- a reinforement in a lemma on ldrop allows to prove a lemma on lsx :)

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / etc / leq / leq.etc

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/leq/leq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/leq/leq.etc
index a21457b4ed449d2e9c4eae5137e323f2d7c441d4..095e1ced3765ad8a590868ec495f1db205f84e1a 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/leq/leq.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/leq/leq.etc
@@ -12,54 +12,70 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Basic_2/grammar/lenv_length.ma".
+include "ground_2/ynat/ynat_succ.ma".
+include "basic_2/notation/relations/iso_4.ma".
+include "basic_2/grammar/lenv_length.ma".
 
-(* LOCAL ENVIRONMENT EQUALITY ***********************************************)
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
 
-notation "hvbox( T1 break [ d , break e ] ≈ break T2 )"
-   non associative with precedence 45
-   for @{ 'Eq $T1 $d $e $T2 }.
-
-inductive leq: nat → nat → relation lenv ≝
-| leq_sort: ∀d,e. leq d e (⋆) (⋆)
-| leq_OO:   ∀L1,L2. leq 0 0 L1 L2
-| leq_eq:   ∀L1,L2,I,V,e. leq 0 e L1 L2 →
-            leq 0 (e + 1) (L1. 𝕓{I} V) (L2.𝕓{I} V)
-| leq_skip: ∀L1,L2,I1,I2,V1,V2,d,e.
-            leq d e L1 L2 → leq (d + 1) e (L1. 𝕓{I1} V1) (L2. 𝕓{I2} V2)
+inductive leq: ynat → ynat → relation lenv ≝
+| leq_atom: ∀d,e. leq d e (⋆) (⋆)
+| leq_zero: ∀I,L1,L2,V. leq 0 0 L1 L2 → leq 0 0 (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| leq_pair: ∀I1,I2,L1,L2,V1,V2,e.
+            leq 0 e L1 L2 → leq 0 (⫯e) (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+| leq_succ: ∀I,L1,L2,V,d,e. leq d e L1 L2 → leq (⫯d) e (L1.ⓑ{I}V) (L2.ⓑ{I}V)
 .
 
-interpretation "local environment equality" 'Eq L1 d e L2 = (leq d e L1 L2).
-
-definition leq_repl_dx: ∀S. (lenv → relation S) → Prop ≝ λS,R.
-                        ∀L1,s1,s2. R L1 s1 s2 →
-                        ∀L2,d,e. L1 [d, e]≈ L2 → R L2 s1 s2.
+interpretation
+   "equivalence (local environment)"
+   'Iso d e L1 L2 = (leq d e L1 L2).
 
 (* Basic properties *********************************************************)
 
-lemma TC_leq_repl_dx: ∀S,R. leq_repl_dx S R → leq_repl_dx S (λL. (TC … (R L))).
-#S #R #HR #L1 #s1 #s2 #H elim H -H s2
-[ /3 width=5/
-| #s #s2 #_ #Hs2 #IHs1 #L2 #d #e #HL12
-  lapply (HR … Hs2 … HL12) -HR Hs2 HL12 /3/
-]
+lemma leq_refl: ∀L,d,e. L ≃[d, e] L.
+#L elim L -L /2 width=1 by/
+#L #I #V #IHL #d #e elim (ynat_cases … d) [ | * /2 width=1 by leq_succ/ ]
+elim (ynat_cases … e) [ | * ]
+/2 width=1 by leq_zero, leq_pair/
 qed.
 
-lemma leq_refl: ∀d,e,L. L [d, e] ≈ L.
-#d elim d -d
-[ #e elim e -e // #e #IHe #L elim L -L /2/
-| #d #IHd #e #L elim L -L /2/
-]
-qed.
+lemma leq_sym: ∀L1,L2,d,e. L1 ≃[d, e] L2 → L2 ≃[d, e] L1.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+/2 width=1 by leq_atom, leq_zero, leq_pair, leq_succ/
+qed-.
 
-lemma leq_sym: ∀L1,L2,d,e. L1 [d, e] ≈ L2 → L2 [d, e] ≈ L1.
-#L1 #L2 #d #e #H elim H -H L1 L2 d e /2/
+lemma leq_O_Y: ∀L1,L2. |L1| = |L2| → L1 ≃[0, ∞] L2.
+#L1 elim L1 -L1
+[ #X #H lapply (length_inv_zero_sn … H) -H //
+| #L1 #I1 #V1 #IHL1 #X #H elim (length_inv_pos_sn … H) -H
+  #L2 #I2 #V2 #HL12 #H destruct
+  @(leq_pair … (∞)) /2 width=1 by/ (**) (* explicit constructor *)
+]
 qed.
 
-lemma leq_skip_lt: ∀L1,L2,d,e. L1 [d - 1, e] ≈ L2 → 0 < d →
-                   ∀I1,I2,V1,V2. L1. 𝕓{I1} V1 [d, e] ≈ L2. 𝕓{I2} V2.
+(* Basic forward lemmas *****************************************************)
 
-#L1 #L2 #d #e #HL12 #Hd >(plus_minus_m_m d 1) /2/
-qed.
+lemma leq_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 leq_inv_O2_aux: ∀L1,L2,d,e. L1 ≃[d, e] L2 → e = 0 → L1 = L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e /3 width=1 by eq_f3/
+#I1 #I2 #L1 #L2 #V1 #V2 #e #_ #_ #H elim (ysucc_inv_O_dx … H)
+qed-.
+
+lemma leq_inv_O2: ∀L1,L2,d. L1 ≃[d, 0] L2 → L1 = L2.
+/2 width=4 by leq_inv_O2_aux/ qed-.
+
+fact leq_inv_Y1_aux: ∀L1,L2,d,e. L1 ≃[d, e] L2 → d = ∞ → L1 = L2.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e /3 width=1 by eq_f3/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #e #_ #_ #H destruct
+| #I #L1 #L2 #V #d #e #_ #IHL12 #H lapply (ysucc_inv_Y_dx … H) -H
+  /3 width=1 by eq_f3/
+]
+qed-.
+
+lemma leq_inv_Y1: ∀L1,L2,e. L1 ≃[∞, e] L2 → L1 = L2.
+/2 width=4 by leq_inv_Y1_aux/ qed-.