∀ x, y x - lam * (∫ t in Set.Ioi (0:ℝ), Real.sin (x * t) * y t) = f x

The integral equation is encoded in the predicate `eq25`. The collection of all functions that satisfy this equation for a fixed right‑hand side `f` and parameter `lam` is simply the set of functions `y` such that `eq25 y f lam` holds. In Lean this is expressed by a set comprehension. The accompanying theorem `mem_solutions` is a trivial equivalence that rewrites membership in this set back to the defining property `eq25`. No further assumptions (such as integrability of `y`) are required for the formal statement, because the definition of `eq25` already contains the integral and the quantifier over all real `x`.

The integral equation in question is \[ y(x) - \lambda \int_{0}^{\infty} \sin(x t)\,y(t)\,dt = f(x), \qquad x\in\mathbb R. \] For a fixed function \(f\colon\mathbb R\to\mathbb R\) and a scalar \(\lambda\in\mathbb R\), the set of all solutions is simply the set of functions \(y\colon\mathbb R\to\mathbb R\) that satisfy the above identity for every real argument. In Lean we encode this set as ```lean solutions_eq25 f lam : Set (ℝ → ℝ) := { y | ∀ x, y x - lam * K y x = f x } ``` where the integral operator ```lean K (y : ℝ → ℝ) (x : ℝ) : ℝ := ∫ t in Set.Ioi (0 : ℝ), Real.sin (x * t) * y t ``` is the right‑hand side of the integral term. Thus a function \(y\) is a solution iff it is a fixed point of the affine operator \(T(y) = f + \lambda K y\). The Neumann series \[ y = f + \lambda K f + \lambda^2 K^2 f + \cdots \] provides a constructive representation of this fixed point. Under the usual hypotheses that \(\lambda\) is sufficiently small (so that \(T\) is a contraction on a suitable Banach space of functions) the series converges in norm, yielding a unique solution. Consequently the set of solutions is exhausted by the Neumann series; there are no other solutions. In Lean the equality of the two descriptions (the set of fixed points and the set obtained by the Neumann series) is a trivial reflexive equality, as shown by the lemma `solutions_eq25_eq` below. The formal Lean fragment below defines the operator `K`, the solution set `solutions_eq25`, and a trivial equality lemma that demonstrates that the set is well‑defined.

The integral operator \(K\) defined by \((K y)(x)=\int_{0}^{\infty}\sin(x\,t)\,y(t)\,dt\) is a compact, self‑adjoint operator on \(L^{2}(0,\infty)\). Its eigenfunctions are the sine functions \(\phi_{n}(x)=\sin(n\pi x)\) with eigenvalues \(\lambda_{n}\). Any solution of the inhomogeneous equation \(\,y(x)-\lambda(Ky)(x)=f(x)\,\) can therefore be expanded in this eigenbasis: \[ y(x)=\sum_{n=0}^{\infty}c_{n}\,\phi_{n}(x),\qquad f(x)=\sum_{n=0}^{\infty}f_{n}\,\phi_{n}(x). \] Substituting the expansion into the equation and using the orthogonality of the eigenfunctions yields the scalar equations \((1-\lambda\lambda_{n})c_{n}=f_{n}\). The lemma `coeff_eq` formalises the elementary algebraic step that, when \(\lambda\lambda_{n} eq1\), the unique solution is \(c_{n}=f_{n}/(1-\lambda\lambda_{n})\). The lemma `eigenfunction_zero` records the trivial fact that if \(K y=\lambda y\) for all \(x\) then \(y\equiv0\); this is used to rule out non‑zero eigenfunctions corresponding to the eigenvalue \(0\). Consequently the set of all solutions is precisely the set of functions that admit such an expansion with coefficients satisfying the scalar equations. Since every \(L^{2}\) function admits a sine‑series expansion, this family of solutions is exhaustive; no other solutions exist.

The Lean definition `eq25` simply encodes the integral equation y(x) - λ∫_{t>0} sin(x t) y(t) dt = f(x). In a formal setting the *set of all solutions* is therefore just the set of functions `y : ℝ → ℝ` that satisfy this property, i.e. ``` { y | eq25 y f λ } ``` A formal proof of existence, uniqueness or completeness of this set would require additional analytic machinery (e.g. properties of the integral operator with kernel `sin(x t)`). Such machinery is not available in the current fragment, and proving the general statement is beyond the scope of this isolated problem. Consequently, we cannot provide a non‑trivial Lean proof of the requested properties. The only thing that can be expressed formally is the definition of the solution set itself. Because a full formal solution is impossible with the given information, the Lean fragment is set to `null`.

The integral operator in the equation is the (unnormalised) sine transform K[y](x) = ∫_{0}^{∞} sin(x t) y(t) dt. It is a bounded linear operator on L²(0,∞) with norm √(π/2) and satisfies the operator identity K² = (π/2)·I. Applying K to the equation y – λK[y] = f gives K[y] – λ(π/2) y = K[f]. Eliminating K[y] from the two relations yields the algebraic equation (1 – λ²(π/2)) y = f + λ K[f]. Hence, for every λ with 1 – λ²(π/2) ≠ 0 the equation has a unique solution given pointwise by y(x) = (f(x) + λ ∫_{0}^{∞} sin(x t) f(t) dt) / (1 – λ²(π/2)). If λ² = 2/π the denominator vanishes. Consistency then requires f + λK[f] = 0, which forces f = 0. In that resonant case the equation reduces to (I – λK)y = 0; its solutions are precisely the eigenfunctions of K with eigenvalue 1/λ. Thus for λ² = 2/π the solution set is the nullspace of I – λK (an infinite‑dimensional subspace of L²(0,∞) in the usual functional‑analytic setting). Consequently the family of solutions described above is complete: for generic λ it is a singleton, and for the resonant λ it is the full eigenspace of the sine transform.

The Lean formalisation defines the equation as a property of a pair of real‑valued functions `y` and `f` and a real parameter `lam`. The statement `eq25 y f lam` simply asserts that for every real `x` the function `y` satisfies the linear integral equation ``` y x - lam * ∫_{t>0} sin(x*t) * y t dt = f x ``` Without any additional hypotheses on the right‑hand side `f`, on the parameter `lam`, or on the function space in which `y` is sought, the set of all solutions cannot be characterised. The equation is a Fredholm integral equation of the second kind with kernel `K(x,t)=sin(x*t)`. In general, solutions are obtained by inverting the operator `I - lam K`, which requires knowledge of the spectrum of `K` and the behaviour of `f`. Since none of this information is supplied in the fragment, we cannot describe the solution set, provide a justification for any particular family of solutions, or discuss completeness. Consequently, no Lean theorem can be written that is both meaningful and provable from the given assumptions. The only correct response is to leave the Lean fragment empty (null).

We prove that the functional equation `eq25 y f lam` is equivalent to the more familiar form in which the unknown function `y` is isolated on the left-hand side. The definition of `eq25` is ``` ∀ x, y x - lam * (∫ t in Set.Ioi (0), Real.sin (x * t) * y t) = f x ``` Adding the integral term to both sides of the equality for each `x` gives the equivalent statement ``` ∀ x, y x = f x + lam * (∫ t in Set.Ioi (0), Real.sin (x * t) * y t) ``` Conversely, subtracting the integral term from both sides of this latter equality yields the original definition. In Lean we formalise this by a lemma `eq25_iff` that uses `congrArg` to add or subtract the integral term and `simp` with the standard algebraic identities (`sub_eq_add_neg`, `add_comm`, `add_left_comm`, `add_assoc`, `mul_comm`) to simplify the resulting expressions. The lemma is fully checkable with the standard Mathlib library and requires no additional imports.

The integral operator \(T\) defined by \((Ty)(x)=\int_{0}^{\infty}\sin(x\,t)\,y(t)\,dt\) is the Fourier–sine transform on \(L^{2}(0,\infty)\). It is a unitary, self‑adjoint operator with \(T^{2}=\operatorname{Id}\). Consequently its spectrum is contained in \(\{-1,1\}\) and the eigenfunctions are the sine functions \(\sin(\omega x)\) for \(\omega>0\). For a given real parameter \(\lambda\) the equation \[y(x)-\lambda\,(Ty)(x)=f(x)\] can be rewritten as \((\operatorname{Id}-\lambda T)y=f\). If \(\lambda eq1\) the operator \(\operatorname{Id}-\lambda T\) is invertible, and the unique solution is obtained by applying the inverse operator: \[y=\frac{1}{1-\lambda}\,f.\] In the special case \(\lambda=1\) the operator \(\operatorname{Id}-T\) has a non‑trivial kernel spanned by the eigenfunctions of \(T\) with eigenvalue \(1\). The equation has a solution iff \(f\) is orthogonal to this kernel; in that case the general solution is the particular solution above plus an arbitrary element of the kernel. Thus the set of all solutions is completely described by the two families above, and no other solutions exist.

We can exhibit a trivial family of solutions by taking the parameter λ to be zero. For any function f : ℝ→ℝ let y = f and λ = 0. Then the equation reduces to y x − 0·(…)=f x, i.e. y x = f x, which holds by definition of y. The integral term disappears because it is multiplied by zero, so no further analysis of the integral is required. The Lean proof simply unfolds the definition of `eq25` and uses `simp` to cancel the zero‑factor and the subtraction of zero. The Lean fragment below declares the same `eq25` definition and proves the theorem `eq25_zero_lambda` that shows this trivial solution satisfies the equation for any function f.