file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Nat/Factorial/Basic.lean
|
Nat.ascFactorial_succ
|
[] |
[
247,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.ringHom_ext'
|
[] |
[
789,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.toRingHom_eq_coe
|
[] |
[
685,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
684,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul
|
[
{
"state_after": "case inl\np : ℝ[X]\nx : ℝ\nhx : x < 0\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval x⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue x) x\n\ncase inr.inl\np : ℝ[X]\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval 0⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue 0) 0\n\ncase inr.inr\np : ℝ[X]\nx : ℝ\nhx : 0 < x\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval x⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue x) x",
"state_before": "p : ℝ[X]\nx : ℝ\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval x⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue x) x",
"tactic": "rcases lt_trichotomy x 0 with hx | rfl | hx"
},
{
"state_after": "case inl\np : ℝ[X]\nx : ℝ\nhx : x < 0\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) 0 x",
"state_before": "case inl\np : ℝ[X]\nx : ℝ\nhx : x < 0\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval x⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue x) x",
"tactic": "rw [zero_of_nonpos hx.le, mul_zero]"
},
{
"state_after": "case inl\np : ℝ[X]\nx : ℝ\nhx : x < 0\n⊢ (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) =ᶠ[𝓝 x] fun x => 0",
"state_before": "case inl\np : ℝ[X]\nx : ℝ\nhx : x < 0\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) 0 x",
"tactic": "refine (hasDerivAt_const _ 0).congr_of_eventuallyEq ?_"
},
{
"state_after": "case h\np : ℝ[X]\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\n⊢ Polynomial.eval y⁻¹ p * expNegInvGlue y = 0",
"state_before": "case inl\np : ℝ[X]\nx : ℝ\nhx : x < 0\n⊢ (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) =ᶠ[𝓝 x] fun x => 0",
"tactic": "filter_upwards [gt_mem_nhds hx] with y hy"
},
{
"state_after": "no goals",
"state_before": "case h\np : ℝ[X]\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\n⊢ Polynomial.eval y⁻¹ p * expNegInvGlue y = 0",
"tactic": "rw [zero_of_nonpos hy.le, mul_zero]"
},
{
"state_after": "case inr.inl\np : ℝ[X]\n⊢ Tendsto (slope (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) 0) (𝓝[{0}ᶜ] 0) (𝓝 0)",
"state_before": "case inr.inl\np : ℝ[X]\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval 0⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue 0) 0",
"tactic": "rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]"
},
{
"state_after": "case inr.inl\np : ℝ[X]\nx : ℝ\n⊢ Polynomial.eval x⁻¹ (p * X) * expNegInvGlue x = slope (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) 0 x",
"state_before": "case inr.inl\np : ℝ[X]\n⊢ Tendsto (slope (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) 0) (𝓝[{0}ᶜ] 0) (𝓝 0)",
"tactic": "refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\np : ℝ[X]\nx : ℝ\n⊢ Polynomial.eval x⁻¹ (p * X) * expNegInvGlue x = slope (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) 0 x",
"tactic": "simp [slope_def_field, div_eq_mul_inv, mul_right_comm]"
},
{
"state_after": "case inr.inr\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval x⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue x) x",
"state_before": "case inr.inr\np : ℝ[X]\nx : ℝ\nhx : 0 < x\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval x⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue x) x",
"tactic": "have := ((p.hasDerivAt x⁻¹).mul (hasDerivAt_neg _).exp).comp x (hasDerivAt_inv hx.ne')"
},
{
"state_after": "case h.e'_7\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\n⊢ Polynomial.eval x⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue x =\n (Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹\n\ncase inr.inr.convert_2\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\n⊢ (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) =ᶠ[𝓝 x] (fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv",
"state_before": "case inr.inr\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\n⊢ HasDerivAt (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x)\n (Polynomial.eval x⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue x) x",
"tactic": "convert this.congr_of_eventuallyEq _ using 1"
},
{
"state_after": "case h.e'_7\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\n⊢ (x ^ 2)⁻¹ * (Polynomial.eval x⁻¹ p - Polynomial.eval x⁻¹ (↑derivative p)) * exp (-x⁻¹) =\n -((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + -(Polynomial.eval x⁻¹ p * exp (-x⁻¹))) * (x ^ 2)⁻¹)",
"state_before": "case h.e'_7\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\n⊢ Polynomial.eval x⁻¹ (X ^ 2 * (p - ↑derivative p)) * expNegInvGlue x =\n (Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹",
"tactic": "simp [expNegInvGlue, hx.not_le]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\n⊢ (x ^ 2)⁻¹ * (Polynomial.eval x⁻¹ p - Polynomial.eval x⁻¹ (↑derivative p)) * exp (-x⁻¹) =\n -((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + -(Polynomial.eval x⁻¹ p * exp (-x⁻¹))) * (x ^ 2)⁻¹)",
"tactic": "ring"
},
{
"state_after": "case h\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\ny : ℝ\nhy : 0 < y\n⊢ Polynomial.eval y⁻¹ p * expNegInvGlue y = ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv) y",
"state_before": "case inr.inr.convert_2\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\n⊢ (fun x => Polynomial.eval x⁻¹ p * expNegInvGlue x) =ᶠ[𝓝 x] (fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv",
"tactic": "filter_upwards [lt_mem_nhds hx] with y hy"
},
{
"state_after": "no goals",
"state_before": "case h\np : ℝ[X]\nx : ℝ\nhx : 0 < x\nthis :\n HasDerivAt ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv)\n ((Polynomial.eval x⁻¹ (↑derivative p) * exp (-x⁻¹) + Polynomial.eval x⁻¹ p * (exp (-x⁻¹) * -1)) * -(x ^ 2)⁻¹) x\ny : ℝ\nhy : 0 < y\n⊢ Polynomial.eval y⁻¹ p * expNegInvGlue y = ((fun y => Polynomial.eval y p * exp (-y)) ∘ Inv.inv) y",
"tactic": "simp [expNegInvGlue, hy.not_le]"
}
] |
[
138,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Init/Data/Nat/Lemmas.lean
|
Nat.one_le_bit0
|
[] |
[
206,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
11
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
ContDiffOn.dist
|
[] |
[
211,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Basic.lean
|
IsAlgClosed.exists_pow_nat_eq
|
[
{
"state_after": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\nthis : degree (X ^ n - ↑C x) ≠ 0\n⊢ ∃ z, z ^ n = x",
"state_before": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\n⊢ ∃ z, z ^ n = x",
"tactic": "have : degree (X ^ n - C x) ≠ 0 := by\n rw [degree_X_pow_sub_C hn x]\n exact ne_of_gt (WithBot.coe_lt_coe.2 hn)"
},
{
"state_after": "case intro\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\nthis : degree (X ^ n - ↑C x) ≠ 0\nz : k\nhz : IsRoot (X ^ n - ↑C x) z\n⊢ ∃ z, z ^ n = x",
"state_before": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\nthis : degree (X ^ n - ↑C x) ≠ 0\n⊢ ∃ z, z ^ n = x",
"tactic": "obtain ⟨z, hz⟩ := exists_root (X ^ n - C x) this"
},
{
"state_after": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\n⊢ ↑n ≠ 0",
"state_before": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\n⊢ degree (X ^ n - ↑C x) ≠ 0",
"tactic": "rw [degree_X_pow_sub_C hn x]"
},
{
"state_after": "no goals",
"state_before": "k : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\n⊢ ↑n ≠ 0",
"tactic": "exact ne_of_gt (WithBot.coe_lt_coe.2 hn)"
},
{
"state_after": "case intro\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\nthis : degree (X ^ n - ↑C x) ≠ 0\nz : k\nhz : IsRoot (X ^ n - ↑C x) z\n⊢ z ^ n = x",
"state_before": "case intro\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\nthis : degree (X ^ n - ↑C x) ≠ 0\nz : k\nhz : IsRoot (X ^ n - ↑C x) z\n⊢ ∃ z, z ^ n = x",
"tactic": "use z"
},
{
"state_after": "case intro\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\nthis : degree (X ^ n - ↑C x) ≠ 0\nz : k\nhz : z ^ n - x = 0\n⊢ z ^ n = x",
"state_before": "case intro\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\nthis : degree (X ^ n - ↑C x) ≠ 0\nz : k\nhz : IsRoot (X ^ n - ↑C x) z\n⊢ z ^ n = x",
"tactic": "simp only [eval_C, eval_X, eval_pow, eval_sub, IsRoot.def] at hz"
},
{
"state_after": "no goals",
"state_before": "case intro\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsAlgClosed k\nx : k\nn : ℕ\nhn : 0 < n\nthis : degree (X ^ n - ↑C x) ≠ 0\nz : k\nhz : z ^ n - x = 0\n⊢ z ^ n = x",
"tactic": "exact sub_eq_zero.1 hz"
}
] |
[
92,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Algebra/Hom/GroupAction.lean
|
DistribMulActionHom.toMulActionHom_injective
|
[
{
"state_after": "case a\nM' : Type ?u.113752\nX : Type ?u.113755\ninst✝²³ : SMul M' X\nY : Type ?u.113762\ninst✝²² : SMul M' Y\nZ : Type ?u.113769\ninst✝²¹ : SMul M' Z\nM : Type u_1\ninst✝²⁰ : Monoid M\nA : Type u_2\ninst✝¹⁹ : AddMonoid A\ninst✝¹⁸ : DistribMulAction M A\nA' : Type ?u.113810\ninst✝¹⁷ : AddGroup A'\ninst✝¹⁶ : DistribMulAction M A'\nB : Type u_3\ninst✝¹⁵ : AddMonoid B\ninst✝¹⁴ : DistribMulAction M B\nB' : Type ?u.114112\ninst✝¹³ : AddGroup B'\ninst✝¹² : DistribMulAction M B'\nC : Type ?u.114388\ninst✝¹¹ : AddMonoid C\ninst✝¹⁰ : DistribMulAction M C\nR : Type ?u.114414\ninst✝⁹ : Semiring R\ninst✝⁸ : MulSemiringAction M R\nR' : Type ?u.114441\ninst✝⁷ : Ring R'\ninst✝⁶ : MulSemiringAction M R'\nS : Type ?u.114637\ninst✝⁵ : Semiring S\ninst✝⁴ : MulSemiringAction M S\nS' : Type ?u.114664\ninst✝³ : Ring S'\ninst✝² : MulSemiringAction M S'\nT : Type ?u.114860\ninst✝¹ : Semiring T\ninst✝ : MulSemiringAction M T\nf g : A →+[M] B\nh : ↑f = ↑g\na : A\n⊢ ↑f a = ↑g a",
"state_before": "M' : Type ?u.113752\nX : Type ?u.113755\ninst✝²³ : SMul M' X\nY : Type ?u.113762\ninst✝²² : SMul M' Y\nZ : Type ?u.113769\ninst✝²¹ : SMul M' Z\nM : Type u_1\ninst✝²⁰ : Monoid M\nA : Type u_2\ninst✝¹⁹ : AddMonoid A\ninst✝¹⁸ : DistribMulAction M A\nA' : Type ?u.113810\ninst✝¹⁷ : AddGroup A'\ninst✝¹⁶ : DistribMulAction M A'\nB : Type u_3\ninst✝¹⁵ : AddMonoid B\ninst✝¹⁴ : DistribMulAction M B\nB' : Type ?u.114112\ninst✝¹³ : AddGroup B'\ninst✝¹² : DistribMulAction M B'\nC : Type ?u.114388\ninst✝¹¹ : AddMonoid C\ninst✝¹⁰ : DistribMulAction M C\nR : Type ?u.114414\ninst✝⁹ : Semiring R\ninst✝⁸ : MulSemiringAction M R\nR' : Type ?u.114441\ninst✝⁷ : Ring R'\ninst✝⁶ : MulSemiringAction M R'\nS : Type ?u.114637\ninst✝⁵ : Semiring S\ninst✝⁴ : MulSemiringAction M S\nS' : Type ?u.114664\ninst✝³ : Ring S'\ninst✝² : MulSemiringAction M S'\nT : Type ?u.114860\ninst✝¹ : Semiring T\ninst✝ : MulSemiringAction M T\nf g : A →+[M] B\nh : ↑f = ↑g\n⊢ f = g",
"tactic": "ext a"
},
{
"state_after": "no goals",
"state_before": "case a\nM' : Type ?u.113752\nX : Type ?u.113755\ninst✝²³ : SMul M' X\nY : Type ?u.113762\ninst✝²² : SMul M' Y\nZ : Type ?u.113769\ninst✝²¹ : SMul M' Z\nM : Type u_1\ninst✝²⁰ : Monoid M\nA : Type u_2\ninst✝¹⁹ : AddMonoid A\ninst✝¹⁸ : DistribMulAction M A\nA' : Type ?u.113810\ninst✝¹⁷ : AddGroup A'\ninst✝¹⁶ : DistribMulAction M A'\nB : Type u_3\ninst✝¹⁵ : AddMonoid B\ninst✝¹⁴ : DistribMulAction M B\nB' : Type ?u.114112\ninst✝¹³ : AddGroup B'\ninst✝¹² : DistribMulAction M B'\nC : Type ?u.114388\ninst✝¹¹ : AddMonoid C\ninst✝¹⁰ : DistribMulAction M C\nR : Type ?u.114414\ninst✝⁹ : Semiring R\ninst✝⁸ : MulSemiringAction M R\nR' : Type ?u.114441\ninst✝⁷ : Ring R'\ninst✝⁶ : MulSemiringAction M R'\nS : Type ?u.114637\ninst✝⁵ : Semiring S\ninst✝⁴ : MulSemiringAction M S\nS' : Type ?u.114664\ninst✝³ : Ring S'\ninst✝² : MulSemiringAction M S'\nT : Type ?u.114860\ninst✝¹ : Semiring T\ninst✝ : MulSemiringAction M T\nf g : A →+[M] B\nh : ↑f = ↑g\na : A\n⊢ ↑f a = ↑g a",
"tactic": "exact MulActionHom.congr_fun h a"
}
] |
[
296,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Order/SuccPred/IntervalSucc.lean
|
Antitone.pairwise_disjoint_on_Ioo_pred
|
[] |
[
142,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.exists_isLUB
|
[
{
"state_after": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\n⊢ ∃ x, IsLUB S x",
"state_before": "x y : ℝ\nS : Set ℝ\nhne : Set.Nonempty S\nhbdd : BddAbove S\n⊢ ∃ x, IsLUB S x",
"tactic": "rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩"
},
{
"state_after": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\n⊢ ∃ x, IsLUB S x",
"state_before": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\n⊢ ∃ x, IsLUB S x",
"tactic": "have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by\n cases' exists_int_gt U with k hk\n refine' fun d => ⟨k * d, fun z h => _⟩\n rcases h with ⟨y, yS, hy⟩\n refine' Int.cast_le.1 (hy.trans _)\n push_cast\n exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg"
},
{
"state_after": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\n⊢ ∃ x, IsLUB S x",
"state_before": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\n⊢ ∃ x, IsLUB S x",
"tactic": "choose f hf using fun d : ℕ =>\n Int.exists_greatest_of_bdd (this d) ⟨⌊L * d⌋, L, hL, Int.floor_le _⟩"
},
{
"state_after": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\n⊢ ∃ x, IsLUB S x",
"state_before": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\n⊢ ∃ x, IsLUB S x",
"tactic": "have hf₁ : ∀ n > 0, ∃ y ∈ S, ((f n / n : ℚ) : ℝ) ≤ y := fun n n0 =>\n let ⟨y, yS, hy⟩ := (hf n).1\n ⟨y, yS, by simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy⟩"
},
{
"state_after": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\n⊢ ∃ x, IsLUB S x",
"state_before": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\n⊢ ∃ x, IsLUB S x",
"tactic": "have hf₂ : ∀ n > 0, ∀ y ∈ S, (y - ((n : ℕ) : ℝ)⁻¹) < (f n / n : ℚ) := by\n intro n n0 y yS\n have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)\n simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]\n rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]\n exact ne_of_gt (Nat.cast_pos.2 n0)"
},
{
"state_after": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\n⊢ ∃ x, IsLUB S x",
"state_before": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\n⊢ ∃ x, IsLUB S x",
"tactic": "have hg : IsCauSeq abs (fun n => f n / n : ℕ → ℚ) := by\n intro ε ε0\n suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by\n refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩\n rw [neg_lt, neg_sub]\n exact this _ le_rfl _ ij\n intro j ij k ik\n replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)\n replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)\n have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)\n have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)\n rcases hf₁ _ j0 with ⟨y, yS, hy⟩\n refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)\n simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)"
},
{
"state_after": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\n⊢ ∃ x, IsLUB S x",
"state_before": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\n⊢ ∃ x, IsLUB S x",
"tactic": "let g : CauSeq ℚ abs := ⟨fun n => f n / n, hg⟩"
},
{
"state_after": "case intro.intro.refine'_1\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\n⊢ x ≤ mk g\n\ncase intro.intro.refine'_2\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\ny : ℝ\nh : y ∈ upperBounds S\n⊢ mk g ≤ y",
"state_before": "case intro.intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\n⊢ ∃ x, IsLUB S x",
"tactic": "refine' ⟨mk g, ⟨fun x xS => _, fun y h => _⟩⟩"
},
{
"state_after": "case intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\n⊢ ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\n⊢ ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}",
"tactic": "cases' exists_int_gt U with k hk"
},
{
"state_after": "case intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\nd : ℕ\nz : ℤ\nh : z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\n⊢ z ≤ k * ↑d",
"state_before": "case intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\n⊢ ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}",
"tactic": "refine' fun d => ⟨k * d, fun z h => _⟩"
},
{
"state_after": "case intro.intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\nd : ℕ\nz : ℤ\ny : ℝ\nyS : y ∈ S\nhy : ↑z ≤ y * ↑d\n⊢ z ≤ k * ↑d",
"state_before": "case intro\nx y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\nd : ℕ\nz : ℤ\nh : z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\n⊢ z ≤ k * ↑d",
"tactic": "rcases h with ⟨y, yS, hy⟩"
},
{
"state_after": "case intro.intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\nd : ℕ\nz : ℤ\ny : ℝ\nyS : y ∈ S\nhy : ↑z ≤ y * ↑d\n⊢ y * ↑d ≤ ↑(k * ↑d)",
"state_before": "case intro.intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\nd : ℕ\nz : ℤ\ny : ℝ\nyS : y ∈ S\nhy : ↑z ≤ y * ↑d\n⊢ z ≤ k * ↑d",
"tactic": "refine' Int.cast_le.1 (hy.trans _)"
},
{
"state_after": "case intro.intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\nd : ℕ\nz : ℤ\ny : ℝ\nyS : y ∈ S\nhy : ↑z ≤ y * ↑d\n⊢ y * ↑d ≤ ↑k * ↑d",
"state_before": "case intro.intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\nd : ℕ\nz : ℤ\ny : ℝ\nyS : y ∈ S\nhy : ↑z ≤ y * ↑d\n⊢ y * ↑d ≤ ↑(k * ↑d)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nk : ℤ\nhk : U < ↑k\nd : ℕ\nz : ℤ\ny : ℝ\nyS : y ∈ S\nhy : ↑z ≤ y * ↑d\n⊢ y * ↑d ≤ ↑k * ↑d",
"tactic": "exact mul_le_mul_of_nonneg_right ((hU yS).trans hk.le) d.cast_nonneg"
},
{
"state_after": "no goals",
"state_before": "x y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ S\nhy : ↑(f n) ≤ y * ↑n\n⊢ ↑(↑(f n) / ↑n) ≤ y",
"tactic": "simpa using (div_le_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _)).2 hy"
},
{
"state_after": "x y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ S\n⊢ y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\n⊢ ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)",
"tactic": "intro n n0 y yS"
},
{
"state_after": "x y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ S\nthis : y * ↑n - 1 < ↑(f n)\n⊢ y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)",
"state_before": "x y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ S\n⊢ y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)",
"tactic": "have := (Int.sub_one_lt_floor _).trans_le (Int.cast_le.2 <| (hf n).2 _ ⟨y, yS, Int.floor_le _⟩)"
},
{
"state_after": "x y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ S\nthis : y * ↑n - 1 < ↑(f n)\n⊢ y - (↑n)⁻¹ < ↑(f n) / ↑n",
"state_before": "x y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ S\nthis : y * ↑n - 1 < ↑(f n)\n⊢ y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)",
"tactic": "simp only [Rat.cast_div, Rat.cast_coe_int, Rat.cast_coe_nat, gt_iff_lt]"
},
{
"state_after": "x y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ S\nthis : y * ↑n - 1 < ↑(f n)\n⊢ ↑n ≠ 0",
"state_before": "x y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ S\nthis : y * ↑n - 1 < ↑(f n)\n⊢ y - (↑n)⁻¹ < ↑(f n) / ↑n",
"tactic": "rwa [lt_div_iff (Nat.cast_pos.2 n0 : (_ : ℝ) < _), sub_mul, _root_.inv_mul_cancel]"
},
{
"state_after": "no goals",
"state_before": "x y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ S\nthis : y * ↑n - 1 < ↑(f n)\n⊢ ↑n ≠ 0",
"tactic": "exact ne_of_gt (Nat.cast_pos.2 n0)"
},
{
"state_after": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs ((fun n => ↑(f n) / ↑n) j - (fun n => ↑(f n) / ↑n) i) < ε",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\n⊢ IsCauSeq abs fun n => ↑(f n) / ↑n",
"tactic": "intro ε ε0"
},
{
"state_after": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\n⊢ ∀ (j : ℕ), j ≥ ⌈ε⁻¹⌉₊ → ∀ (k : ℕ), k ≥ ⌈ε⁻¹⌉₊ → ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs ((fun n => ↑(f n) / ↑n) j - (fun n => ↑(f n) / ↑n) i) < ε",
"tactic": "suffices ∀ j ≥ ⌈ε⁻¹⌉₊, ∀ k ≥ ⌈ε⁻¹⌉₊, (f j / j - f k / k : ℚ) < ε by\n refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩\n rw [neg_lt, neg_sub]\n exact this _ le_rfl _ ij"
},
{
"state_after": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj : ℕ\nij : j ≥ ⌈ε⁻¹⌉₊\nk : ℕ\nik : k ≥ ⌈ε⁻¹⌉₊\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\n⊢ ∀ (j : ℕ), j ≥ ⌈ε⁻¹⌉₊ → ∀ (k : ℕ), k ≥ ⌈ε⁻¹⌉₊ → ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"tactic": "intro j ij k ik"
},
{
"state_after": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nik : k ≥ ⌈ε⁻¹⌉₊\nij : ε⁻¹ ≤ ↑j\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj : ℕ\nij : j ≥ ⌈ε⁻¹⌉₊\nk : ℕ\nik : k ≥ ⌈ε⁻¹⌉₊\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"tactic": "replace ij := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ij)"
},
{
"state_after": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nik : k ≥ ⌈ε⁻¹⌉₊\nij : ε⁻¹ ≤ ↑j\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"tactic": "replace ik := le_trans (Nat.le_ceil _) (Nat.cast_le.2 ik)"
},
{
"state_after": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\nj0 : 0 < j\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"tactic": "have j0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ij)"
},
{
"state_after": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\nj0 : 0 < j\nk0 : 0 < k\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\nj0 : 0 < j\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"tactic": "have k0 := Nat.cast_pos.1 ((inv_pos.2 ε0).trans_le ik)"
},
{
"state_after": "case intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\nj0 : 0 < j\nk0 : 0 < k\ny : ℝ\nyS : y ∈ S\nhy : ↑(↑(f j) / ↑j) ≤ y\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\nj0 : 0 < j\nk0 : 0 < k\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"tactic": "rcases hf₁ _ j0 with ⟨y, yS, hy⟩"
},
{
"state_after": "case intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\nj0 : 0 < j\nk0 : 0 < k\ny : ℝ\nyS : y ∈ S\nhy : ↑(↑(f j) / ↑j) ≤ y\n⊢ ↑(↑(f j) / ↑j - ↑(f k) / ↑k) < ↑(↑k)⁻¹",
"state_before": "case intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\nj0 : 0 < j\nk0 : 0 < k\ny : ℝ\nyS : y ∈ S\nhy : ↑(↑(f j) / ↑j) ≤ y\n⊢ ↑(f j) / ↑j - ↑(f k) / ↑k < ε",
"tactic": "refine' lt_of_lt_of_le ((@Rat.cast_lt ℝ _ _ _).1 _) ((inv_le ε0 (Nat.cast_pos.2 k0)).1 ik)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nj k : ℕ\nij : ε⁻¹ ≤ ↑j\nik : ε⁻¹ ≤ ↑k\nj0 : 0 < j\nk0 : 0 < k\ny : ℝ\nyS : y ∈ S\nhy : ↑(↑(f j) / ↑j) ≤ y\n⊢ ↑(↑(f j) / ↑j - ↑(f k) / ↑k) < ↑(↑k)⁻¹",
"tactic": "simpa using sub_lt_iff_lt_add'.2 (lt_of_le_of_lt hy <| sub_lt_iff_lt_add.1 <| hf₂ _ k0 _ yS)"
},
{
"state_after": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nthis : ∀ (j : ℕ), j ≥ ⌈ε⁻¹⌉₊ → ∀ (k : ℕ), k ≥ ⌈ε⁻¹⌉₊ → ↑(f j) / ↑j - ↑(f k) / ↑k < ε\nj : ℕ\nij : j ≥ ⌈ε⁻¹⌉₊\n⊢ -ε < (fun n => ↑(f n) / ↑n) j - (fun n => ↑(f n) / ↑n) ⌈ε⁻¹⌉₊",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nthis : ∀ (j : ℕ), j ≥ ⌈ε⁻¹⌉₊ → ∀ (k : ℕ), k ≥ ⌈ε⁻¹⌉₊ → ↑(f j) / ↑j - ↑(f k) / ↑k < ε\n⊢ ∃ i, ∀ (j : ℕ), j ≥ i → abs ((fun n => ↑(f n) / ↑n) j - (fun n => ↑(f n) / ↑n) i) < ε",
"tactic": "refine' ⟨_, fun j ij => abs_lt.2 ⟨_, this _ ij _ le_rfl⟩⟩"
},
{
"state_after": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nthis : ∀ (j : ℕ), j ≥ ⌈ε⁻¹⌉₊ → ∀ (k : ℕ), k ≥ ⌈ε⁻¹⌉₊ → ↑(f j) / ↑j - ↑(f k) / ↑k < ε\nj : ℕ\nij : j ≥ ⌈ε⁻¹⌉₊\n⊢ (fun n => ↑(f n) / ↑n) ⌈ε⁻¹⌉₊ - (fun n => ↑(f n) / ↑n) j < ε",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nthis : ∀ (j : ℕ), j ≥ ⌈ε⁻¹⌉₊ → ∀ (k : ℕ), k ≥ ⌈ε⁻¹⌉₊ → ↑(f j) / ↑j - ↑(f k) / ↑k < ε\nj : ℕ\nij : j ≥ ⌈ε⁻¹⌉₊\n⊢ -ε < (fun n => ↑(f n) / ↑n) j - (fun n => ↑(f n) / ↑n) ⌈ε⁻¹⌉₊",
"tactic": "rw [neg_lt, neg_sub]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis✝ : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nε : ℚ\nε0 : ε > 0\nthis : ∀ (j : ℕ), j ≥ ⌈ε⁻¹⌉₊ → ∀ (k : ℕ), k ≥ ⌈ε⁻¹⌉₊ → ↑(f j) / ↑j - ↑(f k) / ↑k < ε\nj : ℕ\nij : j ≥ ⌈ε⁻¹⌉₊\n⊢ (fun n => ↑(f n) / ↑n) ⌈ε⁻¹⌉₊ - (fun n => ↑(f n) / ↑n) j < ε",
"tactic": "exact this _ le_rfl _ ij"
},
{
"state_after": "case intro.intro.refine'_1\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nxz : z < x\n⊢ z ≤ mk g",
"state_before": "case intro.intro.refine'_1\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\n⊢ x ≤ mk g",
"tactic": "refine' le_of_forall_ge_of_dense fun z xz => _"
},
{
"state_after": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nxz : z < x\nK : ℕ\nhK : (x - z)⁻¹ < ↑K\n⊢ z ≤ mk g",
"state_before": "case intro.intro.refine'_1\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nxz : z < x\n⊢ z ≤ mk g",
"tactic": "cases' exists_nat_gt (x - z)⁻¹ with K hK"
},
{
"state_after": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nxz : z < x\nK : ℕ\nhK : (x - z)⁻¹ < ↑K\nn : ℕ\nnK : n ≥ K\n⊢ z ≤ ↑(↑g n)",
"state_before": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nxz : z < x\nK : ℕ\nhK : (x - z)⁻¹ < ↑K\n⊢ z ≤ mk g",
"tactic": "refine' le_mk_of_forall_le ⟨K, fun n nK => _⟩"
},
{
"state_after": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nK : ℕ\nhK : (x - z)⁻¹ < ↑K\nn : ℕ\nnK : n ≥ K\nxz : 0 < x - z\n⊢ z ≤ ↑(↑g n)",
"state_before": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nxz : z < x\nK : ℕ\nhK : (x - z)⁻¹ < ↑K\nn : ℕ\nnK : n ≥ K\n⊢ z ≤ ↑(↑g n)",
"tactic": "replace xz := sub_pos.2 xz"
},
{
"state_after": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nK n : ℕ\nnK : n ≥ K\nxz : 0 < x - z\nhK : (x - z)⁻¹ ≤ ↑n\n⊢ z ≤ ↑(↑g n)",
"state_before": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nK : ℕ\nhK : (x - z)⁻¹ < ↑K\nn : ℕ\nnK : n ≥ K\nxz : 0 < x - z\n⊢ z ≤ ↑(↑g n)",
"tactic": "replace hK := hK.le.trans (Nat.cast_le.2 nK)"
},
{
"state_after": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nK n : ℕ\nnK : n ≥ K\nxz : 0 < x - z\nhK : (x - z)⁻¹ ≤ ↑n\nn0 : 0 < n\n⊢ z ≤ ↑(↑g n)",
"state_before": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nK n : ℕ\nnK : n ≥ K\nxz : 0 < x - z\nhK : (x - z)⁻¹ ≤ ↑n\n⊢ z ≤ ↑(↑g n)",
"tactic": "have n0 : 0 < n := Nat.cast_pos.1 ((inv_pos.2 xz).trans_le hK)"
},
{
"state_after": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nK n : ℕ\nnK : n ≥ K\nxz : 0 < x - z\nhK : (x - z)⁻¹ ≤ ↑n\nn0 : 0 < n\n⊢ z ≤ x - (↑n)⁻¹",
"state_before": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nK n : ℕ\nnK : n ≥ K\nxz : 0 < x - z\nhK : (x - z)⁻¹ ≤ ↑n\nn0 : 0 < n\n⊢ z ≤ ↑(↑g n)",
"tactic": "refine' le_trans _ (hf₂ _ n0 _ xS).le"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1.intro\nx✝ y : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\nx : ℝ\nxS : x ∈ S\nz : ℝ\nK n : ℕ\nnK : n ≥ K\nxz : 0 < x - z\nhK : (x - z)⁻¹ ≤ ↑n\nn0 : 0 < n\n⊢ z ≤ x - (↑n)⁻¹",
"tactic": "rwa [le_sub_comm, inv_le (Nat.cast_pos.2 n0 : (_ : ℝ) < _) xz]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2\nx y✝ : ℝ\nS : Set ℝ\nL : ℝ\nhL : L ∈ S\nU : ℝ\nhU : U ∈ upperBounds S\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} ∧ ∀ (z : ℤ), z ∈ {m | ∃ y, y ∈ S ∧ ↑m ≤ y * ↑d} → z ≤ f d\nhf₁ : ∀ (n : ℕ), n > 0 → ∃ y, y ∈ S ∧ ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ (n : ℕ), n > 0 → ∀ (y : ℝ), y ∈ S → y - (↑n)⁻¹ < ↑(↑(f n) / ↑n)\nhg : IsCauSeq abs fun n => ↑(f n) / ↑n\ng : CauSeq ℚ abs := { val := fun n => ↑(f n) / ↑n, property := hg }\ny : ℝ\nh : y ∈ upperBounds S\n⊢ mk g ≤ y",
"tactic": "exact\n mk_le_of_forall_le\n ⟨1, fun n n1 =>\n let ⟨x, xS, hx⟩ := hf₁ _ n1\n le_trans hx (h xS)⟩"
}
] |
[
730,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
682,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
MeasureTheory.AEEqFun.toGerm_injective
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.443960\nδ : Type ?u.443963\ninst✝³ : MeasurableSpace α\nμ ν : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf g : α →ₘ[μ] β\nH : toGerm f = toGerm g\n⊢ ↑↑f = ↑↑g",
"tactic": "rwa [← toGerm_eq, ← toGerm_eq]"
}
] |
[
387,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Dynamics/Ergodic/Ergodic.lean
|
Ergodic.ae_empty_or_univ_of_ae_le_preimage'
|
[
{
"state_after": "case h_fin\nα : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : MeasureTheory.Measure α\nhf : Ergodic f\nhs : MeasurableSet s\nhs' : s ≤ᶠ[ae μ] f ⁻¹' s\nh_fin : ↑↑μ s ≠ ⊤\n⊢ ↑↑μ (f ⁻¹' s) ≠ ⊤\n\nα : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : MeasureTheory.Measure α\nhf : Ergodic f\nhs : MeasurableSet s\nhs' : s ≤ᶠ[ae μ] f ⁻¹' s\nh_fin : ↑↑μ (f ⁻¹' s) ≠ ⊤\n⊢ s =ᶠ[ae μ] ∅ ∨ s =ᶠ[ae μ] univ",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : MeasureTheory.Measure α\nhf : Ergodic f\nhs : MeasurableSet s\nhs' : s ≤ᶠ[ae μ] f ⁻¹' s\nh_fin : ↑↑μ s ≠ ⊤\n⊢ s =ᶠ[ae μ] ∅ ∨ s =ᶠ[ae μ] univ",
"tactic": "replace h_fin : μ (f ⁻¹' s) ≠ ∞"
},
{
"state_after": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : MeasureTheory.Measure α\nhf : Ergodic f\nhs : MeasurableSet s\nhs' : s ≤ᶠ[ae μ] f ⁻¹' s\nh_fin : ↑↑μ (f ⁻¹' s) ≠ ⊤\n⊢ f ⁻¹' s =ᶠ[ae μ] s",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : MeasureTheory.Measure α\nhf : Ergodic f\nhs : MeasurableSet s\nhs' : s ≤ᶠ[ae μ] f ⁻¹' s\nh_fin : ↑↑μ (f ⁻¹' s) ≠ ⊤\n⊢ s =ᶠ[ae μ] ∅ ∨ s =ᶠ[ae μ] univ",
"tactic": "refine' hf.quasiErgodic.ae_empty_or_univ' hs _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : MeasureTheory.Measure α\nhf : Ergodic f\nhs : MeasurableSet s\nhs' : s ≤ᶠ[ae μ] f ⁻¹' s\nh_fin : ↑↑μ (f ⁻¹' s) ≠ ⊤\n⊢ f ⁻¹' s =ᶠ[ae μ] s",
"tactic": "exact (ae_eq_of_ae_subset_of_measure_ge hs' (hf.measure_preimage hs).le hs h_fin).symm"
},
{
"state_after": "no goals",
"state_before": "case h_fin\nα : Type u_1\nm : MeasurableSpace α\nf : α → α\ns : Set α\nμ : MeasureTheory.Measure α\nhf : Ergodic f\nhs : MeasurableSet s\nhs' : s ≤ᶠ[ae μ] f ⁻¹' s\nh_fin : ↑↑μ s ≠ ⊤\n⊢ ↑↑μ (f ⁻¹' s) ≠ ⊤",
"tactic": "rwa [hf.measure_preimage hs]"
}
] |
[
151,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.smul_mulVec_assoc
|
[
{
"state_after": "case h\nl : Type ?u.851375\nm : Type u_3\nn : Type u_1\no : Type ?u.851384\nm' : o → Type ?u.851389\nn' : o → Type ?u.851394\nR : Type u_2\nS : Type ?u.851400\nα : Type v\nβ : Type w\nγ : Type ?u.851407\ninst✝⁴ : NonUnitalNonAssocSemiring α\ninst✝³ : Fintype n\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R α\ninst✝ : IsScalarTower R α α\na : R\nA : Matrix m n α\nb : n → α\nx✝ : m\n⊢ mulVec (a • A) b x✝ = (a • mulVec A b) x✝",
"state_before": "l : Type ?u.851375\nm : Type u_3\nn : Type u_1\no : Type ?u.851384\nm' : o → Type ?u.851389\nn' : o → Type ?u.851394\nR : Type u_2\nS : Type ?u.851400\nα : Type v\nβ : Type w\nγ : Type ?u.851407\ninst✝⁴ : NonUnitalNonAssocSemiring α\ninst✝³ : Fintype n\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R α\ninst✝ : IsScalarTower R α α\na : R\nA : Matrix m n α\nb : n → α\n⊢ mulVec (a • A) b = a • mulVec A b",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nl : Type ?u.851375\nm : Type u_3\nn : Type u_1\no : Type ?u.851384\nm' : o → Type ?u.851389\nn' : o → Type ?u.851394\nR : Type u_2\nS : Type ?u.851400\nα : Type v\nβ : Type w\nγ : Type ?u.851407\ninst✝⁴ : NonUnitalNonAssocSemiring α\ninst✝³ : Fintype n\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R α\ninst✝ : IsScalarTower R α α\na : R\nA : Matrix m n α\nb : n → α\nx✝ : m\n⊢ mulVec (a • A) b x✝ = (a • mulVec A b) x✝",
"tactic": "apply smul_dotProduct"
}
] |
[
1745,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1742,
1
] |
Mathlib/Order/BoundedOrder.lean
|
subsingleton_of_bot_eq_top
|
[] |
[
683,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
682,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.subset_vsub
|
[] |
[
1560,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1558,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Ioo_mem_nhdsWithin_Ioi'
|
[] |
[
403,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Continuous.if_le
|
[] |
[
663,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/Algebra/Ring/Regular.lean
|
isRegular_iff_ne_zero'
|
[
{
"state_after": "α : Type u_1\ninst✝² : Nontrivial α\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : NoZeroDivisors α\nh : IsRegular 0\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : Nontrivial α\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : NoZeroDivisors α\nk : α\nh : IsRegular k\n⊢ k ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : Nontrivial α\ninst✝¹ : NonUnitalNonAssocRing α\ninst✝ : NoZeroDivisors α\nh : IsRegular 0\n⊢ False",
"tactic": "exact not_not.mpr h.left not_isLeftRegular_zero"
}
] |
[
49,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.isDiag_iff_mem_range_diag
|
[] |
[
461,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
460,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.prod_mono_set'
|
[] |
[
414,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/ModelTheory/Substructures.lean
|
FirstOrder.Language.Hom.comp_codRestrict
|
[] |
[
813,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
811,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.sub_top
|
[] |
[
1108,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1108,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Adjunction.homEquiv_apply_eq
|
[
{
"state_after": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A ⟶ B\n⊢ f = ↑(homEquiv adj A B).symm (↑(homEquiv adj A B) f)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A ⟶ B\ng : A ⟶ G.obj B\nh : ↑(homEquiv adj A B) f = g\n⊢ f = ↑(homEquiv adj A B).symm g",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A ⟶ B\n⊢ f = ↑(homEquiv adj A B).symm (↑(homEquiv adj A B) f)",
"tactic": "simp"
},
{
"state_after": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\ng : A ⟶ G.obj B\n⊢ ↑(homEquiv adj A B) (↑(homEquiv adj A B).symm g) = g",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\nf : F.obj A ⟶ B\ng : A ⟶ G.obj B\nh : f = ↑(homEquiv adj A B).symm g\n⊢ ↑(homEquiv adj A B) f = g",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nA : C\nB : D\ng : A ⟶ G.obj B\n⊢ ↑(homEquiv adj A B) (↑(homEquiv adj A B).symm g) = g",
"tactic": "simp"
}
] |
[
223,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Logic/Embedding/Basic.lean
|
Function.Embedding.coe_quotientOut
|
[] |
[
246,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Data/PNat/Xgcd.lean
|
PNat.XgcdType.reduce_isReduced'
|
[] |
[
374,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
LinearMap.toSpanSingleton_one
|
[] |
[
927,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
926,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
BoundedContinuousFunction.coeFn_toLp
|
[] |
[
1656,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1654,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.iSup_div
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nf✝ f : ι → ℝ≥0\na : ℝ≥0\n⊢ (⨆ (i : ι), f i) / a = ⨆ (i : ι), f i / a",
"tactic": "simp only [div_eq_mul_inv, iSup_mul]"
}
] |
[
973,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
972,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.closure_mono
|
[] |
[
430,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.ratio_le_opNorm
|
[] |
[
278,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/RingTheory/ReesAlgebra.lean
|
adjoin_monomial_eq_reesAlgebra
|
[
{
"state_after": "case a\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\n⊢ Algebra.adjoin R ↑(Submodule.map (monomial 1) I) ≤ reesAlgebra I\n\ncase a\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\n⊢ reesAlgebra I ≤ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\n⊢ Algebra.adjoin R ↑(Submodule.map (monomial 1) I) = reesAlgebra I",
"tactic": "apply le_antisymm"
},
{
"state_after": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\n⊢ ↑(Submodule.map (monomial 1) I) ⊆ ↑(reesAlgebra I)",
"state_before": "case a\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\n⊢ Algebra.adjoin R ↑(Submodule.map (monomial 1) I) ≤ reesAlgebra I",
"tactic": "apply Algebra.adjoin_le _"
},
{
"state_after": "case intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nr : R\nhr : r ∈ ↑I\n⊢ ↑(monomial 1) r ∈ ↑(reesAlgebra I)",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\n⊢ ↑(Submodule.map (monomial 1) I) ⊆ ↑(reesAlgebra I)",
"tactic": "rintro _ ⟨r, hr, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nr : R\nhr : r ∈ ↑I\n⊢ ↑(monomial 1) r ∈ ↑(reesAlgebra I)",
"tactic": "exact reesAlgebra.monomial_mem.mpr (by rwa [pow_one])"
},
{
"state_after": "no goals",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nr : R\nhr : r ∈ ↑I\n⊢ r ∈ I ^ 1",
"tactic": "rwa [pow_one]"
},
{
"state_after": "case a\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\np : R[X]\nhp : p ∈ reesAlgebra I\n⊢ p ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"state_before": "case a\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\n⊢ reesAlgebra I ≤ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"tactic": "intro p hp"
},
{
"state_after": "case a\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\np : R[X]\nhp : p ∈ reesAlgebra I\n⊢ ∑ i in support p, ↑(monomial i) (coeff p i) ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"state_before": "case a\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\np : R[X]\nhp : p ∈ reesAlgebra I\n⊢ p ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"tactic": "rw [p.as_sum_support]"
},
{
"state_after": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\np : R[X]\nhp : p ∈ reesAlgebra I\n⊢ ∀ (x : ℕ), x ∈ support p → ↑(monomial x) (coeff p x) ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"state_before": "case a\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\np : R[X]\nhp : p ∈ reesAlgebra I\n⊢ ∑ i in support p, ↑(monomial i) (coeff p i) ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"tactic": "apply Subalgebra.sum_mem _ _"
},
{
"state_after": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\np : R[X]\nhp : p ∈ reesAlgebra I\ni : ℕ\n⊢ ↑(monomial i) (coeff p i) ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\np : R[X]\nhp : p ∈ reesAlgebra I\n⊢ ∀ (x : ℕ), x ∈ support p → ↑(monomial x) (coeff p x) ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"tactic": "rintro i -"
},
{
"state_after": "no goals",
"state_before": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\np : R[X]\nhp : p ∈ reesAlgebra I\ni : ℕ\n⊢ ↑(monomial i) (coeff p i) ∈ Algebra.adjoin R ↑(Submodule.map (monomial 1) I)",
"tactic": "exact monomial_mem_adjoin_monomial (hp i)"
}
] |
[
111,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/RingTheory/WittVector/Defs.lean
|
WittVector.constantCoeff_wittMul
|
[
{
"state_after": "p : ℕ\nR : Type ?u.121832\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nn : ℕ\n⊢ ↑constantCoeff (X 0 * X 1) = 0",
"state_before": "p : ℕ\nR : Type ?u.121832\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nn : ℕ\n⊢ ↑constantCoeff (wittMul p n) = 0",
"tactic": "apply constantCoeff_wittStructureInt p _ _ n"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type ?u.121832\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R\nn : ℕ\n⊢ ↑constantCoeff (X 0 * X 1) = 0",
"tactic": "simp only [MulZeroClass.mul_zero, RingHom.map_mul, constantCoeff_X]"
}
] |
[
308,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.integral_add_compl
|
[] |
[
167,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sup_sInf_le_iInf_sup
|
[] |
[
1922,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1921,
1
] |
src/lean/Init/Control/StateCps.lean
|
StateCpsT.runK_set
|
[] |
[
55,
139
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
55,
9
] |
Mathlib/RingTheory/Subring/Pointwise.lean
|
Subring.smul_mem_pointwise_smul_iff
|
[] |
[
114,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.norm_of_nonneg
|
[] |
[
692,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
691,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.Equiv.refl
|
[] |
[
608,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
607,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Products.lean
|
CategoryTheory.Limits.fan_mk_proj
|
[] |
[
91,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Data/Nat/Size.lean
|
Nat.shiftl'_ne_zero_left
|
[
{
"state_after": "no goals",
"state_before": "b : Bool\nm : ℕ\nh : m ≠ 0\nn : ℕ\n⊢ shiftl' b m n ≠ 0",
"tactic": "induction n <;> simp [bit_ne_zero, shiftl', *]"
}
] |
[
63,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
Submodule.existsUnique_from_graph
|
[
{
"state_after": "case refine'_1\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\n⊢ ∃ x, (a, x) ∈ g\n\ncase refine'_2\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\n⊢ ∀ (y₁ y₂ : F), (a, y₁) ∈ g → (a, y₂) ∈ g → y₁ = y₂",
"state_before": "R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\n⊢ ∃! b, (a, b) ∈ g",
"tactic": "refine' exists_unique_of_exists_of_unique _ _"
},
{
"state_after": "case refine'_2\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ y₂ : F\nhy₁ : (a, y₁) ∈ g\nhy₂ : (a, y₂) ∈ g\n⊢ y₁ = y₂",
"state_before": "case refine'_2\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\n⊢ ∀ (y₁ y₂ : F), (a, y₁) ∈ g → (a, y₂) ∈ g → y₁ = y₂",
"tactic": "intro y₁ y₂ hy₁ hy₂"
},
{
"state_after": "case refine'_2\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ y₂ : F\nhy₁ : (a, y₁) ∈ g\nhy₂ : (a, y₂) ∈ g\nhy : (0, y₁ - y₂) ∈ g\n⊢ y₁ = y₂",
"state_before": "case refine'_2\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ y₂ : F\nhy₁ : (a, y₁) ∈ g\nhy₂ : (a, y₂) ∈ g\n⊢ y₁ = y₂",
"tactic": "have hy : ((0 : E), y₁ - y₂) ∈ g := by\n convert g.sub_mem hy₁ hy₂\n exact (sub_self _).symm"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ y₂ : F\nhy₁ : (a, y₁) ∈ g\nhy₂ : (a, y₂) ∈ g\nhy : (0, y₁ - y₂) ∈ g\n⊢ y₁ = y₂",
"tactic": "exact sub_eq_zero.mp (hg hy (by simp))"
},
{
"state_after": "case a\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\n⊢ (∃ x, (a, x) ∈ g) ↔ a ∈ map (LinearMap.fst R E F) g",
"state_before": "case refine'_1\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\n⊢ ∃ x, (a, x) ∈ g",
"tactic": "convert ha"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\n⊢ (∃ x, (a, x) ∈ g) ↔ a ∈ map (LinearMap.fst R E F) g",
"tactic": "simp"
},
{
"state_after": "case h.e'_4.h.e'_3\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ y₂ : F\nhy₁ : (a, y₁) ∈ g\nhy₂ : (a, y₂) ∈ g\n⊢ 0 = ((a, y₁) - (a, y₂)).fst",
"state_before": "R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ y₂ : F\nhy₁ : (a, y₁) ∈ g\nhy₂ : (a, y₂) ∈ g\n⊢ (0, y₁ - y₂) ∈ g",
"tactic": "convert g.sub_mem hy₁ hy₂"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.e'_3\nR : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ y₂ : F\nhy₁ : (a, y₁) ∈ g\nhy₂ : (a, y₂) ∈ g\n⊢ 0 = ((a, y₁) - (a, y₂)).fst",
"tactic": "exact (sub_self _).symm"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.618707\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ {x : E × F}, x ∈ g → x.fst = 0 → x.snd = 0\na : E\nha : a ∈ map (LinearMap.fst R E F) g\ny₁ y₂ : F\nhy₁ : (a, y₁) ∈ g\nhy₂ : (a, y₂) ∈ g\nhy : (0, y₁ - y₂) ∈ g\n⊢ (0, y₁ - y₂).fst = 0",
"tactic": "simp"
}
] |
[
936,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
926,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
HasFDerivAt.lim
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\n⊢ Tendsto (fun n => c n • (c n)⁻¹ • v) l (𝓝 v)",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\n⊢ Tendsto (fun n => c n • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (↑f' v))",
"tactic": "refine' (hasFDerivWithinAt_univ.2 hf).lim _ univ_mem hc _"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\nU : Set E\nhU : U ∈ 𝓝 v\n⊢ U ∈ map (fun n => c n • (c n)⁻¹ • v) l",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\n⊢ Tendsto (fun n => c n • (c n)⁻¹ • v) l (𝓝 v)",
"tactic": "intro U hU"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\nU : Set E\nhU : U ∈ 𝓝 v\ny : α\nhy : c y ≠ 0\n⊢ (fun n => c n • (c n)⁻¹ • v) y ∈ U",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\nU : Set E\nhU : U ∈ 𝓝 v\n⊢ U ∈ map (fun n => c n • (c n)⁻¹ • v) l",
"tactic": "refine' (eventually_ne_of_tendsto_norm_atTop hc (0 : 𝕜)).mono fun y hy => _"
},
{
"state_after": "case h.e'_4\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\nU : Set E\nhU : U ∈ 𝓝 v\ny : α\nhy : c y ≠ 0\n⊢ (fun n => c n • (c n)⁻¹ • v) y = v",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\nU : Set E\nhU : U ∈ 𝓝 v\ny : α\nhy : c y ≠ 0\n⊢ (fun n => c n • (c n)⁻¹ • v) y ∈ U",
"tactic": "convert mem_of_mem_nhds hU"
},
{
"state_after": "case h.e'_4\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\nU : Set E\nhU : U ∈ 𝓝 v\ny : α\nhy : c y ≠ 0\n⊢ c y • (c y)⁻¹ • v = v",
"state_before": "case h.e'_4\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\nU : Set E\nhU : U ∈ 𝓝 v\ny : α\nhy : c y ≠ 0\n⊢ (fun n => c n • (c n)⁻¹ • v) y = v",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.219286\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.219381\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nhf : HasFDerivAt f f' x\nv : E\nα : Type u_4\nc : α → 𝕜\nl : Filter α\nhc : Tendsto (fun n => ‖c n‖) l atTop\nU : Set E\nhU : U ∈ 𝓝 v\ny : α\nhy : c y ≠ 0\n⊢ c y • (c y)⁻¹ • v = v",
"tactic": "rw [← mul_smul, mul_inv_cancel hy, one_smul]"
}
] |
[
469,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biproduct.matrix_π
|
[
{
"state_after": "case w\nJ : Type\ninst✝⁴ : Fintype J\nK : Type\ninst✝³ : Fintype K\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasFiniteBiproducts C\nf : J → C\ng : K → C\nm : (j : J) → (k : K) → f j ⟶ g k\nk : K\nj✝ : J\n⊢ ι (fun j => f j) j✝ ≫ matrix m ≫ π g k = ι (fun j => f j) j✝ ≫ desc fun j => m j k",
"state_before": "J : Type\ninst✝⁴ : Fintype J\nK : Type\ninst✝³ : Fintype K\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasFiniteBiproducts C\nf : J → C\ng : K → C\nm : (j : J) → (k : K) → f j ⟶ g k\nk : K\n⊢ matrix m ≫ π g k = desc fun j => m j k",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w\nJ : Type\ninst✝⁴ : Fintype J\nK : Type\ninst✝³ : Fintype K\nC : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasFiniteBiproducts C\nf : J → C\ng : K → C\nm : (j : J) → (k : K) → f j ⟶ g k\nk : K\nj✝ : J\n⊢ ι (fun j => f j) j✝ ≫ matrix m ≫ π g k = ι (fun j => f j) j✝ ≫ desc fun j => m j k",
"tactic": "simp [biproduct.matrix]"
}
] |
[
833,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
830,
1
] |
Mathlib/CategoryTheory/MorphismProperty.lean
|
CategoryTheory.MorphismProperty.StableUnderComposition.diagonal
|
[
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type ?u.77182\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh₁ : MorphismProperty.diagonal P f\nh₂ : MorphismProperty.diagonal P g\n⊢ MorphismProperty.diagonal P (f ≫ g)",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.77182\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\n⊢ StableUnderComposition (MorphismProperty.diagonal P)",
"tactic": "introv X h₁ h₂"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nD : Type ?u.77182\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh₁ : MorphismProperty.diagonal P f\nh₂ : MorphismProperty.diagonal P g\n⊢ P (pullback.diagonal f ≫ (pullbackDiagonalMapIdIso f f g).inv ≫ pullback.snd)",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.77182\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh₁ : MorphismProperty.diagonal P f\nh₂ : MorphismProperty.diagonal P g\n⊢ MorphismProperty.diagonal P (f ≫ g)",
"tactic": "rw [diagonal_iff, pullback.diagonal_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.77182\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh₁ : MorphismProperty.diagonal P f\nh₂ : MorphismProperty.diagonal P g\n⊢ P (pullback.diagonal f ≫ (pullbackDiagonalMapIdIso f f g).inv ≫ pullback.snd)",
"tactic": "exact hP _ _ h₁ (by simpa [hP'.cancel_left_isIso] using hP''.snd _ _ h₂)"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nD : Type ?u.77182\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nP : MorphismProperty C\nhP : StableUnderComposition P\nhP' : RespectsIso P\nhP'' : StableUnderBaseChange P\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nh₁ : MorphismProperty.diagonal P f\nh₂ : MorphismProperty.diagonal P g\n⊢ P ((pullbackDiagonalMapIdIso f f g).inv ≫ pullback.snd)",
"tactic": "simpa [hP'.cancel_left_isIso] using hP''.snd _ _ h₂"
}
] |
[
546,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingHom.cancel_right
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.48360\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.48372\ninst✝⁷ : NonAssocSemiring α\ninst✝⁶ : Preorder α\ninst✝⁵ : NonAssocSemiring β\ninst✝⁴ : Preorder β\ninst✝³ : NonAssocSemiring γ\ninst✝² : Preorder γ\ninst✝¹ : NonAssocSemiring δ\ninst✝ : Preorder δ\nf₁ f₂ : β →+*o γ\ng : α →+*o β\nhg : Surjective ↑g\nh : f₁ = f₂\n⊢ OrderRingHom.comp f₁ g = OrderRingHom.comp f₂ g",
"tactic": "rw [h]"
}
] |
[
334,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Data/Real/Irrational.lean
|
irrational_nat_sub_iff
|
[] |
[
573,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
572,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_multiset_count_of_subset
|
[
{
"state_after": "ι : Type ?u.595257\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : Multiset α\ns : Finset α\n⊢ Multiset.toFinset m ⊆ s → Multiset.prod m = ∏ i in s, i ^ Multiset.count i m",
"state_before": "ι : Type ?u.595257\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : Multiset α\ns : Finset α\nhs : Multiset.toFinset m ⊆ s\n⊢ Multiset.prod m = ∏ i in s, i ^ Multiset.count i m",
"tactic": "revert hs"
},
{
"state_after": "ι : Type ?u.595257\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : Multiset α\ns : Finset α\nl : List α\n⊢ Multiset.toFinset (Quot.mk Setoid.r l) ⊆ s →\n Multiset.prod (Quot.mk Setoid.r l) = ∏ i in s, i ^ Multiset.count i (Quot.mk Setoid.r l)",
"state_before": "ι : Type ?u.595257\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : Multiset α\ns : Finset α\n⊢ Multiset.toFinset m ⊆ s → Multiset.prod m = ∏ i in s, i ^ Multiset.count i m",
"tactic": "refine' Quot.induction_on m fun l => _"
},
{
"state_after": "ι : Type ?u.595257\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : Multiset α\ns : Finset α\nl : List α\n⊢ Multiset.toFinset ↑l ⊆ s → List.prod l = ∏ x in s, x ^ List.count x l",
"state_before": "ι : Type ?u.595257\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : Multiset α\ns : Finset α\nl : List α\n⊢ Multiset.toFinset (Quot.mk Setoid.r l) ⊆ s →\n Multiset.prod (Quot.mk Setoid.r l) = ∏ i in s, i ^ Multiset.count i (Quot.mk Setoid.r l)",
"tactic": "simp only [quot_mk_to_coe'', coe_prod, coe_count]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.595257\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝² : CommMonoid β\ninst✝¹ : DecidableEq α\ninst✝ : CommMonoid α\nm : Multiset α\ns : Finset α\nl : List α\n⊢ Multiset.toFinset ↑l ⊆ s → List.prod l = ∏ x in s, x ^ List.count x l",
"tactic": "apply prod_list_count_of_subset l s"
}
] |
[
1339,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1334,
1
] |
Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean
|
MeasureTheory.Integrable.comp_mul_right'
|
[] |
[
178,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.trailingDegree_zero
|
[] |
[
88,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Topology/MetricSpace/ThickenedIndicator.lean
|
thickenedIndicatorAux_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\nx : α\nx_in_E : x ∈ E\n⊢ thickenedIndicatorAux δ E x = 1",
"tactic": "simp [thickenedIndicatorAux, infEdist_zero_of_mem x_in_E, tsub_zero]"
}
] |
[
89,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
differentiable_neg
|
[] |
[
281,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.inv_def
|
[] |
[
414,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.length_eq_zero_of_bot_eq_bot_of_top_eq_top_of_length_eq_zero
|
[
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : bot s₁ = bot s₂\nht : top s₁ = top s₂\nhs₁ : s₁.length = 0\nthis : bot s₁ = top s₁\n⊢ s₂.length = 0",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : bot s₁ = bot s₂\nht : top s₁ = top s₂\nhs₁ : s₁.length = 0\n⊢ s₂.length = 0",
"tactic": "have : s₁.bot = s₁.top := congr_arg s₁ (Fin.ext (by simp [hs₁]))"
},
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : bot s₁ = bot s₂\nht : top s₁ = top s₂\nhs₁ : s₁.length = 0\nthis✝ : bot s₁ = top s₁\nthis : Fin.last s₂.length = 0\n⊢ s₂.length = 0",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : bot s₁ = bot s₂\nht : top s₁ = top s₂\nhs₁ : s₁.length = 0\nthis : bot s₁ = top s₁\n⊢ s₂.length = 0",
"tactic": "have : Fin.last s₂.length = (0 : Fin s₂.length.succ) :=\n s₂.injective (hb.symm.trans (this.trans ht)).symm"
},
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : bot s₁ = bot s₂\nht : top s₁ = top s₂\nhs₁ : s₁.length = 0\nthis✝ : bot s₁ = top s₁\nthis : ↑(Fin.last s₂.length) = ↑0\n⊢ s₂.length = 0",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : bot s₁ = bot s₂\nht : top s₁ = top s₂\nhs₁ : s₁.length = 0\nthis✝ : bot s₁ = top s₁\nthis : Fin.last s₂.length = 0\n⊢ s₂.length = 0",
"tactic": "rw [Fin.ext_iff] at this"
},
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : bot s₁ = bot s₂\nht : top s₁ = top s₂\nhs₁ : s₁.length = 0\nthis✝ : bot s₁ = top s₁\nthis : ↑(Fin.last s₂.length) = ↑0\n⊢ s₂.length = 0",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nhb : bot s₁ = bot s₂\nht : top s₁ = top s₂\nhs₁ : s₁.length = 0\n⊢ ↑0 = ↑(Fin.last s₁.length)",
"tactic": "simp [hs₁]"
}
] |
[
723,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
716,
1
] |
Mathlib/Topology/Order/Basic.lean
|
IsGLB.mem_of_isClosed
|
[] |
[
2096,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2094,
1
] |
Mathlib/Data/Set/Intervals/Disjoint.lean
|
Set.iUnion_Ico_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort u\nα : Type v\nβ : Type w\ninst✝¹ : Preorder α\na✝ b c : α\ninst✝ : NoMaxOrder α\na : α\n⊢ (⋃ (b : α), Ico a b) = Ici a",
"tactic": "simp only [← Ici_inter_Iio, ← inter_iUnion, iUnion_Iio, inter_univ]"
}
] |
[
108,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Algebra/Order/Group/MinMax.lean
|
min_inv_inv'
|
[] |
[
42,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
ContinuousLinearMap.adjointAux_apply
|
[] |
[
81,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mul_smul
|
[
{
"state_after": "case a.h\nl : Type u_4\nm : Type u_3\nn : Type u_1\no : Type ?u.257857\nm' : o → Type ?u.257862\nn' : o → Type ?u.257867\nR : Type u_2\nS : Type ?u.257873\nα : Type v\nβ : Type w\nγ : Type ?u.257880\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : Mul α\ninst✝³ : Fintype n\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R α\ninst✝ : SMulCommClass R α α\nM : Matrix m n α\na : R\nN : Matrix n l α\ni✝ : m\nx✝ : l\n⊢ (M ⬝ (a • N)) i✝ x✝ = (a • M ⬝ N) i✝ x✝",
"state_before": "l : Type u_4\nm : Type u_3\nn : Type u_1\no : Type ?u.257857\nm' : o → Type ?u.257862\nn' : o → Type ?u.257867\nR : Type u_2\nS : Type ?u.257873\nα : Type v\nβ : Type w\nγ : Type ?u.257880\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : Mul α\ninst✝³ : Fintype n\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R α\ninst✝ : SMulCommClass R α α\nM : Matrix m n α\na : R\nN : Matrix n l α\n⊢ M ⬝ (a • N) = a • M ⬝ N",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type u_4\nm : Type u_3\nn : Type u_1\no : Type ?u.257857\nm' : o → Type ?u.257862\nn' : o → Type ?u.257867\nR : Type u_2\nS : Type ?u.257873\nα : Type v\nβ : Type w\nγ : Type ?u.257880\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : Mul α\ninst✝³ : Fintype n\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R α\ninst✝ : SMulCommClass R α α\nM : Matrix m n α\na : R\nN : Matrix n l α\ni✝ : m\nx✝ : l\n⊢ (M ⬝ (a • N)) i✝ x✝ = (a • M ⬝ N) i✝ x✝",
"tactic": "apply dotProduct_smul"
}
] |
[
979,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
976,
1
] |
Mathlib/Data/List/OfFn.lean
|
List.ofFn_get
|
[
{
"state_after": "α : Type u\na : α\nl : List α\n⊢ (get (a :: l) 0 :: ofFn fun i => get (a :: l) (Fin.succ i)) = a :: l",
"state_before": "α : Type u\na : α\nl : List α\n⊢ ofFn (get (a :: l)) = a :: l",
"tactic": "rw [ofFn_succ]"
},
{
"state_after": "case e_tail\nα : Type u\na : α\nl : List α\n⊢ (ofFn fun i => get (a :: l) (Fin.succ i)) = l",
"state_before": "α : Type u\na : α\nl : List α\n⊢ (get (a :: l) 0 :: ofFn fun i => get (a :: l) (Fin.succ i)) = a :: l",
"tactic": "congr"
},
{
"state_after": "case e_tail\nα : Type u\na : α\nl : List α\n⊢ (ofFn fun i => get (a :: l) (Fin.succ i)) = l",
"state_before": "case e_tail\nα : Type u\na : α\nl : List α\n⊢ (ofFn fun i => get (a :: l) (Fin.succ i)) = l",
"tactic": "simp only [Fin.val_succ]"
},
{
"state_after": "no goals",
"state_before": "case e_tail\nα : Type u\na : α\nl : List α\n⊢ (ofFn fun i => get (a :: l) (Fin.succ i)) = l",
"tactic": "exact ofFn_get l"
}
] |
[
193,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.iUnion_univ_pi_of_monotone
|
[] |
[
1562,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1559,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.mul_apply
|
[] |
[
198,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Dynamics/OmegaLimit.lean
|
mapsTo_omegaLimit
|
[] |
[
112,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
tendstoLocallyUniformly_iff_filter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\n⊢ TendstoLocallyUniformly F f p ↔ ∀ (x : α), TendstoUniformlyOnFilter F f p (𝓝 x)",
"tactic": "simpa [← tendstoLocallyUniformlyOn_univ, ← nhdsWithin_univ] using\n @tendstoLocallyUniformlyOn_iff_filter _ _ _ _ F f univ p _"
}
] |
[
771,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
768,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
ContinuousLinearMap.integral_comp_L1_comm
|
[] |
[
1123,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1121,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
lt_mul_of_one_lt_of_lt'
|
[] |
[
925,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
922,
1
] |
Mathlib/Data/FunLike/Basic.lean
|
FunLike.congr_arg
|
[] |
[
221,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
11
] |
Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean
|
Subalgebra.pointwise_smul_toSubsemiring
|
[] |
[
96,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_subset
|
[
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.1411\nβ : Type u_2\nβ' : Type ?u.1417\nγ : Type u_3\nγ' : Type ?u.1423\nδ : Type ?u.1426\nδ' : Type ?u.1429\nε : Type ?u.1432\nε' : Type ?u.1435\nζ : Type ?u.1438\nζ' : Type ?u.1441\nν : Type ?u.1444\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc c' : γ\nd d' : δ\nhs : s ⊆ s'\nht : t ⊆ t'\na : α\nb : β\nha : a ∈ s\nhb : b ∈ t\n⊢ f a b ∈ image2 f s' t'",
"state_before": "α : Type u_1\nα' : Type ?u.1411\nβ : Type u_2\nβ' : Type ?u.1417\nγ : Type u_3\nγ' : Type ?u.1423\nδ : Type ?u.1426\nδ' : Type ?u.1429\nε : Type ?u.1432\nε' : Type ?u.1435\nζ : Type ?u.1438\nζ' : Type ?u.1441\nν : Type ?u.1444\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nhs : s ⊆ s'\nht : t ⊆ t'\n⊢ image2 f s t ⊆ image2 f s' t'",
"tactic": "rintro _ ⟨a, b, ha, hb, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nα' : Type ?u.1411\nβ : Type u_2\nβ' : Type ?u.1417\nγ : Type u_3\nγ' : Type ?u.1423\nδ : Type ?u.1426\nδ' : Type ?u.1429\nε : Type ?u.1432\nε' : Type ?u.1435\nζ : Type ?u.1438\nζ' : Type ?u.1441\nν : Type ?u.1444\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na✝ a' : α\nb✝ b' : β\nc c' : γ\nd d' : δ\nhs : s ⊆ s'\nht : t ⊆ t'\na : α\nb : β\nha : a ∈ s\nhb : b ∈ t\n⊢ f a b ∈ image2 f s' t'",
"tactic": "exact mem_image2_of_mem (hs ha) (ht hb)"
}
] |
[
61,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
CategoryTheory.Limits.biproduct.map_matrix
|
[
{
"state_after": "case w.w\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nJ : Type\ninst✝³ : Fintype J\nf✝ : J → C\ninst✝² : HasBiproduct f✝\nK : Type\ninst✝¹ : Fintype K\ninst✝ : HasFiniteBiproducts C\nf g : J → C\nh : K → C\nm : (k : J) → f k ⟶ g k\nn : (j : J) → (k : K) → g j ⟶ h k\nj✝¹ : J\nj✝ : K\n⊢ (ι (fun b => f b) j✝¹ ≫ map m ≫ matrix n) ≫ π (fun k => h k) j✝ =\n (ι (fun b => f b) j✝¹ ≫ matrix fun j k => m j ≫ n j k) ≫ π (fun k => h k) j✝",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nJ : Type\ninst✝³ : Fintype J\nf✝ : J → C\ninst✝² : HasBiproduct f✝\nK : Type\ninst✝¹ : Fintype K\ninst✝ : HasFiniteBiproducts C\nf g : J → C\nh : K → C\nm : (k : J) → f k ⟶ g k\nn : (j : J) → (k : K) → g j ⟶ h k\n⊢ map m ≫ matrix n = matrix fun j k => m j ≫ n j k",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w.w\nC : Type u\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\nJ : Type\ninst✝³ : Fintype J\nf✝ : J → C\ninst✝² : HasBiproduct f✝\nK : Type\ninst✝¹ : Fintype K\ninst✝ : HasFiniteBiproducts C\nf g : J → C\nh : K → C\nm : (k : J) → f k ⟶ g k\nn : (j : J) → (k : K) → g j ⟶ h k\nj✝¹ : J\nj✝ : K\n⊢ (ι (fun b => f b) j✝¹ ≫ map m ≫ matrix n) ≫ π (fun k => h k) j✝ =\n (ι (fun b => f b) j✝¹ ≫ matrix fun j k => m j ≫ n j k) ≫ π (fun k => h k) j✝",
"tactic": "simp"
}
] |
[
270,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.ofDual_sup
|
[] |
[
497,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Data/Set/Intervals/Group.lean
|
Set.sub_mem_Ico_iff_left
|
[] |
[
108,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.rescale_rescale
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nf : PowerSeries R\na b : R\nn : ℕ\n⊢ ↑(coeff R n) (↑(rescale b) (↑(rescale a) f)) = ↑(coeff R n) (↑(rescale (a * b)) f)",
"state_before": "R : Type u_1\ninst✝ : CommSemiring R\nf : PowerSeries R\na b : R\n⊢ ↑(rescale b) (↑(rescale a) f) = ↑(rescale (a * b)) f",
"tactic": "ext n"
},
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nf : PowerSeries R\na b : R\nn : ℕ\n⊢ b ^ n * (a ^ n * ↑(coeff R n) f) = (a * b) ^ n * ↑(coeff R n) f",
"state_before": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nf : PowerSeries R\na b : R\nn : ℕ\n⊢ ↑(coeff R n) (↑(rescale b) (↑(rescale a) f)) = ↑(coeff R n) (↑(rescale (a * b)) f)",
"tactic": "simp_rw [coeff_rescale]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nf : PowerSeries R\na b : R\nn : ℕ\n⊢ b ^ n * (a ^ n * ↑(coeff R n) f) = (a * b) ^ n * ↑(coeff R n) f",
"tactic": "rw [mul_pow, mul_comm _ (b ^ n), mul_assoc]"
}
] |
[
1794,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1790,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
Ordinal.toNatOrdinal_symm_eq
|
[] |
[
155,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Data/Int/ModEq.lean
|
Int.modEq_and_modEq_iff_modEq_mul
|
[
{
"state_after": "m✝ n✝ a✝ b✝ c d a b m n : ℤ\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m ∣ b - a ∧ n ∣ b - a\n⊢ a ≡ b [ZMOD m * n]",
"state_before": "m✝ n✝ a✝ b✝ c d a b m n : ℤ\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : a ≡ b [ZMOD m] ∧ a ≡ b [ZMOD n]\n⊢ a ≡ b [ZMOD m * n]",
"tactic": "rw [modEq_iff_dvd, modEq_iff_dvd] at h"
},
{
"state_after": "m✝ n✝ a✝ b✝ c d a b m n : ℤ\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m ∣ b - a ∧ n ∣ b - a\n⊢ natAbs m * natAbs n ∣ natAbs (b - a)",
"state_before": "m✝ n✝ a✝ b✝ c d a b m n : ℤ\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m ∣ b - a ∧ n ∣ b - a\n⊢ a ≡ b [ZMOD m * n]",
"tactic": "rw [modEq_iff_dvd, ← natAbs_dvd, ← dvd_natAbs, coe_nat_dvd, natAbs_mul]"
},
{
"state_after": "no goals",
"state_before": "m✝ n✝ a✝ b✝ c d a b m n : ℤ\nhmn : Nat.coprime (natAbs m) (natAbs n)\nh : m ∣ b - a ∧ n ∣ b - a\n⊢ natAbs m * natAbs n ∣ natAbs (b - a)",
"tactic": "refine' hmn.mul_dvd_of_dvd_of_dvd _ _ <;> rw [← coe_nat_dvd, natAbs_dvd, dvd_natAbs] <;>\n tauto"
}
] |
[
277,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Init/Logic.lean
|
Implies.trans
|
[] |
[
28,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
27,
15
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.lt_lift_iff
|
[] |
[
831,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
826,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_finset_product
|
[
{
"state_after": "ι : Type ?u.317398\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nr : Finset (γ × α)\ns : Finset γ\nt : γ → Finset α\nh : ∀ (p : γ × α), p ∈ r ↔ p.fst ∈ s ∧ p.snd ∈ t p.fst\nf : γ × α → β\n⊢ ∏ p in r, f p = ∏ x in Finset.sigma s t, f (x.fst, x.snd)",
"state_before": "ι : Type ?u.317398\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nr : Finset (γ × α)\ns : Finset γ\nt : γ → Finset α\nh : ∀ (p : γ × α), p ∈ r ↔ p.fst ∈ s ∧ p.snd ∈ t p.fst\nf : γ × α → β\n⊢ ∏ p in r, f p = ∏ c in s, ∏ a in t c, f (c, a)",
"tactic": "refine' Eq.trans _ (prod_sigma s t fun p => f (p.1, p.2))"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.317398\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\nr : Finset (γ × α)\ns : Finset γ\nt : γ → Finset α\nh : ∀ (p : γ × α), p ∈ r ↔ p.fst ∈ s ∧ p.snd ∈ t p.fst\nf : γ × α → β\n⊢ ∏ p in r, f p = ∏ x in Finset.sigma s t, f (x.fst, x.snd)",
"tactic": "exact\n prod_bij' (fun p _hp => ⟨p.1, p.2⟩) (fun p => mem_sigma.mpr ∘ (h p).mp)\n (fun p hp => congr_arg f Prod.mk.eta.symm) (fun p _hp => (p.1, p.2))\n (fun p => (h (p.1, p.2)).mpr ∘ mem_sigma.mp) (fun p _hp => Prod.mk.eta) fun p _hp => p.eta"
}
] |
[
606,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
599,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
powCoprime_inv
|
[] |
[
993,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
991,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.comap_lift_eq2
|
[] |
[
125,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
IsPrincipalIdealRing.of_surjective
|
[] |
[
345,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.not_mem_bot
|
[] |
[
193,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
lcm_mul_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\na b c : α\n⊢ lcm (b * a) (c * a) = lcm b c * ↑normalize a",
"tactic": "simp only [mul_comm, lcm_mul_left]"
}
] |
[
835,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
834,
1
] |
Mathlib/Analysis/Convex/Integral.lean
|
Convex.average_mem
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.245931\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : Convex ℝ s\nhsc : IsClosed s\nhμ : μ ≠ 0\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : IsProbabilityMeasure ((↑↑μ univ)⁻¹ • μ)\n⊢ (⨍ (x : α), f x ∂μ) ∈ s",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.245931\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : Convex ℝ s\nhsc : IsClosed s\nhμ : μ ≠ 0\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\n⊢ (⨍ (x : α), f x ∂μ) ∈ s",
"tactic": "have : IsProbabilityMeasure ((μ univ)⁻¹ • μ) := isProbabilityMeasureSmul hμ"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.245931\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : Convex ℝ s\nhsc : IsClosed s\nhμ : μ ≠ 0\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : IsProbabilityMeasure ((↑↑μ univ)⁻¹ • μ)\n⊢ (↑↑μ univ)⁻¹ • μ ≪ μ",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.245931\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : Convex ℝ s\nhsc : IsClosed s\nhμ : μ ≠ 0\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : IsProbabilityMeasure ((↑↑μ univ)⁻¹ • μ)\n⊢ (⨍ (x : α), f x ∂μ) ∈ s",
"tactic": "refine' hs.integral_mem hsc (ae_mono' _ hfs) hfi.to_average"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.245931\nm0 : MeasurableSpace α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\nμ : MeasureTheory.Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nC : ℝ\ninst✝ : IsFiniteMeasure μ\nhs : Convex ℝ s\nhsc : IsClosed s\nhμ : μ ≠ 0\nhfs : ∀ᵐ (x : α) ∂μ, f x ∈ s\nhfi : Integrable f\nthis : IsProbabilityMeasure ((↑↑μ univ)⁻¹ • μ)\n⊢ (↑↑μ univ)⁻¹ • μ ≪ μ",
"tactic": "exact AbsolutelyContinuous.smul (refl _) _"
}
] |
[
94,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Std/Data/List/Lemmas.lean
|
List.infix_of_mem_join
|
[] |
[
1707,
78
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1702,
1
] |
Mathlib/Order/CompleteBooleanAlgebra.lean
|
PUnit.sInf_eq
|
[] |
[
406,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Algebra/Order/CompleteField.lean
|
LinearOrderedField.inducedMap_nonneg
|
[] |
[
215,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Data/List/Forall2.lean
|
List.forall₂_cons
|
[
{
"state_after": "case cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.620\nδ : Type ?u.623\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na : α\nb : β\nl₁ : List α\nl₂ : List β\na✝¹ : R a b\na✝ : Forall₂ R l₁ l₂\n⊢ R a b ∧ Forall₂ R l₁ l₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.620\nδ : Type ?u.623\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na : α\nb : β\nl₁ : List α\nl₂ : List β\nh : Forall₂ R (a :: l₁) (b :: l₂)\n⊢ R a b ∧ Forall₂ R l₁ l₂",
"tactic": "cases' h with h₁ h₂"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.620\nδ : Type ?u.623\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\na : α\nb : β\nl₁ : List α\nl₂ : List β\na✝¹ : R a b\na✝ : Forall₂ R l₁ l₂\n⊢ R a b ∧ Forall₂ R l₁ l₂",
"tactic": "constructor <;> assumption"
}
] |
[
39,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.snd_pow_of_smul_comm
|
[
{
"state_after": "R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\n⊢ List.sum (List.map (fun i => fst x ^ (Nat.pred n - i + i) • snd x) (List.range n)) = n • fst x ^ Nat.pred n • snd x",
"state_before": "R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\n⊢ snd (x ^ n) = n • fst x ^ Nat.pred n • snd x",
"tactic": "simp_rw [snd_pow_eq_sum, this, smul_smul, ← pow_add]"
},
{
"state_after": "R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nn : ℕ\n⊢ op (fst x ^ n) • snd x = fst x ^ n • snd x",
"state_before": "R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : op (fst x) • snd x = fst x • snd x\n⊢ ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x",
"tactic": "intro n"
},
{
"state_after": "case zero\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : op (fst x) • snd x = fst x • snd x\n⊢ op (fst x ^ Nat.zero) • snd x = fst x ^ Nat.zero • snd x\n\ncase succ\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nn : ℕ\nih : op (fst x ^ n) • snd x = fst x ^ n • snd x\n⊢ op (fst x ^ Nat.succ n) • snd x = fst x ^ Nat.succ n • snd x",
"state_before": "R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nn : ℕ\n⊢ op (fst x ^ n) • snd x = fst x ^ n • snd x",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : op (fst x) • snd x = fst x • snd x\n⊢ op (fst x ^ Nat.zero) • snd x = fst x ^ Nat.zero • snd x",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nn : ℕ\nih : op (fst x ^ n) • snd x = fst x ^ n • snd x\n⊢ op (fst x ^ Nat.succ n) • snd x = fst x ^ Nat.succ n • snd x",
"tactic": "rw [pow_succ', MulOpposite.op_mul, mul_smul, mul_smul, ← h,\n smul_comm (_ : R) (op x.fst) x.snd, ih]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\n⊢ List.sum (List.map (fun i => fst x ^ (Nat.pred 0 - i + i) • snd x) (List.range 0)) = 0 • fst x ^ Nat.pred 0 • snd x",
"tactic": "rw [Nat.pred_zero, pow_zero, List.range_zero, zero_smul, List.map_nil, List.sum_nil]"
},
{
"state_after": "R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\n⊢ List.sum (List.map (fun i => fst x ^ (n - i + i) • snd x) (List.range (Nat.succ n))) = Nat.succ n • fst x ^ n • snd x",
"state_before": "R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\n⊢ List.sum (List.map (fun i => fst x ^ (Nat.pred (Nat.succ n) - i + i) • snd x) (List.range (Nat.succ n))) =\n Nat.succ n • fst x ^ Nat.pred (Nat.succ n) • snd x",
"tactic": "simp_rw [Nat.pred_succ]"
},
{
"state_after": "case refine'_1\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\n⊢ ∀ (x_1 : M), x_1 ∈ List.map (fun i => fst x ^ (n - i + i) • snd x) (List.range (Nat.succ n)) → x_1 = fst x ^ n • snd x\n\ncase refine'_2\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\n⊢ List.length (List.map (fun i => fst x ^ (n - i + i) • snd x) (List.range (Nat.succ n))) • fst x ^ n • snd x =\n Nat.succ n • fst x ^ n • snd x",
"state_before": "R : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\n⊢ List.sum (List.map (fun i => fst x ^ (n - i + i) • snd x) (List.range (Nat.succ n))) = Nat.succ n • fst x ^ n • snd x",
"tactic": "refine' (List.sum_eq_card_nsmul _ (x.fst ^ n • x.snd) _).trans _"
},
{
"state_after": "case refine'_1\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\nm : M\nhm : m ∈ List.map (fun i => fst x ^ (n - i + i) • snd x) (List.range (Nat.succ n))\n⊢ m = fst x ^ n • snd x",
"state_before": "case refine'_1\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\n⊢ ∀ (x_1 : M), x_1 ∈ List.map (fun i => fst x ^ (n - i + i) • snd x) (List.range (Nat.succ n)) → x_1 = fst x ^ n • snd x",
"tactic": "rintro m hm"
},
{
"state_after": "case refine'_1\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\nm : M\nhm : ∃ a, a < Nat.succ n ∧ fst x ^ (n - a + a) • snd x = m\n⊢ m = fst x ^ n • snd x",
"state_before": "case refine'_1\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\nm : M\nhm : m ∈ List.map (fun i => fst x ^ (n - i + i) • snd x) (List.range (Nat.succ n))\n⊢ m = fst x ^ n • snd x",
"tactic": "simp_rw [List.mem_map, List.mem_range] at hm"
},
{
"state_after": "case refine'_1.intro.intro\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn i : ℕ\nhi : i < Nat.succ n\n⊢ fst x ^ (n - i + i) • snd x = fst x ^ n • snd x",
"state_before": "case refine'_1\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\nm : M\nhm : ∃ a, a < Nat.succ n ∧ fst x ^ (n - a + a) • snd x = m\n⊢ m = fst x ^ n • snd x",
"tactic": "obtain ⟨i, hi, rfl⟩ := hm"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn i : ℕ\nhi : i < Nat.succ n\n⊢ fst x ^ (n - i + i) • snd x = fst x ^ n • snd x",
"tactic": "rw [tsub_add_cancel_of_le (Nat.lt_succ_iff.mp hi)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u\nM : Type v\ninst✝⁴ : Monoid R\ninst✝³ : AddMonoid M\ninst✝² : DistribMulAction R M\ninst✝¹ : DistribMulAction Rᵐᵒᵖ M\ninst✝ : SMulCommClass R Rᵐᵒᵖ M\nx : tsze R M\nn✝ : ℕ\nh : op (fst x) • snd x = fst x • snd x\nthis : ∀ (n : ℕ), op (fst x ^ n) • snd x = fst x ^ n • snd x\nn : ℕ\n⊢ List.length (List.map (fun i => fst x ^ (n - i + i) • snd x) (List.range (Nat.succ n))) • fst x ^ n • snd x =\n Nat.succ n • fst x ^ n • snd x",
"tactic": "rw [List.length_map, List.length_range]"
}
] |
[
628,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
609,
1
] |
Mathlib/Algebra/Module/Zlattice.lean
|
Zspan.fract_apply
|
[] |
[
117,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.Tape.mk'_nth_nat
|
[
{
"state_after": "no goals",
"state_before": "Γ : Type u_1\ninst✝ : Inhabited Γ\nL R : ListBlank Γ\nn : ℕ\n⊢ nth (mk' L R) ↑n = ListBlank.nth R n",
"tactic": "rw [← Tape.right₀_nth, Tape.mk'_right₀]"
}
] |
[
629,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
627,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.le_comap_map
|
[] |
[
1662,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1661,
1
] |
Mathlib/LinearAlgebra/FreeModule/PID.lean
|
eq_bot_of_generator_maximal_map_eq_zero
|
[
{
"state_after": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬map ϕ N < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\n⊢ ∀ (x : M), x ∈ N → x = 0",
"state_before": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬map ϕ N < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\n⊢ N = ⊥",
"tactic": "rw [Submodule.eq_bot_iff]"
},
{
"state_after": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬map ϕ N < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\nx : M\nhx : x ∈ N\n⊢ x = 0",
"state_before": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬map ϕ N < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\n⊢ ∀ (x : M), x ∈ N → x = 0",
"tactic": "intro x hx"
},
{
"state_after": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬map ϕ N < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\nx : M\nhx : x ∈ N\ni : ι\n⊢ ↑(↑b.repr x) i = ↑(↑b.repr 0) i",
"state_before": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬map ϕ N < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\nx : M\nhx : x ∈ N\n⊢ x = 0",
"tactic": "refine' b.ext_elem fun i ↦ _"
},
{
"state_after": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬⊥ < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\nx : M\nhx : x ∈ N\ni : ι\n⊢ ↑(↑b.repr x) i = ↑(↑b.repr 0) i",
"state_before": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬map ϕ N < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\nx : M\nhx : x ∈ N\ni : ι\n⊢ ↑(↑b.repr x) i = ↑(↑b.repr 0) i",
"tactic": "rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ"
},
{
"state_after": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬⊥ < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\nx : M\nhx : x ∈ N\ni : ι\n⊢ ↑(↑b.repr x) i = 0",
"state_before": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬⊥ < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\nx : M\nhx : x ∈ N\ni : ι\n⊢ ↑(↑b.repr x) i = ↑(↑b.repr 0) i",
"tactic": "rw [LinearEquiv.map_zero, Finsupp.zero_apply]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_1\nb✝ b : Basis ι R M\nN : Submodule R M\nϕ : M →ₗ[R] R\nhϕ : ∀ (ψ : M →ₗ[R] R), ¬⊥ < map ψ N\ninst✝ : IsPrincipal (map ϕ N)\nhgen : generator (map ϕ N) = 0\nx : M\nhx : x ∈ N\ni : ι\n⊢ ↑(↑b.repr x) i = 0",
"tactic": "exact\n (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _\n ⟨x, hx, rfl⟩"
}
] |
[
77,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sign_pi_add
|
[
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ sign (↑π + θ) = -sign θ",
"tactic": "rw [add_comm, sign_add_pi]"
}
] |
[
880,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
880,
1
] |
Mathlib/Algebra/Lie/Abelian.lean
|
LieModule.coe_maxTrivEquiv_apply
|
[] |
[
199,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Data/Multiset/Lattice.lean
|
Multiset.inf_ndinsert
|
[
{
"state_after": "α : Type u_1\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\ninst✝ : DecidableEq α\na : α\ns : Multiset α\n⊢ ∀ (a_1 : α), a_1 ∈ ndinsert a s ↔ a_1 ∈ a ::ₘ s",
"state_before": "α : Type u_1\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\ninst✝ : DecidableEq α\na : α\ns : Multiset α\n⊢ inf (ndinsert a s) = a ⊓ inf s",
"tactic": "rw [← inf_dedup, dedup_ext.2, inf_dedup, inf_cons]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : SemilatticeInf α\ninst✝¹ : OrderTop α\ninst✝ : DecidableEq α\na : α\ns : Multiset α\n⊢ ∀ (a_1 : α), a_1 ∈ ndinsert a s ↔ a_1 ∈ a ::ₘ s",
"tactic": "simp"
}
] |
[
177,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/Deprecated/Submonoid.lean
|
Multiplicative.isSubmonoid_iff
|
[] |
[
81,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean
|
SimpleGraph.farFromTriangleFree.mono
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\n𝕜 : Type u_2\ninst✝¹ : Fintype α\ninst✝ : LinearOrderedField 𝕜\nG H : SimpleGraph α\nε δ : 𝕜\nn : ℕ\ns : Finset α\nhε : FarFromTriangleFree G ε\nh : δ ≤ ε\n⊢ δ * ↑(Fintype.card α ^ 2) ≤ ε * ↑(Fintype.card α ^ 2)",
"tactic": "gcongr"
}
] |
[
58,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.sub_bind
|
[] |
[
262,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Algebra/Module/Basic.lean
|
Units.neg_smul
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.106759\nR : Type u_1\nk : Type ?u.106765\nS : Type ?u.106768\nM : Type u_2\nM₂ : Type ?u.106774\nM₃ : Type ?u.106777\nι : Type ?u.106780\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr s : R\nx✝ y : M\nu : Rˣ\nx : M\n⊢ -u • x = -(u • x)",
"tactic": "rw [Units.smul_def, Units.val_neg, _root_.neg_smul, Units.smul_def]"
}
] |
[
312,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Algebra/Algebra/Bilinear.lean
|
LinearMap.commute_mulLeft_right
|
[
{
"state_after": "case h\nR : Type u_2\nA : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : SMulCommClass R A A\ninst✝ : IsScalarTower R A A\na b c : A\n⊢ ↑(mulLeft R a * mulRight R b) c = ↑(mulRight R b * mulLeft R a) c",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : SMulCommClass R A A\ninst✝ : IsScalarTower R A A\na b : A\n⊢ Commute (mulLeft R a) (mulRight R b)",
"tactic": "ext c"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_2\nA : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : SMulCommClass R A A\ninst✝ : IsScalarTower R A A\na b c : A\n⊢ ↑(mulLeft R a * mulRight R b) c = ↑(mulRight R b * mulLeft R a) c",
"tactic": "exact (mul_assoc a c b).symm"
}
] |
[
140,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
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