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Mathlib/Order/Synonym.lean | OrderDual.ofDual_lt_ofDual | [] | [
109,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/Data/PNat/Basic.lean | PNat.pos_of_div_pos | [
{
"state_after": "n : ℕ+\na : ℕ\nh : a ∣ ↑n\n⊢ a ≠ 0",
"state_before": "n : ℕ+\na : ℕ\nh : a ∣ ↑n\n⊢ 0 < a",
"tactic": "apply pos_iff_ne_zero.2"
},
{
"state_after": "n : ℕ+\na : ℕ\nh : a ∣ ↑n\nhzero : a = 0\n⊢ False",
"state_before": "n : ℕ+\na : ℕ\nh : a ∣ ↑n\n⊢ a ≠ 0",
"tactic": "intro hzero"
},
{
"state_after": "n : ℕ+\na : ℕ\nh : 0 ∣ ↑n\nhzero : a = 0\n⊢ False",
"state_before": "n : ℕ+\na : ℕ\nh : a ∣ ↑n\nhzero : a = 0\n⊢ False",
"tactic": "rw [hzero] at h"
},
{
"state_after": "no goals",
"state_before": "n : ℕ+\na : ℕ\nh : 0 ∣ ↑n\nhzero : a = 0\n⊢ False",
"tactic": "exact PNat.ne_zero n (eq_zero_of_zero_dvd h)"
}
] | [
450,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
446,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.pred_lt_pred | [] | [
56,
41
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
54,
1
] |
Mathlib/Order/UpperLower/Basic.lean | isLowerSet_iUnion₂ | [] | [
144,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
1
] |
Mathlib/Order/Heyting/Basic.lean | compl_sup_compl_le | [] | [
926,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
925,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean | exists_ratio_hasDerivAt_eq_ratio_slope | [
{
"state_after": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"state_before": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"tactic": "let h x := (g b - g a) * f x - (f b - f a) * g x"
},
{
"state_after": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"state_before": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"tactic": "have hI : h a = h b := by simp only; ring"
},
{
"state_after": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"state_before": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"tactic": "let h' x := (g b - g a) * f' x - (f b - f a) * g' x"
},
{
"state_after": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"state_before": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"tactic": "have hhh' : ∀ x ∈ Ioo a b, HasDerivAt h (h' x) x := fun x hx =>\n ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a))"
},
{
"state_after": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\nhhc : ContinuousOn h (Icc a b)\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"state_before": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"tactic": "have hhc : ContinuousOn h (Icc a b) :=\n (continuousOn_const.mul hfc).sub (continuousOn_const.mul hgc)"
},
{
"state_after": "case intro.intro\nE : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\nhhc : ContinuousOn h (Icc a b)\nc : ℝ\ncmem : c ∈ Ioo a b\nhc : h' c = 0\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"state_before": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\nhhc : ContinuousOn h (Icc a b)\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"tactic": "rcases exists_hasDerivAt_eq_zero h h' hab hhc hI hhh' with ⟨c, cmem, hc⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nE : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\nhI : h a = h b\nh' : ℝ → ℝ := fun x => (g b - g a) * f' x - (f b - f a) * g' x\nhhh' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt h (h' x) x\nhhc : ContinuousOn h (Icc a b)\nc : ℝ\ncmem : c ∈ Ioo a b\nhc : h' c = 0\n⊢ ∃ c, c ∈ Ioo a b ∧ (g b - g a) * f' c = (f b - f a) * g' c",
"tactic": "exact ⟨c, cmem, sub_eq_zero.1 hc⟩"
},
{
"state_after": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\n⊢ (g b - g a) * f a - (f b - f a) * g a = (g b - g a) * f b - (f b - f a) * g b",
"state_before": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\n⊢ h a = h b",
"tactic": "simp only"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.307798\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.307894\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhff' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt f (f' x) x\nhfd : DifferentiableOn ℝ f (Ioo a b)\ng g' : ℝ → ℝ\nhgc : ContinuousOn g (Icc a b)\nhgg' : ∀ (x : ℝ), x ∈ Ioo a b → HasDerivAt g (g' x) x\nhgd : DifferentiableOn ℝ g (Ioo a b)\nh : ℝ → ℝ := fun x => (g b - g a) * f x - (f b - f a) * g x\n⊢ (g b - g a) * f a - (f b - f a) * g a = (g b - g a) * f b - (f b - f a) * g b",
"tactic": "ring"
}
] | [
724,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
714,
1
] |
Mathlib/Topology/ContinuousFunction/Algebra.lean | ContinuousMap.coe_smul | [] | [
612,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
611,
1
] |
Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | MeasureTheory.FiniteMeasure.smul_testAgainstNN_apply | [
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\ninst✝⁵ : MeasurableSpace Ω\nR : Type ?u.54872\ninst✝⁴ : SMul R ℝ≥0\ninst✝³ : SMul R ℝ≥0∞\ninst✝² : IsScalarTower R ℝ≥0 ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : TopologicalSpace Ω\nc : ℝ≥0\nμ : FiniteMeasure Ω\nf : Ω →ᵇ ℝ≥0\n⊢ testAgainstNN (c • μ) f = c • testAgainstNN μ f",
"tactic": "simp only [testAgainstNN, toMeasure_smul, smul_eq_mul, ← ENNReal.smul_toNNReal, ENNReal.smul_def,\n lintegral_smul_measure]"
}
] | [
378,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
375,
1
] |
Mathlib/Order/CompleteLattice.lean | sInf_sup_le_iInf_sup | [] | [
1932,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1931,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean | Nat.choose_eq_factorial_div_factorial | [
{
"state_after": "n k : ℕ\nhk : k ≤ n\n⊢ choose n k = choose n k * (k ! * (n - k)!) / (k ! * (n - k)!)",
"state_before": "n k : ℕ\nhk : k ≤ n\n⊢ choose n k = n ! / (k ! * (n - k)!)",
"tactic": "rw [← choose_mul_factorial_mul_factorial hk, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "n k : ℕ\nhk : k ≤ n\n⊢ choose n k = choose n k * (k ! * (n - k)!) / (k ! * (n - k)!)",
"tactic": "exact (mul_div_left _ (mul_pos (factorial_pos _) (factorial_pos _))).symm"
}
] | [
171,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/Order/Filter/Pi.lean | Filter.pi_mem_pi_iff | [] | [
106,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
1
] |
Mathlib/Computability/TMToPartrec.lean | Turing.PartrecToTM2.tr_respects | [] | [
1693,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1691,
1
] |
Mathlib/Algebra/Order/Pointwise.lean | sInf_inv | [
{
"state_after": "α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ (⨅ (a : α) (_ : a ∈ s), a⁻¹) = (sSup s)⁻¹",
"state_before": "α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ sInf s⁻¹ = (sSup s)⁻¹",
"tactic": "rw [← image_inv, sInf_image]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : CompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns✝ t s : Set α\n⊢ (⨅ (a : α) (_ : a ∈ s), a⁻¹) = (sSup s)⁻¹",
"tactic": "exact ((OrderIso.inv α).map_sSup _).symm"
}
] | [
73,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Analysis/Complex/Basic.lean | Complex.reClm_apply | [] | [
272,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
271,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.ext_iff | [] | [
362,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean | QuasiconvexOn.convex | [] | [
96,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Data/List/Rotate.lean | List.nthLe_rotate' | [] | [
307,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
303,
1
] |
Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.mem_sInf | [] | [
223,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
1
] |
Mathlib/MeasureTheory/Measure/Regular.lean | MeasureTheory.Measure.InnerRegular.exists_subset_lt_add | [
{
"state_after": "case inl\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : InnerRegular μ p q\nh0 : p ∅\nhU : q U\nhμU : ↑↑μ U ≠ ⊤\nhε : ε ≠ 0\nh₀ : ↑↑μ U = 0\n⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε\n\ncase inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : InnerRegular μ p q\nh0 : p ∅\nhU : q U\nhμU : ↑↑μ U ≠ ⊤\nhε : ε ≠ 0\nh₀ : ↑↑μ U ≠ 0\n⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε",
"state_before": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : InnerRegular μ p q\nh0 : p ∅\nhU : q U\nhμU : ↑↑μ U ≠ ⊤\nhε : ε ≠ 0\n⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε",
"tactic": "cases' eq_or_ne (μ U) 0 with h₀ h₀"
},
{
"state_after": "case inl\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : InnerRegular μ p q\nh0 : p ∅\nhU : q U\nhμU : ↑↑μ U ≠ ⊤\nhε : ε ≠ 0\nh₀ : ↑↑μ U = 0\n⊢ ↑↑μ U < ↑↑μ ∅ + ε",
"state_before": "case inl\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : InnerRegular μ p q\nh0 : p ∅\nhU : q U\nhμU : ↑↑μ U ≠ ⊤\nhε : ε ≠ 0\nh₀ : ↑↑μ U = 0\n⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε",
"tactic": "refine' ⟨∅, empty_subset _, h0, _⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : InnerRegular μ p q\nh0 : p ∅\nhU : q U\nhμU : ↑↑μ U ≠ ⊤\nhε : ε ≠ 0\nh₀ : ↑↑μ U = 0\n⊢ ↑↑μ U < ↑↑μ ∅ + ε",
"tactic": "rwa [measure_empty, h₀, zero_add, pos_iff_ne_zero]"
},
{
"state_after": "case inr.intro.intro.intro\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : InnerRegular μ p q\nh0 : p ∅\nhU : q U\nhμU : ↑↑μ U ≠ ⊤\nhε : ε ≠ 0\nh₀ : ↑↑μ U ≠ 0\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhrK : ↑↑μ U - ε < ↑↑μ K\n⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε",
"state_before": "case inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : InnerRegular μ p q\nh0 : p ∅\nhU : q U\nhμU : ↑↑μ U ≠ ⊤\nhε : ε ≠ 0\nh₀ : ↑↑μ U ≠ 0\n⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε",
"tactic": "rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : InnerRegular μ p q\nh0 : p ∅\nhU : q U\nhμU : ↑↑μ U ≠ ⊤\nhε : ε ≠ 0\nh₀ : ↑↑μ U ≠ 0\nK : Set α\nhKU : K ⊆ U\nhKc : p K\nhrK : ↑↑μ U - ε < ↑↑μ K\n⊢ ∃ K, K ⊆ U ∧ p K ∧ ↑↑μ U < ↑↑μ K + ε",
"tactic": "exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩"
}
] | [
169,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
163,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | HasStrictFDerivAt.of_local_left_inverse | [
{
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhfg : ∀ᶠ (y : F) in 𝓝 a, f (g y) = y\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\n⊢ HasStrictFDerivAt g (↑(ContinuousLinearEquiv.symm f')) a",
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhg : ContinuousAt g a\nhf : HasStrictFDerivAt f (↑f') (g a)\nhfg : ∀ᶠ (y : F) in 𝓝 a, f (g y) = y\n⊢ HasStrictFDerivAt g (↑(ContinuousLinearEquiv.symm f')) a",
"tactic": "replace hg := hg.prod_map' hg"
},
{
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\n⊢ HasStrictFDerivAt g (↑(ContinuousLinearEquiv.symm f')) a",
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhfg : ∀ᶠ (y : F) in 𝓝 a, f (g y) = y\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\n⊢ HasStrictFDerivAt g (↑(ContinuousLinearEquiv.symm f')) a",
"tactic": "replace hfg := hfg.prod_mk_nhds hfg"
},
{
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\nthis :\n (fun p => g p.fst - g p.snd - ↑(ContinuousLinearEquiv.symm f') (p.fst - p.snd)) =O[𝓝 (a, a)] fun p =>\n ↑f' (g p.fst - g p.snd) - (p.fst - p.snd)\n⊢ HasStrictFDerivAt g (↑(ContinuousLinearEquiv.symm f')) a",
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\n⊢ HasStrictFDerivAt g (↑(ContinuousLinearEquiv.symm f')) a",
"tactic": "have :\n (fun p : F × F => g p.1 - g p.2 - f'.symm (p.1 - p.2)) =O[𝓝 (a, a)] fun p : F × F =>\n f' (g p.1 - g p.2) - (p.1 - p.2) := by\n refine' ((f'.symm : F →L[𝕜] E).isBigO_comp _ _).congr (fun x => _) fun _ => rfl\n simp"
},
{
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\nthis :\n (fun p => g p.fst - g p.snd - ↑(ContinuousLinearEquiv.symm f') (p.fst - p.snd)) =O[𝓝 (a, a)] fun p =>\n ↑f' (g p.fst - g p.snd) - (p.fst - p.snd)\n⊢ (fun p => ↑f' (g p.fst - g p.snd) - (p.fst - p.snd)) =o[𝓝 (a, a)] fun p => p.fst - p.snd",
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\nthis :\n (fun p => g p.fst - g p.snd - ↑(ContinuousLinearEquiv.symm f') (p.fst - p.snd)) =O[𝓝 (a, a)] fun p =>\n ↑f' (g p.fst - g p.snd) - (p.fst - p.snd)\n⊢ HasStrictFDerivAt g (↑(ContinuousLinearEquiv.symm f')) a",
"tactic": "refine' this.trans_isLittleO _"
},
{
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\n⊢ (fun p => ↑f' (g p.fst - g p.snd) - (p.fst - p.snd)) =o[𝓝 (a, a)] fun p => p.fst - p.snd",
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\nthis :\n (fun p => g p.fst - g p.snd - ↑(ContinuousLinearEquiv.symm f') (p.fst - p.snd)) =O[𝓝 (a, a)] fun p =>\n ↑f' (g p.fst - g p.snd) - (p.fst - p.snd)\n⊢ (fun p => ↑f' (g p.fst - g p.snd) - (p.fst - p.snd)) =o[𝓝 (a, a)] fun p => p.fst - p.snd",
"tactic": "clear this"
},
{
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"tactic": "refine'\n ((hf.comp_tendsto hg).symm.congr' (hfg.mono _) (eventually_of_forall fun _ => rfl)).trans_isBigO\n _"
},
{
"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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\nx : F × F\n⊢ ↑↑(ContinuousLinearEquiv.symm f') (↑f' (g x.fst - g x.snd) - (x.fst - x.snd)) =\n g x.fst - g x.snd - ↑(ContinuousLinearEquiv.symm f') (x.fst - x.snd)",
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"tactic": "refine' ((f'.symm : F →L[𝕜] E).isBigO_comp _ _).congr (fun x => _) fun _ => rfl"
},
{
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},
{
"state_after": "case refine'_1.intro\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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\np : F × F\nhp1 : f (g p.fst) = p.fst\nhp2 : f (g p.snd) = p.snd\n⊢ (fun x =>\n ↑↑f' (((fun p => (g p.fst, g p.snd)) x).fst - ((fun p => (g p.fst, g p.snd)) x).snd) -\n (f ((fun p => (g p.fst, g p.snd)) x).fst - f ((fun p => (g p.fst, g p.snd)) x).snd))\n p =\n ↑f' (g p.fst - g p.snd) - (p.fst - p.snd)",
"state_before": "case refine'_1\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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\n⊢ ∀ (x : F × F),\n f (g x.fst) = x.fst ∧ f (g x.snd) = x.snd →\n (fun x =>\n ↑↑f' (((fun p => (g p.fst, g p.snd)) x).fst - ((fun p => (g p.fst, g p.snd)) x).snd) -\n (f ((fun p => (g p.fst, g p.snd)) x).fst - f ((fun p => (g p.fst, g p.snd)) x).snd))\n x =\n ↑f' (g x.fst - g x.snd) - (x.fst - x.snd)",
"tactic": "rintro p ⟨hp1, hp2⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro\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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\np : F × F\nhp1 : f (g p.fst) = p.fst\nhp2 : f (g p.snd) = p.snd\n⊢ (fun x =>\n ↑↑f' (((fun p => (g p.fst, g p.snd)) x).fst - ((fun p => (g p.fst, g p.snd)) x).snd) -\n (f ((fun p => (g p.fst, g p.snd)) x).fst - f ((fun p => (g p.fst, g p.snd)) x).snd))\n p =\n ↑f' (g p.fst - g p.snd) - (p.fst - p.snd)",
"tactic": "simp [hp1, hp2]"
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{
"state_after": "case refine'_2\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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\n⊢ ∀ (x : F × F),\n f (g x.fst) = x.fst ∧ f (g x.snd) = x.snd →\n ((fun p => f p.fst - f p.snd) ∘ fun p => (g p.fst, g p.snd)) x = (fun p => p.fst - p.snd) x",
"state_before": "case refine'_2\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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\n⊢ (fun x => ((fun p => p.fst - p.snd) ∘ fun p => (g p.fst, g p.snd)) x) =O[𝓝 (a, a)] fun p => p.fst - p.snd",
"tactic": "refine (hf.isBigO_sub_rev.comp_tendsto hg).congr' (eventually_of_forall fun _ => rfl)\n (hfg.mono ?_)"
},
{
"state_after": "case refine'_2.intro\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.375202\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.375297\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\nf : E → F\nf' : E ≃L[𝕜] F\ng : F → E\na : F\nhf : HasStrictFDerivAt f (↑f') (g a)\nhg : ContinuousAt (fun p => (g p.fst, g p.snd)) (a, a)\nhfg : ∀ᶠ (p : F × F) in 𝓝 (a, a), f (g p.fst) = p.fst ∧ f (g p.snd) = p.snd\np : F × F\nhp1 : f (g p.fst) = p.fst\nhp2 : f (g p.snd) = p.snd\n⊢ ((fun p => f p.fst - f p.snd) ∘ fun p => (g p.fst, g p.snd)) p = (fun p => p.fst - p.snd) p",
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"tactic": "rintro p ⟨hp1, hp2⟩"
}
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Mathlib/Logic/Equiv/Fin.lean | finRotate_of_lt | [
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{
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/RingTheory/TensorProduct.lean | Module.endTensorEndAlgHom_apply | [
{
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Mathlib/LinearAlgebra/FreeModule/Rank.lean | rank_finsupp_self' | [
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1
] |
Mathlib/Algebra/Order/CompleteField.lean | LinearOrderedField.inducedMap_one | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.19004\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.19013\ninst✝³ : LinearOrderedField α\ninst✝² : ConditionallyCompleteLinearOrderedField β\ninst✝¹ : ConditionallyCompleteLinearOrderedField γ\ninst✝ : Archimedean α\n⊢ inducedMap α β 1 = 1",
"tactic": "exact_mod_cast inducedMap_rat α β 1"
}
] | [
209,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
209,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.deleteEdges_deleteEdges | [
{
"state_after": "case Adj.h.h.a\nι : Sort ?u.155613\n𝕜 : Type ?u.155616\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ns s' : Set (Sym2 V)\nx✝¹ x✝ : V\n⊢ Adj (deleteEdges (deleteEdges G s) s') x✝¹ x✝ ↔ Adj (deleteEdges G (s ∪ s')) x✝¹ x✝",
"state_before": "ι : Sort ?u.155613\n𝕜 : Type ?u.155616\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ns s' : Set (Sym2 V)\n⊢ deleteEdges (deleteEdges G s) s' = deleteEdges G (s ∪ s')",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a\nι : Sort ?u.155613\n𝕜 : Type ?u.155616\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ns s' : Set (Sym2 V)\nx✝¹ x✝ : V\n⊢ Adj (deleteEdges (deleteEdges G s) s') x✝¹ x✝ ↔ Adj (deleteEdges G (s ∪ s')) x✝¹ x✝",
"tactic": "simp [and_assoc, not_or]"
}
] | [
1135,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1132,
1
] |
Mathlib/LinearAlgebra/Isomorphisms.lean | LinearMap.quotKerEquivRange_apply_mk | [] | [
57,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
] |
Mathlib/Analysis/SpecialFunctions/Exp.lean | Complex.continuous_exp | [
{
"state_after": "case h'.h.e'_4\nz y✝ x✝ : ℝ\nx y : ℂ\na✝ : ‖y - x‖ < 1\n⊢ 2 * ‖exp x‖ * dist y x = (1 + 1) * ‖exp x‖ * ‖y - x‖",
"state_before": "z y✝ x✝ : ℝ\nx y : ℂ\n⊢ dist y x < 1 → dist (exp y) (exp x) ≤ 2 * ‖exp x‖ * dist y x",
"tactic": "convert locally_lipschitz_exp zero_le_one le_rfl x y using 2"
},
{
"state_after": "case h'.h.e'_4.e_a.e_a\nz y✝ x✝ : ℝ\nx y : ℂ\na✝ : ‖y - x‖ < 1\n⊢ 2 = 1 + 1",
"state_before": "case h'.h.e'_4\nz y✝ x✝ : ℝ\nx y : ℂ\na✝ : ‖y - x‖ < 1\n⊢ 2 * ‖exp x‖ * dist y x = (1 + 1) * ‖exp x‖ * ‖y - x‖",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h'.h.e'_4.e_a.e_a\nz y✝ x✝ : ℝ\nx y : ℂ\na✝ : ‖y - x‖ < 1\n⊢ 2 = 1 + 1",
"tactic": "ring"
}
] | [
77,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean | Set.ordConnected_dual | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3064\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns✝ t s : Set α\nh : OrdConnected (↑ofDual ⁻¹' s)\n⊢ OrdConnected s",
"tactic": "simpa only [ordConnected_def] using h.dual"
}
] | [
100,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.add_congr | [] | [
1773,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1771,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.coe_add | [] | [
790,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
789,
1
] |
Mathlib/Data/Num/Lemmas.lean | ZNum.cast_lt | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedRing α\nm n : ZNum\n⊢ ↑m < ↑n ↔ m < n",
"tactic": "rw [← cast_to_int m, ← cast_to_int n, Int.cast_lt, lt_to_int]"
}
] | [
1407,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1406,
1
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean | DirectSum.coe_decompose_mul_of_right_mem_of_le | [
{
"state_after": "case intro\nι : Type u_3\nR : Type ?u.367118\nA : Type u_1\nσ : Type u_2\ninst✝⁸ : Semiring A\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : CanonicallyOrderedAddMonoid ι\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝³ : GradedRing 𝒜\na : A\nn i : ι\ninst✝² : Sub ι\ninst✝¹ : OrderedSub ι\ninst✝ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : i ≤ n\nb : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (a * ↑b)) n) = ↑(↑(↑(decompose 𝒜) a) (n - i)) * ↑b",
"state_before": "ι : Type u_3\nR : Type ?u.367118\nA : Type u_1\nσ : Type u_2\ninst✝⁸ : Semiring A\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : CanonicallyOrderedAddMonoid ι\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝³ : GradedRing 𝒜\na b : A\nn i : ι\ninst✝² : Sub ι\ninst✝¹ : OrderedSub ι\ninst✝ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nb_mem : b ∈ 𝒜 i\nh : i ≤ n\n⊢ ↑(↑(↑(decompose 𝒜) (a * b)) n) = ↑(↑(↑(decompose 𝒜) a) (n - i)) * b",
"tactic": "lift b to 𝒜 i using b_mem"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_3\nR : Type ?u.367118\nA : Type u_1\nσ : Type u_2\ninst✝⁸ : Semiring A\ninst✝⁷ : DecidableEq ι\ninst✝⁶ : CanonicallyOrderedAddMonoid ι\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝³ : GradedRing 𝒜\na : A\nn i : ι\ninst✝² : Sub ι\ninst✝¹ : OrderedSub ι\ninst✝ : ContravariantClass ι ι (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\nh : i ≤ n\nb : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (a * ↑b)) n) = ↑(↑(↑(decompose 𝒜) a) (n - i)) * ↑b",
"tactic": "rwa [decompose_mul, decompose_coe, coe_mul_of_apply_of_le]"
}
] | [
324,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
321,
1
] |
Mathlib/Computability/TuringMachine.lean | Turing.TM2to1.addBottom_map | [
{
"state_after": "no goals",
"state_before": "K : Type ?u.595576\ninst✝² : DecidableEq K\nΓ : K → Type ?u.595590\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL : ListBlank ((k : K) → Option (Γ k))\n⊢ default.snd = default",
"tactic": "rfl"
},
{
"state_after": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL : ListBlank ((k : K) → Option (Γ k))\n⊢ ListBlank.cons (ListBlank.head L)\n (ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n (ListBlank.tail L))) =\n L",
"state_before": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL : ListBlank ((k : K) → Option (Γ k))\n⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) } (addBottom L) = L",
"tactic": "simp only [addBottom, ListBlank.map_cons]"
},
{
"state_after": "case h.e'_2.h.e'_4\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL : ListBlank ((k : K) → Option (Γ k))\n⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) =\n ListBlank.tail L",
"state_before": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL : ListBlank ((k : K) → Option (Γ k))\n⊢ ListBlank.cons (ListBlank.head L)\n (ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) }\n (ListBlank.tail L))) =\n L",
"tactic": "convert ListBlank.cons_head_tail L"
},
{
"state_after": "case h.e'_2.h.e'_4\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL L' : ListBlank ((k : K) → Option (Γ k))\n⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } L') =\n L'",
"state_before": "case h.e'_2.h.e'_4\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL : ListBlank ((k : K) → Option (Γ k))\n⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.tail L)) =\n ListBlank.tail L",
"tactic": "generalize ListBlank.tail L = L'"
},
{
"state_after": "case h.e'_2.h.e'_4\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL L' : ListBlank ((k : K) → Option (Γ k))\nl : List ((k : K) → Option (Γ k))\n⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.mk l)) =\n ListBlank.mk l",
"state_before": "case h.e'_2.h.e'_4\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL L' : ListBlank ((k : K) → Option (Γ k))\n⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } L') =\n L'",
"tactic": "refine' L'.induction_on fun l ↦ _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_4\nK : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type ?u.595593\ninst✝¹ : Inhabited Λ\nσ : Type ?u.595599\ninst✝ : Inhabited σ\nL L' : ListBlank ((k : K) → Option (Γ k))\nl : List ((k : K) → Option (Γ k))\n⊢ ListBlank.map { f := Prod.snd, map_pt' := (_ : default.snd = default.snd) }\n (ListBlank.map { f := Prod.mk false, map_pt' := (_ : (false, default) = (false, default)) } (ListBlank.mk l)) =\n ListBlank.mk l",
"tactic": "simp"
}
] | [
2378,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2373,
1
] |
Mathlib/CategoryTheory/Over.lean | CategoryTheory.Over.iteratedSliceBackward_forget_forget | [] | [
287,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
285,
1
] |
Mathlib/Data/Sigma/Basic.lean | Sigma.subtype_ext_iff | [] | [
87,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
] |
Mathlib/Data/PFun.lean | PFun.id_apply | [] | [
574,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
573,
1
] |
Mathlib/Data/Sym/Sym2.lean | Sym2.other_spec' | [
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.101059\nγ : Type ?u.101062\ninst✝ : DecidableEq α\na x✝ y✝ : α\nh : a ∈ Quotient.mk (Rel.setoid α) (x✝, y✝)\n⊢ Quotient.mk (Rel.setoid α) (a, Mem.other' h) = Quotient.mk (Rel.setoid α) (x✝, y✝)",
"state_before": "α : Type u_1\nβ : Type ?u.101059\nγ : Type ?u.101062\ninst✝ : DecidableEq α\na : α\nz : Sym2 α\nh : a ∈ z\n⊢ Quotient.mk (Rel.setoid α) (a, Mem.other' h) = z",
"tactic": "induction z using Sym2.ind"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.101059\nγ : Type ?u.101062\ninst✝ : DecidableEq α\na x✝ y✝ : α\nh : a ∈ Quotient.mk (Rel.setoid α) (x✝, y✝)\nh' : a = x✝ ∨ a = y✝\n⊢ Quotient.mk (Rel.setoid α) (a, Mem.other' h) = Quotient.mk (Rel.setoid α) (x✝, y✝)",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.101059\nγ : Type ?u.101062\ninst✝ : DecidableEq α\na x✝ y✝ : α\nh : a ∈ Quotient.mk (Rel.setoid α) (x✝, y✝)\n⊢ Quotient.mk (Rel.setoid α) (a, Mem.other' h) = Quotient.mk (Rel.setoid α) (x✝, y✝)",
"tactic": "have h' := mem_iff.mp h"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.101059\nγ : Type ?u.101062\ninst✝ : DecidableEq α\na x✝ y✝ : α\nh : a ∈ Quotient.mk (Rel.setoid α) (x✝, y✝)\nh' : a = x✝ ∨ a = y✝\n⊢ Quotient.mk (Rel.setoid α) (a, Mem.other' h) = Quotient.mk (Rel.setoid α) (x✝, y✝)",
"tactic": "aesop (add norm unfold [Quotient.rec, Quot.rec]) (rule_sets [Sym2])"
}
] | [
706,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
703,
1
] |
Mathlib/Data/Complex/Exponential.lean | Real.cos_sq' | [
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ cos x ^ 2 = 1 - sin x ^ 2",
"tactic": "rw [← sin_sq_add_cos_sq x, add_sub_cancel']"
}
] | [
1298,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1298,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean | CategoryTheory.GrothendieckTopology.transitive | [] | [
136,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
Mathlib/Algebra/Group/Basic.lean | mul_left_eq_self | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.14402\nβ : Type ?u.14405\nG : Type ?u.14408\nM : Type u\ninst✝ : RightCancelMonoid M\na b : M\n⊢ a * b = b ↔ a * b = 1 * b",
"tactic": "rw [one_mul]"
}
] | [
213,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
] |
Mathlib/Order/Filter/Extr.lean | IsMinOn.sub | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\ninst✝ : OrderedAddCommGroup β\nf g : α → β\na : α\ns : Set α\nl : Filter α\nhf : IsMinOn f s a\nhg : IsMaxOn g s a\n⊢ IsMinOn (fun x => f x - g x) s a",
"tactic": "simpa only [sub_eq_add_neg] using hf.add hg.neg"
}
] | [
515,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
514,
1
] |
Mathlib/Order/Monotone/Monovary.lean | Antivary.dual | [] | [
204,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
] |
Mathlib/Computability/Ackermann.lean | exists_lt_ack_of_nat_primrec | [
{
"state_after": "case zero\nf : ℕ → ℕ\n⊢ ∃ m, ∀ (n : ℕ), (fun x => 0) n < ack m n\n\ncase succ\nf : ℕ → ℕ\n⊢ ∃ m, ∀ (n : ℕ), succ n < ack m n\n\ncase left\nf : ℕ → ℕ\n⊢ ∃ m, ∀ (n : ℕ), (fun n => (unpair n).fst) n < ack m n\n\ncase right\nf : ℕ → ℕ\n⊢ ∃ m, ∀ (n : ℕ), (fun n => (unpair n).snd) n < ack m n\n\ncase pair\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : ∃ m, ∀ (n : ℕ), f n < ack m n\nIHg : ∃ m, ∀ (n : ℕ), g n < ack m n\n⊢ ∃ m, ∀ (n : ℕ), (fun n => pair (f n) (g n)) n < ack m n\n\ncase comp\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : ∃ m, ∀ (n : ℕ), f n < ack m n\nIHg : ∃ m, ∀ (n : ℕ), g n < ack m n\n⊢ ∃ m, ∀ (n : ℕ), (fun n => f (g n)) n < ack m n\n\ncase prec\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : ∃ m, ∀ (n : ℕ), f n < ack m n\nIHg : ∃ m, ∀ (n : ℕ), g n < ack m n\n⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n",
"state_before": "f : ℕ → ℕ\nhf : Nat.Primrec f\n⊢ ∃ m, ∀ (n : ℕ), f n < ack m n",
"tactic": "induction' hf with f g hf hg IHf IHg f g hf hg IHf IHg f g hf hg IHf IHg"
},
{
"state_after": "case pair.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\n⊢ ∃ m, ∀ (n : ℕ), (fun n => pair (f n) (g n)) n < ack m n\n\ncase comp.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\n⊢ ∃ m, ∀ (n : ℕ), (fun n => f (g n)) n < ack m n\n\ncase prec.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\n⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n",
"state_before": "case pair\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : ∃ m, ∀ (n : ℕ), f n < ack m n\nIHg : ∃ m, ∀ (n : ℕ), g n < ack m n\n⊢ ∃ m, ∀ (n : ℕ), (fun n => pair (f n) (g n)) n < ack m n\n\ncase comp\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : ∃ m, ∀ (n : ℕ), f n < ack m n\nIHg : ∃ m, ∀ (n : ℕ), g n < ack m n\n⊢ ∃ m, ∀ (n : ℕ), (fun n => f (g n)) n < ack m n\n\ncase prec\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : ∃ m, ∀ (n : ℕ), f n < ack m n\nIHg : ∃ m, ∀ (n : ℕ), g n < ack m n\n⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n",
"tactic": "all_goals cases' IHf with a ha; cases' IHg with b hb"
},
{
"state_after": "no goals",
"state_before": "case zero\nf : ℕ → ℕ\n⊢ ∃ m, ∀ (n : ℕ), (fun x => 0) n < ack m n",
"tactic": "exact ⟨0, ack_pos 0⟩"
},
{
"state_after": "case succ\nf : ℕ → ℕ\nn : ℕ\n⊢ succ n < ack 1 n",
"state_before": "case succ\nf : ℕ → ℕ\n⊢ ∃ m, ∀ (n : ℕ), succ n < ack m n",
"tactic": "refine' ⟨1, fun n => _⟩"
},
{
"state_after": "case succ\nf : ℕ → ℕ\nn : ℕ\n⊢ 1 + n < ack 1 n",
"state_before": "case succ\nf : ℕ → ℕ\nn : ℕ\n⊢ succ n < ack 1 n",
"tactic": "rw [succ_eq_one_add]"
},
{
"state_after": "no goals",
"state_before": "case succ\nf : ℕ → ℕ\nn : ℕ\n⊢ 1 + n < ack 1 n",
"tactic": "apply add_lt_ack"
},
{
"state_after": "case left\nf : ℕ → ℕ\nn : ℕ\n⊢ (fun n => (unpair n).fst) n < ack 0 n",
"state_before": "case left\nf : ℕ → ℕ\n⊢ ∃ m, ∀ (n : ℕ), (fun n => (unpair n).fst) n < ack m n",
"tactic": "refine' ⟨0, fun n => _⟩"
},
{
"state_after": "case left\nf : ℕ → ℕ\nn : ℕ\n⊢ (fun n => (unpair n).fst) n ≤ n",
"state_before": "case left\nf : ℕ → ℕ\nn : ℕ\n⊢ (fun n => (unpair n).fst) n < ack 0 n",
"tactic": "rw [ack_zero, lt_succ_iff]"
},
{
"state_after": "no goals",
"state_before": "case left\nf : ℕ → ℕ\nn : ℕ\n⊢ (fun n => (unpair n).fst) n ≤ n",
"tactic": "exact unpair_left_le n"
},
{
"state_after": "case right\nf : ℕ → ℕ\nn : ℕ\n⊢ (fun n => (unpair n).snd) n < ack 0 n",
"state_before": "case right\nf : ℕ → ℕ\n⊢ ∃ m, ∀ (n : ℕ), (fun n => (unpair n).snd) n < ack m n",
"tactic": "refine' ⟨0, fun n => _⟩"
},
{
"state_after": "case right\nf : ℕ → ℕ\nn : ℕ\n⊢ (fun n => (unpair n).snd) n ≤ n",
"state_before": "case right\nf : ℕ → ℕ\nn : ℕ\n⊢ (fun n => (unpair n).snd) n < ack 0 n",
"tactic": "rw [ack_zero, lt_succ_iff]"
},
{
"state_after": "no goals",
"state_before": "case right\nf : ℕ → ℕ\nn : ℕ\n⊢ (fun n => (unpair n).snd) n ≤ n",
"tactic": "exact unpair_right_le n"
},
{
"state_after": "case prec.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHg : ∃ m, ∀ (n : ℕ), g n < ack m n\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\n⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n",
"state_before": "case prec\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHf : ∃ m, ∀ (n : ℕ), f n < ack m n\nIHg : ∃ m, ∀ (n : ℕ), g n < ack m n\n⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n",
"tactic": "cases' IHf with a ha"
},
{
"state_after": "case prec.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\n⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n",
"state_before": "case prec.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\nIHg : ∃ m, ∀ (n : ℕ), g n < ack m n\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\n⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n",
"tactic": "cases' IHg with b hb"
},
{
"state_after": "case pair.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nn : ℕ\n⊢ max (f n) (g n) ≤ ack (max a b) n",
"state_before": "case pair.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\n⊢ ∃ m, ∀ (n : ℕ), (fun n => pair (f n) (g n)) n < ack m n",
"tactic": "refine'\n ⟨max a b + 3, fun n =>\n (pair_lt_max_add_one_sq _ _).trans_le <|\n (pow_le_pow_of_le_left (add_le_add_right _ _) 2).trans <|\n ack_add_one_sq_lt_ack_add_three _ _⟩"
},
{
"state_after": "case pair.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nn : ℕ\n⊢ max (f n) (g n) ≤ max (ack a n) (ack b n)",
"state_before": "case pair.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nn : ℕ\n⊢ max (f n) (g n) ≤ ack (max a b) n",
"tactic": "rw [max_ack_left]"
},
{
"state_after": "no goals",
"state_before": "case pair.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nn : ℕ\n⊢ max (f n) (g n) ≤ max (ack a n) (ack b n)",
"tactic": "exact max_le_max (ha n).le (hb n).le"
},
{
"state_after": "no goals",
"state_before": "case comp.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\n⊢ ∃ m, ∀ (n : ℕ), (fun n => f (g n)) n < ack m n",
"tactic": "exact\n ⟨max a b + 2, fun n =>\n (ha _).trans <| (ack_strictMono_right a <| hb n).trans <| ack_ack_lt_ack_max_add_two a b n⟩"
},
{
"state_after": "no goals",
"state_before": "case prec.intro.intro\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nthis : ∀ {m n : ℕ}, rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n⊢ ∃ m, ∀ (n : ℕ), unpaired (fun z n => rec (f z) (fun y IH => g (pair z (pair y IH))) n) n < ack m n",
"tactic": "exact ⟨max a b + 9, fun n => this.trans_le <| ack_mono_right _ <| unpair_add_le n⟩"
},
{
"state_after": "f✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\n⊢ rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)",
"state_before": "f✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\n⊢ ∀ {m n : ℕ}, rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)",
"tactic": "intro m n"
},
{
"state_after": "case zero\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm : ℕ\n⊢ rec (f m) (fun y IH => g (pair m (pair y IH))) zero < ack (max a b + 9) (m + zero)\n\ncase succ\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n⊢ rec (f m) (fun y IH => g (pair m (pair y IH))) (succ n) < ack (max a b + 9) (m + succ n)",
"state_before": "f✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\n⊢ rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)",
"tactic": "induction' n with n IH"
},
{
"state_after": "f✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm : ℕ\n⊢ max a b < max a b + 9",
"state_before": "case zero\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm : ℕ\n⊢ rec (f m) (fun y IH => g (pair m (pair y IH))) zero < ack (max a b + 9) (m + zero)",
"tactic": "apply (ha m).trans (ack_strictMono_left m <| (le_max_left a b).trans_lt _)"
},
{
"state_after": "no goals",
"state_before": "f✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm : ℕ\n⊢ max a b < max a b + 9",
"tactic": "linarith"
},
{
"state_after": "case succ\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n⊢ g (pair m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) < ack (max a b + 9) (m + succ n)",
"state_before": "case succ\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n⊢ rec (f m) (fun y IH => g (pair m (pair y IH))) (succ n) < ack (max a b + 9) (m + succ n)",
"tactic": "simp only [ge_iff_le]"
},
{
"state_after": "f✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n⊢ ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) ≤ ack (max a b + 9) (m + succ n)",
"state_before": "case succ\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n⊢ g (pair m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) < ack (max a b + 9) (m + succ n)",
"tactic": "apply (hb _).trans ((ack_pair_lt _ _ _).trans_le _)"
},
{
"state_after": "case inl\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : ?m.125091 < m\n⊢ ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) ≤ ack (max a b + 9) (m + succ n)\n\ncase inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ ?m.125091\n⊢ ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) ≤ ack (max a b + 9) (m + succ n)",
"state_before": "f✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\n⊢ ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) ≤ ack (max a b + 9) (m + succ n)",
"tactic": "cases' lt_or_le _ m with h₁ h₁"
},
{
"state_after": "case inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n⊢ ack (b + 4) (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) ≤ ack (max a b + 9) (m + succ n)",
"state_before": "case inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n⊢ ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) ≤ ack (max a b + 9) (m + succ n)",
"tactic": "rw [max_eq_right h₁]"
},
{
"state_after": "case inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n⊢ ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) ≤ ack (max a b + 9) (m + succ n)",
"state_before": "case inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n⊢ ack (b + 4) (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) ≤ ack (max a b + 9) (m + succ n)",
"tactic": "apply (ack_pair_lt _ _ _).le.trans"
},
{
"state_after": "case inr.inl\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : ?m.125452 < n\n⊢ ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) ≤ ack (max a b + 9) (m + succ n)\n\ncase inr.inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : n ≤ ?m.125452\n⊢ ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) ≤ ack (max a b + 9) (m + succ n)",
"state_before": "case inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\n⊢ ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) ≤ ack (max a b + 9) (m + succ n)",
"tactic": "cases' lt_or_le _ n with h₂ h₂"
},
{
"state_after": "case inr.inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : n ≤ rec (f m) (fun y IH => g (pair m (pair y IH))) n\n⊢ ack (b + 4 + 4) (rec (f m) (fun y IH => g (pair m (pair y IH))) n) ≤ ack (max a b + 9) (m + succ n)",
"state_before": "case inr.inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : n ≤ rec (f m) (fun y IH => g (pair m (pair y IH))) n\n⊢ ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) ≤ ack (max a b + 9) (m + succ n)",
"tactic": "rw [max_eq_right h₂]"
},
{
"state_after": "case inr.inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : n ≤ rec (f m) (fun y IH => g (pair m (pair y IH))) n\n⊢ ack (b + 4 + 4) (ack (max a b + 9) (m + n)) ≤ ack (max a b + 9) (m + succ n)",
"state_before": "case inr.inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : n ≤ rec (f m) (fun y IH => g (pair m (pair y IH))) n\n⊢ ack (b + 4 + 4) (rec (f m) (fun y IH => g (pair m (pair y IH))) n) ≤ ack (max a b + 9) (m + succ n)",
"tactic": "apply (ack_strictMono_right _ IH).le.trans"
},
{
"state_after": "case inr.inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : n ≤ rec (f m) (fun y IH => g (pair m (pair y IH))) n\n⊢ ack (b + (4 + 4)) (ack (max a b + 8 + 1) (m + n)) ≤ ack (max a b + 8) (ack (max a b + 8 + 1) (m + n))",
"state_before": "case inr.inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : n ≤ rec (f m) (fun y IH => g (pair m (pair y IH))) n\n⊢ ack (b + 4 + 4) (ack (max a b + 9) (m + n)) ≤ ack (max a b + 9) (m + succ n)",
"tactic": "rw [add_succ m, add_succ _ 8, succ_eq_add_one, succ_eq_add_one,\n ack_succ_succ (_ + 8), add_assoc]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : n ≤ rec (f m) (fun y IH => g (pair m (pair y IH))) n\n⊢ ack (b + (4 + 4)) (ack (max a b + 8 + 1) (m + n)) ≤ ack (max a b + 8) (ack (max a b + 8 + 1) (m + n))",
"tactic": "exact ack_mono_left _ (Nat.add_le_add (le_max_right a b) le_rfl)"
},
{
"state_after": "case inl\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m\n⊢ ack (b + 4) m ≤ ack (max a b + 9) (m + succ n)",
"state_before": "case inl\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : ?m.125091 < m\n⊢ ack (b + 4) (max m (pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n))) ≤ ack (max a b + 9) (m + succ n)",
"tactic": "rw [max_eq_left h₁.le]"
},
{
"state_after": "no goals",
"state_before": "case inl\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m\n⊢ ack (b + 4) m ≤ ack (max a b + 9) (m + succ n)",
"tactic": "exact ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num)\n (self_le_add_right m _)"
},
{
"state_after": "no goals",
"state_before": "f✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n) < m\n⊢ 4 ≤ 9",
"tactic": "norm_num"
},
{
"state_after": "case inr.inl\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n\n⊢ ack (b + (4 + 4)) n ≤ ack (max a b + 9) (m + succ n)",
"state_before": "case inr.inl\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : ?m.125452 < n\n⊢ ack (b + 4 + 4) (max n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)) ≤ ack (max a b + 9) (m + succ n)",
"tactic": "rw [max_eq_left h₂.le, add_assoc]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nf✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n\n⊢ ack (b + (4 + 4)) n ≤ ack (max a b + 9) (m + succ n)",
"tactic": "exact\n ack_le_ack (Nat.add_le_add (le_max_right a b) <| by norm_num)\n ((le_succ n).trans <| self_le_add_left _ _)"
},
{
"state_after": "no goals",
"state_before": "f✝ f g : ℕ → ℕ\nhf : Nat.Primrec f\nhg : Nat.Primrec g\na : ℕ\nha : ∀ (n : ℕ), f n < ack a n\nb : ℕ\nhb : ∀ (n : ℕ), g n < ack b n\nm n : ℕ\nIH : rec (f m) (fun y IH => g (pair m (pair y IH))) n < ack (max a b + 9) (m + n)\nh₁ : m ≤ pair n (rec (f m) (fun y IH => g (pair m (pair y IH))) n)\nh₂ : rec (f m) (fun y IH => g (pair m (pair y IH))) n < n\n⊢ 4 + 4 ≤ 9",
"tactic": "norm_num"
}
] | [
380,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean | smul_ite | [
{
"state_after": "no goals",
"state_before": "M : Type u_2\nN : Type ?u.14217\nG : Type ?u.14220\nA : Type ?u.14223\nB : Type ?u.14226\nα : Type u_1\nβ : Type ?u.14232\nγ : Type ?u.14235\nδ : Type ?u.14238\ninst✝¹ : SMul M α\np : Prop\ninst✝ : Decidable p\na : M\nb₁ b₂ : α\n⊢ (a • if p then b₁ else b₂) = if p then a • b₁ else a • b₂",
"tactic": "split_ifs <;> rfl"
}
] | [
462,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
461,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | HomogeneousIdeal.toIdeal_iSup₂ | [
{
"state_after": "no goals",
"state_before": "ι : Type u_3\nσ : Type u_4\nR : Type ?u.163506\nA : Type u_5\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nκ : Sort u_1\nκ' : κ → Sort u_2\ns : (i : κ) → κ' i → HomogeneousIdeal 𝒜\n⊢ toIdeal (⨆ (i : κ) (j : κ' i), s i j) = ⨆ (i : κ) (j : κ' i), toIdeal (s i j)",
"tactic": "simp_rw [toIdeal_iSup]"
}
] | [
408,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
406,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean | Set.infsep_pair_eq_toReal | [
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.57659\ninst✝ : PseudoEMetricSpace α\nx y : α\ns : Set α\nhxy : x = y\n⊢ infsep {x, y} = ENNReal.toReal (edist x y)\n\ncase neg\nα : Type u_1\nβ : Type ?u.57659\ninst✝ : PseudoEMetricSpace α\nx y : α\ns : Set α\nhxy : ¬x = y\n⊢ infsep {x, y} = ENNReal.toReal (edist x y)",
"state_before": "α : Type u_1\nβ : Type ?u.57659\ninst✝ : PseudoEMetricSpace α\nx y : α\ns : Set α\n⊢ infsep {x, y} = ENNReal.toReal (edist x y)",
"tactic": "by_cases hxy : x = y"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.57659\ninst✝ : PseudoEMetricSpace α\nx y : α\ns : Set α\nhxy : x = y\n⊢ infsep {y, y} = ENNReal.toReal (edist y y)",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.57659\ninst✝ : PseudoEMetricSpace α\nx y : α\ns : Set α\nhxy : x = y\n⊢ infsep {x, y} = ENNReal.toReal (edist x y)",
"tactic": "rw [hxy]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.57659\ninst✝ : PseudoEMetricSpace α\nx y : α\ns : Set α\nhxy : x = y\n⊢ infsep {y, y} = ENNReal.toReal (edist y y)",
"tactic": "simp only [infsep_singleton, pair_eq_singleton, edist_self, ENNReal.zero_toReal]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.57659\ninst✝ : PseudoEMetricSpace α\nx y : α\ns : Set α\nhxy : ¬x = y\n⊢ infsep {x, y} = ENNReal.toReal (edist x y)",
"tactic": "rw [infsep, einfsep_pair hxy]"
}
] | [
389,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
385,
1
] |
Mathlib/Order/WithBot.lean | WithTop.ofDual_apply_coe | [] | [
690,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
689,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean | Filter.Tendsto.lineMap | [] | [
312,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
309,
1
] |
Mathlib/Order/Bounds/Basic.lean | mem_lowerBounds_image2_of_mem_upperBounds | [] | [
1464,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1462,
1
] |
Mathlib/Data/Fintype/Pi.lean | Fintype.piFinset_subsingleton | [] | [
72,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/Data/List/Infix.lean | List.isPrefix.map | [
{
"state_after": "case nil\nα : Type u_1\nβ : Type u_2\nl l₁ l₂✝ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂✝\nf : α → β\nl₂ : List α\nh : [] <+: l₂\n⊢ List.map f [] <+: List.map f l₂\n\ncase cons\nα : Type u_1\nβ : Type u_2\nl l₁ l₂✝ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂✝\nf : α → β\nhd : α\ntl : List α\nhl : ∀ {l₂ : List α}, tl <+: l₂ → List.map f tl <+: List.map f l₂\nl₂ : List α\nh : hd :: tl <+: l₂\n⊢ List.map f (hd :: tl) <+: List.map f l₂",
"state_before": "α : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nh : l₁ <+: l₂\nf : α → β\n⊢ List.map f l₁ <+: List.map f l₂",
"tactic": "induction' l₁ with hd tl hl generalizing l₂"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type u_2\nl l₁ l₂✝ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂✝\nf : α → β\nl₂ : List α\nh : [] <+: l₂\n⊢ List.map f [] <+: List.map f l₂",
"tactic": "simp only [nil_prefix, map_nil]"
},
{
"state_after": "case cons.nil\nα : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂\nf : α → β\nhd : α\ntl : List α\nhl : ∀ {l₂ : List α}, tl <+: l₂ → List.map f tl <+: List.map f l₂\nh : hd :: tl <+: []\n⊢ List.map f (hd :: tl) <+: List.map f []\n\ncase cons.cons\nα : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂\nf : α → β\nhd : α\ntl : List α\nhl : ∀ {l₂ : List α}, tl <+: l₂ → List.map f tl <+: List.map f l₂\nhd₂ : α\ntl₂ : List α\nh : hd :: tl <+: hd₂ :: tl₂\n⊢ List.map f (hd :: tl) <+: List.map f (hd₂ :: tl₂)",
"state_before": "case cons\nα : Type u_1\nβ : Type u_2\nl l₁ l₂✝ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂✝\nf : α → β\nhd : α\ntl : List α\nhl : ∀ {l₂ : List α}, tl <+: l₂ → List.map f tl <+: List.map f l₂\nl₂ : List α\nh : hd :: tl <+: l₂\n⊢ List.map f (hd :: tl) <+: List.map f l₂",
"tactic": "cases' l₂ with hd₂ tl₂"
},
{
"state_after": "no goals",
"state_before": "case cons.nil\nα : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂\nf : α → β\nhd : α\ntl : List α\nhl : ∀ {l₂ : List α}, tl <+: l₂ → List.map f tl <+: List.map f l₂\nh : hd :: tl <+: []\n⊢ List.map f (hd :: tl) <+: List.map f []",
"tactic": "simpa only using eq_nil_of_prefix_nil h"
},
{
"state_after": "case cons.cons\nα : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂\nf : α → β\nhd : α\ntl : List α\nhl : ∀ {l₂ : List α}, tl <+: l₂ → List.map f tl <+: List.map f l₂\nhd₂ : α\ntl₂ : List α\nh : hd = hd₂ ∧ tl <+: tl₂\n⊢ List.map f (hd :: tl) <+: List.map f (hd₂ :: tl₂)",
"state_before": "case cons.cons\nα : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂\nf : α → β\nhd : α\ntl : List α\nhl : ∀ {l₂ : List α}, tl <+: l₂ → List.map f tl <+: List.map f l₂\nhd₂ : α\ntl₂ : List α\nh : hd :: tl <+: hd₂ :: tl₂\n⊢ List.map f (hd :: tl) <+: List.map f (hd₂ :: tl₂)",
"tactic": "rw [cons_prefix_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case cons.cons\nα : Type u_1\nβ : Type u_2\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\nh✝ : l₁ <+: l₂\nf : α → β\nhd : α\ntl : List α\nhl : ∀ {l₂ : List α}, tl <+: l₂ → List.map f tl <+: List.map f l₂\nhd₂ : α\ntl₂ : List α\nh : hd = hd₂ ∧ tl <+: tl₂\n⊢ List.map f (hd :: tl) <+: List.map f (hd₂ :: tl₂)",
"tactic": "simp only [List.map_cons, h, prefix_cons_inj, hl, map]"
}
] | [
290,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
284,
1
] |
src/lean/Init/SimpLemmas.lean | Bool.and_self | [
{
"state_after": "no goals",
"state_before": "b : Bool\n⊢ (b && b) = b",
"tactic": "cases b <;> rfl"
}
] | [
114,
87
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
114,
9
] |
Mathlib/Algebra/Ring/Defs.lean | ite_mul | [
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nR : Type x\nα : Type u_1\ninst✝¹ : Mul α\nP : Prop\ninst✝ : Decidable P\na b c : α\n⊢ (if P then a else b) * c = if P then a * c else b * c",
"tactic": "split_ifs <;> rfl"
}
] | [
205,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
204,
1
] |
Mathlib/Logic/Relation.lean | Relation.TransGen.lift' | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.23283\nδ : Type ?u.23286\nr : α → α → Prop\na✝ b✝ c d : α\np : β → β → Prop\na b : α\nf : α → β\nh : ∀ (a b : α), r a b → TransGen p (f a) (f b)\nhab : TransGen r a b\n⊢ TransGen p (f a) (f b)",
"tactic": "simpa [transGen_idem] using hab.lift f h"
}
] | [
484,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
481,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearEquiv.preimage_symm_preimage | [] | [
2133,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2131,
11
] |
Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | ZMod.neg_one_pow_div_two_of_one_mod_four | [
{
"state_after": "n : ℕ\nhn : n % 4 = 1\n⊢ ↑χ₄ ↑1 = 1",
"state_before": "n : ℕ\nhn : n % 4 = 1\n⊢ (-1) ^ (n / 2) = 1",
"tactic": "rw [← χ₄_eq_neg_one_pow (Nat.odd_of_mod_four_eq_one hn), ← nat_cast_mod, hn]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n % 4 = 1\n⊢ ↑χ₄ ↑1 = 1",
"tactic": "rfl"
}
] | [
124,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
1
] |
Mathlib/Topology/ContinuousOn.lean | frequently_nhdsWithin_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.711\nγ : Type ?u.714\nδ : Type ?u.717\ninst✝ : TopologicalSpace α\nz : α\ns : Set α\np : α → Prop\n⊢ (∃ᶠ (x : α) in 𝓝[s] z, p x) ↔ ∃ᶠ (x : α) in 𝓝 z, p x ∧ x ∈ s",
"tactic": "simp only [Filter.Frequently, eventually_nhdsWithin_iff, not_and']"
}
] | [
59,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
57,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | CategoryTheory.Limits.mono_of_isLimit_fork | [] | [
838,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
837,
1
] |
Std/Data/Nat/Lemmas.lean | Nat.mod_two_eq_zero_or_one | [
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ 2 > 0",
"tactic": "simp"
}
] | [
244,
21
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
241,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean | Set.preimage_neg_Ioc | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ -Ioc a b = Ico (-b) (-a)",
"tactic": "simp [← Ioi_inter_Iic, ← Ici_inter_Iio, inter_comm]"
}
] | [
163,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
162,
1
] |
Mathlib/Analysis/SpecialFunctions/Exponential.lean | hasFDerivAt_exp_smul_const' | [] | [
391,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
387,
1
] |
Mathlib/Order/LocallyFinite.lean | Finset.coe_Iio | [] | [
435,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
434,
1
] |
Mathlib/Data/Fin/Basic.lean | Fin.rev_involutive | [] | [
455,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
454,
1
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean | DirectSum.coe_decompose_mul_of_right_mem_of_not_le | [
{
"state_after": "case intro\nι : Type u_3\nR : Type ?u.348167\nA : Type u_1\nσ : Type u_2\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : CanonicallyOrderedAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\na : A\nn i : ι\nh : ¬i ≤ n\nb : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (a * ↑b)) n) = 0",
"state_before": "ι : Type u_3\nR : Type ?u.348167\nA : Type u_1\nσ : Type u_2\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : CanonicallyOrderedAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\na b : A\nn i : ι\nb_mem : b ∈ 𝒜 i\nh : ¬i ≤ n\n⊢ ↑(↑(↑(decompose 𝒜) (a * b)) n) = 0",
"tactic": "lift b to 𝒜 i using b_mem"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type u_3\nR : Type ?u.348167\nA : Type u_1\nσ : Type u_2\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : CanonicallyOrderedAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\na : A\nn i : ι\nh : ¬i ≤ n\nb : { x // x ∈ 𝒜 i }\n⊢ ↑(↑(↑(decompose 𝒜) (a * ↑b)) n) = 0",
"tactic": "rwa [decompose_mul, decompose_coe, coe_mul_of_apply_of_not_le]"
}
] | [
310,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
307,
1
] |
Mathlib/Deprecated/Group.lean | IsGroupHom.map_one | [] | [
295,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
294,
1
] |
Mathlib/Data/PEquiv.lean | PEquiv.ofSet_eq_some_iff | [] | [
239,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
237,
1
] |
Mathlib/Data/Finset/Sort.lean | Finset.orderEmbOfCardLe_mem | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.61043\ninst✝ : LinearOrder α\ns : Finset α\nk : ℕ\nh : k ≤ card s\na : Fin k\n⊢ ↑(orderEmbOfCardLe s h) a ∈ s",
"tactic": "simp only [orderEmbOfCardLe, RelEmbedding.coe_trans, Finset.orderEmbOfFin_mem,\n Function.comp_apply]"
}
] | [
254,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
251,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | TopCat.Presheaf.SheafConditionEqualizerProducts.fork_π_app_walkingParallelPair_one | [] | [
139,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/Data/Fintype/Basic.lean | Fintype.univ_bool | [] | [
903,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
902,
1
] |
Mathlib/Data/Multiset/Powerset.lean | Multiset.powersetLen_zero_left | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Multiset α\nl : List α\n⊢ powersetLen 0 (Quotient.mk (isSetoid α) l) = {0}",
"tactic": "simp [powersetLen_coe']"
}
] | [
247,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
246,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.rel_eq_refl | [] | [
2680,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2679,
1
] |
Mathlib/RingTheory/FinitePresentation.lean | Algebra.FinitePresentation.quotient | [
{
"state_after": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI : Ideal A\nh : Ideal.FG I\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\n⊢ FinitePresentation R (A ⧸ I)",
"state_before": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI : Ideal A\nh : Ideal.FG I\nhfp : FinitePresentation R A\n⊢ FinitePresentation R (A ⧸ I)",
"tactic": "obtain ⟨n, f, hf⟩ := hfp"
},
{
"state_after": "case intro.intro.refine'_1\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI : Ideal A\nh : Ideal.FG I\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\n⊢ Surjective ↑(AlgHom.comp (Ideal.Quotient.mkₐ R I) f)\n\ncase intro.intro.refine'_2\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI : Ideal A\nh : Ideal.FG I\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\n⊢ Ideal.FG (RingHom.ker ↑(AlgHom.comp (Ideal.Quotient.mkₐ R I) f))",
"state_before": "case intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI : Ideal A\nh : Ideal.FG I\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\n⊢ FinitePresentation R (A ⧸ I)",
"tactic": "refine' ⟨n, (Ideal.Quotient.mkₐ R I).comp f, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI : Ideal A\nh : Ideal.FG I\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\n⊢ Surjective ↑(AlgHom.comp (Ideal.Quotient.mkₐ R I) f)",
"tactic": "exact (Ideal.Quotient.mkₐ_surjective R I).comp hf.1"
},
{
"state_after": "case intro.intro.refine'_2\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI : Ideal A\nh : Ideal.FG I\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\n⊢ Ideal.FG (RingHom.ker ↑(Ideal.Quotient.mkₐ R I))",
"state_before": "case intro.intro.refine'_2\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI : Ideal A\nh : Ideal.FG I\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\n⊢ Ideal.FG (RingHom.ker ↑(AlgHom.comp (Ideal.Quotient.mkₐ R I) f))",
"tactic": "refine' Ideal.fg_ker_comp _ _ hf.2 _ hf.1"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nI : Ideal A\nh : Ideal.FG I\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] A\nhf : Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\n⊢ Ideal.FG (RingHom.ker ↑(Ideal.Quotient.mkₐ R I))",
"tactic": "simp [h]"
}
] | [
137,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | CategoryTheory.Limits.WidePushout.ι_desc | [
{
"state_after": "no goals",
"state_before": "J : Type w\nC : Type u\ninst✝² : Category C\nD : Type u_1\ninst✝¹ : Category D\nB : D\nobjs : J → D\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : D\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\nj : J\n⊢ ι arrows j ≫ desc f fs w = fs j",
"tactic": "simp only [colimit.ι_desc, WidePushoutShape.mkCocone_pt, WidePushoutShape.mkCocone_ι_app]"
}
] | [
422,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
421,
1
] |
Mathlib/Data/Fintype/Basic.lean | exists_seq_of_forall_finset_exists | [
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"tactic": "classical\n have : Nonempty α := by\n rcases h ∅ (by simp) with ⟨y, _⟩\n exact ⟨y⟩\n choose! F hF using h\n have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩\n set f := seqOfForallFinsetExistsAux P r h' with hf\n have A : ∀ n : ℕ, P (f n) := by\n intro n\n induction' n using Nat.strong_induction_on with n IH\n have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2\n rw [hf, seqOfForallFinsetExistsAux]\n exact\n (Classical.choose_spec\n (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n))))\n (by simp [IH'])).1\n refine' ⟨f, A, fun m n hmn => _⟩\n conv_rhs => rw [hf]\n rw [seqOfForallFinsetExistsAux]\n apply\n (Classical.choose_spec\n (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n)))) (by simp [A])).2\n exact Finset.mem_image.2 ⟨⟨m, hmn⟩, Finset.mem_univ _, rfl⟩"
},
{
"state_after": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nthis : Nonempty α\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"tactic": "have : Nonempty α := by\n rcases h ∅ (by simp) with ⟨y, _⟩\n exact ⟨y⟩"
},
{
"state_after": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\nthis : Nonempty α\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"tactic": "choose! F hF using h"
},
{
"state_after": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"tactic": "have h' : ∀ s : Finset α, ∃ y, (∀ x ∈ s, P x) → P y ∧ ∀ x ∈ s, r x y := fun s => ⟨F s, hF s⟩"
},
{
"state_after": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"tactic": "set f := seqOfForallFinsetExistsAux P r h' with hf"
},
{
"state_after": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"tactic": "have A : ∀ n : ℕ, P (f n) := by\n intro n\n induction' n using Nat.strong_induction_on with n IH\n have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2\n rw [hf, seqOfForallFinsetExistsAux]\n exact\n (Classical.choose_spec\n (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n))))\n (by simp [IH'])).1"
},
{
"state_after": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\nm n : ℕ\nhmn : m < n\n⊢ r (f m) (f n)",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\n⊢ ∃ f, (∀ (n : ℕ), P (f n)) ∧ ∀ (m n : ℕ), m < n → r (f m) (f n)",
"tactic": "refine' ⟨f, A, fun m n hmn => _⟩"
},
{
"state_after": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\nm n : ℕ\nhmn : m < n\n⊢ r (f m) (seqOfForallFinsetExistsAux P r h' n)",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\nm n : ℕ\nhmn : m < n\n⊢ r (f m) (f n)",
"tactic": "conv_rhs => rw [hf]"
},
{
"state_after": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\nm n : ℕ\nhmn : m < n\n⊢ r (f m)\n (Classical.choose\n (_ :\n ∃ y,\n (∀ (x : α), x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ → P x) →\n P y ∧ ∀ (x : α), x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ → r x y))",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\nm n : ℕ\nhmn : m < n\n⊢ r (f m) (seqOfForallFinsetExistsAux P r h' n)",
"tactic": "rw [seqOfForallFinsetExistsAux]"
},
{
"state_after": "case a\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\nm n : ℕ\nhmn : m < n\n⊢ f m ∈ image (fun i => f ↑i) univ",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\nm n : ℕ\nhmn : m < n\n⊢ r (f m)\n (Classical.choose\n (_ :\n ∃ y,\n (∀ (x : α), x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ → P x) →\n P y ∧ ∀ (x : α), x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ → r x y))",
"tactic": "apply\n (Classical.choose_spec\n (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n)))) (by simp [A])).2"
},
{
"state_after": "no goals",
"state_before": "case a\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\nm n : ℕ\nhmn : m < n\n⊢ f m ∈ image (fun i => f ↑i) univ",
"tactic": "exact Finset.mem_image.2 ⟨⟨m, hmn⟩, Finset.mem_univ _, rfl⟩"
},
{
"state_after": "case intro\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\ny : α\nh✝ : P y ∧ ∀ (x : α), x ∈ ∅ → r x y\n⊢ Nonempty α",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\n⊢ Nonempty α",
"tactic": "rcases h ∅ (by simp) with ⟨y, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\ny : α\nh✝ : P y ∧ ∀ (x : α), x ∈ ∅ → r x y\n⊢ Nonempty α",
"tactic": "exact ⟨y⟩"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nh : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → ∃ y, P y ∧ ∀ (x : α), x ∈ s → r x y\n⊢ ∀ (x : α), x ∈ ∅ → P x",
"tactic": "simp"
},
{
"state_after": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nn : ℕ\n⊢ P (f n)",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\n⊢ ∀ (n : ℕ), P (f n)",
"tactic": "intro n"
},
{
"state_after": "case h\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nn : ℕ\nIH : ∀ (m : ℕ), m < n → P (f m)\n⊢ P (f n)",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nn : ℕ\n⊢ P (f n)",
"tactic": "induction' n using Nat.strong_induction_on with n IH"
},
{
"state_after": "case h\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nn : ℕ\nIH : ∀ (m : ℕ), m < n → P (f m)\nIH' : ∀ (x : Fin n), P (f ↑x)\n⊢ P (f n)",
"state_before": "case h\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nn : ℕ\nIH : ∀ (m : ℕ), m < n → P (f m)\n⊢ P (f n)",
"tactic": "have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2"
},
{
"state_after": "case h\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nn : ℕ\nIH : ∀ (m : ℕ), m < n → P (f m)\nIH' : ∀ (x : Fin n), P (f ↑x)\n⊢ P\n (Classical.choose\n (_ :\n ∃ y,\n (∀ (x : α), x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ → P x) →\n P y ∧ ∀ (x : α), x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ → r x y))",
"state_before": "case h\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nn : ℕ\nIH : ∀ (m : ℕ), m < n → P (f m)\nIH' : ∀ (x : Fin n), P (f ↑x)\n⊢ P (f n)",
"tactic": "rw [hf, seqOfForallFinsetExistsAux]"
},
{
"state_after": "no goals",
"state_before": "case h\nα✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nn : ℕ\nIH : ∀ (m : ℕ), m < n → P (f m)\nIH' : ∀ (x : Fin n), P (f ↑x)\n⊢ P\n (Classical.choose\n (_ :\n ∃ y,\n (∀ (x : α), x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ → P x) →\n P y ∧ ∀ (x : α), x ∈ image (fun i => seqOfForallFinsetExistsAux P r h' ↑i) univ → r x y))",
"tactic": "exact\n (Classical.choose_spec\n (h' (Finset.image (fun i : Fin n => f i) (Finset.univ : Finset (Fin n))))\n (by simp [IH'])).1"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nn : ℕ\nIH : ∀ (m : ℕ), m < n → P (f m)\nIH' : ∀ (x : Fin n), P (f ↑x)\n⊢ ∀ (x : α), x ∈ image (fun i => f ↑i) univ → P x",
"tactic": "simp [IH']"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.153481\nβ : Type ?u.153484\nγ : Type ?u.153487\nα : Type u_1\nP : α → Prop\nr : α → α → Prop\nthis : Nonempty α\nF : Finset α → α\nhF : ∀ (s : Finset α), (∀ (x : α), x ∈ s → P x) → P (F s) ∧ ∀ (x : α), x ∈ s → r x (F s)\nh' : ∀ (s : Finset α), ∃ y, (∀ (x : α), x ∈ s → P x) → P y ∧ ∀ (x : α), x ∈ s → r x y\nf : ℕ → α := seqOfForallFinsetExistsAux P r h'\nhf : f = seqOfForallFinsetExistsAux P r h'\nA : ∀ (n : ℕ), P (f n)\nm n : ℕ\nhmn : m < n\n⊢ ∀ (x : α), x ∈ image (fun i => f ↑i) univ → P x",
"tactic": "simp [A]"
}
] | [
1242,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1217,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean | MeasureTheory.integrableOn_zero | [] | [
118,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
117,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean | lp.norm_const_smul_le | [
{
"state_after": "case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : 0 ≠ 0\nf : { x // x ∈ lp E 0 }\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖\n\ncase inr.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖\n\ncase inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "rcases p.trichotomy with (rfl | rfl | hp)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : 0 ≠ 0\nf : { x // x ∈ lp E 0 }\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "exact absurd rfl hp"
},
{
"state_after": "case inr.inl.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : IsEmpty α\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖\n\ncase inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"state_before": "case inr.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "cases isEmpty_or_nonempty α"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "have hcf := lp.isLUB_norm (c • f)"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB ((fun b => ‖c‖ * b) '' Set.range fun i => ‖↑f i‖) (‖c‖ * ‖f‖)\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "have hfc := (lp.isLUB_norm f).mul_left (norm_nonneg c)"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB ((fun b => ‖c‖ * b) '' Set.range fun i => ‖↑f i‖) (‖c‖ * ‖f‖)\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "simp_rw [← Set.range_comp, Function.comp] at hfc"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\n⊢ ‖c‖ * ‖f‖ ∈ upperBounds (Set.range fun i => ‖↑(c • f) i‖)",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "refine' hcf.right _"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\nthis : ‖c‖ * ‖f‖ ∈ upperBounds (Set.range fun x => ‖c‖ * ‖↑f x‖)\n⊢ ‖c‖ * ‖f‖ ∈ upperBounds (Set.range fun i => ‖↑(c • f) i‖)",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\n⊢ ‖c‖ * ‖f‖ ∈ upperBounds (Set.range fun i => ‖↑(c • f) i‖)",
"tactic": "have := hfc.left"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\nthis : ∀ (a : α), ‖c‖ * ‖↑f a‖ ≤ ‖c‖ * ‖f‖\n⊢ ∀ (a : α), ‖↑(c • f) a‖ ≤ ‖c‖ * ‖f‖",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\nthis : ‖c‖ * ‖f‖ ∈ upperBounds (Set.range fun x => ‖c‖ * ‖↑f x‖)\n⊢ ‖c‖ * ‖f‖ ∈ upperBounds (Set.range fun i => ‖↑(c • f) i‖)",
"tactic": "simp_rw [mem_upperBounds, Set.mem_range, forall_exists_index, forall_apply_eq_imp_iff'] at this⊢"
},
{
"state_after": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\nthis : ∀ (a : α), ‖c‖ * ‖↑f a‖ ≤ ‖c‖ * ‖f‖\na : α\n⊢ ‖↑(c • f) a‖ ≤ ‖c‖ * ‖f‖",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\nthis : ∀ (a : α), ‖c‖ * ‖↑f a‖ ≤ ‖c‖ * ‖f‖\n⊢ ∀ (a : α), ‖↑(c • f) a‖ ≤ ‖c‖ * ‖f‖",
"tactic": "intro a"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inr\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : Nonempty α\nhcf : IsLUB (Set.range fun i => ‖↑(c • f) i‖) ‖c • f‖\nhfc : IsLUB (Set.range fun x => ‖c‖ * ‖↑f x‖) (‖c‖ * ‖f‖)\nthis : ∀ (a : α), ‖c‖ * ‖↑f a‖ ≤ ‖c‖ * ‖f‖\na : α\n⊢ ‖↑(c • f) a‖ ≤ ‖c‖ * ‖f‖",
"tactic": "exact (norm_smul_le _ _).trans (this a)"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nc : 𝕜\nhp : ⊤ ≠ 0\nf : { x // x ∈ lp E ⊤ }\nh✝ : IsEmpty α\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "simp [lp.eq_zero' f]"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p } := { nnnorm := fun f => { val := ‖f‖, property := (_ : 0 ≤ ‖f‖) } }\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "letI inst : NNNorm (lp E p) := ⟨fun f => ⟨‖f‖, norm_nonneg' _⟩⟩"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p } := { nnnorm := fun f => { val := ‖f‖, property := (_ : 0 ≤ ‖f‖) } }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p } := { nnnorm := fun f => { val := ‖f‖, property := (_ : 0 ≤ ‖f‖) } }\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "have coe_nnnorm : ∀ f : lp E p, ↑‖f‖₊ = ‖f‖ := fun _ => rfl"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p } := { nnnorm := fun f => { val := ‖f‖, property := (_ : 0 ≤ ‖f‖) } }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ (‖c‖₊ * ‖f‖₊) ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p } := { nnnorm := fun f => { val := ‖f‖, property := (_ : 0 ≤ ‖f‖) } }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "suffices ‖c • f‖₊ ^ p.toReal ≤ (‖c‖₊ * ‖f‖₊) ^ p.toReal by\n rwa [NNReal.rpow_le_rpow_iff hp] at this"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ (‖c‖₊ * ‖f‖₊) ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p } := { nnnorm := fun f => { val := ‖f‖, property := (_ : 0 ≤ ‖f‖) } }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ (‖c‖₊ * ‖f‖₊) ^ ENNReal.toReal p",
"tactic": "clear_value inst"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ (‖c‖₊ * ‖f‖₊) ^ ENNReal.toReal p",
"tactic": "rw [NNReal.mul_rpow]"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhLHS : HasSum (fun i => ‖↑(c • f) i‖ ^ ENNReal.toReal p) (‖c • f‖ ^ ENNReal.toReal p)\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p",
"tactic": "have hLHS := lp.hasSum_norm hp (c • f)"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhLHS : HasSum (fun i => ‖↑(c • f) i‖ ^ ENNReal.toReal p) (‖c • f‖ ^ ENNReal.toReal p)\nhRHS :\n HasSum (fun i => ‖c‖ ^ ENNReal.toReal p * ‖↑f i‖ ^ ENNReal.toReal p) (‖c‖ ^ ENNReal.toReal p * ‖f‖ ^ ENNReal.toReal p)\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhLHS : HasSum (fun i => ‖↑(c • f) i‖ ^ ENNReal.toReal p) (‖c • f‖ ^ ENNReal.toReal p)\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p",
"tactic": "have hRHS := (lp.hasSum_norm hp f).mul_left (‖c‖ ^ p.toReal)"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhLHS : HasSum (fun i => ‖↑(c • f) i‖ ^ ENNReal.toReal p) (‖c • f‖ ^ ENNReal.toReal p)\nhRHS :\n HasSum (fun i => ‖c‖ ^ ENNReal.toReal p * ‖↑f i‖ ^ ENNReal.toReal p) (‖c‖ ^ ENNReal.toReal p * ‖f‖ ^ ENNReal.toReal p)\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p",
"tactic": "simp_rw [← coe_nnnorm, ← _root_.coe_nnnorm, ← NNReal.coe_rpow, ← NNReal.coe_mul,\n NNReal.hasSum_coe] at hRHS hLHS"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\ni : α\n⊢ ‖↑(c • f) i‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖↑f i‖₊ ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\n⊢ ‖c • f‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p",
"tactic": "refine' hasSum_mono hLHS hRHS fun i => _"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\ni : α\n⊢ ‖↑(c • f) i‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖↑f i‖₊ ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\ni : α\n⊢ ‖↑(c • f) i‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖↑f i‖₊ ^ ENNReal.toReal p",
"tactic": "dsimp only"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\ni : α\n⊢ ‖↑(c • f) i‖₊ ^ ENNReal.toReal p ≤ (‖c‖₊ * ‖↑f i‖₊) ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\ni : α\n⊢ ‖↑(c • f) i‖₊ ^ ENNReal.toReal p ≤ ‖c‖₊ ^ ENNReal.toReal p * ‖↑f i‖₊ ^ ENNReal.toReal p",
"tactic": "rw [← NNReal.mul_rpow]"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\ni : α\n⊢ ‖c • ↑f i‖₊ ^ ENNReal.toReal p ≤ (‖c‖₊ * ‖↑f i‖₊) ^ ENNReal.toReal p",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\ni : α\n⊢ ‖↑(c • f) i‖₊ ^ ENNReal.toReal p ≤ (‖c‖₊ * ‖↑f i‖₊) ^ ENNReal.toReal p",
"tactic": "rw [lp.coeFn_smul, Pi.smul_apply]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nhRHS :\n HasSum (fun a => ‖c‖₊ ^ ENNReal.toReal p * ‖↑f a‖₊ ^ ENNReal.toReal p)\n (‖c‖₊ ^ ENNReal.toReal p * ‖f‖₊ ^ ENNReal.toReal p)\nhLHS : HasSum (fun a => ‖↑(c • f) a‖₊ ^ ENNReal.toReal p) (‖c • f‖₊ ^ ENNReal.toReal p)\ni : α\n⊢ ‖c • ↑f i‖₊ ^ ENNReal.toReal p ≤ (‖c‖₊ * ‖↑f i‖₊) ^ ENNReal.toReal p",
"tactic": "exact NNReal.rpow_le_rpow (nnnorm_smul_le _ _) ENNReal.toReal_nonneg"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝⁶ : (i : α) → NormedAddCommGroup (E i)\n𝕜 : Type u_3\n𝕜' : Type ?u.424487\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : (i : α) → Module 𝕜 (E i)\ninst✝² : (i : α) → Module 𝕜' (E i)\ninst✝¹ : ∀ (i : α), BoundedSMul 𝕜 (E i)\ninst✝ : ∀ (i : α), BoundedSMul 𝕜' (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : { x // x ∈ lp E p }\nhp : 0 < ENNReal.toReal p\ninst : NNNorm { x // x ∈ lp E p } := { nnnorm := fun f => { val := ‖f‖, property := (_ : 0 ≤ ‖f‖) } }\ncoe_nnnorm : ∀ (f : { x // x ∈ lp E p }), ↑‖f‖₊ = ‖f‖\nthis : ‖c • f‖₊ ^ ENNReal.toReal p ≤ (‖c‖₊ * ‖f‖₊) ^ ENNReal.toReal p\n⊢ ‖c • f‖ ≤ ‖c‖ * ‖f‖",
"tactic": "rwa [NNReal.rpow_le_rpow_iff hp] at this"
}
] | [
689,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
660,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean | Commute.cast_nat_mul_self | [] | [
1120,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1119,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean | HasStrictFDerivAt.snd | [] | [
245,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
243,
11
] |
Mathlib/Algebra/Group/Basic.lean | div_mul_comm | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.41688\nG : Type ?u.41691\ninst✝ : DivisionCommMonoid α\na b c d : α\n⊢ a / b * c = c / b * a",
"tactic": "simp"
}
] | [
575,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
575,
1
] |
Mathlib/Tactic/IntervalCases.lean | Mathlib.Tactic.IntervalCases.of_not_le_left | [] | [
134,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
Mathlib/Data/Nat/Prime.lean | Nat.Primes.coe_nat_injective | [] | [
779,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
778,
1
] |
Mathlib/Probability/Integration.lean | ProbabilityTheory.lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun | [] | [
116,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Algebra/RingQuot.lean | RingQuot.one_quot | [
{
"state_after": "no goals",
"state_before": "R : Type u₁\ninst✝³ : Semiring R\nS : Type u₂\ninst✝² : CommSemiring S\nA : Type u₃\ninst✝¹ : Semiring A\ninst✝ : Algebra S A\nr : R → R → Prop\n⊢ { toQuot := Quot.mk (Rel r) 1 } = RingQuot.one r",
"tactic": "rw [one_def]"
}
] | [
233,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
232,
1
] |
Mathlib/Algebra/IndicatorFunction.lean | Set.mulIndicator_apply | [
{
"state_after": "α : Type u_1\nβ : Type ?u.1047\nι : Type ?u.1050\nM : Type u_2\nN : Type ?u.1056\ninst✝² : One M\ninst✝¹ : One N\ns✝ t : Set α\nf✝ g : α → M\na✝ : α\ns : Set α\nf : α → M\na : α\ninst✝ : Decidable (a ∈ s)\n⊢ (let x := a;\n if x ∈ s then f x else 1) =\n if a ∈ s then f a else 1",
"state_before": "α : Type u_1\nβ : Type ?u.1047\nι : Type ?u.1050\nM : Type u_2\nN : Type ?u.1056\ninst✝² : One M\ninst✝¹ : One N\ns✝ t : Set α\nf✝ g : α → M\na✝ : α\ns : Set α\nf : α → M\na : α\ninst✝ : Decidable (a ∈ s)\n⊢ mulIndicator s f a = if a ∈ s then f a else 1",
"tactic": "unfold mulIndicator"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1047\nι : Type ?u.1050\nM : Type u_2\nN : Type ?u.1056\ninst✝² : One M\ninst✝¹ : One N\ns✝ t : Set α\nf✝ g : α → M\na✝ : α\ns : Set α\nf : α → M\na : α\ninst✝ : Decidable (a ∈ s)\n⊢ (let x := a;\n if x ∈ s then f x else 1) =\n if a ∈ s then f a else 1",
"tactic": "split_ifs with h <;> simp"
}
] | [
74,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | one_lt_div | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.76204\nα : Type u_1\nβ : Type ?u.76210\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nhb : 0 < b\n⊢ 1 < a / b ↔ b < a",
"tactic": "rw [lt_div_iff hb, one_mul]"
}
] | [
430,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
430,
1
] |
Mathlib/Data/Polynomial/Splits.lean | Polynomial.splits_of_degree_eq_one | [] | [
103,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Data/List/Perm.lean | List.subperm_singleton_iff | [
{
"state_after": "no goals",
"state_before": "α✝ : Type uu\nβ : Type vv\nl₁ l₂ : List α✝\ninst✝ : DecidableEq α✝\nα : Type u_1\nl : List α\na : α\nx✝ : [a] <+~ l\ns : List α\nhla : s ~ [a]\nh : s <+ l\n⊢ a ∈ l",
"tactic": "rwa [perm_singleton.mp hla, singleton_sublist] at h"
}
] | [
933,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
931,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Icc_eq_empty_of_lt | [] | [
104,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Mathlib/Order/Heyting/Basic.lean | fst_sdiff | [] | [
133,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
132,
1
] |
Mathlib/Deprecated/Submonoid.lean | IsSubmonoid.iInter | [] | [
102,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
98,
1
] |
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.Formula.realize_rel | [
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.235577\nP : Type ?u.235580\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\nφ ψ : Formula L α\nv : α → M\nk : ℕ\nR : Relations L k\nts : Fin k → Term L α\n⊢ (RelMap R fun i => Term.realize (Sum.elim v default) (Term.relabel Sum.inl (ts i))) ↔\n RelMap R fun i => Term.realize v (ts i)",
"tactic": "simp"
}
] | [
643,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
641,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean | Metric.continuous_infDist_pt | [] | [
575,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
574,
1
] |
Mathlib/CategoryTheory/Monoidal/End.lean | CategoryTheory.ε_inv_app_obj | [
{
"state_after": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ 𝟙 ((F.obj (𝟙_ M)).obj ((F.obj n).obj X)) =\n ((LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X) ≫\n F.ε.app ((F.obj n).obj X)",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ (MonoidalFunctor.εIso F).inv.app ((F.obj n).obj X) =\n (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X",
"tactic": "rw [← cancel_mono (F.ε.app ((F.obj n).obj X)), ε_inv_hom_app]"
},
{
"state_after": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ 𝟙 ((F.obj (𝟙_ M)).obj ((F.obj n).obj X)) = 𝟙 ((F.obj n ⊗ F.obj (𝟙_ M)).obj X)",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ 𝟙 ((F.obj (𝟙_ M)).obj ((F.obj n).obj X)) =\n ((LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X) ≫\n F.ε.app ((F.obj n).obj X)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\nM : Type u_2\ninst✝¹ : Category M\ninst✝ : MonoidalCategory M\nF : MonoidalFunctor M (C ⥤ C)\nn : M\nX : C\n⊢ 𝟙 ((F.obj (𝟙_ M)).obj ((F.obj n).obj X)) = 𝟙 ((F.obj n ⊗ F.obj (𝟙_ M)).obj X)",
"tactic": "rfl"
}
] | [
235,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
231,
1
] |
Mathlib/Data/Complex/Exponential.lean | Complex.sin_sub | [
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ sin (x - y) = sin x * cos y - cos x * sin y",
"tactic": "simp [sub_eq_add_neg, sin_add, sin_neg, cos_neg]"
}
] | [
869,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
868,
1
] |
Mathlib/Data/Seq/Computation.lean | Computation.results_bind | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nm n : ℕ\nh1 : Results s a m\nh2 : Results (f a) b n\nthis : a ∈ s\n⊢ Results (bind s f) b (n + m)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nm n : ℕ\nh1 : Results s a m\nh2 : Results (f a) b n\n⊢ Results (bind s f) b (n + m)",
"tactic": "have := h1.mem"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\n⊢ ∀ {m : ℕ}, Results s a m → Results (bind s f) b (n + m)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nm n : ℕ\nh1 : Results s a m\nh2 : Results (f a) b n\nthis : a ∈ s\n⊢ Results (bind s f) b (n + m)",
"tactic": "revert m"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\n⊢ ∀ {m : ℕ}, Results (pure a) a m → Results (bind (pure a) f) b (n + m)\n\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\n⊢ ∀ (s : Computation α),\n (∀ {m : ℕ}, Results s a m → Results (bind s f) b (n + m)) →\n ∀ {m : ℕ}, Results (think s) a m → Results (bind (think s) f) b (n + m)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\n⊢ ∀ {m : ℕ}, Results s a m → Results (bind s f) b (n + m)",
"tactic": "apply memRecOn this _ fun s IH => _"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\nm✝ : ℕ\nh1 : Results (pure a) a m✝\n⊢ Results (bind (pure a) f) b (n + m✝)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\n⊢ ∀ {m : ℕ}, Results (pure a) a m → Results (bind (pure a) f) b (n + m)",
"tactic": "intro _ h1"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\nm✝ : ℕ\nh1 : Results (pure a) a m✝\n⊢ Results (f a) b (n + m✝)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\nm✝ : ℕ\nh1 : Results (pure a) a m✝\n⊢ Results (bind (pure a) f) b (n + m✝)",
"tactic": "rw [ret_bind]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\nm✝ : ℕ\nh1 : Results (pure a) a m✝\n⊢ Results (f a) b (n + 0)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\nm✝ : ℕ\nh1 : Results (pure a) a m✝\n⊢ Results (f a) b (n + m✝)",
"tactic": "rw [h1.len_unique (results_pure _)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\nm✝ : ℕ\nh1 : Results (pure a) a m✝\n⊢ Results (f a) b (n + 0)",
"tactic": "exact h2"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1 : Results (think s✝) a m✝\n⊢ Results (bind (think s✝) f) b (n + m✝)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\n⊢ ∀ (s : Computation α),\n (∀ {m : ℕ}, Results s a m → Results (bind s f) b (n + m)) →\n ∀ {m : ℕ}, Results (think s) a m → Results (bind (think s) f) b (n + m)",
"tactic": "intro _ h3 _ h1"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1 : Results (think s✝) a m✝\n⊢ Results (think (bind s✝ f)) b (n + m✝)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1 : Results (think s✝) a m✝\n⊢ Results (bind (think s✝) f) b (n + m✝)",
"tactic": "rw [think_bind]"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1 : Results (think s✝) a m✝\nm' : ℕ\nh : Results s✝ a m' ∧ m✝ = m' + 1\n⊢ Results (think (bind s✝ f)) b (n + m✝)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1 : Results (think s✝) a m✝\n⊢ Results (think (bind s✝ f)) b (n + m✝)",
"tactic": "cases' of_results_think h1 with m' h"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1✝ : Results (think s✝) a m✝\nm' : ℕ\nh1 : Results s✝ a m'\ne : m✝ = m' + 1\n⊢ Results (think (bind s✝ f)) b (n + m✝)",
"state_before": "case intro\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1 : Results (think s✝) a m✝\nm' : ℕ\nh : Results s✝ a m' ∧ m✝ = m' + 1\n⊢ Results (think (bind s✝ f)) b (n + m✝)",
"tactic": "cases' h with h1 e"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1✝ : Results (think s✝) a m✝\nm' : ℕ\nh1 : Results s✝ a m'\ne : m✝ = m' + 1\n⊢ Results (think (bind s✝ f)) b (n + (m' + 1))",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1✝ : Results (think s✝) a m✝\nm' : ℕ\nh1 : Results s✝ a m'\ne : m✝ = m' + 1\n⊢ Results (think (bind s✝ f)) b (n + m✝)",
"tactic": "rw [e]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nγ : Type w\ns : Computation α\nf : α → Computation β\na : α\nb : β\nn : ℕ\nh2 : Results (f a) b n\nthis : a ∈ s\ns✝ : Computation α\nh3 : ∀ {m : ℕ}, Results s✝ a m → Results (bind s✝ f) b (n + m)\nm✝ : ℕ\nh1✝ : Results (think s✝) a m✝\nm' : ℕ\nh1 : Results s✝ a m'\ne : m✝ = m' + 1\n⊢ Results (think (bind s✝ f)) b (n + (m' + 1))",
"tactic": "exact results_think (h3 h1)"
}
] | [
805,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
792,
1
] |
Mathlib/Analysis/Normed/Group/AddTorsor.lean | nndist_vadd_cancel_left | [] | [
104,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsor.lean | nndist_midpoint_midpoint_le | [] | [
276,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
274,
1
] |
Mathlib/CategoryTheory/Abelian/RightDerived.lean | CategoryTheory.Abelian.Functor.preserves_exact_of_preservesFiniteLimits_of_mono | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : PreservesFiniteLimits F\ninst✝ : Mono f\nex : Exact f g\n⊢ F.map f ≫ F.map g = 0",
"tactic": "simp [← Functor.map_comp, ex.w]"
}
] | [
198,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
195,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | ENNReal.one_lt_rpow_of_pos_of_lt_one_of_neg | [
{
"state_after": "case intro\nz : ℝ\nhz : z < 0\nx : ℝ≥0\nhx1 : 0 < ↑x\nhx2 : ↑x < 1\n⊢ 1 < ↑x ^ z",
"state_before": "x : ℝ≥0∞\nz : ℝ\nhx1 : 0 < x\nhx2 : x < 1\nhz : z < 0\n⊢ 1 < x ^ z",
"tactic": "lift x to ℝ≥0 using ne_of_lt (lt_of_lt_of_le hx2 le_top)"
},
{
"state_after": "case intro\nz : ℝ\nhz : z < 0\nx : ℝ≥0\nhx1 : 0 < x\nhx2 : x < 1\n⊢ 1 < ↑x ^ z",
"state_before": "case intro\nz : ℝ\nhz : z < 0\nx : ℝ≥0\nhx1 : 0 < ↑x\nhx2 : ↑x < 1\n⊢ 1 < ↑x ^ z",
"tactic": "simp only [coe_lt_one_iff, coe_pos] at hx1 hx2⊢"
},
{
"state_after": "no goals",
"state_before": "case intro\nz : ℝ\nhz : z < 0\nx : ℝ≥0\nhx1 : 0 < x\nhx2 : x < 1\n⊢ 1 < ↑x ^ z",
"tactic": "simp [coe_rpow_of_ne_zero (ne_of_gt hx1), NNReal.one_lt_rpow_of_pos_of_lt_one_of_neg hx1 hx2 hz]"
}
] | [
711,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
707,
1
] |
Mathlib/Order/Hom/Bounded.lean | BoundedOrderHom.coe_comp_orderHom | [] | [
659,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
657,
1
] |
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