File: Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean

full_name: iteratedDerivWithin_add
code: theorem iteratedDerivWithin_add (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) :
    iteratedDerivWithin n (f + g) s x =
      iteratedDerivWithin n f s x + iteratedDerivWithin n g s x 
goal: iteratedDerivWithin n (f + g) s x = iteratedDerivWithin n f s x + iteratedDerivWithin n g s x

full_name: iteratedDerivWithin_congr
code: theorem iteratedDerivWithin_congr (hfg : Set.EqOn f g s) :
    Set.EqOn (iteratedDerivWithin n f s) (iteratedDerivWithin n g s) s 
goal: Set.EqOn (iteratedDerivWithin n f s) (iteratedDerivWithin n g s) s

full_name: iteratedDerivWithin_const_add
code: theorem iteratedDerivWithin_const_add (hn : 0 < n) (c : F) :
    iteratedDerivWithin n (fun z => c + f z) s x = iteratedDerivWithin n f s x 
goal: iteratedDerivWithin n (fun z => c + f z) s x = iteratedDerivWithin n f s x

full_name: iteratedDerivWithin_const_neg
code: theorem iteratedDerivWithin_const_neg (hn : 0 < n) (c : F) :
    iteratedDerivWithin n (fun z => c - f z) s x = iteratedDerivWithin n (fun z => -f z) s x 
goal: iteratedDerivWithin n (fun z => c - f z) s x = iteratedDerivWithin n (fun z => -f z) s x

full_name: iteratedDerivWithin_const_smul
code: theorem iteratedDerivWithin_const_smul (c : R) (hf : ContDiffOn 𝕜 n f s) :
    iteratedDerivWithin n (c • f) s x = c • iteratedDerivWithin n f s x 
goal: iteratedDerivWithin n (c • f) s x = c • iteratedDerivWithin n f s x

full_name: iteratedDerivWithin_const_mul
code: theorem iteratedDerivWithin_const_mul (c : 𝕜) {f : 𝕜 → 𝕜} (hf : ContDiffOn 𝕜 n f s) :
    iteratedDerivWithin n (fun z => c * f z) s x = c * iteratedDerivWithin n f s x 
goal: iteratedDerivWithin n (fun z => c * f z) s x = c * iteratedDerivWithin n f s x

full_name: iteratedDerivWithin_neg
code: theorem iteratedDerivWithin_neg :
    iteratedDerivWithin n (-f) s x = -iteratedDerivWithin n f s x 
goal: iteratedDerivWithin n (-f) s x = -iteratedDerivWithin n f s x

full_name: iteratedDerivWithin_neg'
code: theorem iteratedDerivWithin_neg' :
    iteratedDerivWithin n (fun z => -f z) s x = -iteratedDerivWithin n f s x 
goal: iteratedDerivWithin n (fun z => -f z) s x = -iteratedDerivWithin n f s x

full_name: iteratedDerivWithin_sub
code: theorem iteratedDerivWithin_sub (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) :
    iteratedDerivWithin n (f - g) s x =
      iteratedDerivWithin n f s x - iteratedDerivWithin n g s x 
goal: iteratedDerivWithin n (f - g) s x = iteratedDerivWithin n f s x - iteratedDerivWithin n g s x

full_name: iteratedDeriv_const_smul
code: theorem iteratedDeriv_const_smul {n : ℕ} {f : 𝕜 → F} (h : ContDiff 𝕜 n f) (c : 𝕜) :
    iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n • iteratedDeriv n f (c * x) 
goal: (iteratedDeriv n fun x => f (c * x)) = fun x => c ^ n • iteratedDeriv n f (c * x)

full_name: iteratedDeriv_const_mul
code: theorem iteratedDeriv_const_mul {n : ℕ} {f : 𝕜 → 𝕜} (h : ContDiff 𝕜 n f) (c : 𝕜) :
    iteratedDeriv n (fun x => f (c * x)) = fun x => c ^ n * iteratedDeriv n f (c * x) 
goal: (iteratedDeriv n fun x => f (c * x)) = fun x => c ^ n * iteratedDeriv n f (c * x)

full_name: iteratedDeriv_neg
code: lemma iteratedDeriv_neg (n : ℕ) (f : 𝕜 → F) (a : 𝕜) :
    iteratedDeriv n (fun x ↦ -(f x)) a = -(iteratedDeriv n f a) 
goal: iteratedDeriv n (fun x => -f x) a = -iteratedDeriv n f a

full_name: iteratedDeriv_comp_neg
code: lemma iteratedDeriv_comp_neg (n : ℕ) (f : 𝕜 → F) (a : 𝕜) :
    iteratedDeriv n (fun x ↦ f (-x)) a = (-1 : 𝕜) ^ n • iteratedDeriv n f (-a) 
goal: iteratedDeriv n (fun x => f (-x)) a = (-1) ^ n • iteratedDeriv n f (-a)