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reals.v
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reals.v
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Set Warnings "-notation-overridden,-ambiguous-paths".
Require Export ssreflect ssrbool ssrfun rationals.
Definition 𝐑 :=
{α in P ℚ | α ≠ ∅ ∧ α ≠ ℚ ∧ (∀ p q : Q, p ∈ α → q < p → q ∈ α) ∧
∀ p : Q, p ∈ α → ∃ r : Q, p < r ∧ r ∈ α}.
Definition R := elts 𝐑.
Definition IRS (a : R) := elt_to_set a : set.
Coercion IRS : R >-> set.
Lemma Dedekind_cut_0 : ∀ (α : R) (p : set), p ∈ α → p ∈ ℚ.
Proof.
move=> [α /[dup] /Specify_classification
[/Powerset_classification H _] ?] ? /H //.
Qed.
Lemma Dedekind_cut_1 : ∀ α : R, ∅ ≠ α.
Proof.
move=> [α /[dup] /Specify_classification [?]]
[] /[swap] _ /[swap] ? /[swap] -> //.
Qed.
Lemma Dedekind_cut_2 : ∀ (α : R) (p q : Q), p ∈ α → q < p → q ∈ α.
Proof.
move=> [α /[dup] /Specify_classification [? [? [? []]]]] //.
Qed.
Lemma Dedekind_cut_3 : ∀ (α : R) (p : Q), p ∈ α → ∃ r : Q, p < r ∧ r ∈ α.
Proof.
move=> [α /[dup] /Specify_classification [? [? [? []]]]] //.
Qed.
Lemma Dedekind_cut_4 : ∀ α : R, ∀ p q : Q, p ∈ α → q ∉ α → p < q.
Proof.
move=> [α /[dup] /Specify_classification [? [? [? [? ?]]]]] *.
apply (ordered_rings.lt_not_ge ℚ_ring_order) => [[/= ? | ?]]; subst; eauto.
Qed.
Lemma Dedekind_cut_5 : ∀ α : R, ∀ r s : Q, r ∉ α → r < s → s ∉ α.
Proof.
move=> [α /[dup] /Specify_classification
[? [? [? [H ?]]]] ?] ? ? ? /H /[apply] //.
Qed.
Lemma Dedekind_cut_6 : ∀ a : R, ∃ q : Q, q ∉ a.
Proof.
move=> [α /[dup] /Specify_classification
[/Powerset_classification H
[? [/[dup] ? /neq_sym /not_proper_subset_inhab]]]]
[[] /Subset_equality /(_ H) // | ? [H0 ?] _ ?].
by exists (mkSet H0 : Q).
Qed.
Declare Scope R_scope.
Delimit Scope R_scope with R.
Open Scope R_scope.
Bind Scope R_scope with R.
Definition lt (a b : R) := a ⊊ b.
Infix "<" := lt : R_scope.
Notation "a > b" := (b < a) (only parsing) : R_scope.
Definition le a b := a < b ∨ a = b.
Infix "≤" := le : R_scope.
Notation "a ≥ b" := (b ≤ a) (only parsing) : R_scope.
Notation "a ≤ b < c" := (a ≤ b ∧ b < c) (at level 70, b at next level): R_scope.
Notation "a < b ≤ c" := (a < b ∧ b ≤ c) (at level 70, b at next level): R_scope.
Notation "a ≤ b ≤ c" := (a ≤ b ∧ b ≤ c) (at level 70, b at next level): R_scope.
Theorem le_is_subset : ∀ a b, a ≤ b ↔ a ⊂ b.
Proof.
rewrite /le /lt /proper_subset => a b.
split => [[[H H0] | H] | H]; subst; auto using Set_is_subset.
case (classic (a = b)); intuition eauto using set_proj_injective.
Qed.
Theorem lt_trans : ∀ a b c : R, a < b → b < c → a < c.
Proof.
move=> a b c [H H0] [H1 H2].
split => [z H3 | ]; eauto.
move: H1 => /[swap] <- /Subset_equality /(_ H) /(@eq_sym set) //.
Qed.
Theorem lt_trichotomy : ∀ a b : R, a < b ∨ a = b ∨ b < a.
Proof.
move=> a b.
(case (classic (a < b)); case (classic (a = b)); auto) => H H0.
have [p [/[dup] /Dedekind_cut_0 H1 H2 H3]]: ∃ p, p ∈ a ∧ p ∉ b
by eauto using not_proper_subset_inhab, set_proj_injective.
(apply or_intror, or_intror, conj => [q /[dup] /Dedekind_cut_0 H4 | ]);
rewrite (reify H1) ? (reify H4) in H2 H3 |-*;
eauto using Dedekind_cut_2, Dedekind_cut_4, set_proj_injective.
Qed.
Theorem T : ∀ a b : R, a < b ∧ a ≠ b ∧ ¬ b < a
∨ ¬ a < b ∧ a = b ∧ ¬ b < a
∨ ¬ a < b ∧ a ≠ b ∧ b < a.
Proof.
(move: lt_trichotomy => /[swap] a /[swap] b /(_ a b) [[H H0] | [-> | [H H0]]];
[left | right; left | right; right]); repeat split;
auto; try (contradict H0; subst; auto; move: H0 =>
[] /Subset_equality /(_ H) -> //); move=> [H0 []] //.
Qed.
Theorem lub : ∀ A, A ⊂ 𝐑 → A ≠ ∅ → (∃ β : R, ∀ α : R, α ∈ A → α ≤ β) →
∃ γ : R, (∀ α : R, α ∈ A → α ≤ γ) ∧
∀ δ : R, (∀ α : R, α ∈ A → α ≤ δ) → γ ≤ δ.
Proof.
move=> A H /Nonempty_classification [z /[dup] H0 /H H1]
[[b /[dup] /Specify_classification [H2 [H3 [H4 [H5 H6]]]] B] H7].
have H8: ⋃ A ∈ 𝐑.
{ apply Specify_classification.
(repeat split) =>
[ | | H8 | p q /Union_classification
[x [/[dup] H8 /H /Specify_classification
[H9 [H10 [H11 [H12 H13]]]] H14]] /H12 ? |
p /Union_classification
[x [/[dup] H8 /H /Specify_classification
[H9 [H10 [H11 [H12 H13]]]] /H13 [r [H14 H15]]]]].
- apply Powerset_classification => x /Union_classification [a [/H H8]].
by apply (Dedekind_cut_0 (mkSet H8)).
- move: (Dedekind_cut_1 (mkSet H1)) => /neq_sym.
rewrite ? Nonempty_classification => [[x H8]].
apply /ex_intro /Union_classification; eauto.
- move: H2 H4 => /Powerset_classification /Subset_equality -> //.
rewrite -H8 => x /Union_classification [y [/[dup] H9 /H H10]].
move: (H7 (mkSet H10) H9) => [[] /[swap] _ /[apply] | [] ->] //.
- apply Union_classification; eauto.
- eapply ex_intro, conj, Union_classification; eauto. }
exists (mkSet H8).
split => [α H9 | δ H9]; rewrite /le.
- case (classic (α = (mkSet H8))); auto; left; split => [x H11 | ] /=.
+ apply Union_classification; eauto.
+ contradict H10.
by apply set_proj_injective.
- case (T (mkSet H8) δ) => [H10 | [H10 | [H10 [H11 H12]]]]; try tauto.
have /= [s [/Union_classification [a /[swap] H13 [/[dup] /H H14]]]]:
(∃ s, s ∈ (mkSet H8) ∧ s ∉ δ)
by eauto using not_proper_subset_inhab, set_proj_injective.
rewrite (reify H14) => /H9.
rewrite /le => H16 H17.
have []: ¬ δ < (mkSet H14) by (move: (T (mkSet H14) δ); tauto).
split => [x H18 | ].
+ move: (H17) (H18) => /[dup] /(Dedekind_cut_0 (mkSet H14))
H19 /[swap] /Dedekind_cut_0 H20.
rewrite (reify H19) (reify H20) => ?.
eauto using Dedekind_cut_2, Dedekind_cut_4.
+ move: H13 => /[swap] -> [] //.
Qed.
Definition iqr_set (q : Q) := {x of type ℚ | (x < q)%Q}.
Theorem iqr_in : ∀ q, iqr_set q ∈ 𝐑.
Proof.
move=> q.
rewrite Specify_classification Powerset_classification
Nonempty_classification.
have H: (q + 1 ∈ ℚ) by eauto using elts_in_set.
(((repeat split) => [z /Specify_classification [] | | H0 |
p x /[swap] H0 /Specify_classification [H1] |
p /Specify_classification [H0]] //);
first exists (q - 1); rewrite -? H0 ? Specify_classification ? despecify) =>
[ | | | /lt_dense [r []] ? ?]; eauto using elts_in_set, rationals.lt_trans.
- rewrite -[rationals.lt]/(ordered_rings.lt ℚ_ring_order)
(ordered_rings.lt_shift ℚ_ring_order) /=.
replace (q + -(q - 1)) with 1 by field.
eauto using elts_in_set, (ordered_rings.zero_lt_1 ℚ_ring_order : 0 < 1)%Q.
- rewrite -H0 Specify_classification despecify in H.
move: H => [] _ /(lt_antisym ℚ_ring_order) [].
eauto using (lt_succ ℚ_ring_order).
- eapply ex_intro, conj, Specify_classification, conj;
rewrite ? despecify; eauto using elts_in_set.
Qed.
Definition IQR (q : Q) := (mkSet (iqr_in q)) : R.
Definition zero := IQR 0.
Coercion IQR : Q >-> R.
Notation "0" := zero : R_scope.
Definition add_set (α β : R) := {x in ℚ | ∃ r s, x = r + s ∧ r ∈ α ∧ s ∈ β}.
Lemma not_Q_subset : ∀ α : R, ¬ ℚ ⊊ α.
Proof.
move: Dedekind_cut_0 => /[swap] α /(_ α) H [/Subset_equality] /(_ H) //.
Qed.
Lemma not_Q_eq : ∀ α : R, (α : set) ≠ ℚ.
Proof.
move=> [α /[dup] /Specify_classification [_ [_ [_]]]] //.
Qed.
Theorem add_in : ∀ α β, add_set α β ∈ 𝐑.
Proof.
move=> α β.
(rewrite Specify_classification Powerset_classification
Nonempty_classification); (repeat split) =>
[z /Specify_classification [] | | |
p q /Specify_classification [H [r [s [/set_proj_injective -> [H0 H1]]]]]
/(O1 (-s)) H2 | p /Specify_classification
[H [r [s [/set_proj_injective ->
[/Dedekind_cut_3 [t [H0 H1]] H2]]]]]] //.
- move: (Dedekind_cut_1 α) (Dedekind_cut_1 β) => /neq_sym =>
/Nonempty_classification [x /[dup] /Dedekind_cut_0 H H0]
/neq_sym /Nonempty_classification [y /[dup] /Dedekind_cut_0 H1 H2].
exists (mkSet H + mkSet H1).
apply Specify_classification.
split; eauto using elts_in_set.
- move: (not_proper_subset_inhab ℚ α) (not_proper_subset_inhab ℚ β) =>
[ | | r' [H H0] [ | | s' [H1 H2]]]; auto using not_Q_subset, not_Q_eq
=> /Subset_equality_iff [H3 H4].
move: (elts_in_set ((mkSet H) + (mkSet H1))) => /H4 =>
/Specify_classification [H5 [r [s [/set_proj_injective H6 [H7 H8]]]]].
contradiction (lt_irrefl ℚ_ring_order (r + s)).
rewrite -{2}H6.
apply (lt_cross_add ℚ_ring_order) => /=; eauto using Dedekind_cut_4.
- apply Specify_classification.
split; eauto using elts_in_set.
exists (-s + q), s.
ring_simplify in H2.
intuition (eauto using Dedekind_cut_2 || f_equal; ring).
- exists (t+s).
rewrite Specify_classification {1}(rationals.A1 t s) (rationals.A1 r).
intuition eauto using elts_in_set, O1.
Qed.
Definition add (a b : R) := mkSet (add_in a b) : R.
Infix "+" := add : R_scope.
Theorem A1 : ∀ a b, a + b = b + a.
Proof.
rewrite /add => a b.
apply set_proj_injective, Extensionality => /= z.
rewrite ? Specify_classification.
split => [[H [r [s [H0 [H1 H2]]]]] | [H [r [s [H0 [H1 H2]]]]]];
repeat split; auto; exists s, r; rewrite A1; repeat split; auto.
Qed.
Theorem A2 : ∀ a b c, a + (b + c) = (a + b) + c.
Proof.
rewrite /add => a b c.
apply set_proj_injective, Extensionality => z.
rewrite ? Specify_classification.
(split => [[H [r [s [H0 [H1 /Specify_classification
[H2 [t [u [/set_proj_injective H3 [H4 H5]]]]]]]]]]
| [H [r [s [H0 [/Specify_classification
[H1 [t [u [/set_proj_injective H2 [H3 H4]]]]]
H5]]]]]]); subst; split; auto;
[exists (r + t)%Q, u | exists t, (u + s)%Q]; rewrite Specify_classification;
repeat split; eauto using elts_in_set; f_equal; ring.
Qed.
Theorem A3 : ∀ a, a + 0 = a.
Proof.
rewrite /add /zero => α.
apply set_proj_injective, Extensionality => z.
split => [/Specify_classification
[H [r [s [-> [H0 /Specify_classification [H1]]]]]] |
/[dup] /Dedekind_cut_0 H].
- rewrite despecify => H2.
eapply Dedekind_cut_2; eauto.
have {2}->: r = (r + 0)%Q by ring.
auto using O1.
- rewrite Specify_classification (reify H) => /Dedekind_cut_3 [r [H0 H1]].
split; auto.
exists r, ((mkSet H) + -r)%Q.
repeat split; auto; rewrite 1 ? (rationals.A1 _ (-r)) ? (rationals.A2);
rewrite ? (rationals.A4) ? (rationals.A3) // ? Specify_classification
? despecify -(A4 r) ? (rationals.A1 _ (-r)).
eauto using elts_in_set, O1.
Qed.
Definition neg_set (α : R) :=
{p of type ℚ | ∃ r : Q, (0 < r)%Q ∧ (- p - r)%Q ∉ α}.
Theorem neg_in : ∀ a, neg_set a ∈ 𝐑.
Proof.
move=> α.
rewrite Specify_classification Powerset_classification
Nonempty_classification.
(repeat split) => [z /Specify_classification [] | | | p q | p ] //.
- move: (elts_in_set α) => /Specify_classification =>
[[/Powerset_classification H [H0 [/neq_sym H1 [H2 H3]]]]].
elim (not_proper_subset_inhab ℚ α) =>
[s [H4 H5] | [] /Subset_equality /(_ H) | ] //.
set (σ := mkSet H4 : Q).
exists (-σ - 1 : Q).
rewrite Specify_classification despecify.
split; eauto using elts_in_set.
exists 1.
repeat split; auto using (ordered_rings.zero_lt_1 ℚ_ring_order : 0 < 1)%Q.
by have ->: (-(-σ - 1) - 1)%Q = σ by ring.
- move: (Dedekind_cut_1 α) => /neq_sym /Nonempty_classification
[s /[dup] /Dedekind_cut_0 H H0].
set (σ := (mkSet H) : Q) => H1.
move: (elts_in_set (-σ)) => /[dup]; simpl in *.
rewrite -{2 4}H1 => H2.
rewrite Specify_classification despecify => [[H3 [r [H4 []]]]].
apply (Dedekind_cut_2 _ σ); auto.
rewrite -[rationals.lt]/(ordered_rings.lt ℚ_ring_order)
(ordered_rings.lt_shift ℚ_ring_order) /=.
by have ->: (σ + -(--σ - r))%Q = r by ring.
- rewrite ? Specify_classification ? despecify => [[H [r [H1 H2]]]] H3.
split; eauto using elts_in_set.
exists r.
repeat split; auto.
contradict H2.
apply (Dedekind_cut_2 _ (-q - r)); auto.
rewrite /sub ? (rationals.A1 _ (-r)).
by apply O1, (neg_neg_lt ℚ_ring_order).
- rewrite ? Specify_classification ? despecify =>
[[H [r [/[dup] H0 /(O1 p) /[swap] H1]]]].
rewrite rationals.A1 rationals.A3 => /lt_dense [t [H2 H3]].
exists t.
rewrite Specify_classification despecify.
repeat split; eauto using elts_in_set.
exists (p + r - t).
repeat split; auto.
+ by apply (ordered_rings.lt_shift ℚ_ring_order) in H3.
+ by have ->: (-t - (p + r - t) = -p - r) by ring.
Qed.
Definition neg (a : R) := mkSet (neg_in a) : R.
Notation "- a" := (neg a) : R_scope.
Theorem cut_archimedean : ∀ (α : R) (b : Q),
(0 < b)%Q → ∃ n : Z, n * b ∈ α ∧ (n + 1) * b ∉ α.
Proof.
move=> α b H.
move: (elts_in_set α) => /Specify_classification =>
[[/Powerset_classification H0 [/Nonempty_classification
[x /[dup] /Dedekind_cut_0 H1]]]].
rewrite ? (reify H1) => [H2 [H3 [H4 H5]]].
elim (Q_archimedean (mkSet H1) b) => [k [H6 H7] | ] //.
elim (WOP (λ m, (k + m)%Z * b ∉ α)) => [n [H8 H9] | m H8 | ].
- exists (k + (n + -1))%Z.
rewrite ? IZQ_add.
split; last by have ->: (k + (n + -1) + 1 = k + n)%Z by ring.
apply NNPP => /[dup] H10 /H9 /(le_not_gt ℤ_order) [].
rewrite (ordered_rings.lt_shift ℤ_order) /=.
have ->: (n + -(n + -1) = 1)%Z by ring.
auto using integers.zero_lt_1.
- apply NNPP.
contradict H8.
move: H8 => /(le_not_gt ℤ_order) => [[H8 | ->]].
+ apply (H4 (mkSet H1)); auto; (case: H6 => [H6 | H6]);
rewrite -IZQ_add -[rationals.lt]/(ordered_rings.lt ℚ_ring_order);
ring_simplify; [apply (ordered_rings.lt_trans ℚ_ring_order _ (k*b)) | ];
auto; (rewrite -? H6 (ordered_rings.lt_shift ℚ_ring_order) /=);
(have ->: (k * b + -(k * b + m * b) = -m * b)%Q by field);
apply O2; auto; by rewrite (lt_neg_0 ℤ_order) /= IZQ_lt -IZQ_neg in H8.
+ rewrite -> (A3_r ℤ).
case: H6 => [H6 | ->]; eauto.
- elim: (not_proper_subset_inhab ℚ α) =>
[z [H8 H9] | [/Subset_equality /(_ H0) H8 H9] | ]; auto.
elim: (Q_archimedean (mkSet H8) b) => [m [H10 H11] | ] //.
exists (m - k + 1)%Z.
have ->: (k + (m - k + 1) = m + 1)%Z by ring.
apply (Dedekind_cut_5 α (mkSet H8)); auto.
by rewrite -IZQ_add.
Qed.
Theorem A4 : ∀ a, a + -a = 0.
Proof.
move=> α.
apply set_proj_injective, Extensionality => z.
split => [/Specify_classification [H [r [p [-> [] H1]]]] |
/Specify_classification [H]].
- rewrite ? Specify_classification ? despecify =>
[[H2 [q [/[dup] H3 /(O1 (-p - q)) H4 H5]]]].
ring_simplify in H4.
(have: (-p ∉ α)%Q by eauto using Dedekind_cut_5) =>
/Dedekind_cut_4 => /(_ _ H1) /(O1 p) H6.
rewrite A4 rationals.A1 in H6.
eauto using elts_in_set.
- rewrite (reify H) despecify => H0.
set (v := mkSet H : Q).
set (w := (-v * 2^-1)%Q).
have H1: (0 < 2)%Z by apply (ordered_rings.zero_lt_2 ℤ_order).
have H2: (0 < w)%Q.
{ apply rationals.O2; try now apply (lt_neg_0 ℚ_ring_order).
now apply (O4 ℚ_order), IZQ_lt. }
elim: (cut_archimedean α w) => [n [H3 H4] | ] //.
apply Specify_classification.
split; auto.
exists (n * w)%Q, (-(n + 2) * w)%Q.
repeat split; auto; rewrite /IQS /w ? Specify_classification ? despecify.
+ f_equal.
ring_simplify.
rewrite -M2 inv_l 1 ? M1 ? M3 // => /IZQ_eq H5.
contradiction (ordered_rings.lt_irrefl ℤ_order 0%Z).
by rewrite -{2}H5.
+ split; eauto using elts_in_set.
exists w.
repeat split; auto.
have ->: (-(-(n + 2) * w) - w = n * w + 2 * w + -w)%Q by ring.
have ->: ((2 : Q) = 1 / 1 + 1 / 1)%Q; first rewrite add_wf;
auto using integers.zero_ne_1;
rewrite /IZQ ? Qequiv ? (integers.M1 2) ? integers.M3 ? D1 ? M3
-? rationals.A2 ? A4 ? (rationals.A1 w) ? rationals.A3
-1 ? {2}(M3 w) -? D1 //; auto using integers.zero_ne_1.
Qed.
Theorem A3_l : ∀ a, 0 + a = a.
Proof.
eauto using A1, A3, eq_trans.
Qed.
Theorem A4_l : ∀ a, -a + a = 0.
Proof.
eauto using A1, A4, eq_trans.
Qed.
Theorem O1 : ∀ a b c, b < c → a + b < a + c.
Proof.
move=> a b c [H H0].
split => [z /Specify_classification [H1 [r [s [H2 [H3 /H H4]]]]] | ].
- apply Specify_classification.
intuition eauto.
- move: H0 => /[swap] /set_proj_injective /(f_equal (add (-a))).
rewrite ? A2 ? (A1 (-a)) ? A4 ? A3_l => -> //.
Qed.
Theorem lt_irrefl : ∀ a, ¬ a < a.
Proof.
move=> a [H H0] //.
Qed.
Theorem lt_antisym : ∀ a b, a < b → ¬ b < a.
Proof.
move=> a b /lt_trans /[apply] /lt_irrefl //.
Qed.
Local Definition mul_pos_set (a b : R) :=
{x of type ℚ | (∃ r s : Q, r ∈ a ∧ s ∈ b ∧ 0 < r ∧ 0 < s ∧ x ≤ r * s)%Q}.
Definition one : R := IQR 1.
Notation "1" := one : R_scope.
Notation "- 1" := (neg one) : R_scope.
Theorem pos_nonempty : ∀ a, 0 < a → ∃ c : Q, (0 < c)%Q ∧ c ∈ a.
Proof.
move=> a /proper_subset_inhab [c [/[dup] /Dedekind_cut_0 H H0 H1]].
have H2: ¬ (mkSet H < 0)%Q by
rewrite Specify_classification (reify H) despecify in H1; auto.
(case (classic (mkSet H = 0%Q)) => [ | H3]);
first by elim: (Dedekind_cut_3 a (mkSet H) H0) => [x] /[swap] ->; eauto.
exists (mkSet H).
split; auto.
apply (ordered_rings.lt_not_ge ℚ_ring_order) => [[ | ]] //.
Qed.
Local Theorem mul_pos_in : ∀ a b, 0 < a → 0 < b → mul_pos_set a b ∈ 𝐑.
Proof.
move=> a b /[dup] H /pos_nonempty [c [H0 H1]]
/[dup] H2 /pos_nonempty [d [H3 H4]].
rewrite Specify_classification Powerset_classification
Nonempty_classification.
(repeat split) => [x | | /Subset_equality_iff [H5 H6] | p q | p].
- rewrite Specify_classification => [[]] //.
- exists (c * d - 1).
rewrite Specify_classification despecify.
split; eauto using elts_in_set.
exists c, d.
repeat split; auto.
left.
apply lt_sub_pos, (ordered_rings.zero_lt_1 ℚ_ring_order).
- move: (Dedekind_cut_6 a) (Dedekind_cut_6 b) => [c' H7] [d' H8].
have: c' * d' ∈ mul_pos_set a b by eauto using elts_in_set.
rewrite Specify_classification despecify
-[rationals.le]/(ordered_rings.le ℚ_ring_order)
=> [[H9 [r [s [H10 [H11 [H12 [H13]]]]]]]].
rewrite (le_not_gt ℚ_ring_order) => [[]].
eapply (lt_cross_mul ℚ_ring_order) => /=; eauto using Dedekind_cut_4.
- rewrite ? Specify_classification ? despecify =>
[[H5 [r [s [H6 [H7 [H8 [H9 H10]]]]]]]]
/[dup] H11 /(@or_introl (q < p)%Q (q = p)) H12.
eauto 9 using (ordered_rings.le_trans ℚ_ring_order :
∀ a b c : Q, a ≤ b → b ≤ c → a ≤ c)%Q, elts_in_set.
- rewrite Specify_classification despecify =>
[[H5 [r [s [/Dedekind_cut_3 [ρ [H6 H7]]
[/Dedekind_cut_3 [σ [H8 H9]] [H10 [H11 H12]]]]]]]].
repeat esplit.
+ eapply (ordered_rings.le_lt_trans ℚ_ring_order),
(lt_cross_mul ℚ_ring_order); eauto.
+ rewrite Specify_classification despecify /rationals.le /ordered_rings.le.
eauto 11 using elts_in_set, rationals.lt_trans.
Qed.
Local Definition mul_pos (a b : R) : R.
Proof.
case: (excluded_middle_informative (0 < a))
(excluded_middle_informative (0 < b)) =>
H [H0 | H0]; first exact (mkSet (mul_pos_in a b H H0)); exact 0.
Defined.
Local Infix "·" := mul_pos (at level 40) : R_scope.
Local Theorem M1_pos : ∀ a b, 0 < a → 0 < b → a · b = b · a.
Proof.
rewrite /mul_pos /mul_pos_set => a b H H0.
repeat elim excluded_middle_informative => * //.
apply set_proj_injective, Extensionality => z.
rewrite ? Specify_classification ? despecify.
(split => [[H1] | [H1]]; rewrite ? (reify H1) ? despecify) =>
[[r [s [H2 [H3 [H4 [H5 H6]]]]]] | [r [s [H2 [H3 [H4 [H5 H6]]]]]]];
split; auto; exists s, r; rewrite M1; split; auto.
Qed.
Local Theorem O2_pos : ∀ a b, 0 < a → 0 < b → 0 < a · b.
Proof.
rewrite /mul_pos => a b /[dup] ? /pos_nonempty
[c [/[dup] ? /lt_dense [c' [? ?]] ?]] /[dup] ?
/pos_nonempty [d [/[dup] ? /lt_dense [d' [? ?]] ?]].
repeat elim excluded_middle_informative => * //.
split => [z /Specify_classification [H] | H].
- rewrite (reify H) Specify_classification ? despecify
/rationals.le /ordered_rings.le /=.
eauto 11 using O2, rationals.lt_trans.
- (have: (c' * d')%Q ∈ 0; first rewrite H;
rewrite Specify_classification despecify /rationals.le /ordered_rings.le)
=> /= [ | [] _ /(lt_not_ge ℚ_ring_order) []]; last by apply or_introl, O2.
eauto 10 using elts_in_set, rationals.lt_trans, or_introl,
(lt_cross_mul ℚ_ring_order : ∀ a b c d,
0 < a → 0 < c → a < b → c < d → a * c < b * d)%Q.
Qed.
Local Theorem M2_pos :
∀ a b c, 0 < a → 0 < b → 0 < c → a · (b · c) = (a · b) · c.
Proof.
rewrite /mul_pos => a b c /[dup] H /O2_pos H0 /[dup] /H0 {}H0 /[dup] H1
/O2_pos H2 /[dup] /H2 {}H2 H3.
repeat destruct excluded_middle_informative; try tauto;
try (contradict n; unfold mul_pos in *;
repeat destruct excluded_middle_informative; tauto).
apply set_proj_injective, Extensionality => z.
rewrite ? Specify_classification.
(split => [[H4] | [H4]]; rewrite (reify H4) ? despecify /= =>
[[ρ [τ [H5 [H6 [H7 [H8 H9]]]]]]]); [move: (H6) | move: (H5)];
rewrite Specify_classification despecify =>
[[H10 [r [s [H11 [H12 [H13 [H14 H15]]]]]]]]; split;
eauto using elts_in_set; [exists (ρ * r)%Q, s | exists r, (s * τ)%Q];
rewrite ? Specify_classification ? despecify; repeat split;
eauto using O2, elts_in_set; [exists ρ, r | | exists s, τ | ];
try (repeat split; auto; by right);
[apply (mul_le_l ℚ_ring_order ρ) in H15; auto; rewrite -M2 |
apply (mul_le_r ℚ_ring_order _ _ τ) in H15; auto; rewrite M2];
eapply (ordered_rings.le_trans ℚ_ring_order); eauto.
Qed.
Theorem zero_ne_1 : 1 ≠ 0.
Proof.
rewrite /zero /one => H.
inversion H as [H0].
move: H0 (ordered_rings.zero_lt_1 ℚ_ring_order) =>
/Subset_equality_iff [H0 H1] /lt_dense [c [H2 H3]].
contradiction (ordered_rings.lt_antisym ℚ_ring_order c 0%Q).
move: (H0 c).
rewrite ? Specify_classification ? despecify => [[]]; eauto using elts_in_set.
Qed.
Theorem zero_lt_1 : 0 < 1.
Proof.
split => [z | /set_proj_injective /(@eq_sym R) /zero_ne_1] //.
rewrite ? Specify_classification => [[]] H.
rewrite ? (reify H) ? despecify =>
/rationals.lt_trans /(_ (ordered_rings.zero_lt_1 ℚ_ring_order)) //.
Qed.
Local Theorem M3_pos : ∀ a, 0 < a → 1 · a = a.
Proof.
rewrite /mul_pos => a H.
repeat elim excluded_middle_informative => *; try (move: zero_lt_1; tauto).
apply set_proj_injective, Extensionality => z.
((split => [ | /[dup] /Dedekind_cut_0]; rewrite Specify_classification) =>
[[H0] | H0]; rewrite (reify H0) despecify) =>
[[r [s [/Specify_classification [H1] /[swap] [[H2 [H3 [/[swap] H4]]]]]]] |
/[dup] H1 /Dedekind_cut_3 [t [H2 H3]]].
- rewrite despecify => /[swap] /(O3 ℚ_ring_order s) /[apply] H6.
eapply Dedekind_cut_2; eauto.
rewrite ? (M1 r s) /= -? (rationals.M1 1%Q) ? rationals.M3 in H4 H6.
eapply (le_lt_trans ℚ_ring_order); eauto.
- case: (classic (0 < (mkSet H0))%Q) => [H4 | ].
+ split; eauto using elts_in_set.
exists ((mkSet H0) * t^-1)%Q, t.
have H5: (t ≠ 0)%Q by eapply (lt_neq ℚ_ring_order),
rationals.lt_trans; eauto.
repeat split; eauto using rationals.lt_trans.
* rewrite Specify_classification despecify.
split; eauto using elts_in_set.
rewrite -(inv_l t) ? (M1 (mkSet H0)); auto.
apply (O3 ℚ_ring_order); try apply (inv_lt ℚ_order) => /=; auto.
eauto using rationals.lt_trans.
* apply O2, (inv_lt ℚ_order) => /=; eauto using rationals.lt_trans.
* rewrite -M2 inv_l 1 ? M1 ? M3 /rationals.le /ordered_rings.le; auto.
+ rewrite -(le_not_gt ℚ_ring_order)
-[(ordered_rings.le ℚ_ring_order)]/(rationals.le) => H4.
move: zero_lt_1 H => /pos_nonempty [c [H5 H6]] /pos_nonempty [d [H7 H8]].
repeat esplit; eauto using elts_in_set.
eapply or_introl, (le_lt_trans ℚ_ring_order) => /=; eauto using O2.
Qed.
Local Definition inv_pos_set (α : R) :=
{p of type ℚ | ∃ r : Q, 1 < r ∧ (p ≤ 0 ∨ (0 < p ∧ (p * r)^-1 ∉ α))}%Q.
Local Theorem inv_pos_in : ∀ a, 0 < a → inv_pos_set a ∈ 𝐑.
Proof.
move=> a /[dup] H /pos_nonempty [c [/[dup] H0 /[dup] /(lt_neq ℚ_ring_order)
H1 /(inv_lt ℚ_order) /= H2 H3]].
rewrite Specify_classification Powerset_classification
Nonempty_classification.
(repeat split) => [x /Specify_classification [] | | H4 | p q | p] //.
- exists 0%Q.
rewrite Specify_classification despecify /rationals.le /ordered_rings.le.
eauto 6 using zero_lt_1, one_lt_2, rationals.lt_trans, elts_in_set.
- have: c^-1 ∈ ℚ by eauto using elts_in_set.
rewrite -H4 Specify_classification despecify =>
[[H5 [r [/[dup] H6 /[dup] /(ordered_fields.inv_lt_1 ℚ_order) /=
H7 /(one_lt ℚ_ring_order) /= /[dup] H8
/(lt_neq ℚ_ring_order) H9 [[/= ? | H10] | [? H11]]]]]].
+ move: H2 => /(ordered_rings.lt_antisym ℚ_ring_order) [] //.
+ move: H10 H2 -> => /(ordered_rings.lt_antisym ℚ_ring_order) //.
+ contradict H11.
eapply Dedekind_cut_2; eauto.
have ->: (c^-1 * r)^-1 = r^-1 * c by (field; auto).
rewrite -{2}(rationals.M3 c).
apply (O3_r ℚ_ring_order) => /=; auto.
- (rewrite ? Specify_classification ? despecify) =>
[[H4 [r [/[dup] H5 /(one_lt ℚ_ring_order) /= H6
[H7 H8 | [H7 H8] /[dup] H9 /(ordered_rings.O3 ℚ_ring_order r)
/= H10]]]]]; split; eauto using elts_in_set;
exists r; repeat split; auto.
+ eapply or_introl, or_introl, (lt_le_trans ℚ_ring_order); eauto.
+ (elim (classic (q ≤ 0)%Q); try tauto) => /lt_not_ge /= H11.
eapply or_intror, conj, Dedekind_cut_5; eauto.
(rewrite -[rationals.lt]/(ordered_fields.lt ℚ_order) -lt_cross_inv /=);
rewrite ? (M1 _ r); eauto using O3, O2, rationals.lt_trans.
- rewrite Specify_classification despecify =>
[[? [r [/[dup] ? /[dup] /lt_dense
[c' [/[dup] ? /(one_lt ℚ_ring_order) /= /[dup] /[dup]
? /(inv_lt ℚ_order) /= ? /(lt_neq ℚ_ring_order)
/= ? ?]] /(one_lt ℚ_ring_order) /= /[dup]
/(lt_neq ℚ_ring_order) /= ? /[dup] ?
/(inv_lt ℚ_order) /= ? [[/= ? | ?] | [? ?]]]]]].
+ exists 0%Q.
rewrite Specify_classification despecify.
eauto 6 using elts_in_set, one_lt_2, (le_refl ℚ_ring_order 0 : 0 ≤ 0)%Q.
+ subst.
move: (Dedekind_cut_6 a) => [c'' H14].
exists (c''^-1 * r^-1 * r^-1)%Q.
have /[dup] /(inv_lt ℚ_order) /= *: (0 < c'')%Q.
{ eapply Dedekind_cut_4; eauto.
move: H => /pos_nonempty [d] [] /Dedekind_cut_2 /[apply] //. }
rewrite Specify_classification despecify.
repeat split; eauto using elts_in_set, O2.
exists r.
repeat (split; try right); auto using O2.
eapply Dedekind_cut_5; eauto.
rewrite -M2 inv_l // -(M1 1) M3 inv_mul inv_inv -{1}(M3 c'') -(M1 c'').
apply (O3 ℚ_ring_order); auto.
+ exists (p * r * c'^-1)%Q.
rewrite -{1}(M3 p) -(M1 p) -M2 Specify_classification despecify.
repeat split; eauto using elts_in_set.
* apply (O3 ℚ_ring_order), (lt_div ℚ_order); auto.
* exists c'.
rewrite -? M2 inv_l // -(M1 1) M3.
auto 6 using O2.
Qed.
Definition inv_pos (a : R) : R.
Proof.
case (excluded_middle_informative (0 < a)) => [H | H].
- exact (mkSet (inv_pos_in _ H)).
- exact 0.
Defined.
Local Notation "a '^-1' " :=
(inv_pos a) (at level 30, format "a '^-1'") : R_scope.
Lemma pos_not_in_0 : ∀ x : Q, (0 < x)%Q → x ∉ 0.
Proof.
move=> x H /Specify_classification [H0].
rewrite despecify => /(ordered_rings.lt_antisym ℚ_ring_order) //.
Qed.
Theorem inv_lt : ∀ a, 0 < a → 0 < a^-1.
Proof.
rewrite /lt /inv_pos => a /[dup] H /pos_nonempty [c [H0 H1]].
split => [z /Specify_classification [H2] | ].
- elim excluded_middle_informative => // ?.
rewrite Specify_classification ? (reify H2) ? despecify
/rationals.le /ordered_rings.le.
eauto 6 using one_lt_2.
- elim excluded_middle_informative => // H2 H3.
elim: (Dedekind_cut_6 a) => x H4.
have /[dup] /(inv_lt ℚ_order) /= /[swap]: (0 < 2)%Q =>
[ | /[dup] H5 /(ordered_rings.lt_neq ℚ_ring_order) H6].
{ rewrite -IZQ_add.
apply (ordered_rings.zero_lt_2 ℚ_ring_order). }
(have /[dup] /(inv_lt ℚ_order) /= /[swap]: (0 < x)%Q by
eauto using Dedekind_cut_2, Dedekind_cut_4) =>
H7 /O2 /[apply] /[dup] H8 /pos_not_in_0 [].
rewrite H3 /inv_pos /inv_pos_set Specify_classification despecify.
split; eauto using elts_in_set.
exists 2%Q.
rewrite -M2 inv_l // -(M1 1) M3 inv_inv.
auto using one_lt_2.
Qed.
Theorem pow_archimedean : ∀ (a : R) (r : Q),
0 < a → (1 < r)%Q → ∃ n : Z, (r^n)%Q ∈ a ∧ (r^(n + 1))%Q ∉ a.
Proof.
move=> a r /[dup] H /pos_nonempty [c [H0 H1]] /[dup]
/(ordered_rings.one_lt ℚ_ring_order) /= H2.
move: (Dedekind_cut_6 a) => [q H3] /[dup] /(neg_pow_archimedean c r H0)
=> [[m [H4 H5]]] /[dup] H6 /(pos_pow_archimedean q r) [n [H7 H8]].
move: (WOP (λ x, (r^(m + x))%Q ∉ a)) => [x H9 | | x [H9 H10]].
- apply (lt_not_ge ℤ_order) => /= [[/= H10 | H10]].
+ rewrite -[rationals.pow]/(fields.pow ℚ) (pow_add_r ℚ) /= in H9;
auto using (lt_neq ℚ_ring_order).
eapply H9, Dedekind_cut_2; eauto.
rewrite -(M3 c) (M1 1).
apply (lt_cross_mul ℚ_ring_order); auto; try by apply (pow_pos ℚ_order).
by apply (pow_lt_1 ℚ_order).
+ rewrite H10 integers.A1 integers.A3 in H9.
eauto using Dedekind_cut_2.
- exists (n + -m)%Z.
have ->: (m + (n + -m) = n)%Z by ring.
eauto using Dedekind_cut_5.
- exists (m + (x + -1))%Z.
split; last by have ->: (m + (x + -1) + 1 = m + x)%Z by ring.
apply NNPP => /H10 /le_not_gt [] /=.
have {2}->: (x = x + 1 + -1)%Z by ring.
apply (O1_r ℤ_order), lt_succ.
Qed.
Local Theorem M4_pos : ∀ a, 0 < a → a^-1 · a = 1.
Proof.
rewrite /mul_pos => a H.
apply set_proj_injective, Extensionality => z.
(repeat case excluded_middle_informative) => // => [ | /inv_lt] // {}H H0.
rewrite ? Specify_classification.
(split => [[H1] | [H1]]; rewrite ? (reify H1) ? despecify) =>
[[r [s [/[swap] [[H2 [H3 [H4 H5]]]]]]] | H2].
- rewrite /inv_pos.
case excluded_middle_informative => // {}H.
rewrite Specify_classification despecify /= =>
[[H6 [q [/[dup] H7 /[dup] /(one_lt ℚ_ring_order) /= /[dup]
/(lt_neq ℚ_ring_order) H8 H9 /(inv_lt_1 ℚ_order) /=
/(_ H9) H10 [/le_lt_trans /(_ H3)
/ordered_rings.lt_irrefl |
[/[dup] H11 /(lt_neq ℚ_ring_order) H12
/(Dedekind_cut_4 _ _ _ H2) H13]]]]]] //.
split; auto.
eapply (ordered_rings.lt_trans ℚ_ring_order); eauto.
eapply le_lt_trans; eauto => /=.
have ->: (q^-1 = r * (r * q)^-1)%Q by field.
by apply (O3 ℚ_ring_order).
- split; auto.
move: (H) (H0) (H2) => /pos_nonempty [c [H3 H4]] /pos_nonempty [d [H5 H6]].
rewrite -{1} (inv_inv (mkSet H1)) => /(inv_lt_1 ℚ_order) /=.
(case (classic (0 < (mkSet H1))%Q) =>
[/[dup] H7 /(ordered_fields.inv_lt ℚ_order) /= /[dup] H8 /[swap]
/[apply] /[dup] H9 /square_in_interval
[r [/[dup] H10 /(lt_neq ℚ_ring_order) H11
[/[dup] /(square_ge_1 ℚ_ring_order _ H10) /=
/[dup] H12 /(pow_archimedean _ _ H) [n [H13 H14]] H15 H16]]] |
/(le_not_gt ℚ_ring_order) H7]);
last by rewrite /rationals.le /ordered_rings.le;
eauto 10 using le_lt_trans, (ordered_rings.O2 ℚ_ring_order).
exists (r^(-(n+2))), (r^n).
(repeat split; auto; try now apply (pow_pos ℚ_order));
rewrite /inv_pos /pow -? [rationals.mul]/(rings.mul ℚ) -? pow_add_r //.
+ case excluded_middle_informative => // {}H.
rewrite Specify_classification despecify.
split; eauto using elts_in_set.
exists r.
split; auto.
right.
split; first by apply (pow_pos ℚ_order).
rewrite -inv_mul -? inv_pow -pow_mul_r /pow -[rationals.mul]/(rings.mul ℚ)
-pow_add_r //.
by have ->: (-(n + 2) * -1 + -1 = n + 1)%Z by ring.
+ have ->: (-(n + 2) + n = -2%Z)%Z by ring.
rewrite -/pow (neg_pow r 2%Z) -(inv_inv (mkSet H1)).
apply or_introl, (lt_cross_inv ℚ_order (r^2)%Q); auto;
rewrite /pow /fields.pow INZ_add add_1_r pow_nonneg pow_2_r //=.
by apply O2.
Qed.
Local Theorem D1_pos :
∀ a b c, 0 < a → 0 < b → 0 < c → (a + b) · c = a · c + b · c.
Proof.
rewrite /mul_pos /add_set /mul_pos_set => a b c H /[dup] H0 /(O1 a) /[swap].
move: A3 (H) -> => /lt_trans /[swap] H1 /[apply] H2.
repeat destruct excluded_middle_informative; try tauto;
try (contradict n; unfold mul_pos in *;
repeat destruct excluded_middle_informative; tauto).
apply set_proj_injective, Extensionality => z.
(split => /Specify_classification [H3];
rewrite ? Specify_classification ? (reify H3) ? despecify) =>
[[r [s [/Specify_classification
[H4 [r' [s' [/set_proj_injective -> [H5 H6]]]]]
[H7 [/[dup] H8 /(O0_opp ℚ_ring_order) H9 [H10 H11]]]]]] |
[ac [bc [/set_proj_injective H4 [H5 H6]]]]];
repeat split; auto; set (ζ := mkSet H3 : Q).
- move: (pos_nonempty a H) (pos_nonempty b H0) => [t [H12 ?]] [t' [H14 ?]].
move: (lt_dense2 t (r'+s') H12 H8) (lt_dense2 t' (r'+s') H14 H8) =>
[k [? [? ?]]] [k' [? [? ?]]].
(case: (classic (0 < r')%Q) (classic (0 < s')%Q) =>
[? | ?] [? | ?]; try tauto);
[ exists (ζ + -s' * s)%Q, (s' * s)%Q | exists (ζ + -k' * s)%Q, (k' * s)%Q |
exists (k * s)%Q, (ζ + -k * s)%Q ];
rewrite /IQS ? Specify_classification ? despecify;
repeat split; eauto 8 using elts_in_set, Dedekind_cut_2,
(le_refl ℚ_ring_order : ∀ a, a ≤ a)%Q; try (f_equal; ring).
+ exists r', s.
repeat split; auto.
move: H11 => /(add_le_l ℚ_ring_order) /= /(_ (-s' * s)%Q).
rewrite -[(ordered_rings.le ℚ_ring_order)]/(rationals.le) => H11.
ring_simplify in H11; ring_simplify.
by have ->: (ζ - s' * s = - s' * s + ζ)%Q by ring.
+ exists (r' + s' + -k')%Q, s.
have ->: ((r' + s' + -k')*s = (r' + s') * s + -k' * s)%Q by ring.
(repeat split; auto; last by apply add_le_r);
rewrite -? [rationals.lt]/(ordered_rings.lt ℚ_ring_order)
-? (ordered_rings.lt_shift ℚ_ring_order) //.
eapply (Dedekind_cut_2 _ r'), (lt_shift ℚ_ring_order); auto => /=.
have ->: (r' + -(r' + s' + -k') = k' + -s')%Q by ring.
eapply (lt_shift ℚ_ring_order s'), (le_lt_trans ℚ_ring_order); eauto.
by apply le_not_gt.
+ exists (r' + s' + -k)%Q, s.
have ->: ((r' + s' + -k) * s = (r' + s') * s + -k * s)%Q by ring.
(repeat split; auto; last by apply add_le_r);
rewrite -? [rationals.lt]/(ordered_rings.lt ℚ_ring_order)
-? (ordered_rings.lt_shift ℚ_ring_order) //.
eapply (Dedekind_cut_2 _ s'), (lt_shift ℚ_ring_order); auto => /=.
have ->: (s' + -(r' + s' + -k) = k + -r')%Q by ring.
eapply (lt_shift ℚ_ring_order r'), (le_lt_trans ℚ_ring_order); eauto.
by apply le_not_gt.
- move: H6 H5 => /Specify_classification /[swap] /Specify_classification.
rewrite ? despecify => [[H5 [a' [c' [H6 [H7 [H8 [H9 H10]]]]]]]]
[H11 [b' [c'' [H12 [H13 [H14 [H15 H16]]]]]]].
exists (a' + b')%Q, (ordered_rings.max ℚ_ring_order c' c'').
rewrite Specify_classification.
repeat split; eauto using O0, elts_in_set; rewrite /ζ ? H4 ? rationals.D1;
try by case: (max_eq ℚ_ring_order c' c'') => -> //.
apply le_cross_add; fold rationals.le; auto;
eapply (ordered_rings.le_trans ℚ_ring_order); eauto;
apply mul_le_l; simpl; auto; fold rationals.le;
[apply max_le_l | apply max_le_r].
Qed.
Definition mul (a b : R) : R.
Proof.
case (excluded_middle_informative (0 < a)),
(excluded_middle_informative (0 < b)).
- exact (a · b).
- exact (If (0 = b) then 0 else (-(a · (-b)))).
- exact (If (0 = a) then 0 else (-((-a) · b))).
- exact (If (0 = b) then 0 else If (0 = a) then 0 else ((-a) · (-b))).
Defined.
Infix "*" := mul : R_scope.
Theorem mul_0_r : ∀ a, a * 0 = 0.
Proof.
rewrite /mul => a.
(repeat case excluded_middle_informative => // [ | _]) => /lt_irrefl //.
Qed.
Theorem mul_0_l : ∀ a, 0 * a = 0.
Proof.
rewrite /mul => a.
(repeat case excluded_middle_informative => //) => [_ | _ _] /lt_irrefl //.
Qed.
Local Theorem R_mul_pos_pos : ∀ a b, 0 < a → 0 < b → a * b = a · b.
Proof.
rewrite /mul => a b.
repeat case excluded_middle_informative => //.
Qed.
Local Theorem R_mul_pos_neg : ∀ a b, 0 < a → b < 0 → a * b = -(a · -b).
Proof.
rewrite /mul => a b.
(repeat case excluded_middle_informative => //) => [/lt_antisym | <-] //.
Qed.
Local Theorem R_mul_neg_pos : ∀ a b, a < 0 → 0 < b → a * b = -(-a · b).
Proof.
rewrite /mul => a b.
(repeat case excluded_middle_informative => //) => [_ /lt_antisym | <-] //.
Qed.
Local Theorem R_mul_neg_neg : ∀ a b, a < 0 → b < 0 → a * b = (-a · -b).
Proof.
rewrite /mul => a b.
(repeat case excluded_middle_informative => //) =>
[/lt_antisym | <- | _ _ | <- | _ | <- | <- ] // /lt_antisym //.
Qed.
Theorem lt_shift : ∀ a b, a < b ↔ 0 < b + -a.
Proof.
(split => [/(O1 (-a)) | /(O1 a)]);
rewrite ? A4_l ? A4 ? A3 A1 // -A2 A4_l A3 //.
Qed.
Theorem lt_neg_0 : ∀ a, a < 0 ↔ 0 < -a.
Proof.
move=> a.
by rewrite lt_shift A3_l.
Qed.
Theorem neg_neg : ∀ a, - - a = a.
Proof.
move=> a.
by rewrite -{2}(A3 a) -(A4 (-a)) A2 A4 A1 A3.
Qed.
Theorem neg_lt_0 : ∀ a, -a < 0 ↔ 0 < a.
Proof.
move=> a.
rewrite lt_shift neg_neg A3_l //.
Qed.
Theorem M1 : ∀ a b, a * b = b * a.
Proof.
rewrite /mul => a b.
move: (T 0 a) (T 0 b) => *.
(repeat case excluded_middle_informative => * //);
rewrite ? (M1_pos a) ? (M1_pos b) // 1 ? M1_pos // -lt_neg_0; tauto.
Qed.
Theorem M2 : ∀ a b c, a * (b * c) = (a * b) * c.
Proof.
(move=> a b c; move: O2_pos (T 0 a) (T 0 b) (T 0 c) =>
? [[? _] | [[? [<- _]] | [_ [_ /[dup] ? /lt_neg_0 ?]]]]
[[? _] | [[? [<- _]] | [_ [_ /[dup] ? /lt_neg_0 ?]]]]
[[? _] | [[? [<- _]] | [_ [_ /[dup] ? /lt_neg_0 ?]]]];
repeat rewrite ? mul_0_l ? mul_0_r //);
[ rewrite ? R_mul_pos_pos |
rewrite ? (R_mul_pos_neg _ c) ? (R_mul_pos_pos _ b) ? R_mul_pos_neg |
rewrite ? (R_mul_pos_neg a) ? (R_mul_neg_pos_c) ? (R_mul_neg_pos) |
rewrite ? (R_mul_neg_neg _ c) ? (R_mul_pos_neg _ b) ? R_mul_pos_pos |
rewrite ? (R_mul_neg_pos a) ? (R_mul_pos_pos b) ? R_mul_neg_pos |
rewrite (R_mul_pos_neg b) ? (R_mul_neg_pos _ b) ? R_mul_neg_neg |
rewrite ? (R_mul_neg_neg a) ? (R_mul_neg_pos b) ? R_mul_pos_pos |
rewrite (R_mul_neg_pos _ (b * c)) ? (R_mul_pos_neg (a * b))
? R_mul_neg_neg ]; rewrite ? lt_neg_0 ? neg_neg ? M2_pos; auto.
Qed.
Theorem M3 : ∀ a, 1 * a = a.
Proof.
move: zero_lt_1 T => H /[swap] a =>
/(_ 0 a) [[? _] | [[? [<- _]] | [_ [_ /[dup] ? /lt_neg_0 ?]]]].
- rewrite R_mul_pos_pos ? M3_pos //.
- rewrite mul_0_r //.
- rewrite R_mul_pos_neg ? lt_neg_0 ? M3_pos ? neg_neg //.
Qed.
Theorem O0 : ∀ a b, 0 < a → 0 < b → 0 < a + b.
Proof.
move=> a b /lt_trans H /(O1 a).
rewrite A3 => /H //.
Qed.
Theorem neg_add : ∀ a b, - (a + b) = - a + - b.
Proof.
move=> a b.
rewrite -(A3 (-a + -b)) -(A4 (a + b)) A2 (A1 a) A2 -A2
-(A2 (-a)) A4_l A3 A2 A4_l A3_l //.
Qed.
Local Lemma D1_opp : ∀ a b c, 0 < a → b < 0 → (a + b) * c = a * c + b * c.
Proof.
move=> a b c H /[dup] H0 /lt_neg_0 H1.
have H2: a = (a + b + -b) by rewrite -A2 A4 A3.
have H3: -b = -(a + b) + a by rewrite neg_add A1 A2 A4 A3_l.
(move: (T 0 c) => [[? _] | [[? [<- _]] | [_ [_ /[dup] ? /lt_neg_0 ?]]]];
rewrite ? mul_0_r ? A3 // {2}H2);
move: (T 0 (a + b)) => [[? _] | [[? [<- _]] | [_ [_ /[dup] ? /lt_neg_0 ?]]]].
- rewrite ? (R_mul_pos_pos (a + b + -b)) ? (D1_pos (a + b))
-? H2 // R_mul_pos_pos // R_mul_neg_pos // -A2 A4 A3 //.
- rewrite mul_0_l A3_l R_mul_pos_pos // R_mul_neg_pos // A4 //.
- rewrite -H2 R_mul_neg_pos // R_mul_pos_pos // R_mul_neg_pos // H3
D1_pos // ? neg_add (A1 (a · c)) -A2 A4_l A3 //.
- rewrite R_mul_pos_neg // R_mul_pos_neg ? (D1_pos _ (-b))
-? H2 // R_mul_neg_neg // neg_add -A2 A4_l A3 //.
- rewrite A3_l mul_0_l R_mul_pos_neg ? R_mul_neg_neg // A4_l //.
- rewrite -H2 R_mul_neg_neg // R_mul_pos_neg // R_mul_neg_neg // H3
D1_pos // (A1 (- (a · -c))) -A2 A4 A3 //.
Qed.