[{"type":"thinking","thinking":"Line 4198: `∫ (x : ℝ) in L..2.5024, pinchF' 2.5024 ≤ ∫ (x : ℝ) in L..2.5024, pinchF' x` — the goal has ∫ of the CONSTANT pinchF' 2.5024 — and integral_mono_on's conclusion ∫ f ≤ ∫ g with f = ?m.893 — the ← intervalIntegral.integral_const turned the (b−a)•c into ∫ (pinchF' 2.5024) — as a CONSTANT function ✓ — but the goal is ∫ (const) ≤ ∫ pinchF' — and I applied integral_mono_on with (ContinuousOn.intervalIntegrable continuousOn_const) as hg — hg should be the pinchF'-integrable (the RIGHT side), and hf should be the constant's integrability! I swapped them: integral_mono_on (hab)(hf)(hg)(hfg) : ∫ f ≤ ∫ g — the goal is ∫ const ≤ ∫ pinchF' — so f = const, g = pinchF': hf = const-integrable, hg = pinchF'-integrable. My apply had hf = `ContinuousOn.intervalIntegrable (by mono pinchF'...)` (pinchF'!) and hg = const — SWAPPED. And in the FIRST branch: goal ∫ pinchF' ≤ ∫ (fun _ => 0): f = pinchF', g = const: hf = pinchF'-integrable, hg = const — my first-branch structure had hf = pinchF'-mono ✓ hg = const ✓ correct there — and the first branch's error is different (4166 linarith).\n\nFor 4198 (second branch): swap the two integrability args:\n```\n      apply intervalIntegral.integral_mono_on hcase.le\n        (ContinuousOn.intervalIntegrable continuousOn_const)\n        (ContinuousOn.intervalIntegrable (by\n          apply ContinuousOn.mono (continuousOn_pinchF' hLL)\n          rw [Set.uIcc_of_le hcase.le]\n          exact Set.Icc_subset_Ici_self))\n      · intro x hx\n        ...\n```\n\nFor 4166 (first branch): the linarith failure with h1 not shown — let me look at the actual text at 4160-4167 now. From the read earlier (after my mono_on restructure):\n```\n      · intro x hx\n        rw [Set.mem_Icc] at hx\n        have h1 : pinchF' x ≤ pinchF' 2.5024 := by\n          apply hanti (by linarith [h12le] : (2.5024 : ℝ) ∈ Set.Ici (Real.log 12))\n            (by linarith [hx.1, h12le] : x ∈ Set.Ici (Real.log 12)) hx.1\n        show pinchF' x ≤ (0 : ℝ)\n        linarith [h1, hsign]\n```\nThe linarith at 4167:16 — but the error was at 4166:26 AND 4167:16 (two linariths). 4166:26 is the `show pinchF' x ≤ (0 : ℝ)` line?? no — col 26 of line 4166 is inside h1's apply: `(by linarith [hx.1, h12le] : x ∈ Set.Ici (Real.log 12))` — col 26 ≈ `hx.1, h12le]` — the linarith [hx.1, h12le] for x ∈ Ici (log 12): hx.1 : 2.5024 ≤ x and h12le : log 12 ≤ 2.5024: log 12 ≤ 2.5024 ≤ x ⟹ log 12 ≤ x ✓ linarith [hx.1, h12le] should close... — WAIT — x ∈ Set.Ici (Real.log 12) is a SET MEMBERSHIP: x ∈ Ici (log 12) ⟺ log 12 ≤ x — linarith needs the membership UNPACKED: `by linarith [hx.1, h12le] : x ∈ Set.Ici (Real.log 12)` — the goal is a membership `x ∈ Set.Ici (Real.log 12)` — linarith works on ≤-goals: membership in Ici unfolds to `Real.log 12 ≤ x` — is that automatic for linarith? The goal `x ∈ Set.Ici (Real.log 12)` — linarith can see through Set membership for Ici? Usually needs `Set.mem_Ici.mpr` or `show Real.log 12 ≤ x`. The earlier working versions: `(by linarith [hcase] : (2.5024 : ℝ) ∈ Set.Ici (Real.log 12))` — hmm that also should have the same issue... — the error at 4166:26 IS that inner linarith: `by linarith [hx.1, h12le] : x ∈ Set.Ici (Real.log 12)` — with the context showing ⊢ False from the linarith. So linarith can't see through the Ici membership here?! But wait — earlier in the pinch I had `(by linarith [hcase] : (2.5024 : ℝ) ∈ Set.Ici (Real.log 12))` — and hcase : 2.5024 ≤ L doesn't give log 12 ≤ 2.5024!! — that's what h12le is for — and in THAT spot linarith [hcase] would fail too (no log 12 ≤ 2.5024 relation!). Hmm — but the error reported was at 4166:26 (the x-membership linarith) not the 2.5024-membership one... Set.Ici membership: `x ∈ Set.Ici a` — is it defeq to `a ≤ x`? Set.Ici a = {b | a ≤ b} — membership: x ∈ {b | a ≤ b} = (a ≤ x) ✓ DEFEQ — linarith should see through it via whnf... — linarith does unfold? It preprocesses hypotheses, and the GOAL `x ∈ Set.Ici (Real.log 12)` — maybe linarith doesn't unfold Ici. Use `show Real.log 12 ≤ x` or `Set.mem_Ici.mpr (by linarith [...])`. Fix all such membership-by-linarith spots with explicit `Set.mem_Ici.mpr`:\n```\n(by Set.mem_Ici.mpr (by linarith [hx.1, h12le]) : x ∈ Set.Ici (Real.log 12))\n```\nHmm — actually the cleaner: `show x ∈ Set.Ici (Real.log 12) from by linarith [hx.1, h12le]`? — the linarith inside sees the goal x ∈ Ici — same issue. Let me just: `by have h9 : Real.log 12 ≤ x := by linarith [hx.1, h12le]; exact h9` — the exact h9 : log 12 ≤ x — and the goal x ∈ Ici (log 12) — defeq ✓ exact works (whnf). Or `by linarith [hx.1, h12le] : (2.5024 : ℝ) ≤ x`?? no the goal is membership. Use Set.mem_Ici.mpr — hmm does that exist: `Set.mem_Ici : a ≤ b ↔ a ∈ Set.Ici b`?? — `Set.mem_Ici : a ≤ b ↔ a ∈ Ici b`? — I recall `Set.mem_Ici : a ≤ b ↔ a ∈ Set.Ici b` ✓ — so Set.mem_Ici.mpr (h : log 12 ≤ x) : x ∈ Set.Ici (Real.log 12) ✓.\n\nApply Set.mem_Ici.mpr to the 3 membership spots (2 in branch 1, 1 in branch 2's hfg — the second branch's exact hanti (by linarith [hx.2, h12le] : (2.5024 : ℝ) ∈ Set.Ici (Real.log 12)) (by linarith [hx.1, hL] : x ∈ Set.Ici (Real.log 12)) hx.2 — same membership-linarith issue at both).","signature":""},{"type":"text","text":"Two fixes: swapped integrability args in branch 2, and `Set.mem_Ici.mpr` for the membership goals."},{"type":"tool_use","id":"Bash_299","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1 && python - <<'EOF'\np = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\n\n# membership proofs via Set.mem_Ici.mpr\nsrc = src.replace(\"\"\"          apply hanti (by linarith [h12le] : (2.5024 : ℝ) ∈ Set.Ici (Real.log 12))\n            (by linarith [hx.1, h12le] : x ∈ Set.Ici (Real.log 12)) hx.1\"\"\", \"\"\"          apply hanti (Set.mem_Ici.mpr (by linarith [h12le] : (Real.log 12 : ℝ) ≤ 2.5024))\n            (Set.mem_Ici.mpr (by linarith [hx.1, h12le] : (Real.log 12 : ℝ) ≤ x)) hx.1\"\"\")\n\nsrc = src.replace(\"\"\"        exact hanti (by linarith [hx.2, h12le] : (2.5024 : ℝ) ∈ Set.Ici (Real.log 12))\n          (by linarith [hx.1, hL] : x ∈ Set.Ici (Real.log 12)) hx.2\"\"\", \"\"\"        exact hanti (Set.mem_Ici.mpr (by linarith [hx.2, h12le] : (Real.log 12 : ℝ) ≤ 2.5024))\n          (Set.mem_Ici.mpr (by linarith [hx.1, hL] : (Real.log 12 : ℝ) ≤ x)) hx.2\"\"\")\n\n# branch 2 hbound: swap the integrability args (const first, pinchF' second)\nsrc = src.replace(\"\"\"      apply intervalIntegral.integral_mono_on hcase.le\n        (ContinuousOn.intervalIntegrable (by\n          apply ContinuousOn.mono (continuousOn_pinchF' hLL)\n          rw [Set.uIcc_of_le hcase.le]\n          exact Set.Icc_subset_Ici_self))\n        (ContinuousOn.intervalIntegrable continuousOn_const)\"\"\", \"\"\"      apply intervalIntegral.integral_mono_on hcase.le\n        (ContinuousOn.intervalIntegrable continuousOn_const)\n        (ContinuousOn.intervalIntegrable (by\n          apply ContinuousOn.mono (continuousOn_pinchF' hLL)\n          rw [Set.uIcc_of_le hcase.le]\n          exact Set.Icc_subset_Ici_self))\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"done\")\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiTightR.lean 2>&1 | head -30","description":"Fix memberships and swapped integrability args; recompile","timeout":420000},"extra_content":null}]