(* *)
(**************************************************************************)
-include "ground_2A/ynat/ynat_lt.ma".
+include "ground/ynat/ynat_lt.ma".
include "basic_2A/notation/relations/midiso_4.ma".
include "basic_2A/grammar/lenv_length.ma".
| lreq_zero: ∀I1,I2,L1,L2,V1,V2.
lreq 0 0 L1 L2 → lreq 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
| lreq_pair: ∀I,L1,L2,V,m. lreq 0 m L1 L2 →
- lreq 0 (⫯m) (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+ lreq 0 (â\86\91m) (L1.ⓑ{I}V) (L2.ⓑ{I}V)
| lreq_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
- lreq l m L1 L2 â\86\92 lreq (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+ lreq l m L1 L2 â\86\92 lreq (â\86\91l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
.
interpretation
(* Basic properties *********************************************************)
-lemma lreq_pair_lt: â\88\80I,L1,L2,V,m. L1 ⩬[0, â«°m] L2 → 0 < m →
+lemma lreq_pair_lt: â\88\80I,L1,L2,V,m. L1 ⩬[0, â\86\93m] L2 → 0 < m →
L1.ⓑ{I}V ⩬[0, m] L2.ⓑ{I}V.
#I #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lreq_pair/
qed.
-lemma lreq_succ_lt: â\88\80I1,I2,L1,L2,V1,V2,l,m. L1 ⩬[â«°l, m] L2 → 0 < l →
+lemma lreq_succ_lt: â\88\80I1,I2,L1,L2,V1,V2,l,m. L1 ⩬[â\86\93l, m] L2 → 0 < l →
L1.ⓑ{I1}V1 ⩬[l, m] L2. ⓑ{I2}V2.
#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lreq_succ/
qed.
fact lreq_inv_pair1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
∀J,K1,W. L1 = K1.ⓑ{J}W → l = 0 → 0 < m →
- â\88\83â\88\83K2. K1 ⩬[0, â«°m] K2 & L2 = K2.ⓑ{J}W.
+ â\88\83â\88\83K2. K1 ⩬[0, â\86\93m] K2 & L2 = K2.ⓑ{J}W.
#L1 #L2 #l #m * -L1 -L2 -l -m
[ #l #m #J #K1 #W #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J #K1 #W #_ #_ #H
qed-.
lemma lreq_inv_pair1: ∀I,K1,L2,V,m. K1.ⓑ{I}V ⩬[0, m] L2 → 0 < m →
- â\88\83â\88\83K2. K1 ⩬[0, â«°m] K2 & L2 = K2.ⓑ{I}V.
+ â\88\83â\88\83K2. K1 ⩬[0, â\86\93m] K2 & L2 = K2.ⓑ{I}V.
/2 width=6 by lreq_inv_pair1_aux/ qed-.
lemma lreq_inv_pair: ∀I1,I2,L1,L2,V1,V2,m. L1.ⓑ{I1}V1 ⩬[0, m] L2.ⓑ{I2}V2 → 0 < m →
- â\88§â\88§ L1 ⩬[0, â«°m] L2 & I1 = I2 & V1 = V2.
+ â\88§â\88§ L1 ⩬[0, â\86\93m] L2 & I1 = I2 & V1 = V2.
#I1 #I2 #L1 #L2 #V1 #V2 #m #H #Hm elim (lreq_inv_pair1 … H) -H //
#Y #HL12 #H destruct /2 width=1 by and3_intro/
qed-.
fact lreq_inv_succ1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
- â\88\83â\88\83J2,K2,W2. K1 ⩬[â«°l, m] K2 & L2 = K2.ⓑ{J2}W2.
+ â\88\83â\88\83J2,K2,W2. K1 ⩬[â\86\93l, m] K2 & L2 = K2.ⓑ{J2}W2.
#L1 #L2 #l #m * -L1 -L2 -l -m
[ #l #m #J1 #K1 #W1 #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
qed-.
lemma lreq_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⩬[l, m] L2 → 0 < l →
- â\88\83â\88\83I2,K2,V2. K1 ⩬[â«°l, m] K2 & L2 = K2.ⓑ{I2}V2.
+ â\88\83â\88\83I2,K2,V2. K1 ⩬[â\86\93l, m] K2 & L2 = K2.ⓑ{I2}V2.
/2 width=5 by lreq_inv_succ1_aux/ qed-.
lemma lreq_inv_atom2: ∀L1,l,m. L1 ⩬[l, m] ⋆ → L1 = ⋆.
qed-.
lemma lreq_inv_succ: ∀I1,I2,L1,L2,V1,V2,l,m. L1.ⓑ{I1}V1 ⩬[l, m] L2.ⓑ{I2}V2 → 0 < l →
- L1 ⩬[⫰l, m] L2.
+ L1 ⩬[â\86\93l, m] L2.
#I1 #I2 #L1 #L2 #V1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … H) -H //
#Z #Y #X #HL12 #H destruct //
qed-.
qed-.
lemma lreq_inv_pair2: ∀I,K2,L1,V,m. L1 ⩬[0, m] K2.ⓑ{I}V → 0 < m →
- â\88\83â\88\83K1. K1 ⩬[0, â«°m] K2 & L1 = K1.ⓑ{I}V.
+ â\88\83â\88\83K1. K1 ⩬[0, â\86\93m] K2 & L1 = K1.ⓑ{I}V.
#I #K2 #L1 #V #m #H #Hm elim (lreq_inv_pair1 … (lreq_sym … H)) -H
/3 width=3 by lreq_sym, ex2_intro/
qed-.
lemma lreq_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⩬[l, m] K2.ⓑ{I2}V2 → 0 < l →
- â\88\83â\88\83I1,K1,V1. K1 ⩬[â«°l, m] K2 & L1 = K1.ⓑ{I1}V1.
+ â\88\83â\88\83I1,K1,V1. K1 ⩬[â\86\93l, m] K2 & L1 = K1.ⓑ{I1}V1.
#I2 #K2 #L1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … (lreq_sym … H)) -H
/3 width=5 by lreq_sym, ex2_3_intro/
qed-.