(* Basic properties *********************************************************)
+lemma tw_pos: ∀T. 1 ≤ #[T].
+#T elim T -T /2/
+qed.
+
+(* Basic eliminators ********************************************************)
+
axiom tw_wf_ind: ∀R:term→Prop.
(∀T2. (∀T1. #[T1] < #[T2] → R T1) → R T2) →
∀T. R T.
(* *)
(**************************************************************************)
+include "Basic-2/grammar/lenv_weight.ma".
include "Basic-2/grammar/leq.ma".
include "Basic-2/substitution/lift.ma".
]
qed.
-lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e.
+lemma drop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
+#L1 #L2 #d #e #H elim H -H L1 L2 d e // normalize
+[ /2/
+| #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
+ >(tw_lift … HV21) -HV21 /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_atom1 … H) -H #H destruct
]
qed.
-lemma drop_inv_atom1: ∀d,e,L2. ⋆ [d, e] ≫ L2 → L2 = ⋆.
+lemma ltps_inv_atom1: ∀d,e,L2. ⋆ [d, e] ≫ L2 → L2 = ⋆.
/2 width=5/ qed.
fact ltps_inv_tps21_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → d = 0 → 0 < e →
]
qed.
-lemma drop_inv_atom2: ∀d,e,L1. L1 [d, e] ≫ ⋆ → L1 = ⋆.
+lemma ltps_inv_atom2: ∀d,e,L1. L1 [d, e] ≫ ⋆ → L1 = ⋆.
/2 width=5/ qed.
fact ltps_inv_tps22_aux: ∀d,e,L1,L2. L1 [d, e] ≫ L2 → d = 0 → 0 < e →
(* *)
(**************************************************************************)
+include "Basic-2/grammar/cl_weight.ma".
include "Basic-2/substitution/drop.ma".
(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
lemma tps_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≫ T2 → T1 = T2.
/2 width=6/ qed.
+(* Basic forward lemmas *****************************************************)
+
+lemma tps_fwd_tw: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → #[T1] ≤ #[T2].
+#L #T1 #T2 #d #e #H elim H normalize -H L T1 T2 d e
+[ //
+| /2/
+| /3 by monotonic_le_plus_l, le_plus/ (**) (* just /3/ is too slow *)
+| /3 by monotonic_le_plus_l, le_plus/ (**) (* just /3/ is too slow *)
+]
+qed.
+
(* Basic-1: removed theorems 25:
subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
interpretation "partial unfold (term)"
'PSubstStar L T1 d e T2 = (tpss d e L T1 T2).
-(* Basic properties *********************************************************)
+(* Basic eliminators ********************************************************)
lemma tpss_ind: ∀d,e,L,T1. ∀R: term → Prop. R T1 →
- (∀T,T2. L ⊢ T1 [d, e] ≫* T → L ⊢ T[d, e] ≫ T2 → R T → R T2) →
+ (∀T,T2. L ⊢ T1 [d, e] ≫* T → L ⊢ T [d, e] ≫ T2 → R T → R T2) →
∀T2. L ⊢ T1 [d, e] ≫* T2 → R T2.
#d #e #L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
qed.
+(* Basic properties *********************************************************)
+
+lemma tpss_strap: ∀L,T1,T,T2,d,e.
+ L ⊢ T1 [d, e] ≫ T → L ⊢ T [d, e] ≫* T2 → L ⊢ T1 [d, e] ≫* T2.
+/2/ qed.
+
lemma tpss_leq_repl_dx: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≫* T2 →
∀L2. L1 [d, e] ≈ L2 → L2 ⊢ T1 [d, e] ≫* T2.
/3/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "Basic-2/substitution/ltps_tps.ma".
+include "Basic-2/unfold/tpss_tpss.ma".
+
+(* PARTIAL UNFOLD ON TERMS **************************************************)
+
+(* Properties concerning parallel substitution on local environments ********)
+
+lemma ltps_tpss_conf_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2.
+ d1 + e1 ≤ d2 → L0 [d1, e1] ≫ L1 →
+ L0 ⊢ T2 [d2, e2] ≫* U2 → L1 ⊢ T2 [d2, e2] ≫* U2.
+#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HL01 #H @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltps_tps_conf_ge … HU2 … HL01 ?) -HU2 HL01 /2/
+qed.
+
+lemma ltps_tpss_conf: ∀L0,L1,T2,U2,d1,e1,d2,e2.
+ L0 [d1, e1] ≫ L1 → L0 ⊢ T2 [d2, e2] ≫* U2 →
+ ∃∃T. L1 ⊢ T2 [d2, e2] ≫* T & L1 ⊢ U2 [d1, e1] ≫* T.
+#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #HL01 #H @(tpss_ind … H) -U2
+[ /3/
+| #U #U2 #_ #HU2 * #T #HT2 #HUT
+ elim (ltps_tps_conf … HU2 … HL01) -HU2 HL01 #W #HUW #HU2W
+ elim (tpss_strip_eq … HUT … HUW) -U
+ /3 width=5 by ex2_1_intro, step, tpss_strap/ (**) (* just /3 width=5/ is too slow *)
+]
+qed.
+
+lemma ltps_tpss_trans_ge: ∀L0,L1,T2,U2,d1,e1,d2,e2.
+ d1 + e1 ≤ d2 → L1 [d1, e1] ≫ L0 →
+ L0 ⊢ T2 [d2, e2] ≫* U2 → L1 ⊢ T2 [d2, e2] ≫* U2.
+#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde1d2 #HL10 #H @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU
+lapply (ltps_tps_trans_ge … HU2 … HL10 ?) -HU2 HL10 /2/
+qed.
+
+lemma ltps_tpss_trans_down: ∀L0,L1,T2,U2,d1,e1,d2,e2. d2 + e2 ≤ d1 →
+ L1 [d1, e1] ≫ L0 → L0 ⊢ T2 [d2, e2] ≫* U2 →
+ ∃∃T. L1 ⊢ T2 [d2, e2] ≫* T & L0 ⊢ T [d1, e1] ≫* U2.
+#L0 #L1 #T2 #U2 #d1 #e1 #d2 #e2 #Hde2d1 #HL10 #H @(tpss_ind … H) -U2
+[ /3/
+| #U #U2 #_ #HU2 * #T #HT2 #HTU
+ elim (tpss_strap1_down … HTU … HU2 ?) -U // #U #HTU #HU2
+ elim (ltps_tps_trans … HTU … HL10) -HTU HL10 #W #HTW #HWU
+ @(ex2_1_intro … W) /2/ (**) (* /3 width=5/ does not work as in ltps_tpss_conf *)
+]
+qed.
+
+fact ltps_tps_trans_eq_aux: ∀Y1,X2,L1,T2,U2,d,e.
+ L1 ⊢ T2 [d, e] ≫ U2 → ∀L0. L0 [d, e] ≫ L1 →
+ Y1 = L1 → X2 = T2 → L0 ⊢ T2 [d, e] ≫* U2.
+#Y1 #X2 @(cw_wf_ind … Y1 X2) -Y1 X2 #Y1 #X2 #IH
+#L1 #T2 #U2 #d #e * -L1 T2 U2 d e
+[ //
+| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L0 #HL10 #H1 #H2 destruct -Y1 X2;
+ lapply (drop_fwd_lw … HLK1) normalize #H1
+ elim (ltps_drop_trans_be … HL10 … HLK1 ? ?) -HL10 HLK1 [2,3: /2/ ] #X #H #HLK0
+ elim (ltps_inv_tps22 … H ?) -H [2: /2/ ] #K0 #V0 #HK01 #HV01 #H destruct -X;
+ lapply (tps_fwd_tw … HV01) #H2
+ lapply (transitive_le (#[K1] + #[V0]) … H1) -H1 [ /2/ ] -H2 #H
+ lapply (IH … HV01 … HK01 ? ?) -IH HV01 HK01 [1,3: // |2,4: skip | /2/ | /3 width=6/ ]
+| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #L0 #HL0 #H1 #H2 destruct -Y1 X2;
+ lapply (tps_leq_repl_dx … HT12 (L. 𝕓{I} V1) ?) -HT12 /2/ #HT12
+ lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3,5: // |2,4: skip ] #HV12
+ lapply (IH … HT12 (L0. 𝕓{I} V1) ? ? ?) -IH HT12 [1,3,5: /2/ |2,4: skip | normalize // ] -HL0 #HT12
+ lapply (tpss_leq_repl_dx … HT12 (L0. 𝕓{I} V2) ?) -HT12 /2/
+| #L #I #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #L0 #HL0 #H1 #H2 destruct -Y1 X2;
+ lapply (IH … HV12 … HL0 ? ?) -HV12 [1,3,5: // |2,4: skip ]
+ lapply (IH … HT12 … HL0 ? ?) -IH HT12 [1,3,5: // |2,4: skip ] -HL0 /2/
+]
+qed.
+
+lemma ltps_tps_trans_eq: ∀L1,T2,U2,d,e. L1 ⊢ T2 [d, e] ≫ U2 →
+ ∀L0. L0 [d, e] ≫ L1 → L0 ⊢ T2 [d, e] ≫* U2.
+/2 width=5/ qed.
+
+lemma ltps_tpss_trans_eq: ∀L0,L1,T2,U2,d,e. L0 [d, e] ≫ L1 →
+ L1 ⊢ T2 [d, e] ≫* U2 → L0 ⊢ T2 [d, e] ≫* U2.
+#L0 #L1 #T2 #U2 #d #e #HL01 #H @(tpss_ind … H) -U2 //
+#U #U2 #_ #HU2 #IHU @(tpss_trans_eq … IHU) /2/
+qed.
(* Advanced properties ******************************************************)
-lemma tpss_trans_down_strap1: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫* T0 →
- ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 →
- ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫* T2.
+lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 [d1, e1] ≫* T1 →
+ ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 →
+ ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T2 [d1, e1] ≫* T.
+/3/ qed.
+
+lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 [d1, e1] ≫* T1 →
+ ∀L2,T2,d2,e2. L2 ⊢ T0 [d2, e2] ≫ T2 →
+ (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+ ∃∃T. L2 ⊢ T1 [d2, e2] ≫ T & L1 ⊢ T2 [d1, e1] ≫* T.
+/3/ qed.
+
+lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫* T0 →
+ ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫* T2.
+/3/ qed.
+
+lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫ T0 →
+ ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫* T2 → d2 + e2 ≤ d1 →
+ ∃∃T. L ⊢ T1 [d2, e2] ≫* T & L ⊢ T [d1, e1] ≫ T2.
/3/ qed.
lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫* T2 →
[ /2/
| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
elim (tps_split_up … HT12 … Hdi Hide) -HT12 Hide #T0 #HT0 #HT02
- elim (tpss_
trans_down_strap1 … HT3 … HT0 ?) -T [2: <plus_minus_m_m_comm // ]
+ elim (tpss_
strap1_down … HT3 … HT0 ?) -T [2: <plus_minus_m_m_comm // ]
/3 width=7 by ex2_1_intro, step/ (**) (* just /3 width=7/ is too slow *)
]
qed.
lemma TC_strip2: ∀A,R1,R2. confluent A R1 R2 →
∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a0 a1 →
- ∃∃a. R1 a2 a & TC … R2 a1 a.
+ ∃∃a. TC … R2 a1 a & R1 a2 a.
#A #R1 #R2 #HR12 #a0 #a2 #H elim H -H a2
[ #a2 #Ha02 #a1 #Ha01
elim (HR12 … Ha01 … Ha02) -HR12 a0 /3/
| #a #a2 #_ #Ha2 #IHa0 #a1 #Ha01
- elim (IHa0 … Ha01) -IHa0 Ha01 a0 #a0 #Ha0 #Ha21
+ elim (IHa0 … Ha01) -IHa0 Ha01 a0 #a0 #Ha10 #Ha0
elim (HR12 … Ha0 … Ha2) -HR12 a /4/
]
qed.
]
qed.
+lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
+ R a1 a → TC … R a a2 → TC … R a1 a2.
+/3/ qed.
+
lemma TC_strap1: ∀A,R1,R2. transitive A R1 R2 →
∀a1,a0. TC … R1 a1 a0 → ∀a2. R2 a0 a2 →
∃∃a. R2 a1 a & TC … R1 a a2.
lemma TC_strap2: ∀A,R1,R2. transitive A R1 R2 →
∀a0,a2. TC … R2 a0 a2 → ∀a1. R1 a1 a0 →
- ∃∃a. R1 a a2 & TC … R2 a1 a.
+ ∃∃a. TC … R2 a1 a & R1 a a2.
#A #R1 #R2 #HR12 #a0 #a2 #H elim H -H a2
[ #a2 #Ha02 #a1 #Ha10
elim (HR12 … Ha10 … Ha02) -HR12 a0 /3/
| #a #a2 #_ #Ha02 #IHa #a1 #Ha10
- elim (IHa … Ha10) -a0 #a0 #Ha0 #Ha10
+ elim (IHa … Ha10) -a0 #a0 #Ha10 #Ha0
elim (HR12 … Ha0 … Ha02) -HR12 a /4/
]
qed.
[ #a0 #Ha10 #a2 #Ha02
elim (TC_strap2 … HR12 … Ha02 … Ha10) -HR12 a0 /3/
| #a #a0 #_ #Ha0 #IHa #a2 #Ha02
- elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 a0 #a0 #Ha02 #Ha0
+ elim (TC_strap2 … HR12 … Ha02 … Ha0) -HR12 a0 #a0 #Ha0 #Ha02
elim (IHa … Ha0) -a /4/
]
qed.
projects / helm.git / commitdiff