[{"type":"thinking","thinking":"Line 1101-1104: in the y = s branch, after `subst hye` — wait: hye : y = s; `subst hye` — subst with hye : y = s eliminates... s is the lemma's outer variable — subst can only eliminate a local hypothesis variable, and s IS a local variable (the lemma's argument) — but subst hye : y = s tries to eliminate y (a local) or s (a local) — if it eliminates S (rewriting s := y everywhere), then all my later references to `s` are unknown! The error: unknown identifier s after subst — so subst eliminated s, replacing it with y. Then `𝓝 s` etc. fail. Fix: use `subst y` explicitly? `subst hye` picks based on the equation orientation... For hye : y = s, `subst hye` eliminates y if possible (y local ✓ — subst prefers... apparently it eliminated s). Use `subst y`?? subst takes the hypothesis or variable name: `subst y` eliminates y (needs y = s or s = y: hye : y = s ✓). Hmm — actually `subst hye` where hye : y = s: eliminates the VARIABLE y by default (subst h with h : x = t eliminates x)... it eliminated s instead?? The errors show s unknown — so s was eliminated. Because... both are locals; Lean's subst h with h : y = s tries to eliminate the LHS variable y — but maybe y couldn't be eliminated (y is... y IS local). Hmm whatever — the safe fix: don't subst; instead work with y and convert at the end. Replace `subst hye` with `rw [hye]`-style... The branch goal is Gentire y = W y with hye : y = s: I want to show Gentire y = W y via the limUnder at s. Restructure: \n```lean\n  by_cases hye : y = s\n  · -- the center: limUnder = W s\n    haveI : ... (no haveI needed)\n    have hWcont : Tendsto W (𝓝[≠] s) (𝓝 (W s)) := hW.continuousAt.mono_left nhdsWithin_le_nhds\n    have hlim : Tendsto Fexpr (𝓝[≠] s) (𝓝 (W s)) := Tendsto.congr' hWeq.symm hWcont\n    have hval : limUnder (𝓝[≠] s) Fexpr = W s :=\n      tendsto_nhds_unique (tendsto_nhds_limUnder ⟨W s, hlim⟩) hlim\n    rw [hye]\n    exact hval  -- goal Gentire s = W s; hval : limUnder ... = W s: defeq Gentire s ≡ limUnder ✓\n```\nrw [hye] : goal Gentire y = W y with hye : y = s: rewrites y → s: goal Gentire s = W s ✓ then hval (defeq ascription ✓). But wait — hye : y = s: rw [hye] rewrites y := s in the goal ✓ safe (no dependency issues for the goal).\n\nAlso the hWeq.symm error: `hWeq.symm` — hWeq's TYPE was shown as `∃ a ∈ 𝓝 y, ...` — because after the FIRST compile error (unknown s), hWeq got mis-elaborated... no — hWeq : ∀ᶠ y in 𝓝[≠] s, Fexpr y = W y is fine; the error \"Exists.symm\" at hWeq — because hWeq got shadowed/mis-scoped after the subst removed s... The hWeq in context printed as the weird ∃-form — that's `Filter.EventuallyEq` UNFOLDED display? EventuallyEq f l g := ∀ᶠ x in l, f x = g x — the displayed `∃ a ∈ 𝓝 y, ∃ b ∈ 𝓟 {y}ᶜ, ...` is the Eventually (𝓝[≠]) unfolded — odd but that's just the display of the filter membership. The `.symm` failing with Exists.symm means hWeq elaborated to an ∃ — because... the subst chaos. Fix the subst and it should resolve.\n\nAlso `hWeq.symm` — EventuallyEq.symm : f =ᶠ[l] g → g =ᶠ[l] f ✓ exists.","signature":""},{"type":"text","text":"The `subst hye` eliminated `s` instead of `y`. Fixing with a direct rewrite approach."},{"type":"tool_use","id":"Edit_214","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiFEntire.lean","old_string":"  by_cases hye : y = s\n  · subst hye\n    have hWcont : Tendsto W (𝓝[≠] s) (𝓝 (W s)) :=\n      hW.continuousAt.mono_left nhdsWithin_le_nhds\n    have hlim : Tendsto Fexpr (𝓝[≠] s) (𝓝 (W s)) := Tendsto.congr' hWeq.symm hWcont\n    show limUnder (𝓝[≠] s) Fexpr = W s\n    exact tendsto_nhds_unique (tendsto_nhds_limUnder ⟨W s, hlim⟩) hlim\n  · have hygood : ¬ badPoint y := hgood y hδ₂y hye\n    rw [gentire_eq_Fexpr hygood]\n    exact hWeqBall y hε₁y hye","new_string":"  by_cases hye : y = s\n  · have hWcont : Tendsto W (𝓝[≠] s) (𝓝 (W s)) :=\n      hW.continuousAt.mono_left nhdsWithin_le_nhds\n    have hlim : Tendsto Fexpr (𝓝[≠] s) (𝓝 (W s)) := Tendsto.congr' hWeq.symm hWcont\n    have hval : limUnder (𝓝[≠] s) Fexpr = W s :=\n      tendsto_nhds_unique (tendsto_nhds_limUnder ⟨W s, hlim⟩) hlim\n    rw [hye]\n    exact hval\n  · have hygood : ¬ badPoint y := hgood y hδ₂y hye\n    rw [gentire_eq_Fexpr hygood]\n    exact hWeqBall y hε₁y hye"},"extra_content":null}]