file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/List/ProdSigma.lean
|
List.sigma_nil
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7878\nσ : α → Type u_2\nhead✝ : α\nl : List α\n⊢ (List.sigma (head✝ :: l) fun a => []) = []",
"tactic": "simp [sigma_cons, sigma_nil]"
}
] |
[
80,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.HasBasis.lift'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.28763\nι✝ : Sort ?u.28766\nf f₁ f₂ : Filter α\nh h₁ h₂ : Set α → Set β\nι : Sort u_1\np : ι → Prop\ns : ι → Set α\nhf : HasBasis f p s\nhh : Monotone h\nt : Set β\n⊢ (∃ i, p i ∧ ∃ x, True ∧ h (s i) ⊆ t) ↔ ∃ i, p i ∧ (h ∘ s) i ⊆ t",
"tactic": "simp only [exists_const, true_and, comp]"
}
] |
[
261,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iInf_pair
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nβ₂ : Type ?u.142048\nγ : Type ?u.142051\nι : Sort ?u.142054\nι' : Sort ?u.142057\nκ : ι → Sort ?u.142062\nκ' : ι' → Sort ?u.142067\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na✝ b✝ : α\nf : β → α\na b : β\n⊢ (⨅ (x : β) (_ : x ∈ {a, b}), f x) = f a ⊓ f b",
"tactic": "rw [iInf_insert, iInf_singleton]"
}
] |
[
1470,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1469,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
UniformSpace.replaceTopology_eq
|
[] |
[
387,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.toSubalgebra_subtype
|
[] |
[
167,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
AntivaryOn.sum_mul_eq_sum_comp_perm_mul_iff
|
[] |
[
439,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
tendsto_one_plus_div_pow_exp
|
[
{
"state_after": "no goals",
"state_before": "t : ℝ\n⊢ ∀ (x : ℕ), ((fun x => (1 + t / x) ^ x) ∘ Nat.cast) x = (1 + t / ↑x) ^ ↑x",
"tactic": "simp"
}
] |
[
627,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Mathlib/CategoryTheory/Abelian/Pseudoelements.lean
|
CategoryTheory.Abelian.Pseudoelement.zero_morphism_ext'
|
[] |
[
290,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Sbtw.left_ne
|
[] |
[
247,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Logic/Nonempty.lean
|
subsingleton_of_not_nonempty
|
[] |
[
171,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Order/Atoms.lean
|
OrderIso.isCoatom_iff
|
[] |
[
812,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
810,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.biInf_le_nhds
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.69959\nβ : Type ?u.69962\nγ : Type ?u.69965\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ 𝓟 (Icc (⊤ - 1) (⊤ + 1)) ≤ 𝓝 ⊤",
"tactic": "simpa only [← coe_one, top_sub_coe, top_add, Icc_self, principal_singleton] using pure_le_nhds _"
}
] |
[
267,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.card_image₂_singleton_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nα' : Type ?u.50799\nβ : Type u_1\nβ' : Type ?u.50805\nγ : Type u_2\nγ' : Type ?u.50811\nδ : Type ?u.50814\nδ' : Type ?u.50817\nε : Type ?u.50820\nε' : Type ?u.50823\nζ : Type ?u.50826\nζ' : Type ?u.50829\nν : Type ?u.50832\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\nhf : Injective (f a)\n⊢ card (image₂ f {a} t) = card t",
"tactic": "rw [image₂_singleton_left, card_image_of_injective _ hf]"
}
] |
[
258,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.Poly.denote_append
|
[
{
"state_after": "no goals",
"state_before": "ctx : Context\np q : Poly\n⊢ denote ctx (p ++ q) = denote ctx p + denote ctx q",
"tactic": "match p with\n| [] => simp\n| (k, v) :: p => simp [denote_append]"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\np q : Poly\n⊢ denote ctx ([] ++ q) = denote ctx [] + denote ctx q",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\np✝ q : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\n⊢ denote ctx ((k, v) :: p ++ q) = denote ctx ((k, v) :: p) + denote ctx q",
"tactic": "simp [denote_append]"
}
] |
[
300,
40
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
297,
1
] |
Mathlib/Data/LazyList/Basic.lean
|
LazyList.mem_nil
|
[] |
[
252,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
VectorPrebundle.coordChange_apply
|
[] |
[
895,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
891,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.monic_zero_iff_subsingleton'
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\na✝ : Subsingleton R\n⊢ (∀ (f g : R[X]), f = g) ∧ ∀ (a b : R), a = b",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\n⊢ Subsingleton R → (∀ (f g : R[X]), f = g) ∧ ∀ (a b : R), a = b",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\na✝ : Subsingleton R\n⊢ (∀ (f g : R[X]), f = g) ∧ ∀ (a b : R), a = b",
"tactic": "simp"
}
] |
[
51,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.exists_coprime'
|
[] |
[
289,
57
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
287,
1
] |
Mathlib/Combinatorics/Composition.lean
|
List.map_length_splitWrtCompositionAux
|
[
{
"state_after": "case nil\nn : ℕ\nα : Type u_1\nl : List α\nh : True\n⊢ map length (splitWrtCompositionAux l []) = []\n\ncase cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\n⊢ map length (splitWrtCompositionAux l (n :: ns)) = n :: ns",
"state_before": "n : ℕ\nα : Type u_1\nns : List ℕ\n⊢ ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns",
"tactic": "induction' ns with n ns IH <;> intro l h <;> simp at h"
},
{
"state_after": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\n⊢ map length (splitWrtCompositionAux l (n :: ns)) = n :: ns",
"state_before": "case nil\nn : ℕ\nα : Type u_1\nl : List α\nh : True\n⊢ map length (splitWrtCompositionAux l []) = []\n\ncase cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\n⊢ map length (splitWrtCompositionAux l (n :: ns)) = n :: ns",
"tactic": ". simp"
},
{
"state_after": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ map length (splitWrtCompositionAux l (n :: ns)) = n :: ns",
"state_before": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\n⊢ map length (splitWrtCompositionAux l (n :: ns)) = n :: ns",
"tactic": "have := le_trans (Nat.le_add_right _ _) h"
},
{
"state_after": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ map length (take n l :: splitWrtCompositionAux (drop n l) ns) = n :: ns",
"state_before": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ map length (splitWrtCompositionAux l (n :: ns)) = n :: ns",
"tactic": "simp only [splitWrtCompositionAux_cons, this]"
},
{
"state_after": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ length (take n l) :: map length (splitWrtCompositionAux (drop n l) ns) = n :: ns",
"state_before": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ map length (take n l :: splitWrtCompositionAux (drop n l) ns) = n :: ns",
"tactic": "dsimp"
},
{
"state_after": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ n ≤ length l\n\ncase cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ sum ns ≤ length l - n",
"state_before": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ length (take n l) :: map length (splitWrtCompositionAux (drop n l) ns) = n :: ns",
"tactic": "rw [length_take, IH] <;> simp [length_drop]"
},
{
"state_after": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ sum ns ≤ length l - n",
"state_before": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ n ≤ length l\n\ncase cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ sum ns ≤ length l - n",
"tactic": ". assumption"
},
{
"state_after": "no goals",
"state_before": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ sum ns ≤ length l - n",
"tactic": ". exact le_tsub_of_add_le_left h"
},
{
"state_after": "no goals",
"state_before": "case nil\nn : ℕ\nα : Type u_1\nl : List α\nh : True\n⊢ map length (splitWrtCompositionAux l []) = []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ n ≤ length l",
"tactic": "assumption"
},
{
"state_after": "no goals",
"state_before": "case cons\nn✝ : ℕ\nα : Type u_1\nn : ℕ\nns : List ℕ\nIH : ∀ {l : List α}, sum ns ≤ length l → map length (splitWrtCompositionAux l ns) = ns\nl : List α\nh : n + sum ns ≤ length l\nthis : n ≤ length l\n⊢ sum ns ≤ length l - n",
"tactic": "exact le_tsub_of_add_le_left h"
}
] |
[
701,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
693,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
Localization.mk_algebraMap
|
[
{
"state_after": "R : Type u_2\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2882462\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type ?u.2882629\ninst✝² : CommSemiring P\nA : Type u_1\ninst✝¹ : CommSemiring A\ninst✝ : Algebra A R\nm : A\n⊢ ↑(algebraMap ((fun x => R) m) (Localization M)) (↑(algebraMap A R) m) = ↑(algebraMap A (Localization M)) m",
"state_before": "R : Type u_2\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2882462\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type ?u.2882629\ninst✝² : CommSemiring P\nA : Type u_1\ninst✝¹ : CommSemiring A\ninst✝ : Algebra A R\nm : A\n⊢ mk (↑(algebraMap A R) m) 1 = ↑(algebraMap A (Localization M)) m",
"tactic": "rw [mk_eq_mk', mk'_eq_iff_eq_mul, Submonoid.coe_one, map_one, mul_one]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type ?u.2882462\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nP : Type ?u.2882629\ninst✝² : CommSemiring P\nA : Type u_1\ninst✝¹ : CommSemiring A\ninst✝ : Algebra A R\nm : A\n⊢ ↑(algebraMap ((fun x => R) m) (Localization M)) (↑(algebraMap A R) m) = ↑(algebraMap A (Localization M)) m",
"tactic": "rfl"
}
] |
[
1040,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1038,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.IsWf.mono
|
[] |
[
199,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.bsup_eq_blsub_or_succ_bsup_eq_blsub
|
[
{
"state_after": "α : Type ?u.353871\nβ : Type ?u.353874\nγ : Type ?u.353877\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ sup (familyOfBFamily o f) = lsub (familyOfBFamily o f) ∨ succ (sup (familyOfBFamily o f)) = lsub (familyOfBFamily o f)",
"state_before": "α : Type ?u.353871\nβ : Type ?u.353874\nγ : Type ?u.353877\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ bsup o f = blsub o f ∨ succ (bsup o f) = blsub o f",
"tactic": "rw [← sup_eq_bsup, ← lsub_eq_blsub]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.353871\nβ : Type ?u.353874\nγ : Type ?u.353877\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\n⊢ sup (familyOfBFamily o f) = lsub (familyOfBFamily o f) ∨ succ (sup (familyOfBFamily o f)) = lsub (familyOfBFamily o f)",
"tactic": "exact sup_eq_lsub_or_sup_succ_eq_lsub _"
}
] |
[
1837,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1834,
1
] |
Mathlib/AlgebraicTopology/DoldKan/PInfty.lean
|
AlgebraicTopology.DoldKan.PInfty_f_comp_QInfty_f
|
[
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f PInfty n ≫ HomologicalComplex.Hom.f (𝟙 K[X] - PInfty) n = 0",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f PInfty n ≫ HomologicalComplex.Hom.f QInfty n = 0",
"tactic": "dsimp only [QInfty]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ HomologicalComplex.Hom.f PInfty n ≫ HomologicalComplex.Hom.f (𝟙 K[X] - PInfty) n = 0",
"tactic": "simp only [HomologicalComplex.sub_f_apply, HomologicalComplex.id_f, comp_sub, comp_id,\n PInfty_f_idem, sub_self]"
}
] |
[
135,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/CategoryTheory/Sites/DenseSubsite.lean
|
CategoryTheory.CoverDense.sheaf_eq_amalgamation
|
[] |
[
150,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.comap_eq_top_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nrc : RingHomClass F R S\nf : F\nI✝ J : Ideal R\nK L : Ideal S\nG : Type ?u.781465\nrcg : RingHomClass G S R\nI : Ideal S\nh : I = ⊤\n⊢ comap f I = ⊤",
"tactic": "rw [h, comap_top]"
}
] |
[
1446,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1444,
1
] |
Mathlib/GroupTheory/SpecificGroups/Quaternion.lean
|
QuaternionGroup.orderOf_xa
|
[
{
"state_after": "n : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\n⊢ orderOf (xa i) = 2 ^ 2",
"state_before": "n : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\n⊢ orderOf (xa i) = 4",
"tactic": "change _ = 2 ^ 2"
},
{
"state_after": "n : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\n⊢ orderOf (xa i) = 2 ^ 2",
"state_before": "n : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\n⊢ orderOf (xa i) = 2 ^ 2",
"tactic": "haveI : Fact (Nat.Prime 2) := Fact.mk Nat.prime_two"
},
{
"state_after": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\n⊢ ¬xa i ^ 2 ^ 1 = 1\n\ncase hfin\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\n⊢ xa i ^ 2 ^ (1 + 1) = 1",
"state_before": "n : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\n⊢ orderOf (xa i) = 2 ^ 2",
"tactic": "apply orderOf_eq_prime_pow"
},
{
"state_after": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh : xa i ^ 2 ^ 1 = 1\n⊢ False",
"state_before": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\n⊢ ¬xa i ^ 2 ^ 1 = 1",
"tactic": "intro h"
},
{
"state_after": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh : a ↑n = 1\n⊢ False",
"state_before": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh : xa i ^ 2 ^ 1 = 1\n⊢ False",
"tactic": "simp only [pow_one, xa_sq] at h"
},
{
"state_after": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh' : ↑n = 0\n⊢ False",
"state_before": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh : a ↑n = 1\n⊢ False",
"tactic": "injection h with h'"
},
{
"state_after": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh' : ZMod.val ↑n = ZMod.val 0\n⊢ False",
"state_before": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh' : ↑n = 0\n⊢ False",
"tactic": "apply_fun ZMod.val at h'"
},
{
"state_after": "no goals",
"state_before": "case hnot\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\nh' : ZMod.val ↑n / n = ZMod.val 0 / n\n⊢ False",
"tactic": "simp only [ZMod.val_nat_cast, ZMod.val_zero, Nat.zero_div, Nat.mod_mul_left_div_self,\n Nat.div_self (NeZero.pos n)] at h'"
},
{
"state_after": "no goals",
"state_before": "case hfin\nn : ℕ\ninst✝ : NeZero n\ni : ZMod (2 * n)\nthis : Fact (Nat.Prime 2)\n⊢ xa i ^ 2 ^ (1 + 1) = 1",
"tactic": "norm_num"
}
] |
[
226,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_preimage_ofInjective_subset_range
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ns✝ t : Set α\na b x y : α\nf : α → β\ns : Set β\nH : Function.Injective f\nhs : s ⊆ range f\n⊢ ncard (f ⁻¹' s) = ncard s",
"tactic": "rw [← ncard_image_ofInjective _ H, image_preimage_eq_iff.mpr hs]"
}
] |
[
276,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.IsImage.image_eq
|
[] |
[
405,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.union_smul_inter_subset_union
|
[] |
[
226,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_linear_lt_degree_C_mul_X_sq
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.1018519\nha : a ≠ 0\n⊢ degree (↑C b * X + ↑C c) < degree (↑C a * X ^ 2)",
"tactic": "simpa only [degree_C_mul_X_pow 2 ha] using degree_linear_lt"
}
] |
[
1185,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1183,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
MultilinearMap.mkPiAlgebraFin_apply_const
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type u'\nn : ℕ\nM : Fin (Nat.succ n) → Type v\nM₁ : ι → Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\nM' : Type v'\ninst✝⁸ : CommSemiring R\ninst✝⁷ : (i : ι) → AddCommMonoid (M₁ i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → AddCommMonoid (M i)\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : (i : Fin (Nat.succ n)) → Module R (M i)\ninst✝³ : (i : ι) → Module R (M₁ i)\ninst✝² : Module R M₂\nf f' : MultilinearMap R M₁ M₂\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\na : A\n⊢ (↑(MultilinearMap.mkPiAlgebraFin R n A) fun x => a) = a ^ n",
"tactic": "simp"
}
] |
[
1032,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1031,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Valid'.rotateL_lemma₄
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.313834\ninst✝ : Preorder α\na b : ℕ\nH3 : 2 * b ≤ 9 * a + 3\n⊢ 3 * b ≤ 16 * a + 9",
"tactic": "linarith"
}
] |
[
1232,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1231,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.mul_nonsing_inv_cancel_left
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.206113\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\nB : Matrix n m α\nh : IsUnit (det A)\n⊢ A ⬝ (A⁻¹ ⬝ B) = B",
"tactic": "simp [← Matrix.mul_assoc, mul_nonsing_inv A h]"
}
] |
[
331,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.injOn_iff_injective
|
[] |
[
684,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
682,
1
] |
Mathlib/AlgebraicTopology/SplitSimplicialObject.lean
|
SimplicialObject.Split.comp_f
|
[] |
[
457,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/ModelTheory/Satisfiability.lean
|
Cardinal.Categorical.isComplete
|
[
{
"state_after": "case intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\n⊢ T ⊨ᵇ φ ∨ T ⊨ᵇ Formula.not φ",
"state_before": "L : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\n⊢ T ⊨ᵇ φ ∨ T ⊨ᵇ Formula.not φ",
"tactic": "obtain ⟨_, _⟩ := Theory.exists_model_card_eq ⟨hS.some, hT hS.some⟩ κ h1 h2"
},
{
"state_after": "case intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\n⊢ (∀ (M : Theory.ModelType T), ↑M ⊨ φ) ∨ ∀ (M : Theory.ModelType T), ↑M ⊨ Formula.not φ",
"state_before": "case intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\n⊢ T ⊨ᵇ φ ∨ T ⊨ᵇ Formula.not φ",
"tactic": "rw [Theory.models_sentence_iff, Theory.models_sentence_iff]"
},
{
"state_after": "case intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\ncon : ¬((∀ (M : Theory.ModelType T), ↑M ⊨ φ) ∨ ∀ (M : Theory.ModelType T), ↑M ⊨ Formula.not φ)\n⊢ False",
"state_before": "case intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\n⊢ (∀ (M : Theory.ModelType T), ↑M ⊨ φ) ∨ ∀ (M : Theory.ModelType T), ↑M ⊨ Formula.not φ",
"tactic": "by_contra con"
},
{
"state_after": "case intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\ncon : (∃ M, ¬↑M ⊨ φ) ∧ ∃ M, ¬↑M ⊨ Formula.not φ\n⊢ False",
"state_before": "case intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\ncon : ¬((∀ (M : Theory.ModelType T), ↑M ⊨ φ) ∨ ∀ (M : Theory.ModelType T), ↑M ⊨ Formula.not φ)\n⊢ False",
"tactic": "push_neg at con"
},
{
"state_after": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ¬↑MT ⊨ Formula.not φ\n⊢ False",
"state_before": "case intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\ncon : (∃ M, ¬↑M ⊨ φ) ∧ ∃ M, ¬↑M ⊨ Formula.not φ\n⊢ False",
"tactic": "obtain ⟨⟨MF, hMF⟩, MT, hMT⟩ := con"
},
{
"state_after": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\n⊢ False",
"state_before": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ¬↑MT ⊨ Formula.not φ\n⊢ False",
"tactic": "rw [Sentence.realize_not, Classical.not_not] at hMT"
},
{
"state_after": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\n⊢ ↑MF ⊨ φ",
"state_before": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\n⊢ False",
"tactic": "refine' hMF _"
},
{
"state_after": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis : Infinite ↑MT\n⊢ ↑MF ⊨ φ",
"state_before": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\n⊢ ↑MF ⊨ φ",
"tactic": "haveI := hT MT"
},
{
"state_after": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis✝ : Infinite ↑MT\nthis : Infinite ↑MF\n⊢ ↑MF ⊨ φ",
"state_before": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis : Infinite ↑MT\n⊢ ↑MF ⊨ φ",
"tactic": "haveI := hT MF"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis✝ : Infinite ↑MT\nthis : Infinite ↑MF\nNT : Bundled (Structure L)\nMNT : ↑MT ≅[L] ↑NT\nhNT : (#↑NT) = κ\n⊢ ↑MF ⊨ φ",
"state_before": "case intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis✝ : Infinite ↑MT\nthis : Infinite ↑MF\n⊢ ↑MF ⊨ φ",
"tactic": "obtain ⟨NT, MNT, hNT⟩ := exists_elementarilyEquivalent_card_eq L MT κ h1 h2"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis✝ : Infinite ↑MT\nthis : Infinite ↑MF\nNT : Bundled (Structure L)\nMNT : ↑MT ≅[L] ↑NT\nhNT : (#↑NT) = κ\nNF : Bundled (Structure L)\nMNF : ↑MF ≅[L] ↑NF\nhNF : (#↑NF) = κ\n⊢ ↑MF ⊨ φ",
"state_before": "case intro.intro.intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis✝ : Infinite ↑MT\nthis : Infinite ↑MF\nNT : Bundled (Structure L)\nMNT : ↑MT ≅[L] ↑NT\nhNT : (#↑NT) = κ\n⊢ ↑MF ⊨ φ",
"tactic": "obtain ⟨NF, MNF, hNF⟩ := exists_elementarilyEquivalent_card_eq L MF κ h1 h2"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis✝ : Infinite ↑MT\nthis : Infinite ↑MF\nNT : Bundled (Structure L)\nMNT : ↑MT ≅[L] ↑NT\nhNT : (#↑NT) = κ\nNF : Bundled (Structure L)\nMNF : ↑MF ≅[L] ↑NF\nhNF : (#↑NF) = κ\nTF : ↑(ElementarilyEquivalent.toModel T MNT) ≃[L] ↑(ElementarilyEquivalent.toModel T MNF)\n⊢ ↑MF ⊨ φ",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis✝ : Infinite ↑MT\nthis : Infinite ↑MF\nNT : Bundled (Structure L)\nMNT : ↑MT ≅[L] ↑NT\nhNT : (#↑NT) = κ\nNF : Bundled (Structure L)\nMNF : ↑MF ≅[L] ↑NF\nhNF : (#↑NF) = κ\n⊢ ↑MF ⊨ φ",
"tactic": "obtain ⟨TF⟩ := h (MNT.toModel T) (MNF.toModel T) hNT hNF"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nL : Language\nκ : Cardinal\nT : Theory L\nh : Categorical κ T\nh1 : ℵ₀ ≤ κ\nh2 : lift (card L) ≤ lift κ\nhS : Theory.IsSatisfiable T\nhT : ∀ (M : Theory.ModelType T), Infinite ↑M\nφ : Sentence L\nw✝ : Theory.ModelType T\nh✝ : (#↑w✝) = κ\nMF : Theory.ModelType T\nhMF : ¬↑MF ⊨ φ\nMT : Theory.ModelType T\nhMT : ↑MT ⊨ φ\nthis✝ : Infinite ↑MT\nthis : Infinite ↑MF\nNT : Bundled (Structure L)\nMNT : ↑MT ≅[L] ↑NT\nhNT : (#↑NT) = κ\nNF : Bundled (Structure L)\nMNF : ↑MF ≅[L] ↑NF\nhNF : (#↑NF) = κ\nTF : ↑(ElementarilyEquivalent.toModel T MNT) ≃[L] ↑(ElementarilyEquivalent.toModel T MNF)\n⊢ ↑MF ⊨ φ",
"tactic": "exact\n ((MNT.realize_sentence φ).trans\n ((TF.realize_sentence φ).trans (MNF.realize_sentence φ).symm)).1 hMT"
}
] |
[
674,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
656,
1
] |
Mathlib/RingTheory/Algebraic.lean
|
polynomial_smul_apply'
|
[] |
[
434,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
OmegaCompletePartialOrder.Chain.mem_map
|
[] |
[
124,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Analysis/Complex/PhragmenLindelof.lean
|
PhragmenLindelof.eqOn_quadrant_II
|
[] |
[
528,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
CharP.int_cast_eq_zero_iff
|
[
{
"state_after": "case inl\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nh : a < 0\n⊢ ↑a = 0 ↔ ↑p ∣ a\n\ncase inr.inl\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\n⊢ ↑0 = 0 ↔ ↑p ∣ 0\n\ncase inr.inr\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nh : 0 < a\n⊢ ↑a = 0 ↔ ↑p ∣ a",
"state_before": "R : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\n⊢ ↑a = 0 ↔ ↑p ∣ a",
"tactic": "rcases lt_trichotomy a 0 with (h | rfl | h)"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nh : a < 0\n⊢ ↑(-a) = 0 ↔ ↑p ∣ -a",
"state_before": "case inl\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nh : a < 0\n⊢ ↑a = 0 ↔ ↑p ∣ a",
"tactic": "rw [← neg_eq_zero, ← Int.cast_neg, ← dvd_neg]"
},
{
"state_after": "case inl.intro\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nh : a < 0\nb : ℕ\n⊢ ↑↑b = 0 ↔ ↑p ∣ ↑b",
"state_before": "case inl\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nh : a < 0\n⊢ ↑(-a) = 0 ↔ ↑p ∣ -a",
"tactic": "lift -a to ℕ using neg_nonneg.mpr (le_of_lt h) with b"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nh : a < 0\nb : ℕ\n⊢ ↑↑b = 0 ↔ ↑p ∣ ↑b",
"tactic": "rw [Int.cast_ofNat, CharP.cast_eq_zero_iff R p, Int.coe_nat_dvd]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\n⊢ ↑0 = 0 ↔ ↑p ∣ 0",
"tactic": "simp only [Int.cast_zero, eq_self_iff_true, dvd_zero]"
},
{
"state_after": "case inr.inr.intro\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nb : ℕ\nh✝ h : 0 < ↑b\n⊢ ↑↑b = 0 ↔ ↑p ∣ ↑b",
"state_before": "case inr.inr\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nh : 0 < a\n⊢ ↑a = 0 ↔ ↑p ∣ a",
"tactic": "lift a to ℕ using le_of_lt h with b"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.intro\nR : Type u_1\ninst✝¹ : AddGroupWithOne R\np : ℕ\ninst✝ : CharP R p\na : ℤ\nb : ℕ\nh✝ h : 0 < ↑b\n⊢ ↑↑b = 0 ↔ ↑p ∣ ↑b",
"tactic": "rw [Int.cast_ofNat, CharP.cast_eq_zero_iff R p, Int.coe_nat_dvd]"
}
] |
[
141,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.erase_subset
|
[] |
[
1942,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1941,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.pure_sets
|
[] |
[
1972,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1971,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.centroid_map
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_4\nV : Type u_5\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type u_1\ns₂ : Finset ι₂\ne : ι₂ ↪ ι\np : ι → P\n⊢ centroid k (map e s₂) p = centroid k s₂ (p ∘ ↑e)",
"tactic": "simp [centroid_def, affineCombination_map, centroidWeights]"
}
] |
[
889,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
888,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.ne_and_ne_of_swap_mul_apply_ne_self
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\nhy : (if ↑f y = x then ↑f x else if y = x then x else ↑f y) ≠ y\n⊢ ↑f y ≠ y ∧ y ≠ x",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\nhy : ↑(swap x (↑f x) * f) y ≠ y\n⊢ ↑f y ≠ y ∧ y ≠ x",
"tactic": "simp only [swap_apply_def, mul_apply, f.injective.eq_iff] at *"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\nhy : (if ↑f y = x then ↑f x else if y = x then x else ↑f y) ≠ y\nh : ↑f y = x\n⊢ ↑f y ≠ y ∧ y ≠ x\n\ncase neg\nα : Type u_1\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\nhy : (if ↑f y = x then ↑f x else if y = x then x else ↑f y) ≠ y\nh : ¬↑f y = x\n⊢ ↑f y ≠ y ∧ y ≠ x",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\nhy : (if ↑f y = x then ↑f x else if y = x then x else ↑f y) ≠ y\n⊢ ↑f y ≠ y ∧ y ≠ x",
"tactic": "by_cases h : f y = x"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\nhy : (if ↑f y = x then ↑f x else if y = x then x else ↑f y) ≠ y\nh : ↑f y = x\n⊢ ↑f y ≠ y ∧ y ≠ x",
"tactic": "constructor <;> intro <;> simp_all only [if_true, eq_self_iff_true, not_true, Ne.def]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\nf : Perm α\nx y : α\nhy : (if ↑f y = x then ↑f x else if y = x then x else ↑f y) ≠ y\nh : ¬↑f y = x\n⊢ ↑f y ≠ y ∧ y ≠ x",
"tactic": "split_ifs at hy with h h <;> try { subst x } <;> try { simp [*] at * }"
}
] |
[
249,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.indicator_nonpos_le_indicator
|
[] |
[
895,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
893,
1
] |
Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean
|
AlgebraicTopology.DoldKan.N₁Γ₀_hom_app
|
[
{
"state_after": "C : 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)).inv ≫\n (toKaroubi (ChainComplex C ℕ)).map (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.hom.app K =\n (Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).inv ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀NondegComplexIso K).hom",
"tactic": "change (N₁Γ₀.app K).hom = _"
},
{
"state_after": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\n⊢ ((Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).symm ≪≫\n (toKaroubi (ChainComplex C ℕ)).mapIso (Γ₀NondegComplexIso K)).hom =\n (Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).inv ≫\n (toKaroubi (ChainComplex C ℕ)).map (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).hom =\n (Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).inv ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀NondegComplexIso K).hom",
"tactic": "simp only [N₁Γ₀_app]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝² : Category C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\n⊢ ((Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).symm ≪≫\n (toKaroubi (ChainComplex C ℕ)).mapIso (Γ₀NondegComplexIso K)).hom =\n (Splitting.toKaroubiNondegComplexIsoN₁ (Γ₀.splitting K)).inv ≫\n (toKaroubi (ChainComplex C ℕ)).map (Γ₀NondegComplexIso K).hom",
"tactic": "rfl"
}
] |
[
91,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.cons_right_injective
|
[] |
[
117,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/LinearAlgebra/Isomorphisms.lean
|
LinearMap.quotKerEquivRange_symm_apply_image
|
[] |
[
63,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.lcm_one_left
|
[
{
"state_after": "no goals",
"state_before": "m : Nat\n⊢ lcm 1 m = m",
"tactic": "simp [lcm]"
}
] |
[
189,
70
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
189,
9
] |
Mathlib/Algebra/Ring/Defs.lean
|
mul_one_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : NonAssocRing α\na b : α\n⊢ a * (1 - b) = a - a * b",
"tactic": "rw [mul_sub, mul_one]"
}
] |
[
419,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.diam_image_le_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.284548\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns✝ t : Set α\nd : ℝ≥0∞\nf : β → α\ns : Set β\n⊢ diam (f '' s) ≤ d ↔ ∀ (x : β), x ∈ s → ∀ (y : β), y ∈ s → edist (f x) (f y) ≤ d",
"tactic": "simp only [diam_le_iff, ball_image_iff]"
}
] |
[
884,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
882,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.isUnit_iff_singleton
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.318482\nα : Type u_1\nβ : Type ?u.318488\nγ : Type ?u.318491\nδ : Type ?u.318494\nε : Type ?u.318497\ninst✝² : Group α\ninst✝¹ : DivisionMonoid β\ninst✝ : MonoidHomClass F α β\nm : F\nf g f₁ g₁ : Filter α\nf₂ g₂ : Filter β\n⊢ IsUnit f ↔ ∃ a, f = pure a",
"tactic": "simp only [isUnit_iff, Group.isUnit, and_true_iff]"
}
] |
[
887,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
886,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_add
|
[] |
[
220,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/LinearAlgebra/Matrix/Determinant.lean
|
Matrix.det_eq_zero_of_column_eq_zero
|
[
{
"state_after": "m : Type ?u.1632406\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\nj : n\nh : ∀ (i : n), A i j = 0\n⊢ det Aᵀ = 0",
"state_before": "m : Type ?u.1632406\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\nj : n\nh : ∀ (i : n), A i j = 0\n⊢ det A = 0",
"tactic": "rw [← det_transpose]"
},
{
"state_after": "no goals",
"state_before": "m : Type ?u.1632406\nn : Type u_1\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\nR : Type v\ninst✝ : CommRing R\nA : Matrix n n R\nj : n\nh : ∀ (i : n), A i j = 0\n⊢ det Aᵀ = 0",
"tactic": "exact det_eq_zero_of_row_eq_zero j h"
}
] |
[
373,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Algebra/Lie/Semisimple.lean
|
LieAlgebra.center_eq_bot_of_semisimple
|
[
{
"state_after": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh : ∀ (I : LieIdeal R L), IsLieAbelian { x // x ∈ ↑I } → I = ⊥\n⊢ center R L = ⊥",
"state_before": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh : IsSemisimple R L\n⊢ center R L = ⊥",
"tactic": "rw [isSemisimple_iff_no_abelian_ideals] at h"
},
{
"state_after": "case a\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh : ∀ (I : LieIdeal R L), IsLieAbelian { x // x ∈ ↑I } → I = ⊥\n⊢ IsLieAbelian { x // x ∈ ↑(center R L) }",
"state_before": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh : ∀ (I : LieIdeal R L), IsLieAbelian { x // x ∈ ↑I } → I = ⊥\n⊢ center R L = ⊥",
"tactic": "apply h"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh : ∀ (I : LieIdeal R L), IsLieAbelian { x // x ∈ ↑I } → I = ⊥\n⊢ IsLieAbelian { x // x ∈ ↑(center R L) }",
"tactic": "infer_instance"
}
] |
[
83,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/NumberTheory/Padics/PadicNorm.lean
|
padicNorm.sum_le'
|
[
{
"state_after": "case inl\np : ℕ\nhp : Fact (Nat.Prime p)\nα : Type u_1\nF : α → ℚ\nt : ℚ\nht : 0 ≤ t\nhF : ∀ (i : α), i ∈ ∅ → padicNorm p (F i) ≤ t\n⊢ padicNorm p (∑ i in ∅, F i) ≤ t\n\ncase inr\np : ℕ\nhp : Fact (Nat.Prime p)\nα : Type u_1\nF : α → ℚ\nt : ℚ\ns : Finset α\nhF : ∀ (i : α), i ∈ s → padicNorm p (F i) ≤ t\nht : 0 ≤ t\nhs : Finset.Nonempty s\n⊢ padicNorm p (∑ i in s, F i) ≤ t",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nα : Type u_1\nF : α → ℚ\nt : ℚ\ns : Finset α\nhF : ∀ (i : α), i ∈ s → padicNorm p (F i) ≤ t\nht : 0 ≤ t\n⊢ padicNorm p (∑ i in s, F i) ≤ t",
"tactic": "obtain rfl | hs := Finset.eq_empty_or_nonempty s"
},
{
"state_after": "no goals",
"state_before": "case inl\np : ℕ\nhp : Fact (Nat.Prime p)\nα : Type u_1\nF : α → ℚ\nt : ℚ\nht : 0 ≤ t\nhF : ∀ (i : α), i ∈ ∅ → padicNorm p (F i) ≤ t\n⊢ padicNorm p (∑ i in ∅, F i) ≤ t",
"tactic": "simp [ht]"
},
{
"state_after": "no goals",
"state_before": "case inr\np : ℕ\nhp : Fact (Nat.Prime p)\nα : Type u_1\nF : α → ℚ\nt : ℚ\ns : Finset α\nhF : ∀ (i : α), i ∈ s → padicNorm p (F i) ≤ t\nht : 0 ≤ t\nhs : Finset.Nonempty s\n⊢ padicNorm p (∑ i in s, F i) ≤ t",
"tactic": "exact sum_le hs hF"
}
] |
[
356,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
352,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.symm_trans_self
|
[
{
"state_after": "case H\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.36194\nδ : Type ?u.36197\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nx✝ : β\n⊢ ↑(Homeomorph.trans (Homeomorph.symm h) h) x✝ = ↑(Homeomorph.refl β) x✝",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.36194\nδ : Type ?u.36197\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\n⊢ Homeomorph.trans (Homeomorph.symm h) h = Homeomorph.refl β",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.36194\nδ : Type ?u.36197\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\nx✝ : β\n⊢ ↑(Homeomorph.trans (Homeomorph.symm h) h) x✝ = ↑(Homeomorph.refl β) x✝",
"tactic": "apply apply_symm_apply"
}
] |
[
168,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Data/List/Dedup.lean
|
List.Nodup.dedup
|
[] |
[
84,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
11
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineSubspace.subtypeₐᵢ_toAffineMap
|
[] |
[
304,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Data/Nat/Interval.lean
|
Nat.card_Icc
|
[] |
[
98,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
|
Complex.exp_inj_of_neg_pi_lt_of_le_pi
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\nhx₁ : -π < x.im\nhx₂ : x.im ≤ π\nhy₁ : -π < y.im\nhy₂ : y.im ≤ π\nhxy : exp x = exp y\n⊢ x = y",
"tactic": "rw [← log_exp hx₁ hx₂, ← log_exp hy₁ hy₂, hxy]"
}
] |
[
70,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.card_erase_of_mem
|
[] |
[
1106,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1105,
1
] |
Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
|
Profinite.exists_locallyConstant_finite_nonempty
|
[
{
"state_after": "J : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\n⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "J : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\n⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "inhabit α"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\n⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "J : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\n⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "obtain ⟨j, gg, h⟩ := exists_locallyConstant_finite_aux _ hC f"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\n⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro.intro\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\n⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "let ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\n⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"state_before": "case intro.intro\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\n⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "let σ : (α → Fin 2) → α := fun f => if h : ∃ a : α, ι a = f then h.choose else default"
},
{
"state_after": "case intro.intro\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\n⊢ f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) (LocallyConstant.map σ gg)",
"state_before": "case intro.intro\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\n⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g",
"tactic": "refine' ⟨j, gg.map σ, _⟩"
},
{
"state_after": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ ↑f x = ↑(LocallyConstant.comap ((forget Profinite).map (C.π.app j)) (LocallyConstant.map σ gg)) x",
"state_before": "case intro.intro\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\n⊢ f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) (LocallyConstant.map σ gg)",
"tactic": "ext x"
},
{
"state_after": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ ↑f x = (↑(LocallyConstant.map σ gg) ∘ (forget Profinite).map (C.π.1 j)) x",
"state_before": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ ↑f x = ↑(LocallyConstant.comap ((forget Profinite).map (C.π.app j)) (LocallyConstant.map σ gg)) x",
"tactic": "rw [LocallyConstant.coe_comap ((forget Profinite).map _) _ (C.π.app j).continuous]"
},
{
"state_after": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ ↑f x =\n if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg ((forget Profinite).map (C.π.1 j) x) then Exists.choose h\n else default",
"state_before": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ ↑f x = (↑(LocallyConstant.map σ gg) ∘ (forget Profinite).map (C.π.1 j)) x",
"tactic": "dsimp"
},
{
"state_after": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ↑f x =\n if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg ((forget Profinite).map (C.π.1 j) x) then Exists.choose h\n else default",
"state_before": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ ↑f x =\n if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg ((forget Profinite).map (C.π.1 j) x) then Exists.choose h\n else default",
"tactic": "have h1 : ι (f x) = gg (C.π.app j x) := by\n change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _\n rw [h, LocallyConstant.coe_comap ((forget Profinite).map _) _ (C.π.app j).continuous]\n rfl"
},
{
"state_after": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ↑f x =\n if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg ((forget Profinite).map (C.π.1 j) x) then Exists.choose h\n else default",
"state_before": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ↑f x =\n if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg ((forget Profinite).map (C.π.1 j) x) then Exists.choose h\n else default",
"tactic": "have h2 : ∃ a : α, ι a = gg (C.π.app j x) := ⟨f x, h1⟩"
},
{
"state_after": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ↑f x = Exists.choose h2",
"state_before": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ↑f x =\n if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg ((forget Profinite).map (C.π.1 j) x) then Exists.choose h\n else default",
"tactic": "rw [dif_pos h2]"
},
{
"state_after": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ι (↑f x) = ι (Exists.choose h2)\n\ncase intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ Function.Injective ι",
"state_before": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ↑f x = Exists.choose h2",
"tactic": "apply_fun ι"
},
{
"state_after": "J : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ ↑(LocallyConstant.map (fun a b => if a = b then 0 else 1) f) x = ↑gg ((forget Profinite).map (C.π.app j) x)",
"state_before": "J : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)",
"tactic": "change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _"
},
{
"state_after": "J : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ (↑gg ∘ (forget Profinite).map (C.π.1 j)) x = ↑gg ((forget Profinite).map (C.π.app j) x)",
"state_before": "J : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ ↑(LocallyConstant.map (fun a b => if a = b then 0 else 1) f) x = ↑gg ((forget Profinite).map (C.π.app j) x)",
"tactic": "rw [h, LocallyConstant.coe_comap ((forget Profinite).map _) _ (C.π.app j).continuous]"
},
{
"state_after": "no goals",
"state_before": "J : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\n⊢ (↑gg ∘ (forget Profinite).map (C.π.1 j)) x = ↑gg ((forget Profinite).map (C.π.app j) x)",
"tactic": "rfl"
},
{
"state_after": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)",
"state_before": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ι (↑f x) = ι (Exists.choose h2)",
"tactic": "rw [h2.choose_spec]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)",
"tactic": "exact h1"
},
{
"state_after": "case intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\n⊢ a = b",
"state_before": "case intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\n⊢ Function.Injective ι",
"tactic": "intro a b hh"
},
{
"state_after": "case intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhhh : ι a a = ι b a\n⊢ a = b",
"state_before": "case intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\n⊢ a = b",
"tactic": "have hhh := congr_fun hh a"
},
{
"state_after": "case intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhhh : (if a = a then 0 else 1) = if b = a then 0 else 1\n⊢ a = b",
"state_before": "case intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhhh : ι a a = ι b a\n⊢ a = b",
"tactic": "dsimp at hhh"
},
{
"state_after": "case intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhhh : 0 = if b = a then 0 else 1\n⊢ a = b",
"state_before": "case intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhhh : (if a = a then 0 else 1) = if b = a then 0 else 1\n⊢ a = b",
"tactic": "rw [if_pos rfl] at hhh"
},
{
"state_after": "case intro.intro.h.inj.inl\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhh1 : b = a\nhhh : 0 = 0\n⊢ a = b\n\ncase intro.intro.h.inj.inr\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhh1 : ¬b = a\nhhh : 0 = 1\n⊢ a = b",
"state_before": "case intro.intro.h.inj\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhhh : 0 = if b = a then 0 else 1\n⊢ a = b",
"tactic": "split_ifs at hhh with hh1"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h.inj.inl\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhh1 : b = a\nhhh : 0 = 0\n⊢ a = b",
"tactic": "exact hh1.symm"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.h.inj.inr\nJ : Type u\ninst✝³ : SmallCategory J\ninst✝² : IsCofiltered J\nF : J ⥤ Profinite\nC : Cone F\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : Nonempty α\nhC : IsLimit C\nf : LocallyConstant (↑C.pt.toCompHaus.toTop) α\ninhabited_h : Inhabited α\nj : J\ngg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)\nh :\n LocallyConstant.map (fun a b => if a = b then 0 else 1) f =\n LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg\nι : α → α → Fin 2 := fun a b => if a = b then 0 else 1\nσ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then Exists.choose h else default\nx : ↑C.pt.toCompHaus.toTop\nh1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)\nh2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)\na b : α\nhh : ι a = ι b\nhh1 : ¬b = a\nhhh : 0 = 1\n⊢ a = b",
"tactic": "exact False.elim (bot_ne_top hhh)"
}
] |
[
212,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/FieldTheory/IsAlgClosed/Basic.lean
|
IsAlgClosed.exists_root
|
[] |
[
82,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.exists_mul_lt_of_lt_op_norm
|
[
{
"state_after": "case intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.744106\nE : Type u_3\nEₗ : Type ?u.744112\nF : Type u_4\nFₗ : Type ?u.744118\nG : Type ?u.744121\nGₗ : Type ?u.744124\n𝓕 : Type ?u.744127\ninst✝¹⁶ : SeminormedAddCommGroup E\ninst✝¹⁵ : SeminormedAddCommGroup Eₗ\ninst✝¹⁴ : SeminormedAddCommGroup F\ninst✝¹³ : SeminormedAddCommGroup Fₗ\ninst✝¹² : SeminormedAddCommGroup G\ninst✝¹¹ : SeminormedAddCommGroup Gₗ\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜₂\ninst✝⁸ : NontriviallyNormedField 𝕜₃\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedSpace 𝕜 Eₗ\ninst✝⁵ : NormedSpace 𝕜₂ F\ninst✝⁴ : NormedSpace 𝕜 Fₗ\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\nr : ℝ≥0\nhr : ↑r < ‖f‖\n⊢ ∃ x, ↑r * ‖x‖ < ‖↑f x‖",
"state_before": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.744106\nE : Type u_3\nEₗ : Type ?u.744112\nF : Type u_4\nFₗ : Type ?u.744118\nG : Type ?u.744121\nGₗ : Type ?u.744124\n𝓕 : Type ?u.744127\ninst✝¹⁶ : SeminormedAddCommGroup E\ninst✝¹⁵ : SeminormedAddCommGroup Eₗ\ninst✝¹⁴ : SeminormedAddCommGroup F\ninst✝¹³ : SeminormedAddCommGroup Fₗ\ninst✝¹² : SeminormedAddCommGroup G\ninst✝¹¹ : SeminormedAddCommGroup Gₗ\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜₂\ninst✝⁸ : NontriviallyNormedField 𝕜₃\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedSpace 𝕜 Eₗ\ninst✝⁵ : NormedSpace 𝕜₂ F\ninst✝⁴ : NormedSpace 𝕜 Fₗ\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\nr : ℝ\nhr₀ : 0 ≤ r\nhr : r < ‖f‖\n⊢ ∃ x, r * ‖x‖ < ‖↑f x‖",
"tactic": "lift r to ℝ≥0 using hr₀"
},
{
"state_after": "no goals",
"state_before": "case intro\n𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type ?u.744106\nE : Type u_3\nEₗ : Type ?u.744112\nF : Type u_4\nFₗ : Type ?u.744118\nG : Type ?u.744121\nGₗ : Type ?u.744124\n𝓕 : Type ?u.744127\ninst✝¹⁶ : SeminormedAddCommGroup E\ninst✝¹⁵ : SeminormedAddCommGroup Eₗ\ninst✝¹⁴ : SeminormedAddCommGroup F\ninst✝¹³ : SeminormedAddCommGroup Fₗ\ninst✝¹² : SeminormedAddCommGroup G\ninst✝¹¹ : SeminormedAddCommGroup Gₗ\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜₂\ninst✝⁸ : NontriviallyNormedField 𝕜₃\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedSpace 𝕜 Eₗ\ninst✝⁵ : NormedSpace 𝕜₂ F\ninst✝⁴ : NormedSpace 𝕜 Fₗ\ninst✝³ : NormedSpace 𝕜₃ G\ninst✝² : NormedSpace 𝕜 Gₗ\nσ₁₂ : 𝕜 →+* 𝕜₂\nσ₂₃ : 𝕜₂ →+* 𝕜₃\nσ₁₃ : 𝕜 →+* 𝕜₃\ninst✝¹ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝ : RingHomIsometric σ₁₂\nf : E →SL[σ₁₂] F\nr : ℝ≥0\nhr : ↑r < ‖f‖\n⊢ ∃ x, ↑r * ‖x‖ < ‖↑f x‖",
"tactic": "exact f.exists_mul_lt_apply_of_lt_op_nnnorm hr"
}
] |
[
516,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
isBoundedBilinearMap_inner
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.3418113\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nx y : E\n⊢ inner (r • x, y).fst (r • x, y).snd = r • inner (x, y).fst (x, y).snd",
"tactic": "simp only [← algebraMap_smul 𝕜 r x, algebraMap_eq_ofReal, inner_smul_real_left]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.3418113\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nx y : E\n⊢ inner (x, r • y).fst (x, r • y).snd = r • inner (x, y).fst (x, y).snd",
"tactic": "simp only [← algebraMap_smul 𝕜 r y, algebraMap_eq_ofReal, inner_smul_real_right]"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.3418113\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\ninst✝ : NormedSpace ℝ E\nx y : E\n⊢ ‖inner (x, y).fst (x, y).snd‖ ≤ ‖x‖ * ‖y‖",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.3418113\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\ninst✝ : NormedSpace ℝ E\nx y : E\n⊢ ‖inner (x, y).fst (x, y).snd‖ ≤ 1 * ‖x‖ * ‖y‖",
"tactic": "rw [one_mul]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.3418113\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\ninst✝ : NormedSpace ℝ E\nx y : E\n⊢ ‖inner (x, y).fst (x, y).snd‖ ≤ ‖x‖ * ‖y‖",
"tactic": "exact norm_inner_le_norm x y"
}
] |
[
1870,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1859,
1
] |
Mathlib/Topology/Algebra/ContinuousAffineMap.lean
|
ContinuousLinearMap.coe_toContinuousAffineMap
|
[] |
[
283,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.mem_inf_of_left
|
[] |
[
426,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
425,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.Monic.leadingCoeff
|
[] |
[
93,
5
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
Subfield.comap_comap
|
[] |
[
501,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
499,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.MemUnion.elim
|
[] |
[
751,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
749,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.coe_nat_val
|
[] |
[
295,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
WithBot.coe_iInf
|
[] |
[
160,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
OrderIso.tendsto_atTop_iff
|
[
{
"state_after": "ι : Type ?u.53584\nι' : Type ?u.53587\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nl : Filter γ\nf : γ → α\ne : α ≃o β\n⊢ Tendsto (fun x => ↑e (f x)) l atTop ↔ Tendsto (↑e ∘ f) l atTop",
"state_before": "ι : Type ?u.53584\nι' : Type ?u.53587\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nl : Filter γ\nf : γ → α\ne : α ≃o β\n⊢ Tendsto (fun x => ↑e (f x)) l atTop ↔ Tendsto f l atTop",
"tactic": "rw [← e.comap_atTop, tendsto_comap_iff]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.53584\nι' : Type ?u.53587\nα : Type u_2\nβ : Type u_3\nγ : Type u_1\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nl : Filter γ\nf : γ → α\ne : α ≃o β\n⊢ Tendsto (fun x => ↑e (f x)) l atTop ↔ Tendsto (↑e ∘ f) l atTop",
"tactic": "rfl"
}
] |
[
432,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Analysis/Calculus/FDerivSymmetric.lean
|
Convex.isLittleO_alternate_sum_square
|
[
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have A : (1 : ℝ) / 2 ∈ Ioc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have B : (1 : ℝ) / 2 ∈ Icc (0 : ℝ) 1 := ⟨by norm_num, by norm_num⟩"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have C : ∀ w : E, (2 : ℝ) • w = 2 • w := fun w => by simp only [two_smul]"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have h2v2w : x + (2 : ℝ) • v + (2 : ℝ) • w ∈ interior s := by\n convert s_conv.interior.add_smul_sub_mem h4v h4w B using 1\n simp only [smul_sub, smul_smul, one_div, add_sub_add_left_eq_sub, mul_add, add_smul]\n norm_num\n simp only [show (4 : ℝ) = (2 : ℝ) + (2 : ℝ) by norm_num, _root_.add_smul]\n abel"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have h2vww : x + (2 • v + w) + w ∈ interior s := by\n convert h2v2w using 1\n simp only [two_smul]\n abel"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have h2v : x + (2 : ℝ) • v ∈ interior s := by\n convert s_conv.add_smul_sub_mem_interior xs h4v A using 1\n simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]\n norm_num"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have h2w : x + (2 : ℝ) • w ∈ interior s := by\n convert s_conv.add_smul_sub_mem_interior xs h4w A using 1\n simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]\n norm_num"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have hvw : x + (v + w) ∈ interior s := by\n convert s_conv.add_smul_sub_mem_interior xs h2v2w A using 1\n simp only [smul_smul, one_div, add_sub_cancel', add_right_inj, smul_add, smul_sub]\n norm_num\n abel"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have h2vw : x + (2 • v + w) ∈ interior s := by\n convert s_conv.interior.add_smul_sub_mem h2v h2v2w B using 1\n simp only [smul_add, smul_sub, smul_smul, ← C]\n norm_num\n abel"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have hvww : x + (v + w) + w ∈ interior s := by\n convert s_conv.interior.add_smul_sub_mem h2w h2v2w B using 1\n rw [one_div, add_sub_add_right_eq_sub, add_sub_cancel', inv_smul_smul₀ two_ne_zero, two_smul]\n abel"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have TA1 := s_conv.taylor_approx_two_segment hf xs hx h2vw h2vww"
},
{
"state_after": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nTA2 :\n (fun h =>\n f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "have TA2 := s_conv.taylor_approx_two_segment hf xs hx hvw hvww"
},
{
"state_after": "case h.e'_7\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nTA2 :\n (fun h =>\n f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =\n fun x_1 =>\n f (x + x_1 • (2 • v + w) + x_1 • w) - f (x + x_1 • (2 • v + w)) - x_1 • ↑(f' x) w -\n x_1 ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (x_1 ^ 2 / 2) • ↑(↑f'' w) w -\n (f (x + x_1 • (v + w) + x_1 • w) - f (x + x_1 • (v + w)) - x_1 • ↑(f' x) w - x_1 ^ 2 • ↑(↑f'' (v + w)) w -\n (x_1 ^ 2 / 2) • ↑(↑f'' w) w)",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nTA2 :\n (fun h =>\n f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2",
"tactic": "convert TA1.sub TA2 using 1"
},
{
"state_after": "case h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nTA2 :\n (fun h =>\n f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nh : ℝ\n⊢ f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w =\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w -\n (f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w)",
"state_before": "case h.e'_7\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nTA2 :\n (fun h =>\n f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\n⊢ (fun h =>\n f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w) =\n fun x_1 =>\n f (x + x_1 • (2 • v + w) + x_1 • w) - f (x + x_1 • (2 • v + w)) - x_1 • ↑(f' x) w -\n x_1 ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (x_1 ^ 2 / 2) • ↑(↑f'' w) w -\n (f (x + x_1 • (v + w) + x_1 • w) - f (x + x_1 • (v + w)) - x_1 • ↑(f' x) w - x_1 ^ 2 • ↑(↑f'' (v + w)) w -\n (x_1 ^ 2 / 2) • ↑(↑f'' w) w)",
"tactic": "ext h"
},
{
"state_after": "case h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nTA2 :\n (fun h =>\n f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nh : ℝ\n⊢ f (x + h • v + h • v + h • w + h • w) + f (x + h • v + h • w) - f (x + h • v + h • v + h • w) -\n f (x + h • v + h • w + h • w) -\n h ^ 2 • ↑(↑f'' v) w =\n f (x + h • v + h • v + h • w + h • w) - f (x + h • v + h • v + h • w) - h • ↑(f' x) w -\n (h ^ 2 • ↑(↑f'' v) w + h ^ 2 • ↑(↑f'' v) w + h ^ 2 • ↑(↑f'' w) w) -\n (h ^ 2 / 2) • ↑(↑f'' w) w -\n (f (x + h • v + h • w + h • w) - f (x + h • v + h • w) - h • ↑(f' x) w -\n (h ^ 2 • ↑(↑f'' v) w + h ^ 2 • ↑(↑f'' w) w) -\n (h ^ 2 / 2) • ↑(↑f'' w) w)",
"state_before": "case h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nTA2 :\n (fun h =>\n f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nh : ℝ\n⊢ f (x + h • (2 • v + 2 • w)) + f (x + h • (v + w)) - f (x + h • (2 • v + w)) - f (x + h • (v + 2 • w)) -\n h ^ 2 • ↑(↑f'' v) w =\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w -\n (f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w)",
"tactic": "simp only [two_smul, smul_add, ← add_assoc, ContinuousLinearMap.map_add,\n ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul',\n ContinuousLinearMap.map_smul]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7.h\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\nhvww : x + (v + w) + w ∈ interior s\nTA1 :\n (fun h =>\n f (x + h • (2 • v + w) + h • w) - f (x + h • (2 • v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (2 • v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nTA2 :\n (fun h =>\n f (x + h • (v + w) + h • w) - f (x + h • (v + w)) - h • ↑(f' x) w - h ^ 2 • ↑(↑f'' (v + w)) w -\n (h ^ 2 / 2) • ↑(↑f'' w) w) =o[𝓝[Ioi 0] 0]\n fun h => h ^ 2\nh : ℝ\n⊢ f (x + h • v + h • v + h • w + h • w) + f (x + h • v + h • w) - f (x + h • v + h • v + h • w) -\n f (x + h • v + h • w + h • w) -\n h ^ 2 • ↑(↑f'' v) w =\n f (x + h • v + h • v + h • w + h • w) - f (x + h • v + h • v + h • w) - h • ↑(f' x) w -\n (h ^ 2 • ↑(↑f'' v) w + h ^ 2 • ↑(↑f'' v) w + h ^ 2 • ↑(↑f'' w) w) -\n (h ^ 2 / 2) • ↑(↑f'' w) w -\n (f (x + h • v + h • w + h • w) - f (x + h • v + h • w) - h • ↑(f' x) w -\n (h ^ 2 • ↑(↑f'' v) w + h ^ 2 • ↑(↑f'' w) w) -\n (h ^ 2 / 2) • ↑(↑f'' w) w)",
"tactic": "abel"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\n⊢ 0 < 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\n⊢ 1 / 2 ≤ 1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\n⊢ 0 ≤ 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\n⊢ 1 / 2 ≤ 1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w✝ : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w✝ ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nw : E\n⊢ 2 • w = 2 • w",
"tactic": "simp only [two_smul]"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ x + 2 • v + 2 • w = x + 4 • v + (1 / 2) • (x + 4 • w - (x + 4 • v))",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ x + 2 • v + 2 • w ∈ interior s",
"tactic": "convert s_conv.interior.add_smul_sub_mem h4v h4w B using 1"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ x + 2 • v + 2 • w = x + 4 • v + ((2⁻¹ * 4) • w - (2⁻¹ * 4) • v)",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ x + 2 • v + 2 • w = x + 4 • v + (1 / 2) • (x + 4 • w - (x + 4 • v))",
"tactic": "simp only [smul_sub, smul_smul, one_div, add_sub_add_left_eq_sub, mul_add, add_smul]"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ x + 2 • v + 2 • w = x + 4 • v + (2 • w - 2 • v)",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ x + 2 • v + 2 • w = x + 4 • v + ((2⁻¹ * 4) • w - (2⁻¹ * 4) • v)",
"tactic": "norm_num"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ x + 2 • v + 2 • w = x + (2 • v + 2 • v) + (2 • w - 2 • v)",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ x + 2 • v + 2 • w = x + 4 • v + (2 • w - 2 • v)",
"tactic": "simp only [show (4 : ℝ) = (2 : ℝ) + (2 : ℝ) by norm_num, _root_.add_smul]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ x + 2 • v + 2 • w = x + (2 • v + 2 • v) + (2 • w - 2 • v)",
"tactic": "abel"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\n⊢ 4 = 2 + 2",
"tactic": "norm_num"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ x + (2 • v + w) + w = x + 2 • v + 2 • w",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ x + (2 • v + w) + w ∈ interior s",
"tactic": "convert h2v2w using 1"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ x + (v + v + w) + w = x + (v + v) + (w + w)",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ x + (2 • v + w) + w = x + 2 • v + 2 • w",
"tactic": "simp only [two_smul]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\n⊢ x + (v + v + w) + w = x + (v + v) + (w + w)",
"tactic": "abel"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ x + 2 • v = x + (1 / 2) • (x + 4 • v - x)",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ x + 2 • v ∈ interior s",
"tactic": "convert s_conv.add_smul_sub_mem_interior xs h4v A using 1"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ 2 • v = (2⁻¹ * 4) • v",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ x + 2 • v = x + (1 / 2) • (x + 4 • v - x)",
"tactic": "simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\n⊢ 2 • v = (2⁻¹ * 4) • v",
"tactic": "norm_num"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ x + 2 • w = x + (1 / 2) • (x + 4 • w - x)",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ x + 2 • w ∈ interior s",
"tactic": "convert s_conv.add_smul_sub_mem_interior xs h4w A using 1"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ 2 • w = (2⁻¹ * 4) • w",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ x + 2 • w = x + (1 / 2) • (x + 4 • w - x)",
"tactic": "simp only [smul_smul, one_div, add_sub_cancel', add_right_inj]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\n⊢ 2 • w = (2⁻¹ * 4) • w",
"tactic": "norm_num"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ x + (v + w) = x + (1 / 2) • (x + 2 • v + 2 • w - x)",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ x + (v + w) ∈ interior s",
"tactic": "convert s_conv.add_smul_sub_mem_interior xs h2v2w A using 1"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ v + w = 2⁻¹ • x + (2⁻¹ * 2) • v + (2⁻¹ * 2) • w - 2⁻¹ • x",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ x + (v + w) = x + (1 / 2) • (x + 2 • v + 2 • w - x)",
"tactic": "simp only [smul_smul, one_div, add_sub_cancel', add_right_inj, smul_add, smul_sub]"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ v + w = (1 / 2) • x + v + w - (1 / 2) • x",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ v + w = 2⁻¹ • x + (2⁻¹ * 2) • v + (2⁻¹ * 2) • w - 2⁻¹ • x",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\n⊢ v + w = (1 / 2) • x + v + w - (1 / 2) • x",
"tactic": "abel"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ x + (2 • v + w) = x + 2 • v + (1 / 2) • (x + 2 • v + 2 • w - (x + 2 • v))",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ x + (2 • v + w) ∈ interior s",
"tactic": "convert s_conv.interior.add_smul_sub_mem h2v h2v2w B using 1"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ x + (2 • v + w) = x + 2 • v + ((1 / 2) • x + (1 / 2 * 2) • v + (1 / 2 * 2) • w - ((1 / 2) • x + (1 / 2 * 2) • v))",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ x + (2 • v + w) = x + 2 • v + (1 / 2) • (x + 2 • v + 2 • w - (x + 2 • v))",
"tactic": "simp only [smul_add, smul_sub, smul_smul, ← C]"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ x + (2 • v + w) = x + 2 • v + w",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ x + (2 • v + w) = x + 2 • v + ((1 / 2) • x + (1 / 2 * 2) • v + (1 / 2 * 2) • w - ((1 / 2) • x + (1 / 2 * 2) • v))",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\n⊢ x + (2 • v + w) = x + 2 • v + w",
"tactic": "abel"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ x + (v + w) + w = x + 2 • w + (1 / 2) • (x + 2 • v + 2 • w - (x + 2 • w))",
"state_before": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ x + (v + w) + w ∈ interior s",
"tactic": "convert s_conv.interior.add_smul_sub_mem h2w h2v2w B using 1"
},
{
"state_after": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ x + (v + w) + w = x + (w + w) + v",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ x + (v + w) + w = x + 2 • w + (1 / 2) • (x + 2 • v + 2 • w - (x + 2 • w))",
"tactic": "rw [one_div, add_sub_add_right_eq_sub, add_sub_cancel', inv_smul_smul₀ two_ne_zero, two_smul]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nE : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set E\ns_conv : Convex ℝ s\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ (x : E), x ∈ interior s → HasFDerivAt f (f' x) x\nx : E\nxs : x ∈ s\nhx : HasFDerivWithinAt f' f'' (interior s) x\nv w : E\nh4v : x + 4 • v ∈ interior s\nh4w : x + 4 • w ∈ interior s\nA : 1 / 2 ∈ Ioc 0 1\nB : 1 / 2 ∈ Icc 0 1\nC : ∀ (w : E), 2 • w = 2 • w\nh2v2w : x + 2 • v + 2 • w ∈ interior s\nh2vww : x + (2 • v + w) + w ∈ interior s\nh2v : x + 2 • v ∈ interior s\nh2w : x + 2 • w ∈ interior s\nhvw : x + (v + w) ∈ interior s\nh2vw : x + (2 • v + w) ∈ interior s\n⊢ x + (v + w) + w = x + (w + w) + v",
"tactic": "abel"
}
] |
[
229,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/ModelTheory/Semantics.lean
|
FirstOrder.Language.Formula.realize_not
|
[] |
[
617,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.finite_of_finite_preimage
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set β\nh : Set.Finite (f ⁻¹' s)\nhs : s ⊆ range f\n⊢ Set.Finite (f '' (f ⁻¹' s))",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set β\nh : Set.Finite (f ⁻¹' s)\nhs : s ⊆ range f\n⊢ Set.Finite s",
"tactic": "rw [← image_preimage_eq_of_subset hs]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nf : α → β\ns : Set β\nh : Set.Finite (f ⁻¹' s)\nhs : s ⊆ range f\n⊢ Set.Finite (f '' (f ⁻¹' s))",
"tactic": "exact Finite.image f h"
}
] |
[
882,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
879,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
MulAction.zpow_smul_mod_minimalPeriod
|
[] |
[
632,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
628,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_compl_mul_prod
|
[] |
[
491,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.IsNormal.bsup_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.366130\nβ : Type ?u.366133\nγ : Type ?u.366136\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\nH : IsNormal f\no : Ordinal\nh : IsLimit o\n⊢ (Ordinal.bsup o fun x x_1 => f x) = f o",
"tactic": "rw [← IsNormal.bsup.{u, u, v} H (fun x _ => x) h.1, bsup_id_limit h.2]"
}
] |
[
1964,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1962,
1
] |
Mathlib/SetTheory/Cardinal/Finite.lean
|
Nat.card_zmod
|
[
{
"state_after": "case zero\nα : Type ?u.13639\nβ : Type ?u.13642\n⊢ Nat.card (ZMod zero) = zero\n\ncase succ\nα : Type ?u.13639\nβ : Type ?u.13642\nn✝ : ℕ\n⊢ Nat.card (ZMod (succ n✝)) = succ n✝",
"state_before": "α : Type ?u.13639\nβ : Type ?u.13642\nn : ℕ\n⊢ Nat.card (ZMod n) = n",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type ?u.13639\nβ : Type ?u.13642\n⊢ Nat.card (ZMod zero) = zero",
"tactic": "exact @Nat.card_eq_zero_of_infinite _ Int.infinite"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type ?u.13639\nβ : Type ?u.13642\nn✝ : ℕ\n⊢ Nat.card (ZMod (succ n✝)) = succ n✝",
"tactic": "rw [Nat.card_eq_fintype_card, ZMod.card]"
}
] |
[
129,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PullbackCone.snd_colimit_cocone
|
[] |
[
1156,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1155,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
tendstoLocallyUniformlyOn_sUnion
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nS : Set (Set α)\nhS : ∀ (s : Set α), s ∈ S → IsOpen s\nh : ∀ (s : Set α), s ∈ S → TendstoLocallyUniformlyOn F f p s\n⊢ TendstoLocallyUniformlyOn F f p (⋃ (i : Set α) (_ : i ∈ S), i)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nS : Set (Set α)\nhS : ∀ (s : Set α), s ∈ S → IsOpen s\nh : ∀ (s : Set α), s ∈ S → TendstoLocallyUniformlyOn F f p s\n⊢ TendstoLocallyUniformlyOn F f p (⋃₀ S)",
"tactic": "rw [sUnion_eq_biUnion]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ninst✝ : TopologicalSpace α\nS : Set (Set α)\nhS : ∀ (s : Set α), s ∈ S → IsOpen s\nh : ∀ (s : Set α), s ∈ S → TendstoLocallyUniformlyOn F f p s\n⊢ TendstoLocallyUniformlyOn F f p (⋃ (i : Set α) (_ : i ∈ S), i)",
"tactic": "exact tendstoLocallyUniformlyOn_biUnion hS h"
}
] |
[
672,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
1
] |
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_mem_of_mem
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl : List α\nx✝ x : α\nh : x ∈ []\n⊢ ↑(formPerm []) x ∈ []\n\ncase cons\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ x y : α\nl : List α\nh : x ∈ y :: l\n⊢ ↑(formPerm (y :: l)) x ∈ y :: l",
"state_before": "α : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ x : α\nl : List α\nh : x ∈ l\n⊢ ↑(formPerm l) x ∈ l",
"tactic": "cases' l with y l"
},
{
"state_after": "case cons.nil\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝ : List α\nx✝¹ x✝ y✝ : α\nl : List α\nh✝ : x✝ ∈ y✝ :: l\nx y : α\nh : x ∈ [y]\n⊢ ↑(formPerm [y]) x ∈ [y]\n\ncase cons.cons\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\n⊢ ↑(formPerm (y :: z :: l)) x ∈ y :: z :: l",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝ : List α\nx✝ x y : α\nl : List α\nh : x ∈ y :: l\n⊢ ↑(formPerm (y :: l)) x ∈ y :: l",
"tactic": "induction' l with z l IH generalizing x y"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl : List α\nx✝ x : α\nh : x ∈ []\n⊢ ↑(formPerm []) x ∈ []",
"tactic": "simp at h"
},
{
"state_after": "no goals",
"state_before": "case cons.nil\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝ : List α\nx✝¹ x✝ y✝ : α\nl : List α\nh✝ : x✝ ∈ y✝ :: l\nx y : α\nh : x ∈ [y]\n⊢ ↑(formPerm [y]) x ∈ [y]",
"tactic": "simpa using h"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\n⊢ ↑(formPerm (y :: z :: l)) x ∈ y :: z :: l\n\ncase neg\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : ¬x ∈ z :: l\n⊢ ↑(formPerm (y :: z :: l)) x ∈ y :: z :: l",
"state_before": "case cons.cons\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\n⊢ ↑(formPerm (y :: z :: l)) x ∈ y :: z :: l",
"tactic": "by_cases hx : x ∈ z :: l"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\n⊢ (if ↑(formPerm (z :: l)) x = y then z else if ↑(formPerm (z :: l)) x = z then y else ↑(formPerm (z :: l)) x) ∈\n y :: z :: l",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\n⊢ ↑(formPerm (y :: z :: l)) x ∈ y :: z :: l",
"tactic": "rw [formPerm_cons_cons, mul_apply, swap_apply_def]"
},
{
"state_after": "case pos.inl\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝¹ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝ : ↑(formPerm (z :: l)) x = y\n⊢ z ∈ y :: z :: l\n\ncase pos.inr.inl\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ↑(formPerm (z :: l)) x = z\n⊢ y ∈ y :: z :: l\n\ncase pos.inr.inr\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ¬↑(formPerm (z :: l)) x = z\n⊢ ↑(formPerm (z :: l)) x ∈ y :: z :: l",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\n⊢ (if ↑(formPerm (z :: l)) x = y then z else if ↑(formPerm (z :: l)) x = z then y else ↑(formPerm (z :: l)) x) ∈\n y :: z :: l",
"tactic": "split_ifs"
},
{
"state_after": "case pos.inr.inl\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ↑(formPerm (z :: l)) x = z\n⊢ y ∈ y :: z :: l\n\ncase pos.inr.inr\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ¬↑(formPerm (z :: l)) x = z\n⊢ ↑(formPerm (z :: l)) x ∈ y :: z :: l",
"state_before": "case pos.inl\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝¹ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝ : ↑(formPerm (z :: l)) x = y\n⊢ z ∈ y :: z :: l\n\ncase pos.inr.inl\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ↑(formPerm (z :: l)) x = z\n⊢ y ∈ y :: z :: l\n\ncase pos.inr.inr\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ¬↑(formPerm (z :: l)) x = z\n⊢ ↑(formPerm (z :: l)) x ∈ y :: z :: l",
"tactic": ". simp [IH _ _ hx]"
},
{
"state_after": "case pos.inr.inr\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ¬↑(formPerm (z :: l)) x = z\n⊢ ↑(formPerm (z :: l)) x ∈ y :: z :: l",
"state_before": "case pos.inr.inl\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ↑(formPerm (z :: l)) x = z\n⊢ y ∈ y :: z :: l\n\ncase pos.inr.inr\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ¬↑(formPerm (z :: l)) x = z\n⊢ ↑(formPerm (z :: l)) x ∈ y :: z :: l",
"tactic": ". simp"
},
{
"state_after": "no goals",
"state_before": "case pos.inr.inr\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ¬↑(formPerm (z :: l)) x = z\n⊢ ↑(formPerm (z :: l)) x ∈ y :: z :: l",
"tactic": ". simpa [*] using IH _ _ hx"
},
{
"state_after": "no goals",
"state_before": "case pos.inl\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝¹ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝ : ↑(formPerm (z :: l)) x = y\n⊢ z ∈ y :: z :: l",
"tactic": "simp [IH _ _ hx]"
},
{
"state_after": "no goals",
"state_before": "case pos.inr.inl\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ↑(formPerm (z :: l)) x = z\n⊢ y ∈ y :: z :: l",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case pos.inr.inr\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝² : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : x ∈ z :: l\nh✝¹ : ¬↑(formPerm (z :: l)) x = y\nh✝ : ¬↑(formPerm (z :: l)) x = z\n⊢ ↑(formPerm (z :: l)) x ∈ y :: z :: l",
"tactic": "simpa [*] using IH _ _ hx"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nhx : ¬x ∈ z :: l\nh : x = y\n⊢ ↑(formPerm (y :: z :: l)) x ∈ y :: z :: l",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nh : x ∈ y :: z :: l\nhx : ¬x ∈ z :: l\n⊢ ↑(formPerm (y :: z :: l)) x ∈ y :: z :: l",
"tactic": "replace h : x = y := Or.resolve_right (mem_cons.1 h) hx"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.4870\ninst✝ : DecidableEq α\nl✝¹ : List α\nx✝¹ x✝ y✝ : α\nl✝ : List α\nh✝ : x✝ ∈ y✝ :: l✝\nz : α\nl : List α\nIH : ∀ (x y : α), x ∈ y :: l → ↑(formPerm (y :: l)) x ∈ y :: l\nx y : α\nhx : ¬x ∈ z :: l\nh : x = y\n⊢ ↑(formPerm (y :: z :: l)) x ∈ y :: z :: l",
"tactic": "simp [formPerm_apply_of_not_mem _ _ hx, ← h]"
}
] |
[
102,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/MeasureTheory/Measure/Regular.lean
|
MeasurableSet.exists_isClosed_diff_lt
|
[
{
"state_after": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.75620\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\nμ : Measure α\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : WeaklyRegular μ\nA : Set α\nhA : MeasurableSet A\nh'A : ↑↑μ A ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nF : Set α\nhFA : F ⊆ A\nhFc : IsClosed F\nhF : ↑↑μ A < ↑↑μ F + ε\n⊢ ∃ F, F ⊆ A ∧ IsClosed F ∧ ↑↑μ (A \\ F) < ε",
"state_before": "α : Type u_1\nβ : Type ?u.75620\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\nμ : Measure α\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : WeaklyRegular μ\nA : Set α\nhA : MeasurableSet A\nh'A : ↑↑μ A ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ F, F ⊆ A ∧ IsClosed F ∧ ↑↑μ (A \\ F) < ε",
"tactic": "rcases hA.exists_isClosed_lt_add h'A hε with ⟨F, hFA, hFc, hF⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\nβ : Type ?u.75620\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\nμ : Measure α\ninst✝¹ : OpensMeasurableSpace α\ninst✝ : WeaklyRegular μ\nA : Set α\nhA : MeasurableSet A\nh'A : ↑↑μ A ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nF : Set α\nhFA : F ⊆ A\nhFc : IsClosed F\nhF : ↑↑μ A < ↑↑μ F + ε\n⊢ ∃ F, F ⊆ A ∧ IsClosed F ∧ ↑↑μ (A \\ F) < ε",
"tactic": "exact\n ⟨F, hFA, hFc,\n measure_diff_lt_of_lt_add hFc.measurableSet hFA (ne_top_of_le_ne_top h'A <| measure_mono hFA)\n hF⟩"
}
] |
[
596,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
589,
1
] |
Mathlib/Data/List/Basic.lean
|
List.append_left_injective
|
[] |
[
413,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Icc_subset_Iic_self
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.52044\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\ninst✝ : LocallyFiniteOrderBot α\n⊢ Icc a b ⊆ Iic b",
"tactic": "simpa [← coe_subset] using Set.Icc_subset_Iic_self"
}
] |
[
427,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.mul_subset_mul
|
[] |
[
257,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.bijective_iff_bijOn_univ
|
[] |
[
1007,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1000,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
modelWithCornersSelf_coe
|
[] |
[
383,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.zpow_eq_one
|
[
{
"state_after": "M : Type ?u.2599937\nN : Type ?u.2599940\nG : Type u_1\nR : Type ?u.2599946\nS : Type ?u.2599949\nF : Type ?u.2599952\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\n⊢ ζ ^ k = 1",
"state_before": "M : Type ?u.2599937\nN : Type ?u.2599940\nG : Type u_1\nR : Type ?u.2599946\nS : Type ?u.2599949\nF : Type ?u.2599952\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\n⊢ ζ ^ ↑k = 1",
"tactic": "rw [zpow_ofNat]"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.2599937\nN : Type ?u.2599940\nG : Type u_1\nR : Type ?u.2599946\nS : Type ?u.2599949\nF : Type ?u.2599952\ninst✝² : CommMonoid M\ninst✝¹ : CommMonoid N\ninst✝ : DivisionCommMonoid G\nk l : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\n⊢ ζ ^ k = 1",
"tactic": "exact h.pow_eq_one"
}
] |
[
558,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/MeasureTheory/Covering/Besicovitch.lean
|
Besicovitch.ae_tendsto_measure_inter_div
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.1407422\ninst✝¹⁰ : MetricSpace α\nβ : Type u\ninst✝⁹ : SecondCountableTopology α\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : OpensMeasurableSpace α\ninst✝⁶ : HasBesicovitchCovering α\ninst✝⁵ : MetricSpace β\ninst✝⁴ : MeasurableSpace β\ninst✝³ : BorelSpace β\ninst✝² : SecondCountableTopology β\ninst✝¹ : HasBesicovitchCovering β\nμ : MeasureTheory.Measure β\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set β\n⊢ ∀ᵐ (x : β) ∂Measure.restrict μ s,\n Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 1)",
"tactic": "filter_upwards [VitaliFamily.ae_tendsto_measure_inter_div (Besicovitch.vitaliFamily μ) s] with x\n hx using hx.comp (tendsto_filterAt μ x)"
}
] |
[
1194,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1190,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.image_seq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.274141\nι' : Sort ?u.274144\nι₂ : Sort ?u.274147\nκ : ι → Sort ?u.274152\nκ₁ : ι → Sort ?u.274157\nκ₂ : ι → Sort ?u.274162\nκ' : ι' → Sort ?u.274167\nf : β → γ\ns : Set (α → β)\nt : Set α\n⊢ f '' seq s t = seq ((fun x x_1 => x ∘ x_1) f '' s) t",
"tactic": "rw [← singleton_seq, ← singleton_seq, seq_seq, image_singleton]"
}
] |
[
1989,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1987,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.infinite_of_finite_compl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝ : Infinite α\ns : Set α\nhs : Set.Finite (sᶜ)\nh : Set.Finite s\n⊢ Set.Finite univ",
"tactic": "simpa using hs.union h"
}
] |
[
1300,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1299,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_at_one
|
[
{
"state_after": "case h.e'_2.h.e'_6\nR : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\n⊢ 1 = ↑f 1",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\n⊢ eval₂ f 1 p = ↑f (eval 1 p)",
"tactic": "convert eval₂_at_apply (p := p) f 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_6\nR : Type u\nS✝ : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q r : R[X]\nx : R\nS : Type u_1\ninst✝ : Semiring S\nf : R →+* S\n⊢ 1 = ↑f 1",
"tactic": "simp"
}
] |
[
341,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Topology/Algebra/Module/FiniteDimension.lean
|
LinearMap.coe_toContinuousLinearMap_symm
|
[] |
[
312,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.infs_sups_subset_left
|
[] |
[
410,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.iUnion_restrict
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\n⊢ Prepartition.iUnion (restrict π J) = ↑J ∩ Prepartition.iUnion π",
"tactic": "simp [restrict, ← inter_iUnion, ← iUnion_def]"
}
] |
[
527,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/Analysis/Calculus/Deriv/Comp.lean
|
HasDerivAt.comp_hasFDerivAt
|
[] |
[
145,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
Submodule.finrank_add_inf_finrank_orthogonal'
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.998953\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK K₁ K₂ : Submodule 𝕜 E\ninst✝ : FiniteDimensional 𝕜 { x // x ∈ K₂ }\nh : K₁ ≤ K₂\nn : ℕ\nh_dim : finrank 𝕜 { x // x ∈ K₁ } + n = finrank 𝕜 { x // x ∈ K₂ }\n⊢ finrank 𝕜 { x // x ∈ K₁ } + finrank 𝕜 { x // x ∈ K₁ᗮ ⊓ K₂ } = finrank 𝕜 { x // x ∈ K₁ } + n",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.998953\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK K₁ K₂ : Submodule 𝕜 E\ninst✝ : FiniteDimensional 𝕜 { x // x ∈ K₂ }\nh : K₁ ≤ K₂\nn : ℕ\nh_dim : finrank 𝕜 { x // x ∈ K₁ } + n = finrank 𝕜 { x // x ∈ K₂ }\n⊢ finrank 𝕜 { x // x ∈ K₁ᗮ ⊓ K₂ } = n",
"tactic": "rw [← add_right_inj (finrank 𝕜 K₁)]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.998953\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK K₁ K₂ : Submodule 𝕜 E\ninst✝ : FiniteDimensional 𝕜 { x // x ∈ K₂ }\nh : K₁ ≤ K₂\nn : ℕ\nh_dim : finrank 𝕜 { x // x ∈ K₁ } + n = finrank 𝕜 { x // x ∈ K₂ }\n⊢ finrank 𝕜 { x // x ∈ K₁ } + finrank 𝕜 { x // x ∈ K₁ᗮ ⊓ K₂ } = finrank 𝕜 { x // x ∈ K₁ } + n",
"tactic": "simp [Submodule.finrank_add_inf_finrank_orthogonal h, h_dim]"
}
] |
[
1103,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1099,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.