← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (6221 chars)
I see the issues now:
1. **Line 505-511 (h1)**: `Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt x h2` — comp_hasDerivAt (hl : HasFDerivAt l l' (f x)) (hf : HasDerivAt f f' x) : HasDerivAt (l ∘ f) (l' f') x. With hl := ofRealCLM.hasFDerivAt : HasFDerivAt ⇑ofRealCLM ofRealCLM ?x — the point ?x must unify with (f x) where f := Real.Gamma — l := ofRealCLM, f := Real.Gamma — the l' f' := ofRealCLM (deriv Real.Gamma x) ✓. So h3 : HasDerivAt (⇑ofRealCLM ∘ Real.Gamma) (ofRealCLM (deriv Real.Gamma x)) x. Then rw [ofRealCLM_apply] at h3 — rewrites ofRealCLM (deriv Real.Gamma x) → ↑(deriv Real.Gamma x) ✓. Then heq rewrites the function ⇑ofRealCLM ∘ Real.Gamma → fun y => ↑(Real.Gamma y) — via funext + comp_apply + ofRealCLM_apply ✓. So h3 should be the right type. The error at 511 `exact h3`: "h3 has type HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x but expected HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x" — WAIT — the expected is h2's type!! That means the error is at `comp_hasDerivAt x h2`?? — no — the error at 511:6 is at `exact h3` — and the expected type is h1's STATED type — HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x??? — but h1's stated type (line 504) is HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x — that's NOT what the error shows as expected. The error's expected: HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x — the ℝ-Gamma! — so the error is NOT at h1's exact — it's at the `h2` USAGE — line 505: `Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt x h2` — the error message "h3 has type ... but expected HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x" — hmm — no. The error at 511:6: "Type mismatch: h3 has type HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x but is expected to have type HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x" — at 511 which is `exact h3` — and h1's stated type is the ↑(Gamma) form — but the error says the expected is the un-cast form — IMPOSSIBLE unless h1's statement got mis-parsed... OH WAIT. I see it — the `simp at hx0` at line 500/522 — `rw [hmeq] at hx0; simp at hx0` — hx0 : 0 < x with hmeq : x = −↑m — rw [hmeq] at hx0 : 0 < −↑m — simp at hx0 — simplifies to hx0 : ??? — simp might turn 0 < −↑m into ↑m < 0 — fine. Not the issue.
The error's expected type `HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x` — Real.Gamma y — hmm — this is h2's type — and h3's type shown is the CORRECT h1 type — so the mismatch is INSIDE the h1 proof at `comp_hasDerivAt x h2`? — if ofRealCLM.hasFDerivAt's point metavariable ?x couldn't unify with (Real.Gamma x)... the error would be about that.
You know what — line 511 `exact h3` with h3 : HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x and h1's stated type — line 504: `HasDerivAt (fun y : ℝ => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x` — IDENTICAL — so the 511 error's "expected type" being different means line 511 is NOT h1's exact — the file shifted! The error at "511:6" — hmm the current file's line 511 IS `exact h3` per the read. But the error's expected type is h2-like...
OHHH. I bet the error is from an EARLIER compile (stale line numbers from the previous iteration's output)?? No — the last compile output was fresh. Line 511:6 in the fresh output — "h3 has type ... but expected HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x" — hmm wait — actually let me recheck: could `Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt x h2` produce the error because the RESULT type unification assigned h3's type from the h2 form first and then my rewrites... no.
Actually — maybe the error ISN'T at `exact h3` — the "511:6" might be at `rw [heq] at h3`?? no — 511 is `exact h3` per the read (line 505-511: h3 line 505, rw 506, heq 507-509, rw 510, exact 511 ✓). So at `exact h3`: h3's type after the two rws: HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x — and the expected (h1's statement) — the error claims expected = HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x — that means h1's STATEMENT got elaborated with ↑(Real.Gamma y) → ... no wait — maybe the issue: `↑(Real.Gamma y)` in h1's statement — the ↑ — and `↑(deriv Real.Gamma x)` — hmm — the expected type in the error shows NO casts — as if the casts got stripped... I wonder if the error is actually that the elaboration of h1's statement produced `fun y => Real.Gamma y` because the `↑(Real.Gamma y)` failed to elaborate as ℂ — no...
I'm going in circles. Let me just look at the full error output again for 505-517 carefully. From the last compile:
```
NguyenRH1/LiDigamma.lean:511:6: error: Type mismatch
h3
has type
HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x
but is expected to have type
HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x
```
Hmm wait — maybe the issue is at the PREVIOUS error line 505 — no error at 505. So at 511: exact h3 against h1's stated type. The stated type at line 504: `HasDerivAt (fun y : ℝ => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x` — but the error's expected type shows `Real.Gamma y` (no cast!) — that means in h1's statement, `↑(Real.Gamma y)` elaborated to just `Real.Gamma y`?!?! That happens if the expected type of the HasDerivAt statement's function position is ℝ → ℝ?? no — HasDerivAt (fun y : ℝ => ↑(Real.Gamma y)) — the ↑(·) : ℂ — the function is ℝ → ℂ — hmm.
WAIT. Actually — I bet the error positions in the last output are from the compile BEFORE my last edits (the output might have been from a cached/stale run? no — the command recompiled).
Let me try a completely different approach to hderiv — avoiding the comp gymnastics: prove `deriv Complex.Gamma ↑x = ↑(deriv Real.Gamma x)` via the EventuallyEq route: the functions Complex.Gamma ∘ ofReal and ofReal ∘ Real.Gamma are EQUAL (Gamma_ofReal pointwise); so deriv (Gamma∘ofReal) x = deriv (ofReal∘Real.Gamma) x (congruence of equal functions); LHS = deriv Gamma ↑x (via comp h9/h7); RHS = ↑(deriv Real.Gamma x) (via the CLM comp h3). The pieces are what I have — the issue is just the comp forms. Let me probe them in isolation — write a small probe with the two comp derivations and iterate quickly on THAT instead of the big file.
Let me probe the two comp-derivation patterns in isolation.
tool: Bash ok
input
{
"command": "cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && export PATH=\"$HOME/.elan/bin:$PATH\" && cat > probe.lean << 'EOF'\nimport Mathlib\nopen Complex\nexample (x : \u211d) (g : \u211d \u2192 \u211d) (g' : \u211d) (hg : HasDerivAt g g' x) :\n HasDerivAt (fun y : \u211d => (\u2191(g y) : \u2102)) (\u2191g') x := by\n have h3 := Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt x hg\n rw [Complex.ofRealCLM_apply] at h3\n have heq : (Complex.ofRealCLM \u2218 g) = fun y : \u211d => (\u2191(g y) : \u2102) := by\n funext y\n rw [Function.comp_apply, Complex.ofRealCLM_apply]\n rw [heq] at h3\n exact h3\nEOF\nlake env lean probe.lean 2>&1 | head -20",
"description": "Probe the CLM comp pattern isolated",
"timeout": 300000
}
result 2026-08-22 07:55:41Z → 2026-08-26 08:29:29Z
[master 4bc3c09] ii'-6-DL-4 COMPLETE (session 040): the far-right vertical drop Lean-PROVED — kFun_right_vertical_norm_le_exp_sqrt_half_add_one (the w=2sqrt(n) kernel variant: ||kFun n (2sqrt(n)+it)|| <= e^{sqrt(n)/2}+1 for n>=16, t^2>=n) and the capstone vIntegral_Jlag_right_norm_le: ||VIntegral (Jlag n) (2sqrt(n)) T U|| <= (U-T)(e^{sqrt(n)/2}+1)*2^{-(2sqrt(n)-2)}*dirichletC2 — exponentially tiny, NO perimeter. VRightBound charged IIMaj/2 (provable with dirichletC2 <= 1; IIMaj/4 = documented tightening path). Full build GREEN (8773 jobs); audit: 23 decls, 3 standard axioms only, 0 sorryAx. E-034 re-validated at VRightBound=IIMaj/2: ALL CHECKS PASS (margins 13160/88911/2.458e26). Next: DL-5 (total right-horizontal bound + lower side).
8 files changed, 209 insertions(+), 22 deletions(-)
4bc3c09 ii'-6-DL-4 COMPLETE (session 040): the far-right vertical drop Lean-PROVED — kFun_right_vertical_norm_le_exp_sqrt_half_add_one (the w=2sqrt(n) kernel variant: ||kFun n (2sqrt(n)+it)|| <= e^{sqrt(n)/2}+1 for n>=16, t^2>=n) and the capstone vIntegral_Jlag_right_norm_le: ||VIntegral (Jlag n) (2sqrt(n)) T U|| <= (U-T)(e^{sqrt(n)/2}+1)*2^{-(2sqrt(n)-2)}*dirichletC2 — exponentially tiny, NO perimeter. VRightBound charged IIMaj/2 (provable with dirichletC2 <= 1; IIMaj/4 = documented tightening path). Full build GREEN (8773 jobs); audit: 23 decls, 3 standard axioms only, 0 sorryAx. E-034 re-validated at VRightBound=IIMaj/2: ALL CHECKS PASS (margins 13160/88911/2.458e26). Next: DL-5 (total right-horizontal bound + lower side).
raw content_json (database row)
[{"type":"thinking","thinking":"I see the issues now:\n\n1. **Line 505-511 (h1)**: `Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt x h2` — comp_hasDerivAt (hl : HasFDerivAt l l' (f x)) (hf : HasDerivAt f f' x) : HasDerivAt (l ∘ f) (l' f') x. With hl := ofRealCLM.hasFDerivAt : HasFDerivAt ⇑ofRealCLM ofRealCLM ?x — the point ?x must unify with (f x) where f := Real.Gamma — l := ofRealCLM, f := Real.Gamma — the l' f' := ofRealCLM (deriv Real.Gamma x) ✓. So h3 : HasDerivAt (⇑ofRealCLM ∘ Real.Gamma) (ofRealCLM (deriv Real.Gamma x)) x. Then rw [ofRealCLM_apply] at h3 — rewrites ofRealCLM (deriv Real.Gamma x) → ↑(deriv Real.Gamma x) ✓. Then heq rewrites the function ⇑ofRealCLM ∘ Real.Gamma → fun y => ↑(Real.Gamma y) — via funext + comp_apply + ofRealCLM_apply ✓. So h3 should be the right type. The error at 511 `exact h3`: \"h3 has type HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x but expected HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x\" — WAIT — the expected is h2's type!! That means the error is at `comp_hasDerivAt x h2`?? — no — the error at 511:6 is at `exact h3` — and the expected type is h1's STATED type — HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x??? — but h1's stated type (line 504) is HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x — that's NOT what the error shows as expected. The error's expected: HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x — the ℝ-Gamma! — so the error is NOT at h1's exact — it's at the `h2` USAGE — line 505: `Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt x h2` — the error message \"h3 has type ... but expected HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x\" — hmm — no. The error at 511:6: \"Type mismatch: h3 has type HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x but is expected to have type HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x\" — at 511 which is `exact h3` — and h1's stated type is the ↑(Gamma) form — but the error says the expected is the un-cast form — IMPOSSIBLE unless h1's statement got mis-parsed... OH WAIT. I see it — the `simp at hx0` at line 500/522 — `rw [hmeq] at hx0; simp at hx0` — hx0 : 0 < x with hmeq : x = −↑m — rw [hmeq] at hx0 : 0 < −↑m — simp at hx0 — simplifies to hx0 : ??? — simp might turn 0 < −↑m into ↑m < 0 — fine. Not the issue.\n\nThe error's expected type `HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x` — Real.Gamma y — hmm — this is h2's type — and h3's type shown is the CORRECT h1 type — so the mismatch is INSIDE the h1 proof at `comp_hasDerivAt x h2`? — if ofRealCLM.hasFDerivAt's point metavariable ?x couldn't unify with (Real.Gamma x)... the error would be about that. \n\nYou know what — line 511 `exact h3` with h3 : HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x and h1's stated type — line 504: `HasDerivAt (fun y : ℝ => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x` — IDENTICAL — so the 511 error's \"expected type\" being different means line 511 is NOT h1's exact — the file shifted! The error at \"511:6\" — hmm the current file's line 511 IS `exact h3` per the read. But the error's expected type is h2-like... \n\nOHHH. I bet the error is from an EARLIER compile (stale line numbers from the previous iteration's output)?? No — the last compile output was fresh. Line 511:6 in the fresh output — \"h3 has type ... but expected HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x\" — hmm wait — actually let me recheck: could `Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt x h2` produce the error because the RESULT type unification assigned h3's type from the h2 form first and then my rewrites... no.\n\nActually — maybe the error ISN'T at `exact h3` — the \"511:6\" might be at `rw [heq] at h3`?? no — 511 is `exact h3` per the read (line 505-511: h3 line 505, rw 506, heq 507-509, rw 510, exact 511 ✓). So at `exact h3`: h3's type after the two rws: HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x — and the expected (h1's statement) — the error claims expected = HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x — that means h1's STATEMENT got elaborated with ↑(Real.Gamma y) → ... no wait — maybe the issue: `↑(Real.Gamma y)` in h1's statement — the ↑ — and `↑(deriv Real.Gamma x)` — hmm — the expected type in the error shows NO casts — as if the casts got stripped... I wonder if the error is actually that the elaboration of h1's statement produced `fun y => Real.Gamma y` because the `↑(Real.Gamma y)` failed to elaborate as ℂ — no...\n\nI'm going in circles. Let me just look at the full error output again for 505-517 carefully. From the last compile:\n```\nNguyenRH1/LiDigamma.lean:511:6: error: Type mismatch\n h3\nhas type\n HasDerivAt (fun y => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x\nbut is expected to have type\n HasDerivAt (fun y => Real.Gamma y) (deriv Real.Gamma x) x\n```\nHmm wait — maybe the issue is at the PREVIOUS error line 505 — no error at 505. So at 511: exact h3 against h1's stated type. The stated type at line 504: `HasDerivAt (fun y : ℝ => ↑(Real.Gamma y)) (↑(deriv Real.Gamma x)) x` — but the error's expected type shows `Real.Gamma y` (no cast!) — that means in h1's statement, `↑(Real.Gamma y)` elaborated to just `Real.Gamma y`?!?! That happens if the expected type of the HasDerivAt statement's function position is ℝ → ℝ?? no — HasDerivAt (fun y : ℝ => ↑(Real.Gamma y)) — the ↑(·) : ℂ — the function is ℝ → ℂ — hmm.\n\nWAIT. Actually — I bet the error positions in the last output are from the compile BEFORE my last edits (the output might have been from a cached/stale run? no — the command recompiled).\n\nLet me try a completely different approach to hderiv — avoiding the comp gymnastics: prove `deriv Complex.Gamma ↑x = ↑(deriv Real.Gamma x)` via the EventuallyEq route: the functions Complex.Gamma ∘ ofReal and ofReal ∘ Real.Gamma are EQUAL (Gamma_ofReal pointwise); so deriv (Gamma∘ofReal) x = deriv (ofReal∘Real.Gamma) x (congruence of equal functions); LHS = deriv Gamma ↑x (via comp h9/h7); RHS = ↑(deriv Real.Gamma x) (via the CLM comp h3). The pieces are what I have — the issue is just the comp forms. Let me probe them in isolation — write a small probe with the two comp derivations and iterate quickly on THAT instead of the big file.","signature":""},{"type":"text","text":"Let me probe the two comp-derivation patterns in isolation."},{"type":"tool_use","id":"Bash_374","name":"Bash","input":{"command":"cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && export PATH=\"$HOME/.elan/bin:$PATH\" && cat > probe.lean << 'EOF'\nimport Mathlib\nopen Complex\nexample (x : ℝ) (g : ℝ → ℝ) (g' : ℝ) (hg : HasDerivAt g g' x) :\n HasDerivAt (fun y : ℝ => (↑(g y) : ℂ)) (↑g') x := by\n have h3 := Complex.ofRealCLM.hasFDerivAt.comp_hasDerivAt x hg\n rw [Complex.ofRealCLM_apply] at h3\n have heq : (Complex.ofRealCLM ∘ g) = fun y : ℝ => (↑(g y) : ℂ) := by\n funext y\n rw [Function.comp_apply, Complex.ofRealCLM_apply]\n rw [heq] at h3\n exact h3\nEOF\nlake env lean probe.lean 2>&1 | head -20","description":"Probe the CLM comp pattern isolated","timeout":300000},"extra_content":null}]