+++ /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/permutation".
-
-include "nat/compare.ma".
-include "nat/sigma_and_pi.ma".
-
-definition injn: (nat \to nat) \to nat \to Prop \def
-\lambda f:nat \to nat.\lambda n:nat.\forall i,j:nat.
-i \le n \to j \le n \to f i = f j \to i = j.
-
-theorem injn_Sn_n: \forall f:nat \to nat. \forall n:nat.
-injn f (S n) \to injn f n.unfold injn.
-intros.apply H.
-apply le_S.assumption.
-apply le_S.assumption.
-assumption.
-qed.
-
-theorem injective_to_injn: \forall f:nat \to nat. \forall n:nat.
-injective nat nat f \to injn f n.
-unfold injective.unfold injn.intros.apply H.assumption.
-qed.
-
-definition permut : (nat \to nat) \to nat \to Prop
-\def \lambda f:nat \to nat. \lambda m:nat.
-(\forall i:nat. i \le m \to f i \le m )\land injn f m.
-
-theorem permut_O_to_eq_O: \forall h:nat \to nat.
-permut h O \to (h O) = O.
-intros.unfold permut in H.
-elim H.apply sym_eq.apply le_n_O_to_eq.
-apply H1.apply le_n.
-qed.
-
-theorem permut_S_to_permut: \forall f:nat \to nat. \forall m:nat.
-permut f (S m) \to f (S m) = (S m) \to permut f m.
-unfold permut.intros.
-elim H.
-split.intros.
-cut (f i < S m \lor f i = S m).
-elim Hcut.
-apply le_S_S_to_le.assumption.
-apply False_ind.
-apply (not_le_Sn_n m).
-cut ((S m) = i).
-rewrite > Hcut1.assumption.
-apply H3.apply le_n.apply le_S.assumption.
-rewrite > H5.assumption.
-apply le_to_or_lt_eq.apply H2.apply le_S.assumption.
-apply (injn_Sn_n f m H3).
-qed.
-
-(* transpositions *)
-
-definition transpose : nat \to nat \to nat \to nat \def
-\lambda i,j,n:nat.
-match eqb n i with
- [ true \Rightarrow j
- | false \Rightarrow
- match eqb n j with
- [ true \Rightarrow i
- | false \Rightarrow n]].
-
-lemma transpose_i_j_i: \forall i,j:nat. transpose i j i = j.
-intros.unfold transpose.
-rewrite > (eqb_n_n i).simplify. reflexivity.
-qed.
-
-lemma transpose_i_j_j: \forall i,j:nat. transpose i j j = i.
-intros.unfold transpose.
-apply (eqb_elim j i).simplify.intro.assumption.
-rewrite > (eqb_n_n j).simplify.
-intros. reflexivity.
-qed.
-
-theorem transpose_i_i: \forall i,n:nat. (transpose i i n) = n.
-intros.unfold transpose.
-apply (eqb_elim n i).
-intro.simplify.apply sym_eq. assumption.
-intro.simplify.reflexivity.
-qed.
-
-theorem transpose_i_j_j_i: \forall i,j,n:nat.
-transpose i j n = transpose j i n.
-intros.unfold transpose.
-apply (eqb_elim n i).
-apply (eqb_elim n j).
-intros. simplify.rewrite < H. rewrite < H1.
-reflexivity.
-intros.simplify.reflexivity.
-apply (eqb_elim n j).
-intros.simplify.reflexivity.
-intros.simplify.reflexivity.
-qed.
-
-theorem transpose_transpose: \forall i,j,n:nat.
-(transpose i j (transpose i j n)) = n.
-intros.unfold transpose. unfold transpose.
-apply (eqb_elim n i).simplify.
-intro.
-apply (eqb_elim j i).
-simplify.intros.rewrite > H. rewrite > H1.reflexivity.
-rewrite > (eqb_n_n j).simplify.intros.
-apply sym_eq.
-assumption.
-apply (eqb_elim n j).simplify.
-rewrite > (eqb_n_n i).intros.simplify.
-apply sym_eq. assumption.
-simplify.intros.
-rewrite > (not_eq_to_eqb_false n i H1).
-rewrite > (not_eq_to_eqb_false n j H).
-simplify.reflexivity.
-qed.
-
-theorem injective_transpose : \forall i,j:nat.
-injective nat nat (transpose i j).
-unfold injective.
-intros.
-rewrite < (transpose_transpose i j x).
-rewrite < (transpose_transpose i j y).
-apply eq_f.assumption.
-qed.
-
-variant inj_transpose: \forall i,j,n,m:nat.
-transpose i j n = transpose i j m \to n = m \def
-injective_transpose.
-
-theorem permut_transpose: \forall i,j,n:nat. i \le n \to j \le n \to
-permut (transpose i j) n.
-unfold permut.intros.
-split.unfold transpose.
-intros.
-elim (eqb i1 i).simplify.assumption.
-elim (eqb i1 j).simplify.assumption.
-simplify.assumption.
-apply (injective_to_injn (transpose i j) n).
-apply injective_transpose.
-qed.
-
-theorem permut_fg: \forall f,g:nat \to nat. \forall n:nat.
-permut f n \to permut g n \to permut (\lambda m.(f(g m))) n.
-unfold permut. intros.
-elim H.elim H1.
-split.intros.simplify.apply H2.
-apply H4.assumption.
-simplify.intros.
-apply H5.assumption.assumption.
-apply H3.apply H4.assumption.apply H4.assumption.
-assumption.
-qed.
-
-theorem permut_transpose_l:
-\forall f:nat \to nat. \forall m,i,j:nat.
-i \le m \to j \le m \to permut f m \to permut (\lambda n.transpose i j (f n)) m.
-intros.apply (permut_fg (transpose i j) f m ? ?).
-apply permut_transpose.assumption.assumption.
-assumption.
-qed.
-
-theorem permut_transpose_r:
-\forall f:nat \to nat. \forall m,i,j:nat.
-i \le m \to j \le m \to permut f m \to permut (\lambda n.f (transpose i j n)) m.
-intros.apply (permut_fg f (transpose i j) m ? ?).
-assumption.apply permut_transpose.assumption.assumption.
-qed.
-
-theorem eq_transpose : \forall i,j,k,n:nat. \lnot j=i \to
- \lnot i=k \to \lnot j=k \to
-transpose i j n = transpose i k (transpose k j (transpose i k n)).
-(* uffa: triplo unfold? *)
-intros.unfold transpose.unfold transpose.unfold transpose.
-apply (eqb_elim n i).intro.
-simplify.rewrite > (eqb_n_n k).
-simplify.rewrite > (not_eq_to_eqb_false j i H).
-rewrite > (not_eq_to_eqb_false j k H2).
-reflexivity.
-intro.apply (eqb_elim n j).
-intro.
-cut (\lnot n = k).
-cut (\lnot n = i).
-rewrite > (not_eq_to_eqb_false n k Hcut).
-simplify.
-rewrite > (not_eq_to_eqb_false n k Hcut).
-rewrite > (eq_to_eqb_true n j H4).
-simplify.
-rewrite > (not_eq_to_eqb_false k i).
-rewrite > (eqb_n_n k).
-simplify.reflexivity.
-unfold Not.intro.apply H1.apply sym_eq.assumption.
-assumption.
-unfold Not.intro.apply H2.apply (trans_eq ? ? n).
-apply sym_eq.assumption.assumption.
-intro.apply (eqb_elim n k).intro.
-simplify.
-rewrite > (not_eq_to_eqb_false i k H1).
-rewrite > (not_eq_to_eqb_false i j).
-simplify.
-rewrite > (eqb_n_n i).
-simplify.assumption.
-unfold Not.intro.apply H.apply sym_eq.assumption.
-intro.simplify.
-rewrite > (not_eq_to_eqb_false n k H5).
-rewrite > (not_eq_to_eqb_false n j H4).
-simplify.
-rewrite > (not_eq_to_eqb_false n i H3).
-rewrite > (not_eq_to_eqb_false n k H5).
-simplify.reflexivity.
-qed.
-
-theorem permut_S_to_permut_transpose: \forall f:nat \to nat.
-\forall m:nat. permut f (S m) \to permut (\lambda n.transpose (f (S m)) (S m)
-(f n)) m.
-unfold permut.intros.
-elim H.
-split.intros.simplify.unfold transpose.
-apply (eqb_elim (f i) (f (S m))).
-intro.apply False_ind.
-cut (i = (S m)).
-apply (not_le_Sn_n m).
-rewrite < Hcut.assumption.
-apply H2.apply le_S.assumption.apply le_n.assumption.
-intro.simplify.
-apply (eqb_elim (f i) (S m)).
-intro.
-cut (f (S m) \lt (S m) \lor f (S m) = (S m)).
-elim Hcut.apply le_S_S_to_le.assumption.
-apply False_ind.apply H4.rewrite > H6.assumption.
-apply le_to_or_lt_eq.apply H1.apply le_n.
-intro.simplify.
-cut (f i \lt (S m) \lor f i = (S m)).
-elim Hcut.apply le_S_S_to_le.assumption.
-apply False_ind.apply H5.assumption.
-apply le_to_or_lt_eq.apply H1.apply le_S.assumption.
-unfold injn.intros.
-apply H2.apply le_S.assumption.apply le_S.assumption.
-apply (inj_transpose (f (S m)) (S m)).
-apply H5.
-qed.
-
-(* bounded bijectivity *)
-
-definition bijn : (nat \to nat) \to nat \to Prop \def
-\lambda f:nat \to nat. \lambda n. \forall m:nat. m \le n \to
-ex nat (\lambda p. p \le n \land f p = m).
-
-theorem eq_to_bijn: \forall f,g:nat\to nat. \forall n:nat.
-(\forall i:nat. i \le n \to (f i) = (g i)) \to
-bijn f n \to bijn g n.
-intros 4.unfold bijn.
-intros.elim (H1 m).
-apply (ex_intro ? ? a).
-rewrite < (H a).assumption.
-elim H3.assumption.assumption.
-qed.
-
-theorem bijn_Sn_n: \forall f:nat \to nat. \forall n:nat.
-bijn f (S n) \to f (S n) = (S n) \to bijn f n.
-unfold bijn.intros.elim (H m).
-elim H3.
-apply (ex_intro ? ? a).split.
-cut (a < S n \lor a = S n).
-elim Hcut.apply le_S_S_to_le.assumption.
-apply False_ind.
-apply (not_le_Sn_n n).
-rewrite < H1.rewrite < H6.rewrite > H5.assumption.
-apply le_to_or_lt_eq.assumption.assumption.
-apply le_S.assumption.
-qed.
-
-theorem bijn_n_Sn: \forall f:nat \to nat. \forall n:nat.
-bijn f n \to f (S n) = (S n) \to bijn f (S n).
-unfold bijn.intros.
-cut (m < S n \lor m = S n).
-elim Hcut.
-elim (H m).
-elim H4.
-apply (ex_intro ? ? a).split.
-apply le_S.assumption.assumption.
-apply le_S_S_to_le.assumption.
-apply (ex_intro ? ? (S n)).
-split.apply le_n.
-rewrite > H3.assumption.
-apply le_to_or_lt_eq.assumption.
-qed.
-
-theorem bijn_fg: \forall f,g:nat\to nat. \forall n:nat.
-bijn f n \to bijn g n \to bijn (\lambda p.f(g p)) n.
-unfold bijn.
-intros.simplify.
-elim (H m).elim H3.
-elim (H1 a).elim H6.
-apply (ex_intro ? ? a1).
-split.assumption.
-rewrite > H8.assumption.
-assumption.assumption.
-qed.
-
-theorem bijn_transpose : \forall n,i,j. i \le n \to j \le n \to
-bijn (transpose i j) n.
-intros.unfold bijn.unfold transpose.intros.
-cut (m = i \lor \lnot m = i).
-elim Hcut.
-apply (ex_intro ? ? j).
-split.assumption.
-apply (eqb_elim j i).
-intro.simplify.rewrite > H3.rewrite > H4.reflexivity.
-rewrite > (eqb_n_n j).simplify.
-intros. apply sym_eq.assumption.
-cut (m = j \lor \lnot m = j).
-elim Hcut1.
-apply (ex_intro ? ? i).
-split.assumption.
-rewrite > (eqb_n_n i).simplify.
-apply sym_eq. assumption.
-apply (ex_intro ? ? m).
-split.assumption.
-rewrite > (not_eq_to_eqb_false m i).
-rewrite > (not_eq_to_eqb_false m j).
-simplify. reflexivity.
-assumption.
-assumption.
-apply (decidable_eq_nat m j).
-apply (decidable_eq_nat m i).
-qed.
-
-theorem bijn_transpose_r: \forall f:nat\to nat.\forall n,i,j. i \le n \to j \le n \to
-bijn f n \to bijn (\lambda p.f (transpose i j p)) n.
-intros.
-apply (bijn_fg f ?).assumption.
-apply (bijn_transpose n i j).assumption.assumption.
-qed.
-
-theorem bijn_transpose_l: \forall f:nat\to nat.\forall n,i,j. i \le n \to j \le n \to
-bijn f n \to bijn (\lambda p.transpose i j (f p)) n.
-intros.
-apply (bijn_fg ? f).
-apply (bijn_transpose n i j).assumption.assumption.
-assumption.
-qed.
-
-theorem permut_to_bijn: \forall n:nat.\forall f:nat\to nat.
-permut f n \to bijn f n.
-intro.
-elim n.unfold bijn.intros.
-apply (ex_intro ? ? m).
-split.assumption.
-apply (le_n_O_elim m ? (\lambda p. f p = p)).
-assumption.unfold permut in H.
-elim H.apply sym_eq. apply le_n_O_to_eq.apply H2.apply le_n.
-apply (eq_to_bijn (\lambda p.
-(transpose (f (S n1)) (S n1)) (transpose (f (S n1)) (S n1) (f p))) f).
-intros.apply transpose_transpose.
-apply (bijn_fg (transpose (f (S n1)) (S n1))).
-apply bijn_transpose.
-unfold permut in H1.
-elim H1.apply H2.apply le_n.apply le_n.
-apply bijn_n_Sn.
-apply H.
-apply permut_S_to_permut_transpose.
-assumption.unfold transpose.
-rewrite > (eqb_n_n (f (S n1))).simplify.reflexivity.
-qed.
-
-let rec invert_permut n f m \def
- match eqb m (f n) with
- [true \Rightarrow n
- |false \Rightarrow
- match n with
- [O \Rightarrow O
- |(S p) \Rightarrow invert_permut p f m]].
-
-theorem invert_permut_f: \forall f:nat \to nat. \forall n,m:nat.
-m \le n \to injn f n\to invert_permut n f (f m) = m.
-intros 4.
-elim H.
-apply (nat_case1 m).
-intro.simplify.
-rewrite > (eqb_n_n (f O)).simplify.reflexivity.
-intros.simplify.
-rewrite > (eqb_n_n (f (S m1))).simplify.reflexivity.
-simplify.
-rewrite > (not_eq_to_eqb_false (f m) (f (S n1))).
-simplify.apply H2.
-apply injn_Sn_n. assumption.
-unfold Not.intro.absurd (m = S n1).
-apply H3.apply le_S.assumption.apply le_n.assumption.
-unfold Not.intro.
-apply (not_le_Sn_n n1).rewrite < H5.assumption.
-qed.
-
-theorem injective_invert_permut: \forall f:nat \to nat. \forall n:nat.
-permut f n \to injn (invert_permut n f) n.
-intros.
-unfold injn.intros.
-cut (bijn f n).
-unfold bijn in Hcut.
-generalize in match (Hcut i H1).intro.
-generalize in match (Hcut j H2).intro.
-elim H4.elim H6.
-elim H5.elim H9.
-rewrite < H8.
-rewrite < H11.
-apply eq_f.
-rewrite < (invert_permut_f f n a).
-rewrite < (invert_permut_f f n a1).
-rewrite > H8.
-rewrite > H11.
-assumption.assumption.
-unfold permut in H.elim H. assumption.
-assumption.
-unfold permut in H.elim H. assumption.
-apply permut_to_bijn.assumption.
-qed.
-
-theorem permut_invert_permut: \forall f:nat \to nat. \forall n:nat.
-permut f n \to permut (invert_permut n f) n.
-intros.unfold permut.split.
-intros.simplify.elim n.
-simplify.elim (eqb i (f O)).simplify.apply le_n.simplify.apply le_n.
-simplify.elim (eqb i (f (S n1))).simplify.apply le_n.
-simplify.apply le_S. assumption.
-apply injective_invert_permut.assumption.
-qed.
-
-theorem f_invert_permut: \forall f:nat \to nat. \forall n,m:nat.
-m \le n \to permut f n\to f (invert_permut n f m) = m.
-intros.
-apply (injective_invert_permut f n H1).
-unfold permut in H1.elim H1.
-apply H2.
-cut (permut (invert_permut n f) n).unfold permut in Hcut.
-elim Hcut.apply H4.assumption.
-apply permut_invert_permut.assumption.assumption.
-apply invert_permut_f.
-cut (permut (invert_permut n f) n).unfold permut in Hcut.
-elim Hcut.apply H2.assumption.
-apply permut_invert_permut.assumption.
-unfold permut in H1.elim H1.assumption.
-qed.
-
-theorem permut_n_to_eq_n: \forall h:nat \to nat.\forall n:nat.
-permut h n \to (\forall m:nat. m < n \to h m = m) \to h n = n.
-intros.unfold permut in H.elim H.
-cut (invert_permut n h n < n \lor invert_permut n h n = n).
-elim Hcut.
-rewrite < (f_invert_permut h n n) in \vdash (? ? ? %).
-apply eq_f.
-rewrite < (f_invert_permut h n n) in \vdash (? ? % ?).
-apply H1.assumption.apply le_n.assumption.apply le_n.assumption.
-rewrite < H4 in \vdash (? ? % ?).
-apply (f_invert_permut h).apply le_n.assumption.
-apply le_to_or_lt_eq.
-cut (permut (invert_permut n h) n).
-unfold permut in Hcut.elim Hcut.
-apply H4.apply le_n.
-apply permut_invert_permut.assumption.
-qed.
-
-theorem permut_n_to_le: \forall h:nat \to nat.\forall k,n:nat.
-k \le n \to permut h n \to (\forall m:nat. m < k \to h m = m) \to
-\forall j. k \le j \to j \le n \to k \le h j.
-intros.unfold permut in H1.elim H1.
-cut (h j < k \lor \not(h j < k)).
-elim Hcut.absurd (k \le j).assumption.
-apply lt_to_not_le.
-cut (h j = j).rewrite < Hcut1.assumption.
-apply H6.apply H5.assumption.assumption.
-apply H2.assumption.
-apply not_lt_to_le.assumption.
-apply (decidable_lt (h j) k).
-qed.
-
-(* applications *)
-
-let rec map_iter_i k (g:nat \to nat) f (i:nat) \def
- match k with
- [ O \Rightarrow g i
- | (S k) \Rightarrow f (g (S (k+i))) (map_iter_i k g f i)].
-
-theorem eq_map_iter_i: \forall g1,g2:nat \to nat.
-\forall f:nat \to nat \to nat. \forall n,i:nat.
-(\forall m:nat. i\le m \to m \le n+i \to g1 m = g2 m) \to
-map_iter_i n g1 f i = map_iter_i n g2 f i.
-intros 5.elim n.simplify.apply H.apply le_n.
-apply le_n.simplify.apply eq_f2.apply H1.simplify.
-apply le_S.apply le_plus_n.simplify.apply le_n.
-apply H.intros.apply H1.assumption.simplify.apply le_S.assumption.
-qed.
-
-(* map_iter examples *)
-
-theorem eq_map_iter_i_sigma: \forall g:nat \to nat. \forall n,m:nat.
-map_iter_i n g plus m = sigma n g m.
-intros.elim n.simplify.reflexivity.
-simplify.
-apply eq_f.assumption.
-qed.
-
-theorem eq_map_iter_i_pi: \forall g:nat \to nat. \forall n,m:nat.
-map_iter_i n g times m = pi n g m.
-intros.elim n.simplify.reflexivity.
-simplify.
-apply eq_f.assumption.
-qed.
-
-theorem eq_map_iter_i_fact: \forall n:nat.
-map_iter_i n (\lambda m.m) times (S O) = (S n)!.
-intros.elim n.
-simplify.reflexivity.
-change with
-(((S n1)+(S O))*(map_iter_i n1 (\lambda m.m) times (S O)) = (S(S n1))*(S n1)!).
-rewrite < plus_n_Sm.rewrite < plus_n_O.
-apply eq_f.assumption.
-qed.
-
-theorem eq_map_iter_i_transpose_l : \forall f:nat\to nat \to nat.associative nat f \to
-symmetric2 nat nat f \to \forall g:nat \to nat. \forall n,k:nat.
-map_iter_i (S k) g f n = map_iter_i (S k) (\lambda m. g (transpose (k+n) (S k+n) m)) f n.
-intros.apply (nat_case1 k).
-intros.simplify.
-change with
-(f (g (S n)) (g n) =
-f (g (transpose n (S n) (S n))) (g (transpose n (S n) n))).
-rewrite > transpose_i_j_i.
-rewrite > transpose_i_j_j.
-apply H1.
-intros.
-change with
-(f (g (S (S (m+n)))) (f (g (S (m+n))) (map_iter_i m g f n)) =
-f (g (transpose (S m + n) (S (S m) + n) (S (S m)+n)))
-(f (g (transpose (S m + n) (S (S m) + n) (S m+n)))
-(map_iter_i m (\lambda m1. g (transpose (S m+n) (S (S m)+n) m1)) f n))).
-rewrite > transpose_i_j_i.
-rewrite > transpose_i_j_j.
-rewrite < H.
-rewrite < H.
-rewrite < (H1 (g (S m + n))).
-apply eq_f.
-apply eq_map_iter_i.
-intros.simplify.unfold transpose.
-rewrite > (not_eq_to_eqb_false m1 (S m+n)).
-rewrite > (not_eq_to_eqb_false m1 (S (S m)+n)).
-simplify.
-reflexivity.
-apply (lt_to_not_eq m1 (S ((S m)+n))).
-unfold lt.apply le_S_S.change with (m1 \leq S (m+n)).apply le_S.assumption.
-apply (lt_to_not_eq m1 (S m+n)).
-simplify.unfold lt.apply le_S_S.assumption.
-qed.
-
-theorem eq_map_iter_i_transpose_i_Si : \forall f:nat\to nat \to nat.associative nat f \to
-symmetric2 nat nat f \to \forall g:nat \to nat. \forall n,k,i:nat. n \le i \to i \le k+n \to
-map_iter_i (S k) g f n = map_iter_i (S k) (\lambda m. g (transpose i (S i) m)) f n.
-intros 6.elim k.cut (i=n).
-rewrite > Hcut.
-apply (eq_map_iter_i_transpose_l f H H1 g n O).
-apply antisymmetric_le.assumption.assumption.
-cut (i < S n1 + n \lor i = S n1 + n).
-elim Hcut.
-change with
-(f (g (S (S n1)+n)) (map_iter_i (S n1) g f n) =
-f (g (transpose i (S i) (S (S n1)+n))) (map_iter_i (S n1) (\lambda m. g (transpose i (S i) m)) f n)).
-apply eq_f2.unfold transpose.
-rewrite > (not_eq_to_eqb_false (S (S n1)+n) i).
-rewrite > (not_eq_to_eqb_false (S (S n1)+n) (S i)).
-simplify.reflexivity.
-simplify.unfold Not.intro.
-apply (lt_to_not_eq i (S n1+n)).assumption.
-apply inj_S.apply sym_eq. assumption.
-simplify.unfold Not.intro.
-apply (lt_to_not_eq i (S (S n1+n))).simplify.unfold lt.
-apply le_S_S.assumption.
-apply sym_eq. assumption.
-apply H2.assumption.apply le_S_S_to_le.
-assumption.
-rewrite > H5.
-apply (eq_map_iter_i_transpose_l f H H1 g n (S n1)).
-apply le_to_or_lt_eq.assumption.
-qed.
-
-theorem eq_map_iter_i_transpose:
-\forall f:nat\to nat \to nat.
-associative nat f \to symmetric2 nat nat f \to \forall n,k,o:nat.
-\forall g:nat \to nat. \forall i:nat. n \le i \to S (o + i) \le S k+n \to
-map_iter_i (S k) g f n = map_iter_i (S k) (\lambda m. g (transpose i (S(o + i)) m)) f n.
-intros 6.
-apply (nat_elim1 o).
-intro.
-apply (nat_case m ?).
-intros.
-apply (eq_map_iter_i_transpose_i_Si ? H H1).
-exact H3.apply le_S_S_to_le.assumption.
-intros.
-apply (trans_eq ? ? (map_iter_i (S k) (\lambda m. g (transpose i (S(m1 + i)) m)) f n)).
-apply H2.
-unfold lt. apply le_n.assumption.
-apply (trans_le ? (S(S (m1+i)))).
-apply le_S.apply le_n.assumption.
-apply (trans_eq ? ? (map_iter_i (S k) (\lambda m. g
-(transpose i (S(m1 + i)) (transpose (S(m1 + i)) (S(S(m1 + i))) m))) f n)).
-apply (H2 O ? ? (S(m1+i))).
-unfold lt.apply le_S_S.apply le_O_n.
-apply (trans_le ? i).assumption.
-change with (i \le (S m1)+i).apply le_plus_n.
-exact H4.
-apply (trans_eq ? ? (map_iter_i (S k) (\lambda m. g
-(transpose i (S(m1 + i))
-(transpose (S(m1 + i)) (S(S(m1 + i)))
-(transpose i (S(m1 + i)) m)))) f n)).
-apply (H2 m1).
-unfold lt. apply le_n.assumption.
-apply (trans_le ? (S(S (m1+i)))).
-apply le_S.apply le_n.assumption.
-apply eq_map_iter_i.
-intros.apply eq_f.
-apply sym_eq. apply eq_transpose.
-unfold Not. intro.
-apply (not_le_Sn_n i).
-rewrite < H7 in \vdash (? ? %).
-apply le_S_S.apply le_S.
-apply le_plus_n.
-unfold Not. intro.
-apply (not_le_Sn_n i).
-rewrite > H7 in \vdash (? ? %).
-apply le_S_S.
-apply le_plus_n.
-unfold Not. intro.
-apply (not_eq_n_Sn (S m1+i)).
-apply sym_eq.assumption.
-qed.
-
-theorem eq_map_iter_i_transpose1: \forall f:nat\to nat \to nat.associative nat f \to
-symmetric2 nat nat f \to \forall n,k,i,j:nat.
-\forall g:nat \to nat. n \le i \to i < j \to j \le S k+n \to
-map_iter_i (S k) g f n = map_iter_i (S k) (\lambda m. g (transpose i j m)) f n.
-intros.
-simplify in H3.
-cut ((S i) < j \lor (S i) = j).
-elim Hcut.
-cut (j = S ((j - (S i)) + i)).
-rewrite > Hcut1.
-apply (eq_map_iter_i_transpose f H H1 n k (j - (S i)) g i).
-assumption.
-rewrite < Hcut1.assumption.
-rewrite > plus_n_Sm.
-apply plus_minus_m_m.apply lt_to_le.assumption.
-rewrite < H5.
-apply (eq_map_iter_i_transpose_i_Si f H H1 g).
-simplify.
-assumption.apply le_S_S_to_le.
-apply (trans_le ? j).assumption.assumption.
-apply le_to_or_lt_eq.assumption.
-qed.
-
-theorem eq_map_iter_i_transpose2: \forall f:nat\to nat \to nat.associative nat f \to
-symmetric2 nat nat f \to \forall n,k,i,j:nat.
-\forall g:nat \to nat. n \le i \to i \le (S k+n) \to n \le j \to j \le (S k+n) \to
-map_iter_i (S k) g f n = map_iter_i (S k) (\lambda m. g (transpose i j m)) f n.
-intros.
-apply (nat_compare_elim i j).
-intro.apply (eq_map_iter_i_transpose1 f H H1 n k i j g H2 H6 H5).
-intro.rewrite > H6.
-apply eq_map_iter_i.intros.
-rewrite > (transpose_i_i j).reflexivity.
-intro.
-apply (trans_eq ? ? (map_iter_i (S k) (\lambda m:nat.g (transpose j i m)) f n)).
-apply (eq_map_iter_i_transpose1 f H H1 n k j i g H4 H6 H3).
-apply eq_map_iter_i.
-intros.apply eq_f.apply transpose_i_j_j_i.
-qed.
-
-theorem permut_to_eq_map_iter_i:\forall f:nat\to nat \to nat.associative nat f \to
-symmetric2 nat nat f \to \forall k,n:nat.\forall g,h:nat \to nat.
-permut h (k+n) \to (\forall m:nat. m \lt n \to h m = m) \to
-map_iter_i k g f n = map_iter_i k (\lambda m.g(h m)) f n.
-intros 4.elim k.
-simplify.rewrite > (permut_n_to_eq_n h).reflexivity.assumption.assumption.
-apply (trans_eq ? ? (map_iter_i (S n) (\lambda m.g ((transpose (h (S n+n1)) (S n+n1)) m)) f n1)).
-unfold permut in H3.
-elim H3.
-apply (eq_map_iter_i_transpose2 f H H1 n1 n ? ? g).
-apply (permut_n_to_le h n1 (S n+n1)).
-apply le_plus_n.assumption.assumption.apply le_plus_n.apply le_n.
-apply H5.apply le_n.apply le_plus_n.apply le_n.
-apply (trans_eq ? ? (map_iter_i (S n) (\lambda m.
-(g(transpose (h (S n+n1)) (S n+n1)
-(transpose (h (S n+n1)) (S n+n1) (h m)))) )f n1)).
-change with
-(f (g (transpose (h (S n+n1)) (S n+n1) (S n+n1)))
-(map_iter_i n (\lambda m.
-g (transpose (h (S n+n1)) (S n+n1) m)) f n1)
-=
-f
-(g(transpose (h (S n+n1)) (S n+n1)
-(transpose (h (S n+n1)) (S n+n1) (h (S n+n1)))))
-(map_iter_i n
-(\lambda m.
-(g(transpose (h (S n+n1)) (S n+n1)
-(transpose (h (S n+n1)) (S n+n1) (h m))))) f n1)).
-apply eq_f2.apply eq_f.
-rewrite > transpose_i_j_j.
-rewrite > transpose_i_j_i.
-rewrite > transpose_i_j_j.reflexivity.
-apply (H2 n1 (\lambda m.(g(transpose (h (S n+n1)) (S n+n1) m)))).
-apply permut_S_to_permut_transpose.
-assumption.
-intros.
-unfold transpose.
-rewrite > (not_eq_to_eqb_false (h m) (h (S n+n1))).
-rewrite > (not_eq_to_eqb_false (h m) (S n+n1)).
-simplify.apply H4.assumption.
-rewrite > H4.
-apply lt_to_not_eq.apply (trans_lt ? n1).assumption.
-simplify.unfold lt.apply le_S_S.apply le_plus_n.assumption.
-unfold permut in H3.elim H3.
-simplify.unfold Not.intro.
-apply (lt_to_not_eq m (S n+n1)).apply (trans_lt ? n1).assumption.
-simplify.unfold lt.apply le_S_S.apply le_plus_n.
-unfold injn in H7.
-apply (H7 m (S n+n1)).apply (trans_le ? n1).
-apply lt_to_le.assumption.apply le_plus_n.apply le_n.
-assumption.
-apply eq_map_iter_i.intros.
-rewrite > transpose_transpose.reflexivity.
-qed.
\ No newline at end of file
projects / helm.git / blobdiff