(* *)
(**************************************************************************)
-include "ground_2/ynat/ynat_succ.ma".
+include "ground_2/ynat/ynat_lt.ma".
include "basic_2/notation/relations/iso_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)"
+ 'Iso d e L1 L2 = (leq d e L1 L2).
(* Basic properties *********************************************************)
+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_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_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.
+
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/
+#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_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: ∀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/
+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_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_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_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.
+qed-.
-(* Basic forward lemmas *****************************************************)
+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-.
-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 //
+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-.
-(* Basic inversion lemmas ***************************************************)
+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-.
-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)
+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 → |L2| = |L1|.
+#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-.
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