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Mathlib/Algebra/Hom/Freiman.lean | FreimanHom.inv_comp | [] | [
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Mathlib/Data/Set/Pointwise/SMul.lean | Set.smul_set_iInter₂_subset | [] | [
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Mathlib/Data/Fintype/BigOperators.lean | Fintype.card_eq_sum_ones | [] | [
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Mathlib/RingTheory/RootsOfUnity/Basic.lean | IsPrimitiveRoot.pow_iff_coprime | [
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Mathlib/Logic/Equiv/Defs.lean | Equiv.symm_symm | [
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Mathlib/Topology/SubsetProperties.lean | compactSpace_of_finite_subfamily_closed | [
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Mathlib/Data/Set/Image.lean | Function.Injective.mem_range_iff_exists_unique | [] | [
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Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.comap_map_eq | [
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Mathlib/Topology/Maps.lean | Inducing.map_nhds_eq | [] | [
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Mathlib/Data/Fin/Tuple/Basic.lean | Fin.snoc_update | [
{
"state_after": "case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\n⊢ snoc (update p i y) x j = update (snoc p x) (↑castSucc i) y j",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\n⊢ snoc (update p i y) x = update (snoc p x) (↑castSucc i) y",
"tactic": "ext j"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\n⊢ snoc (update p i y) x j = update (snoc p x) (↑castSucc i) y j\n\ncase neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ¬↑j < n\n⊢ snoc (update p i y) x j = update (snoc p x) (↑castSucc i) y j",
"state_before": "case h\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\n⊢ snoc (update p i y) x j = update (snoc p x) (↑castSucc i) y j",
"tactic": "by_cases h : j.val < n"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\n⊢ (if h : ↑j < n then _root_.cast (_ : α (↑castSucc (castLT j h)) = α j) (update p i y (castLT j h))\n else _root_.cast (_ : α (last n) = α j) x) =\n update (snoc p x) (↑castSucc i) y j",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\n⊢ snoc (update p i y) x j = update (snoc p x) (↑castSucc i) y j",
"tactic": "rw [snoc]"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\n⊢ (if h_1 : True then\n _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n)))\n else _root_.cast (_ : α (last n) = α j) x) =\n update (snoc p x) (↑castSucc i) y j",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\n⊢ (if h : ↑j < n then _root_.cast (_ : α (↑castSucc (castLT j h)) = α j) (update p i y (castLT j h))\n else _root_.cast (_ : α (last n) = α j) x) =\n update (snoc p x) (↑castSucc i) y j",
"tactic": "simp only [h]"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n))) =\n update (snoc p x) (↑castSucc i) y j",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\n⊢ (if h_1 : True then\n _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n)))\n else _root_.cast (_ : α (last n) = α j) x) =\n update (snoc p x) (↑castSucc i) y j",
"tactic": "simp only [dif_pos]"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n))) =\n update (snoc p x) (↑castSucc i) y j\n\ncase neg\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : ¬j = ↑castSucc i\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n))) =\n update (snoc p x) (↑castSucc i) y j",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n))) =\n update (snoc p x) (↑castSucc i) y j",
"tactic": "by_cases h' : j = castSucc i"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n))) =\n update (snoc p x) (↑castSucc i) y j",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n))) =\n update (snoc p x) (↑castSucc i) y j",
"tactic": "have C1 : α (castSucc i) = α j := by rw [h']"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n))) =\n update (snoc p x) (↑castSucc i) y j",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n))) =\n update (snoc p x) (↑castSucc i) y j",
"tactic": "have C2 : α (castSucc i) = α (castSucc (castLT j h)) := by rw [castSucc_castLT, h']"
},
{
"state_after": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\nE2 : update p i y (castLT j h) = _root_.cast C2 y\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (_root_.cast C2 y) = _root_.cast C1 y",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\nE2 : update p i y (castLT j h) = _root_.cast C2 y\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (update p i y (castLT j (_ : ↑j < n))) =\n update (snoc p x) (↑castSucc i) y j",
"tactic": "rw [E1, E2]"
},
{
"state_after": "no goals",
"state_before": "case pos\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\nE2 : update p i y (castLT j h) = _root_.cast C2 y\n⊢ _root_.cast (_ : α (↑castSucc (castLT j (_ : ↑j < n))) = α j) (_root_.cast C2 y) = _root_.cast C1 y",
"tactic": "exact eq_rec_compose (Eq.trans C2.symm C1) C2 y"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\n⊢ α (↑castSucc i) = α j",
"tactic": "rw [h']"
},
{
"state_after": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nthis : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y\n⊢ update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\n⊢ update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y",
"tactic": "have : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y := by simp"
},
{
"state_after": "case h.e'_2.h.e'_5\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nthis : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y\n⊢ ↑castSucc i = j\n\ncase h.e'_2.h.e'_6\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nthis : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y\ne_5✝ : ↑castSucc i = j\n⊢ HEq y (_root_.cast C1 y)",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nthis : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y\n⊢ update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y",
"tactic": "convert this"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\n⊢ update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_5\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nthis : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y\n⊢ ↑castSucc i = j",
"tactic": "exact h'.symm"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nthis : update (snoc p x) j (_root_.cast C1 y) j = _root_.cast C1 y\ne_5✝ : ↑castSucc i = j\n⊢ HEq y (_root_.cast C1 y)",
"tactic": "exact heq_of_cast_eq (congr_arg α (Eq.symm h')) rfl"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\n⊢ α (↑castSucc i) = α (↑castSucc (castLT j h))",
"tactic": "rw [castSucc_castLT, h']"
},
{
"state_after": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\nthis : update p (castLT j h) (_root_.cast C2 y) (castLT j h) = _root_.cast C2 y\n⊢ update p i y (castLT j h) = _root_.cast C2 y",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\n⊢ update p i y (castLT j h) = _root_.cast C2 y",
"tactic": "have : update p (castLT j h) (_root_.cast C2 y) (castLT j h) = _root_.cast C2 y := by simp"
},
{
"state_after": "case h.e'_2.h.e'_5\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\nthis : update p (castLT j h) (_root_.cast C2 y) (castLT j h) = _root_.cast C2 y\n⊢ i = castLT j h\n\ncase h.e'_2.h.e'_6\nm n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\nthis : update p (castLT j h) (_root_.cast C2 y) (castLT j h) = _root_.cast C2 y\ne_5✝ : i = castLT j h\n⊢ HEq y (_root_.cast C2 y)",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\nthis : update p (castLT j h) (_root_.cast C2 y) (castLT j h) = _root_.cast C2 y\n⊢ update p i y (castLT j h) = _root_.cast C2 y",
"tactic": "convert this"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : j = ↑castSucc i\nC1 : α (↑castSucc i) = α j\nE1 : update (snoc p x) (↑castSucc i) y j = _root_.cast C1 y\nC2 : α (↑castSucc i) = α (↑castSucc (castLT j h))\n⊢ update p (castLT j h) (_root_.cast C2 y) (castLT j h) = _root_.cast C2 y",
"tactic": "simp"
},
{
"state_after": "no goals",
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},
{
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},
{
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"tactic": "have : ¬castLT j h = i := by\n intro E\n apply h'\n rw [← E, castSucc_castLT]"
},
{
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"tactic": "simp [h', this, snoc, h]"
},
{
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"tactic": "intro E"
},
{
"state_after": "m n : ℕ\nα : Fin (n + 1) → Type u\nx : α (last n)\nq : (i : Fin (n + 1)) → α i\np : (i : Fin n) → α (↑castSucc i)\ni : Fin n\ny : α (↑castSucc i)\nz : α (last n)\nj : Fin (n + 1)\nh : ↑j < n\nh' : ¬j = ↑castSucc i\nE : castLT j h = i\n⊢ j = ↑castSucc i",
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},
{
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"tactic": "rw [← E, castSucc_castLT]"
},
{
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"tactic": "rw [eq_last_of_not_lt h]"
},
{
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"tactic": "simp [Ne.symm (ne_of_lt (castSucc_lt_last i))]"
}
] | [
528,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Order/WithBot.lean | WithBot.none_lt_some | [] | [
271,
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Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | balancedCore_empty | [] | [
81,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
80,
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Mathlib/Topology/Algebra/Group/Basic.lean | Homeomorph.mulRight_symm | [
{
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"tactic": "rfl"
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122,
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Mathlib/Algebra/Hom/NonUnitalAlg.lean | NonUnitalAlgHom.inr_apply | [] | [
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Mathlib/Analysis/Convex/Measure.lean | Convex.add_haar_frontier | [
{
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{
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"tactic": "intro s hs hx hb"
},
{
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"tactic": "replace hb : μ (interior s) ≠ ∞"
},
{
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"tactic": "exact (hb.mono interior_subset).measure_lt_top.ne"
},
{
"state_after": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\n⊢ ↑↑μ (closure s) ≤ ↑↑μ (interior s)",
"state_before": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\n⊢ ↑↑μ (frontier s) = 0",
"tactic": "suffices μ (closure s) ≤ μ (interior s) by\n rwa [frontier, measure_diff interior_subset_closure isOpen_interior.measurableSet hb,\n tsub_eq_zero_iff_le]"
},
{
"state_after": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\n⊢ ↑↑μ (closure s) ≤ ↑↑μ (interior s)",
"state_before": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\n⊢ ↑↑μ (closure s) ≤ ↑↑μ (interior s)",
"tactic": "set d : ℕ := FiniteDimensional.finrank ℝ E"
},
{
"state_after": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\nthis : ∀ (r : ℝ≥0), 1 < r → ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\n⊢ ↑↑μ (closure s) ≤ ↑↑μ (interior s)",
"state_before": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\n⊢ ↑↑μ (closure s) ≤ ↑↑μ (interior s)",
"tactic": "have : ∀ r : ℝ≥0, 1 < r → μ (closure s) ≤ ↑(r ^ d) * μ (interior s) := by\n intro r hr\n refine' (measure_mono <|\n hs.closure_subset_image_homothety_interior_of_one_lt hx r hr).trans_eq _\n rw [add_haar_image_homothety, ← NNReal.coe_pow, NNReal.abs_eq, ENNReal.ofReal_coe_nnreal]"
},
{
"state_after": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\nthis✝ : ∀ (r : ℝ≥0), 1 < r → ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\nthis : ∀ᶠ (r : ℝ≥0) in 𝓝[Ioi 1] 1, ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\n⊢ ↑↑μ (closure s) ≤ ↑↑μ (interior s)",
"state_before": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\nthis : ∀ (r : ℝ≥0), 1 < r → ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\n⊢ ↑↑μ (closure s) ≤ ↑↑μ (interior s)",
"tactic": "have : ∀ᶠ (r : ℝ≥0) in 𝓝[>] 1, μ (closure s) ≤ ↑(r ^ d) * μ (interior s) :=\n mem_of_superset self_mem_nhdsWithin this"
},
{
"state_after": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\nthis✝ : ∀ (r : ℝ≥0), 1 < r → ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\nthis : ∀ᶠ (r : ℝ≥0) in 𝓝[Ioi 1] 1, ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\n⊢ Tendsto (fun c => ↑(c ^ d) * ↑↑μ (interior s)) (𝓝[Ioi 1] 1) (𝓝 (↑↑μ (interior s)))",
"state_before": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\nthis✝ : ∀ (r : ℝ≥0), 1 < r → ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\nthis : ∀ᶠ (r : ℝ≥0) in 𝓝[Ioi 1] 1, ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\n⊢ ↑↑μ (closure s) ≤ ↑↑μ (interior s)",
"tactic": "refine' ge_of_tendsto _ this"
},
{
"state_after": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\nthis✝ : ∀ (r : ℝ≥0), 1 < r → ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\nthis : ∀ᶠ (r : ℝ≥0) in 𝓝[Ioi 1] 1, ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\n⊢ ((fun x => x * ↑↑μ (interior s)) ∘ ENNReal.some ∘ fun a => a ^ d) 1 = ↑↑μ (interior s)",
"state_before": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\nthis✝ : ∀ (r : ℝ≥0), 1 < r → ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\nthis : ∀ᶠ (r : ℝ≥0) in 𝓝[Ioi 1] 1, ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\n⊢ Tendsto (fun c => ↑(c ^ d) * ↑↑μ (interior s)) (𝓝[Ioi 1] 1) (𝓝 (↑↑μ (interior s)))",
"tactic": "refine' (((ENNReal.continuous_mul_const hb).comp\n (ENNReal.continuous_coe.comp (continuous_pow d))).tendsto' _ _ _).mono_left nhdsWithin_le_nhds"
},
{
"state_after": "no goals",
"state_before": "case H\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\nthis✝ : ∀ (r : ℝ≥0), 1 < r → ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\nthis : ∀ᶠ (r : ℝ≥0) in 𝓝[Ioi 1] 1, ↑↑μ (closure s) ≤ ↑(r ^ d) * ↑↑μ (interior s)\n⊢ ((fun x => x * ↑↑μ (interior s)) ∘ ENNReal.some ∘ fun a => a ^ d) 1 = ↑↑μ (interior s)",
"tactic": "simp"
},
{
"state_after": "case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nhspan : affineSpan ℝ s ≠ ⊤\n⊢ frontier s ⊆ ↑(affineSpan ℝ s)",
"state_before": "case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nhspan : affineSpan ℝ s ≠ ⊤\n⊢ ↑↑μ (frontier s) = 0",
"tactic": "refine' measure_mono_null _ (add_haar_affineSubspace _ _ hspan)"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nhspan : affineSpan ℝ s ≠ ⊤\n⊢ frontier s ⊆ ↑(affineSpan ℝ s)",
"tactic": "exact frontier_subset_closure.trans\n (closure_minimal (subset_affineSpan _ _) (affineSpan ℝ s).closed_of_finiteDimensional)"
},
{
"state_after": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\n⊢ ↑↑μ (frontier s) = 0",
"state_before": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\n⊢ ↑↑μ (frontier s) = 0",
"tactic": "let B : ℕ → Set E := fun n => ball x (n + 1)"
},
{
"state_after": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\nthis : ↑↑μ (⋃ (n : ℕ), frontier (s ∩ B n)) = 0\n⊢ ↑↑μ (frontier s) = 0",
"state_before": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\n⊢ ↑↑μ (frontier s) = 0",
"tactic": "have : μ (⋃ n : ℕ, frontier (s ∩ B n)) = 0 := by\n refine' measure_iUnion_null fun n =>\n H _ (hs.inter (convex_ball _ _)) _ (bounded_ball.mono (inter_subset_right _ _))\n rw [interior_inter, isOpen_ball.interior_eq]\n exact ⟨hx, mem_ball_self (add_pos_of_nonneg_of_pos n.cast_nonneg zero_lt_one)⟩"
},
{
"state_after": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\nthis : ↑↑μ (⋃ (n : ℕ), frontier (s ∩ B n)) = 0\ny : E\nhy : y ∈ frontier s\n⊢ y ∈ ⋃ (n : ℕ), frontier (s ∩ B n)",
"state_before": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\nthis : ↑↑μ (⋃ (n : ℕ), frontier (s ∩ B n)) = 0\n⊢ ↑↑μ (frontier s) = 0",
"tactic": "refine' measure_mono_null (fun y hy => _) this"
},
{
"state_after": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\ny : E\nhy : y ∈ frontier s\n⊢ y ∈ ⋃ (n : ℕ), frontier (s ∩ B n)",
"state_before": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\nthis : ↑↑μ (⋃ (n : ℕ), frontier (s ∩ B n)) = 0\ny : E\nhy : y ∈ frontier s\n⊢ y ∈ ⋃ (n : ℕ), frontier (s ∩ B n)",
"tactic": "clear this"
},
{
"state_after": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\ny : E\nhy : y ∈ frontier s\nN : ℕ := ⌊dist y x⌋₊\n⊢ y ∈ ⋃ (n : ℕ), frontier (s ∩ B n)",
"state_before": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\ny : E\nhy : y ∈ frontier s\n⊢ y ∈ ⋃ (n : ℕ), frontier (s ∩ B n)",
"tactic": "set N : ℕ := ⌊dist y x⌋₊"
},
{
"state_after": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\ny : E\nhy : y ∈ frontier s\nN : ℕ := ⌊dist y x⌋₊\n⊢ y ∈ frontier (s ∩ B N)",
"state_before": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\ny : E\nhy : y ∈ frontier s\nN : ℕ := ⌊dist y x⌋₊\n⊢ y ∈ ⋃ (n : ℕ), frontier (s ∩ B n)",
"tactic": "refine' mem_iUnion.2 ⟨N, _⟩"
},
{
"state_after": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\ny : E\nhy : y ∈ frontier s\nN : ℕ := ⌊dist y x⌋₊\nhN : y ∈ B N\n⊢ y ∈ frontier (s ∩ B N)",
"state_before": "case inr.intro\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\ny : E\nhy : y ∈ frontier s\nN : ℕ := ⌊dist y x⌋₊\n⊢ y ∈ frontier (s ∩ B N)",
"tactic": "have hN : y ∈ B N := by simp [Nat.lt_floor_add_one]"
},
{
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"tactic": "suffices : y ∈ frontier (s ∩ B N) ∩ B N"
},
{
"state_after": "case this\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\ny : E\nhy : y ∈ frontier s\nN : ℕ := ⌊dist y x⌋₊\nhN : y ∈ B N\n⊢ y ∈ frontier (s ∩ B N) ∩ B N",
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},
{
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"tactic": "rw [frontier_inter_open_inter isOpen_ball]"
},
{
"state_after": "no goals",
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"tactic": "exact ⟨hy, hN⟩"
},
{
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"tactic": "refine' measure_iUnion_null fun n =>\n H _ (hs.inter (convex_ball _ _)) _ (bounded_ball.mono (inter_subset_right _ _))"
},
{
"state_after": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\nn : ℕ\n⊢ x ∈ interior s ∩ ball x (↑n + 1)",
"state_before": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nH : ∀ (t : Set E), Convex ℝ t → x ∈ interior t → Metric.Bounded t → ↑↑μ (frontier t) = 0\nB : ℕ → Set E := fun n => ball x (↑n + 1)\nn : ℕ\n⊢ x ∈ interior (s ∩ B n)",
"tactic": "rw [interior_inter, isOpen_ball.interior_eq]"
},
{
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"tactic": "exact ⟨hx, mem_ball_self (add_pos_of_nonneg_of_pos n.cast_nonneg zero_lt_one)⟩"
},
{
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"tactic": "simp [Nat.lt_floor_add_one]"
},
{
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"tactic": "rwa [frontier, measure_diff interior_subset_closure isOpen_interior.measurableSet hb,\n tsub_eq_zero_iff_le]"
},
{
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"tactic": "intro r hr"
},
{
"state_after": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : MeasureTheory.Measure E\ninst✝ : IsAddHaarMeasure μ\ns✝ : Set E\nhs✝ : Convex ℝ s✝\nx : E\nhx✝ : x ∈ interior s✝\ns : Set E\nhs : Convex ℝ s\nhx : x ∈ interior s\nhb : ↑↑μ (interior s) ≠ ⊤\nd : ℕ := finrank ℝ E\nr : ℝ≥0\nhr : 1 < r\n⊢ ↑↑μ (↑(AffineMap.homothety x ↑r) '' interior s) = ↑(r ^ d) * ↑↑μ (interior s)",
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"tactic": "refine' (measure_mono <|\n hs.closure_subset_image_homothety_interior_of_one_lt hx r hr).trans_eq _"
},
{
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"tactic": "rw [add_haar_image_homothety, ← NNReal.coe_pow, NNReal.abs_eq, ENNReal.ofReal_coe_nnreal]"
}
] | [
85,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
36,
1
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Mathlib/Data/Nat/Cast/Basic.lean | ext_nat' | [
{
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"tactic": "intro n"
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{
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"tactic": "induction n with\n| zero => simp_rw [Nat.zero_eq, map_zero f, map_zero g]\n| succ n ihn =>\n simp [Nat.succ_eq_add_one, h, ihn]"
},
{
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"tactic": "simp_rw [Nat.zero_eq, map_zero f, map_zero g]"
},
{
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"tactic": "simp [Nat.succ_eq_add_one, h, ihn]"
}
] | [
208,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
202,
1
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Mathlib/Data/Int/ModEq.lean | Int.modEq_iff_dvd | [
{
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},
{
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"tactic": "simp [emod_eq_emod_iff_emod_sub_eq_zero, dvd_iff_emod_eq_zero]"
}
] | [
99,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
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Mathlib/Topology/List.lean | Vector.continuousAt_removeNth | [
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"tactic": "rw [ContinuousAt, removeNth, tendsto_subtype_rng]"
},
{
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"state_before": "α : Type u_1\nβ : Type ?u.68450\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\nl : List α\nhl : List.length l = n + 1\n⊢ Tendsto (fun x => ↑(removeNth i x)) (𝓝 { val := l, property := hl })\n (𝓝\n ↑(match { val := l, property := hl } with\n | { val := l, property := p } =>\n { val := List.removeNth l ↑i, property := (_ : List.length (List.removeNth l ↑i) = n + 1 - 1) }))",
"tactic": "simp only [Vector.removeNth_val]"
},
{
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"state_before": "α : Type u_1\nβ : Type ?u.68450\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\nl : List α\nhl : List.length l = n + 1\n⊢ Tendsto (fun x => List.removeNth ↑x ↑i) (𝓝 { val := l, property := hl }) (𝓝 (List.removeNth l ↑i))",
"tactic": "exact Tendsto.comp List.tendsto_removeNth continuousAt_subtype_val"
}
] | [
223,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
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Mathlib/Data/Finset/Lattice.lean | Finset.min'_mem | [
{
"state_after": "F : Type ?u.332825\nα : Type u_1\nβ : Type ?u.332831\nγ : Type ?u.332834\nι : Type ?u.332837\nκ : Type ?u.332840\ninst✝ : LinearOrder α\ns : Finset α\nH : Finset.Nonempty s\nx : α\n⊢ inf s WithTop.some = inf s (WithTop.some ∘ fun x => x)",
"state_before": "F : Type ?u.332825\nα : Type u_1\nβ : Type ?u.332831\nγ : Type ?u.332834\nι : Type ?u.332837\nκ : Type ?u.332840\ninst✝ : LinearOrder α\ns : Finset α\nH : Finset.Nonempty s\nx : α\n⊢ Finset.min s = ↑(min' s H)",
"tactic": "simp only [Finset.min, min', id_eq, coe_inf']"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.332825\nα : Type u_1\nβ : Type ?u.332831\nγ : Type ?u.332834\nι : Type ?u.332837\nκ : Type ?u.332840\ninst✝ : LinearOrder α\ns : Finset α\nH : Finset.Nonempty s\nx : α\n⊢ inf s WithTop.some = inf s (WithTop.some ∘ fun x => x)",
"tactic": "rfl"
}
] | [
1325,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1324,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean | Rel.abs_edgeDensity_sub_edgeDensity_le_one_sub_mul | [
{
"state_after": "𝕜 : Type ?u.36968\nι : Type ?u.36971\nκ : Type ?u.36974\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ edgeDensity r s₁ t₁ - edgeDensity r s₂ t₂ ≤ 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"state_before": "𝕜 : Type ?u.36968\nι : Type ?u.36971\nκ : Type ?u.36974\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ abs (edgeDensity r s₂ t₂ - edgeDensity r s₁ t₁) ≤ 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"tactic": "refine' abs_sub_le_iff.2 ⟨edgeDensity_sub_edgeDensity_le_one_sub_mul r hs ht hs₂ ht₂, _⟩"
},
{
"state_after": "𝕜 : Type ?u.36968\nι : Type ?u.36971\nκ : Type ?u.36974\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ edgeDensity (fun x y => ¬r x y) s₂ t₂ - edgeDensity (fun x y => ¬r x y) s₁ t₁ ≤\n 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"state_before": "𝕜 : Type ?u.36968\nι : Type ?u.36971\nκ : Type ?u.36974\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ edgeDensity r s₁ t₁ - edgeDensity r s₂ t₂ ≤ 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"tactic": "rw [← add_sub_cancel (edgeDensity r s₁ t₁) (edgeDensity (fun x y ↦ ¬r x y) s₁ t₁),\n ← add_sub_cancel (edgeDensity r s₂ t₂) (edgeDensity (fun x y ↦ ¬r x y) s₂ t₂),\n edgeDensity_add_edgeDensity_compl _ (hs₂.mono hs) (ht₂.mono ht),\n edgeDensity_add_edgeDensity_compl _ hs₂ ht₂, sub_sub_sub_cancel_left]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.36968\nι : Type ?u.36971\nκ : Type ?u.36974\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns s₁ s₂ : Finset α\nt t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\nhs : s₂ ⊆ s₁\nht : t₂ ⊆ t₁\nhs₂ : Finset.Nonempty s₂\nht₂ : Finset.Nonempty t₂\n⊢ edgeDensity (fun x y => ¬r x y) s₂ t₂ - edgeDensity (fun x y => ¬r x y) s₁ t₁ ≤\n 1 - ↑(card s₂) / ↑(card s₁) * (↑(card t₂) / ↑(card t₁))",
"tactic": "exact edgeDensity_sub_edgeDensity_le_one_sub_mul _ hs ht hs₂ ht₂"
}
] | [
215,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
207,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean | CategoryTheory.Presieve.extend_restrict | [
{
"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\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\n⊢ FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R ≤ (generate R).arrows) x) = x",
"state_before": "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\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.Compatible x\n⊢ FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R ≤ (generate R).arrows) x) = x",
"tactic": "rw [compatible_iff_sieveCompatible] at t"
},
{
"state_after": "case h.h.h\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\nx✝¹ : C\nx✝ : x✝¹ ⟶ X\nh : (generate R).arrows x✝\n⊢ FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R ≤ (generate R).arrows) x) x✝ h = x x✝ h",
"state_before": "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\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\n⊢ FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R ≤ (generate R).arrows) x) = x",
"tactic": "funext _ _ h"
},
{
"state_after": "case h.h.h\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\nx✝¹ : C\nx✝ : x✝¹ ⟶ X\nh : (generate R).arrows x✝\n⊢ x\n (Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫\n Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫ g = x✝))\n (_ :\n (generate R).arrows\n (Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫\n Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫ g = x✝))) =\n x x✝ h",
"state_before": "case h.h.h\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\nx✝¹ : C\nx✝ : x✝¹ ⟶ X\nh : (generate R).arrows x✝\n⊢ FamilyOfElements.sieveExtend (FamilyOfElements.restrict (_ : R ≤ (generate R).arrows) x) x✝ h = x x✝ h",
"tactic": "apply (t _ _ _).symm.trans"
},
{
"state_after": "case h.h.h.e_f\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\nx✝¹ : C\nx✝ : x✝¹ ⟶ X\nh : (generate R).arrows x✝\n⊢ Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫\n Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫ g = x✝) =\n x✝",
"state_before": "case h.h.h\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\nx✝¹ : C\nx✝ : x✝¹ ⟶ X\nh : (generate R).arrows x✝\n⊢ x\n (Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫\n Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫ g = x✝))\n (_ :\n (generate R).arrows\n (Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫\n Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫ g = x✝))) =\n x x✝ h",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.e_f\nC : Type u₁\ninst✝ : Category C\nP Q U : Cᵒᵖ ⥤ Type w\nX Y : C\nS : Sieve X\nR : Presieve X\nJ J₂ : GrothendieckTopology C\nx : FamilyOfElements P (generate R).arrows\nt : FamilyOfElements.SieveCompatible x\nx✝¹ : C\nx✝ : x✝¹ ⟶ X\nh : (generate R).arrows x✝\n⊢ Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫\n Exists.choose (_ : ∃ g, R g ∧ Exists.choose (_ : ∃ h_1 g, R g ∧ h_1 ≫ g = x✝) ≫ g = x✝) =\n x✝",
"tactic": "exact h.choose_spec.choose_spec.choose_spec.2"
}
] | [
256,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
] |
Mathlib/Order/BoundedOrder.lean | not_isMax_bot | [] | [
420,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
419,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean | Padic.div_nat_pos | [
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ↑padicNormE\nn : ℕ\n⊢ 0 < ↑n + 1",
"tactic": "exact_mod_cast succ_pos _"
}
] | [
683,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
682,
9
] |
Mathlib/Algebra/FreeMonoid/Basic.lean | FreeMonoid.ofList_toList | [] | [
68,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
68,
1
] |
Mathlib/Geometry/Manifold/Instances/Real.lean | range_quadrant | [] | [
101,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/Data/Nat/GCD/Basic.lean | Nat.pow_dvd_pow_iff | [
{
"state_after": "a b n : ℕ\nn0 : 0 < n\nh : a ^ n ∣ b ^ n\n⊢ a ∣ b",
"state_before": "a b n : ℕ\nn0 : 0 < n\n⊢ a ^ n ∣ b ^ n ↔ a ∣ b",
"tactic": "refine' ⟨fun h => _, fun h => pow_dvd_pow_of_dvd h _⟩"
},
{
"state_after": "case inl\na b n : ℕ\nn0 : 0 < n\nh : a ^ n ∣ b ^ n\ng0 : gcd a b = 0\n⊢ a ∣ b\n\ncase inr\na b n : ℕ\nn0 : 0 < n\nh : a ^ n ∣ b ^ n\ng0 : gcd a b > 0\n⊢ a ∣ b",
"state_before": "a b n : ℕ\nn0 : 0 < n\nh : a ^ n ∣ b ^ n\n⊢ a ∣ b",
"tactic": "cases' Nat.eq_zero_or_pos (gcd a b) with g0 g0"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\nh : (a' * g) ^ n ∣ (b' * g) ^ n\ng0 : gcd (a' * g) (b' * g) > 0\n⊢ a' * g ∣ b' * g",
"state_before": "case inr\na b n : ℕ\nn0 : 0 < n\nh : a ^ n ∣ b ^ n\ng0 : gcd a b > 0\n⊢ a ∣ b",
"tactic": "rcases exists_coprime' g0 with ⟨g, a', b', g0', co, rfl, rfl⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\nh : a' ^ n * g ^ n ∣ b' ^ n * g ^ n\ng0 : gcd (a' * g) (b' * g) > 0\n⊢ a' * g ∣ b' * g",
"state_before": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\nh : (a' * g) ^ n ∣ (b' * g) ^ n\ng0 : gcd (a' * g) (b' * g) > 0\n⊢ a' * g ∣ b' * g",
"tactic": "rw [mul_pow, mul_pow] at h"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\ng0 : gcd (a' * g) (b' * g) > 0\nh : a' ^ n ∣ b' ^ n\n⊢ a' * g ∣ b' * g",
"state_before": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\nh : a' ^ n * g ^ n ∣ b' ^ n * g ^ n\ng0 : gcd (a' * g) (b' * g) > 0\n⊢ a' * g ∣ b' * g",
"tactic": "replace h := Nat.dvd_of_mul_dvd_mul_right (pow_pos g0' _) h"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\ng0 : gcd (a' * g) (b' * g) > 0\nh : a' ^ n ∣ b' ^ n\nthis : a' ^ succ 0 ∣ a' ^ n\n⊢ a' * g ∣ b' * g",
"state_before": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\ng0 : gcd (a' * g) (b' * g) > 0\nh : a' ^ n ∣ b' ^ n\n⊢ a' * g ∣ b' * g",
"tactic": "have := pow_dvd_pow a' n0"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\ng0 : gcd (a' * g) (b' * g) > 0\nh : a' ^ n ∣ b' ^ n\nthis : a' ∣ 1\n⊢ a' * g ∣ b' * g",
"state_before": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\ng0 : gcd (a' * g) (b' * g) > 0\nh : a' ^ n ∣ b' ^ n\nthis : a' ^ succ 0 ∣ a' ^ n\n⊢ a' * g ∣ b' * g",
"tactic": "rw [pow_one, (co.pow n n).eq_one_of_dvd h] at this"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.intro.intro.intro\nn : ℕ\nn0 : 0 < n\ng : ℕ\na' b' : ℕ\ng0' : 0 < g\nco : coprime a' b'\ng0 : gcd (a' * g) (b' * g) > 0\nh : a' ^ n ∣ b' ^ n\nthis : a' ∣ 1\n⊢ a' * g ∣ b' * g",
"tactic": "simp [eq_one_of_dvd_one this]"
},
{
"state_after": "no goals",
"state_before": "case inl\na b n : ℕ\nn0 : 0 < n\nh : a ^ n ∣ b ^ n\ng0 : gcd a b = 0\n⊢ a ∣ b",
"tactic": "simp [eq_zero_of_gcd_eq_zero_right g0]"
}
] | [
268,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
259,
1
] |
Mathlib/Algebra/Tropical/Basic.lean | Tropical.inf_eq_add | [] | [
317,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
316,
1
] |
Mathlib/Data/Finset/Lattice.lean | Finset.inf_def | [] | [
324,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
] |
Mathlib/Topology/Sequences.lean | isSeqClosed_iff | [] | [
113,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
112,
1
] |
Mathlib/Data/Complex/Exponential.lean | Complex.ofReal_tanh | [] | [
715,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
714,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean | Ideal.comap_top | [] | [
1440,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1439,
1
] |
Mathlib/Data/Int/GCD.lean | Nat.xgcd_zero_left | [
{
"state_after": "no goals",
"state_before": "s t : ℤ\nr' : ℕ\ns' t' : ℤ\n⊢ xgcdAux 0 s t r' s' t' = (r', s', t')",
"tactic": "simp [xgcdAux]"
}
] | [
58,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | csInf_Ioo | [] | [
745,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
744,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.rescale_mk | [
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nf : ℕ → R\na : R\nn✝ : ℕ\n⊢ ↑(coeff R n✝) (↑(rescale a) (mk f)) = ↑(coeff R n✝) (mk fun n => a ^ n * f n)",
"state_before": "R : Type u_1\ninst✝ : CommSemiring R\nf : ℕ → R\na : R\n⊢ ↑(rescale a) (mk f) = mk fun n => a ^ n * f n",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\ninst✝ : CommSemiring R\nf : ℕ → R\na : R\nn✝ : ℕ\n⊢ ↑(coeff R n✝) (↑(rescale a) (mk f)) = ↑(coeff R n✝) (mk fun n => a ^ n * f n)",
"tactic": "rw [coeff_rescale, coeff_mk, coeff_mk]"
}
] | [
1787,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1785,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean | MeasureTheory.weightedSMul_smul | [
{
"state_after": "no goals",
"state_before": "α : Type u_3\nE : Type ?u.60121\nF : Type u_2\n𝕜 : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nc : 𝕜\ns : Set α\nx : F\n⊢ ↑(weightedSMul μ s) (c • x) = c • ↑(weightedSMul μ s) x",
"tactic": "simp_rw [weightedSMul_apply, smul_comm]"
}
] | [
233,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
231,
1
] |
Mathlib/Data/Real/NNReal.lean | Set.OrdConnected.preimage_coe_nnreal_real | [] | [
1022,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1021,
1
] |
Mathlib/FieldTheory/Finite/Polynomial.lean | MvPolynomial.degrees_indicator | [
{
"state_after": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\n⊢ degrees (∏ n : σ, (1 - (X n - ↑C (c n)) ^ (Fintype.card K - 1))) ≤ ∑ s : σ, (Fintype.card K - 1) • {s}",
"state_before": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\n⊢ degrees (indicator c) ≤ ∑ s : σ, (Fintype.card K - 1) • {s}",
"tactic": "rw [indicator]"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees (1 - (X s - ↑C (c s)) ^ (Fintype.card K - 1)) ≤ (Fintype.card K - 1) • {s}",
"state_before": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\n⊢ degrees (∏ n : σ, (1 - (X n - ↑C (c n)) ^ (Fintype.card K - 1))) ≤ ∑ s : σ, (Fintype.card K - 1) • {s}",
"tactic": "refine' le_trans (degrees_prod _ _) (Finset.sum_le_sum fun s _ => _)"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees 1 ⊔ degrees ((X s - ↑C (c s)) ^ (Fintype.card K - 1)) ≤ (Fintype.card K - 1) • {s}",
"state_before": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees (1 - (X s - ↑C (c s)) ^ (Fintype.card K - 1)) ≤ (Fintype.card K - 1) • {s}",
"tactic": "refine' le_trans (degrees_sub _ _) _"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees ((X s - ↑C (c s)) ^ (Fintype.card K - 1)) ≤ (Fintype.card K - 1) • {s}",
"state_before": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees 1 ⊔ degrees ((X s - ↑C (c s)) ^ (Fintype.card K - 1)) ≤ (Fintype.card K - 1) • {s}",
"tactic": "rw [degrees_one, ← bot_eq_zero, bot_sup_eq]"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees (X s - ↑C (c s)) ≤ {s}",
"state_before": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees ((X s - ↑C (c s)) ^ (Fintype.card K - 1)) ≤ (Fintype.card K - 1) • {s}",
"tactic": "refine' le_trans (degrees_pow _ _) (nsmul_le_nsmul_of_le_right _ _)"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees (X s) ⊔ degrees (↑C (c s)) ≤ {s}",
"state_before": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees (X s - ↑C (c s)) ≤ {s}",
"tactic": "refine' le_trans (degrees_sub _ _) _"
},
{
"state_after": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees (X s) ≤ {s}",
"state_before": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees (X s) ⊔ degrees (↑C (c s)) ≤ {s}",
"tactic": "rw [degrees_C, ← bot_eq_zero, sup_bot_eq]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_2\nσ : Type u_1\ninst✝² : Fintype K\ninst✝¹ : Fintype σ\ninst✝ : CommRing K\nc : σ → K\ns : σ\nx✝ : s ∈ Finset.univ\n⊢ degrees (X s) ≤ {s}",
"tactic": "exact degrees_X' _"
}
] | [
93,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/CategoryTheory/Limits/Types.lean | CategoryTheory.Limits.Types.Image.lift_fac | [
{
"state_after": "case h\nJ : Type v\ninst✝ : SmallCategory J\nα β : Type u\nf : α ⟶ β\nF' : MonoFactorisation f\nx : Image f\n⊢ (lift F' ≫ F'.m) x = ι f x",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nα β : Type u\nf : α ⟶ β\nF' : MonoFactorisation f\n⊢ lift F' ≫ F'.m = ι f",
"tactic": "funext x"
},
{
"state_after": "case h\nJ : Type v\ninst✝ : SmallCategory J\nα β : Type u\nf : α ⟶ β\nF' : MonoFactorisation f\nx : Image f\n⊢ (F'.e ≫ F'.m) ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) (_ : ↑x ∈ Set.range f)) = ι f x",
"state_before": "case h\nJ : Type v\ninst✝ : SmallCategory J\nα β : Type u\nf : α ⟶ β\nF' : MonoFactorisation f\nx : Image f\n⊢ (lift F' ≫ F'.m) x = ι f x",
"tactic": "change (F'.e ≫ F'.m) _ = _"
},
{
"state_after": "case h\nJ : Type v\ninst✝ : SmallCategory J\nα β : Type u\nf : α ⟶ β\nF' : MonoFactorisation f\nx : Image f\n⊢ ↑x = ι f x",
"state_before": "case h\nJ : Type v\ninst✝ : SmallCategory J\nα β : Type u\nf : α ⟶ β\nF' : MonoFactorisation f\nx : Image f\n⊢ (F'.e ≫ F'.m) ↑(Classical.indefiniteDescription (fun x_1 => f x_1 = ↑x) (_ : ↑x ∈ Set.range f)) = ι f x",
"tactic": "rw [F'.fac, (Classical.indefiniteDescription _ x.2).2]"
},
{
"state_after": "no goals",
"state_before": "case h\nJ : Type v\ninst✝ : SmallCategory J\nα β : Type u\nf : α ⟶ β\nF' : MonoFactorisation f\nx : Image f\n⊢ ↑x = ι f x",
"tactic": "rfl"
}
] | [
529,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
525,
1
] |
Std/Data/Int/Lemmas.lean | Int.add_one_le_iff | [] | [
846,
63
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
846,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.coe_toReal | [
{
"state_after": "case intro\nx : ℝ\nhx : ↑x ≠ ⊤\nh'x : ↑x ≠ ⊥\n⊢ ↑(toReal ↑x) = ↑x",
"state_before": "x : EReal\nhx : x ≠ ⊤\nh'x : x ≠ ⊥\n⊢ ↑(toReal x) = x",
"tactic": "lift x to ℝ using ⟨hx, h'x⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nx : ℝ\nhx : ↑x ≠ ⊤\nh'x : ↑x ≠ ⊥\n⊢ ↑(toReal ↑x) = ↑x",
"tactic": "rfl"
}
] | [
415,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
413,
1
] |
Mathlib/Data/Polynomial/Laurent.lean | ext | [] | [
97,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | Complex.exp_add_pi_mul_I | [] | [
1393,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1392,
1
] |
Mathlib/Algebra/Order/UpperLower.lean | LowerSet.one_mul | [
{
"state_after": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns✝ t : Set α\na : α\ns : LowerSet α\n⊢ (⋃ (a : α) (_ : a ∈ Set.Iic 1), a • ↑s) ⊆ ↑s",
"state_before": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns✝ t : Set α\na : α\ns : LowerSet α\n⊢ Set.Iic 1 * ↑s ⊆ ↑s",
"tactic": "rw [← smul_eq_mul, ← Set.iUnion_smul_set]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns✝ t : Set α\na : α\ns : LowerSet α\n⊢ (⋃ (a : α) (_ : a ∈ Set.Iic 1), a • ↑s) ⊆ ↑s",
"tactic": "exact Set.iUnion₂_subset fun _ ↦ s.lower.smul_subset"
}
] | [
248,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
244,
9
] |
Mathlib/Data/Nat/Interval.lean | Nat.Iio_eq_range | [
{
"state_after": "case h.a\na b✝ c b x : ℕ\n⊢ x ∈ Iio b ↔ x ∈ range b",
"state_before": "a b c : ℕ\n⊢ Iio = range",
"tactic": "ext (b x)"
},
{
"state_after": "no goals",
"state_before": "case h.a\na b✝ c b x : ℕ\n⊢ x ∈ Iio b ↔ x ∈ range b",
"tactic": "rw [mem_Iio, mem_range]"
}
] | [
85,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
83,
1
] |
Mathlib/Order/Disjoint.lean | Codisjoint.top_le | [] | [
317,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
316,
1
] |
Mathlib/Data/Bool/Count.lean | List.Chain'.two_mul_count_bool_eq_ite | [
{
"state_after": "case pos\nl : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) l\nb : Bool\nh2 : Even (length l)\n⊢ 2 * count b l =\n if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1\n\ncase neg\nl : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) l\nb : Bool\nh2 : ¬Even (length l)\n⊢ 2 * count b l =\n if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1",
"state_before": "l : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) l\nb : Bool\n⊢ 2 * count b l =\n if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1",
"tactic": "by_cases h2 : Even (length l)"
},
{
"state_after": "no goals",
"state_before": "case pos\nl : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) l\nb : Bool\nh2 : Even (length l)\n⊢ 2 * count b l =\n if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1",
"tactic": "rw [if_pos h2, hl.two_mul_count_bool_of_even h2]"
},
{
"state_after": "case neg.nil\nb : Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) []\nh2 : ¬Even (length [])\n⊢ 2 * count b [] =\n if Even (length []) then length [] else if (some b == head? []) = true then length [] + 1 else length [] - 1\n\ncase neg.cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) (x :: l)\nh2 : ¬Even (length (x :: l))\n⊢ 2 * count b (x :: l) =\n if Even (length (x :: l)) then length (x :: l)\n else if (some b == head? (x :: l)) = true then length (x :: l) + 1 else length (x :: l) - 1",
"state_before": "case neg\nl : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) l\nb : Bool\nh2 : ¬Even (length l)\n⊢ 2 * count b l =\n if Even (length l) then length l else if (some b == head? l) = true then length l + 1 else length l - 1",
"tactic": "cases' l with x l"
},
{
"state_after": "case neg.cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) (x :: l)\nh2 : ¬Even (length (x :: l))\n⊢ (2 * count b l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1",
"state_before": "case neg.cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) (x :: l)\nh2 : ¬Even (length (x :: l))\n⊢ 2 * count b (x :: l) =\n if Even (length (x :: l)) then length (x :: l)\n else if (some b == head? (x :: l)) = true then length (x :: l) + 1 else length (x :: l) - 1",
"tactic": "simp only [if_neg h2, count_cons', mul_add, head?, Option.mem_some_iff, @eq_comm _ x]"
},
{
"state_after": "case neg.cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) (x :: l)\nh2 : Even (length l)\n⊢ (2 * count b l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1",
"state_before": "case neg.cons\nb x : Bool\nl : List Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) (x :: l)\nh2 : ¬Even (length (x :: l))\n⊢ (2 * count b l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1",
"tactic": "rw [length_cons, Nat.even_add_one, not_not] at h2"
},
{
"state_after": "case neg.cons\nb x : Bool\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x ≠ x_1) l\n⊢ (length l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1",
"state_before": "case neg.cons\nb x : Bool\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x ≠ x_1) l\n⊢ (2 * count b l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1",
"tactic": "rw [hl.two_mul_count_bool_of_even h2]"
},
{
"state_after": "no goals",
"state_before": "case neg.cons\nb x : Bool\nl : List Bool\nh2 : Even (length l)\nhl : Chain' (fun x x_1 => x ≠ x_1) l\n⊢ (length l + 2 * if b = x then 1 else 0) =\n if (some b == some x) = true then length (x :: l) + 1 else length (x :: l) - 1",
"tactic": "cases b <;> cases x <;> split_ifs <;> simp <;> contradiction"
},
{
"state_after": "no goals",
"state_before": "case neg.nil\nb : Bool\nhl : Chain' (fun x x_1 => x ≠ x_1) []\nh2 : ¬Even (length [])\n⊢ 2 * count b [] =\n if Even (length []) then length [] else if (some b == head? []) = true then length [] + 1 else length [] - 1",
"tactic": "exact (h2 even_zero).elim"
}
] | [
120,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.conj_im | [] | [
345,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
344,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean | inv_mul_mul_self | [
{
"state_after": "case pos\nα : Type ?u.25133\nM₀ : Type ?u.25136\nG₀ : Type u_1\nM₀' : Type ?u.25142\nG₀' : Type ?u.25145\nF : Type ?u.25148\nF' : Type ?u.25151\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : a = 0\n⊢ a⁻¹ * a * a = a\n\ncase neg\nα : Type ?u.25133\nM₀ : Type ?u.25136\nG₀ : Type u_1\nM₀' : Type ?u.25142\nG₀' : Type ?u.25145\nF : Type ?u.25148\nF' : Type ?u.25151\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : ¬a = 0\n⊢ a⁻¹ * a * a = a",
"state_before": "α : Type ?u.25133\nM₀ : Type ?u.25136\nG₀ : Type u_1\nM₀' : Type ?u.25142\nG₀' : Type ?u.25145\nF : Type ?u.25148\nF' : Type ?u.25151\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\n⊢ a⁻¹ * a * a = a",
"tactic": "by_cases h : a = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.25133\nM₀ : Type ?u.25136\nG₀ : Type u_1\nM₀' : Type ?u.25142\nG₀' : Type ?u.25145\nF : Type ?u.25148\nF' : Type ?u.25151\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : a = 0\n⊢ a⁻¹ * a * a = a",
"tactic": "rw [h, inv_zero, mul_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.25133\nM₀ : Type ?u.25136\nG₀ : Type u_1\nM₀' : Type ?u.25142\nG₀' : Type ?u.25145\nF : Type ?u.25148\nF' : Type ?u.25151\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\nh : ¬a = 0\n⊢ a⁻¹ * a * a = a",
"tactic": "rw [inv_mul_cancel h, one_mul]"
}
] | [
368,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
365,
1
] |
Mathlib/Algebra/Order/Module.lean | lt_of_smul_lt_smul_of_nonpos | [
{
"state_after": "k : Type u_2\nM : Type u_1\nN : Type ?u.22212\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nh : -c • b < -c • a\nhc : c ≤ 0\n⊢ b < a",
"state_before": "k : Type u_2\nM : Type u_1\nN : Type ?u.22212\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nh : c • a < c • b\nhc : c ≤ 0\n⊢ b < a",
"tactic": "rw [← neg_neg c, neg_smul, neg_smul (-c), neg_lt_neg_iff] at h"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nM : Type u_1\nN : Type ?u.22212\ninst✝³ : OrderedRing k\ninst✝² : OrderedAddCommGroup M\ninst✝¹ : Module k M\ninst✝ : OrderedSMul k M\na b : M\nc : k\nh : -c • b < -c • a\nhc : c ≤ 0\n⊢ b < a",
"tactic": "exact lt_of_smul_lt_smul_of_nonneg h (neg_nonneg_of_nonpos hc)"
}
] | [
73,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | ContDiffOn.derivWithin | [
{
"state_after": "case none\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : none + 1 ≤ n\n⊢ ContDiffOn 𝕜 none (derivWithin f₂ s₂) s₂\n\ncase some\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nval✝ : ℕ\nhmn : some val✝ + 1 ≤ n\n⊢ ContDiffOn 𝕜 (some val✝) (derivWithin f₂ s₂) s₂",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : m + 1 ≤ n\n⊢ ContDiffOn 𝕜 m (derivWithin f₂ s₂) s₂",
"tactic": "cases m"
},
{
"state_after": "case none\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : ⊤ + 1 ≤ n\n⊢ ContDiffOn 𝕜 none (derivWithin f₂ s₂) s₂",
"state_before": "case none\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : none + 1 ≤ n\n⊢ ContDiffOn 𝕜 none (derivWithin f₂ s₂) s₂",
"tactic": "change ∞ + 1 ≤ n at hmn"
},
{
"state_after": "case none\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : ⊤ + 1 ≤ n\nthis : n = ⊤\n⊢ ContDiffOn 𝕜 none (derivWithin f₂ s₂) s₂",
"state_before": "case none\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : ⊤ + 1 ≤ n\n⊢ ContDiffOn 𝕜 none (derivWithin f₂ s₂) s₂",
"tactic": "have : n = ∞ := by simpa using hmn"
},
{
"state_after": "case none\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 ⊤ f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : ⊤ + 1 ≤ n\nthis : n = ⊤\n⊢ ContDiffOn 𝕜 none (derivWithin f₂ s₂) s₂",
"state_before": "case none\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : ⊤ + 1 ≤ n\nthis : n = ⊤\n⊢ ContDiffOn 𝕜 none (derivWithin f₂ s₂) s₂",
"tactic": "rw [this] at hf"
},
{
"state_after": "no goals",
"state_before": "case none\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 ⊤ f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : ⊤ + 1 ≤ n\nthis : n = ⊤\n⊢ ContDiffOn 𝕜 none (derivWithin f₂ s₂) s₂",
"tactic": "exact ((contDiffOn_top_iff_derivWithin hs).1 hf).2"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nhmn : ⊤ + 1 ≤ n\n⊢ n = ⊤",
"tactic": "simpa using hmn"
},
{
"state_after": "case some\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nval✝ : ℕ\nhmn : ↑(Nat.succ val✝) ≤ n\n⊢ ContDiffOn 𝕜 (some val✝) (derivWithin f₂ s₂) s₂",
"state_before": "case some\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nval✝ : ℕ\nhmn : some val✝ + 1 ≤ n\n⊢ ContDiffOn 𝕜 (some val✝) (derivWithin f₂ s₂) s₂",
"tactic": "change (Nat.succ _ : ℕ∞) ≤ n at hmn"
},
{
"state_after": "no goals",
"state_before": "case some\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3032059\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nn : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nhf : ContDiffOn 𝕜 n f₂ s₂\nhs : UniqueDiffOn 𝕜 s₂\nval✝ : ℕ\nhmn : ↑(Nat.succ val✝) ≤ n\n⊢ ContDiffOn 𝕜 (some val✝) (derivWithin f₂ s₂) s₂",
"tactic": "exact ((contDiffOn_succ_iff_derivWithin hs).1 (hf.of_le hmn)).2"
}
] | [
2129,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2121,
11
] |
Mathlib/Algebra/Hom/GroupAction.lean | DistribMulActionHom.id_apply | [
{
"state_after": "no goals",
"state_before": "M' : Type ?u.147246\nX : Type ?u.147249\ninst✝²³ : SMul M' X\nY : Type ?u.147256\ninst✝²² : SMul M' Y\nZ : Type ?u.147263\ninst✝²¹ : SMul M' Z\nM : Type u_2\ninst✝²⁰ : Monoid M\nA : Type u_1\ninst✝¹⁹ : AddMonoid A\ninst✝¹⁸ : DistribMulAction M A\nA' : Type ?u.147304\ninst✝¹⁷ : AddGroup A'\ninst✝¹⁶ : DistribMulAction M A'\nB : Type ?u.147580\ninst✝¹⁵ : AddMonoid B\ninst✝¹⁴ : DistribMulAction M B\nB' : Type ?u.147606\ninst✝¹³ : AddGroup B'\ninst✝¹² : DistribMulAction M B'\nC : Type ?u.147882\ninst✝¹¹ : AddMonoid C\ninst✝¹⁰ : DistribMulAction M C\nR : Type ?u.147908\ninst✝⁹ : Semiring R\ninst✝⁸ : MulSemiringAction M R\nR' : Type ?u.147935\ninst✝⁷ : Ring R'\ninst✝⁶ : MulSemiringAction M R'\nS : Type ?u.148131\ninst✝⁵ : Semiring S\ninst✝⁴ : MulSemiringAction M S\nS' : Type ?u.148158\ninst✝³ : Ring S'\ninst✝² : MulSemiringAction M S'\nT : Type ?u.148354\ninst✝¹ : Semiring T\ninst✝ : MulSemiringAction M T\nx : A\n⊢ ↑(DistribMulActionHom.id M) x = x",
"tactic": "rfl"
}
] | [
332,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
331,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean | exists_Ioo_extr_on_Icc | [
{
"state_after": "f f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"state_before": "f f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "have ne : (Icc a b).Nonempty := nonempty_Icc.2 (le_of_lt hab)"
},
{
"state_after": "case intro.intro\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"state_before": "f f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "obtain ⟨c, cmem, cle⟩ : ∃ c ∈ Icc a b, ∀ x ∈ Icc a b, f c ≤ f x :=\n isCompact_Icc.exists_forall_le ne hfc"
},
{
"state_after": "case intro.intro.intro.intro\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"state_before": "case intro.intro\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "obtain ⟨C, Cmem, Cge⟩ : ∃ C ∈ Icc a b, ∀ x ∈ Icc a b, f x ≤ f C :=\n isCompact_Icc.exists_forall_ge ne hfc"
},
{
"state_after": "case pos\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c\n\ncase neg\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : ¬f c = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"state_before": "case intro.intro.intro.intro\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "by_cases hc : f c = f a"
},
{
"state_after": "case pos\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : f C = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c\n\ncase neg\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : ¬f C = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"state_before": "case pos\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "by_cases hC : f C = f a"
},
{
"state_after": "case pos\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : f C = f a\nthis : ∀ (x : ℝ), x ∈ Icc a b → f x = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"state_before": "case pos\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : f C = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "have : ∀ x ∈ Icc a b, f x = f a := fun x hx => le_antisymm (hC ▸ Cge x hx) (hc ▸ cle x hx)"
},
{
"state_after": "case pos.intro\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : f C = f a\nthis : ∀ (x : ℝ), x ∈ Icc a b → f x = f a\nc' : ℝ\nhc' : c' ∈ Ioo a b\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"state_before": "case pos\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : f C = f a\nthis : ∀ (x : ℝ), x ∈ Icc a b → f x = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "rcases nonempty_Ioo.2 hab with ⟨c', hc'⟩"
},
{
"state_after": "case pos.intro\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : f C = f a\nthis : ∀ (x : ℝ), x ∈ Icc a b → f x = f a\nc' : ℝ\nhc' : c' ∈ Ioo a b\nx : ℝ\nhx : x ∈ Icc a b\n⊢ x ∈ {x | (fun x => f c' ≤ f x) x}",
"state_before": "case pos.intro\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : f C = f a\nthis : ∀ (x : ℝ), x ∈ Icc a b → f x = f a\nc' : ℝ\nhc' : c' ∈ Ioo a b\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "refine ⟨c', hc', Or.inl fun x hx ↦ ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : f C = f a\nthis : ∀ (x : ℝ), x ∈ Icc a b → f x = f a\nc' : ℝ\nhc' : c' ∈ Ioo a b\nx : ℝ\nhx : x ∈ Icc a b\n⊢ x ∈ {x | (fun x => f c' ≤ f x) x}",
"tactic": "simp only [mem_setOf_eq, this x hx, this c' (Ioo_subset_Icc_self hc'), le_rfl]"
},
{
"state_after": "case neg.refine'_1\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : ¬f C = f a\n⊢ a = C → f C = f a\n\ncase neg.refine'_2\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : ¬f C = f a\n⊢ C = b → f C = f a",
"state_before": "case neg\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : ¬f C = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "refine' ⟨C, ⟨lt_of_le_of_ne Cmem.1 <| mt _ hC, lt_of_le_of_ne Cmem.2 <| mt _ hC⟩, Or.inr Cge⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : ¬f C = f a\n⊢ a = C → f C = f a\n\ncase neg.refine'_2\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : ¬f C = f a\n⊢ C = b → f C = f a",
"tactic": "exacts [fun h => by rw [h], fun h => by rw [h, hfI]]"
},
{
"state_after": "no goals",
"state_before": "f f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : ¬f C = f a\nh : a = C\n⊢ f C = f a",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "f f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : f c = f a\nhC : ¬f C = f a\nh : C = b\n⊢ f C = f a",
"tactic": "rw [h, hfI]"
},
{
"state_after": "case neg.refine'_1\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : ¬f c = f a\n⊢ a = c → f c = f a\n\ncase neg.refine'_2\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : ¬f c = f a\n⊢ c = b → f c = f a",
"state_before": "case neg\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : ¬f c = f a\n⊢ ∃ c, c ∈ Ioo a b ∧ IsExtrOn f (Icc a b) c",
"tactic": "refine' ⟨c, ⟨lt_of_le_of_ne cmem.1 <| mt _ hc, lt_of_le_of_ne cmem.2 <| mt _ hc⟩, Or.inl cle⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.refine'_1\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : ¬f c = f a\n⊢ a = c → f c = f a\n\ncase neg.refine'_2\nf f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : ¬f c = f a\n⊢ c = b → f c = f a",
"tactic": "exacts [fun h => by rw [h], fun h => by rw [h, hfI]]"
},
{
"state_after": "no goals",
"state_before": "f f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : ¬f c = f a\nh : a = c\n⊢ f c = f a",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "f f' : ℝ → ℝ\na b : ℝ\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI : f a = f b\nne : Set.Nonempty (Icc a b)\nc : ℝ\ncmem : c ∈ Icc a b\ncle : ∀ (x : ℝ), x ∈ Icc a b → f c ≤ f x\nC : ℝ\nCmem : C ∈ Icc a b\nCge : ∀ (x : ℝ), x ∈ Icc a b → f x ≤ f C\nhc : ¬f c = f a\nh : c = b\n⊢ f c = f a",
"tactic": "rw [h, hfI]"
}
] | [
287,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
269,
1
] |
Mathlib/Algebra/Hom/Group.lean | MonoidHom.div_apply | [] | [
1688,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1687,
1
] |
Mathlib/Algebra/Module/Submodule/Pointwise.lean | Submodule.pointwise_smul_toAddSubmonoid | [] | [
220,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
1
] |
Mathlib/Topology/Sets/Closeds.lean | TopologicalSpace.Closeds.coe_finset_sup | [] | [
149,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
147,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.nsmul_singleton | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.91193\nγ : Type ?u.91196\na : α\nn : ℕ\n⊢ n • {a} = replicate n a",
"tactic": "rw [← replicate_one, nsmul_replicate, mul_one]"
}
] | [
965,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
964,
1
] |
Mathlib/LinearAlgebra/Finrank.lean | finrank_span_finset_le_card | [
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Finset V\n⊢ Finset.card (Set.toFinset ↑s) = Finset.card s",
"tactic": "simp"
}
] | [
321,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
318,
1
] |
Std/Data/Array/Init/Lemmas.lean | Array.pop_data | [] | [
188,
83
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
188,
9
] |
Mathlib/Algebra/Order/Group/Defs.lean | mul_inv_le_iff_le_mul' | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : CommGroup α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c d : α\n⊢ a * b⁻¹ ≤ c ↔ a ≤ b * c",
"tactic": "rw [← inv_mul_le_iff_le_mul, mul_comm]"
}
] | [
536,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
535,
1
] |
Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | CategoryTheory.MonoidalCategory.id_tensor_rightUnitor_inv | [
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nX Y : C\n⊢ 𝟙 X ⊗ (ρ_ Y).inv = (ρ_ (X ⊗ Y)).inv ≫ (α_ X Y tensorUnit').hom",
"tactic": "coherence"
}
] | [
50,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
49,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean | smul_sphere' | [
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\ny : E\n⊢ y ∈ c • sphere x r ↔ y ∈ sphere (c • x) (‖c‖ * r)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\n⊢ c • sphere x r = sphere (c • x) (‖c‖ * r)",
"tactic": "ext y"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\ny : E\n⊢ c⁻¹ • y ∈ sphere x r ↔ y ∈ sphere (c • x) (‖c‖ * r)",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\ny : E\n⊢ y ∈ c • sphere x r ↔ y ∈ sphere (c • x) (‖c‖ * r)",
"tactic": "rw [mem_smul_set_iff_inv_smul_mem₀ hc]"
},
{
"state_after": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\ny : E\n⊢ c⁻¹ • y ∈ sphere (c⁻¹ • c • x) r ↔ y ∈ sphere (c • x) (‖c‖ * r)",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\ny : E\n⊢ c⁻¹ • y ∈ sphere x r ↔ y ∈ sphere (c • x) (‖c‖ * r)",
"tactic": "conv_lhs => rw [← inv_smul_smul₀ hc x]"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\ny : E\n⊢ c⁻¹ • y ∈ sphere (c⁻¹ • c • x) r ↔ y ∈ sphere (c • x) (‖c‖ * r)",
"tactic": "simp only [mem_sphere, dist_smul₀, norm_inv, ← div_eq_inv_mul, div_eq_iff (norm_pos_iff.2 hc).ne',\n mul_comm r]"
}
] | [
107,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Mathlib/Data/Nat/Factorial/DoubleFactorial.lean | Nat.doubleFactorial_two_mul | [
{
"state_after": "n : ℕ\n⊢ (2 * n + 2) * (2 ^ n * n !) = 2 ^ n * 2 * ((n + 1) * n !)",
"state_before": "n : ℕ\n⊢ (2 * (n + 1))‼ = 2 ^ (n + 1) * (n + 1)!",
"tactic": "rw [mul_add, mul_one, doubleFactorial_add_two, factorial, pow_succ, doubleFactorial_two_mul _,\n succ_eq_add_one]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ (2 * n + 2) * (2 ^ n * n !) = 2 ^ n * 2 * ((n + 1) * n !)",
"tactic": "ring"
}
] | [
61,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
56,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Real.lean | dist_le_tsum_dist_of_tendsto₀ | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nh : Summable fun n => dist (f n) (f (Nat.succ n))\nha : Tendsto f atTop (𝓝 a)\n⊢ dist (f 0) a ≤ ∑' (n : ℕ), dist (f n) (f (Nat.succ n))",
"tactic": "simpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0"
}
] | [
98,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.inter_smul_subset | [] | [
1395,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1394,
1
] |
Mathlib/Combinatorics/Additive/Behrend.lean | Behrend.lower_bound_le_one | [
{
"state_after": "case inl\nα : Type ?u.881020\nβ : Type ?u.881023\nn d k : ℕ\nx : Fin n → ℕ\nhN : 1 ≤ 1\nhN' : 1 ≤ 4096\n⊢ ↑1 * exp (-4 * Real.sqrt (log ↑1)) ≤ 1\n\ncase inr\nα : Type ?u.881020\nβ : Type ?u.881023\nn d k N : ℕ\nx : Fin n → ℕ\nhN✝ : 1 ≤ N\nhN' : N ≤ 4096\nhN : 1 < N\n⊢ ↑N * exp (-4 * Real.sqrt (log ↑N)) ≤ 1",
"state_before": "α : Type ?u.881020\nβ : Type ?u.881023\nn d k N : ℕ\nx : Fin n → ℕ\nhN : 1 ≤ N\nhN' : N ≤ 4096\n⊢ ↑N * exp (-4 * Real.sqrt (log ↑N)) ≤ 1",
"tactic": "obtain rfl | hN := hN.eq_or_lt"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.881020\nβ : Type ?u.881023\nn d k : ℕ\nx : Fin n → ℕ\nhN : 1 ≤ 1\nhN' : 1 ≤ 4096\n⊢ ↑1 * exp (-4 * Real.sqrt (log ↑1)) ≤ 1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type ?u.881020\nβ : Type ?u.881023\nn d k N : ℕ\nx : Fin n → ℕ\nhN✝ : 1 ≤ N\nhN' : N ≤ 4096\nhN : 1 < N\n⊢ ↑N * exp (-4 * Real.sqrt (log ↑N)) ≤ 1",
"tactic": "exact lower_bound_le_one' hN hN'"
}
] | [
549,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
545,
1
] |
Mathlib/Algebra/IndicatorFunction.lean | Set.mulIndicator_self_mul_compl | [] | [
440,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
438,
1
] |
Mathlib/Data/List/Basic.lean | List.enum_get? | [
{
"state_after": "ι : Type ?u.431551\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\n⊢ List α → ℕ → True",
"state_before": "ι : Type ?u.431551\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\n⊢ ∀ (l : List α) (n : ℕ), get? (enum l) n = (fun a => (n, a)) <$> get? l n",
"tactic": "simp only [enum, enumFrom_get?, zero_add]"
},
{
"state_after": "ι : Type ?u.431551\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l✝ : List α\nn✝ : ℕ\n⊢ True",
"state_before": "ι : Type ?u.431551\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\n⊢ List α → ℕ → True",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.431551\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l✝ : List α\nn✝ : ℕ\n⊢ True",
"tactic": "trivial"
}
] | [
3832,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3831,
1
] |
src/lean/Init/Control/Lawful.lean | StateT.seqRight_eq | [
{
"state_after": "case h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\n⊢ ∀ (s : σ), run (SeqRight.seqRight x fun x => y) s = run (Seq.seq (const α id <$> x) fun x => y) s",
"state_before": "m : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\n⊢ (SeqRight.seqRight x fun x => y) = Seq.seq (const α id <$> x) fun x => y",
"tactic": "apply ext"
},
{
"state_after": "case h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\n⊢ run (SeqRight.seqRight x fun x => y) s = run (Seq.seq (const α id <$> x) fun x => y) s",
"state_before": "case h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\n⊢ ∀ (s : σ), run (SeqRight.seqRight x fun x => y) s = run (Seq.seq (const α id <$> x) fun x => y) s",
"tactic": "intro s"
},
{
"state_after": "case h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\n⊢ (do\n let p ← run x s\n run y p.snd) =\n do\n let x ← run x s\n let a ← run y x.snd\n pure (a.fst, a.snd)",
"state_before": "case h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\n⊢ run (SeqRight.seqRight x fun x => y) s = run (Seq.seq (const α id <$> x) fun x => y) s",
"tactic": "simp [map_eq_pure_bind, const]"
},
{
"state_after": "case h.h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\n⊢ ∀ (a : α × σ),\n run y a.snd = do\n let a ← run y a.snd\n pure (a.fst, a.snd)",
"state_before": "case h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\n⊢ (do\n let p ← run x s\n run y p.snd) =\n do\n let x ← run x s\n let a ← run y x.snd\n pure (a.fst, a.snd)",
"tactic": "apply bind_congr"
},
{
"state_after": "case h.h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\np : α × σ\n⊢ run y p.snd = do\n let a ← run y p.snd\n pure (a.fst, a.snd)",
"state_before": "case h.h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\n⊢ ∀ (a : α × σ),\n run y a.snd = do\n let a ← run y a.snd\n pure (a.fst, a.snd)",
"tactic": "intro p"
},
{
"state_after": "case h.h.mk\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\nfst✝ : α\nsnd✝ : σ\n⊢ run y (fst✝, snd✝).snd = do\n let a ← run y (fst✝, snd✝).snd\n pure (a.fst, a.snd)",
"state_before": "case h.h\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\np : α × σ\n⊢ run y p.snd = do\n let a ← run y p.snd\n pure (a.fst, a.snd)",
"tactic": "cases p"
},
{
"state_after": "no goals",
"state_before": "case h.h.mk\nm : Type u_1 → Type u_2\nσ α β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nx : StateT σ m α\ny : StateT σ m β\ns : σ\nfst✝ : α\nsnd✝ : σ\n⊢ run y (fst✝, snd✝).snd = do\n let a ← run y (fst✝, snd✝).snd\n pure (a.fst, a.snd)",
"tactic": "simp [Prod.eta]"
}
] | [
292,
18
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
288,
1
] |
Mathlib/Data/List/Basic.lean | List.map_erase | [
{
"state_after": "ι : Type ?u.425891\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nfinj : Injective f\na : α\nl : List α\nthis : Eq a = Eq (f a) ∘ f\n⊢ map f (List.erase l a) = List.erase (map f l) (f a)",
"state_before": "ι : Type ?u.425891\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nfinj : Injective f\na : α\nl : List α\n⊢ map f (List.erase l a) = List.erase (map f l) (f a)",
"tactic": "have this : Eq a = Eq (f a) ∘ f := by ext b; simp [finj.eq_iff]"
},
{
"state_after": "ι : Type ?u.425891\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nfinj : Injective f\na : α\nl : List α\nthis : Eq a = Eq (f a) ∘ f\n⊢ map f (eraseP (fun b => decide (f a = f b)) l) = map f (eraseP ((fun b => decide (f a = b)) ∘ f) l)",
"state_before": "ι : Type ?u.425891\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nfinj : Injective f\na : α\nl : List α\nthis : Eq a = Eq (f a) ∘ f\n⊢ map f (List.erase l a) = List.erase (map f l) (f a)",
"tactic": "simp [erase_eq_eraseP, erase_eq_eraseP, eraseP_map, this]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.425891\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nfinj : Injective f\na : α\nl : List α\nthis : Eq a = Eq (f a) ∘ f\n⊢ map f (eraseP (fun b => decide (f a = f b)) l) = map f (eraseP ((fun b => decide (f a = b)) ∘ f) l)",
"tactic": "rfl"
},
{
"state_after": "case h.a\nι : Type ?u.425891\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nfinj : Injective f\na : α\nl : List α\nb : α\n⊢ a = b ↔ (Eq (f a) ∘ f) b",
"state_before": "ι : Type ?u.425891\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nfinj : Injective f\na : α\nl : List α\n⊢ Eq a = Eq (f a) ∘ f",
"tactic": "ext b"
},
{
"state_after": "no goals",
"state_before": "case h.a\nι : Type ?u.425891\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → β\nfinj : Injective f\na : α\nl : List α\nb : α\n⊢ a = b ↔ (Eq (f a) ∘ f) b",
"tactic": "simp [finj.eq_iff]"
}
] | [
3746,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3743,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | edist_lt_top | [] | [
343,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
342,
1
] |
Mathlib/Algebra/Parity.lean | Even.add_odd | [
{
"state_after": "case intro.intro\nF : Type ?u.81616\nα : Type u_1\nβ : Type ?u.81622\nR : Type ?u.81625\ninst✝¹ : Semiring α\ninst✝ : Semiring β\nm n : α\n⊢ Odd (m + m + (2 * n + 1))",
"state_before": "F : Type ?u.81616\nα : Type u_1\nβ : Type ?u.81622\nR : Type ?u.81625\ninst✝¹ : Semiring α\ninst✝ : Semiring β\nm n : α\n⊢ Even m → Odd n → Odd (m + n)",
"tactic": "rintro ⟨m, rfl⟩ ⟨n, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nF : Type ?u.81616\nα : Type u_1\nβ : Type ?u.81622\nR : Type ?u.81625\ninst✝¹ : Semiring α\ninst✝ : Semiring β\nm n : α\n⊢ Odd (m + m + (2 * n + 1))",
"tactic": "exact ⟨m + n, by rw [mul_add, ← two_mul, add_assoc]⟩"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.81616\nα : Type u_1\nβ : Type ?u.81622\nR : Type ?u.81625\ninst✝¹ : Semiring α\ninst✝ : Semiring β\nm n : α\n⊢ m + m + (2 * n + 1) = 2 * (m + n) + 1",
"tactic": "rw [mul_add, ← two_mul, add_assoc]"
}
] | [
332,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
330,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | CategoryTheory.Limits.terminal.hom_ext | [] | [
251,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
] |
Mathlib/Order/Heyting/Hom.lean | map_hnot | [
{
"state_after": "no goals",
"state_before": "F : Type u_3\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.37721\nδ : Type ?u.37724\ninst✝² : CoheytingAlgebra α\ninst✝¹ : CoheytingAlgebra β\ninst✝ : CoheytingHomClass F α β\nf : F\na : α\n⊢ ↑f (￢a) = ￢↑f a",
"tactic": "rw [← top_sdiff', ← top_sdiff', map_sdiff, map_top]"
}
] | [
208,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
] |
Mathlib/Topology/MetricSpace/Isometry.lean | Isometry.closedEmbedding | [] | [
214,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
212,
1
] |
Mathlib/Algebra/Free.lean | FreeSemigroup.lift_comp_of | [] | [
546,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
546,
1
] |
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.Formula.realize_equivSentence_symm_con | [
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\n⊢ BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ)\n (fun x => ↑(Language.con L (↑(Equiv.sumEmpty α Empty) x))) default ↔\n M ⊨ φ",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\n⊢ (Realize (↑equivSentence.symm φ) fun a => ↑(Language.con L a)) ↔ M ⊨ φ",
"tactic": "simp only [equivSentence, Equiv.symm_symm, Equiv.coe_trans, Realize,\n BoundedFormula.realize_relabelEquiv, Function.comp]"
},
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\n⊢ BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ)\n (fun x => ↑(Language.con L (↑(Equiv.sumEmpty α Empty) x))) default ↔\n BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ) (Sum.elim (fun a => ↑(Language.con L a)) default)\n default",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\n⊢ BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ)\n (fun x => ↑(Language.con L (↑(Equiv.sumEmpty α Empty) x))) default ↔\n M ⊨ φ",
"tactic": "refine' _root_.trans _ BoundedFormula.realize_constantsVarsEquiv"
},
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\n⊢ BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ)\n (fun x => ↑(Language.con L (↑(Equiv.sumEmpty α Empty) x))) default =\n BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ) (Sum.elim (fun a => ↑(Language.con L a)) default)\n default",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\n⊢ BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ)\n (fun x => ↑(Language.con L (↑(Equiv.sumEmpty α Empty) x))) default ↔\n BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ) (Sum.elim (fun a => ↑(Language.con L a)) default)\n default",
"tactic": "rw [iff_iff_eq]"
},
{
"state_after": "case e__v.h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\nval✝ : α\n⊢ ↑(Language.con L (↑(Equiv.sumEmpty α Empty) (Sum.inl val✝))) =\n Sum.elim (fun a => ↑(Language.con L a)) default (Sum.inl val✝)\n\ncase e__v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\na : Empty\n⊢ ↑(Language.con L (↑(Equiv.sumEmpty α Empty) (Sum.inr a))) =\n Sum.elim (fun a => ↑(Language.con L a)) default (Sum.inr a)",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\n⊢ BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ)\n (fun x => ↑(Language.con L (↑(Equiv.sumEmpty α Empty) x))) default =\n BoundedFormula.Realize (↑BoundedFormula.constantsVarsEquiv φ) (Sum.elim (fun a => ↑(Language.con L a)) default)\n default",
"tactic": "congr with (_ | a)"
},
{
"state_after": "case e__v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\na : Empty\n⊢ ↑(Language.con L (↑(Equiv.sumEmpty α Empty) (Sum.inr a))) =\n Sum.elim (fun a => ↑(Language.con L a)) default (Sum.inr a)",
"state_before": "case e__v.h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\nval✝ : α\n⊢ ↑(Language.con L (↑(Equiv.sumEmpty α Empty) (Sum.inl val✝))) =\n Sum.elim (fun a => ↑(Language.con L a)) default (Sum.inl val✝)\n\ncase e__v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\na : Empty\n⊢ ↑(Language.con L (↑(Equiv.sumEmpty α Empty) (Sum.inr a))) =\n Sum.elim (fun a => ↑(Language.con L a)) default (Sum.inr a)",
"tactic": ". simp"
},
{
"state_after": "no goals",
"state_before": "case e__v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\na : Empty\n⊢ ↑(Language.con L (↑(Equiv.sumEmpty α Empty) (Sum.inr a))) =\n Sum.elim (fun a => ↑(Language.con L a)) default (Sum.inr a)",
"tactic": ". cases a"
},
{
"state_after": "no goals",
"state_before": "case e__v.h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\nval✝ : α\n⊢ ↑(Language.con L (↑(Equiv.sumEmpty α Empty) (Sum.inl val✝))) =\n Sum.elim (fun a => ↑(Language.con L a)) default (Sum.inl val✝)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case e__v.h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.257040\nP : Type ?u.257043\ninst✝⁴ : Structure L M\ninst✝³ : Structure L N\ninst✝² : Structure L P\nα : Type u'\nβ : Type v'\nn : ℕ\ninst✝¹ : Structure (L[[α]]) M\ninst✝ : LHom.IsExpansionOn (lhomWithConstants L α) M\nφ : Sentence (L[[α]])\na : Empty\n⊢ ↑(Language.con L (↑(Equiv.sumEmpty α Empty) (Sum.inr a))) =\n Sum.elim (fun a => ↑(Language.con L a)) default (Sum.inr a)",
"tactic": "cases a"
}
] | [
745,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
736,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.IsCycle.extendDomain | [
{
"state_after": "case intro.intro\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\n⊢ IsCycle (extendDomain g f)",
"state_before": "ι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\n⊢ IsCycle g → IsCycle (extendDomain g f)",
"tactic": "rintro ⟨a, ha, ha'⟩"
},
{
"state_after": "case intro.intro.refine'_1\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\n⊢ ↑(extendDomain g f) ↑(↑f a) ≠ ↑(↑f a)\n\ncase intro.intro.refine'_2\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\nb : β\nhb : ↑(extendDomain g f) b ≠ b\n⊢ SameCycle (extendDomain g f) (↑(↑f a)) b",
"state_before": "case intro.intro\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\n⊢ IsCycle (extendDomain g f)",
"tactic": "refine' ⟨f a, _, fun b hb => _⟩"
},
{
"state_after": "case intro.intro.refine'_2\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\nb : β\nhb : ↑(extendDomain g f) b ≠ b\nh : b = ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))\n⊢ SameCycle (extendDomain g f) (↑(↑f a)) b",
"state_before": "case intro.intro.refine'_2\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\nb : β\nhb : ↑(extendDomain g f) b ≠ b\n⊢ SameCycle (extendDomain g f) (↑(↑f a)) b",
"tactic": "have h : b = f (f.symm ⟨b, of_not_not <| hb ∘ extendDomain_apply_not_subtype _ _⟩) := by\n rw [apply_symm_apply, Subtype.coe_mk]"
},
{
"state_after": "case intro.intro.refine'_2\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\nb : β\nhb✝ : ↑(extendDomain g f) b ≠ b\nhb :\n ↑(extendDomain g f) ↑(↑f (↑f.symm { val := b, property := (_ : p b) })) ≠\n ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))\nh : b = ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))\n⊢ SameCycle (extendDomain g f) ↑(↑f a) ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))",
"state_before": "case intro.intro.refine'_2\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\nb : β\nhb : ↑(extendDomain g f) b ≠ b\nh : b = ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))\n⊢ SameCycle (extendDomain g f) (↑(↑f a)) b",
"tactic": "rw [h] at hb⊢"
},
{
"state_after": "case intro.intro.refine'_2\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\nb : β\nhb✝ : ↑(extendDomain g f) b ≠ b\nh : b = ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))\nhb : ↑g (↑f.symm { val := b, property := (_ : p b) }) ≠ ↑f.symm { val := b, property := (_ : p b) }\n⊢ SameCycle (extendDomain g f) ↑(↑f a) ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))",
"state_before": "case intro.intro.refine'_2\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\nb : β\nhb✝ : ↑(extendDomain g f) b ≠ b\nhb :\n ↑(extendDomain g f) ↑(↑f (↑f.symm { val := b, property := (_ : p b) })) ≠\n ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))\nh : b = ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))\n⊢ SameCycle (extendDomain g f) ↑(↑f a) ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))",
"tactic": "simp only [extendDomain_apply_image, Subtype.coe_injective.ne_iff, f.injective.ne_iff] at hb"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_2\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\nb : β\nhb✝ : ↑(extendDomain g f) b ≠ b\nh : b = ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))\nhb : ↑g (↑f.symm { val := b, property := (_ : p b) }) ≠ ↑f.symm { val := b, property := (_ : p b) }\n⊢ SameCycle (extendDomain g f) ↑(↑f a) ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))",
"tactic": "exact (ha' hb).extendDomain"
},
{
"state_after": "case intro.intro.refine'_1\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\n⊢ ↑(↑f (↑g a)) ≠ ↑(↑f a)",
"state_before": "case intro.intro.refine'_1\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\n⊢ ↑(extendDomain g f) ↑(↑f a) ≠ ↑(↑f a)",
"tactic": "rw [extendDomain_apply_image]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.refine'_1\nι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\n⊢ ↑(↑f (↑g a)) ≠ ↑(↑f a)",
"tactic": "exact Subtype.coe_injective.ne (f.injective.ne ha)"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.344602\nα : Type u_2\nβ : Type u_1\nf✝ g : Perm α\nx y : α\np : β → Prop\ninst✝ : DecidablePred p\nf : α ≃ Subtype p\na : α\nha : ↑g a ≠ a\nha' : ∀ ⦃y : α⦄, ↑g y ≠ y → SameCycle g a y\nb : β\nhb : ↑(extendDomain g f) b ≠ b\n⊢ b = ↑(↑f (↑f.symm { val := b, property := (_ : p b) }))",
"tactic": "rw [apply_symm_apply, Subtype.coe_mk]"
}
] | [
337,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
327,
11
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | intervalIntegral.integral_comp_smul_deriv'' | [
{
"state_after": "ι : Type ?u.1904826\n𝕜 : Type ?u.1904829\nE : Type u_1\nF : Type ?u.1904835\nA : Type ?u.1904838\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ IntegrableOn g (f '' [[a, b]])",
"state_before": "ι : Type ?u.1904826\n𝕜 : Type ?u.1904829\nE : Type u_1\nF : Type ?u.1904835\nA : Type ?u.1904838\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ (∫ (x : ℝ) in a..b, f' x • (g ∘ f) x) = ∫ (u : ℝ) in f a..f b, g u",
"tactic": "refine'\n integral_comp_smul_deriv''' hf hff' (hg.mono <| image_subset _ Ioo_subset_Icc_self) _\n (hf'.smul (hg.comp hf <| subset_preimage_image f _)).integrableOn_Icc"
},
{
"state_after": "ι : Type ?u.1904826\n𝕜 : Type ?u.1904829\nE : Type u_1\nF : Type ?u.1904835\nA : Type ?u.1904838\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]\n⊢ IntegrableOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]",
"state_before": "ι : Type ?u.1904826\n𝕜 : Type ?u.1904829\nE : Type u_1\nF : Type ?u.1904835\nA : Type ?u.1904838\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ IntegrableOn g (f '' [[a, b]])",
"tactic": "rw [hf.image_uIcc] at hg ⊢"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1904826\n𝕜 : Type ?u.1904829\nE : Type u_1\nF : Type ?u.1904835\nA : Type ?u.1904838\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\nf✝¹ : ℝ → E\ng' g✝ φ : ℝ → ℝ\nf✝ f'✝ : ℝ → E\na b : ℝ\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ (x : ℝ), x ∈ Ioo (min a b) (max a b) → HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]\n⊢ IntegrableOn g [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]]",
"tactic": "exact hg.integrableOn_Icc"
}
] | [
1407,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1399,
1
] |
Mathlib/Tactic/NormNum/Basic.lean | Mathlib.Meta.NormNum.isInt_cast | [
{
"state_after": "case mk.refl\nR : Type u_1\ninst✝ : Ring R\nm : ℤ\n⊢ IsInt (↑↑m) m",
"state_before": "R : Type u_1\ninst✝ : Ring R\nn m : ℤ\n⊢ IsInt n m → IsInt (↑n) m",
"tactic": "rintro ⟨⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.refl\nR : Type u_1\ninst✝ : Ring R\nm : ℤ\n⊢ IsInt (↑↑m) m",
"tactic": "exact ⟨rfl⟩"
}
] | [
87,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean | LinearMap.toPMap_apply | [] | [
640,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
639,
1
] |
Mathlib/Order/Chain.lean | IsChain.mono_rel | [] | [
81,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
79,
1
] |
Std/Data/Int/Lemmas.lean | Int.mul_self_le_mul_self | [] | [
1234,
47
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1233,
11
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean | IsPrimitiveRoot.mem_rootsOfUnity | [] | [
488,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
486,
11
] |
Mathlib/Topology/Homeomorph.lean | Homeomorph.comp_isOpenMap_iff' | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.72743\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nf : β → γ\n⊢ IsOpenMap (f ∘ ↑h) → IsOpenMap f",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.72743\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nf : β → γ\n⊢ IsOpenMap (f ∘ ↑h) ↔ IsOpenMap f",
"tactic": "refine' ⟨_, fun hf => hf.comp h.isOpenMap⟩"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.72743\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nf : β → γ\nhf : IsOpenMap (f ∘ ↑h)\n⊢ IsOpenMap f",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.72743\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nf : β → γ\n⊢ IsOpenMap (f ∘ ↑h) → IsOpenMap f",
"tactic": "intro hf"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.72743\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nf : β → γ\nhf : IsOpenMap (f ∘ ↑h)\n⊢ IsOpenMap ((f ∘ ↑h) ∘ ↑(Homeomorph.symm h))",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.72743\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nf : β → γ\nhf : IsOpenMap (f ∘ ↑h)\n⊢ IsOpenMap f",
"tactic": "rw [← Function.comp.right_id f, ← h.self_comp_symm, ← Function.comp.assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type ?u.72743\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nf : β → γ\nhf : IsOpenMap (f ∘ ↑h)\n⊢ IsOpenMap ((f ∘ ↑h) ∘ ↑(Homeomorph.symm h))",
"tactic": "exact hf.comp h.symm.isOpenMap"
}
] | [
447,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
443,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean | Zsqrtd.norm_eq_zero | [
{
"state_after": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : norm a = 0\n⊢ a.re = 0.re ∧ a.im = 0.im",
"state_before": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\n⊢ norm a = 0 ↔ a = 0",
"tactic": "refine' ⟨fun ha => ext.mpr _, fun h => by rw [h, norm_zero]⟩"
},
{
"state_after": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re - d * a.im * a.im = 0\n⊢ a.re = 0.re ∧ a.im = 0.im",
"state_before": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : norm a = 0\n⊢ a.re = 0.re ∧ a.im = 0.im",
"tactic": "dsimp only [norm] at ha"
},
{
"state_after": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\n⊢ a.re = 0.re ∧ a.im = 0.im",
"state_before": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re - d * a.im * a.im = 0\n⊢ a.re = 0.re ∧ a.im = 0.im",
"tactic": "rw [sub_eq_zero] at ha"
},
{
"state_after": "case pos\nd : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : 0 ≤ d\n⊢ a.re = 0.re ∧ a.im = 0.im\n\ncase neg\nd : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : ¬0 ≤ d\n⊢ a.re = 0.re ∧ a.im = 0.im",
"state_before": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\n⊢ a.re = 0.re ∧ a.im = 0.im",
"tactic": "by_cases h : 0 ≤ d"
},
{
"state_after": "no goals",
"state_before": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nh : a = 0\n⊢ norm a = 0",
"tactic": "rw [h, norm_zero]"
},
{
"state_after": "case pos.intro\nd' : ℕ\nh_nonsquare : ∀ (n : ℤ), ↑d' ≠ n * n\na : ℤ√↑d'\nha : a.re * a.re = ↑d' * a.im * a.im\nh : 0 ≤ ↑d'\n⊢ a.re = 0.re ∧ a.im = 0.im",
"state_before": "case pos\nd : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : 0 ≤ d\n⊢ a.re = 0.re ∧ a.im = 0.im",
"tactic": "obtain ⟨d', rfl⟩ := Int.eq_ofNat_of_zero_le h"
},
{
"state_after": "case pos.intro\nd' : ℕ\nh_nonsquare : ∀ (n : ℤ), ↑d' ≠ n * n\na : ℤ√↑d'\nha : a.re * a.re = ↑d' * a.im * a.im\nh : 0 ≤ ↑d'\nthis : Nonsquare d'\n⊢ a.re = 0.re ∧ a.im = 0.im",
"state_before": "case pos.intro\nd' : ℕ\nh_nonsquare : ∀ (n : ℤ), ↑d' ≠ n * n\na : ℤ√↑d'\nha : a.re * a.re = ↑d' * a.im * a.im\nh : 0 ≤ ↑d'\n⊢ a.re = 0.re ∧ a.im = 0.im",
"tactic": "haveI : Nonsquare d' := ⟨fun n h => h_nonsquare n <| by exact_mod_cast h⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nd' : ℕ\nh_nonsquare : ∀ (n : ℤ), ↑d' ≠ n * n\na : ℤ√↑d'\nha : a.re * a.re = ↑d' * a.im * a.im\nh : 0 ≤ ↑d'\nthis : Nonsquare d'\n⊢ a.re = 0.re ∧ a.im = 0.im",
"tactic": "exact divides_sq_eq_zero_z ha"
},
{
"state_after": "no goals",
"state_before": "d' : ℕ\nh_nonsquare : ∀ (n : ℤ), ↑d' ≠ n * n\na : ℤ√↑d'\nha : a.re * a.re = ↑d' * a.im * a.im\nh✝ : 0 ≤ ↑d'\nn : ℕ\nh : d' = n * n\n⊢ ↑d' = ↑n * ↑n",
"tactic": "exact_mod_cast h"
},
{
"state_after": "case neg\nd : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : d < 0\n⊢ a.re = 0.re ∧ a.im = 0.im",
"state_before": "case neg\nd : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : ¬0 ≤ d\n⊢ a.re = 0.re ∧ a.im = 0.im",
"tactic": "push_neg at h"
},
{
"state_after": "case neg\nd : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : d < 0\n⊢ a.re * a.re = 0",
"state_before": "case neg\nd : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : d < 0\n⊢ a.re = 0.re ∧ a.im = 0.im",
"tactic": "suffices a.re * a.re = 0 by\n rw [eq_zero_of_mul_self_eq_zero this] at ha⊢\n simpa only [true_and_iff, or_self_right, zero_re, zero_im, eq_self_iff_true, zero_eq_mul,\n mul_zero, mul_eq_zero, h.ne, false_or_iff, or_self_iff] using ha"
},
{
"state_after": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : d < 0\n⊢ a.re * a.re ≤ 0",
"state_before": "case neg\nd : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : d < 0\n⊢ a.re * a.re = 0",
"tactic": "apply _root_.le_antisymm _ (mul_self_nonneg _)"
},
{
"state_after": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : d < 0\n⊢ d * (a.im * a.im) ≤ 0",
"state_before": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : d < 0\n⊢ a.re * a.re ≤ 0",
"tactic": "rw [ha, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : d < 0\n⊢ d * (a.im * a.im) ≤ 0",
"tactic": "exact mul_nonpos_of_nonpos_of_nonneg h.le (mul_self_nonneg _)"
},
{
"state_after": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : 0 * 0 = d * a.im * a.im\nh : d < 0\nthis : a.re * a.re = 0\n⊢ 0 = 0.re ∧ a.im = 0.im",
"state_before": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : a.re * a.re = d * a.im * a.im\nh : d < 0\nthis : a.re * a.re = 0\n⊢ a.re = 0.re ∧ a.im = 0.im",
"tactic": "rw [eq_zero_of_mul_self_eq_zero this] at ha⊢"
},
{
"state_after": "no goals",
"state_before": "d : ℤ\nh_nonsquare : ∀ (n : ℤ), d ≠ n * n\na : ℤ√d\nha : 0 * 0 = d * a.im * a.im\nh : d < 0\nthis : a.re * a.re = 0\n⊢ 0 = 0.re ∧ a.im = 0.im",
"tactic": "simpa only [true_and_iff, or_self_right, zero_re, zero_im, eq_self_iff_true, zero_eq_mul,\n mul_zero, mul_eq_zero, h.ne, false_or_iff, or_self_iff] using ha"
}
] | [
1026,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1011,
1
] |
Mathlib/CategoryTheory/Localization/Construction.lean | CategoryTheory.Localization.Construction.morphismProperty_is_top' | [] | [
268,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
264,
1
] |
Std/Data/List/Lemmas.lean | List.foldl_map | [
{
"state_after": "no goals",
"state_before": "β₁ : Type u_1\nβ₂ : Type u_2\nα : Type u_3\nf : β₁ → β₂\ng : α → β₂ → α\nl : List β₁\ninit : α\n⊢ foldl g init (map f l) = foldl (fun x y => g x (f y)) init l",
"tactic": "induction l generalizing init <;> simp [*]"
}
] | [
1411,
45
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1409,
1
] |
Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | coe_gramSchmidtBasis | [] | [
262,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
] |
Mathlib/Topology/Connected.lean | connectedComponent_eq_iff_mem | [] | [
678,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
676,
1
] |
Mathlib/Algebra/Lie/Basic.lean | LieHom.ext | [] | [
385,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
384,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean | Lagrange.values_eq_on_of_interpolate_eq | [
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝¹ : Field F\nι : Type u_1\ninst✝ : DecidableEq ι\ns t : Finset ι\ni j : ι\nv r r' : ι → F\nhvs : Set.InjOn v ↑s\nhrr' : ↑(interpolate s v) r = ↑(interpolate s v) r'\nx✝ : ι\nhi : x✝ ∈ s\n⊢ r x✝ = r' x✝",
"tactic": "rw [← eval_interpolate_at_node r hvs hi, hrr', eval_interpolate_at_node r' hvs hi]"
}
] | [
361,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
359,
1
] |
Mathlib/Algebra/Group/OrderSynonym.lean | toLex_mul | [] | [
254,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | MeasureTheory.FinStronglyMeasurable.sup | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.278636\nι : Type ?u.278639\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ ↑↑μ (support ↑((fun n => FinStronglyMeasurable.approx hf n ⊔ FinStronglyMeasurable.approx hg n) n)) < ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.278636\nι : Type ?u.278639\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\n⊢ FinStronglyMeasurable (f ⊔ g) μ",
"tactic": "refine'\n ⟨fun n => hf.approx n ⊔ hg.approx n, fun n => _, fun x =>\n (hf.tendsto_approx x).sup_right_nhds (hg.tendsto_approx x)⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.278636\nι : Type ?u.278639\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ ↑↑μ\n ((support fun x => ↑(FinStronglyMeasurable.approx hf n) x) ∪\n support fun x => ↑(FinStronglyMeasurable.approx hg n) x) <\n ⊤",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.278636\nι : Type ?u.278639\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ ↑↑μ (support ↑((fun n => FinStronglyMeasurable.approx hf n ⊔ FinStronglyMeasurable.approx hg n) n)) < ⊤",
"tactic": "refine' (measure_mono (support_sup _ _)).trans_lt _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.278636\nι : Type ?u.278639\ninst✝⁴ : Countable ι\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ ↑↑μ\n ((support fun x => ↑(FinStronglyMeasurable.approx hf n) x) ∪\n support fun x => ↑(FinStronglyMeasurable.approx hg n) x) <\n ⊤",
"tactic": "exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩"
}
] | [
1116,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1110,
11
] |
Mathlib/Order/Basic.lean | lt_of_not_le | [] | [
472,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
471,
1
] |
Mathlib/Data/Rel.lean | Rel.core_univ | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.23811\nr : Rel α β\n⊢ ∀ (x : α), x ∈ core r Set.univ ↔ x ∈ Set.univ",
"tactic": "simp [mem_core]"
}
] | [
236,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/Order/Basic.lean | Pi.le_def | [] | [
798,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
796,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean | Set.smul_set_union | [] | [
370,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
369,
1
] |
Mathlib/Analysis/Complex/ReImTopology.lean | Complex.closure_setOf_re_lt | [
{
"state_after": "no goals",
"state_before": "a : ℝ\n⊢ closure {z | z.re < a} = {z | z.re ≤ a}",
"tactic": "simpa only [closure_Iio] using closure_preimage_re (Iio a)"
}
] | [
118,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
117,
1
] |
Mathlib/Order/RelClasses.lean | IsTotal.isTrichotomous | [] | [
128,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
11
] |
Mathlib/Data/Complex/Basic.lean | Complex.I_zpow_bit1 | [
{
"state_after": "no goals",
"state_before": "n : ℤ\n⊢ I ^ bit1 n = (-1) ^ n * I",
"tactic": "rw [zpow_bit1', I_mul_I]"
}
] | [
776,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
776,
1
] |
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