- ldrop is now drop as in basic_1

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / grammar / leq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma
index f7f23760d8594c36af97c4b1c5986cd343798b0d..a898abdb566f65941eb7d63cb572bdf85838a536 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma
@@ -13,7 +13,7 @@
 (**************************************************************************)
 
 include "ground_2/ynat/ynat_lt.ma".
-include "basic_2/notation/relations/iso_4.ma".
+include "basic_2/notation/relations/midiso_4.ma".
 include "basic_2/grammar/lenv_length.ma".
 
 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
@@ -25,31 +25,31 @@ inductive leq: relation4 ynat ynat lenv lenv ≝
 | leq_pair: ∀I,L1,L2,V,e. leq 0 e L1 L2 →
             leq 0 (⫯e) (L1.ⓑ{I}V) (L2.ⓑ{I}V)
 | leq_succ: ∀I1,I2,L1,L2,V1,V2,d,e.
-            leq d e L1 L2 → leq (⫯d) e (L1. ⓑ{I1}V1) (L2. ⓑ{I2} V2)
+            leq d e L1 L2 → leq (⫯d) e (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
 .
 
 interpretation
   "equivalence (local environment)"
-  'Iso d e L1 L2 = (leq d e L1 L2).
+  'MidIso d e L1 L2 = (leq d e L1 L2).
 
 (* Basic properties *********************************************************)
 
-lemma leq_pair_lt: â\88\80I,L1,L2,V,e. L1 â\89\83[0, ⫰e] L2 → 0 < e →
-                   L1.â\93\91{I}V â\89\83[0, e] L2.ⓑ{I}V.
+lemma leq_pair_lt: â\88\80I,L1,L2,V,e. L1 ⩬[0, ⫰e] L2 → 0 < e →
+                   L1.â\93\91{I}V ⩬[0, e] L2.ⓑ{I}V.
 #I #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 … He) /2 width=1 by leq_pair/
 qed.
 
-lemma leq_succ_lt: â\88\80I1,I2,L1,L2,V1,V2,d,e. L1 â\89\83[⫰d, e] L2 → 0 < d →
-                   L1.â\93\91{I1}V1 â\89\83[d, e] L2. ⓑ{I2}V2.
+lemma leq_succ_lt: â\88\80I1,I2,L1,L2,V1,V2,d,e. L1 ⩬[⫰d, e] L2 → 0 < d →
+                   L1.â\93\91{I1}V1 ⩬[d, e] L2. ⓑ{I2}V2.
 #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd <(ylt_inv_O1 … Hd) /2 width=1 by leq_succ/
 qed.
 
-lemma leq_pair_O_Y: â\88\80L1,L2. L1 â\89\83[0, ∞] L2 →
-                    â\88\80I,V. L1.â\93\91{I}V â\89\83[0, ∞] L2.ⓑ{I}V.
+lemma leq_pair_O_Y: â\88\80L1,L2. L1 ⩬[0, ∞] L2 →
+                    â\88\80I,V. L1.â\93\91{I}V ⩬[0, ∞] L2.ⓑ{I}V.
 #L1 #L2 #HL12 #I #V lapply (leq_pair I … V … HL12) -HL12 //
 qed.
 
-lemma leq_refl: â\88\80L,d,e. L â\89\83[d, e] L.
+lemma leq_refl: â\88\80L,d,e. L ⩬[d, e] L.
 #L elim L -L //
 #L #I #V #IHL #d elim (ynat_cases … d) [| * #x ]
 #Hd destruct /2 width=1 by leq_succ/
@@ -57,7 +57,7 @@ lemma leq_refl: ∀L,d,e. L ≃[d, e] L.
 #He destruct /2 width=1 by leq_zero, leq_pair/
 qed.
 
-lemma leq_O2: â\88\80L1,L2,d. |L1| = |L2| â\86\92 L1 â\89\83[d, yinj 0] L2.
+lemma leq_O2: â\88\80L1,L2,d. |L1| = |L2| â\86\92 L1 ⩬[d, yinj 0] L2.
 #L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ]
 * // [1,3: #L2 #I2 #V2 ] #d normalize
 [1,3: <plus_n_Sm #H destruct ]
@@ -73,7 +73,7 @@ qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact leq_inv_atom1_aux: â\88\80L1,L2,d,e. L1 â\89\83[d, e] L2 → L1 = ⋆ → L2 = ⋆.
+fact leq_inv_atom1_aux: â\88\80L1,L2,d,e. L1 ⩬[d, e] L2 → L1 = ⋆ → L2 = ⋆.
 #L1 #L2 #d #e * -L1 -L2 -d -e //
 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
 | #I #L1 #L2 #V #e #_ #H destruct
@@ -81,12 +81,12 @@ fact leq_inv_atom1_aux: ∀L1,L2,d,e. L1 ≃[d, e] L2 → L1 = ⋆ → L2 = ⋆.
 ]
 qed-.
 
-lemma leq_inv_atom1: â\88\80L2,d,e. â\8b\86 â\89\83[d, e] L2 → L2 = ⋆.
+lemma leq_inv_atom1: â\88\80L2,d,e. â\8b\86 ⩬[d, e] L2 → L2 = ⋆.
 /2 width=5 by leq_inv_atom1_aux/ qed-.
 
-fact leq_inv_zero1_aux: â\88\80L1,L2,d,e. L1 â\89\83[d, e] L2 →
+fact leq_inv_zero1_aux: â\88\80L1,L2,d,e. L1 ⩬[d, e] L2 →
                         ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → e = 0 →
-                        â\88\83â\88\83J2,K2,W2. K1 â\89\83[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
+                        â\88\83â\88\83J2,K2,W2. K1 ⩬[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
 #L1 #L2 #d #e * -L1 -L2 -d -e
 [ #d #e #J1 #K1 #W1 #H destruct
 | #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
@@ -98,13 +98,13 @@ fact leq_inv_zero1_aux: ∀L1,L2,d,e. L1 ≃[d, e] L2 →
 ]
 qed-.
 
-lemma leq_inv_zero1: â\88\80I1,K1,L2,V1. K1.â\93\91{I1}V1 â\89\83[0, 0] L2 →
-                     â\88\83â\88\83I2,K2,V2. K1 â\89\83[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
+lemma leq_inv_zero1: â\88\80I1,K1,L2,V1. K1.â\93\91{I1}V1 ⩬[0, 0] L2 →
+                     â\88\83â\88\83I2,K2,V2. K1 ⩬[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
 /2 width=9 by leq_inv_zero1_aux/ qed-.
 
-fact leq_inv_pair1_aux: â\88\80L1,L2,d,e. L1 â\89\83[d, e] L2 →
+fact leq_inv_pair1_aux: â\88\80L1,L2,d,e. L1 ⩬[d, e] L2 →
                         ∀J,K1,W. L1 = K1.ⓑ{J}W → d = 0 → 0 < e →
-                        â\88\83â\88\83K2. K1 â\89\83[0, ⫰e] K2 & L2 = K2.ⓑ{J}W.
+                        â\88\83â\88\83K2. K1 ⩬[0, ⫰e] K2 & L2 = K2.ⓑ{J}W.
 #L1 #L2 #d #e * -L1 -L2 -d -e
 [ #d #e #J #K1 #W #H destruct
 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J #K1 #W #_ #_ #H
@@ -116,19 +116,19 @@ fact leq_inv_pair1_aux: ∀L1,L2,d,e. L1 ≃[d, e] L2 →
 ]
 qed-.
 
-lemma leq_inv_pair1: â\88\80I,K1,L2,V,e. K1.â\93\91{I}V â\89\83[0, e] L2 → 0 < e →
-                     â\88\83â\88\83K2. K1 â\89\83[0, ⫰e] K2 & L2 = K2.ⓑ{I}V.
+lemma leq_inv_pair1: â\88\80I,K1,L2,V,e. K1.â\93\91{I}V ⩬[0, e] L2 → 0 < e →
+                     â\88\83â\88\83K2. K1 ⩬[0, ⫰e] K2 & L2 = K2.ⓑ{I}V.
 /2 width=6 by leq_inv_pair1_aux/ qed-.
 
-lemma leq_inv_pair: â\88\80I1,I2,L1,L2,V1,V2,e. L1.â\93\91{I1}V1 â\89\83[0, e] L2.ⓑ{I2}V2 → 0 < e →
-                    â\88§â\88§ L1 â\89\83[0, ⫰e] L2 & I1 = I2 & V1 = V2.
+lemma leq_inv_pair: â\88\80I1,I2,L1,L2,V1,V2,e. L1.â\93\91{I1}V1 ⩬[0, e] L2.ⓑ{I2}V2 → 0 < e →
+                    â\88§â\88§ L1 ⩬[0, ⫰e] L2 & I1 = I2 & V1 = V2.
 #I1 #I2 #L1 #L2 #V1 #V2 #e #H #He elim (leq_inv_pair1 … H) -H //
 #Y #HL12 #H destruct /2 width=1 by and3_intro/
 qed-.
 
-fact leq_inv_succ1_aux: â\88\80L1,L2,d,e. L1 â\89\83[d, e] L2 →
+fact leq_inv_succ1_aux: â\88\80L1,L2,d,e. L1 ⩬[d, e] L2 →
                         ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < d →
-                        â\88\83â\88\83J2,K2,W2. K1 â\89\83[⫰d, e] K2 & L2 = K2.ⓑ{J2}W2.
+                        â\88\83â\88\83J2,K2,W2. K1 ⩬[⫰d, e] K2 & L2 = K2.ⓑ{J2}W2.
 #L1 #L2 #d #e * -L1 -L2 -d -e
 [ #d #e #J1 #K1 #W1 #H destruct
 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
@@ -140,47 +140,47 @@ fact leq_inv_succ1_aux: ∀L1,L2,d,e. L1 ≃[d, e] L2 →
 ]
 qed-.
 
-lemma leq_inv_succ1: â\88\80I1,K1,L2,V1,d,e. K1.â\93\91{I1}V1 â\89\83[d, e] L2 → 0 < d →
-                     â\88\83â\88\83I2,K2,V2. K1 â\89\83[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
+lemma leq_inv_succ1: â\88\80I1,K1,L2,V1,d,e. K1.â\93\91{I1}V1 ⩬[d, e] L2 → 0 < d →
+                     â\88\83â\88\83I2,K2,V2. K1 ⩬[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
 /2 width=5 by leq_inv_succ1_aux/ qed-.
 
-lemma leq_inv_atom2: â\88\80L1,d,e. L1 â\89\83[d, e] ⋆ → L1 = ⋆.
+lemma leq_inv_atom2: â\88\80L1,d,e. L1 ⩬[d, e] ⋆ → L1 = ⋆.
 /3 width=3 by leq_inv_atom1, leq_sym/
 qed-.
 
-lemma leq_inv_succ: â\88\80I1,I2,L1,L2,V1,V2,d,e. L1.â\93\91{I1}V1 â\89\83[d, e] L2.ⓑ{I2}V2 → 0 < d →
-                    L1 â\89\83[⫰d, e] L2.
+lemma leq_inv_succ: â\88\80I1,I2,L1,L2,V1,V2,d,e. L1.â\93\91{I1}V1 ⩬[d, e] L2.ⓑ{I2}V2 → 0 < d →
+                    L1 ⩬[⫰d, e] L2.
 #I1 #I2 #L1 #L2 #V1 #V2 #d #e #H #Hd elim (leq_inv_succ1 … H) -H //
 #Z #Y #X #HL12 #H destruct //
 qed-.
 
-lemma leq_inv_zero2: â\88\80I2,K2,L1,V2. L1 â\89\83[0, 0] K2.ⓑ{I2}V2 →
-                     â\88\83â\88\83I1,K1,V1. K1 â\89\83[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
+lemma leq_inv_zero2: â\88\80I2,K2,L1,V2. L1 ⩬[0, 0] K2.ⓑ{I2}V2 →
+                     â\88\83â\88\83I1,K1,V1. K1 ⩬[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
 #I2 #K2 #L1 #V2 #H elim (leq_inv_zero1 … (leq_sym … H)) -H 
 /3 width=5 by leq_sym, ex2_3_intro/
 qed-.
 
-lemma leq_inv_pair2: â\88\80I,K2,L1,V,e. L1 â\89\83[0, e] K2.ⓑ{I}V → 0 < e →
-                     â\88\83â\88\83K1. K1 â\89\83[0, ⫰e] K2 & L1 = K1.ⓑ{I}V.
+lemma leq_inv_pair2: â\88\80I,K2,L1,V,e. L1 ⩬[0, e] K2.ⓑ{I}V → 0 < e →
+                     â\88\83â\88\83K1. K1 ⩬[0, ⫰e] K2 & L1 = K1.ⓑ{I}V.
 #I #K2 #L1 #V #e #H #He elim (leq_inv_pair1 … (leq_sym … H)) -H
 /3 width=3 by leq_sym, ex2_intro/
 qed-.
 
-lemma leq_inv_succ2: â\88\80I2,K2,L1,V2,d,e. L1 â\89\83[d, e] K2.ⓑ{I2}V2 → 0 < d →
-                     â\88\83â\88\83I1,K1,V1. K1 â\89\83[⫰d, e] K2 & L1 = K1.ⓑ{I1}V1.
+lemma leq_inv_succ2: â\88\80I2,K2,L1,V2,d,e. L1 ⩬[d, e] K2.ⓑ{I2}V2 → 0 < d →
+                     â\88\83â\88\83I1,K1,V1. K1 ⩬[⫰d, e] K2 & L1 = K1.ⓑ{I1}V1.
 #I2 #K2 #L1 #V2 #d #e #H #Hd elim (leq_inv_succ1 … (leq_sym … H)) -H 
 /3 width=5 by leq_sym, ex2_3_intro/
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma leq_fwd_length: â\88\80L1,L2,d,e. L1 â\89\83[d, e] L2 → |L1| = |L2|.
+lemma leq_fwd_length: â\88\80L1,L2,d,e. L1 ⩬[d, e] L2 → |L1| = |L2|.
 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize //
 qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-fact leq_inv_O_Y_aux: â\88\80L1,L2,d,e. L1 â\89\83[d, e] L2 → d = 0 → e = ∞ → L1 = L2.
+fact leq_inv_O_Y_aux: â\88\80L1,L2,d,e. L1 ⩬[d, e] L2 → d = 0 → e = ∞ → L1 = L2.
 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e //
 [ #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #_ #H destruct
 | /4 width=1 by eq_f3, ysucc_inv_Y_dx/
@@ -188,5 +188,5 @@ fact leq_inv_O_Y_aux: ∀L1,L2,d,e. L1 ≃[d, e] L2 → d = 0 → e = ∞ → L1
 ]
 qed-.
 
-lemma leq_inv_O_Y: â\88\80L1,L2. L1 â\89\83[0, ∞] L2 → L1 = L2.
+lemma leq_inv_O_Y: â\88\80L1,L2. L1 ⩬[0, ∞] L2 → L1 = L2.
 /2 width=5 by leq_inv_O_Y_aux/ qed-.