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Mathlib/RingTheory/Ideal/Quotient.lean | Ideal.exists_sub_one_mem_and_mem | [
{
"state_after": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ ∃ r x, r ∈ f j",
"state_before": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\n⊢ ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j",
"tactic": "intro j hjs hji"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nhf : f i ⊔ f j = ⊤\n⊢ ∃ r x, r ∈ f j",
"state_before": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ ∃ r x, r ∈ f j",
"tactic": "specialize hf i his j hjs hji.symm"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nhf : ∃ y, y ∈ f i ∧ ∃ z, z ∈ f j ∧ y + z = 1\n⊢ ∃ r x, r ∈ f j",
"state_before": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nhf : f i ⊔ f j = ⊤\n⊢ ∃ r x, r ∈ f j",
"tactic": "rw [eq_top_iff_one, Submodule.mem_sup] at hf"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ ∃ r x, r ∈ f j",
"state_before": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nhf : ∃ y, y ∈ f i ∧ ∃ z, z ∈ f j ∧ y + z = 1\n⊢ ∃ r x, r ∈ f j",
"tactic": "rcases hf with ⟨r, hri, s, hsj, hrs⟩"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ 1 - r - 1 ∈ f i\n\ncase intro.intro.intro.intro.refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ 1 - r ∈ f j",
"state_before": "case intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ ∃ r x, r ∈ f j",
"tactic": "refine' ⟨1 - r, _, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ -r ∈ f i",
"state_before": "case intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ 1 - r - 1 ∈ f i",
"tactic": "rw [sub_right_comm, sub_self, zero_sub]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ -r ∈ f i",
"tactic": "exact (f i).neg_mem hri"
},
{
"state_after": "case intro.intro.intro.intro.refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ s ∈ f j",
"state_before": "case intro.intro.intro.intro.refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ 1 - r ∈ f j",
"tactic": "rw [← hrs, add_sub_cancel']"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns✝ : Finset ι\nf : ι → Ideal R\ni : ι\nhis : i ∈ s✝\nj : ι\nhjs : j ∈ s✝\nhji : j ≠ i\nr : R\nhri : r ∈ f i\ns : R\nhsj : s ∈ f j\nhrs : r + s = 1\n⊢ s ∈ f j",
"tactic": "exact hsj"
},
{
"state_after": "case intro.intro\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∃ r, r - 1 ∈ f i ∧ ∀ (j : ι), j ∈ s → j ≠ i → r ∈ f j",
"state_before": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis✝ : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\nthis : ∃ g, (∀ (j : ι), g j - 1 ∈ f i) ∧ ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∃ r, r - 1 ∈ f i ∧ ∀ (j : ι), j ∈ s → j ≠ i → r ∈ f j",
"tactic": "rcases this with ⟨g, hgi, hgj⟩"
},
{
"state_after": "case intro.intro\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∏ x in Finset.erase s i, g x - 1 ∈ f i ∧ ∀ (j : ι), j ∈ s → j ≠ i → ∏ x in Finset.erase s i, g x ∈ f j",
"state_before": "case intro.intro\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∃ r, r - 1 ∈ f i ∧ ∀ (j : ι), j ∈ s → j ≠ i → r ∈ f j",
"tactic": "use ∏ x in s.erase i, g x"
},
{
"state_after": "case intro.intro.left\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∏ x in Finset.erase s i, g x - 1 ∈ f i\n\ncase intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∀ (j : ι), j ∈ s → j ≠ i → ∏ x in Finset.erase s i, g x ∈ f j",
"state_before": "case intro.intro\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∏ x in Finset.erase s i, g x - 1 ∈ f i ∧ ∀ (j : ι), j ∈ s → j ≠ i → ∏ x in Finset.erase s i, g x ∈ f j",
"tactic": "constructor"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\n⊢ ∃ g, (∀ (j : ι), g j - 1 ∈ f i) ∧ ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j",
"state_before": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\n⊢ ∃ g, (∀ (j : ι), g j - 1 ∈ f i) ∧ ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j",
"tactic": "choose g hg1 hg2 using this"
},
{
"state_after": "case refine'_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\n⊢ (fun j => if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) j - 1 ∈ f i\n\ncase refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\n⊢ j ∈ s → j ≠ i → (fun j => if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) j ∈ f j",
"state_before": "R : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\n⊢ ∃ g, (∀ (j : ι), g j - 1 ∈ f i) ∧ ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j",
"tactic": "refine' ⟨fun j => if H : j ∈ s ∧ j ≠ i then g j H.1 H.2 else 1, fun j => _, fun j => _⟩"
},
{
"state_after": "case refine'_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\n⊢ (if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) - 1 ∈ f i",
"state_before": "case refine'_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\n⊢ (fun j => if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) j - 1 ∈ f i",
"tactic": "dsimp only"
},
{
"state_after": "case refine'_1.inl\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nh : j ∈ s ∧ j ≠ i\n⊢ g j (_ : j ∈ s) (_ : j ≠ i) - 1 ∈ f i\n\ncase refine'_1.inr\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nh : ¬(j ∈ s ∧ j ≠ i)\n⊢ 1 - 1 ∈ f i",
"state_before": "case refine'_1\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\n⊢ (if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) - 1 ∈ f i",
"tactic": "split_ifs with h"
},
{
"state_after": "case refine'_1.inr\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nh : ¬(j ∈ s ∧ j ≠ i)\n⊢ 0 ∈ f i",
"state_before": "case refine'_1.inr\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nh : ¬(j ∈ s ∧ j ≠ i)\n⊢ 1 - 1 ∈ f i",
"tactic": "rw [sub_self]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inr\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nh : ¬(j ∈ s ∧ j ≠ i)\n⊢ 0 ∈ f i",
"tactic": "exact (f i).zero_mem"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inl\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nh : j ∈ s ∧ j ≠ i\n⊢ g j (_ : j ∈ s) (_ : j ≠ i) - 1 ∈ f i",
"tactic": "apply hg1"
},
{
"state_after": "case refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ (fun j => if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) j ∈ f j",
"state_before": "case refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\n⊢ j ∈ s → j ≠ i → (fun j => if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) j ∈ f j",
"tactic": "intro hjs hji"
},
{
"state_after": "case refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ (if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) ∈ f j",
"state_before": "case refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ (fun j => if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) j ∈ f j",
"tactic": "dsimp only"
},
{
"state_after": "case refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ g j (_ : j ∈ s) (_ : j ≠ i) ∈ f j\n\ncase refine'_2.hc\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ j ∈ s ∧ j ≠ i",
"state_before": "case refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ (if H : j ∈ s ∧ j ≠ i then g j (_ : j ∈ s) (_ : j ≠ i) else 1) ∈ f j",
"tactic": "rw [dif_pos]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.hc\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ j ∈ s ∧ j ≠ i",
"tactic": "exact ⟨hjs, hji⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\ng : (j : ι) → j ∈ s → j ≠ i → R\nhg1 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 - 1 ∈ f i\nhg2 : ∀ (j : ι) (a : j ∈ s) (a_1 : j ≠ i), g j a a_1 ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ g j (_ : j ∈ s) (_ : j ≠ i) ∈ f j",
"tactic": "apply hg2"
},
{
"state_after": "case intro.intro.left\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∏ x in Finset.erase s i, ↑(Quotient.mk (f i)) (g x) = 1",
"state_before": "case intro.intro.left\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∏ x in Finset.erase s i, g x - 1 ∈ f i",
"tactic": "rw [← Ideal.Quotient.mk_eq_mk_iff_sub_mem, map_one, map_prod]"
},
{
"state_after": "case intro.intro.left.h\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∀ (x : ι), x ∈ Finset.erase s i → ↑(Quotient.mk (f i)) (g x) = 1",
"state_before": "case intro.intro.left\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∏ x in Finset.erase s i, ↑(Quotient.mk (f i)) (g x) = 1",
"tactic": "apply Finset.prod_eq_one"
},
{
"state_after": "case intro.intro.left.h\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nx✝ : ι\na✝ : x✝ ∈ Finset.erase s i\n⊢ ↑(Quotient.mk (f i)) (g x✝) = 1",
"state_before": "case intro.intro.left.h\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∀ (x : ι), x ∈ Finset.erase s i → ↑(Quotient.mk (f i)) (g x) = 1",
"tactic": "intros"
},
{
"state_after": "case intro.intro.left.h\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nx✝ : ι\na✝ : x✝ ∈ Finset.erase s i\n⊢ g x✝ - 1 ∈ f i",
"state_before": "case intro.intro.left.h\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nx✝ : ι\na✝ : x✝ ∈ Finset.erase s i\n⊢ ↑(Quotient.mk (f i)) (g x✝) = 1",
"tactic": "rw [← RingHom.map_one, Ideal.Quotient.mk_eq_mk_iff_sub_mem]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.left.h\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nx✝ : ι\na✝ : x✝ ∈ Finset.erase s i\n⊢ g x✝ - 1 ∈ f i",
"tactic": "apply hgi"
},
{
"state_after": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ ∏ x in Finset.erase s i, g x ∈ f j",
"state_before": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\n⊢ ∀ (j : ι), j ∈ s → j ≠ i → ∏ x in Finset.erase s i, g x ∈ f j",
"tactic": "intro j hjs hji"
},
{
"state_after": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ ∏ x in Finset.erase s i, ↑(Quotient.mk (f j)) (g x) = 0",
"state_before": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ ∏ x in Finset.erase s i, g x ∈ f j",
"tactic": "rw [← Quotient.eq_zero_iff_mem, map_prod]"
},
{
"state_after": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis✝ : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nthis : CommMonoidWithZero (R ⧸ f j) := CommSemiring.toCommMonoidWithZero\n⊢ ∏ x in Finset.erase s i, ↑(Quotient.mk (f j)) (g x) = 0",
"state_before": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\n⊢ ∏ x in Finset.erase s i, ↑(Quotient.mk (f j)) (g x) = 0",
"tactic": "letI : CommMonoidWithZero (R ⧸ f j) := CommSemiring.toCommMonoidWithZero"
},
{
"state_after": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis✝ : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nthis : CommMonoidWithZero (R ⧸ f j) := CommSemiring.toCommMonoidWithZero\n⊢ ↑(Quotient.mk (f j)) (g j) = 0",
"state_before": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis✝ : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nthis : CommMonoidWithZero (R ⧸ f j) := CommSemiring.toCommMonoidWithZero\n⊢ ∏ x in Finset.erase s i, ↑(Quotient.mk (f j)) (g x) = 0",
"tactic": "refine' Finset.prod_eq_zero (Finset.mem_erase_of_ne_of_mem hji hjs) _"
},
{
"state_after": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis✝ : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nthis : CommMonoidWithZero (R ⧸ f j) := CommSemiring.toCommMonoidWithZero\n⊢ g j ∈ f j",
"state_before": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis✝ : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nthis : CommMonoidWithZero (R ⧸ f j) := CommSemiring.toCommMonoidWithZero\n⊢ ↑(Quotient.mk (f j)) (g j) = 0",
"tactic": "rw [Quotient.eq_zero_iff_mem]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.right\nR : Type u\ninst✝ : CommRing R\nI : Ideal R\na b : R\nS ι : Type v\ns : Finset ι\nf : ι → Ideal R\nhf : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → f i ⊔ f j = ⊤\ni : ι\nhis : i ∈ s\nthis✝ : ∀ (j : ι), j ∈ s → j ≠ i → ∃ r x, r ∈ f j\ng : ι → R\nhgi : ∀ (j : ι), g j - 1 ∈ f i\nhgj : ∀ (j : ι), j ∈ s → j ≠ i → g j ∈ f j\nj : ι\nhjs : j ∈ s\nhji : j ≠ i\nthis : CommMonoidWithZero (R ⧸ f j) := CommSemiring.toCommMonoidWithZero\n⊢ g j ∈ f j",
"tactic": "exact hgj j hjs hji"
}
] | [
446,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
405,
1
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Mathlib/Data/Nat/Lattice.lean | Nat.iInf_lt_succ | [] | [
190,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
189,
1
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Mathlib/Logic/Basic.lean | ne_of_apply_ne | [] | [
510,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
509,
1
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Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | CategoryTheory.Limits.kernel_not_iso_of_nonzero | [
{
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"tactic": "skip"
},
{
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}
] | [
397,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
394,
1
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Mathlib/GroupTheory/Perm/Cycle/Type.lean | Equiv.Perm.IsThreeCycle.sign | [
{
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"tactic": "rw [Equiv.Perm.sign_of_cycleType, h.cycleType]"
},
{
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}
] | [
599,
6
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597,
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src/lean/Init/Data/List/Control.lean | List.forM_nil | [] | [
191,
6
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
190,
9
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Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | CategoryTheory.Limits.hasBinaryBiproducts_of_finite_biproducts | [] | [
1239,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1234,
1
] |
Mathlib/Data/Sum/Order.lean | Sum.inl_mono | [] | [
188,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/Order/RelIso/Basic.lean | RelEmbedding.wellFounded | [] | [
400,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
399,
11
] |
Mathlib/Topology/MetricSpace/Basic.lean | Prod.dist_eq | [] | [
1751,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1751,
1
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Mathlib/CategoryTheory/Preadditive/Mat.lean | CategoryTheory.Mat.comp_apply | [] | [
616,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
614,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean | measurable_of_restrict_of_restrict_compl | [] | [
606,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
604,
1
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Mathlib/LinearAlgebra/BilinearForm.lean | BilinForm.ne_zero_of_not_isOrtho_self | [] | [
773,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
772,
1
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Mathlib/Data/Int/Order/Basic.lean | Int.toNat_le | [
{
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"state_before": "a✝ b : ℤ\nn✝ : ℕ\na : ℤ\nn : ℕ\n⊢ toNat a ≤ n ↔ a ≤ ↑n",
"tactic": "rw [ofNat_le.symm, toNat_eq_max, max_le_iff]"
},
{
"state_after": "no goals",
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"tactic": "exact and_iff_left (ofNat_zero_le _)"
}
] | [
502,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
501,
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Mathlib/MeasureTheory/Function/LocallyIntegrable.lean | MeasureTheory.LocallyIntegrableOn.mono | [] | [
54,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
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Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | map_lt_lineMap_iff_slope_lt_slope_left | [] | [
232,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
229,
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Mathlib/Algebra/GCDMonoid/Multiset.lean | Multiset.gcd_dedup | [
{
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"tactic": "simp"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : gcd (dedup s) = gcd s\nh : a ∈ s\n⊢ gcd s = GCDMonoid.gcd a (gcd s)",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : gcd (dedup s) = gcd s\n⊢ gcd (dedup (a ::ₘ s)) = gcd (a ::ₘ s)",
"tactic": "by_cases h : a ∈ s <;> simp [IH, h]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : gcd (dedup s) = gcd s\nh : a ∈ s\n⊢ fold GCDMonoid.gcd 0 s = GCDMonoid.gcd a (fold GCDMonoid.gcd 0 s)",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : gcd (dedup s) = gcd s\nh : a ∈ s\n⊢ gcd s = GCDMonoid.gcd a (gcd s)",
"tactic": "unfold gcd"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : gcd (dedup s) = gcd s\nh : a ∈ s\n⊢ GCDMonoid.gcd a (fold GCDMonoid.gcd 0 (erase s a)) = GCDMonoid.gcd (↑normalize a) (fold GCDMonoid.gcd 0 (erase s a))",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : gcd (dedup s) = gcd s\nh : a ∈ s\n⊢ fold GCDMonoid.gcd 0 s = GCDMonoid.gcd a (fold GCDMonoid.gcd 0 s)",
"tactic": "rw [← cons_erase h, fold_cons_left, ← gcd_assoc, gcd_same]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : gcd (dedup s) = gcd s\nh : a ∈ s\n⊢ GCDMonoid.gcd a (fold GCDMonoid.gcd 0 (erase s a)) = GCDMonoid.gcd (↑normalize a) (fold GCDMonoid.gcd 0 (erase s a))",
"tactic": "apply (associated_normalize _).gcd_eq_left"
}
] | [
206,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/Topology/Algebra/Module/FiniteDimension.lean | LinearEquiv.toLinearEquiv_toContinuousLinearEquiv | [
{
"state_after": "case h\n𝕜 : Type u\nhnorm : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹⁸ : AddCommGroup E\ninst✝¹⁷ : Module 𝕜 E\ninst✝¹⁶ : TopologicalSpace E\ninst✝¹⁵ : TopologicalAddGroup E\ninst✝¹⁴ : ContinuousSMul 𝕜 E\nF : Type w\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : TopologicalSpace F\ninst✝¹⁰ : TopologicalAddGroup F\ninst✝⁹ : ContinuousSMul 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\ninst✝² : T2Space E\ninst✝¹ : T2Space F\ninst✝ : FiniteDimensional 𝕜 E\ne : E ≃ₗ[𝕜] F\nx : E\n⊢ ↑(toContinuousLinearEquiv e).toLinearEquiv x = ↑e x",
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"tactic": "ext x"
},
{
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}
] | [
394,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
391,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | NormedCommGroup.tendsto_nhds_one | [
{
"state_after": "no goals",
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}
] | [
744,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
742,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.darts_append | [
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w : V\np : Walk G u v\np' : Walk G v w\n⊢ darts (append p p') = darts p ++ darts p'",
"tactic": "induction p <;> simp [*]"
}
] | [
709,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
707,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean | Trivialization.symm_coe_proj | [] | [
587,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
585,
1
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Mathlib/ModelTheory/Substructures.lean | FirstOrder.Language.Substructure.comap_surjective_of_injective | [] | [
572,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
571,
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Mathlib/RepresentationTheory/Action.lean | Action.rightDual_ρ | [
{
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"state_before": "V : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : RightRigidCategory V\nh : ↑H\n⊢ ↑Xᘁ.ρ h = ↑X.ρ h⁻¹ᘁ",
"tactic": "rw [← SingleObj.inv_as_inv]"
},
{
"state_after": "no goals",
"state_before": "V : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : RightRigidCategory V\nh : ↑H\n⊢ ↑Xᘁ.ρ h = ↑X.ρ (inv h)ᘁ\n\ncase x\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : RightRigidCategory V\nh : ↑H\n⊢ SingleObj H.1\n\ncase y\nV : Type (u + 1)\ninst✝² : LargeCategory V\nG : MonCat\ninst✝¹ : MonoidalCategory V\nH : GroupCat\nX : Action V ((forget₂ GroupCat MonCat).obj H)\ninst✝ : RightRigidCategory V\nh : ↑H\n⊢ SingleObj H.1",
"tactic": "rfl"
}
] | [
748,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
747,
1
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Mathlib/Analysis/Convex/Between.lean | wbtw_one_zero_iff | [
{
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}
] | [
453,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
452,
1
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Mathlib/Algebra/Order/Group/Abs.lean | le_abs_self | [] | [
61,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
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Mathlib/Order/Concept.lean | extentClosure_empty | [] | [
92,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
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Mathlib/Data/Polynomial/Basic.lean | Polynomial.lhom_ext' | [] | [
825,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
823,
1
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Mathlib/RingTheory/PowerSeries/Basic.lean | MvPolynomial.coe_eq_one_iff | [
{
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}
] | [
1154,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1154,
1
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Mathlib/CategoryTheory/MorphismProperty.lean | CategoryTheory.MorphismProperty.diagonal_iff | [] | [
527,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
526,
1
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Mathlib/Logic/Equiv/LocalEquiv.lean | LocalEquiv.ofSet_target | [] | [
658,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
657,
1
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Mathlib/RingTheory/FinitePresentation.lean | AlgHom.FinitePresentation.comp_surjective | [] | [
516,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
514,
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Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.ElementarilyEquivalent.trans | [] | [
1107,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1106,
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Mathlib/Analysis/NormedSpace/AffineIsometry.lean | AffineIsometryEquiv.coe_one | [] | [
605,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
604,
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Mathlib/CategoryTheory/Sites/SheafOfTypes.lean | CategoryTheory.Presieve.isAmalgamation_sieveExtend | [
{
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"tactic": "intro Y f hf"
},
{
"state_after": "C : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y✝ : C\nS : Sieve X\nR✝ : Presieve X\nJ J₂ : GrothendieckTopology C\nR : Presieve X\nx : FamilyOfElements P R\nt : P.obj X.op\nht : FamilyOfElements.IsAmalgamation x t\nY : C\nf : Y ⟶ X\nhf : (generate R).arrows f\n⊢ P.map f.op t =\n P.map (Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f)).op\n (x (Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f))\n (_ : R (Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h g, R g ∧ h ≫ g = f) ≫ g = f))))",
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"tactic": "dsimp [FamilyOfElements.sieveExtend]"
},
{
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"tactic": "rw [← ht _, ← FunctorToTypes.map_comp_apply, ← op_comp, hf.choose_spec.choose_spec.choose_spec.2]"
}
] | [
408,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
404,
1
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Mathlib/Topology/UniformSpace/UniformConvergence.lean | TendstoLocallyUniformlyOn.congr | [
{
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{
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},
{
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},
{
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"tactic": "filter_upwards [h]with i hi y hy using hg i hy.1 ▸ hi y hy.2"
}
] | [
790,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
785,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | MeasureTheory.SimpleFunc.nearestPt_zero | [] | [
86,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
] |
Mathlib/Algebra/Order/Floor.lean | Int.ceil_zero | [
{
"state_after": "no goals",
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"tactic": "rw [← cast_zero, ceil_intCast]"
}
] | [
1196,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1196,
1
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Mathlib/Data/Real/ENNReal.lean | ENNReal.sub_eq_sInf | [] | [
1096,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1095,
1
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Mathlib/Geometry/Euclidean/Basic.lean | EuclideanGeometry.reflection_eq_self_iff | [
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ ↑(↑(orthogonalProjection s) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s) p) = p ↔ ↑(↑(orthogonalProjection s) p) = p",
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"tactic": "rw [← orthogonalProjection_eq_self_iff, reflection_apply]"
},
{
"state_after": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ ↑(↑(orthogonalProjection s) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s) p) = p → ↑(↑(orthogonalProjection s) p) = p\n\ncase mpr\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ ↑(↑(orthogonalProjection s) p) = p → ↑(↑(orthogonalProjection s) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s) p) = p",
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},
{
"state_after": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\nh : ↑(↑(orthogonalProjection s) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s) p) = p\n⊢ ↑(↑(orthogonalProjection s) p) = p",
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"tactic": "intro h"
},
{
"state_after": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\nh : 2 = 0 ∨ ↑(↑(orthogonalProjection s) p) -ᵥ p = 0\n⊢ ↑(↑(orthogonalProjection s) p) = p",
"state_before": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\nh : ↑(↑(orthogonalProjection s) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s) p) = p\n⊢ ↑(↑(orthogonalProjection s) p) = p",
"tactic": "rw [← @vsub_eq_zero_iff_eq V, vadd_vsub_assoc, ← two_smul ℝ (↑(orthogonalProjection s p) -ᵥ p),\n smul_eq_zero] at h"
},
{
"state_after": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\nh : ↑(↑(orthogonalProjection s) p) = p\n⊢ ↑(↑(orthogonalProjection s) p) = p",
"state_before": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\nh : 2 = 0 ∨ ↑(↑(orthogonalProjection s) p) -ᵥ p = 0\n⊢ ↑(↑(orthogonalProjection s) p) = p",
"tactic": "norm_num at h"
},
{
"state_after": "no goals",
"state_before": "case mp\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\nh : ↑(↑(orthogonalProjection s) p) = p\n⊢ ↑(↑(orthogonalProjection s) p) = p",
"tactic": "exact h"
},
{
"state_after": "case mpr\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\nh : ↑(↑(orthogonalProjection s) p) = p\n⊢ ↑(↑(orthogonalProjection s) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s) p) = p",
"state_before": "case mpr\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\n⊢ ↑(↑(orthogonalProjection s) p) = p → ↑(↑(orthogonalProjection s) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s) p) = p",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case mpr\nV : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty { x // x ∈ s }\ninst✝ : CompleteSpace { x // x ∈ direction s }\np : P\nh : ↑(↑(orthogonalProjection s) p) = p\n⊢ ↑(↑(orthogonalProjection s) p) -ᵥ p +ᵥ ↑(↑(orthogonalProjection s) p) = p",
"tactic": "simp [h]"
}
] | [
594,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
584,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean | Finset.weightedVSubOfPoint_insert | [
{
"state_after": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.53031\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\nw : ι → k\np : ι → P\ni : ι\n⊢ ∑ i_1 in insert i s, w i_1 • (p i_1 -ᵥ p i) = ∑ i_1 in s, w i_1 • (p i_1 -ᵥ p i)",
"state_before": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.53031\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\nw : ι → k\np : ι → P\ni : ι\n⊢ ↑(weightedVSubOfPoint (insert i s) p (p i)) w = ↑(weightedVSubOfPoint s p (p i)) w",
"tactic": "rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply]"
},
{
"state_after": "case h\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.53031\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\nw : ι → k\np : ι → P\ni : ι\n⊢ w i • (p i -ᵥ p i) = 0",
"state_before": "k : Type u_3\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.53031\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\nw : ι → k\np : ι → P\ni : ι\n⊢ ∑ i_1 in insert i s, w i_1 • (p i_1 -ᵥ p i) = ∑ i_1 in s, w i_1 • (p i_1 -ᵥ p i)",
"tactic": "apply sum_insert_zero"
},
{
"state_after": "no goals",
"state_before": "case h\nk : Type u_3\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.53031\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\nw : ι → k\np : ι → P\ni : ι\n⊢ w i • (p i -ᵥ p i) = 0",
"tactic": "rw [vsub_self, smul_zero]"
}
] | [
164,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | DifferentiableOn.csin | [] | [
399,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
398,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean | Associates.mem_factorSet_top | [
{
"state_after": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\np : Associates α\nhp : Irreducible p\n⊢ FactorSetMem p ⊤",
"state_before": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\np : Associates α\nhp : Irreducible p\n⊢ p ∈ ⊤",
"tactic": "dsimp only [Membership.mem]"
},
{
"state_after": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\np : Associates α\nhp : Irreducible p\n⊢ if hp : Irreducible p then BfactorSetMem { val := p, property := hp } ⊤ else False",
"state_before": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\np : Associates α\nhp : Irreducible p\n⊢ FactorSetMem p ⊤",
"tactic": "dsimp only [FactorSetMem]"
},
{
"state_after": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\np : Associates α\nhp : Irreducible p\n⊢ BfactorSetMem { val := p, property := hp } ⊤",
"state_before": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\np : Associates α\nhp : Irreducible p\n⊢ if hp : Irreducible p then BfactorSetMem { val := p, property := hp } ⊤ else False",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\np : Associates α\nhp : Irreducible p\n⊢ BfactorSetMem { val := p, property := hp } ⊤",
"tactic": "exact trivial"
}
] | [
1335,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1334,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.zero_smul_finset_subset | [] | [
2117,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2116,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean | Pell.xy_coprime | [
{
"state_after": "a : ℕ\na1 : 1 < a\nn k : ℕ\nx✝ : Nat.Prime k\nkx : k ∣ xn a1 n\nky : k ∣ yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n⊢ k ∣ 1",
"state_before": "a : ℕ\na1 : 1 < a\nn k : ℕ\nx✝ : Nat.Prime k\nkx : k ∣ xn a1 n\nky : k ∣ yn a1 n\n⊢ k ∣ 1",
"tactic": "let p := pell_eq a1 n"
},
{
"state_after": "a : ℕ\na1 : 1 < a\nn k : ℕ\nx✝ : Nat.Prime k\nkx : k ∣ xn a1 n\nky : k ∣ yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n⊢ k ∣ xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n",
"state_before": "a : ℕ\na1 : 1 < a\nn k : ℕ\nx✝ : Nat.Prime k\nkx : k ∣ xn a1 n\nky : k ∣ yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n⊢ k ∣ 1",
"tactic": "rw [← p]"
},
{
"state_after": "no goals",
"state_before": "a : ℕ\na1 : 1 < a\nn k : ℕ\nx✝ : Nat.Prime k\nkx : k ∣ xn a1 n\nky : k ∣ yn a1 n\np : xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n = 1 := pell_eq a1 n\n⊢ k ∣ xn a1 n * xn a1 n - Pell.d a1 * yn a1 n * yn a1 n",
"tactic": "exact Nat.dvd_sub (le_of_lt <| Nat.lt_of_sub_eq_succ p) (kx.mul_left _) (ky.mul_left _)"
}
] | [
401,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
397,
1
] |
Mathlib/Algebra/FreeAlgebra.lean | FreeAlgebra.lift_ι_apply | [
{
"state_after": "R : Type u_2\ninst✝² : CommSemiring R\nX : Type u_3\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\nx : X\n⊢ ↑(↑{ toFun := FreeAlgebra.liftAux R, invFun := fun F => ↑F ∘ ι R,\n left_inv := (_ : ∀ (f : X → A), (fun F => ↑F ∘ ι R) (FreeAlgebra.liftAux R f) = f),\n right_inv := (_ : ∀ (F : FreeAlgebra R X →ₐ[R] A), FreeAlgebra.liftAux R ((fun F => ↑F ∘ ι R) F) = F) }\n f)\n (Quot.mk (Rel R X) (Pre.of x)) =\n f x",
"state_before": "R : Type u_2\ninst✝² : CommSemiring R\nX : Type u_3\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\nx : X\n⊢ ↑(↑(lift R) f) (ι R x) = f x",
"tactic": "rw [ι_def, lift]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝² : CommSemiring R\nX : Type u_3\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\nx : X\n⊢ ↑(↑{ toFun := FreeAlgebra.liftAux R, invFun := fun F => ↑F ∘ ι R,\n left_inv := (_ : ∀ (f : X → A), (fun F => ↑F ∘ ι R) (FreeAlgebra.liftAux R f) = f),\n right_inv := (_ : ∀ (F : FreeAlgebra R X →ₐ[R] A), FreeAlgebra.liftAux R ((fun F => ↑F ∘ ι R) F) = F) }\n f)\n (Quot.mk (Rel R X) (Pre.of x)) =\n f x",
"tactic": "rfl"
}
] | [
349,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
347,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.image_mul_product | [] | [
319,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean | LieSubalgebra.span_empty | [] | [
735,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
734,
1
] |
Mathlib/Data/Nat/Parity.lean | Nat.div_two_mul_two_of_even | [] | [
227,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
226,
1
] |
Mathlib/Algebra/Lie/Submodule.lean | LieSubalgebra.exists_lieIdeal_coe_eq_iff | [
{
"state_after": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nK : LieSubalgebra R L\n⊢ (∀ (x m : L), m ∈ K.toSubmodule → ⁅x, m⁆ ∈ K.toSubmodule) ↔ ∀ (x y : L), y ∈ K → ⁅x, y⁆ ∈ K",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nK : LieSubalgebra R L\n⊢ (∃ I, ↑R L I = K) ↔ ∀ (x y : L), y ∈ K → ⁅x, y⁆ ∈ K",
"tactic": "simp only [← coe_to_submodule_eq_iff, LieIdeal.coe_to_lieSubalgebra_to_submodule,\n Submodule.exists_lieSubmodule_coe_eq_iff L]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nK : LieSubalgebra R L\n⊢ (∀ (x m : L), m ∈ K.toSubmodule → ⁅x, m⁆ ∈ K.toSubmodule) ↔ ∀ (x y : L), y ∈ K → ⁅x, y⁆ ∈ K",
"tactic": "simp only [mem_coe_submodule]"
}
] | [
327,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean | smul_inv | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : Group α\ninst✝³ : MulAction α β\ninst✝² : Group β\ninst✝¹ : SMulCommClass α β β\ninst✝ : IsScalarTower α β β\nc : α\nx : β\n⊢ (c • x)⁻¹ = c⁻¹ • x⁻¹",
"tactic": "rw [inv_eq_iff_mul_eq_one, smul_mul_smul, mul_right_inv, mul_right_inv, one_smul]"
}
] | [
125,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/MeasureTheory/Measure/AEMeasurable.lean | AEMeasurable.add_measure | [] | [
130,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/Data/Nat/PartENat.lean | PartENat.lt_add_one_iff_lt | [
{
"state_after": "x y : PartENat\nhx : x ≠ ⊤\nh : x ≤ y\n⊢ x < y + 1",
"state_before": "x y : PartENat\nhx : x ≠ ⊤\n⊢ x < y + 1 ↔ x ≤ y",
"tactic": "refine ⟨le_of_lt_add_one, fun h => ?_⟩"
},
{
"state_after": "case intro\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh : ↑m ≤ y\n⊢ ↑m < y + 1",
"state_before": "x y : PartENat\nhx : x ≠ ⊤\nh : x ≤ y\n⊢ x < y + 1",
"tactic": "rcases ne_top_iff.mp hx with ⟨m, rfl⟩"
},
{
"state_after": "case intro.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nh : ↑m ≤ ⊤\n⊢ ↑m < ⊤ + 1\n\ncase intro.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nn : ℕ\nh : ↑m ≤ ↑n\n⊢ ↑m < ↑n + 1",
"state_before": "case intro\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh : ↑m ≤ y\n⊢ ↑m < y + 1",
"tactic": "induction' y using PartENat.casesOn with n"
},
{
"state_after": "case intro.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nn : ℕ\nh : ↑m ≤ ↑n\n⊢ m < n + 1",
"state_before": "case intro.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nn : ℕ\nh : ↑m ≤ ↑n\n⊢ ↑m < ↑n + 1",
"tactic": "norm_cast"
},
{
"state_after": "case intro.a.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nn : ℕ\nh : ↑m ≤ ↑n\n⊢ m ≤ n",
"state_before": "case intro.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nn : ℕ\nh : ↑m ≤ ↑n\n⊢ m < n + 1",
"tactic": "apply Nat.lt_succ_of_le"
},
{
"state_after": "no goals",
"state_before": "case intro.a.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nn : ℕ\nh : ↑m ≤ ↑n\n⊢ m ≤ n",
"tactic": "norm_cast at h"
},
{
"state_after": "case intro.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nh : ↑m ≤ ⊤\n⊢ ↑m < ⊤",
"state_before": "case intro.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nh : ↑m ≤ ⊤\n⊢ ↑m < ⊤ + 1",
"tactic": "rw [top_add]"
},
{
"state_after": "no goals",
"state_before": "case intro.a\ny : PartENat\nm : ℕ\nhx : ↑m ≠ ⊤\nh✝ : ↑m ≤ y\nh : ↑m ≤ ⊤\n⊢ ↑m < ⊤",
"tactic": "apply natCast_lt_top"
}
] | [
522,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
515,
1
] |
Mathlib/Algebra/DirectSum/Module.lean | DirectSum.coeLinearMap_of | [] | [
324,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
322,
1
] |
Std/Data/Int/Lemmas.lean | Int.toNat_sub | [
{
"state_after": "m n : Nat\n⊢ toNat (subNatNat m n) = m - n",
"state_before": "m n : Nat\n⊢ toNat (↑m - ↑n) = m - n",
"tactic": "rw [← Int.subNatNat_eq_coe]"
},
{
"state_after": "case refine_1\nm n✝ i n : Nat\n⊢ (fun m n i => toNat i = m - n) (n + i) n ↑i\n\ncase refine_2\nm n✝ i n : Nat\n⊢ (fun m n i => toNat i = m - n) n (n + i + 1) -[i+1]",
"state_before": "m n : Nat\n⊢ toNat (subNatNat m n) = m - n",
"tactic": "refine subNatNat_elim m n (fun m n i => toNat i = m - n) (fun i n => ?_) (fun i n => ?_)"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nm n✝ i n : Nat\n⊢ (fun m n i => toNat i = m - n) (n + i) n ↑i",
"tactic": "exact (Nat.add_sub_cancel_left ..).symm"
},
{
"state_after": "case refine_2\nm n✝ i n : Nat\n⊢ toNat -[i+1] = n - (n + i + 1)",
"state_before": "case refine_2\nm n✝ i n : Nat\n⊢ (fun m n i => toNat i = m - n) n (n + i + 1) -[i+1]",
"tactic": "dsimp"
},
{
"state_after": "case refine_2\nm n✝ i n : Nat\n⊢ toNat -[i+1] = 0",
"state_before": "case refine_2\nm n✝ i n : Nat\n⊢ toNat -[i+1] = n - (n + i + 1)",
"tactic": "rw [Nat.add_assoc, Nat.sub_eq_zero_of_le (Nat.le_add_right ..)]"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nm n✝ i n : Nat\n⊢ toNat -[i+1] = 0",
"tactic": "rfl"
}
] | [
518,
80
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
514,
1
] |
Mathlib/Data/Option/Basic.lean | Option.join_pmap_eq_pmap_join | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.13574\nδ : Type ?u.13577\np : α → Prop\nf✝ : (a : α) → p a → β\nx✝ : Option α\nf : (a : α) → p a → β\nx : Option (Option α)\nH : ∀ (a : Option α), a ∈ x → ∀ (a_2 : α), a_2 ∈ a → p a_2\n⊢ join (pmap (pmap f) x H) = pmap f (join x) (_ : ∀ (a : α), a ∈ join x → p a)",
"tactic": "rcases x with (_ | _ | x) <;> simp"
}
] | [
259,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
257,
1
] |
Mathlib/Order/SuccPred/Limit.lean | Order.isPredLimit_of_pred_ne | [] | [
340,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
339,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.range_val | [] | [
2980,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2979,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean | Zsqrtd.ofNat_re | [] | [
283,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Mathlib/Data/Sum/Basic.lean | Sum.getLeft_inr | [] | [
100,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
9
] |
Mathlib/Topology/MetricSpace/Baire.lean | dense_iUnion_interior_of_closed | [] | [
349,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
347,
1
] |
Mathlib/Order/Lattice.lean | toDual_inf | [] | [
963,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
962,
1
] |
Mathlib/Algebra/QuadraticDiscriminant.lean | quadratic_eq_zero_iff | [
{
"state_after": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x✝ : K\nha : a ≠ 0\ns : K\nh : discrim a b c = s * s\nx : K\n⊢ s = 2 * a * x + b ∨ s = -(2 * a * x + b) ↔ x = (-b + s) / (2 * a) ∨ x = (-b - s) / (2 * a)",
"state_before": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x✝ : K\nha : a ≠ 0\ns : K\nh : discrim a b c = s * s\nx : K\n⊢ a * x * x + b * x + c = 0 ↔ x = (-b + s) / (2 * a) ∨ x = (-b - s) / (2 * a)",
"tactic": "rw [quadratic_eq_zero_iff_discrim_eq_sq ha, h, sq, mul_self_eq_mul_self_iff]"
},
{
"state_after": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x✝ : K\nha : a ≠ 0\ns : K\nh : discrim a b c = s * s\nx : K\n⊢ s = 2 * a * x + b ∨ s = -b + -(2 * a * x) ↔ x * (2 * a) = -b + s ∨ x * (2 * a) = -b - s",
"state_before": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x✝ : K\nha : a ≠ 0\ns : K\nh : discrim a b c = s * s\nx : K\n⊢ s = 2 * a * x + b ∨ s = -(2 * a * x + b) ↔ x = (-b + s) / (2 * a) ∨ x = (-b - s) / (2 * a)",
"tactic": "field_simp"
},
{
"state_after": "case h₁\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x✝ : K\nha : a ≠ 0\ns : K\nh : discrim a b c = s * s\nx : K\n⊢ s = 2 * a * x + b ↔ x * (2 * a) = -b + s\n\ncase h₂\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x✝ : K\nha : a ≠ 0\ns : K\nh : discrim a b c = s * s\nx : K\n⊢ s = -b + -(2 * a * x) ↔ x * (2 * a) = -b - s",
"state_before": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x✝ : K\nha : a ≠ 0\ns : K\nh : discrim a b c = s * s\nx : K\n⊢ s = 2 * a * x + b ∨ s = -b + -(2 * a * x) ↔ x * (2 * a) = -b + s ∨ x * (2 * a) = -b - s",
"tactic": "apply or_congr"
},
{
"state_after": "no goals",
"state_before": "case h₁\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x✝ : K\nha : a ≠ 0\ns : K\nh : discrim a b c = s * s\nx : K\n⊢ s = 2 * a * x + b ↔ x * (2 * a) = -b + s",
"tactic": "constructor <;> intro h' <;> linear_combination -h'"
},
{
"state_after": "no goals",
"state_before": "case h₂\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c x✝ : K\nha : a ≠ 0\ns : K\nh : discrim a b c = s * s\nx : K\n⊢ s = -b + -(2 * a * x) ↔ x * (2 * a) = -b - s",
"tactic": "constructor <;> intro h' <;> linear_combination h'"
}
] | [
95,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.vsub_subset_vsub_right | [] | [
1527,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1526,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean | Polynomial.leadingCoeff_divByMonic_of_monic | [
{
"state_after": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\n⊢ leadingCoeff (p /ₘ q) = leadingCoeff p",
"state_before": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n⊢ leadingCoeff (p /ₘ q) = leadingCoeff p",
"tactic": "nontriviality"
},
{
"state_after": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\nh : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ leadingCoeff (p /ₘ q) = leadingCoeff p",
"state_before": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\n⊢ leadingCoeff (p /ₘ q) = leadingCoeff p",
"tactic": "have h : q.leadingCoeff * (p /ₘ q).leadingCoeff ≠ 0 := by\n simpa [divByMonic_eq_zero_iff hmonic, hmonic.leadingCoeff,\n Nat.WithBot.one_le_iff_zero_lt] using hdegree"
},
{
"state_after": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\nh : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ leadingCoeff (p /ₘ q) = leadingCoeff (p %ₘ q + q * (p /ₘ q))",
"state_before": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\nh : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ leadingCoeff (p /ₘ q) = leadingCoeff p",
"tactic": "nth_rw 2 [← modByMonic_add_div p hmonic]"
},
{
"state_after": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\nh : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ degree (p %ₘ q) < degree (q * (p /ₘ q))",
"state_before": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\nh : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ leadingCoeff (p /ₘ q) = leadingCoeff (p %ₘ q + q * (p /ₘ q))",
"tactic": "rw [leadingCoeff_add_of_degree_lt, leadingCoeff_monic_mul hmonic]"
},
{
"state_after": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\nh : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ degree (p %ₘ q) < degree p",
"state_before": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\nh : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ degree (p %ₘ q) < degree (q * (p /ₘ q))",
"tactic": "rw [degree_mul' h, degree_add_divByMonic hmonic hdegree]"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\nh : leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0\n⊢ degree (p %ₘ q) < degree p",
"tactic": "exact (degree_modByMonic_lt p hmonic).trans_le hdegree"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type u\nS : Type v\nT : Type w\na b : R✝\nn : ℕ\ninst✝² : CommRing R✝\ninst✝¹ : IsDomain R✝\np✝ q✝ : R✝[X]\nR : Type u\ninst✝ : CommRing R\np q : R[X]\nhmonic : Monic q\nhdegree : degree q ≤ degree p\n✝ : Nontrivial R\n⊢ leadingCoeff q * leadingCoeff (p /ₘ q) ≠ 0",
"tactic": "simpa [divByMonic_eq_zero_iff hmonic, hmonic.leadingCoeff,\n Nat.WithBot.one_le_iff_zero_lt] using hdegree"
}
] | [
1014,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1005,
1
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean | Polynomial.natDegree_lt_coeff_mul | [] | [
163,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/Order/CompactlyGenerated.lean | CompleteLattice.IsSupClosedCompact.wellFounded | [
{
"state_after": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ False",
"state_before": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\n⊢ WellFounded fun x x_1 => x > x_1",
"tactic": "refine' RelEmbedding.wellFounded_iff_no_descending_seq.mpr ⟨fun a => _⟩"
},
{
"state_after": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ sSup (Set.range ↑a) ∈ Set.range ↑a",
"state_before": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ False",
"tactic": "suffices sSup (Set.range a) ∈ Set.range a by\n obtain ⟨n, hn⟩ := Set.mem_range.mp this\n have h' : sSup (Set.range a) < a (n + 1) := by\n change _ > _\n simp [← hn, a.map_rel_iff]\n apply lt_irrefl (a (n + 1))\n apply lt_of_le_of_lt _ h'\n apply le_sSup\n apply Set.mem_range_self"
},
{
"state_after": "case x\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ Set.Nonempty (Set.range ↑a)\n\ncase a\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ ∀ (a_1 : α), a_1 ∈ Set.range ↑a → ∀ (b : α), b ∈ Set.range ↑a → a_1 ⊔ b ∈ Set.range ↑a",
"state_before": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ sSup (Set.range ↑a) ∈ Set.range ↑a",
"tactic": "apply h (Set.range a)"
},
{
"state_after": "case intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\n⊢ False",
"state_before": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\n⊢ False",
"tactic": "obtain ⟨n, hn⟩ := Set.mem_range.mp this"
},
{
"state_after": "case intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\nh' : sSup (Set.range ↑a) < ↑a (n + 1)\n⊢ False",
"state_before": "case intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\n⊢ False",
"tactic": "have h' : sSup (Set.range a) < a (n + 1) := by\n change _ > _\n simp [← hn, a.map_rel_iff]"
},
{
"state_after": "case intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\nh' : sSup (Set.range ↑a) < ↑a (n + 1)\n⊢ ↑a (n + 1) < ↑a (n + 1)",
"state_before": "case intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\nh' : sSup (Set.range ↑a) < ↑a (n + 1)\n⊢ False",
"tactic": "apply lt_irrefl (a (n + 1))"
},
{
"state_after": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\nh' : sSup (Set.range ↑a) < ↑a (n + 1)\n⊢ ↑a (n + 1) ≤ sSup (Set.range ↑a)",
"state_before": "case intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\nh' : sSup (Set.range ↑a) < ↑a (n + 1)\n⊢ ↑a (n + 1) < ↑a (n + 1)",
"tactic": "apply lt_of_le_of_lt _ h'"
},
{
"state_after": "case a\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\nh' : sSup (Set.range ↑a) < ↑a (n + 1)\n⊢ ↑a (n + 1) ∈ Set.range ↑a",
"state_before": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\nh' : sSup (Set.range ↑a) < ↑a (n + 1)\n⊢ ↑a (n + 1) ≤ sSup (Set.range ↑a)",
"tactic": "apply le_sSup"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\nh' : sSup (Set.range ↑a) < ↑a (n + 1)\n⊢ ↑a (n + 1) ∈ Set.range ↑a",
"tactic": "apply Set.mem_range_self"
},
{
"state_after": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\n⊢ ↑a (n + 1) > sSup (Set.range ↑a)",
"state_before": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\n⊢ sSup (Set.range ↑a) < ↑a (n + 1)",
"tactic": "change _ > _"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nthis : sSup (Set.range ↑a) ∈ Set.range ↑a\nn : ℕ\nhn : ↑a n = sSup (Set.range ↑a)\n⊢ ↑a (n + 1) > sSup (Set.range ↑a)",
"tactic": "simp [← hn, a.map_rel_iff]"
},
{
"state_after": "case x\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ ↑a 37 ∈ Set.range ↑a",
"state_before": "case x\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ Set.Nonempty (Set.range ↑a)",
"tactic": "use a 37"
},
{
"state_after": "no goals",
"state_before": "case x\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ ↑a 37 ∈ Set.range ↑a",
"tactic": "apply Set.mem_range_self"
},
{
"state_after": "case a.intro.intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nx : α\nm : ℕ\nhm : ↑a m = x\ny : α\nn : ℕ\nhn : ↑a n = y\n⊢ x ⊔ y ∈ Set.range ↑a",
"state_before": "case a\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\n⊢ ∀ (a_1 : α), a_1 ∈ Set.range ↑a → ∀ (b : α), b ∈ Set.range ↑a → a_1 ⊔ b ∈ Set.range ↑a",
"tactic": "rintro x ⟨m, hm⟩ y ⟨n, hn⟩"
},
{
"state_after": "case a.intro.intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nx : α\nm : ℕ\nhm : ↑a m = x\ny : α\nn : ℕ\nhn : ↑a n = y\n⊢ ↑a (m ⊔ n) = x ⊔ y",
"state_before": "case a.intro.intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nx : α\nm : ℕ\nhm : ↑a m = x\ny : α\nn : ℕ\nhn : ↑a n = y\n⊢ x ⊔ y ∈ Set.range ↑a",
"tactic": "use m ⊔ n"
},
{
"state_after": "case a.intro.intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nx : α\nm : ℕ\nhm : ↑a m = x\ny : α\nn : ℕ\nhn : ↑a n = y\n⊢ ↑a (m ⊔ n) = ↑a m ⊔ ↑a n",
"state_before": "case a.intro.intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nx : α\nm : ℕ\nhm : ↑a m = x\ny : α\nn : ℕ\nhn : ↑a n = y\n⊢ ↑a (m ⊔ n) = x ⊔ y",
"tactic": "rw [← hm, ← hn]"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro\nι : Sort ?u.27269\nα : Type u_1\ninst✝ : CompleteLattice α\nf : ι → α\nh : IsSupClosedCompact α\na : (fun x x_1 => x > x_1) ↪r fun x x_1 => x > x_1\nx : α\nm : ℕ\nhm : ↑a m = x\ny : α\nn : ℕ\nhn : ↑a n = y\n⊢ ↑a (m ⊔ n) = ↑a m ⊔ ↑a n",
"tactic": "apply RelHomClass.map_sup a"
}
] | [
246,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
228,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | MeasureTheory.intervalIntegral_tendsto_integral_Iic | [
{
"state_after": "ι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\n⊢ Tendsto (fun i => ∫ (x : ℝ) in a i..b, f x ∂μ) l (𝓝 (∫ (x : ℝ) in Iic b, f x ∂μ))",
"state_before": "ι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\n⊢ Tendsto (fun i => ∫ (x : ℝ) in a i..b, f x ∂μ) l (𝓝 (∫ (x : ℝ) in Iic b, f x ∂μ))",
"tactic": "let φ i := Ioi (a i)"
},
{
"state_after": "ι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\nhφ : AECover (Measure.restrict μ (Iic b)) l φ\n⊢ Tendsto (fun i => ∫ (x : ℝ) in a i..b, f x ∂μ) l (𝓝 (∫ (x : ℝ) in Iic b, f x ∂μ))",
"state_before": "ι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\n⊢ Tendsto (fun i => ∫ (x : ℝ) in a i..b, f x ∂μ) l (𝓝 (∫ (x : ℝ) in Iic b, f x ∂μ))",
"tactic": "have hφ : AECover (μ.restrict <| Iic b) l φ := aecover_Ioi ha"
},
{
"state_after": "ι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\nhφ : AECover (Measure.restrict μ (Iic b)) l φ\n⊢ (fun i => ∫ (x : ℝ) in φ i, f x ∂Measure.restrict μ (Iic b)) =ᶠ[l] fun i => ∫ (x : ℝ) in a i..b, f x ∂μ",
"state_before": "ι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\nhφ : AECover (Measure.restrict μ (Iic b)) l φ\n⊢ Tendsto (fun i => ∫ (x : ℝ) in a i..b, f x ∂μ) l (𝓝 (∫ (x : ℝ) in Iic b, f x ∂μ))",
"tactic": "refine' (hφ.integral_tendsto_of_countably_generated hfi).congr' _"
},
{
"state_after": "case h\nι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\nhφ : AECover (Measure.restrict μ (Iic b)) l φ\ni : ι\nhai : a i ≤ b\n⊢ (∫ (x : ℝ) in Ioi (a i), f x ∂Measure.restrict μ (Iic b)) = ∫ (x : ℝ) in a i..b, f x ∂μ",
"state_before": "ι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\nhφ : AECover (Measure.restrict μ (Iic b)) l φ\n⊢ (fun i => ∫ (x : ℝ) in φ i, f x ∂Measure.restrict μ (Iic b)) =ᶠ[l] fun i => ∫ (x : ℝ) in a i..b, f x ∂μ",
"tactic": "filter_upwards [ha.eventually (eventually_le_atBot <| b)] with i hai"
},
{
"state_after": "case h\nι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\nhφ : AECover (Measure.restrict μ (Iic b)) l φ\ni : ι\nhai : a i ≤ b\n⊢ (∫ (x : ℝ) in φ i ∩ Iic b, f x ∂μ) = ∫ (x : ℝ) in Ioc (a i) b, f x ∂μ",
"state_before": "case h\nι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\nhφ : AECover (Measure.restrict μ (Iic b)) l φ\ni : ι\nhai : a i ≤ b\n⊢ (∫ (x : ℝ) in Ioi (a i), f x ∂Measure.restrict μ (Iic b)) = ∫ (x : ℝ) in a i..b, f x ∂μ",
"tactic": "rw [intervalIntegral.integral_of_le hai, Measure.restrict_restrict (hφ.measurableSet i)]"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_2\nE : Type u_1\nμ : Measure ℝ\nl : Filter ι\ninst✝³ : IsCountablyGenerated l\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b✝ : ι → ℝ\nf : ℝ → E\nb : ℝ\nhfi : IntegrableOn f (Iic b)\nha : Tendsto a l atBot\nφ : ι → Set ℝ := fun i => Ioi (a i)\nhφ : AECover (Measure.restrict μ (Iic b)) l φ\ni : ι\nhai : a i ≤ b\n⊢ (∫ (x : ℝ) in φ i ∩ Iic b, f x ∂μ) = ∫ (x : ℝ) in Ioc (a i) b, f x ∂μ",
"tactic": "rfl"
}
] | [
642,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
634,
1
] |
Mathlib/Topology/MetricSpace/Polish.lean | Equiv.polishSpace_induced | [] | [
169,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
166,
1
] |
Mathlib/LinearAlgebra/Dimension.lean | LinearMap.lift_rank_comp_le_right | [
{
"state_after": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1114972\ninst✝⁸ : Ring K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V''\ninst✝ : Module K V''\ng : V →ₗ[K] V'\nf : V' →ₗ[K] V''\n⊢ lift (Module.rank K { x // x ∈ Submodule.map f (range g) }) ≤ lift (Module.rank K { x // x ∈ range g })",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1114972\ninst✝⁸ : Ring K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V''\ninst✝ : Module K V''\ng : V →ₗ[K] V'\nf : V' →ₗ[K] V''\n⊢ lift (rank (comp f g)) ≤ lift (rank g)",
"tactic": "rw [rank, rank, LinearMap.range_comp]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.1114972\ninst✝⁸ : Ring K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module K V₁\ninst✝³ : AddCommGroup V'\ninst✝² : Module K V'\ninst✝¹ : AddCommGroup V''\ninst✝ : Module K V''\ng : V →ₗ[K] V'\nf : V' →ₗ[K] V''\n⊢ lift (Module.rank K { x // x ∈ Submodule.map f (range g) }) ≤ lift (Module.rank K { x // x ∈ range g })",
"tactic": "exact lift_rank_map_le _ _"
}
] | [
1341,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1339,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean | ContinuousMultilinearMap.norm_map_cons_le | [
{
"state_after": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : ContinuousMultilinearMap 𝕜 Ei G\nx : Ei 0\nm : (i : Fin n) → Ei (succ i)\n⊢ ‖f‖ * (‖Fin.cons x m 0‖ * ∏ i : Fin n, ‖Fin.cons x m (succ i)‖) = ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : ContinuousMultilinearMap 𝕜 Ei G\nx : Ei 0\nm : (i : Fin n) → Ei (succ i)\n⊢ ‖f‖ * ∏ i : Fin (n + 1), ‖Fin.cons x m i‖ = ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖",
"tactic": "rw [prod_univ_succ]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : ContinuousMultilinearMap 𝕜 Ei G\nx : Ei 0\nm : (i : Fin n) → Ei (succ i)\n⊢ ‖f‖ * (‖Fin.cons x m 0‖ * ∏ i : Fin n, ‖Fin.cons x m (succ i)‖) = ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖",
"tactic": "simp [mul_assoc]"
}
] | [
1307,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1301,
1
] |
Mathlib/Data/List/Cycle.lean | Cycle.Subsingleton.congr | [
{
"state_after": "case h\nα : Type u_1\ns : Cycle α\nh✝ : Subsingleton s\nl : List α\nh : Subsingleton (Quot.mk Setoid.r l)\n⊢ ∀ ⦃x : α⦄, x ∈ Quot.mk Setoid.r l → ∀ ⦃y : α⦄, y ∈ Quot.mk Setoid.r l → x = y",
"state_before": "α : Type u_1\ns : Cycle α\nh : Subsingleton s\n⊢ ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → x = y",
"tactic": "induction' s using Quot.inductionOn with l"
},
{
"state_after": "case h\nα : Type u_1\ns : Cycle α\nh✝ : Subsingleton s\nl : List α\nh : l = [] ∨ ∃ a, l = [a]\n⊢ ∀ ⦃x : α⦄, x ∈ Quot.mk Setoid.r l → ∀ ⦃y : α⦄, y ∈ Quot.mk Setoid.r l → x = y",
"state_before": "case h\nα : Type u_1\ns : Cycle α\nh✝ : Subsingleton s\nl : List α\nh : Subsingleton (Quot.mk Setoid.r l)\n⊢ ∀ ⦃x : α⦄, x ∈ Quot.mk Setoid.r l → ∀ ⦃y : α⦄, y ∈ Quot.mk Setoid.r l → x = y",
"tactic": "simp only [length_subsingleton_iff, length_coe, mk_eq_coe, le_iff_lt_or_eq, Nat.lt_add_one_iff,\n length_eq_zero, length_eq_one, Nat.not_lt_zero, false_or_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ns : Cycle α\nh✝ : Subsingleton s\nl : List α\nh : l = [] ∨ ∃ a, l = [a]\n⊢ ∀ ⦃x : α⦄, x ∈ Quot.mk Setoid.r l → ∀ ⦃y : α⦄, y ∈ Quot.mk Setoid.r l → x = y",
"tactic": "rcases h with (rfl | ⟨z, rfl⟩) <;> simp"
}
] | [
614,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
609,
1
] |
Mathlib/Algebra/Tropical/Basic.lean | Tropical.succ_nsmul | [
{
"state_after": "case zero\nR✝ : Type u\ninst✝² : LinearOrderedAddCommMonoidWithTop R✝\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\n⊢ (Nat.zero + 1) • x = x\n\ncase succ\nR✝ : Type u\ninst✝² : LinearOrderedAddCommMonoidWithTop R✝\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\nn : ℕ\nIH : (n + 1) • x = x\n⊢ (Nat.succ n + 1) • x = x",
"state_before": "R✝ : Type u\ninst✝² : LinearOrderedAddCommMonoidWithTop R✝\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\nn : ℕ\n⊢ (n + 1) • x = x",
"tactic": "induction' n with n IH"
},
{
"state_after": "no goals",
"state_before": "case zero\nR✝ : Type u\ninst✝² : LinearOrderedAddCommMonoidWithTop R✝\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\n⊢ (Nat.zero + 1) • x = x",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nR✝ : Type u\ninst✝² : LinearOrderedAddCommMonoidWithTop R✝\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\nx : Tropical R\nn : ℕ\nIH : (n + 1) • x = x\n⊢ (Nat.succ n + 1) • x = x",
"tactic": "rw [add_nsmul, IH, one_nsmul, add_self]"
}
] | [
571,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
568,
1
] |
Mathlib/Data/Fintype/Sum.lean | Set.MapsTo.exists_equiv_extend_of_card_eq | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"tactic": "classical\n let s' : Finset α := s.toFinset\n have hfst' : s'.image f ⊆ t := by simpa [← Finset.coe_subset] using hfst\n have hfs' : Set.InjOn f s' := by simpa using hfs\n obtain ⟨g, hg⟩ := Finset.exists_equiv_extend_of_card_eq hαt hfst' hfs'\n refine' ⟨g, fun i hi => _⟩\n apply hg\n simpa using hi"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"tactic": "let s' : Finset α := s.toFinset"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"tactic": "have hfst' : s'.image f ⊆ t := by simpa [← Finset.coe_subset] using hfst"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\nhfs' : InjOn f ↑s'\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"tactic": "have hfs' : Set.InjOn f s' := by simpa using hfs"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\nhfs' : InjOn f ↑s'\ng : α ≃ { x // x ∈ t }\nhg : ∀ (i : α), i ∈ s' → ↑(↑g i) = f i\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\nhfs' : InjOn f ↑s'\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"tactic": "obtain ⟨g, hg⟩ := Finset.exists_equiv_extend_of_card_eq hαt hfst' hfs'"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\nhfs' : InjOn f ↑s'\ng : α ≃ { x // x ∈ t }\nhg : ∀ (i : α), i ∈ s' → ↑(↑g i) = f i\ni : α\nhi : i ∈ s\n⊢ ↑(↑g i) = f i",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\nhfs' : InjOn f ↑s'\ng : α ≃ { x // x ∈ t }\nhg : ∀ (i : α), i ∈ s' → ↑(↑g i) = f i\n⊢ ∃ g, ∀ (i : α), i ∈ s → ↑(↑g i) = f i",
"tactic": "refine' ⟨g, fun i hi => _⟩"
},
{
"state_after": "case intro.a\nα : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\nhfs' : InjOn f ↑s'\ng : α ≃ { x // x ∈ t }\nhg : ∀ (i : α), i ∈ s' → ↑(↑g i) = f i\ni : α\nhi : i ∈ s\n⊢ i ∈ s'",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\nhfs' : InjOn f ↑s'\ng : α ≃ { x // x ∈ t }\nhg : ∀ (i : α), i ∈ s' → ↑(↑g i) = f i\ni : α\nhi : i ∈ s\n⊢ ↑(↑g i) = f i",
"tactic": "apply hg"
},
{
"state_after": "no goals",
"state_before": "case intro.a\nα : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\nhfs' : InjOn f ↑s'\ng : α ≃ { x // x ∈ t }\nhg : ∀ (i : α), i ∈ s' → ↑(↑g i) = f i\ni : α\nhi : i ∈ s\n⊢ i ∈ s'",
"tactic": "simpa using hi"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\n⊢ Finset.image f s' ⊆ t",
"tactic": "simpa [← Finset.coe_subset] using hfst"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝ : Fintype α\nt : Finset β\nhαt : Fintype.card α = card t\ns : Set α\nf : α → β\nhfst : MapsTo f s ↑t\nhfs : InjOn f s\ns' : Finset α := toFinset s\nhfst' : Finset.image f s' ⊆ t\n⊢ InjOn f ↑s'",
"tactic": "simpa using hfs"
}
] | [
118,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/Algebra/Module/Injective.lean | Module.Baer.extensionOfMax_adjoin.aux1 | [
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\nmem1 : ↑x ∈ ↑(supExtensionOfMaxSingleton i f y)\n⊢ ∃ a b, ↑x = ↑a + b • y",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\n⊢ ∃ a b, ↑x = ↑a + b • y",
"tactic": "have mem1 : x.1 ∈ (_ : Set _) := x.2"
},
{
"state_after": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\nmem1 : ↑x ∈ ↑(extensionOfMax i f).toLinearPMap.domain + ↑(Submodule.span R {y})\n⊢ ∃ a b, ↑x = ↑a + b • y",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\nmem1 : ↑x ∈ ↑(supExtensionOfMaxSingleton i f y)\n⊢ ∃ a b, ↑x = ↑a + b • y",
"tactic": "rw [Submodule.coe_sup] at mem1"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\na b : N\na_mem : a ∈ ↑(extensionOfMax i f).toLinearPMap.domain\nb_mem : b ∈ Submodule.span R {y}\neq1 : (fun x x_1 => x + x_1) a b = ↑x\n⊢ ∃ a b, ↑x = ↑a + b • y",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\nmem1 : ↑x ∈ ↑(extensionOfMax i f).toLinearPMap.domain + ↑(Submodule.span R {y})\n⊢ ∃ a b, ↑x = ↑a + b • y",
"tactic": "rcases mem1 with ⟨a, b, a_mem, b_mem : b ∈ (Submodule.span R _ : Submodule R N), eq1⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\na b : N\na_mem : a ∈ ↑(extensionOfMax i f).toLinearPMap.domain\nb_mem : ∃ a, a • y = b\neq1 : (fun x x_1 => x + x_1) a b = ↑x\n⊢ ∃ a b, ↑x = ↑a + b • y",
"state_before": "case intro.intro.intro.intro\nR : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\na b : N\na_mem : a ∈ ↑(extensionOfMax i f).toLinearPMap.domain\nb_mem : b ∈ Submodule.span R {y}\neq1 : (fun x x_1 => x + x_1) a b = ↑x\n⊢ ∃ a b, ↑x = ↑a + b • y",
"tactic": "rw [Submodule.mem_span_singleton] at b_mem"
},
{
"state_after": "case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\na b : N\na_mem : a ∈ ↑(extensionOfMax i f).toLinearPMap.domain\neq1 : (fun x x_1 => x + x_1) a b = ↑x\nz : R\neq2 : z • y = b\n⊢ ∃ a b, ↑x = ↑a + b • y",
"state_before": "case intro.intro.intro.intro\nR : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\na b : N\na_mem : a ∈ ↑(extensionOfMax i f).toLinearPMap.domain\nb_mem : ∃ a, a • y = b\neq1 : (fun x x_1 => x + x_1) a b = ↑x\n⊢ ∃ a b, ↑x = ↑a + b • y",
"tactic": "rcases b_mem with ⟨z, eq2⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nR : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\na b : N\na_mem : a ∈ ↑(extensionOfMax i f).toLinearPMap.domain\neq1 : (fun x x_1 => x + x_1) a b = ↑x\nz : R\neq2 : z • y = b\n⊢ ∃ a b, ↑x = ↑a + b • y",
"tactic": "exact ⟨⟨a, a_mem⟩, z, by rw [← eq1, ← eq2]⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝⁷ : Ring R\nQ : TypeMax\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM N : Type (max u v)\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ↑i)\ny : N\nx : { x // x ∈ supExtensionOfMaxSingleton i f y }\na b : N\na_mem : a ∈ ↑(extensionOfMax i f).toLinearPMap.domain\neq1 : (fun x x_1 => x + x_1) a b = ↑x\nz : R\neq2 : z • y = b\n⊢ ↑x = ↑{ val := a, property := a_mem } + z • y",
"tactic": "rw [← eq1, ← eq2]"
}
] | [
268,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
9
] |
Mathlib/Data/Setoid/Partition.lean | Setoid.card_classes_ker_le | [
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1538\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype β\nf : α → β\ninst✝ : Fintype ↑(classes (ker f))\n⊢ Fintype.card ↑(classes (ker f)) ≤ Fintype.card β",
"tactic": "classical exact\n le_trans (Set.card_le_of_subset (classes_ker_subset_fiber_set f)) (Fintype.card_range_le _)"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.1538\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype β\nf : α → β\ninst✝ : Fintype ↑(classes (ker f))\n⊢ Fintype.card ↑(classes (ker f)) ≤ Fintype.card β",
"tactic": "exact\nle_trans (Set.card_le_of_subset (classes_ker_subset_fiber_set f)) (Fintype.card_range_le _)"
}
] | [
95,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
92,
1
] |
Mathlib/Algebra/Associated.lean | Associates.bot_eq_one | [] | [
786,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
785,
1
] |
Mathlib/RingTheory/Polynomial/Chebyshev.lean | Polynomial.Chebyshev.U_eq_X_mul_U_add_T | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.12813\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ U R (0 + 1) = X * U R 0 + T R (0 + 1)",
"tactic": "simp only [T, U, two_mul, mul_one]"
},
{
"state_after": "R : Type u_1\nS : Type ?u.12813\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ 2 * X * (2 * X) - 1 = X * (2 * X) + (2 * X * X - 1)",
"state_before": "R : Type u_1\nS : Type ?u.12813\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ U R (1 + 1) = X * U R 1 + T R (1 + 1)",
"tactic": "simp only [T, U]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.12813\ninst✝¹ : CommRing R\ninst✝ : CommRing S\n⊢ 2 * X * (2 * X) - 1 = X * (2 * X) + (2 * X * X - 1)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.12813\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ U R (n + 2 + 1) = 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1))",
"tactic": "rw [U_add_two, U_eq_X_mul_U_add_T n, U_eq_X_mul_U_add_T (n + 1), U_eq_X_mul_U_add_T n]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.12813\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ 2 * X * (X * U R (n + 1) + T R (n + 2)) - (X * U R n + T R (n + 1)) =\n X * (2 * X * U R (n + 1) - U R n) + (2 * X * T R (n + 2) - T R (n + 1))",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.12813\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nn : ℕ\n⊢ X * (2 * X * U R (n + 1) - U R n) + (2 * X * T R (n + 2) - T R (n + 1)) = X * U R (n + 2) + T R (n + 2 + 1)",
"tactic": "simp only [U_add_two, T_add_two]"
}
] | [
135,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
Mathlib/Order/SuccPred/Basic.lean | Order.lt_succ_iff_eq_or_lt_of_not_isMax | [] | [
445,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
444,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | Seminorm.const_isBounded | [
{
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{
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{
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{
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"tactic": "exact ⟨C, le_trans h (smul_le_smul (Finset.sup_le fun _ _ => le_rfl) le_rfl)⟩"
}
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243,
38
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237,
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Mathlib/Data/Int/Bitwise.lean | Int.bodd_zero | [] | [
31,
6
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30,
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Mathlib/CategoryTheory/Sites/Closed.lean | CategoryTheory.topology_eq_iff_same_sheaves | [
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"tactic": "intro P"
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"tactic": "rfl"
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"tactic": "intro h"
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{
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"tactic": "apply le_antisymm"
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{
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"tactic": "apply le_topology_of_closedSieves_isSheaf"
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"tactic": "rw [h]"
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{
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"tactic": "apply classifier_isSheaf"
}
] | [
287,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
274,
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Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsLittleO.smul | [
{
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"tactic": "intros"
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{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.584464\nE : Type ?u.584467\nF : Type ?u.584470\nG : Type ?u.584473\nE' : Type u_4\nF' : Type u_5\nG' : Type ?u.584482\nE'' : Type ?u.584485\nF'' : Type ?u.584488\nG'' : Type ?u.584491\nR : Type ?u.584494\nR' : Type ?u.584497\n𝕜 : Type u_2\n𝕜' : Type u_3\ninst✝¹⁴ : Norm E\ninst✝¹³ : Norm F\ninst✝¹² : Norm G\ninst✝¹¹ : SeminormedAddCommGroup E'\ninst✝¹⁰ : SeminormedAddCommGroup F'\ninst✝⁹ : SeminormedAddCommGroup G'\ninst✝⁸ : NormedAddCommGroup E''\ninst✝⁷ : NormedAddCommGroup F''\ninst✝⁶ : NormedAddCommGroup G''\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : SeminormedRing R'\ninst✝³ : NormedField 𝕜\ninst✝² : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\ninst✝¹ : NormedSpace 𝕜 E'\ninst✝ : NormedSpace 𝕜' F'\nk₁ : α → 𝕜\nk₂ : α → 𝕜'\nh₁ : k₁ =o[l] k₂\nh₂ : f' =o[l] g'\nx✝ : α\n⊢ ‖k₂ x✝‖ * ‖g' x✝‖ = ‖k₂ x✝ • g' x✝‖",
"tactic": "simp only [norm_smul]"
}
] | [
1777,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1774,
1
] |
Mathlib/Algebra/Module/Submodule/Bilinear.lean | Submodule.map₂_sup_left | [] | [
133,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
125,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.smul_extend | [
{
"state_after": "case h\nα : Type u_2\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\nR : Type u_1\ninst✝³ : Zero R\ninst✝² : SMulWithZero R ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : NoZeroSMulDivisors R ℝ≥0∞\nc : R\nhc : c ≠ 0\ns : α\n⊢ (c • extend m) s = extend (fun s h => c • m s h) s",
"state_before": "α : Type u_2\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\nR : Type u_1\ninst✝³ : Zero R\ninst✝² : SMulWithZero R ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : NoZeroSMulDivisors R ℝ≥0∞\nc : R\nhc : c ≠ 0\n⊢ c • extend m = extend fun s h => c • m s h",
"tactic": "ext1 s"
},
{
"state_after": "case h\nα : Type u_2\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\nR : Type u_1\ninst✝³ : Zero R\ninst✝² : SMulWithZero R ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : NoZeroSMulDivisors R ℝ≥0∞\nc : R\nhc : c ≠ 0\ns : α\n⊢ (c • ⨅ (h : P s), m s h) = ⨅ (h : P s), c • m s h",
"state_before": "case h\nα : Type u_2\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\nR : Type u_1\ninst✝³ : Zero R\ninst✝² : SMulWithZero R ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : NoZeroSMulDivisors R ℝ≥0∞\nc : R\nhc : c ≠ 0\ns : α\n⊢ (c • extend m) s = extend (fun s h => c • m s h) s",
"tactic": "dsimp [extend]"
},
{
"state_after": "case pos\nα : Type u_2\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\nR : Type u_1\ninst✝³ : Zero R\ninst✝² : SMulWithZero R ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : NoZeroSMulDivisors R ℝ≥0∞\nc : R\nhc : c ≠ 0\ns : α\nh : P s\n⊢ (c • ⨅ (h : P s), m s h) = ⨅ (h : P s), c • m s h\n\ncase neg\nα : Type u_2\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\nR : Type u_1\ninst✝³ : Zero R\ninst✝² : SMulWithZero R ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : NoZeroSMulDivisors R ℝ≥0∞\nc : R\nhc : c ≠ 0\ns : α\nh : ¬P s\n⊢ (c • ⨅ (h : P s), m s h) = ⨅ (h : P s), c • m s h",
"state_before": "case h\nα : Type u_2\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\nR : Type u_1\ninst✝³ : Zero R\ninst✝² : SMulWithZero R ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : NoZeroSMulDivisors R ℝ≥0∞\nc : R\nhc : c ≠ 0\ns : α\n⊢ (c • ⨅ (h : P s), m s h) = ⨅ (h : P s), c • m s h",
"tactic": "by_cases h : P s"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_2\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\nR : Type u_1\ninst✝³ : Zero R\ninst✝² : SMulWithZero R ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : NoZeroSMulDivisors R ℝ≥0∞\nc : R\nhc : c ≠ 0\ns : α\nh : P s\n⊢ (c • ⨅ (h : P s), m s h) = ⨅ (h : P s), c • m s h",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_2\nP : α → Prop\nm : (s : α) → P s → ℝ≥0∞\nR : Type u_1\ninst✝³ : Zero R\ninst✝² : SMulWithZero R ℝ≥0∞\ninst✝¹ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\ninst✝ : NoZeroSMulDivisors R ℝ≥0∞\nc : R\nhc : c ≠ 0\ns : α\nh : ¬P s\n⊢ (c • ⨅ (h : P s), m s h) = ⨅ (h : P s), c • m s h",
"tactic": "simp [h, ENNReal.smul_top, hc]"
}
] | [
1335,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1328,
1
] |
Mathlib/ModelTheory/Syntax.lean | FirstOrder.Language.Term.relabel_comp_relabel | [] | [
135,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
] |
Mathlib/Analysis/PSeries.lean | NNReal.summable_rpow | [
{
"state_after": "no goals",
"state_before": "p : ℝ\n⊢ (Summable fun n => ↑n ^ p) ↔ p < -1",
"tactic": "simp [← NNReal.summable_coe]"
}
] | [
281,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
280,
1
] |
Mathlib/Algebra/Module/Basic.lean | Convex.combo_eq_smul_sub_add | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.85204\nR : Type u_1\nk : Type ?u.85210\nS : Type ?u.85213\nM : Type u_2\nM₂ : Type ?u.85219\nM₃ : Type ?u.85222\nι : Type ?u.85225\ninst✝² : Semiring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\na b : R\nh : a + b = 1\n⊢ a • x + b • y = b • y - b • x + (a • x + b • x)",
"tactic": "abel"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.85204\nR : Type u_1\nk : Type ?u.85210\nS : Type ?u.85213\nM : Type u_2\nM₂ : Type ?u.85219\nM₃ : Type ?u.85222\nι : Type ?u.85225\ninst✝² : Semiring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\na b : R\nh : a + b = 1\n⊢ b • y - b • x + (a • x + b • x) = b • (y - x) + x",
"tactic": "rw [smul_sub, Convex.combo_self h]"
}
] | [
282,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
278,
1
] |
Mathlib/Topology/Basic.lean | isClosed_empty | [] | [
208,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
9
] |
Mathlib/CategoryTheory/Limits/EssentiallySmall.lean | CategoryTheory.Limits.hasColimitsOfShape_of_essentiallySmall | [] | [
39,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
37,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean | Finsupp.liftAddHom_apply_single | [] | [
489,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
487,
1
] |
Mathlib/Algebra/CharP/Basic.lean | CharP.char_is_prime_or_zero | [] | [
551,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
547,
1
] |
Mathlib/Topology/Constructions.lean | openEmbedding_sigmaMk | [] | [
1489,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1487,
1
] |
Mathlib/Analysis/Convex/Hull.lean | convexHull_singleton | [] | [
131,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | Asymptotics.isEquivalent_of_tendsto_one' | [] | [
222,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
220,
1
] |
Mathlib/Data/Finset/Image.lean | Equiv.finsetCongr_symm | [] | [
796,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
795,
1
] |
Mathlib/Topology/LocallyFinite.lean | LocallyFinite.comp_injOn | [
{
"state_after": "ι : Type u_1\nι' : Type u_3\nα : Type ?u.1343\nX : Type u_2\nY : Type ?u.1349\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf g✝ : ι → Set X\ng : ι' → ι\nhf : LocallyFinite f\nhg : InjOn g {i | Set.Nonempty (f (g i))}\nx : X\nt : Set X\nhtx : t ∈ 𝓝 x\nhtf : Set.Finite {i | Set.Nonempty (f i ∩ t)}\n⊢ ∃ t, t ∈ 𝓝 x ∧ Set.Finite {i | Set.Nonempty ((f ∘ g) i ∩ t)}",
"state_before": "ι : Type u_1\nι' : Type u_3\nα : Type ?u.1343\nX : Type u_2\nY : Type ?u.1349\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf g✝ : ι → Set X\ng : ι' → ι\nhf : LocallyFinite f\nhg : InjOn g {i | Set.Nonempty (f (g i))}\nx : X\n⊢ ∃ t, t ∈ 𝓝 x ∧ Set.Finite {i | Set.Nonempty ((f ∘ g) i ∩ t)}",
"tactic": "let ⟨t, htx, htf⟩ := hf x"
},
{
"state_after": "ι : Type u_1\nι' : Type u_3\nα : Type ?u.1343\nX : Type u_2\nY : Type ?u.1349\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf g✝ : ι → Set X\ng : ι' → ι\nhf : LocallyFinite f\nhg : InjOn g {i | Set.Nonempty (f (g i))}\nx : X\nt : Set X\nhtx : t ∈ 𝓝 x\nhtf : Set.Finite {i | Set.Nonempty (f i ∩ t)}\n⊢ InjOn (fun i => g i) ((fun i => g i) ⁻¹' {i | Set.Nonempty (f i ∩ t)})",
"state_before": "ι : Type u_1\nι' : Type u_3\nα : Type ?u.1343\nX : Type u_2\nY : Type ?u.1349\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf g✝ : ι → Set X\ng : ι' → ι\nhf : LocallyFinite f\nhg : InjOn g {i | Set.Nonempty (f (g i))}\nx : X\nt : Set X\nhtx : t ∈ 𝓝 x\nhtf : Set.Finite {i | Set.Nonempty (f i ∩ t)}\n⊢ ∃ t, t ∈ 𝓝 x ∧ Set.Finite {i | Set.Nonempty ((f ∘ g) i ∩ t)}",
"tactic": "refine ⟨t, htx, htf.preimage <| ?_⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type u_3\nα : Type ?u.1343\nX : Type u_2\nY : Type ?u.1349\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf g✝ : ι → Set X\ng : ι' → ι\nhf : LocallyFinite f\nhg : InjOn g {i | Set.Nonempty (f (g i))}\nx : X\nt : Set X\nhtx : t ∈ 𝓝 x\nhtf : Set.Finite {i | Set.Nonempty (f i ∩ t)}\n⊢ InjOn (fun i => g i) ((fun i => g i) ⁻¹' {i | Set.Nonempty (f i ∩ t)})",
"tactic": "exact hg.mono fun i (hi : Set.Nonempty _) => hi.left"
}
] | [
54,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
50,
1
] |
Mathlib/CategoryTheory/Extensive.lean | CategoryTheory.FinitaryExtensive.isPullback_initial_to_binaryCofan | [] | [
273,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
270,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | le_mul_of_one_le_of_le_of_nonneg | [] | [
930,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
928,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Subalgebra.mem_of_span_eq_top_of_smul_pow_mem | [] | [
1464,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1460,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean | FreeAbelianGroup.seq_sub | [] | [
320,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
] |
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