X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fiteration.ma;fp=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fiteration.ma;h=0000000000000000000000000000000000000000;hb=f79585fe2f19c4a545938e189439d87b2611a47a;hp=404cbbded789be21dd043010069ad4adf0c94c76;hpb=ff5d15158c83a1f45d78daf99f22de83aed3eab0;p=helm.git diff --git a/helm/software/matita/library/nat/iteration.ma b/helm/software/matita/library/nat/iteration.ma deleted file mode 100644 index 404cbbded..000000000 --- a/helm/software/matita/library/nat/iteration.ma +++ /dev/null @@ -1,904 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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 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. -