MathClasses.theory.cut_minus
Require
orders.semirings.
Require Import
Ring abstract_algebra interfaces.additional_operations
interfaces.orders orders.minmax.
Section cut_minus_properties.
Context `{FullPseudoSemiRingOrder R} `{!TrivialApart R}
`{∀ x y, Decision (x = y)} `{!CutMinusSpec R cm}.
Local Existing Instance pseudo_srorder_semiring.
Add Ring SR: (rings.stdlib_semiring_theory R).
Hint Resolve (@orders.le_flip R _ _).
Global Instance cut_minus_proper: Proper ((=) ==> (=) ==> (=)) cut_minus | 1.
Proof.
intros x₁ x₂ E y₁ y₂ F.
destruct (total (≤) x₂ y₂).
rewrite cut_minus_0, cut_minus_0; try easy. now rewrite E, F.
apply (right_cancellation (+) y₂).
rewrite cut_minus_le by easy.
rewrite <-E, <-F. apply cut_minus_le. now rewrite E, F.
Qed.
Global Instance cut_minus_mor_1: ∀ x : R, Setoid_Morphism (x ∸) | 0.
Proof. split; try apply _. Qed.
Global Instance cut_minus_mor_2: ∀ x : R, Setoid_Morphism (∸ x) | 0.
Proof. split; try apply _. solve_proper. Qed.
Hint Resolve (cut_minus_0).
Hint Resolve (cut_minus_le).
Lemma cut_minus_diag x : x ∸ x = 0.
Proof. now apply cut_minus_0. Qed.
Lemma cut_minus_nonneg_0_r x : 0 ≤ x → x ∸ 0 = x.
Proof. intros E. rewrite <-(rings.plus_0_r (x ∸ 0)). auto. Qed.
Lemma cut_minus_0_l x : 0 ≤ x → 0 ∸ x = 0.
Proof. auto. Qed.
Lemma cut_minus_nonpos_0_r x : x ≤ 0 → x ∸ 0 = 0.
Proof. auto. Qed.
Lemma cut_minus_nonneg x y : 0 ≤ x ∸ y.
Proof.
destruct (total (≤) x y) as [E|E].
apply orders.eq_le. symmetry. now auto.
apply (order_reflecting (+ y)).
now rewrite cut_minus_le; ring_simplify.
Qed.
Lemma cut_minus_le_r x y : y ≤ x → x ∸ y + y = x.
Proof cut_minus_le x y.
Lemma cut_minus_le_l x y : y ≤ x → y + (x ∸ y) = x.
Proof. rewrite commutativity. now apply cut_minus_le. Qed.
Lemma cut_minus_le_trans x y z : y ≤ x → z ≤ y → (x ∸ y) + (y ∸ z) = x ∸ z.
Proof.
intros. apply (right_cancellation (+) z).
rewrite <-associativity, !cut_minus_le; try easy.
now transitivity y.
Qed.
Hint Resolve cut_minus_le_trans.
Lemma cut_minus_plus_distr x₁ x₂ y₁ y₂ :
y₁ ≤ x₁ → y₂ ≤ x₂ → (x₁ ∸ y₁) + (x₂ ∸ y₂) = (x₁ + x₂) ∸ (y₁ + y₂).
Proof.
intros E F. apply (right_cancellation (+) (y₁ + y₂)).
rewrite cut_minus_le.
setoid_replace (x₁ ∸ y₁ + (x₂ ∸ y₂) + (y₁ + y₂)) with (((x₁ ∸ y₁) + y₁) + ((x₂ ∸ y₂) + y₂)) by ring.
now rewrite !cut_minus_le.
now apply semirings.plus_le_compat.
Qed.
Lemma cut_minus_mult_distr_l x y z : 0 ≤ x → x × (y ∸ z) = x × y ∸ x × z.
Proof.
intros E. destruct (total (≤) y z).
rewrite !cut_minus_0; trivial.
ring.
now apply (maps.order_preserving_nonneg (.*.) x).
apply (right_cancellation (+) (x × z)).
rewrite <-distribute_l, !cut_minus_le; try easy.
now apply (maps.order_preserving_nonneg (.*.) x).
Qed.
Lemma cut_minus_mult_distr_r x y z : 0 ≤ x → (y ∸ z) × x = y × x ∸ z × x.
Proof.
intros E. rewrite 3!(commutativity _ x).
now apply cut_minus_mult_distr_l.
Qed.
Lemma cut_minus_plus_rev_l x y z : y ∸ z = (x + y) ∸ (x + z).
Proof.
destruct (total (≤) y z).
rewrite !cut_minus_0; intuition.
apply (right_cancellation (+) (x + z)).
transitivity ((y ∸ z + z) + x); try ring.
rewrite !cut_minus_le; try easy; try ring.
now apply (order_preserving (x +)).
Qed.
Lemma cut_minus_plus_rev_r x y z : y ∸ z = (y + x) ∸ (z + x).
Proof. rewrite !(commutativity _ x). now apply cut_minus_plus_rev_l. Qed.
Lemma cut_minus_plus_r x y z : 0 ≤ z → x ∸ (y + z) = (x ∸ y) ∸ z.
Proof.
intros E. case (total (≤) x y); intros Exy.
rewrite (cut_minus_0 x y) by easy.
rewrite cut_minus_0_l, cut_minus_0; try easy.
now apply semirings.plus_le_compat_r.
rewrite (cut_minus_plus_rev_r y (x ∸ y) z).
now rewrite cut_minus_le, commutativity.
Qed.
Lemma cut_minus_plus_l x y z : 0 ≤ y → x ∸ (y + z) = (x ∸ z) ∸ y.
Proof. rewrite commutativity. now apply cut_minus_plus_r. Qed.
Lemma cut_minus_plus_toggle1 x y z : x ≤ y → z ≤ y → (y ∸ x) + (x ∸ z) = (y ∸ z) + (z ∸ x).
Proof.
intros. destruct (total (≤) x z).
rewrite (cut_minus_0 x z), cut_minus_le_trans by easy. ring.
rewrite (cut_minus_0 z x), cut_minus_le_trans by easy. ring.
Qed.
Lemma cut_minus_plus_toggle2 x y z : y ≤ x → y ≤ z → (x ∸ z) + (z ∸ y) = (z ∸ x) + (x ∸ y).
Proof.
intros. destruct (total (≤) x z).
rewrite (cut_minus_0 x z), cut_minus_le_trans by easy. ring.
rewrite (cut_minus_0 z x) by easy. ring_simplify. now auto.
Qed.
Lemma cut_minus_plus_toggle3 x₁ x₂ y₁ y₂ : x₁ ≤ y₁ → y₂ ≤ x₂
→ (y₁ ∸ x₁) + ((x₁ + x₂) ∸ (y₁ + y₂)) = (x₂ ∸ y₂) + ((y₁ + y₂) ∸ (x₁ + x₂)).
Proof.
intros. destruct (total (≤) (x₁ + x₂) (y₁ + y₂)).
rewrite (cut_minus_0 (x₁ + x₂) (y₁ + y₂)) by easy.
rewrite cut_minus_plus_distr by easy.
setoid_replace (x₂ + (y₁ + y₂)) with (y₁ + (x₂ + y₂)) by ring.
setoid_replace (y₂ + (x₁ + x₂)) with (x₁ + (x₂ + y₂)) by ring.
rewrite <-cut_minus_plus_rev_r. ring.
rewrite (cut_minus_0 (y₁ + y₂) (x₁ + x₂)) by easy.
rewrite cut_minus_plus_distr by easy.
setoid_replace (y₁ + (x₁ + x₂)) with (x₂ + (x₁ + y₁)) by ring.
setoid_replace (x₁ + (y₁ + y₂)) with (y₂ + (x₁ + y₁)) by ring.
rewrite <-cut_minus_plus_rev_r. ring.
Qed.
Lemma cut_minus_0_plus_toggle x : x + (0 ∸ x) = x ∸ 0.
Proof.
destruct (total (≤) 0 x).
rewrite (cut_minus_0 0 x), (cut_minus_nonneg_0_r x) by easy. ring.
rewrite (cut_minus_0 x 0), commutativity; auto.
Qed.
Lemma cut_minus_0_le x y : x ≤ y → (y ∸ x) + (x ∸ 0) + (0 ∸ y) = (y ∸ 0) + (0 ∸ x).
Proof.
intros. rewrite <-!cut_minus_0_plus_toggle.
apply (right_cancellation (+) x).
setoid_replace (y ∸ x + (x + (0 ∸ x)) + (0 ∸ y) + x) with ((y ∸ x + x) + (x + (0 ∸ x)) + (0 ∸ y)) by ring.
rewrite (cut_minus_le y x) by easy. ring.
Qed.
Section min.
Context `{∀ x y : R, Decision (x ≤ y)}.
Lemma cut_minus_min1 x y z : x ∸ (y ⊓ z) = x ∸ y + ((x ⊓ y) ∸ z).
Proof with eauto; try ring.
unfold meet, min, sort.
case (decide_rel (≤) x y); case (decide_rel (≤) y z); intros F G; simpl.
rewrite (cut_minus_0 x z)...
rewrite (cut_minus_0 x y)...
rewrite (cut_minus_0 y z)...
symmetry...
Qed.
Lemma cut_minus_min2 x y z : y ∸ z + ((y ⊓ z) ∸ x) = y ∸ x + ((x ⊓ y) ∸ z).
Proof.
rewrite <-cut_minus_min1.
rewrite (commutativity x y), <-cut_minus_min1.
now rewrite commutativity.
Qed.
Lemma cut_minus_min3 x y z : (x + (y ⊓ z)) ∸ ((x + y) ⊓ (x + z)) = ((x + y) ⊓ (x + z)) ∸ (x + (y ⊓ z)).
Proof with auto; try reflexivity.
destruct (total (≤) y z) as [G1|G1].
rewrite (lattices.meet_l y z), (lattices.meet_l (x + y) (x + z))...
rewrite (lattices.meet_r y z), (lattices.meet_r (x + y) (x + z))...
Qed.
End min.
Context `{Negate R} `{!Ring R}.
Add Ring R: (rings.stdlib_ring_theory R).
Lemma cut_minus_ring_minus (x y : R) : y ≤ x → x ∸ y = x - y.
Proof. intros. apply (right_cancellation (+) y). ring_simplify. now auto. Qed.
Lemma cut_minus_negate (x : R) : x ≤ 0 → 0 ∸ x = -x.
Proof. intros. now rewrite <-(rings.plus_0_l (-x)), cut_minus_ring_minus. Qed.
End cut_minus_properties.
Section cut_minus_default.
Context `{Ring R} `{!SemiRingOrder o} `{∀ x y : R, Decision (x ≤ y)}.
Global Instance default_cut_minus: CutMinus R | 10 := λ x y, if decide_rel (≤) x y then 0 else x - y.
Add Ring R2: (rings.stdlib_ring_theory R).
Global Instance: CutMinusSpec R default_cut_minus.
Proof.
split; unfold cut_minus, default_cut_minus; intros x y ?.
case (decide_rel (≤) x y); intros.
ring_simplify. now apply (antisymmetry (≤)).
ring.
now case (decide_rel (≤) x y).
Qed.
End cut_minus_default.
Typeclasses Opaque default_cut_minus.
Section order_preserving.
Context `{FullPseudoSemiRingOrder A} `{!TrivialApart A} `{!CutMinusSpec A cmA} `{∀ x y : A, Decision (x = y)}
`{FullPseudoSemiRingOrder B} `{!TrivialApart B} `{!CutMinusSpec B cmB} `{∀ x y : B, Decision (x = y)}
{f : A → B} `{!OrderPreserving f} `{!SemiRing_Morphism f}.
Local Existing Instance pseudo_srorder_semiring.
Lemma preserves_cut_minus x y : f (x ∸ y) = f x ∸ f y.
Proof.
destruct (total (≤) x y) as [E|E].
rewrite (cut_minus_0 x y E), (cut_minus_0 (f x) (f y)).
apply rings.preserves_0.
now apply (order_preserving _).
apply (left_cancellation (+) (f y)). rewrite 2!(commutativity (f y)).
rewrite <-rings.preserves_plus.
rewrite (cut_minus_le x y E), (cut_minus_le (f x) (f y)).
reflexivity.
now apply (order_preserving _).
Qed.
End order_preserving.
orders.semirings.
Require Import
Ring abstract_algebra interfaces.additional_operations
interfaces.orders orders.minmax.
Section cut_minus_properties.
Context `{FullPseudoSemiRingOrder R} `{!TrivialApart R}
`{∀ x y, Decision (x = y)} `{!CutMinusSpec R cm}.
Local Existing Instance pseudo_srorder_semiring.
Add Ring SR: (rings.stdlib_semiring_theory R).
Hint Resolve (@orders.le_flip R _ _).
Global Instance cut_minus_proper: Proper ((=) ==> (=) ==> (=)) cut_minus | 1.
Proof.
intros x₁ x₂ E y₁ y₂ F.
destruct (total (≤) x₂ y₂).
rewrite cut_minus_0, cut_minus_0; try easy. now rewrite E, F.
apply (right_cancellation (+) y₂).
rewrite cut_minus_le by easy.
rewrite <-E, <-F. apply cut_minus_le. now rewrite E, F.
Qed.
Global Instance cut_minus_mor_1: ∀ x : R, Setoid_Morphism (x ∸) | 0.
Proof. split; try apply _. Qed.
Global Instance cut_minus_mor_2: ∀ x : R, Setoid_Morphism (∸ x) | 0.
Proof. split; try apply _. solve_proper. Qed.
Hint Resolve (cut_minus_0).
Hint Resolve (cut_minus_le).
Lemma cut_minus_diag x : x ∸ x = 0.
Proof. now apply cut_minus_0. Qed.
Lemma cut_minus_nonneg_0_r x : 0 ≤ x → x ∸ 0 = x.
Proof. intros E. rewrite <-(rings.plus_0_r (x ∸ 0)). auto. Qed.
Lemma cut_minus_0_l x : 0 ≤ x → 0 ∸ x = 0.
Proof. auto. Qed.
Lemma cut_minus_nonpos_0_r x : x ≤ 0 → x ∸ 0 = 0.
Proof. auto. Qed.
Lemma cut_minus_nonneg x y : 0 ≤ x ∸ y.
Proof.
destruct (total (≤) x y) as [E|E].
apply orders.eq_le. symmetry. now auto.
apply (order_reflecting (+ y)).
now rewrite cut_minus_le; ring_simplify.
Qed.
Lemma cut_minus_le_r x y : y ≤ x → x ∸ y + y = x.
Proof cut_minus_le x y.
Lemma cut_minus_le_l x y : y ≤ x → y + (x ∸ y) = x.
Proof. rewrite commutativity. now apply cut_minus_le. Qed.
Lemma cut_minus_le_trans x y z : y ≤ x → z ≤ y → (x ∸ y) + (y ∸ z) = x ∸ z.
Proof.
intros. apply (right_cancellation (+) z).
rewrite <-associativity, !cut_minus_le; try easy.
now transitivity y.
Qed.
Hint Resolve cut_minus_le_trans.
Lemma cut_minus_plus_distr x₁ x₂ y₁ y₂ :
y₁ ≤ x₁ → y₂ ≤ x₂ → (x₁ ∸ y₁) + (x₂ ∸ y₂) = (x₁ + x₂) ∸ (y₁ + y₂).
Proof.
intros E F. apply (right_cancellation (+) (y₁ + y₂)).
rewrite cut_minus_le.
setoid_replace (x₁ ∸ y₁ + (x₂ ∸ y₂) + (y₁ + y₂)) with (((x₁ ∸ y₁) + y₁) + ((x₂ ∸ y₂) + y₂)) by ring.
now rewrite !cut_minus_le.
now apply semirings.plus_le_compat.
Qed.
Lemma cut_minus_mult_distr_l x y z : 0 ≤ x → x × (y ∸ z) = x × y ∸ x × z.
Proof.
intros E. destruct (total (≤) y z).
rewrite !cut_minus_0; trivial.
ring.
now apply (maps.order_preserving_nonneg (.*.) x).
apply (right_cancellation (+) (x × z)).
rewrite <-distribute_l, !cut_minus_le; try easy.
now apply (maps.order_preserving_nonneg (.*.) x).
Qed.
Lemma cut_minus_mult_distr_r x y z : 0 ≤ x → (y ∸ z) × x = y × x ∸ z × x.
Proof.
intros E. rewrite 3!(commutativity _ x).
now apply cut_minus_mult_distr_l.
Qed.
Lemma cut_minus_plus_rev_l x y z : y ∸ z = (x + y) ∸ (x + z).
Proof.
destruct (total (≤) y z).
rewrite !cut_minus_0; intuition.
apply (right_cancellation (+) (x + z)).
transitivity ((y ∸ z + z) + x); try ring.
rewrite !cut_minus_le; try easy; try ring.
now apply (order_preserving (x +)).
Qed.
Lemma cut_minus_plus_rev_r x y z : y ∸ z = (y + x) ∸ (z + x).
Proof. rewrite !(commutativity _ x). now apply cut_minus_plus_rev_l. Qed.
Lemma cut_minus_plus_r x y z : 0 ≤ z → x ∸ (y + z) = (x ∸ y) ∸ z.
Proof.
intros E. case (total (≤) x y); intros Exy.
rewrite (cut_minus_0 x y) by easy.
rewrite cut_minus_0_l, cut_minus_0; try easy.
now apply semirings.plus_le_compat_r.
rewrite (cut_minus_plus_rev_r y (x ∸ y) z).
now rewrite cut_minus_le, commutativity.
Qed.
Lemma cut_minus_plus_l x y z : 0 ≤ y → x ∸ (y + z) = (x ∸ z) ∸ y.
Proof. rewrite commutativity. now apply cut_minus_plus_r. Qed.
Lemma cut_minus_plus_toggle1 x y z : x ≤ y → z ≤ y → (y ∸ x) + (x ∸ z) = (y ∸ z) + (z ∸ x).
Proof.
intros. destruct (total (≤) x z).
rewrite (cut_minus_0 x z), cut_minus_le_trans by easy. ring.
rewrite (cut_minus_0 z x), cut_minus_le_trans by easy. ring.
Qed.
Lemma cut_minus_plus_toggle2 x y z : y ≤ x → y ≤ z → (x ∸ z) + (z ∸ y) = (z ∸ x) + (x ∸ y).
Proof.
intros. destruct (total (≤) x z).
rewrite (cut_minus_0 x z), cut_minus_le_trans by easy. ring.
rewrite (cut_minus_0 z x) by easy. ring_simplify. now auto.
Qed.
Lemma cut_minus_plus_toggle3 x₁ x₂ y₁ y₂ : x₁ ≤ y₁ → y₂ ≤ x₂
→ (y₁ ∸ x₁) + ((x₁ + x₂) ∸ (y₁ + y₂)) = (x₂ ∸ y₂) + ((y₁ + y₂) ∸ (x₁ + x₂)).
Proof.
intros. destruct (total (≤) (x₁ + x₂) (y₁ + y₂)).
rewrite (cut_minus_0 (x₁ + x₂) (y₁ + y₂)) by easy.
rewrite cut_minus_plus_distr by easy.
setoid_replace (x₂ + (y₁ + y₂)) with (y₁ + (x₂ + y₂)) by ring.
setoid_replace (y₂ + (x₁ + x₂)) with (x₁ + (x₂ + y₂)) by ring.
rewrite <-cut_minus_plus_rev_r. ring.
rewrite (cut_minus_0 (y₁ + y₂) (x₁ + x₂)) by easy.
rewrite cut_minus_plus_distr by easy.
setoid_replace (y₁ + (x₁ + x₂)) with (x₂ + (x₁ + y₁)) by ring.
setoid_replace (x₁ + (y₁ + y₂)) with (y₂ + (x₁ + y₁)) by ring.
rewrite <-cut_minus_plus_rev_r. ring.
Qed.
Lemma cut_minus_0_plus_toggle x : x + (0 ∸ x) = x ∸ 0.
Proof.
destruct (total (≤) 0 x).
rewrite (cut_minus_0 0 x), (cut_minus_nonneg_0_r x) by easy. ring.
rewrite (cut_minus_0 x 0), commutativity; auto.
Qed.
Lemma cut_minus_0_le x y : x ≤ y → (y ∸ x) + (x ∸ 0) + (0 ∸ y) = (y ∸ 0) + (0 ∸ x).
Proof.
intros. rewrite <-!cut_minus_0_plus_toggle.
apply (right_cancellation (+) x).
setoid_replace (y ∸ x + (x + (0 ∸ x)) + (0 ∸ y) + x) with ((y ∸ x + x) + (x + (0 ∸ x)) + (0 ∸ y)) by ring.
rewrite (cut_minus_le y x) by easy. ring.
Qed.
Section min.
Context `{∀ x y : R, Decision (x ≤ y)}.
Lemma cut_minus_min1 x y z : x ∸ (y ⊓ z) = x ∸ y + ((x ⊓ y) ∸ z).
Proof with eauto; try ring.
unfold meet, min, sort.
case (decide_rel (≤) x y); case (decide_rel (≤) y z); intros F G; simpl.
rewrite (cut_minus_0 x z)...
rewrite (cut_minus_0 x y)...
rewrite (cut_minus_0 y z)...
symmetry...
Qed.
Lemma cut_minus_min2 x y z : y ∸ z + ((y ⊓ z) ∸ x) = y ∸ x + ((x ⊓ y) ∸ z).
Proof.
rewrite <-cut_minus_min1.
rewrite (commutativity x y), <-cut_minus_min1.
now rewrite commutativity.
Qed.
Lemma cut_minus_min3 x y z : (x + (y ⊓ z)) ∸ ((x + y) ⊓ (x + z)) = ((x + y) ⊓ (x + z)) ∸ (x + (y ⊓ z)).
Proof with auto; try reflexivity.
destruct (total (≤) y z) as [G1|G1].
rewrite (lattices.meet_l y z), (lattices.meet_l (x + y) (x + z))...
rewrite (lattices.meet_r y z), (lattices.meet_r (x + y) (x + z))...
Qed.
End min.
Context `{Negate R} `{!Ring R}.
Add Ring R: (rings.stdlib_ring_theory R).
Lemma cut_minus_ring_minus (x y : R) : y ≤ x → x ∸ y = x - y.
Proof. intros. apply (right_cancellation (+) y). ring_simplify. now auto. Qed.
Lemma cut_minus_negate (x : R) : x ≤ 0 → 0 ∸ x = -x.
Proof. intros. now rewrite <-(rings.plus_0_l (-x)), cut_minus_ring_minus. Qed.
End cut_minus_properties.
Section cut_minus_default.
Context `{Ring R} `{!SemiRingOrder o} `{∀ x y : R, Decision (x ≤ y)}.
Global Instance default_cut_minus: CutMinus R | 10 := λ x y, if decide_rel (≤) x y then 0 else x - y.
Add Ring R2: (rings.stdlib_ring_theory R).
Global Instance: CutMinusSpec R default_cut_minus.
Proof.
split; unfold cut_minus, default_cut_minus; intros x y ?.
case (decide_rel (≤) x y); intros.
ring_simplify. now apply (antisymmetry (≤)).
ring.
now case (decide_rel (≤) x y).
Qed.
End cut_minus_default.
Typeclasses Opaque default_cut_minus.
Section order_preserving.
Context `{FullPseudoSemiRingOrder A} `{!TrivialApart A} `{!CutMinusSpec A cmA} `{∀ x y : A, Decision (x = y)}
`{FullPseudoSemiRingOrder B} `{!TrivialApart B} `{!CutMinusSpec B cmB} `{∀ x y : B, Decision (x = y)}
{f : A → B} `{!OrderPreserving f} `{!SemiRing_Morphism f}.
Local Existing Instance pseudo_srorder_semiring.
Lemma preserves_cut_minus x y : f (x ∸ y) = f x ∸ f y.
Proof.
destruct (total (≤) x y) as [E|E].
rewrite (cut_minus_0 x y E), (cut_minus_0 (f x) (f y)).
apply rings.preserves_0.
now apply (order_preserving _).
apply (left_cancellation (+) (f y)). rewrite 2!(commutativity (f y)).
rewrite <-rings.preserves_plus.
rewrite (cut_minus_le x y E), (cut_minus_le (f x) (f y)).
reflexivity.
now apply (order_preserving _).
Qed.
End order_preserving.