From 80f85c7fb53791fd72c0e64450c3c87d8f30b84d Mon Sep 17 00:00:00 2001 From: Cristian Armentano Date: Sun, 14 Oct 2007 14:06:05 +0000 Subject: [PATCH] Theorem sigma_p_knm changed into generic_iter_p_knm. 2 specific versions in nat/iteration2.ma and Z/sigma_p.ma --- matita/library/Z/sigma_p.ma | 203 ++++++++++- matita/library/nat/generic_iter_p.ma | 492 +++++++++++++++++++++++++++ matita/library/nat/iteration2.ma | 34 +- 3 files changed, 724 insertions(+), 5 deletions(-) diff --git a/matita/library/Z/sigma_p.ma b/matita/library/Z/sigma_p.ma index 8b4c87d3e..71340ac7c 100644 --- a/matita/library/Z/sigma_p.ma +++ b/matita/library/Z/sigma_p.ma @@ -314,7 +314,39 @@ apply eq_sigma_p qed. -(* sigma from n1*m1 to n2*m2 *) +theorem sigma_p_knm: +\forall g: nat \to Z. +\forall h2:nat \to nat \to nat. +\forall h11,h12:nat \to nat. +\forall k,n,m. +\forall p1,p21:nat \to bool. +\forall p22:nat \to nat \to bool. +(\forall x. x < k \to p1 x = true \to +p21 (h11 x) = true \land p22 (h11 x) (h12 x) = true +\land h2 (h11 x) (h12 x) = x +\land (h11 x) < n \land (h12 x) < m) \to +(\forall i,j. i < n \to j < m \to p21 i = true \to p22 i j = true \to +p1 (h2 i j) = true \land +h11 (h2 i j) = i \land h12 (h2 i j) = j +\land h2 i j < k) \to +sigma_p k p1 g= +sigma_p n p21 (\lambda x:nat.sigma_p m (p22 x) (\lambda y. g (h2 x y))). +intros. +unfold sigma_p. +unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)). +apply iter_p_gen_knm + [ apply symmetricZPlus + |apply associative_Zplus + | intro. + apply (Zplus_z_OZ a) + | exact h11 + | exact h12 + | assumption + | assumption + ] +qed. + + theorem sigma_p2_eq: \forall g: nat \to nat \to Z. \forall h11,h12,h21,h22: nat \to nat \to nat. @@ -332,8 +364,165 @@ p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true sigma_p n1 p11 (\lambda x:nat .sigma_p m1 (p12 x) (\lambda y. g x y)) = sigma_p n2 p21 (\lambda x:nat .sigma_p m2 (p22 x) (\lambda y. g (h11 x y) (h12 x y))). intros. +unfold sigma_p. +unfold sigma_p in \vdash (? ? (? ? ? ? (\lambda x:?.%) ? ?) ?). +unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)). + +apply(iter_p_gen_2_eq Z OZ Zplus ? ? ? g h11 h12 h21 h22 n1 m1 n2 m2 p11 p21 p12 p22) +[ apply symmetricZPlus +| apply associative_Zplus +| intro. + apply (Zplus_z_OZ a) +| assumption +| assumption +] +qed. + + + + +(* + + + + + rewrite < sigma_p2'. -rewrite < sigma_p2'. +letin ha:= (\lambda x,y.(((h11 x y)*m1) + (h12 x y))). +letin ha12:= (\lambda x.(h21 (x/m1) (x \mod m1))). +letin ha22:= (\lambda x.(h22 (x/m1) (x \mod m1))). + +apply (trans_eq ? ? +(sigma_p n2 p21 (\lambda x:nat. sigma_p m2 (p22 x) + (\lambda y:nat.(g (((h11 x y)*m1+(h12 x y))/m1) (((h11 x y)*m1+(h12 x y))\mod m1)) ) ) )) +[ + apply (sigma_p_knm (\lambda e. (g (e/m1) (e \mod m1))) ha ha12 ha22);intros + [ elim (and_true ? ? H3). + cut(O \lt m1) + [ cut(x/m1 < n1) + [ cut((x \mod m1) < m1) + [ elim (H1 ? ? Hcut1 Hcut2 H4 H5). + elim H6.clear H6. + elim H8.clear H8. + elim H6.clear H6. + elim H8.clear H8. + split + [ split + [ split + [ split + [ assumption + | assumption + ] + | rewrite > H11. + rewrite > H10. + apply sym_eq. + apply div_mod. + assumption + ] + | assumption + ] + | assumption + ] + | apply lt_mod_m_m. + assumption + ] + | apply (lt_times_n_to_lt m1) + [ assumption + | apply (le_to_lt_to_lt ? x) + [ apply (eq_plus_to_le ? ? (x \mod m1)). + apply div_mod. + assumption + | assumption + ] + ] + ] + | apply not_le_to_lt.unfold.intro. + generalize in match H2. + apply (le_n_O_elim ? H6). + rewrite < times_n_O. + apply le_to_not_lt. + apply le_O_n. + ] + | elim (H ? ? H2 H3 H4 H5). + elim H6.clear H6. + elim H8.clear H8. + elim H6.clear H6. + elim H8.clear H8. + cut(((h11 i j)*m1 + (h12 i j))/m1 = (h11 i j)) + [ cut(((h11 i j)*m1 + (h12 i j)) \mod m1 = (h12 i j)) + [ split + [ split + [ split + [ apply true_to_true_to_andb_true + [ rewrite > Hcut. + assumption + | rewrite > Hcut1. + rewrite > Hcut. + assumption + ] + | rewrite > Hcut1. + rewrite > Hcut. + assumption + ] + | rewrite > Hcut1. + rewrite > Hcut. + assumption + ] + | cut(O \lt m1) + [ cut(O \lt n1) + [ apply (lt_to_le_to_lt ? ((h11 i j)*m1 + m1) ) + [ apply (lt_plus_r). + assumption + | rewrite > sym_plus. + rewrite > (sym_times (h11 i j) m1). + rewrite > times_n_Sm. + rewrite > sym_times. + apply (le_times_l). + assumption + ] + | apply not_le_to_lt.unfold.intro. + generalize in match H9. + apply (le_n_O_elim ? H8). + apply le_to_not_lt. + apply le_O_n + ] + | apply not_le_to_lt.unfold.intro. + generalize in match H7. + apply (le_n_O_elim ? H8). + apply le_to_not_lt. + apply le_O_n + ] + ] + | rewrite > (mod_plus_times m1 (h11 i j) (h12 i j)). + reflexivity. + assumption + ] + | rewrite > (div_plus_times m1 (h11 i j) (h12 i j)). + reflexivity. + assumption + ] + ] +| apply (eq_sigma_p1) + [ intros. reflexivity + | intros. + apply (eq_sigma_p1) + [ intros. reflexivity + | intros. + rewrite > (div_plus_times) + [ rewrite > (mod_plus_times) + [ reflexivity + | elim (H x x1 H2 H4 H3 H5). + assumption + ] + | elim (H x x1 H2 H4 H3 H5). + assumption + ] + ] + ] +] +qed. + +rewrite < sigma_p2' in \vdash (? ? ? %). apply sym_eq. letin h := (\lambda x.(h11 (x/m2) (x\mod m2))*m1 + (h12 (x/m2) (x\mod m2))). letin h1 := (\lambda x.(h21 (x/m1) (x\mod m1))*m2 + (h22 (x/m1) (x\mod m1))). @@ -357,7 +546,10 @@ apply (trans_eq ? ? [apply sym_eq. apply div_plus_times. assumption - |autobatch + | + apply sym_eq. + apply mod_plus_times. + assumption ] |apply lt_mod_m_m. assumption @@ -625,4 +817,7 @@ apply (trans_eq ? ? ] ] ] -qed. \ No newline at end of file +qed. +*) + + diff --git a/matita/library/nat/generic_iter_p.ma b/matita/library/nat/generic_iter_p.ma index c424f82e0..3a9adc231 100644 --- a/matita/library/nat/generic_iter_p.ma +++ b/matita/library/nat/generic_iter_p.ma @@ -1142,4 +1142,496 @@ elim n ] qed. +(* old version - proved without theorem iter_p_gen_knm +theorem iter_p_gen_2_eq: +\forall A:Type. +\forall baseA: A. +\forall plusA: A \to A \to A. +(symmetric A plusA) \to +(associative A plusA) \to +(\forall a:A.(plusA a baseA) = a)\to +\forall g: nat \to nat \to A. +\forall h11,h12,h21,h22: nat \to nat \to nat. +\forall n1,m1,n2,m2. +\forall p11,p21:nat \to bool. +\forall p12,p22:nat \to nat \to bool. +(\forall i,j. i < n2 \to j < m2 \to p21 i = true \to p22 i j = true \to +p11 (h11 i j) = true \land p12 (h11 i j) (h12 i j) = true +\land h21 (h11 i j) (h12 i j) = i \land h22 (h11 i j) (h12 i j) = j +\land h11 i j < n1 \land h12 i j < m1) \to +(\forall i,j. i < n1 \to j < m1 \to p11 i = true \to p12 i j = true \to +p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true +\land h11 (h21 i j) (h22 i j) = i \land h12 (h21 i j) (h22 i j) = j +\land (h21 i j) < n2 \land (h22 i j) < m2) \to +iter_p_gen n1 p11 A + (\lambda x:nat .iter_p_gen m1 (p12 x) A (\lambda y. g x y) baseA plusA) + baseA plusA = +iter_p_gen n2 p21 A + (\lambda x:nat .iter_p_gen m2 (p22 x) A (\lambda y. g (h11 x y) (h12 x y)) baseA plusA ) + baseA plusA. +intros. +rewrite < (iter_p_gen2' ? ? ? ? ? ? ? ? H H1 H2). +rewrite < (iter_p_gen2' ? ? ? ? ? ? ? ? H H1 H2). +apply sym_eq. +letin h := (\lambda x.(h11 (x/m2) (x\mod m2))*m1 + (h12 (x/m2) (x\mod m2))). +letin h1 := (\lambda x.(h21 (x/m1) (x\mod m1))*m2 + (h22 (x/m1) (x\mod m1))). +apply (trans_eq ? ? + (iter_p_gen (n2*m2) (\lambda x:nat.p21 (x/m2)\land p22 (x/m2) (x\mod m2)) A + (\lambda x:nat.g ((h x)/m1) ((h x)\mod m1)) baseA plusA )) + [clear h.clear h1. + apply eq_iter_p_gen1 + [intros.reflexivity + |intros. + cut (O < m2) + [cut (x/m2 < n2) + [cut (x \mod m2 < m2) + [elim (and_true ? ? H6). + elim (H3 ? ? Hcut1 Hcut2 H7 H8). + elim H9.clear H9. + elim H11.clear H11. + elim H9.clear H9. + elim H11.clear H11. + apply eq_f2 + [apply sym_eq. + apply div_plus_times. + assumption + | apply sym_eq. + apply mod_plus_times. + assumption + ] + |apply lt_mod_m_m. + assumption + ] + |apply (lt_times_n_to_lt m2) + [assumption + |apply (le_to_lt_to_lt ? x) + [apply (eq_plus_to_le ? ? (x \mod m2)). + apply div_mod. + assumption + |assumption + ] + ] + ] + |apply not_le_to_lt.unfold.intro. + generalize in match H5. + apply (le_n_O_elim ? H7). + rewrite < times_n_O. + apply le_to_not_lt. + apply le_O_n + ] + ] + |apply (eq_iter_p_gen_gh ? ? ? H H1 H2 ? h h1);intros + [cut (O < m2) + [cut (i/m2 < n2) + [cut (i \mod m2 < m2) + [elim (and_true ? ? H6). + elim (H3 ? ? Hcut1 Hcut2 H7 H8). + elim H9.clear H9. + elim H11.clear H11. + elim H9.clear H9. + elim H11.clear H11. + cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))/m1 = + h11 (i/m2) (i\mod m2)) + [cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))\mod m1 = + h12 (i/m2) (i\mod m2)) + [rewrite > Hcut3. + rewrite > Hcut4. + rewrite > H9. + rewrite > H15. + reflexivity + |apply mod_plus_times. + assumption + ] + |apply div_plus_times. + assumption + ] + |apply lt_mod_m_m. + assumption + ] + |apply (lt_times_n_to_lt m2) + [assumption + |apply (le_to_lt_to_lt ? i) + [apply (eq_plus_to_le ? ? (i \mod m2)). + apply div_mod. + assumption + |assumption + ] + ] + ] + |apply not_le_to_lt.unfold.intro. + generalize in match H5. + apply (le_n_O_elim ? H7). + rewrite < times_n_O. + apply le_to_not_lt. + apply le_O_n + ] + |cut (O < m2) + [cut (i/m2 < n2) + [cut (i \mod m2 < m2) + [elim (and_true ? ? H6). + elim (H3 ? ? Hcut1 Hcut2 H7 H8). + elim H9.clear H9. + elim H11.clear H11. + elim H9.clear H9. + elim H11.clear H11. + cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))/m1 = + h11 (i/m2) (i\mod m2)) + [cut ((h11 (i/m2) (i\mod m2)*m1+h12 (i/m2) (i\mod m2))\mod m1 = + h12 (i/m2) (i\mod m2)) + [rewrite > Hcut3. + rewrite > Hcut4. + rewrite > H13. + rewrite > H14. + apply sym_eq. + apply div_mod. + assumption + |apply mod_plus_times. + assumption + ] + |apply div_plus_times. + assumption + ] + |apply lt_mod_m_m. + assumption + ] + |apply (lt_times_n_to_lt m2) + [assumption + |apply (le_to_lt_to_lt ? i) + [apply (eq_plus_to_le ? ? (i \mod m2)). + apply div_mod. + assumption + |assumption + ] + ] + ] + |apply not_le_to_lt.unfold.intro. + generalize in match H5. + apply (le_n_O_elim ? H7). + rewrite < times_n_O. + apply le_to_not_lt. + apply le_O_n + ] + |cut (O < m2) + [cut (i/m2 < n2) + [cut (i \mod m2 < m2) + [elim (and_true ? ? H6). + elim (H3 ? ? Hcut1 Hcut2 H7 H8). + elim H9.clear H9. + elim H11.clear H11. + elim H9.clear H9. + elim H11.clear H11. + apply lt_times_plus_times + [assumption|assumption] + |apply lt_mod_m_m. + assumption + ] + |apply (lt_times_n_to_lt m2) + [assumption + |apply (le_to_lt_to_lt ? i) + [apply (eq_plus_to_le ? ? (i \mod m2)). + apply div_mod. + assumption + |assumption + ] + ] + ] + |apply not_le_to_lt.unfold.intro. + generalize in match H5. + apply (le_n_O_elim ? H7). + rewrite < times_n_O. + apply le_to_not_lt. + apply le_O_n + ] + |cut (O < m1) + [cut (j/m1 < n1) + [cut (j \mod m1 < m1) + [elim (and_true ? ? H6). + elim (H4 ? ? Hcut1 Hcut2 H7 H8). + elim H9.clear H9. + elim H11.clear H11. + elim H9.clear H9. + elim H11.clear H11. + cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))/m2 = + h21 (j/m1) (j\mod m1)) + [cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))\mod m2 = + h22 (j/m1) (j\mod m1)) + [rewrite > Hcut3. + rewrite > Hcut4. + rewrite > H9. + rewrite > H15. + reflexivity + |apply mod_plus_times. + assumption + ] + |apply div_plus_times. + assumption + ] + |apply lt_mod_m_m. + assumption + ] + |apply (lt_times_n_to_lt m1) + [assumption + |apply (le_to_lt_to_lt ? j) + [apply (eq_plus_to_le ? ? (j \mod m1)). + apply div_mod. + assumption + |assumption + ] + ] + ] + |apply not_le_to_lt.unfold.intro. + generalize in match H5. + apply (le_n_O_elim ? H7). + rewrite < times_n_O. + apply le_to_not_lt. + apply le_O_n + ] + |cut (O < m1) + [cut (j/m1 < n1) + [cut (j \mod m1 < m1) + [elim (and_true ? ? H6). + elim (H4 ? ? Hcut1 Hcut2 H7 H8). + elim H9.clear H9. + elim H11.clear H11. + elim H9.clear H9. + elim H11.clear H11. + cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))/m2 = + h21 (j/m1) (j\mod m1)) + [cut ((h21 (j/m1) (j\mod m1)*m2+h22 (j/m1) (j\mod m1))\mod m2 = + h22 (j/m1) (j\mod m1)) + [rewrite > Hcut3. + rewrite > Hcut4. + rewrite > H13. + rewrite > H14. + apply sym_eq. + apply div_mod. + assumption + |apply mod_plus_times. + assumption + ] + |apply div_plus_times. + assumption + ] + |apply lt_mod_m_m. + assumption + ] + |apply (lt_times_n_to_lt m1) + [assumption + |apply (le_to_lt_to_lt ? j) + [apply (eq_plus_to_le ? ? (j \mod m1)). + apply div_mod. + assumption + |assumption + ] + ] + ] + |apply not_le_to_lt.unfold.intro. + generalize in match H5. + apply (le_n_O_elim ? H7). + rewrite < times_n_O. + apply le_to_not_lt. + apply le_O_n + ] + |cut (O < m1) + [cut (j/m1 < n1) + [cut (j \mod m1 < m1) + [elim (and_true ? ? H6). + elim (H4 ? ? Hcut1 Hcut2 H7 H8). + elim H9.clear H9. + elim H11.clear H11. + elim H9.clear H9. + elim H11.clear H11. + apply (lt_times_plus_times ? ? ? m2) + [assumption|assumption] + |apply lt_mod_m_m. + assumption + ] + |apply (lt_times_n_to_lt m1) + [assumption + |apply (le_to_lt_to_lt ? j) + [apply (eq_plus_to_le ? ? (j \mod m1)). + apply div_mod. + assumption + |assumption + ] + ] + ] + |apply not_le_to_lt.unfold.intro. + generalize in match H5. + apply (le_n_O_elim ? H7). + rewrite < times_n_O. + apply le_to_not_lt. + apply le_O_n + ] + ] + ] +qed.*) + + +theorem iter_p_gen_2_eq: +\forall A:Type. +\forall baseA: A. +\forall plusA: A \to A \to A. +(symmetric A plusA) \to +(associative A plusA) \to +(\forall a:A.(plusA a baseA) = a)\to +\forall g: nat \to nat \to A. +\forall h11,h12,h21,h22: nat \to nat \to nat. +\forall n1,m1,n2,m2. +\forall p11,p21:nat \to bool. +\forall p12,p22:nat \to nat \to bool. +(\forall i,j. i < n2 \to j < m2 \to p21 i = true \to p22 i j = true \to +p11 (h11 i j) = true \land p12 (h11 i j) (h12 i j) = true +\land h21 (h11 i j) (h12 i j) = i \land h22 (h11 i j) (h12 i j) = j +\land h11 i j < n1 \land h12 i j < m1) \to +(\forall i,j. i < n1 \to j < m1 \to p11 i = true \to p12 i j = true \to +p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true +\land h11 (h21 i j) (h22 i j) = i \land h12 (h21 i j) (h22 i j) = j +\land (h21 i j) < n2 \land (h22 i j) < m2) \to +iter_p_gen n1 p11 A + (\lambda x:nat .iter_p_gen m1 (p12 x) A (\lambda y. g x y) baseA plusA) + baseA plusA = +iter_p_gen n2 p21 A + (\lambda x:nat .iter_p_gen m2 (p22 x) A (\lambda y. g (h11 x y) (h12 x y)) baseA plusA ) + baseA plusA. + +intros. +rewrite < (iter_p_gen2' ? ? ? ? ? ? ? ? H H1 H2). +letin ha:= (\lambda x,y.(((h11 x y)*m1) + (h12 x y))). +letin ha12:= (\lambda x.(h21 (x/m1) (x \mod m1))). +letin ha22:= (\lambda x.(h22 (x/m1) (x \mod m1))). + +apply (trans_eq ? ? +(iter_p_gen n2 p21 A (\lambda x:nat. iter_p_gen m2 (p22 x) A + (\lambda y:nat.(g (((h11 x y)*m1+(h12 x y))/m1) (((h11 x y)*m1+(h12 x y))\mod m1))) baseA plusA ) baseA plusA)) +[ + apply (iter_p_gen_knm A baseA plusA H H1 H2 (\lambda e. (g (e/m1) (e \mod m1))) ha ha12 ha22);intros + [ elim (and_true ? ? H6). + cut(O \lt m1) + [ cut(x/m1 < n1) + [ cut((x \mod m1) < m1) + [ elim (H4 ? ? Hcut1 Hcut2 H7 H8). + elim H9.clear H9. + elim H11.clear H11. + elim H9.clear H9. + elim H11.clear H11. + split + [ split + [ split + [ split + [ assumption + | assumption + ] + | rewrite > H14. + rewrite > H13. + apply sym_eq. + apply div_mod. + assumption + ] + | assumption + ] + | assumption + ] + | apply lt_mod_m_m. + assumption + ] + | apply (lt_times_n_to_lt m1) + [ assumption + | apply (le_to_lt_to_lt ? x) + [ apply (eq_plus_to_le ? ? (x \mod m1)). + apply div_mod. + assumption + | assumption + ] + ] + ] + | apply not_le_to_lt.unfold.intro. + generalize in match H5. + apply (le_n_O_elim ? H9). + rewrite < times_n_O. + apply le_to_not_lt. + apply le_O_n. + ] + | elim (H3 ? ? H5 H6 H7 H8). + elim H9.clear H9. + elim H11.clear H11. + elim H9.clear H9. + elim H11.clear H11. + cut(((h11 i j)*m1 + (h12 i j))/m1 = (h11 i j)) + [ cut(((h11 i j)*m1 + (h12 i j)) \mod m1 = (h12 i j)) + [ split + [ split + [ split + [ apply true_to_true_to_andb_true + [ rewrite > Hcut. + assumption + | rewrite > Hcut1. + rewrite > Hcut. + assumption + ] + | rewrite > Hcut1. + rewrite > Hcut. + assumption + ] + | rewrite > Hcut1. + rewrite > Hcut. + assumption + ] + | cut(O \lt m1) + [ cut(O \lt n1) + [ apply (lt_to_le_to_lt ? ((h11 i j)*m1 + m1) ) + [ apply (lt_plus_r). + assumption + | rewrite > sym_plus. + rewrite > (sym_times (h11 i j) m1). + rewrite > times_n_Sm. + rewrite > sym_times. + apply (le_times_l). + assumption + ] + | apply not_le_to_lt.unfold.intro. + generalize in match H12. + apply (le_n_O_elim ? H11). + apply le_to_not_lt. + apply le_O_n + ] + | apply not_le_to_lt.unfold.intro. + generalize in match H10. + apply (le_n_O_elim ? H11). + apply le_to_not_lt. + apply le_O_n + ] + ] + | rewrite > (mod_plus_times m1 (h11 i j) (h12 i j)). + reflexivity. + assumption + ] + | rewrite > (div_plus_times m1 (h11 i j) (h12 i j)). + reflexivity. + assumption + ] + ] +| apply (eq_iter_p_gen1) + [ intros. reflexivity + | intros. + apply (eq_iter_p_gen1) + [ intros. reflexivity + | intros. + rewrite > (div_plus_times) + [ rewrite > (mod_plus_times) + [ reflexivity + | elim (H3 x x1 H5 H7 H6 H8). + assumption + ] + | elim (H3 x x1 H5 H7 H6 H8). + assumption + ] + ] + ] +] +qed. + + + + diff --git a/matita/library/nat/iteration2.ma b/matita/library/nat/iteration2.ma index e00bb4420..0230362e7 100644 --- a/matita/library/nat/iteration2.ma +++ b/matita/library/nat/iteration2.ma @@ -611,4 +611,36 @@ apply iter_p_gen_knm |assumption ] qed. - \ No newline at end of file + + +theorem sigma_p2_eq: +\forall g: nat \to nat \to nat. +\forall h11,h12,h21,h22: nat \to nat \to nat. +\forall n1,m1,n2,m2. +\forall p11,p21:nat \to bool. +\forall p12,p22:nat \to nat \to bool. +(\forall i,j. i < n2 \to j < m2 \to p21 i = true \to p22 i j = true \to +p11 (h11 i j) = true \land p12 (h11 i j) (h12 i j) = true +\land h21 (h11 i j) (h12 i j) = i \land h22 (h11 i j) (h12 i j) = j +\land h11 i j < n1 \land h12 i j < m1) \to +(\forall i,j. i < n1 \to j < m1 \to p11 i = true \to p12 i j = true \to +p21 (h21 i j) = true \land p22 (h21 i j) (h22 i j) = true +\land h11 (h21 i j) (h22 i j) = i \land h12 (h21 i j) (h22 i j) = j +\land (h21 i j) < n2 \land (h22 i j) < m2) \to +sigma_p n1 p11 (\lambda x:nat .sigma_p m1 (p12 x) (\lambda y. g x y)) = +sigma_p n2 p21 (\lambda x:nat .sigma_p m2 (p22 x) (\lambda y. g (h11 x y) (h12 x y))). +intros. +unfold sigma_p. +unfold sigma_p in \vdash (? ? (? ? ? ? (\lambda x:?.%) ? ?) ?). +unfold sigma_p in \vdash (? ? ? (? ? ? ? (\lambda x:?.%) ? ?)). + +apply(iter_p_gen_2_eq nat O plus ? ? ? g h11 h12 h21 h22 n1 m1 n2 m2 p11 p21 p12 p22) +[ apply symmetricIntPlus +| apply associative_plus +| intro. + rewrite < (plus_n_O). + reflexivity +| assumption +| assumption +] +qed. \ No newline at end of file -- 2.39.2