statement stringlengths 1 1.51k | proof stringlengths 0 19.7k | type stringclasses 22
values | symbolic_name stringlengths 1 47 | library stringclasses 40
values | filename stringclasses 367
values | imports listlengths 0 43 | deps listlengths 0 64 | docstring stringlengths 0 4.03k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
Bernstein (n i:nat) {struct n}: (i <= n) -> cpoly_cring R | :=
match n return (i <= n) -> cpoly_cring R with
O => fun _ => [1]
|S n' =>
match i return (i <= S n') -> cpoly_cring R with
O => fun _ => ([1][-]_X_)[*](Bernstein (Nat.le_0_l n'))
|S i' => fun p =>
match (le_lt_eq_dec _ _ p) with
| left p' => ([1][-]_X_)[*](Bernstein (proj1 (Nat.lt_succ_r _ _) p'))[... | Fixpoint | Bernstein | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"cpoly_cring"
] | [Bernstein n i] is the ith element of the n dimensional Bernstein basis | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
Bernstein_inv1 : forall n i (H:i < n) (H0:S i <= S n),
Bernstein H0[=]([1][-]_X_)[*](Bernstein (proj1 (Nat.lt_succ_r _ _) (proj1 (Nat.succ_lt_mono _ _) H)))[+]_X_[*](Bernstein (le_S_n _ _ H0)). | Proof.
intros n i H H0.
simpl (Bernstein H0).
destruct (le_lt_eq_dec _ _ H0).
replace (proj1 (Nat.lt_succ_r (S i) n) l) with (proj1 (Nat.lt_succ_r _ _) (proj1 (Nat.succ_lt_mono _ _) H)) by apply le_irrelevent.
reflexivity.
exfalso; lia.
Qed. | Lemma | Bernstein_inv1 | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"H0",
"apply",
"le_irrelevent"
] | These lemmas provide an induction principle for polynomials using the Bernstien basis | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
Bernstein_inv2 : forall n (H:S n <= S n),
Bernstein H[=]_X_[*](Bernstein (le_S_n _ _ H)). | Proof.
intros n H.
simpl (Bernstein H).
destruct (le_lt_eq_dec _ _ H).
exfalso; lia.
replace (proj1 (Nat.lt_succ_r n n) H) with (le_S_n n n H) by apply le_irrelevent.
reflexivity.
Qed. | Lemma | Bernstein_inv2 | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"apply",
"le_irrelevent"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Bernstein_ind : forall n i (H:i<=n) (P : nat -> nat -> cpoly_cring R -> Prop),
P 0 0 [1] ->
(forall n p, P n 0 p -> P (S n) 0 (([1][-]_X_)[*]p)) ->
(forall n p, P n n p -> P (S n) (S n) (_X_[*]p)) ->
(forall i n p q, (i < n) -> P n i p -> P n (S i) q -> P (S n) (S i) (([1][-]_X_)[*]q[+]_X_[*]p)) ->
P n i (Bernstein H). | Proof.
intros n i H P H0 H1 H2 H3.
revert n i H.
induction n; intros [|i] H.
apply H0.
exfalso; auto with *.
apply H1.
apply IHn.
simpl.
destruct (le_lt_eq_dec (S i) (S n)).
apply H3; auto with *.
inversion e.
revert H.
rewrite H5.
intros H.
apply H2.
auto with *.
Qed. | Lemma | Bernstein_ind | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"H0",
"H1",
"H2",
"H3",
"apply",
"cpoly_cring"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
partitionOfUnity : forall n, @Sumx (cpoly_cring R) _ (fun i H => Bernstein (proj1 (Nat.lt_succ_r i n) H)) [=][1]. | Proof.
induction n.
reflexivity.
set (A:=(fun (i : nat) (H : i < S n) => Bernstein (proj1 (Nat.lt_succ_r i n) H))) in *.
set (B:=(fun i => ([1][-]_X_)[*](part_tot_nat_fun (cpoly_cring R) _ A i)[+]_X_[*]match i with O => [0] | S i' => (part_tot_nat_fun _ _ A i') end)).
rewrite -> (fun a b => Sumx_Sum0 _ a b B).
... | Lemma | partitionOfUnity | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"Hi",
"Hj",
"Sum",
"Sum0",
"Sum0_plus_Sum0",
"Sum_first",
"Sum_last",
"Sum_shift",
"Sumx",
"Sumx_Sum0",
"Sumx_to_Sum",
"apply",
"cpoly_cring",
"eq_reflexive",
"le_irrelevent",
"mult_distr_sum0_lft",
"part_tot_nat_fun"
] | [1] important property of the Bernstein basis is that its elements form a partition of unity | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
RaiseDegreeA : forall n i (H:i<=n), (nring (S n))[*]_X_[*]Bernstein H[=](nring (S i))[*]Bernstein (le_n_S _ _ H). | Proof.
induction n.
intros [|i] H; [|exfalso; lia].
repeat split; ring.
intros i H.
change (nring (S (S n)):cpoly_cring R) with (nring (S n)[+][1]:cpoly_cring R).
rstepl (nring (S n)[*]_X_[*]Bernstein H[+]_X_[*]Bernstein H).
destruct i as [|i].
simpl (Bernstein H) at 1.
rstepl (([1][-]_X_)[*](nring (S n)[*... | Lemma | RaiseDegreeA | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"Bernstein_inv1",
"Bernstein_inv2",
"apply",
"cpoly_cring",
"le_irrelevent",
"nring",
"repeat",
"split"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
RaiseDegreeB : forall n i (H:i<=n), (nring (S n))[*]([1][-]_X_)[*]Bernstein H[=](nring (S n - i))[*]Bernstein (le_S _ _ H). | Proof.
induction n.
intros [|i] H; [|exfalso; lia].
repeat split; ring.
intros i H.
change (nring (S (S n)):cpoly_cring R) with (nring (S n)[+][1]:cpoly_cring R).
set (X0:=([1][-](@cpoly_var R))) in *.
rstepl (nring (S n)[*]X0[*]Bernstein H[+]X0[*]Bernstein H).
destruct i as [|i].
simpl (Bernstein H) at 1.
... | Lemma | RaiseDegreeB | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"Bernstein_inv1",
"apply",
"cpoly_cring",
"cpoly_var",
"le_irrelevent",
"nring",
"repeat",
"split"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
RaiseDegree : forall n i (H: i<=n),
(nring (S n))[*]Bernstein H[=](nring (S n - i))[*]Bernstein (le_S _ _ H)[+](nring (S i))[*]Bernstein (le_n_S _ _ H). | Proof.
intros n i H.
rstepl ((nring (S n))[*]([1][-]_X_)[*]Bernstein H[+](nring (S n))[*]_X_[*]Bernstein H).
rewrite RaiseDegreeA, RaiseDegreeB. reflexivity.
Qed. | Lemma | RaiseDegree | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"RaiseDegreeA",
"RaiseDegreeB",
"nring"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
evalBernsteinBasisH (n i:nat) (v:Vector.t R i) : i <= n -> cpoly_cring R | :=
match v in Vector.t _ i return i <= n -> cpoly_cring R with
|Vector.nil => fun _ => [0]
|Vector.cons a i' v' =>
match n as n return (S i' <= n) -> cpoly_cring R with
| O => fun p => False_rect _ (Nat.nle_succ_0 _ p)
| S n' => fun p => _C_ a[*]Bernstein (le_S_n _ _ p)[+]evalBernsteinBasisH v' (Nat.lt_le_incl _ ... | Fixpoint | evalBernsteinBasisH | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"Vector",
"cpoly_cring"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
evalBernsteinBasis (n:nat) (v:Vector.t R n) : cpoly_cring R | :=
evalBernsteinBasisH v (Nat.le_refl n). | Definition | evalBernsteinBasis | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Vector",
"cpoly_cring",
"evalBernsteinBasisH",
"le_refl"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Vbinary : forall (n : nat), Vector.t A n -> Vector.t A n -> Vector.t A n. | Proof.
induction n as [| n h]; intros v v0.
apply Vector.nil.
inversion v as [| a n0 H0 H1]; inversion v0 as [| a0 n1 H2 H3].
exact (Vector.cons (g a a0) n (h H0 H2)).
Defined. | Lemma | Vbinary | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"H0",
"H1",
"H2",
"H3",
"Vector",
"apply"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Vid n : Vector.t A n -> Vector.t A n | :=
match n with
| O => fun _ => Vector.nil
| S n' => fun v : Vector.t A (S n') => Vector.cons (Vector.hd v) _ (Vector.tl v)
end. | Definition | Vid | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Vector"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Vid_eq : forall (n:nat) (v:Vector.t A n), v = Vid v. | Proof.
destruct v; auto.
Qed. | Lemma | Vid_eq | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Vector",
"Vid"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
VSn_eq :
forall (n : nat) (v : Vector.t A (S n)), v = Vector.cons (Vector.hd v) _ (Vector.tl v). | Proof.
intros.
exact (Vid_eq v).
Qed. | Lemma | VSn_eq | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Vector",
"Vid_eq"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
V0_eq : forall (v : Vector.t A 0), v = Vector.nil. | Proof.
intros.
exact (Vid_eq v).
Qed. | Lemma | V0_eq | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Vector",
"Vid_eq"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
evalBernsteinBasisPlus : forall n (v1 v2: Vector.t R n),
evalBernsteinBasis (Vbinary (fun (x y:R)=>x[+]y) v1 v2)[=]evalBernsteinBasis v1[+]evalBernsteinBasis v2. | Proof.
unfold evalBernsteinBasis.
intros n.
generalize (Nat.le_refl n).
generalize n at 1 3 4 6 7 9 11.
intros i.
induction i.
intros l v1 v2.
rewrite (V0_eq v1), (V0_eq v2). ring.
intros l v1 v2.
destruct n as [|n].
exfalso; auto with *.
rewrite (VSn_eq v1), (VSn_eq v2).
simpl.
rewrite IHi.
rewrite... | Lemma | evalBernsteinBasisPlus | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"V0_eq",
"VSn_eq",
"Vbinary",
"Vector",
"c_plus",
"evalBernsteinBasis",
"le_refl"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
evalBernsteinBasisConst : forall n c,
evalBernsteinBasis (Vector.const c (S n))[=]_C_ c. | Proof.
intros n c.
stepr (evalBernsteinBasis (Vector.const c (S n))[+]_C_ c[*]Sum (S n) n (part_tot_nat_fun _ _ (fun (i : nat) (H : i < S n) => Bernstein (proj1 (Nat.lt_succ_r i n) H)))).
rewrite -> Sum_empty by auto with *.
ring.
unfold evalBernsteinBasis.
generalize (Nat.le_refl (S n)).
generalize (S n) at 1... | Lemma | evalBernsteinBasisConst | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"H'",
"Sum",
"Sum_empty",
"Sum_first",
"Sumx_to_Sum",
"Vector",
"apply",
"const",
"cpoly_cring",
"evalBernsteinBasis",
"evalBernsteinBasisH",
"le_irrelevent",
"le_refl",
"part_tot_nat_fun",
"partitionOfUnity"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
BernsteinBasisTimesXH (n i:nat) (v:Vector.t R i) : i <= n -> Vector.t R (S i) | :=
match v in Vector.t _ i return i <= n -> Vector.t R (S i) with
| Vector.nil => fun _ => Vector.cons [0] _ Vector.nil
| Vector.cons a i' v' => match n as n return S i' <= n -> Vector.t R (S (S i')) with
| O => fun p => False_rect _ (Nat.nle_succ_0 _ p)
| S n' => fun p => Vector.cons (eta(Qred (i#P_of_succ_nat n')... | Fixpoint | BernsteinBasisTimesXH | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Vector",
"eta"
] | To convert a polynomial to the Bernstein basis, we need to know how to
multiply a bernstein basis element by [_X_] can convert it to the Bernstein basis.
At this point we must work with rational coeffients. So we assume there is a
ring homomorphism from [Q] to R | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
BernsteinBasisTimesX (n:nat) (v:Vector.t R n) : Vector.t R (S n) | :=
BernsteinBasisTimesXH v (Nat.le_refl n). | Definition | BernsteinBasisTimesX | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"BernsteinBasisTimesXH",
"Vector",
"le_refl"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
evalBernsteinBasisTimesX : forall n (v:Vector.t R n),
evalBernsteinBasis (BernsteinBasisTimesX v)[=]_X_[*]evalBernsteinBasis v. | Proof.
intros n.
unfold evalBernsteinBasis, BernsteinBasisTimesX.
generalize (Nat.le_refl (S n)) (Nat.le_refl n).
generalize n at 1 3 5 7 9 11.
intros i.
induction i.
intros l l0 v.
rewrite (V0_eq v).
simpl.
rewrite <- c_zero. ring.
intros l l0 v.
destruct n as [|n].
exfalso; auto with *.
rewrite (VS... | Lemma | evalBernsteinBasisTimesX | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"BernsteinBasisTimesX",
"Hn",
"Q_as_CField",
"Qap",
"RHcompose",
"RaiseDegreeA",
"V0_eq",
"VSn_eq",
"Vector",
"Zpos",
"apply",
"c_mult",
"c_zero",
"cpoly_cring",
"csbf_wd",
"csf_wd",
"div_1",
"eta",
"evalBernsteinBasis",
"le_irrelevent",
"le_refl",
"mult_asso... | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
BernsteinCoefficents (p:cpoly_cring R) : sigT (Vector.t R) | :=
match p with
| cpoly_zero _ => existT _ _ Vector.nil
| cpoly_linear _ c p' =>
let (n', b') := (BernsteinCoefficents p') in
existT _ _ (Vbinary (fun (x y:R)=>x[+]y) (Vector.const c _) (BernsteinBasisTimesX b'))
end. | Fixpoint | BernsteinCoefficents | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"BernsteinBasisTimesX",
"Vbinary",
"Vector",
"b'",
"const",
"cpoly_cring"
] | Convert a polynomial to the Bernstein basis | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
evalBernsteinCoefficents : forall p, (let (n,b) := BernsteinCoefficents p in evalBernsteinBasis b)[=]p. | Proof.
induction p.
reflexivity.
simpl.
destruct (BernsteinCoefficents p).
rewrite -> evalBernsteinBasisPlus.
rewrite -> evalBernsteinBasisConst.
rewrite -> evalBernsteinBasisTimesX.
rewrite -> IHp.
rewrite -> poly_linear.
ring.
Qed. | Lemma | evalBernsteinCoefficents | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"BernsteinCoefficents",
"evalBernsteinBasis",
"evalBernsteinBasisConst",
"evalBernsteinBasisPlus",
"evalBernsteinBasisTimesX",
"poly_linear"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
BernsteinNonNeg : forall x:F, [0] [<=] x -> x [<=] [1] ->
forall n i (p:Nat.le i n), [0][<=](Bernstein F p)!x. | Proof.
intros x Hx0 Hx1.
induction n.
intros i p.
simpl (Bernstein F p).
autorewrite with apply.
auto with *.
intros [|i] p; simpl (Bernstein F p).
autorewrite with apply.
auto with *.
destruct (le_lt_eq_dec (S i) (S n) p); autorewrite with apply; auto with *.
Qed. | Lemma | BernsteinNonNeg | algebra | algebra/Bernstein.v | [
"CoRN.algebra.CPolynomials",
"CoRN.algebra.CSums",
"CoRN.tactics.Rational",
"CoRN.model.ordfields.Qordfield",
"CoRN.algebra.COrdFields2",
"CoRN.algebra.CRing_Homomorphisms",
"Coq.Vectors.Vector",
"Vector.VectorNotations",
"Coq",
"Lia"
] | [
"Bernstein",
"Hx0",
"apply",
"le"
] | A second important property of the Bernstein polynomials is that they
are all non-negative on the unit interval. | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
is_CAbGroup (G : CGroup) | := commutes (csg_op (c:=G)). | Definition | is_CAbGroup | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"CGroup",
"commutes"
] | * Abelian Groups
Now we introduce commutativity and add some results. | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
CAbGroup : Type | :=
{cag_crr : CGroup;
cag_proof : is_CAbGroup cag_crr}. | Record | CAbGroup | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"CGroup",
"cag_crr",
"is_CAbGroup"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
cag_crr : CAbGroup >-> CGroup. | Coercion | cag_crr | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"CAbGroup",
"CGroup"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | ||
CAbGroup_is_CAbGroup : is_CAbGroup G. | Proof.
elim G; auto.
Qed. | Lemma | CAbGroup_is_CAbGroup | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"is_CAbGroup"
] | %\begin{convention}% Let [G] be an Abelian Group.
%\end{convention}% | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
cag_commutes : commutes (csg_op (c:=G)). | Proof.
exact CAbGroup_is_CAbGroup.
Qed. | Lemma | cag_commutes | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"CAbGroup_is_CAbGroup",
"commutes"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
cag_commutes_unfolded : forall x y : G, x[+]y [=] y[+]x. | Proof cag_commutes. | Lemma | cag_commutes_unfolded | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"cag_commutes"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
subcrr : CGroup | := Build_SubCGroup _ _ Punit op_pres_P inv_pres_P. | Let | subcrr | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Build_SubCGroup",
"CGroup"
] | %\begin{convention}%
Let [G] be an Abelian Group and [P] be a ([CProp]-valued) predicate on [G]
that contains [Zero] and is closed under [[+]] and [[--]].
%\end{convention}% | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
isabgrp_scrr : is_CAbGroup subcrr. | Proof.
red in |- *. intros x y. case x. case y. intros. simpl in |- *. apply cag_commutes_unfolded.
Qed. | Lemma | isabgrp_scrr | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"apply",
"cag_commutes_unfolded",
"is_CAbGroup",
"subcrr"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Build_SubCAbGroup : CAbGroup | := Build_CAbGroup subcrr isabgrp_scrr. | Definition | Build_SubCAbGroup | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"CAbGroup",
"isabgrp_scrr",
"subcrr"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
cag_op_inv : forall x y : G, [--] (x[+]y) [=] [--]x[+] [--]y. | Proof.
intros x y.
astepr ([--]y[+] [--]x).
apply cg_inv_op.
Qed. | Lemma | cag_op_inv | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"apply",
"astepr",
"cg_inv_op"
] | %\begin{convention}% Let [G] be an Abelian Group.
%\end{convention}% | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
assoc_1 : forall x y z : G, x[-] (y[-]z) [=] x[-]y[+]z. | Proof.
intros x y z; unfold cg_minus in |- *.
astepr (x[+]([--]y[+]z)).
Step_final (x[+]([--]y[+] [--][--]z)).
Qed. | Lemma | assoc_1 | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"astepr",
"cg_minus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
minus_plus : forall x y z : G, x[-] (y[+]z) [=] x[-]y[-]z. | Proof.
intros x y z.
unfold cg_minus in |- *.
Step_final (x[+]([--]y[+] [--]z)).
Qed. | Lemma | minus_plus | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"cg_minus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
op_lft_resp_ap : forall x y z : G, y [#] z -> x[+]y [#] x[+]z. | Proof.
intros x y z H.
astepl (y[+]x).
astepr (z[+]x).
apply op_rht_resp_ap; assumption.
Qed. | Lemma | op_lft_resp_ap | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"apply",
"astepl",
"astepr",
"op_rht_resp_ap"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
cag_ap_cancel_lft : forall x y z : G, x[+]y [#] x[+]z -> y [#] z. | Proof.
intros x y z H.
apply ap_symmetric_unfolded.
apply cg_ap_cancel_rht with x.
apply ap_symmetric_unfolded.
astepl (x[+]y).
astepr (x[+]z).
auto.
Qed. | Lemma | cag_ap_cancel_lft | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"ap_symmetric_unfolded",
"apply",
"astepl",
"astepr",
"cg_ap_cancel_rht"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
plus_cancel_ap_lft : forall x y z : G, z[+]x [#] z[+]y -> x [#] y. | Proof.
intros x y z H.
apply cag_ap_cancel_lft with z.
assumption.
Qed. | Lemma | plus_cancel_ap_lft | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"apply",
"cag_ap_cancel_lft"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
plus_lext : forall x y z : S, plus x z [#] plus y z -> x [#] y. | Hypothesis | plus_lext | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"plus"
] | %\begin{convention}%
Let [S] be a Setoid and [unit:S], [plus:S->S->S] and [inv] a unary
setoid operation on [S].
Assume that [plus] is commutative, associative and `left-strongly-extensional
([(plus x z) [#] (plus y z) -> x [#] y]), that [unit] is a left-unit
for [plus] and [(inv x)] is a right-inverse of [x] w.r.t.%\%... | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
plus_lunit : forall x : S, plus unit x [=] x. | Hypothesis | plus_lunit | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"plus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | ||
plus_comm : forall x y : S, plus x y [=] plus y x. | Hypothesis | plus_comm | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"plus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | ||
plus_assoc : associative plus. | Hypothesis | plus_assoc | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"associative",
"plus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | ||
inv_inv : forall x : S, plus x (inv x) [=] unit. | Hypothesis | inv_inv | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"inv",
"plus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | ||
plus_rext : forall x y z : S, plus x y [#] plus x z -> y [#] z. | Proof.
intros x y z H.
apply plus_lext with x.
astepl (plus x y).
astepr (plus x z).
auto.
Qed. | Lemma | plus_rext | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"apply",
"astepl",
"astepr",
"plus",
"plus_lext"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
plus_runit : forall x : S, plus x unit [=] x. | Proof.
intro x.
Step_final (plus unit x).
Qed. | Lemma | plus_runit | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"plus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
plus_is_fun : bin_fun_strext _ _ _ plus. | Proof.
intros x x' y y' H.
elim (ap_cotransitive_unfolded _ _ _ H (plus x y')); intro H'.
right; apply plus_lext with x.
astepl (plus x y); astepr (plus x y'); auto.
left; eauto.
Qed. | Lemma | plus_is_fun | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"H'",
"ap_cotransitive_unfolded",
"apply",
"astepl",
"astepr",
"bin_fun_strext",
"plus",
"plus_lext"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
inv_inv' : forall x : S, plus (inv x) x [=] unit. | Proof.
intro.
Step_final (plus x (inv x)).
Qed. | Lemma | inv_inv' | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"inv",
"plus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
plus_fun : CSetoid_bin_op S | := Build_CSetoid_bin_fun _ _ _ plus plus_is_fun. | Definition | plus_fun | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"CSetoid_bin_op",
"plus",
"plus_is_fun"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Build_CSemiGroup' : CSemiGroup. | Proof.
apply Build_CSemiGroup with S plus_fun.
exact plus_assoc.
Defined. | Definition | Build_CSemiGroup' | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"CSemiGroup",
"apply",
"plus_assoc",
"plus_fun"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Build_CMonoid' : CMonoid. | Proof.
apply Build_CMonoid with Build_CSemiGroup' unit.
apply Build_is_CMonoid.
exact plus_runit.
exact plus_lunit.
Defined. | Definition | Build_CMonoid' | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Build_CSemiGroup'",
"CMonoid",
"apply",
"plus_lunit",
"plus_runit"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Build_CGroup' : CGroup. | Proof.
apply Build_CGroup with Build_CMonoid' inv.
split.
auto.
apply inv_inv'.
Defined. | Definition | Build_CGroup' | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Build_CMonoid'",
"CGroup",
"apply",
"inv",
"inv_inv'",
"split"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Build_CAbGroup' : CAbGroup. | Proof.
apply Build_CAbGroup with Build_CGroup'.
exact plus_comm.
Defined. | Definition | Build_CAbGroup' | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Build_CGroup'",
"CAbGroup",
"apply",
"plus_comm"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
nmult (a:G) (n:nat) {struct n} : G | :=
match n with
| O => [0]
| S p => a[+]nmult a p
end. | Fixpoint | nmult | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
nmult_wd : forall (x y:G) (n m:nat), (x [=] y) -> n = m -> nmult x n [=] nmult y m. | Proof.
simple induction n; intros.
rewrite <- H0; algebra.
rewrite <- H1; simpl in |- *; algebra.
Qed. | Lemma | nmult_wd | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"H0",
"H1",
"algebra",
"nmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
nmult_one : forall x:G, nmult x 1 [=] x. | Proof.
simpl in |- *; algebra.
Qed. | Lemma | nmult_one | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"algebra",
"nmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
nmult_Zero : forall n:nat, nmult [0] n [=] [0]. | Proof.
intro n.
induction n.
algebra.
simpl in |- *; Step_final (([0]:G)[+][0]).
Qed. | Lemma | nmult_Zero | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"algebra",
"nmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
nmult_plus : forall m n x, nmult x m[+]nmult x n [=] nmult x (m + n). | Proof.
simple induction m.
simpl in |- *; algebra.
clear m; intro m.
intros.
simpl in |- *. Step_final (x[+](nmult x m[+]nmult x n)).
Qed. | Lemma | nmult_plus | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"algebra",
"nmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
nmult_mult : forall n m x, nmult (nmult x m) n [=] nmult x (m * n). | Proof.
simple induction n.
intro. rewrite Nat.mul_0_r. algebra.
clear n; intros.
simpl in |- *.
rewrite Nat.mul_comm. simpl in |- *.
eapply eq_transitive_unfolded.
2: apply nmult_plus.
rewrite Nat.mul_comm. algebra.
Qed. | Lemma | nmult_mult | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"algebra",
"apply",
"eq_transitive_unfolded",
"nmult",
"nmult_plus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
nmult_inv : forall n x, nmult [--]x n [=] [--] (nmult x n). | Proof.
intro; induction n; simpl in |- *.
algebra.
intros.
Step_final ([--]x[+] [--](nmult x n)).
Qed. | Lemma | nmult_inv | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"algebra",
"nmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
nmult_plus' : forall n x y, nmult x n[+]nmult y n [=] nmult (x[+]y) n. | Proof.
intro; induction n; simpl in |- *; intros.
algebra.
astepr (x[+]y[+](nmult x n[+]nmult y n)).
astepr (x[+](y[+](nmult x n[+]nmult y n))).
astepr (x[+](y[+]nmult x n[+]nmult y n)).
astepr (x[+](nmult x n[+]y[+]nmult y n)).
Step_final (x[+](nmult x n[+](y[+]nmult y n))).
Qed. | Lemma | nmult_plus' | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"algebra",
"astepr",
"nmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
zmult a z | := caseZ_diff z (fun n m => nmult a n[-]nmult a m). | Definition | zmult | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"caseZ_diff",
"nmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
zmult_char : forall (m n:nat) z, z = (m - n)%Z ->
forall x, zmult x z [=] nmult x m[-]nmult x n. | Proof.
simple induction z; intros.
simpl in |- *.
replace m with n. Step_final ([0]:G). auto with zarith.
simpl in |- *.
astepl (nmult x (nat_of_P p)).
apply cg_cancel_rht with (nmult x n).
astepr (nmult x m).
astepl (nmult x (nat_of_P p + n)).
apply nmult_wd; algebra.
rewrite <- convert_is_POS i... | Lemma | zmult_char | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"algebra",
"apply",
"astepl",
"astepr",
"cg_cancel_lft",
"cg_cancel_rht",
"cg_minus",
"convert_is_POS",
"min_convert_is_NEG",
"nat_of_P",
"nmult",
"nmult_wd",
"un_op_wd_unfolded",
"zmult"
] | Lemma Zeq_imp_nat_eq : forall m n:nat, m = n -> m = n.
auto.
intro m; induction m.
intro n; induction n; auto.
intro; induction n.
intro. inversion H.
intros.
rewrite (IHm n).
auto.
repeat rewrite inj_S in H.
auto with zarith.
Qed. | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
zmult_wd : forall (x y:G) (n m:Z), (x [=] y) -> n = m -> zmult x n [=] zmult y m. | Proof.
do 3 intro.
case n; intros; inversion H0.
algebra.
unfold zmult in |- *.
simpl in |- *.
astepl (nmult x (nat_of_P p)); Step_final (nmult y (nat_of_P p)).
simpl in |- *.
astepl [--](nmult x (nat_of_P p)).
Step_final [--](nmult y (nat_of_P p)).
Qed. | Lemma | zmult_wd | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"H0",
"Step_final",
"algebra",
"astepl",
"nat_of_P",
"nmult",
"zmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
zmult_one : forall x:G, zmult x 1 [=] x. | Proof.
simpl in |- *; algebra.
Qed. | Lemma | zmult_one | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"algebra",
"zmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
zmult_min_one : forall x:G, zmult x (-1) [=] [--]x. | Proof.
intros; simpl in |- *; Step_final ([0][-]x).
Qed. | Lemma | zmult_min_one | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"zmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
zmult_zero : forall x:G, zmult x 0 [=] [0]. | Proof.
simpl in |- *; algebra.
Qed. | Lemma | zmult_zero | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"algebra",
"zmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
zmult_Zero : forall k:Z, zmult [0] k [=] [0]. | Proof.
intro; induction k; simpl in |- *.
algebra.
Step_final (([0]:G)[-][0]).
Step_final (([0]:G)[-][0]).
Qed. | Lemma | zmult_Zero | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"algebra",
"zmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
zmult_plus : forall m n x, zmult x m[+]zmult x n [=] zmult x (m + n). | Proof.
intros; case m; case n; intros.
simpl in |- *; Step_final ([0][+]([0][-][0]):G).
simpl in |- *; Step_final ([0][+](nmult x (nat_of_P p)[-][0])).
simpl in |- *; Step_final ([0][+]([0][-]nmult x (nat_of_P p))).
simpl in |- *; Step_final (nmult x (nat_of_P p)[-][0][+][0]).
simpl ... | Lemma | zmult_plus | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"Zpos",
"apply",
"astepl",
"astepr",
"convert_is_POS",
"eq_symmetric_unfolded",
"min_convert_is_NEG",
"nat_of_P",
"nmult",
"nmult_plus",
"un_op_wd_unfolded",
"zmult",
"zmult_char"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
zmult_mult : forall m n x, zmult (zmult x m) n [=] zmult x (m * n). | Proof.
simple induction m; simple induction n; simpl in |- *; intros.
Step_final ([0][-][0][+]([0]:G)).
astepr ([0]:G). astepl (nmult ([0][-][0]) (nat_of_P p)).
Step_final (nmult [0] (nat_of_P p)).
astepr [--]([0]:G). astepl [--](nmult ([0][-][0]) (nat_of_P p)).
Step_final [--](n... | Lemma | zmult_mult | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"Step_final",
"algebra",
"apply",
"astepl",
"astepr",
"eq_transitive_unfolded",
"nat_of_P",
"nmult",
"nmult_inv",
"nmult_mult",
"un_op_wd_unfolded",
"zmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
zmult_plus' : forall z x y, zmult x z[+]zmult y z [=] zmult (x[+]y) z. | Proof.
intro z; pattern z in |- *.
apply nats_Z_ind.
intro n; case n.
intros; simpl in |- *. Step_final (([0]:G)[+]([0][-][0])).
clear n; intros.
rewrite POS_anti_convert; simpl in |- *. set (p := nat_of_P (P_of_succ_nat n)) in *.
astepl (nmult x p[+]nmult y p). Step_final (nmult (x[+]y) p).
intro n; ca... | Lemma | zmult_plus' | algebra | algebra/CAbGroups.v | [
"CoRN.algebra.CGroups",
"CGroups.coercions"
] | [
"NEG_anti_convert",
"POS_anti_convert",
"Step_final",
"apply",
"astepl",
"astepr",
"nat_of_P",
"nats_Z_ind",
"nmult",
"zmult"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
is_CAbMonoid (G : CMonoid) | := commutes (csg_op (c:=G)). | Definition | is_CAbMonoid | algebra | algebra/CAbMonoids.v | [
"CoRN.algebra.CMonoids",
"CoRN.util.SetoidPermutation",
"Coq.Setoids.Setoid",
"Coq.Classes.Morphisms"
] | [
"CMonoid",
"commutes"
] | * Abelian Monoids
Now we introduce commutativity and add some results. | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
CAbMonoid : Type | :=
{cam_crr :> CMonoid;
cam_proof : is_CAbMonoid cam_crr}. | Record | CAbMonoid | algebra | algebra/CAbMonoids.v | [
"CoRN.algebra.CMonoids",
"CoRN.util.SetoidPermutation",
"Coq.Setoids.Setoid",
"Coq.Classes.Morphisms"
] | [
"CMonoid",
"is_CAbMonoid"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
CAbMonoid_is_CAbMonoid : is_CAbMonoid M. | Proof.
elim M; auto.
Qed. | Lemma | CAbMonoid_is_CAbMonoid | algebra | algebra/CAbMonoids.v | [
"CoRN.algebra.CMonoids",
"CoRN.util.SetoidPermutation",
"Coq.Setoids.Setoid",
"Coq.Classes.Morphisms"
] | [
"is_CAbMonoid"
] | %\begin{convention}% Let [M] be an abelian monoid.
%\end{convention}% | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
cam_commutes : commutes (csg_op (c:=M)). | Proof.
exact CAbMonoid_is_CAbMonoid.
Qed. | Lemma | cam_commutes | algebra | algebra/CAbMonoids.v | [
"CoRN.algebra.CMonoids",
"CoRN.util.SetoidPermutation",
"Coq.Setoids.Setoid",
"Coq.Classes.Morphisms"
] | [
"CAbMonoid_is_CAbMonoid",
"commutes"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
cam_commutes_unfolded : forall x y : M, x[+]y [=] y[+]x. | Proof cam_commutes. | Lemma | cam_commutes_unfolded | algebra | algebra/CAbMonoids.v | [
"CoRN.algebra.CMonoids",
"CoRN.util.SetoidPermutation",
"Coq.Setoids.Setoid",
"Coq.Classes.Morphisms"
] | [
"cam_commutes"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
cm_Sum_AbMonoid_Proper: forall {M: CAbMonoid},
Proper (SetoidPermutation (@st_eq M) ==> @st_eq M) cm_Sum. | Proof.
repeat intro.
apply cm_Sum_Proper.
apply cam_proof.
assumption.
Qed. | Instance | cm_Sum_AbMonoid_Proper | algebra | algebra/CAbMonoids.v | [
"CoRN.algebra.CMonoids",
"CoRN.util.SetoidPermutation",
"Coq.Setoids.Setoid",
"Coq.Classes.Morphisms"
] | [
"CAbMonoid",
"SetoidPermutation",
"apply",
"cm_Sum",
"cm_Sum_Proper",
"repeat"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
subcrr : CMonoid | := Build_SubCMonoid _ _ Punit op_pres_P. | Let | subcrr | algebra | algebra/CAbMonoids.v | [
"CoRN.algebra.CMonoids",
"CoRN.util.SetoidPermutation",
"Coq.Setoids.Setoid",
"Coq.Classes.Morphisms"
] | [
"Build_SubCMonoid",
"CMonoid"
] | %\begin{convention}%
Let [M] be an Abelian Monoid and [P] be a ([CProp]-valued) predicate on [M]
that contains [Zero] and is closed under [[+]] and [[--]].
%\end{convention}% | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
isabgrp_scrr : is_CAbMonoid subcrr. | Proof.
red in |- *. intros x y. case x. case y. intros.
simpl in |- *. apply cam_commutes_unfolded.
Qed. | Lemma | isabgrp_scrr | algebra | algebra/CAbMonoids.v | [
"CoRN.algebra.CMonoids",
"CoRN.util.SetoidPermutation",
"Coq.Setoids.Setoid",
"Coq.Classes.Morphisms"
] | [
"apply",
"cam_commutes_unfolded",
"is_CAbMonoid",
"subcrr"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Build_SubCAbMonoid : CAbMonoid | := Build_CAbMonoid _ isabgrp_scrr. | Definition | Build_SubCAbMonoid | algebra | algebra/CAbMonoids.v | [
"CoRN.algebra.CMonoids",
"CoRN.util.SetoidPermutation",
"Coq.Setoids.Setoid",
"Coq.Classes.Morphisms"
] | [
"CAbMonoid",
"isabgrp_scrr"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_Set | := CauchySeq F. | Definition | R_Set | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"CauchySeq"
] | ** Setoid Structure
[R_Set] is the setoid of Cauchy sequences over [F]; given two sequences
[x,y] over [F], we say that [x] is smaller than [y] if from some point
onwards [(y n) [-] (x n)] is greater than some fixed, positive
[e]. Apartness of two sequences means that one of them is smaller
than the other, equality i... | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
R_lt (x y : R_Set) | := {N : nat |
{e : F | [0] [<] e | forall n, N <= n -> e [<=] CS_seq _ y n[-]CS_seq _ x n}}. | Definition | R_lt | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"R_Set"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_ap (x y : R_Set) | := R_lt x y or R_lt y x. | Definition | R_ap | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"R_Set",
"R_lt"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_eq (x y : R_Set) | := Not (R_ap x y). | Definition | R_eq | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"Not",
"R_Set",
"R_ap"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_lt_cotrans : cotransitive R_lt. | Proof.
red in |- *.
intros x y.
elim x; intros x_ px.
elim y; intros y_ py.
intros Hxy z.
elim z; intros z_ pz.
elim Hxy; intros N H.
elim H; clear Hxy H; intros e He HN.
simpl in HN.
set (e3 := e [/]ThreeNZ) in *.
cut ([0] [<] e3); [ intro He3 | unfold e3 in |- *; apply pos_div_three; auto ].
set (e6 := e ... | Lemma | R_lt_cotrans | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"He",
"Hn",
"R_lt",
"apply",
"astepl",
"cg_minus",
"cotransitive",
"leEq_transitive",
"leEq_wdl",
"le_trans",
"less_cotransitive_unfolded",
"less_leEq",
"less_leEq_trans",
"less_wdl",
"max",
"minus_resp_leEq_rht",
"plus_resp_leEq_both",
"pos_div_six",
"pos_div_three",
"pos_div_... | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_ap_cotrans : cotransitive R_ap. | Proof.
red in |- *; intros x y Hxy z.
elim Hxy; intro H; elim (R_lt_cotrans _ _ H z); unfold R_ap in |- *; auto.
Qed. | Lemma | R_ap_cotrans | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"R_ap",
"R_lt_cotrans",
"cotransitive"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_ap_symmetric : Csymmetric R_ap. | Proof.
red in |- *; intros x y Hxy.
elim Hxy; unfold R_ap in |- *; auto.
Qed. | Lemma | R_ap_symmetric | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"Csymmetric",
"R_ap"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_lt_irreflexive : irreflexive R_lt. | Proof.
red in |- *; intros x Hx.
elim Hx; intros N HN.
elim HN; clear Hx HN; intros e He HN.
apply (ap_irreflexive_unfolded _ (x N)).
apply less_imp_ap.
apply less_leEq_trans with (x N[+]e).
astepl (x N[+][0]); apply plus_resp_less_lft; auto.
apply shift_plus_leEq'; auto with arith.
Qed. | Lemma | R_lt_irreflexive | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"He",
"Hx",
"R_lt",
"ap_irreflexive_unfolded",
"apply",
"astepl",
"irreflexive",
"less_imp_ap",
"less_leEq_trans",
"plus_resp_less_lft",
"shift_plus_leEq'"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_ap_irreflexive : irreflexive R_ap. | Proof.
red in |- *; intros x Hx.
elim (R_lt_irreflexive x).
elim Hx; auto.
Qed. | Lemma | R_ap_irreflexive | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"Hx",
"R_ap",
"R_lt_irreflexive",
"irreflexive"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_ap_eq_tight : tight_apart R_eq R_ap. | Proof.
split; auto.
Qed. | Lemma | R_ap_eq_tight | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"R_ap",
"R_eq",
"split",
"tight_apart"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_CSetoid : CSetoid. | Proof.
apply Build_CSetoid with R_Set R_eq R_ap.
split.
exact R_ap_irreflexive.
exact R_ap_symmetric.
exact R_ap_cotrans.
exact R_ap_eq_tight.
Defined. | Definition | R_CSetoid | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"Build_CSetoid",
"CSetoid",
"R_Set",
"R_ap",
"R_ap_cotrans",
"R_ap_eq_tight",
"R_ap_irreflexive",
"R_ap_symmetric",
"R_eq",
"apply",
"split"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_plus (x y : R_CSetoid) : R_CSetoid | :=
Build_CauchySeq _ _ (CS_seq_plus F _ _ (CS_proof _ x) (CS_proof _ y)). | Definition | R_plus | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"CS_seq_plus",
"R_CSetoid"
] | ** Group Structure
The group structure is just the expected one; the lemmas which
are specifically proved are just the necessary ones to get the group axioms. | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
R_zero | := Build_CauchySeq _ _ (CS_seq_const F [0]). | Definition | R_zero | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"CS_seq_const"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_plus_lft_ext : forall x y z, R_plus x z [#] R_plus y z -> x [#] y. | Proof.
intros x y z Hxy.
elim Hxy; clear Hxy; intro H; [ left | right ]; elim H; intros N HN; elim HN;
clear H HN; intros e He HN; exists N; exists e; auto; intros n Hn; simpl in HN.
rstepr (CS_seq _ y n[+]CS_seq _ z n[-] (CS_seq _ x n[+]CS_seq _ z n)); auto.
rstepr (CS_seq _ x n[+]CS_seq _ z n[-] (CS_seq _ y n... | Lemma | R_plus_lft_ext | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"He",
"Hn",
"R_plus"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_plus_assoc : associative R_plus. | Proof.
intros x y z Hap.
elim Hap; clear Hap; intro H; elim H; intros N HN; elim HN; clear H HN;
intros e He HN; simpl in HN; apply (less_irreflexive_unfolded _ e).
apply leEq_less_trans with (CS_seq _ x N[+]CS_seq _ y N[+]CS_seq _ z N[-]
(CS_seq _ x N[+] (CS_seq _ y N[+]CS_seq _ z N))); auto.
rstepl ([0]:... | Lemma | R_plus_assoc | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"Hap",
"He",
"R_plus",
"apply",
"associative",
"leEq_less_trans",
"less_irreflexive_unfolded"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_zero_lft_unit : forall x, R_plus R_zero x [=] x. | Proof.
intro x; intro x_ap.
apply (R_lt_irreflexive x).
elim x_ap; clear x_ap; intro x_lt; elim x_lt; intros N H; elim H;
clear x_lt H; intros e He HN; exists N; exists e; auto; simpl in HN; intros n Hn.
astepr (CS_seq _ x n[-] ([0][+]CS_seq _ x n)); auto.
astepr ([0][+]CS_seq _ x n[-]CS_seq _ x n); auto.
Qed. | Lemma | R_zero_lft_unit | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"He",
"Hn",
"R_lt_irreflexive",
"R_plus",
"R_zero",
"apply",
"astepr"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_plus_comm : forall x y, R_plus x y [=] R_plus y x. | Proof.
intros x y Hxy.
elim Hxy; clear Hxy; intro H; elim H; intros N HN; elim HN; clear H HN;
intros e He HN; simpl in HN; apply (less_irreflexive_unfolded _ e).
apply leEq_less_trans with (CS_seq _ y N[+]CS_seq _ x N[-] (CS_seq _ x N[+]CS_seq _ y N)); auto.
rstepl ([0]:F); auto.
apply leEq_less_trans with (... | Lemma | R_plus_comm | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"He",
"R_plus",
"apply",
"leEq_less_trans",
"less_irreflexive_unfolded"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_inv (x : R_CSetoid) : R_CSetoid | :=
Build_CauchySeq _ _ (CS_seq_inv F _ (CS_proof _ x)). | Definition | R_inv | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"CS_seq_inv",
"R_CSetoid"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_inv_is_inv : forall x, R_plus x (R_inv x) [=] R_zero. | Proof.
intro x; intro x_ap.
apply (R_lt_irreflexive R_zero).
elim x_ap; clear x_ap; intro x_lt; elim x_lt; intros N H; elim H;
clear x_lt H; intros e He HN; exists N; exists e; auto; simpl in HN; intros n Hn.
simpl in |- *; astepr ([0][-] (CS_seq _ x n[+][--] (CS_seq _ x n))); auto.
simpl in |- *; astepr (CS_s... | Lemma | R_inv_is_inv | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"He",
"Hn",
"R_inv",
"R_lt_irreflexive",
"R_plus",
"R_zero",
"apply",
"astepr"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
R_inv_ext : un_op_strext _ R_inv. | Proof.
intros x y Hxy.
elim Hxy; clear Hxy; intro x_lt; [ right | left ]; elim x_lt; intros N H;
elim H; clear x_lt H; intros e He HN; exists N; exists e; auto; simpl in HN; intros n Hn.
rstepr ([--] (CS_seq _ y n) [-][--] (CS_seq _ x n)); auto.
rstepr ([--] (CS_seq _ x n) [-][--] (CS_seq _ y n)); auto.
Qed. | Lemma | R_inv_ext | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"He",
"Hn",
"R_inv",
"un_op_strext"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f | |
Rinv : CSetoid_un_op R_CSetoid. | Proof.
red in |- *.
apply Build_CSetoid_un_op with R_inv.
exact R_inv_ext.
Defined. | Definition | Rinv | algebra | algebra/Cauchy_COF.v | [
"CoRN.algebra.COrdCauchy",
"CoRN.tactics.RingReflection"
] | [
"Build_CSetoid_un_op",
"CSetoid_un_op",
"R_CSetoid",
"R_inv",
"R_inv_ext",
"apply"
] | https://github.com/coq-community/corn | ada7c0b497ff15dd67cf7932c6f20e143a2aee2f |
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