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
-(\forall j:nat. f i = f j \to i = j)).
+(\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 O.apply sym_eq.apply le_n_O_to_eq.
-assumption.apply le_n.
+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 i.
-elim H (S m).split.
-cut f i < S m \lor f i = S m.
+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.
+apply (not_le_Sn_n m).
+cut ((S m) = i).
rewrite > Hcut1.assumption.
-apply H6.rewrite > H7.assumption.
-apply le_to_or_lt_eq.assumption.
-assumption.apply le_n.
-apply le_S.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 *)
lemma transpose_i_j_i: \forall i,j:nat. transpose i j i = j.
intros.unfold transpose.
-rewrite > eqb_n_n i.simplify. reflexivity.
+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.
+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.
+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.
+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.
+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.
+apply (eqb_elim n i).simplify.
intro.
-apply eqb_elim j i.
+apply (eqb_elim j i).
simplify.intros.rewrite > H. rewrite > H1.reflexivity.
-rewrite > eqb_n_n j.simplify.intros.
+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 (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.
+rewrite > (not_eq_to_eqb_false n i H1).
+rewrite > (not_eq_to_eqb_false n j H).
simplify.reflexivity.
qed.
injective nat nat (transpose i j).
unfold injective.
intros.
-rewrite < transpose_transpose i j x.
-rewrite < transpose_transpose i j y.
+rewrite < (transpose_transpose i j x).
+rewrite < (transpose_transpose i j y).
apply eq_f.assumption.
qed.
permut (transpose i j) n.
unfold permut.intros.
split.unfold transpose.
-elim eqb i1 i.simplify.assumption.
-elim eqb i1 j.simplify.assumption.
+intros.
+elim (eqb i1 i).simplify.assumption.
+elim (eqb i1 j).simplify.assumption.
simplify.assumption.
-apply inj_transpose.
+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.
-split.simplify.elim H1 i.
-elim H (g i).assumption.assumption.assumption.
-intro.simplify.intros.
-elim H1 i.apply H5.
-elim H (g i).
-apply H7.assumption.assumption.assumption.
+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 ? ?.
+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 ? ?.
+intros.apply (permut_fg f (transpose i j) m ? ?).
assumption.apply permut_transpose.assumption.assumption.
qed.
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.
+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.apply (eqb_elim n j).
intro.
-cut \lnot n = k.
-cut \lnot n = i.
-rewrite > not_eq_to_eqb_false n k Hcut.
+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.
+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.
+rewrite > (not_eq_to_eqb_false k i).
+rewrite > (eqb_n_n k).
simplify.reflexivity.
-simplify.intro.apply H1.apply sym_eq.assumption.
+unfold Not.intro.apply H1.apply sym_eq.assumption.
assumption.
-simplify.intro.apply H2.apply trans_eq ? ? n.
+unfold Not.intro.apply H2.apply (trans_eq ? ? n).
apply sym_eq.assumption.assumption.
-intro.apply eqb_elim n k.intro.
+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.
+rewrite > (not_eq_to_eqb_false i k H1).
+rewrite > (not_eq_to_eqb_false i j).
simplify.
-rewrite > eqb_n_n i.
+rewrite > (eqb_n_n i).
simplify.assumption.
-simplify.intro.apply H.apply sym_eq.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.
+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.
+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.
-intros.
-cut permut (\lambda n.transpose (f (S m)) (S m) (f n)) (S m).
-apply permut_S_to_permut.assumption.
-simplify.
-apply eqb_elim (f (S m)) (f (S m)).simplify.
-intro.reflexivity.
-intro.apply False_ind.apply H1.reflexivity.
-apply permut_transpose_l.
-unfold permut in H.
-elim H (S m).
-assumption.apply le_n.
-apply le_n.
-assumption.
+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 *)
(\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.
+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.
+unfold bijn.intros.elim (H m).
elim H3.
-apply ex_intro ? ? a.split.
-cut a < S n \lor a = S n.
+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.
+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.
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.
+cut (m < S n \lor m = S n).
elim Hcut.
-elim H m.
+elim (H m).
elim H4.
-apply ex_intro ? ? a.split.
+apply (ex_intro ? ? a).split.
apply le_S.assumption.assumption.
apply le_S_S_to_le.assumption.
-apply ex_intro ? ? (S n).
+apply (ex_intro ? ? (S n)).
split.apply le_n.
rewrite > H3.assumption.
apply le_to_or_lt_eq.assumption.
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.
+elim (H m).elim H3.
+elim (H1 a).elim H6.
+apply (ex_intro ? ? a1).
split.assumption.
rewrite > H8.assumption.
assumption.assumption.
theorem bijn_transpose : \forall n,i,j. i \le n \to j \le n \to
bijn (transpose i j) n.
-intros.simplify.intros.
-cut m = i \lor \lnot m = i.
+intros.unfold bijn.unfold transpose.intros.
+cut (m = i \lor \lnot m = i).
elim Hcut.
-apply ex_intro ? ? j.
+apply (ex_intro ? ? j).
split.assumption.
-apply eqb_elim j i.
+apply (eqb_elim j i).
intro.simplify.rewrite > H3.rewrite > H4.reflexivity.
-rewrite > eqb_n_n j.simplify.
+rewrite > (eqb_n_n j).simplify.
intros. apply sym_eq.assumption.
-cut m = j \lor \lnot m = j.
+cut (m = j \lor \lnot m = j).
elim Hcut1.
-apply ex_intro ? ? i.
+apply (ex_intro ? ? i).
split.assumption.
-rewrite > eqb_n_n i.simplify.
+rewrite > (eqb_n_n i).simplify.
apply sym_eq. assumption.
-apply ex_intro ? ? m.
+apply (ex_intro ? ? m).
split.assumption.
-rewrite > not_eq_to_eqb_false m i.
-rewrite > not_eq_to_eqb_false m j.
+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.
+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.
+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.
+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.simplify.intros.
-apply ex_intro ? ? m.
+elim n.unfold bijn.intros.
+apply (ex_intro ? ? m).
split.assumption.
-apply le_n_O_elim m ? (\lambda p. f p = p).
-assumption.
-elim H O.apply sym_eq. apply le_n_O_to_eq.assumption.
-apply le_n.
-apply eq_to_bijn (\lambda p.
-(transpose (f (S n1)) (S n1)) (transpose (f (S n1)) (S n1) (f p))) f.
+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_fg (transpose (f (S n1)) (S n1))).
apply bijn_transpose.
-simplify in H1.
-elim H1 (S n1).assumption.apply le_n.apply le_n.
+unfold permut in H1.
+elim H1.apply H2.apply le_n.apply le_n.
apply bijn_n_Sn.
-apply H ?.
+apply H.
apply permut_S_to_permut_transpose.
-assumption.simplify.
-rewrite > eqb_n_n (f (S n1)).simplify.reflexivity.
+assumption.unfold transpose.
+rewrite > (eqb_n_n (f (S n1))).simplify.reflexivity.
qed.
let rec invert_permut n f m \def
|(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 (\forall p:nat. f m = f p \to m = p) \to invert_permut n f (f m) = m.
+m \le n \to injn f n\to invert_permut n f (f m) = m.
intros 4.
elim H.
-apply nat_case1 m.
+apply (nat_case1 m).
intro.simplify.
-rewrite > eqb_n_n (f O).simplify.reflexivity.
+rewrite > (eqb_n_n (f O)).simplify.reflexivity.
intros.simplify.
-rewrite > eqb_n_n (f (S m1)).simplify.reflexivity.
+rewrite > (eqb_n_n (f (S m1))).simplify.reflexivity.
simplify.
-rewrite > not_eq_to_eqb_false (f m) (f (S n1)).
-simplify.apply H2.assumption.
-simplify.intro.absurd m = S n1.
-apply H3.assumption.simplify.intro.
-apply not_le_Sn_n n1.rewrite < H5.assumption.
+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.
+(* uffa: lo ha espanso troppo *)
+change with (invert_permut n f (f (invert_permut n f m)) = invert_permut n f m).
+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 n (g:nat \to nat) f (a:nat) \def
- match n with
- [ O \Rightarrow a
- | (S p) \Rightarrow f (g p) (map_iter p g f a)].
-
-theorem eq_map_iter: \forall g1,g2:nat \to nat.
-\forall f:nat \to nat \to nat. \forall n,a:nat.
-(\forall m:nat. m \lt n \to g1 m = g2 m) \to
-map_iter n g1 f a = map_iter n g2 f a.
-intros 5.elim n.simplify.reflexivity.
-simplify.apply eq_f2.apply H1.simplify.apply le_n.
-apply H.intros.apply H1.apply trans_lt ? n1.assumption.
-simplify.apply le_n.
+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_sigma: \forall g:nat \to nat. \forall n:nat.
-map_iter n g plus O = sigma n g.
+
+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_pi: \forall g:nat \to nat. \forall n:nat.
-map_iter n g times (S O) = pi n g.
+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_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,i,a:nat. i \lt n \to
-map_iter (S n) g f a = map_iter (S n) (\lambda m. g (transpose i (S i) m)) f a.
-intros 5.elim n.apply False_ind.apply not_le_Sn_O i H2.
-simplify in H3.
-cut i < n1 \lor i = n1.
-change with
-f (g (S n1)) (map_iter (S n1) g f a) =
-f (g (transpose i (S i) (S n1))) (map_iter (S n1) (\lambda m. g (transpose i (S i) m)) f a).
-elim Hcut.
-cut g (transpose i (S i) (S n1)) = g (S n1).
-rewrite > Hcut1.
-rewrite < H2 i a H4.reflexivity.
-unfold transpose.
-rewrite > not_eq_to_eqb_false (S n1) i.
-rewrite > not_eq_to_eqb_false (S n1) (S i).
+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.
-simplify.intro.
-apply lt_to_not_eq i n1 H4.
-apply injective_S i.apply sym_eq.assumption.
-simplify. intro.
-apply lt_to_not_eq i (S n1).
-apply trans_lt ? n1.assumption.simplify.apply le_n.
-apply sym_eq.assumption.
-(* interesting case *)
-rewrite > H4.
-change with f (g (S n1)) (f (g n1) (map_iter n1 g f a)) =
-f (g (transpose n1 (S n1) (S n1)))
-(f (g (transpose n1 (S n1) n1))
-(map_iter n1 (\lambda m. g (transpose n1 (S n1) m)) f a)).
-cut (g (transpose n1 (S n1) (S n1))) = g n1.
-cut (g (transpose n1 (S n1) n1)) = g (S n1).
-cut map_iter n1 (\lambda m. g (transpose n1 (S n1) m)) f a = map_iter n1 g f a.
-rewrite > Hcut1.rewrite > Hcut2.rewrite > Hcut3.
+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 > H1 (g (S n1)).
-rewrite > H.reflexivity.
-apply eq_map_iter.
-intros.unfold transpose.
-rewrite > not_eq_to_eqb_false m n1.
-rewrite > not_eq_to_eqb_false m (S n1).
+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. intro.apply lt_to_not_eq m (S n1).
-apply trans_lt ? n1.assumption.simplify. apply le_n.
+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.
-simplify. intro.apply lt_to_not_eq m n1.
-assumption.assumption.
-unfold transpose.
-rewrite > eqb_n_n n1.simplify.reflexivity.
-unfold transpose.
-rewrite > not_eq_to_eqb_false.simplify.
-rewrite > eqb_n_n n1.simplify.reflexivity.
-simplify. intro.
-apply not_eq_n_Sn n1. apply sym_eq.assumption.
-apply le_to_or_lt_eq.
-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_transpose: \forall f:nat\to nat \to nat.associative nat f \to
-symmetric2 nat nat f \to \forall n,a,k:nat.
-\forall g:nat \to nat. \forall i:nat. S (k + i) \le n \to
-map_iter (S n) g f a = map_iter (S n) (\lambda m. g (transpose i (S(k + i)) m)) f a.
+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 k.
+apply (nat_elim1 o).
intro.
-apply nat_case m ?.
+apply (nat_case m ?).
intros.
-apply eq_map_iter_transpose_i_Si f H H1 g n i.
-exact H3.
+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 (S n) (\lambda m. g (transpose i (S(m1 + i)) m)) f a).
-apply H2 m1 ? g.
-simplify. apply le_n.
-apply trans_le ? (S(S (m1+i))).
+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 (S n) (\lambda m. g
-(transpose i (S(m1 + i)) (transpose (S(m1 + i)) (S(S(m1 + i))) m))) f a).
-apply H2 O ? ? (S(m1+i)) ?.
-simplify.apply le_S_S.apply le_O_n.
-exact H3.
-apply trans_eq ? ? (map_iter (S n) (\lambda m. g
+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 a).
-apply H2 m1 ?.
-simplify. apply le_n.
-apply trans_le ? (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.
+apply eq_map_iter_i.
intros.apply eq_f.
apply sym_eq. apply eq_transpose.
-simplify. intro.
-apply not_le_Sn_n i.
-rewrite < H5 in \vdash (? ? %).
+unfold Not. intro.
+apply (not_le_Sn_n i).
+rewrite < H7 in \vdash (? ? %).
apply le_S_S.apply le_S.
apply le_plus_n.
-simplify. intro.
-apply not_le_Sn_n i.
-rewrite > H5 in \vdash (? ? %).
+unfold Not. intro.
+apply (not_le_Sn_n i).
+rewrite > H7 in \vdash (? ? %).
apply le_S_S.
apply le_plus_n.
-simplify. intro.
-apply not_eq_n_Sn (S m1+i).
+unfold Not. intro.
+apply (not_eq_n_Sn (S m1+i)).
apply sym_eq.assumption.
qed.
-theorem eq_map_iter_transpose1: \forall f:nat\to nat \to nat.associative nat f \to
-symmetric2 nat nat f \to \forall n,a,i,j:nat.
-\forall g:nat \to nat. i < j \to j \le n \to
-map_iter (S n) g f a = map_iter (S n) (\lambda m. g (transpose i j m)) f a.
+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 H2.
-cut (S i) < j \lor (S i) = j.
+simplify in H3.
+cut ((S i) < j \lor (S i) = j).
elim Hcut.
-cut j = S ((j - (S i)) + i).
+cut (j = S ((j - (S i)) + i)).
rewrite > Hcut1.
-apply eq_map_iter_transpose f H H1 n a (j - (S i)) g i.
+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 < H4.
-apply eq_map_iter_transpose_i_Si f H H1 g.
+rewrite < H5.
+apply (eq_map_iter_i_transpose_i_Si f H H1 g).
simplify.
-apply trans_le ? j.assumption.assumption.
+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_transpose2: \forall f:nat\to nat \to nat.associative nat f \to
-symmetric2 nat nat f \to \forall n,a,i,j:nat.
-\forall g:nat \to nat. i \le n \to j \le n \to
-map_iter (S n) g f a = map_iter (S n) (\lambda m. g (transpose i j m)) f a.
+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_transpose1 f H H1 n a i j g H4 H3.
-intro.rewrite > H4.
-apply eq_map_iter.intros.
-rewrite > transpose_i_i j.reflexivity.
+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 (S n) (\lambda m:nat.g (transpose j i m)) f a).
-apply eq_map_iter_transpose1 f H H1 n a j i g H4 H2.
-apply eq_map_iter.
+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:\forall f:nat\to nat \to nat.associative nat f \to
-symmetric2 nat nat f \to \forall a,n:nat.\forall g,h:nat \to nat.
-permut h n \to
-map_iter (S n) g f a = map_iter (S n) (\lambda m.g(h m)) f a.
-intros 5.elim n.
-simplify.rewrite > permut_O_to_eq_O h.reflexivity.assumption.
-apply trans_eq ? ? (map_iter (S(S n1)) (\lambda m.g ((transpose (h (S n1)) (S n1)) m)) f a).
-apply eq_map_iter_transpose2 f H H1 (S n1) a ? ? g.
+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 (S n1).assumption.apply le_n.apply le_n.
-apply trans_eq ? ? (map_iter (S(S n1)) (\lambda m.
-(g(transpose (h (S n1)) (S n1)
-(transpose (h (S n1)) (S n1) (h m)))) )f a).
+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 n1)) (S n1) (S n1)))
-(map_iter (S n1) (\lambda m.
-g (transpose (h (S n1)) (S n1) m)) f a)
+(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 n1)) (S n1)
-(transpose (h (S n1)) (S n1) (h (S n1)))))
-(map_iter (S n1)
+(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 n1)) (S n1)
-(transpose (h (S n1)) (S n1) (h m))))) f a).
+(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 (\lambda m.(g(transpose (h (S n1)) (S n1) m))).
+apply (H2 n1 (\lambda m.(g(transpose (h (S n+n1)) (S n+n1) m)))).
apply permut_S_to_permut_transpose.
assumption.
-apply eq_map_iter.intros.
-rewrite > transpose_transpose.reflexivity.
-qed.
-
-theorem eq_sigma_transpose_i_Si: \forall n,i:nat. \forall f:nat \to nat. i \lt n \to
-sigma (S n) f = sigma (S n) (\lambda m. f (transpose i (S i) m)).
-intro.elim n.apply False_ind.apply not_le_Sn_O i H.
-simplify in H1.
-cut i < n1 \lor i = n1.
-change with
-(f (S n1))+(sigma (S n1) f) =
-(f (transpose i (S i) (S n1)))+(sigma (S n1) (\lambda m. f (transpose i (S i) m))).
-elim Hcut.
-cut f (transpose i (S i) (S n1)) = f (S n1).
-rewrite > Hcut1.
-rewrite < H i f.reflexivity.
-assumption.unfold transpose.
-rewrite > not_eq_to_eqb_false (S n1) i.
-rewrite > not_eq_to_eqb_false (S n1) (S i).
-simplify.reflexivity.
-simplify.intro.
-apply lt_to_not_eq i n1 H2.
-apply injective_S i.apply sym_eq.assumption.
-simplify. intro.
-apply lt_to_not_eq i (S n1).
-apply trans_lt ? n1.assumption.simplify.apply le_n.
-apply sym_eq.assumption.
-rewrite > H2.
-change with (f (S n1))+((f n1)+(sigma n1 f)) =
-(f (transpose n1 (S n1) (S n1)))+
-((f (transpose n1 (S n1) n1))+
-(sigma n1 (\lambda m. f (transpose n1 (S n1) m)))).
-cut (f (transpose n1 (S n1) (S n1))) = f n1.
-cut (f (transpose n1 (S n1) n1)) = f (S n1).
-cut sigma n1 (\lambda m. f (transpose n1 (S n1) m)) = sigma n1 f.
-rewrite > Hcut1.rewrite > Hcut2.rewrite > Hcut3.
-rewrite < assoc_plus.rewrite > sym_plus (f (S n1)).
-rewrite > assoc_plus.reflexivity.
-apply eq_sigma.
-intros.unfold transpose.
-rewrite > not_eq_to_eqb_false m n1.
-rewrite > not_eq_to_eqb_false m (S n1).
-simplify.reflexivity.
-simplify. intro.apply lt_to_not_eq m (S n1).
-apply trans_lt ? n1.assumption.simplify. apply le_n.
-assumption.
-simplify. intro.apply lt_to_not_eq m n1.
-assumption.assumption.
-unfold transpose.
-rewrite > eqb_n_n n1.simplify.reflexivity.
-unfold transpose.
-rewrite > not_eq_to_eqb_false.simplify.
-rewrite > eqb_n_n n1.simplify.reflexivity.
-simplify. intro.
-apply not_eq_n_Sn n1. apply sym_eq.assumption.
-apply le_to_or_lt_eq.
-apply le_S_S_to_le.assumption.
-qed.
-
-theorem eq_sigma_transpose: \forall n,k:nat. \forall f:nat \to nat.
-\forall i. S (k + i) \le n \to
-sigma (S n) f = sigma (S n) (\lambda m. f (transpose i (S(k + i)) m)).
-intros 2.
-apply nat_elim1 k.
-intro.
-apply nat_case m ?.
-intros.
-apply eq_sigma_transpose_i_Si n i f.
-exact H1.
intros.
-apply trans_eq ? ? (sigma (S n) (\lambda m. f (transpose i (S(m1 + i)) m))).
-apply H m1 ? f ?.
-simplify. apply le_n.
-apply trans_le ? (S(S (m1+i))).
-apply le_S.apply le_n.assumption.
-apply trans_eq ? ? (sigma (S n) (\lambda m. f
-(transpose i (S(m1 + i)) (transpose (S(m1 + i)) (S(S(m1 + i))) m)))).
-apply H O ? ? (S(m1+i)) ?.
-simplify.apply le_S_S.apply le_O_n.
-exact H1.
-apply trans_eq ? ? (sigma (S n) (\lambda m. f
-(transpose i (S(m1 + i))
-(transpose (S(m1 + i)) (S(S(m1 + i)))
-(transpose i (S(m1 + i)) m))))).
-apply H m1 ? ? i.
-simplify. apply le_n.
-apply trans_le ? (S(S (m1+i))).
-apply le_S.apply le_n.assumption.
-apply eq_sigma.
-intros.apply eq_f.
-apply sym_eq. apply eq_transpose.
-simplify. intro.
-apply not_le_Sn_n i.
-rewrite < H3 in \vdash (? ? %).
-apply le_S_S.apply le_S.
-apply le_plus_n.
-simplify. intro.
-apply not_le_Sn_n i.
-rewrite > H3 in \vdash (? ? %).
-apply le_S_S.
-apply le_plus_n.
-simplify. intro.
-apply not_eq_n_Sn (S m1+i).
-apply sym_eq.assumption.
-qed.
-
-(*
-theorem bad: \forall n:nat. \forall f:nat \to nat.
-permut f n \to sigma (S n) f = sigma (S n) (\lambda n.n).
-intro.elim n.
-simplify.unfold permut in H.elim H O.
-rewrite < (le_n_O_to_eq (f O)).simplify. reflexivity.apply le_n.
-cut permut (\lambda n.transpose (f (S n1)) (S n1) (f n)) n1.
-apply trans_eq ? ?
-(sigma (S(S n1)) (\lambda n.transpose (f (S n1)) (S n1) (f n))).
-*)
\ No newline at end of file
+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