+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
-(* ||A|| E.Tassi, S.Zacchiroli *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU Lesser General Public License Version 2.1 *)
-(* *)
-(**************************************************************************)
-
-set "baseuri" "cic:/matita/nat/iteration.ma".
-
-include "nat/permutation.ma".
-include "nat/count.ma".
-
-lemma le_to_le_to_eq: \forall n,m. n \le m \to m \le n \to n = m.
-apply nat_elim2
- [intros.apply le_n_O_to_eq.assumption
- |intros.apply sym_eq.apply le_n_O_to_eq.assumption
- |intros.apply eq_f.apply H
- [apply le_S_S_to_le.assumption
- |apply le_S_S_to_le.assumption
- ]
- ]
-qed.
-
-let rec map_iter_P n p (g:nat \to nat) (a:nat) f \def
- match n with
- [ O \Rightarrow a
- | (S k) \Rightarrow
- match p (S k) with
- [true \Rightarrow f (g (S k)) (map_iter_P k p g a f)
- |false \Rightarrow map_iter_P k p g a f]
- ].
-
-theorem eq_map_iter_P: \forall g1,g2:nat \to nat.
-\forall p:nat \to bool.
-\forall f:nat \to nat \to nat. \forall a,n:nat.
-(\forall m:nat. m \le n \to g1 m = g2 m) \to
-map_iter_P n p g1 a f = map_iter_P n p g2 a f.
-intros 6.elim n
- [simplify.reflexivity.
- |simplify.elim (p (S n1))
- [simplify.apply eq_f2
- [apply H1.apply le_n
- |simplify.apply H.intros.apply H1.
- apply le_S.assumption
- ]
- |simplify.apply H.intros.apply H1.
- apply le_S.assumption
- ]
- ]
-qed.
-
-(* useful since simply simpifies too much *)
-
-theorem map_iter_P_O: \forall p.\forall g.\forall f. \forall a:nat.
-map_iter_P O p g a f = a.
-intros.simplify.reflexivity.
-qed.
-
-theorem map_iter_P_S_true: \forall p.\forall g.\forall f. \forall a,n:nat.
-p (S n) = true \to
-map_iter_P (S n) p g a f = f (g (S n)) (map_iter_P n p g a f).
-intros.simplify.rewrite > H.reflexivity.
-qed.
-
-theorem map_iter_P_S_false: \forall p.\forall g.\forall f. \forall a,n:nat.
-p (S n) = false \to
-map_iter_P (S n) p g a f = map_iter_P n p g a f.
-intros.simplify.rewrite > H.reflexivity.
-qed.
-
-(* map_iter examples *)
-definition Pi_P \def \lambda p. \lambda n.
-map_iter_P n p (\lambda n.n) (S O) times.
-
-lemma Pi_P_S: \forall n.\forall p.
-Pi_P p (S n) =
- match p (S n) with
- [true \Rightarrow (S n)*(Pi_P p n)
- |false \Rightarrow (Pi_P p n)
- ].
-intros.reflexivity.
-qed.
-
-lemma lt_O_Pi_P: \forall n.\forall p.
-O < Pi_P p n.
-intros.elim n
- [simplify.apply le_n
- |rewrite > Pi_P_S.
- elim p (S n1)
- [change with (O < (S n1)*(Pi_P p n1)).
- rewrite >(times_n_O n1).
- apply lt_times[apply le_n|assumption]
- | assumption
- ]
- ]
-qed.
-
-let rec card n p \def
- match n with
- [O \Rightarrow O
- |(S m) \Rightarrow
- (bool_to_nat (p (S m))) + (card m p)].
-
-lemma a_times_Pi_P: \forall p. \forall a,n.
-exp a (card n p) * Pi_P p n = map_iter_P n p (\lambda n.a*n) (S O) times.
-intros.elim n
- [simplify.reflexivity
- |simplify.apply (bool_elim ? (p (S n1)))
- [intro.
- change with
- (a*exp a (card n1 p) * ((S n1) * (Pi_P p n1)) =
- a*(S n1)*map_iter_P n1 p (\lambda n.a*n) (S O) times).
- rewrite < H.
- auto
- |intro.assumption
- ]
- ]
-qed.
-
-definition permut_p \def \lambda f. \lambda p:nat\to bool. \lambda n.
-\forall i. i \le n \to p i = true \to ((f i \le n \land p (f i) = true)
-\land (\forall j. p j = true \to j \le n \to i \neq j \to (f i \neq f j))).
-
-definition extentional_eq_n \def \lambda f,g:nat \to nat.\lambda n.
-\forall x. x \le n \to f x = g x.
-
-lemma extentional_eq_n_to_permut_p: \forall f,g. \forall p. \forall n.
-extentional_eq_n f g n\to permut_p f p n \to permut_p g p n.
-intros.unfold permut_p.
-intros.
-elim (H1 i H2 H3).
-split
- [elim H4.split
- [rewrite < (H i H2).assumption
- |rewrite < (H i H2).assumption
- ]
- |intros.
- unfold.intro.apply (H5 j H6 H7 H8).
- rewrite > (H i H2).
- rewrite > (H j H7).assumption
- ]
-qed.
-
-theorem permut_p_compose: \forall f,g.\forall p.\forall n.
-permut_p f p n \to permut_p g p n \to permut_p (compose ? ? ? g f) p n.
-intros.unfold permut_p.intros.
-elim (H i H2 H3).
-elim H4.
-elim (H1 (f i) H6 H7).
-elim H8.
-split
- [split
- [unfold compose.assumption
- |unfold compose.rewrite < H11.reflexivity
- ]
- |intros.
- unfold compose.
- apply (H9 (f j))
- [elim (H j H13 H12).elim H15.rewrite < H18.reflexivity
- |elim (H j H13 H12).elim H15.assumption.
- |apply (H5 j H12 H13 H14)
- ]
- ]
-qed.
-
-theorem permut_p_S_to_permut_p: \forall f.\forall p.\forall n.
-permut_p f p (S n) \to (f (S n)) = (S n) \to permut_p f p n.
-intros.
-unfold permut_p.
-intros.
-split
- [elim (H i (le_S i n H2) H3).split
- [elim H4.
- elim (le_to_or_lt_eq (f i) (S n))
- [apply le_S_S_to_le.assumption
- |absurd (f i = (S n))
- [assumption
- |rewrite < H1.
- apply H5
- [rewrite < H8.assumption
- |apply le_n
- |unfold.intro.rewrite > H8 in H2.
- apply (not_le_Sn_n n).rewrite < H9.assumption
- ]
- ]
- |assumption
- ]
- |elim H4.assumption
- ]
- |intros.
- elim (H i (le_S i n H2) H3).
- apply H8
- [assumption|apply le_S.assumption|assumption]
- ]
-qed.
-
-lemma permut_p_transpose: \forall p.\forall i,j,n.
-i \le n \to j \le n \to p i = p j \to
-permut_p (transpose i j) p n.
-unfold permut_p.intros.
-split
- [split
- [unfold transpose.
- apply (eqb_elim i1 i)
- [intro.
- apply (eqb_elim i1 j)
- [simplify.intro.assumption
- |simplify.intro.assumption
- ]
- |intro.
- apply (eqb_elim i1 j)
- [simplify.intro.assumption
- |simplify.intro.assumption
- ]
- ]
- |unfold transpose.
- apply (eqb_elim i1 i)
- [intro.
- apply (eqb_elim i1 j)
- [simplify.intro.rewrite < H6.assumption
- |simplify.intro.rewrite < H2.rewrite < H5.assumption
- ]
- |intro.
- apply (eqb_elim i1 j)
- [simplify.intro.rewrite > H2.rewrite < H6.assumption
- |simplify.intro.assumption
- ]
- ]
- ]
- |intros.unfold Not.
- intro.apply H7.
- apply (injective_transpose ? ? ? ? H8).
- ]
-qed.
-
-theorem eq_map_iter_P_k: \forall f,g.\forall p.\forall a,k,n:nat.
-p (S n-k) = true \to (\forall i. (S n)-k < i \to i \le (S n) \to (p i) = false) \to
-map_iter_P (S n) p g a f = map_iter_P (S n-k) p g a f.
-intros 5.
-elim k 3
- [rewrite < minus_n_O.reflexivity
- |apply (nat_case n1)
- [intros.
- rewrite > map_iter_P_S_false
- [reflexivity
- |apply H2[simplify.apply lt_O_S.|apply le_n]
- ]
- |intros.
- rewrite > map_iter_P_S_false
- [rewrite > (H m H1)
- [reflexivity
- |intros.
- apply (H2 i H3).
- apply le_S.
- assumption
- ]
- |apply H2[auto|apply le_n]
- ]
- ]
- ]
-qed.
-
-theorem eq_map_iter_p_a: \forall p.\forall f.\forall g. \forall a,n:nat.
-(\forall i.i \le n \to p i = false) \to map_iter_P n p g a f = a.
-intros 5.
-elim n
- [simplify.reflexivity
- |rewrite > map_iter_P_S_false
- [apply H.
- intros.
- apply H1.apply le_S.assumption
- |apply H1.apply le_n
- ]
- ]
-qed.
-
-theorem eq_map_iter_p_transpose: \forall p.\forall f.associative nat f \to
-symmetric2 nat nat f \to \forall g. \forall a,k,n:nat. k < n \to
-(p (S n) = true) \to (p (n-k)) = true \to (\forall i. n-k < i \to i \le n \to (p i) = false)
-\to map_iter_P (S n) p g a f = map_iter_P (S n) p (\lambda m. g (transpose (n-k) (S n) m)) a f.
-intros 8.
-apply (nat_case n)
- [intro.absurd (k < O)
- [assumption|apply le_to_not_lt.apply le_O_n]
- |intros.
- rewrite > (map_iter_P_S_true ? ? ? ? ? H3).
- rewrite > (map_iter_P_S_true ? ? ? ? ? H3).
- rewrite > (eq_map_iter_P_k ? ? ? ? ? ? H4 H5).
- rewrite > (eq_map_iter_P_k ? ? ? ? ? ? H4 H5).
- generalize in match H4.
- rewrite > minus_Sn_m
- [intro.
- rewrite > (map_iter_P_S_true ? ? ? ? ? H6).
- rewrite > (map_iter_P_S_true ? ? ? ? ? H6).
- rewrite > transpose_i_j_j.
- rewrite > transpose_i_j_i.
- cut (map_iter_P (m-k) p g a f =
- map_iter_P (m-k) p (\lambda x.g (transpose (S(m-k)) (S(S m)) x)) a f)
- [rewrite < Hcut.
- rewrite < H.
- rewrite < H1 in \vdash (? ? (? % ?) ?).
- rewrite > H.
- reflexivity
- |apply eq_map_iter_P.
- intros.unfold transpose.
- cut (eqb m1 (S (m-k)) =false)
- [cut (eqb m1 (S (S m)) =false)
- [rewrite > Hcut.
- rewrite > Hcut1.
- reflexivity
- |apply not_eq_to_eqb_false.
- apply lt_to_not_eq.
- apply (le_to_lt_to_lt ? m)
- [apply (trans_le ? (m-k))
- [assumption|auto]
- |apply le_S.apply le_n
- ]
- ]
- |apply not_eq_to_eqb_false.
- apply lt_to_not_eq.
- unfold.auto
- ]
- ]
- |apply le_S_S_to_le.assumption
- ]
- ]
-qed.
-
-theorem eq_map_iter_p_transpose1: \forall p.\forall f.associative nat f \to
-symmetric2 nat nat f \to \forall g. \forall a,k1,k2,n:nat. O < k1 \to k1 < k2 \to k2 \le n \to
-(p k1) = true \to (p k2) = true \to (\forall i. k1 < i \to i < k2 \to (p i) = false)
-\to map_iter_P n p g a f = map_iter_P n p (\lambda m. g (transpose k1 k2 m)) a f.
-intros 10.
-elim n 2
- [absurd (k2 \le O)
- [assumption|apply lt_to_not_le.apply (trans_lt ? k1 ? H2 H3)]
- |apply (eqb_elim (S n1) k2)
- [intro.
- rewrite < H4.
- intros.
- cut (k1 = n1 - (n1 -k1))
- [rewrite > Hcut.
- apply (eq_map_iter_p_transpose p f H H1 g a (n1-k1))
- [cut (k1 \le n1)[auto|auto]
- |assumption
- |rewrite < Hcut.assumption
- |rewrite < Hcut.intros.
- apply (H9 i H10).unfold.auto
- ]
- |apply sym_eq.
- apply plus_to_minus.
- auto.
- ]
- |intros.
- cut ((S n1) \neq k1)
- [apply (bool_elim ? (p (S n1)))
- [intro.
- rewrite > map_iter_P_S_true
- [rewrite > map_iter_P_S_true
- [cut ((transpose k1 k2 (S n1)) = (S n1))
- [rewrite > Hcut1.
- apply eq_f.
- apply (H3 H5)
- [elim (le_to_or_lt_eq ? ? H6)
- [auto
- |absurd (S n1=k2)[apply sym_eq.assumption|assumption]
- ]
- |assumption
- |assumption
- |assumption
- ]
- |unfold transpose.
- rewrite > (not_eq_to_eqb_false ? ? Hcut).
- rewrite > (not_eq_to_eqb_false ? ? H4).
- reflexivity
- ]
- |assumption
- ]
- |assumption
- ]
- |intro.
- rewrite > map_iter_P_S_false
- [rewrite > map_iter_P_S_false
- [apply (H3 H5)
- [elim (le_to_or_lt_eq ? ? H6)
- [auto
- |absurd (S n1=k2)[apply sym_eq.assumption|assumption]
- ]
- |assumption
- |assumption
- |assumption
- ]
- |assumption
- ]
- |assumption
- ]
- ]
- |unfold.intro.
- absurd (k1 < k2)
- [assumption
- |apply le_to_not_lt.
- rewrite < H10.
- assumption
- ]
- ]
- ]
- ]
-qed.
-
-lemma decidable_n:\forall p.\forall n.
-(\forall m. m \le n \to (p m) = false) \lor
-(\exists m. m \le n \land (p m) = true \land
-\forall i. m < i \to i \le n \to (p i) = false).
-intros.
-elim n
- [apply (bool_elim ? (p O))
- [intro.right.
- apply (ex_intro ? ? O).
- split
- [split[apply le_n|assumption]
- |intros.absurd (O<i)[assumption|apply le_to_not_lt.assumption]
- ]
- |intro.left.
- intros.apply (le_n_O_elim m H1).assumption
- ]
- |apply (bool_elim ? (p (S n1)))
- [intro.right.
- apply (ex_intro ? ? (S n1)).
- split
- [split[apply le_n|assumption]
- |intros.absurd (S n1<i)[assumption|apply le_to_not_lt.assumption]
- ]
- |elim H
- [left.
- intros.
- elim (le_to_or_lt_eq m (S n1) H3)
- [apply H1.apply le_S_S_to_le.assumption
- |rewrite > H4.assumption
- ]
- |right.
- elim H1.elim H3.elim H4.
- apply (ex_intro ? ? a).
- split
- [split[apply le_S.assumption|assumption]
- |intros.elim (le_to_or_lt_eq i (S n1) H9)
- [apply H5[assumption|apply le_S_S_to_le.assumption]
- |rewrite > H10.assumption
- ]
- ]
- ]
- ]
- ]
-qed.
-
-lemma decidable_n1:\forall p.\forall n,j. j \le n \to (p j)=true \to
-(\forall m. j < m \to m \le n \to (p m) = false) \lor
-(\exists m. j < m \land m \le n \land (p m) = true \land
-\forall i. m < i \to i \le n \to (p i) = false).
-intros.
-elim (decidable_n p n)
- [absurd ((p j)=true)
- [assumption
- |unfold.intro.
- apply not_eq_true_false.
- rewrite < H3.
- apply H2.assumption
- ]
- |elim H2.clear H2.
- apply (nat_compare_elim j a)
- [intro.
- right.
- apply (ex_intro ? ? a).
- elim H3.clear H3.
- elim H4.clear H4.
- split
- [split
- [split
- [assumption|assumption]
- |assumption
- ]
- |assumption
- ]
- |intro.
- rewrite > H2.
- left.
- elim H3 2.assumption
- |intro.
- absurd (p j = true)
- [assumption
- |unfold.intro.
- apply not_eq_true_false.
- rewrite < H4.
- elim H3.clear H3.
- apply (H6 j H2).assumption
- ]
- ]
- ]
-qed.
-
-lemma decidable_n2:\forall p.\forall n,j. j \le n \to (p j)=true \to
-(\forall m. j < m \to m \le n \to (p m) = false) \lor
-(\exists m. j < m \land m \le n \land (p m) = true \land
-\forall i. j < i \to i < m \to (p i) = false).
-intros 3.
-elim n
- [left.
- apply (le_n_O_elim j H).intros.
- absurd (m \le O)
- [assumption|apply lt_to_not_le.assumption]
- |elim (le_to_or_lt_eq ? ? H1)
- [cut (j \le n1)
- [elim (H Hcut H2)
- [apply (bool_elim ? (p (S n1)))
- [intro.
- right.
- apply (ex_intro ? ? (S n1)).
- split
- [split
- [split
- [assumption|apply le_n]
- |assumption
- ]
- |intros.
- apply (H4 i H6).
- apply le_S_S_to_le.
- assumption
- ]
- |intro.
- left.
- intros.
- elim (le_to_or_lt_eq ? ? H7)
- [apply H4
- [assumption|apply le_S_S_to_le.assumption]
- |rewrite > H8.assumption
- ]
- ]
- |right.
- elim H4.clear H4.
- elim H5.clear H5.
- elim H4.clear H4.
- elim H5.clear H5.
- apply (ex_intro ? ? a).
- split
- [split
- [split[assumption|apply le_S.assumption]
- |assumption
- ]
- |assumption
- ]
- ]
- |apply le_S_S_to_le.
- assumption
- ]
- |left.
- intros.
- absurd (j < m)
- [assumption
- |apply le_to_not_lt.
- rewrite > H3.
- assumption
- ]
- ]
- ]
-qed.
-
-(* tutti d spostare *)
-theorem lt_minus_to_lt_plus:
-\forall n,m,p. n - m < p \to n < m + p.
-intros 2.
-apply (nat_elim2 ? ? ? ? n m)
- [simplify.intros.auto.
- |intros 2.rewrite < minus_n_O.
- intro.assumption
- |intros.
- simplify.
- cut (n1 < m1+p)
- [auto
- |apply H.
- apply H1
- ]
- ]
-qed.
-
-theorem lt_plus_to_lt_minus:
-\forall n,m,p. m \le n \to n < m + p \to n - m < p.
-intros 2.
-apply (nat_elim2 ? ? ? ? n m)
- [simplify.intros 3.
- apply (le_n_O_elim ? H).
- simplify.intros.assumption
- |simplify.intros.assumption.
- |intros.
- simplify.
- apply H
- [apply le_S_S_to_le.assumption
- |apply le_S_S_to_le.apply H2
- ]
- ]
-qed.
-
-theorem minus_m_minus_mn: \forall n,m. n\le m \to n=m-(m-n).
-intros.
-apply sym_eq.
-apply plus_to_minus.
-auto.
-qed.
-
-theorem eq_map_iter_p_transpose2: \forall p.\forall f.associative nat f \to
-symmetric2 nat nat f \to \forall g. \forall a,k,n:nat. O < k \to k \le n \to
-(p (S n) = true) \to (p k) = true
-\to map_iter_P (S n) p g a f = map_iter_P (S n) p (\lambda m. g (transpose k (S n) m)) a f.
-intros 10.
-cut (k = (S n)-(S n -k))
- [generalize in match H3.clear H3.
- generalize in match g.
- generalize in match H2.clear H2.
- rewrite > Hcut.
- (*generalize in match Hcut.clear Hcut.*)
- (* generalize in match H3.clear H3.*)
- (* something wrong here
- rewrite > Hcut in \vdash (?\rarr ?\rarr %). *)
- apply (nat_elim1 (S n - k)).
- intros.
- elim (decidable_n2 p n (S n -m) H4 H6)
- [apply (eq_map_iter_p_transpose1 p f H H1 f1 a)
- [assumption.
- |unfold.auto.
- |apply le_n
- |assumption
- |assumption
- |intros.apply H7
- [assumption|apply le_S_S_to_le.assumption]
- ]
- |elim H7.clear H7.
- elim H8.clear H8.
- elim H7.clear H7.
- elim H8.clear H8.
- apply (trans_eq ? ?
- (map_iter_P (S n) p (\lambda i.f1 (transpose a1 (S n) (transpose (S n -m) a1 i))) a f))
- [apply (trans_eq ? ?
- (map_iter_P (S n) p (\lambda i.f1 (transpose a1 (S n) i)) a f))
- [cut (a1 = (S n -(S n -a1)))
- [rewrite > Hcut1.
- apply H2
- [apply lt_plus_to_lt_minus
- [apply le_S.assumption
- |rewrite < sym_plus.
- apply lt_minus_to_lt_plus.
- assumption
- ]
- |rewrite < Hcut1.
- apply (trans_lt ? (S n -m))[assumption|assumption]
- |rewrite < Hcut1.assumption
- |assumption
- |rewrite < Hcut1.assumption
- ]
- |apply minus_m_minus_mn.
- apply le_S.assumption
- ]
- |apply (eq_map_iter_p_transpose1 p f H H1)
- [assumption
- |assumption
- |apply le_S.assumption
- |assumption
- |assumption
- |assumption
- ]
- ]
- |apply (trans_eq ? ?
- (map_iter_P (S n) p (\lambda i.f1 (transpose a1 (S n) (transpose (S n -m) a1 (transpose (S n -(S n -a1)) (S n) i)))) a f))
- [cut (a1 = (S n) -(S n -a1))
- [apply H2
- [apply lt_plus_to_lt_minus
- [apply le_S.assumption
- |rewrite < sym_plus.
- apply lt_minus_to_lt_plus.
- assumption
- ]
- |rewrite < Hcut1.
- apply (trans_lt ? (S n -m))[assumption|assumption]
- |rewrite < Hcut1.assumption
- |assumption
- |rewrite < Hcut1.assumption
- ]
- |apply minus_m_minus_mn.
- apply le_S.assumption
- ]
- |apply eq_map_iter_P.
- cut (a1 = (S n) -(S n -a1))
- [intros.
- apply eq_f.
- rewrite < Hcut1.
- rewrite < transpose_i_j_j_i.
- rewrite > (transpose_i_j_j_i (S n -m)).
- rewrite > (transpose_i_j_j_i a1 (S n)).
- rewrite > (transpose_i_j_j_i (S n -m)).
- apply sym_eq.
- apply eq_transpose
- [unfold.intro.
- apply (not_le_Sn_n n).
- rewrite < H12.assumption
- |unfold.intro.
- apply (not_le_Sn_n n).
- rewrite > H12.assumption
- |unfold.intro.
- apply (not_le_Sn_n a1).
- rewrite < H12 in \vdash (? (? %) ?).assumption
- ]
- |apply minus_m_minus_mn.
- apply le_S.assumption
- ]
- ]
- ]
- ]
- |apply minus_m_minus_mn.
- apply le_S.assumption
- ]
-qed.
-
-theorem eq_map_iter_p_transpose3: \forall p.\forall f.associative nat f \to
-symmetric2 nat nat f \to \forall g. \forall a,k,n:nat. O < k \to k \le (S n) \to
-(p (S n) = true) \to (p k) = true
-\to map_iter_P (S n) p g a f = map_iter_P (S n) p (\lambda m. g (transpose k (S n) m)) a f.
-intros.
-elim (le_to_or_lt_eq ? ? H3)
- [apply (eq_map_iter_p_transpose2 p f H H1 g a k n H2)
- [apply le_S_S_to_le.assumption|assumption|assumption]
- |rewrite > H6.
- apply eq_map_iter_P.
- intros.
- apply eq_f.apply sym_eq. apply transpose_i_i.
- ]
-qed.
-
-lemma permut_p_O: \forall p.\forall h.\forall n.
-permut_p h p n \to p O = false \to \forall m. (S m) \le n \to p (S m) = true \to O < h(S m).
-intros.unfold permut_p in H.
-apply not_le_to_lt.unfold.intro.
-elim (H (S m) H2 H3).
-elim H5.
-absurd (p (h (S m)) = true)
- [assumption
- |apply (le_n_O_elim ? H4).
- unfold.intro.
- apply not_eq_true_false.
- rewrite < H9.rewrite < H1.reflexivity
- ]
-qed.
-
-theorem eq_map_iter_p_permut: \forall p.\forall f.associative nat f \to
-symmetric2 nat nat f \to \forall n.\forall g. \forall h.\forall a:nat.
-permut_p h p n \to p O = false \to
-map_iter_P n p g a f = map_iter_P n p (compose ? ? ? g h) a f .
-intros 5.
-elim n
- [simplify.reflexivity
- |apply (bool_elim ? (p (S n1)))
- [intro.
- apply (trans_eq ? ? (map_iter_P (S n1) p (\lambda m.g ((transpose (h (S n1)) (S n1)) m)) a f))
- [unfold permut_p in H3.
- elim (H3 (S n1) (le_n ?) H5).
- elim H6. clear H6.
- apply (eq_map_iter_p_transpose3 p f H H1 g a (h(S n1)) n1)
- [apply (permut_p_O ? ? ? H3 H4)
- [apply le_n|assumption]
- |assumption
- |assumption
- |assumption
- ]
- |apply (trans_eq ? ? (map_iter_P (S n1) p (\lambda m.
- (g(transpose (h (S n1)) (S n1)
- (transpose (h (S n1)) (S n1) (h m)))) ) a f))
- [rewrite > (map_iter_P_S_true ? ? ? ? ? H5).
- rewrite > (map_iter_P_S_true ? ? ? ? ? H5).
- apply eq_f2
- [rewrite > transpose_i_j_j.
- rewrite > transpose_i_j_i.
- rewrite > transpose_i_j_j.
- reflexivity
- |apply (H2 (\lambda m.(g(transpose (h (S n1)) (S n1) m))) ?)
- [unfold.intros.
- split
- [split
- [simplify.
- unfold permut_p in H3.
- elim (H3 i (le_S ? ? H6) H7).
- elim H8. clear H8.
- elim (le_to_or_lt_eq ? ? H10)
- [unfold transpose.
- rewrite > (not_eq_to_eqb_false ? ? (lt_to_not_eq ? ? H8)).
- cut (h i \neq h (S n1))
- [rewrite > (not_eq_to_eqb_false ? ? Hcut).
- simplify.
- apply le_S_S_to_le.
- assumption
- |apply H9
- [apply H5
- |apply le_n
- |apply lt_to_not_eq.
- unfold.apply le_S_S.assumption
- ]
- ]
- |rewrite > H8.
- apply (eqb_elim (S n1) (h (S n1)))
- [intro.
- absurd (h i = h (S n1))
- [rewrite > H8.
- assumption
- |apply H9
- [assumption
- |apply le_n
- |apply lt_to_not_eq.
- unfold.apply le_S_S.assumption
- ]
- ]
- |intro.
- unfold transpose.
- rewrite > (not_eq_to_eqb_false ? ? H12).
- rewrite > (eq_to_eqb_true ? ? (refl_eq ? (S n1))).
- simplify.
- elim (H3 (S n1) (le_n ? ) H5).
- elim H13.clear H13.
- elim (le_to_or_lt_eq ? ? H15)
- [apply le_S_S_to_le.assumption
- |apply False_ind.
- apply H12.
- apply sym_eq.assumption
- ]
- ]
- ]
- |simplify.
- unfold permut_p in H3.
- unfold transpose.
- apply (eqb_elim (h i) (S n1))
- [intro.
- apply (eqb_elim (h i) (h (S n1)))
- [intro.simplify.assumption
- |intro.simplify.
- elim (H3 (S n1) (le_n ? ) H5).
- elim H10. assumption
- ]
- |intro.
- apply (eqb_elim (h i) (h (S n1)))
- [intro.simplify.assumption
- |intro.simplify.
- elim (H3 i (le_S ? ? H6) H7).
- elim H10. assumption
- ]
- ]
- ]
- |simplify.intros.unfold Not.intro.
- unfold permut_p in H3.
- elim (H3 i (le_S i ? H6) H7).
- apply (H13 j H8 (le_S j ? H9) H10).
- apply (injective_transpose ? ? ? ? H11)
- ]
- |assumption
- ]
- ]
- |apply eq_map_iter_P.
- intros.
- rewrite > transpose_transpose.reflexivity
- ]
- ]
- |intro.
- rewrite > (map_iter_P_S_false ? ? ? ? ? H5).
- rewrite > (map_iter_P_S_false ? ? ? ? ? H5).
- apply H2
- [unfold permut_p.
- unfold permut_p in H3.
- intros.
- elim (H3 i (le_S i ? H6) H7).
- elim H8.
- split
- [split
- [elim (le_to_or_lt_eq ? ? H10)
- [apply le_S_S_to_le.assumption
- |absurd (p (h i) = true)
- [assumption
- |rewrite > H12.
- rewrite > H5.
- unfold.intro.apply not_eq_true_false.
- apply sym_eq.assumption
- ]
- ]
- |assumption
- ]
- |intros.
- apply H9
- [assumption|apply (le_S ? ? H13)|assumption]
- ]
- |assumption
- ]
- ]
- ]
-qed.
-
projects / helm.git / blobdiff