[{"type":"thinking","thinking":"The probes were run last session but I don't have their outputs. Rather than re-derive, the journal records the signatures. Let me get exact signatures via a quick probe run — but building Scratch.lean takes a while (Mathlib import ~ minutes, though oleans are cached; `lake env lean Scratch.lean` should be quick since Mathlib is built). Let me run it in the package directory to get exact signatures. Actually the journal already records them:\n\n- `iteratedDerivWithin_tsum` (probed: (m) (IsOpen s) (x∈s) (∀ t∈s, Summable) (∀ k, 1≤k→k≤m → SummableLocallyUniformlyOn ...) (∀ n k r, k≤m→r∈s → DifferentiableAt (iteratedDerivWithin k (f n) s) r))\n- `SummableLocallyUniformlyOn_of_locally_bounded` (probed: (IsOpen s) → (∀ K ⊆ s, IsCompact K → ∃ u, Summable u ∧ ∀ n k ∈ K, ‖f n k‖ ≤ u n))\n- `iteratedDeriv_neg (n) (f) (a)` explicit args\n- `AnalyticAt.contDiffAt` exists\n- `Equiv.tsum_eq e f : ∑' a, f (e a) = ∑' b, f b`\n- `iteratedDeriv_const` — signature probably `iteratedDeriv_const (c : F) (n : ℕ) : iteratedDeriv n (fun _ => c) = if n = 0 then fun _ => c else 0` — used at LiDigamma:703 as `rw [iteratedDeriv_const, if_neg (by omega : k ≠ 0)]`.\n\nHmm, in LiDigamma:703 the usage is `rw [iteratedDeriv_const, if_neg ...]` — after `iteratedDeriv_add hcf hcg`, goal contains `iteratedDeriv k (fun s => 1/((m:ℂ)+1)) s`. So `iteratedDeriv_const` rewrites to an `if`.\n\nWait actually let me check: maybe it's `iteratedDeriv_const (c) (n)`? The rw usage suggests it's an equation `iteratedDeriv n (fun _ => c) x = if n = 0 then c else 0` or similar producing an `ite`. Let me just run the probe to get exact signatures. Since Mathlib is fully built, `lake env lean Scratch.lean` should take a minute or two. Worth it for accuracy.\n\nAlso need `iteratedDeriv_add` signature and `iteratedDerivWithin_of_isOpen`. And `eventually_nhdsWithin_iff`.\n\nLet me run the probe in background/foreground. Package dir: D:/Python/Quantum/research/lean/NguyenRH1. Command: `export PATH=\"$HOME/.elan/bin:$PATH\" && cd ... && lake env lean Scratch.lean`. On Windows with git-bash presumably. Shell is \"unknown (use Unix shell syntax)\". OK.\n\nActually, let me add the additional checks I need to Scratch.lean first, then run once:\n- @iteratedDeriv_add\n- @iteratedDeriv_const\n- @iteratedDerivWithin_of_isOpen\n- @iteratedDeriv_const_mul\n- @eventually_nhdsWithin_iff (already there)\n- @iteratedDerivWithin_tsum (already)\n- @hasDerivAt_zpow\n- @iteratedDerivWithin_zero\n- @differentiableAt_const? not needed.\n\nLet me update Scratch.lean and run the probe.","signature":""},{"type":"text","text":"I have the pattern. Let me re-run the probe to confirm exact signatures before writing §5/§6."},{"type":"tool_use","id":"Edit_11","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean","old_string":"#check @iteratedDeriv_const\n#check @iteratedDerivWithin_neg\n#check @inv_pow","new_string":"#check @iteratedDeriv_const\n#check @iteratedDerivWithin_neg\n#check @inv_pow\n#check @iteratedDeriv_add\n#check @iteratedDerivWithin_of_isOpen\n#check @iteratedDeriv_const_mul\n#check @hasDerivAt_zpow\n#check @iteratedDerivWithin_zero\n#check @iteratedDeriv_zero\n#check @AnalyticAt.differentiableAt\n#check @differentiableAt_const\n#check @ContDiffAt\n#check @Nat.cast_factorial\n#check @Nat.factorial_ne_zero\n#check @pow_add\n#check @inv_eq_of_mul_eq_one_right\n#check @zpow_neg_one\n#check @zero_sub\n#check @one_div\n#check @inv_pow\n#check @Complex.normSq\n#check @Filter.EventuallyEq.filter_mono\n#check @nhdsWithin_le_nhds\n#check @eventuallyEq_nhdsWithin_of_eventuallyEq_nhds\n#check @Filter.EventuallyEq.mul\n#check @Metric.closedBall\n#check @isCompact_closedBall\n#check @Set.Summable\n#check @iteratedDerivWithin_congr_set"},"extra_content":null}]