sn3/props sn3_change
sn3/props sn3_gen_def
sn3/props sn3_cdelta
-sn3/props sn3_beta
sn3/props sn3_appl_lref
sn3/props sn3_appl_abbr
sn3/props sn3_appl_cast
sn3/props sn3_appls_cast
sn3/props sn3_appls_bind
sn3/props sn3_appls_beta
-sn3/props sn3_abbr
sn3/props sn3_appls_abbr
sn3/props sns3_lifts
#L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
#T #T2 #_ #HT2 #IHT2 /3 width=5/
qed.
+
+lemma cpr_abbr: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀T1,T2. L. ⓓV1 ⊢ T1 ➡ T2 →
+ L ⊢ ⓓV1. T1 ➡* ⓓV2. T2.
+#L #V1 #V2 #HV12 #T1 #T2 #HT12
+@(cprs_strap2 … (ⓓV1.T2)) /2 width=1/ /3 width=1/
+qed.
(* Basic forward lemmas *****************************************************)
-fact csn_fwd_flat2_aux: ∀L,U. L ⊢ ⬇* U → ∀I,V,T. U = ⓕ{I} V. T → L ⊢ ⬇* T.
+fact csn_fwd_flat_dx_aux: ∀L,U. L ⊢ ⬇* U → ∀I,V,T. U = ⓕ{I} V. T → L ⊢ ⬇* T.
#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
@csn_intro #T2 #HLT2 #HT2
@(IH (ⓕ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
qed.
(* Basic_1: was: sn3_gen_flat *)
-lemma csn_fwd_flat2: ∀I,L,V,T. L ⊢ ⬇* ⓕ{I} V. T → L ⊢ ⬇* T.
+lemma csn_fwd_flat_dx: ∀I,L,V,T. L ⊢ ⬇* ⓕ{I} V. T → L ⊢ ⬇* T.
/2 width=5/ qed-.
-(*
-sn3/fwd sn3_gen_bind
-sn3/fwd sn3_gen_head
-*)
-
(* Basic_1: removed theorems 3: sn3_gen_cflat sn3_cflat sn3_bind *)
--- /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/reducibility/cpr_cpr.ma".
+include "Basic_2/computation/csn.ma".
+
+(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
+
+(* Advanced forvard lemmas **************************************************)
+
+fact csn_fwd_pair_sn_aux: ∀L,U. L ⊢ ⬇* U → ∀I,V,T. U = ②{I} V. T → L ⊢ ⬇* V.
+#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csn_intro #V2 #HLV2 #HV2
+@(IH (②{I} V2. T)) -IH // /2 width=1/ -HLV2 #H destruct /2 width=1/
+qed.
+
+(* Basic_1: was: sn3_gen_head *)
+lemma csn_fwd_pair_sn: ∀I,L,V,T. L ⊢ ⬇* ②{I} V. T → L ⊢ ⬇* V.
+/2 width=5/ qed.
+
+fact csn_fwd_bind_dx_aux: ∀L,U. L ⊢ ⬇* U →
+ ∀I,V,T. U = ⓑ{I} V. T → L. ⓑ{I} V ⊢ ⬇* T.
+#L #U #H elim H -H #U0 #_ #IH #I #V #T #H destruct
+@csn_intro #T2 #HLT2 #HT2
+@(IH (ⓑ{I} V. T2)) -IH // /2 width=1/ -HLT2 #H destruct /2 width=1/
+qed.
+
+(* Basic_1: was: sn3_gen_bind *)
+lemma csn_fwd_bind_dx: ∀I,L,V,T. L ⊢ ⬇* ⓑ{I} V. T → L. ⓑ{I} V ⊢ ⬇* T.
+/2 width=3/ qed.
"context-sensitive strong normalization (term)"
'SNStar L T = (csns L T).
-notation "hvbox( L ⊢ ⬇ * * T )"
- non associative with precedence 45
- for @{ 'SNStar $L $T }.
-
(* Basic eliminators ********************************************************)
lemma csns_ind: ∀L. ∀R:predicate term.
(**************************************************************************)
include "Basic_2/reducibility/lcpr_cpr.ma".
-include "Basic_2/computation/cprs_lcpr.ma".
-include "Basic_2/computation/csn_cprs.ma".
+include "Basic_2/computation/cprs_cprs.ma".
include "Basic_2/computation/csn_lift.ma".
+include "Basic_2/computation/csn_cpr.ma".
+include "Basic_2/computation/csn_cprs.ma".
(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
(* Advanced properties ******************************************************)
-lemma csn_lcpr_trans: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T. L1 ⊢ ⬇* T → L2 ⊢ ⬇* T.
+lemma csn_lcpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T. L1 ⊢ ⬇* T → L2 ⊢ ⬇* T.
#L1 #L2 #HL12 #T #H @(csn_ind_cprs … H) -T #T #_ #IHT
@csn_intro #T0 #HLT0 #HT0
@IHT /2 width=2/ -IHT -HT0 /2 width=3/
qed.
lemma csn_abbr: ∀L,V. L ⊢ ⬇* V → ∀T. L. ⓓV ⊢ ⬇* T → L ⊢ ⬇* ⓓV. T.
-#L #V #HV @(csn_ind … HV) -V #V #_ #IHV #T #HT @(csn_ind_cprs … HT) -T #T #HT #IHT
+#L #V #HV elim HV -V #V #_ #IHV #T #HT @(csn_ind_cprs … HT) -T #T #HT #IHT
@csn_intro #X #H1 #H2
elim (cpr_inv_abbr1 … H1) -H1 *
[ #V0 #V1 #T1 #HLV0 #HLV01 #HLT1 #H destruct
lapply (ltpr_cpr_trans (L. ⓓV) … HLT1) /2 width=1/ -V0 #HLT1
elim (eq_false_inv_tpair … H2) -H2
[ #HV1 @IHV // /2 width=1/ -HV1
- @(csn_lcpr_trans (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/
+ @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -HLV1 /2 width=3/
| -IHV -HLV1 * #H destruct /3 width=1/
]
| -IHV -IHT -H2 #T0 #HT0 #HLT0
lapply (csn_inv_lift … HT … HT0) -HT /2 width=3/
]
qed.
+
+fact csn_appl_beta_aux: ∀L,W. L ⊢ ⬇* W → ∀U. L ⊢ ⬇* U →
+ ∀V,T. U = ⓓV. T → L ⊢ ⬇* ⓐV. ⓛW. T.
+#L #W #H elim H -W #W #_ #IHW #X #H @(csn_ind_cprs … H) -X #X #HVT #IHVT #V #T #H destruct
+lapply (csn_fwd_pair_sn … HVT) #HV
+lapply (csn_fwd_bind_dx … HVT) #HT -HVT
+@csn_intro #X #H #H2
+elim (cpr_inv_appl1 … H) -H *
+[ #V0 #Y #HLV0 #H #H0 destruct
+ elim (cpr_inv_abst1 … H Abbr V) -H #W0 #T0 #HLW0 #HLT0 #H destruct
+ elim (eq_false_inv_beta … H2) -H2
+ [ -IHVT #HW0 @IHW -IHW [1,5: // |3: skip ] -HLW0 /2 width=1/ -HW0
+ @csn_abbr /2 width=3/ -HV
+ @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/
+ | -IHW -HLW0 -HV -HT * #H #HVT0 destruct
+ @(IHVT … HVT0) -IHVT -HVT0 // /2 width=1/
+ ]
+| -IHW -IHVT -H2 #V0 #W0 #T0 #T1 #HLV0 #HLT01 #H1 #H2 destruct
+ lapply (lcpr_cpr_trans (L. ⓓV) … HLT01) -HLT01 /2 width=1/ #HLT01
+ @csn_abbr /2 width=3/ -HV
+ @(csn_lcpr_conf (L. ⓓV)) /2 width=1/ -V0 /2 width=3/
+| -IHW -IHVT -HV -HT -H2 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #_ #H destruct
+]
+qed.
+
+(* Basic_1: was: sn3_beta *)
+lemma csn_appl_beta: ∀L,W. L ⊢ ⬇* W → ∀V,T. L ⊢ ⬇* (ⓓV. T) → (**)
+ L ⊢ ⬇* ⓐV. ⓛW. T.
+/2 width=3/ qed.
(* CONTEXT-SENSITIVE STRONGLY NORMALIZING TERMS *****************************)
-(* Advanced properties ******************************************************)
-
-lemma csn_acp: acp cpr (eq …) (csn …).
-@mk_acp
-[ /2 width=1/
-| /2 width=3/
-| /2 width=5/
-| @cnf_lift
-]
-qed.
-
-lemma csn_abst: ∀L,W. L ⊢ ⬇* W → ∀I,V,T. L. ⓑ{I} V ⊢ ⬇* T → L ⊢ ⬇* ⓛW. T.
-#L #W #HW elim HW -W #W #_ #IHW #I #V #T #HT @(csn_ind … HT) -T #T #HT #IHT
-@csn_intro #X #H1 #H2
-elim (cpr_inv_abst1 … H1 I V) -H1
-#W0 #T0 #HLW0 #HLT0 #H destruct
-elim (eq_false_inv_tpair … H2) -H2
-[ /3 width=5/
-| -HLW0 * #H destruct /3 width=1/
-]
-qed.
-
(* Relocation properties ****************************************************)
(* Basic_1: was: sn3_lift *)
@(IHT1 … HLT10) // -L1 -L2 #H destruct
>(lift_inj … HT0 … HT21) in HT2; -T0 /2 width=1/
qed.
+
+(* Advanced properties ******************************************************)
+
+lemma csn_acp: acp cpr (eq …) (csn …).
+@mk_acp
+[ /2 width=1/
+| /2 width=3/
+| /2 width=5/
+| @cnf_lift
+]
+qed.
+
+(* Basic_1: was: sn3_abbr *)
+lemma csn_lref_abbr: ∀L,K,V,i. ⇩[0, i] L ≡ K. ⓓV → K ⊢ ⬇* V → L ⊢ ⬇* #i.
+#L #K #V #i #HLK #HV
+@csn_intro #X #H #Hi
+elim (cpr_inv_lref1 … H) -H
+[ #H destruct elim (Hi ?) //
+| -Hi * #K0 #V0 #V1 #HLK0 #HV01 #HV1 #_
+ lapply (ldrop_mono … HLK0 … HLK) -HLK #H destruct
+ lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #HLK
+ @(csn_lift … HLK HV1) -HLK -HV1
+ @(csn_cpr_trans … HV) -HV
+ @(cpr_intro … HV01) -HV01 //
+]
+qed.
+
+lemma csn_abst: ∀L,W. L ⊢ ⬇* W → ∀I,V,T. L. ⓑ{I} V ⊢ ⬇* T → L ⊢ ⬇* ⓛW. T.
+#L #W #HW elim HW -W #W #_ #IHW #I #V #T #HT @(csn_ind … HT) -T #T #HT #IHT
+@csn_intro #X #H1 #H2
+elim (cpr_inv_abst1 … H1 I V) -H1
+#W0 #T0 #HLW0 #HLT0 #H destruct
+elim (eq_false_inv_tpair … H2) -H2
+[ /3 width=5/
+| -HLW0 * #H destruct /3 width=1/
+]
+qed.
@or_intror @conj // #HT12 destruct /2 width=1/
qed-.
+lemma eq_false_inv_beta: ∀V1,V2,W1,W2,T1,T2.
+ (ⓐV1. ⓛW1. T1 = ⓐV2. ⓛW2 .T2 →False) →
+ (W1 = W2 → False) ∨
+ (W1 = W2 ∧ (ⓓV1. T1 = ⓓV2. T2 → False)).
+#V1 #V2 #W1 #W2 #T1 #T2 #H
+elim (eq_false_inv_tpair … H) -H
+[ #HV12 elim (term_eq_dec W1 W2) /3 width=1/
+ #H destruct @or_intror @conj // #H destruct /2 width=1/
+| * #H1 #H2 destruct
+ elim (eq_false_inv_tpair … H2) -H2 /3 width=1/
+ * #H #HT12 destruct
+ @or_intror @conj // #H destruct /2 width=1/
+]
+qed.
+
(* Basic_1: removed theorems 3:
not_void_abst not_abbr_void not_abst_void
*)
non associative with precedence 45
for @{ 'SN $L $T }.
+notation "hvbox( L ⊢ ⬇ * * T )"
+ non associative with precedence 45
+ for @{ 'SNStar $L $T }.
+
notation "hvbox( ⦃ L, break T ⦄ break [ R ] ϵ break 〚 A 〛 )"
non associative with precedence 45
for @{ 'InEInt $R $L $T $A }.
elim (tpss_inv_flat1 … HU2) -HU2 #V2 #T2 #HV2 #HT2 #H destruct /4 width=5/
qed-.
-(* Basic_1: removed theorems 5:
- pr2_head_1 pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans
+(* Basic_1: removed theorems 4:
+ pr2_head_2 pr2_cflat pr2_gen_cflat clear_pr2_trans
Basic_1: removed local theorems 3:
pr2_free_free pr2_free_delta pr2_delta_delta
*)
lemma cpr_bind_sn: ∀I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
L ⊢ ⓑ{I} V1. T1 ➡ ⓑ{I} V2. T2.
-#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12
+#I #L #V1 #V2 #T1 #T2 * #V #HV1 #HV2 #HT12
@ex2_1_intro [2: @(tpr_delta … HV1 HT12) | skip ] /2 width=3/ (* /3 width=5/ is too slow *)
qed.
@ex2_1_intro [2: @(tpr_delta … HV12 HT1 HT0) | skip | /2 width=1/ ] (**) (* /3 width=5/ is too slow *)
qed.
+
+(* Basic_1: was only: pr2_head_1 *)
+lemma cpr_pair_sn: ∀I,L,V1,V2,T1,T2. L ⊢ V1 ➡ V2 → T1 ➡ T2 →
+ L ⊢ ②{I} V1. T1 ➡ ②{I} V2. T2.
+* /2 width=1/ /3 width=1/
+qed.
+
(* Advanced forward lemmas **************************************************)
lemma cpr_shift_fwd: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L @ T1 ➡ L @ T2.
(* CONTEXT-SENSITIVE PARALLEL REDUCTION ON TERMS ****************************)
-(* Unfold properties ********************************************************)
+(* Properties concerning parallel unfold on terms ***************************)
(* Note: we could invoke tpss_weak_all instead of ltpr_fwd_length *)
(* Basic_1: was only: pr2_subst1 *)
(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-(* Basic_1: was: wcpr0_ldrop *)
+(* Basic_1: was: wcpr0_drop *)
lemma ltpr_ldrop_conf: ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀L2. L1 ➡ L2 →
∃∃K2. ⇩[d, e] L2 ≡ K2 & K1 ➡ K2.
#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
]
qed.
-(* Basic_1: was: wcpr0_ldrop_back *)
+(* Basic_1: was: wcpr0_drop_back *)
lemma ldrop_ltpr_trans: ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. K1 ➡ K2 →
∃∃L2. ⇩[d, e] L2 ≡ K2 & L1 ➡ L2.
#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
elim (IHT12 … HL12) -L2 /3 width=5/
]
qed.
+
+lemma ltpr_tps_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ▶ T2 → ∀L2. L1 ➡ L2 →
+ ∃∃T. L2 ⊢ T1 [d, e] ▶ T & T2 ➡ T.
+#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
+[ /2 width=3/
+| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #HLK2 #H
+ elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
+ elim (lift_total V2 0 (i+1)) #W2 #HVW2
+ lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/
+| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
+ elim (IHV12 … HL12) -IHV12 #V #HV1 #HV2
+ elim (IHT12 (L2.ⓑ{I}V) ?) /2 width=1/ -L1 /3 width=5/
+| #L1 #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #HL12
+ elim (IHV12 … HL12) -IHV12
+ elim (IHT12 … HL12) -L1 /3 width=5/
+]
+qed.
(* LOCAL ENVIRONMENT SLICING ************************************************)
-(* Basic_1: includes: ldrop_skip_bind *)
+(* Basic_1: includes: drop_skip_bind *)
inductive ldrop: nat → nat → relation lenv ≝
| ldrop_atom : ∀d,e. ldrop d e (⋆) (⋆)
| ldrop_pair : ∀L,I,V. ldrop 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
]
qed.
-(* Basic_1: was: ldrop_gen_refl *)
+(* Basic_1: was: drop_gen_refl *)
lemma ldrop_inv_refl: ∀L1,L2. ⇩[0, 0] L1 ≡ L2 → L1 = L2.
/2 width=5/ qed-.
]
qed.
-(* Basic_1: was: ldrop_gen_sort *)
+(* Basic_1: was: drop_gen_sort *)
lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆.
/2 width=5/ qed-.
(0 < e ∧ ⇩[0, e - 1] K ≡ L2).
/2 width=3/ qed-.
-(* Basic_1: was: ldrop_gen_ldrop *)
+(* Basic_1: was: drop_gen_drop *)
lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
⇩[0, e] K. ⓑ{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2.
#e #K #I #V #L2 #H #He
]
qed.
-(* Basic_1: was: ldrop_gen_skip_l *)
+(* Basic_1: was: drop_gen_skip_l *)
lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
⇧[d - 1, e] V2 ≡ V1 &
]
qed.
-(* Basic_1: was: ldrop_gen_skip_r *)
+(* Basic_1: was: drop_gen_skip_r *)
lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ⇩[d, e] L1 ≡ K2. ⓑ{I} V2 → 0 < d →
∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & ⇧[d - 1, e] V2 ≡ V1 &
L1 = K1. ⓑ{I} V1.
(* Basic properties *********************************************************)
-(* Basic_1: was by definition: ldrop_refl *)
+(* Basic_1: was by definition: drop_refl *)
lemma ldrop_refl: ∀L. ⇩[0, 0] L ≡ L.
#L elim L -L //
qed.
(* Basic forvard lemmas *****************************************************)
-(* Basic_1: was: ldrop_S *)
+(* Basic_1: was: drop_S *)
lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. ⓑ{I2} V2 →
⇩[O, e + 1] L1 ≡ K2.
#L1 elim L1 -L1
qed-.
(* Basic_1: removed theorems 49:
- ldrop_skip_flat
+ drop_skip_flat
cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
- ldrop_clear ldrop_clear_O ldrop_clear_S
+ drop_clear drop_clear_O drop_clear_S
clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
- getl_clear_bind getl_clear_conf getl_dec getl_ldrop getl_ldrop_conf_lt
- getl_ldrop_conf_ge getl_conf_ge_ldrop getl_ldrop_conf_rev
- ldrop_getl_trans_lt ldrop_getl_trans_le ldrop_getl_trans_ge
- getl_ldrop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+ getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
+ getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
+ drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
+ getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
*)
(* Main properties **********************************************************)
-(* Basic_1: was: ldrop_mono *)
+(* Basic_1: was: drop_mono *)
theorem ldrop_mono: ∀d,e,L,L1. ⇩[d, e] L ≡ L1 →
∀L2. ⇩[d, e] L ≡ L2 → L1 = L2.
#d #e #L #L1 #H elim H -d -e -L -L1
]
qed-.
-(* Basic_1: was: ldrop_conf_ge *)
+(* Basic_1: was: drop_conf_ge *)
theorem ldrop_conf_ge: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
⇩[0, e2 - e1] L1 ≡ L2.
]
qed.
-(* Basic_1: was: ldrop_conf_lt *)
+(* Basic_1: was: drop_conf_lt *)
theorem ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
∀e2,K2,I,V2. ⇩[0, e2] L ≡ K2. ⓑ{I} V2 →
e2 < d1 → let d ≝ d1 - e2 - 1 in
]
qed.
-(* Basic_1: was: ldrop_trans_le *)
+(* Basic_1: was: drop_trans_le *)
theorem ldrop_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.
]
qed.
-(* Basic_1: was: ldrop_trans_ge *)
+(* Basic_1: was: drop_trans_ge *)
theorem ldrop_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 -d1 -e1 -L1 -L
#e1 #e1 #e2 >commutative_plus /2 width=5/
qed.
-(* Basic_1: was: ldrop_conf_rev *)
+(* Basic_1: was: drop_conf_rev *)
axiom ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L →
∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1.
/2 width=3/ qed-.
(* Basic_1: removed theorems 27:
- csubst0_clear_O csubst0_ldrop_lt csubst0_ldrop_gt csubst0_ldrop_eq
+ csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
- csubst0_ldrop_gt_back csubst0_ldrop_eq_back csubst0_ldrop_lt_back
+ csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
csubst0_gen_sort csubst0_gen_head csubst0_getl_ge csubst0_getl_lt
csubst0_gen_S_bind_2 csubst0_getl_ge_back csubst0_getl_lt_back
csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
projects / helm.git / commitdiff