| {"name":"SciLean.Scalar.sin.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.HasAdjDiff_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : SciLean.HasAdjDiff C fun u => SciLean.Scalar.sin (x u)"} | |
| {"name":"SciLean.Scalar.cos.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.CDifferentiable_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : SciLean.CDifferentiable C fun w => SciLean.Scalar.cos (x w)"} | |
| {"name":"SciLean.Scalar.tanh.arg_x.revDeriv_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.revDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : (SciLean.revDeriv C fun u => SciLean.Scalar.tanh (x u)) = fun u =>\n let xdx := SciLean.revDeriv C x u;\n (SciLean.Scalar.tanh xdx.1, fun dy => xdx.2 ((starRingEnd C) (1 - SciLean.Scalar.tanh xdx.1 ^ 2) * dy))"} | |
| {"name":"SciLean.Scalar.cos.arg_x.ceriv_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.ceriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.cderiv C fun w => SciLean.Scalar.cos (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let s := -SciLean.Scalar.sin xdx.1;\n xdx.2 * s"} | |
| {"name":"SciLean.Scalar.cos.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.HasAdjDiff_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : SciLean.HasAdjDiff C fun u => SciLean.Scalar.cos (x u)"} | |
| {"name":"SciLean.Scalar.sin.arg_x.revDeriv_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.revDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : (SciLean.revDeriv C fun u => SciLean.Scalar.sin (x u)) = fun u =>\n let xdx := SciLean.revDeriv C x u;\n (SciLean.Scalar.sin xdx.1, fun dy => xdx.2 ((starRingEnd C) (SciLean.Scalar.cos xdx.1) * dy))"} | |
| {"name":"SciLean.Scalar.sin.arg_x.ceriv_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.ceriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.cderiv C fun w => SciLean.Scalar.sin (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let c := SciLean.Scalar.cos xdx.1;\n xdx.2 * c"} | |
| {"name":"SciLean.Scalar.cos.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.fwdDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.fwdDeriv C fun w => SciLean.Scalar.cos (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let s := SciLean.Scalar.sin xdx.1;\n let c := SciLean.Scalar.cos xdx.1;\n (c, xdx.2 * -s)"} | |
| {"name":"SciLean.Scalar.tanh.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.CDifferentiable_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : SciLean.CDifferentiable C fun w => SciLean.Scalar.tanh (x w)"} | |
| {"name":"SciLean.Scalar.tanh.arg_x.ceriv_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.ceriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.cderiv C fun w => SciLean.Scalar.tanh (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let t := SciLean.Scalar.tanh xdx.1;\n let dt := 1 - t ^ 2;\n xdx.2 * dt"} | |
| {"name":"SciLean.Scalar.tanh.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.fwdDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.fwdDeriv C fun w => SciLean.Scalar.tanh (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let t := SciLean.Scalar.tanh xdx.1;\n let dt := 1 - t ^ 2;\n (t, xdx.2 * dt)"} | |
| {"name":"SciLean.Scalar.tanh.arg_x.HasAdjDiff_rule","declaration":"theorem SciLean.Scalar.tanh.arg_x.HasAdjDiff_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : SciLean.HasAdjDiff C fun u => SciLean.Scalar.tanh (x u)"} | |
| {"name":"SciLean.Scalar.sin.arg_x.fwdDeriv_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.fwdDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : (SciLean.fwdDeriv C fun w => SciLean.Scalar.sin (x w)) = fun w dw =>\n let xdx := SciLean.fwdDeriv C x w dw;\n let s := SciLean.Scalar.sin xdx.1;\n let c := SciLean.Scalar.cos xdx.1;\n (s, xdx.2 * c)"} | |
| {"name":"SciLean.Scalar.cos.arg_x.revDeriv_rule","declaration":"theorem SciLean.Scalar.cos.arg_x.revDeriv_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {U : Type u_2} [SciLean.SemiInnerProductSpace C U] (x : U → C) (hx : SciLean.HasAdjDiff C x) : (SciLean.revDeriv C fun u => SciLean.Scalar.cos (x u)) = fun u =>\n let xdx := SciLean.revDeriv C x u;\n (SciLean.Scalar.cos xdx.1, fun dy => xdx.2 (-(starRingEnd C) (SciLean.Scalar.sin xdx.1) * dy))"} | |
| {"name":"SciLean.Scalar.sin.arg_x.CDifferentiable_rule","declaration":"theorem SciLean.Scalar.sin.arg_x.CDifferentiable_rule {R : Type u_3} {C : Type u_1} [SciLean.Scalar R C] {W : Type u_2} [SciLean.Vec C W] (x : W → C) (hx : SciLean.CDifferentiable C x) : SciLean.CDifferentiable C fun w => SciLean.Scalar.sin (x w)"} | |