lemma sameF_upto_le: ∀A,f,g,n,m.
n ≤m → sameF_upto m A f g → sameF_upto n A f g.
-#A #f #g #n #m #lenm #samef #i #ltin @samef /2/
+#A #f #g #n #m #lenm #samef #i #ltin @samef /2 by lt_to_le_to_lt/
qed.
lemma sameF_p_le: ∀A,p,f,g,n,m.
n ≤m → sameF_p m p A f g → sameF_p n p A f g.
-#A #p #f #g #n #m #lenm #samef #i #ltin #pi @samef /2/
+#A #p #f #g #n #m #lenm #samef #i #ltin #pi @samef /2 by lt_to_le_to_lt/
qed.
(*
for @{'bigop ($b-$a) $op $nil (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
(λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
-(* notation "\big [ op , nil ]_{( term 50) a ≤ ident j < b | p } f"
+(* notation "\big [ op , nil ]_{( term 55) a ≤ ident j < b | p } f"
with precedence 80
for @{\big[$op,$nil]_{${ident j} < ($b-$a) | ((λ${ident j}.$p) (${ident j}+$a))}((λ${ident j}.$f)(${ident j}+$a))}.
*)
[@same_bigop #i #lti // >(not_le_to_leb_false …) /2/
|#j #leup #Hind >bigop_Sfalse >(le_to_leb_true … leup) //
] qed.
+
+theorem pad_bigop1: ∀k,n,p,B,nil,op.∀f:nat→B. n ≤ k →
+ (∀i. n ≤ i → i < k → p i = false) →
+ \big[op,nil]_{i < n | p i}(f i)
+ = \big[op,nil]_{i < k | p i}(f i).
+#k #n #p #B #nil #op #f #lenk (elim lenk)
+ [#_ @same_bigop #i #lti //
+ |#j #leup #Hind #Hfalse >bigop_Sfalse
+ [@Hind #i #leni #ltij @Hfalse // @le_S //
+ |@Hfalse //
+ ]
+ ]
+qed.
+
+theorem bigop_false: ∀n,B,nil,op.∀f:nat→B.
+ \big[op,nil]_{i < n | false }(f i) = nil.
+#n #B #nil #op #f elim n // #n1 #Hind
+>bigop_Sfalse //
+qed.
record Aop (A:Type[0]) (nil:A) : Type[0] ≝
{op :2> A → A → A;
nilr:∀a. op a nil = a;
assoc: ∀a,b,c.op a (op b c) = op (op a b) c
}.
+
+theorem pad_bigop_nil: ∀k,n,p,B,nil.∀op:Aop B nil.∀f:nat→B. n ≤ k →
+ (∀i. n ≤ i → i < k → p i = false ∨ f i = nil) →
+ \big[op,nil]_{i < n | p i}(f i)
+ = \big[op,nil]_{i < k | p i}(f i).
+#k #n #p #B #nil #op #f #lenk (elim lenk)
+ [#_ @same_bigop #i #lti //
+ |#j #leup #Hind #Hfalse cases (true_or_false (p j)) #Hpj
+ [>bigop_Strue //
+ cut (f j = nil)
+ [cases (Hfalse j leup (le_n … )) // >Hpj #H destruct (H)] #Hfj
+ >Hfj >nill @Hind #i #leni #ltij
+ cases (Hfalse i leni (le_S … ltij)) /2/
+ |>bigop_Sfalse // @Hind #i #leni #ltij
+ cases (Hfalse i leni (le_S … ltij)) /2/
+ ]
+ ]
+qed.
theorem bigop_sum: ∀k1,k2,p1,p2,B.∀nil.∀op:Aop B nil.∀f,g:nat→B.
op (\big[op,nil]_{i<k1|p1 i}(f i)) \big[op,nil]_{i<k2|p2 i}(g i) =
qed.
lemma plus_minus1: ∀a,b,c. c ≤ b → a + (b -c) = a + b -c.
-#a #b #c #lecb @sym_eq @plus_to_minus >(commutative_plus c)
+#a #b #c #lecb @sym_eq @plus_to_minus >(commutative_plus c)
>associative_plus <plus_minus_m_m //
qed.
\big[op,nil]_{i∈[0,n[ |p i}(f i) = \big[op,nil]_{i < n|p i}(f i).
#n #p #B #nil #op #f <minus_n_O @same_bigop //
qed.
-
+
+theorem bigop_I_gen: ∀a,b,p,B.∀nil.∀op:Aop B nil.∀f:nat→B. a ≤b →
+\big[op,nil]_{i∈[a,b[ |p i}(f i) = \big[op,nil]_{i < b|leb a i ∧ p i}(f i).
+#a #b elim b // -b #b #Hind #p #B #nil #op #f #lea
+cut (∀a,b. a ≤ b → S b - a = S (b -a))
+ [#a #b cases a // #a1 #lta1 normalize >eq_minus_S_pred >S_pred
+ /2 by lt_plus_to_minus_r/] #Hcut
+cases (le_to_or_lt_eq … lea) #Ha
+ [cases (true_or_false (p b)) #Hcase
+ [>bigop_Strue [2: >Hcase >(le_to_leb_true a b) // @le_S_S_to_le @Ha]
+ >(Hcut … (le_S_S_to_le … Ha))
+ >bigop_Strue
+ [@eq_f2
+ [@eq_f <plus_minus_m_m [//|@le_S_S_to_le //] @Hind
+ |@Hind @le_S_S_to_le //
+ ]
+ |<plus_minus_m_m // @le_S_S_to_le //
+ ]
+ |>bigop_Sfalse [2: >Hcase cases (leb a b)//]
+ >(Hcut … (le_S_S_to_le … Ha)) >bigop_Sfalse
+ [@Hind @le_S_S_to_le // | <plus_minus_m_m // @le_S_S_to_le //]
+ ]
+ |<Ha <minus_n_n whd in ⊢ (??%?); <(bigop_false a B nil op f) in ⊢ (??%?);
+ @same_bigop // #i #ltia >(not_le_to_leb_false a i) // @lt_to_not_le //
+ ]
+qed.
+
theorem bigop_sumI: ∀a,b,c,p,B.∀nil.∀op:Aop B nil.∀f:nat→B.
a ≤ b → b ≤ c →
\big[op,nil]_{i∈[a,c[ |p i}(f i) =
theorem bigop_prod: ∀k1,k2,p1,p2,B.∀nil.∀op:Aop B nil.∀f: nat →nat → B.
\big[op,nil]_{x<k1|p1 x}(\big[op,nil]_{i<k2|p2 x i}(f x i)) =
- \big[op,nil]_{i<k1*k2|andb (p1 (div i k2)) (p2 (div i k2) (i \mod k2))}
- (f (div i k2) (i \mod k2)).
+ \big[op,nil]_{i<k1*k2|andb (p1 (i/k2)) (p2 (i/k2) (i \mod k2))}
+ (f (i/k2) (i \mod k2)).
#k1 #k2 #p1 #p2 #B #nil #op #f (elim k1) //
#n #Hind cases(true_or_false (p1 n)) #Hp1
[>bigop_Strue // >Hind >bigop_sum @same_bigop
- #i #lti @leb_elim // #lei cut (i = n*k2+(i-n*k2)) /2/
+ #i #lti @leb_elim // #lei cut (i = n*k2+(i-n*k2)) /2 by plus_minus/
#eqi [|#H] >eqi in ⊢ (???%);
- >div_plus_times /2/ >Hp1 >(mod_plus_times …) /2/
+ >div_plus_times /2 by monotonic_lt_minus_l/
+ >Hp1 >(mod_plus_times …) /2 by refl, monotonic_lt_minus_l, eq_f/
|>bigop_Sfalse // >Hind >(pad_bigop (S n*k2)) // @same_bigop
- #i #lti @leb_elim // #lei cut (i = n*k2+(i-n*k2)) /2/
- #eqi >eqi in ⊢ (???%); >div_plus_times /2/
+ #i #lti @leb_elim // #lei cut (i = n*k2+(i-n*k2)) /2 by plus_minus/
+ #eqi >eqi in ⊢ (???%); >div_plus_times
+ /2 by refl, monotonic_lt_minus_l, trans_eq/
]
qed.
lemma sub_lt: ∀A,e,p,n,m. n ≤ m →
sub_hk (λx.x) (λx.x) A (mk_range A e n p) (mk_range A e m p).
-#A #e #f #n #m #lenm #i #lti #fi % // % /2/
+#A #e #f #n #m #lenm #i #lti #fi % // % /2 by lt_to_le_to_lt/
qed.
theorem transitive_sub: ∀h1,k1,h2,k2,A,I,J,K.
]
qed.
+(* commutation *)
+theorem bigop_commute: ∀n,m,p11,p12,p21,p22,B.∀nil.∀op:ACop B nil.∀f.
+0 < n → 0 < m →
+(∀i,j. i < n → j < m → (p11 i ∧ p12 i j) = (p21 j ∧ p22 i j)) →
+\big[op,nil]_{i<n|p11 i}(\big[op,nil]_{j<m|p12 i j}(f i j)) =
+ \big[op,nil]_{j<m|p21 j}(\big[op,nil]_{i<n|p22 i j}(f i j)).
+#n #m #p11 #p12 #p21 #p22 #B #nil #op #f #posn #posm #Heq
+>bigop_prod >bigop_prod @bigop_iso
+%{(λi.(i\mod m)*n + i/m)} %{(λi.(i\mod n)*m + i/n)} %
+ [%
+ [#i #lti #Heq (* whd in ⊢ (???(?(?%?)?)); *) @eq_f2
+ [@sym_eq @mod_plus_times /2 by lt_times_to_lt_div/
+ |@sym_eq @div_plus_times /2 by lt_times_to_lt_div/
+ ]
+ |#i #lti #Hi
+ cut ((i\mod m*n+i/m)\mod n=i/m)
+ [@mod_plus_times @lt_times_to_lt_div //] #H1
+ cut ((i\mod m*n+i/m)/n=i \mod m)
+ [@div_plus_times @lt_times_to_lt_div //] #H2
+ %[%[@(lt_to_le_to_lt ? (i\mod m*n+n))
+ [whd >plus_n_Sm @monotonic_le_plus_r @lt_times_to_lt_div //
+ |>commutative_plus @(le_times (S(i \mod m)) m n n) // @lt_mod_m_m //
+ ]
+ |lapply (Heq (i/m) (i \mod m) ??)
+ [@lt_mod_m_m // |@lt_times_to_lt_div //|>Hi >H1 >H2 //]
+ ]
+ |>H1 >H2 //
+ ]
+ ]
+ |#i #lti #Hi
+ cut ((i\mod n*m+i/n)\mod m=i/n)
+ [@mod_plus_times @lt_times_to_lt_div //] #H1
+ cut ((i\mod n*m+i/n)/m=i \mod n)
+ [@div_plus_times @lt_times_to_lt_div //] #H2
+ %[%[@(lt_to_le_to_lt ? (i\mod n*m+m))
+ [whd >plus_n_Sm @monotonic_le_plus_r @lt_times_to_lt_div //
+ |>commutative_plus @(le_times (S(i \mod n)) n m m) // @lt_mod_m_m //
+ ]
+ |lapply (Heq (i \mod n) (i/n) ??)
+ [@lt_times_to_lt_div // |@lt_mod_m_m // |>Hi >H1 >H2 //]
+ ]
+ |>H1 >H2 //
+ ]
+ ]
+qed.
+
(* distributivity *)
record Dop (A:Type[0]) (nil:A): Type[0] ≝
|>bigop_Sfalse // >bigop_Sfalse //
]
qed.
-
-(* Sigma e Pi
-
-
-notation "Σ_{ ident i < n | p } f"
- with precedence 80
-for @{'bigop $n plus 0 (λ${ident i}.p) (λ${ident i}. $f)}.
-
-notation "Σ_{ ident i < n } f"
- with precedence 80
-for @{'bigop $n plus 0 (λ${ident i}.true) (λ${ident i}. $f)}.
-
-notation "Σ_{ ident j ∈ [a,b[ } f"
- with precedence 80
-for @{'bigop ($b-$a) plus 0 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
- (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
-
-notation "Σ_{ ident j ∈ [a,b[ | p } f"
- with precedence 80
-for @{'bigop ($b-$a) plus 0 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
- (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
-
-notation "Π_{ ident i < n | p} f"
- with precedence 80
-for @{'bigop $n times 1 (λ${ident i}.$p) (λ${ident i}. $f)}.
-
-notation "Π_{ ident i < n } f"
- with precedence 80
-for @{'bigop $n times 1 (λ${ident i}.true) (λ${ident i}. $f)}.
-
-notation "Π_{ ident j ∈ [a,b[ } f"
- with precedence 80
-for @{'bigop ($b-$a) times 1 (λ${ident j}.((λ${ident j}.true) (${ident j}+$a)))
- (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
-
-notation "Π_{ ident j ∈ [a,b[ | p } f"
- with precedence 80
-for @{'bigop ($b-$a) times 1 (λ${ident j}.((λ${ident j}.$p) (${ident j}+$a)))
- (λ${ident j}.((λ${ident j}.$f)(${ident j}+$a)))}.
-
-*)
-(*
-
-definition p_ord_times \def
-\lambda p,m,x.
- match p_ord x p with
- [pair q r \Rightarrow r*m+q].
-
-theorem eq_p_ord_times: \forall p,m,x.
-p_ord_times p m x = (ord_rem x p)*m+(ord x p).
-intros.unfold p_ord_times. unfold ord_rem.
-unfold ord.
-elim (p_ord x p).
-reflexivity.
-qed.
-
-theorem div_p_ord_times:
-\forall p,m,x. ord x p < m \to p_ord_times p m x / m = ord_rem x p.
-intros.rewrite > eq_p_ord_times.
-apply div_plus_times.
-assumption.
-qed.
-
-theorem mod_p_ord_times:
-\forall p,m,x. ord x p < m \to p_ord_times p m x \mod m = ord x p.
-intros.rewrite > eq_p_ord_times.
-apply mod_plus_times.
-assumption.
-qed.
-
-lemma lt_times_to_lt_O: \forall i,n,m:nat. i < n*m \to O < m.
-intros.
-elim (le_to_or_lt_eq O ? (le_O_n m))
- [assumption
- |apply False_ind.
- rewrite < H1 in H.
- rewrite < times_n_O in H.
- apply (not_le_Sn_O ? H)
- ]
-qed.
-
-theorem iter_p_gen_knm:
-\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 A.
-\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
-iter_p_gen k p1 A g baseA plusA =
-iter_p_gen n p21 A (\lambda x:nat.iter_p_gen m (p22 x) A (\lambda y. g (h2 x y)) baseA plusA) baseA plusA.
-intros.
-rewrite < (iter_p_gen2' n m p21 p22 ? ? ? ? H H1 H2).
-apply sym_eq.
-apply (eq_iter_p_gen_gh A baseA plusA H H1 H2 g ? (\lambda x.(h11 x)*m+(h12 x)))
- [intros.
- elim (H4 (i/m) (i \mod m));clear H4
- [elim H7.clear H7.
- elim H4.clear H4.
- assumption
- |apply (lt_times_to_lt_div ? ? ? H5)
- |apply lt_mod_m_m.
- apply (lt_times_to_lt_O ? ? ? H5)
- |apply (andb_true_true ? ? H6)
- |apply (andb_true_true_r ? ? H6)
- ]
- |intros.
- elim (H4 (i/m) (i \mod m));clear H4
- [elim H7.clear H7.
- elim H4.clear H4.
- rewrite > H10.
- rewrite > H9.
- apply sym_eq.
- apply div_mod.
- apply (lt_times_to_lt_O ? ? ? H5)
- |apply (lt_times_to_lt_div ? ? ? H5)
- |apply lt_mod_m_m.
- apply (lt_times_to_lt_O ? ? ? H5)
- |apply (andb_true_true ? ? H6)
- |apply (andb_true_true_r ? ? H6)
- ]
- |intros.
- elim (H4 (i/m) (i \mod m));clear H4
- [elim H7.clear H7.
- elim H4.clear H4.
- assumption
- |apply (lt_times_to_lt_div ? ? ? H5)
- |apply lt_mod_m_m.
- apply (lt_times_to_lt_O ? ? ? H5)
- |apply (andb_true_true ? ? H6)
- |apply (andb_true_true_r ? ? H6)
- ]
- |intros.
- elim (H3 j H5 H6).
- elim H7.clear H7.
- elim H9.clear H9.
- elim H7.clear H7.
- rewrite > div_plus_times
- [rewrite > mod_plus_times
- [rewrite > H9.
- rewrite > H12.
- reflexivity.
- |assumption
- ]
- |assumption
- ]
- |intros.
- elim (H3 j H5 H6).
- elim H7.clear H7.
- elim H9.clear H9.
- elim H7.clear H7.
- rewrite > div_plus_times
- [rewrite > mod_plus_times
- [assumption
- |assumption
- ]
- |assumption
- ]
- |intros.
- elim (H3 j H5 H6).
- elim H7.clear H7.
- elim H9.clear H9.
- elim H7.clear H7.
- apply (lt_to_le_to_lt ? ((h11 j)*m+m))
- [apply monotonic_lt_plus_r.
- assumption
- |rewrite > sym_plus.
- change with ((S (h11 j)*m) \le n*m).
- apply monotonic_le_times_l.
- assumption
- ]
- ]
-qed.
-
-theorem iter_p_gen_divides:
-\forall A:Type.
-\forall baseA: A.
-\forall plusA: A \to A \to A.
-\forall n,m,p:nat.O < n \to prime p \to Not (divides p n) \to
-\forall g: nat \to A.
-(symmetric A plusA) \to (associative A plusA) \to (\forall a:A.(plusA a baseA) = a)
-
-\to
-
-iter_p_gen (S (n*(exp p m))) (\lambda x.divides_b x (n*(exp p m))) A g baseA plusA =
-iter_p_gen (S n) (\lambda x.divides_b x n) A
- (\lambda x.iter_p_gen (S m) (\lambda y.true) A (\lambda y.g (x*(exp p y))) baseA plusA) baseA plusA.
-intros.
-cut (O < p)
- [rewrite < (iter_p_gen2 ? ? ? ? ? ? ? ? H3 H4 H5).
- apply (trans_eq ? ?
- (iter_p_gen (S n*S m) (\lambda x:nat.divides_b (x/S m) n) A
- (\lambda x:nat.g (x/S m*(p)\sup(x\mod S m))) baseA plusA) )
- [apply sym_eq.
- apply (eq_iter_p_gen_gh ? ? ? ? ? ? g ? (p_ord_times p (S m)))
- [ assumption
- | assumption
- | assumption
- |intros.
- lapply (divides_b_true_to_lt_O ? ? H H7).
- apply divides_to_divides_b_true
- [rewrite > (times_n_O O).
- apply lt_times
- [assumption
- |apply lt_O_exp.assumption
- ]
- |apply divides_times
- [apply divides_b_true_to_divides.assumption
- |apply (witness ? ? (p \sup (m-i \mod (S m)))).
- rewrite < exp_plus_times.
- apply eq_f.
- rewrite > sym_plus.
- apply plus_minus_m_m.
- autobatch by le_S_S_to_le, lt_mod_m_m, lt_O_S;
- ]
- ]
- |intros.
- lapply (divides_b_true_to_lt_O ? ? H H7).
- unfold p_ord_times.
- rewrite > (p_ord_exp1 p ? (i \mod (S m)) (i/S m))
- [change with ((i/S m)*S m+i \mod S m=i).
- apply sym_eq.
- apply div_mod.
- apply lt_O_S
- |assumption
- |unfold Not.intro.
- apply H2.
- apply (trans_divides ? (i/ S m))
- [assumption|
- apply divides_b_true_to_divides;assumption]
- |apply sym_times.
- ]
- |intros.
- apply le_S_S.
- apply le_times
- [apply le_S_S_to_le.
- change with ((i/S m) < S n).
- apply (lt_times_to_lt_l m).
- apply (le_to_lt_to_lt ? i);[2:assumption]
- autobatch by eq_plus_to_le, div_mod, lt_O_S.
- |apply le_exp
- [assumption
- |apply le_S_S_to_le.
- apply lt_mod_m_m.
- apply lt_O_S
- ]
- ]
- |intros.
- cut (ord j p < S m)
- [rewrite > div_p_ord_times
- [apply divides_to_divides_b_true
- [apply lt_O_ord_rem
- [elim H1.assumption
- |apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- ]
- |cut (n = ord_rem (n*(exp p m)) p)
- [rewrite > Hcut2.
- apply divides_to_divides_ord_rem
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord_rem.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |assumption
- ]
- |cut (m = ord (n*(exp p m)) p)
- [apply le_S_S.
- rewrite > Hcut1.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |intros.
- cut (ord j p < S m)
- [rewrite > div_p_ord_times
- [rewrite > mod_p_ord_times
- [rewrite > sym_times.
- apply sym_eq.
- apply exp_ord
- [elim H1.assumption
- |apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- ]
- |cut (m = ord (n*(exp p m)) p)
- [apply le_S_S.
- rewrite > Hcut2.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |assumption
- ]
- |cut (m = ord (n*(exp p m)) p)
- [apply le_S_S.
- rewrite > Hcut1.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |apply sym_times
- ]
- ]
- ]
- |intros.
- rewrite > eq_p_ord_times.
- rewrite > sym_plus.
- apply (lt_to_le_to_lt ? (S m +ord_rem j p*S m))
- [apply lt_plus_l.
- apply le_S_S.
- cut (m = ord (n*(p \sup m)) p)
- [rewrite > Hcut1.
- apply divides_to_le_ord
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- |unfold ord.
- rewrite > sym_times.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |reflexivity
- ]
- ]
- |change with (S (ord_rem j p)*S m \le S n*S m).
- apply le_times_l.
- apply le_S_S.
- cut (n = ord_rem (n*(p \sup m)) p)
- [rewrite > Hcut1.
- apply divides_to_le
- [apply lt_O_ord_rem
- [elim H1.assumption
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- ]
- |apply divides_to_divides_ord_rem
- [apply (divides_b_true_to_lt_O ? ? ? H7).
- rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |rewrite > (times_n_O O).
- apply lt_times
- [assumption|apply lt_O_exp.assumption]
- |assumption
- |apply divides_b_true_to_divides.
- assumption
- ]
- ]
- |unfold ord_rem.
- rewrite > sym_times.
- rewrite > (p_ord_exp1 p ? m n)
- [reflexivity
- |assumption
- |assumption
- |reflexivity
- ]
- ]
- ]
- ]
- |apply eq_iter_p_gen
-
- [intros.
- elim (divides_b (x/S m) n);reflexivity
- |intros.reflexivity
- ]
- ]
-|elim H1.apply lt_to_le.assumption
-]
-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
- ]
- | unfold ha.
- unfold ha12.
- unfold ha22.
- 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
- ]
- | unfold ha.
- unfold ha12.
- rewrite > Hcut1.
- rewrite > Hcut.
- assumption
- ]
- | unfold ha.
- unfold ha22.
- rewrite > Hcut1.
- rewrite > Hcut.
- assumption
- ]
- | cut(O \lt m1)
- [ cut(O \lt n1)
- [ apply (lt_to_le_to_lt ? ((h11 i j)*m1 + m1) )
- [ unfold ha.
- 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.
-
-theorem iter_p_gen_iter_p_gen:
-\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 n,m.
-\forall p11,p21:nat \to bool.
-\forall p12,p22:nat \to nat \to bool.
-(\forall x,y. x < n \to y < m \to
- (p11 x \land p12 x y) = (p21 y \land p22 y x)) \to
-iter_p_gen n p11 A
- (\lambda x:nat.iter_p_gen m (p12 x) A (\lambda y. g x y) baseA plusA)
- baseA plusA =
-iter_p_gen m p21 A
- (\lambda y:nat.iter_p_gen n (p22 y) A (\lambda x. g x y) baseA plusA )
- baseA plusA.
-intros.
-apply (iter_p_gen_2_eq A baseA plusA H H1 H2 (\lambda x,y. g x y) (\lambda x,y.y) (\lambda x,y.x) (\lambda x,y.y) (\lambda x,y.x)
- n m m n p11 p21 p12 p22)
- [intros.split
- [split
- [split
- [split
- [split
- [apply (andb_true_true ? (p12 j i)).
- rewrite > H3
- [rewrite > H6.rewrite > H7.reflexivity
- |assumption
- |assumption
- ]
- |apply (andb_true_true_r (p11 j)).
- rewrite > H3
- [rewrite > H6.rewrite > H7.reflexivity
- |assumption
- |assumption
- ]
- ]
- |reflexivity
- ]
- |reflexivity
- ]
- |assumption
- ]
- |assumption
- ]
- |intros.split
- [split
- [split
- [split
- [split
- [apply (andb_true_true ? (p22 j i)).
- rewrite < H3
- [rewrite > H6.rewrite > H7.reflexivity
- |assumption
- |assumption
- ]
- |apply (andb_true_true_r (p21 j)).
- rewrite < H3
- [rewrite > H6.rewrite > H7.reflexivity
- |assumption
- |assumption
- ]
- ]
- |reflexivity
- ]
- |reflexivity
- ]
- |assumption
- ]
- |assumption
- ]
- ]
-qed. *)
\ No newline at end of file
projects / helm.git / blobdiff