[{"type":"thinking","thinking":"Now the contDiffOn_m* lemmas — update each to the k-shift infrastructure:\n\n- contDiffOn_mFblock: `contDiffOn_const.mul (hF.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift_Ioi1)` → `contDiffOn_const.mul (hF.comp (contDiffOn_ksqrt_shift 1.5 (by norm_num)) (mapsTo_ksqrt_shift_Ioi1 1.5 (by norm_num)))`\n- contDiffOn_mTL1b: `contDiffOn_const.mul (contDiffOn_TL1.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)` → with k-versions.\n- mTL2b, mL1sqb, mL1L2b, mL2sqb, mL1b, mL2b similarly.\n- contDiffOn_mIdb: `contDiffOn_const.mul contDiffOn_sqrt_shift` → `contDiffOn_const.mul (contDiffOn_ksqrt_shift 1.5 (by norm_num))`.\n- contDiffOn_mIIM: unchanged (mIIM's body changed but the proof is constant-agnostic — `unfold mIIM; rw [hId]; exact contDiffOn_const.mul hcomp` — hmm wait, the proof at 1886–1897: `unfold mIIM; rw [hId]; exact contDiffOn_const.mul hcomp` — hId : IIMaj = fun m => psiTilde (√m): after rw, goal ContDiffOn (fun m => (−1.5)·psiTilde(√m))?? — hmm, rw [hId] rewrites IIMaj occurrences: the goal after unfold mIIM is ContDiffOn ℝ 2 (fun m => (-1.5) * IIMaj m) (Set.Ioi 20) — rw [hId] rewrites IIMaj (the constant) to the lambda — goal becomes ContDiffOn (fun m => (-1.5)*(fun m => psiTilde (√m)) m)... beta-redex — the old proof handled it (worked with −1). Should work identically. ✓ No change needed.","signature":""},{"type":"text","text":"Now updating the block contDiffOn lemmas to the k-shift infrastructure."},{"type":"tool_use","id":"Edit_86","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"theorem contDiffOn_mFblock (F : ℝ → ℝ) (hF : ContDiffOn ℝ 2 F (Set.Ioi 1)) :\n    ContDiffOn ℝ 2 (mFblock F) (Set.Ioi 20) :=\n  contDiffOn_const.mul\n    (hF.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift_Ioi1)\n\ntheorem contDiffOn_mTL1b : ContDiffOn ℝ 2 mTL1b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_TL1.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mTL2b : ContDiffOn ℝ 2 mTL2b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_TL2.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mIdb : ContDiffOn ℝ 2 mIdb (Set.Ioi 20) :=\n  contDiffOn_const.mul contDiffOn_sqrt_shift\n\ntheorem contDiffOn_mL1sqb : ContDiffOn ℝ 2 mL1sqb (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L1sq.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mL1L2b : ContDiffOn ℝ 2 mL1L2b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L1L2.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mL2sqb : ContDiffOn ℝ 2 mL2sqb (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L2sq.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mL1b : ContDiffOn ℝ 2 mL1b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L1.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)\n\ntheorem contDiffOn_mL2b : ContDiffOn ℝ 2 mL2b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L2.comp contDiffOn_sqrt_shift mapsTo_sqrt_shift)","new_string":"theorem contDiffOn_mFblock (F : ℝ → ℝ) (hF : ContDiffOn ℝ 2 F (Set.Ioi 1)) :\n    ContDiffOn ℝ 2 (mFblock F) (Set.Ioi 20) :=\n  contDiffOn_const.mul\n    (hF.comp (contDiffOn_ksqrt_shift 1.5 (by norm_num))\n      (mapsTo_ksqrt_shift_Ioi1 1.5 (by norm_num)))\n\ntheorem contDiffOn_mTL1b : ContDiffOn ℝ 2 mTL1b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_TL1.comp (contDiffOn_ksqrt_shift 1.5 (by norm_num))\n    (mapsTo_ksqrt_shift 1.5 (by norm_num)))\n\ntheorem contDiffOn_mTL2b : ContDiffOn ℝ 2 mTL2b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_TL2.comp (contDiffOn_ksqrt_shift 1.5 (by norm_num))\n    (mapsTo_ksqrt_shift 1.5 (by norm_num)))\n\ntheorem contDiffOn_mIdb : ContDiffOn ℝ 2 mIdb (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_ksqrt_shift 1.5 (by norm_num))\n\ntheorem contDiffOn_mL1sqb : ContDiffOn ℝ 2 mL1sqb (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L1sq.comp (contDiffOn_ksqrt_shift 1.5 (by norm_num))\n    (mapsTo_ksqrt_shift 1.5 (by norm_num)))\n\ntheorem contDiffOn_mL1L2b : ContDiffOn ℝ 2 mL1L2b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L1L2.comp (contDiffOn_ksqrt_shift 1.5 (by norm_num))\n    (mapsTo_ksqrt_shift 1.5 (by norm_num)))\n\ntheorem contDiffOn_mL2sqb : ContDiffOn ℝ 2 mL2sqb (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L2sq.comp (contDiffOn_ksqrt_shift 1.5 (by norm_num))\n    (mapsTo_ksqrt_shift 1.5 (by norm_num)))\n\ntheorem contDiffOn_mL1b : ContDiffOn ℝ 2 mL1b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L1.comp (contDiffOn_ksqrt_shift 1.5 (by norm_num))\n    (mapsTo_ksqrt_shift 1.5 (by norm_num)))\n\ntheorem contDiffOn_mL2b : ContDiffOn ℝ 2 mL2b (Set.Ioi 20) :=\n  contDiffOn_const.mul (contDiffOn_L2.comp (contDiffOn_ksqrt_shift 1.5 (by norm_num))\n    (mapsTo_ksqrt_shift 1.5 (by norm_num)))"},"extra_content":null},{"type":"tool_use","id":"Edit_87","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"  have hsqrt : ContinuousOn Real.sqrt (Set.Ici 20) := Real.continuous_sqrt.continuousOn\n  have hT : ContinuousOn (fun m => Real.sqrt m + 1) (Set.Ici 20) :=\n    hsqrt.add continuousOn_const\n  have hmaps : Set.MapsTo (fun m => Real.sqrt m + 1) (Set.Ici 20) (Set.Ioi (-1)) := by\n    intro x hx\n    have h : 20 ≤ x := hx\n    have hnn := Real.sqrt_nonneg x\n    exact Set.mem_Ioi.mpr (by linarith)","new_string":"  have hsqrt : ContinuousOn Real.sqrt (Set.Ici 20) := Real.continuous_sqrt.continuousOn\n  have hT : ContinuousOn (fun m => 1.5 * Real.sqrt m + 1) (Set.Ici 20) :=\n    (hsqrt.const_mul 1.5).add continuousOn_const\n  have hmaps : Set.MapsTo (fun m => 1.5 * Real.sqrt m + 1) (Set.Ici 20) (Set.Ioi (-1)) := by\n    intro x hx\n    have h : 20 ≤ x := hx\n    have hnn := Real.sqrt_nonneg x\n    exact Set.mem_Ioi.mpr (by linarith [mul_pos (by norm_num : (0:ℝ) < 1.5)\n      (Real.sqrt_pos.mpr (by linarith : (0:ℝ) < x))])"},"extra_content":null}]