`Compfiles/Imo1988P4.lean`

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International Mathematical Olympiad 1988, Problem 4

Show that the set of real numbers $x$ which satisfy the inequality $$\sum_{k=1}^{70} \frac{k}{x - k} \ge \frac{5}{4}$$ is a union of disjoint intervals, the sum of whose lengths is $1988$.

Imports (2)

Imported by 1 file

Compfiles.lean

abbrev Imo1988P4.singularities line 33 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: Finset.Icc, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.
abbrev Imo1988P4.f line 37 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: HSub.hSub, Finset.sum, Nat, Imo1988P4.singularities, HDiv.hDiv, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.
abbrev Imo1988P4.S line 39 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real
Proof uses: Set.ofPred, LE.le, HDiv.hDiv, OfNat.ofNat, Finset.sum, Nat, Imo1988P4.singularities, HSub.hSub
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
instance Imo1988P4.(instance at line 41) line 41 source
abbrev Imo1988P4.last line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities
Proof uses: Subtype.mk, OfNat.ofNat
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Imo1988P4.s_nonempty line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset.Nonempty, Nat, Imo1988P4.singularities
Proof uses: Exists.intro, Membership.mem, Finset, Imo1988P4.last, Iff.mp, Eq, Insert.insert, Finset.insert_eq_self, rfl
5 instances, coercions and auxiliary constants left out.
lemma Imo1988P4.f_aligned line 47 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Set, Real, Set.ofPred, LE.le, OfNat.ofNat, Imo1988P4.f, Imo1988P4.S
Proof uses: of_eq_true, Eq.trans, HDiv.hDiv, Finset.sum, Nat, Imo1988P4.singularities, HSub.hSub, True, congrFun', congrArg, funext, eq_self
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.div_lt_div' line 52 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, LT.lt, OfNat.ofNat, GT.gt, Iff, HDiv.hDiv
Proof uses: Eq.mpr, Neg.neg, id, Eq, congrArg, Eq.symm, propext, neg_lt_neg_iff, HAdd.hAdd, div_neg, div_lt_div_iff_of_pos_left, Iff.mpr, Left.neg_pos_iff, Iff.rfl
78 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.anti_io line 55 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: StrictAntiOn, Real, Imo1988P4.f, Set.Iio, OfNat.ofNat
Proof uses: StrictAntiOn.add_const, Finset.sum, Nat, Imo1988P4.singularities, HDiv.hDiv, HSub.hSub, HAdd.hAdd, Membership.mem, Set, LT.lt, Finset.sum_lt_sum_of_nonempty, Imo1988P4.s_nonempty, Finset, LE.le, Iff.mpr, Nat.one_le_cast, And.left, Eq.mp, And, Eq.mpr and 55 more
105 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.anti_oi line 61 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: StrictAntiOn, Real, Imo1988P4.f, Set.Ioi, OfNat.ofNat, Nat
Proof uses: StrictAntiOn.add_const, Finset.sum, Imo1988P4.singularities, HDiv.hDiv, HSub.hSub, HAdd.hAdd, Membership.mem, Set, LT.lt, Finset.sum_lt_sum_of_nonempty, Imo1988P4.s_nonempty, Finset, LE.le, Iff.mpr, Nat.cast_le_ofNat, And.right, Eq.mp, And, Eq.mpr, id and 9 more
96 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.component_anti_oo line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, StrictAntiOn, Real, HDiv.hDiv, HSub.hSub, Set.Ioo, HAdd.hAdd, OfNat.ofNat
Proof uses: Subtype.mk, LT.lt, LE.le, cast, Eq.symm, Eq.trans, implies_congr, Eq.refl, congrFun', congrArg, Nat.cast_one, Iff.mpr, Order.add_one_le_iff, Set, dite, Eq.mpr, id, Eq, propext, div_lt_div_iff_of_pos_left and 61 more
158 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.anti_oo line 80 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, StrictAntiOn, Real, Imo1988P4.f, Set.Ioo, HAdd.hAdd, OfNat.ofNat
Proof uses: StrictAntiOn.add_const, Finset.sum, HDiv.hDiv, HSub.hSub, Set, LT.lt, Finset.sum_lt_sum_of_nonempty, Imo1988P4.s_nonempty, Imo1988P4.component_anti_oo, Subtype.mk, Neg.neg
58 instances, coercions and auxiliary constants left out.
lemma Imo1988P4.component_continuous line 90 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, ContinuousOn, Real, HDiv.hDiv, HSub.hSub, Union.union, Set, Set.Iio, Set.Ioi
Proof uses: Eq.mpr, Subtype.mk, And, id, Eq, congrArg, propext, continuousOn_union_iff_of_isOpen, isOpen_Iio, isOpen_Ioi, And.intro, ContinuousOn.div₀, continuousOn_const, ContinuousOn.sub, continuousOn_id
66 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.continuous_oi line 98 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: ContinuousOn, Real, Imo1988P4.f, Set.Ioi, OfNat.ofNat, Nat
Proof uses: ContinuousOn.add, Finset.sum, Imo1988P4.singularities, HDiv.hDiv, HSub.hSub, Neg.neg, continuousOn_finsetSum, Membership.mem, Finset, ContinuousOn.mono, Union.union, Set, Set.Iio, Subtype.mk, Imo1988P4.component_continuous, Eq.mpr, LE.le, id, Eq, Eq.trans and 10 more
71 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.continuous_oo line 104 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, ContinuousOn, Real, Imo1988P4.f, Set.Ioo, HAdd.hAdd, OfNat.ofNat
Proof uses: ContinuousOn.add, Finset.sum, HDiv.hDiv, HSub.hSub, Neg.neg, continuousOn_finsetSum, ContinuousOn.mono, Union.union, Set, Set.Iio, Subtype.mk, Set.Ioi, Imo1988P4.component_continuous, Eq.mpr, LE.le, LT.lt, id, Eq, Eq.trans, Not and 13 more
83 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.component_atBot each singularity (at the integers) has the left side approaching bot and right side approaching top line 118 source
Source comment
each singularity (at the integers) has the left side approaching bot and right side approaching top

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Filter.Tendsto, Real, HDiv.hDiv, HSub.hSub, nhdsWithin, Set.Iio, Filter.atBot
Proof uses: Eq.mpr, Filter.Eventually, LE.le, id, Eq, congrArg, propext, Filter.tendsto_atBot, Exists, Set, And, Filter, Filter.eventually_iff_exists_mem, LT.lt, OfNat.ofNat, Iff.mpr, Nat.cast_pos, And.left, Eq.mp, dite and 87 more
165 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
```
lemma component_atBot (k: singularities) : Tendsto (fun (x: ℝ) => k / (x - k)) (𝓝[<] k) atBot := by
…
```
Mentions: Imo1988P4.singularities
lemma Imo1988P4.component_atTop line 140 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Filter.Tendsto, Real, HDiv.hDiv, HSub.hSub, nhdsWithin, Set.Ioi, Filter.atTop
Proof uses: Eq.mpr, Filter.Eventually, LE.le, id, Eq, congrArg, propext, Filter.tendsto_atTop, Exists, Set, And, Filter, Filter.eventually_iff_exists_mem, OfNat.ofNat, Subtype.mk, Eq.mp, dite, LT.lt, Exists.intro, Set.Ioo and 61 more
151 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.f_atTop line 156 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Filter.Tendsto, Real, Imo1988P4.f, Filter.atTop, nhds, Neg.neg, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: id, Membership.mem, Finset, Imo1988P4.singularities, HSub.hSub, Eq.mpr, LE.le, GT.gt, Exists, GE.ge, LT.lt, Dist.dist, Eq, forall_congr, Finset.Icc, Eq.trans, And, implies_congr, Exists.intro, HAdd.hAdd and 22 more
135 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.f_atBot line 171 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Filter.Tendsto, Real, Imo1988P4.f, Filter.atBot, nhds, Neg.neg, HDiv.hDiv, OfNat.ofNat, Nat
Proof uses: id, Membership.mem, Finset, Imo1988P4.singularities, HSub.hSub, Eq.mpr, LE.le, Eq, forall_congr, Finset.Icc, Eq.trans, And, implies_congr, Eq.refl, GT.gt, Filter.Eventually, LT.lt, Dist.dist, abs, congrFun' and 25 more
129 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.erase_continuousOn line 187 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, ContinuousOn, Real, Finset.sum, SDiff.sdiff, Singleton.singleton, HDiv.hDiv, HSub.hSub, Set.Ioo, OfNat.ofNat, HAdd.hAdd
Proof uses: continuousOn_finsetSum, Eq.mpr, And, Not, Eq, id, forall_congr, implies_congr, Eq.trans, congrArg, Eq.refl, ContinuousOn.mono, Union.union, Set, Set.Iio, Subtype.mk, Set.Ioi, Imo1988P4.component_continuous, LT.lt, Or and 10 more
111 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.erase_continuousAt line 199 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, ContinuousAt, Real, HSub.hSub, Finset.sum, SDiff.sdiff, Singleton.singleton, HDiv.hDiv, OfNat.ofNat
Proof uses: ContinuousAt.add, Neg.neg, ContinuousOn.continuousAt, Set.Ioo, HAdd.hAdd, Imo1988P4.erase_continuousOn, Iff.mpr, Set, Filter, nhds, Exists, And, LE.le, IsOpen, mem_nhds_iff, Exists.intro, And.intro, le_refl, isOpen_Ioo, of_eq_true and 7 more
107 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.f_singularities_nhdsLT_atBot line 202 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Filter.Tendsto, Real, Imo1988P4.f, nhdsWithin, Set.Iio, Filter.atBot
Proof uses: Eq.mpr, HAdd.hAdd, HDiv.hDiv, HSub.hSub, Finset.sum, SDiff.sdiff, Singleton.singleton, OfNat.ofNat, id, Eq, congrArg, of_eq_true, Eq.trans, True, funext, congrFun', Finset.sum_sdiff_eq_sub, LE.le, Finset.sum_singleton, add_sub_cancel and 13 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.f_singularities_nhdsGT_atTop line 207 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Filter.Tendsto, Real, Imo1988P4.f, nhdsWithin, Set.Ioi, Filter.atTop
Proof uses: Eq.mpr, HAdd.hAdd, HDiv.hDiv, HSub.hSub, Finset.sum, SDiff.sdiff, Singleton.singleton, OfNat.ofNat, id, Eq, congrArg, of_eq_true, Eq.trans, True, funext, congrFun', Finset.sum_sdiff_eq_sub, LE.le, Finset.sum_singleton, add_sub_cancel and 13 more
54 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.root_Ioi we can have a stronger condition by replacing ∃ with ∃! line 220 source
Source comment
we can have a stronger condition by replacing ∃ with ∃! from combining the antitone condition

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Real, And, Membership.mem, Set, Set.Ioi, OfNat.ofNat, Nat, Eq, Imo1988P4.f
Proof uses: LE.le, Filter, nhdsWithin, Filter.principal, Eq.mpr, id, Set.Ioo, LT.lt, congrArg, funext, congrFun', Exists.intro, HAdd.hAdd, And.intro, lt_add_one, Set.Ioo_subset_Ioi_self, Filter.atTop, Iff.mpr, Filter.le_principal_iff, Filter.Ioi_mem_atTop and 21 more
119 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma root_Ioi : ∃ x ∈ Set.Ioi 70, f x = 0 := by
…
```
Mentions: Imo1988P4.f
lemma Imo1988P4.root_Ioo line 231 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Ne, OfNat.ofNat, Exists, Real, And, Set, Set.Ioo, HAdd.hAdd, Eq, Imo1988P4.f
Proof uses: LE.le, Filter, nhdsWithin, Set.Ioi, Filter.principal, Eq.mpr, id, LT.lt, congrArg, funext, congrFun', Exists.intro, And.intro, of_eq_true, Eq.trans, True, le_refl, Set.Iio, lt_add_one, Subtype.mk and 68 more
186 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
def Imo1988P4.r line 249 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, NNReal
Proof uses: dite, Eq, OfNat.ofNat, Subtype.mk, Real, LE.le, Exists.choose, And, Set, Set.Ioi, Imo1988P4.f, Imo1988P4.root_Ioi, Not, Set.Ioo, HAdd.hAdd, Imo1988P4.root_Ioo
22 instances, coercions and auxiliary constants left out.
lemma Imo1988P4.r_root line 253 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Eq, Real, Imo1988P4.f, Imo1988P4.r, OfNat.ofNat
Proof uses: dite, Eq.mpr, Subtype.mk, LE.le, Exists.choose, And, Set, Set.Ioi, Imo1988P4.root_Ioi, id, congrFun', congrArg, NNReal, Eq.trans, Not, Set.Ioo, HAdd.hAdd, Imo1988P4.root_Ioo, Exists, funext and 13 more
29 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.r_bounds line 260 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Ne, OfNat.ofNat, Real, Set, Set.Ioo, HAdd.hAdd, Imo1988P4.r
Proof uses: Eq.mpr, And, LT.lt, Subtype.mk, LE.le, Exists.choose, Eq, Imo1988P4.f, Exists, Imo1988P4.root_Ioo, of_eq_false, eq_false, funext, congrFun', congrArg, Exists.choose.congr_simp, id, Eq.trans, NNReal, dite and 7 more
37 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.r_bounds₁ line 264 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Ne, OfNat.ofNat, LT.lt, NNReal, Imo1988P4.r, HAdd.hAdd
Proof uses: And.right, Real, Iff.mp, Set, Set.Ioo, And, Set.mem_Ioo, Imo1988P4.r_bounds
32 instances, coercions and auxiliary constants left out.
lemma Imo1988P4.r_bounds₂ line 267 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, LT.lt, NNReal, Imo1988P4.r
Proof uses: dite, Eq, OfNat.ofNat, Eq.mpr, Subtype.mk, Real, LE.le, Exists.choose, And, Set, Set.Ioi, Imo1988P4.f, Imo1988P4.root_Ioi, id, congr, congrArg, Eq.trans, Not, Set.Ioo, HAdd.hAdd and 19 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.r_not_singularity line 274 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Ne, NNReal, Imo1988P4.r
Proof uses: dite, LE.le, Eq.mpr, id, Eq, Eq.trans, LT.lt, Imo1988P4.r_bounds₂, Not, OfNat.ofNat, Subtype.mk, Eq.mp, And, HAdd.hAdd, Imo1988P4.r_bounds₁, congrFun', congrArg, Eq.symm, Nat.cast_one
55 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.r_le_r_70 line 286 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, LE.le, NNReal, Imo1988P4.r, Imo1988P4.last
Proof uses: dite, Subtype.mk, Eq, OfNat.ofNat, le_refl, Eq.symm, Not, le_of_lt, Trans.trans, LT.lt, HAdd.hAdd, Imo1988P4.r_bounds₁, Eq.mpr, id, Eq.trans, congrFun', congrArg, Nat.cast_one, Eq.mp, Finset.Icc and 4 more
45 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.r_inj line 295 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Function.Injective, Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, NNReal, Imo1988P4.r
Proof uses: Eq, Mathlib.Tactic.Contrapose.contrapose₁, id, Not, Ne, LT.lt, Or, Classical.em, Subtype.mk, HAdd.hAdd, OfNat.ofNat, Imo1988P4.r_bounds₁, Eq.mp, Finset.Icc, And, LE.le, congrArg, propext, Finset.mem_Icc, ne_of_lt and 11 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 17 steps
Open to load the checked before-and-after goal states.
abbrev Imo1988P4.rset line 308 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Real
Proof uses: Finset.image, Subtype, Nat, Membership.mem, Imo1988P4.singularities, Function.comp, NNReal, Imo1988P4.r, Finset.univ
5 instances, coercions and auxiliary constants left out.
lemma Imo1988P4.rset_card line 310 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Finset.card, Real, Imo1988P4.rset, OfNat.ofNat
Proof uses: Finset.card_image_of_injective, Subtype, Membership.mem, Finset, Imo1988P4.singularities, Function.comp, NNReal, Imo1988P4.r, Finset.univ, Function.Injective.comp, NNReal.coe_injective, Imo1988P4.r_inj
6 instances, coercions and auxiliary constants left out.
lemma Imo1988P4.singularities_sum line 313 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, Finset.sum, Fin, OfNat.ofNat, Nat, NNReal, Finset.univ, Imo1988P4.r, Subtype.mk, Membership.mem, Finset, Imo1988P4.singularities, HAdd.hAdd, Imo1988P4.rset
Proof uses: Eq.mpr, Finset.image, Subtype, Function.comp, id, congrArg, Imo1988P4.rset.eq_1, Finset.sum_image, Function.Injective.injOn, Function.Injective.comp, NNReal.coe_injective, Imo1988P4.r_inj, Finset.attach, congrFun', NNReal.coe_sum, Eq.trans, congr, Eq.refl
29 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def Imo1988P4.I the endpoints don't matter in the volume, so we don't need to know if they are in the solution or not line 323 source
Source comment
the endpoints don't matter in the volume, so we don't need to know if they are in the solution or not

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Set, Real
Proof uses: ite, LE.le, OfNat.ofNat, Imo1988P4.f, Set.Icc, Set.Ioc, Imo1988P4.r
13 instances, coercions and auxiliary constants left out.
```
def I (n: singularities) := (if 0 ≤ f n then Set.Icc else Set.Ioc) (n: ℝ) (r n)
```
Mentions: Imo1988P4.singularities, Imo1988P4.f, Imo1988P4.r
lemma Imo1988P4.I_bounds line 325 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Ne, OfNat.ofNat, LE.le, Set, Real, Imo1988P4.I, Set.Ico, HAdd.hAdd
Proof uses: LT.lt, NNReal, Imo1988P4.r, Imo1988P4.r_bounds₁, id, Imo1988P4.f, Decidable, ite, Set.Icc, Set.Ioc, Not, Eq.mpr, Eq, congrFun', congrArg, ite_eq_right, LE.le.trans, Set.Ioc_subset_Icc_self, Set.Icc_subset_Ico_right, cast and 6 more
56 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.I_bounds' line 331 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LE.le, Set, Real, Imo1988P4.I, Subtype.mk, Nat, Membership.mem, Finset, Imo1988P4.singularities, OfNat.ofNat, Iff.mp, Eq, Insert.insert, Finset.insert_eq_self, rfl, Set.Ici
Proof uses: id, Imo1988P4.f, Decidable, ite, Set.Icc, Set.Ioc, Imo1988P4.r, Not, Eq.mpr, congrFun', congrArg, ite_eq_right, LE.le.trans, Set.Ioc_subset_Icc_self, Set.Icc_subset_Ici_self, ite_eq_left
36 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.I_bounds_lower line 336 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Real, Set, Imo1988P4.I, LE.le
Proof uses: OfNat.ofNat, Imo1988P4.f, Decidable, Not, Eq.mp, Set.Ioc, Imo1988P4.r, And, LT.lt, ite, Set.Icc, congrFun', congrArg, ite_eq_right, Imo1988P4.I.eq_1, ite_eq_left
26 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.I_bounds_upper line 339 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Real, Set, Imo1988P4.I, LE.le, Imo1988P4.r
Proof uses: OfNat.ofNat, Imo1988P4.f, Decidable, Not, Eq.mp, Set.Ioc, And, LT.lt, ite, Set.Icc, congrFun', congrArg, ite_eq_right, Imo1988P4.I.eq_1, ite_eq_left
24 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.ico_stack line 342 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Set, Real, Set.Ico, OfNat.ofNat, Nat, Set.iUnion, Subtype, Membership.mem, Finset, Imo1988P4.singularities, HAdd.hAdd
Proof uses: Eq.mpr, Iff, And, LE.le, LT.lt, Exists, id, Eq.trans, forall_congr, congr, congrArg, Subtype.mk, funext, propext, exists_prop_congr, Iff.of_eq, Eq.refl, Finset.Icc, Iff.intro, Exists.intro and 19 more
74 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.I_connected line 350 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Set.OrdConnected, Real, Imo1988P4.I
Proof uses: id, LE.le, OfNat.ofNat, Imo1988P4.f, Decidable, ite, Set, Set.Icc, Set.Ioc, Imo1988P4.r, Not, Eq.mpr, Eq, congrArg, ite_eq_right, ite_eq_left
17 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.I_disjoint line 355 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Ne, Disjoint, Set, Real, Imo1988P4.I
Proof uses: LT.lt, Not, Or, Classical.em, Subtype.mk, LE.le, Set.Ico, HAdd.hAdd, OfNat.ofNat, Imo1988P4.I_bounds, Eq.mp, Finset.Icc, And, congrArg, propext, Finset.mem_Icc, Imo1988P4.singularities.eq_1, Eq, Max.max, max_eq_right_of_lt and 34 more
117 instances, coercions and auxiliary constants left out.

Tactic trace: 27 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.I_compl line 372 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Set, Real, Compl.compl, Set.iUnion, Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Imo1988P4.I, Union.union, Set.Iio, OfNat.ofNat, SDiff.sdiff, Set.Ico, HAdd.hAdd, Set.Ioi, Imo1988P4.r, Imo1988P4.last
Proof uses: Set.ext, Iff.intro, dite, Or.inl, Or.inr, Eq.mpr, Exists, id, Eq.mp, congrArg, Imo1988P4.ico_stack, Exists.intro, And.intro, Not, Eq.trans, Set.iInter, congrFun', Set.compl_iUnion, forall_congr, LT.lt and 106 more
132 instances, coercions and auxiliary constants left out.

Tactic trace: 66 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.I_aligned line 408 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Set, Real, Set.iUnion, Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Imo1988P4.I, Imo1988P4.S
Proof uses: Eq.mpr, Iff, id, Iff.intro, Exists, congrArg, propext, Set.mem_iUnion, forall_exists_index, StrictAntiOn, Imo1988P4.f, Set.Ioc, Imo1988P4.r, dite, OfNat.ofNat, StrictAntiOn.mono, Set.Ioi, Imo1988P4.anti_oi, Set.Ioc_subset_Ioi_self, Not and 162 more
170 instances, coercions and auxiliary constants left out.

Tactic trace: 89 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.I_volume line 456 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Eq, ENNReal, MeasureTheory.Measure, Real, Set, MeasureTheory.MeasureSpace.volume, Imo1988P4.I, HSub.hSub, Imo1988P4.r
Proof uses: id, LE.le, OfNat.ofNat, Imo1988P4.f, Decidable, ite, Set.Icc, Set.Ioc, Not, Eq.mpr, congrFun', congrArg, ite_eq_right, ENNReal.ofReal, Real.volume_Ioc, Eq.refl, ite_eq_left, Real.volume_Icc
37 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
abbrev Imo1988P4.M line 467 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real
Proof uses: Finset.prod, Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Finset.univ, HSub.hSub, Polynomial.X, RingHom, Polynomial.C
18 instances, coercions and auxiliary constants left out.
abbrev Imo1988P4.M' line 469 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Polynomial, Real
Proof uses: Finset.prod, SDiff.sdiff, Singleton.singleton, HSub.hSub, Polynomial.X, RingHom, Polynomial.C
22 instances, coercions and auxiliary constants left out.
abbrev Imo1988P4.p line 471 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Polynomial, Real
Proof uses: HAdd.hAdd, Imo1988P4.M, HMul.hMul, RingHom, Polynomial.C, Neg.neg, HDiv.hDiv, OfNat.ofNat, Finset.sum, Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Finset.univ, Imo1988P4.M'
24 instances, coercions and auxiliary constants left out.
lemma Imo1988P4.M_sum line 473 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, Finset.sum, Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Finset.attach, OfNat.ofNat
Proof uses: Eq.mpr, id, Eq.trans, congrFun', congrArg, of_decide_eq_true, Bool, Decidable.decide, Bool.true, Eq.refl
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.M_natDegree line 475 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Polynomial.natDegree, Real, Imo1988P4.M, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Finset.prod, Subtype, Membership.mem, Finset, Imo1988P4.singularities, Polynomial, Finset.univ, HSub.hSub, Polynomial.X, RingHom, Polynomial.C, Finset.card, HAdd.hAdd, Polynomial.natDegree_finsetProd_X_sub_C_eq_card, Finset.attach and 3 more
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.M'_natDegree line 477 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Eq, Polynomial.natDegree, Real, Imo1988P4.M', OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, Finset.prod, Polynomial, SDiff.sdiff, Singleton.singleton, HSub.hSub, Polynomial.X, RingHom, Polynomial.C, Finset.card, Polynomial.natDegree_finsetProd_X_sub_C_eq_card, Finset.erase, Finset.sdiff_singleton_eq_erase, Finset.card_erase_of_mem, Nat.card_Icc, eq_self
29 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.M_principal_term' line 479 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LT.lt, Nat, Polynomial.natDegree, Real, HMul.hMul, Polynomial, RingHom, Polynomial.C, Neg.neg, HDiv.hDiv, OfNat.ofNat, Finset.sum, Subtype, Membership.mem, Finset, Imo1988P4.singularities, Finset.univ, Imo1988P4.M', Imo1988P4.M
Proof uses: Eq.mpr, id, Eq, congrArg, Polynomial.natDegree_C_mul, Not.intro, Mathlib.Tactic.Linarith.lt_irrefl, Eq.mp, HAdd.hAdd, inferInstance, AddMonoidWithOne, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, Ring, Int.negOfNat, Nat.rawCast, Mathlib.Meta.NormNum.instAddMonoidWithOne, Mathlib.Tactic.Ring.Common.mul_congr and 68 more
125 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.p_natDegree line 484 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, Polynomial.natDegree, Real, Imo1988P4.p, OfNat.ofNat
Proof uses: Eq.mpr, Imo1988P4.M, id, congrArg, HAdd.hAdd, Polynomial, HMul.hMul, RingHom, Polynomial.C, Neg.neg, HDiv.hDiv, Finset.sum, Subtype, Membership.mem, Finset, Imo1988P4.singularities, Finset.univ, Imo1988P4.M', Polynomial.natDegree_add_eq_left_of_degree_lt, Polynomial.degree_lt_degree and 2 more
26 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.p_ne_zero line 488 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Ne, Polynomial, Real, Imo1988P4.p, OfNat.ofNat
Proof uses: Iff.mp, LE.le, WithBot, Nat, Polynomial.degree, Polynomial.zero_le_degree_iff, LT.lt.le, LT.lt, Polynomial.natDegree, Polynomial.natDegree_pos_iff_degree_pos, Eq.mpr, id, Eq, congrArg, Imo1988P4.p_natDegree
16 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
example Imo1988P4.(example at line 490) line 490 source
lemma Imo1988P4.p_leadingCoeff line 494 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, Polynomial.leadingCoeff, Imo1988P4.p, OfNat.ofNat
Proof uses: Eq.mpr, Finsupp, Nat, Polynomial.coeff, Polynomial.natDegree, id, congrArg, Polynomial.leadingCoeff.eq_1, Imo1988P4.p_natDegree, HAdd.hAdd, Imo1988P4.M, HMul.hMul, Polynomial, RingHom, Polynomial.C, Neg.neg, HDiv.hDiv, Finset.sum, Subtype, Membership.mem and 47 more
83 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.p_nextCoeff line 508 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, Polynomial.nextCoeff, Imo1988P4.p, Neg.neg, OfNat.ofNat, Nat
Proof uses: Eq.mpr, Finsupp, Polynomial.coeff, HSub.hSub, Polynomial.natDegree, id, congrArg, Polynomial.nextCoeff_of_natDegree_pos, lt_of_not_ge, LE.le, Mathlib.Tactic.Linarith.lt_irrefl, Int, Eq.mp, LT.lt, HAdd.hAdd, HMul.hMul, Mathlib.Tactic.Ring.of_eq, Mathlib.Tactic.Ring.Common.add_congr, Int.rawCast, inferInstance and 116 more
164 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.p_roots_card_le_rset_card line 522 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LE.le, Nat, Multiset.card, Real, Polynomial.roots, Imo1988P4.p, Finset.val, Imo1988P4.rset
Proof uses: LE.le.trans, Polynomial.natDegree, Polynomial.card_roots', Eq.mpr, Finset.card, id, Eq, congrArg, Finset.card_val, OfNat.ofNat, Imo1988P4.rset_card, Imo1988P4.p_natDegree, le_refl
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.p_root_of_r_root line 525 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Subtype, Nat, Membership.mem, Finset, Imo1988P4.singularities, Polynomial.IsRoot, Real, Imo1988P4.p, Imo1988P4.r
Proof uses: id, Eq.mpr, HAdd.hAdd, Polynomial, Imo1988P4.M, HMul.hMul, RingHom, Polynomial.C, Neg.neg, HDiv.hDiv, OfNat.ofNat, Finset.sum, Finset.univ, Imo1988P4.M', Eq, Finset.prod, Finset.attach, HSub.hSub, SDiff.sdiff, Singleton.singleton and 124 more
167 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.rset_le_p_roots line 549 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: LE.le, Multiset, Real, Finset.val, Imo1988P4.rset, Polynomial.roots, Imo1988P4.p
Proof uses: Iff.mpr, HasSubset.Subset, Finset.val_le_iff_val_subset, Membership.mem, Nat, Exists, Finset, Imo1988P4.singularities, And, Subtype, Multiset.attach, Subtype.mk, Eq, Imo1988P4.r, Eq.mp, Eq.trans, Multiset.map, Multiset.dedup, congrFun', congrArg and 13 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.rset_eq_p_roots line 556 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Multiset, Real, Finset.val, Imo1988P4.rset, Polynomial.roots, Imo1988P4.p
Proof uses: Multiset.eq_of_le_of_card_le, Imo1988P4.rset_le_p_roots, Imo1988P4.p_roots_card_le_rset_card
4 instances, coercions and auxiliary constants left out.
lemma Imo1988P4.p_eq_X_sub_C_rset_prod line 559 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Polynomial, Real, Imo1988P4.p, Finset.prod, Imo1988P4.rset, HSub.hSub, Polynomial.X, RingHom, Polynomial.C
Proof uses: HMul.hMul, Polynomial.leadingCoeff, Multiset.prod, Multiset.map, Polynomial.roots, Polynomial.C_leadingCoeff_mul_prod_multiset_X_sub_C, Eq.mpr, Nat, Multiset.card, Polynomial.natDegree, Finset.val, id, congrArg, Multiset, Eq.symm, Imo1988P4.rset_eq_p_roots, Finset.card, Finset.card_val, OfNat.ofNat, Imo1988P4.rset_card and 20 more
46 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma Imo1988P4.roots_sum line 565 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Real, Finset.sum, Imo1988P4.rset, OfNat.ofNat, Nat
Proof uses: Polynomial.nextCoeff, Finset.prod, Polynomial, HSub.hSub, Polynomial.X, RingHom, Polynomial.C, Neg.neg, Eq.mpr, Imo1988P4.p, id, congrArg, Eq.symm, Imo1988P4.p_eq_X_sub_C_rset_prod, Imo1988P4.p_nextCoeff, Eq.mp, Eq.trans, congrFun', Polynomial.prod_X_sub_C_nextCoeff
42 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
problem Imo1988P4.imo1988_p4 line 574 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Nat, Fin, Set, Real, And, Set.OrdConnected, Pairwise, Disjoint, Eq, Set.iUnion, Set.ofPred, LE.le, HDiv.hDiv, OfNat.ofNat, Finset.sum, Finset.Icc, HSub.hSub, ENNReal, Finset.univ and 2 more
Proof uses: Exists.intro, Imo1988P4.I, Subtype.mk, Membership.mem, Finset, Imo1988P4.singularities, HAdd.hAdd, And.intro, Imo1988P4.I_connected, Ne, Imo1988P4.I_disjoint, Eq.mpr, Not, id, congrArg, Fin.val_ne_of_ne, Subtype, Function.Surjective.iUnion_comp, Fin.mk, Eq.mp and 30 more
117 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.