[{"type":"thinking","thinking":"1. **Line 312**: `neg_eq_of_add_eq_zero_left hh : -↑n = k` — wrong direction (hh : k + ↑n = 0 gives... the lemma `neg_eq_of_add_eq_zero_left : a + b = 0 → -b = a`?? In this Mathlib the arg order: neg_eq_of_add_eq_zero_left (h : a + b = 0) : -b = a. So from hh : k + ↑n = 0 I get -↑n = k; I need k = -↑n: use `eq_neg_of_add_eq_zero_left hh : k = -↑n`? `eq_neg_of_add_eq_zero_left : a + b = 0 → a = -b` ✓ (used earlier at line 97 successfully). Fix.\n\n2. **Line 317**: `rw [Complex.norm_intCast, Int.norm_eq_abs]` — the target `‖↑k + ↑↑n‖ = |↑k + ↑↑n|` — the pattern ‖↑?n‖ doesn't match because the argument is a SUM ↑k + ↑↑n not a single cast. The h3 rewrite turned (k:ℂ) − −↑n into ↑k + ↑↑n (not a single cast ↑((k+↑n:ℤ))!). My h3 was stated `(k : ℂ) - -(n : ℂ) = ((k + (n : ℤ)) : ℂ)` but the rewrite produced `↑k + ↑↑n`?? The error shows h3 : ↑k − −↑n = ↑k + ↑↑n — so `push_cast; ring` normalized ((k + (n:ℤ)):ℂ) to ↑k + ↑↑n — push_cast distributes the cast! So the cast-whole thing got split by push_cast. Then norm_intCast can't fire on a sum. Fix: don't push_cast in h3; instead: `have h3 : (k : ℂ) - -(n : ℂ) = ((k + (n : ℤ) : ℤ) : ℂ) := by rw [← Int.cast_add, ← Int.cast_neg]...`? Cleaner: ((k + (n:ℤ) : ℤ) : ℂ) = ↑k + ↑n by Int.cast_add (as a rewrite to USE in reverse): I want to REPLACE ↑k + ↑↑n by the single cast: \n```lean\n    have h3 : (k : ℂ) - -(n : ℂ) = ((k + (n : ℤ)) : ℂ) := by\n      rw [Int.cast_add]  -- goal: ↑k − −↑n = ↑k + ↑↑n: then simp [Int.cast_natCast]... : push_cast; ring — that's what happened. The equation is TRUE; the issue is only that after rw [h3] at h2 the form is split. Instead: rw [h3] at h2 — the rewrite uses h3's LHS → RHS: replaces (k:ℂ) − −↑n with the SINGLE cast ✓!! The error shows h3's RHS is ↑k + ↑↑n — because push_cast in h3's PROOF normalized the STATEMENT too? No — `by push_cast; ring` — push_cast acts on the GOAL of the have, normalizing BOTH sides of the equation: the goal ((k:ℂ) − −↑n) = ((k + ↑n:ℤ):ℂ): push_cast normalizes the casts: RHS ↑((k + ↑n : ℤ)) → ↑k + ↑↑n (Int.cast_add pushes outward) — and push_cast CHANGES h3's statement to the normalized form! Right — `push_cast` rewrites the goal, so the resulting proof has the normalized type ↑k − −↑n = ↑k + ↑↑n.\nFix: prove h3 with the cast preserved: `have h3 : (k : ℂ) - -(n : ℂ) = ((k + (n : ℤ)) : ℂ) := by rw [Int.cast_add, Int.cast_neg, Int.cast_natCast]; ring` — wait: Int.cast_add : ((a + b : ℤ):ℂ) = ↑a + ↑b — rewriting RHS ↑((k+↑n)) with Int.cast_add splits it again! I want the REVERSE: prove LHS = RHS by converting LHS to the cast form: `by push_cast [← Int.cast_add]; ring`? push_cast with reverse rules... Simplest: symm + the cast lemmas forward:\n```lean\n    have h3 : (k : ℂ) - -(n : ℂ) = ((k + (n : ℤ)) : ℂ) := by\n      rw [Int.cast_add, Int.cast_natCast, Int.cast_neg, Int.cast_natCast]\n      ring\n```\nInt.cast_add : (↑(a + b) : ?) = ↑a + ↑b — rewrites the RHS occurrence ((k + ↑n : ℤ):ℂ) → (k:ℂ) + (↑(↑n:ℤ):ℂ)... then Int.cast_natCast : ((↑n : ℤ):ℂ) = ↑n ✓; goal becomes ↑k − −↑n = ↑k + ↑n: ring ✓. And h3's STATEMENT remains the single-cast form ✓ (rw inside the proof changes the goal only). Wait no — the rewrites happen in the PROOF's goal: goal is the equation; rw [Int.cast_add] rewrites in the goal: the equation's RHS ↑((k + ↑n : ℤ)) becomes ↑k + ↑↑n — the GOAL changes but h3's TYPE is fixed as stated (the goal starts AS the statement; rewrites transform the goal; the final proof has the stated type ✓✓). Right — h3's type is the statement; push_cast in the proof is fine, BUT the earlier problem: after `push_cast; ring` the elaborated proof term has... no wait — h3's type is whatever I wrote after `:`! The error message showed `h3 : ↑k - -↑n = ↑k + ↑↑n` — meaning MY STATEMENT was already normalized?? I wrote `have h3 : (k : ℂ) - -(n : ℂ) = ((k + (n : ℤ)) : ℂ) := by push_cast; ring` — and the error display shows h3 : ↑k − −↑n = ↑k + ↑↑n — the DISPLAY normalizes?? No — Lean displays h3's type as written... unless push_cast at the END leaves the proof... Hmm, actually the error display at 317 shows `h3 : ↑k - -↑n = ↑k + ↑↑n` — that's h3's type as ELABORATED — meaning my written statement ((k + (n:ℤ)):ℂ) got parsed/elaborated to ↑k + ↑↑n?? OH — I see: in the edit I wrote `((k + (n : ℤ)) : ℂ)` — the cast `((k + (n:ℤ)) : ℂ)` where k : ℤ and (n : ℤ): so k + (n:ℤ) is a ℤ-sum, cast to ℂ — single cast ✓... but the error shows the RHS of h3 as ↑k + ↑↑n. Hmm wait, looking at the error context more carefully: it shows `h3 : ↑k - -↑n = ↑k + ↑↑n` — so h3's type after elaboration is the split form. That means the elaborator parsed `((k + (n : ℤ)) : ℂ)` with the coercion OUTER... no. Whatever — the fix: make h3's RHS unambiguous and use the rw-chain proof:\n\nActually the real issue might be: `push_cast` in the proof TRIES to close the goal and normalizes it FIRST, and then... no, h3's type is fixed at the `:`. UNLESS the `show`-style... I wrote `have h3 : TYPE := by ...` — the type is fixed ✓. The display ↑k + ↑↑n must then be how Lean PRINTS ((k + (n:ℤ)):ℂ)?? No...\n\nHold on, re-read the error: \"Did not find an occurrence of the pattern ‖↑?n‖ in the target expression ‖↑k + ↑↑n‖ = |↑k + ↑↑n|\" — the TARGET (goal of h5) is ‖↑k + ↑↑n‖ = |↑k + ↑↑n| — MY h5 statement was `‖((k + (n : ℤ)) : ℂ)‖ = (|(k + (n : ℤ))| : ℝ)` — displayed as ‖↑k + ↑↑n‖ = |↑k + ↑↑n| — so the DISPLAY of ((k + (n:ℤ)):ℂ) is ↑k + ↑↑n?! That means... my written form was parsed with the cast pushed? No wait — `((k + (n : ℤ)) : ℂ)`: k : ℤ, (n:ℤ) : ℤ, sum in ℤ, then cast: ↑((k + ↑n)) — the printer might display it as ↑k + ↑↑n?? No, the printer shows what's there.\n\nHmm — OR: h5's goal after `rw [h3] at h2` — no the error is in h5's proof: `rw [Complex.norm_intCast, Int.norm_eq_abs]` failing to find ‖↑?n‖ in ‖↑k + ↑↑n‖ = |↑k + ↑↑n|. So the GOAL (h5's statement) displays as ‖↑k + ↑↑n‖ — so MY STATEMENT `((k + (n : ℤ)) : ℂ)` was elaborated to ↑k + ↑↑n — i.e. the coercion is being distributed at PARSE time?? OH WAIT. I see it now: in my edit I wrote `((k + (n : ℤ)) : ℂ)` — but earlier the sed... no. Hmm — actually: is it possible that `push_cast` in h3's proof modified h3's statement?? NO — proofs don't modify statements.\n\nLet me look at what I actually wrote — in the factored lemma I wrote:\n```\n    have h3 : (k : ℂ) - -(n : ℂ) = ((k + (n : ℤ)) : ℂ) := by push_cast; ring\n    rw [h3] at h2\n    have h4 : (1 : ℝ) ≤ ‖((k + (n : ℤ)) : ℂ)‖ := by\n      have h5 : ‖((k + (n : ℤ)) : ℂ)‖ = (|(k + (n : ℤ))| : ℝ) := by\n        rw [Complex.norm_intCast, Int.norm_eq_abs]\n```\nThe error context shows `h3 : ↑k - -↑n = ↑k + ↑↑n` — hmm so h3's stored type IS the split form. Why? Because `push_cast; ring` — push_cast normalized the GOAL (which is h3's statement), and when the tactic block finishes, the proof term is checked against the ORIGINAL statement — but the DISPLAY in the error context shows... the context display shows the STATEMENT AS STORED — which for a `have h : T := by tac` is T itself, unmodified. So T = ↑k − −↑n = ↑k + ↑↑n — meaning `((k + (n : ℤ)) : ℂ)` ELABORATED to ↑k + ↑↑n!! \n\nOH. I bet `((k + (n : ℤ)) : ℂ)` — the outer cast is being parsed as a TYPE ASCRIPTION: `((k + (n : ℤ)) : ℂ)` = \"the term k + (n:ℤ), ascribed type ℂ\" — which inserts a coercion to make it typecheck: k + (n:ℤ) : ℂ forces the + to be ℂ-addition with k coerced: (↑k : ℂ) + ((n:ℤ) : ℂ) — i.e. the elaborator type-checks `k + (n : ℤ)` AT TYPE ℂ by inserting coercions AT THE LEAVES (since + is overloaded, the expected type ℂ propagates INTO the sum, coercing k and (n:ℤ) separately)! That's the coercion-at-expected-type behavior: `↑(k + ↑n)` (one cast) vs `↑k + ↑↑n` (elaborated with expected type). To force the single outer cast: `(((k + (n : ℤ)) : ℤ) : ℂ)` — pin the inner type: then k + (n:ℤ) : ℤ with k : ℤ ✓ no leaf coercion, then one cast ✓. That's the fix: double-ascribe: `(((k + (n : ℤ)) : ℤ) : ℂ)`. Hmm — (n : ℤ) with n : ℕ is a cast; the sum k + (n:ℤ) : ℤ ✓; outer ((· : ℤ) : ℂ) single Int.cast ✓.\n\nSame for h5: `‖(((k + (n : ℤ)) : ℤ) : ℂ)‖ = (|(k + (n : ℤ))| : ℝ)` — hmm |(k + (n:ℤ))| : the abs in ℤ then cast to ℝ: `((|k + (n:ℤ)| : ℤ) : ℝ)` — pin similarly. Then rw [Complex.norm_intCast]: pattern ‖(↑?n : ℂ)‖ matches ‖↑(k+↑n)‖ ✓ gives ‖(k+↑n : ℤ)‖ (ℤ norm); Int.norm_eq_abs : ‖(m:ℤ)‖ = |m|?? form: `Int.norm_eq_abs (n : ℤ) : ‖n‖ = |n|` where |n| : ℝ?? From Normed/Group/Int.lean:33-39: `theorem norm_natCast (n : ℕ) : ‖(n : ℤ)‖ = n` and line 39 `|(n : ℝ)|` via Int.cast_abs — the ℤ-norm ‖n‖ for the ℤ-as-normed-group is |n| as a REAL: `Int.norm_eq_abs : ‖(n:ℤ)‖ = |n|`?? the |n| there has type ℝ via OfNat... Let me just try and adjust from the error.\n\nActually — simpler: sidestep the whole norm-of-int-cast: from h2 : ‖↑k + ↑↑n‖ < 1/2 (the split form is FINE for the contradiction!): k + n ≠ 0 (since k ≠ −n) ⟹ |k + n| ≥ 1 (integers) ⟹ ‖↑k + ↑↑n‖ = ‖((k+n : ℤ):ℂ)‖... the split form: ‖(↑k + ↑↑n : ℂ)‖ = |(k : ℝ) + (n : ℝ)| via ofReal: (↑k + ↑↑n : ℂ) = (((k:ℝ) + (n:ℝ)) : ℂ): push_cast ring — then RCLike.norm_ofReal: ‖((x:ℝ):ℂ)‖ = |x| ✓: so:\n```lean\n      have h4 : (1 : ℝ) ≤ ‖(k : ℂ) + ((n : ℤ) : ℂ)‖ := by\n        rw [show (k : ℂ) + ((n : ℤ) : ℂ) = (((k : ℝ) + (n : ℝ)) : ℂ) by push_cast; ring,\n          RCLike.norm_ofReal]\n        have h5 : (1 : ℝ) ≤ |(k : ℝ) + (n : ℝ)| := by\n          rw [show (k : ℝ) + (n : ℝ) = ((k + (n : ℤ) : ℝ)) by push_cast; ring,\n            ← Int.cast_abs]  -- |↑(k+n)| = ↑|k+n|\n          exact_mod_cast h1  -- (1:ℝ) ≤ ↑|k+n|: from h1 : 1 ≤ |k+n| in ℤ: Int.cast_le: exact_mod_cast ✓\n        exact h5\n      linarith [h2... ]\n```\nHmm wait h2's form after rw [h3]: h2 : ‖↑k + ↑↑n‖ < 1/2 (the split form — since h3's elaborated RHS is split ✓ consistent!). So h4's LHS matches h2's atom ✓ then linarith [h2, h4]?? h4 : 1 ≤ ‖↑k + ↑↑n‖ and h2 : ‖↑k + ↑↑n‖ < 1/2 ⟹ False ✓ — but the GOAL at that point is False (we're inside by_contra? no — my structure: by_cases hk0 : k = −↑n: the negative branch must produce False... wait no — the lemma's goal is w ∈ integerComplement and after intro/obtain/rfl we're proving False?? integerComplement membership = w ∈ (range)ᶜ = range → False: yes goal False ✓. So the negative branch: linarith [h2, h4] closes False ✓.\n\nLet me rewrite the factored lemma cleanly:\n\n```lean\nlemma integerComplement_of_mem_ball_neg_natCast {n : ℕ} {w : ℂ}\n    (hw : w ∈ Metric.ball (-(n : ℂ)) (1 / 2)) (hwn : w ≠ -(n : ℂ)) :\n    w ∈ Complex.integerComplement := by\n  intro hk\n  obtain ⟨k, rfl⟩ := hk\n  have h2 : dist (k : ℂ) (-(n : ℂ)) < 1 / 2 := Metric.mem_ball.mp hw\n  rw [Complex.dist_eq] at h2\n  by_cases hk0 : k = -(n : ℤ)\n  · exact hwn (by rw [hk0]; push_cast; ring)\n  · have h1 : (1 : ℤ) ≤ |k + (n : ℤ)| := Int.one_le_abs (by\n      intro hh\n      exact hk0 (eq_neg_of_add_eq_zero_left hh))\n    have h3 : (k : ℂ) - -(n : ℂ) = (k : ℂ) + ((n : ℤ) : ℂ) := by push_cast; ring\n    rw [h3] at h2\n    have h4 : (1 : ℝ) ≤ ‖(k : ℂ) + ((n : ℤ) : ℂ)‖ := by\n      rw [show (k : ℂ) + ((n : ℤ) : ℂ) = (((k : ℝ) + (n : ℝ)) : ℂ) by push_cast; ring,\n        RCLike.norm_ofReal]\n      have h5 : (1 : ℝ) ≤ |(k : ℝ) + (n : ℝ)| := by\n        have e : |(k : ℝ) + (n : ℝ)| = ((|k + (n : ℤ)| : ℤ) : ℝ) := by\n          rw [show (k : ℝ) + (n : ℝ) = ((k + (n : ℤ) : ℝ)) by push_cast; ring,\n            Int.cast_abs]\n        rw [e]\n        exact_mod_cast h1\n      exact h5\n    linarith\n```\nCheck pieces:\n- `eq_neg_of_add_eq_zero_left hh` : from hh : k + ↑n = 0 gives k = −↑n ✓ (used at line 97 before ✓).\n- h3's statement `(k:ℂ) - −↑n = ↑k + ↑(↑n:ℤ)`: by push_cast; ring — the statement elaborates: RHS `((n : ℤ) : ℂ)` pinned single cast ✓ I wrote ((n : ℤ) : ℂ) — wait in the code above I wrote `(k : ℂ) + ((n : ℤ) : ℂ)` ✓ pinned. push_cast would normalize it to ↑k + ↑n (nat) — fine for the proof.\n- The show: (↑k:ℂ) + ↑(↑n:ℤ) = (((k:ℝ) + (n:ℝ)):ℂ): push_cast; ring ✓.\n- RCLike.norm_ofReal rewrites ‖((x:ℝ):ℂ)‖ → |x| ✓ goal becomes |↑k + ↑n| ≥ 1... wait after the rw-chain the goal is |(k:ℝ) + (n:ℝ)| ≥ ... no — the goal after rw [show..., RCLike.norm_ofReal] is `(1:ℝ) ≤ |(k:ℝ) + (n:ℝ)|` — then h5 proves exactly that ✓ `exact h5`.\n- h5: e : |(k:ℝ)+(n:ℝ)| = ((|k + (n:ℤ)| : ℤ):ℝ): the show: (k:ℝ)+(n:ℝ) = ((k + (n:ℤ) : ℝ))?? careful: (k : ℝ) with k : ℤ — Int.cast; `show (k:ℝ) + (n:ℝ) = ((k + (n:ℤ) : ℤ) : ℝ) by push_cast; ring` — hmm the RHS: ((k + (n:ℤ) : ℤ) : ℝ) — pin ℤ then cast to ℝ ✓; push_cast normalizes RHS to ↑k + ↑n ✓ ring closes. Then Int.cast_abs : ((|m| : ℤ) : ℝ) = |(m : ℝ)|?? Int.cast_abs : (↑|n| : ℝ) = |↑n| — so `← Int.cast_abs` rewrites ((k+(n:ℤ):ℝ))-abs... wait: after the show-rw, e's goal: |((k + (n:ℤ) : ℤ) : ℝ)| = ((|k + (n:ℤ)| : ℤ) : ℝ) — that's Int.cast_abs reversed: Int.cast_abs : ((|m|:ℤ):ℝ) = |(m:ℝ)| — so `rw [← Int.cast_abs]`?? goal LHS |↑m| = RHS ↑|m|: rw [← Int.cast_abs] turns ↑|m| into |↑m|... the goal is |↑m| = ↑|m|: rw [Int.cast_abs] rewrites RHS ↑|m| → |↑m| giving |↑m| = |↑m| ✓ rfl-closed. So: `rw [show ..., Int.cast_abs]` and the goal closes ✓ (rw auto-rfl). \n  Hmm wait — the abs on |(k + (n:ℤ) : ℝ)| — abs in ℝ of the cast ✓.\n  Then exact_mod_cast h1 : goal (1:ℝ) ≤ ↑|k + ↑n| from h1 : (1:ℤ) ≤ |k + ↑n|: Int.cast_le ✓ exact_mod_cast ✓.\n- linarith [h2...]: goal False: h2 : ‖↑k + ↑↑n‖ < 1/2, h4 : 1 ≤ ‖↑k + ↑↑n‖ — same atom ✓ linarith ✓. Just `linarith` (both in context) ✓.\n\nAlso fix the hwn final: goal (↑k:ℂ) = −(↑n:ℂ) after obtain-rfl — hwn : (↑k:ℂ) ≠ −↑n: `exact hwn (by rw [hk0]; push_cast; ring)`: goal (↑k:ℂ) = −(↑n:ℂ): rw [hk0] — hk0 : k = −(n:ℤ): rewrites k: ((−(n:ℤ) : ℤ):ℂ) = −(↑n:ℂ): push_cast: LHS −↑↑n... = −↑n: ring ✓.\n\n3. **Line 495** (add_nat_add_one_ne_zero_of_mem_ball_neg's h4): same norm_ofReal pattern issue: `‖↑↑n − (↑↑j + 1)‖ = |↑n − (↑j + 1)|` — the show I wrote: `(n : ℂ) - ((j : ℂ) + 1) = (((n : ℝ) - ((j : ℝ) + 1)) : ℂ)` — the RHS cast of an ℝ expression ✓ RCLike.norm_ofReal should match ‖((x:ℝ):ℂ)‖... the error shows the target as ‖↑↑n − (↑↑j + 1)‖ = ... — meaning the show's LHS `(n:ℂ) − ((j:ℂ)+1)` elaborated with leaf-coercions: ↑↑n − (↑↑j + 1) — fine that's what h2 has ✓ — but the rewrite `rw [show (n:ℂ) − ((j:ℂ)+1) = (((n:ℝ) − ((j:ℝ)+1)):ℂ) by push_cast; ring, RCLike.norm_ofReal] at h2` — the error says RCLike.norm_ofReal pattern ‖↑?r‖ not found in ‖↑↑n − (↑↑j+1)‖ = |↑n − (↑j+1)| — i.e. the SHOW-rewrite did NOT fire first (its LHS pattern `(n:ℂ) − ((j:ℂ)+1)` — hmm the elaborated LHS of my show is ↑↑n − (↑↑j + 1)... and h2's content is ‖↑↑n − (↑↑j+1)‖ — should match syntactically? The elaborated forms might differ in coercion insertion points (Nat.cast vs OfNat etc). The error target shows the ABS already applied on RHS (|↑n − (↑j+1)|)?? — confusing. Let me restructure to avoid the mixed-cast show: use the re/sub norm approach instead: ‖x‖ ≥ |re x|:\n```lean\n    -- |re(↑n − (↑j+1))| = |n − (j+1)| ≥ 1\n    have h5 : (1 : ℝ) ≤ ‖((n : ℂ) - ((j : ℂ) + 1))‖ := by\n      have hre : ((n : ℂ) - ((j : ℂ) + 1)).re = (n : ℝ) - ((j : ℝ) + 1) := by\n        simp [Complex.sub_re, Complex.natCast_re, Complex.one_re, Complex.add_re]\n      have h6 : |((n : ℂ) - ((j : ℂ) + 1)).re| ≤ ‖(n : ℂ) - ((j : ℂ) + 1)‖ := Complex.abs_re_le_norm _\n      rw [hre] at h6\n      have h7 : (1 : ℝ) ≤ |(n : ℝ) - ((j : ℝ) + 1)| := by\n        ...the integer-separation (from hne : n ≠ j + 1): |↑n − (↑j + 1)| = |(n − j − 1 : ℤ)-cast| ≥ 1:\n        have h8 : (1 : ℤ) ≤ |(n : ℤ) - ((j : ℤ) + 1)| := Int.one_le_abs (sub_ne_zero.mpr hne-int)\n        -- cast: |↑n − (↑j+1)| in ℝ = ↑|↑n − (↑j+1)| (ℤ): Int.cast_abs, push_cast\n        have h9 : |(n : ℝ) - ((j : ℝ) + 1)| = ((|(n : ℤ) - ((j : ℤ) + 1)| : ℤ) : ℝ) := by\n          rw [show (n : ℝ) - ((j : ℝ) + 1) = (((n : ℤ) - ((j : ℤ) + 1) : ℤ) : ℝ) by push_cast; ring,\n            Int.cast_abs]\n        rw [h9]\n        exact_mod_cast h8\n      exact h7.trans h6\n    linarith [h2, h5]  -- h2 : ‖↑n − (↑j+1)‖ < 1/2: contra\n```\nHmm wait h2's form: h2 : ‖↑n − (↑j + 1)‖ < 1/2 where the term is `(n:ℂ) − ((j:ℂ)+1)` — h5's LHS `‖((n:ℂ) − ((j:ℂ)+1))‖` — same? The elaboration: `(n : ℂ) - ((j : ℂ) + 1)` — n : ℕ cast ✓ (j:ℂ)+1 ✓ — same term ✓ hopefully syntactically identical.\n\nhne-int : (n:ℤ) − ((j:ℤ)+1) ≠ 0: from hne : n ≠ j+1: `sub_ne_zero.mpr`: (n:ℤ) − ((j:ℤ)+1) ≠ 0 ⟺ (n:ℤ) ≠ (j:ℤ)+1: from hne (ℕ): exact_mod_cast: `have hne' : (n:ℤ) ≠ (j:ℤ) + 1 := by exact_mod_cast hne` — hne : n ≠ j + 1 (ℕ): cast: (n:ℤ) ≠ ((j+1 : ℕ):ℤ): push_cast: = ↑j + 1 ✓ exact_mod_cast handles ✓.\n\nThis re-route avoids RCLike.norm_ofReal on cast sums entirely (uses re + abs_re_le_norm ✓ banked pattern).\n\n4. **Line 689** (digamma_eq_neg_inv_add_Hsub final): `rw [hdig, Hsub, hsplit]` — after rw [hdig] the goal has `∑' (n:ℕ), (1/(↑n+1) − 1/(w+↑n+1))` (the RAW digamma_series tsum, note the bound variable printed as n — collision with our n!) and rw [hsplit] fails because hsplit's LHS is `digammaSeriesPart w` (the def) while the goal has the raw tsum. Also `rw [Hsub]` — Hsub is a def — rw [Hsub]?? defs can't be rw'd directly — need `show` or simp only [Hsub]. Fix: don't rw Hsub; instead:\n```lean\n  rw [hdig]\n  -- goal: −γ − 1/w + ∑' j, (1/(j+1) − 1/(w+j+1)) = −1/(w+n) + Hsub n w\n  show -(Real.eulerMascheroniConstant : ℂ) - 1 / w + digammaSeriesPart w\n      = -1 / (w + (n : ℂ)) + Hsub n w\n  rw [hsplit, Hsub]?? — Hsub unfold: Hsub n w = −γ − 1/w + sub + 1/n: `show ... = -1/(w+↑n) + (-(γ:ℂ) - 1/w + digammaSeriesPartSub n w + 1/↑n)` via defeq (Hsub n w unfolds ✓):\n  show -(Real.eulerMascheroniConstant : ℂ) - 1 / w + digammaSeriesPart w\n      = -1 / (w + (n : ℂ)) + (-(Real.eulerMascheroniConstant : ℂ) - 1 / w\n        + digammaSeriesPartSub n w + 1 / (n : ℂ))\n  rw [hsplit]\n  ring\n```\nThe first show: LHS = −γ − 1/w + ∑'... vs my show's −γ − 1/w + digammaSeriesPart w — defeq (digammaSeriesPart w ≡ the tsum ✓) ✓. RHS: −1/(w+↑n) + Hsub n w vs show's — Hsub unfolds ✓ defeq. The binder-name shadowing (∑' (n:ℕ) with our n : ℕ) — display only, no problem. Then rw [hsplit]: pattern digammaSeriesPart w present ✓ → −γ − 1/w + (sub + (1/↑n − 1/(w+↑n))) = ...: ring ✓ (1/(w+↑n) atoms cancel).\n\n5. **Line 695** (Hsub_analyticAt hA): `analyticAt_const` can't synthesize the point x — because I left `_` as the point: `AnalyticAt ℂ (fun _ : ℂ => −γ) _` — the underscore point can't be inferred from the eventual application (hA.sub hB needs the point = −↑n). Fix: ascribe the point: `(fun _ : ℂ => ...) (−(n:ℂ))`:\n```lean\n  have hA : AnalyticAt ℂ (fun _ : ℂ => -(Real.eulerMascheroniConstant : ℂ)) (-(n : ℂ)) := analyticAt_const\n```\nSimilarly hD's `_` → (−(n:ℂ)). And hB's `simp; positivity` — goal after apply AnalyticAt.div analyticAt_const analyticAt_id: g x ≠ 0 with g = id: id (−↑n) ≠ 0 i.e. −↑n ≠ 0: `simp; positivity`? simp reduces −↑n ≠ 0 to n ≠ 0 (neg_ne_zero, Nat.cast_ne_zero...) then n ≥ 1: `by simp; positivity` — hmm positivity proves 0 < something not ne. Use: `(by simp [hn.ne']...)`: −(↑n:ℂ) ≠ 0 ⟺ ↑n ≠ 0 ⟺ n ≠ 0: hn : 1 ≤ n: `by simp only [ne_eq, neg_eq_zero, Nat.cast_eq_zero]; omega`? omega proves n ≠ 0 from 1 ≤ n — as a tactic goal n ≠ 0: omega ✓. So:\n```lean\n  have hB : AnalyticAt ℂ (fun w : ℂ => 1 / w) (-(n : ℂ)) := by\n    apply AnalyticAt.div analyticAt_const analyticAt_id\n    simp only [ne_eq, neg_eq_zero, Nat.cast_eq_zero]\n    omega\n```\nWait — the hg0 goal form: analyticAt_id's value at the point: id (−↑n) ≠ 0 — displays as −↑n ≠ 0 after beta/id-unfold? `analyticAt_id : AnalyticAt 𝕜 id x` — the g := id: g x = id x — norm via simp only [id]?? Let me use `Function.id_def`... simpler: use (show (-(n:ℂ)) ≠ 0 by ...) with the defeq id x ≡ x: hg0 : id (−↑n) ≠ 0 — the ascription `(show (-(n:ℂ)) ≠ 0 by ...)` defeq ✓:\n```lean\n    apply AnalyticAt.div analyticAt_const analyticAt_id (show (-(n : ℂ)) ≠ 0 by\n      simp only [ne_eq, neg_eq_zero, Nat.cast_eq_zero]\n      omega)\n```\nWait the div signature order: AnalyticAt.div (hf) (hg) (hg0)? check: Constructions.lean:1115: `theorem AnalyticAt.div {f g : E → 𝕝} {x : E} (hf : AnalyticAt 𝕜 f x) (hg : AnalyticAt 𝕜 g x) (hgx : g x ≠ 0) : AnalyticAt 𝕜 (f / g) x`? Hmm — but f/g is Pi-div: the goal fun w => 1/w vs (1/id)... apply AnalyticAt.div on goal AnalyticAt ℂ (fun w => 1/w) (−↑n): pattern AnalyticAt (f / g) x vs fun w => 1/w: HOU f/g =?= fun w => 1/w: f := 1 (const), g := id ✓ via Pi.div_apply reverse — apply's unification is up to... this worked for the earlier hC at 0? That one had apply AnalyticAt.div analyticAt_const (g := fun s => s − 1)... with g pinned. For g := id: apply AnalyticAt.div analyticAt_const analyticAt_id (show...) — the hg0's type g x ≠ 0 with g := id — fine. But will `apply AnalyticAt.div` unify the goal lambda fun w => 1/w with f/g? f / g = fun x => f x / g x (Pi instHpDiv): fun w => 1/w =?= fun x => f x / g x: f w := 1 (const lambda), g w := w ✓ this HOU is pattern-y enough (worked in §4 for hC : fun s => 1/s at 1 — that one compiled ✓ (`analyticAt_const.div analyticAt_id (by norm_num)` in Fexpr_eventually_one — yes it compiled). So fine.\n\nSimilarly Fexpr_eventually_neg_two_mul's hC had `simp; positivity` — replace with the same show-form.\n\n6. Also Fexpr_eventually_neg_two_mul's hC1: `apply AnalyticAt.div analyticAt_const _ (by ...)` — pin g := fun s => s − 1 like hC-at-0.\n\nLet me apply all these fixes now.","signature":""},{"type":"text","text":"Applying the fix batch (cast-pinning, the re-norm route for integer separation, def-unfold shows, ascribed points)."},{"type":"tool_use","id":"Edit_177","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiFEntire.lean","old_string":"  intro hk\n  obtain ⟨k, rfl⟩ := hk\n  have h2 : dist (k : ℂ) (-(n : ℂ)) < 1 / 2 := Metric.mem_ball.mp hw\n  rw [Complex.dist_eq] at h2\n  by_cases hk0 : k = -(n : ℤ)\n  · exact hwn (by rw [hk0]; push_cast; ring)\n  · have h1 : (1 : ℤ) ≤ |k + (n : ℤ)| := Int.one_le_abs (by\n      intro hh\n      exact hk0 (neg_eq_of_add_eq_zero_left hh))\n    have h3 : (k : ℂ) - -(n : ℂ) = ((k + (n : ℤ)) : ℂ) := by push_cast; ring\n    rw [h3] at h2\n    have h4 : (1 : ℝ) ≤ ‖((k + (n : ℤ)) : ℂ)‖ := by\n      have h5 : ‖((k + (n : ℤ)) : ℂ)‖ = (|(k + (n : ℤ))| : ℝ) := by\n        rw [Complex.norm_intCast, Int.norm_eq_abs]\n      rw [h5]\n      exact_mod_cast h1\n    linarith","new_string":"  intro hk\n  obtain ⟨k, rfl⟩ := hk\n  have h2 : dist (k : ℂ) (-(n : ℂ)) < 1 / 2 := Metric.mem_ball.mp hw\n  rw [Complex.dist_eq] at h2\n  by_cases hk0 : k = -(n : ℤ)\n  · exact hwn (by rw [hk0]; push_cast; ring)\n  · have h1 : (1 : ℤ) ≤ |k + (n : ℤ)| := Int.one_le_abs (by\n      intro hh\n      exact hk0 (eq_neg_of_add_eq_zero_left hh))\n    have h3 : (k : ℂ) - -(n : ℂ) = (k : ℂ) + ((n : ℤ) : ℂ) := by push_cast; ring\n    rw [h3] at h2\n    have h4 : (1 : ℝ) ≤ ‖(k : ℂ) + ((n : ℤ) : ℂ)‖ := by\n      rw [show (k : ℂ) + ((n : ℤ) : ℂ) = (((k : ℝ) + (n : ℝ)) : ℂ) by push_cast; ring,\n        RCLike.norm_ofReal]\n      have h5 : (1 : ℝ) ≤ |(k : ℝ) + (n : ℝ)| := by\n        have e : |(k : ℝ) + (n : ℝ)| = ((|k + (n : ℤ)| : ℤ) : ℝ) := by\n          rw [show (k : ℝ) + (n : ℝ) = (((k + (n : ℤ)) : ℤ) : ℝ) by push_cast; ring,\n            Int.cast_abs]\n        rw [e]\n        exact_mod_cast h1\n      exact h5\n    linarith"},"extra_content":null}]