module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.CharZero.Defs | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 35
} | {
"line": 53,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : AddGroupWithOne R\nH : ∀ (n : ℕ), ↑n = 0 → n = 0\nm n : ℕ\nh : ↑m = ↑n\n⊢ m = n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Nat.recAux",
"AddMonoid.toAddSemigroup",
"AddGroupWithOne.toAddGroup",
... | [] | induction m generalizing n with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Module.Defs | {
"line": 81,
"column": 45
} | {
"line": 81,
"column": 90
} | {
"line": 83,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ 2 • x = x + x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
"Ad... | [] | rw [← one_add_one_eq_two, add_smul, one_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.Defs | {
"line": 81,
"column": 45
} | {
"line": 81,
"column": 90
} | {
"line": 83,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ 2 • x = x + x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
"Ad... | [] | rw [← one_add_one_eq_two, add_smul, one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Defs | {
"line": 81,
"column": 45
} | {
"line": 81,
"column": 90
} | {
"line": 83,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx : M\n⊢ 2 • x = x + x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
"Ad... | [] | rw [← one_add_one_eq_two, add_smul, one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Defs | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 55
} | {
"line": 348,
"column": 56
} | [
{
"pp": "α : Type u\ninst✝ : NonUnitalNonAssocRing α\na b c : α\n⊢ a * (b - c) = a * b - a * c",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssocRing.toAddCommGroup",
"sub_eq_... | [
"α : Type u\ninst✝ : NonUnitalNonAssocRing α\na b c : α\n⊢ a * (b + -c) = a * b + a * -c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Defs | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 55
} | {
"line": 353,
"column": 56
} | [
{
"pp": "α : Type u\ninst✝ : NonUnitalNonAssocRing α\na b c : α\n⊢ (a - b) * c = a * c - b * c",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssocRing.toAddCommGroup",
"sub_eq_... | [
"α : Type u\ninst✝ : NonUnitalNonAssocRing α\na b c : α\n⊢ (a + -b) * c = a * c + -b * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Defs | {
"line": 365,
"column": 59
} | {
"line": 365,
"column": 83
} | {
"line": 367,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : NonAssocRing α\na b : α\n⊢ a * (b - 1) = a * b - a",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"mul_sub",
"AddGroupWithOne.toAddGroup",
"congrArg",
"NonUnitalNonAssocRing.toAdd... | [] | by rw [mul_sub, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Defs | {
"line": 369,
"column": 59
} | {
"line": 369,
"column": 83
} | {
"line": 371,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : NonAssocRing α\na b : α\n⊢ a * (1 - b) = a - a * b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"mul_sub",
"AddGroupWithOne.toAddGroup",
"congrArg",
"NonUnitalNonAssocRing.toAdd... | [] | by rw [mul_sub, mul_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Even | {
"line": 157,
"column": 30
} | {
"line": 157,
"column": 41
} | {
"line": 157,
"column": 42
} | [
{
"pp": "case mp\nα : Type u_2\ninst✝ : DivisionMonoid α\na : α\nh : IsSquare a⁻¹\n⊢ IsSquare a",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nα : Type u_2\ninst✝ : DivisionMonoid α\na : α\nh : IsSquare a⁻¹\n⊢ IsSquare a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Even | {
"line": 157,
"column": 30
} | {
"line": 157,
"column": 41
} | {
"line": 157,
"column": 42
} | [
{
"pp": "case mpr\nα : Type u_2\ninst✝ : DivisionMonoid α\na : α\nh : IsSquare a\n⊢ IsSquare a⁻¹",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mpr\nα : Type u_2\ninst✝ : DivisionMonoid α\na : α\nh : IsSquare a\n⊢ IsSquare a⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Sqrt | {
"line": 30,
"column": 46
} | {
"line": 30,
"column": 71
} | {
"line": 30,
"column": 72
} | [
{
"pp": "n : ℕ\n⊢ n.sqrt ^ 2 ≤ n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Nat.pow_two",
"instMulNat",
"Nat.sqrt",
"instOfNatNat",
"LE.le",
"instLENat",
"in... | [
"n : ℕ\n⊢ n.sqrt * n.sqrt ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Sqrt | {
"line": 32,
"column": 58
} | {
"line": 32,
"column": 83
} | {
"line": 32,
"column": 84
} | [
{
"pp": "n : ℕ\n⊢ n < n.sqrt.succ ^ 2",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Nat.pow_two",
"instMulNat",
"Nat.sqrt",
"instOfNatNat",
"instNatPowNat",
"instHAdd... | [
"n : ℕ\n⊢ n < (n.sqrt + 1) * (n.sqrt + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Sqrt | {
"line": 41,
"column": 46
} | {
"line": 41,
"column": 76
} | {
"line": 41,
"column": 77
} | [
{
"pp": "m n : ℕ\n⊢ m ≤ n.sqrt ↔ m ^ 2 ≤ n",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Nat.pow_two",
"instMulNat",
"Nat.sqrt",
"instOfNatNat",
"LE.le",
"instLENat",... | [
"m n : ℕ\n⊢ m ≤ n.sqrt ↔ m * m ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Sqrt | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 32
} | {
"line": 57,
"column": 33
} | [
{
"pp": "n a : ℕ\n⊢ a = n.sqrt ↔ a ^ 2 ≤ n ∧ n < (a + 1) ^ 2",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Nat.pow_two",
"instMulNat",
"Nat.sqrt",
"instOfNatNat",
"LE.le",
... | [
"n a : ℕ\n⊢ a = n.sqrt ↔ a * a ≤ n ∧ n < (a + 1) * (a + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Sqrt | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 27
} | {
"line": 75,
"column": 28
} | [
{
"pp": "a n : ℕ\nh : a ≤ n + n\n⊢ (n ^ 2 + a).sqrt = n",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Nat.pow_two",
"instMulNat",
"Nat.sqrt",
"instOfNatNat",
"instNatPowNat... | [
"a n : ℕ\nh : a ≤ n + n\n⊢ (n * n + a).sqrt = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Sqrt | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 32
} | {
"line": 110,
"column": 33
} | [
{
"pp": "x : ℕ\n⊢ (∃ n, n ^ 2 = x) ↔ x.sqrt ^ 2 = x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Exists",
"id",
"Nat.pow_two",
"instMulNat",
"Nat.sqrt",
"instOfNatNat",
"i... | [
"x : ℕ\n⊢ (∃ n, n * n = x) ↔ x.sqrt * x.sqrt = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Sqrt | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 32
} | {
"line": 132,
"column": 33
} | [
{
"pp": "m n : ℕ\n⊢ m ^ 2 < n → n < (m + 1) ^ 2 → ¬∃ t, t ^ 2 = n",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Exists",
"id",
"Nat.pow_two",
"instMulNat",
"instOfNatNat",
"instN... | [
"m n : ℕ\n⊢ m * m < n → n < (m + 1) * (m + 1) → ¬∃ t, t * t = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Int.Even | {
"line": 89,
"column": 12
} | {
"line": 89,
"column": 23
} | {
"line": 89,
"column": 24
} | [
{
"pp": "case zero\n⊢ IsSquare (sign 0) ↔ 0 ≤ 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Std.IsLinearPreorder.toIsPreorder",
"Std.instReflLeOfIsPreorder",
"Lean.Grind.instIsLinearOrderInt",
"iff_true",
"Std.le_refl._si... | [
"case zero\n⊢ IsSquare 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Basic | {
"line": 268,
"column": 36
} | {
"line": 268,
"column": 53
} | {
"line": 268,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝¹ : DivisionMonoid R\ninst✝ : HasDistribNeg R\na b : R\n⊢ b / -a = b * (1 / -a)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"InvOneClass.toOne",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝¹ : DivisionMonoid R\ninst✝ : HasDistribNeg R\na b : R\n⊢ b / -a = b * (-a)⁻¹"
] | ← inv_eq_one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Hom.Instances | {
"line": 87,
"column": 25
} | {
"line": 87,
"column": 40
} | {
"line": 87,
"column": 41
} | [
{
"pp": "M : Type uM\nN : Type uN\nP : Type uP\nQ : Type uQ\ninst✝¹ : MulOneClass M\ninst✝ : CommGroup N\nf : M →* N\nn : ℤ\nx y : M\n⊢ (⇑f ^ n) (x * y) = (⇑f ^ n) x * (⇑f ^ n) y",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunL... | [] | simp [mul_zpow] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Hom.Instances | {
"line": 87,
"column": 25
} | {
"line": 87,
"column": 40
} | {
"line": 87,
"column": 41
} | [
{
"pp": "M : Type uM\nN : Type uN\nP : Type uP\nQ : Type uQ\ninst✝¹ : MulOneClass M\ninst✝ : CommGroup N\nf : M →* N\nn : ℤ\nx y : M\n⊢ (⇑f ^ n) (x * y) = (⇑f ^ n) x * (⇑f ^ n) y",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunL... | [] | simp [mul_zpow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Hom.Instances | {
"line": 87,
"column": 25
} | {
"line": 87,
"column": 40
} | {
"line": 87,
"column": 41
} | [
{
"pp": "M : Type uM\nN : Type uN\nP : Type uP\nQ : Type uQ\ninst✝¹ : MulOneClass M\ninst✝ : CommGroup N\nf : M →* N\nn : ℤ\nx y : M\n⊢ (⇑f ^ n) (x * y) = (⇑f ^ n) x * (⇑f ^ n) y",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunL... | [] | simp [mul_zpow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Commute | {
"line": 42,
"column": 12
} | {
"line": 42,
"column": 53
} | {
"line": 43,
"column": 2
} | [
{
"pp": "M₀ : Type u_1\ninst✝ : MonoidWithZero M₀\nr : M₀\n⊢ r⁻¹ʳ ^ 0 = (r ^ 0)⁻¹ʳ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"id",
"Ring.inverse",
"instOfNatNat",
"NPow.toPow... | [] | rw [pow_zero, pow_zero, Ring.inverse_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.GroupWithZero.Commute | {
"line": 42,
"column": 12
} | {
"line": 42,
"column": 53
} | {
"line": 43,
"column": 2
} | [
{
"pp": "M₀ : Type u_1\ninst✝ : MonoidWithZero M₀\nr : M₀\n⊢ r⁻¹ʳ ^ 0 = (r ^ 0)⁻¹ʳ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"id",
"Ring.inverse",
"instOfNatNat",
"NPow.toPow... | [] | rw [pow_zero, pow_zero, Ring.inverse_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Commute | {
"line": 42,
"column": 12
} | {
"line": 42,
"column": 53
} | {
"line": 43,
"column": 2
} | [
{
"pp": "M₀ : Type u_1\ninst✝ : MonoidWithZero M₀\nr : M₀\n⊢ r⁻¹ʳ ^ 0 = (r ^ 0)⁻¹ʳ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"id",
"Ring.inverse",
"instOfNatNat",
"NPow.toPow... | [] | rw [pow_zero, pow_zero, Ring.inverse_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 43,
"column": 15
} | {
"line": 43,
"column": 26
} | {
"line": 43,
"column": 27
} | [
{
"pp": "M₀ : Type u_2\ninst✝ : MonoidWithZero M₀\nu : M₀ˣ\na : M₀\nh : a * ↑u = 0\n⊢ a = 0",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M₀ : Type u_2\ninst✝ : MonoidWithZero M₀\nu : M₀ˣ\na : M₀\nh : a * ↑u = 0\n⊢ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 47,
"column": 15
} | {
"line": 47,
"column": 26
} | {
"line": 47,
"column": 27
} | [
{
"pp": "M₀ : Type u_2\ninst✝ : MonoidWithZero M₀\nu : M₀ˣ\na : M₀\nh : ↑u * a = 0\n⊢ a = 0",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M₀ : Type u_2\ninst✝ : MonoidWithZero M₀\nu : M₀ˣ\na : M₀\nh : ↑u * a = 0\n⊢ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 173,
"column": 41
} | {
"line": 173,
"column": 52
} | {
"line": 173,
"column": 53
} | [
{
"pp": "M₀ : Type u_2\ninst✝¹ : MonoidWithZero M₀\ninst✝ : Nontrivial M₀\nx : M₀\n⊢ x⁻¹ʳ ≠ 0 → IsUnit x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"IsUnit",
"id",
"Ne",
"Ring.inverse",
"MonoidWithZero.toMulZeroOneClass",
"Zero.toOfNat0",
"Mul... | [
"M₀ : Type u_2\ninst✝¹ : MonoidWithZero M₀\ninst✝ : Nontrivial M₀\nx : M₀\n⊢ ¬x⁻¹ʳ = 0 → IsUnit x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 13
} | {
"line": 253,
"column": 14
} | [
{
"pp": "G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\n⊢ a = 0 ∨ ∃ u, a = ↑u",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"congrArg",
"Exists",
"Units",
"id",
"Ne",
"Units.ex... | [
"G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\n⊢ a = 0 ∨ ¬a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Semiconj | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 35
} | {
"line": 81,
"column": 36
} | [
{
"pp": "R : Type u\ninst✝ : NonUnitalNonAssocRing R\na x y x' y' : R\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n⊢ SemiconjBy a (x - x') (y - y')",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAsso... | [
"R : Type u\ninst✝ : NonUnitalNonAssocRing R\na x y x' y' : R\nh : SemiconjBy a x y\nh' : SemiconjBy a x' y'\n⊢ SemiconjBy a (x + -x') (y + -y')"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Semiconj | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 35
} | {
"line": 86,
"column": 36
} | [
{
"pp": "R : Type u\ninst✝ : NonUnitalNonAssocRing R\na b x y : R\nha : SemiconjBy a x y\nhb : SemiconjBy b x y\n⊢ SemiconjBy (a - b) x y",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssocRing.toAddCo... | [
"R : Type u\ninst✝ : NonUnitalNonAssocRing R\na b x y : R\nha : SemiconjBy a x y\nhb : SemiconjBy b x y\n⊢ SemiconjBy (a + -b) x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 35
} | {
"line": 387,
"column": 36
} | [
{
"pp": "G₀ : Type u_3\ninst✝ : GroupWithZero G₀\nm n : ℕ\na : G₀\nha : a ≠ 0\nh : n ≤ m\nh1 : m - n + n = m\nh2 : a ^ (m - n) * a ^ n = a ^ m\n⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G₀ : Type u_3\ninst✝ : GroupWithZero G₀\nm n : ℕ\na : G₀\nha : a ≠ 0\nh : n ≤ m\nh1 : m - n + n = m\nh2 : a ^ (m - n) * a ^ n = a ^ m\n⊢ a ^ (m - n) = a ^ m * (a ^ n)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 13
} | {
"line": 404,
"column": 14
} | [
{
"pp": "G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nm n : ℕ\n⊢ a ^ (↑m - ↑n) = a ^ m / a ^ n",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nm n : ℕ\n⊢ a ^ (↑m - ↑n) = a ^ m / a ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 13
} | {
"line": 407,
"column": 14
} | [
{
"pp": "G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (↑n - 1) = a ^ n / a",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (↑n - 1) = a ^ n / a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 13
} | {
"line": 410,
"column": 14
} | [
{
"pp": "G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (1 - ↑n) = a / a ^ n",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G₀ : Type u_3\ninst✝ : GroupWithZero G₀\na : G₀\nha : a ≠ 0\nn : ℕ\n⊢ a ^ (1 - ↑n) = a / a ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Units.Basic | {
"line": 466,
"column": 6
} | {
"line": 466,
"column": 18
} | {
"line": 466,
"column": 19
} | [
{
"pp": "G₀ : Type u_3\ninst✝ : CommGroupWithZero G₀\nb d a c : G₀\nhb : b ≠ 0\nhd : d ≠ 0\nh : a / b = c / d\n⊢ a * d = c * b",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"HMul.hMul",
"MulZeroClas... | [
"G₀ : Type u_3\ninst✝ : CommGroupWithZero G₀\nb d a c : G₀\nhb : b ≠ 0\nhd : d ≠ 0\nh : a / b = c / d\n⊢ a * 1 * d = c * b"
] | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 95,
"column": 21
} | {
"line": 95,
"column": 32
} | {
"line": 95,
"column": 33
} | [
{
"pp": "α : Type u_3\ninst✝ : NonAssocRing α\nn : ℕ\nx : α\n⊢ Commute (↑↑n) x",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"Commute",
"id",
"NonUni... | [
"α : Type u_3\ninst✝ : NonAssocRing α\nn : ℕ\nx : α\n⊢ Commute (↑n) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 80
} | {
"line": 98,
"column": 6
} | [
{
"pp": "α : Type u_3\ninst✝ : NonAssocRing α\nn : ℕ\nx : α\n⊢ Commute (↑-[n+1]) x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | [
"α : Type u_3\ninst✝ : NonAssocRing α\nn : ℕ\nx : α\n⊢ Commute (↑(n + 1)) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 237,
"column": 4
} | {
"line": 239,
"column": 55
} | {
"line": 241,
"column": 0
} | [
{
"pp": "case negSucc\nM : Type u_5\ninst✝ : Monoid M\nf g : ℤ →* M\nh_neg_one : f (-1) = g (-1)\nh_nat : f.comp ↑ofNatHom = g.comp ↑ofNatHom\nx : ℕ\n⊢ f -[x+1] = g -[x+1]",
"ppTerm": "?negSucc",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.ofNatHom",
"MulOne.toOne",
"... | [] | rw [Int.negSucc_eq, ← neg_one_mul, f.map_mul, g.map_mul]
congr 1
exact mod_cast (DFunLike.congr_fun h_nat (x + 1) :) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 237,
"column": 4
} | {
"line": 239,
"column": 55
} | {
"line": 241,
"column": 0
} | [
{
"pp": "case negSucc\nM : Type u_5\ninst✝ : Monoid M\nf g : ℤ →* M\nh_neg_one : f (-1) = g (-1)\nh_nat : f.comp ↑ofNatHom = g.comp ↑ofNatHom\nx : ℕ\n⊢ f -[x+1] = g -[x+1]",
"ppTerm": "?negSucc",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.ofNatHom",
"MulOne.toOne",
"... | [] | rw [Int.negSucc_eq, ← neg_one_mul, f.map_mul, g.map_mul]
congr 1
exact mod_cast (DFunLike.congr_fun h_nat (x + 1) :) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 314,
"column": 71
} | {
"line": 314,
"column": 86
} | {
"line": 314,
"column": 87
} | [
{
"pp": "F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝¹ : CommGroup α\ninst✝ : AddCommGroup β\na b : α\nn : Multiplicative ℤ\n⊢ ((zpowersHom α).toFun (a * b)) n = ((zpowersHom α).toFun a * (zpowersHom α).toFun b) n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Mono... | [] | simp [mul_zpow] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 314,
"column": 71
} | {
"line": 314,
"column": 86
} | {
"line": 314,
"column": 87
} | [
{
"pp": "F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝¹ : CommGroup α\ninst✝ : AddCommGroup β\na b : α\nn : Multiplicative ℤ\n⊢ ((zpowersHom α).toFun (a * b)) n = ((zpowersHom α).toFun a * (zpowersHom α).toFun b) n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Mono... | [] | simp [mul_zpow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Cast.Lemmas | {
"line": 314,
"column": 71
} | {
"line": 314,
"column": 86
} | {
"line": 314,
"column": 87
} | [
{
"pp": "F : Type u_1\nι : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝¹ : CommGroup α\ninst✝ : AddCommGroup β\na b : α\nn : Multiplicative ℤ\n⊢ ((zpowersHom α).toFun (a * b)) n = ((zpowersHom α).toFun a * (zpowersHom α).toFun b) n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Mono... | [] | simp [mul_zpow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Field.Defs | {
"line": 195,
"column": 65
} | {
"line": 195,
"column": 93
} | {
"line": 197,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionSemiring K\nq : ℚ≥0\n⊢ q • 1 = ↑q",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
"HMul.hMul",
"congrArg",
"NNRat.smulDivisionSe... | [] | rw [NNRat.smul_def, mul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Field.Defs | {
"line": 195,
"column": 65
} | {
"line": 195,
"column": 93
} | {
"line": 197,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionSemiring K\nq : ℚ≥0\n⊢ q • 1 = ↑q",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
"HMul.hMul",
"congrArg",
"NNRat.smulDivisionSe... | [] | rw [NNRat.smul_def, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Field.Defs | {
"line": 195,
"column": 65
} | {
"line": 195,
"column": 93
} | {
"line": 197,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : DivisionSemiring K\nq : ℚ≥0\n⊢ q • 1 = ↑q",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
"HMul.hMul",
"congrArg",
"NNRat.smulDivisionSe... | [] | rw [NNRat.smul_def, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Parity | {
"line": 137,
"column": 31
} | {
"line": 137,
"column": 48
} | {
"line": 137,
"column": 48
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : Semiring α\ninst✝² : Semiring β\ninst✝¹ : FunLike F α β\ninst✝ : RingHomClass F α β\nf : F\na : α\n⊢ f (2 * a + 1) = 2 * f a + 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | by simp [two_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Parity | {
"line": 149,
"column": 2
} | {
"line": 151,
"column": 51
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Semiring α\na : α\nn : ℕ\nha : Odd a\n⊢ Odd (a ^ n)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"pow_succ",
"Odd",
... | [] | induction n with
| zero => simp [pow_zero]
| succ n hrec => rw [pow_succ]; exact hrec.mul ha | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Ring.Parity | {
"line": 149,
"column": 2
} | {
"line": 151,
"column": 51
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Semiring α\na : α\nn : ℕ\nha : Odd a\n⊢ Odd (a ^ n)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"pow_succ",
"Odd",
... | [] | induction n with
| zero => simp [pow_zero]
| succ n hrec => rw [pow_succ]; exact hrec.mul ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Parity | {
"line": 149,
"column": 2
} | {
"line": 151,
"column": 51
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Semiring α\na : α\nn : ℕ\nha : Odd a\n⊢ Odd (a ^ n)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"pow_succ",
"Odd",
... | [] | induction n with
| zero => simp [pow_zero]
| succ n hrec => rw [pow_succ]; exact hrec.mul ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Parity | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 52
} | {
"line": 278,
"column": 53
} | [
{
"pp": "n : ℕ\n⊢ ∃ k, n = 2 * k ∨ n = 2 * k + 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Exists",
"id",
"instMulNat",
"instOfNatNat",
"instHAdd",
"_private.Mathlib.Algebra.Ring.Parity.0.Nat.even_or_odd'._simp_1... | [
"n : ℕ\n⊢ (∃ x, n = 2 * x) ∨ ∃ x, n = 2 * x + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Parity | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 32
} | {
"line": 343,
"column": 33
} | [
{
"pp": "case succ\nk : ℕ\n⊢ 2 ∣ (k + 1) * (k + 1 - 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"HSub.hSub",
"id",
"instSubNat",
"CommMagma.toM... | [
"case succ\nk : ℕ\n⊢ 2 ∣ k * (k + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Parity | {
"line": 380,
"column": 54
} | {
"line": 380,
"column": 71
} | {
"line": 380,
"column": 71
} | [
{
"pp": "α : Type u_4\nf : α → α\nn : ℕ\nhf : Involutive f\nhne : f ≠ id\nH : f^[n] = id\nhn : Odd n\n⊢ f = id",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"Function.Involutive.iterate_odd",
"Odd",
"Eq.mp",
"id",
"Nat.iterate",
"Nat"... | [
"α : Type u_4\nf : α → α\nn : ℕ\nhf : Involutive f\nhne : f ≠ id\nH : f = id\nhn : Odd n\n⊢ f = id"
] | hf.iterate_odd hn | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Parity | {
"line": 392,
"column": 43
} | {
"line": 392,
"column": 54
} | {
"line": 392,
"column": 55
} | [
{
"pp": "R : Type u_4\ninst✝¹ : Monoid R\ninst✝ : HasDistribNeg R\nn : ℕ\nh : Odd n\n⊢ ¬Even n",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Nat.not_even_iff_odd._simp_1",
"Eq.mpr",
"Odd",
"id",
"Nat",
"Even",
"instAddNat",
"Nat.instSemiri... | [
"R : Type u_4\ninst✝¹ : Monoid R\ninst✝ : HasDistribNeg R\nn : ℕ\nh : Odd n\n⊢ Odd n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monoid.Unbundled.Basic | {
"line": 338,
"column": 10
} | {
"line": 338,
"column": 21
} | {
"line": 338,
"column": 22
} | [
{
"pp": "case inr.inl\nα : Type u_1\ninst✝³ : Mul α\ninst✝² : LinearOrder α\na b d : α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\nh : a * b = a * d\n⊢ a = a ∧ b = d",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"And... | [
"case inr.inl\nα : Type u_1\ninst✝³ : Mul α\ninst✝² : LinearOrder α\na b d : α\ninst✝¹ : MulLeftStrictMono α\ninst✝ : MulRightStrictMono α\nh : a * b = a * d\n⊢ b = d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 39
} | {
"line": 53,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝³ : MulOneClass α\ninst✝² : Preorder α\ninst✝¹ : ExistsMulOfLE α\na b : α\ninst✝ : MulLeftReflectLE α\nh : a ≤ b\n⊢ ∃ c, 1 ≤ c ∧ a * c = b",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Preorder.toLE",
"Exists... | [
"α : Type u\ninst✝³ : MulOneClass α\ninst✝² : Preorder α\ninst✝¹ : ExistsMulOfLE α\na : α\ninst✝ : MulLeftReflectLE α\nc : α\nh : a ≤ a * c\n⊢ ∃ c_1, 1 ≤ c_1 ∧ a * c_1 = a * c"
] | obtain ⟨c, rfl⟩ := exists_mul_of_le h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.Monoid.Unbundled.Basic | {
"line": 1398,
"column": 2
} | {
"line": 1398,
"column": 28
} | {
"line": 1398,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MulOneClass α\ninst✝ : LE α\na b : α\n⊢ 1 * a ≤ 1 * b → a ≤ b",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"id",
"MulOne.toMul",
"LE.le",
"MulOneClass.toMulO... | [
"α : Type u_1\ninst✝¹ : MulOneClass α\ninst✝ : LE α\na b : α\n⊢ a ≤ b → a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Unbundled.Basic | {
"line": 473,
"column": 2
} | {
"line": 473,
"column": 35
} | {
"line": 473,
"column": 36
} | [
{
"pp": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : MulRightMono α\na b c : α\n⊢ a / c ≤ b / c ↔ a ≤ b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [
"α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : MulRightMono α\na b c : α\n⊢ a * c⁻¹ ≤ b * c⁻¹ ↔ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Unbundled.Basic | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 35
} | {
"line": 543,
"column": 36
} | [
{
"pp": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LE α\ninst✝ : MulLeftMono α\na b c d : α\n⊢ a / b ≤ c / d ↔ a * d ≤ c * b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
... | [
"α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LE α\ninst✝ : MulLeftMono α\na b c d : α\n⊢ a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monoid.Unbundled.MinMax | {
"line": 84,
"column": 2
} | {
"line": 85,
"column": 36
} | {
"line": 87,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"congrArg"... | [] | contrapose!
exact fun h => mul_le_mul' h.1 h.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Monoid.Unbundled.MinMax | {
"line": 84,
"column": 2
} | {
"line": 85,
"column": 36
} | {
"line": 87,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : Mul α\ninst✝¹ : MulLeftMono α\ninst✝ : MulRightMono α\na₁ a₂ b₁ b₂ : α\n⊢ a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"congrArg"... | [] | contrapose!
exact fun h => mul_le_mul' h.1 h.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Unbundled.Basic | {
"line": 597,
"column": 2
} | {
"line": 597,
"column": 35
} | {
"line": 597,
"column": 36
} | [
{
"pp": "α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : MulRightStrictMono α\na b c : α\n⊢ a / c < b / c ↔ a < b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congr... | [
"α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : MulRightStrictMono α\na b c : α\n⊢ a * c⁻¹ < b * c⁻¹ ↔ a < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Unbundled.Basic | {
"line": 618,
"column": 53
} | {
"line": 619,
"column": 69
} | {
"line": 621,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝² : Group α\ninst✝¹ : LT α\ninst✝ : MulRightStrictMono α\na b c : α\n⊢ a < c / b ↔ a * b < c",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congr... | [] | by
rw [← mul_lt_mul_iff_right b, div_eq_mul_inv, inv_mul_cancel_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Group.Unbundled.Basic | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 35
} | {
"line": 663,
"column": 36
} | [
{
"pp": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LT α\ninst✝ : MulLeftStrictMono α\na b c d : α\n⊢ a / b < c / d ↔ a * d < c * b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",... | [
"α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LT α\ninst✝ : MulLeftStrictMono α\na b c d : α\n⊢ a * b⁻¹ < c * d⁻¹ ↔ a * d < c * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Unbundled.Basic | {
"line": 725,
"column": 41
} | {
"line": 725,
"column": 52
} | {
"line": 725,
"column": 53
} | [
{
"pp": "α : Type u\ninst✝² : Group α\ninst✝¹ : LinearOrder α\ninst✝ : MulLeftMono α\na b : α\nh : ∀ (ε : α), 1 < ε → a < b * ε\nh₁ : b < a\n⊢ a < a",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"PartialOrder.toPreorder",
... | [
"α : Type u\ninst✝² : Group α\ninst✝¹ : LinearOrder α\ninst✝ : MulLeftMono α\na b : α\nh : ∀ (ε : α), 1 < ε → a < b * ε\nh₁ : b < a\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monoid.Defs | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 28
} | {
"line": 106,
"column": 29
} | [
{
"pp": "case inr\nα : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : LinearOrder α\nhmul : ∀ (a b c : α), b < c → a * b < a * c\na b : α\nh✝ : a ≤ b\nc : α\nh : a < b\n⊢ a * c ≤ b * c",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nα : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : LinearOrder α\nhmul : ∀ (a b c : α), b < c → a * b < a * c\na b : α\nh✝ : a ≤ b\nc : α\nh : a < b\n⊢ a * c ≤ b * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Defs | {
"line": 49,
"column": 39
} | {
"line": 49,
"column": 50
} | {
"line": 49,
"column": 51
} | [
{
"pp": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedMonoid α\na b c : α\nbc : a * b ≤ a * c\n⊢ b ≤ c",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝² : CommGroup α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedMonoid α\na b c : α\nbc : a * b ≤ a * c\n⊢ b ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Defs | {
"line": 50,
"column": 40
} | {
"line": 50,
"column": 51
} | {
"line": 50,
"column": 52
} | [
{
"pp": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedMonoid α\na b c : α\nbc : b * a ≤ c * a\n⊢ b ≤ c",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝² : CommGroup α\ninst✝¹ : Preorder α\ninst✝ : IsOrderedMonoid α\na b c : α\nbc : b * a ≤ c * a\n⊢ b ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Defs | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 54
} | {
"line": 111,
"column": 54
} | [
{
"pp": "α : Type u\ninst✝³ : CommGroup α\ninst✝² : LinearOrder α\ninst✝¹ : IsOrderedMonoid α\na : α\ninst✝ : Nontrivial α\n⊢ ∀ (a : α), ∃ b, b < a",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"instHDiv",
"div_lt_self_iff",
"InvOn... | [] | obtain ⟨y, hy⟩ : ∃ a : α, 1 < a := exists_one_lt'
exact fun a => ⟨a / y, (div_lt_self_iff a).mpr hy⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Defs | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 54
} | {
"line": 111,
"column": 54
} | [
{
"pp": "α : Type u\ninst✝³ : CommGroup α\ninst✝² : LinearOrder α\ninst✝¹ : IsOrderedMonoid α\na : α\ninst✝ : Nontrivial α\n⊢ ∀ (a : α), ∃ b, b < a",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"instHDiv",
"div_lt_self_iff",
"InvOn... | [] | obtain ⟨y, hy⟩ : ∃ a : α, 1 < a := exists_one_lt'
exact fun a => ⟨a / y, (div_lt_self_iff a).mpr hy⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Defs | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 42
} | {
"line": 100,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : ZeroLEOneClass R\nmul_nonneg : ∀ (a b : R), 0 ≤ a → 0 ≤ b → 0 ≤ a * b\na : R\nha : 0 ≤ a\nb c : R\nhbc : b ≤ c\n⊢ a * b ≤ a * c",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"used... | [
"R : Type u\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : ZeroLEOneClass R\nmul_nonneg : ∀ (a b : R), 0 ≤ a → 0 ≤ b → 0 ≤ a * b\na : R\nha : 0 ≤ a\nb c : R\nhbc : b ≤ c\n⊢ a * b ≤ a * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Defs | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 42
} | {
"line": 102,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : ZeroLEOneClass R\nmul_nonneg : ∀ (a b : R), 0 ≤ a → 0 ≤ b → 0 ≤ a * b\na : R\nha : 0 ≤ a\nb c : R\nhbc : b ≤ c\n⊢ b * a ≤ c * a",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"used... | [
"R : Type u\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : ZeroLEOneClass R\nmul_nonneg : ∀ (a b : R), 0 ≤ a → 0 ≤ b → 0 ≤ a * b\na : R\nha : 0 ≤ a\nb c : R\nhbc : b ≤ c\n⊢ b * a ≤ c * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Defs | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 39
} | {
"line": 108,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : Nontrivial R\nmul_pos : ∀ (a b : R), 0 < a → 0 < b → 0 < a * b\na : R\nha : 0 < a\nb c : R\nhbc : b < c\n⊢ a * b < a * c",
"ppTerm": "?m.32",
"assigned": false,
"usedConst... | [
"R : Type u\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : Nontrivial R\nmul_pos : ∀ (a b : R), 0 < a → 0 < b → 0 < a * b\na : R\nha : 0 < a\nb c : R\nhbc : b < c\n⊢ a * b < a * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Defs | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 39
} | {
"line": 110,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : Nontrivial R\nmul_pos : ∀ (a b : R), 0 < a → 0 < b → 0 < a * b\na : R\nha : 0 < a\nb c : R\nhbc : b < c\n⊢ b * a < c * a",
"ppTerm": "?m.39",
"assigned": false,
"usedConst... | [
"R : Type u\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : ZeroLEOneClass R\ninst✝ : Nontrivial R\nmul_pos : ∀ (a b : R), 0 < a → 0 < b → 0 < a * b\na : R\nha : 0 < a\nb c : R\nhbc : b < c\n⊢ b * a < c * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Defs | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 18
} | {
"line": 175,
"column": 19
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\na b c : R\nh : a + b + b * c ≤ c\n⊢ 1 + a ≤ (1 - b) * (1 + c)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Ring.toNonAssocRing",
"AddGroupWithOne.toAdd... | [
"R : Type u\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\na b c : R\nh : a + b + b * c ≤ c\n⊢ 1 + a ≤ 1 + c - b * (1 + c)"
] | one_sub_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nha : a ≤ 0\nhb : b ≤ 0\n⊢ 0 ≤ a * b",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nha : a ≤ 0\nhb : b ≤ 0\n⊢ 0 ≤ a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Defs | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 18
} | {
"line": 183,
"column": 19
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\na b c : R\nh : b + b * c ≤ a + c\n⊢ 1 - a ≤ (1 - b) * (1 + c)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Ring.toNonAssocRing",
"AddGroupWithOne.toAdd... | [
"R : Type u\ninst✝² : Ring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\na b c : R\nh : b + b * c ≤ a + c\n⊢ 1 - a ≤ 1 + c - b * (1 + c)"
] | one_sub_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 93,
"column": 47
} | {
"line": 93,
"column": 68
} | {
"line": 95,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Preorder R\na b c d : R\ninst✝⁴ : ExistsAddOfLE R\ninst✝³ : MulPosMono R\ninst✝² : PosMulMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nhca : c ≤ a\nhbd : b ≤ d\nhc : 0 ≤ c\nhb : b ≤ 0\n⊢ c * b ≤ c * d",
"ppTerm": "?m.50",
"assigned": tru... | [] | by gcongr; assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 28
} | {
"line": 124,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nhb : b ≤ 0\nh : a ≤ 1\n⊢ b ≤ a * b",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nhb : b ≤ 0\nh : a ≤ 1\n⊢ b ≤ a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 28
} | {
"line": 130,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nhb : b ≤ 0\nh : 1 ≤ a\n⊢ a * b ≤ b",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nhb : b ≤ 0\nh : 1 ≤ a\n⊢ a * b ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 28
} | {
"line": 136,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : PosMulMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nha : a ≤ 0\nh : b ≤ 1\n⊢ a ≤ a * b",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : PosMulMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nha : a ≤ 0\nh : b ≤ 1\n⊢ a ≤ a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 28
} | {
"line": 142,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : PosMulMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nha : a ≤ 0\nh : 1 ≤ b\n⊢ a * b ≤ a",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : Preorder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : PosMulMono R\ninst✝¹ : AddRightMono R\ninst✝ : AddRightReflectLE R\nha : a ≤ 0\nh : 1 ≤ b\n⊢ a * b ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 29
} | {
"line": 231,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\na b : R\nha : a < 0\nhb : b < 0\n⊢ 0 < a * b",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\na b : R\nha : a < 0\nhb : b < 0\n⊢ 0 < a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 28
} | {
"line": 237,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\nhb : b < 0\nh : a < 1\n⊢ b < a * b",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\nhb : b < 0\nh : a < 1\n⊢ b < a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 28
} | {
"line": 243,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\nhb : b < 0\nh : 1 < a\n⊢ a * b < b",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : MulPosStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\nhb : b < 0\nh : 1 < a\n⊢ a * b < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 28
} | {
"line": 249,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : PosMulStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\nha : a < 0\nh : b < 1\n⊢ a < a * b",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : PosMulStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\nha : a < 0\nh : b < 1\n⊢ a < a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 28
} | {
"line": 255,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : PosMulStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\nha : a < 0\nh : 1 < b\n⊢ a * b < a",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"R : Type u\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\na b : R\ninst✝³ : ExistsAddOfLE R\ninst✝² : PosMulStrictMono R\ninst✝¹ : AddRightStrictMono R\ninst✝ : AddRightReflectLT R\nha : a < 0\nh : 1 < b\n⊢ a * b < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 29
} | {
"line": 42,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : PosMulStrictMono α\nha : 0 < a\nhb : 0 < b\n⊢ 0 < a * b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : PosMulStrictMono α\nha : 0 < a\nhb : 0 < b\n⊢ 0 < a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 336,
"column": 31
} | {
"line": 336,
"column": 39
} | {
"line": 336,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\na b : R\ninst✝¹ : MulPosStrictMono R\ninst✝ : PosMulStrictMono R\nhab : 0 ≤ a * b\n⊢ (0 ≤ a → b < 0) → ¬(a ≤ 0 → 0 < b)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
... | [
"R : Type u\ninst✝³ : Semiring R\ninst✝² : LinearOrder R\na b : R\ninst✝¹ : MulPosStrictMono R\ninst✝ : PosMulStrictMono R\nhab : 0 ≤ a * b\nab : 0 ≤ a → b < 0\n⊢ ¬(a ≤ 0 → 0 < b)"
] | intro ab | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 29
} | {
"line": 47,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : PosMulStrictMono α\nha : 0 < a\nhb : b < 0\n⊢ a * b < 0",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : PosMulStrictMono α\nha : 0 < a\nhb : b < 0\n⊢ a * b < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 51,
"column": 28
} | {
"line": 51,
"column": 39
} | {
"line": 51,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝³ : MulZeroClass α\na b : α\ninst✝² : Preorder α\ninst✝¹ : PosMulStrictMono α\ninst✝ : PosMulReflectLT α\nh : 0 < a\n⊢ 0 < a * b ↔ 0 < b",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝³ : MulZeroClass α\na b : α\ninst✝² : Preorder α\ninst✝¹ : PosMulStrictMono α\ninst✝ : PosMulReflectLT α\nh : 0 < a\n⊢ 0 < a * b ↔ 0 < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 29
} | {
"line": 55,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : MulPosStrictMono α\nha : 0 < a\nhb : 0 < b\n⊢ 0 < a * b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : MulPosStrictMono α\nha : 0 < a\nhb : 0 < b\n⊢ 0 < a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 29
} | {
"line": 58,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : MulPosStrictMono α\nha : a < 0\nhb : 0 < b\n⊢ a * b < 0",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : MulPosStrictMono α\nha : a < 0\nhb : 0 < b\n⊢ a * b < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 62,
"column": 28
} | {
"line": 62,
"column": 39
} | {
"line": 62,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝³ : MulZeroClass α\na b : α\ninst✝² : Preorder α\ninst✝¹ : MulPosStrictMono α\ninst✝ : MulPosReflectLT α\nh : 0 < b\n⊢ 0 < a * b ↔ 0 < a",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝³ : MulZeroClass α\na b : α\ninst✝² : Preorder α\ninst✝¹ : MulPosStrictMono α\ninst✝ : MulPosReflectLT α\nh : 0 < b\n⊢ 0 < a * b ↔ 0 < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 29
} | {
"line": 66,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : PosMulMono α\nha : 0 ≤ a\nhb : 0 ≤ b\n⊢ 0 ≤ a * b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : PosMulMono α\nha : 0 ≤ a\nhb : 0 ≤ b\n⊢ 0 ≤ a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 29
} | {
"line": 71,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : PosMulMono α\nha : 0 ≤ a\nhb : b ≤ 0\n⊢ a * b ≤ 0",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : PosMulMono α\nha : 0 ≤ a\nhb : b ≤ 0\n⊢ a * b ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 29
} | {
"line": 75,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : MulPosMono α\nha : 0 ≤ a\nhb : 0 ≤ b\n⊢ 0 ≤ a * b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : MulPosMono α\nha : 0 ≤ a\nhb : 0 ≤ b\n⊢ 0 ≤ a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.GroupWithZero.Basic | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 29
} | {
"line": 78,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : MulPosMono α\nha : a ≤ 0\nhb : 0 ≤ b\n⊢ a * b ≤ 0",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : MulZeroClass α\na b : α\ninst✝¹ : Preorder α\ninst✝ : MulPosMono α\nha : a ≤ 0\nhb : 0 ≤ b\n⊢ a * b ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 13
} | {
"line": 367,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\nb c : R\ninst✝ : MulPosStrictMono R\nh : 0 < c\n⊢ 0 ≤ b * c ↔ 0 ≤ b",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\nb c : R\ninst✝ : MulPosStrictMono R\nh : 0 < c\n⊢ 0 ≤ b * c ↔ 0 ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.Unbundled.Basic | {
"line": 485,
"column": 21
} | {
"line": 485,
"column": 57
} | {
"line": 485,
"column": 58
} | [
{
"pp": "R : Type u\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\na b : R\ninst✝ : PosMulStrictMono R\nh : ∀ (n : ℕ), 0 ≤ a * b ^ n\n⊢ 0 ≤ a",
"ppTerm": "?m.48",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\na b : R\ninst✝ : PosMulStrictMono R\nh : ∀ (n : ℕ), 0 ≤ a * b ^ n\n⊢ 0 ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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