X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fgrammar%2Fleq.ma;h=a898abdb566f65941eb7d63cb572bdf85838a536;hb=52e675f555f559c047d5449db7fc89a51b977d35;hp=095e1ced3765ad8a590868ec495f1db205f84e1a;hpb=1cc9a122e904468085545bb53543cc1a0f2f48f5;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma
index 095e1ced3..a898abdb5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma
@@ -12,70 +12,181 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/ynat/ynat_succ.ma".
-include "basic_2/notation/relations/iso_4.ma".
+include "ground_2/ynat/ynat_lt.ma".
+include "basic_2/notation/relations/midiso_4.ma".
 include "basic_2/grammar/lenv_length.ma".
 
 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
 
-inductive leq: ynat → ynat → relation lenv ≝
+inductive leq: relation4 ynat ynat lenv 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)
+| leq_zero: ∀I1,I2,L1,L2,V1,V2.
+            leq 0 0 L1 L2 → leq 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+| 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)
 .
 
 interpretation
-   "equivalence (local environment)"
-   'Iso d e L1 L2 = (leq d e L1 L2).
+  "equivalence (local environment)"
+  'MidIso d e L1 L2 = (leq d e L1 L2).
 
 (* Basic properties *********************************************************)
 
-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/
+lemma leq_pair_lt: ∀I,L1,L2,V,e. L1 ⩬[0, ⫰e] L2 → 0 < e →
+                   L1.ⓑ{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_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_succ_lt: ∀I1,I2,L1,L2,V1,V2,d,e. L1 ⩬[⫰d, e] L2 → 0 < d →
+                   L1.ⓑ{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_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 *)
-]
+lemma leq_pair_O_Y: ∀L1,L2. L1 ⩬[0, ∞] L2 →
+                    ∀I,V. L1.ⓑ{I}V ⩬[0, ∞] L2.ⓑ{I}V.
+#L1 #L2 #HL12 #I #V lapply (leq_pair I … V … HL12) -HL12 //
 qed.
 
-(* Basic forward lemmas *****************************************************)
+lemma leq_refl: ∀L,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/
+#e elim (ynat_cases … e) [| * #x ]
+#He destruct /2 width=1 by leq_zero, leq_pair/
+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 //
+lemma leq_O2: ∀L1,L2,d. |L1| = |L2| → 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 ]
+#H lapply (injective_plus_l … H) -H #HL12
+elim (ynat_cases d) /3 width=1 by leq_zero/
+* /3 width=1 by leq_succ/
+qed.
+
+lemma leq_sym: ∀d,e. symmetric … (leq d e).
+#d #e #L1 #L2 #H elim H -L1 -L2 -d -e
+/2 width=1 by leq_zero, leq_pair, leq_succ/
 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)
+fact leq_inv_atom1_aux: ∀L1,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
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #H destruct
+]
+qed-.
+
+lemma leq_inv_atom1: ∀L2,d,e. ⋆ ⩬[d, e] L2 → L2 = ⋆.
+/2 width=5 by leq_inv_atom1_aux/ qed-.
+
+fact leq_inv_zero1_aux: ∀L1,L2,d,e. L1 ⩬[d, e] L2 →
+                        ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → d = 0 → e = 0 →
+                        ∃∃J2,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
+  /2 width=5 by ex2_3_intro/
+| #I #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_ #H
+  elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
+  elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma leq_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⩬[0, 0] L2 →
+                     ∃∃I2,K2,V2. K1 ⩬[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=9 by leq_inv_zero1_aux/ qed-.
+
+fact leq_inv_pair1_aux: ∀L1,L2,d,e. L1 ⩬[d, e] L2 →
+                        ∀J,K1,W. L1 = K1.ⓑ{J}W → d = 0 → 0 < e →
+                        ∃∃K2. 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
+  elim (ylt_yle_false … H) //
+| #I #L1 #L2 #V #e #HL12 #J #K1 #W #H #_ #_ destruct
+  /2 width=3 by ex2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J #K1 #W #_ #H
+  elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma leq_inv_pair1: ∀I,K1,L2,V,e. K1.ⓑ{I}V ⩬[0, e] L2 → 0 < e →
+                     ∃∃K2. K1 ⩬[0, ⫰e] K2 & L2 = K2.ⓑ{I}V.
+/2 width=6 by leq_inv_pair1_aux/ qed-.
+
+lemma leq_inv_pair: ∀I1,I2,L1,L2,V1,V2,e. L1.ⓑ{I1}V1 ⩬[0, e] L2.ⓑ{I2}V2 → 0 < e →
+                    ∧∧ 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: ∀L1,L2,d,e. L1 ⩬[d, e] L2 →
+                        ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < d →
+                        ∃∃J2,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
+  elim (ylt_yle_false … H) //
+| #I #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
+  elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
+  /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+lemma leq_inv_succ1: ∀I1,K1,L2,V1,d,e. K1.ⓑ{I1}V1 ⩬[d, e] L2 → 0 < d →
+                     ∃∃I2,K2,V2. K1 ⩬[⫰d, e] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=5 by leq_inv_succ1_aux/ qed-.
+
+lemma leq_inv_atom2: ∀L1,d,e. L1 ⩬[d, e] ⋆ → L1 = ⋆.
+/3 width=3 by leq_inv_atom1, leq_sym/
+qed-.
+
+lemma leq_inv_succ: ∀I1,I2,L1,L2,V1,V2,d,e. L1.ⓑ{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: ∀I2,K2,L1,V2. L1 ⩬[0, 0] K2.ⓑ{I2}V2 →
+                     ∃∃I1,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: ∀I,K2,L1,V,e. L1 ⩬[0, e] K2.ⓑ{I}V → 0 < e →
+                     ∃∃K1. 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: ∀I2,K2,L1,V2,d,e. L1 ⩬[d, e] K2.ⓑ{I2}V2 → 0 < d →
+                     ∃∃I1,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: ∀L1,L2,d,e. L1 ⩬[d, e] L2 → |L1| = |L2|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize //
 qed-.
 
-lemma leq_inv_O2: ∀L1,L2,d. L1 ≃[d, 0] L2 → L1 = L2.
-/2 width=4 by leq_inv_O2_aux/ qed-.
+(* Advanced inversion lemmas ************************************************)
 
-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/
+fact leq_inv_O_Y_aux: ∀L1,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/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #_ #H elim (ysucc_inv_O_dx … H)
 ]
 qed-.
 
-lemma leq_inv_Y1: ∀L1,L2,e. L1 ≃[∞, e] L2 → L1 = L2.
-/2 width=4 by leq_inv_Y1_aux/ qed-.
+lemma leq_inv_O_Y: ∀L1,L2. L1 ⩬[0, ∞] L2 → L1 = L2.
+/2 width=5 by leq_inv_O_Y_aux/ qed-.