Theorem: Probly.flip.arg_x.randFwdDeriv_rule (Probly/Flip.lean)

State before:
W : Type u_1
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : NormedSpace ℝ W
inst✝¹¹ : MeasurableSpace W
X : Type ?u.33480
inst✝¹⁰ : NormedAddCommGroup X
inst✝⁹ : NormedSpace ℝ X
inst✝⁸ : MeasurableSpace X
Y : Type ?u.33597
inst✝⁷ : NormedAddCommGroup Y
inst✝⁶ : NormedSpace ℝ Y
inst✝⁵ : MeasurableSpace Y
Z : Type ?u.33714
inst✝⁴ : NormedAddCommGroup Z
inst✝³ : NormedSpace ℝ Z
inst✝² : MeasurableSpace Z
α : Type ?u.33831
inst✝¹ : MeasurableSpace α
β : Type ?u.33837
inst✝ : MeasurableSpace β
x : W → ℝ
hf : Differentiable ℝ x
⊢ (randFwdDeriv fun w => flip (x w)) = fun w dw =>
    let x' := x w;
    let dx' := (fderiv ℝ x w) dw;
    { val := flip x', dval := dx' • dflip }
Tactic:
unfold randFwdDeriv
State after:
W : Type u_1
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : NormedSpace ℝ W
inst✝¹¹ : MeasurableSpace W
X : Type ?u.33480
inst✝¹⁰ : NormedAddCommGroup X
inst✝⁹ : NormedSpace ℝ X
inst✝⁸ : MeasurableSpace X
Y : Type ?u.33597
inst✝⁷ : NormedAddCommGroup Y
inst✝⁶ : NormedSpace ℝ Y
inst✝⁵ : MeasurableSpace Y
Z : Type ?u.33714
inst✝⁴ : NormedAddCommGroup Z
inst✝³ : NormedSpace ℝ Z
inst✝² : MeasurableSpace Z
α : Type ?u.33831
inst✝¹ : MeasurableSpace α
β : Type ?u.33837
inst✝ : MeasurableSpace β
x : W → ℝ
hf : Differentiable ℝ x
⊢ (fun x_1 dx => { val := (fun w => flip (x w)) x_1, dval := randDeriv (fun w => flip (x w)) x_1 dx }) = fun w dw =>
    let x' := x w;
    let dx' := (fderiv ℝ x w) dw;
    { val := flip x', dval := dx' • dflip }

State before:
W : Type u_1
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : NormedSpace ℝ W
inst✝¹¹ : MeasurableSpace W
X : Type ?u.33480
inst✝¹⁰ : NormedAddCommGroup X
inst✝⁹ : NormedSpace ℝ X
inst✝⁸ : MeasurableSpace X
Y : Type ?u.33597
inst✝⁷ : NormedAddCommGroup Y
inst✝⁶ : NormedSpace ℝ Y
inst✝⁵ : MeasurableSpace Y
Z : Type ?u.33714
inst✝⁴ : NormedAddCommGroup Z
inst✝³ : NormedSpace ℝ Z
inst✝² : MeasurableSpace Z
α : Type ?u.33831
inst✝¹ : MeasurableSpace α
β : Type ?u.33837
inst✝ : MeasurableSpace β
x : W → ℝ
hf : Differentiable ℝ x
⊢ (fun x_1 dx => { val := (fun w => flip (x w)) x_1, dval := randDeriv (fun w => flip (x w)) x_1 dx }) = fun w dw =>
    let x' := x w;
    let dx' := (fderiv ℝ x w) dw;
    { val := flip x', dval := dx' • dflip }
Tactic:
simp (disch:= first | apply hf | sorry) [rand_simp]
State after:
W : Type u_1
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : NormedSpace ℝ W
inst✝¹¹ : MeasurableSpace W
X : Type ?u.33480
inst✝¹⁰ : NormedAddCommGroup X
inst✝⁹ : NormedSpace ℝ X
inst✝⁸ : MeasurableSpace X
Y : Type ?u.33597
inst✝⁷ : NormedAddCommGroup Y
inst✝⁶ : NormedSpace ℝ Y
inst✝⁵ : MeasurableSpace Y
Z : Type ?u.33714
inst✝⁴ : NormedAddCommGroup Z
inst✝³ : NormedSpace ℝ Z
inst✝² : MeasurableSpace Z
α : Type ?u.33831
inst✝¹ : MeasurableSpace α
β : Type ?u.33837
inst✝ : MeasurableSpace β
x : W → ℝ
hf : Differentiable ℝ x
⊢ (fun x_1 dx => { val := flip (x x_1), dval := randDeriv flip (x x_1) ((fderiv ℝ (fun x_2 => x x_2) x_1) dx) }) =
    fun w dw => { val := flip (x w), dval := (fderiv ℝ x w) dw • dflip }

State before:
W : Type u_1
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : NormedSpace ℝ W
inst✝¹¹ : MeasurableSpace W
X : Type ?u.33480
inst✝¹⁰ : NormedAddCommGroup X
inst✝⁹ : NormedSpace ℝ X
inst✝⁸ : MeasurableSpace X
Y : Type ?u.33597
inst✝⁷ : NormedAddCommGroup Y
inst✝⁶ : NormedSpace ℝ Y
inst✝⁵ : MeasurableSpace Y
Z : Type ?u.33714
inst✝⁴ : NormedAddCommGroup Z
inst✝³ : NormedSpace ℝ Z
inst✝² : MeasurableSpace Z
α : Type ?u.33831
inst✝¹ : MeasurableSpace α
β : Type ?u.33837
inst✝ : MeasurableSpace β
x : W → ℝ
hf : Differentiable ℝ x
⊢ (fun x_1 dx => { val := flip (x x_1), dval := randDeriv flip (x x_1) ((fderiv ℝ (fun x_2 => x x_2) x_1) dx) }) =
    fun w dw => { val := flip (x w), dval := (fderiv ℝ x w) dw • dflip }
Tactic:
rw[flip.arg_x.randDeriv_rule _ differentiable_id']
State after:
W : Type u_1
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : NormedSpace ℝ W
inst✝¹¹ : MeasurableSpace W
X : Type ?u.33480
inst✝¹⁰ : NormedAddCommGroup X
inst✝⁹ : NormedSpace ℝ X
inst✝⁸ : MeasurableSpace X
Y : Type ?u.33597
inst✝⁷ : NormedAddCommGroup Y
inst✝⁶ : NormedSpace ℝ Y
inst✝⁵ : MeasurableSpace Y
Z : Type ?u.33714
inst✝⁴ : NormedAddCommGroup Z
inst✝³ : NormedSpace ℝ Z
inst✝² : MeasurableSpace Z
α : Type ?u.33831
inst✝¹ : MeasurableSpace α
β : Type ?u.33837
inst✝ : MeasurableSpace β
x : W → ℝ
hf : Differentiable ℝ x
⊢ (fun x_1 dx =>
      { val := flip (x x_1),
        dval :=
          (fun w dw =>
              let dx' := (fderiv ℝ (fun x => x) w) dw;
              dx' • dflip)
            (x x_1) ((fderiv ℝ (fun x_2 => x x_2) x_1) dx) }) =
    fun w dw => { val := flip (x w), dval := (fderiv ℝ x w) dw • dflip }

State before:
W : Type u_1
inst✝¹³ : NormedAddCommGroup W
inst✝¹² : NormedSpace ℝ W
inst✝¹¹ : MeasurableSpace W
X : Type ?u.33480
inst✝¹⁰ : NormedAddCommGroup X
inst✝⁹ : NormedSpace ℝ X
inst✝⁸ : MeasurableSpace X
Y : Type ?u.33597
inst✝⁷ : NormedAddCommGroup Y
inst✝⁶ : NormedSpace ℝ Y
inst✝⁵ : MeasurableSpace Y
Z : Type ?u.33714
inst✝⁴ : NormedAddCommGroup Z
inst✝³ : NormedSpace ℝ Z
inst✝² : MeasurableSpace Z
α : Type ?u.33831
inst✝¹ : MeasurableSpace α
β : Type ?u.33837
inst✝ : MeasurableSpace β
x : W → ℝ
hf : Differentiable ℝ x
⊢ (fun x_1 dx =>
      { val := flip (x x_1),
        dval :=
          (fun w dw =>
              let dx' := (fderiv ℝ (fun x => x) w) dw;
              dx' • dflip)
            (x x_1) ((fderiv ℝ (fun x_2 => x x_2) x_1) dx) }) =
    fun w dw => { val := flip (x w), dval := (fderiv ℝ x w) dw • dflip }
Tactic:
simp
State after:
no goals