← Session 21 — Tier ii′ planning + launch: close SfBoundProp in Lean (D-064)
thinking (6300 chars)
Progress but still issues:
1. `Neg ℕ` — in my `show` I wrote `-(↑i - 1)` where ↑i : ℤ?? — the `show` I wrote has `fun w : ℂ => -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-(↑i - 1))` — the exponents: `-(↑i - 1)` — the ↑i got elaborated as ℕ (the Gprim def's `(-((i : ℤ) - 1))` — in my show I wrote `(-(↑i - 1))` — the ↑i is... i : ℕ, `↑i` with target ℤ from the zpow — but the `-(↑i - 1)` — the `1` and `-` might elaborate in ℕ! In the Gprim def I wrote `(w - 1) ^ (-((i : ℤ) - 1))` — explicit (i : ℤ). In my show I wrote `(-(↑i - 1))` without the type ascription — elaborates in ℕ (Neg ℕ fails). Fix: use `(-((i : ℤ) - 1))` exactly as in Gprim's def.
2. The `show` defeq failure: target `DifferentiableAt ℂ (fun w => ∑ i, f i w) w` vs my show `DifferentiableAt ℂ (∑ i, fun w => f i w) w` — NOT definitionally equal (the sum is over functions vs the lambda outside). Finset.sum_apply : (∑ i ∈ s, f i) x = ∑ i ∈ s, f i x — as functions: (fun w => ∑ i, f i w) vs (∑ i, fun w => f i w) — these ARE equal by Finset.sum_apply pointwise, but as function TERMS they differ — `show` needs defeq — funext-level, not defeq. So `show` fails. Instead: rewrite the goal first with `simp only [Finset.sum_apply]`? — the goal `DifferentiableAt ℂ (fun w => ∑ i, f i w) w` — simp only [Finset.sum_apply] rewrites `(∑ i, f i) w` → `∑ i, f i w` — the goal already has the applied form... Hmm — the issue: after `unfold Gprim`, the goal is `DifferentiableAt ℂ (fun w => ∑ i ∈ Icc 2 (n+1), f i w) w`?? NO — Gprim n w = ∑ i, f i w — the goal is DifferentiableAt ℂ (Gprim n) w — unfold Gprim gives DifferentiableAt ℂ (fun w => ∑ i ∈ ..., f i w) w? — unfold Gprim in the GOAL `DifferentiableAt ℂ (Gprim n) w`: Gprim n is a FUNCTION ℂ → ℂ; unfold gives `DifferentiableAt ℂ (fun w => ∑ i ∈ ..., f i w) w`?? — Gprim n = fun w => ∑ i ∈ ..., (fun w => ...) — wait Gprim's def: `Gprim (n : ℕ) (w : ℂ) : ℂ := ∑ i ∈ ..., -(A_i/(i−1)) * (w−1)^...` — so Gprim n = fun w => ∑ i, term i w — after unfold, goal: DifferentiableAt ℂ (fun w => ∑ i, term i w) w. And DifferentiableAt.sum expects `DifferentiableAt 𝕜 (∑ i ∈ u, A i) x` where A i : E → F — unify `∑ i ∈ u, ?A i` with `fun w => ∑ i, term i w` — eta: ?A i := fun w => term i w ✓ — the unifier FAILED earlier because of the form... The robust way: `apply DifferentiableAt.sum _ (fun i hi => ?_)`? no — the issue is matching `∑ i ∈ u, A i` with `fun w => ∑ i, f i w` — higher-order pattern: `∑ i ∈ u, A i` (a function) vs `fun w => ∑ i, f i w` — unifier sets A i := fun w => f i w ✓ should work via eta... it didn't. Use `Finset.sum_apply` to convert the goal's lambda into the sum-of-functions form FIRST:
```
unfold Gprim
rw [show (fun w : ℂ => ∑ i ∈ Finset.Icc 2 (n + 1), -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-((i : ℤ) - 1))) =
∑ i ∈ Finset.Icc 2 (n + 1), (fun w : ℂ => -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-((i : ℤ) - 1))) from by
funext w
simp [Finset.sum_apply]]
apply DifferentiableAt.sum
intro i hi
...
```
`Finset.sum_apply : (∑ i ∈ s, f i) x = ∑ i ∈ s, f i x` — funext: (fun w => ∑ i, f i w) = (∑ i, fun w => f i w)? — funext w: goal (fun w => ∑ i, f i w) w = (∑ i, fun w => f i w) w — LHS beta → ∑ i, f i w; RHS → ∑ i, (fun w => f i w) w = ∑ i, f i w ✓ — simp [Finset.sum_apply] normalizes ✓.
Hmm — actually wait — is Gprim n itself (as a function) already in the "sum of functions" form? `Gprim n` unfolds to `fun w => ∑ i ∈ ..., -(Acoeff n i / (↑i-1)) * (w-1)^...` — the body is ∑ i, (term with w) — so as a function it's `fun w => ∑ i, term i w` — to apply DifferentiableAt.sum I need it as `∑ i, (fun w => term i w)` — same thing via sum_apply. OK.
3. tail empty case: `ContinuousAt (fun w => 0) w` — after simp at empty: goal `ContinuousAt (fun w => ∑ i ∈ ∅, ...) w` = ContinuousAt (fun w => 0) w — `simp` reduced to that but didn't close: `continuousAt_const` — add it: `| empty => simp [continuousAt_const]`? — `simp` should close ContinuousAt (fun w => 0) w via continuousAt_const — it didn't. Use `exact continuousAt_const`.
4. `Finset.sum_insert has` — the `has` got type ℕ (rename_i a s renamed things: `has` is now the a ∉ s proof? The error: `has` has type ℕ — the rename_i a s renamed the WRONG variables: in `Finset.induction`, the insert case binders are (a : α) (s : Finset α) (has : a ∉ s) (hs : p s) — my `| insert has hs =>` pattern: has := a (the element!), hs := s — WRONG order. The case pattern: `| insert a s has hs` — wait the induction's insert case: `∀ ⦃a : α⦄ {s : Finset α}, a ∉ s → p s → p (insert a s)` — binders: a, s, has, hs — pattern `| insert has hs =>` binds has := a, hs := s?? — with two names for four binders, Lean assigns has := a, hs := s, and the remaining two get inaccessible names ✓ that's why `has` was ℕ. Fix: `| insert a s has hs =>` and use has for ∉ and hs for the IH:
```
| insert a s has hs =>
rw [Finset.sum_insert has]
exact ContinuousAt.add hs (hterm a)
```
And remove rename_i.
5. `Set.sdiff_subset_sdiff_left (Set.subset_univ ?m)` — gives `?m \ ?r ⊆ univ \ ?r` — I want Rectangle \ {1} ⊆ {1}ᶜ — {1}ᶜ = univ \ {1} ✓ — so I need sdiff_subset_sdiff_left (h : Rectangle ⊆ univ) : Rectangle \ {1} ⊆ univ \ {1} ✓ — the expected type `z.Rectangle w \ {1} ⊆ {1}ᶜ` — {1}ᶜ vs univ \ {1} — sdiff notation: {1}ᶜ is compl, univ \ {1} is sdiff — same set, different head — convert: `Set.sdiff_subset_sdiff_left (Set.subset_univ _)` produces `z.Rectangle w \ {1} ⊆ Set.univ \ {1}` — need `⊆ {1}ᶜ` — univ \ {1} = {1}ᶜ defeq? `Set.compl_eq_univ_diff`? — use `show z.Rectangle w \ {1} ⊆ {1}ᶜ; exact ...`? Or `.mono` with the univ-diff version — DifferentiableOn.mono expects `z.Rectangle w \ {1} ⊆ {1}ᶜ` — give: `Set.subset_univ _ |> Set.sdiff_subset_sdiff_left` produces univ\{1} — hmm — write it as: `(Set.sdiff_subset_sdiff_left (Set.subset_univ _))` with expected type via `show`:
```
exact hopen.mono (t := z.Rectangle w \ {1}) (by
intro s hs
exact ⟨Set.mem_univ _, hs.2⟩)
```
Cleaner: prove the subset by hand: Rectangle \ {1} ⊆ {1}ᶜ: intro s hs; exact hs.2 — hs : s ∈ Rectangle \ {1} — hs.2 : s ∉ {1} — {1}ᶜ membership: s ∈ {1}ᶜ ⟺ s ∉ {1} ✓ `exact hs.2`. So:
```
exact hopen.mono (by intro s hs; exact hs.2)
```
`DifferentiableOn.mono (h : DifferentiableOn f t) (hs : s ⊆ t) : DifferentiableOn f s` — hopen.mono (subset-proof) ✓.
Let me apply all these fixes.
Several structural fixes needed. Applying them now.
tool: Bash
input
{
"command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. Gprim_differentiableAt: proper function-form conversion\ns = s.replace(\"\"\" unfold Gprim\n show DifferentiableAt \u2102 (\u2211 i \u2208 Finset.Icc 2 (n + 1),\n fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-(\u2191i - 1))) w\n apply DifferentiableAt.sum\n intro i hi\n apply DifferentiableAt.mul (differentiableAt_const _)\"\"\",\n\"\"\" unfold Gprim\n rw [show (fun w : \u2102 => \u2211 i \u2208 Finset.Icc 2 (n + 1),\n -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-((i : \u2124) - 1))) =\n \u2211 i \u2208 Finset.Icc 2 (n + 1),\n (fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-((i : \u2124) - 1))) from by\n funext w\n simp [Finset.sum_apply]]\n apply DifferentiableAt.sum\n intro i hi\n apply DifferentiableAt.mul (differentiableAt_const _)\"\"\")\n\n# 2. deriv_Gprim: same conversion\ns = s.replace(\"\"\" show deriv (\u2211 i \u2208 Finset.Icc 2 (n + 1),\n fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-(\u2191i - 1))) w = _\n rw [deriv_sum (fun i hi => ?_)]\"\"\",\n\"\"\" unfold Gprim\n rw [show (fun w : \u2102 => \u2211 i \u2208 Finset.Icc 2 (n + 1),\n -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-((i : \u2124) - 1))) =\n \u2211 i \u2208 Finset.Icc 2 (n + 1),\n (fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-((i : \u2124) - 1))) from by\n funext w\n simp [Finset.sum_apply]]\n rw [deriv_sum (fun i hi => ?_)]\"\"\")\n\n# 3. tail induction fixes\ns = s.replace(\"\"\" induction s using Finset.induction with\n | empty => simp\n | insert has hs =>\n rename_i a s\n rw [Finset.sum_insert has]\n exact ContinuousAt.add hs (hterm a)\n exact this _\"\"\",\n\"\"\" induction s using Finset.induction with\n | empty => exact continuousAt_const\n | insert a s has hs =>\n rw [Finset.sum_insert has]\n exact ContinuousAt.add hs (hterm a)\n exact this _\"\"\")\n\n# 4. subset by hand\ns = s.replace(\"exact hopen.mono (Set.sdiff_subset_sdiff_left (Set.subset_univ _))\",\n \"exact hopen.mono (by intro s hs; exact hs.2)\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
"description": "Apply structural fixes and recompile",
"timeout": 420000
}
raw content_json (database row)
[{"type":"thinking","thinking":"Progress but still issues:\n\n1. `Neg ℕ` — in my `show` I wrote `-(↑i - 1)` where ↑i : ℤ?? — the `show` I wrote has `fun w : ℂ => -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-(↑i - 1))` — the exponents: `-(↑i - 1)` — the ↑i got elaborated as ℕ (the Gprim def's `(-((i : ℤ) - 1))` — in my show I wrote `(-(↑i - 1))` — the ↑i is... i : ℕ, `↑i` with target ℤ from the zpow — but the `-(↑i - 1)` — the `1` and `-` might elaborate in ℕ! In the Gprim def I wrote `(w - 1) ^ (-((i : ℤ) - 1))` — explicit (i : ℤ). In my show I wrote `(-(↑i - 1))` without the type ascription — elaborates in ℕ (Neg ℕ fails). Fix: use `(-((i : ℤ) - 1))` exactly as in Gprim's def.\n\n2. The `show` defeq failure: target `DifferentiableAt ℂ (fun w => ∑ i, f i w) w` vs my show `DifferentiableAt ℂ (∑ i, fun w => f i w) w` — NOT definitionally equal (the sum is over functions vs the lambda outside). Finset.sum_apply : (∑ i ∈ s, f i) x = ∑ i ∈ s, f i x — as functions: (fun w => ∑ i, f i w) vs (∑ i, fun w => f i w) — these ARE equal by Finset.sum_apply pointwise, but as function TERMS they differ — `show` needs defeq — funext-level, not defeq. So `show` fails. Instead: rewrite the goal first with `simp only [Finset.sum_apply]`? — the goal `DifferentiableAt ℂ (fun w => ∑ i, f i w) w` — simp only [Finset.sum_apply] rewrites `(∑ i, f i) w` → `∑ i, f i w` — the goal already has the applied form... Hmm — the issue: after `unfold Gprim`, the goal is `DifferentiableAt ℂ (fun w => ∑ i ∈ Icc 2 (n+1), f i w) w`?? NO — Gprim n w = ∑ i, f i w — the goal is DifferentiableAt ℂ (Gprim n) w — unfold Gprim gives DifferentiableAt ℂ (fun w => ∑ i ∈ ..., f i w) w? — unfold Gprim in the GOAL `DifferentiableAt ℂ (Gprim n) w`: Gprim n is a FUNCTION ℂ → ℂ; unfold gives `DifferentiableAt ℂ (fun w => ∑ i ∈ ..., f i w) w`?? — Gprim n = fun w => ∑ i ∈ ..., (fun w => ...) — wait Gprim's def: `Gprim (n : ℕ) (w : ℂ) : ℂ := ∑ i ∈ ..., -(A_i/(i−1)) * (w−1)^...` — so Gprim n = fun w => ∑ i, term i w — after unfold, goal: DifferentiableAt ℂ (fun w => ∑ i, term i w) w. And DifferentiableAt.sum expects `DifferentiableAt 𝕜 (∑ i ∈ u, A i) x` where A i : E → F — unify `∑ i ∈ u, ?A i` with `fun w => ∑ i, term i w` — eta: ?A i := fun w => term i w ✓ — the unifier FAILED earlier because of the form... The robust way: `apply DifferentiableAt.sum _ (fun i hi => ?_)`? no — the issue is matching `∑ i ∈ u, A i` with `fun w => ∑ i, f i w` — higher-order pattern: `∑ i ∈ u, A i` (a function) vs `fun w => ∑ i, f i w` — unifier sets A i := fun w => f i w ✓ should work via eta... it didn't. Use `Finset.sum_apply` to convert the goal's lambda into the sum-of-functions form FIRST:\n```\n unfold Gprim\n rw [show (fun w : ℂ => ∑ i ∈ Finset.Icc 2 (n + 1), -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-((i : ℤ) - 1))) =\n ∑ i ∈ Finset.Icc 2 (n + 1), (fun w : ℂ => -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-((i : ℤ) - 1))) from by\n funext w\n simp [Finset.sum_apply]]\n apply DifferentiableAt.sum\n intro i hi\n ...\n```\n`Finset.sum_apply : (∑ i ∈ s, f i) x = ∑ i ∈ s, f i x` — funext: (fun w => ∑ i, f i w) = (∑ i, fun w => f i w)? — funext w: goal (fun w => ∑ i, f i w) w = (∑ i, fun w => f i w) w — LHS beta → ∑ i, f i w; RHS → ∑ i, (fun w => f i w) w = ∑ i, f i w ✓ — simp [Finset.sum_apply] normalizes ✓.\n\nHmm — actually wait — is Gprim n itself (as a function) already in the \"sum of functions\" form? `Gprim n` unfolds to `fun w => ∑ i ∈ ..., -(Acoeff n i / (↑i-1)) * (w-1)^...` — the body is ∑ i, (term with w) — so as a function it's `fun w => ∑ i, term i w` — to apply DifferentiableAt.sum I need it as `∑ i, (fun w => term i w)` — same thing via sum_apply. OK.\n\n3. tail empty case: `ContinuousAt (fun w => 0) w` — after simp at empty: goal `ContinuousAt (fun w => ∑ i ∈ ∅, ...) w` = ContinuousAt (fun w => 0) w — `simp` reduced to that but didn't close: `continuousAt_const` — add it: `| empty => simp [continuousAt_const]`? — `simp` should close ContinuousAt (fun w => 0) w via continuousAt_const — it didn't. Use `exact continuousAt_const`.\n\n4. `Finset.sum_insert has` — the `has` got type ℕ (rename_i a s renamed things: `has` is now the a ∉ s proof? The error: `has` has type ℕ — the rename_i a s renamed the WRONG variables: in `Finset.induction`, the insert case binders are (a : α) (s : Finset α) (has : a ∉ s) (hs : p s) — my `| insert has hs =>` pattern: has := a (the element!), hs := s — WRONG order. The case pattern: `| insert a s has hs` — wait the induction's insert case: `∀ ⦃a : α⦄ {s : Finset α}, a ∉ s → p s → p (insert a s)` — binders: a, s, has, hs — pattern `| insert has hs =>` binds has := a, hs := s?? — with two names for four binders, Lean assigns has := a, hs := s, and the remaining two get inaccessible names ✓ that's why `has` was ℕ. Fix: `| insert a s has hs =>` and use has for ∉ and hs for the IH:\n```\n | insert a s has hs =>\n rw [Finset.sum_insert has]\n exact ContinuousAt.add hs (hterm a)\n```\nAnd remove rename_i.\n\n5. `Set.sdiff_subset_sdiff_left (Set.subset_univ ?m)` — gives `?m \\ ?r ⊆ univ \\ ?r` — I want Rectangle \\ {1} ⊆ {1}ᶜ — {1}ᶜ = univ \\ {1} ✓ — so I need sdiff_subset_sdiff_left (h : Rectangle ⊆ univ) : Rectangle \\ {1} ⊆ univ \\ {1} ✓ — the expected type `z.Rectangle w \\ {1} ⊆ {1}ᶜ` — {1}ᶜ vs univ \\ {1} — sdiff notation: {1}ᶜ is compl, univ \\ {1} is sdiff — same set, different head — convert: `Set.sdiff_subset_sdiff_left (Set.subset_univ _)` produces `z.Rectangle w \\ {1} ⊆ Set.univ \\ {1}` — need `⊆ {1}ᶜ` — univ \\ {1} = {1}ᶜ defeq? `Set.compl_eq_univ_diff`? — use `show z.Rectangle w \\ {1} ⊆ {1}ᶜ; exact ...`? Or `.mono` with the univ-diff version — DifferentiableOn.mono expects `z.Rectangle w \\ {1} ⊆ {1}ᶜ` — give: `Set.subset_univ _ |> Set.sdiff_subset_sdiff_left` produces univ\\{1} — hmm — write it as: `(Set.sdiff_subset_sdiff_left (Set.subset_univ _))` with expected type via `show`: \n```\n exact hopen.mono (t := z.Rectangle w \\ {1}) (by\n intro s hs\n exact ⟨Set.mem_univ _, hs.2⟩)\n```\nCleaner: prove the subset by hand: Rectangle \\ {1} ⊆ {1}ᶜ: intro s hs; exact hs.2 — hs : s ∈ Rectangle \\ {1} — hs.2 : s ∉ {1} — {1}ᶜ membership: s ∈ {1}ᶜ ⟺ s ∉ {1} ✓ `exact hs.2`. So:\n```\n exact hopen.mono (by intro s hs; exact hs.2)\n```\n`DifferentiableOn.mono (h : DifferentiableOn f t) (hs : s ⊆ t) : DifferentiableOn f s` — hopen.mono (subset-proof) ✓.\n\nLet me apply all these fixes.","signature":""},{"type":"text","text":"Several structural fixes needed. Applying them now."},{"type":"tool_use","id":"Bash_271","name":"Bash","input":{"command":"cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. Gprim_differentiableAt: proper function-form conversion\ns = s.replace(\"\"\" unfold Gprim\n show DifferentiableAt ℂ (∑ i ∈ Finset.Icc 2 (n + 1),\n fun w : ℂ => -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-(↑i - 1))) w\n apply DifferentiableAt.sum\n intro i hi\n apply DifferentiableAt.mul (differentiableAt_const _)\"\"\",\n\"\"\" unfold Gprim\n rw [show (fun w : ℂ => ∑ i ∈ Finset.Icc 2 (n + 1),\n -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-((i : ℤ) - 1))) =\n ∑ i ∈ Finset.Icc 2 (n + 1),\n (fun w : ℂ => -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-((i : ℤ) - 1))) from by\n funext w\n simp [Finset.sum_apply]]\n apply DifferentiableAt.sum\n intro i hi\n apply DifferentiableAt.mul (differentiableAt_const _)\"\"\")\n\n# 2. deriv_Gprim: same conversion\ns = s.replace(\"\"\" show deriv (∑ i ∈ Finset.Icc 2 (n + 1),\n fun w : ℂ => -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-(↑i - 1))) w = _\n rw [deriv_sum (fun i hi => ?_)]\"\"\",\n\"\"\" unfold Gprim\n rw [show (fun w : ℂ => ∑ i ∈ Finset.Icc 2 (n + 1),\n -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-((i : ℤ) - 1))) =\n ∑ i ∈ Finset.Icc 2 (n + 1),\n (fun w : ℂ => -(Acoeff n i / (↑i - 1)) * (w - 1) ^ (-((i : ℤ) - 1))) from by\n funext w\n simp [Finset.sum_apply]]\n rw [deriv_sum (fun i hi => ?_)]\"\"\")\n\n# 3. tail induction fixes\ns = s.replace(\"\"\" induction s using Finset.induction with\n | empty => simp\n | insert has hs =>\n rename_i a s\n rw [Finset.sum_insert has]\n exact ContinuousAt.add hs (hterm a)\n exact this _\"\"\",\n\"\"\" induction s using Finset.induction with\n | empty => exact continuousAt_const\n | insert a s has hs =>\n rw [Finset.sum_insert has]\n exact ContinuousAt.add hs (hterm a)\n exact this _\"\"\")\n\n# 4. subset by hand\ns = s.replace(\"exact hopen.mono (Set.sdiff_subset_sdiff_left (Set.subset_univ _))\",\n \"exact hopen.mono (by intro s hs; exact hs.2)\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40","description":"Apply structural fixes and recompile","timeout":420000},"extra_content":null}]