module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Complex.Order | {
"line": 83,
"column": 12
} | {
"line": 83,
"column": 17
} | {
"line": 83,
"column": 17
} | [
{
"pp": "z : ℂ\n⊢ z.re ^ 2 - z.im ^ 2 ≤ 0 ∧ (z.re = 0 ∨ z.im = 0) → z.re = 0",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"IsOrderedRing.toPosMulMono",
"Real.instZero... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Complex.Order | {
"line": 83,
"column": 12
} | {
"line": 83,
"column": 17
} | {
"line": 83,
"column": 17
} | [
{
"pp": "z : ℂ\n⊢ z.re ^ 2 - z.im ^ 2 ≤ 0 ∧ (z.re = 0 ∨ z.im = 0) → z.re = 0",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"IsOrderedRing.toPosMulMono",
"Real.instZero... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Order | {
"line": 83,
"column": 12
} | {
"line": 83,
"column": 17
} | {
"line": 83,
"column": 17
} | [
{
"pp": "z : ℂ\n⊢ z.re ^ 2 - z.im ^ 2 ≤ 0 ∧ (z.re = 0 ∨ z.im = 0) → z.re = 0",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"IsOrderedRing.toPosMulMono",
"Real.instZero... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 204,
"column": 46
} | {
"line": 204,
"column": 51
} | {
"line": 204,
"column": 51
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\nf : ι → α\nhf : 0 ≤ f\n⊢ ∀ i ∈ univ, 0 ≤ f i",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Finset.univ",
"Finset",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 204,
"column": 46
} | {
"line": 204,
"column": 51
} | {
"line": 204,
"column": 51
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\nf : ι → α\nhf : 0 ≤ f\n⊢ ∀ i ∈ univ, 0 ≤ f i",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Finset.univ",
"Finset",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 204,
"column": 46
} | {
"line": 204,
"column": 51
} | {
"line": 204,
"column": 51
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\nf : ι → α\nhf : 0 ≤ f\n⊢ ∀ i ∈ univ, 0 ≤ f i",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Finset.univ",
"Finset",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 7
} | {
"line": 207,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\nf : ι → α\nhf : 0 ≤ f\n⊢ (∀ i ∈ univ, f i = 0) ↔ f = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 208,
"column": 46
} | {
"line": 208,
"column": 51
} | {
"line": 208,
"column": 51
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\nf : ι → α\nhf : f ≤ 0\n⊢ ∀ i ∈ univ, f i ≤ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Finset.univ",
"Finset",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 208,
"column": 46
} | {
"line": 208,
"column": 51
} | {
"line": 208,
"column": 51
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\nf : ι → α\nhf : f ≤ 0\n⊢ ∀ i ∈ univ, f i ≤ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Finset.univ",
"Finset",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 208,
"column": 46
} | {
"line": 208,
"column": 51
} | {
"line": 208,
"column": 51
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\nf : ι → α\nhf : f ≤ 0\n⊢ ∀ i ∈ univ, f i ≤ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Finset.univ",
"Finset",... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.Expect | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 7
} | {
"line": 211,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : Fintype ι\ninst✝³ : AddCommMonoid α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Module ℚ≥0 α\nf : ι → α\nhf : f ≤ 0\n⊢ (∀ i ∈ univ, f i = 0) ↔ f = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 785,
"column": 4
} | {
"line": 786,
"column": 17
} | {
"line": 788,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type u_5\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : SeminormedAddCommGroup W\ninst✝⁵ : SeminormedAddCommGroup V₁\ninst✝⁴ : SeminormedAddCommGroup V₂\ninst✝³ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V W\nW₁ : Type u_6\nW₂ : Type ... | [] | obtain ⟨C, _C_pos, hC⟩ := φ.bound
exact ⟨C, hC⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 785,
"column": 4
} | {
"line": 786,
"column": 17
} | {
"line": 788,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nV₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type u_5\ninst✝⁷ : SeminormedAddCommGroup V\ninst✝⁶ : SeminormedAddCommGroup W\ninst✝⁵ : SeminormedAddCommGroup V₁\ninst✝⁴ : SeminormedAddCommGroup V₂\ninst✝³ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V W\nW₁ : Type u_6\nW₂ : Type ... | [] | obtain ⟨C, _C_pos, hC⟩ := φ.bound
exact ⟨C, hC⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.Unitary | {
"line": 43,
"column": 38
} | {
"line": 43,
"column": 41
} | {
"line": 43,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Monoid R\ninst✝ : StarMul R\nU B : R\nx✝¹ : U ∈ {U | star U * U = 1 ∧ U * star U = 1}\nx✝ : B ∈ {U | star U * U = 1 ∧ U * star U = 1}\nhA₁ : star U * U = 1\nhA₂ : U * star U = 1\nhB₁ : star B * B = 1\nhB₂ : B * star B = 1\n⊢ star B * B = 1",
"ppTerm": "?m.135",
"assigned"... | [
"R : Type u_1\ninst✝¹ : Monoid R\ninst✝ : StarMul R\nU B : R\nx✝¹ : U ∈ {U | star U * U = 1 ∧ U * star U = 1}\nx✝ : B ∈ {U | star U * U = 1 ∧ U * star U = 1}\nhA₁ : star U * U = 1\nhA₂ : U * star U = 1\nhB₁ : star B * B = 1\nhB₂ : B * star B = 1\n⊢ 1 = 1"
] | hB₁ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Basic | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 11
} | {
"line": 482,
"column": 0
} | [
{
"pp": "x y : ℂ\nh : RCLike.re x ≤ RCLike.re y ∧ RCLike.im x = RCLike.im y\n⊢ x.re ≤ y.re",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
"AddMonoid.toAddZeroClass",
"AddMonoid.toZ... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 410,
"column": 87
} | {
"line": 411,
"column": 22
} | {
"line": 413,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : A\n⊢ ↑(ℑ a) = -I • 2⁻¹ • (a - star a)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | by
simp [imaginaryPart] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.OpenPartialHomeomorph.Basic | {
"line": 219,
"column": 2
} | {
"line": 220,
"column": 72
} | {
"line": 221,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\n⊢ IsOpenEmbedding (e.source.restrict ↑e)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"continuous_subtype_val",
"ContinuousOn.comp_continuous",
... | [
"X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nV : Set ↑e.source\nhV : IsOpen[instTopologicalSpaceSubtype] V\n⊢ IsOpen[inst✝] (e.source.restrict ↑e '' V)"
] | refine .of_continuous_injective_isOpenMap (e.continuousOn.comp_continuous
continuous_subtype_val Subtype.prop) e.injOn.injective fun V hV ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.RCLike.Basic | {
"line": 856,
"column": 51
} | {
"line": 856,
"column": 56
} | {
"line": 858,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ 0 ≤ re z ∧ im z = 0 ↔ ∃ x ≥ 0, re ↑x = re z ∧ im ↑x = im z",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"RCLike.ofReal_re",
"AddMonoid.toAddSemigroup",
"Real.instZero",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 859,
"column": 48
} | {
"line": 859,
"column": 53
} | {
"line": 861,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ 0 < re z ∧ im z = 0 ↔ ∃ x > 0, re ↑x = re z ∧ im ↑x = im z",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"RCLike.ofReal_re",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddM... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 862,
"column": 51
} | {
"line": 862,
"column": 56
} | {
"line": 864,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ re z ≤ 0 ∧ im z = 0 ↔ ∃ x ≤ 0, re ↑x = re z ∧ im ↑x = im z",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"RCLike.ofReal_re",
"AddMonoid.toAddSemigroup",
"Real.instZero",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 865,
"column": 48
} | {
"line": 865,
"column": 53
} | {
"line": 867,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ re z < 0 ∧ im z = 0 ↔ ∃ x < 0, re ↑x = re z ∧ im ↑x = im z",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"RCLike.ofReal_re",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddM... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 915,
"column": 2
} | {
"line": 915,
"column": 11
} | {
"line": 917,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\nh : re x ≤ re y ∧ im x = im y\n⊢ re x ≤ re y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
"AddMonoid.toAddZeroClass",
"AddMonoid... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 147,
"column": 2
} | {
"line": 150,
"column": 20
} | {
"line": 152,
"column": 0
} | [
{
"pp": "x y : ℂ\n⊢ cosh (x + y) = cosh x * cosh y + sinh x * sinh y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
... | [] | rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
mul_assoc, two_cosh, ← mul_assoc, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
mul_left_comm, two_cosh, mul_left_comm, two_sinh]
exact cosh_add_aux | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 147,
"column": 2
} | {
"line": 150,
"column": 20
} | {
"line": 152,
"column": 0
} | [
{
"pp": "x y : ℂ\n⊢ cosh (x + y) = cosh x * cosh y + sinh x * sinh y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
... | [] | rw [← mul_right_inj' (two_ne_zero' ℂ), two_cosh, exp_add, neg_add, exp_add, eq_comm, mul_add, ←
mul_assoc, two_cosh, ← mul_assoc, two_sinh, ← mul_right_inj' (two_ne_zero' ℂ), mul_add,
mul_left_comm, two_cosh, mul_left_comm, two_sinh]
exact cosh_add_aux | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Exponential | {
"line": 389,
"column": 6
} | {
"line": 389,
"column": 58
} | {
"line": 390,
"column": 6
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\nn : ℕ\nhn : 0 < n\nj : ℕ\nhj : j ≥ n\n⊢ ‖∑ m ∈ range j with n ≤ m, x ^ m / ↑m.factorial‖ = ‖∑ m ∈ range j with n ≤ m, x ^ n * (x ^ (m - n) / ↑m.factorial)‖",
"ppTerm": "?m.237",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"instHDiv",
... | [
"x : ℂ\nhx : ‖x‖ ≤ 1\nn : ℕ\nhn : 0 < n\nj : ℕ\nhj : j ≥ n\nm : ℕ\nhm : m ∈ {m ∈ range j | n ≤ m}\n⊢ x ^ m / ↑m.factorial = x ^ n * (x ^ (m - n) / ↑m.factorial)"
] | refine congr_arg norm (sum_congr rfl fun m hm => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.Exponential | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 59
} | {
"line": 482,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ ‖cexp x‖ ≤ Real.exp ‖x‖",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"congrArg",
"sub_zero",
"HEq.refl",
"Real.instDivInvMonoid",
"Real.instSub",
... | [] | convert norm_exp_sub_sum_le_exp_norm_sub_sum x 0 <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Complex.Exponential | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 59
} | {
"line": 482,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ ‖cexp x‖ ≤ Real.exp ‖x‖",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"congrArg",
"sub_zero",
"HEq.refl",
"Real.instDivInvMonoid",
"Real.instSub",
... | [] | convert norm_exp_sub_sum_le_exp_norm_sub_sum x 0 <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Exponential | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 59
} | {
"line": 482,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ ‖cexp x‖ ≤ Real.exp ‖x‖",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"congrArg",
"sub_zero",
"HEq.refl",
"Real.instDivInvMonoid",
"Real.instSub",
... | [] | convert norm_exp_sub_sum_le_exp_norm_sub_sum x 0 <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Exponential | {
"line": 492,
"column": 6
} | {
"line": 492,
"column": 58
} | {
"line": 493,
"column": 6
} | [
{
"pp": "x : ℂ\nn j : ℕ\nhj : j ≥ n\n⊢ ‖∑ m ∈ Ico n j, x ^ m / ↑m.factorial‖ = ‖∑ m ∈ Ico n j, x ^ n * (x ^ (m - n) / ↑m.factorial)‖",
"ppTerm": "?m.196",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"instHDiv",
"HMul.hMul",
"Finset",
"HSub.hSub",
... | [
"x : ℂ\nn j : ℕ\nhj : j ≥ n\nm : ℕ\nhm : m ∈ Ico n j\n⊢ x ^ m / ↑m.factorial = x ^ n * (x ^ (m - n) / ↑m.factorial)"
] | refine congr_arg norm (sum_congr rfl fun m hm => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.Exponential | {
"line": 493,
"column": 10
} | {
"line": 493,
"column": 17
} | {
"line": 493,
"column": 17
} | [
{
"pp": "x : ℂ\nn j : ℕ\nhj : j ≥ n\nm : ℕ\nhm : m ∈ Ico n j\n⊢ x ^ m / ↑m.factorial = x ^ n * (x ^ (m - n) / ↑m.factorial)",
"ppTerm": "?m.355",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"Finset",
"Preorder.toLE",
"Nat.instLocallyFiniteOrder",
... | [
"x : ℂ\nn j : ℕ\nhj : j ≥ n\nm : ℕ\nhm : n ≤ m ∧ m < j\n⊢ x ^ m / ↑m.factorial = x ^ n * (x ^ (m - n) / ↑m.factorial)"
] | mem_Ico | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Field.Power | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 83
} | {
"line": 45,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : α\nk : ℤ\nh : k + k ≠ 0\n⊢ 0 < a ^ (k + k) ↔ a ≠ 0",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"IsDomain.to_noZeroDivisors",... | [] | rw [zpow_add' (by simp [em']), mul_self_pos, zpow_ne_zero_iff (by simpa using h)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 488,
"column": 46
} | {
"line": 488,
"column": 97
} | {
"line": 490,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ sin x ^ 2 = 1 - cos x ^ 2",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.cos",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"Complex.sin",
"HSub.hSub",
"AddZeroClass.toAddZero",
... | [] | by rw [← sin_sq_add_cos_sq x, add_sub_cancel_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 524,
"column": 2
} | {
"line": 525,
"column": 37
} | {
"line": 527,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ (cexp x).im = Real.exp x.re * Real.sin x.im",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Eq.mpr",
"Real",
"Complex.mul_re",
"HMul.hMul",
"Complex.cos",
"sub_self",
"Real.instZero",
"Real.instAddMon... | [] | rw [exp_eq_exp_re_mul_sin_add_cos]
simp [exp_ofReal_re, sin_ofReal_re] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 524,
"column": 2
} | {
"line": 525,
"column": 37
} | {
"line": 527,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ (cexp x).im = Real.exp x.re * Real.sin x.im",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Eq.mpr",
"Real",
"Complex.mul_re",
"HMul.hMul",
"Complex.cos",
"sub_self",
"Real.instZero",
"Real.instAddMon... | [] | rw [exp_eq_exp_re_mul_sin_add_cos]
simp [exp_ofReal_re, sin_ofReal_re] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 278,
"column": 4
} | {
"line": 279,
"column": 77
} | {
"line": 280,
"column": 2
} | [
{
"pp": "case inl\nb c : ℝ\nhb : 0 < b\n⊢ Tendsto (fun x ↦ (b * rexp x + c) / x ^ 0) atTop atTop",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Filter.Tendsto.const_mul_atTop",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCo... | [] | simp only [pow_zero, div_one]
exact (tendsto_exp_atTop.const_mul_atTop hb).atTop_add tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 278,
"column": 4
} | {
"line": 279,
"column": 77
} | {
"line": 280,
"column": 2
} | [
{
"pp": "case inl\nb c : ℝ\nhb : 0 < b\n⊢ Tendsto (fun x ↦ (b * rexp x + c) / x ^ 0) atTop atTop",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Filter.Tendsto.const_mul_atTop",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCo... | [] | simp only [pow_zero, div_one]
exact (tendsto_exp_atTop.const_mul_atTop hb).atTop_add tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Quotient | {
"line": 310,
"column": 36
} | {
"line": 310,
"column": 47
} | {
"line": 310,
"column": 47
} | [
{
"pp": "case inr\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : ∀ x ∈ S, f x = 0\nx : M ⧸ S\nh : 0 < ‖f‖\n⊢ ¬‖x‖ < ‖(lift S f.toAddMonoidHom hf) x‖ / ‖f‖",
"ppTerm": "?inr",
"assigned": true,
"used... | [
"case inr\nM : Type u_1\nN : Type u_2\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : SeminormedAddCommGroup N\nS : AddSubgroup M\nf : NormedAddGroupHom M N\nhf : ∀ x ∈ S, f x = 0\nx : M ⧸ S\nh : 0 < ‖f‖\n⊢ ¬∃ m, ↑m = x ∧ ‖m‖ < ‖(lift S f.toAddMonoidHom hf) x‖ / ‖f‖"
] | norm_lt_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 501,
"column": 12
} | {
"line": 505,
"column": 58
} | {
"line": 505,
"column": 58
} | [
{
"pp": "x : ℝ\nhx₁ : -π < x\nhx₂ : x < π\nh : sin x = 0\n⊢ x = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.instZero",
"PartialOrder.toPreorder",
"id",
"Ne",
"Real.sin_pos_of_pos_of_lt_pi",
"Or.casesOn",
... | [] | by
contrapose! h
cases h.lt_or_gt with
| inl h0 => exact (sin_neg_of_neg_of_neg_pi_lt h0 hx₁).ne
| inr h0 => exact (sin_pos_of_pos_of_lt_pi h0 hx₂).ne' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 153,
"column": 27
} | {
"line": 153,
"column": 42
} | {
"line": 153,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\nx y : E\n⊢ {x} * (closedBall y δ)⁻¹ = closedBall (x * y⁻¹) δ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivIn... | [
"E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\nx y : E\n⊢ {x} * closedBall y⁻¹ δ = closedBall (x * y⁻¹) δ"
] | inv_closedBall, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Module.RCLike.Real | {
"line": 56,
"column": 41
} | {
"line": 56,
"column": 50
} | {
"line": 57,
"column": 4
} | [
{
"pp": "case h\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nx y : E\nh : r ∈ Icc 0 1\n⊢ 0 ≤ r",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Preorder.toLE",
"LE.le",
"Real.instOne",
"And.left"... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 661,
"column": 13
} | {
"line": 666,
"column": 67
} | {
"line": 668,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ sqrtTwoAddSeries 0 (n + 1) < 2",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.sqrtTwoAddSeries.eq_2",
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"lt_sub_iff_add_lt'",
"FloorRing.... | [] | by
refine lt_of_lt_of_le ?_ (sqrt_sq zero_lt_two.le).le
rw [sqrtTwoAddSeries, sqrt_lt_sqrt_iff, ← lt_sub_iff_add_lt']
· refine (sqrtTwoAddSeries_lt_two n).trans_le ?_
norm_num
· exact add_nonneg zero_le_two (sqrtTwoAddSeries_zero_nonneg n) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 718,
"column": 4
} | {
"line": 719,
"column": 12
} | {
"line": 720,
"column": 2
} | [
{
"pp": "⊢ cos (π / 4) = cos (π / 2 ^ 2)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Real.cos",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.inst... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 718,
"column": 4
} | {
"line": 719,
"column": 12
} | {
"line": 720,
"column": 2
} | [
{
"pp": "⊢ cos (π / 4) = cos (π / 2 ^ 2)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Real.cos",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.inst... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 725,
"column": 4
} | {
"line": 726,
"column": 12
} | {
"line": 727,
"column": 2
} | [
{
"pp": "⊢ sin (π / 4) = sin (π / 2 ^ 2)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNa... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 725,
"column": 4
} | {
"line": 726,
"column": 12
} | {
"line": 727,
"column": 2
} | [
{
"pp": "⊢ sin (π / 4) = sin (π / 2 ^ 2)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNa... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 732,
"column": 4
} | {
"line": 733,
"column": 12
} | {
"line": 734,
"column": 2
} | [
{
"pp": "⊢ cos (π / 8) = cos (π / 2 ^ 3)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Real.cos",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.inst... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 732,
"column": 4
} | {
"line": 733,
"column": 12
} | {
"line": 734,
"column": 2
} | [
{
"pp": "⊢ cos (π / 8) = cos (π / 2 ^ 3)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Real.cos",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.inst... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 739,
"column": 4
} | {
"line": 740,
"column": 12
} | {
"line": 741,
"column": 2
} | [
{
"pp": "⊢ sin (π / 8) = sin (π / 2 ^ 3)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNa... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 739,
"column": 4
} | {
"line": 740,
"column": 12
} | {
"line": 741,
"column": 2
} | [
{
"pp": "⊢ sin (π / 8) = sin (π / 2 ^ 3)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNa... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 746,
"column": 4
} | {
"line": 747,
"column": 12
} | {
"line": 748,
"column": 2
} | [
{
"pp": "⊢ cos (π / 16) = cos (π / 2 ^ 4)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Real.cos",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.ins... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 746,
"column": 4
} | {
"line": 747,
"column": 12
} | {
"line": 748,
"column": 2
} | [
{
"pp": "⊢ cos (π / 16) = cos (π / 2 ^ 4)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Real.cos",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.ins... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 753,
"column": 4
} | {
"line": 754,
"column": 12
} | {
"line": 755,
"column": 2
} | [
{
"pp": "⊢ sin (π / 16) = sin (π / 2 ^ 4)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfN... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 753,
"column": 4
} | {
"line": 754,
"column": 12
} | {
"line": 755,
"column": 2
} | [
{
"pp": "⊢ sin (π / 16) = sin (π / 2 ^ 4)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfN... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 760,
"column": 4
} | {
"line": 761,
"column": 12
} | {
"line": 762,
"column": 2
} | [
{
"pp": "⊢ cos (π / 32) = cos (π / 2 ^ 5)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Real.cos",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.ins... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 760,
"column": 4
} | {
"line": 761,
"column": 12
} | {
"line": 762,
"column": 2
} | [
{
"pp": "⊢ cos (π / 32) = cos (π / 2 ^ 5)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Real.cos",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.ins... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 767,
"column": 4
} | {
"line": 768,
"column": 12
} | {
"line": 769,
"column": 2
} | [
{
"pp": "⊢ sin (π / 32) = sin (π / 2 ^ 5)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfN... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 767,
"column": 4
} | {
"line": 768,
"column": 12
} | {
"line": 769,
"column": 2
} | [
{
"pp": "⊢ sin (π / 32) = sin (π / 2 ^ 5)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Meta.NormNum.isNat_eq_true",
"Mathlib.Meta.NormNum.IsNatPowT.run",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfN... | [] | congr
norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 775,
"column": 2
} | {
"line": 779,
"column": 44
} | {
"line": 780,
"column": 2
} | [
{
"pp": "⊢ cos (π / 3) = 1 / 2",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mp... | [
"h₁ : (2 * cos (π / 3) - 1) ^ 2 * (2 * cos (π / 3) + 2) = 0\n⊢ cos (π / 3) = 1 / 2"
] | have h₁ : (2 * cos (π / 3) - 1) ^ 2 * (2 * cos (π / 3) + 2) = 0 := by
have : cos (3 * (π / 3)) = cos π := by
congr 1
ring
linarith [cos_pi, cos_three_mul (π / 3)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 41
} | {
"line": 194,
"column": 2
} | [
{
"pp": "case neg\nk : ℕ\nr : ℝ\nhr : |r| < 1\nh0 : ¬r = 0\nhr' : 1 < |r|⁻¹\n⊢ Tendsto (fun n ↦ ↑n ^ k * r ^ n) atTop (𝓝 0)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"No... | [
"case neg\nk : ℕ\nr : ℝ\nhr : |r| < 1\nh0 : ¬r = 0\nhr' : 1 < |r|⁻¹\n⊢ Tendsto (fun x ↦ ‖↑x ^ k * r ^ x‖) atTop (𝓝 0)"
] | rw [tendsto_zero_iff_norm_tendsto_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 68
} | {
"line": 285,
"column": 0
} | [
{
"pp": "α : Type u_1\nR✝ : Type u_2\nS : Type u_3\ninst✝⁷ : Field R✝\ninst✝⁶ : Field S\ninst✝⁵ : LinearOrder S\nv w : AbsoluteValue R✝ S\ninst✝⁴ : TopologicalSpace S\ninst✝³ : IsStrictOrderedRing S\ninst✝² : Archimedean S\n_i : OrderTopology S\nR : Type u_4\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nx : ... | [] | exact h1.of_norm_bounded_eventually_nat (eventually_norm_pow_le x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.PathConnected | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 28
} | {
"line": 227,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y z : X\nF : Set X\nhxy : JoinedIn F x y\nhyz : JoinedIn F y z\n⊢ JoinedIn F x z",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"JoinedIn",
"Membership.mem",
"JoinedIn.mem",
"And.casesOn",
"And",
"Set.... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nx y z : X\nF : Set X\nhxy : JoinedIn F x y\nhyz : JoinedIn F y z\nhx : x ∈ F\nhy : y ∈ F\n⊢ JoinedIn F x z"
] | obtain ⟨hx, hy⟩ := hxy.mem | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Connected.PathConnected | {
"line": 553,
"column": 2
} | {
"line": 556,
"column": 56
} | {
"line": 557,
"column": 2
} | [
{
"pp": "case mp\nX : Type u_1\ninst✝ : TopologicalSpace X\nthis : Setoid X := ⋯\n⊢ PathConnectedSpace X → Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"pathSetoid",
"ZerothHomotopy",
"Quot.in... | [
"case mpr\nX : Type u_1\ninst✝ : TopologicalSpace X\nthis : Setoid X := ⋯\n⊢ Nonempty (ZerothHomotopy X) ∧ Subsingleton (ZerothHomotopy X) → PathConnectedSpace X"
] | · intro h
refine ⟨(nonempty_quotient_iff _).mpr h.1, ⟨?_⟩⟩
rintro ⟨x⟩ ⟨y⟩
exact Quotient.sound (PathConnectedSpace.joined x y) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Connected.PathConnected | {
"line": 619,
"column": 2
} | {
"line": 619,
"column": 26
} | {
"line": 620,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nt : Set Y\nhs : IsPathConnected s\nht : IsPathConnected t\n⊢ IsPathConnected (s ×ˢ t)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"IsPathConnected",
... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nt : Set Y\nhs : IsPathConnected s\nht : IsPathConnected t\n⊢ (s ×ˢ t).Nonempty ∧ ∀ x ∈ s ×ˢ t, ∀ y ∈ s ×ˢ t, JoinedIn (s ×ˢ t) x y"
] | rw [isPathConnected_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Instances.AddCircle.Real | {
"line": 82,
"column": 79
} | {
"line": 87,
"column": 94
} | {
"line": 89,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\n⊢ Injective ⇑toAddCircle",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"Real.partialOrder",
"Real",... | [] | by
intro x y hxy
have : (0 : ℝ) < N := Nat.cast_pos.mpr (NeZero.pos _)
rwa [toAddCircle_apply, toAddCircle_apply, AddCircle.coe_eq_coe_iff_of_mem_Ico,
div_left_inj' this.ne', Nat.cast_inj, (val_injective N).eq_iff] at hxy <;>
exact ⟨by positivity, by simpa only [zero_add, div_lt_one this, Nat.cast_lt] using... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Path | {
"line": 313,
"column": 54
} | {
"line": 317,
"column": 10
} | {
"line": 319,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y z : X\nγ₁ : Path x y\nγ₂ : Path y z\nt : ℝ\nht : 1 / 2 ≤ t\n⊢ (γ₁.trans γ₂).extend t = γ₂.extend (2 * t - 1)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.... | [] | by
conv_lhs => rw [← sub_sub_cancel 1 t]
rw [← extend_symm_apply, trans_symm, extend_trans_of_le_half _ _ (by linarith), extend_symm_apply]
congr 1
linarith | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 52,
"column": 23
} | {
"line": 52,
"column": 38
} | {
"line": 52,
"column": 39
} | [
{
"pp": "case e_s\np x t : ℝ\nht : t ≠ 0\nm : ℝ\n⊢ m ∈ {m | ↑m = ↑(t * x)} ↔ m ∈ t • {m | ↑m = ↑x}",
"ppTerm": "?e_s",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"Semiring.toModule",
"AddSubgroup.zmul... | [
"case e_s\np x t : ℝ\nht : t ≠ 0\nm : ℝ\n⊢ m ∈ {m | m - t * x ∈ { carrier := range fun x ↦ x • (t * p), add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }} ↔\n m ∈ t • {m | m - x ∈ { carrier := range fun x ↦ x • p, add_mem' := ⋯, zero_mem' := ⋯, neg_mem' := ⋯ }}"
] | eq_iff_sub_mem, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 423,
"column": 35
} | {
"line": 423,
"column": 40
} | {
"line": 423,
"column": 40
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc (-(π / 2)) (π / 2)\n⊢ -(π / 2) ≤ θ",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Preorder.toLE",
"Membership.mem",
"Eq.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 423,
"column": 35
} | {
"line": 423,
"column": 40
} | {
"line": 423,
"column": 40
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc (-(π / 2)) (π / 2)\n⊢ -(π / 2) ≤ θ",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Preorder.toLE",
"Membership.mem",
"Eq.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 423,
"column": 35
} | {
"line": 423,
"column": 40
} | {
"line": 423,
"column": 40
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc (-(π / 2)) (π / 2)\n⊢ -(π / 2) ≤ θ",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Preorder.toLE",
"Membership.mem",
"Eq.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 423,
"column": 46
} | {
"line": 423,
"column": 51
} | {
"line": 423,
"column": 51
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc (-(π / 2)) (π / 2)\n⊢ θ ≤ π / 2",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Preorder.toLE",
"Membership.mem",
"Eq.mp"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 423,
"column": 46
} | {
"line": 423,
"column": 51
} | {
"line": 423,
"column": 51
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc (-(π / 2)) (π / 2)\n⊢ θ ≤ π / 2",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Preorder.toLE",
"Membership.mem",
"Eq.mp"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 423,
"column": 46
} | {
"line": 423,
"column": 51
} | {
"line": 423,
"column": 51
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc (-(π / 2)) (π / 2)\n⊢ θ ≤ π / 2",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Preorder.toLE",
"Membership.mem",
"Eq.mp"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 424,
"column": 36
} | {
"line": 424,
"column": 41
} | {
"line": 424,
"column": 41
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ -1 ≤ x",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 424,
"column": 36
} | {
"line": 424,
"column": 41
} | {
"line": 424,
"column": 41
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ -1 ≤ x",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 424,
"column": 36
} | {
"line": 424,
"column": 41
} | {
"line": 424,
"column": 41
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ -1 ≤ x",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 424,
"column": 47
} | {
"line": 424,
"column": 52
} | {
"line": 424,
"column": 52
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ x ≤ 1",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"R... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 424,
"column": 47
} | {
"line": 424,
"column": 52
} | {
"line": 424,
"column": 52
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ x ≤ 1",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"R... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 424,
"column": 47
} | {
"line": 424,
"column": 52
} | {
"line": 424,
"column": 52
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ x ≤ 1",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"R... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 458,
"column": 35
} | {
"line": 458,
"column": 40
} | {
"line": 458,
"column": 40
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc 0 π\n⊢ 0 ≤ θ",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"Real.instZero",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"And",
"Set.Icc",
"Set.mem_Icc._sim... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 458,
"column": 35
} | {
"line": 458,
"column": 40
} | {
"line": 458,
"column": 40
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc 0 π\n⊢ 0 ≤ θ",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"Real.instZero",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"And",
"Set.Icc",
"Set.mem_Icc._sim... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 458,
"column": 35
} | {
"line": 458,
"column": 40
} | {
"line": 458,
"column": 40
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc 0 π\n⊢ 0 ≤ θ",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"Real.instZero",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"And",
"Set.Icc",
"Set.mem_Icc._sim... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 458,
"column": 46
} | {
"line": 458,
"column": 51
} | {
"line": 458,
"column": 51
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc 0 π\n⊢ θ ≤ π",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"Real.instZero",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"And",
"Set.Icc",
"Set.mem_Icc._sim... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 458,
"column": 46
} | {
"line": 458,
"column": 51
} | {
"line": 458,
"column": 51
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc 0 π\n⊢ θ ≤ π",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"Real.instZero",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"And",
"Set.Icc",
"Set.mem_Icc._sim... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 458,
"column": 46
} | {
"line": 458,
"column": 51
} | {
"line": 458,
"column": 51
} | [
{
"pp": "x y θ : ℝ\nhθ : θ ∈ Icc 0 π\n⊢ θ ≤ π",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"Real.instZero",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"And",
"Set.Icc",
"Set.mem_Icc._sim... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 459,
"column": 36
} | {
"line": 459,
"column": 41
} | {
"line": 459,
"column": 41
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ -1 ≤ x",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 459,
"column": 36
} | {
"line": 459,
"column": 41
} | {
"line": 459,
"column": 41
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ -1 ≤ x",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 459,
"column": 36
} | {
"line": 459,
"column": 41
} | {
"line": 459,
"column": 41
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ -1 ≤ x",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 459,
"column": 47
} | {
"line": 459,
"column": 52
} | {
"line": 459,
"column": 52
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ x ≤ 1",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"R... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 459,
"column": 47
} | {
"line": 459,
"column": 52
} | {
"line": 459,
"column": 52
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ x ≤ 1",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"R... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 459,
"column": 47
} | {
"line": 459,
"column": 52
} | {
"line": 459,
"column": 52
} | [
{
"pp": "x✝ y x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ x ≤ 1",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"id",
"LE.le",
"Real.instOne",
"And",
"Set.Icc",
"Set.mem_Icc._simp_1",
"R... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 397,
"column": 8
} | {
"line": 397,
"column": 19
} | {
"line": 397,
"column": 20
} | [
{
"pp": "case inr\nψ : Angle\n⊢ |(ψ + ↑π).sin| = |ψ.sin|",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.lattice",
"Real.Angle",
"Real.Angle.coe",
"abs",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | [
"case inr\nψ : Angle\n⊢ |(-ψ.sin)| = |ψ.sin|"
] | sin_add_pi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 420,
"column": 6
} | {
"line": 420,
"column": 24
} | {
"line": 420,
"column": 25
} | [
{
"pp": "θ ψ : ℝ\n⊢ toIcoMod two_pi_pos ψ θ - θ = 2 * π * ↑(-toIcoDiv two_pi_pos ψ θ)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"instHSMul",
"Real.instArchimedean",
"Real.pi",
"HMul.hMul",
"congrArg",
... | [
"θ ψ : ℝ\n⊢ -toIcoDiv two_pi_pos ψ θ • (2 * π) = 2 * π * ↑(-toIcoDiv two_pi_pos ψ θ)"
] | toIcoMod_sub_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 355,
"column": 4
} | {
"line": 357,
"column": 40
} | {
"line": 358,
"column": 4
} | [
{
"pp": "case inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ z.arg ≤ π / 2 ↔ False",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"not_le",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
... | [
"case inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ z.re ≠ 0",
"case inr.inl\nz : ℂ\nhre : z.re < 0\nhim : 0 ≤ z.im\n⊢ 0 < ‖z‖"
] | rw [iff_false, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub,
Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ←
abs_of_nonneg him, abs_im_lt_norm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 431,
"column": 4
} | {
"line": 432,
"column": 65
} | {
"line": 433,
"column": 2
} | [
{
"pp": "case inr.inl.inr.inr\nx : ℂ\nhi : x.im = 0\nhr : 0 < x.re\n⊢ (-↑x.re).arg = (↑x.re).arg - π ↔ 0 < (↑x.re).im ∨ (↑x.re).im = 0 ∧ (↑x.re).re < 0",
"ppTerm": "?inr.inl.inr.inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtrac... | [] | · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)]
simp [hr.not_gt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 737,
"column": 18
} | {
"line": 737,
"column": 29
} | {
"line": 737,
"column": 30
} | [
{
"pp": "θ : Angle\n⊢ SignType.sign (θ + ↑π).sin = -SignType.sign θ.sin",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"Real.instZero",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | [
"θ : Angle\n⊢ SignType.sign (-θ.sin) = -SignType.sign θ.sin"
] | sin_add_pi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 868,
"column": 41
} | {
"line": 868,
"column": 46
} | {
"line": 869,
"column": 6
} | [
{
"pp": "a b : Angle\nha : a.sign ≠ 0\nh : a.sign = b.sign\nh2 : a = b + ↑π\n⊢ a.sign = (b + ↑π).sign",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"False",
"Real.pi",
"Real.Angle",
"eq_false",
"Real.Angle.coe",
"congrArg",
"AddCommGroup.toAddCom... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 871,
"column": 6
} | {
"line": 871,
"column": 11
} | {
"line": 872,
"column": 2
} | [
{
"pp": "case mp.inr\na b : Angle\nha : a.sign ≠ 0\nh : a.sign = b.sign\nh2 : a = b + ↑π\nthis✝ : a.sign = -b.sign\nthis : (a.sign = b.sign) = (-b.sign = b.sign)\n⊢ a = b",
"ppTerm": "?mp.inr",
"assigned": true,
"usedConstants": [
"False",
"Real.pi",
"Real.Angle",
"eq_false",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 873,
"column": 4
} | {
"line": 873,
"column": 9
} | {
"line": 875,
"column": 0
} | [
{
"pp": "case mpr\na b : Angle\nha : a.sign ≠ 0\nh✝ : a.sign = b.sign\nh : a = b\n⊢ a = b ∨ a = b + ↑π",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Real.pi",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddC... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 94,
"column": 48
} | {
"line": 94,
"column": 53
} | {
"line": 96,
"column": 0
} | [
{
"pp": "b : ℝ\nhb : 1 < b\n⊢ 1 < b",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hb"
],
"usedGoals": []
}
] | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 94,
"column": 48
} | {
"line": 94,
"column": 53
} | {
"line": 96,
"column": 0
} | [
{
"pp": "b : ℝ\nhb : 1 < b\n⊢ 1 < b",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hb"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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