[{"type":"thinking","thinking":"Two errors in oddHarm_eq's succ branch:\n1. Line 216 \"No goals\" — the `show (∑ k ∈ Finset.Icc 1 (K+1), ...) = _` worked? No wait, error at 219: my second `show` has `2 * (↑K + 1) + 1` but goal has `2 * ↑(K+1) + 1` — cast placement differs. And error at 216 says \"No goals to be solved\" at... line 216 is the first `show`? Let me count: the `show` before rw [Finset.sum_Icc_succ_top] is unnecessary — the goal after `| succ K ih =>` is already `oddHarm (K+1) = ...` which needs unfolding to see the sum. My first `show (∑ k ∈ Finset.Icc 1 (K+1), ...) = _` apparently succeeded (it's defeq since oddHarm is a def), then rw gave the goal with `2 * ↑(K + 1) + 1` (cast of K+1 as ℕ then...). Hmm the goal shows `2 * (↑K + 1) + 1`?? The error message says target has `1 / (2 * ↑(K + 1) + 1)` while my show has `1 / (2 * (↑K + 1) + 1)`. ↑(K+1) vs (↑K + 1): defeq but not syntactic; `show` checks defeq though! It said \"not definitionally equal\" — probably because of the RHS `= ?m.374` metavariable — my `show ... = _` can't unify a metavariable by defeq check? The issue: `show X = _` — the `_` creates a metavariable assigned to the actual RHS — should work... but the error suggests the defeq check failed on the LHS part.\n\nSimplify: drop the two `show` lines. After rw [Finset.sum_Icc_succ_top ...], the goal is `∑ k ∈ Icc 1 K, f k + f (K+1) = ↑(harmonic(2*(K+1)+1)) − ...`. Then `rw [show (∑ k ∈ Finset.Icc 1 K, (1:ℝ)/(2*k+1)) = oddHarm K from rfl, ih, h1, h2]` — wait, after rw [ih], the goal has the sum replaced by oddHarm K? rw [←] direction... I want to rewrite `∑ k ∈ Icc 1 K, 1/(2↑k+1)` to `oddHarm K` — that's `show ... = oddHarm K from rfl` used with rw: rw [this] replaces occurrences of LHS with RHS ✓ (oddHarm K is definitionally the sum; the show proves an Eq by rfl — but is it rfl? oddHarm K unfolds to exactly `∑ k ∈ Finset.Icc 1 K, 1 / (2*↑k+1)` ✓ should be rfl with default transparency since oddHarm is a regular def — `rfl` proofs by `Eq.refl` require syntactic-after-unfold... `from rfl` uses `rfl` which checks defeq ✓ works for defs).\n\nAlso `push_cast` after — the f (K+1) term: `1 / (2 * ↑(K+1) + 1)` — push_cast turns ↑(K+1) into ↑K + 1 ✓. So reorder: do the rewrites, THEN push_cast, then rw [e3] (e3's LHS `1/(2*↑K+2)` — after push_cast, the h1/h2 terms appear as `1/(2*↑K+2)` ✓ h1 was stated with (K:ℝ) casts so push_cast inside h1's proof... h1/h2 are already proved; their statements use (K:ℝ) — after rw [h1, h2] the goal contains 1/(2*↑K+2), 1/(2*↑K+3), 1/(↑K+1) directly ✓. The remaining sum term f(K+1) = 1/(2*↑(K+1)+1): push_cast at goal → 1/(2*(↑K+1)+1) — ring-normalizes to 1/(2↑K+3) ✓ same atom as h1's after ring normalization — ring handles.\n\nNew succ branch:\n```lean\n  | succ K ih =>\n    have h1 : ... := ...\n    have h2 : ... := ...\n    have e3 : ... := ...\n    show (∑ k ∈ Finset.Icc 1 (K + 1), (1 : ℝ) / (2 * k + 1)) = _\n    rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ K + 1) (fun k => (1 : ℝ) / (2 * k + 1)),\n      show (∑ k ∈ Finset.Icc 1 K, (1 : ℝ) / (2 * k + 1)) = oddHarm K from rfl, ih, h1, h2]\n    push_cast\n    rw [e3]\n    ring\n```\nBut wait — error 1 was \"No goals to be solved\" at 216:6 — that suggests the FIRST show at line ~216 closed the goal?? Or an earlier rw closed it and subsequent tactic had no goals. Hmm, 216 is likely the `rw [Finset.sum_Icc_succ_top ...]` line or the first show. If first `show ... = _` — with metavariable RHS — Lean assigns the RHS metav to the actual RHS, and if LHS is defeq to actual LHS, show succeeds with goal unchanged. \"No goals\" would mean the PREVIOUS tactic closed everything. Possibly `simp [oddHarm, harmonic_succ]` in zero branch... no, branches are separate.\n\nActually the line numbers: 216:6 \"No goals to be solved\" and 219:4 'show' failed. Line 219 is the second show. Line 216 might be the first `show (∑ k ∈ Finset.Icc 1 (K + 1), ...) = _`?? If that show errored \"No goals\", then something before it closed the goal — that'd be `rw [e3]`? No, order: h1, h2, e3, show, rw sum_Icc, show, rw..., push_cast, rw e3, ring. Hmm 216:6 column 6 — tactics inside the succ branch are indented 4, their sub-tactics 6. h1's proof body tactics are at column 6! So 216:6 is inside h1's or e3's proof — \"No goals to be solved\": e.g. in h1: rw [hs, harmonic_succ, harmonic_succ]; push_cast; ring — maybe push_cast closes the goal and then `ring` has no goals. Yes! In h1's proof, after the rewrites, push_cast might fully solve it (push_cast tries to close goals), making `ring` report \"No goals\". Similarly in h2 or e3 (field_simp might close, then ring errors).\n\nFix: reorder to `push_cast; ring` → use `<;>` or just try `ring` first then push_cast... simplest: combine: `rw [...]; push_cast; ring` — if push_cast closes it, remove ring. I can't know without testing. Use `push_cast` then `try ring`? Hmm — but if push_cast DOESN'T close, need ring. `push_cast; ring` errors only in the case push_cast closes it. Actually \"No goals to be solved\" is an error that fails the tactic block. Safe pattern: `rw [...]; push_cast; ring` where final is `ring` — replace with `<;> ring`? no. Use: `rw [...]; push_cast; (try ring)`? Hmm, but if push_cast leaves a goal that ring must close, (try ring) still closes it; if push_cast already closed, try ring is fine. But sloppy. Alternatively `push_cast` is not supposed to fully close... it did apparently in h1 (216 is in h1's body? h1 spans lines ~200-207...). Eh — just make the endings robust: replace `push_cast; ring` with `push_cast; ring` → if error, switch order: `ring_nf`? Let me just try `push_cast; ring` variants per error lines.\n\nActually, cleaner: in h1 use `norm_cast` style: after rw [hs, harmonic_succ, harmonic_succ], the goal is over ℝ with ℚ casts: `push_cast` normalizes casts; then `ring`. The error \"No goals\" at 216:6 — let me count actual lines in current file. Lines: succ branch starts ~199. h1 at 200: `have h1 : ...` lines 200-201 statement, proof lines 202-205: hs, rw, push_cast, ring at columns 6. Line 205ish = `ring`. Hmm 216 would be in e3 (field_simp; ring). e3: have e3 (line ~210-213): hne, field_simp, ring — field_simp probably closed it (field_simp + the equation 1/(2K+2) = (1/(K+1))/2 — after clearing denominators: 2*(K+1) = 2*(K+1)? field_simp proves by normalization and can close refl goals, then `ring` errors no-goals).\n\nWait but if field_simp closed e3, that's success for e3; the error is just the stray `ring`. Then error 2 at 219:4 is the second `show` as analyzed.\n\nFixes:\n1. h1: `rw [hs, harmonic_succ, harmonic_succ]; push_cast; ring` — might have same issue if push_cast closes. Make endings `push_cast; ring` → `push_cast; ring` risky everywhere. I'll change all three to end with a combined tactic that tolerates: actually simplest: in e3, remove `ring` (field_simp closes or not — if not, error \"unsolved goals\" tells me). Let me instead write e3 as: `have e3 : ... := by field_simp; ring` → change to: `by rw [show (2*(K:ℝ)+2) = 2*((K:ℝ)+1) by ring, one_div, mul_inv]; ring`. Deterministic. For h1/h2 keep push_cast; ring — no \"no goals\" error reported there (216 was only one error; but if h1's ring also errors after fixing e3...). h1's proof: after push_cast, is the goal closed? h1: ↑(harmonic(2K+1)) + ↑(↑(2K+2))⁻¹ + ↑(↑(2K+3))⁻¹ = ↑(harmonic(2K+1)) + 1/(2↑K+2) + 1/(2↑K+3) — push_cast normalizes: ↑(↑(N:ℕ):ℚ)⁻¹ → (↑N:ℝ)⁻¹... hmm actually harmonic_succ gives terms (↑(N+1) : ℚ)⁻¹ inside harmonic sum — the cast ↑...⁻¹ to ℝ: Rat.cast_inv etc. Then RHS has 1/(2↑K+2). push_cast normalizes (↑(2K+2):ℝ)⁻¹ to... it won't unfold ↑N with N=2K+2 into 2↑K+2? push_cast DOES: Nat.cast_add, Nat.cast_mul push casts inside: ↑(2*K+2) → 2*↑K+2 ✓ (norm_cast moves casts outward... \"push_cast pushes casts inward toward the leaves\": ↑(2*K+2) becomes 2*↑K+2 ✓). Then both sides identical → can push_cast close by rfl? push_cast calls simp, which closes goals that become rfl ✓ possible! Then `ring` errors. So h1/h2 need the same robustness.\n\nRobust universal fix: end those proofs with `push_cast; ring` → replace by `push_cast; ring_nf` — same issue. Use sequencing with `<;>`? no. Use `first | ring | push_cast; ring`? Overkill. Cleanest: `push_cast` then `ring` where I EXPECT push_cast not to close — but evidence says it might. Alternative: skip push_cast; use `norm_num [Rat.cast_add, ...]`? Ugh.\n\nDeterministic approach for h1: prove via Rat-level identity then cast:\n```lean\n    have h1 : (harmonic (2 * (K + 1) + 1) : ℝ) = ... := by\n      have hs : 2 * (K + 1) + 1 = (2 * K + 1) + 1 + 1 := by ring\n      have hr : harmonic (2 * (K + 1) + 1) =\n          harmonic (2 * K + 1) + (↑(2 * K + 2))⁻¹ + (↑(2 * K + 3))⁻¹ := by\n        rw [hs, harmonic_succ, harmonic_succ]\n        ring\n      rw [hr]\n      push_cast\n      ring\n```\nhr is a ℚ identity: after the two harmonic_succ rewrites, goal: H + ↑(2K+2)⁻¹ + ... wait LHS harmonic((2K+1)+1+1) → rw gives harmonic(2K+1)+↑(2K+1+1)⁻¹+↑(2K+1+1+1)⁻¹?? order: outer rewrite: harmonic(((2K+1)+1)+1) = harmonic((2K+1)+1) + ↑((2K+1)+1+1)⁻¹; then second: harmonic((2K+1)+1) → harmonic(2K+1) + ↑(2K+1+1)⁻¹. Result: H + ↑(2K+1+1)⁻¹ + ↑(2K+1+1+1)⁻¹ = RHS H + ↑(2K+2)⁻¹ + ↑(2K+3)⁻¹: `ring` in ℚ? Atoms: ↑(2K+1+1) vs ↑(2K+2) — Nat casts of different nat expressions; ring works on the LEAVES: ↑(2K+1+1) — is 2K+1+1 normalized? In ℚ, ↑(2K+1+1): Nat.cast of literal expression — ring might normalize Nat.cast applications via Nat.cast_add? `ring` does handle Nat.cast semiring-normalization: it can push casts... `ring` normalizes in commutative (semi)rings and handles Nat.cast by moving it outward? I believe `ring` can close ↑(2*K+1+1) = ↑(2*K+2) — hmm, these are casts of ℕ terms; ring's preprocessing includes `push_cast`-like atom normalization? Not reliably.\n\nSimplest robust: end h1/h2/e3 with `<;> try ring`? No...\n\nUse `ring_nf` as final with allowance: actually there's a clean idiom: `push_cast; ring` fails only if push_cast closes goal. `push_cast` closing is fine — then don't call ring. To be safe in both worlds: `push_cast; try ring`? Hmm wait — actually if push_cast closes the goal, `ring` gets \"no goals\" ERROR. With `try ring`, no error ✓. If push_cast leaves goal, ring must close (try ring attempts; if ring fails, unsolved goals error surfaces anyway) ✓. It's slightly sloppy but fine for our private research package... the project has no linter against try. But hmm — could `try ring` mask a genuine failure? If ring fails, goals remain → build errors at end of block (\"unsolved goals\") ✓ still caught. OK but actually cleaner: `push_cast` followed by `ring` → replace with just `norm_cast` then... no. Use: \n\nh1: `rw [hs, harmonic_succ, harmonic_succ]; push_cast; ring` → `rw [hs, harmonic_succ, harmonic_succ]; push_cast; ring` — I'll just test. Instead of over-thinking, make all three endings `push_cast; ring` and e3 deterministic, then build and fix what errors.\n\nActually e3 rewrite plan is deterministic:\n```lean\n    have e3 : (1 : ℝ) / (2 * (K : ℝ) + 2) = (1 / ((K : ℝ) + 1)) / 2 := by\n      have h4 : (2 : ℝ) * (K : ℝ) + 2 = 2 * ((K : ℝ) + 1) := by ring\n      rw [h4, one_div, mul_inv, one_div, div_eq_mul_inv]\n      ring\n```\nGoal after rw: 2⁻¹ * (↑K+1)⁻¹ = (↑K+1)⁻¹ * 2⁻¹ → ring closes ✓ deterministic. Hmm `mul_inv : (a*b)⁻¹ = a⁻¹ * b⁻¹` ✓. div_eq_mul_inv : a/b = a * b⁻¹ ✓ rewrites RHS (1/(K+1))/2 → (1/(K+1)) * 2⁻¹; and LHS 1/(2*(K+1)) was rewritten to (2*(K+1))⁻¹ by the first one_div, then mul_inv → 2⁻¹ * (K+1)⁻¹. But the second `one_div` rewrites (1/(K+1)) → (K+1)⁻¹ ✓. Then ring ✓.\n\nAnd the second `show` in succ branch: drop it entirely — after rw [sum_Icc_succ_top], do the oddHarm fold rw directly (the show from rfl works on the subterm regardless of the f(K+1) cast form), then ih/h1/h2, then push_cast (normalizes ↑(K+1)), rw [e3]?? after push_cast, does `1/(2*↑K+2)` still appear syntactically for e3's rw? h1 was rw'd in, contributing `1/(2*↑K+2)` literally ✓ push_cast normalizes casts INSIDE — `1/(2*↑K+2)` has no ℚ casts, stays ✓. e3 rw ✓ then ring.\n\nBut wait — will push_cast alter `1/(2*↑K+2)`'s form? It's ℝ division of ℝ atoms; push_cast only moves type casts. ↑(K:ℝ) is already at leaves. ✓ stays.\n\nOrder concern: rw [h1] — h1's LHS is `(harmonic (2*(K+1)+1) : ℝ)` — the goal has `↑(harmonic (2*(K+1)+1))` ✓ rewrites. But after rw [sum_Icc_succ_top...] goal's RHS still `↑(harmonic (2*(K+1)+1)) - ↑(harmonic (K+1))/2 - 1` ✓ rw [h1, h2] fine.\n\nOne more: `show (∑ k ∈ Finset.Icc 1 (K + 1), (1 : ℝ) / (2 * k + 1)) = _` as first tactic in succ branch — needed to unfold oddHarm (K+1) so sum_Icc_succ_top can rewrite. Defeq ✓ should work (the earlier error was the SECOND show). Keep it.\n\nWait, actually the rw [show (∑ ...) = oddHarm K from rfl] — the term `oddHarm K` in ih: ih : oddHarm K = ↑(harmonic(2*K+1)) − ...; rw [ih] needs `oddHarm K` to appear — after folding rw it does ✓.\n\nLet me rewrite the whole oddHarm_eq succ branch.","signature":""},{"type":"text","text":"Fixing `oddHarm_eq`: the `show` cast placement and a `field_simp`-closes-goal issue."},{"type":"tool_use","id":"Edit_58","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"  | succ K ih =>\n    have h1 : (harmonic (2 * (K + 1) + 1) : ℝ) =\n        (harmonic (2 * K + 1) : ℝ) + 1 / (2 * (K : ℝ) + 2) + 1 / (2 * (K : ℝ) + 3) := by\n      have hs : 2 * (K + 1) + 1 = (2 * K + 1) + 1 + 1 := by ring\n      rw [hs, harmonic_succ, harmonic_succ]\n      push_cast\n      ring\n    have h2 : (harmonic (K + 1) : ℝ) = (harmonic K : ℝ) + 1 / ((K : ℝ) + 1) := by\n      rw [harmonic_succ]\n      push_cast\n      ring\n    have e3 : (1 : ℝ) / (2 * (K : ℝ) + 2) = (1 / ((K : ℝ) + 1)) / 2 := by\n      have hne : (K : ℝ) + 1 ≠ 0 := by positivity\n      field_simp\n      ring\n    show (∑ k ∈ Finset.Icc 1 (K + 1), (1 : ℝ) / (2 * k + 1)) = _\n    rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ K + 1) (fun k => (1 : ℝ) / (2 * k + 1))]\n    show (∑ k ∈ Finset.Icc 1 K, (1 : ℝ) / (2 * k + 1)) + 1 / (2 * ((K : ℝ) + 1) + 1) = _\n    rw [show (∑ k ∈ Finset.Icc 1 K, (1 : ℝ) / (2 * k + 1)) = oddHarm K from rfl, ih, h1, h2]\n    push_cast\n    rw [e3]\n    ring","new_string":"  | succ K ih =>\n    have h1 : (harmonic (2 * (K + 1) + 1) : ℝ) =\n        (harmonic (2 * K + 1) : ℝ) + 1 / (2 * (K : ℝ) + 2) + 1 / (2 * (K : ℝ) + 3) := by\n      have hs : 2 * (K + 1) + 1 = (2 * K + 1) + 1 + 1 := by ring\n      rw [hs, harmonic_succ, harmonic_succ]\n      push_cast\n      ring\n    have h2 : (harmonic (K + 1) : ℝ) = (harmonic K : ℝ) + 1 / ((K : ℝ) + 1) := by\n      rw [harmonic_succ]\n      push_cast\n      ring\n    have e3 : (1 : ℝ) / (2 * (K : ℝ) + 2) = (1 / ((K : ℝ) + 1)) / 2 := by\n      have h4 : (2 : ℝ) * (K : ℝ) + 2 = 2 * ((K : ℝ) + 1) := by ring\n      rw [h4, one_div, mul_inv, one_div, div_eq_mul_inv]\n      ring\n    show (∑ k ∈ Finset.Icc 1 (K + 1), (1 : ℝ) / (2 * k + 1)) = _\n    rw [Finset.sum_Icc_succ_top (by omega : 1 ≤ K + 1) (fun k => (1 : ℝ) / (2 * k + 1)),\n      show (∑ k ∈ Finset.Icc 1 K, (1 : ℝ) / (2 * k + 1)) = oddHarm K from rfl, ih, h1, h2]\n    push_cast\n    rw [e3]\n    ring"},"extra_content":null}]