[{"type":"thinking","thinking":"More errors:\n\n**Error A (line 390, \"No goals to be solved\")**: in hdm's second rintro branch — after `simp only [Finset.mem_image, Set.Finite.mem_toFinset]`, the goal `∃ ρ, ρ ∈ S ∧ f ρ = x ↔ ∃ ρ, ρ ∈ S ∧ f ρ = x`?? The simp only may have fully closed the goal (both sides identical → simp closes by rfl), leaving my constructor/rintro with no goals. Actually the error is at 390:6 which is the second `rintro` branch or the `constructor`? Line 390 — let me count: hdm block starts ~374. The `constructor` then two rintro branches. \"No goals to be solved\" at 390 — likely simp only already closed the goal (it can prove iff by reflexivity when both sides become syntactically equal), so `constructor` errors \"no goals\". Fix: remove the constructor/rintros, keep only simp only — but will simp only close it? Finset.mem_image: (x ∈ image f t ↔ ∃ a ∈ t, f a = x); Set.Finite.mem_toFinset rewrites hSfin.toFinset membership to S membership. After simp, both sides: LHS ∃ ρ, ρ ∈ S ∧ ‖...‖ = x; RHS same ✓ simp closes iff-rfl? simp only closes goals of form `a ↔ a`? It applies `Iff.rfl`? simp only uses rfl-closure for the remaining goal — yes simp only tries `rfl` at the end for both Eq and Iff? Hmm, simp only closes goals that become `True` or syntactic reflexivity — Iff.rfl is applied by simp's rfl-closer? I believe simp only can close `a ↔ a` via rfl. The error says \"No goals\" at the line with `· rintro` (first bullet) — meaning simp only DID close it. So remove constructor + both bullets.\n\n**Error B (line 403)**: `obtain ⟨ρ, hρ, rfl⟩ := (hdm _).mp (Finset.min'_mem d hdne)` — rfl pattern on ‖↑ρ − ↑ρ₀‖ = d.min' hdne: subst fails because the equation is `f ρ = min'` (value = expression, and the expression isn't a local var). rfl-pattern subst only works when one side is a local variable. Here RHS is `d.min' hdne` — not a variable ⟹ subst fails. Fix: name the equality: `obtain ⟨ρ, hρ, heq⟩ := ...; rw [← heq]; ...` goal: d.min' hdne ≤ 1; heq : ‖↑ρ − ↑ρ₀‖ = d.min' hdne; rw [← heq]: goal ‖↑ρ − ↑ρ₀‖ ≤ 1; then from hρ.1 (closedBall): ‖↑ρ − ↑ρ₀‖ ≤ 1 ✓ via dist_eq_norm. But hρ : ρ ∈ S — S is a set-def (set S := {...}): membership ρ ∈ S unfolds to the pair? S := {ρ | ↑ρ ∈ closedBall ∧ ρ ≠ ρ₀} — membership defeq to the And ✓ so hρ.1 works? With `set`, S is a local definition; hρ : ρ ∈ S — hρ.1 : needs S membership to be an And — defeq but dot-notation .1 requires the TYPE to be an And structure syntactically? Anonymous constructor projection `.1` on hρ : ρ ∈ S — Lean unfolds S (a let-ish local def from set) — `set` introduces S as a local definition with equation hS; ρ ∈ S doesn't unfold automatically for `.1`?? Dot notation hρ.1 : unfolds the type to whnf — S is a local def (fvar with value) — whnf unfolds ✓ should work. But to be safe: `rw [hS] at hρ`? hS : S = {...}: rewriting S in hρ : ρ ∈ S — no dependency issues (hρ not depended on) ✓ gives hρ : ρ ∈ {ρ | ...} defeq And: then hρ.1 ✓. Hmm — actually simpler to avoid set for S too... but S is used in several places. Try hρ.1 first; if it fails, add the unfold.\n\nWait — actually in the previous (working) build, `hρ.2` was used in hdpos: `fun h => hρ.2 (Subtype.ext h)` — hρ : ρ ∈ S (after the mem_toFinset rw — no wait, in the new hdm version hρ comes from the iff directly: ∃ ρ ∈ S, ... — hρ : ρ ∈ S). hρ.2 : ρ ≠ ρ₀ — needs the And unfold. Since the elaborator got past hdpos (error is at 403 = hmin1's obtain-rfl), hρ.2 worked ✓ so .1 will too.\n\n**Error C (line 468)**: seriesFun_eq_sub_add's last calc step: `_ = seriesFunSub ρ₀ s + seriesTerm s ρ₀ := by rw [hsingle]` — after rw [hsingle], goal: ∑' (seriesTermSub ρ₀ s) + seriesTerm s ρ₀ = seriesFunSub ρ₀ s + seriesTerm s ρ₀ — the tsum vs seriesFunSub defeq but not syntactic; `rw [hsingle]` leaves the goal unsolved (\"unsolved goals\"). Fix: close with `show` defeq or: after rw [hsingle], the remaining is defeq-rfl: add `; rfl`? rfl tactic works syntactic-up-to-defeq? `rfl` tactic (Eq.refl) uses defeq ✓. But in calc, the step proof is a term/tactic block: `:= by rw [hsingle]` then goal remains → add rfl... hmm `by rw [hsingle]` ends with unsolved goal error. Use `:= by rw [hsingle]; rfl`? If after rw the goal is `∑' sub + seriesTerm s ρ₀ = seriesFunSub ρ₀ s + seriesTerm s ρ₀` — rfl closes by defeq (seriesFunSub unfolds) ✓. Hmm, but will `rw [hsingle]` even fire? hsingle's LHS `∑' ρ, if ρ = ρ₀ then seriesTerm s ρ else 0` must appear in the goal — the goal at that step: `∑' ρ, seriesTermSub ρ₀ s ρ + ∑' ρ, (if ρ = ρ₀ then seriesTerm s ρ else 0) = seriesFunSub ρ₀ s + seriesTerm s ρ₀` ✓ fires. Then rfl. Alternatively restructure: `:= by rw [hsingle]` → replace with `:= by rw [hsingle]; rfl`.\n\nHmm wait — actually simpler: `calc ... _ = seriesFunSub ρ₀ s + seriesTerm s ρ₀ := by rw [hsingle]` — the first summand ∑' ρ, seriesTermSub ρ₀ s ρ IS seriesFunSub ρ₀ s by definition (rfl) — does `rw` close goals up to rfl? rw tries rfl after rewriting ✓ — rw's closing rfl is `Eq.refl` with defeq check?? rw closes the goal if it becomes `a = a` syntactically after rewriting (it tries `rfl` which allows some defeq — semireducible defeq). seriesFunSub ρ₀ s vs ∑' ρ, seriesTermSub ρ₀ s ρ — unfolding seriesFunSub (a plain def, semireducible) — rw's post-rfl uses reducible defeq only?? The error confirms it didn't close. Fix: `by rw [hsingle]; rfl` — rfl tactic uses full defeq ✓... actually rfl tactic uses `Eq.refl` checked by isDefEq with default transparency ✓ unfolds plain defs fine.\n\n**Error D (line 516)**: in seriesFunSub_analyticAt hfu, the `ρ = ρ₀` branch: `rw [seriesTermSub, if_pos h0, norm_zero]` — wait the error: `mul_nonneg (mul_nonneg ?m hM0) hg0` — hg0 has ρ₀ but goal has ρ — because `rw [if_pos h0]` rewrote the if-condition but the goal still mentions ρ in `u ρ`... The goal: `‖if ρ = ρ₀ then 0 else seriesTerm s ρ‖ ≤ u ρ` — after rw [seriesTermSub, if_pos h0, norm_zero]: goal `0 ≤ u ρ`. My hg0 was stated for ρ₀ and I tried to prove `0 ≤ u ρ`... u ρ is a set-application: `u ρ` with u := fun ρ => 17/6*M*(mult ↑ρ/(1+γ ↑ρ)) — goal `0 ≤ u ρ` — the exact proof term must be of type `0 ≤ u ρ` which is defeq to `0 ≤ 17/6*M*(mult ↑ρ/...)`. My term mul_nonneg (mul_nonneg _ hM0) hg0 with hg0 about ρ₀ — wrong object (should be ρ). Wait I wrote hg0 with (zeroMult ↑ρ₀) — because after if_pos h0 the goal... hmm I intended the goal to become about ρ₀ via subst? h0 : ρ = ρ₀ — if_pos h0 : (if ρ = ρ₀ then a else b) = a — rewrites the if to 0 but does NOT substitute ρ := ρ₀ elsewhere (u ρ stays). So the bound needed is at ρ. Fix: state hg0 for ρ not ρ₀. But wait — `rw [seriesTermSub, ...]`: seriesTermSub is a def — rw [seriesTermSub]?? That fails like before! But the error indicates it got past that to the mul_nonneg application — so rw [seriesTermSub] worked? For a def with pattern variables, `rw [seriesTermSub]` uses the auto-generated equational lemma `seriesTermSub.eq_...`?? Hmm — in Lean4 Mathlib, `rw [foo]` for a def foo DOES work via the equation lemma (rw can unfold definitions since Lean 4.? — `rw [seriesTermSub]` unfolds if seriesTermSub is defined by equation compiler giving `seriesTermSub.eq_def`... Actually Lean 4 `rw [f]` where f is a definition tries `f.eq_def` or unfolds — it worked per the error location. OK.\n\nSo fix: hg0 should be about ρ: \n```lean\n    · rw [seriesTermSub, if_pos h0, norm_zero]\n      have hg0 : (0 : ℝ) ≤ (zeroMult ρ : ℝ) / (1 + Complex.normSq (gammaOf (ρ : ℂ))) := by\n        have hnn := Complex.normSq_nonneg (gammaOf (ρ : ℂ))\n        positivity\n      exact mul_nonneg (mul_nonneg (by norm_num) hM0) hg0\n```\nBut hold on — the goal is `0 ≤ u ρ` (a beta-redex application of the set-defined u). `exact mul_nonneg ...` — exact checks defeq: u ρ ≡ 17/6*M*(mult/(1+γ²)) ✓ accepted.\n\nHmm wait, but with h0 : ρ = ρ₀, does `exact` still work — yes, u ρ is the goal regardless.\n\nActually cleaner: use the hdm-style... no, simple fix suffices.\n\n**Error E (line 550)**: `rw [if_pos h0]` failed: \"did not find pattern if ρ = ρ₀ then _ else _ in target `DifferentiableAt ℂ (fun s => if ρ = ρ₀ then 0 else seriesTerm s ρ) s`\" — the if is INSIDE the lambda; rw [if_pos h0] can't rewrite under a binder! Right — rw doesn't go under binders. The target after `show DifferentiableAt ℂ (fun s => if ρ = ρ₀ then (0:ℂ) else seriesTerm s ρ) s`: I need to show this equals the constant function: the lambda is (fun s => if ...) with h0 : ρ = ρ₀ — the if-condition doesn't involve s, so the function is fun s => 0, i.e. constant. Rewrite: `simp only [if_pos h0]`? simp only [if_pos h0] — hmm if_pos : (c → ...) style: `if_pos h0 : (if ρ = ρ₀ then a else b) = a` — as a simp rule with h0 : ρ = ρ₀ — simp CAN rewrite under binders ✓. So: `simp only [if_pos h0]` — hmm if_pos's statement: `if_pos {c : Prop} [Decidable c] (h : c) : (if c then a else b) = a`? Something like that — as simp lemma with h0 provided it works under binders ✓. Then goal DifferentiableAt ℂ (fun s => 0) s → differentiableAt_const. Or one shot: `exact differentiableAt_const 0` after showing defeq — the lambda fun s => (if ρ = ρ₀ then 0 else ...) with h0 : ρ = ρ₀ is defeq to fun s => 0? The if with a PROOF-dependent condition: if_pos reduction is propositional (needs the Decidable instance to reduce — if the instance is `Classical.propDecidable` it won't reduce definitionally... but with h0 : ρ = ρ₀ the decidable instance `decEq` on the subtype — Subtype decidable equality — `if ρ = ρ₀ then ...` with h0 proof: defeq of `if h then a else b` to `a` requires the Decidable instance to be `isTrue` — for Classical.propDecidable it's NOT defeq. So use simp only [if_pos h0].\n\nWait — but actually how did the goal become `DifferentiableAt ℂ (fun s => if ρ = ρ₀ then 0 else seriesTerm s ρ) s`? I wrote `show DifferentiableAt ℂ (fun s => if ρ = ρ₀ then (0:ℂ) else seriesTerm s ρ) s` — matched (the original goal was DifferentiableAt ℂ (fun s => seriesTermSub ρ₀ s ρ) s — defeq via unfolding seriesTermSub ✓). Then simp only [if_pos h0] rewrites inside ✓, then exact differentiableAt_const _.\n\nSimilarly in the else branch: `show ...; rw [if_neg h0]` — same binder problem! The neg branch: goal DifferentiableAt ℂ (fun s => if ρ = ρ₀ then 0 else seriesTerm s ρ) s: use `simp only [if_neg h0]` then show the seriesTerm form... after simp only [if_neg h0]: DifferentiableAt ℂ (fun s => seriesTerm s ρ) s — then my next `show DifferentiableAt ℂ (fun s => (zeroMult ρ:ℂ)*(...)) s` unfolds seriesTerm ✓ then the exact term.\n\nSame issue exists in hfu's branches? There: `rw [seriesTermSub, if_pos h0, norm_zero]` — the goal was `‖seriesTermSub ρ₀ s ρ‖ ≤ u ρ` — seriesTermSub ρ₀ s ρ is an APPLICATION (no binder) ⟹ rw [seriesTermSub] unfolds it to ‖if ρ = ρ₀ then 0 else seriesTerm s ρ‖ — the if not under a binder ✓ rw [if_pos h0] fine ✓.\n\nAnd in seriesTermSub_norm_le: `by_cases h : ρ = ρ₀ <;> simp [seriesTermSub, h]` — simp fine under no binders ✓ (that one elaborated OK? It appeared before line 468 errors... the reported errors were at 390, 403, 468, 516, 550 — seriesTermSub_norm_le at ~line 430ish passed ✓).\n\nAlso seriesFun_eq_sub_add's tsum_congr branch: `by_cases h : ρ = ρ₀ <;> simp [seriesTermSub, h]` — goal: seriesTerm s ρ = seriesTermSub ρ₀ s ρ + (if ρ = ρ₀ then seriesTerm s ρ else 0) — no binders ✓ fine.\n\nAnd single_summable's `rw [if_neg ...]` — goal there: after apply hρ, goal: (if ρ = ρ₀ then seriesTerm s ρ else 0) = 0 — no binder ✓ fine (it elaborated? it's before 468 — line ~450 — passed presumably).\n\nAlso fix error A: remove constructor+bullets in hdm (simp closes).\n\nLet me apply all fixes.","signature":""},{"type":"text","text":"More fixes: simp closes the hdm iff itself; the `rfl`-subst in hmin1 needs a named equality; the calc step needs a closing `rfl`; hg0 must be about ρ; and `rw [if_pos]` can't go under a binder (use `simp only`)."},{"type":"tool_use","id":"Edit_54","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSeries.lean","old_string":"      intro x\n      rw [hd]\n      simp only [Finset.mem_image, Set.Finite.mem_toFinset]\n      constructor\n      · rintro ⟨ρ, hρ, rfl⟩; exact ⟨ρ, hρ, rfl⟩\n      · rintro ⟨ρ, hρ, rfl⟩; exact ⟨ρ, hρ, rfl⟩","new_string":"      intro x\n      rw [hd]\n      simp only [Finset.mem_image, Set.Finite.mem_toFinset]"},"extra_content":null}]