--- /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/rafter_3.ma".
+include "ground_2/relocation/trace_at.ma".
+
+(* RELOCATION TRACE *********************************************************)
+
+inductive after: relation3 trace trace trace ≝
+ | after_empty: after (◊) (◊) (◊)
+ | after_true : ∀cs1,cs2,cs. after cs1 cs2 cs →
+ ∀b. after (Ⓣ @ cs1) (b @ cs2) (b @ cs)
+ | after_false: ∀cs1,cs2,cs. after cs1 cs2 cs →
+ after (Ⓕ @ cs1) cs2 (Ⓕ @ cs).
+
+interpretation "composition (trace)"
+ 'RAfter cs1 cs2 cs = (after cs1 cs2 cs).
+
+(* Basic properties *********************************************************)
+
+lemma after_length: ∀cs1,cs2. ∥cs1∥ = |cs2| →
+ ∃∃cs. cs1 ⊚ cs2 ≡ cs & |cs| = |cs1| & ∥cs∥ = ∥cs2∥.
+#cs1 elim cs1 -cs1
+[ #cs2 #H >(length_inv_zero_sn … H) -cs2 /2 width=4 by after_empty, ex3_intro/
+| * #cs1 #IH #cs2 #Hcs
+ [ elim (length_inv_succ_sn … Hcs) -Hcs
+ #tl #b #Hcs #H destruct
+ ]
+ elim (IH … Hcs) -IH -Hcs
+ #cs #Hcs #H1 #H2 [ @(ex3_intro … (b@cs)) | @(ex3_intro … (Ⓕ@cs)) ] /2 width=1 by after_true, after_false, colength_cons/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact after_inv_empty1_aux: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → cs1 = ◊ →
+ cs2 = ◊ ∧ cs = ◊.
+#cs1 #cs2 #cs * -cs1 -cs2 -cs
+[ /2 width=1 by conj/
+| #cs1 #cs2 #cs #_ #b #H destruct
+| #cs1 #cs2 #cs #_ #H destruct
+]
+qed-.
+
+lemma after_inv_empty1: ∀cs2,cs. ◊ ⊚ cs2 ≡ cs → cs2 = ◊ ∧ cs = ◊.
+/2 width=3 by after_inv_empty1_aux/ qed-.
+
+fact after_inv_true1_aux: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → ∀tl1. cs1 = Ⓣ @ tl1 →
+ ∃∃tl2,tl,b. cs2 = b @ tl2 & cs = b @ tl & tl1 ⊚ tl2 ≡ tl.
+#cs1 #cs2 #cs * -cs1 -cs2 -cs
+[ #tl1 #H destruct
+| #cs1 #cs2 #cs #H12 #b #tl1 #H destruct
+ /2 width=6 by ex3_3_intro/
+| #cs1 #cs2 #cs #_ #tl1 #H destruct
+]
+qed-.
+
+lemma after_inv_true1: ∀tl1,cs2,cs. (Ⓣ @ tl1) ⊚ cs2 ≡ cs →
+ ∃∃tl2,tl,b. cs2 = b @ tl2 & cs = b @ tl & tl1 ⊚ tl2 ≡ tl.
+/2 width=3 by after_inv_true1_aux/ qed-.
+
+fact after_inv_false1_aux: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → ∀tl1. cs1 = Ⓕ @ tl1 →
+ ∃∃tl. cs = Ⓕ @ tl & tl1 ⊚ cs2 ≡ tl.
+#cs1 #cs2 #cs * -cs1 -cs2 -cs
+[ #tl1 #H destruct
+| #cs1 #cs2 #cs #_ #b #tl1 #H destruct
+| #cs1 #cs2 #cs #H12 #tl1 #H destruct
+ /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma after_inv_false1: ∀tl1,cs2,cs. (Ⓕ @ tl1) ⊚ cs2 ≡ cs →
+ ∃∃tl. cs = Ⓕ @ tl & tl1 ⊚ cs2 ≡ tl.
+/2 width=3 by after_inv_false1_aux/ qed-.
+
+fact after_inv_empty3_aux: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → cs = ◊ →
+ cs1 = ◊ ∧ cs2 = ◊.
+#cs1 #cs2 #cs * -cs1 -cs2 -cs
+[ /2 width=1 by conj/
+| #cs1 #cs2 #cs #_ #b #H destruct
+| #cs1 #cs2 #cs #_ #H destruct
+]
+qed-.
+
+lemma after_inv_empty3: ∀cs1,cs2. cs1 ⊚ cs2 ≡ ◊ → cs1 = ◊ ∧ cs2 = ◊.
+/2 width=3 by after_inv_empty3_aux/ qed-.
+
+fact after_inv_inh3_aux: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → ∀tl,b. cs = b @ tl →
+ (∃∃tl1,tl2. cs1 = Ⓣ @ tl1 & cs2 = b @ tl2 & tl1 ⊚ tl2 ≡ tl) ∨
+ ∃∃tl1. cs1 = Ⓕ @ tl1 & b = Ⓕ & tl1 ⊚ cs2 ≡ tl.
+#cs1 #cs2 #cs * -cs1 -cs2 -cs
+[ #tl #b #H destruct
+| #cs1 #cs2 #cs #H12 #b0 #tl #b #H destruct
+ /3 width=5 by ex3_2_intro, or_introl/
+| #cs1 #cs2 #cs #H12 #tl #b #H destruct
+ /3 width=3 by ex3_intro, or_intror/
+]
+qed-.
+
+lemma after_inv_inh3: ∀cs1,cs2,tl,b. cs1 ⊚ cs2 ≡ b @ tl →
+ (∃∃tl1,tl2. cs1 = Ⓣ @ tl1 & cs2 = b @ tl2 & tl1 ⊚ tl2 ≡ tl) ∨
+ ∃∃tl1. cs1 = Ⓕ @ tl1 & b = Ⓕ & tl1 ⊚ cs2 ≡ tl.
+/2 width=3 by after_inv_inh3_aux/ qed-.
+
+lemma after_inv_true3: ∀cs1,cs2,tl. cs1 ⊚ cs2 ≡ Ⓣ @ tl →
+ ∃∃tl1,tl2. cs1 = Ⓣ @ tl1 & cs2 = Ⓣ @ tl2 & tl1 ⊚ tl2 ≡ tl.
+#cs1 #cs2 #tl #H elim (after_inv_inh3 … H) -H // *
+#tl1 #_ #H destruct
+qed-.
+
+lemma after_inv_false3: ∀cs1,cs2,tl. cs1 ⊚ cs2 ≡ Ⓕ @ tl →
+ (∃∃tl1,tl2. cs1 = Ⓣ @ tl1 & cs2 = Ⓕ @ tl2 & tl1 ⊚ tl2 ≡ tl) ∨
+ ∃∃tl1. cs1 = Ⓕ @ tl1 & tl1 ⊚ cs2 ≡ tl.
+#cs1 #cs2 #tl #H elim (after_inv_inh3 … H) -H /2 width=1 by or_introl/ * /3 width=3 by ex2_intro, or_intror/
+qed-.
+
+lemma after_inv_length: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs →
+ ∧∧ ∥cs1∥ = |cs2| & |cs| = |cs1| & ∥cs∥ = ∥cs2∥.
+#cs1 #cs2 #cs #H elim H -cs1 -cs2 -cs /2 width=1 by and3_intro/
+#cs1 #cs2 #cs #_ [ * ] * /2 width=1 by and3_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma after_at_fwd: ∀cs1,cs2,cs. cs2 ⊚ cs1 ≡ cs → ∀i1,i. @⦃i1, cs⦄ ≡ i →
+ ∃∃i2. @⦃i1, cs1⦄ ≡ i2 & @⦃i2, cs2⦄ ≡ i.
+#cs1 #cs2 #cs #H elim H -cs1 -cs2 -cs
+[ #i1 #i #H elim (at_inv_empty … H) -H
+ #H1 #H2 destruct /2 width=3 by at_empty, ex2_intro/
+| #cs1 #cs2 #cs #_ * #IH #i1 #i #H
+ [ elim (at_inv_true … H) -H *
+ [ -IH #H1 #H2 destruct /2 width=3 by at_zero, ex2_intro/
+ | #j1 #j #H1 #H2 #Hj1 destruct
+ elim (IH … Hj1) -IH -Hj1 /3 width=3 by at_succ, ex2_intro/
+ ]
+ | elim (at_inv_false … H) -H
+ #j #H #Hj destruct
+ elim (IH … Hj) -IH -Hj /3 width=3 by at_succ, at_false, ex2_intro/
+ ]
+| #cs1 #cs2 #cs #_ #IH #i1 #i #H elim (at_inv_false … H) -H
+ #j #H #Hj destruct
+ elim (IH … Hj) -IH -Hj /3 width=3 by at_false, ex2_intro/
+]
+qed-.
+
+lemma after_at1_fwd: ∀cs1,cs2,cs. cs2 ⊚ cs1 ≡ cs → ∀i1,i2. @⦃i1, cs1⦄ ≡ i2 →
+ ∃∃i. @⦃i2, cs2⦄ ≡ i & @⦃i1, cs⦄ ≡ i.
+#cs1 #cs2 #cs #H elim H -cs1 -cs2 -cs
+[ #i1 #i2 #H elim (at_inv_empty … H) -H
+ #H1 #H2 destruct /2 width=3 by at_empty, ex2_intro/
+| #cs1 #cs2 #cs #_ * #IH #i1 #i2 #H
+ [ elim (at_inv_true … H) -H *
+ [ -IH #H1 #H2 destruct /2 width=3 by at_zero, ex2_intro/
+ | #j1 #j2 #H1 #H2 #Hj12 destruct
+ elim (IH … Hj12) -IH -Hj12 /3 width=3 by at_succ, ex2_intro/
+ ]
+ | elim (at_inv_false … H) -H
+ #j2 #H #Hj2 destruct
+ elim (IH … Hj2) -IH -Hj2 /3 width=3 by at_succ, at_false, ex2_intro/
+ ]
+| #cs1 #cs2 #cs #_ #IH #i1 #i2 #H elim (IH … H) -IH -H
+ /3 width=3 by at_false, ex2_intro/
+]
+qed-.
+
+lemma after_fwd_at: ∀cs1,cs2,cs. cs2 ⊚ cs1 ≡ cs →
+ ∀i1,i2,i. @⦃i1, cs1⦄ ≡ i2 → @⦃i2, cs2⦄ ≡ i → @⦃i1, cs⦄ ≡ i.
+#cs1 #cs2 #cs #Hcs #i1 #i2 #i #Hi1 #Hi2 elim (after_at1_fwd … Hcs … Hi1) -cs1
+#j #H #Hj >(at_mono … Hi2 … H) -i2 //
+qed-.
+
+lemma after_fwd_at1: ∀cs1,cs2,cs. cs2 ⊚ cs1 ≡ cs →
+ ∀i1,i2,i. @⦃i1, cs⦄ ≡ i → @⦃i2, cs2⦄ ≡ i → @⦃i1, cs1⦄ ≡ i2.
+#cs1 #cs2 #cs #Hcs #i1 #i2 #i #Hi1 #Hi2 elim (after_at_fwd … Hcs … Hi1) -cs
+#j1 #Hij1 #H >(at_inj … Hi2 … H) -i //
+qed-.
+
+lemma after_fwd_at2: ∀cs1,cs2,cs. cs2 ⊚ cs1 ≡ cs →
+ ∀i1,i2,i. @⦃i1, cs⦄ ≡ i → @⦃i1, cs1⦄ ≡ i2 → @⦃i2, cs2⦄ ≡ i.
+#cs1 #cs2 #cs #H elim H -cs1 -cs2 -cs
+[ #i1 #i2 #i #Hi #Hi1 elim (at_inv_empty … Hi1) -Hi1 //
+| #cs1 #cs2 #cs #_ * #IH #i1 #i2 #i #Hi #Hi1
+ [ elim (at_inv_true … Hi1) -Hi1 *
+ [ -IH #H1 #H2 destruct >(at_inv_true_zero_sn … Hi) -i //
+ | #j1 #j2 #H1 #H2 #Hj12 destruct elim (at_inv_true_succ_sn … Hi) -Hi
+ #j #H #Hj1 destruct /3 width=3 by at_succ/
+ ]
+ | elim (at_inv_false … Hi1) -Hi1
+ #j2 #H #Hj2 destruct elim (at_inv_false … Hi) -Hi
+ #j #H #Hj destruct /3 width=3 by at_succ/
+ ]
+| #cs1 #cs2 #cs #_ #IH #i1 #i2 #i #Hi #Hi2 elim (at_inv_false … Hi) -Hi
+ #j #H #Hj destruct /3 width=3 by at_false/
+]
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem after_trans1: ∀cs1,cs2,cs0. cs1 ⊚ cs2 ≡ cs0 →
+ ∀cs3, cs4. cs0 ⊚ cs3 ≡ cs4 →
+ ∃∃cs. cs2 ⊚ cs3 ≡ cs & cs1 ⊚ cs ≡ cs4.
+#cs1 #cs2 #cs0 #H elim H -cs1 -cs2 -cs0
+[ #cs3 #cs4 #H elim (after_inv_empty1 … H) -H
+ #H1 #H2 destruct /2 width=3 by ex2_intro, after_empty/
+| #cs1 #cs2 #cs0 #_ * #IH #cs3 #cs4 #H
+ [ elim (after_inv_true1 … H) -H
+ #tl3 #tl4 #b #H1 #H2 #Htl destruct
+ elim (IH … Htl) -cs0
+ /3 width=3 by ex2_intro, after_true/
+ | elim (after_inv_false1 … H) -H
+ #tl4 #H #Htl destruct
+ elim (IH … Htl) -cs0
+ /3 width=3 by ex2_intro, after_true, after_false/
+ ]
+| #cs1 #cs2 #cs0 #_ #IH #cs3 #cs4 #H
+ elim (after_inv_false1 … H) -H
+ #tl4 #H #Htl destruct
+ elim (IH … Htl) -cs0
+ /3 width=3 by ex2_intro, after_false/
+]
+qed-.
+
+theorem after_trans2: ∀cs1,cs0,cs4. cs1 ⊚ cs0 ≡ cs4 →
+ ∀cs2, cs3. cs2 ⊚ cs3 ≡ cs0 →
+ ∃∃cs. cs1 ⊚ cs2 ≡ cs & cs ⊚ cs3 ≡ cs4.
+#cs1 #cs0 #cs4 #H elim H -cs1 -cs0 -cs4
+[ #cs2 #cs3 #H elim (after_inv_empty3 … H) -H
+ #H1 #H2 destruct /2 width=3 by ex2_intro, after_empty/
+| #cs1 #cs0 #cs4 #_ #b #IH #cs2 #cs3 #H elim (after_inv_inh3 … H) -H *
+ [ #tl2 #tl3 #H1 #H2 #Htl destruct
+ elim (IH … Htl) -cs0
+ /3 width=3 by ex2_intro, after_true/
+ | #tl2 #H1 #H2 #Htl destruct
+ elim (IH … Htl) -cs0
+ /3 width=3 by ex2_intro, after_true, after_false/
+ ]
+| #cs1 #cs0 #cs4 #_ #IH #cs2 #cs3 #H elim (IH … H) -cs0
+ /3 width=3 by ex2_intro, after_false/
+]
+qed-.
+
+theorem after_mono: ∀cs1,cs2,x. cs1 ⊚ cs2 ≡ x → ∀y. cs1 ⊚ cs2 ≡ y → x = y.
+#cs1 #cs2 #x #H elim H -cs1 -cs2 -x
+[ #y #H elim (after_inv_empty1 … H) -H //
+| #cs1 #cs #x #_ #b #IH #y #H elim (after_inv_true1 … H) -H
+ #tl #tly #b0 #H1 #H2 #Htl destruct >(IH … Htl) -tl -cs1 -x //
+| #cs1 #cs2 #x #_ #IH #y #H elim (after_inv_false1 … H) -H
+ #tly #H #Htl destruct >(IH … Htl) -cs1 -cs2 -x //
+]
+qed-.
+
+theorem after_inj: ∀cs1,x,cs. cs1 ⊚ x ≡ cs → ∀y. cs1 ⊚ y ≡ cs → x = y.
+#cs1 #x #cs #H elim H -cs1 -x -cs
+[ #y #H elim (after_inv_empty1 … H) -H //
+| #cs1 #x #cs #_ #b #IH #y #H elim (after_inv_true1 … H) -H
+ #tly #tl #b0 #H1 #H2 #Htl destruct >(IH … Htl) -tl -cs1 -x //
+| #cs1 #x #cs #_ #IH #y #H elim (after_inv_false1 … H) -H
+ #tl #H #Htl destruct >(IH … Htl) -tl -cs1 -x //
+]
+qed-.
projects / helm.git / blobdiff