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|---|---|---|---|---|---|---|
Mathlib/Init/Data/Ordering/Lemmas.lean | Ordering.ite_eq_gt_distrib | [
{
"state_after": "no goals",
"state_before": "c : Prop\ninst✝ : Decidable c\na b : Ordering\n⊢ ((if c then a else b) = gt) = if c then a = gt else b = gt",
"tactic": "by_cases c <;> simp [*]"
}
]
| [
39,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
37,
1
]
|
Mathlib/Dynamics/PeriodicPts.lean | Function.minimalPeriod_id | []
| [
370,
77
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
1
]
|
Std/Data/Nat/Lemmas.lean | Nat.lt_add_of_pos_right | []
| [
90,
26
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
89,
11
]
|
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | tendsto_one_plus_div_rpow_exp | [
{
"state_after": "t : ℝ\n⊢ (exp ∘ fun x => x * log (1 + t / x)) =ᶠ[atTop] fun x => (1 + t / x) ^ x",
"state_before": "t : ℝ\n⊢ Tendsto (fun x => (1 + t / x) ^ x) atTop (𝓝 (exp t))",
"tactic": "apply ((Real.continuous_exp.tendsto _).comp (tendsto_mul_log_one_plus_div_atTop t)).congr' _"
},
{
"state_after": "t : ℝ\nh₁ : 1 / 2 < 1\n⊢ (exp ∘ fun x => x * log (1 + t / x)) =ᶠ[atTop] fun x => (1 + t / x) ^ x",
"state_before": "t : ℝ\n⊢ (exp ∘ fun x => x * log (1 + t / x)) =ᶠ[atTop] fun x => (1 + t / x) ^ x",
"tactic": "have h₁ : (1 : ℝ) / 2 < 1 := by linarith"
},
{
"state_after": "t : ℝ\nh₁ : 1 / 2 < 1\nh₂ : Tendsto (fun x => 1 + t / x) atTop (𝓝 1)\n⊢ (exp ∘ fun x => x * log (1 + t / x)) =ᶠ[atTop] fun x => (1 + t / x) ^ x",
"state_before": "t : ℝ\nh₁ : 1 / 2 < 1\n⊢ (exp ∘ fun x => x * log (1 + t / x)) =ᶠ[atTop] fun x => (1 + t / x) ^ x",
"tactic": "have h₂ : Tendsto (fun x : ℝ => 1 + t / x) atTop (𝓝 1) := by\n simpa using (tendsto_inv_atTop_zero.const_mul t).const_add 1"
},
{
"state_after": "t : ℝ\nh₁ : 1 / 2 < 1\nh₂ : Tendsto (fun x => 1 + t / x) atTop (𝓝 1)\nx : ℝ\nhx : 1 / 2 ≤ 1 + t / x\n⊢ (exp ∘ fun x => x * log (1 + t / x)) x = (fun x => (1 + t / x) ^ x) x",
"state_before": "t : ℝ\nh₁ : 1 / 2 < 1\nh₂ : Tendsto (fun x => 1 + t / x) atTop (𝓝 1)\n⊢ (exp ∘ fun x => x * log (1 + t / x)) =ᶠ[atTop] fun x => (1 + t / x) ^ x",
"tactic": "refine' (eventually_ge_of_tendsto_gt h₁ h₂).mono fun x hx => _"
},
{
"state_after": "t : ℝ\nh₁ : 1 / 2 < 1\nh₂ : Tendsto (fun x => 1 + t / x) atTop (𝓝 1)\nx : ℝ\nhx : 1 / 2 ≤ 1 + t / x\nhx' : 0 < 1 + t / x\n⊢ (exp ∘ fun x => x * log (1 + t / x)) x = (fun x => (1 + t / x) ^ x) x",
"state_before": "t : ℝ\nh₁ : 1 / 2 < 1\nh₂ : Tendsto (fun x => 1 + t / x) atTop (𝓝 1)\nx : ℝ\nhx : 1 / 2 ≤ 1 + t / x\n⊢ (exp ∘ fun x => x * log (1 + t / x)) x = (fun x => (1 + t / x) ^ x) x",
"tactic": "have hx' : 0 < 1 + t / x := by linarith"
},
{
"state_after": "no goals",
"state_before": "t : ℝ\nh₁ : 1 / 2 < 1\nh₂ : Tendsto (fun x => 1 + t / x) atTop (𝓝 1)\nx : ℝ\nhx : 1 / 2 ≤ 1 + t / x\nhx' : 0 < 1 + t / x\n⊢ (exp ∘ fun x => x * log (1 + t / x)) x = (fun x => (1 + t / x) ^ x) x",
"tactic": "simp [mul_comm x, exp_mul, exp_log hx']"
},
{
"state_after": "no goals",
"state_before": "t : ℝ\n⊢ 1 / 2 < 1",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "t : ℝ\nh₁ : 1 / 2 < 1\n⊢ Tendsto (fun x => 1 + t / x) atTop (𝓝 1)",
"tactic": "simpa using (tendsto_inv_atTop_zero.const_mul t).const_add 1"
},
{
"state_after": "no goals",
"state_before": "t : ℝ\nh₁ : 1 / 2 < 1\nh₂ : Tendsto (fun x => 1 + t / x) atTop (𝓝 1)\nx : ℝ\nhx : 1 / 2 ≤ 1 + t / x\n⊢ 0 < 1 + t / x",
"tactic": "linarith"
}
]
| [
621,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
613,
1
]
|
Mathlib/RingTheory/Finiteness.lean | Submodule.FG.stablizes_of_iSup_eq | [
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\n⊢ ∃ n, M' = ↑N n",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] P\nM' : Submodule R M\nhM' : FG M'\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\n⊢ ∃ n, M' = ↑N n",
"tactic": "obtain ⟨S, hS⟩ := hM'"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nthis : ∀ (s : { x // x ∈ S }), ∃ n, ↑s ∈ ↑N n\n⊢ ∃ n, M' = ↑N n",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\n⊢ ∃ n, M' = ↑N n",
"tactic": "have : ∀ s : S, ∃ n, (s : M) ∈ N n := fun s =>\n (Submodule.mem_iSup_of_chain N s).mp\n (by\n rw [H, ← hS]\n exact Submodule.subset_span s.2)"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ ∃ n, M' = ↑N n",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nthis : ∀ (s : { x // x ∈ S }), ∃ n, ↑s ∈ ↑N n\n⊢ ∃ n, M' = ↑N n",
"tactic": "choose f hf using this"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ M' = ↑N (Finset.sup (Finset.attach S) f)",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ ∃ n, M' = ↑N n",
"tactic": "use S.attach.sup f"
},
{
"state_after": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ M' ≤ ↑N (Finset.sup (Finset.attach S) f)\n\ncase intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ ↑N (Finset.sup (Finset.attach S) f) ≤ M'",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ M' = ↑N (Finset.sup (Finset.attach S) f)",
"tactic": "apply le_antisymm"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\ns : { x // x ∈ S }\n⊢ ↑s ∈ span R ↑S",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\ns : { x // x ∈ S }\n⊢ ↑s ∈ ⨆ (k : ℕ), ↑N k",
"tactic": "rw [H, ← hS]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\ns : { x // x ∈ S }\n⊢ ↑s ∈ span R ↑S",
"tactic": "exact Submodule.subset_span s.2"
},
{
"state_after": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ span R ↑S ≤ ↑N (Finset.sup (Finset.attach S) f)",
"state_before": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ M' ≤ ↑N (Finset.sup (Finset.attach S) f)",
"tactic": "conv_lhs => rw [← hS]"
},
{
"state_after": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ ↑S ⊆ ↑(↑N (Finset.sup (Finset.attach S) f))",
"state_before": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ span R ↑S ≤ ↑N (Finset.sup (Finset.attach S) f)",
"tactic": "rw [Submodule.span_le]"
},
{
"state_after": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\ns : M\nhs : s ∈ ↑S\n⊢ s ∈ ↑(↑N (Finset.sup (Finset.attach S) f))",
"state_before": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ ↑S ⊆ ↑(↑N (Finset.sup (Finset.attach S) f))",
"tactic": "intro s hs"
},
{
"state_after": "no goals",
"state_before": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\ns : M\nhs : s ∈ ↑S\n⊢ s ∈ ↑(↑N (Finset.sup (Finset.attach S) f))",
"tactic": "exact N.2 (Finset.le_sup <| S.mem_attach ⟨s, hs⟩) (hf _)"
},
{
"state_after": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ ↑N (Finset.sup (Finset.attach S) f) ≤ iSup ↑N",
"state_before": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ ↑N (Finset.sup (Finset.attach S) f) ≤ M'",
"tactic": "rw [← H]"
},
{
"state_after": "no goals",
"state_before": "case intro.a\nR : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nP : Type ?u.311147\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf✝ : M →ₗ[R] P\nM' : Submodule R M\nN : ℕ →o Submodule R M\nH : iSup ↑N = M'\nS : Finset M\nhS : span R ↑S = M'\nf : { x // x ∈ S } → ℕ\nhf : ∀ (s : { x // x ∈ S }), ↑s ∈ ↑N (f s)\n⊢ ↑N (Finset.sup (Finset.attach S) f) ≤ iSup ↑N",
"tactic": "exact le_iSup _ _"
}
]
| [
388,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
372,
1
]
|
Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.rangeS_subtype | []
| [
1241,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1240,
1
]
|
Std/Data/Int/Lemmas.lean | Int.subNatNat_add_left | [
{
"state_after": "m n : Nat\n⊢ (match m - (m + n) with\n | 0 => ofNat (m + n - m)\n | succ k => -[k+1]) =\n ↑n",
"state_before": "m n : Nat\n⊢ subNatNat (m + n) m = ↑n",
"tactic": "unfold subNatNat"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\n⊢ (match m - (m + n) with\n | 0 => ofNat (m + n - m)\n | succ k => -[k+1]) =\n ↑n",
"tactic": "rw [Nat.sub_eq_zero_of_le (Nat.le_add_right ..), Nat.add_sub_cancel_left, ofNat_eq_coe]"
}
]
| [
116,
90
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
114,
1
]
|
Mathlib/Algebra/Lie/Semisimple.lean | LieAlgebra.ad_ker_eq_bot_of_semisimple | [
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\n⊢ LieHom.ker (ad R L) = ⊥",
"tactic": "simp"
}
]
| [
121,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
]
|
Mathlib/Analysis/Convex/Caratheodory.lean | Caratheodory.affineIndependent_minCardFinsetOfMemConvexHull | [
{
"state_after": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\n⊢ AffineIndependent 𝕜 Subtype.val",
"state_before": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\n⊢ AffineIndependent 𝕜 Subtype.val",
"tactic": "let k := (minCardFinsetOfMemConvexHull hx).card - 1"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\n⊢ AffineIndependent 𝕜 Subtype.val",
"state_before": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\n⊢ AffineIndependent 𝕜 Subtype.val",
"tactic": "have hk : (minCardFinsetOfMemConvexHull hx).card = k + 1 :=\n (Nat.succ_pred_eq_of_pos (Finset.card_pos.mpr (minCardFinsetOfMemConvexHull_nonempty hx))).symm"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\n⊢ AffineIndependent 𝕜 Subtype.val",
"tactic": "classical\nby_contra h\nobtain ⟨p, hp⟩ := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull hx)\nhave contra := minCardFinsetOfMemConvexHull_card_le_card hx (Set.Subset.trans\n (Finset.erase_subset (p : E) (minCardFinsetOfMemConvexHull hx))\n (minCardFinsetOfMemConvexHull_subseteq hx)) hp\nrw [← not_lt] at contra\napply contra\nerw [card_erase_of_mem p.2, hk]\nexact lt_add_one _"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\n⊢ False",
"state_before": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\n⊢ AffineIndependent 𝕜 Subtype.val",
"tactic": "by_contra h"
},
{
"state_after": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\n⊢ False",
"state_before": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\n⊢ False",
"tactic": "obtain ⟨p, hp⟩ := mem_convexHull_erase h (mem_minCardFinsetOfMemConvexHull hx)"
},
{
"state_after": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\ncontra : card (minCardFinsetOfMemConvexHull hx) ≤ card (erase (minCardFinsetOfMemConvexHull hx) ↑p)\n⊢ False",
"state_before": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\n⊢ False",
"tactic": "have contra := minCardFinsetOfMemConvexHull_card_le_card hx (Set.Subset.trans\n (Finset.erase_subset (p : E) (minCardFinsetOfMemConvexHull hx))\n (minCardFinsetOfMemConvexHull_subseteq hx)) hp"
},
{
"state_after": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\ncontra : ¬card (erase (minCardFinsetOfMemConvexHull hx) ↑p) < card (minCardFinsetOfMemConvexHull hx)\n⊢ False",
"state_before": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\ncontra : card (minCardFinsetOfMemConvexHull hx) ≤ card (erase (minCardFinsetOfMemConvexHull hx) ↑p)\n⊢ False",
"tactic": "rw [← not_lt] at contra"
},
{
"state_after": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\ncontra : ¬card (erase (minCardFinsetOfMemConvexHull hx) ↑p) < card (minCardFinsetOfMemConvexHull hx)\n⊢ card (erase (minCardFinsetOfMemConvexHull hx) ↑p) < card (minCardFinsetOfMemConvexHull hx)",
"state_before": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\ncontra : ¬card (erase (minCardFinsetOfMemConvexHull hx) ↑p) < card (minCardFinsetOfMemConvexHull hx)\n⊢ False",
"tactic": "apply contra"
},
{
"state_after": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\ncontra : ¬card (erase (minCardFinsetOfMemConvexHull hx) ↑p) < card (minCardFinsetOfMemConvexHull hx)\n⊢ k + 1 - 1 < k + 1",
"state_before": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\ncontra : ¬card (erase (minCardFinsetOfMemConvexHull hx) ↑p) < card (minCardFinsetOfMemConvexHull hx)\n⊢ card (erase (minCardFinsetOfMemConvexHull hx) ↑p) < card (minCardFinsetOfMemConvexHull hx)",
"tactic": "erw [card_erase_of_mem p.2, hk]"
},
{
"state_after": "no goals",
"state_before": "case intro\n𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\nk : ℕ := card (minCardFinsetOfMemConvexHull hx) - 1\nhk : card (minCardFinsetOfMemConvexHull hx) = k + 1\nh : ¬AffineIndependent 𝕜 Subtype.val\np : ↑↑(minCardFinsetOfMemConvexHull hx)\nhp : x ∈ ↑(convexHull 𝕜).toOrderHom ↑(erase (minCardFinsetOfMemConvexHull hx) ↑p)\ncontra : ¬card (erase (minCardFinsetOfMemConvexHull hx) ↑p) < card (minCardFinsetOfMemConvexHull hx)\n⊢ k + 1 - 1 < k + 1",
"tactic": "exact lt_add_one _"
}
]
| [
148,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
]
|
Mathlib/Analysis/Calculus/MeanValue.lean | constant_of_derivWithin_zero | [
{
"state_after": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.49446\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → E\na b : ℝ\nhdiff : DifferentiableOn ℝ f (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ico a b → derivWithin f (Icc a b) x = 0\nH : ∀ (x : ℝ), x ∈ Ico a b → ‖derivWithin f (Icc a b) x‖ ≤ 0\n⊢ ∀ (x : ℝ), x ∈ Icc a b → f x = f a",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.49446\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → E\na b : ℝ\nhdiff : DifferentiableOn ℝ f (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ico a b → derivWithin f (Icc a b) x = 0\n⊢ ∀ (x : ℝ), x ∈ Icc a b → f x = f a",
"tactic": "have H : ∀ x ∈ Ico a b, ‖derivWithin f (Icc a b) x‖ ≤ 0 := by\n simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.49446\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → E\na b : ℝ\nhdiff : DifferentiableOn ℝ f (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ico a b → derivWithin f (Icc a b) x = 0\nH : ∀ (x : ℝ), x ∈ Ico a b → ‖derivWithin f (Icc a b) x‖ ≤ 0\n⊢ ∀ (x : ℝ), x ∈ Icc a b → f x = f a",
"tactic": "simpa only [zero_mul, norm_le_zero_iff, sub_eq_zero] using fun x hx =>\n norm_image_sub_le_of_norm_deriv_le_segment hdiff H x hx"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.49446\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → E\na b : ℝ\nhdiff : DifferentiableOn ℝ f (Icc a b)\nhderiv : ∀ (x : ℝ), x ∈ Ico a b → derivWithin f (Icc a b) x = 0\n⊢ ∀ (x : ℝ), x ∈ Ico a b → ‖derivWithin f (Icc a b) x‖ ≤ 0",
"tactic": "simpa only [norm_le_zero_iff] using fun x hx => hderiv x hx"
}
]
| [
409,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
404,
1
]
|
Mathlib/Topology/Basic.lean | continuousAt_def | []
| [
1593,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1592,
1
]
|
Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.Ico_inter_Ico_of_le | []
| [
217,
62
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
216,
1
]
|
Mathlib/Analysis/Normed/Group/Basic.lean | AntilipschitzWith.mul_lipschitzWith | [
{
"state_after": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\n⊢ AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x => f x * g x",
"state_before": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\n⊢ AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x => f x * g x",
"tactic": "letI : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace hf.edist_ne_top"
},
{
"state_after": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\nx y : α\n⊢ dist x y ≤ ↑(Kf⁻¹ - Kg)⁻¹ * dist (f x * g x) (f y * g y)",
"state_before": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\n⊢ AntilipschitzWith (Kf⁻¹ - Kg)⁻¹ fun x => f x * g x",
"tactic": "refine' AntilipschitzWith.of_le_mul_dist fun x y => _"
},
{
"state_after": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\nx y : α\n⊢ dist x y ≤ dist (f x * g x) (f y * g y) / ↑(Kf⁻¹ - Kg)",
"state_before": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\nx y : α\n⊢ dist x y ≤ ↑(Kf⁻¹ - Kg)⁻¹ * dist (f x * g x) (f y * g y)",
"tactic": "rw [NNReal.coe_inv, ← _root_.div_eq_inv_mul]"
},
{
"state_after": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\nx y : α\n⊢ dist x y * ↑(Kf⁻¹ - Kg) ≤ dist (f x * g x) (f y * g y)",
"state_before": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\nx y : α\n⊢ dist x y ≤ dist (f x * g x) (f y * g y) / ↑(Kf⁻¹ - Kg)",
"tactic": "rw [le_div_iff (NNReal.coe_pos.2 <| tsub_pos_iff_lt.2 hK)]"
},
{
"state_after": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\nx y : α\n⊢ ↑Kf⁻¹ * dist x y - ↑Kg * dist x y ≤ dist (f x * g x) (f y * g y)",
"state_before": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\nx y : α\n⊢ dist x y * ↑(Kf⁻¹ - Kg) ≤ dist (f x * g x) (f y * g y)",
"tactic": "rw [mul_comm, NNReal.coe_sub hK.le, _root_.sub_mul]"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.1208856\n𝕜 : Type ?u.1208859\nα : Type u_1\nι : Type ?u.1208865\nκ : Type ?u.1208868\nE : Type u_2\nF : Type ?u.1208874\nG : Type ?u.1208877\ninst✝² : SeminormedCommGroup E\ninst✝¹ : SeminormedCommGroup F\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : PseudoEMetricSpace α\nK Kf Kg : ℝ≥0\nf g : α → E\nhf : AntilipschitzWith Kf f\nhg : LipschitzWith Kg g\nhK : Kg < Kf⁻¹\nthis : PseudoMetricSpace α := PseudoEMetricSpace.toPseudoMetricSpace (_ : ∀ (x y : α), edist x y ≠ ⊤)\nx y : α\n⊢ ↑Kf⁻¹ * dist x y - ↑Kg * dist x y ≤ dist (f x * g x) (f y * g y)",
"tactic": "calc\n ↑Kf⁻¹ * dist x y - Kg * dist x y ≤ dist (f x) (f y) - dist (g x) (g y) :=\n sub_le_sub (hf.mul_le_dist x y) (hg.dist_le_mul x y)\n _ ≤ _ := le_trans (le_abs_self _) (abs_dist_sub_le_dist_mul_mul _ _ _ _)"
}
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1898,
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Data/Matrix/Basic.lean | Matrix.sub_mulVec | [
{
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"tactic": "simp [sub_eq_add_neg, add_mulVec, neg_mulVec]"
}
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1917,
99
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | borel_eq_generateFrom_Ici | []
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175,
41
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
174,
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Mathlib/Order/UpperLower/Basic.lean | upperClosure_prod | [
{
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{
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"tactic": "simp [Prod.le_def, @and_and_and_comm _ (_ ∈ t)]"
}
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1752,
50
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1749,
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Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.mem_think | [
{
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"tactic": "constructor <;> intro h"
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{
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{
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"tactic": "injections"
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{
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"tactic": "apply Stream'.mem_cons_of_mem _ h"
}
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932,
38
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
925,
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|
Mathlib/Analysis/NormedSpace/Complemented.lean | ContinuousLinearMap.equivProdOfSurjectiveOfIsCompl_toLinearEquiv | []
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72,
68
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
69,
1
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|
Mathlib/MeasureTheory/Function/SpecialFunctions/Basic.lean | Real.measurable_exp | []
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33,
28
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
32,
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|
Mathlib/Order/RelClasses.lean | IsAsymm.swap | []
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88,
38
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
87,
1
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|
Mathlib/Data/TwoPointing.lean | TwoPointing.prop_fst | []
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166,
6
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
165,
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|
Mathlib/Topology/FiberBundle/Trivialization.lean | Trivialization.eqOn | []
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357,
88
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
357,
11
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|
Mathlib/Data/Part.lean | Part.right_dom_of_mul_dom | []
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729,
89
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729,
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Mathlib/FieldTheory/IntermediateField.lean | IntermediateField.coe_sum | [
{
"state_after": "case empty\nK : Type u_3\nL : Type u_2\nL' : Type ?u.104812\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Field L'\ninst✝² : Algebra K L\ninst✝¹ : Algebra K L'\nS : IntermediateField K L\nι : Type u_1\ninst✝ : Fintype ι\nf : ι → { x // x ∈ S }\n⊢ ↑(∑ i in ∅, f i) = ∑ i in ∅, ↑(f i)\n\ncase insert\nK : Type u_3\nL : Type u_2\nL' : Type ?u.104812\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Field L'\ninst✝² : Algebra K L\ninst✝¹ : Algebra K L'\nS : IntermediateField K L\nι : Type u_1\ninst✝ : Fintype ι\nf : ι → { x // x ∈ S }\ni : ι\ns : Finset ι\nhi : ¬i ∈ s\nH : ↑(∑ i in s, f i) = ∑ i in s, ↑(f i)\n⊢ ↑(∑ i in insert i s, f i) = ∑ i in insert i s, ↑(f i)",
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{
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{
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"tactic": "rw [Finset.sum_insert hi, AddMemClass.coe_add, H, Finset.sum_insert hi]"
}
]
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354,
78
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350,
1
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Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.LiftRel.swap_lem | [
{
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{
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"tactic": "rw [← LiftRelO.swap, Computation.LiftRel.swap]"
},
{
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"tactic": "apply liftRel_destruct h"
}
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27
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Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.Integrable.hasFiniteIntegral | []
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461,
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Mathlib/Order/Chain.lean | ChainClosure.isChain | [
{
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{
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{
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"tactic": "case union s hs h =>\n exact fun c₁ ⟨t₁, ht₁, (hc₁ : c₁ ∈ t₁)⟩ c₂ ⟨t₂, ht₂, (hc₂ : c₂ ∈ t₂)⟩ hneq =>\n ((hs _ ht₁).total <| hs _ ht₂).elim (fun ht => h t₂ ht₂ (ht hc₁) hc₂ hneq) fun ht =>\n h t₁ ht₁ hc₁ (ht hc₂) hneq"
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{
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},
{
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"tactic": "exact fun c₁ ⟨t₁, ht₁, (hc₁ : c₁ ∈ t₁)⟩ c₂ ⟨t₂, ht₂, (hc₂ : c₂ ∈ t₂)⟩ hneq =>\n ((hs _ ht₁).total <| hs _ ht₂).elim (fun ht => h t₂ ht₂ (ht hc₁) hc₂ hneq) fun ht =>\n h t₁ ht₁ hc₁ (ht hc₂) hneq"
}
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| [
270,
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
264,
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Mathlib/Computability/RegularExpressions.lean | RegularExpression.map_id | [
{
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432,
39
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
426,
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Mathlib/Data/Set/Lattice.lean | Set.disjoint_iUnion₂_right | []
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2096,
21
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2094,
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Mathlib/Algebra/Order/Monoid/WithTop.lean | WithTop.untop_one' | []
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59,
6
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
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Mathlib/Data/Finset/Interval.lean | Finset.card_Ioo_finset | [
{
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"tactic": "rw [card_Ioo_eq_card_Icc_sub_two, card_Icc_finset h]"
}
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119,
55
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
118,
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Mathlib/Order/Filter/Basic.lean | Filter.sInf_neBot_of_directed | []
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934,
89
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Mathlib/Topology/UniformSpace/UniformEmbedding.lean | uniformInducing_id | [
{
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}
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68,
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Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.to_openEmbedding | [
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{
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"tactic": "simpa only [h, subset_univ, mfld_simps] using e.image_open_of_open hU"
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1275,
74
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1266,
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Mathlib/Order/Filter/Bases.lean | Filter.HasBasis.eq_generate | [
{
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"tactic": "rw [← h.isBasis.filter_eq_generate, h.filter_eq]"
}
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328,
51
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
327,
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Mathlib/Topology/Separation.lean | SeparatedNhds.empty_left | []
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155,
23
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154,
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Mathlib/CategoryTheory/SingleObj.lean | MonoidHom.comp_toFunctor | []
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186,
6
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184,
1
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|
Mathlib/Data/Int/Cast/Lemmas.lean | eq_intCast' | []
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250,
57
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248,
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Mathlib/Algebra/Opposites.lean | AddOpposite.op_inv | []
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412,
6
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411,
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Mathlib/Data/Finset/Pointwise.lean | Finset.union_mul_inter_subset | []
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774,
37
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773,
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Mathlib/Data/Real/ENNReal.lean | ENNReal.not_top_le_coe | []
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659,
61
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659,
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|
Mathlib/CategoryTheory/Sites/InducedTopology.lean | CategoryTheory.LocallyCoverDense.pushforward_cover_iff_cover_pullback | [
{
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70,
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Algebra/Module/Submodule/Lattice.lean | Submodule.finset_inf_coe | [
{
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{
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{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u_2\nS : Type ?u.133565\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Type u_1\ns : Finset ι\np : ι → Submodule R M\nthis : DecidableEq ι := Classical.decEq ι\n⊢ ↑(Finset.inf ∅ p) = ⋂ (i : ι) (_ : i ∈ ∅), ↑(p i)",
"tactic": "simp"
},
{
"state_after": "case refine'_2\nR : Type u_2\nS : Type ?u.133565\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Type u_1\ns✝ : Finset ι\np : ι → Submodule R M\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\ns : Finset ι\nx✝ : ¬i ∈ s\nih : ↑(Finset.inf s p) = ⋂ (i : ι) (_ : i ∈ s), ↑(p i)\n⊢ (↑(p i) ∩ ⋂ (i : ι) (_ : i ∈ s), ↑(p i)) = ⋂ (i_1 : ι) (_ : i_1 ∈ insert i s), ↑(p i_1)",
"state_before": "case refine'_2\nR : Type u_2\nS : Type ?u.133565\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Type u_1\ns✝ : Finset ι\np : ι → Submodule R M\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\ns : Finset ι\nx✝ : ¬i ∈ s\nih : ↑(Finset.inf s p) = ⋂ (i : ι) (_ : i ∈ s), ↑(p i)\n⊢ ↑(Finset.inf (insert i s) p) = ⋂ (i_1 : ι) (_ : i_1 ∈ insert i s), ↑(p i_1)",
"tactic": "rw [Finset.inf_insert, inf_coe, ih]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_2\nS : Type ?u.133565\nM : Type u_3\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q : Submodule R M\nι : Type u_1\ns✝ : Finset ι\np : ι → Submodule R M\nthis : DecidableEq ι := Classical.decEq ι\ni : ι\ns : Finset ι\nx✝ : ¬i ∈ s\nih : ↑(Finset.inf s p) = ⋂ (i : ι) (_ : i ∈ s), ↑(p i)\n⊢ (↑(p i) ∩ ⋂ (i : ι) (_ : i ∈ s), ↑(p i)) = ⋂ (i_1 : ι) (_ : i_1 ∈ insert i s), ↑(p i_1)",
"tactic": "simp"
}
]
| [
254,
9
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
]
|
Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.lt_nfp | [
{
"state_after": "f : Ordinal → Ordinal\na b : Ordinal\n⊢ a < (fun a => sup fun n => (f^[n]) a) b ↔ ∃ n, a < (f^[n]) b",
"state_before": "f : Ordinal → Ordinal\na b : Ordinal\n⊢ a < nfp f b ↔ ∃ n, a < (f^[n]) b",
"tactic": "rw [← sup_iterate_eq_nfp]"
},
{
"state_after": "no goals",
"state_before": "f : Ordinal → Ordinal\na b : Ordinal\n⊢ a < (fun a => sup fun n => (f^[n]) a) b ↔ ∃ n, a < (f^[n]) b",
"tactic": "exact lt_sup"
}
]
| [
449,
15
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
447,
1
]
|
Mathlib/FieldTheory/Minpoly/Basic.lean | minpoly.natDegree_pos | [
{
"state_after": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\n⊢ natDegree (minpoly A x) ≠ 0",
"state_before": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\n⊢ 0 < natDegree (minpoly A x)",
"tactic": "rw [pos_iff_ne_zero]"
},
{
"state_after": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n⊢ False",
"state_before": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\n⊢ natDegree (minpoly A x) ≠ 0",
"tactic": "intro ndeg_eq_zero"
},
{
"state_after": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\neq_one : minpoly A x = 1\n⊢ False",
"state_before": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n⊢ False",
"tactic": "have eq_one : minpoly A x = 1 := by\n rw [eq_C_of_natDegree_eq_zero ndeg_eq_zero]\n convert C_1 (R := A)\n simpa only [ndeg_eq_zero.symm] using (monic hx).leadingCoeff"
},
{
"state_after": "no goals",
"state_before": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\neq_one : minpoly A x = 1\n⊢ False",
"tactic": "simpa only [eq_one, AlgHom.map_one, one_ne_zero] using aeval A x"
},
{
"state_after": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n⊢ ↑C (coeff (minpoly A x) 0) = 1",
"state_before": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n⊢ minpoly A x = 1",
"tactic": "rw [eq_C_of_natDegree_eq_zero ndeg_eq_zero]"
},
{
"state_after": "case h.e'_2.h.e'_6\nA : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n⊢ coeff (minpoly A x) 0 = 1",
"state_before": "A : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n⊢ ↑C (coeff (minpoly A x) 0) = 1",
"tactic": "convert C_1 (R := A)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6\nA : Type u_2\nB : Type u_1\nB' : Type ?u.58941\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nhx : IsIntegral A x\nndeg_eq_zero : natDegree (minpoly A x) = 0\n⊢ coeff (minpoly A x) 0 = 1",
"tactic": "simpa only [ndeg_eq_zero.symm] using (monic hx).leadingCoeff"
}
]
| [
194,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
]
|
Mathlib/Logic/Equiv/Set.lean | Equiv.symm_image_image | []
| [
79,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
]
|
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | isLeast_csInf | [
{
"state_after": "α : Type u_1\nβ : Type ?u.69153\nγ : Type ?u.69156\nι : Sort ?u.69159\ninst✝¹ : ConditionallyCompleteLinearOrder α\ns t : Set α\na b : α\ninst✝ : IsWellOrder α fun x x_1 => x < x_1\nhs : Set.Nonempty s\n⊢ IsLeast s (argminOn id (_ : WellFounded fun x x_1 => x < x_1) s hs)",
"state_before": "α : Type u_1\nβ : Type ?u.69153\nγ : Type ?u.69156\nι : Sort ?u.69159\ninst✝¹ : ConditionallyCompleteLinearOrder α\ns t : Set α\na b : α\ninst✝ : IsWellOrder α fun x x_1 => x < x_1\nhs : Set.Nonempty s\n⊢ IsLeast s (sInf s)",
"tactic": "rw [sInf_eq_argmin_on hs]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.69153\nγ : Type ?u.69156\nι : Sort ?u.69159\ninst✝¹ : ConditionallyCompleteLinearOrder α\ns t : Set α\na b : α\ninst✝ : IsWellOrder α fun x x_1 => x < x_1\nhs : Set.Nonempty s\n⊢ IsLeast s (argminOn id (_ : WellFounded fun x x_1 => x < x_1) s hs)",
"tactic": "exact ⟨argminOn_mem _ _ _ _, fun a ha => argminOn_le id _ _ ha⟩"
}
]
| [
975,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
973,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | CategoryTheory.Limits.isIso_ι_of_isInitial | [
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ colimit.ι F j ≫ colimit.desc F (coconeOfDiagramInitial I F) = 𝟙 (F.obj j)",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ colimit.ι F j ≫ colimit.desc F (coconeOfDiagramInitial I F) = 𝟙 (F.obj j) ∧\n colimit.desc F (coconeOfDiagramInitial I F) ≫ colimit.ι F j = 𝟙 (colimit F)",
"tactic": "refine ⟨?_, by ext; simp⟩"
},
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ colimit.ι F j ≫ colimit.desc F (coconeOfDiagramInitial I F) = 𝟙 (F.obj j)",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ colimit.ι F j ≫ colimit.desc F (coconeOfDiagramInitial I F) = 𝟙 (F.obj j)",
"tactic": "dsimp"
},
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ inv (𝟙 (F.obj j)) = 𝟙 (F.obj j)",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ colimit.ι F j ≫ colimit.desc F (coconeOfDiagramInitial I F) = 𝟙 (F.obj j)",
"tactic": "simp only [colimit.ι_desc, coconeOfDiagramInitial_pt, coconeOfDiagramInitial_ι_app,\nFunctor.const_obj_obj, IsInitial.to_self, Functor.map_id]"
},
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ Classical.choose (_ : ∃ inv, 𝟙 (F.obj j) ≫ inv = 𝟙 (F.obj j) ∧ inv ≫ 𝟙 (F.obj j) = 𝟙 (F.obj j)) = 𝟙 (F.obj j)",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ inv (𝟙 (F.obj j)) = 𝟙 (F.obj j)",
"tactic": "dsimp [inv]"
},
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ Classical.choose (_ : ∃ x, (fun x => x = 𝟙 (F.obj j)) x) = 𝟙 (F.obj j)",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ Classical.choose (_ : ∃ inv, 𝟙 (F.obj j) ≫ inv = 𝟙 (F.obj j) ∧ inv ≫ 𝟙 (F.obj j) = 𝟙 (F.obj j)) = 𝟙 (F.obj j)",
"tactic": "simp only [Category.id_comp, Category.comp_id, and_self]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ Classical.choose (_ : ∃ x, (fun x => x = 𝟙 (F.obj j)) x) = 𝟙 (F.obj j)",
"tactic": "apply @Classical.choose_spec _ (fun x => x = 𝟙 F.obj j) _"
},
{
"state_after": "case w\nC : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\nj✝ : J\n⊢ colimit.ι F j✝ ≫ colimit.desc F (coconeOfDiagramInitial I F) ≫ colimit.ι F j = colimit.ι F j✝ ≫ 𝟙 (colimit F)",
"state_before": "C : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\n⊢ colimit.desc F (coconeOfDiagramInitial I F) ≫ colimit.ι F j = 𝟙 (colimit F)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u₁\ninst✝³ : Category C\nJ : Type u\ninst✝² : Category J\nj : J\nI : IsInitial j\nF : J ⥤ C\ninst✝¹ : HasColimit F\ninst✝ : ∀ (i j : J) (f : i ⟶ j), IsIso (F.map f)\nj✝ : J\n⊢ colimit.ι F j✝ ≫ colimit.desc F (coconeOfDiagramInitial I F) ≫ colimit.ι F j = colimit.ι F j✝ ≫ 𝟙 (colimit F)",
"tactic": "simp"
}
]
| [
748,
5
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
740,
1
]
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.nonpos_iff_eq_zero' | []
| [
1091,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1090,
1
]
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.bfamilyOfFamily_typein | []
| [
1150,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1148,
1
]
|
Mathlib/Topology/ContinuousOn.lean | nhdsWithin_mono | []
| [
165,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
]
|
Mathlib/Algebra/CharP/Basic.lean | ringChar.eq_iff | []
| [
230,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
229,
1
]
|
Mathlib/Logic/Function/Basic.lean | Function.bijective_iff_has_inverse | []
| [
512,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
510,
1
]
|
Mathlib/Computability/Partrec.lean | Nat.Partrec.prec' | [
{
"state_after": "no goals",
"state_before": "f g h : ℕ →. ℕ\nhf : Partrec f\nhg : Partrec g\nhh : Partrec h\na s : ℕ\n⊢ (s ∈\n (Seq.seq (Nat.pair <$> Part.some a) fun x => f a) >>=\n unpaired fun a n =>\n Nat.rec (g a)\n (fun y IH => do\n let i ← IH\n h (Nat.pair a (Nat.pair y i)))\n n) ↔\n s ∈\n Part.bind (f a) fun n =>\n Nat.rec (g a)\n (fun y IH => do\n let i ← IH\n h (Nat.pair a (Nat.pair y i)))\n n",
"tactic": "simp [Seq.seq]"
}
]
| [
215,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
]
|
Mathlib/Data/Polynomial/Div.lean | Polynomial.degree_add_divByMonic | [
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\n⊢ degree q + degree (p /ₘ q) = degree p",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n⊢ degree q + degree (p /ₘ q) = degree p",
"tactic": "nontriviality R"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\n⊢ degree q + degree (p /ₘ q) = degree p",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\n⊢ degree q + degree (p /ₘ q) = degree p",
"tactic": "have hdiv0 : p /ₘ q ≠ 0 := by rwa [Ne.def, divByMonic_eq_zero_iff hq, not_lt]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\nhlc : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ degree q + degree (p /ₘ q) = degree p",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\n⊢ degree q + degree (p /ₘ q) = degree p",
"tactic": "have hlc : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0 := by\n rwa [Monic.def.1 hq, one_mul, Ne.def, leadingCoeff_eq_zero]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\nhlc : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\nhmod : degree (p %ₘ q) < degree (q * (p /ₘ q))\n⊢ degree q + degree (p /ₘ q) = degree p",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\nhlc : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ degree q + degree (p /ₘ q) = degree p",
"tactic": "have hmod : degree (p %ₘ q) < degree (q * (p /ₘ q)) :=\n calc\n degree (p %ₘ q) < degree q := degree_modByMonic_lt _ hq\n _ ≤ _ := by\n rw [degree_mul' hlc, degree_eq_natDegree hq.ne_zero, degree_eq_natDegree hdiv0, ←\n Nat.cast_add, Nat.cast_withBot, Nat.cast_withBot, WithBot.coe_le_coe]\n exact Nat.le_add_right _ _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\nhlc : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\nhmod : degree (p %ₘ q) < degree (q * (p /ₘ q))\n⊢ degree q + degree (p /ₘ q) = degree p",
"tactic": "calc\n degree q + degree (p /ₘ q) = degree (q * (p /ₘ q)) := Eq.symm (degree_mul' hlc)\n _ = degree (p %ₘ q + q * (p /ₘ q)) := (degree_add_eq_right_of_degree_lt hmod).symm\n _ = _ := congr_arg _ (modByMonic_add_div _ hq)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\n⊢ p /ₘ q ≠ 0",
"tactic": "rwa [Ne.def, divByMonic_eq_zero_iff hq, not_lt]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\n⊢ leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0",
"tactic": "rwa [Monic.def.1 hq, one_mul, Ne.def, leadingCoeff_eq_zero]"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\nhlc : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ natDegree q ≤ natDegree q + natDegree (p /ₘ q)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\nhlc : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ degree q ≤ degree (q * (p /ₘ q))",
"tactic": "rw [degree_mul' hlc, degree_eq_natDegree hq.ne_zero, degree_eq_natDegree hdiv0, ←\n Nat.cast_add, Nat.cast_withBot, Nat.cast_withBot, WithBot.coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np q : R[X]\nhq : Monic q\nh : degree q ≤ degree p\n✝ : Nontrivial R\nhdiv0 : p /ₘ q ≠ 0\nhlc : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ natDegree q ≤ natDegree q + natDegree (p /ₘ q)",
"tactic": "exact Nat.le_add_right _ _"
}
]
| [
281,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
265,
1
]
|
Mathlib/MeasureTheory/Group/Prod.lean | MeasureTheory.measure_mul_measure_eq | [
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns✝ : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : ↑↑ν s ≠ 0\nh3s : ↑↑ν s ≠ ⊤\nh1 :\n (↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) =\n ∫⁻ (x : G), indicator t (fun x => 1) x ∂ν\n⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns✝ : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : ↑↑ν s ≠ 0\nh3s : ↑↑ν s ≠ ⊤\n⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t",
"tactic": "have h1 :=\n measure_lintegral_div_measure ν ν hs h2s h3s (t.indicator fun _ => 1)\n (measurable_const.indicator ht)"
},
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns✝ : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : ↑↑ν s ≠ 0\nh3s : ↑↑ν s ≠ ⊤\nh1 :\n (↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) =\n ∫⁻ (x : G), indicator t (fun x => 1) x ∂ν\nh2 :\n (↑↑μ s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) =\n ∫⁻ (x : G), indicator t (fun x => 1) x ∂μ\n⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns✝ : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : ↑↑ν s ≠ 0\nh3s : ↑↑ν s ≠ ⊤\nh1 :\n (↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) =\n ∫⁻ (x : G), indicator t (fun x => 1) x ∂ν\n⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t",
"tactic": "have h2 :=\n measure_lintegral_div_measure μ ν hs h2s h3s (t.indicator fun _ => 1)\n (measurable_const.indicator ht)"
},
{
"state_after": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns✝ : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : ↑↑ν s ≠ 0\nh3s : ↑↑ν s ≠ ⊤\nh1 : (↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) = ↑↑ν t\nh2 : (↑↑μ s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) = ↑↑μ t\n⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns✝ : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : ↑↑ν s ≠ 0\nh3s : ↑↑ν s ≠ ⊤\nh1 :\n (↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) =\n ∫⁻ (x : G), indicator t (fun x => 1) x ∂ν\nh2 :\n (↑↑μ s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) =\n ∫⁻ (x : G), indicator t (fun x => 1) x ∂μ\n⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t",
"tactic": "rw [lintegral_indicator _ ht, set_lintegral_one] at h1 h2"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns✝ : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\ns t : Set G\nhs : MeasurableSet s\nht : MeasurableSet t\nh2s : ↑↑ν s ≠ 0\nh3s : ↑↑ν s ≠ ⊤\nh1 : (↑↑ν s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) = ↑↑ν t\nh2 : (↑↑μ s * ∫⁻ (y : G), indicator t (fun x => 1) y⁻¹ / ↑↑ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) = ↑↑μ t\n⊢ ↑↑μ s * ↑↑ν t = ↑↑ν s * ↑↑μ t",
"tactic": "rw [← h1, mul_left_comm, h2]"
}
]
| [
348,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
]
|
Mathlib/Algebra/Hom/Group.lean | MonoidWithZeroHom.ext | []
| [
662,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
661,
1
]
|
Mathlib/RingTheory/Localization/InvSubmonoid.lean | IsLocalization.smul_toInvSubmonoid | [
{
"state_after": "case h.e'_2.h.e\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.131549\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nm : { x // x ∈ M }\n⊢ HSMul.hSMul m = HMul.hMul (↑(algebraMap R S) ↑m)",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.131549\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nm : { x // x ∈ M }\n⊢ m • ↑(↑(toInvSubmonoid M S) m) = 1",
"tactic": "convert mul_toInvSubmonoid M S m"
},
{
"state_after": "case h.e'_2.h.e.h\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.131549\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nm : { x // x ∈ M }\nx✝ : S\n⊢ m • x✝ = ↑(algebraMap R S) ↑m * x✝",
"state_before": "case h.e'_2.h.e\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.131549\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nm : { x // x ∈ M }\n⊢ HSMul.hSMul m = HMul.hMul (↑(algebraMap R S) ↑m)",
"tactic": "ext"
},
{
"state_after": "case h.e'_2.h.e.h\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.131549\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nm : { x // x ∈ M }\nx✝ : S\n⊢ m • x✝ = ↑m • x✝",
"state_before": "case h.e'_2.h.e.h\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.131549\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nm : { x // x ∈ M }\nx✝ : S\n⊢ m • x✝ = ↑(algebraMap R S) ↑m * x✝",
"tactic": "rw [← Algebra.smul_def]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e.h\nR : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type ?u.131549\ninst✝¹ : CommRing P\ninst✝ : IsLocalization M S\nm : { x // x ∈ M }\nx✝ : S\n⊢ m • x✝ = ↑m • x✝",
"tactic": "rfl"
}
]
| [
87,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
]
|
Mathlib/Topology/Algebra/UniformGroup.lean | uniformEmbedding_translate_mul | [
{
"state_after": "α : Type u_1\nβ : Type ?u.112462\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\na : α\n⊢ 𝓤 α = 𝓤 α\n\nα : Type u_1\nβ : Type ?u.112462\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\na : α\n⊢ Function.Injective fun x => (x.fst * a, x.snd * a)",
"state_before": "α : Type u_1\nβ : Type ?u.112462\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\na : α\n⊢ comap (fun x => (x.fst * a, x.snd * a)) (𝓤 α) = 𝓤 α",
"tactic": "nth_rewrite 1 [← uniformity_translate_mul a, comap_map]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.112462\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\na : α\n⊢ Function.Injective fun x => (x.fst * a, x.snd * a)",
"state_before": "α : Type u_1\nβ : Type ?u.112462\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\na : α\n⊢ 𝓤 α = 𝓤 α\n\nα : Type u_1\nβ : Type ?u.112462\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\na : α\n⊢ Function.Injective fun x => (x.fst * a, x.snd * a)",
"tactic": "rfl"
},
{
"state_after": "case mk.mk\nα : Type u_1\nβ : Type ?u.112462\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\na p₁ p₂ q₁ q₂ : α\n⊢ (fun x => (x.fst * a, x.snd * a)) (p₁, p₂) = (fun x => (x.fst * a, x.snd * a)) (q₁, q₂) → (p₁, p₂) = (q₁, q₂)",
"state_before": "α : Type u_1\nβ : Type ?u.112462\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\na : α\n⊢ Function.Injective fun x => (x.fst * a, x.snd * a)",
"tactic": "rintro ⟨p₁, p₂⟩ ⟨q₁, q₂⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nα : Type u_1\nβ : Type ?u.112462\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : UniformGroup α\na p₁ p₂ q₁ q₂ : α\n⊢ (fun x => (x.fst * a, x.snd * a)) (p₁, p₂) = (fun x => (x.fst * a, x.snd * a)) (q₁, q₂) → (p₁, p₂) = (q₁, q₂)",
"tactic": "simp only [Prod.mk.injEq, mul_left_inj, imp_self]"
}
]
| [
189,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
183,
1
]
|
Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.val_eq_coe | []
| [
180,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
]
|
Mathlib/Data/List/MinMax.lean | List.argAux_self | []
| [
67,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
]
|
Mathlib/Analysis/NormedSpace/LinearIsometry.lean | LinearIsometryEquiv.coe_ofEq_apply | []
| [
1179,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1178,
1
]
|
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.sum_add_distrib' | []
| [
894,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
892,
1
]
|
Mathlib/Data/Set/Function.lean | Set.SurjOn.mapsTo_compl | []
| [
877,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
874,
1
]
|
Mathlib/GroupTheory/FreeGroup.lean | FreeGroup.map_eq_lift | [
{
"state_after": "no goals",
"state_before": "α : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\nβ : Type v\nf : α → β\nx✝ y : FreeGroup α\nx : α\n⊢ ↑(↑lift (of ∘ f)) (of x) = of (f x)",
"tactic": "simp"
}
]
| [
848,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
847,
1
]
|
Mathlib/GroupTheory/Submonoid/Pointwise.lean | AddSubmonoid.pointwise_smul_le_iff | []
| [
434,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
433,
1
]
|
Mathlib/Data/Finset/Basic.lean | Finset.mem_erase | []
| [
1860,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1859,
1
]
|
Mathlib/NumberTheory/ArithmeticFunction.lean | Nat.ArithmeticFunction.ppow_zero | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : ArithmeticFunction R\n⊢ ppow f 0 = ↑ζ",
"tactic": "rw [ppow, dif_pos rfl]"
}
]
| [
551,
89
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
551,
1
]
|
Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | SimpleGraph.dart_card_eq_sum_degrees | [
{
"state_after": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype (Sym2 V)\nthis : DecidableEq V\n⊢ Fintype.card (Dart G) = ∑ v : V, degree G v",
"state_before": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype (Sym2 V)\n⊢ Fintype.card (Dart G) = ∑ v : V, degree G v",
"tactic": "haveI := Classical.decEq V"
},
{
"state_after": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype (Sym2 V)\nthis : DecidableEq V\n⊢ card univ = ∑ x : V, card (filter (fun d => d.fst = x) univ)",
"state_before": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype (Sym2 V)\nthis : DecidableEq V\n⊢ Fintype.card (Dart G) = ∑ v : V, degree G v",
"tactic": "simp only [← card_univ, ← dart_fst_fiber_card_eq_degree]"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype (Sym2 V)\nthis : DecidableEq V\n⊢ card univ = ∑ x : V, card (filter (fun d => d.fst = x) univ)",
"tactic": "exact card_eq_sum_card_fiberwise (by simp)"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nG : SimpleGraph V\ninst✝² : Fintype V\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype (Sym2 V)\nthis : DecidableEq V\n⊢ ∀ (x : Dart G), x ∈ univ → x.fst ∈ univ",
"tactic": "simp"
}
]
| [
81,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
]
|
Mathlib/Analysis/Convex/Topology.lean | Convex.combo_interior_closure_subset_interior | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.39791\n𝕜 : Type u_2\nE : Type u_1\ninst✝⁵ : LinearOrderedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalAddGroup E\ninst✝ : ContinuousConstSMul 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\na b : 𝕜\nha : 0 < a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ interior (a • s) + closure (b • s) = interior (a • s) + b • s",
"tactic": "rw [isOpen_interior.add_closure (b • s)]"
}
]
| [
125,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
117,
1
]
|
Mathlib/SetTheory/Ordinal/Exponential.lean | Ordinal.right_le_opow | []
| [
177,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
]
|
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | Matrix.mul_inv_cancel_right_of_invertible | []
| [
356,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
355,
1
]
|
Mathlib/Topology/Bases.lean | TopologicalSpace.IsSeparable.image | [
{
"state_after": "case intro.intro\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Continuous f\nc : Set α\nc_count : Set.Countable c\nhc : s ⊆ _root_.closure c\n⊢ IsSeparable (f '' s)",
"state_before": "α : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\ns : Set α\nhs : IsSeparable s\nf : α → β\nhf : Continuous f\n⊢ IsSeparable (f '' s)",
"tactic": "rcases hs with ⟨c, c_count, hc⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Continuous f\nc : Set α\nc_count : Set.Countable c\nhc : s ⊆ _root_.closure c\n⊢ f '' s ⊆ _root_.closure (f '' c)",
"state_before": "case intro.intro\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Continuous f\nc : Set α\nc_count : Set.Countable c\nhc : s ⊆ _root_.closure c\n⊢ IsSeparable (f '' s)",
"tactic": "refine' ⟨f '' c, c_count.image _, _⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Continuous f\nc : Set α\nc_count : Set.Countable c\nhc : s ⊆ _root_.closure c\n⊢ s ⊆ f ⁻¹' _root_.closure (f '' c)",
"state_before": "case intro.intro\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Continuous f\nc : Set α\nc_count : Set.Countable c\nhc : s ⊆ _root_.closure c\n⊢ f '' s ⊆ _root_.closure (f '' c)",
"tactic": "rw [image_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nt : TopologicalSpace α\nβ : Type u_1\ninst✝ : TopologicalSpace β\ns : Set α\nf : α → β\nhf : Continuous f\nc : Set α\nc_count : Set.Countable c\nhc : s ⊆ _root_.closure c\n⊢ s ⊆ f ⁻¹' _root_.closure (f '' c)",
"tactic": "exact hc.trans (closure_subset_preimage_closure_image hf)"
}
]
| [
438,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
433,
1
]
|
Mathlib/Topology/Homotopy/Path.lean | Path.Homotopic.map | []
| [
290,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
288,
8
]
|
Mathlib/Data/ZMod/Parity.lean | ZMod.ne_zero_iff_odd | [
{
"state_after": "case mpr\nn : ℕ\n⊢ ¬↑n ≠ 0 → ¬Odd n",
"state_before": "case mpr\nn : ℕ\n⊢ Odd n → ↑n ≠ 0",
"tactic": "contrapose"
},
{
"state_after": "no goals",
"state_before": "case mpr\nn : ℕ\n⊢ ¬↑n ≠ 0 → ¬Odd n",
"tactic": "simp [eq_zero_iff_even]"
}
]
| [
39,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
36,
1
]
|
Mathlib/Topology/MetricSpace/Baire.lean | dense_of_mem_residual | []
| [
253,
14
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
251,
1
]
|
Mathlib/Data/Polynomial/Coeff.lean | Polynomial.coeff_smul | [
{
"state_after": "case ofFinsupp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\ninst✝ : SMulZeroClass S R\nr : S\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff (r • { toFinsupp := toFinsupp✝ }) n = r • coeff { toFinsupp := toFinsupp✝ } n",
"state_before": "R : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝¹ : Semiring R\np✝ q r✝ : R[X]\ninst✝ : SMulZeroClass S R\nr : S\np : R[X]\nn : ℕ\n⊢ coeff (r • p) n = r • coeff p n",
"tactic": "rcases p with ⟨⟩"
},
{
"state_after": "case ofFinsupp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\ninst✝ : SMulZeroClass S R\nr : S\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ↑(r • toFinsupp✝) n = r • ↑toFinsupp✝ n",
"state_before": "case ofFinsupp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\ninst✝ : SMulZeroClass S R\nr : S\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ coeff (r • { toFinsupp := toFinsupp✝ }) n = r • coeff { toFinsupp := toFinsupp✝ } n",
"tactic": "simp_rw [← ofFinsupp_smul, coeff]"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp\nR : Type u\nS : Type v\na b : R\nn✝ m : ℕ\ninst✝¹ : Semiring R\np q r✝ : R[X]\ninst✝ : SMulZeroClass S R\nr : S\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ ↑(r • toFinsupp✝) n = r • ↑toFinsupp✝ n",
"tactic": "exact Finsupp.smul_apply _ _ _"
}
]
| [
64,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
]
|
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.op_norm_le_of_lipschitz | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.274096\nE : Type u_3\nEₗ : Type ?u.274102\nF : Type u_4\nFₗ : Type ?u.274108\nG : Type ?u.274111\nGₗ : Type ?u.274114\n𝓕 : Type ?u.274117\ninst✝¹⁵ : SeminormedAddCommGroup E\ninst✝¹⁴ : SeminormedAddCommGroup Eₗ\ninst✝¹³ : SeminormedAddCommGroup F\ninst✝¹² : SeminormedAddCommGroup Fₗ\ninst✝¹¹ : SeminormedAddCommGroup G\ninst✝¹⁰ : SeminormedAddCommGroup Gₗ\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜₂\ninst✝⁷ : NontriviallyNormedField 𝕜₃\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 Eₗ\ninst✝⁴ : NormedSpace 𝕜₂ F\ninst✝³ : NormedSpace 𝕜 Fₗ\ninst✝² : NormedSpace 𝕜₃ G\ninst✝¹ : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nf : E →SL[σ₁₂] F\nK : ℝ≥0\nhf : LipschitzWith K ↑f\nx : E\n⊢ ‖↑f x‖ ≤ ↑K * ‖x‖",
"tactic": "simpa only [dist_zero_right, f.map_zero] using hf.dist_le_mul x 0"
}
]
| [
176,
70
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
174,
1
]
|
Mathlib/Data/Finset/Lattice.lean | Finset.inf'_induction | []
| [
966,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
964,
1
]
|
Mathlib/GroupTheory/Index.lean | Subgroup.relindex_bot_right | [
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\n⊢ relindex H ⊥ = 1",
"tactic": "rw [relindex, subgroupOf_bot_eq_top, index_top]"
}
]
| [
267,
100
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
267,
1
]
|
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | ContDiff.sin | []
| [
1036,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1035,
1
]
|
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.continuousWithinAt_iff_continuousWithinAt_comp_right | []
| [
1143,
75
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1139,
1
]
|
Mathlib/Data/Polynomial/Basic.lean | Polynomial.support_C_mul_X_pow' | [
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\nc : R\n⊢ support (↑C c * X ^ n) ⊆ {n}",
"tactic": "simpa only [C_mul_X_pow_eq_monomial] using support_monomial' n c"
}
]
| [
858,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
857,
1
]
|
Mathlib/Data/UnionFind.lean | UFModel.Agrees.set | [
{
"state_after": "case mk\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\n⊢ Agrees (Array.set arr i x) f m'",
"state_before": "α : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\nn : ℕ\nm : Fin n → β\nH : Agrees arr f m\ni : Fin (Array.size arr)\nx : α\nm' : Fin n → β\nhm₁ : ∀ (j : Fin n), ↑j ≠ ↑i → m' j = m j\nhm₂ : ∀ (h : ↑i < n), f x = m' { val := ↑i, isLt := h }\n⊢ Agrees (Array.set arr i x) f m'",
"tactic": "cases H"
},
{
"state_after": "case mk\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\n⊢ f (Array.get (Array.set arr i x) { val := j, isLt := hj₁ }) = m' { val := j, isLt := hj₂ }",
"state_before": "case mk\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\n⊢ Agrees (Array.set arr i x) f m'",
"tactic": "refine mk' (by simp) fun j hj₁ hj₂ ↦ ?_"
},
{
"state_after": "case mk\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\n⊢ f (Array.set arr i x)[j] = m' { val := j, isLt := hj₂ }",
"state_before": "case mk\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\n⊢ f (Array.get (Array.set arr i x) { val := j, isLt := hj₁ }) = m' { val := j, isLt := hj₂ }",
"tactic": "suffices f (Array.set arr i x)[j] = m' ⟨j, hj₂⟩ by simp_all [Array.get_set]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ↑i = j\n⊢ f (Array.set arr i x)[j] = m' { val := j, isLt := hj₂ }\n\ncase neg\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ¬↑i = j\n⊢ f (Array.set arr i x)[j] = m' { val := j, isLt := hj₂ }",
"state_before": "case mk\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\n⊢ f (Array.set arr i x)[j] = m' { val := j, isLt := hj₂ }",
"tactic": "by_cases h : i = j"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\n⊢ Array.size arr = Array.size (Array.set arr i x)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nthis : f (Array.set arr i x)[j] = m' { val := j, isLt := hj₂ }\n⊢ f (Array.get (Array.set arr i x) { val := j, isLt := hj₁ }) = m' { val := j, isLt := hj₂ }",
"tactic": "simp_all [Array.get_set]"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nhj₁ : ↑i < Array.size (Array.set arr i x)\nhj₂ : ↑i < Array.size arr\n⊢ f (Array.set arr i x)[↑i] = m' { val := ↑i, isLt := hj₂ }",
"state_before": "case pos\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ↑i = j\n⊢ f (Array.set arr i x)[j] = m' { val := j, isLt := hj₂ }",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nhj₁ : ↑i < Array.size (Array.set arr i x)\nhj₂ : ↑i < Array.size arr\n⊢ f (Array.set arr i x)[↑i] = m' { val := ↑i, isLt := hj₂ }",
"tactic": "rw [Array.get_set_eq, ← hm₂]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ¬↑i = j\n⊢ f arr[j] = (fun i => f (Array.get arr i)) { val := j, isLt := hj₂ }\n\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ¬↑i = j\n⊢ j < Array.size arr\n\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ¬↑i = j\n⊢ j < Array.size arr",
"state_before": "case neg\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ¬↑i = j\n⊢ f (Array.set arr i x)[j] = m' { val := j, isLt := hj₂ }",
"tactic": "rw [arr.get_set_ne _ _ _ h, hm₁ ⟨j, _⟩ (Ne.symm h)]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ¬↑i = j\n⊢ f arr[j] = (fun i => f (Array.get arr i)) { val := j, isLt := hj₂ }\n\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ¬↑i = j\n⊢ j < Array.size arr\n\nα : Type u_1\nβ : Sort u_2\nf : α → β\narr : Array α\ni : Fin (Array.size arr)\nx : α\nm' : Fin (Array.size arr) → β\nhm₂ : ∀ (h : ↑i < Array.size arr), f x = m' { val := ↑i, isLt := h }\nhm₁ : ∀ (j : Fin (Array.size arr)), ↑j ≠ ↑i → m' j = (fun i => f (Array.get arr i)) j\nj : ℕ\nhj₁ : j < Array.size (Array.set arr i x)\nhj₂ : j < Array.size arr\nh : ¬↑i = j\n⊢ j < Array.size arr",
"tactic": "rfl"
}
]
| [
113,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
1
]
|
Mathlib/Order/CompleteLattice.lean | sInf_eq_top | []
| [
549,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
548,
1
]
|
Mathlib/LinearAlgebra/Ray.lean | Module.Ray.map_neg | [
{
"state_after": "case h\nR : Type u_1\ninst✝⁴ : StrictOrderedCommRing R\nM : Type u_2\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nx y : M\nf : M ≃ₗ[R] N\ng : M\nhg : g ≠ 0\n⊢ ↑(map f) (-rayOfNeZero R g hg) = -↑(map f) (rayOfNeZero R g hg)",
"state_before": "R : Type u_1\ninst✝⁴ : StrictOrderedCommRing R\nM : Type u_2\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nx y : M\nf : M ≃ₗ[R] N\nv : Ray R M\n⊢ ↑(map f) (-v) = -↑(map f) v",
"tactic": "induction' v using Module.Ray.ind with g hg"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝⁴ : StrictOrderedCommRing R\nM : Type u_2\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nx y : M\nf : M ≃ₗ[R] N\ng : M\nhg : g ≠ 0\n⊢ ↑(map f) (-rayOfNeZero R g hg) = -↑(map f) (rayOfNeZero R g hg)",
"tactic": "simp"
}
]
| [
498,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
496,
11
]
|
Mathlib/Topology/MetricSpace/HausdorffDistance.lean | Metric.infDist_singleton | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.59247\nα : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\ns t u : Set α\nx y : α\nΦ : α → β\n⊢ infDist x {y} = dist x y",
"tactic": "simp [infDist, dist_edist]"
}
]
| [
505,
86
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
505,
1
]
|
Mathlib/Data/Finset/Preimage.lean | Finset.preimage_subset | []
| [
110,
96
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
]
|
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | DifferentiableWithinAt.cexp | []
| [
140,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
1
]
|
Mathlib/SetTheory/Game/PGame.lean | PGame.relabel_moveRight' | []
| [
1162,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1160,
1
]
|
Mathlib/Data/Rat/NNRat.lean | Rat.toNNRat_add | [
{
"state_after": "no goals",
"state_before": "p q : ℚ\nhq : 0 ≤ q\nhp : 0 ≤ p\n⊢ ↑(toNNRat (q + p)) = ↑(toNNRat q + toNNRat p)",
"tactic": "simp [toNNRat, hq, hp, add_nonneg]"
}
]
| [
379,
53
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
378,
1
]
|
Mathlib/Data/Num/Lemmas.lean | Num.cast_pos | []
| [
290,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
289,
1
]
|
Mathlib/Topology/Algebra/Module/Multilinear.lean | ContinuousMultilinearMap.map_add | []
| [
123,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
]
|
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.smul_inv | []
| [
2204,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2203,
1
]
|
Mathlib/RingTheory/Ideal/Quotient.lean | Ideal.Quotient.eq_zero_iff_mem | []
| [
137,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
]
|
Mathlib/RingTheory/RootsOfUnity/Basic.lean | rootsOfUnityEquivNthRoots_apply | []
| [
225,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
223,
1
]
|
Mathlib/Data/List/Card.lean | List.card_nil | []
| [
79,
57
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
9
]
|
Mathlib/Computability/Primrec.lean | Primrec₂.right | []
| [
434,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
433,
1
]
|
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_to_list | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.290155\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nf : α → β\n⊢ List.prod (List.map f (toList s)) = Finset.prod s f",
"tactic": "rw [Finset.prod, ← Multiset.coe_prod, ← Multiset.coe_map, Finset.coe_toList]"
}
]
| [
430,
79
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
429,
1
]
|
Mathlib/Order/Bounded.lean | Set.bounded_ge_Icc | []
| [
274,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
]
|
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