| ltpss_atom : ∀d,e. ltpss d e (⋆) (⋆)
| ltpss_pair : ∀L,I,V. ltpss 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
| ltpss_tpss2: ∀L1,L2,I,V1,V2,e.
- ltpss 0 e L1 L2 → L2 ⊢ V1 [0, e] ▶* V2 →
+ ltpss 0 e L1 L2 → L2 ⊢ V1 ▶* [0, e] V2 →
ltpss 0 (e + 1) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
| ltpss_tpss1: ∀L1,L2,I,V1,V2,d,e.
- ltpss d e L1 L2 → L2 ⊢ V1 [d, e] ▶* V2 →
+ ltpss d e L1 L2 → L2 ⊢ V1 ▶* [d, e] V2 →
ltpss (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
.
(* Basic properties *********************************************************)
lemma ltpss_tps2: ∀L1,L2,I,V1,V2,e.
- L1 [0, e] ▶* L2 → L2 ⊢ V1 [0, e] ▶ V2 →
- L1. ⓑ{I} V1 [0, e + 1] ▶* L2. ⓑ{I} V2.
+ L1 ▶* [0, e] L2 → L2 ⊢ V1 ▶ [0, e] V2 →
+ L1. ⓑ{I} V1 ▶* [0, e + 1] L2. ⓑ{I} V2.
/3 width=1/ qed.
lemma ltpss_tps1: ∀L1,L2,I,V1,V2,d,e.
- L1 [d, e] ▶* L2 → L2 ⊢ V1 [d, e] ▶ V2 →
- L1. ⓑ{I} V1 [d + 1, e] ▶* L2. ⓑ{I} V2.
+ L1 ▶* [d, e] L2 → L2 ⊢ V1 ▶ [d, e] V2 →
+ L1. ⓑ{I} V1 ▶* [d + 1, e] L2. ⓑ{I} V2.
/3 width=1/ qed.
lemma ltpss_tpss2_lt: ∀L1,L2,I,V1,V2,e.
- L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶* V2 →
- 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
+ L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶* [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
#L1 #L2 #I #V1 #V2 #e #HL12 #HV12 #He
>(plus_minus_m_m e 1) /2 width=1/
qed.
lemma ltpss_tpss1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 [d - 1, e] ▶* L2 → L2 ⊢ V1 [d - 1, e] ▶* V2 →
- 0 < d → L1. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
+ L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶* [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
#L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #Hd
>(plus_minus_m_m d 1) /2 width=1/
qed.
lemma ltpss_tps2_lt: ∀L1,L2,I,V1,V2,e.
- L1 [0, e - 1] ▶* L2 → L2 ⊢ V1 [0, e - 1] ▶ V2 →
- 0 < e → L1. ⓑ{I} V1 [0, e] ▶* L2. ⓑ{I} V2.
+ L1 ▶* [0, e - 1] L2 → L2 ⊢ V1 ▶ [0, e - 1] V2 →
+ 0 < e → L1. ⓑ{I} V1 ▶* [0, e] L2. ⓑ{I} V2.
/3 width=1/ qed.
lemma ltpss_tps1_lt: ∀L1,L2,I,V1,V2,d,e.
- L1 [d - 1, e] ▶* L2 → L2 ⊢ V1 [d - 1, e] ▶ V2 →
- 0 < d → L1. ⓑ{I} V1 [d, e] ▶* L2. ⓑ{I} V2.
+ L1 ▶* [d - 1, e] L2 → L2 ⊢ V1 ▶ [d - 1, e] V2 →
+ 0 < d → L1. ⓑ{I} V1 ▶* [d, e] L2. ⓑ{I} V2.
/3 width=1/ qed.
(* Basic_1: was by definition: csubst1_refl *)
-lemma ltpss_refl: ∀L,d,e. L [d, e] ▶* L.
+lemma ltpss_refl: ∀L,d,e. L ▶* [d, e] L.
#L elim L -L //
#L #I #V #IHL * /2 width=1/ * /2 width=1/
qed.
-lemma ltpss_weak_all: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → L1 [0, |L2|] ▶* L2.
+lemma ltpss_weak: ∀L1,L2,d1,e1. L1 ▶* [d1, e1] L2 →
+ ∀d2,e2. d2 ≤ d1 → d1 + e1 ≤ d2 + e2 → L1 ▶* [d2, e2] L2.
+#L1 #L2 #d1 #e1 #H elim H -L1 -L2 -d1 -e1 //
+[ #L1 #L2 #I #V1 #V2 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd2 #Hde2
+ lapply (le_n_O_to_eq … Hd2) #H destruct normalize in Hde2;
+ lapply (lt_to_le_to_lt 0 … Hde2) // #He2
+ lapply (le_plus_to_minus_r … Hde2) -Hde2 /3 width=5/
+| #L1 #L2 #I #V1 #V2 #d1 #e1 #_ #HV12 #IHL12 #d2 #e2 #Hd21 #Hde12
+ >plus_plus_comm_23 in Hde12; #Hde12
+ elim (le_to_or_lt_eq 0 d2 ?) // #H destruct
+ [ lapply (le_plus_to_minus_r … Hde12) -Hde12 <plus_minus // #Hde12
+ lapply (le_plus_to_minus … Hd21) -Hd21 #Hd21 /3 width=5/
+ | -Hd21 normalize in Hde12;
+ lapply (lt_to_le_to_lt 0 … Hde12) // #He2
+ lapply (le_plus_to_minus_r … Hde12) -Hde12 /3 width=5/
+ ]
+]
+qed.
+
+lemma ltpss_weak_all: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → L1 ▶* [0, |L2|] L2.
#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
// /3 width=2/ /3 width=3/
qed.
(* Basic forward lemmas *****************************************************)
-lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 [d, e] ▶* L2 → |L1| = |L2|.
+lemma ltpss_fwd_length: ∀L1,L2,d,e. L1 ▶* [d, e] L2 → |L1| = |L2|.
#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
normalize //
qed-.
(* Basic inversion lemmas ***************************************************)
-fact ltpss_inv_refl_O2_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → e = 0 → L1 = L2.
+fact ltpss_inv_refl_O2_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → e = 0 → L1 = L2.
#d #e #L1 #L2 #H elim H -d -e -L1 -L2 //
[ #L1 #L2 #I #V1 #V2 #e #_ #_ #_ >commutative_plus normalize #H destruct
| #L1 #L2 #I #V1 #V2 #d #e #_ #HV12 #IHL12 #He destruct
]
qed.
-lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 [d, 0] ▶* L2 → L1 = L2.
+lemma ltpss_inv_refl_O2: ∀d,L1,L2. L1 ▶* [d, 0] L2 → L1 = L2.
/2 width=4/ qed-.
fact ltpss_inv_atom1_aux: ∀d,e,L1,L2.
- L1 [d, e] ▶* L2 → L1 = ⋆ → L2 = ⋆.
+ L1 ▶* [d, e] L2 → L1 = ⋆ → L2 = ⋆.
#d #e #L1 #L2 * -d -e -L1 -L2
[ //
| #L #I #V #H destruct
]
qed.
-lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ [d, e] ▶* L2 → L2 = ⋆.
+lemma ltpss_inv_atom1: ∀d,e,L2. ⋆ ▶* [d, e] L2 → L2 = ⋆.
/2 width=5/ qed-.
-fact ltpss_inv_tpss21_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → d = 0 → 0 < e →
+fact ltpss_inv_tpss21_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 [0, e - 1] ▶* K2 &
- K2 ⊢ V1 [0, e - 1] ▶* V2 &
+ ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
L2 = K2. ⓑ{I} V2.
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #_ #K1 #I #V1 #H destruct
]
qed.
-lemma ltpss_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 [0, e] ▶* L2 → 0 < e →
- ∃∃K2,V2. K1 [0, e - 1] ▶* K2 & K2 ⊢ V1 [0, e - 1] ▶* V2 &
+lemma ltpss_inv_tpss21: ∀e,K1,I,V1,L2. K1. ⓑ{I} V1 ▶* [0, e] L2 → 0 < e →
+ ∃∃K2,V2. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
L2 = K2. ⓑ{I} V2.
/2 width=5/ qed-.
-fact ltpss_inv_tpss11_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → 0 < d →
+fact ltpss_inv_tpss11_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 [d - 1, e] ▶* K2 &
- K2 ⊢ V1 [d - 1, e] ▶* V2 &
+ ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
L2 = K2. ⓑ{I} V2.
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #I #K1 #V1 #H destruct
]
qed.
-lemma ltpss_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 [d, e] ▶* L2 → 0 < d →
- ∃∃K2,V2. K1 [d - 1, e] ▶* K2 &
- K2 ⊢ V1 [d - 1, e] ▶* V2 &
+lemma ltpss_inv_tpss11: ∀d,e,I,K1,V1,L2. K1. ⓑ{I} V1 ▶* [d, e] L2 → 0 < d →
+ ∃∃K2,V2. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
L2 = K2. ⓑ{I} V2.
/2 width=3/ qed-.
fact ltpss_inv_atom2_aux: ∀d,e,L1,L2.
- L1 [d, e] ▶* L2 → L2 = ⋆ → L1 = ⋆.
+ L1 ▶* [d, e] L2 → L2 = ⋆ → L1 = ⋆.
#d #e #L1 #L2 * -d -e -L1 -L2
[ //
| #L #I #V #H destruct
]
qed.
-lemma ltpss_inv_atom2: ∀d,e,L1. L1 [d, e] ▶* ⋆ → L1 = ⋆.
+lemma ltpss_inv_atom2: ∀d,e,L1. L1 ▶* [d, e] ⋆ → L1 = ⋆.
/2 width=5/ qed-.
-fact ltpss_inv_tpss22_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → d = 0 → 0 < e →
+fact ltpss_inv_tpss22_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → d = 0 → 0 < e →
∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 [0, e - 1] ▶* K2 &
- K2 ⊢ V1 [0, e - 1] ▶* V2 &
+ ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
L1 = K1. ⓑ{I} V1.
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #_ #K1 #I #V1 #H destruct
]
qed.
-lemma ltpss_inv_tpss22: ∀e,L1,K2,I,V2. L1 [0, e] ▶* K2. ⓑ{I} V2 → 0 < e →
- ∃∃K1,V1. K1 [0, e - 1] ▶* K2 & K2 ⊢ V1 [0, e - 1] ▶* V2 &
+lemma ltpss_inv_tpss22: ∀e,L1,K2,I,V2. L1 ▶* [0, e] K2. ⓑ{I} V2 → 0 < e →
+ ∃∃K1,V1. K1 ▶* [0, e - 1] K2 &
+ K2 ⊢ V1 ▶* [0, e - 1] V2 &
L1 = K1. ⓑ{I} V1.
/2 width=5/ qed-.
-fact ltpss_inv_tpss12_aux: ∀d,e,L1,L2. L1 [d, e] ▶* L2 → 0 < d →
+fact ltpss_inv_tpss12_aux: ∀d,e,L1,L2. L1 ▶* [d, e] L2 → 0 < d →
∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 [d - 1, e] ▶* K2 &
- K2 ⊢ V1 [d - 1, e] ▶* V2 &
- L1 = K1. ⓑ{I} V1.
+ ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
+ L1 = K1. ⓑ{I} V1.
#d #e #L1 #L2 * -d -e -L1 -L2
[ #d #e #_ #I #K2 #V2 #H destruct
| #L #I #V #H elim (lt_refl_false … H)
]
qed.
-lemma ltpss_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 [d, e] ▶* K2. ⓑ{I} V2 → 0 < d →
- ∃∃K1,V1. K1 [d - 1, e] ▶* K2 &
- K2 ⊢ V1 [d - 1, e] ▶* V2 &
+lemma ltpss_inv_tpss12: ∀L1,K2,I,V2,d,e. L1 ▶* [d, e] K2. ⓑ{I} V2 → 0 < d →
+ ∃∃K1,V1. K1 ▶* [d - 1, e] K2 &
+ K2 ⊢ V1 ▶* [d - 1, e] V2 &
L1 = K1. ⓑ{I} V1.
/2 width=3/ qed-.
-(* Basic_1: removed theorems 27:
+(* Basic_1: removed theorems 28:
csubst0_clear_O csubst0_drop_lt csubst0_drop_gt csubst0_drop_eq
csubst0_clear_O_back csubst0_clear_S csubst0_clear_trans
csubst0_drop_gt_back csubst0_drop_eq_back csubst0_drop_lt_back
csubst0_snd_bind csubst0_fst_bind csubst0_both_bind
csubst1_head csubst1_flat csubst1_gen_head
csubst1_getl_ge csubst1_getl_lt csubst1_getl_ge_back getl_csubst1
-
+ fsubst0_gen_base
*)
projects / helm.git / blobdiff