+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/notation/relations/rat_3.ma".
-include "ground_2/relocation/trace.ma".
-
-(* RELOCATION TRACE *********************************************************)
-
-inductive at: trace → relation nat ≝
- | at_empty: at (◊) 0 0
- | at_zero : ∀cs. at (Ⓣ @ cs) 0 0
- | at_succ : ∀cs,i1,i2. at cs i1 i2 → at (Ⓣ @ cs) (⫯i1) (⫯i2)
- | at_false: ∀cs,i1,i2. at cs i1 i2 → at (Ⓕ @ cs) i1 (⫯i2).
-
-interpretation "relocation (trace)"
- 'RAt i1 cs i2 = (at cs i1 i2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact at_inv_empty_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → cs = ◊ → i1 = 0 ∧ i2 = 0.
-#cs #i1 #i2 * -cs -i1 -i2 /2 width=1 by conj/
-#cs #i1 #i2 #_ #H destruct
-qed-.
-
-lemma at_inv_empty: ∀i1,i2. @⦃i1, ◊⦄ ≡ i2 → i1 = 0 ∧ i2 = 0.
-/2 width=5 by at_inv_empty_aux/ qed-.
-
-lemma at_inv_empty_zero_sn: ∀i. @⦃0, ◊⦄ ≡ i → i = 0.
-#i #H elim (at_inv_empty … H) -H //
-qed-.
-
-lemma at_inv_empty_zero_dx: ∀i. @⦃i, ◊⦄ ≡ 0 → i = 0.
-#i #H elim (at_inv_empty … H) -H //
-qed-.
-
-lemma at_inv_empty_succ_sn: ∀i1,i2. @⦃⫯i1, ◊⦄ ≡ i2 → ⊥.
-#i1 #i2 #H elim (at_inv_empty … H) -H #H1 #H2 destruct
-qed-.
-
-lemma at_inv_empty_succ_dx: ∀i1,i2. @⦃i1, ◊⦄ ≡ ⫯i2 → ⊥.
-#i1 #i2 #H elim (at_inv_empty … H) -H #H1 #H2 destruct
-qed-.
-
-fact at_inv_true_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → ∀tl. cs = Ⓣ @ tl →
- (i1 = 0 ∧ i2 = 0) ∨
- ∃∃j1,j2. i1 = ⫯j1 & i2 = ⫯j2 & @⦃j1, tl⦄ ≡ j2.
-#cs #i1 #i2 * -cs -i1 -i2
-[2,3,4: #cs [2,3: #i1 #i2 #Hij ] ] #tl #H destruct
-/3 width=5 by ex3_2_intro, or_introl, or_intror, conj/
-qed-.
-
-lemma at_inv_true: ∀cs,i1,i2. @⦃i1, Ⓣ @ cs⦄ ≡ i2 →
- (i1 = 0 ∧ i2 = 0) ∨
- ∃∃j1,j2. i1 = ⫯j1 & i2 = ⫯j2 & @⦃j1, cs⦄ ≡ j2.
-/2 width=3 by at_inv_true_aux/ qed-.
-
-lemma at_inv_true_zero_sn: ∀cs,i. @⦃0, Ⓣ @ cs⦄ ≡ i → i = 0.
-#cs #i #H elim (at_inv_true … H) -H * //
-#j1 #j2 #H destruct
-qed-.
-
-lemma at_inv_true_zero_dx: ∀cs,i. @⦃i, Ⓣ @ cs⦄ ≡ 0 → i = 0.
-#cs #i #H elim (at_inv_true … H) -H * //
-#j1 #j2 #_ #H destruct
-qed-.
-
-lemma at_inv_true_succ_sn: ∀cs,i1,i2. @⦃⫯i1, Ⓣ @ cs⦄ ≡ i2 →
- ∃∃j2. i2 = ⫯j2 & @⦃i1, cs⦄ ≡ j2.
-#cs #i1 #i2 #H elim (at_inv_true … H) -H *
-[ #H destruct
-| #j1 #j2 #H1 #H2 destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma at_inv_true_succ_dx: ∀cs,i1,i2. @⦃i1, Ⓣ @ cs⦄ ≡ ⫯i2 →
- ∃∃j1. i1 = ⫯j1 & @⦃j1, cs⦄ ≡ i2.
-#cs #i1 #i2 #H elim (at_inv_true … H) -H *
-[ #_ #H destruct
-| #j1 #j2 #H1 #H2 destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma at_inv_true_succ: ∀cs,i1,i2. @⦃⫯i1, Ⓣ @ cs⦄ ≡ ⫯i2 →
- @⦃i1, cs⦄ ≡ i2.
-#cs #i1 #i2 #H elim (at_inv_true … H) -H *
-[ #H destruct
-| #j1 #j2 #H1 #H2 destruct //
-]
-qed-.
-
-lemma at_inv_true_O_S: ∀cs,i. @⦃0, Ⓣ @ cs⦄ ≡ ⫯i → ⊥.
-#cs #i #H elim (at_inv_true … H) -H *
-[ #_ #H destruct
-| #j1 #j2 #H destruct
-]
-qed-.
-
-lemma at_inv_true_S_O: ∀cs,i. @⦃⫯i, Ⓣ @ cs⦄ ≡ 0 → ⊥.
-#cs #i #H elim (at_inv_true … H) -H *
-[ #H destruct
-| #j1 #j2 #_ #H destruct
-]
-qed-.
-
-fact at_inv_false_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → ∀tl. cs = Ⓕ @ tl →
- ∃∃j2. i2 = ⫯j2 & @⦃i1, tl⦄ ≡ j2.
-#cs #i1 #i2 * -cs -i1 -i2
-[2,3,4: #cs [2,3: #i1 #i2 #Hij ] ] #tl #H destruct
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma at_inv_false: ∀cs,i1,i2. @⦃i1, Ⓕ @ cs⦄ ≡ i2 →
- ∃∃j2. i2 = ⫯j2 & @⦃i1, cs⦄ ≡ j2.
-/2 width=3 by at_inv_false_aux/ qed-.
-
-lemma at_inv_false_S: ∀cs,i1,i2. @⦃i1, Ⓕ @ cs⦄ ≡ ⫯i2 → @⦃i1, cs⦄ ≡ i2.
-#cs #i1 #i2 #H elim (at_inv_false … H) -H
-#j2 #H destruct //
-qed-.
-
-lemma at_inv_false_O: ∀cs,i. @⦃i, Ⓕ @ cs⦄ ≡ 0 → ⊥.
-#cs #i #H elim (at_inv_false … H) -H
-#j2 #H destruct
-qed-.
-
-lemma at_inv_le: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → i1 ≤ ∥cs∥ ∧ i2 ≤ |cs|.
-#cs #i1 #i2 #H elim H -cs -i1 -i2 /2 width=1 by conj/
-#cs #i1 #i2 #_ * /3 width=1 by le_S_S, conj/
-qed-.
-
-lemma at_inv_gt1: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → ∥cs∥ < i1 → ⊥.
-#cs #i1 #i2 #H elim (at_inv_le … H) -H /2 width=4 by lt_le_false/
-qed-.
-
-lemma at_inv_gt2: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → |cs| < i2 → ⊥.
-#cs #i1 #i2 #H elim (at_inv_le … H) -H /2 width=4 by lt_le_false/
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Note: lemma 250 *)
-lemma at_le: ∀cs,i1. i1 ≤ ∥cs∥ →
- ∃∃i2. @⦃i1, cs⦄ ≡ i2 & i2 ≤ |cs|.
-#cs elim cs -cs
-[ #i1 #H <(le_n_O_to_eq … H) -i1 /2 width=3 by at_empty, ex2_intro/
-| * #cs #IH
- [ * /2 width=3 by at_zero, ex2_intro/
- #i1 #H lapply (le_S_S_to_le … H) -H
- #H elim (IH … H) -IH -H /3 width=3 by at_succ, le_S_S, ex2_intro/
- | #i1 #H elim (IH … H) -IH -H /3 width=3 by at_false, le_S_S, ex2_intro/
- ]
-]
-qed-.
-
-lemma at_top: ∀cs. @⦃∥cs∥, cs⦄ ≡ |cs|.
-#cs elim cs -cs // * /2 width=1 by at_succ, at_false/
-qed.
-
-lemma at_monotonic: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → ∀j1. j1 < i1 →
- ∃∃j2. @⦃j1, cs⦄ ≡ j2 & j2 < i2.
-#cs #i1 #i2 #H elim H -cs -i1 -i2
-[ #j1 #H elim (lt_zero_false … H)
-| #cs #j1 #H elim (lt_zero_false … H)
-| #cs #i1 #i2 #Hij #IH * /2 width=3 by ex2_intro, at_zero/
- #j1 #H lapply (lt_S_S_to_lt … H) -H
- #H elim (IH … H) -i1
- #j2 #Hj12 #H /3 width=3 by le_S_S, ex2_intro, at_succ/
-| #cs #i1 #i2 #_ #IH #j1 #H elim (IH … H) -i1
- /3 width=3 by le_S_S, ex2_intro, at_false/
-]
-qed-.
-
-lemma at_dec: ∀cs,i1,i2. Decidable (@⦃i1, cs⦄ ≡ i2).
-#cs elim cs -cs [ | * #cs #IH ]
-[ * [2: #i1 ] * [2,4: #i2 ]
- [4: /2 width=1 by at_empty, or_introl/
- |*: @or_intror #H elim (at_inv_empty … H) #H1 #H2 destruct
- ]
-| * [2: #i1 ] * [2,4: #i2 ]
- [ elim (IH i1 i2) -IH
- /4 width=1 by at_inv_true_succ, at_succ, or_introl, or_intror/
- | -IH /3 width=3 by at_inv_true_O_S, or_intror/
- | -IH /3 width=3 by at_inv_true_S_O, or_intror/
- | -IH /2 width=1 by or_introl, at_zero/
- ]
-| #i1 * [2: #i2 ]
- [ elim (IH i1 i2) -IH
- /4 width=1 by at_inv_false_S, at_false, or_introl, or_intror/
- | -IH /3 width=3 by at_inv_false_O, or_intror/
- ]
-]
-qed-.
-
-lemma is_at_dec: ∀cs,i2. Decidable (∃i1. @⦃i1, cs⦄ ≡ i2).
-#cs elim cs -cs
-[ * /3 width=2 by ex_intro, or_introl/
- #i2 @or_intror * /2 width=3 by at_inv_empty_succ_dx/
-| * #cs #IH * [2,4: #i2 ]
- [ elim (IH i2) -IH
- [ * /4 width=2 by at_succ, ex_intro, or_introl/
- | #H @or_intror * #x #Hx
- elim (at_inv_true_succ_dx … Hx) -Hx
- /3 width=2 by ex_intro/
- ]
- | elim (IH i2) -IH
- [ * /4 width=2 by at_false, ex_intro, or_introl/
- | #H @or_intror * /4 width=2 by at_inv_false_S, ex_intro/
- ]
- | /3 width=2 by at_zero, ex_intro, or_introl/
- | @or_intror * /2 width=3 by at_inv_false_O/
- ]
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma at_increasing: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → i1 ≤ i2.
-#cs #i1 elim i1 -i1 //
-#j1 #IHi #i2 #H elim (at_monotonic … H j1) -H
-/3 width=3 by le_to_lt_to_lt/
-qed-.
-
-lemma at_increasing_strict: ∀cs,i1,i2. @⦃i1, Ⓕ @ cs⦄ ≡ i2 →
- i1 < i2 ∧ @⦃i1, cs⦄ ≡ ⫰i2.
-#cs #i1 #i2 #H elim (at_inv_false … H) -H
-#j2 #H #Hj2 destruct /4 width=2 by conj, at_increasing, le_S_S/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem at_mono: ∀cs,i,i1. @⦃i, cs⦄ ≡ i1 → ∀i2. @⦃i, cs⦄ ≡ i2 → i1 = i2.
-#cs #i #i1 #H elim H -cs -i -i1
-[2,3,4: #cs [2,3: #i #i1 #_ #IH ] ] #i2 #H
-[ elim (at_inv_true_succ_sn … H) -H
- #j2 #H destruct #H >(IH … H) -cs -i -i1 //
-| elim (at_inv_false … H) -H
- #j2 #H destruct #H >(IH … H) -cs -i -i1 //
-| /2 width=2 by at_inv_true_zero_sn/
-| /2 width=1 by at_inv_empty_zero_sn/
-]
-qed-.
-
-theorem at_inj: ∀cs,i1,i. @⦃i1, cs⦄ ≡ i → ∀i2. @⦃i2, cs⦄ ≡ i → i1 = i2.
-#cs #i1 #i #H elim H -cs -i1 -i
-[2,3,4: #cs [ |2,3: #i1 #i #_ #IH ] ] #i2 #H
-[ /2 width=2 by at_inv_true_zero_dx/
-| elim (at_inv_true_succ_dx … H) -H
- #j2 #H destruct #H >(IH … H) -cs -i1 -i //
-| elim (at_inv_false … H) -H
- #j #H destruct #H >(IH … H) -cs -i1 -j //
-| /2 width=1 by at_inv_empty_zero_dx/
-]
-qed-.
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