[{"type":"thinking","thinking":"Important findings — the signatures differ from my assumptions:\n\n1. `intervalIntegrable_const [IsLocallyFiniteMeasure μ] {c : E} : IntervalIntegrable (fun _ => c) μ a b` ✓ (no args needed).\n\n2. `Continuous.intervalIntegrable (hu : Continuous u) (a b : ℝ) : IntervalIntegrable u μ a b` ✓.\n\n3. `norm_integral_le_integral_norm (h : a ≤ b) : ‖∫ x in a..b, f x ∂μ‖ ≤ ∫ x in a..b, ‖f x‖ ∂μ` ✓ (namespace: these are inside `namespace intervalIntegral` presumably — the grep stripped the namespace. The file likely has `namespace intervalIntegral`. So full names: `intervalIntegral.norm_integral_le_integral_norm` etc.)\n\n4. `integral_const [CompleteSpace E] (c : E) : ∫ _ in a..b, c = (b - a) • c` ✓.\n\n5. `integral_comp_neg : (∫ x in a..b, f (-x)) = ∫ x in -b..-a, f x` — note: it takes NO explicit args (f, a, b implicit or section variables). Section variables — likely `variable {f : ℝ → E} {a b : ℝ}`. So usage: `intervalIntegral.integral_comp_neg` with expected-type unification.\n\n6. `integral_congr {a b : ℝ} (h : EqOn f g [[a, b]]) : ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ` — note `[[a, b]]` is uIcc notation. EqOn on uIcc.\n\n7. `integral_add_adjacent_intervals (hab) (hbc) : (∫ a..b) + (∫ b..c) = ∫ a..c` ✓ (use .symm for my split direction).\n\n8. **`integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) : (∫ u in a..b, f u ∂μ) ≤ ∫ u in a..b, g u ∂μ`** — CRITICAL DIFFERENCE: this version takes ONLY the pointwise hypothesis! The `hab : a ≤ b`, `hf`, `hg` are section variables (the proof body references `hab`, `hf.1`, `hg.1`). So the signature is probably:\n\n```\ntheorem integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) : ...\n```\nwith `variable (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b)` included via `include` or as part of the theorem... Let me look at more context around line 1432 to see the variable declarations. The proof uses hab, hf, hg — these must be section variables that are included. Since they're used in the proof, they're auto-included in the theorem signature. So the full signature likely: `theorem integral_mono_on (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) (h : ∀ x ∈ Icc a b, f x ≤ g x) : ...` — wait but the grep shows `theorem integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) :` as the first explicit binder. So hab/hf/hg come BEFORE as section variables, meaning the argument order is: hab hf hg h? No — section variables appear in the order they're declared, before explicit binders... Actually section variables are inserted in the order of first use? No — `variable` declarations are included in the order declared when used. The signature would be `(hab : a ≤ b) (hf : ...) (hg : ...) (h : ...)` if declared in that order.\n\nLet me read the context above line 1432 to see the variable declarations.\n\nAlso need to double check the namespace — whether these are in `namespace intervalIntegral`.\n\nLet me read the file sections.","signature":""},{"type":"text","text":"Signatures found. Let me check the section variables and namespace context around `integral_mono_on`."},{"type":"tool_use","id":"Bash_15","name":"Bash","input":{"command":"grep -n \"^variable\\|^namespace\\|^section\\|^end\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/MeasureTheory/Integral/IntervalIntegral/Basic.lean\" | sed -n '1,80p'","description":"Show variable/namespace declarations in IntervalIntegral Basic.lean"},"extra_content":null}]