{"name":"hahn_banach","declaration":"/-- Let $H_0$ be a subgroup of $G$.  Then every homomorphism $\\phi: H_0 \\to G'$ can be extended to a\nhomomorphism $\\tilde \\phi: G \\to G'$. -/\ntheorem hahn_banach {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [AddCommGroup G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (H₀ : AddSubgroup G) (φ : ↥H₀ →+ G') : ∃ φ', ∀ (x : ↥H₀), φ x = φ' ↑x"}
{"name":"homomorphism_pfr","declaration":"/-- Let $f: G \\to G'$ be a function, and let $S$ denote the set\n$$ S := \\{ f(x+y)-f(x)-f(y): x,y \\in G \\}.$$\nThen there exists a homomorphism $\\phi: G \\to G'$ such that\n$$ |\\{f(x) - \\phi(x)\\}| \\leq |S|^{12}. $$ -/\ntheorem homomorphism_pfr {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [Fintype G] [AddCommGroup G'] [Fintype G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (f : G → G') (S : Set G') (hS : ∀ (x y : G), f (x + y) - f x - f y ∈ S) : ∃ φ T, Nat.card ↑T ≤ Nat.card ↑S ^ 12 ∧ ∀ (x : G), f x - φ x ∈ T"}
{"name":"goursat","declaration":"/-- Let $H$ be a subgroup of $G \\times G'$.  Then there exists a subgroup $H_0$ of $G$, a\nsubgroup $H_1$ of $G'$, and a homomorphism $\\phi: G \\to G'$ such that\n$$ H := \\{ (x, \\phi(x) + y): x \\in H_0, y \\in H_1 \\}.$$\nIn particular, $|H| = |H_0| |H_1|$. -/\ntheorem goursat {G : Type u_1} {G' : Type u_2} [AddCommGroup G] [AddCommGroup G'] [ElementaryAddCommGroup G 2] [ElementaryAddCommGroup G' 2] (H : AddSubgroup (G × G')) : ∃ H₀ H₁ φ, (∀ (x : G × G'), x ∈ H ↔ x.1 ∈ H₀ ∧ x.2 - φ x.1 ∈ H₁) ∧ Nat.card ↥H = Nat.card ↥H₀ * Nat.card ↥H₁"}