[{"type":"thinking","thinking":"Now I see: the goal after ring-normalization has the RHS containing `((fun t => ...) + fun t => ...) x` — a Pi.add APPLICATION not reduced: `((fun t => exp(...)) + fun t => exp(...)) x` vs LHS's expanded `exp(...) + exp(...)`. The HasDerivAt.add produced the SUM function as Pi.add `(f + g)` — and hva.add hvb's function is `f + g` (Pi.add), while hvb/hva are lambdas. So in h, the function is `(fun t => t+1) * ((fun t => eA) + fun t => eB)` — Pi instances — and the goal wants pointwise. The ring tactic normalized everything except that Pi.add application (it treats ((f+g) x) as an atom!). So the value-equality isn't ring-provable because one side has the unreduced application atom.\n\nFix: beta/Pi reduce h before convert: add `simp only [Pi.add_apply, Pi.mul_apply] at h`? The Pi.add instance: `(f + g) x` reduces via `Pi.add_apply` simp lemma ✓. And similarly the function parts. So after building h: `simp only [Pi.add_apply, Pi.mul_apply] at h` — that turns (f+g) x → f x + g x everywhere in h (both function-lambda body applications... wait the Pi.add is INSIDE a lambda: fun y => 4.56 * ((f + g) y) — simp only [Pi.add_apply] rewrites (f+g) y → f y + g y ✓ under binders ✓.\n\nAlso HasDerivAt.mul's function: (f * g) as Pi.mul too ✓ same treatment. Then convert h using 2 — functions match syntactically (hopefully) or via ring-free rfl; value goal → ring.\n\nAlternatively skip convert: after simp, `exact h`? The derivative value still needs algebra massage (1*(eA+eB) + (x+1)*(eA*−a + eB*−b) = target). So keep convert+ring after the simp.\n\nSame issue likely in deriv2_psiTilde's h2 (hw.add hz, hu.mul hg, .const_mul) → add simp only [Pi.add_apply, Pi.mul_apply, Pi.sub_apply] at h there too. hg used .sub → Pi.sub_apply.\n\nAlso error 987: rw [Real.log_pow]; ring. And 1002: fix the inner-expand show.\n\nApply all fixes.","signature":""},{"type":"text","text":"The issue: `HasDerivAt.add/mul` leave `Pi.add`/`Pi.mul` applications that `ring` treats as atoms. Fix with `simp only [Pi.add_apply, ...]`, plus the other two errors."},{"type":"tool_use","id":"Edit_168","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"    have e1 : (2 : ℝ) ^ (2 : ℝ) = 4 := by\n      rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2),\n        show Real.log 2 * 2 = Real.log (2 ^ 2) by rw [Real.log_pow]; norm_num,\n        Real.exp_log (by norm_num : (0 : ℝ) < 2 ^ 2)]\n      norm_num\n    rw [e2, e1]","new_string":"    have e1 : (2 : ℝ) ^ (2 : ℝ) = 4 := by\n      rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2),\n        show Real.log 2 * 2 = Real.log (2 ^ 2) by rw [Real.log_pow]; ring,\n        Real.exp_log (by norm_num : (0 : ℝ) < 2 ^ 2)]\n      norm_num\n    rw [e2, e1]"},"extra_content":null}]