text
stringlengths 1.3k
3.62M
|
|---|
(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(* Contribution to the Coq Library V6.3 (July 1999) *)
(****************************************************************************)
(* The Calculus of Inductive Constructions *)
(* *)
(* Projet Coq *)
(* *)
(* INRIA ENS-CNRS *)
(* Rocquencourt Lyon *)
(* *)
(* Coq V5.10 *)
(* *)
(****************************************************************************)
(* Gilbreath.v *)
(****************************************************************************)
(* G. Huet - V5.8 Nov. 1994 *)
(* ported V5.10 June 1995 *)
Require Import Bool.
Require Import Words.
Require Import Alternate.
Require Import Opposite.
Require Import Paired.
Require Import Shuffle.
(****************************)
(* The Gilbreath card trick *)
(****************************)
Section Context.
Variable x : word.
Hypothesis Even_x : even x.
Variable b : bool. (* witness for (alternate x) *)
Hypothesis A : alt b x.
Variable u v : word.
Hypothesis C : conc u v x.
Variable w : word.
Hypothesis S : shuffle u v w.
Lemma Alt_u : alt b u.
Proof.
(* Goal: alt b u *)
apply alt_conc_l with v x.
(* Goal: conc u v x *)
(* Goal: alt b x *)
apply C.
(* Goal: alt b x *)
apply A.
Qed.
Section Case1_.
Hypothesis Odd_u : odd u.
Lemma Not_even_u : ~ even u.
Proof.
(* Goal: not (even u) *)
red in |- *; intro.
(* Goal: False *)
elim not_odd_and_even with u; trivial.
Qed.
Lemma Odd_v : odd v.
Proof.
(* Goal: even v *)
elim even_conc with u v x.
(* Goal: forall _ : and (even u) (alt b v), alt b v *)
(* Goal: conc u v x *)
(* Goal: alt b x *)
intro H; elim H; trivial.
(* Goal: forall _ : and (even u) (alt b v), alt (negb b) v *)
(* Goal: conc u v x *)
(* Goal: alt b x *)
intro H; elim H; intro; elim Not_even_u; trivial.
(* Goal: conc u v x *)
(* Goal: alt b x *)
apply C.
(* Goal: even x *)
apply Even_x.
Qed.
Remark Alt_v_neg : alt (negb b) v.
Proof.
(* Goal: alt b v *)
elim alt_conc_r with u v x b.
(* Goal: forall _ : and (even u) (alt b v), alt b v *)
(* Goal: conc u v x *)
(* Goal: alt b x *)
intro H; elim H; trivial.
(* Goal: forall _ : and (even u) (alt b v), alt (negb b) v *)
(* Goal: conc u v x *)
(* Goal: alt b x *)
intro H; elim H; intro; elim Not_even_u; trivial.
(* Goal: conc u v x *)
(* Goal: alt b x *)
apply C.
(* Goal: alt b x *)
apply A.
Qed.
Lemma Opp_uv : opposite u v.
Proof.
(* Goal: opposite u v *)
apply alt_neg_opp with b.
(* Goal: odd u *)
(* Goal: alt b u *)
(* Goal: odd v *)
(* Goal: alt (negb b) v *)
apply Odd_u.
(* Goal: alt b u *)
apply Alt_u.
(* Goal: odd v *)
(* Goal: forall _ : and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)))), paired w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply Odd_v.
(* Goal: alt (negb b) v *)
(* Goal: paired w *)
(* Goal: forall _ : and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)))), paired w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply Alt_v_neg.
Qed.
Lemma Case1 : paired w.
Proof.
(* Goal: paired_rot b w *)
elim Shuffling with u v w b.
simple induction 1; simple induction 2; simple induction 1;
simple induction 2; intros P1 P2.
(* Goal: paired_rot b w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply P1.
(* Goal: alt (negb b) v *)
(* Goal: paired w *)
(* Goal: forall _ : and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)))), paired w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply Alt_v_neg.
(* Goal: paired w *)
(* Goal: forall _ : and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)))), paired w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
elim not_odd_and_even with v; trivial.
(* Goal: odd v *)
(* Goal: forall _ : and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)))), paired w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply Odd_v.
(* Goal: forall _ : and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)))), paired w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
simple induction 1; intro; elim Not_even_u; trivial.
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply S.
(* Goal: alt b u *)
apply Alt_u.
Qed.
End Case1_.
Section Case2_.
Hypothesis Even_u : even u.
Lemma Not_odd_u : ~ odd u.
Proof.
(* Goal: False *)
red in |- *; intro; elim not_odd_and_even with u; trivial.
Qed.
Lemma Even_v : even v.
Proof.
(* Goal: even v *)
elim even_conc with u v x.
(* Goal: forall _ : and (odd u) (alt (negb b) v), alt b v *)
(* Goal: forall _ : and (even u) (alt b v), alt b v *)
(* Goal: conc u v x *)
(* Goal: alt b x *)
intro H; elim H; intro; elim Not_odd_u; trivial.
(* Goal: forall _ : and (even u) (alt b v), alt b v *)
(* Goal: conc u v x *)
(* Goal: alt b x *)
intro H; elim H; trivial.
(* Goal: conc u v x *)
(* Goal: alt b x *)
apply C.
(* Goal: even x *)
apply Even_x.
Qed.
Remark Alt_v : alt b v.
Proof.
(* Goal: alt b v *)
elim alt_conc_r with u v x b.
(* Goal: forall _ : and (odd u) (alt (negb b) v), alt b v *)
(* Goal: forall _ : and (even u) (alt b v), alt b v *)
(* Goal: conc u v x *)
(* Goal: alt b x *)
intro H; elim H; intro; elim Not_odd_u; trivial.
(* Goal: forall _ : and (even u) (alt b v), alt b v *)
(* Goal: conc u v x *)
(* Goal: alt b x *)
intro H; elim H; trivial.
(* Goal: conc u v x *)
(* Goal: alt b x *)
apply C.
(* Goal: alt b x *)
apply A.
Qed.
Lemma Not_opp_uv : ~ opposite u v.
Proof.
(* Goal: not (opposite u v) *)
apply alt_not_opp with b.
(* Goal: alt b u *)
apply Alt_u.
(* Goal: alt b v *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply Alt_v.
Qed.
Lemma Case2 : paired (rotate w).
Proof.
(* Goal: paired (rotate w) *)
apply paired_rotate with b.
(* Goal: paired_rot b w *)
elim Shuffling with u v w b.
(* Goal: forall _ : and (odd u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired w) (forall _ : alt b v, paired_bet b w))) (and (even v) (and (forall _ : alt b v, paired_odd_l b w) (forall _ : alt (negb b) v, paired_odd_r (negb b) w)))), paired_rot b w *)
(* Goal: forall _ : and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)))), paired_rot b w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
simple induction 1; intro; elim Not_odd_u; trivial.
(* Goal: forall _ : and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)))), paired_rot b w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
simple induction 1; simple induction 2.
(* Goal: paired w *)
(* Goal: forall _ : and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)))), paired w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
simple induction 1; intros; elim not_odd_and_even with v; trivial.
(* Goal: even v *)
(* Goal: forall _ : and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)), paired_rot b w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply Even_v.
(* Goal: forall _ : and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w)), paired_rot b w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
simple induction 1; simple induction 2; intros P1 P2.
(* Goal: paired_rot b w *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply P1.
(* Goal: alt b v *)
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply Alt_v.
(* Goal: shuffle u v w *)
(* Goal: alt b u *)
apply S.
(* Goal: alt b u *)
apply Alt_u.
Qed.
End Case2_.
(* We recall from the prelude the definition of the conditional :
Definition IF := [P,Q,R:Prop](P /\ Q) \/ ((~P) /\ R)
Syntax IF "IF _ then _ else _" *)
Lemma Main : IF opposite u v then paired w else paired (rotate w).
Proof.
(* Goal: IF_then_else (opposite u v) (paired w) (paired (rotate w)) *)
unfold IF_then_else in |- *; elim odd_or_even with u; intros.
(* (odd u) : Case 1 *)
(* Goal: or (and (opposite u v) (paired w)) (and (not (opposite u v)) (paired (rotate w))) *)
(* Goal: or (and (opposite u v) (paired w)) (and (not (opposite u v)) (paired (rotate w))) *)
left; split.
(* Goal: opposite u v *)
(* Goal: paired w *)
(* Goal: or (and (opposite u v) (paired w)) (and (not (opposite u v)) (paired (rotate w))) *)
apply Opp_uv; trivial.
(* Goal: paired w *)
(* Goal: or (and (opposite u v) (paired w)) (and (not (opposite u v)) (paired (rotate w))) *)
apply Case1; trivial.
(* (even u) : Case 2 *)
(* Goal: or (and (opposite u v) (paired w)) (and (not (opposite u v)) (paired (rotate w))) *)
right; split.
(* Goal: not (opposite u v) *)
(* Goal: paired (rotate w) *)
apply Not_opp_uv; trivial.
(* Goal: paired (rotate w) *)
apply Case2; trivial.
Qed.
End Context.
(*********************)
(* Gilbreath's trick *)
(*********************)
Theorem Gilbreath :
forall x : word,
even x ->
alternate x ->
forall u v : word,
conc u v x ->
forall w : word,
shuffle u v w -> IF opposite u v then paired w else paired (rotate w).
Proof.
(* Goal: forall (x : word) (_ : even x) (_ : alternate x) (u v : word) (_ : conc u v x) (w : word) (_ : shuffle u v w), IF_then_else (opposite u v) (paired w) (paired (rotate w)) *)
simple induction 2; intros. (* Existential intro *)
(* Goal: IF_then_else (opposite u v) (paired w) (paired (rotate w)) *)
apply Main with x b; trivial.
Qed.
|
(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(* Contribution to the Coq Library V6.3 (July 1999) *)
(****************************************************************************)
(* The Calculus of Inductive Constructions *)
(* *)
(* Projet Coq *)
(* *)
(* INRIA ENS-CNRS *)
(* Rocquencourt Lyon *)
(* *)
(* Coq V5.10 *)
(* *)
(****************************************************************************)
(* Paired.v *)
(****************************************************************************)
(* G. Huet - V5.8 Nov. 1994 *)
(* ported V5.10 June 1995 *)
Require Import Bool.
Require Import Words.
(****************)
(* Paired words *)
(****************)
(* (paired w) == w = [b1 ~b1 b2 ~b2 ... bn ~bn] *)
Inductive paired : word -> Prop :=
| paired_empty : paired empty
| paired_bit :
forall w : word,
paired w -> forall b : bool, paired (bit (negb b) (bit b w)).
(* paired_odd_l b w == w = [b b1 ~b1 b2 ~b2 ... bn ~bn] *)
Definition paired_odd_l (b : bool) (w : word) := paired (bit (negb b) w).
Lemma paired_odd_l_intro :
forall (b : bool) (w : word), paired w -> paired_odd_l b (bit b w).
Proof.
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_l (negb b) w), paired (bit b w) *)
unfold paired_odd_l in |- *; intros.
(* Goal: paired (bit (negb (negb b)) (bit (negb b) (append w0 (single (negb (negb b)))))) *)
apply paired_bit; trivial.
Qed.
Lemma paired_odd_l_elim :
forall (b : bool) (w : word), paired_odd_l (negb b) w -> paired (bit b w).
Proof.
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_l (negb b) w), paired (bit b w) *)
unfold paired_odd_l in |- *; intros.
(* Goal: paired (bit b w) *)
rewrite (negb_intro b); trivial.
Qed.
(* paired_odd_r b w == w = [b1 ~b1 b2 ~b2 ... bn ~bn ~b] *)
Definition paired_odd_r (b : bool) (w : word) := paired (append w (single b)).
(* paired_rot b w == w = [b b2 ~b2 ... bn ~bn ~b] *)
Inductive paired_rot : bool -> word -> Prop :=
| paired_rot_empty : forall b : bool, paired_rot b empty
| paired_rot_bit :
forall (b : bool) (w : word),
paired_odd_r b w -> paired_rot b (bit b w).
Lemma paired_odd_r_from_rot :
forall (w : word) (b : bool),
paired_rot b w -> paired_odd_r b (bit (negb b) w).
Proof.
(* Goal: forall (w : word) (b : bool) (_ : paired_rot b w), paired (rotate w) *)
simple induction 1.
(* Goal: forall b : bool, paired_odd_r b (bit (negb b) empty) *)
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
intro; unfold paired_odd_r in |- *; simpl in |- *.
(* Goal: paired (bit (negb b0) (single b0)) *)
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
unfold single in |- *; apply paired_bit.
(* Goal: paired empty *)
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
apply paired_empty.
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
intros b0 b' w'; unfold paired_odd_r in |- *; intros.
(* Goal: paired (append (bit (negb b0) (bit b0 b')) (single b0)) *)
simpl in |- *; apply paired_bit; auto.
Qed.
(* paired_bet b w == w = [b b2 ~b2 ... bn ~bn b] *)
Inductive paired_bet (b : bool) : word -> Prop :=
paired_bet_bit :
forall w : word, paired_odd_r (negb b) w -> paired_bet b (bit b w).
Lemma paired_odd_r_from_bet :
forall (b : bool) (w : word),
paired_bet (negb b) w -> paired_odd_r b (bit b w).
Proof.
(* Goal: forall (b : bool) (w : word) (_ : paired_bet (negb b) w), paired_odd_r b (bit b w) *)
intros b w pb.
(* Goal: paired_odd_r b (bit b w) *)
rewrite (negb_intro b).
(* Goal: paired_odd_r (negb (negb b)) (bit (negb (negb b)) w) *)
elim pb.
(* Goal: forall (w : word) (_ : paired_odd_r (negb (negb b)) w), paired_odd_r (negb (negb b)) (bit (negb (negb b)) (bit (negb b) w)) *)
unfold paired_odd_r in |- *. (* Unfolds twice *)
(* Goal: forall (w : word) (_ : paired (append w (single (negb (negb b))))), paired (append (bit (negb (negb b)) (bit (negb b) w)) (single (negb (negb b)))) *)
intros; simpl in |- *.
(* Goal: paired (bit (negb (negb b)) (bit (negb b) (append w0 (single (negb (negb b)))))) *)
apply paired_bit; trivial.
Qed.
(************)
(* Rotation *)
(************)
Definition rotate (w : word) : word :=
match w with
| empty => empty
| bit b w => append w (single b)
end.
Lemma paired_rotate :
forall (w : word) (b : bool), paired_rot b w -> paired (rotate w).
Proof.
(* Goal: forall (w : word) (b : bool) (_ : paired_rot b w), paired (rotate w) *)
simple induction 1.
(* Goal: paired empty *)
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired_odd_r b (bit (negb b) (bit b w)) *)
intro; simpl in |- *; apply paired_empty.
(* Goal: forall (b : bool) (w : word) (_ : paired_odd_r b w), paired (rotate (bit b w)) *)
intros b' w'; simpl in |- *.
(* Goal: forall _ : paired_odd_r b' w', paired (append w' (single b')) *)
unfold paired_odd_r in |- *; trivial.
Qed.
|
(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(* Contribution to the Coq Library V6.3 (July 1999) *)
(****************************************************************************)
(* The Calculus of Inductive Constructions *)
(* *)
(* Projet Coq *)
(* *)
(* INRIA ENS-CNRS *)
(* Rocquencourt Lyon *)
(* *)
(* Coq V5.10 *)
(* *)
(****************************************************************************)
(* Words.v *)
(****************************************************************************)
(* G. Huet - V5.8 Nov. 1994 *)
(* ported V5.10 June 1995 *)
Require Import Bool.
(*****************)
(* Boolean words *)
(*****************)
Inductive word : Set :=
| empty : word
| bit : bool -> word -> word.
(* Remark : word ~ bool list *)
(* word concatenation : logical definition *)
Inductive conc : word -> word -> word -> Prop :=
| conc_empty : forall v : word, conc empty v v
| conc_bit :
forall (u v w : word) (b : bool),
conc u v w -> conc (bit b u) v (bit b w).
(* word concatenation : functional definition *)
Fixpoint append (u : word) : word -> word :=
fun v : word =>
match u with
| empty => v
| bit b w => bit b (append w v)
end.
(* Relating the two definitions; unused below *)
Lemma conc_append : forall u v w : word, conc u v w -> w = append u v.
Proof.
(* Goal: forall (u v w : word) (_ : conc u v w), @eq word w (append u v) *)
simple induction 1; simpl in |- *; trivial.
(* Goal: forall (u v w : word) (b : bool) (_ : conc u v w) (_ : @eq word w (append u v)), @eq word (bit b w) (bit b (append u v)) *)
simple induction 2; trivial.
Qed.
(* Associativity of append; unused below *)
Lemma assoc_append :
forall u v w : word, append u (append v w) = append (append u v) w.
Proof.
(* Goal: forall u v w : word, @eq word (append u (append v w)) (append (append u v) w) *)
simple induction u; simpl in |- *; intros; auto.
(* Goal: @eq word (bit b (append w (append v w0))) (bit b (append (append w v) w0)) *)
rewrite H; trivial.
Qed.
(**************)
(* Singletons *)
(**************)
Definition single (b : bool) := bit b empty.
(*********************)
(* Parities of words *)
(*********************)
Inductive odd : word -> Prop :=
even_odd : forall w : word, even w -> forall b : bool, odd (bit b w)
with even : word -> Prop :=
| even_empty : even empty
| odd_even : forall w : word, odd w -> forall b : bool, even (bit b w).
Hint Resolve odd_even even_empty even_odd.
Lemma not_odd_empty : ~ odd empty.
Proof.
(* Goal: not (odd empty) *)
unfold not in |- *; intro od.
(* Goal: False *)
inversion od.
Qed.
Hint Resolve not_odd_empty.
Lemma inv_odd : forall (w : word) (b : bool), odd (bit b w) -> even w.
Proof.
(* Goal: forall (w : word) (b : bool) (_ : odd (bit b w)), even w *)
intros w b od.
(* Goal: even w *)
inversion od; trivial.
Qed.
Lemma inv_even : forall (w : word) (b : bool), even (bit b w) -> odd w.
Proof.
(* Goal: forall (w : word) (b : bool) (_ : even (bit b w)), odd w *)
intros w b ev.
(* Goal: odd w *)
inversion ev; trivial.
Qed.
(**********************)
(* (odd w) + (even w) *)
(**********************)
Lemma odd_or_even : forall w : word, odd w \/ even w.
Proof.
(* Goal: forall w : word, or (odd w) (even w) *)
simple induction w; auto.
(* Goal: forall (u v w : word) (_ : conc u v w), or (and (odd w) (or (and (odd u) (even v)) (and (even u) (odd v)))) (and (even w) (or (and (odd u) (odd v)) (and (even u) (even v)))) *)
simple induction 1; intros.
(* Goal: or (odd (bit b w0)) (even (bit b w0)) *)
(* Goal: or (odd (bit b w0)) (even (bit b w0)) *)
right; auto.
(* Goal: or (odd (bit b w0)) (even (bit b w0)) *)
left; auto.
Qed.
Lemma not_odd_and_even : forall w : word, odd w -> even w -> False.
Proof.
(* Goal: forall (w : word) (_ : odd w) (_ : even w), False *)
simple induction w; intros.
(* Goal: False *)
(* Goal: False *)
elim not_odd_empty; trivial.
(* Goal: False *)
apply H.
(* Goal: odd w0 *)
(* Goal: even w0 *)
apply inv_even with b; trivial.
(* Goal: even w0 *)
apply inv_odd with b; trivial.
Qed.
(************************)
(* Parities of subwords *)
(************************)
Lemma odd_even_conc :
forall u v w : word,
conc u v w ->
odd w /\ (odd u /\ even v \/ even u /\ odd v) \/
even w /\ (odd u /\ odd v \/ even u /\ even v).
Proof.
(* Goal: forall (u v w : word) (_ : conc u v w), or (and (odd w) (or (and (odd u) (even v)) (and (even u) (odd v)))) (and (even w) (or (and (odd u) (odd v)) (and (even u) (even v)))) *)
simple induction 1; intros.
(* Goal: or (and (odd v0) (or (and (odd empty) (even v0)) (and (even empty) (odd v0)))) (and (even v0) (or (and (odd empty) (odd v0)) (and (even empty) (even v0)))) *)
(* Goal: or (and (odd (bit b w0)) (or (and (odd (bit b u0)) (even v0)) (and (even (bit b u0)) (odd v0)))) (and (even (bit b w0)) (or (and (odd (bit b u0)) (odd v0)) (and (even (bit b u0)) (even v0)))) *)
elim (odd_or_even v0); auto.
(* Goal: or (and (odd (bit b w0)) (or (and (odd (bit b u0)) (even v0)) (and (even (bit b u0)) (odd v0)))) (and (even (bit b w0)) (or (and (odd (bit b u0)) (odd v0)) (and (even (bit b u0)) (even v0)))) *)
elim H1; [ right | left ]; intuition.
Qed.
Lemma even_conc :
forall u v w : word,
conc u v w -> even w -> odd u /\ odd v \/ even u /\ even v.
Proof.
(* Goal: forall (u v w : word) (_ : conc u v w) (_ : even w), or (and (odd u) (odd v)) (and (even u) (even v)) *)
intros u v w c e; elim odd_even_conc with u v w; intros.
(* Goal: or (and (odd u) (odd v)) (and (even u) (even v)) *)
(* Goal: or (and (odd u) (odd v)) (and (even u) (even v)) *)
(* Goal: conc u v w *)
elim H; intro o; elim not_odd_and_even with w; auto.
(* Goal: or (and (odd u) (odd v)) (and (even u) (even v)) *)
(* Goal: conc u v w *)
elim H; intros; trivial.
(* Goal: conc u v w *)
trivial.
Qed.
|
(* This program is free software; you can redistribute it and/or *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1 *)
(* of the License, or (at your option) any later version. *)
(* *)
(* This program is distributed in the hope that it will be useful, *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *)
(* GNU General Public License for more details. *)
(* *)
(* You should have received a copy of the GNU Lesser General Public *)
(* License along with this program; if not, write to the Free *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA *)
(* Contribution to the Coq Library V6.3 (July 1999) *)
(****************************************************************************)
(* The Calculus of Inductive Constructions *)
(* *)
(* Projet Coq *)
(* *)
(* INRIA ENS-CNRS *)
(* Rocquencourt Lyon *)
(* *)
(* Coq V5.10 *)
(* *)
(****************************************************************************)
(* Shuffle.v *)
(****************************************************************************)
(* G. Huet - V5.8 Nov. 1994 *)
(* ported V5.10 June 1995 *)
Require Import Bool.
Require Import Words.
Require Import Alternate.
Require Import Opposite.
Require Import Paired.
(***********************)
(* Shuffling two words *)
(***********************)
Inductive shuffle : word -> word -> word -> Prop :=
| shuffle_empty : shuffle empty empty empty
| shuffle_bit_left :
forall u v w : word,
shuffle u v w -> forall b : bool, shuffle (bit b u) v (bit b w)
| shuffle_bit_right :
forall u v w : word,
shuffle u v w -> forall b : bool, shuffle u (bit b v) (bit b w).
(***********************)
(* The shuffling lemma *)
(***********************)
Lemma Shuffling :
forall u v w : word,
shuffle u v w ->
forall b : bool,
alt b u ->
odd u /\
(odd v /\ (alt (negb b) v -> paired w) /\ (alt b v -> paired_bet b w) \/
even v /\
(alt b v -> paired_odd_l b w) /\
(alt (negb b) v -> paired_odd_r (negb b) w)) \/
even u /\
(odd v /\
(alt (negb b) v -> paired_odd_r b w) /\ (alt b v -> paired_odd_l b w) \/
even v /\ (alt b v -> paired_rot b w) /\ (alt (negb b) v -> paired w)).
Proof.
(* Goal: forall (u v w : word) (_ : shuffle u v w) (b : bool) (_ : alt b u), or (and (odd u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired w) (forall _ : alt b v, paired_bet b w))) (and (even v) (and (forall _ : alt b v, paired_odd_l b w) (forall _ : alt (negb b) v, paired_odd_r (negb b) w))))) (and (even u) (or (and (odd v) (and (forall _ : alt (negb b) v, paired_odd_r b w) (forall _ : alt b v, paired_odd_l b w))) (and (even v) (and (forall _ : alt b v, paired_rot b w) (forall _ : alt (negb b) v, paired w))))) *)
simple induction 1; intros.
(* 0. empty case *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
right.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
right.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) *)
(* Goal: alt b0 u0 *)
split; intro.
(* Goal: paired_rot b empty *)
(* Goal: paired empty *)
(* Goal: or (and (odd (bit b u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired (bit b w0)) (forall _ : alt b0 v0, paired_bet b0 (bit b w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) v0, paired_odd_r (negb b0) (bit b w0)))))) (and (even (bit b u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired_odd_r b0 (bit b w0)) (forall _ : alt b0 v0, paired_odd_l b0 (bit b w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) v0, paired (bit b w0)))))) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply paired_rot_empty.
(* Goal: paired empty *)
(* Goal: or (and (odd (bit b u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired (bit b w0)) (forall _ : alt b0 v0, paired_bet b0 (bit b w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) v0, paired_odd_r (negb b0) (bit b w0)))))) (and (even (bit b u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired_odd_r b0 (bit b w0)) (forall _ : alt b0 v0, paired_odd_l b0 (bit b w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) v0, paired (bit b w0)))))) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply paired_empty.
(* 1. u before v *)
(* Goal: or (and (odd (bit b u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired (bit b w0)) (forall _ : alt b0 v0, paired_bet b0 (bit b w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) v0, paired_odd_r (negb b0) (bit b w0)))))) (and (even (bit b u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired_odd_r b0 (bit b w0)) (forall _ : alt b0 v0, paired_odd_l b0 (bit b w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) v0, paired (bit b w0)))))) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
elim (alt_eq b0 b u0); trivial.
(* Goal: or (and (odd (bit b0 u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired (bit b0 w0)) (forall _ : alt b0 v0, paired_bet b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired_odd_r (negb b0) (bit b0 w0)))))) (and (even (bit b0 u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired_odd_r b0 (bit b0 w0)) (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_rot b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired (bit b0 w0)))))) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
elim (H1 (negb b0)); intros.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 1.1. *) right.
(* Goal: and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0))))) *)
(* Goal: alt b0 u0 *)
elim H3; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
elim H5; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H6; intros.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 1.1.1. *) left.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H8; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) *)
(* Goal: alt b0 u0 *)
split; intro.
(* Goal: paired_odd_r b0 (bit b0 w0) *)
(* Goal: paired_odd_l b0 (bit b0 w0) *)
(* Goal: or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired_odd_r b0 (bit b0 w0)) (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_rot b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired (bit b0 w0)))) *)
(* Goal: or (and (odd (bit b0 u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired (bit b0 w0)) (forall _ : alt b0 v0, paired_bet b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired_odd_r (negb b0) (bit b0 w0)))))) (and (even (bit b0 u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired_odd_r b0 (bit b0 w0)) (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_rot b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired (bit b0 w0)))))) *)
(* Goal: alt (negb b0) u0 *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply paired_odd_r_from_bet; auto.
(* Goal: paired_odd_l b0 (bit b0 w0) *)
(* Goal: or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired_odd_r b0 (bit b0 w0)) (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_rot b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired (bit b0 w0)))) *)
(* Goal: or (and (odd (bit b0 u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired (bit b0 w0)) (forall _ : alt b0 v0, paired_bet b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired_odd_r (negb b0) (bit b0 w0)))))) (and (even (bit b0 u0)) (or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired_odd_r b0 (bit b0 w0)) (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_rot b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired (bit b0 w0)))))) *)
(* Goal: alt (negb b0) u0 *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply paired_odd_l_intro; apply H9; rewrite (negb_elim b0); auto.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H6; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H8; intros.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 1.1.2. *) right.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) *)
(* Goal: alt b0 u0 *)
split; intro.
apply paired_rot_bit; rewrite (negb_intro b0); apply H10;
rewrite (negb_elim b0); auto.
(* Goal: paired (bit b0 w0) *)
(* Goal: paired_bet b0 (bit b0 w0) *)
(* Goal: or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired (bit b0 w0)) (forall _ : alt b0 v0, paired_bet b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired_odd_r (negb b0) (bit b0 w0)))) *)
(* Goal: alt (negb b0) u0 *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply paired_odd_l_elim; auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 1.2. *) left.
(* Goal: and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0))))) *)
(* Goal: alt b0 u0 *)
elim H3; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
elim H5; intros.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 1.2.1. *) left.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H6; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H8; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) *)
(* Goal: alt b0 u0 *)
split; intro.
(* Goal: paired (bit b0 w0) *)
(* Goal: paired_bet b0 (bit b0 w0) *)
(* Goal: or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired (bit b0 w0)) (forall _ : alt b0 v0, paired_bet b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired_odd_r (negb b0) (bit b0 w0)))) *)
(* Goal: alt (negb b0) u0 *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply paired_odd_l_elim; auto.
(* Goal: paired_bet b0 (bit b0 w0) *)
(* Goal: or (and (odd v0) (and (forall _ : alt (negb b0) v0, paired (bit b0 w0)) (forall _ : alt b0 v0, paired_bet b0 (bit b0 w0)))) (and (even v0) (and (forall _ : alt b0 v0, paired_odd_l b0 (bit b0 w0)) (forall _ : alt (negb b0) v0, paired_odd_r (negb b0) (bit b0 w0)))) *)
(* Goal: alt (negb b0) u0 *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply paired_bet_bit; apply H9; rewrite (negb_elim b0); auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 1.2.2. *) right.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H6; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H8; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) *)
(* Goal: alt b0 u0 *)
split; intro.
(* Goal: paired_odd_l b0 (bit b0 w0) *)
(* Goal: paired_odd_r (negb b0) (bit b0 w0) *)
(* Goal: alt (negb b0) u0 *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply paired_odd_l_intro; apply H10; rewrite (negb_elim b0); auto.
(* Goal: paired_odd_r (negb b0) (bit b0 w0) *)
(* Goal: alt (negb b0) u0 *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
pattern b0 at 2 in |- *; rewrite (negb_intro b0).
(* Goal: paired_odd_r (negb b0) (bit (negb (negb b0)) w0) *)
(* Goal: alt (negb b0) u0 *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply paired_odd_r_from_rot; auto.
(* Goal: alt (negb b0) u0 *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
apply alt_neg_intro with b; trivial.
(* 2. v before u *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
elim (H1 b0); intros.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 2.1. *) left.
(* Goal: and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0))))) *)
(* Goal: alt b0 u0 *)
elim H3; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
elim H5; intros.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 2.1.1. *) right.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H6; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H8; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) *)
(* Goal: alt b0 u0 *)
split; intro.
(* Goal: paired_odd_l b0 (bit b w0) *)
(* Goal: alt b0 u0 *)
elim (alt_eq b0 b v0); trivial.
(* Goal: paired_odd_l b0 (bit b0 w0) *)
(* Goal: paired_odd_r (negb b0) (bit b w0) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
apply paired_odd_l_intro; apply H9; apply alt_neg_intro with b; auto.
(* Goal: paired_odd_r b0 (bit b w0) *)
(* Goal: paired_odd_l b0 (bit b w0) *)
(* Goal: alt b0 u0 *)
elim (alt_eq (negb b0) b v0); trivial.
apply paired_odd_r_from_bet; rewrite (negb_elim b0); apply H10;
apply alt_neg_elim with b; auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 2.1.2. *) left.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H6; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H8; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) *)
(* Goal: alt b0 u0 *)
split; intro.
(* Goal: paired (bit (negb b0) w0) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
apply paired_odd_l_elim.
(* Goal: paired_odd_r b0 (bit b w0) *)
(* Goal: paired_odd_l b0 (bit b w0) *)
(* Goal: alt b0 u0 *)
elim (alt_eq (negb b0) b v0); trivial.
(* Goal: paired_odd_l (negb (negb b0)) w0 *)
(* Goal: paired_bet b0 (bit b w0) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
rewrite (negb_elim b0).
(* Goal: paired_odd_l b0 w0 *)
(* Goal: paired_bet b0 (bit b w0) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
apply H9.
(* Goal: alt b0 v0 *)
(* Goal: paired_bet b0 (bit b w0) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
rewrite (negb_intro b0).
(* Goal: alt (negb (negb b0)) v0 *)
(* Goal: paired_bet b0 (bit b w0) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
apply alt_neg_intro with b; auto.
(* Goal: paired_odd_l b0 (bit b w0) *)
(* Goal: alt b0 u0 *)
elim (alt_eq b0 b v0); trivial.
(* Goal: alt (negb (negb b0)) v0 *)
(* Goal: paired_bet b0 (bit b w0) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
apply paired_bet_bit; apply H10; apply alt_neg_intro with b; auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 2.2. *) right.
(* Goal: and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0))))) *)
(* Goal: alt b0 u0 *)
elim H3; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
elim H5; intros.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 2.2.1. *) right.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H6; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H8; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) *)
(* Goal: alt b0 u0 *)
split; intro.
(* Goal: paired_odd_l b0 (bit b w0) *)
(* Goal: alt b0 u0 *)
elim (alt_eq b0 b v0); trivial.
(* Goal: alt (negb (negb b0)) v0 *)
(* Goal: paired_bet b0 (bit b w0) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
apply paired_rot_bit; apply H9; apply alt_neg_intro with b; auto.
(* Goal: paired_odd_r b0 (bit b w0) *)
(* Goal: paired_odd_l b0 (bit b w0) *)
(* Goal: alt b0 u0 *)
elim (alt_eq (negb b0) b v0); trivial.
(* Goal: paired (bit (negb b0) w0) *)
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
apply paired_odd_l_elim.
rewrite (negb_elim b0); apply H10; rewrite (negb_intro b0);
(* Goal: alt (negb (negb b0)) v0 *)
(* Goal: paired_bet b0 (bit b w0) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
apply alt_neg_intro with b; auto.
(* Goal: or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))) *)
(* Goal: alt b0 u0 *)
(* 2.2.2. *) left.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H6; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
elim H8; intros.
(* Goal: and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0))) *)
(* Goal: alt b0 u0 *)
split; auto.
(* Goal: and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) *)
(* Goal: alt b0 u0 *)
split; intro.
(* Goal: paired_odd_r b0 (bit b w0) *)
(* Goal: paired_odd_l b0 (bit b w0) *)
(* Goal: alt b0 u0 *)
elim (alt_eq (negb b0) b v0); trivial.
apply paired_odd_r_from_rot; apply H9; rewrite (negb_intro b0);
(* Goal: alt (negb (negb b0)) v0 *)
(* Goal: paired_bet b0 (bit b w0) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
apply alt_neg_intro with b; auto.
(* Goal: paired_odd_l b0 (bit b w0) *)
(* Goal: alt b0 u0 *)
elim (alt_eq b0 b v0); trivial.
(* Goal: alt (negb (negb b0)) v0 *)
(* Goal: paired_bet b0 (bit b w0) *)
(* Goal: or (and (odd u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)) (forall _ : alt b0 (bit b v0), paired_bet b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired_odd_r (negb b0) (bit b w0)))))) (and (even u0) (or (and (odd (bit b v0)) (and (forall _ : alt (negb b0) (bit b v0), paired_odd_r b0 (bit b w0)) (forall _ : alt b0 (bit b v0), paired_odd_l b0 (bit b w0)))) (and (even (bit b v0)) (and (forall _ : alt b0 (bit b v0), paired_rot b0 (bit b w0)) (forall _ : alt (negb b0) (bit b v0), paired (bit b w0)))))) *)
(* Goal: alt b0 u0 *)
apply paired_odd_l_intro; apply H10; apply alt_neg_intro with b; auto.
(* Goal: alt b0 u0 *)
trivial.
Qed.
|