scilean-declarations/SciLean.Core.FunctionTransformations.RevDeriv.jsonl

{"name":"Inner.inner.arg_a0a1.revDerivUpdate_rule","declaration":"theorem Inner.inner.arg_a0a1.revDerivUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff R f) (hg : SciLean.HasAdjDiff R g) : (SciLean.revDerivUpdate R fun x => ⟪f x, g x⟫_R) = fun x =>\n  let y₁df := SciLean.revDerivUpdate R f x;\n  let y₂dg := SciLean.revDerivUpdate R g x;\n  (⟪y₁df.1, y₂dg.1⟫_R, fun dr dx => y₂dg.2 (dr • y₁df.1) (y₁df.2 (dr • y₂dg.1) dx))"}
{"name":"Prod.fst.arg_self.revDerivProjUpdate_rule","declaration":"theorem Prod.fst.arg_self.revDerivProjUpdate_rule {K : Type} [RCLike K] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {X' : Type} {Xi : Type} {XI : Xi → Type} [SciLean.StructType X' Xi XI] [DecidableEq Xi] [SciLean.SemiInnerProductSpace K X'] [(i : Xi) → SciLean.SemiInnerProductSpace K (XI i)] {W : Type} [SciLean.SemiInnerProductSpace K W] (f : W → X' × Y) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProjUpdate K Xi fun x => (f x).1) = fun w =>\n  let xydf := SciLean.revDerivProjUpdate K (Xi ⊕ Unit) f w;\n  (xydf.1.1, fun i dxy dw => xydf.2 (Sum.inl i) dxy dw)"}
{"name":"HSMul.hSMul.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HSMul.hSMul.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y Yi YI] [SciLean.IndexType Yi] [SciLean.LawfulIndexType Yi] [DecidableEq Yi] [SciLean.SemiHilbert K Y] [(i : Yi) → SciLean.SemiHilbert K (YI i)] [SciLean.SemiInnerProductSpaceStruct K Y Yi YI] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K Yi fun x => f x • g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDerivProjUpdate K Yi g x;\n  (ydf.1 • zdg.1, fun i dy dx =>\n    let dk := ⟪SciLean.structProj zdg.1 i, dy⟫_K;\n    let dy' := (starRingEnd K) ydf.1 • dy;\n    let dx := zdg.2 i dy' dx;\n    ydf.2 dk dx)"}
{"name":"SciLean.revDeriv'","declaration":"def SciLean.revDeriv' (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : Y × Y ⊸[K] X"}
{"name":"SciLean.revDerivUpdate","declaration":"def SciLean.revDerivUpdate (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : Y × (Y → X → X)"}
{"name":"HSub.hSub.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HSub.hSub.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') (g : X → Y') (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K Yi fun x => f x - g x) = fun x =>\n  let ydf := SciLean.revDerivProjUpdate K Yi f x;\n  let ydg := SciLean.revDerivProjUpdate K Yi g x;\n  (ydf.1 - ydg.1, fun i dy dx =>\n    let dx := ydf.2 i dy dx;\n    let dy' := -dy;\n    ydg.2 i dy' dx)"}
{"name":"SciLean.norm₂.arg_x.revDerivProj_rule_at","declaration":"theorem SciLean.norm₂.arg_x.revDerivProj_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt R f x) (hx : f x ≠ 0) : SciLean.revDerivProj R Unit (fun x => ‖f x‖₂[R]) x =\n  let ydf := SciLean.revDeriv R f x;\n  let ynorm := ‖ydf.1‖₂[R];\n  (ynorm, fun x dr => (ynorm⁻¹ * dr) • ydf.2 ydf.1)"}
{"name":"Prod.mk.arg_fstsnd.revDerivUpdate_rule","declaration":"theorem Prod.mk.arg_fstsnd.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (g : X → Y) (f : X → Z) (hg : SciLean.HasAdjDiff K g) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => (g x, f x)) = fun x =>\n  let ydg := SciLean.revDerivUpdate K g x;\n  let zdf := SciLean.revDerivUpdate K f x;\n  ((ydg.1, zdf.1), fun dyz dx =>\n    let dx := ydg.2 dyz.1 dx;\n    zdf.2 dyz.2 dx)"}
{"name":"SciLean.gradient","declaration":"def SciLean.gradient (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : Y → X"}
{"name":"HSub.hSub.arg_a0a1.revDeriv_rule","declaration":"theorem HSub.hSub.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f x - g x) = fun x =>\n  let ydf := SciLean.revDeriv K f x;\n  let ydg := SciLean.revDerivUpdate K g x;\n  (ydf.1 - ydg.1, fun dy =>\n    let dx := ydf.2 dy;\n    let dy' := -dy;\n    ydg.2 dy' dx)"}
{"name":"HPow.hPow.arg_a0.revDerivProjUpdate_rule","declaration":"def HPow.hPow.arg_a0.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (n : ℕ) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProjUpdate K Unit fun x => f x ^ n) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let y' := ↑n * (starRingEnd K) ydf.1 ^ (n - 1);\n  (ydf.1 ^ n, fun x dy dx => ydf.2 (y' * dy) dx)"}
{"name":"Prod.snd.arg_self.revDerivUpdate_rule","declaration":"theorem Prod.snd.arg_self.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => (f x).2) = fun x =>\n  let yzdf := SciLean.revDerivProjUpdate K (Unit ⊕ Unit) f x;\n  (yzdf.1.2, fun dy dx => yzdf.2 (Sum.inr ()) dy dx)"}
{"name":"SciLean.revDerivProj.apply_rule","declaration":"theorem SciLean.revDerivProj.apply_rule (K : Type u_5) (I : Type u_1) [RCLike K] {ι : Type u_2} [SciLean.IndexType ι] [DecidableEq ι] {E : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [DecidableEq I] (i : ι) : (SciLean.revDerivProj K I fun f => f i) = fun f => (f i, fun j dxj => SciLean.oneHot (i, j) dxj)"}
{"name":"dite.arg_te.revDerivProj_rule","declaration":"theorem dite.arg_te.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (c : Prop) [dec : Decidable c] (t : c → X → Y') (e : ¬c → X → Y') : (SciLean.revDerivProj K Yi fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n  if p : c then SciLean.revDerivProj K Yi (t p) y else SciLean.revDerivProj K Yi (e p) y"}
{"name":"SciLean.revDeriv.comp_rule","declaration":"theorem SciLean.revDeriv.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f (g x)) = fun x =>\n  let ydg := SciLean.revDeriv K g x;\n  let zdf := SciLean.revDeriv K f ydg.1;\n  (zdf.1, fun dz =>\n    let dy := zdf.2 dz;\n    ydg.2 dy)"}
{"name":"dite.arg_te.revDerivProjUpdate_rule","declaration":"theorem dite.arg_te.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (c : Prop) [dec : Decidable c] (t : c → X → Y') (e : ¬c → X → Y') : (SciLean.revDerivProjUpdate K Yi fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n  if p : c then SciLean.revDerivProjUpdate K Yi (t p) y else SciLean.revDerivProjUpdate K Yi (e p) y"}
{"name":"SciLean.revDerivProjUpdate.let_rule","declaration":"theorem SciLean.revDerivProjUpdate.let_rule (K : Type u_1) (I : Type u_5) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_4} {EI : I → Type u_6} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → Y → E) (g : X → Y) (hf : SciLean.HasAdjDiff K fun x =>\n  match x with\n  | (x, y) => f x y) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K I fun x =>\n    let y := g x;\n    f x y) =\n  fun x =>\n  let ydg' := SciLean.revDerivUpdate K g x;\n  let zdf' :=\n    SciLean.revDerivProjUpdate K I\n      (fun x =>\n        match x with\n        | (x, y) => f x y)\n      (x, ydg'.1);\n  (zdf'.1, fun i dei dx =>\n    let dxy := zdf'.2 i dei (dx, 0);\n    ydg'.2 dxy.2 dxy.1)"}
{"name":"dite.arg_te.revDeriv_rule","declaration":"theorem dite.arg_te.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) : (SciLean.revDeriv K fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n  if p : c then SciLean.revDeriv K (t p) y else SciLean.revDeriv K (e p) y"}
{"name":"SciLean.revDerivProjUpdate.apply_rule","declaration":"theorem SciLean.revDerivProjUpdate.apply_rule (K : Type u_5) (I : Type u_1) [RCLike K] {ι : Type u_2} [SciLean.IndexType ι] [DecidableEq ι] {E : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [DecidableEq I] (i : ι) : (SciLean.revDerivProjUpdate K I fun f => f i) = fun f =>\n  (f i, fun j dxj df i' => if i = i' then SciLean.structModify j (fun xj => xj + dxj) (df i') else df i')"}
{"name":"SciLean.revDerivProj","declaration":"def SciLean.revDerivProj (K : Type u_1) (I : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {E : Type u_4} {EI : I → Type u_5} [SciLean.StructType E I EI] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [DecidableEq I] (f : X → E) (x : X) : E × ((i : I) → EI i → X)"}
{"name":"Neg.neg.arg_a0.revDerivProj_rule","declaration":"theorem Neg.neg.arg_a0.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') : (SciLean.revDerivProj K Yi fun x => -f x) = fun x =>\n  let ydf := SciLean.revDerivProj K Yi f x;\n  (-ydf.1, fun i dy =>\n    let dy' := -dy;\n    ydf.2 i dy')"}
{"name":"ite.arg_te.revDerivProjUpdate_rule","declaration":"theorem ite.arg_te.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (c : Prop) [dec : Decidable c] (t : X → Y') (e : X → Y') : (SciLean.revDerivProjUpdate K Yi fun x => if c then t x else e x) = fun y =>\n  if c then SciLean.revDerivProjUpdate K Yi t y else SciLean.revDerivProjUpdate K Yi e y"}
{"name":"Prod.fst.arg_self.revDerivUpdate_rule","declaration":"theorem Prod.fst.arg_self.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => (f x).1) = fun x =>\n  let yzdf := SciLean.revDerivProjUpdate K (Unit ⊕ Unit) f x;\n  (yzdf.1.1, fun dy dx => yzdf.2 (Sum.inl ()) dy dx)"}
{"name":"Prod.snd.arg_self.revDerivProj_rule","declaration":"theorem Prod.snd.arg_self.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] {W : Type} [SciLean.SemiInnerProductSpace K W] (f : W → X × Y') (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K Yi fun x => (f x).2) = fun w =>\n  let xydf := SciLean.revDerivProj K (Unit ⊕ Yi) f w;\n  (xydf.1.2, fun i dxy => xydf.2 (Sum.inr i) dxy)"}
{"name":"SciLean.scalarGradient","declaration":"def SciLean.scalarGradient (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] (f : X → K) (x : X) : X"}
{"name":"Prod.fst.arg_self.revDeriv_rule","declaration":"theorem Prod.fst.arg_self.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => (f x).1) = fun x =>\n  let yzdf := SciLean.revDerivProj K (Unit ⊕ Unit) f x;\n  (yzdf.1.1, fun dy => yzdf.2 (Sum.inl ()) dy)"}
{"name":"HDiv.hDiv.arg_a0a1.revDeriv_rule","declaration":"theorem HDiv.hDiv.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) (hx : SciLean.fpropParam (∀ (x : X), g x ≠ 0)) : (SciLean.revDeriv K fun x => f x / g x) = fun x =>\n  let ydf := SciLean.revDeriv K f x;\n  let zdg := SciLean.revDerivUpdate K g x;\n  (ydf.1 / zdg.1, fun dx' =>\n    (1 / (starRingEnd K) zdg.1 ^ 2) • zdg.2 (-(starRingEnd K) ydf.1 • dx') ((starRingEnd K) zdg.1 • ydf.2 dx'))"}
{"name":"Neg.neg.arg_a0.revDerivProjUpdate_rule","declaration":"theorem Neg.neg.arg_a0.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') : (SciLean.revDerivProjUpdate K Yi fun x => -f x) = fun x =>\n  let ydf := SciLean.revDerivProjUpdate K Yi f x;\n  (-ydf.1, fun i dy dx =>\n    let dy' := -dy;\n    ydf.2 i dy' dx)"}
{"name":"ite.arg_te.revDerivProj_rule","declaration":"theorem ite.arg_te.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (c : Prop) [dec : Decidable c] (t : X → Y') (e : X → Y') : (SciLean.revDerivProj K Yi fun x => if c then t x else e x) = fun y =>\n  if c then SciLean.revDerivProj K Yi t y else SciLean.revDerivProj K Yi e y"}
{"name":"HSMul.hSMul.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HSMul.hSMul.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiHilbert K Y] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f x • g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDerivUpdate K g x;\n  (ydf.1 • zdg.1, fun dy dx =>\n    let dk := ⟪zdg.1, dy⟫_K;\n    let dy' := (starRingEnd K) ydf.1 • dy;\n    let dx := zdg.2 dy' dx;\n    ydf.2 dk dx)"}
{"name":"HPow.hPow.arg_a0.revDerivProj_rule","declaration":"def HPow.hPow.arg_a0.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (n : ℕ) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K Unit fun x => f x ^ n) = fun x =>\n  let ydf := SciLean.revDeriv K f x;\n  let y' := ↑n * (starRingEnd K) ydf.1 ^ (n - 1);\n  (ydf.1 ^ n, fun x dx' => ydf.2 (y' * dx'))"}
{"name":"HMul.hMul.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HMul.hMul.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f x * g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDerivUpdate K g x;\n  (ydf.1 * zdg.1, fun dx' dx =>\n    let dx₁ := (starRingEnd K) zdg.1 * dx';\n    let dx₂ := (starRingEnd K) ydf.1 * dx';\n    let dx := zdg.2 dx₂ dx;\n    ydf.2 dx₁ dx)"}
{"name":"SciLean.revDerivUpdate.comp_rule","declaration":"theorem SciLean.revDerivUpdate.comp_rule (K : Type u_1) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_3} [SciLean.SemiInnerProductSpace K Z] (f : Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f (g x)) = fun x =>\n  let ydg := SciLean.revDerivUpdate K g x;\n  let zdf := SciLean.revDeriv K f ydg.1;\n  (zdf.1, fun dz dx =>\n    let dy := zdf.2 dz;\n    ydg.2 dy dx)"}
{"name":"HSMul.hSMul.arg_a0a1.revDerivProj_rule","declaration":"theorem HSMul.hSMul.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y Yi YI] [SciLean.IndexType Yi] [SciLean.LawfulIndexType Yi] [DecidableEq Yi] [SciLean.SemiHilbert K Y] [(i : Yi) → SciLean.SemiHilbert K (YI i)] [SciLean.SemiInnerProductSpaceStruct K Y Yi YI] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K Yi fun x => f x • g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDerivProj K Yi g x;\n  (ydf.1 • zdg.1, fun i dy =>\n    let dk := ⟪SciLean.structProj zdg.1 i, dy⟫_K;\n    let dx := zdg.2 i dy;\n    let dx := (starRingEnd K) ydf.1 • dx;\n    ydf.2 dk dx)"}
{"name":"SciLean.revDerivUpdate.pi_rule","declaration":"theorem SciLean.revDerivUpdate.pi_rule (K : Type u_1) (I : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {EI : I → Type u_2} [SciLean.IndexType I] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → (i : I) → EI i) (hf : ∀ (i : I), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivUpdate K fun x i => f x i) = fun x =>\n  let xdf := fun i => SciLean.revDerivUpdate K (fun x => f x i) x;\n  (fun i => (xdf i).1, fun dy dx => LeanColls.fold (SciLean.IndexType.univ I) (fun dx i => (xdf i).2 (dy i) dx) dx)"}
{"name":"SciLean.revDerivProjUpdate_fst","declaration":"theorem SciLean.revDerivProjUpdate_fst (K : Type u_5) (I : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {E : Type u_1} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → E) (x : X) : (SciLean.revDerivProjUpdate K I f x).1 = f x"}
{"name":"SciLean.revDerivProj_fst","declaration":"theorem SciLean.revDerivProj_fst (K : Type u_5) (I : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {E : Type u_1} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → E) (x : X) : (SciLean.revDerivProj K I f x).1 = f x"}
{"name":"SciLean.revDeriv.let_rule","declaration":"theorem SciLean.revDeriv.let_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (f : X → Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K fun xy => f xy.1 xy.2) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x =>\n    let y := g x;\n    f x y) =\n  fun x =>\n  let ydg := SciLean.revDerivUpdate K g x;\n  let zdf := SciLean.revDeriv K (fun xy => f xy.1 xy.2) (x, ydg.1);\n  (zdf.1, fun dz =>\n    let dxdy := zdf.2 dz;\n    let dx := ydg.2 dxdy.2 dxdy.1;\n    dx)"}
{"name":"SciLean.revDeriv.id_rule","declaration":"theorem SciLean.revDeriv.id_rule (K : Type u_2) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] : (SciLean.revDeriv K fun x => x) = fun x => (x, fun dx => dx)"}
{"name":"SciLean.revDerivProj.const_rule","declaration":"theorem SciLean.revDerivProj.const_rule (K : Type u_5) (I : Type u_1) [RCLike K] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (x : E) : (SciLean.revDerivProj K I fun x_1 => x) = fun x_1 => (x, fun i de => 0)"}
{"name":"SciLean.revDerivProjUpdate_snd_zero'","declaration":"theorem SciLean.revDerivProjUpdate_snd_zero' (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (dy : Y) : (SciLean.revDerivUpdate K f x).2 dy 0 = (SciLean.revDeriv K f x).2 dy"}
{"name":"Prod.snd.arg_self.revDeriv_rule","declaration":"theorem Prod.snd.arg_self.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (f : X → Y × Z) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => (f x).2) = fun x =>\n  let yzdf := SciLean.revDerivProj K (Unit ⊕ Unit) f x;\n  (yzdf.1.2, fun dy => yzdf.2 (Sum.inr ()) dy)"}
{"name":"HDiv.hDiv.arg_a0a1.revDerivProj_rule","declaration":"theorem HDiv.hDiv.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) (hx : ∀ (x : X), g x ≠ 0) : (SciLean.revDerivProj K Unit fun x => f x / g x) = fun x =>\n  let ydf := SciLean.revDeriv K f x;\n  let zdg := SciLean.revDerivUpdate K g x;\n  (ydf.1 / zdg.1, fun x dx' =>\n    (1 / (starRingEnd K) zdg.1 ^ 2) • zdg.2 (-(starRingEnd K) ydf.1 • dx') ((starRingEnd K) zdg.1 • ydf.2 dx'))"}
{"name":"SciLean.Norm2.norm2.arg_a0.revDeriv_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.revDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) : (SciLean.revDeriv R fun x => ‖f x‖₂²) = fun x =>\n  let ydf := SciLean.revDeriv R f x;\n  let ynorm2 := ‖ydf.1‖₂²;\n  (ynorm2, fun dr => (2 * dr) • ydf.2 ydf.1)"}
{"name":"Neg.neg.arg_a0.revDeriv_rule","declaration":"theorem Neg.neg.arg_a0.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : SciLean.revDeriv K (fun x => -f x) x =\n  let ydf := SciLean.revDeriv K f x;\n  (-ydf.1, fun dy =>\n    let dx := ydf.2 dy;\n    -dx)"}
{"name":"HMul.hMul.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HMul.hMul.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K Unit fun x => f x * g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDerivUpdate K g x;\n  (ydf.1 * zdg.1, fun x dy dx =>\n    let dy₁ := (starRingEnd K) zdg.1 * dy;\n    let dy₂ := (starRingEnd K) ydf.1 * dy;\n    let dx := zdg.2 dy₂ dx;\n    ydf.2 dy₁ dx)"}
{"name":"SciLean.revDeriv","declaration":"def SciLean.revDeriv (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : Y × (Y → X)"}
{"name":"Neg.neg.arg_a0.revDerivUpdate_rule","declaration":"theorem Neg.neg.arg_a0.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) : (SciLean.revDerivUpdate K fun x => -f x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  (-ydf.1, fun dy dx =>\n    let dy' := -dy;\n    ydf.2 dy' dx)"}
{"name":"SciLean.revDeriv.pi_rule","declaration":"theorem SciLean.revDeriv.pi_rule (K : Type u_1) (I : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {EI : I → Type u_2} [SciLean.IndexType I] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → (i : I) → EI i) (hf : ∀ (i : I), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDeriv K fun x i => f x i) = fun x =>\n  let xdf := fun i => SciLean.revDerivUpdate K (fun x => f x i) x;\n  (fun i => (xdf i).1, fun dy => LeanColls.fold (SciLean.IndexType.univ I) (fun dx i => (xdf i).2 (dy i) dx) 0)"}
{"name":"dite.arg_te.revDerivUpdate_rule","declaration":"theorem dite.arg_te.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : c → X → Y) (e : ¬c → X → Y) : (SciLean.revDerivUpdate K fun x => if x_1 : c then t x_1 x else e x_1 x) = fun y =>\n  if p : c then SciLean.revDerivUpdate K (t p) y else SciLean.revDerivUpdate K (e p) y"}
{"name":"Prod.snd.arg_self.revDerivProjUpdate_rule","declaration":"theorem Prod.snd.arg_self.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] {W : Type} [SciLean.SemiInnerProductSpace K W] (f : W → X × Y') (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProjUpdate K Yi fun x => (f x).2) = fun w =>\n  let xydf := SciLean.revDerivProjUpdate K (Unit ⊕ Yi) f w;\n  (xydf.1.2, fun i dxy dw => xydf.2 (Sum.inr i) dxy dw)"}
{"name":"SciLean.revDerivProj.id_rule","declaration":"theorem SciLean.revDerivProj.id_rule (K : Type u_4) (I : Type u_1) [RCLike K] {E : Type u_2} {EI : I → Type u_3} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] : (SciLean.revDerivProj K I fun x => x) = fun x => (x, fun i de => SciLean.oneHot i de)"}
{"name":"SciLean.revDerivProj_snd_zero","declaration":"theorem SciLean.revDerivProj_snd_zero (K : Type u_5) (I : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {E : Type u_2} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → E) (x : X) (i : I) : (SciLean.revDerivProj K I f x).2 i 0 = 0"}
{"name":"SciLean.revDerivProjUpdate.comp_rule","declaration":"theorem SciLean.revDerivProjUpdate.comp_rule (K : Type u_1) (I : Type u_5) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_3} {EI : I → Type u_6} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : Y → E) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K I fun x => f (g x)) = fun x =>\n  let ydg' := SciLean.revDerivUpdate K g x;\n  let zdf' := SciLean.revDerivProj K I f ydg'.1;\n  (zdf'.1, fun i de dx => ydg'.2 (zdf'.2 i de) dx)"}
{"name":"SciLean.norm₂.arg_x.revDeriv_rule","declaration":"theorem SciLean.norm₂.arg_x.revDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.revDeriv R fun x => ‖f x‖₂[R]) = fun x =>\n  let ydf := SciLean.revDeriv R f x;\n  let ynorm := ‖ydf.1‖₂[R];\n  (ynorm, fun dr => (ynorm⁻¹ * dr) • ydf.2 ydf.1)"}
{"name":"SciLean.revDerivProjUpdate.const_rule","declaration":"theorem SciLean.revDerivProjUpdate.const_rule (K : Type u_5) (I : Type u_1) [RCLike K] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_3} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (x : E) : (SciLean.revDerivProjUpdate K I fun x_1 => x) = fun x_1 => (x, fun i de dx => dx)"}
{"name":"IndexType.sum.arg_f.revDerivProj_rule","declaration":"theorem IndexType.sum.arg_f.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] {ι : Type} [SciLean.IndexType ι] [DecidableEq ι] (f : X → ι → Y') (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivProj K Yi fun x =>  ∑ i, f x i) = fun x =>\n  let ydf := SciLean.revDerivProjUpdate K (ι × Yi) f x;\n  ( ∑ i, ydf.1 i, fun j dy => LeanColls.fold (SciLean.IndexType.univ ι) (fun dx i => ydf.2 (i, j) dy dx) 0)"}
{"name":"SciLean.norm₂.arg_x.revDerivProj_rule","declaration":"theorem SciLean.norm₂.arg_x.revDerivProj_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.revDerivProj R Unit fun x => ‖f x‖₂[R]) = fun x =>\n  let ydf := SciLean.revDeriv R f x;\n  let ynorm := ‖ydf.1‖₂[R];\n  (ynorm, fun x dr => (ynorm⁻¹ * dr) • ydf.2 ydf.1)"}
{"name":"HAdd.hAdd.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f x + g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let ydg := SciLean.revDerivUpdate K g x;\n  (ydf.1 + ydg.1, fun dy dx =>\n    let dx := ydf.2 dy dx;\n    ydg.2 dy dx)"}
{"name":"Prod.mk.arg_fstsnd.revDerivProj_rule","declaration":"theorem Prod.mk.arg_fstsnd.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] {Z' : Type} {Zi : Type} {ZI : Zi → Type} [SciLean.StructType Z' Zi ZI] [DecidableEq Zi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] [SciLean.SemiInnerProductSpace K Z'] [(i : Zi) → SciLean.SemiInnerProductSpace K (ZI i)] (g : X → Y') (f : X → Z') (hg : SciLean.HasAdjDiff K g) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K (Yi ⊕ Zi) fun x => (g x, f x)) = fun x =>\n  let ydg := SciLean.revDerivProj K Yi g x;\n  let zdf := SciLean.revDerivProj K Zi f x;\n  ((ydg.1, zdf.1), fun i dyz =>\n    match i, dyz with\n    | Sum.inl j, dyz => ydg.2 j dyz\n    | Sum.inr j, dyz => zdf.2 j dyz)"}
{"name":"ite.arg_te.revDerivUpdate_rule","declaration":"theorem ite.arg_te.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) : (SciLean.revDerivUpdate K fun x => if c then t x else e x) = fun y =>\n  if c then SciLean.revDerivUpdate K t y else SciLean.revDerivUpdate K e y"}
{"name":"SciLean.revDerivUpdate_snd_zero'","declaration":"theorem SciLean.revDerivUpdate_snd_zero' (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (dy : Y) : (SciLean.revDerivUpdate K f x).2 dy 0 = (SciLean.revDeriv K f x).2 dy"}
{"name":"SciLean.revDerivUpdate_fst","declaration":"theorem SciLean.revDerivUpdate_fst (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : (SciLean.revDerivUpdate K f x).1 = f x"}
{"name":"Prod.mk.arg_fstsnd.revDerivProjUpdate_rule","declaration":"theorem Prod.mk.arg_fstsnd.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] {Z' : Type} {Zi : Type} {ZI : Zi → Type} [SciLean.StructType Z' Zi ZI] [DecidableEq Zi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] [SciLean.SemiInnerProductSpace K Z'] [(i : Zi) → SciLean.SemiInnerProductSpace K (ZI i)] (g : X → Y') (f : X → Z') (hg : SciLean.HasAdjDiff K g) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProjUpdate K (Yi ⊕ Zi) fun x => (g x, f x)) = fun x =>\n  let ydg := SciLean.revDerivProjUpdate K Yi g x;\n  let zdf := SciLean.revDerivProjUpdate K Zi f x;\n  ((ydg.1, zdf.1), fun i dyz dx =>\n    match i, dyz with\n    | Sum.inl j, dyz => ydg.2 j dyz dx\n    | Sum.inr j, dyz => zdf.2 j dyz dx)"}
{"name":"HMul.hMul.arg_a0a1.revDeriv_rule","declaration":"theorem HMul.hMul.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f x * g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDeriv K g x;\n  (ydf.1 * zdg.1, fun dx' =>\n    let dx₁ := (starRingEnd K) zdg.1 * dx';\n    let dx₂ := (starRingEnd K) ydf.1 * dx';\n    let dx := zdg.2 dx₂;\n    ydf.2 dx₁ dx)"}
{"name":"Inner.inner.arg_a0a1.revDeriv_rule","declaration":"theorem Inner.inner.arg_a0a1.revDeriv_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff R f) (hg : SciLean.HasAdjDiff R g) : (SciLean.revDeriv R fun x => ⟪f x, g x⟫_R) = fun x =>\n  let y₁df := SciLean.revDeriv R f x;\n  let y₂dg := SciLean.revDerivUpdate R g x;\n  (⟪y₁df.1, y₂dg.1⟫_R, fun dr => y₂dg.2 (dr • y₁df.1) (y₁df.2 (dr • y₂dg.1)))"}
{"name":"SciLean.revDerivUpdate_snd_zero","declaration":"theorem SciLean.revDerivUpdate_snd_zero (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) (dx : X) : (SciLean.revDerivUpdate K f x).2 0 dx = dx"}
{"name":"SciLean.norm₂.arg_x.revDerivUpdate_rule_at","declaration":"theorem SciLean.norm₂.arg_x.revDerivUpdate_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt R f x) (hx : f x ≠ 0) : SciLean.revDerivUpdate R (fun x => ‖f x‖₂[R]) x =\n  let ydf := SciLean.revDerivUpdate R f x;\n  let ynorm := ‖ydf.1‖₂[R];\n  (ynorm, fun dr dx => ydf.2 ((ynorm⁻¹ * dr) • ydf.1) dx)"}
{"name":"SciLean.revDerivProjUpdate_snd_zero","declaration":"theorem SciLean.revDerivProjUpdate_snd_zero (K : Type u_5) (I : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {E : Type u_2} {EI : I → Type u_4} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → E) (x : X) (dx : X) (i : I) : (SciLean.revDerivProjUpdate K I f x).2 i 0 dx = dx"}
{"name":"HAdd.hAdd.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') (g : X → Y') (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProjUpdate K Yi fun x => f x + g x) = fun x =>\n  let ydf := SciLean.revDerivProjUpdate K Yi f x;\n  let ydg := SciLean.revDerivProjUpdate K Yi g x;\n  (ydf.1 + ydg.1, fun i dy dx =>\n    let dx := ydf.2 i dy dx;\n    ydg.2 i dy dx)"}
{"name":"SciLean.revDerivProjUpdate.pi_rule","declaration":"theorem SciLean.revDerivProjUpdate.pi_rule (K : Type u_1) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_3} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivProjUpdate K Unit fun x i => f x i) = fun x =>\n  let ydf := fun i => SciLean.revDerivUpdate K (fun x => f x i) x;\n  (fun i => (ydf i).1, fun x df dx => LeanColls.fold (SciLean.IndexType.univ ι) (fun dx i => (ydf i).2 (df i) dx) dx)"}
{"name":"HPow.hPow.arg_a0.revDerivUpdate_rule","declaration":"def HPow.hPow.arg_a0.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (n : ℕ) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivUpdate K fun x => f x ^ n) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let y' := ↑n * (starRingEnd K) ydf.1 ^ (n - 1);\n  (ydf.1 ^ n, fun dy dx => ydf.2 (y' * dy) dx)"}
{"name":"SciLean.revDeriv_fst","declaration":"theorem SciLean.revDeriv_fst (K : Type u_3) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_1} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : (SciLean.revDeriv K f x).1 = f x"}
{"name":"ite.arg_te.revDeriv_rule","declaration":"theorem ite.arg_te.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (c : Prop) [dec : Decidable c] (t : X → Y) (e : X → Y) : (SciLean.revDeriv K fun x => if c then t x else e x) = fun y =>\n  if c then SciLean.revDeriv K t y else SciLean.revDeriv K e y"}
{"name":"HSub.hSub.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HSub.hSub.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x => f x - g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let ydg := SciLean.revDerivUpdate K g x;\n  (ydf.1 - ydg.1, fun dy dx =>\n    let dx := ydf.2 dy dx;\n    let dy' := -dy;\n    ydg.2 dy' dx)"}
{"name":"SciLean.norm₂.arg_x.revDerivUpdate_rule","declaration":"theorem SciLean.norm₂.arg_x.revDerivUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.revDerivUpdate R fun x => ‖f x‖₂[R]) = fun x =>\n  let ydf := SciLean.revDerivUpdate R f x;\n  let ynorm := ‖ydf.1‖₂[R];\n  (ynorm, fun dr dx => ydf.2 ((ynorm⁻¹ * dr) • ydf.1) dx)"}
{"name":"HSMul.hSMul.arg_a0a1.revDeriv_rule","declaration":"theorem HSMul.hSMul.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiHilbert K Y] (f : X → K) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f x • g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDeriv K g x;\n  (ydf.1 • zdg.1, fun dy' =>\n    let dk := ⟪zdg.1, dy'⟫_K;\n    let dx := zdg.2 dy';\n    let dx := (starRingEnd K) ydf.1 • dx;\n    ydf.2 dk dx)"}
{"name":"HAdd.hAdd.arg_a0a1.revDerivProj_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') (g : X → Y') (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K Yi fun x => f x + g x) = fun x =>\n  let ydf := SciLean.revDerivProj K Yi f x;\n  let ydg := SciLean.revDerivProjUpdate K Yi g x;\n  (ydf.1 + ydg.1, fun i dy =>\n    let dx := ydf.2 i dy;\n    ydg.2 i dy dx)"}
{"name":"HMul.hMul.arg_a0a1.revDerivProj_rule","declaration":"theorem HMul.hMul.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K Unit fun x => f x * g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDeriv K g x;\n  (ydf.1 * zdg.1, fun x dy =>\n    let dy₁ := (starRingEnd K) zdg.1 * dy;\n    let dy₂ := (starRingEnd K) ydf.1 * dy;\n    let dx := zdg.2 dy₂;\n    ydf.2 dy₁ dx)"}
{"name":"SciLean.Norm2.norm2.arg_a0.revDerivProj_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.revDerivProj_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) : (SciLean.revDerivProj R Unit fun x => ‖f x‖₂²) = fun x =>\n  let ydf := SciLean.revDeriv R f x;\n  let ynorm2 := ‖ydf.1‖₂²;\n  (ynorm2, fun x dr => (2 * dr) • ydf.2 ydf.1)"}
{"name":"SciLean.Norm2.norm2.arg_a0.revDerivUpdate_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.revDerivUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) : (SciLean.revDerivUpdate R fun x => ‖f x‖₂²) = fun x =>\n  let ydf := SciLean.revDerivUpdate R f x;\n  let ynorm2 := ‖ydf.1‖₂²;\n  (ynorm2, fun dr dx => ydf.2 ((2 * dr) • ydf.1) dx)"}
{"name":"SciLean.norm₂.arg_x.revDeriv_rule_at","declaration":"theorem SciLean.norm₂.arg_x.revDeriv_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt R f x) (hx : f x ≠ 0) : SciLean.revDeriv R (fun x => ‖f x‖₂[R]) x =\n  let ydf := SciLean.revDeriv R f x;\n  let ynorm := ‖ydf.1‖₂[R];\n  (ynorm, fun dr => (ynorm⁻¹ * dr) • ydf.2 ydf.1)"}
{"name":"SciLean.norm₂.arg_x.revDerivProjUpdate_rule_at","declaration":"theorem SciLean.norm₂.arg_x.revDerivProjUpdate_rule_at {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (x : X) (hf : SciLean.HasAdjDiffAt R f x) (hx : f x ≠ 0) : SciLean.revDerivProjUpdate R Unit (fun x => ‖f x‖₂[R]) x =\n  let ydf := SciLean.revDerivUpdate R f x;\n  let ynorm := ‖ydf.1‖₂[R];\n  (ynorm, fun x dr dx => ydf.2 ((ynorm⁻¹ * dr) • ydf.1) dx)"}
{"name":"SciLean.revDerivProj.pi_rule","declaration":"theorem SciLean.revDerivProj.pi_rule (K : Type u_1) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {ι : Type u_3} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivProj K Unit fun x i => f x i) = fun x =>\n  let ydf := fun i => SciLean.revDerivUpdate K (fun x => f x i) x;\n  (fun i => (ydf i).1, fun x df => LeanColls.fold (SciLean.IndexType.univ ι) (fun dx i => (ydf i).2 (df i) dx) 0)"}
{"name":"SciLean.revDerivProjUpdate.id_rule","declaration":"theorem SciLean.revDerivProjUpdate.id_rule (K : Type u_4) (I : Type u_1) [RCLike K] {E : Type u_2} {EI : I → Type u_3} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] : (SciLean.revDerivProjUpdate K I fun x => x) = fun x => (x, fun i de dx => SciLean.structModify i (fun ei => ei + de) dx)"}
{"name":"SciLean.revDerivUpdate.id_rule","declaration":"theorem SciLean.revDerivUpdate.id_rule (K : Type u_2) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] : (SciLean.revDerivUpdate K fun x => x) = fun x => (x, fun dx' dx => dx + dx')"}
{"name":"Inner.inner.arg_a0a1.revDerivProj_rule","declaration":"theorem Inner.inner.arg_a0a1.revDerivProj_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff R f) (hg : SciLean.HasAdjDiff R g) : (SciLean.revDerivProj R Unit fun x => ⟪f x, g x⟫_R) = fun x =>\n  let y₁df := SciLean.revDeriv R f x;\n  let y₂dg := SciLean.revDerivUpdate R g x;\n  (⟪y₁df.1, y₂dg.1⟫_R, fun x dr => y₂dg.2 (dr • y₁df.1) (y₁df.2 (dr • y₂dg.1)))"}
{"name":"SciLean.revDeriv_snd_zero","declaration":"theorem SciLean.revDeriv_snd_zero (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (x : X) : (SciLean.revDeriv K f x).2 0 = 0"}
{"name":"SciLean.revDeriv.const_rule","declaration":"theorem SciLean.revDeriv.const_rule (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (y : Y) : (SciLean.revDeriv K fun x => y) = fun x => (y, fun x => 0)"}
{"name":"SciLean.revDerivProj.comp_rule","declaration":"theorem SciLean.revDerivProj.comp_rule (K : Type u_1) (I : Type u_5) [RCLike K] {X : Type u_4} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_3} {EI : I → Type u_6} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : Y → E) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K I fun x => f (g x)) = fun x =>\n  let ydg' := SciLean.revDeriv K g x;\n  let zdf' := SciLean.revDerivProj K I f ydg'.1;\n  (zdf'.1, fun i de => ydg'.2 (zdf'.2 i de))"}
{"name":"HPow.hPow.arg_a0.revDeriv_rule","declaration":"def HPow.hPow.arg_a0.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (n : ℕ) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => f x ^ n) = fun x =>\n  let ydf := SciLean.revDeriv K f x;\n  let y' := ↑n * (starRingEnd K) ydf.1 ^ (n - 1);\n  (ydf.1 ^ n, fun dx' => ydf.2 (y' * dx'))"}
{"name":"HDiv.hDiv.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem HDiv.hDiv.arg_a0a1.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) (hx : ∀ (x : X), g x ≠ 0) : (SciLean.revDerivProjUpdate K Unit fun x => f x / g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDerivUpdate K g x;\n  (ydf.1 / zdg.1, fun x dx' dx =>\n    let c := 1 / (starRingEnd K) zdg.1 ^ 2;\n    let a := -(c * (starRingEnd K) ydf.1);\n    let b := c * (starRingEnd K) zdg.1;\n    zdg.2 (a • dx') (ydf.2 (b • dx') dx))"}
{"name":"SciLean.norm₂.arg_x.revDerivProjUpdate_rule","declaration":"theorem SciLean.norm₂.arg_x.revDerivProjUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) (hx : ∀ (x : X), f x ≠ 0) : (SciLean.revDerivProjUpdate R Unit fun x => ‖f x‖₂[R]) = fun x =>\n  let ydf := SciLean.revDerivUpdate R f x;\n  let ynorm := ‖ydf.1‖₂[R];\n  (ynorm, fun x dr dx => ydf.2 ((ynorm⁻¹ * dr) • ydf.1) dx)"}
{"name":"HAdd.hAdd.arg_a0a1.revDeriv_rule","declaration":"theorem HAdd.hAdd.arg_a0a1.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDeriv K fun x => f x + g x) = fun x =>\n  let ydf := SciLean.revDeriv K f x;\n  let ydg := SciLean.revDerivUpdate K g x;\n  (ydf.1 + ydg.1, fun dy =>\n    let dx := ydf.2 dy;\n    ydg.2 dy dx)"}
{"name":"IndexType.sum.arg_f.revDeriv_rule","declaration":"theorem IndexType.sum.arg_f.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {ι : Type} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDeriv K fun x =>  ∑ i, f x i) = fun x =>\n  let ydf := SciLean.revDeriv K f x;\n  ( ∑ i, ydf.1 i, fun dy => ydf.2 fun x => dy)"}
{"name":"SciLean.revDerivUpdate.const_rule","declaration":"theorem SciLean.revDerivUpdate.const_rule (K : Type u_3) [RCLike K] {X : Type u_1} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] (y : Y) : (SciLean.revDerivUpdate K fun x => y) = fun x => (y, fun x dx => dx)"}
{"name":"Prod.fst.arg_self.revDerivProj_rule","declaration":"theorem Prod.fst.arg_self.revDerivProj_rule {K : Type} [RCLike K] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {X' : Type} {Xi : Type} {XI : Xi → Type} [SciLean.StructType X' Xi XI] [DecidableEq Xi] [SciLean.SemiInnerProductSpace K X'] [(i : Xi) → SciLean.SemiInnerProductSpace K (XI i)] {W : Type} [SciLean.SemiInnerProductSpace K W] (f : W → X' × Y) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDerivProj K Xi fun x => (f x).1) = fun w =>\n  let xydf := SciLean.revDerivProj K (Xi ⊕ Unit) f w;\n  (xydf.1.1, fun i dxy => xydf.2 (Sum.inl i) dxy)"}
{"name":"Prod.mk.arg_fstsnd.revDeriv_rule","declaration":"theorem Prod.mk.arg_fstsnd.revDeriv_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {Z : Type} [SciLean.SemiInnerProductSpace K Z] (g : X → Y) (f : X → Z) (hg : SciLean.HasAdjDiff K g) (hf : SciLean.HasAdjDiff K f) : (SciLean.revDeriv K fun x => (g x, f x)) = fun x =>\n  let ydg := SciLean.revDeriv K g x;\n  let zdf := SciLean.revDerivUpdate K f x;\n  ((ydg.1, zdf.1), fun dyz =>\n    let dx := ydg.2 dyz.1;\n    zdf.2 dyz.2 dx)"}
{"name":"Inner.inner.arg_a0a1.revDerivProjUpdate_rule","declaration":"theorem Inner.inner.arg_a0a1.revDerivProjUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (g : X → Y) (hf : SciLean.HasAdjDiff R f) (hg : SciLean.HasAdjDiff R g) : (SciLean.revDerivProjUpdate R Unit fun x => ⟪f x, g x⟫_R) = fun x =>\n  let y₁df := SciLean.revDerivUpdate R f x;\n  let y₂dg := SciLean.revDerivUpdate R g x;\n  (⟪y₁df.1, y₂dg.1⟫_R, fun x dr dx => y₂dg.2 (dr • y₁df.1) (y₁df.2 (dr • y₂dg.1) dx))"}
{"name":"SciLean.Norm2.norm2.arg_a0.revDerivProjUpdate_rule","declaration":"theorem SciLean.Norm2.norm2.arg_a0.revDerivProjUpdate_rule {R : Type u_1} [SciLean.RealScalar R] {X : Type u_2} [SciLean.SemiInnerProductSpace R X] {Y : Type u_3} [SciLean.SemiHilbert R Y] (f : X → Y) (hf : SciLean.HasAdjDiff R f) : (SciLean.revDerivProjUpdate R Unit fun x => ‖f x‖₂²) = fun x =>\n  let ydf := SciLean.revDerivUpdate R f x;\n  let ynorm2 := ‖ydf.1‖₂²;\n  (ynorm2, fun x dr dx => ydf.2 ((2 * dr) • ydf.1) dx)"}
{"name":"SciLean.revDerivProj.let_rule","declaration":"theorem SciLean.revDerivProj.let_rule (K : Type u_1) (I : Type u_5) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {Y : Type u_2} [SciLean.SemiInnerProductSpace K Y] {E : Type u_4} {EI : I → Type u_6} [SciLean.StructType E I EI] [DecidableEq I] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (f : X → Y → E) (g : X → Y) (hf : SciLean.HasAdjDiff K fun x =>\n  match x with\n  | (x, y) => f x y) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K I fun x =>\n    let y := g x;\n    f x y) =\n  fun x =>\n  let ydg' := SciLean.revDerivUpdate K g x;\n  let zdf' :=\n    SciLean.revDerivProj K I\n      (fun x =>\n        match x with\n        | (x, y) => f x y)\n      (x, ydg'.1);\n  (zdf'.1, fun i dei =>\n    let dxy := zdf'.2 i dei;\n    ydg'.2 dxy.2 dxy.1)"}
{"name":"SciLean.revDerivUpdate.apply_rule","declaration":"theorem SciLean.revDerivUpdate.apply_rule (K : Type u_3) (I : Type u_1) [RCLike K] {EI : I → Type u_2} [SciLean.IndexType I] [DecidableEq I] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (i : I) : (SciLean.revDerivUpdate K fun x => x i) = fun x =>\n  (x i, fun dxi dx => SciLean.structModify i (fun dxi' => dxi' + dxi) dx)"}
{"name":"SciLean.revDeriv.apply_rule","declaration":"theorem SciLean.revDeriv.apply_rule (K : Type u_3) (I : Type u_1) [RCLike K] {EI : I → Type u_2} [SciLean.IndexType I] [DecidableEq I] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] (i : I) : (SciLean.revDeriv K fun x => x i) = fun x => (x i, fun dxi => SciLean.oneHot i dxi)"}
{"name":"SciLean.revDerivUpdate.let_rule","declaration":"theorem SciLean.revDerivUpdate.let_rule (K : Type u_1) [RCLike K] {X : Type u_2} [SciLean.SemiInnerProductSpace K X] {Y : Type u_3} [SciLean.SemiInnerProductSpace K Y] {Z : Type u_4} [SciLean.SemiInnerProductSpace K Z] (f : X → Y → Z) (g : X → Y) (hf : SciLean.HasAdjDiff K fun xy => f xy.1 xy.2) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivUpdate K fun x =>\n    let y := g x;\n    f x y) =\n  fun x =>\n  let ydg := SciLean.revDerivUpdate K g x;\n  let zdf := SciLean.revDerivUpdate K (fun xy => f xy.1 xy.2) (x, ydg.1);\n  (zdf.1, fun dz dx =>\n    let dxdy := zdf.2 dz (dx, 0);\n    let dx := ydg.2 dxdy.2 dxdy.1;\n    dx)"}
{"name":"HDiv.hDiv.arg_a0a1.revDerivUpdate_rule","declaration":"theorem HDiv.hDiv.arg_a0a1.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] (f : X → K) (g : X → K) (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) (hx : ∀ (x : X), g x ≠ 0) : (SciLean.revDerivUpdate K fun x => f x / g x) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  let zdg := SciLean.revDerivUpdate K g x;\n  (ydf.1 / zdg.1, fun dx' dx =>\n    let c := 1 / (starRingEnd K) zdg.1 ^ 2;\n    let a := -(c * (starRingEnd K) ydf.1);\n    let b := c * (starRingEnd K) zdg.1;\n    zdg.2 (a • dx') (ydf.2 (b • dx') dx))"}
{"name":"HSub.hSub.arg_a0a1.revDerivProj_rule","declaration":"theorem HSub.hSub.arg_a0a1.revDerivProj_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] (f : X → Y') (g : X → Y') (hf : SciLean.HasAdjDiff K f) (hg : SciLean.HasAdjDiff K g) : (SciLean.revDerivProj K Yi fun x => f x - g x) = fun x =>\n  let ydf := SciLean.revDerivProj K Yi f x;\n  let ydg := SciLean.revDerivProjUpdate K Yi g x;\n  (ydf.1 - ydg.1, fun i dy =>\n    let dx := ydf.2 i dy;\n    let dy' := -dy;\n    ydg.2 i dy' dx)"}
{"name":"SciLean.revDerivProjUpdate","declaration":"def SciLean.revDerivProjUpdate (K : Type u_1) (I : Type u_2) [RCLike K] {X : Type u_3} [SciLean.SemiInnerProductSpace K X] {E : Type u_4} {EI : I → Type u_5} [SciLean.StructType E I EI] [SciLean.SemiInnerProductSpace K E] [(i : I) → SciLean.SemiInnerProductSpace K (EI i)] [DecidableEq I] (f : X → E) (x : X) : E × ((i : I) → EI i → X → X)"}
{"name":"IndexType.sum.arg_f.revDerivProjUpdate_rule","declaration":"theorem IndexType.sum.arg_f.revDerivProjUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y' : Type} {Yi : Type} {YI : Yi → Type} [SciLean.StructType Y' Yi YI] [DecidableEq Yi] [SciLean.SemiInnerProductSpace K Y'] [(i : Yi) → SciLean.SemiInnerProductSpace K (YI i)] {ι : Type} [SciLean.IndexType ι] [DecidableEq ι] (f : X → ι → Y') (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivProjUpdate K Yi fun x =>  ∑ i, f x i) = fun x =>\n  let ydf := SciLean.revDerivProjUpdate K (ι × Yi) f x;\n  ( ∑ i, ydf.1 i, fun j dy dx => LeanColls.fold (SciLean.IndexType.univ ι) (fun dx i => ydf.2 (i, j) dy dx) dx)"}
{"name":"IndexType.sum.arg_f.revDerivUpdate_rule","declaration":"theorem IndexType.sum.arg_f.revDerivUpdate_rule {K : Type} [RCLike K] {X : Type} [SciLean.SemiInnerProductSpace K X] {Y : Type} [SciLean.SemiInnerProductSpace K Y] {ι : Type} [SciLean.IndexType ι] (f : X → ι → Y) (hf : ∀ (i : ι), SciLean.HasAdjDiff K fun x => f x i) : (SciLean.revDerivUpdate K fun x =>  ∑ i, f x i) = fun x =>\n  let ydf := SciLean.revDerivUpdate K f x;\n  ( ∑ i, ydf.1 i, fun dy dx => ydf.2 (fun x => dy) dx)"}