
+ elim (IHV12 … HV20 ?) -IHV12 HV20 //
+ elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T
+ <plus_minus /3 width=5/
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+ elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
+ elim (IHV12 … HV20 ?) -IHV12 HV20 //
+ elim (IHT12 … HT20 ?) -IHT12 HT20 /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 "lambda-delta/syntax/weight.ma".
+include "lambda-delta/substitution/lift.ma".
+
+(* RELOCATION ***************************************************************)
+
+lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #T1 = #T2.
+#d #e #T1 #T2 #H elim H -d e T1 T2; normalize //
+qed.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/drop.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+inductive tps: lenv → term → nat → nat → term → Prop ≝
+| tps_sort : ∀L,k,d,e. tps L (⋆k) d e (⋆k)
+| tps_lref : ∀L,i,d,e. tps L (#i) d e (#i)
+| tps_subst: ∀L,K,V,U1,U2,i,d,e.
+ d ≤ i → i < d + e →
+ ↓[0, i] L ≡ K. 𝕓{Abbr} V → tps K V 0 (d + e - i - 1) U1 →
+ ↑[0, i + 1] U1 ≡ U2 → tps L (#i) d e U2
+| tps_bind : ∀L,I,V1,V2,T1,T2,d,e.
+ tps L V1 d e V2 → tps (L. 𝕓{I} V1) T1 (d + 1) e T2 →
+ tps L (𝕓{I} V1. T1) d e (𝕓{I} V2. T2)
+| tps_flat : ∀L,I,V1,V2,T1,T2,d,e.
+ tps L V1 d e V2 → tps L T1 d e T2 →
+ tps L (𝕗{I} V1. T1) d e (𝕗{I} V2. T2)
+.
+
+interpretation "partial telescopic substritution"
+ 'PSubst L T1 d e T2 = (tps L T1 d e T2).
+
+(* Basic properties *********************************************************)
+
+lemma tps_leq_repl: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫ T2 →
+ ∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫ T2.
+#L1 #T1 #T2 #d #e #H elim H -H L1 T1 T2 d e
+[ //
+| //
+| #L1 #K1 #V #V1 #V2 #i #d #e #Hdi #Hide #HLK1 #_ #HV12 #IHV12 #L2 #HL12
+ elim (drop_leq_drop1 … HL12 … HLK1 ? ?) -HL12 HLK1 // #K2 #HK12 #HLK2
+ @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ] (**) (* /3 width=6/ is too slow *)
+| /4/
+| /3/
+]
+qed.
+
+lemma tps_refl: ∀T,L,d,e. L ⊢ T [d, e] ≫ T.
+#T elim T -T //
+#I elim I -I /2/
+qed.
+
+lemma tps_weak: ∀L,T1,T2,d1,e1. L ⊢ T1 [d1, e1] ≫ T2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 →
+ L ⊢ T1 [d2, e2] ≫ T2.
+#L #T1 #T #d1 #e1 #H elim H -L T1 T d1 e1
+[ //
+| //
+| #L #K #V #V1 #V2 #i #d1 #e1 #Hid1 #Hide1 #HLK #_ #HV12 #IHV12 #d2 #e2 #Hd12 #Hde12
+ lapply (transitive_le … Hd12 … Hid1) -Hd12 Hid1 #Hid2
+ lapply (lt_to_le_to_lt … Hide1 … Hde12) -Hide1 #Hide2
+ @tps_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2/ ] (**) (* /4 width=6/ is too slow *)
+| /4/
+| /4/
+]
+qed.
+
+lemma tps_weak_top: ∀L,T1,T2,d,e.
+ L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [d, |L| - d] ≫ T2.
+#L #T1 #T #d #e #H elim H -L T1 T d e
+[ //
+| //
+| #L #K #V #V1 #V2 #i #d #e #Hdi #_ #HLK #_ #HV12 #IHV12
+ lapply (drop_fwd_drop2_length … HLK) #Hi
+ lapply (le_to_lt_to_lt … Hdi Hi) #Hd
+ lapply (plus_minus_m_m_comm (|L|) d ?) [ /2/ ] -Hd #Hd
+ lapply (drop_fwd_O1_length … HLK) normalize #HKL
+ lapply (tps_weak … IHV12 0 (|L| - i - 1) ? ?) -IHV12 // -HKL /2 width=6/
+| normalize /2/
+| /2/
+]
+qed.
+
+lemma tps_weak_all: ∀L,T1,T2,d,e.
+ L ⊢ T1 [d, e] ≫ T2 → L ⊢ T1 [0, |L|] ≫ T2.
+#L #T1 #T #d #e #HT12
+lapply (tps_weak … HT12 0 (d + e) ? ?) -HT12 // #HT12
+lapply (tps_weak_top … HT12) //
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tps_inv_lref1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. T1 = #i →
+ T2 = #i ∨
+ ∃∃K,V1,V2,i. d ≤ i & i < d + e &
+ ↓[O, i] L ≡ K. 𝕓{Abbr} V1 &
+ K ⊢ V1 [O, d + e - i - 1] ≫ V2 &
+ ↑[O, i + 1] V2 ≡ T2.
+#L #T1 #T2 #d #e * -L T1 T2 d e
+[ #L #k #d #e #i #H destruct
+| /2/
+| #L #K #V1 #V2 #T2 #i #d #e #Hdi #Hide #HLK #HV12 #HVT2 #j #H destruct -i /3 width=9/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+]
+qed.
+
+lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 →
+ T2 = #i ∨
+ ∃∃K,V1,V2,i. d ≤ i & i < d + e &
+ ↓[O, i] L ≡ K. 𝕓{Abbr} V1 &
+ K ⊢ V1 [O, d + e - i - 1] ≫ V2 &
+ ↑[O, i + 1] V2 ≡ T2.
+/2/ qed.
+
+lemma tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+ ∀I,V1,T1. U1 = 𝕓{I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
+ L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
+ U2 = 𝕓{I} V2. T2.
+#d #e #L #U1 #U2 * -d e L U1 U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #i #d #e #I #V1 #T1 #H destruct
+| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
+]
+qed.
+
+lemma tps_inv_bind1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕓{I} V1. T1 [d, e] ≫ U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 &
+ L. 𝕓{I} V1 ⊢ T1 [d + 1, e] ≫ T2 &
+ U2 = 𝕓{I} V2. T2.
+/2/ qed.
+
+lemma tps_inv_flat1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 →
+ ∀I,V1,T1. U1 = 𝕗{I} V1. T1 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
+ U2 = 𝕗{I} V2. T2.
+#d #e #L #U1 #U2 * -d e L U1 U2
+[ #L #k #d #e #I #V1 #T1 #H destruct
+| #L #i #d #e #I #V1 #T1 #H destruct
+| #L #K #V #U1 #U2 #i #d #e #_ #_ #_ #_ #_ #I #V1 #T1 #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #_ #_ #I #V #T #H destruct
+| #L #J #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #I #V #T #H destruct /2 width=5/
+]
+qed.
+
+lemma tps_inv_flat1: ∀d,e,L,I,V1,T1,U2. L ⊢ 𝕗{I} V1. T1 [d, e] ≫ U2 →
+ ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & L ⊢ T1 [d, e] ≫ T2 &
+ U2 = 𝕗{I} V2. T2.
+/2/ qed.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/drop_drop.ma".
+include "lambda-delta/substitution/tps.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Relocation properties ****************************************************)
+
+lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
+ ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
+ ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
+ dt + et ≤ d →
+ L ⊢ U1 [dt, et] ≫ U2.
+#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
+| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
+| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #_ #HV12 #IHV12 #L #U1 #U2 #d #e #HLK #H #HVU2 #Hdetd
+ lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
+ lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct -U1;
+ elim (lift_trans_ge … HV12 … HVU2 ?) -HV12 HVU2 V2 // <minus_plus #V2 #HV12 #HVU2
+ elim (drop_trans_le … HLK … HKV ?) -HLK HKV K /2/ #X #HLK #H
+ elim (drop_inv_skip2 … H ?) -H /2/ -Hid #K #W #HKV #HVW #H destruct -X
+ @tps_subst [4,5,6,8: // |1,2,3: skip | @IHV12 /2 width=6/ ] (**) (* explicit constructor *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+ @tps_bind [ /2 width=6/ | @IHT12 [3,4,5: /2/ |1,2: skip | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+ /3 width=6/
+]
+qed.
+
+lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
+ ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
+ ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
+ d ≤ dt →
+ L ⊢ U1 [dt + e, et] ≫ U2.
+#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
+| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+ lapply (lift_mono … H1 … H2) -H1 H2 #H destruct -U1 //
+| #K #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HKV #HV1 #HV12 #_ #L #U1 #U2 #d #e #HLK #H #HVU2 #Hddt
+ <(arith_c1x ? ? ? e) in HV1 #HV1 (**) (* explicit athmetical rewrite and ?'s *)
+ lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
+ lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct -U1;
+ lapply (lift_trans_be … HV12 … HVU2 ? ?) -HV12 HVU2 V2 /2/ >plus_plus_comm_23 #HV1U2
+ lapply (drop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid #HLKV
+ @tps_subst [4,5: /2/ |6,7,8: // |1,2,3: skip ] (**) (* /3 width=8/ is too slow *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+ @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
+| #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+ /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+ ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+ dt + et ≤ d →
+ ∃∃T2. K ⊢ T1 [dt, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+[ #L #k #dt #et #K #d #e #_ #T1 #H #_
+ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
+| #L #i #dt #et #K #d #e #_ #T1 #H #_
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+| #L #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #_ #HV12 #IHV12 #K #d #e #HLK #T1 #H #Hdetd
+ lapply (lt_to_le_to_lt … Hidet … Hdetd) #Hid
+ lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct -T1;
+ elim (drop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #W #HKL #HKVL #HWV
+ elim (IHV12 … HKVL … HWV ?) -HKVL HWV /2/ -Hdetd #W1 #HW1 #HWV1
+ elim (lift_trans_le … HWV1 … HV12 ?) -HWV1 HV12 V1 // >arith_a2 /3 width=6/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+ elim (IHV12 … HLK … HWV1 ?) -IHV12 //
+ elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @drop_skip // |2: skip ] -HLK HWV1 Hdetd /3 width=5/ (**) (* just /3 width=5/ is too slow *)
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+ elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
+ elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+ ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+ d + e ≤ dt →
+ ∃∃T2. K ⊢ T1 [dt - e, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+[ #L #k #dt #et #K #d #e #_ #T1 #H #_
+ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
+| #L #i #dt #et #K #d #e #_ #T1 #H #_
+ elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+| #L #KV #V #V1 #V2 #i #dt #et #Hdti #Hidet #HLKV #HV1 #HV12 #_ #K #d #e #HLK #T1 #H #Hdedt
+ lapply (transitive_le … Hdedt … Hdti) #Hdei
+ lapply (plus_le_weak … Hdedt) -Hdedt #Hedt
+ lapply (plus_le_weak … Hdei) #Hei
+ <(arith_h1 ? ? ? e ? ?) in HV1 // #HV1
+ lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct -T1;
+ lapply (drop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV
+ elim (lift_split … HV12 d (i - e + 1) ? ? ?) -HV12; [2,3,4: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
+ @ex2_1_intro
+ [2: @tps_subst [4: /2/ |6,7,8: // |1,2,3: skip |5: @arith5 // ]
+ |1: skip
+ | //
+ ] (**) (* explicitc constructors *)
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+ lapply (plus_le_weak … Hdetd) #Hedt
+ elim (IHV12 … HLK … HWV1 ?) -IHV12 // #W2 #HW12 #HWV2
+ elim (IHU12 … HTU1 ?) -IHU12 HTU1 [4: @drop_skip // |2: skip |3: /2/ ]
+ <plus_minus // /3 width=5/
+| #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
+ elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
+ elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
+ L ⊢ U1 [d, e] ≫ U2 → ∀T1. ↑[d, e] T1 ≡ U1 → U1 = U2.
+#L #U1 #U2 #d #e #H elim H -H L U1 U2 d e
+[ //
+| //
+| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #_ #_ #_ #_ #T1 #H
+ elim (lift_inv_lref2 … H) -H * #H
+ [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi H #H
+ elim (lt_refl_false … H)
+ | lapply (lt_to_le_to_lt … Hide … H) -Hide H #H
+ elim (lt_refl_false … H)
+ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
+ >IHV12 // >IHT12 //
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
+ >IHV12 // >IHT12 //
+]
+qed.
+(*
+ Theorem subst0_gen_lift_ge : (u,t1,x:?; i,h,d:?) (subst0 i u (lift h d t1) x) ->
+ (le (plus d h) i) ->
+ (EX t2 | x = (lift h d t2) & (subst0 (minus i h) u t1 t2)).
+
+ Theorem subst0_gen_lift_rev_ge: (t1,v,u2:?; i,h,d:?)
+ (subst0 i v t1 (lift h d u2)) ->
+ (le (plus d h) i) ->
+ (EX u1 | (subst0 (minus i h) v u1 u2) &
+ t1 = (lift h d u1)
+ ).
+
+
+ Theorem subst0_gen_lift_rev_lelt: (t1,v,u2:?; i,h,d:?)
+ (subst0 i v t1 (lift h d u2)) ->
+ (le d i) -> (lt i (plus d h)) ->
+ (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
+*)
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/tps_lift.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Split properties *********************************************************)
+
+lemma tps_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. d ≤ i → i ≤ d + e →
+ ∃∃T. L ⊢ T1 [d, i - d] ≫ T & L ⊢ T [i, d + e - i] ≫ T2.
+#L #T1 #T2 #d #e #H elim H -L T1 T2 d e
+[ /2/
+| /2/
+| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #HV1 #HV12 #IHV12 #j #Hdj #Hjde
+ elim (lt_or_ge i j) #Hij
+ [ -HV1 Hide;
+ lapply (drop_fwd_drop2 … HLK) #HLK'
+ elim (IHV12 (j - i - 1) ? ?) -IHV12; normalize /2/ -Hjde <minus_n_O >arith_b2 // #W1 #HVW1 #HWV1
+ generalize in match HVW1 generalize in match Hij -HVW1 (**) (* rewriting in the premises, rewrites in the goal too *)
+ >(plus_minus_m_m_comm … Hdj) in ⊢ (% → % → ?) -Hdj #Hij' #HVW1
+ elim (lift_total W1 0 (i + 1)) #W2 #HW12
+ lapply (tps_lift_ge … HWV1 … HLK' HW12 HV12 ?) -HWV1 HLK' HV12 // >arith_a2 /3 width=6/
+ | -IHV12 Hdi Hdj;
+ generalize in match HV1 generalize in match Hide -HV1 Hide (**) (* rewriting in the premises, rewrites in the goal too *)
+ >(plus_minus_m_m_comm … Hjde) in ⊢ (% → % → ?) -Hjde #Hide #HV1
+ @ex2_1_intro [2: @tps_lref |1: skip | /2 width=6/ ] (**) (* /3 width=6 is too slow *)
+ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // #V #HV1 #HV2
+ elim (IHT12 (i + 1) ? ?) -IHT12 [2: /2 by arith4/ |3: /2/ ] (* just /2/ is too slow *)
+ -Hdi Hide >arith_c1 >arith_c1x #T #HT1 #HT2
+ @ex2_1_intro [2,3: @tps_bind | skip ] (**) (* explicit constructors *)
+ [3: @HV1 |4: @HT1 |5: // |1,2: skip | /3 width=5/ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hdi #Hide
+ elim (IHV12 i ? ?) -IHV12 // elim (IHT12 i ? ?) -IHT12 //
+ -Hdi Hide /3 width=5/
+]
+qed.
+
+lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+ ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+ d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+ ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tps_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
+lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
+elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
+qed.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/tps_split.ma".
+
+(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+
+(* Main properties **********************************************************)
+(*
+theorem tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
+ L ⊢ T1 [d, e] ≫ T2.
+#L #T1 #T #d #e #H elim H -L T1 T d e
+[ //
+| //
+| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #_ #HV12 #IHV12 #T2 #HVT2
+ lapply (drop_fwd_drop2 … HLK) #H
+ elim (tps_inv_lift1_up … HVT2 … H … HV12 ? ? ?) -HVT2 H HV12 // normalize [2: /2/ ] #V #HV1 #HVT2
+ @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (tps_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
+ @tps_bind /2/ @IHT12 @(tps_leq_repl … HT2) /2/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+ elim (tps_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
+]
+qed.
+*)
+
+axiom tps_conf_subst_subst_lt: ∀L,K1,V1,W1,T1,i1,d,e,T2,K2,V2,W2,i2.
+ ↓[O, i1] L ≡ K1. 𝕓{Abbr} V1 → ↓[O, i2] L≡ K2. 𝕓{Abbr} V2 →
+ K1 ⊢ V1 [O, d + e - i1 - 1] ≫ W1 → K2 ⊢ V2 [O, d + e - i2 - 1] ≫ W2 →
+ ↑[O, i1 + 1] W1 ≡ T1 → ↑[O, i2 + 1] W2 ≡ T2 →
+ d ≤ i1 → i1 < d + e → d ≤ i2 → i2 < d + e → i1 < i2 →
+ ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
+(*
+#L #K1 #V1 #W1 #T1 #i1 #d #e #T2 #K2 #V2 #W2 #i2
+#HLK1 #HLK2 #HVW1 #HVW2 #HWT1 #HWT2 #Hdi1 #Hi1de #Hdi2 #Hi2de #Hi12
+*)
+
+theorem tps_conf: ∀L,T0,T1,d,e. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
+ ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
+#L #T0 #T1 #d #e #H elim H -H L T0 T1 d e
+[ /2/
+| /2/
+| #L #K1 #V1 #W1 #T1 #i1 #d #e #Hdi1 #Hi1de #HLK1 #HVW1 #HWT1 #IHVW1 #T2 #H
+ elim (tps_inv_lref1 … H) -H
+ [ -IHVW1 #HX destruct -T2
+ @ex2_1_intro [2: // | skip ] /2 width=6/ (**) (* /3 width=9/ is slow *)
+ | * #K2 #V2 #W2 #i2 #Hdi2 #Hi2de #HLK2 #HVW2 #HWT2
+ elim (lt_or_eq_or_gt i1 i2) #Hi12
+ [ @tps_conf_subst_subst_lt /width=19/
+ | -HVW1; destruct -i2;
+ lapply (transitive_le ? ? (i1 + 1) Hdi2 ?) -Hdi2 // #Hdi2
+ lapply (drop_mono … HLK1 … HLK2) -HLK1 Hdi1 Hi1de #H destruct -V1 K1;
+ elim (IHVW1 … HVW2) -IHVW1 HVW2 #W #HW1 #HW2
+ lapply (drop_fwd_drop2 … HLK2) -HLK2 #HLK2
+ elim (lift_total W 0 (i1 + 1)) #T #HWT
+ lapply (tps_lift_ge … HW1 … HLK2 HWT1 HWT ?) -HW1 HWT1 //
+ lapply (tps_lift_ge … HW2 … HLK2 HWT2 HWT ?) -HW2 HWT2 HLK2 HWT // normalize #HT2 #HT1
+ lapply (tps_weak … HT1 d e ? ?) -HT1 [ >arith_i2 // | // ]
+ lapply (tps_weak … HT2 d e ? ?) -HT2 [ >arith_i2 // | // ]
+ /2/
+ | @ex2_1_comm @tps_conf_subst_subst_lt /width=19/
+ ]
+ ]
+| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
+ elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
+ elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2
+ elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2
+ @ex2_1_intro
+ [2: @tps_bind [4: @(tps_leq_repl … HT1) /2/ |2: skip ]
+ |1: skip
+ |3: @tps_bind [2: @(tps_leq_repl … HT2) /2/ ]
+ ] // (**) (* /5/ is too slow *)
+| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
+ elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
+ elim (IHV01 … HV02) -IHV01 HV02;
+ elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/
+]
+qed.
+
+(*
+ Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
+ (u1,u:?; i:?) (subst0 i u u1 u2) ->
+ (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t t2)).
+
+ Theorem subst0_subst0_back: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
+ (u1,u:?; i:?) (subst0 i u u2 u1) ->
+ (EX t | (subst0 j u1 t1 t) & (subst0 (S (plus i j)) u t2 t)).
+
+*)
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+(* THE FORMAL SYSTEM λδ - MATITA SOURCE SCRIPTS
+ * Specification started: 2011 April 17
+ * - Patience on me so that I gain peace and perfection! -
+ * [ suggested invocation to start formal specifications with ]
+ *)
+
+include "lambda-delta/ground.ma".
+include "lambda-delta/notation.ma".
+
+(* BINARY ITEMS *************************************************************)
+
+(* binary binding items *)
+inductive bind2: Type[0] ≝
+| Abbr: bind2 (* abbreviation *)
+| Abst: bind2 (* abstraction *)
+.
+
+(* binary non-binding items *)
+inductive flat2: Type[0] ≝
+| Appl: flat2 (* application *)
+| Cast: flat2 (* explicit type annotation *)
+.
+
+(* binary items *)
+inductive item2: Type[0] ≝
+| Bind: bind2 → item2 (* binding item *)
+| Flat: flat2 → item2 (* non-binding item *)
+.
+
+coercion item2_of_bind2: ∀I:bind2.item2 ≝ Bind on _I:bind2 to item2.
+
+coercion item2_of_flat2: ∀I:flat2.item2 ≝ Flat on _I:flat2 to item2.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/syntax/lenv.ma".
+
+(* LENGTH *******************************************************************)
+
+(* the length of a local environment *)
+let rec length L ≝ match L with
+[ LSort ⇒ 0
+| LPair L _ _ ⇒ length L + 1
+].
+
+interpretation "length (local environment)" 'card L = (length L).
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/syntax/term.ma".
+
+(* LOCAL ENVIRONMENTS *******************************************************)
+
+(* local environments *)
+inductive lenv: Type[0] ≝
+| LSort: lenv (* empty *)
+| LPair: lenv → bind2 → term → lenv (* binary binding construction *)
+.
+
+interpretation "sort (local environment)" 'Star = LSort.
+
+interpretation "environment binding construction (binary)" 'DBind L I T = (LPair L I T).
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/ground.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+(* sort hierarchy specifications *)
+record sh: Type[0] ≝ {
+ next: nat → nat; (* next sort in the hierarchy *)
+ next_lt: ∀k. k < next k (* strict monotonicity condition *)
+}.
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/syntax/item.ma".
+
+(* TERMS ********************************************************************)
+
+(* terms *)
+inductive term: Type[0] ≝
+| TSort: nat → term (* sort: starting at 0 *)
+| TLRef: nat → term (* reference by index: starting at 0 *)
+| TPair: item2 → term → term → term (* binary item construction *)
+.
+
+interpretation "sort (term)" 'Star k = (TSort k).
+
+interpretation "local reference (term)" 'Weight i = (TLRef i).
+
+interpretation "term construction (binary)" 'SItem I T1 T2 = (TPair I T1 T2).
+
+interpretation "term binding construction (binary)" 'SBind I T1 T2 = (TPair (Bind I) T1 T2).
+
+interpretation "term flat construction (binary)" 'SFlat I T1 T2 = (TPair (Flat I) T1 T2).
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambda-delta/syntax/lenv.ma".
+
+(* WEIGHTS ******************************************************************)
+
+(* the weight of a term *)
+let rec tw T ≝ match T with
+[ TSort _ ⇒ 1
+| TLRef _ ⇒ 1
+| TPair _ V T ⇒ tw V + tw T + 1
+].
+
+interpretation "weight (term)" 'Weight T = (tw T).
+
+(* the weight of a local environment *)
+let rec lw L ≝ match L with
+[ LSort ⇒ 0
+| LPair L _ V ⇒ lw L + #V
+].
+
+interpretation "weight (local environment)" 'Weight L = (lw L).
+
+(* the weight of a closure *)
+definition cw: lenv → term → ? ≝ λL,T. #L + #T.
+
+interpretation "weight (closure)" 'Weight L T = (cw L T).
+
+axiom tw_wf_ind: ∀P:term→Prop.
+ (∀T2. (∀T1. # T1 < # T2 → P T1) → P T2) →
+ ∀T. P T.
+
+axiom cw_wf_ind: ∀P:lenv→term→Prop.
+ (∀L2,T2. (∀L1,T1. #[L1,T1] < #[L2,T2] → P L1 T1) → P L2 T2) →
+ ∀L,T. P L T.
--- /dev/null
+baseuri=cic:/matita/lambda-delta/
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