← Session 35 — ii′-6-R: Lagarias 6.1 right-tail re-proof (E-032/D-073)
thinking (3379 chars)
Still timeouts at 2190:2 (tactic execution) and 2165 (whnf) — despite the div_lt_div_iff₀ fix. The timeout at 2190:2 is the `rw [div_lt_div_iff₀ ..., one_mul, one_mul]` — hmm, 2190:2 — let me look at what's at 2190 now. And 2165's whnf timeout — the `rw [e]` region in lt_three. These timeouts suggest something deeper is looping — maybe the `rw [div_lt_div_iff₀ ...]` with the `by linarith` args is fine but the subsequent elaboration of the goal with the hbridge/hIIM combination loops.
Wait — maybe the problem is different: the `rw [e]` at lt_three followed by `exact div_lt_div_of_pos_right hb h4n` — hb is the bracket bound whose statement is a HUGE expression (the 4.56·(...) form with all the exp terms) — div_lt_div_of_pos_right hb h4n : bracket/(4n) < (4/3)/(4n) ✓ — the whnf timeout might be div_lt_div_of_pos_right's unification with the huge bracket expression — or the `rw [e]` with e : 1/(3n) = (4/3)/(4n) rewriting the GOAL `deriv (deriv IIMaj) n < 1/(3n)` — the goal's RHS `1/(3n)` → `(4/3)/(4n)` ✓ then div_lt_div_of_pos_right hb h4n proves `bracket/(4n) < (4/3)/(4n)` — the final unification of `deriv (deriv IIMaj) n` with `bracket/(4n)` — hmm — wait — after the chain_rule_sqrt + deriv_psiTilde + deriv2_psiTilde rewrites, the goal's LHS `deriv (deriv IIMaj) n` became the bracket form ALREADY (the rw [hId, chain_rule_sqrt ..., deriv_psiTilde, deriv2_psiTilde] turned it into the bracket) — then `rw [e]` rewrites `1/(3n)` in the goal's RHS to `(4/3)/(4n)` ✓ then div_lt_div_of_pos_right hb h4n : `bracket/(4n) < (4/3)/(4n)` — and the goal is `bracket/(4n) < (4/3)/(4n)`?? — hmm — actually the goal after the first rws is `bracket-expression < 1/(3n)` — then `rw [e]` changes `1/(3n)` to `(4/3)/(4n)`: `bracket < (4/3)/(4n)` — and div_lt_div_of_pos_right hb h4n gives `bracket/(4n) < (4/3)/(4n)` — the goal has `bracket` NOT `bracket/(4n)`!! — wait — after chain_rule_sqrt, the LHS `deriv (deriv IIMaj) n` = `(psiTilde″(√n) − psiTilde′(√n)/√n)/(4n)` — hmm — the chain_rule_sqrt gives `(...)/(4n)` — so the goal is `(psiTilde″ − psiTilde′/√n)/(4n) < 1/(3n)` — the LHS IS a division by 4n ✓ then rw [e] : `X/(4n) < (4/3)/(4n)` where X = the bracket ✓ then div_lt_div_of_pos_right hb h4n : X/(4n) < (4/3)/(4n) ✓ exact ✓.
So the structure is right. The timeouts at whnf/isDefEq — maybe hb's huge expression causes the div_lt_div_of_pos_right's `exact` to loop in isDefEq when unifying X/(4n) with the goal's LHS — the goal's LHS after the rewrites is `(deriv (deriv psiTilde) (√n) − deriv psiTilde (√n)/√n)/(4n)` — and hb : `4.56·(...big...) − 4.56·(...big...)/√n < 4/3` — the X in `X/(4n)` from chain_rule_sqrt is `deriv (deriv psiTilde) (√n) − deriv psiTilde (√n)/√n` — after deriv_psiTilde/deriv2_psiTilde rewrites, X = the psiTilde-deriv closed forms — which after `unfold` inside... — the unification of hb's big form with X might need to unfold psiTilde/iimA/iimB — the whnf loop.
Hmm — but this EXACT structure worked in the ORIGINAL IIMaj_deriv2_lt (with < 1/(2n) and e : 1/(2n) = 2/(4n))! The only differences: e's value (1/(3n) = (4/3)/(4n) vs 1/(2n) = 2/(4n)) and the bracket bound's final constant (4/3 vs 2). The timeouts must be from the elaboration of my new e/hbridge forms... — or something else broke.
Wait — let me look at the ACTUAL error lines: 2190:2 and 2165:0 — line numbers in the CURRENT file. Let me look at what's at those lines.
The timeouts persist — inspecting the exact lines to find what's looping.
tool: Read ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
"offset": 2150,
"limit": 50
}
result 2026-08-26 11:24:29Z → 2026-08-26 11:24:29Z
{
"type": "text",
"file": {
"filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
"content": " have hn0 : (0 : \u211d) < n := by linarith\n have hId : IIMaj = fun m => psiTilde (Real.sqrt m) :=\n funext fun m => IIMaj_eq_psiTilde_sqrt m\n rw [hId, chain_rule_sqrt psiTilde psiTilde_contDiff n hn0, deriv_psiTilde,\n deriv2_psiTilde]\n have hb := iimaj_bracket_bound (Real.sqrt_le_sqrt hn)\n have h4n : (0 : \u211d) < 4 * n := by linarith\n have e : (1 : \u211d) / (3 * n) = (4 / 3) / (4 * n) := by\n have h1 : (0 : \u211d) < 3 * n := by linarith [hn0]\n have h2 : (0 : \u211d) < 4 * n := by linarith [hn0]\n rw [div_eq_div_iff h1.ne' h2.ne']\n ring\n rw [e]\n exact div_lt_div_of_pos_right hb h4n\n\n/-- [PROVED session 011] Margin convexity (D-057, draft \u00a73.1 Step 4):\nfor any twice-differentiable majorant F of the zero count N(0,T] whose\n\u03c8-block \u03c8[F](T) = F\u2033(T) \u2212 F\u2032(T)/(T\u22121) is negative for T \u2265 \u221a20 + 1\n(the D-057 11-block table discharges this for the explicit Trudgian\nmajorant), the margin majorant margin\u0303 is convex on [20, \u221e).\nProof: margin\u0303\u2033(n) = 1/(2n) \u2212 2\u03c8_F/(4n) \u2212 \u03a3w\u1d62\u03c8\u1d62/(4n) \u2212 IIMaj\u2033(n)\n\u2265 1/(2n) \u2212 IIMaj\u2033(n) > 0 (every \u03c8 < 0 by the block table;\nIIMaj\u2033 < 1/(2n) by IIMaj_deriv2_lt).\nTogether with margin\u0303(8\u00d710\u2074) = 16 020 > 0 and margin\u0303\u2032(8\u00d710\u2074) > 0\n(E-028/E-029) this gives margin > 0 on [8\u00d710\u2074, N\u2081]. -/\ntheorem marginTilde_convex (F : \u211d \u2192 \u211d) (hF : ContDiffOn \u211d 2 F (Set.Ioi 1))\n (h\u03c8 : \u2200 T : \u211d, Real.sqrt 20 + 1 \u2264 T \u2192\n deriv (deriv F) T - deriv F T / (T - 1) < 0) :\n ConvexOn \u211d (Set.Ici 20) (marginTilde F) := by\n have hCDfull : ContDiffOn \u211d 2 (marginTilde F) (Set.Ioi 20) := by\n rw [marginTilde_eq_expanded F]\n exact contDiffOn_marginExpanded F hF\n apply convexOn_of_deriv2_nonneg (convex_Ici 20)\n \u00b7 rw [marginTilde_eq_expanded F]\n exact continuousOn_marginExpanded F hF\n \u00b7 rw [interior_Ici]\n exact hCDfull.differentiableOn (by norm_num)\n \u00b7 rw [interior_Ici]\n exact ((hCDfull.deriv_of_isOpen (m := 1) isOpen_Ioi (by norm_num))).differentiableOn\n one_ne_zero\n \u00b7 intro n hn\n rw [interior_Ici] at hn\n have hn20 : 20 < n := hn\n have hn0 : (0 : \u211d) < n := by linarith\n have hn4 : (0 : \u211d) < 4 * n := by linarith\n have hTn : Real.sqrt 20 + 1 \u2264 Real.sqrt n + 1 := by\n have h := Real.sqrt_le_sqrt (show (20 : \u211d) \u2264 n by linarith)\n linarith\n -- diff-fact helpers\n have hdd : \u2200 {P : \u211d \u2192 \u211d}, ContDiffOn \u211d 2 P (Set.Ioi 20) \u2192",
"numLines": 50,
"startLine": 2150,
"totalLines": 2803
}
}
raw content_json (database row)
[{"type":"thinking","thinking":"Still timeouts at 2190:2 (tactic execution) and 2165 (whnf) — despite the div_lt_div_iff₀ fix. The timeout at 2190:2 is the `rw [div_lt_div_iff₀ ..., one_mul, one_mul]` — hmm, 2190:2 — let me look at what's at 2190 now. And 2165's whnf timeout — the `rw [e]` region in lt_three. These timeouts suggest something deeper is looping — maybe the `rw [div_lt_div_iff₀ ...]` with the `by linarith` args is fine but the subsequent elaboration of the goal with the hbridge/hIIM combination loops.\n\nWait — maybe the problem is different: the `rw [e]` at lt_three followed by `exact div_lt_div_of_pos_right hb h4n` — hb is the bracket bound whose statement is a HUGE expression (the 4.56·(...) form with all the exp terms) — div_lt_div_of_pos_right hb h4n : bracket/(4n) < (4/3)/(4n) ✓ — the whnf timeout might be div_lt_div_of_pos_right's unification with the huge bracket expression — or the `rw [e]` with e : 1/(3n) = (4/3)/(4n) rewriting the GOAL `deriv (deriv IIMaj) n < 1/(3n)` — the goal's RHS `1/(3n)` → `(4/3)/(4n)` ✓ then div_lt_div_of_pos_right hb h4n proves `bracket/(4n) < (4/3)/(4n)` — the final unification of `deriv (deriv IIMaj) n` with `bracket/(4n)` — hmm — wait — after the chain_rule_sqrt + deriv_psiTilde + deriv2_psiTilde rewrites, the goal's LHS `deriv (deriv IIMaj) n` became the bracket form ALREADY (the rw [hId, chain_rule_sqrt ..., deriv_psiTilde, deriv2_psiTilde] turned it into the bracket) — then `rw [e]` rewrites `1/(3n)` in the goal's RHS to `(4/3)/(4n)` ✓ then div_lt_div_of_pos_right hb h4n : `bracket/(4n) < (4/3)/(4n)` — and the goal is `bracket/(4n) < (4/3)/(4n)`?? — hmm — actually the goal after the first rws is `bracket-expression < 1/(3n)` — then `rw [e]` changes `1/(3n)` to `(4/3)/(4n)`: `bracket < (4/3)/(4n)` — and div_lt_div_of_pos_right hb h4n gives `bracket/(4n) < (4/3)/(4n)` — the goal has `bracket` NOT `bracket/(4n)`!! — wait — after chain_rule_sqrt, the LHS `deriv (deriv IIMaj) n` = `(psiTilde″(√n) − psiTilde′(√n)/√n)/(4n)` — hmm — the chain_rule_sqrt gives `(...)/(4n)` — so the goal is `(psiTilde″ − psiTilde′/√n)/(4n) < 1/(3n)` — the LHS IS a division by 4n ✓ then rw [e] : `X/(4n) < (4/3)/(4n)` where X = the bracket ✓ then div_lt_div_of_pos_right hb h4n : X/(4n) < (4/3)/(4n) ✓ exact ✓.\n\nSo the structure is right. The timeouts at whnf/isDefEq — maybe hb's huge expression causes the div_lt_div_of_pos_right's `exact` to loop in isDefEq when unifying X/(4n) with the goal's LHS — the goal's LHS after the rewrites is `(deriv (deriv psiTilde) (√n) − deriv psiTilde (√n)/√n)/(4n)` — and hb : `4.56·(...big...) − 4.56·(...big...)/√n < 4/3` — the X in `X/(4n)` from chain_rule_sqrt is `deriv (deriv psiTilde) (√n) − deriv psiTilde (√n)/√n` — after deriv_psiTilde/deriv2_psiTilde rewrites, X = the psiTilde-deriv closed forms — which after `unfold` inside... — the unification of hb's big form with X might need to unfold psiTilde/iimA/iimB — the whnf loop.\n\nHmm — but this EXACT structure worked in the ORIGINAL IIMaj_deriv2_lt (with < 1/(2n) and e : 1/(2n) = 2/(4n))! The only differences: e's value (1/(3n) = (4/3)/(4n) vs 1/(2n) = 2/(4n)) and the bracket bound's final constant (4/3 vs 2). The timeouts must be from the elaboration of my new e/hbridge forms... — or something else broke.\n\nWait — let me look at the ACTUAL error lines: 2190:2 and 2165:0 — line numbers in the CURRENT file. Let me look at what's at those lines.","signature":""},{"type":"text","text":"The timeouts persist — inspecting the exact lines to find what's looping."},{"type":"tool_use","id":"Read_572","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","offset":2150,"limit":50},"extra_content":null}]