file_path
stringlengths 11
79
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stringlengths 2
100
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list | end
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stringclasses 4
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|---|---|---|---|---|---|---|
Mathlib/Computability/Partrec.lean
|
Nat.mem_rfind
|
[
{
"state_after": "p : ℕ →. Bool\nn : ℕ\nx✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m\nh₁ : true ∈ p n\nh₂ : ∀ {m : ℕ}, m < n → false ∈ p m\nm : ℕ\nhm : m ∈ rfind p\n⊢ n ∈ rfind p",
"state_before": "p : ℕ →. Bool\nn : ℕ\nx✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m\nh₁ : true ∈ p n\nh₂ : ∀ {m : ℕ}, m < n → false ∈ p m\n⊢ n ∈ rfind p",
"tactic": "let ⟨m, hm⟩ := dom_iff_mem.1 <| (@rfind_dom p).2 ⟨_, h₁, fun {m} mn => (h₂ mn).fst⟩"
},
{
"state_after": "case inl\np : ℕ →. Bool\nn : ℕ\nx✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m\nh₁ : true ∈ p n\nh₂ : ∀ {m : ℕ}, m < n → false ∈ p m\nm : ℕ\nhm : m ∈ rfind p\nh : m < n\n⊢ n ∈ rfind p\n\ncase inr.inl\np : ℕ →. Bool\nn : ℕ\nx✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m\nh₁ : true ∈ p n\nh₂ : ∀ {m : ℕ}, m < n → false ∈ p m\nm : ℕ\nhm : m ∈ rfind p\nh : m = n\n⊢ n ∈ rfind p\n\ncase inr.inr\np : ℕ →. Bool\nn : ℕ\nx✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m\nh₁ : true ∈ p n\nh₂ : ∀ {m : ℕ}, m < n → false ∈ p m\nm : ℕ\nhm : m ∈ rfind p\nh : n < m\n⊢ n ∈ rfind p",
"state_before": "p : ℕ →. Bool\nn : ℕ\nx✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m\nh₁ : true ∈ p n\nh₂ : ∀ {m : ℕ}, m < n → false ∈ p m\nm : ℕ\nhm : m ∈ rfind p\n⊢ n ∈ rfind p",
"tactic": "rcases lt_trichotomy m n with (h | h | h)"
},
{
"state_after": "no goals",
"state_before": "case inl\np : ℕ →. Bool\nn : ℕ\nx✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m\nh₁ : true ∈ p n\nh₂ : ∀ {m : ℕ}, m < n → false ∈ p m\nm : ℕ\nhm : m ∈ rfind p\nh : m < n\n⊢ n ∈ rfind p",
"tactic": "injection mem_unique (h₂ h) (rfind_spec hm)"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\np : ℕ →. Bool\nn : ℕ\nx✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m\nh₁ : true ∈ p n\nh₂ : ∀ {m : ℕ}, m < n → false ∈ p m\nm : ℕ\nhm : m ∈ rfind p\nh : m = n\n⊢ n ∈ rfind p",
"tactic": "rwa [← h]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\np : ℕ →. Bool\nn : ℕ\nx✝ : true ∈ p n ∧ ∀ {m : ℕ}, m < n → false ∈ p m\nh₁ : true ∈ p n\nh₂ : ∀ {m : ℕ}, m < n → false ∈ p m\nm : ℕ\nhm : m ∈ rfind p\nh : n < m\n⊢ n ∈ rfind p",
"tactic": "injection mem_unique h₁ (rfind_min hm h)"
}
] |
[
112,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.apply_update₂
|
[
{
"state_after": "case pos\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g✝ : (a : α✝) → β✝ a\na : α✝\nb : β✝ a\nι : Sort u_1\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nβ : ι → Sort u_3\nγ : ι → Sort u_4\nf : (i : ι) → α i → β i → γ i\ng : (i : ι) → α i\nh✝ : (i : ι) → β i\ni : ι\nv : α i\nw : β i\nj : ι\nh : j = i\n⊢ f j (update g i v j) (update h✝ i w j) = update (fun k => f k (g k) (h✝ k)) i (f i v w) j\n\ncase neg\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g✝ : (a : α✝) → β✝ a\na : α✝\nb : β✝ a\nι : Sort u_1\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nβ : ι → Sort u_3\nγ : ι → Sort u_4\nf : (i : ι) → α i → β i → γ i\ng : (i : ι) → α i\nh✝ : (i : ι) → β i\ni : ι\nv : α i\nw : β i\nj : ι\nh : ¬j = i\n⊢ f j (update g i v j) (update h✝ i w j) = update (fun k => f k (g k) (h✝ k)) i (f i v w) j",
"state_before": "α✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g✝ : (a : α✝) → β✝ a\na : α✝\nb : β✝ a\nι : Sort u_1\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nβ : ι → Sort u_3\nγ : ι → Sort u_4\nf : (i : ι) → α i → β i → γ i\ng : (i : ι) → α i\nh : (i : ι) → β i\ni : ι\nv : α i\nw : β i\nj : ι\n⊢ f j (update g i v j) (update h i w j) = update (fun k => f k (g k) (h k)) i (f i v w) j",
"tactic": "by_cases h:j = i"
},
{
"state_after": "case pos\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g✝ : (a : α✝) → β✝ a\na : α✝\nb : β✝ a\nι : Sort u_1\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nβ : ι → Sort u_3\nγ : ι → Sort u_4\nf : (i : ι) → α i → β i → γ i\ng : (i : ι) → α i\nh : (i : ι) → β i\ni : ι\nv : α i\nw : β i\n⊢ f i (update g i v i) (update h i w i) = update (fun k => f k (g k) (h k)) i (f i v w) i",
"state_before": "case pos\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g✝ : (a : α✝) → β✝ a\na : α✝\nb : β✝ a\nι : Sort u_1\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nβ : ι → Sort u_3\nγ : ι → Sort u_4\nf : (i : ι) → α i → β i → γ i\ng : (i : ι) → α i\nh✝ : (i : ι) → β i\ni : ι\nv : α i\nw : β i\nj : ι\nh : j = i\n⊢ f j (update g i v j) (update h✝ i w j) = update (fun k => f k (g k) (h✝ k)) i (f i v w) j",
"tactic": "subst j"
},
{
"state_after": "no goals",
"state_before": "case pos\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g✝ : (a : α✝) → β✝ a\na : α✝\nb : β✝ a\nι : Sort u_1\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nβ : ι → Sort u_3\nγ : ι → Sort u_4\nf : (i : ι) → α i → β i → γ i\ng : (i : ι) → α i\nh : (i : ι) → β i\ni : ι\nv : α i\nw : β i\n⊢ f i (update g i v i) (update h i w i) = update (fun k => f k (g k) (h k)) i (f i v w) i",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg\nα✝ : Sort u\nβ✝ : α✝ → Sort v\nα' : Sort w\ninst✝² : DecidableEq α✝\ninst✝¹ : DecidableEq α'\nf✝ g✝ : (a : α✝) → β✝ a\na : α✝\nb : β✝ a\nι : Sort u_1\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nβ : ι → Sort u_3\nγ : ι → Sort u_4\nf : (i : ι) → α i → β i → γ i\ng : (i : ι) → α i\nh✝ : (i : ι) → β i\ni : ι\nv : α i\nw : β i\nj : ι\nh : ¬j = i\n⊢ f j (update g i v j) (update h✝ i w j) = update (fun k => f k (g k) (h✝ k)) i (f i v w) j",
"tactic": "simp [h]"
}
] |
[
657,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
651,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.All_and
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\np q : α → Prop\nt : RBNode α\n⊢ All (fun a => p a ∧ q a) t ↔ All p t ∧ All q t",
"tactic": "induction t <;> simp [*, and_assoc, and_left_comm]"
}
] |
[
28,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
27,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sSup_diff_singleton_bot
|
[] |
[
525,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Basis.lean
|
AffineBasis.ind
|
[] |
[
84,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
coe_nnnorm'
|
[] |
[
888,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
887,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean
|
HasFDerivWithinAt.arsinh
|
[] |
[
255,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Order/Basic.lean
|
strongLT_of_le_of_strongLT
|
[] |
[
842,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/Data/Real/Cardinality.lean
|
Cardinal.mk_Ici_real
|
[] |
[
250,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/CategoryTheory/Subobject/FactorThru.lean
|
CategoryTheory.Subobject.factorThru_add_sub_factorThru_left
|
[
{
"state_after": "case h\nC : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Preadditive C\nX Y : C\nP : Subobject Y\nf g : X ⟶ Y\nw : Factors P (f + g)\nwf : Factors P f\n⊢ (factorThru P (f + g) w - factorThru P f wf) ≫ arrow P = factorThru P g (_ : Factors P g) ≫ arrow P",
"state_before": "C : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Preadditive C\nX Y : C\nP : Subobject Y\nf g : X ⟶ Y\nw : Factors P (f + g)\nwf : Factors P f\n⊢ factorThru P (f + g) w - factorThru P f wf = factorThru P g (_ : Factors P g)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Preadditive C\nX Y : C\nP : Subobject Y\nf g : X ⟶ Y\nw : Factors P (f + g)\nwf : Factors P f\n⊢ (factorThru P (f + g) w - factorThru P f wf) ≫ arrow P = factorThru P g (_ : Factors P g) ≫ arrow P",
"tactic": "simp"
}
] |
[
224,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.trans
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.73759\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.73771\nF' : Type ?u.73774\nG' : Type ?u.73777\nE'' : Type ?u.73780\nF'' : Type ?u.73783\nG'' : Type ?u.73786\nR : Type ?u.73789\nR' : Type ?u.73792\n𝕜 : Type ?u.73795\n𝕜' : Type ?u.73798\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhc : 0 ≤ c\nhfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖\nhgk : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c' * ‖k x‖\n⊢ ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * c' * ‖k x‖",
"state_before": "α : Type u_1\nβ : Type ?u.73759\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.73771\nF' : Type ?u.73774\nG' : Type ?u.73777\nE'' : Type ?u.73780\nF'' : Type ?u.73783\nG'' : Type ?u.73786\nR : Type ?u.73789\nR' : Type ?u.73792\n𝕜 : Type ?u.73795\n𝕜' : Type ?u.73798\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhfg : IsBigOWith c l f g\nhgk : IsBigOWith c' l g k\nhc : 0 ≤ c\n⊢ IsBigOWith (c * c') l f k",
"tactic": "simp only [IsBigOWith_def] at *"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.73759\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.73771\nF' : Type ?u.73774\nG' : Type ?u.73777\nE'' : Type ?u.73780\nF'' : Type ?u.73783\nG'' : Type ?u.73786\nR : Type ?u.73789\nR' : Type ?u.73792\n𝕜 : Type ?u.73795\n𝕜' : Type ?u.73798\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhc : 0 ≤ c\nhfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖\nhgk : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c' * ‖k x‖\nx : α\nhx : ‖f x‖ ≤ c * ‖g x‖\nhx' : ‖g x‖ ≤ c' * ‖k x‖\n⊢ ‖f x‖ ≤ c * c' * ‖k x‖",
"state_before": "α : Type u_1\nβ : Type ?u.73759\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.73771\nF' : Type ?u.73774\nG' : Type ?u.73777\nE'' : Type ?u.73780\nF'' : Type ?u.73783\nG'' : Type ?u.73786\nR : Type ?u.73789\nR' : Type ?u.73792\n𝕜 : Type ?u.73795\n𝕜' : Type ?u.73798\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhc : 0 ≤ c\nhfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖\nhgk : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c' * ‖k x‖\n⊢ ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * c' * ‖k x‖",
"tactic": "filter_upwards [hfg, hgk]with x hx hx'"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.73759\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.73771\nF' : Type ?u.73774\nG' : Type ?u.73777\nE'' : Type ?u.73780\nF'' : Type ?u.73783\nG'' : Type ?u.73786\nR : Type ?u.73789\nR' : Type ?u.73792\n𝕜 : Type ?u.73795\n𝕜' : Type ?u.73798\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhc : 0 ≤ c\nhfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖\nhgk : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c' * ‖k x‖\nx : α\nhx : ‖f x‖ ≤ c * ‖g x‖\nhx' : ‖g x‖ ≤ c' * ‖k x‖\n⊢ ‖f x‖ ≤ c * c' * ‖k x‖",
"tactic": "calc\n ‖f x‖ ≤ c * ‖g x‖ := hx\n _ ≤ c * (c' * ‖k x‖) := by gcongr\n _ = c * c' * ‖k x‖ := (mul_assoc _ _ _).symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.73759\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.73771\nF' : Type ?u.73774\nG' : Type ?u.73777\nE'' : Type ?u.73780\nF'' : Type ?u.73783\nG'' : Type ?u.73786\nR : Type ?u.73789\nR' : Type ?u.73792\n𝕜 : Type ?u.73795\n𝕜' : Type ?u.73798\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhc : 0 ≤ c\nhfg : ∀ᶠ (x : α) in l, ‖f x‖ ≤ c * ‖g x‖\nhgk : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c' * ‖k x‖\nx : α\nhx : ‖f x‖ ≤ c * ‖g x‖\nhx' : ‖g x‖ ≤ c' * ‖k x‖\n⊢ c * ‖g x‖ ≤ c * (c' * ‖k x‖)",
"tactic": "gcongr"
}
] |
[
466,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/Data/List/Basic.lean
|
List.choose_spec
|
[] |
[
3975,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3974,
1
] |
Mathlib/Analysis/Complex/PhragmenLindelof.lean
|
PhragmenLindelof.eq_zero_on_horizontal_strip
|
[] |
[
249,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.natAbs_lt_natAbs_of_nonneg_of_lt
|
[] |
[
1359,
46
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1356,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
eq_star_iff_eq_star
|
[] |
[
109,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
reflection_bot
|
[
{
"state_after": "case h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.748584\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nx✝ : E\n⊢ ↑(reflection ⊥) x✝ = ↑(LinearIsometryEquiv.neg 𝕜) x✝",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.748584\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\n⊢ reflection ⊥ = LinearIsometryEquiv.neg 𝕜",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.748584\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nx✝ : E\n⊢ ↑(reflection ⊥) x✝ = ↑(LinearIsometryEquiv.neg 𝕜) x✝",
"tactic": "simp [reflection_apply]"
}
] |
[
726,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
725,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.set_biInter_insert
|
[] |
[
2054,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2052,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.norm_eq_pow_val
|
[
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ‖Quotient.mk'' f‖ = ↑p ^ (-valuation (Quotient.mk'' f))",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\n⊢ x ≠ 0 → ‖x‖ = ↑p ^ (-valuation x)",
"tactic": "refine Quotient.inductionOn' x fun f hf => ?_"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ↑(PadicSeq.norm f) = ↑p ^ (-PadicSeq.valuation f)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ‖Quotient.mk'' f‖ = ↑p ^ (-valuation (Quotient.mk'' f))",
"tactic": "change (PadicSeq.norm _ : ℝ) = (p : ℝ) ^ (-PadicSeq.valuation _)"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ↑(↑p ^ (-PadicSeq.valuation f)) = ↑p ^ (-PadicSeq.valuation f)\n\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ¬f ≈ 0",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ↑(PadicSeq.norm f) = ↑p ^ (-PadicSeq.valuation f)",
"tactic": "rw [PadicSeq.norm_eq_pow_val]"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ↑(↑p ^ (-PadicSeq.valuation f)) = ↑p ^ (-PadicSeq.valuation f)\n\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ¬f ≈ 0",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ↑(↑p ^ (-PadicSeq.valuation f)) = ↑p ^ (-PadicSeq.valuation f)\n\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ¬f ≈ 0",
"tactic": "change ↑((p : ℚ) ^ (-PadicSeq.valuation f)) = (p : ℝ) ^ (-PadicSeq.valuation f)"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ↑(↑p ^ (-PadicSeq.valuation f)) = ↑p ^ (-PadicSeq.valuation f)",
"tactic": "rw [Rat.cast_zpow, Rat.cast_coe_nat]"
},
{
"state_after": "case hf\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ¬f - 0 ≈ 0",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ¬f ≈ 0",
"tactic": "apply CauSeq.not_limZero_of_not_congr_zero"
},
{
"state_after": "case hf\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\nhf' : f - 0 ≈ 0\n⊢ False",
"state_before": "case hf\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\n⊢ ¬f - 0 ≈ 0",
"tactic": "intro hf'"
},
{
"state_after": "case hf\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\nhf' : f - 0 ≈ 0\n⊢ Quotient.mk'' f = 0",
"state_before": "case hf\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\nhf' : f - 0 ≈ 0\n⊢ False",
"tactic": "apply hf"
},
{
"state_after": "case hf.a\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\nhf' : f - 0 ≈ 0\n⊢ f ≈ const (padicNorm p) 0",
"state_before": "case hf\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\nhf' : f - 0 ≈ 0\n⊢ Quotient.mk'' f = 0",
"tactic": "apply Quotient.sound"
},
{
"state_after": "no goals",
"state_before": "case hf.a\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nf : CauSeq ℚ (padicNorm p)\nhf : Quotient.mk'' f ≠ 0\nhf' : f - 0 ≈ 0\n⊢ f ≈ const (padicNorm p) 0",
"tactic": "simpa using hf'"
}
] |
[
1063,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1052,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.none_eq_top
|
[] |
[
597,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
596,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.singleton_apply
|
[
{
"state_after": "no goals",
"state_before": "ι✝ : Type ?u.609084\nι' : Type ?u.609087\nR✝ : Type ?u.609090\nR₂ : Type ?u.609093\nK : Type ?u.609096\nM : Type ?u.609099\nM' : Type ?u.609102\nM'' : Type ?u.609105\nV : Type u\nV' : Type ?u.609110\ninst✝⁶ : Semiring R✝\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R✝ M\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R✝ M'\nb b₁ : Basis ι✝ R✝ M\ni✝ : ι✝\nc : R✝\nx : M\nι : Type u_1\nR : Type u_2\ninst✝¹ : Unique ι\ninst✝ : Semiring R\ni : ι\n⊢ ↑(Basis.singleton ι R).repr 1 = Finsupp.single i 1",
"tactic": "simp [Basis.singleton]"
}
] |
[
824,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
823,
1
] |
Mathlib/Topology/Algebra/Module/LocallyConvex.lean
|
locallyConvexSpace_iff
|
[] |
[
56,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
Differentiable.neg
|
[] |
[
447,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/Order/SuccPred/LinearLocallyFinite.lean
|
LinearLocallyFiniteOrder.succFn_le_of_lt
|
[
{
"state_after": "ι : Type u_1\ninst✝ : LinearOrder ι\ni j : ι\nhij : i < j\nh : IsGLB (Set.Ioi i) (succFn i)\n⊢ succFn i ≤ j",
"state_before": "ι : Type u_1\ninst✝ : LinearOrder ι\ni j : ι\nhij : i < j\n⊢ succFn i ≤ j",
"tactic": "have h := succFn_spec i"
},
{
"state_after": "ι : Type u_1\ninst✝ : LinearOrder ι\ni j : ι\nhij : i < j\nh : (∀ (x : ι), x ∈ Set.Ioi i → succFn i ≤ x) ∧ succFn i ∈ upperBounds (lowerBounds (Set.Ioi i))\n⊢ succFn i ≤ j",
"state_before": "ι : Type u_1\ninst✝ : LinearOrder ι\ni j : ι\nhij : i < j\nh : IsGLB (Set.Ioi i) (succFn i)\n⊢ succFn i ≤ j",
"tactic": "rw [IsGLB, IsGreatest, mem_lowerBounds] at h"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : LinearOrder ι\ni j : ι\nhij : i < j\nh : (∀ (x : ι), x ∈ Set.Ioi i → succFn i ≤ x) ∧ succFn i ∈ upperBounds (lowerBounds (Set.Ioi i))\n⊢ succFn i ≤ j",
"tactic": "exact h.1 j hij"
}
] |
[
109,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/GroupTheory/Coset.lean
|
QuotientGroup.rightRel_eq
|
[
{
"state_after": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b ↔ b * a⁻¹ ∈ s",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b = (b * a⁻¹ ∈ s)",
"tactic": "simp only [eq_iff_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\n⊢ ∀ (a b : α), Setoid.r a b ↔ b * a⁻¹ ∈ s",
"tactic": "apply rightRel_apply"
}
] |
[
391,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/Order/Basic.lean
|
lt_trans'
|
[] |
[
79,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.choose_mul
|
[
{
"state_after": "no goals",
"state_before": "n k s : ℕ\nhkn : k ≤ n\nhsk : s ≤ k\n⊢ (n - k)! * (k - s)! * s ! ≠ 0",
"tactic": "apply_rules [factorial_ne_zero, mul_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "n k s : ℕ\nhkn : k ≤ n\nhsk : s ≤ k\nh : (n - k)! * (k - s)! * s ! ≠ 0\n⊢ choose n k * choose k s * ((n - k)! * (k - s)! * s !) = choose n k * (choose k s * s ! * (k - s)!) * (n - k)!",
"tactic": "rw [mul_assoc, mul_assoc, mul_assoc, mul_assoc _ s !, mul_assoc, mul_comm (n - k)!,\nmul_comm s !]"
},
{
"state_after": "no goals",
"state_before": "n k s : ℕ\nhkn : k ≤ n\nhsk : s ≤ k\nh : (n - k)! * (k - s)! * s ! ≠ 0\n⊢ choose n k * (choose k s * s ! * (k - s)!) * (n - k)! = n !",
"tactic": "rw [choose_mul_factorial_mul_factorial hsk, choose_mul_factorial_mul_factorial hkn]"
},
{
"state_after": "no goals",
"state_before": "n k s : ℕ\nhkn : k ≤ n\nhsk : s ≤ k\nh : (n - k)! * (k - s)! * s ! ≠ 0\n⊢ n ! = choose n s * s ! * (choose (n - s) (k - s) * (k - s)! * (n - s - (k - s))!)",
"tactic": "rw [choose_mul_factorial_mul_factorial (tsub_le_tsub_right hkn _),\nchoose_mul_factorial_mul_factorial (hsk.trans hkn)]"
},
{
"state_after": "no goals",
"state_before": "n k s : ℕ\nhkn : k ≤ n\nhsk : s ≤ k\nh : (n - k)! * (k - s)! * s ! ≠ 0\n⊢ choose n s * s ! * (choose (n - s) (k - s) * (k - s)! * (n - s - (k - s))!) =\n choose n s * choose (n - s) (k - s) * ((n - k)! * (k - s)! * s !)",
"tactic": "rw [tsub_tsub_tsub_cancel_right hsk, mul_assoc, mul_left_comm s !, mul_assoc,\nmul_comm (k - s)!, mul_comm s !, mul_right_comm, ← mul_assoc]"
}
] |
[
165,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.mem_range_iff
|
[
{
"state_after": "case mp\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nh : y ∈ range ↑f\n⊢ ∃ x, (x, y) ∈ graph f\n\ncase mpr\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nh : ∃ x, (x, y) ∈ graph f\n⊢ y ∈ range ↑f",
"state_before": "R : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\n⊢ y ∈ range ↑f ↔ ∃ x, (x, y) ∈ graph f",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mpr.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nh : (x, y) ∈ graph f\n⊢ y ∈ range ↑f",
"state_before": "case mpr\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nh : ∃ x, (x, y) ∈ graph f\n⊢ y ∈ range ↑f",
"tactic": "cases' h with x h"
},
{
"state_after": "case mpr.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nh : ∃ y_1, ↑y_1 = (x, y).fst ∧ ↑f y_1 = (x, y).snd\n⊢ y ∈ range ↑f",
"state_before": "case mpr.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nh : (x, y) ∈ graph f\n⊢ y ∈ range ↑f",
"tactic": "rw [mem_graph_iff] at h"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : { x // x ∈ f.domain }\nh : ↑x = (x✝, y).fst ∧ ↑f x = (x✝, y).snd\n⊢ y ∈ range ↑f",
"state_before": "case mpr.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nh : ∃ y_1, ↑y_1 = (x, y).fst ∧ ↑f y_1 = (x, y).snd\n⊢ y ∈ range ↑f",
"tactic": "cases' h with x h"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : { x // x ∈ f.domain }\nh : ↑x = (x✝, y).fst ∧ ↑f x = (x✝, y).snd\n⊢ ∃ y_1, ↑f y_1 = y",
"state_before": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : { x // x ∈ f.domain }\nh : ↑x = (x✝, y).fst ∧ ↑f x = (x✝, y).snd\n⊢ y ∈ range ↑f",
"tactic": "rw [Set.mem_range]"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : { x // x ∈ f.domain }\nh : ↑x = (x✝, y).fst ∧ ↑f x = (x✝, y).snd\n⊢ ↑f x = y",
"state_before": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : { x // x ∈ f.domain }\nh : ↑x = (x✝, y).fst ∧ ↑f x = (x✝, y).snd\n⊢ ∃ y_1, ↑f y_1 = y",
"tactic": "use x"
},
{
"state_after": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : { x // x ∈ f.domain }\nh : ↑x = x✝ ∧ ↑f x = y\n⊢ ↑f x = y",
"state_before": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : { x // x ∈ f.domain }\nh : ↑x = (x✝, y).fst ∧ ↑f x = (x✝, y).snd\n⊢ ↑f x = y",
"tactic": "simp only at h"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx✝ : E\nx : { x // x ∈ f.domain }\nh : ↑x = x✝ ∧ ↑f x = y\n⊢ ↑f x = y",
"tactic": "rw [h.2]"
},
{
"state_after": "case mp\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nh : ∃ y_1, ↑f y_1 = y\n⊢ ∃ x, (x, y) ∈ graph f",
"state_before": "case mp\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nh : y ∈ range ↑f\n⊢ ∃ x, (x, y) ∈ graph f",
"tactic": "rw [Set.mem_range] at h"
},
{
"state_after": "case mp.intro.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nhx : x ∈ f.domain\nh : ↑f { val := x, property := hx } = y\n⊢ ∃ x, (x, y) ∈ graph f",
"state_before": "case mp\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nh : ∃ y_1, ↑f y_1 = y\n⊢ ∃ x, (x, y) ∈ graph f",
"tactic": "rcases h with ⟨⟨x, hx⟩, h⟩"
},
{
"state_after": "case mp.intro.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nhx : x ∈ f.domain\nh : ↑f { val := x, property := hx } = y\n⊢ (x, y) ∈ graph f",
"state_before": "case mp.intro.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nhx : x ∈ f.domain\nh : ↑f { val := x, property := hx } = y\n⊢ ∃ x, (x, y) ∈ graph f",
"tactic": "use x"
},
{
"state_after": "case mp.intro.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nhx : x ∈ f.domain\nh : ↑f { val := x, property := hx } = y\n⊢ (x, ↑f { val := x, property := hx }) ∈ graph f",
"state_before": "case mp.intro.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nhx : x ∈ f.domain\nh : ↑f { val := x, property := hx } = y\n⊢ (x, y) ∈ graph f",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.mk\nR : Type u_1\ninst✝⁹ : Ring R\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type u_3\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.585339\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\nM : Type ?u.585855\ninst✝² : Monoid M\ninst✝¹ : DistribMulAction M F\ninst✝ : SMulCommClass R M F\ny✝ : M\nf : E →ₗ.[R] F\ny : F\nx : E\nhx : x ∈ f.domain\nh : ↑f { val := x, property := hx } = y\n⊢ (x, ↑f { val := x, property := hx }) ∈ graph f",
"tactic": "exact f.mem_graph ⟨x, hx⟩"
}
] |
[
873,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
860,
1
] |
Mathlib/Topology/Hom/Open.lean
|
ContinuousOpenMap.cancel_right
|
[] |
[
159,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/LinearAlgebra/Determinant.lean
|
Basis.det_apply
|
[] |
[
535,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/Topology/Basic.lean
|
Dense.mono
|
[] |
[
670,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
1
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
TopologicalSpace.Opens.functor_map_eq_inf
|
[
{
"state_after": "case h\nX : TopCat\nU V : Opens ↑X\n⊢ ↑((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion U)))).obj ((map (inclusion U)).obj V)) = ↑(V ⊓ U)",
"state_before": "X : TopCat\nU V : Opens ↑X\n⊢ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion U)))).obj ((map (inclusion U)).obj V) = V ⊓ U",
"tactic": "ext1"
},
{
"state_after": "case h\nX : TopCat\nU V : Opens ↑X\n⊢ (V.1 ∩ Set.range fun x => (forget TopCat).map (inclusion U) x) = ↑(V ⊓ U)",
"state_before": "case h\nX : TopCat\nU V : Opens ↑X\n⊢ ↑((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion U)))).obj ((map (inclusion U)).obj V)) = ↑(V ⊓ U)",
"tactic": "refine' Set.image_preimage_eq_inter_range.trans _"
},
{
"state_after": "case h\nX : TopCat\nU V : Opens ↑X\n⊢ V.1 ∩ ↑U = ↑(V ⊓ U)",
"state_before": "case h\nX : TopCat\nU V : Opens ↑X\n⊢ (V.1 ∩ Set.range fun x => (forget TopCat).map (inclusion U) x) = ↑(V ⊓ U)",
"tactic": "rw [set_range_forget_map_inclusion U]"
},
{
"state_after": "no goals",
"state_before": "case h\nX : TopCat\nU V : Opens ↑X\n⊢ V.1 ∩ ↑U = ↑(V ⊓ U)",
"tactic": "rfl"
}
] |
[
391,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.of_eq
|
[] |
[
250,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Data/Real/Irrational.lean
|
irrational_rat_add_iff
|
[] |
[
533,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
532,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.mem_iUnion₂_of_mem
|
[] |
[
166,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometryEquiv.coe_constVSub
|
[] |
[
731,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
730,
1
] |
Mathlib/NumberTheory/Padics/Hensel.lean
|
ncs_tendsto_const
|
[
{
"state_after": "case h.e'_3.h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nncs : CauSeq ℤ_[p] norm\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nncs_der_val :\n ∀ (n : ℕ), ‖Polynomial.eval (↑ncs n) (↑Polynomial.derivative F)‖ = ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nx✝ : ℕ\n⊢ ‖Polynomial.eval (↑ncs x✝) (↑Polynomial.derivative F)‖ = ‖Polynomial.eval a (↑Polynomial.derivative F)‖",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nncs : CauSeq ℤ_[p] norm\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nncs_der_val :\n ∀ (n : ℕ), ‖Polynomial.eval (↑ncs n) (↑Polynomial.derivative F)‖ = ‖Polynomial.eval a (↑Polynomial.derivative F)‖\n⊢ Tendsto (fun i => ‖Polynomial.eval (↑ncs i) (↑Polynomial.derivative F)‖) atTop\n (𝓝 ‖Polynomial.eval a (↑Polynomial.derivative F)‖)",
"tactic": "convert @tendsto_const_nhds ℝ ℕ _ _ _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nncs : CauSeq ℤ_[p] norm\nF : Polynomial ℤ_[p]\na : ℤ_[p]\nncs_der_val :\n ∀ (n : ℕ), ‖Polynomial.eval (↑ncs n) (↑Polynomial.derivative F)‖ = ‖Polynomial.eval a (↑Polynomial.derivative F)‖\nx✝ : ℕ\n⊢ ‖Polynomial.eval (↑ncs x✝) (↑Polynomial.derivative F)‖ = ‖Polynomial.eval a (↑Polynomial.derivative F)‖",
"tactic": "rw [ncs_der_val]"
}
] |
[
69,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
9
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
hasDerivAtFilter_const
|
[] |
[
679,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
678,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Filter.compl_mem_hyperfilter_of_finite
|
[] |
[
499,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.TM2to1.trNormal_run
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝² : DecidableEq K\nΓ : K → Type u_2\nΛ : Type u_4\ninst✝¹ : Inhabited Λ\nσ : Type u_3\ninst✝ : Inhabited σ\nk : K\ns : StAct k\nq : Stmt₂\n⊢ trNormal (stRun s q) = goto fun x x => go k s q",
"tactic": "cases s <;> rfl"
}
] |
[
2531,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2529,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.aeval_def
|
[] |
[
190,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/RingTheory/Ideal/Prod.lean
|
Ideal.prod_top_top
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\ninst✝¹ : Ring R\ninst✝ : Ring S\nI I' : Ideal R\nJ J' : Ideal S\n⊢ ∀ (x : R × S), x ∈ prod ⊤ ⊤ ↔ x ∈ ⊤",
"tactic": "simp"
}
] |
[
48,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
Matrix.toBilin_basisFun
|
[
{
"state_after": "case h.H\nR : Type ?u.1302880\nM✝ : Type ?u.1302883\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : AddCommMonoid M✝\ninst✝¹⁵ : Module R M✝\nR₁ : Type ?u.1302919\nM₁ : Type ?u.1302922\ninst✝¹⁴ : Ring R₁\ninst✝¹³ : AddCommGroup M₁\ninst✝¹² : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type ?u.1303534\ninst✝¹¹ : CommSemiring R₂\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : Module R₂ M₂\nR₃ : Type ?u.1303721\nM₃ : Type ?u.1303724\ninst✝⁸ : CommRing R₃\ninst✝⁷ : AddCommGroup M₃\ninst✝⁶ : Module R₃ M₃\nV : Type ?u.1304312\nK : Type ?u.1304315\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB : BilinForm R M✝\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_2\no : Type ?u.1305532\ninst✝² : Fintype n\ninst✝¹ : Fintype o\ninst✝ : DecidableEq n\nb : Basis n R₂ M₂\nM : Matrix n n R₂\nx✝ y✝ : n → R₂\n⊢ bilin (↑(toBilin (Pi.basisFun R₂ n)) M) x✝ y✝ = bilin (↑toBilin' M) x✝ y✝",
"state_before": "R : Type ?u.1302880\nM : Type ?u.1302883\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : AddCommMonoid M\ninst✝¹⁵ : Module R M\nR₁ : Type ?u.1302919\nM₁ : Type ?u.1302922\ninst✝¹⁴ : Ring R₁\ninst✝¹³ : AddCommGroup M₁\ninst✝¹² : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type ?u.1303534\ninst✝¹¹ : CommSemiring R₂\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : Module R₂ M₂\nR₃ : Type ?u.1303721\nM₃ : Type ?u.1303724\ninst✝⁸ : CommRing R₃\ninst✝⁷ : AddCommGroup M₃\ninst✝⁶ : Module R₃ M₃\nV : Type ?u.1304312\nK : Type ?u.1304315\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_2\no : Type ?u.1305532\ninst✝² : Fintype n\ninst✝¹ : Fintype o\ninst✝ : DecidableEq n\nb : Basis n R₂ M₂\n⊢ toBilin (Pi.basisFun R₂ n) = toBilin'",
"tactic": "ext M"
},
{
"state_after": "no goals",
"state_before": "case h.H\nR : Type ?u.1302880\nM✝ : Type ?u.1302883\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : AddCommMonoid M✝\ninst✝¹⁵ : Module R M✝\nR₁ : Type ?u.1302919\nM₁ : Type ?u.1302922\ninst✝¹⁴ : Ring R₁\ninst✝¹³ : AddCommGroup M₁\ninst✝¹² : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type ?u.1303534\ninst✝¹¹ : CommSemiring R₂\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : Module R₂ M₂\nR₃ : Type ?u.1303721\nM₃ : Type ?u.1303724\ninst✝⁸ : CommRing R₃\ninst✝⁷ : AddCommGroup M₃\ninst✝⁶ : Module R₃ M₃\nV : Type ?u.1304312\nK : Type ?u.1304315\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB : BilinForm R M✝\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_2\no : Type ?u.1305532\ninst✝² : Fintype n\ninst✝¹ : Fintype o\ninst✝ : DecidableEq n\nb : Basis n R₂ M₂\nM : Matrix n n R₂\nx✝ y✝ : n → R₂\n⊢ bilin (↑(toBilin (Pi.basisFun R₂ n)) M) x✝ y✝ = bilin (↑toBilin' M) x✝ y✝",
"tactic": "simp only [Matrix.toBilin_apply, Matrix.toBilin'_apply, Pi.basisFun_repr]"
}
] |
[
337,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.transpose_nonsing_inv
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.184059\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B : Matrix n n α\n⊢ A⁻¹ᵀ = Aᵀ⁻¹",
"tactic": "rw [inv_def, inv_def, transpose_smul, det_transpose, adjugate_transpose]"
}
] |
[
293,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.lift_eq_iff_equiv
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nc₁ c₂ : Computation α\nx✝ : LiftRel (fun x x_1 => x = x_1) c₁ c₂\na : α\nh1 : ∀ {a : α}, a ∈ c₁ → ∃ b, b ∈ c₂ ∧ (fun x x_1 => x = x_1) a b\nh2 : ∀ {b : α}, b ∈ c₂ → ∃ a, a ∈ c₁ ∧ (fun x x_1 => x = x_1) a b\na1 : a ∈ c₁\nb : α\nb2 : b ∈ c₂\nab : (fun x x_1 => x = x_1) a b\n⊢ a ∈ c₂",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nc₁ c₂ : Computation α\nx✝ : LiftRel (fun x x_1 => x = x_1) c₁ c₂\na : α\nh1 : ∀ {a : α}, a ∈ c₁ → ∃ b, b ∈ c₂ ∧ (fun x x_1 => x = x_1) a b\nh2 : ∀ {b : α}, b ∈ c₂ → ∃ a, a ∈ c₁ ∧ (fun x x_1 => x = x_1) a b\na1 : a ∈ c₁\n⊢ a ∈ c₂",
"tactic": "let ⟨b, b2, ab⟩ := h1 a1"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nc₁ c₂ : Computation α\nx✝ : LiftRel (fun x x_1 => x = x_1) c₁ c₂\na : α\nh1 : ∀ {a : α}, a ∈ c₁ → ∃ b, b ∈ c₂ ∧ (fun x x_1 => x = x_1) a b\nh2 : ∀ {b : α}, b ∈ c₂ → ∃ a, a ∈ c₁ ∧ (fun x x_1 => x = x_1) a b\na1 : a ∈ c₁\nb : α\nb2 : b ∈ c₂\nab : (fun x x_1 => x = x_1) a b\n⊢ a ∈ c₂",
"tactic": "rwa [ab]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nc₁ c₂ : Computation α\nx✝ : LiftRel (fun x x_1 => x = x_1) c₁ c₂\na : α\nh1 : ∀ {a : α}, a ∈ c₁ → ∃ b, b ∈ c₂ ∧ (fun x x_1 => x = x_1) a b\nh2 : ∀ {b : α}, b ∈ c₂ → ∃ a, a ∈ c₁ ∧ (fun x x_1 => x = x_1) a b\na2 : a ∈ c₂\nb : α\nb1 : b ∈ c₁\nab : (fun x x_1 => x = x_1) b a\n⊢ a ∈ c₁",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nc₁ c₂ : Computation α\nx✝ : LiftRel (fun x x_1 => x = x_1) c₁ c₂\na : α\nh1 : ∀ {a : α}, a ∈ c₁ → ∃ b, b ∈ c₂ ∧ (fun x x_1 => x = x_1) a b\nh2 : ∀ {b : α}, b ∈ c₂ → ∃ a, a ∈ c₁ ∧ (fun x x_1 => x = x_1) a b\na2 : a ∈ c₂\n⊢ a ∈ c₁",
"tactic": "let ⟨b, b1, ab⟩ := h2 a2"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nc₁ c₂ : Computation α\nx✝ : LiftRel (fun x x_1 => x = x_1) c₁ c₂\na : α\nh1 : ∀ {a : α}, a ∈ c₁ → ∃ b, b ∈ c₂ ∧ (fun x x_1 => x = x_1) a b\nh2 : ∀ {b : α}, b ∈ c₂ → ∃ a, a ∈ c₁ ∧ (fun x x_1 => x = x_1) a b\na2 : a ∈ c₂\nb : α\nb1 : b ∈ c₁\nab : (fun x x_1 => x = x_1) b a\n⊢ a ∈ c₁",
"tactic": "rwa [← ab]"
}
] |
[
1062,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1055,
1
] |
Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean
|
DirectSum.toAddMonoidAlgebra_mul
|
[
{
"state_after": "ι : Type u_1\nR : Type ?u.136196\nM : Type u_2\nA : Type ?u.136202\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\ninst✝¹ : Semiring M\ninst✝ : (m : M) → Decidable (m ≠ 0)\nf g : ⨁ (x : ι), M\n⊢ AddMonoidAlgebra.toDirectSum (toAddMonoidAlgebra (f * g)) =\n AddMonoidAlgebra.toDirectSum (toAddMonoidAlgebra f * toAddMonoidAlgebra g)\n\ncase inj\nι : Type u_1\nR : Type ?u.136196\nM : Type u_2\nA : Type ?u.136202\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\ninst✝¹ : Semiring M\ninst✝ : (m : M) → Decidable (m ≠ 0)\nf g : ⨁ (x : ι), M\n⊢ Function.Injective AddMonoidAlgebra.toDirectSum",
"state_before": "ι : Type u_1\nR : Type ?u.136196\nM : Type u_2\nA : Type ?u.136202\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\ninst✝¹ : Semiring M\ninst✝ : (m : M) → Decidable (m ≠ 0)\nf g : ⨁ (x : ι), M\n⊢ toAddMonoidAlgebra (f * g) = toAddMonoidAlgebra f * toAddMonoidAlgebra g",
"tactic": "apply_fun AddMonoidAlgebra.toDirectSum"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nR : Type ?u.136196\nM : Type u_2\nA : Type ?u.136202\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\ninst✝¹ : Semiring M\ninst✝ : (m : M) → Decidable (m ≠ 0)\nf g : ⨁ (x : ι), M\n⊢ AddMonoidAlgebra.toDirectSum (toAddMonoidAlgebra (f * g)) =\n AddMonoidAlgebra.toDirectSum (toAddMonoidAlgebra f * toAddMonoidAlgebra g)",
"tactic": "simp"
},
{
"state_after": "case inj.a\nι : Type u_1\nR : Type ?u.136196\nM : Type u_2\nA : Type ?u.136202\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\ninst✝¹ : Semiring M\ninst✝ : (m : M) → Decidable (m ≠ 0)\nf g : ⨁ (x : ι), M\n⊢ Function.LeftInverse ?inj.g AddMonoidAlgebra.toDirectSum\n\ncase inj.g\nι : Type u_1\nR : Type ?u.136196\nM : Type u_2\nA : Type ?u.136202\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\ninst✝¹ : Semiring M\ninst✝ : (m : M) → Decidable (m ≠ 0)\nf g : ⨁ (x : ι), M\n⊢ (⨁ (x : ι), M) → AddMonoidAlgebra M ι",
"state_before": "case inj\nι : Type u_1\nR : Type ?u.136196\nM : Type u_2\nA : Type ?u.136202\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\ninst✝¹ : Semiring M\ninst✝ : (m : M) → Decidable (m ≠ 0)\nf g : ⨁ (x : ι), M\n⊢ Function.Injective AddMonoidAlgebra.toDirectSum",
"tactic": "apply Function.LeftInverse.injective"
},
{
"state_after": "no goals",
"state_before": "case inj.a\nι : Type u_1\nR : Type ?u.136196\nM : Type u_2\nA : Type ?u.136202\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\ninst✝¹ : Semiring M\ninst✝ : (m : M) → Decidable (m ≠ 0)\nf g : ⨁ (x : ι), M\n⊢ Function.LeftInverse ?inj.g AddMonoidAlgebra.toDirectSum\n\ncase inj.g\nι : Type u_1\nR : Type ?u.136196\nM : Type u_2\nA : Type ?u.136202\ninst✝³ : DecidableEq ι\ninst✝² : AddMonoid ι\ninst✝¹ : Semiring M\ninst✝ : (m : M) → Decidable (m ≠ 0)\nf g : ⨁ (x : ι), M\n⊢ (⨁ (x : ι), M) → AddMonoidAlgebra M ι",
"tactic": "apply AddMonoidAlgebra.toDirectSum_toAddMonoidAlgebra"
}
] |
[
184,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Topology/MetricSpace/Closeds.lean
|
EMetric.NonemptyCompacts.isClosed_in_closeds
|
[
{
"state_after": "α : Type u\ninst✝¹ : EMetricSpace α\ns : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ IsClosed (range NonemptyCompacts.toCloseds)",
"state_before": "α : Type u\ninst✝¹ : EMetricSpace α\ns : Set α\ninst✝ : CompleteSpace α\n⊢ IsClosed (range NonemptyCompacts.toCloseds)",
"tactic": "have :\n range NonemptyCompacts.toCloseds =\n { s : Closeds α | (s : Set α).Nonempty ∧ IsCompact (s : Set α) } := by\n ext s\n refine' ⟨_, fun h => ⟨⟨⟨s, h.2⟩, h.1⟩, Closeds.ext rfl⟩⟩\n rintro ⟨s, hs, rfl⟩\n exact ⟨s.nonempty, s.isCompact⟩"
},
{
"state_after": "α : Type u\ninst✝¹ : EMetricSpace α\ns : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ IsClosed {s | Set.Nonempty ↑s ∧ IsCompact ↑s}",
"state_before": "α : Type u\ninst✝¹ : EMetricSpace α\ns : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ IsClosed (range NonemptyCompacts.toCloseds)",
"tactic": "rw [this]"
},
{
"state_after": "case refine'_1\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ Set.Nonempty ↑s\n\ncase refine'_2\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ IsCompact ↑s",
"state_before": "α : Type u\ninst✝¹ : EMetricSpace α\ns : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ IsClosed {s | Set.Nonempty ↑s ∧ IsCompact ↑s}",
"tactic": "refine' isClosed_of_closure_subset fun s hs => ⟨_, _⟩"
},
{
"state_after": "case h\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\ns : Closeds α\n⊢ s ∈ range NonemptyCompacts.toCloseds ↔ s ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}",
"state_before": "α : Type u\ninst✝¹ : EMetricSpace α\ns : Set α\ninst✝ : CompleteSpace α\n⊢ range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}",
"tactic": "ext s"
},
{
"state_after": "case h\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\ns : Closeds α\n⊢ s ∈ range NonemptyCompacts.toCloseds → s ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}",
"state_before": "case h\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\ns : Closeds α\n⊢ s ∈ range NonemptyCompacts.toCloseds ↔ s ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}",
"tactic": "refine' ⟨_, fun h => ⟨⟨⟨s, h.2⟩, h.1⟩, Closeds.ext rfl⟩⟩"
},
{
"state_after": "case h.intro.refl\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\ns : NonemptyCompacts α\n⊢ NonemptyCompacts.toCloseds s ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}",
"state_before": "case h\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\ns : Closeds α\n⊢ s ∈ range NonemptyCompacts.toCloseds → s ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}",
"tactic": "rintro ⟨s, hs, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.intro.refl\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\ns : NonemptyCompacts α\n⊢ NonemptyCompacts.toCloseds s ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}",
"tactic": "exact ⟨s.nonempty, s.isCompact⟩"
},
{
"state_after": "case refine'_1.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ⊤\n⊢ Set.Nonempty ↑s",
"state_before": "case refine'_1\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ Set.Nonempty ↑s",
"tactic": "rcases mem_closure_iff.1 hs ⊤ ENNReal.coe_lt_top with ⟨t, ht, Dst⟩"
},
{
"state_after": "case refine'_1.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist t s < ⊤\n⊢ Set.Nonempty ↑s",
"state_before": "case refine'_1.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ⊤\n⊢ Set.Nonempty ↑s",
"tactic": "rw [edist_comm] at Dst"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist t s < ⊤\n⊢ Set.Nonempty ↑s",
"tactic": "exact nonempty_of_hausdorffEdist_ne_top ht.1 (ne_of_lt Dst)"
},
{
"state_after": "case refine'_2\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ TotallyBounded ↑s",
"state_before": "case refine'_2\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ IsCompact ↑s",
"tactic": "refine' isCompact_iff_totallyBounded_isComplete.2 ⟨_, s.closed.isComplete⟩"
},
{
"state_after": "case refine'_2\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\n⊢ ∃ t, Set.Finite t ∧ ↑s ⊆ ⋃ (y : α) (_ : y ∈ t), ball y ε",
"state_before": "case refine'_2\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\n⊢ TotallyBounded ↑s",
"tactic": "refine' totallyBounded_iff.2 fun ε (εpos : 0 < ε) => _"
},
{
"state_after": "case refine'_2.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\n⊢ ∃ t, Set.Finite t ∧ ↑s ⊆ ⋃ (y : α) (_ : y ∈ t), ball y ε",
"state_before": "case refine'_2\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\n⊢ ∃ t, Set.Finite t ∧ ↑s ⊆ ⋃ (y : α) (_ : y ∈ t), ball y ε",
"tactic": "rcases mem_closure_iff.1 hs (ε / 2) (ENNReal.half_pos εpos.ne') with ⟨t, ht, Dst⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\n⊢ ∃ t, Set.Finite t ∧ ↑s ⊆ ⋃ (y : α) (_ : y ∈ t), ball y ε",
"state_before": "case refine'_2.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\n⊢ ∃ t, Set.Finite t ∧ ↑s ⊆ ⋃ (y : α) (_ : y ∈ t), ball y ε",
"tactic": "rcases totallyBounded_iff.1 (isCompact_iff_totallyBounded_isComplete.1 ht.2).1 (ε / 2)\n (ENNReal.half_pos εpos.ne') with\n ⟨u, fu, ut⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\nx : α\nhx : x ∈ ↑s\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ u), ball y ε",
"state_before": "case refine'_2.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\n⊢ ∃ t, Set.Finite t ∧ ↑s ⊆ ⋃ (y : α) (_ : y ∈ t), ball y ε",
"tactic": "refine' ⟨u, ⟨fu, fun x hx => _⟩⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\nx : α\nhx : x ∈ ↑s\nz : α\nhz : z ∈ ↑t\nDxz : edist x z < ε / 2\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ u), ball y ε",
"state_before": "case refine'_2.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\nx : α\nhx : x ∈ ↑s\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ u), ball y ε",
"tactic": "rcases exists_edist_lt_of_hausdorffEdist_lt hx Dst with ⟨z, hz, Dxz⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\nx : α\nhx : x ∈ ↑s\nz : α\nhz : z ∈ ↑t\nDxz : edist x z < ε / 2\ny : α\nhy : y ∈ u\nDzy : z ∈ ball y (ε / 2)\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ u), ball y ε",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\nx : α\nhx : x ∈ ↑s\nz : α\nhz : z ∈ ↑t\nDxz : edist x z < ε / 2\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ u), ball y ε",
"tactic": "rcases mem_iUnion₂.1 (ut hz) with ⟨y, hy, Dzy⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis✝ : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\nx : α\nhx : x ∈ ↑s\nz : α\nhz : z ∈ ↑t\nDxz : edist x z < ε / 2\ny : α\nhy : y ∈ u\nDzy : z ∈ ball y (ε / 2)\nthis : edist x y < ε\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ u), ball y ε",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\nx : α\nhx : x ∈ ↑s\nz : α\nhz : z ∈ ↑t\nDxz : edist x z < ε / 2\ny : α\nhy : y ∈ u\nDzy : z ∈ ball y (ε / 2)\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ u), ball y ε",
"tactic": "have : edist x y < ε :=\n calc\n edist x y ≤ edist x z + edist z y := edist_triangle _ _ _\n _ < ε / 2 + ε / 2 := (ENNReal.add_lt_add Dxz Dzy)\n _ = ε := ENNReal.add_halves _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u\ninst✝¹ : EMetricSpace α\ns✝ : Set α\ninst✝ : CompleteSpace α\nthis✝ : range NonemptyCompacts.toCloseds = {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\ns : Closeds α\nhs : s ∈ closure {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nε : ℝ≥0∞\nεpos : 0 < ε\nt : Closeds α\nht : t ∈ {s | Set.Nonempty ↑s ∧ IsCompact ↑s}\nDst : edist s t < ε / 2\nu : Set α\nfu : Set.Finite u\nut : ↑t ⊆ ⋃ (y : α) (_ : y ∈ u), ball y (ε / 2)\nx : α\nhx : x ∈ ↑s\nz : α\nhz : z ∈ ↑t\nDxz : edist x z < ε / 2\ny : α\nhy : y ∈ u\nDzy : z ∈ ball y (ε / 2)\nthis : edist x y < ε\n⊢ x ∈ ⋃ (y : α) (_ : y ∈ u), ball y ε",
"tactic": "exact mem_biUnion hy this"
}
] |
[
300,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Algebra/Associated.lean
|
associated_zero_iff_eq_zero
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.151491\nγ : Type ?u.151494\nδ : Type ?u.151497\ninst✝ : MonoidWithZero α\na : α\nh✝ : a ~ᵤ 0\nu : αˣ\nh : 0 * ↑u = a\n⊢ a = 0",
"state_before": "α : Type u_1\nβ : Type ?u.151491\nγ : Type ?u.151494\nδ : Type ?u.151497\ninst✝ : MonoidWithZero α\na : α\nh : a ~ᵤ 0\n⊢ a = 0",
"tactic": "let ⟨u, h⟩ := h.symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.151491\nγ : Type ?u.151494\nδ : Type ?u.151497\ninst✝ : MonoidWithZero α\na : α\nh✝ : a ~ᵤ 0\nu : αˣ\nh : 0 * ↑u = a\n⊢ a = 0",
"tactic": "simpa using h.symm"
}
] |
[
441,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Std/Data/List/Lemmas.lean
|
List.eraseP_map
|
[
{
"state_after": "no goals",
"state_before": "β : Type u_1\nα : Type u_2\np : α → Bool\nf : β → α\nb : β\nl : List β\n⊢ eraseP p (map f (b :: l)) = map f (eraseP (p ∘ f) (b :: l))",
"tactic": "by_cases h : p (f b) <;> simp [h, eraseP_map f l, eraseP_cons_of_pos]"
}
] |
[
1011,
85
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1009,
1
] |
Mathlib/Topology/Order.lean
|
generateFrom_union
|
[] |
[
940,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
938,
1
] |
Mathlib/Topology/Inseparable.lean
|
Specializes.mem_closed
|
[] |
[
132,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
|
MvQPF.Fix.rec_eq
|
[
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\n⊢ rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\n⊢ rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x)",
"tactic": "have : recF g ∘ fixToW = Fix.rec g := by\n apply funext\n apply Quotient.ind\n intro x\n apply recF_eq_of_wEquiv\n apply wrepr_equiv"
},
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\n⊢ recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> repr x)) = g ((TypeVec.id ::: rec g) <$$> x)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\n⊢ rec g (mk x) = g ((TypeVec.id ::: rec g) <$$> x)",
"tactic": "conv =>\n lhs\n rw [Fix.rec, Fix.mk]\n dsimp"
},
{
"state_after": "case mk\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α ::: Fix F α\nh : repr x = { fst := a, snd := f }\n⊢ recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =\n g ((TypeVec.id ::: rec g) <$$> x)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\n⊢ recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> repr x)) = g ((TypeVec.id ::: rec g) <$$> x)",
"tactic": "cases' h : repr x with a f"
},
{
"state_after": "case mk\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α ::: Fix F α\nh : repr x = { fst := a, snd := f }\n⊢ g (abs ((TypeVec.id ::: recF g) <$$> (TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =\n g ((TypeVec.id ::: rec g) <$$> x)",
"state_before": "case mk\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α ::: Fix F α\nh : repr x = { fst := a, snd := f }\n⊢ recF g (MvPFunctor.wMk' (P F) ((TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =\n g ((TypeVec.id ::: rec g) <$$> x)",
"tactic": "rw [MvPFunctor.map_eq, recF_eq', ← MvPFunctor.map_eq, MvPFunctor.wDest'_wMk']"
},
{
"state_after": "no goals",
"state_before": "case mk\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\nthis : recF g ∘ fixToW = rec g\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α ::: Fix F α\nh : repr x = { fst := a, snd := f }\n⊢ g (abs ((TypeVec.id ::: recF g) <$$> (TypeVec.id ::: fixToW) <$$> { fst := a, snd := f })) =\n g ((TypeVec.id ::: rec g) <$$> x)",
"tactic": "rw [← MvPFunctor.comp_map, abs_map, ← h, abs_repr, ← appendFun_comp, id_comp, this]"
},
{
"state_after": "case h\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\n⊢ ∀ (x : Fix F α), (recF g ∘ fixToW) x = rec g x",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\n⊢ recF g ∘ fixToW = rec g",
"tactic": "apply funext"
},
{
"state_after": "case h.a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\n⊢ ∀ (a : MvPFunctor.W (P F) α), (recF g ∘ fixToW) (Quotient.mk (wSetoid α) a) = rec g (Quotient.mk (wSetoid α) a)",
"state_before": "case h\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\n⊢ ∀ (x : Fix F α), (recF g ∘ fixToW) x = rec g x",
"tactic": "apply Quotient.ind"
},
{
"state_after": "case h.a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx✝ : F (α ::: Fix F α)\nx : MvPFunctor.W (P F) α\n⊢ (recF g ∘ fixToW) (Quotient.mk (wSetoid α) x) = rec g (Quotient.mk (wSetoid α) x)",
"state_before": "case h.a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx : F (α ::: Fix F α)\n⊢ ∀ (a : MvPFunctor.W (P F) α), (recF g ∘ fixToW) (Quotient.mk (wSetoid α) a) = rec g (Quotient.mk (wSetoid α) a)",
"tactic": "intro x"
},
{
"state_after": "case h.a.a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx✝ : F (α ::: Fix F α)\nx : MvPFunctor.W (P F) α\n⊢ WEquiv (fixToW (Quotient.mk (wSetoid α) x)) x",
"state_before": "case h.a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx✝ : F (α ::: Fix F α)\nx : MvPFunctor.W (P F) α\n⊢ (recF g ∘ fixToW) (Quotient.mk (wSetoid α) x) = rec g (Quotient.mk (wSetoid α) x)",
"tactic": "apply recF_eq_of_wEquiv"
},
{
"state_after": "no goals",
"state_before": "case h.a.a\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\ng : F (α ::: β) → β\nx✝ : F (α ::: Fix F α)\nx : MvPFunctor.W (P F) α\n⊢ WEquiv (fixToW (Quotient.mk (wSetoid α) x)) x",
"tactic": "apply wrepr_equiv"
}
] |
[
242,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.one_smul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁶ : CommSemiring R\nR' : Type u_4\ninst✝¹⁵ : Monoid R'\nR'' : Type ?u.221106\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.221118\nQ : Type ?u.221121\nS : Type ?u.221124\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nx : M ⊗[R] N\nthis : ∀ (r : R') (m : M) (n : N), r • m ⊗ₜ[R] n = (r • m) ⊗ₜ[R] n\n⊢ 1 • 0 = 0",
"tactic": "rw [TensorProduct.smul_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁶ : CommSemiring R\nR' : Type u_4\ninst✝¹⁵ : Monoid R'\nR'' : Type ?u.221106\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.221118\nQ : Type ?u.221121\nS : Type ?u.221124\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nx : M ⊗[R] N\nthis : ∀ (r : R') (m : M) (n : N), r • m ⊗ₜ[R] n = (r • m) ⊗ₜ[R] n\nm : M\nn : N\n⊢ 1 • m ⊗ₜ[R] n = m ⊗ₜ[R] n",
"tactic": "rw [this, one_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁶ : CommSemiring R\nR' : Type u_4\ninst✝¹⁵ : Monoid R'\nR'' : Type ?u.221106\ninst✝¹⁴ : Semiring R''\nM : Type u_2\nN : Type u_3\nP : Type ?u.221118\nQ : Type ?u.221121\nS : Type ?u.221124\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : AddCommMonoid P\ninst✝¹⁰ : AddCommMonoid Q\ninst✝⁹ : AddCommMonoid S\ninst✝⁸ : Module R M\ninst✝⁷ : Module R N\ninst✝⁶ : Module R P\ninst✝⁵ : Module R Q\ninst✝⁴ : Module R S\ninst✝³ : DistribMulAction R' M\ninst✝² : Module R'' M\ninst✝¹ : SMulCommClass R R' M\ninst✝ : SMulCommClass R R'' M\nx✝ : M ⊗[R] N\nthis : ∀ (r : R') (m : M) (n : N), r • m ⊗ₜ[R] n = (r • m) ⊗ₜ[R] n\nx y : M ⊗[R] N\nihx : 1 • x = x\nihy : 1 • y = y\n⊢ 1 • (x + y) = x + y",
"tactic": "rw [TensorProduct.smul_add, ihx, ihy]"
}
] |
[
263,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
11
] |
Mathlib/Algebra/Order/WithZero.lean
|
mul_lt_mul₀
|
[] |
[
190,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/Algebra/ContinuedFractions/Translations.lean
|
GeneralizedContinuedFraction.first_continuant_aux_eq_h_one
|
[] |
[
128,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/RingTheory/FreeCommRing.lean
|
FreeCommRing.isSupported_add
|
[] |
[
190,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/CategoryTheory/Monoidal/Mon_.lean
|
Mon_.mul_associator
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ (tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul)) ≫\n (α_ M.X N.X P.X).hom =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))",
"tactic": "simp"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ ((M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ P.mul) ≫ (α_ M.X N.X P.X).hom =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))",
"tactic": "slice_lhs 2 3 => rw [← Category.id_comp P.mul, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫ ((M.mul ⊗ N.mul) ⊗ P.mul)) ≫ (α_ M.X N.X P.X).hom =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))",
"tactic": "slice_lhs 3 4 => rw [associator_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (M.mul ⊗ tensor_μ C (N.X, P.X) (N.X, P.X) ≫ (N.mul ⊗ P.mul))",
"tactic": "slice_rhs 3 4 => rw [← Category.id_comp M.mul, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ (((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X))) ≫\n (M.mul ⊗ N.mul ⊗ P.mul) =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ tensor_μ C (M.X ⊗ N.X, P.X) (M.X ⊗ N.X, P.X) ≫\n (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (P.X ⊗ P.X)) ≫\n (α_ (M.X ⊗ M.X) (N.X ⊗ N.X) (P.X ⊗ P.X)).hom ≫ (M.mul ⊗ N.mul ⊗ P.mul) =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul)",
"tactic": "slice_lhs 1 3 => rw [associator_monoidal]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N P : Mon_ C\n⊢ (((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫ (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X))) ≫\n (M.mul ⊗ N.mul ⊗ P.mul) =\n ((α_ M.X N.X P.X).hom ⊗ (α_ M.X N.X P.X).hom) ≫\n tensor_μ C (M.X, N.X ⊗ P.X) (M.X, N.X ⊗ P.X) ≫\n (𝟙 (M.X ⊗ M.X) ⊗ tensor_μ C (N.X, P.X) (N.X, P.X)) ≫ (M.mul ⊗ N.mul ⊗ P.mul)",
"tactic": "simp only [Category.assoc]"
}
] |
[
460,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean
|
AlgebraicTopology.DoldKan.N₁Γ₀_app
|
[
{
"state_after": "case w\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\n⊢ (N₁Γ₀.app K).hom =\n ((Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).symm ≪≫\n (toKaroubi (ChainComplex C ℕ)).mapIso (Γ₀NondegComplexIso K)).hom",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\n⊢ N₁Γ₀.app K =\n (Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).symm ≪≫\n (toKaroubi (ChainComplex C ℕ)).mapIso (Γ₀NondegComplexIso K)",
"tactic": "ext1"
},
{
"state_after": "case w\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\n⊢ (((Karoubi.Hom.mk PInfty ≫ Split.toKaroubiNondegComplexFunctorIsoN₁.inv.app (Split.mk' (Γ₀.splitting K))) ≫\n Karoubi.Hom.mk (𝟙 (Splitting.nondegComplex (Γ₀.splitting K)))) ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀'CompNondegComplexFunctor.hom.app K)) ≫\n Karoubi.Hom.mk (𝟙 K) =\n (Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).inv ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀NondegComplexIso K).hom",
"state_before": "case w\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\n⊢ (N₁Γ₀.app K).hom =\n ((Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).symm ≪≫\n (toKaroubi (ChainComplex C ℕ)).mapIso (Γ₀NondegComplexIso K)).hom",
"tactic": "dsimp [N₁Γ₀]"
},
{
"state_after": "case w\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\n⊢ Split.toKaroubiNondegComplexFunctorIsoN₁.inv.app (Split.mk' (Γ₀.splitting K)) ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀'CompNondegComplexFunctor.hom.app K) =\n (Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).inv ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀NondegComplexIso K).hom",
"state_before": "case w\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\n⊢ (((Karoubi.Hom.mk PInfty ≫ Split.toKaroubiNondegComplexFunctorIsoN₁.inv.app (Split.mk' (Γ₀.splitting K))) ≫\n Karoubi.Hom.mk (𝟙 (Splitting.nondegComplex (Γ₀.splitting K)))) ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀'CompNondegComplexFunctor.hom.app K)) ≫\n Karoubi.Hom.mk (𝟙 K) =\n (Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).inv ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀NondegComplexIso K).hom",
"tactic": "erw [id_comp, comp_id, comp_id]"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\n⊢ Split.toKaroubiNondegComplexFunctorIsoN₁.inv.app (Split.mk' (Γ₀.splitting K)) ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀'CompNondegComplexFunctor.hom.app K) =\n (Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).inv ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀NondegComplexIso K).hom",
"tactic": "rfl"
}
] |
[
82,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
tendsto_nat_floor_atTop
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.493938\nβ : Type ?u.493941\nι : Type ?u.493944\nα : Type u_1\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\nx : ℕ\n⊢ x ≤ ⌊max 0 (↑x + 1)⌋₊",
"tactic": "simp [Nat.le_floor_iff]"
}
] |
[
574,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
572,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Disjoint.exists_mem_filter_basis
|
[] |
[
656,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
654,
1
] |
Mathlib/Order/SupIndep.lean
|
CompleteLattice.Independent.mono
|
[] |
[
286,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
span_le_span_iff
|
[] |
[
1103,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1101,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
sum_pow_char
|
[] |
[
423,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/MeasureTheory/Function/EssSup.lean
|
meas_lt_essInf
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝³ : ConditionallyCompleteLinearOrder β\nx : β\nf : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : FirstCountableTopology β\ninst✝ : OrderTopology β\nhf : autoParam (IsBoundedUnder (fun x x_1 => x ≥ x_1) (Measure.ae μ) f) _auto✝\n⊢ ↑↑μ {y | ¬essInf f μ ≤ f y} = 0",
"state_before": "α : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝³ : ConditionallyCompleteLinearOrder β\nx : β\nf : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : FirstCountableTopology β\ninst✝ : OrderTopology β\nhf : autoParam (IsBoundedUnder (fun x x_1 => x ≥ x_1) (Measure.ae μ) f) _auto✝\n⊢ ↑↑μ {y | f y < essInf f μ} = 0",
"tactic": "simp_rw [← not_le]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\ninst✝³ : ConditionallyCompleteLinearOrder β\nx : β\nf : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : FirstCountableTopology β\ninst✝ : OrderTopology β\nhf : autoParam (IsBoundedUnder (fun x x_1 => x ≥ x_1) (Measure.ae μ) f) _auto✝\n⊢ ↑↑μ {y | ¬essInf f μ ≤ f y} = 0",
"tactic": "exact ae_essInf_le hf"
}
] |
[
140,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
sdiff_sup_self
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.91987\nα : Type u_1\nβ : Type ?u.91993\ninst✝ : GeneralizedCoheytingAlgebra α\na✝ b✝ c d a b : α\n⊢ b \\ a ⊔ a = b ⊔ a",
"tactic": "rw [sup_comm, sup_sdiff_self, sup_comm]"
}
] |
[
562,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
562,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.continuousAt_extend_symm'
|
[] |
[
858,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
856,
1
] |
Mathlib/Order/Heyting/Boundary.lean
|
Coheyting.boundary_le
|
[] |
[
53,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.IsEquiv.of_eq
|
[] |
[
842,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
Submodule.linear_proj_add_linearProjOfIsCompl_eq_self
|
[
{
"state_after": "R : Type u_2\ninst✝⁹ : Ring R\nE : Type u_1\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.186731\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.187247\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.188210\ninst✝² : Semiring S\nM : Type ?u.188216\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nhpq : IsCompl p q\nx : E\n⊢ ↑(↑(comp (LinearMap.fst R { x // x ∈ p } { x // x ∈ q }) ↑(LinearEquiv.symm (prodEquivOfIsCompl p q hpq))) x) +\n ↑(↑(comp (LinearMap.fst R { x // x ∈ q } { x // x ∈ p })\n ↑(LinearEquiv.symm (prodEquivOfIsCompl q p (_ : IsCompl q p))))\n x) =\n x",
"state_before": "R : Type u_2\ninst✝⁹ : Ring R\nE : Type u_1\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.186731\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.187247\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.188210\ninst✝² : Semiring S\nM : Type ?u.188216\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nhpq : IsCompl p q\nx : E\n⊢ ↑(↑(linearProjOfIsCompl p q hpq) x) + ↑(↑(linearProjOfIsCompl q p (_ : IsCompl q p)) x) = x",
"tactic": "dsimp only [linearProjOfIsCompl]"
},
{
"state_after": "R : Type u_2\ninst✝⁹ : Ring R\nE : Type u_1\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.186731\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.187247\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.188210\ninst✝² : Semiring S\nM : Type ?u.188216\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nhpq : IsCompl p q\nx : E\n⊢ ↑(↑(comp (LinearMap.fst R { x // x ∈ p } { x // x ∈ q }) ↑(LinearEquiv.symm (prodEquivOfIsCompl p q hpq))) x) +\n ↑(↑(comp (LinearMap.fst R { x // x ∈ q } { x // x ∈ p })\n ↑(LinearEquiv.symm\n (LinearEquiv.trans (LinearEquiv.prodComm R { x // x ∈ q } { x // x ∈ p })\n (prodEquivOfIsCompl p q hpq))))\n x) =\n x",
"state_before": "R : Type u_2\ninst✝⁹ : Ring R\nE : Type u_1\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.186731\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.187247\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.188210\ninst✝² : Semiring S\nM : Type ?u.188216\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nhpq : IsCompl p q\nx : E\n⊢ ↑(↑(comp (LinearMap.fst R { x // x ∈ p } { x // x ∈ q }) ↑(LinearEquiv.symm (prodEquivOfIsCompl p q hpq))) x) +\n ↑(↑(comp (LinearMap.fst R { x // x ∈ q } { x // x ∈ p })\n ↑(LinearEquiv.symm (prodEquivOfIsCompl q p (_ : IsCompl q p))))\n x) =\n x",
"tactic": "rw [← prodComm_trans_prodEquivOfIsCompl _ _ hpq]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁹ : Ring R\nE : Type u_1\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module R E\nF : Type ?u.186731\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module R F\nG : Type ?u.187247\ninst✝⁴ : AddCommGroup G\ninst✝³ : Module R G\np q : Submodule R E\nS : Type ?u.188210\ninst✝² : Semiring S\nM : Type ?u.188216\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nm : Submodule S M\nhpq : IsCompl p q\nx : E\n⊢ ↑(↑(comp (LinearMap.fst R { x // x ∈ p } { x // x ∈ q }) ↑(LinearEquiv.symm (prodEquivOfIsCompl p q hpq))) x) +\n ↑(↑(comp (LinearMap.fst R { x // x ∈ q } { x // x ∈ p })\n ↑(LinearEquiv.symm\n (LinearEquiv.trans (LinearEquiv.prodComm R { x // x ∈ q } { x // x ∈ p })\n (prodEquivOfIsCompl p q hpq))))\n x) =\n x",
"tactic": "exact (prodEquivOfIsCompl _ _ hpq).apply_symm_apply x"
}
] |
[
218,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.untrop_add
|
[] |
[
274,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Memℒp.toLp_const_smul
|
[] |
[
541,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Std/Data/List/Basic.lean
|
List.unzip_eq_unzipTR
|
[
{
"state_after": "case h.h.h\nα : Type u_2\nβ : Type u_1\nl : List (α × β)\n⊢ unzip l = unzipTR l",
"state_before": "⊢ @unzip = @unzipTR",
"tactic": "funext α β l"
},
{
"state_after": "case h.h.h\nα : Type u_2\nβ : Type u_1\nl : List (α × β)\n⊢ unzip l = foldr (fun x x_1 => (x.fst :: x_1.fst, x.snd :: x_1.snd)) ([], []) l",
"state_before": "case h.h.h\nα : Type u_2\nβ : Type u_1\nl : List (α × β)\n⊢ unzip l = unzipTR l",
"tactic": "simp [unzipTR]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nα : Type u_2\nβ : Type u_1\nl : List (α × β)\n⊢ unzip l = foldr (fun x x_1 => (x.fst :: x_1.fst, x.snd :: x_1.snd)) ([], []) l",
"tactic": "induction l <;> simp [*]"
}
] |
[
192,
57
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
191,
10
] |
Mathlib/Data/Set/Intervals/Infinite.lean
|
Set.Ici_infinite
|
[] |
[
102,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/CategoryTheory/Monoidal/Mod_.lean
|
Mod_.id_hom'
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nA : Mon_ C\nM✝ M : Mod_ A\n⊢ (𝟙 M).hom = 𝟙 M.X",
"tactic": "rfl"
}
] |
[
86,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.ae_eq_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.101184\nγ : Type ?u.101187\nδ : Type ?u.101190\nι : Type ?u.101193\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\n⊢ (∀ᵐ (x : α) ∂μ, ¬x ∈ s) ↔ ↑↑μ s = 0",
"tactic": "simp only [ae_iff, Classical.not_not, setOf_mem_eq]"
}
] |
[
443,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Algebra/Ring/Defs.lean
|
mul_one_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝² : Add α\ninst✝¹ : MulOneClass α\ninst✝ : LeftDistribClass α\na b : α\n⊢ a * (1 + b) = a + a * b",
"tactic": "rw [mul_add, mul_one]"
}
] |
[
170,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
StructureGroupoid.trans
|
[] |
[
183,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Analysis/Convex/Body.lean
|
ConvexBody.coe_smul
|
[] |
[
143,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Topology/Constructions.lean
|
embedding_subtype_val
|
[] |
[
1003,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.trans_le
|
[] |
[
581,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.SimpleFunc.setToSimpleFunc_congr'
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\n⊢ setToSimpleFunc T (map Prod.fst (pair f g)) = setToSimpleFunc T (map Prod.snd (pair f g))",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\n⊢ setToSimpleFunc T (map Prod.fst (pair f g)) = setToSimpleFunc T (map Prod.snd (pair f g))",
"tactic": "have h_pair : Integrable (f.pair g) μ := integrable_pair hf hg"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\n⊢ ∑ x in SimpleFunc.range (pair f g), ↑(T (↑(pair f g) ⁻¹' {x})) x.fst = setToSimpleFunc T (map Prod.snd (pair f g))",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\n⊢ setToSimpleFunc T (map Prod.fst (pair f g)) = setToSimpleFunc T (map Prod.snd (pair f g))",
"tactic": "rw [map_setToSimpleFunc T h_add h_pair Prod.fst_zero]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\n⊢ ∑ x in SimpleFunc.range (pair f g), ↑(T (↑(pair f g) ⁻¹' {x})) x.fst =\n ∑ x in SimpleFunc.range (pair f g), ↑(T (↑(pair f g) ⁻¹' {x})) x.snd",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\n⊢ ∑ x in SimpleFunc.range (pair f g), ↑(T (↑(pair f g) ⁻¹' {x})) x.fst = setToSimpleFunc T (map Prod.snd (pair f g))",
"tactic": "rw [map_setToSimpleFunc T h_add h_pair Prod.snd_zero]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np✝ : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\np : E × E\nhp : p ∈ SimpleFunc.range (pair f g)\n⊢ ↑(T (↑(pair f g) ⁻¹' {p})) p.fst = ↑(T (↑(pair f g) ⁻¹' {p})) p.snd",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\n⊢ ∑ x in SimpleFunc.range (pair f g), ↑(T (↑(pair f g) ⁻¹' {x})) x.fst =\n ∑ x in SimpleFunc.range (pair f g), ↑(T (↑(pair f g) ⁻¹' {x})) x.snd",
"tactic": "refine' Finset.sum_congr rfl fun p hp => _"
},
{
"state_after": "case intro\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\n⊢ ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).fst =\n ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).snd",
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"tactic": "rcases mem_range.1 hp with ⟨a, rfl⟩"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ↑f a = ↑g a\n⊢ ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).fst =\n ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).snd\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\n⊢ ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).fst =\n ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).snd",
"state_before": "case intro\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\n⊢ ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).fst =\n ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).snd",
"tactic": "by_cases eq : f a = g a"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ↑f a = ↑g a\n⊢ ↑(T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)})) (↑f a) = ↑(T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)})) (↑g a)",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ↑f a = ↑g a\n⊢ ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).fst =\n ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).snd",
"tactic": "dsimp only [pair_apply]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ↑f a = ↑g a\n⊢ ↑(T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)})) (↑f a) = ↑(T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)})) (↑g a)",
"tactic": "rw [eq]"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\nthis : T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = 0\n⊢ ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).fst =\n ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).snd",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\n⊢ ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).fst =\n ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).snd",
"tactic": "have : T (pair f g ⁻¹' {(f a, g a)}) = 0 := by\n have h_eq : T ((⇑(f.pair g)) ⁻¹' {(f a, g a)}) = T (f ⁻¹' {f a} ∩ g ⁻¹' {g a}) := by\n congr; rw [pair_preimage_singleton f g]\n rw [h_eq]\n exact h (f a) (g a) eq"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\nthis : T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = 0\n⊢ ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).fst =\n ↑(T (↑(pair f g) ⁻¹' {↑(pair f g) a})) (↑(pair f g) a).snd",
"tactic": "simp only [this, ContinuousLinearMap.zero_apply, pair_apply]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\nh_eq : T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = T (↑f ⁻¹' {↑f a} ∩ ↑g ⁻¹' {↑g a})\n⊢ T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = 0",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\n⊢ T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = 0",
"tactic": "have h_eq : T ((⇑(f.pair g)) ⁻¹' {(f a, g a)}) = T (f ⁻¹' {f a} ∩ g ⁻¹' {g a}) := by\n congr; rw [pair_preimage_singleton f g]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\nh_eq : T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = T (↑f ⁻¹' {↑f a} ∩ ↑g ⁻¹' {↑g a})\n⊢ T (↑f ⁻¹' {↑f a} ∩ ↑g ⁻¹' {↑g a}) = 0",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\nh_eq : T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = T (↑f ⁻¹' {↑f a} ∩ ↑g ⁻¹' {↑g a})\n⊢ T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = 0",
"tactic": "rw [h_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\nh_eq : T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = T (↑f ⁻¹' {↑f a} ∩ ↑g ⁻¹' {↑g a})\n⊢ T (↑f ⁻¹' {↑f a} ∩ ↑g ⁻¹' {↑g a}) = 0",
"tactic": "exact h (f a) (g a) eq"
},
{
"state_after": "case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\n⊢ ↑(pair f g) ⁻¹' {(↑f a, ↑g a)} = ↑f ⁻¹' {↑f a} ∩ ↑g ⁻¹' {↑g a}",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\n⊢ T (↑(pair f g) ⁻¹' {(↑f a, ↑g a)}) = T (↑f ⁻¹' {↑f a} ∩ ↑g ⁻¹' {↑g a})",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_a\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.191544\nG : Type ?u.191547\n𝕜 : Type ?u.191550\np : ℝ≥0∞\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_add : FinMeasAdditive μ T\nf g : α →ₛ E\nhf : Integrable ↑f\nhg : Integrable ↑g\nh : ∀ (x y : E), x ≠ y → T (↑f ⁻¹' {x} ∩ ↑g ⁻¹' {y}) = 0\nh_pair : Integrable ↑(pair f g)\na : α\nhp : ↑(pair f g) a ∈ SimpleFunc.range (pair f g)\neq : ¬↑f a = ↑g a\n⊢ ↑(pair f g) ⁻¹' {(↑f a, ↑g a)} = ↑f ⁻¹' {↑f a} ∩ ↑g ⁻¹' {↑g a}",
"tactic": "rw [pair_preimage_singleton f g]"
}
] |
[
380,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/Analysis/ODE/PicardLindelof.lean
|
IsPicardLindelof.norm_le₀
|
[] |
[
382,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
|
convexOn_zpow
|
[
{
"state_after": "n : ℕ\n⊢ ConvexOn ℝ (Ioi 0) fun x => x ^ n",
"state_before": "n : ℕ\n⊢ ConvexOn ℝ (Ioi 0) fun x => x ^ ↑n",
"tactic": "simp_rw [zpow_ofNat]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ConvexOn ℝ (Ioi 0) fun x => x ^ n",
"tactic": "exact (convexOn_pow n).subset Ioi_subset_Ici_self (convex_Ioi _)"
},
{
"state_after": "n : ℕ\n⊢ ConvexOn ℝ (Ioi 0) fun x => (x ^ (n + 1))⁻¹",
"state_before": "n : ℕ\n⊢ ConvexOn ℝ (Ioi 0) fun x => x ^ -[n+1]",
"tactic": "simp_rw [zpow_negSucc]"
},
{
"state_after": "n : ℕ\n⊢ ∀ ⦃x : ℝ⦄,\n x ∈ Ioi 0 →\n ∀ ⦃y : ℝ⦄,\n y ∈ Ioi 0 →\n ∀ ⦃a b : ℝ⦄,\n 0 ≤ a →\n 0 ≤ b →\n a + b = 1 →\n (fun x => (x ^ (n + 1))⁻¹) (a • x + b • y) ≤\n a • (fun x => (x ^ (n + 1))⁻¹) x + b • (fun x => (x ^ (n + 1))⁻¹) y",
"state_before": "n : ℕ\n⊢ ConvexOn ℝ (Ioi 0) fun x => (x ^ (n + 1))⁻¹",
"tactic": "refine' ⟨convex_Ioi _, _⟩"
},
{
"state_after": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (fun x => (x ^ (n + 1))⁻¹) (μ • a + ν • b) ≤ μ • (fun x => (x ^ (n + 1))⁻¹) a + ν • (fun x => (x ^ (n + 1))⁻¹) b",
"state_before": "n : ℕ\n⊢ ∀ ⦃x : ℝ⦄,\n x ∈ Ioi 0 →\n ∀ ⦃y : ℝ⦄,\n y ∈ Ioi 0 →\n ∀ ⦃a b : ℝ⦄,\n 0 ≤ a →\n 0 ≤ b →\n a + b = 1 →\n (fun x => (x ^ (n + 1))⁻¹) (a • x + b • y) ≤\n a • (fun x => (x ^ (n + 1))⁻¹) x + b • (fun x => (x ^ (n + 1))⁻¹) y",
"tactic": "rintro a (ha : 0 < a) b (hb : 0 < b) μ ν hμ hν h"
},
{
"state_after": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 1 / (μ * a + ν * b) ^ (n + 1) ≤ (μ * b ^ (n + 1) + ν * a ^ (n + 1)) / (a ^ (n + 1) * b ^ (n + 1))",
"state_before": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (fun x => (x ^ (n + 1))⁻¹) (μ • a + ν • b) ≤ μ • (fun x => (x ^ (n + 1))⁻¹) a + ν • (fun x => (x ^ (n + 1))⁻¹) b",
"tactic": "field_simp [ha.ne', hb.ne']"
},
{
"state_after": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 1 * (a ^ (n + 1) * b ^ (n + 1)) ≤ (μ * b ^ (n + 1) + ν * a ^ (n + 1)) * (μ * a + ν * b) ^ (n + 1)\n\ncase b0\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 0 < (μ * a + ν * b) ^ (n + 1)\n\ncase d0\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 0 < a ^ (n + 1) * b ^ (n + 1)",
"state_before": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 1 / (μ * a + ν * b) ^ (n + 1) ≤ (μ * b ^ (n + 1) + ν * a ^ (n + 1)) / (a ^ (n + 1) * b ^ (n + 1))",
"tactic": "rw [div_le_div_iff]"
},
{
"state_after": "case calc_1\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ ((μ + ν) ^ 2 * (a * b)) ^ (n + 1) ≤ ((μ * b + ν * a) * (μ * a + ν * b)) ^ (n + 1)\n\ncase calc_2\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (μ * b + ν * a) ^ (n + 1) * (μ * a + ν * b) ^ (n + 1) ≤\n (μ * b ^ (n + 1) + ν * a ^ (n + 1)) * (μ * a + ν * b) ^ (n + 1)",
"state_before": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 1 * (a ^ (n + 1) * b ^ (n + 1)) ≤ (μ * b ^ (n + 1) + ν * a ^ (n + 1)) * (μ * a + ν * b) ^ (n + 1)",
"tactic": "calc\n (1 : ℝ) * (a ^ (n + 1) * b ^ (n + 1)) = ((μ + ν) ^ 2 * (a * b)) ^ (n + 1) := by rw [h]; ring\n _ ≤ ((μ * b + ν * a) * (μ * a + ν * b)) ^ (n + 1) := ?_\n _ = (μ * b + ν * a) ^ (n + 1) * (μ * a + ν * b) ^ (n + 1) := by rw [mul_pow]\n _ ≤ (μ * b ^ (n + 1) + ν * a ^ (n + 1)) * (μ * a + ν * b) ^ (n + 1) := ?_"
},
{
"state_after": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 1 * (a ^ (n + 1) * b ^ (n + 1)) = (1 ^ 2 * (a * b)) ^ (n + 1)",
"state_before": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 1 * (a ^ (n + 1) * b ^ (n + 1)) = ((μ + ν) ^ 2 * (a * b)) ^ (n + 1)",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 1 * (a ^ (n + 1) * b ^ (n + 1)) = (1 ^ 2 * (a * b)) ^ (n + 1)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ ((μ * b + ν * a) * (μ * a + ν * b)) ^ (n + 1) = (μ * b + ν * a) ^ (n + 1) * (μ * a + ν * b) ^ (n + 1)",
"tactic": "rw [mul_pow]"
},
{
"state_after": "case calc_1.hab\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (μ + ν) ^ 2 * (a * b) ≤ (μ * b + ν * a) * (μ * a + ν * b)",
"state_before": "case calc_1\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ ((μ + ν) ^ 2 * (a * b)) ^ (n + 1) ≤ ((μ * b + ν * a) * (μ * a + ν * b)) ^ (n + 1)",
"tactic": "gcongr (?_ : ℝ) ^ _"
},
{
"state_after": "case calc_1.hab\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nthis : 0 ≤ μ * ν * (a - b) ^ 2\n⊢ (μ + ν) ^ 2 * (a * b) ≤ (μ * b + ν * a) * (μ * a + ν * b)",
"state_before": "case calc_1.hab\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (μ + ν) ^ 2 * (a * b) ≤ (μ * b + ν * a) * (μ * a + ν * b)",
"tactic": "have : (0 : ℝ) ≤ μ * ν * (a - b) ^ 2 := by positivity"
},
{
"state_after": "no goals",
"state_before": "case calc_1.hab\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nthis : 0 ≤ μ * ν * (a - b) ^ 2\n⊢ (μ + ν) ^ 2 * (a * b) ≤ (μ * b + ν * a) * (μ * a + ν * b)",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 0 ≤ μ * ν * (a - b) ^ 2",
"tactic": "positivity"
},
{
"state_after": "case calc_2.h\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (μ * b + ν * a) ^ (n + 1) ≤ μ * b ^ (n + 1) + ν * a ^ (n + 1)",
"state_before": "case calc_2\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (μ * b + ν * a) ^ (n + 1) * (μ * a + ν * b) ^ (n + 1) ≤\n (μ * b ^ (n + 1) + ν * a ^ (n + 1)) * (μ * a + ν * b) ^ (n + 1)",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case calc_2.h\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ (μ * b + ν * a) ^ (n + 1) ≤ μ * b ^ (n + 1) + ν * a ^ (n + 1)",
"tactic": "apply (convexOn_pow (n + 1)).2 hb.le ha.le hμ hν h"
},
{
"state_after": "case b0\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nthis : 0 < μ * a + ν * b\n⊢ 0 < (μ * a + ν * b) ^ (n + 1)",
"state_before": "case b0\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 0 < (μ * a + ν * b) ^ (n + 1)",
"tactic": "have : 0 < μ * a + ν * b := by cases le_or_lt a b <;> nlinarith"
},
{
"state_after": "no goals",
"state_before": "case b0\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\nthis : 0 < μ * a + ν * b\n⊢ 0 < (μ * a + ν * b) ^ (n + 1)",
"tactic": "positivity"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 0 < μ * a + ν * b",
"tactic": "cases le_or_lt a b <;> nlinarith"
},
{
"state_after": "no goals",
"state_before": "case d0\nn : ℕ\na : ℝ\nha : 0 < a\nb : ℝ\nhb : 0 < b\nμ ν : ℝ\nhμ : 0 ≤ μ\nhν : 0 ≤ ν\nh : μ + ν = 1\n⊢ 0 < a ^ (n + 1) * b ^ (n + 1)",
"tactic": "positivity"
}
] |
[
152,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/CategoryTheory/Bicategory/Coherence.lean
|
CategoryTheory.FreeBicategory.preinclusion_map₂
|
[
{
"state_after": "case up.up\nB : Type u\ninst✝ : Quiver B\na b : B\nf g : Discrete (Path a b)\ndown✝ : f.as = g.as\n⊢ PrelaxFunctor.map₂ (preinclusion B) { down := { down := down✝ } } =\n eqToHom (_ : (↑(preinclusion B)).map f = (↑(preinclusion B)).map g)",
"state_before": "B : Type u\ninst✝ : Quiver B\na b : B\nf g : Discrete (Path a b)\nη : f ⟶ g\n⊢ PrelaxFunctor.map₂ (preinclusion B) η = eqToHom (_ : (↑(preinclusion B)).map f = (↑(preinclusion B)).map g)",
"tactic": "rcases η with ⟨⟨⟩⟩"
},
{
"state_after": "case up.up.refl\nB : Type u\ninst✝ : Quiver B\na b : B\nf : Discrete (Path a b)\ndown✝ : f.as = f.as\n⊢ PrelaxFunctor.map₂ (preinclusion B) { down := { down := down✝ } } =\n eqToHom (_ : (↑(preinclusion B)).map f = (↑(preinclusion B)).map f)",
"state_before": "case up.up\nB : Type u\ninst✝ : Quiver B\na b : B\nf g : Discrete (Path a b)\ndown✝ : f.as = g.as\n⊢ PrelaxFunctor.map₂ (preinclusion B) { down := { down := down✝ } } =\n eqToHom (_ : (↑(preinclusion B)).map f = (↑(preinclusion B)).map g)",
"tactic": "cases Discrete.ext _ _ (by assumption)"
},
{
"state_after": "no goals",
"state_before": "case up.up.refl\nB : Type u\ninst✝ : Quiver B\na b : B\nf : Discrete (Path a b)\ndown✝ : f.as = f.as\n⊢ PrelaxFunctor.map₂ (preinclusion B) { down := { down := down✝ } } =\n eqToHom (_ : (↑(preinclusion B)).map f = (↑(preinclusion B)).map f)",
"tactic": "convert (inclusionPath a b).map_id _"
},
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Quiver B\na b : B\nf g : Discrete (Path a b)\ndown✝ : f.as = g.as\n⊢ ?m.4788.as = ?m.4789.as",
"tactic": "assumption"
}
] |
[
101,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Algebra/Group/UniqueProds.lean
|
UniqueMul.mt
|
[
{
"state_after": "G✝ : Type ?u.271\nH : Type ?u.274\ninst✝² : Mul G✝\ninst✝¹ : Mul H\nA✝ B✝ : Finset G✝\na0✝ b0✝ : G✝\nG : Type u_1\ninst✝ : Mul G\nA B : Finset G\na0 b0 : G\nh : UniqueMul A B a0 b0\nx✝¹ x✝ : G\nha : x✝¹ ∈ A\nhb : x✝ ∈ B\nk : x✝¹ * x✝ = a0 * b0\n⊢ x✝¹ = a0 ∧ x✝ = b0",
"state_before": "G✝ : Type ?u.271\nH : Type ?u.274\ninst✝² : Mul G✝\ninst✝¹ : Mul H\nA✝ B✝ : Finset G✝\na0✝ b0✝ : G✝\nG : Type u_1\ninst✝ : Mul G\nA B : Finset G\na0 b0 : G\nh : UniqueMul A B a0 b0\nx✝¹ x✝ : G\nha : x✝¹ ∈ A\nhb : x✝ ∈ B\nk : x✝¹ ≠ a0 ∨ x✝ ≠ b0\n⊢ x✝¹ * x✝ ≠ a0 * b0",
"tactic": "contrapose! k"
},
{
"state_after": "no goals",
"state_before": "G✝ : Type ?u.271\nH : Type ?u.274\ninst✝² : Mul G✝\ninst✝¹ : Mul H\nA✝ B✝ : Finset G✝\na0✝ b0✝ : G✝\nG : Type u_1\ninst✝ : Mul G\nA B : Finset G\na0 b0 : G\nh : UniqueMul A B a0 b0\nx✝¹ x✝ : G\nha : x✝¹ ∈ A\nhb : x✝ ∈ B\nk : x✝¹ * x✝ = a0 * b0\n⊢ x✝¹ = a0 ∧ x✝ = b0",
"tactic": "exact h ha hb k"
}
] |
[
59,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/GroupTheory/SpecificGroups/Cyclic.lean
|
isCyclic_of_prime_card
|
[
{
"state_after": "case intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"state_before": "α✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"tactic": "obtain ⟨g, hg⟩ : ∃ g : α, g ≠ 1 := Fintype.exists_ne_of_one_lt_card (h.symm ▸ hp.1.one_lt) 1"
},
{
"state_after": "case intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthis : Fintype.card { x // x ∈ zpowers g } ∣ p\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"state_before": "case intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"tactic": "have : Fintype.card (Subgroup.zpowers g) ∣ p := by\n rw [← h]\n apply card_subgroup_dvd_card"
},
{
"state_after": "case intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthis : Fintype.card { x // x ∈ zpowers g } = 1 ∨ Fintype.card { x // x ∈ zpowers g } = p\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"state_before": "case intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthis : Fintype.card { x // x ∈ zpowers g } ∣ p\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"tactic": "rw [Nat.dvd_prime hp.1] at this"
},
{
"state_after": "case intro.inl\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = 1\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g\n\ncase intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = p\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"state_before": "case intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthis : Fintype.card { x // x ∈ zpowers g } = 1 ∨ Fintype.card { x // x ∈ zpowers g } = p\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"tactic": "cases' this with that that"
},
{
"state_after": "α✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\n⊢ Fintype.card { x // x ∈ zpowers g } ∣ Fintype.card α",
"state_before": "α✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\n⊢ Fintype.card { x // x ∈ zpowers g } ∣ p",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\n⊢ Fintype.card { x // x ∈ zpowers g } ∣ Fintype.card α",
"tactic": "apply card_subgroup_dvd_card"
},
{
"state_after": "case intro.inl\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : ∃ x, ∀ (y : { x // x ∈ zpowers g }), y = x\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"state_before": "case intro.inl\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = 1\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"tactic": "rw [Fintype.card_eq_one_iff] at that"
},
{
"state_after": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"state_before": "case intro.inl\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : ∃ x, ∀ (y : { x // x ∈ zpowers g }), y = x\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"tactic": "cases' that with t ht"
},
{
"state_after": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\n⊢ g = 1",
"state_before": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"tactic": "suffices g = 1 by contradiction"
},
{
"state_after": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\nhgt : { val := g, property := (_ : g ∈ zpowers g) } = t\n⊢ g = 1",
"state_before": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\n⊢ g = 1",
"tactic": "have hgt :=\n ht\n ⟨g, by\n change g ∈ Subgroup.zpowers g\n exact Subgroup.mem_zpowers g⟩"
},
{
"state_after": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\nhgt : { val := g, property := (_ : g ∈ zpowers g) } = 1\n⊢ g = 1",
"state_before": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\nhgt : { val := g, property := (_ : g ∈ zpowers g) } = t\n⊢ g = 1",
"tactic": "rw [← ht 1] at hgt"
},
{
"state_after": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\nhgt : { val := g, property := (_ : g ∈ zpowers g) } = { val := 1, property := (_ : 1 ∈ (zpowers g).toSubmonoid) }\n⊢ g = 1",
"state_before": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\nhgt : { val := g, property := (_ : g ∈ zpowers g) } = 1\n⊢ g = 1",
"tactic": "change (⟨_, _⟩ : Subgroup.zpowers g) = ⟨_, _⟩ at hgt"
},
{
"state_after": "no goals",
"state_before": "case intro.inl.intro\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\nhgt : { val := g, property := (_ : g ∈ zpowers g) } = { val := 1, property := (_ : 1 ∈ (zpowers g).toSubmonoid) }\n⊢ g = 1",
"tactic": "simpa using hgt"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\nthis : g = 1\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"tactic": "contradiction"
},
{
"state_after": "α✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\n⊢ g ∈ zpowers g",
"state_before": "α✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\n⊢ g ∈ zpowers g",
"tactic": "change g ∈ Subgroup.zpowers g"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nt : { x // x ∈ zpowers g }\nht : ∀ (y : { x // x ∈ zpowers g }), y = t\n⊢ g ∈ zpowers g",
"tactic": "exact Subgroup.mem_zpowers g"
},
{
"state_after": "case intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = p\n⊢ ∀ (x : α), x ∈ zpowers g",
"state_before": "case intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = p\n⊢ ∃ g, ∀ (x : α), x ∈ zpowers g",
"tactic": "use g"
},
{
"state_after": "case intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = p\nx : α\n⊢ x ∈ zpowers g",
"state_before": "case intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = p\n⊢ ∀ (x : α), x ∈ zpowers g",
"tactic": "intro x"
},
{
"state_after": "case intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = Fintype.card α\nx : α\n⊢ x ∈ zpowers g",
"state_before": "case intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = p\nx : α\n⊢ x ∈ zpowers g",
"tactic": "rw [← h] at that"
},
{
"state_after": "case intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = Fintype.card α\nx : α\n⊢ x ∈ ⊤",
"state_before": "case intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = Fintype.card α\nx : α\n⊢ x ∈ zpowers g",
"tactic": "rw [Subgroup.eq_top_of_card_eq _ that]"
},
{
"state_after": "no goals",
"state_before": "case intro.inr\nα✝ : Type u\na : α✝\ninst✝² : Group α✝\nα : Type u\ninst✝¹ : Group α\ninst✝ : Fintype α\np : ℕ\nhp : Fact (Nat.Prime p)\nh : Fintype.card α = p\ng : α\nhg : g ≠ 1\nthat : Fintype.card { x // x ∈ zpowers g } = Fintype.card α\nx : α\n⊢ x ∈ ⊤",
"tactic": "exact Subgroup.mem_top _"
}
] |
[
144,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.evalRingHom_zero
|
[] |
[
1067,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1066,
1
] |
Mathlib/Analysis/NormedSpace/Banach.lean
|
ContinuousLinearMap.exists_preimage_norm_le
|
[
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\nhC : ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖\n⊢ ∃ C, C > 0 ∧ ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\n⊢ ∃ C, C > 0 ∧ ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "obtain ⟨C, C0, hC⟩ := exists_approx_preimage_norm_le f surj"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\n⊢ ∃ C, C > 0 ∧ ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\nhC : ∀ (y : F), ∃ x, dist (↑f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ C * ‖y‖\n⊢ ∃ C, C > 0 ∧ ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "choose g hg using hC"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\n⊢ ∃ C, C > 0 ∧ ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\n⊢ ∃ C, C > 0 ∧ ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "let h y := y - f (g y)"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\n⊢ ∃ C, C > 0 ∧ ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\n⊢ ∃ C, C > 0 ∧ ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "have hle : ∀ y, ‖h y‖ ≤ 1 / 2 * ‖y‖ := by\n intro y\n rw [← dist_eq_norm, dist_comm]\n exact (hg y).1"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\n⊢ ∃ C, C > 0 ∧ ∀ (y : F), ∃ x, ↑f x = y ∧ ‖x‖ ≤ C * ‖y‖",
"tactic": "refine' ⟨2 * C + 1, by linarith, fun y => _⟩"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "let u n := g ((h^[n]) y)"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "have ule : ∀ n, ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖) := by\n intro n\n apply le_trans (hg _).2 _\n calc\n C * ‖(h^[n]) y‖ ≤ C * ((1 / 2) ^ n * ‖y‖) := mul_le_mul_of_nonneg_left (hnle n) C0\n _ = (1 / 2) ^ n * (C * ‖y‖) := by ring"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "have sNu : Summable fun n => ‖u n‖ := by\n refine' summable_of_nonneg_of_le (fun n => norm_nonneg _) ule _\n exact Summable.mul_right _ (summable_geometric_of_lt_1 (by norm_num) (by norm_num))"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "have su : Summable u := summable_of_summable_norm sNu"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "let x := tsum u"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "have x_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖ :=\n calc\n ‖x‖ ≤ ∑' n, ‖u n‖ := norm_tsum_le_tsum_norm sNu\n _ ≤ ∑' n, (1 / 2) ^ n * (C * ‖y‖) :=\n (tsum_le_tsum ule sNu (Summable.mul_right _ summable_geometric_two))\n _ = (∑' n, (1 / 2) ^ n) * (C * ‖y‖) := tsum_mul_right\n _ = 2 * C * ‖y‖ := by rw [tsum_geometric_two, mul_assoc]\n _ ≤ 2 * C * ‖y‖ + ‖y‖ := (le_add_of_nonneg_right (norm_nonneg y))\n _ = (2 * C + 1) * ‖y‖ := by ring"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "have : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x) := su.hasSum.tendsto_sum_nat"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ : Tendsto (fun n => ↑f (∑ i in Finset.range n, u i)) atTop (𝓝 (↑f x))\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "have L₁ : Tendsto (fun n => f (∑ i in Finset.range n, u i)) atTop (𝓝 (f x)) :=\n (f.continuous.tendsto _).comp this"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ : Tendsto (fun n => ↑f (∑ i in Finset.range n, u i)) atTop (𝓝 (↑f x))\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "simp only [fsumeq] at L₁"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\nL₂ : Tendsto (fun n => y - (h^[n]) y) atTop (𝓝 (y - 0))\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "have L₂ : Tendsto (fun n => y - (h^[n]) y) atTop (𝓝 (y - 0)) := by\n refine' tendsto_const_nhds.sub _\n rw [tendsto_iff_norm_tendsto_zero]\n simp only [sub_zero]\n refine' squeeze_zero (fun _ => norm_nonneg _) hnle _\n rw [← MulZeroClass.zero_mul ‖y‖]\n refine' (_root_.tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds <;> norm_num"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\nL₂ : Tendsto (fun n => y - (h^[n]) y) atTop (𝓝 (y - 0))\nfeq : ↑f x = y - 0\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\nL₂ : Tendsto (fun n => y - (h^[n]) y) atTop (𝓝 (y - 0))\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "have feq : f x = y - 0 := tendsto_nhds_unique L₁ L₂"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\nL₂ : Tendsto (fun n => y - (h^[n]) y) atTop (𝓝 (y - 0))\nfeq : ↑f x = y\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"state_before": "case intro.intro\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\nL₂ : Tendsto (fun n => y - (h^[n]) y) atTop (𝓝 (y - 0))\nfeq : ↑f x = y - 0\n⊢ ∃ x, ↑f x = y ∧ ‖x‖ ≤ (2 * C + 1) * ‖y‖",
"tactic": "rw [sub_zero] at feq"
},
{
"state_after": "no goals",
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"tactic": "intro y"
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"tactic": "rw [← dist_eq_norm, dist_comm]"
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"tactic": "exact (hg y).1"
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{
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"tactic": "linarith"
},
{
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"tactic": "intro n"
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{
"state_after": "case zero\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\n⊢ ‖(h^[Nat.zero]) y‖ ≤ (1 / 2) ^ Nat.zero * ‖y‖\n\ncase succ\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nn : ℕ\nIH : ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\n⊢ ‖(h^[Nat.succ n]) y‖ ≤ (1 / 2) ^ Nat.succ n * ‖y‖",
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"tactic": "induction' n with n IH"
},
{
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"tactic": "simp only [one_div, Nat.zero_eq, one_mul, iterate_zero_apply, pow_zero, le_rfl]"
},
{
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"tactic": "rw [iterate_succ']"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nn : ℕ\nIH : ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\n⊢ 1 / 2 * ‖(h^[n]) y‖ ≤ (1 / 2) ^ Nat.succ n * ‖y‖",
"state_before": "case succ\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nn : ℕ\nIH : ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\n⊢ ‖(h ∘ h^[n]) y‖ ≤ (1 / 2) ^ Nat.succ n * ‖y‖",
"tactic": "apply le_trans (hle _) _"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nn : ℕ\nIH : ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\n⊢ 1 / 2 * ‖(h^[n]) y‖ ≤ 1 / 2 * ((1 / 2) ^ n * ‖y‖)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nn : ℕ\nIH : ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\n⊢ 1 / 2 * ‖(h^[n]) y‖ ≤ (1 / 2) ^ Nat.succ n * ‖y‖",
"tactic": "rw [pow_succ, mul_assoc]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nn : ℕ\nIH : ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\n⊢ 0 ≤ 1 / 2",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nn : ℕ\nIH : ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\n⊢ 1 / 2 * ‖(h^[n]) y‖ ≤ 1 / 2 * ((1 / 2) ^ n * ‖y‖)",
"tactic": "apply mul_le_mul_of_nonneg_left IH"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nn : ℕ\nIH : ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\n⊢ 0 ≤ 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nn : ℕ\n⊢ ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\n⊢ ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)",
"tactic": "intro n"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nn : ℕ\n⊢ C * ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * (C * ‖y‖)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nn : ℕ\n⊢ ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)",
"tactic": "apply le_trans (hg _).2 _"
},
{
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"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nn : ℕ\n⊢ C * ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * (C * ‖y‖)",
"tactic": "calc\n C * ‖(h^[n]) y‖ ≤ C * ((1 / 2) ^ n * ‖y‖) := mul_le_mul_of_nonneg_left (hnle n) C0\n _ = (1 / 2) ^ n * (C * ‖y‖) := by ring"
},
{
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"tactic": "ring"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\n⊢ Summable fun b => (1 / 2) ^ b * (C * ‖y‖)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\n⊢ Summable fun n => ‖u n‖",
"tactic": "refine' summable_of_nonneg_of_le (fun n => norm_nonneg _) ule _"
},
{
"state_after": "no goals",
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"tactic": "exact Summable.mul_right _ (summable_geometric_of_lt_1 (by norm_num) (by norm_num))"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\n⊢ 0 ≤ 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\n⊢ 1 / 2 < 1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\n⊢ (∑' (n : ℕ), (1 / 2) ^ n) * (C * ‖y‖) = 2 * C * ‖y‖",
"tactic": "rw [tsum_geometric_two, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\n⊢ 2 * C * ‖y‖ + ‖y‖ = (2 * C + 1) * ‖y‖",
"tactic": "ring"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nn : ℕ\n⊢ ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\n⊢ ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y",
"tactic": "intro n"
},
{
"state_after": "case zero\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\n⊢ ↑f (∑ i in Finset.range Nat.zero, u i) = y - (h^[Nat.zero]) y\n\ncase succ\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nn : ℕ\nIH : ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\n⊢ ↑f (∑ i in Finset.range (Nat.succ n), u i) = y - (h^[Nat.succ n]) y",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nn : ℕ\n⊢ ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y",
"tactic": "induction' n with n IH"
},
{
"state_after": "no goals",
"state_before": "case zero\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\n⊢ ↑f (∑ i in Finset.range Nat.zero, u i) = y - (h^[Nat.zero]) y",
"tactic": "simp [f.map_zero]"
},
{
"state_after": "no goals",
"state_before": "case succ\n𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nn : ℕ\nIH : ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\n⊢ ↑f (∑ i in Finset.range (Nat.succ n), u i) = y - (h^[Nat.succ n]) y",
"tactic": "rw [sum_range_succ, f.map_add, IH, iterate_succ_apply', sub_add]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun n => (h^[n]) y) atTop (𝓝 0)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun n => y - (h^[n]) y) atTop (𝓝 (y - 0))",
"tactic": "refine' tendsto_const_nhds.sub _"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun e => ‖(h^[e]) y - 0‖) atTop (𝓝 0)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun n => (h^[n]) y) atTop (𝓝 0)",
"tactic": "rw [tendsto_iff_norm_tendsto_zero]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun e => ‖((fun y => y - ↑f (g y))^[e]) y‖) atTop (𝓝 0)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun e => ‖(h^[e]) y - 0‖) atTop (𝓝 0)",
"tactic": "simp only [sub_zero]"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun t => (1 / 2) ^ t * ‖y‖) atTop (𝓝 0)",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun e => ‖((fun y => y - ↑f (g y))^[e]) y‖) atTop (𝓝 0)",
"tactic": "refine' squeeze_zero (fun _ => norm_nonneg _) hnle _"
},
{
"state_after": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun t => (1 / 2) ^ t * ‖y‖) atTop (𝓝 (0 * ‖y‖))",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun t => (1 / 2) ^ t * ‖y‖) atTop (𝓝 0)",
"tactic": "rw [← MulZeroClass.zero_mul ‖y‖]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nsurj : Surjective ↑f\nC : ℝ\nC0 : C ≥ 0\ng : F → E\nhg : ∀ (y : F), dist (↑f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖\nh : F → F := fun y => y - ↑f (g y)\nhle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖\ny : F\nhnle : ∀ (n : ℕ), ‖(h^[n]) y‖ ≤ (1 / 2) ^ n * ‖y‖\nu : ℕ → E := fun n => g ((h^[n]) y)\nule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)\nsNu : Summable fun n => ‖u n‖\nsu : Summable u\nx : E := tsum u\nx_ineq : ‖x‖ ≤ (2 * C + 1) * ‖y‖\nfsumeq : ∀ (n : ℕ), ↑f (∑ i in Finset.range n, u i) = y - (h^[n]) y\nthis : Tendsto (fun n => ∑ i in Finset.range n, u i) atTop (𝓝 x)\nL₁ :\n Tendsto (fun n => y - ((fun y => y - ↑f (g y))^[n]) y) atTop\n (𝓝 (↑f (∑' (n : ℕ), g (((fun y => y - ↑f (g y))^[n]) y))))\n⊢ Tendsto (fun t => (1 / 2) ^ t * ‖y‖) atTop (𝓝 (0 * ‖y‖))",
"tactic": "refine' (_root_.tendsto_pow_atTop_nhds_0_of_lt_1 _ _).mul tendsto_const_nhds <;> norm_num"
}
] |
[
236,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Init/Algebra/Order.lean
|
le_or_lt
|
[] |
[
345,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.reverse_induction_last
|
[
{
"state_after": "n✝ m n : ℕ\nC : Fin (n + 1) → Sort u_1\nh0 : C (last n)\nhs : (i : Fin n) → C (succ i) → C (↑castSucc i)\n⊢ (if hi : last n = last n then _root_.cast (_ : C (last n) = C (last n)) h0\n else\n let j := { val := ↑(last n), isLt := (_ : ↑(last n) < n) };\n let_fun x := (_ : n - ↑(last n) < n + 1 - ↑(last n));\n let_fun hi := (_ : last n = ↑castSucc j);\n _root_.cast (_ : C (↑castSucc j) = C (last n)) (hs j (reverseInduction h0 hs (succ j)))) =\n h0",
"state_before": "n✝ m n : ℕ\nC : Fin (n + 1) → Sort u_1\nh0 : C (last n)\nhs : (i : Fin n) → C (succ i) → C (↑castSucc i)\n⊢ reverseInduction h0 hs (last n) = h0",
"tactic": "rw [reverseInduction]"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\nC : Fin (n + 1) → Sort u_1\nh0 : C (last n)\nhs : (i : Fin n) → C (succ i) → C (↑castSucc i)\n⊢ (if hi : last n = last n then _root_.cast (_ : C (last n) = C (last n)) h0\n else\n let j := { val := ↑(last n), isLt := (_ : ↑(last n) < n) };\n let_fun x := (_ : n - ↑(last n) < n + 1 - ↑(last n));\n let_fun hi := (_ : last n = ↑castSucc j);\n _root_.cast (_ : C (↑castSucc j) = C (last n)) (hs j (reverseInduction h0 hs (succ j)))) =\n h0",
"tactic": "simp"
}
] |
[
1803,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1800,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
Monotone.mul_const'
|
[] |
[
1313,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1312,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.iIndepFun.indepFun_prod
|
[
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"tactic": "classical\n have h_right : f k =\n (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘\n fun a (j : ({k} : Finset ι)) => f j a := rfl\n have h_meas_right : Measurable fun p : ∀ j : ({k} : Finset ι),\n β j => p ⟨k, Finset.mem_singleton_self k⟩ := measurable_pi_apply _\n let s : Finset ι := {i, j}\n have h_left : (fun ω => (f i ω, f j ω)) = (fun p : ∀ l : s, β l =>\n (p ⟨i, Finset.mem_insert_self i _⟩,\n p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘ fun a (j : s) => f j a := by\n ext1 a\n simp only [Prod.mk.inj_iff]\n constructor\n have h_meas_left : Measurable fun p : ∀ l : s, β l =>\n (p ⟨i, Finset.mem_insert_self i _⟩,\n p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) :=\n Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _)\n rw [h_left, h_right]\n refine' (hf_Indep.indepFun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right\n rw [Finset.disjoint_singleton_right]\n simp only [Finset.mem_insert, Finset.mem_singleton, not_or]\n exact ⟨hik.symm, hjk.symm⟩"
},
{
"state_after": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"tactic": "have h_right : f k =\n (fun p : ∀ j : ({k} : Finset ι), β j => p ⟨k, Finset.mem_singleton_self k⟩) ∘\n fun a (j : ({k} : Finset ι)) => f j a := rfl"
},
{
"state_after": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"tactic": "have h_meas_right : Measurable fun p : ∀ j : ({k} : Finset ι),\n β j => p ⟨k, Finset.mem_singleton_self k⟩ := measurable_pi_apply _"
},
{
"state_after": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"tactic": "let s : Finset ι := {i, j}"
},
{
"state_after": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"tactic": "have h_left : (fun ω => (f i ω, f j ω)) = (fun p : ∀ l : s, β l =>\n (p ⟨i, Finset.mem_insert_self i _⟩,\n p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩)) ∘ fun a (j : s) => f j a := by\n ext1 a\n simp only [Prod.mk.inj_iff]\n constructor"
},
{
"state_after": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"tactic": "have h_meas_left : Measurable fun p : ∀ l : s, β l =>\n (p ⟨i, Finset.mem_insert_self i _⟩,\n p ⟨j, Finset.mem_insert_of_mem (Finset.mem_singleton_self _)⟩) :=\n Measurable.prod (measurable_pi_apply _) (measurable_pi_apply _)"
},
{
"state_after": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ IndepFun\n ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘\n fun a j => f (↑j) a)\n ((fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a)",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ IndepFun (fun a => (f i a, f j a)) (f k)",
"tactic": "rw [h_left, h_right]"
},
{
"state_after": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ Disjoint s {k}",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ IndepFun\n ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘\n fun a j => f (↑j) a)\n ((fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a)",
"tactic": "refine' (hf_Indep.indepFun_finset s {k} _ hf_meas).comp h_meas_left h_meas_right"
},
{
"state_after": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ ¬k ∈ s",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ Disjoint s {k}",
"tactic": "rw [Finset.disjoint_singleton_right]"
},
{
"state_after": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ ¬k = i ∧ ¬k = j",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ ¬k ∈ s",
"tactic": "simp only [Finset.mem_insert, Finset.mem_singleton, not_or]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\nh_left :\n (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a\nh_meas_left :\n Measurable fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })\n⊢ ¬k = i ∧ ¬k = j",
"tactic": "exact ⟨hik.symm, hjk.symm⟩"
},
{
"state_after": "case h\nΩ : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\na : Ω\n⊢ (f i a, f j a) =\n ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘\n fun a j => f (↑j) a)\n a",
"state_before": "Ω : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\n⊢ (fun ω => (f i ω, f j ω)) =\n (fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘ fun a j =>\n f (↑j) a",
"tactic": "ext1 a"
},
{
"state_after": "case h\nΩ : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\na : Ω\n⊢ (f i a, f j a) =\n ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘\n fun a j_1 => f (↑j_1) a)\n a",
"state_before": "case h\nΩ : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\na : Ω\n⊢ (f i a, f j a) =\n ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘\n fun a j => f (↑j) a)\n a",
"tactic": "simp only [Prod.mk.inj_iff]"
},
{
"state_after": "no goals",
"state_before": "case h\nΩ : Type u_1\nι✝ : Type ?u.3447382\nβ✝ : Type ?u.3447385\nβ' : Type ?u.3447388\nγ : Type ?u.3447391\nγ' : Type ?u.3447394\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf✝ : Ω → β✝\ng : Ω → β'\ninst✝ : IsProbabilityMeasure μ\nι : Type u_2\nβ : ι → Type u_3\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun m f\nhf_meas : ∀ (i : ι), Measurable (f i)\ni j k : ι\nhik : i ≠ k\nhjk : j ≠ k\nh_right : f k = (fun p => p { val := k, property := (_ : k ∈ {k}) }) ∘ fun a j => f (↑j) a\nh_meas_right : Measurable fun p => p { val := k, property := (_ : k ∈ {k}) }\ns : Finset ι := {i, j}\na : Ω\n⊢ (f i a, f j a) =\n ((fun p => (p { val := i, property := (_ : i ∈ {i, j}) }, p { val := j, property := (_ : j ∈ {i, j}) })) ∘\n fun a j_1 => f (↑j_1) a)\n a",
"tactic": "constructor"
}
] |
[
848,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
823,
1
] |
Mathlib/Data/Nat/SuccPred.lean
|
Nat.succ_iterate
|
[
{
"state_after": "a n : ℕ\n⊢ (succ ∘ succ^[n]) a = succ (a + n)",
"state_before": "a n : ℕ\n⊢ (succ^[n + 1]) a = a + (n + 1)",
"tactic": "rw [Function.iterate_succ', add_succ]"
},
{
"state_after": "no goals",
"state_before": "a n : ℕ\n⊢ (succ ∘ succ^[n]) a = succ (a + n)",
"tactic": "exact congr_arg _ (succ_iterate a n)"
}
] |
[
62,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.cyclicPermutations_rotate
|
[
{
"state_after": "α : Type u\nl✝ l' l : List α\nk : ℕ\nthis : length (cyclicPermutations (rotate l k)) = length (rotate (cyclicPermutations l) k)\nn : ℕ\nhn : n < length (cyclicPermutations (rotate l k))\nhn' : n < length (rotate (cyclicPermutations l) k)\n⊢ nthLe (cyclicPermutations (rotate l k)) n hn = nthLe (rotate (cyclicPermutations l) k) n hn'",
"state_before": "α : Type u\nl✝ l' l : List α\nk : ℕ\nthis : length (cyclicPermutations (rotate l k)) = length (rotate (cyclicPermutations l) k)\n⊢ cyclicPermutations (rotate l k) = rotate (cyclicPermutations l) k",
"tactic": "refine' ext_nthLe this fun n hn hn' => _"
},
{
"state_after": "α : Type u\nl✝ l' l : List α\nk : ℕ\nthis : length (cyclicPermutations (rotate l k)) = length (rotate (cyclicPermutations l) k)\nn : ℕ\nhn : n < length (cyclicPermutations (rotate l k))\nhn' : n < length (rotate (cyclicPermutations l) k)\n⊢ rotate l ((n + k) % length l) =\n nthLe (cyclicPermutations l) ((n + k) % length (cyclicPermutations l))\n (_ : (n + k) % length (cyclicPermutations l) < length (cyclicPermutations l))",
"state_before": "α : Type u\nl✝ l' l : List α\nk : ℕ\nthis : length (cyclicPermutations (rotate l k)) = length (rotate (cyclicPermutations l) k)\nn : ℕ\nhn : n < length (cyclicPermutations (rotate l k))\nhn' : n < length (rotate (cyclicPermutations l) k)\n⊢ nthLe (cyclicPermutations (rotate l k)) n hn = nthLe (rotate (cyclicPermutations l) k) n hn'",
"tactic": "rw [nthLe_rotate, nthLe_cyclicPermutations, rotate_rotate, ← rotate_mod, add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nl✝ l' l : List α\nk : ℕ\nthis : length (cyclicPermutations (rotate l k)) = length (rotate (cyclicPermutations l) k)\nn : ℕ\nhn : n < length (cyclicPermutations (rotate l k))\nhn' : n < length (rotate (cyclicPermutations l) k)\n⊢ rotate l ((n + k) % length l) =\n nthLe (cyclicPermutations l) ((n + k) % length (cyclicPermutations l))\n (_ : (n + k) % length (cyclicPermutations l) < length (cyclicPermutations l))",
"tactic": "cases l <;> simp"
},
{
"state_after": "case nil\nα : Type u\nl l' : List α\nk : ℕ\n⊢ length (cyclicPermutations (rotate [] k)) = length (rotate (cyclicPermutations []) k)\n\ncase cons\nα : Type u\nl l' : List α\nk : ℕ\nhead✝ : α\ntail✝ : List α\n⊢ length (cyclicPermutations (rotate (head✝ :: tail✝) k)) = length (rotate (cyclicPermutations (head✝ :: tail✝)) k)",
"state_before": "α : Type u\nl✝ l' l : List α\nk : ℕ\n⊢ length (cyclicPermutations (rotate l k)) = length (rotate (cyclicPermutations l) k)",
"tactic": "cases l"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nl l' : List α\nk : ℕ\n⊢ length (cyclicPermutations (rotate [] k)) = length (rotate (cyclicPermutations []) k)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nl l' : List α\nk : ℕ\nhead✝ : α\ntail✝ : List α\n⊢ length (cyclicPermutations (rotate (head✝ :: tail✝) k)) = length (rotate (cyclicPermutations (head✝ :: tail✝)) k)",
"tactic": "rw [length_cyclicPermutations_of_ne_nil] <;> simp"
}
] |
[
680,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
Mathlib/Analysis/Convex/Complex.lean
|
convex_halfspace_re_lt
|
[] |
[
23,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
22,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.coe_apply_coe_coe_symm_apply
|
[] |
[
329,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean
|
pow_div_pow_eventuallyEq_atTop
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\n⊢ ∀ (x : 𝕜), 0 < x → (fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\n⊢ (fun x => x ^ p / x ^ q) =ᶠ[atTop] fun x => x ^ (↑p - ↑q)",
"tactic": "apply (eventually_gt_atTop (0 : 𝕜)).mono fun x hx => _"
},
{
"state_after": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\nx : 𝕜\nhx : 0 < x\n⊢ (fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\n⊢ ∀ (x : 𝕜), 0 < x → (fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝ : LinearOrderedField 𝕜\np q : ℕ\nx : 𝕜\nhx : 0 < x\n⊢ (fun x => x ^ p / x ^ q) x = (fun x => x ^ (↑p - ↑q)) x",
"tactic": "simp [zpow_sub₀ hx.ne']"
}
] |
[
48,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
44,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
deriv_fderiv
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n⊢ smulRight 1 (deriv f x) = fderiv 𝕜 f x",
"tactic": "simp [deriv]"
}
] |
[
477,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/Analysis/LocallyConvex/AbsConvex.lean
|
with_gaugeSeminormFamily
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\n⊢ Filter.HasBasis (𝓝 0) (fun s => s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)) id",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\n⊢ WithSeminorms (gaugeSeminormFamily 𝕜 E)",
"tactic": "refine' SeminormFamily.withSeminorms_of_hasBasis _ _"
},
{
"state_after": "case refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s\n⊢ ∃ i', i' ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E) ∧ id i' ⊆ id s\n\ncase refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)\n⊢ ∃ i, (0 ∈ i ∧ IsOpen i ∧ Balanced 𝕜 i ∧ Convex ℝ i) ∧ id i ⊆ id s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\n⊢ Filter.HasBasis (𝓝 0) (fun s => s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)) id",
"tactic": "refine' (nhds_basis_abs_convex_open 𝕜 E).to_hasBasis (fun s hs => _) fun s hs => _"
},
{
"state_after": "case refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)\n⊢ 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s",
"state_before": "case refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)\n⊢ ∃ i, (0 ∈ i ∧ IsOpen i ∧ Balanced 𝕜 i ∧ Convex ℝ i) ∧ id i ⊆ id s",
"tactic": "refine' ⟨s, ⟨_, rfl.subset⟩⟩"
},
{
"state_after": "case refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : ∃ i r x, s = Seminorm.ball (Finset.sup i (gaugeSeminormFamily 𝕜 E)) 0 r\n⊢ 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s",
"state_before": "case refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)\n⊢ 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s",
"tactic": "rw [SeminormFamily.basisSets_iff] at hs"
},
{
"state_after": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\n⊢ 0 ∈ Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r ∧\n IsOpen (Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r) ∧\n Balanced 𝕜 (Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r) ∧\n Convex ℝ (Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r)",
"state_before": "case refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : ∃ i r x, s = Seminorm.ball (Finset.sup i (gaugeSeminormFamily 𝕜 E)) 0 r\n⊢ 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s",
"tactic": "rcases hs with ⟨t, r, hr, rfl⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\n⊢ (0 ∈ ⋂ (i : AbsConvexOpenSets 𝕜 E) (_ : i ∈ t), Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r) ∧\n IsOpen (⋂ (i : AbsConvexOpenSets 𝕜 E) (_ : i ∈ t), Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r) ∧\n Balanced 𝕜 (⋂ (i : AbsConvexOpenSets 𝕜 E) (_ : i ∈ t), Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r) ∧\n Convex ℝ (⋂ (i : AbsConvexOpenSets 𝕜 E) (_ : i ∈ t), Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r)",
"state_before": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\n⊢ 0 ∈ Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r ∧\n IsOpen (Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r) ∧\n Balanced 𝕜 (Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r) ∧\n Convex ℝ (Seminorm.ball (Finset.sup t (gaugeSeminormFamily 𝕜 E)) 0 r)",
"tactic": "rw [Seminorm.ball_finset_sup_eq_iInter _ _ _ hr]"
},
{
"state_after": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\n⊢ IsOpen (Seminorm.ball (gaugeSeminormFamily 𝕜 E S) 0 r)",
"state_before": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\n⊢ (0 ∈ ⋂ (i : AbsConvexOpenSets 𝕜 E) (_ : i ∈ t), Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r) ∧\n IsOpen (⋂ (i : AbsConvexOpenSets 𝕜 E) (_ : i ∈ t), Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r) ∧\n Balanced 𝕜 (⋂ (i : AbsConvexOpenSets 𝕜 E) (_ : i ∈ t), Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r) ∧\n Convex ℝ (⋂ (i : AbsConvexOpenSets 𝕜 E) (_ : i ∈ t), Seminorm.ball (gaugeSeminormFamily 𝕜 E i) 0 r)",
"tactic": "refine'\n ⟨mem_iInter₂.mpr fun _ _ => by simp [Seminorm.mem_ball_zero, hr],\n isOpen_biInter (t.finite_toSet) fun S _ => _,\n balanced_iInter₂ fun _ _ => Seminorm.balanced_ball_zero _ _,\n convex_iInter₂ fun _ _ => Seminorm.convex_ball _ _ _⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\nhr' : r = ‖↑r‖ * 1\n⊢ IsOpen (Seminorm.ball (gaugeSeminormFamily 𝕜 E S) 0 r)",
"state_before": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\n⊢ IsOpen (Seminorm.ball (gaugeSeminormFamily 𝕜 E S) 0 r)",
"tactic": "have hr' : r = ‖(r : 𝕜)‖ * 1 := by simp [abs_of_pos hr]"
},
{
"state_after": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\nhr' : r = ‖↑r‖ * 1\nhr'' : ↑r ≠ 0\n⊢ IsOpen (Seminorm.ball (gaugeSeminormFamily 𝕜 E S) 0 r)",
"state_before": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\nhr' : r = ‖↑r‖ * 1\n⊢ IsOpen (Seminorm.ball (gaugeSeminormFamily 𝕜 E S) 0 r)",
"tactic": "have hr'' : (r : 𝕜) ≠ 0 := by simp [hr.ne']"
},
{
"state_after": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\nhr' : r = ‖↑r‖ * 1\nhr'' : ↑r ≠ 0\n⊢ IsOpen (↑r • ↑S)",
"state_before": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\nhr' : r = ‖↑r‖ * 1\nhr'' : ↑r ≠ 0\n⊢ IsOpen (Seminorm.ball (gaugeSeminormFamily 𝕜 E S) 0 r)",
"tactic": "rw [hr', ← Seminorm.smul_ball_zero hr'', gaugeSeminormFamily_ball]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\nhr' : r = ‖↑r‖ * 1\nhr'' : ↑r ≠ 0\n⊢ IsOpen (↑r • ↑S)",
"tactic": "exact S.coe_isOpen.smul₀ hr''"
},
{
"state_after": "case refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s\n⊢ s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)",
"state_before": "case refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s\n⊢ ∃ i', i' ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E) ∧ id i' ⊆ id s",
"tactic": "refine' ⟨s, ⟨_, rfl.subset⟩⟩"
},
{
"state_after": "case h.e'_4\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s\n⊢ s = Seminorm.ball (gaugeSeminormFamily 𝕜 E { val := s, property := hs }) 0 1",
"state_before": "case refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s\n⊢ s ∈ SeminormFamily.basisSets (gaugeSeminormFamily 𝕜 E)",
"tactic": "convert(gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : 0 ∈ s ∧ IsOpen s ∧ Balanced 𝕜 s ∧ Convex ℝ s\n⊢ s = Seminorm.ball (gaugeSeminormFamily 𝕜 E { val := s, property := hs }) 0 1",
"tactic": "rw [gaugeSeminormFamily_ball, Subtype.coe_mk]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nx✝¹ : AbsConvexOpenSets 𝕜 E\nx✝ : x✝¹ ∈ t\n⊢ 0 ∈ Seminorm.ball (gaugeSeminormFamily 𝕜 E x✝¹) 0 r",
"tactic": "simp [Seminorm.mem_ball_zero, hr]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\n⊢ r = ‖↑r‖ * 1",
"tactic": "simp [abs_of_pos hr]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.88243\nG : Type ?u.88246\nι : Type ?u.88249\ninst✝¹⁰ : IsROrC 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : IsScalarTower ℝ 𝕜 E\ninst✝⁴ : ContinuousSMul ℝ E\ninst✝³ : TopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : SMulCommClass ℝ 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nt : Finset (AbsConvexOpenSets 𝕜 E)\nr : ℝ\nhr : 0 < r\nS : AbsConvexOpenSets 𝕜 E\nx✝ : S ∈ fun i => i ∈ t.val\nhr' : r = ‖↑r‖ * 1\n⊢ ↑r ≠ 0",
"tactic": "simp [hr.ne']"
}
] |
[
186,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Order/RelIso/Set.lean
|
subrel_val
|
[] |
[
63,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.single_injective
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.49306\nγ : Type ?u.49309\nι : Type ?u.49312\nM : Type u_1\nM' : Type ?u.49318\nN : Type ?u.49321\nP : Type ?u.49324\nG : Type ?u.49327\nH : Type ?u.49330\nR : Type ?u.49333\nS : Type ?u.49336\ninst✝ : Zero M\na✝ a' : α\nb : M\na : α\nb₁ b₂ : M\neq : single a b₁ = single a b₂\nthis : ↑(single a b₁) a = ↑(single a b₂) a\n⊢ b₁ = b₂",
"state_before": "α : Type u_2\nβ : Type ?u.49306\nγ : Type ?u.49309\nι : Type ?u.49312\nM : Type u_1\nM' : Type ?u.49318\nN : Type ?u.49321\nP : Type ?u.49324\nG : Type ?u.49327\nH : Type ?u.49330\nR : Type ?u.49333\nS : Type ?u.49336\ninst✝ : Zero M\na✝ a' : α\nb : M\na : α\nb₁ b₂ : M\neq : single a b₁ = single a b₂\n⊢ b₁ = b₂",
"tactic": "have : (single a b₁ : α →₀ M) a = (single a b₂ : α →₀ M) a := by rw [eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.49306\nγ : Type ?u.49309\nι : Type ?u.49312\nM : Type u_1\nM' : Type ?u.49318\nN : Type ?u.49321\nP : Type ?u.49324\nG : Type ?u.49327\nH : Type ?u.49330\nR : Type ?u.49333\nS : Type ?u.49336\ninst✝ : Zero M\na✝ a' : α\nb : M\na : α\nb₁ b₂ : M\neq : single a b₁ = single a b₂\nthis : ↑(single a b₁) a = ↑(single a b₂) a\n⊢ b₁ = b₂",
"tactic": "rwa [single_eq_same, single_eq_same] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.49306\nγ : Type ?u.49309\nι : Type ?u.49312\nM : Type u_1\nM' : Type ?u.49318\nN : Type ?u.49321\nP : Type ?u.49324\nG : Type ?u.49327\nH : Type ?u.49330\nR : Type ?u.49333\nS : Type ?u.49336\ninst✝ : Zero M\na✝ a' : α\nb : M\na : α\nb₁ b₂ : M\neq : single a b₁ = single a b₂\n⊢ ↑(single a b₁) a = ↑(single a b₂) a",
"tactic": "rw [eq]"
}
] |
[
381,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
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