SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\begin{array}{l} \left\| \mathfrak {g} _ {0} ^ {- 1 / 2} \left(D \mathfrak {g} _ {0} \left[ K _ {0} (y), K _ {0} (y) \right] - D \mathfrak {g} _ {0} \left[ K _ {0} \left(y ^ {\prime}\right), K _ {0} \left(y ^ {\prime}\right) \right]\right) \right\| _ {2} \\ \leq 4 r (1 + (1 - r) ^ {2}) ^ {- 2} \{1 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \\ + 4 r (1 + (1 - r) ^ {2}) ^ {- 2} \{1 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \} ^ {2} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \\ \leq 4 r (1 + (1 - r) ^ {2}) ^ {- 2} \left\{2 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \right\} ^ {2} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}}. \\ \end{array}
$$
Conclusion of Step 1. By combining the results of Steps 1.1 to 1.4, it comes that
$$
\begin{array}{l} \left\| D \mathrm {M} _ {0} (y) [ w ] D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} (y), \mathrm {K} _ {0} (y) \right] - D \mathrm {M} _ {0} (y ^ {\prime}) [ w ] D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} (y ^ {\prime}), \mathrm {K} _ {0} (y ^ {\prime}) \right] \right\| _ {\mathfrak {g} _ {0}} \\ = \left\| \mathfrak {g} _ {0} ^ {1 / 2} \left(a _ {1} (y) - a _ {1} \left(y ^ {\prime}\right)\right) \right\| _ {2} \\ \leq \left\{2 r ^ {2} (1 + (1 - r) ^ {2}) ^ {- 2} (1 2 (1 - r) ^ {- 8} + 5 \alpha (\alpha + 1) / 3 (1 - r) ^ {- 6}\right) \\ + 8 r (1 - r) ^ {- 3} \left(1 + (1 - r) ^ {2}\right) ^ {- 2} \left\{2 + 4 r (1 - r) ^ {- 3} \left(1 + (1 - r) ^ {2}\right) ^ {- 1} \right\} ^ {2} \} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \| w \| _ {\mathfrak {g} _ {0}}. \\ \end{array}
$$
Step 2. Let us now bound the second term in (34). For any  $x \in \mathsf{B}_0$ , we denote  $\overline{\mathrm{M}_0(y)} D\mathfrak{g}_0[DK_0(y)[w],K_0(y)]$  by  $a_2(x)$ . We have
$$
\begin{array}{l} \mathfrak {g} _ {0} ^ {1 / 2} \left(a _ {2} (y) - a _ {2} \left(y ^ {\prime}\right)\right) = \left\{\mathfrak {g} _ {0} ^ {1 / 2} \left(\mathrm {M} _ {0} (y) - \mathrm {M} _ {0} \left(y ^ {\prime}\right)\right) \mathfrak {g} _ {0} ^ {1 / 2} \right\} \left\{\mathfrak {g} _ {0} ^ {- 1 / 2} D \mathfrak {g} _ {0} [ D \mathrm {K} _ {0} (y) [ w ], \mathrm {K} _ {0} (y) ] \right\} \tag {44} \\ + \left\{\mathfrak {g} _ {0} ^ {1 / 2} \mathrm {M} _ {0} (y ^ {\prime}) \mathfrak {g} _ {0} ^ {1 / 2} \right\} \left\{\mathfrak {g} _ {0} ^ {- 1 / 2} \left(D \mathfrak {g} _ {0} [ D K _ {0} (y) [ w ], K _ {0} (y) ] - D \mathfrak {g} _ {0} [ D K _ {0} (y ^ {\prime}) [ w ], K _ {0} (y ^ {\prime}) ]\right) \right\}. \\ \end{array}
$$
We now aim to bound each of the four terms that appear in (44).
Step 2.1. First, we have
$$
\begin{array}{l} \left\| \mathfrak {g} _ {0} ^ {1 / 2} \left(\mathrm {M} _ {0} (y) - \mathrm {M} _ {0} \left(y ^ {\prime}\right)\right) \mathfrak {g} _ {0} ^ {1 / 2} \right\| _ {2} = \left\| \mathfrak {g} _ {0} ^ {1 / 2} \left(\mathfrak {g} (y) ^ {- 1} - \mathfrak {g} \left(y ^ {\prime}\right) ^ {- 1}\right) \mathfrak {g} _ {0} ^ {1 / 2} \right\| _ {2} \\ \leq 3 (1 - r) ^ {- 3} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}}. \quad (\text {o n t h e m o d e l o f (1 8)}) \\ \end{array}
$$
Step 2.2. Secondly, we have
$$
\begin{array}{l} \| \mathfrak {g} _ {0} ^ {- 1 / 2} D \mathfrak {g} _ {0} [ D K _ {0} (y) [ w ], K _ {0} (y) ] \| _ {2} \leq 2 \| D K _ {0} (y) [ w ] \| _ {\mathfrak {g} _ {0}} \| M _ {0} (y) ^ {- 1} (y - x ^ {(0)}) \| _ {\mathfrak {g} _ {0} ^ {- 1}} \quad (\text {L e m m a 9 - (a)}) \\ \leq 2 r \left(1 + (1 - r) ^ {2}\right) ^ {- 1} \| D K _ {0} (y) [ w ] \| _ {\mathfrak {g} _ {0}}, \quad (\text {s e e S t e p 1 . 2 .}) \\ \end{array}
$$
$$
\begin{array}{l} \| D K _ {0} (y) [ w ] \| _ {\mathfrak {g} _ {0}} \leq r \| \mathfrak {g} _ {0} ^ {- 1 / 2} (\mathfrak {g} (y) \mathfrak {g} _ {0} ^ {- 1} + I _ {d}) ^ {- 1} \mathfrak {g} _ {0} ^ {1 / 2} \| _ {2} ^ {2} \| \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} (y) ^ {1 / 2} \| _ {2} ^ {2} \| \mathfrak {g} (y) ^ {- 1 / 2} D \mathfrak {g} (y) [ w ] \mathfrak {g} (y) ^ {- 1 / 2} \| _ {2} \\ + \quad \| \mathrm {M} _ {0} (y) ^ {- 1} w \| _ {\mathfrak {g} _ {0} ^ {- 1}}. \\ \end{array}
$$
Using (36) and (38), it comes that
$$
\left\| D K _ {0} (y) [ w ] \right\| _ {\mathfrak {g} _ {0}} \leq \left\{2 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 2} + (1 + (1 - r) ^ {2}) ^ {- 1} \right\} \| w \| _ {\mathfrak {g} _ {0}}.
$$
$$
\| \mathfrak {g} _ {0} ^ {- 1 / 2} D \mathfrak {g} _ {0} [ D K _ {0} (y) [ w ], K _ {0} (y) ] \| _ {2} \leq 2 r (1 + (1 - r) ^ {2}) ^ {- 2} \{1 + 2 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \} \| w \| _ {\mathfrak {g} _ {0}}.
$$
Step 2.3. Thirdly, we have  $\| \mathfrak{g}_0^{1/2} \mathrm{M}_0(y') \mathfrak{g}_0^{1/2} \|_2 = \| \mathfrak{g}_0^{1/2} \mathfrak{g}(y')^{-1} \mathfrak{g}_0^{1/2} + \mathrm{I}_d \|_2 \leq 1 + (1 - r)^{-2}$ .
Step 2.4. Fourthly, using Lemma 9-(a), we have
$$
\begin{array}{l} \left\| \mathfrak {g} _ {0} ^ {- 1 / 2} \left(D \mathfrak {g} _ {0} \left[ D K _ {0} (y) [ w ], K _ {0} (y) \right] - D \mathfrak {g} _ {0} \left[ D K _ {0} \left(y ^ {\prime}\right) [ w ], K _ {0} \left(y ^ {\prime}\right) \right]\right) \right\| _ {2} \\ \leq \left\| D \mathfrak {g} _ {0} \left[ D K _ {0} (y) [ w ] - D K _ {0} \left(y ^ {\prime}\right) [ w ], K _ {0} (y) \right] \right\| _ {\mathfrak {g} _ {0} ^ {- 1}} + \left\| D \mathfrak {g} _ {0} \left[ D K _ {0} (y) [ w ], K _ {0} (y) - K _ {0} \left(y ^ {\prime}\right) \right] \right\| _ {\mathfrak {g} _ {0} ^ {- 1}} \\ + \left\| D \mathfrak {g} _ {0} \left[ D K _ {0} (y) [ w ] - D K _ {0} \left(y ^ {\prime}\right) [ w ], K _ {0} (y) - K _ {0} \left(y ^ {\prime}\right) \right] \right\| _ {\mathfrak {g} _ {0} ^ {- 1}} \\ \leq 2 \| D K _ {0} (y) [ w ] - D K _ {0} \left(y ^ {\prime}\right) [ w ] \| _ {\mathfrak {g} _ {0}} \| K _ {0} (y) \| _ {\mathfrak {g} _ {0}} + 2 \| D K _ {0} (y) [ w ] \| _ {\mathfrak {g} _ {0}} \| K _ {0} (y) - K _ {0} \left(y ^ {\prime}\right) \| _ {\mathfrak {g} _ {0}} \\ + 2 \| D K _ {0} (y) [ w ] - D K _ {0} \left(y ^ {\prime}\right) [ w ] \| _ {\mathfrak {g} _ {0}} \| K _ {0} (y) - K _ {0} \left(y ^ {\prime}\right) \| _ {\mathfrak {g} _ {0}} \\ \leq 2 \left\{\| \mathrm {K} _ {0} (y) \| _ {\mathfrak {g} _ {0}} + \| \mathrm {K} _ {0} (y) - \mathrm {K} _ {0} \left(y ^ {\prime}\right) \| _ {\mathfrak {g} _ {0}} \right\} \| D \mathrm {K} _ {0} (y) [ w ] - D \mathrm {K} _ {0} \left(y ^ {\prime}\right) [ w ] \| _ {\mathfrak {g} _ {0}} \\ + 2 \| D K _ {0} (y) [ w ] \| _ {\mathfrak {g} _ {0}} \| K _ {0} (y) - K _ {0} \left(y ^ {\prime}\right) \| _ {\mathfrak {g} _ {0}}. \\ \end{array}
$$
We recall that the following inequalities hold
$$
\left\| \mathrm {K} _ {0} (y) \right\| _ {\mathfrak {g} _ {0}} \leq r \left(1 + (1 - r) ^ {2}\right) ^ {- 1},
$$
$$
\| D K _ {0} (y) [ w ] \| _ {\mathfrak {g} _ {0}} \leq (1 + (1 - r) ^ {2}) ^ {- 1} \{1 + 2 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \} \| w \| _ {\mathfrak {g} _ {0}},
$$
$$
\| \mathrm {K} _ {0} (y) - \mathrm {K} _ {0} \left(y ^ {\prime}\right) \| _ {\mathfrak {g} _ {0}} \leq (1 + (1 - r) ^ {2}) ^ {- 1} \left\{1 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \right\} \| y ^ {\prime} - y \| _ {\mathfrak {g} _ {0}}.
$$
We are now going to bound  $\| D\mathrm{K}_0(y)[w] - D\mathrm{K}_0(y')[w]\|_{\mathfrak{g}_0}$ . We have
$$
\| D K _ {0} (y) [ w ] - D K _ {0} \left(y ^ {\prime}\right) [ w ] \| _ {\mathfrak {g} _ {0}} \leq \left\| \left(M _ {0} (y) ^ {- 1} - M _ {0} \left(y ^ {\prime}\right) ^ {- 1}\right) w \right\| _ {\mathfrak {g} _ {0} ^ {- 1}} + A _ {1} + A _ {2} + A _ {3} + A _ {4},
$$
$$
\begin{array}{l} A _ {1} = \left\| \left\{\left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} (y) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} - \left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} \left(y ^ {\prime}\right) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} \right\} \right. \\ \times \mathfrak {g} _ {0} ^ {- 1 / 2} D \mathfrak {g} (y) [ w ] \mathfrak {g} _ {0} ^ {- 1 / 2} \left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} (y) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} \mathfrak {g} _ {0} ^ {1 / 2} (y - x ^ {(0)}) \| _ {2}, \\ \end{array}
$$
$$
A _ {2} = \| \left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} (y ^ {\prime}) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} \mathfrak {g} _ {0} ^ {- 1 / 2} \left\{D \mathfrak {g} (y) [ w ] - D \mathfrak {g} (y ^ {\prime}) [ w ] \right\} \mathfrak {g} _ {0} ^ {- 1 / 2} \left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} (y) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} \mathfrak {g} _ {0} ^ {1 / 2} (y - x ^ {(0)}) \| _ {2},
$$
$$
\begin{array}{l} A _ {3} = \left\| \left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} (y ^ {\prime}) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} \mathfrak {g} _ {0} ^ {- 1 / 2} D \mathfrak {g} (y) [ w ] \mathfrak {g} _ {0} ^ {- 1 / 2} \right. \\ \times \left\{\left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} (y) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} - \left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} \left(y ^ {\prime}\right) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} \right\} \mathfrak {g} _ {0} ^ {1 / 2} \left(y - x ^ {(0)}\right) \| _ {2}, \\ \end{array}
$$
$$
A _ {3} = \| \left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} (y ^ {\prime}) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} \mathfrak {g} _ {0} ^ {- 1 / 2} D \mathfrak {g} (y) [ w ] \mathfrak {g} _ {0} ^ {- 1 / 2} \left(\mathrm {I} _ {d} + \mathfrak {g} _ {0} ^ {- 1 / 2} \mathfrak {g} (y ^ {\prime}) \mathfrak {g} _ {0} ^ {- 1 / 2}\right) ^ {- 1} \mathfrak {g} _ {0} ^ {1 / 2} (y - y ^ {\prime}) \| _ {2}.
$$
$$
\begin{array}{l} \left\| \left(\mathrm {M} _ {0} \left(y ^ {\prime}\right) ^ {- 1} - \mathrm {M} _ {0} \left(y ^ {\prime}\right) ^ {- 1}\right) w \right\| _ {\mathfrak {g} _ {0} ^ {- 1}} \leq 4 (1 - r) ^ {- 3} \left(1 + (1 - r) ^ {2}\right) ^ {- 2} \| y ^ {\prime} - y \| _ {\mathfrak {g} _ {0}} \| w \| _ {\mathfrak {g} _ {0}}, (43) \\ \max  \left(A _ {1}, A _ {3}\right) \leq 8 r (1 - r) ^ {- 6} \left(1 + (1 - r) ^ {2}\right) ^ {- 3} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \| w \| _ {\mathfrak {g} _ {0}}, (36, (38), (43)) \\ A _ {2} \leq 5 \alpha (\alpha + 1) / 3 r (1 - r) ^ {- 4} (1 + (1 - r) ^ {2}) ^ {- 2} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \| w \| _ {\mathfrak {g} _ {0}}, ((37), (38))) \\ A _ {4} \leq 2 (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 2} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \| w \| _ {\mathfrak {g} _ {0}}. (36) \\ \end{array}
$$
$$
\begin{array}{l} \left\| \mathfrak {g} _ {0} ^ {- 1 / 2} \left(D \mathfrak {g} _ {0} [ D K _ {0} (y) [ w ], K _ {0} (y) ] - D \mathfrak {g} _ {0} [ D K _ {0} \left(y ^ {\prime}\right) [ w ], K _ {0} \left(y ^ {\prime}\right) ]\right) \right\| _ {2} \\ \leq 2 r (1 + (1 - r) ^ {2}) ^ {- 1} \{3 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1}) \} \\ \times \left\{4 (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 2} + 1 6 r (1 - r) ^ {- 6} (1 + (1 - r) ^ {2}) ^ {- 3} \right. \\ + 5 \alpha (\alpha + 1) / 3 r (1 - r) ^ {- 4} (1 + (1 - r) ^ {2}) ^ {- 2} + 2 (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 2} \} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \| w \| _ {\mathfrak {g} _ {0}} \\ + 2 \{1 + 2 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \} \\ \times (1 + (1 - r) ^ {2}) ^ {- 2} (1 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1}) \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \| w \| _ {\mathfrak {g} _ {0}}. \\ \end{array}
$$
Conclusion of Step 2. By combining the results of Steps 2.1 to 2.4, it comes that
$$
\begin{array}{l} \left\| \mathrm {M} _ {0} (y) D \mathfrak {g} _ {0} \left[ D \mathrm {K} _ {0} (y) [ w ], \mathrm {K} _ {0} (y) \right] - \mathrm {M} _ {0} \left(y ^ {\prime}\right) D \mathfrak {g} _ {0} \left[ D \mathrm {K} _ {0} \left(y ^ {\prime}\right) [ w ], \mathrm {K} _ {0} \left(y ^ {\prime}\right) \right] \right\| _ {\mathfrak {g} _ {0}} \\ = \| \mathfrak {g} _ {0} ^ {1 / 2} \left(a _ {2} (y) - a _ {2} \left(y ^ {\prime}\right)\right) \| _ {2} \\ \leq \left\{6 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 2} [ 1 + 2 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} ] \right. \\ + 2 r [ 1 + (1 - r) ^ {- 2} ] \left(1 + (1 - r) ^ {2}\right) ^ {- 1} \left\{3 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \right\} \\ \times \left\{4 (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 2} + 1 6 r (1 - r) ^ {- 6} (1 + (1 - r) ^ {2}) ^ {- 3} \right. \\ + 5 \alpha (\alpha + 1) / 3 r (1 - r) ^ {- 4} (1 + (1 - r) ^ {2}) ^ {- 2} + 2 (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 2} \} \\ + 2 [ 1 + (1 - r) ^ {- 2} ] \{1 + 2 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \} \\ \times (1 + (1 - r) ^ {2}) ^ {- 2} (1 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1}) \} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \| w \| _ {\mathfrak {g} _ {0}}. \\ \end{array}
$$
Conclusion. Finally, we have  $\| D\phi_0(y)[w] - D\phi_0(y')[w]\|_{\mathfrak{g}_0} \leq c(r)\| y - y'\|_{\mathfrak{g}_0}\| w\|_{\mathfrak{g}_0}$ , where
$$
\begin{array}{l} c (r) \leq 2 4 r ^ {2} (1 - r) ^ {- 1 2} + 1 0 \alpha (\alpha + 1) / 3 r ^ {2} (1 - r) ^ {- 1 0} \\ + 3 2 r (1 - r) ^ {- 7} + 1 2 8 r ^ {2} (1 - r) ^ {- 1 2} + 1 2 8 r ^ {3} (1 - r) ^ {- 1 7} \\ + 1 2 r (1 - r) ^ {- 7} + 2 4 r ^ {2} (1 - r) ^ {- 1 2} \\ + \left\{4 8 r (1 - r) ^ {- 9} + 1 9 2 r ^ {2} (1 - r) ^ {- 1 4} + 2 0 \alpha (\alpha + 1) r ^ {2} (1 - r) ^ {- 1 0} + 2 4 r (1 - r) ^ {- 9} \right\} \{1 + (1 - r) ^ {- 2} \} \\ + \left\{6 4 r ^ {2} (1 - r) ^ {- 1 4} + 2 5 6 r ^ {3} (1 - r) ^ {- 1 9} + 8 0 \alpha (\alpha + 1) / 3 r ^ {3} (1 - r) ^ {- 1 5} + 3 2 r ^ {2} (1 - r) ^ {- 1 4} \right\} \{1 + (1 - r) ^ {- 2} \} \\ + \left\{4 (1 - r) ^ {- 4} + 2 4 r (1 - r) ^ {- 9} + 3 2 r ^ {2} (1 - r) ^ {- 1 4} \right\} \left\{1 + (1 - r) ^ {- 2} \right\}, \\ \end{array}
$$
by combining the results of Step 1 (two first lines) and Step 2 (following lines). Hence, using that  $r \leq 1/11$ , we have
$$
\begin{array}{l} c (r) \leq \max  (2 4 \times 2 5 6, 5 \times 8 0 \alpha (\alpha + 1) / 3) (1 - r) ^ {- 2 1} \\ \leq \max  (6 1 4 4, 4 0 0 \alpha (\alpha + 1) / 3) 1 5 / 2 \\ \leq \max  (4 6 0 8 0, 1 0 0 0 \alpha (\alpha + 1)) \\ \leq 1 / r _ {\alpha} ^ {\star}, \quad (\text {s e e (3 3)}) \\ \end{array}
$$
i.e., for any  $(y, y') \in \mathsf{B}_0 \times \mathsf{B}_0$  and  $w \in \mathbb{R}^d$ , we have
$$
\| D \phi_ {0} (y) [ w ] - D \phi_ {0} (y) [ w ] \| _ {\mathfrak {g} _ {0}} \leq \left(r _ {\alpha} ^ {\star}\right) ^ {- 1} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \| w \| _ {\mathfrak {g} _ {0}},
$$
$$
\left\| D \phi_ {0} (y) - D \phi_ {0} (y) \right\| _ {\mathfrak {g} _ {0}} \leq \left(r _ {\alpha} ^ {\star}\right) ^ {- 1} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}}.
$$
This last inequality finally proves that  $D\phi_0$  is  $(r_{\alpha}^{\star})^{-1}$ -Lipschitz-continuous on  $\mathsf{B}_0$  with respect to  $\| \cdot \|_{\mathfrak{g}_0}$ .
Inequality on  $D\tilde{\phi}_0$ . Elaborating on the smoothness of  $D\phi_0$ , we prove here that  $\| D\tilde{\phi}_0(y)\|_{\mathfrak{g}_0} > 1/2$  for any  $y \in \mathsf{B}_0$ . Let  $y \in \mathsf{B}_0$ , we have
$$
\left\| \mathrm {I} _ {d} \right\| _ {\mathfrak {g} _ {0}} = \left\| D \tilde {\phi} _ {0} (y) + 1 / 2 D \phi_ {0} (y) \right\| _ {\mathfrak {g} _ {0}},
$$
$$
1 \leq \| D \tilde {\phi} _ {0} (y) \| _ {\mathfrak {g} _ {0}} + 1 / 2 \| D \phi_ {0} (y) \| _ {\mathfrak {g} _ {0}},
$$
$$
1 - 1 / 2 \| D \phi_ {0} (y) \| _ {\mathfrak {g} _ {0}} \leq \| D \tilde {\phi} _ {0} (y) \| _ {\mathfrak {g} _ {0}}.
$$
We recall that  $D\phi_0(x^{(0)}) = 0$  and that  $D\phi_0$  is  $(r_{\alpha}^{\star})^{-1}$ -Lipschitz-continuous on  $\mathsf{B}_0$ . Thus, we obtain
$$
\left\| D \phi_ {0} (y) \right\| _ {\mathfrak {g} _ {0}} = \left\| D \phi_ {0} (y) - D \phi_ {0} \left(x ^ {(0)}\right) \right\| _ {\mathfrak {g} _ {0}} \leq \left(r _ {\alpha} ^ {\star}\right) ^ {- 1} \| y - x ^ {(0)} \| _ {\mathfrak {g} _ {0}} <   \left(r _ {\alpha} ^ {\star}\right) ^ {- 1} r \leq 1,
$$
and then  $\| D\tilde{\phi}_0(y)\|_{\mathfrak{g}_0} > 1 / 2$  which proves the result.
Smoothness of  $\phi_0$  and  $\tilde{\phi}_0$ . We prove here that  $\phi_0$  (and thus  $\tilde{\phi}_0$ ) is Lipschitz-continuous on  $\mathsf{B}_0$  with respect to  $\| \cdot \|_{\mathfrak{g}_0}$ . Let  $(y, y') \in \mathsf{B}_0 \times \mathsf{B}_0$ . We have
$$
\begin{array}{l} \left\| \phi_ {0} (y) - \phi_ {0} \left(y ^ {\prime}\right) \right\| _ {\mathfrak {g} _ {0}} \leq \left\| \left\{\mathrm {M} _ {0} (y) - \mathrm {M} _ {0} \left(y ^ {\prime}\right) \right\} D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} (y), \mathrm {K} _ {0} (y) \right] \right\| _ {\mathfrak {g} _ {0}} \tag {45} \\ + \left\| \mathrm {M} _ {0} \left(y ^ {\prime}\right) \left\{D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} (y), \mathrm {K} _ {0} (y) \right] - D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} \left(y ^ {\prime}\right), \mathrm {K} _ {0} \left(y ^ {\prime}\right) \right] \right\} \right\| _ {\mathfrak {g} _ {0}}. \\ \end{array}
$$
To prove the Lipschitz-continuity of  $\phi_0$ , we are going to proceed in two steps, each of them consisting of bounding from above a term of (45).
Step 1. Let us first bound the first term in (45). Since  $r \leq 1/11$ , we have
$$
\begin{array}{l} \| \left\{\mathrm {M} _ {0} (y) - \mathrm {M} _ {0} \left(y ^ {\prime}\right) \right\} D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} (y), \mathrm {K} _ {0} (y) \right] \| _ {\mathfrak {g} _ {0}} \\ \leq \left\| \mathfrak {g} _ {0} ^ {1 / 2} \left\{\mathrm {M} _ {0} (y) - \mathrm {M} _ {0} \left(y ^ {\prime}\right) \right\} \mathfrak {g} _ {0} ^ {1 / 2} \right\| _ {2} \left\| D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} (y), \mathrm {K} _ {0} (y) \right] \right\| _ {\mathfrak {g} _ {0} ^ {- 1}} \\ \leq 3 (1 - r) ^ {- 3} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \times 2 \| \mathrm {K} _ {0} (y) \| _ {\mathfrak {g} _ {0}} ^ {2} \quad (\text {s e e (1 8) a n d L e m m a 9 - (a)}) \\ \leq 6 r ^ {2} (1 - r) ^ {- 3} \left(1 + (1 - r) ^ {2}\right) ^ {- 2} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}}. \tag {see(39)} \\ \end{array}
$$
Step 2. Let us now bound the second term in (45). We have
$$
\begin{array}{l} \left\| \mathrm {M} _ {0} \left(y ^ {\prime}\right) \left\{D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} (y), \mathrm {K} _ {0} (y) \right] - D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} \left(y ^ {\prime}\right), \mathrm {K} _ {0} \left(y ^ {\prime}\right) \right] \right\} \right\| _ {\mathfrak {g} _ {0}} \\ \leq \left\| \mathfrak {g} _ {0} ^ {1 / 2} \mathrm {M} _ {0} \left(y ^ {\prime}\right) \mathfrak {g} _ {0} ^ {1 / 2} \right\| _ {2} \left\| D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} (y), \mathrm {K} _ {0} (y) \right] - D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} \left(y ^ {\prime}\right), \mathrm {K} _ {0} \left(y ^ {\prime}\right) \right] \right\| _ {\mathfrak {g} _ {0} ^ {- 1}} \\ \leq \left\{1 + (1 - r) ^ {- 2} \right\} \left\{2 \| D \mathfrak {g} _ {0} \left[ K _ {0} (y) - K _ {0} \left(y ^ {\prime}\right), K _ {0} (y) \right] \| _ {\mathfrak {g} _ {0} ^ {- 1}} + \| D \mathfrak {g} _ {0} \left[ K _ {0} (y) - K _ {0} \left(y ^ {\prime}\right), K _ {0} (y) - K _ {0} \left(y ^ {\prime}\right) \right] \| _ {\mathfrak {g} _ {0} ^ {- 1}} \right\} \\ \leq \left\{1 + (1 - r) ^ {- 2} \right\} \left\{4 \| \mathrm {K} _ {0} (y) - \mathrm {K} _ {0} \left(y ^ {\prime}\right) \| _ {\mathfrak {g} _ {0}} \| \mathrm {K} _ {0} (y) \| _ {\mathfrak {g} _ {0}} + 2 \| \mathrm {K} _ {0} (y) - \mathrm {K} _ {0} \left(y ^ {\prime}\right) \| _ {\mathfrak {g} _ {0}} ^ {2} \right\} \\ \leq 1 \{1 + (1 - r) ^ {- 2} \} \| \mathrm {K} _ {0} (y) - \mathrm {K} _ {0} (y ^ {\prime}) \| _ {\mathfrak {g} _ {0}} \left\{2 \| \mathrm {K} _ {0} (y) \| _ {\mathfrak {g} _ {0}} + \| \mathrm {K} _ {0} (y) - \mathrm {K} _ {0} (y ^ {\prime}) \| _ {\mathfrak {g} _ {0}} \right\}, \\ \end{array}
$$
where we recall that the following inequalities hold
$$
\left\| \mathrm {K} _ {0} (y) \right\| _ {\mathfrak {g} _ {0}} \leq r (1 + (1 - r) ^ {2}) ^ {- 1},
$$
$$
\left\| \mathrm {K} _ {0} (y) - \mathrm {K} _ {0} \left(y ^ {\prime}\right) \right\| _ {\mathfrak {g} _ {0}} \leq (1 + (1 - r) ^ {2}) ^ {- 1} \left\{1 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \right\} \| y ^ {\prime} - y \| _ {\mathfrak {g} _ {0}}.
$$
$$
\begin{array}{l} \left. \left\| \mathrm {M} _ {0} \left(y ^ {\prime}\right) \left\{D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} (y), \mathrm {K} _ {0} (y) \right] - D \mathfrak {g} _ {0} \left[ \mathrm {K} _ {0} \left(y ^ {\prime}\right), \mathrm {K} _ {0} \left(y ^ {\prime}\right) \right] \right\} \right\| _ {\mathfrak {g} _ {0}} \right. \\ \leq 4 r \left\{1 + (1 - r) ^ {2} \right\} ^ {- 2} \left\{1 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \right\} ^ {2} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}}. \\ \end{array}
$$
Conclusion. Since  $r \leq 1/11$ , we finally have
$$
\begin{array}{l} \left\| \phi_ {0} (y) - \phi_ {0} \left(y ^ {\prime}\right) \right\| _ {\mathfrak {g} _ {0}} \leq \left\{6 r ^ {2} (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 2} \right. \\ 4 r \left\{1 + (1 - r) ^ {2} \right\} ^ {- 2} \left\{1 + 4 r (1 - r) ^ {- 3} (1 + (1 - r) ^ {2}) ^ {- 1} \right\} ^ {2} \left\| y - y ^ {\prime} \right\| _ {\mathfrak {g} _ {0}} \\ \leq \left\{6 r ^ {2} (1 - r) ^ {- 7} + 4 r (1 - r) ^ {- 4} + 3 2 r ^ {2} (1 - r) ^ {- 9} + 6 4 r ^ {3} (1 - r) ^ {- 1 4} \right\} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \\ \leq 4 \times 6 4 r (1 - r) ^ {- 1 4} \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}} \\ \leq 4 8 6 4 / 5 r \| y - y ^ {\prime} \| _ {\mathfrak {g} _ {0}}. \\ \end{array}
$$
Therefore,  $\phi_0$  and  $\tilde{\phi}_0$  are respectively  $(4864 / 5r)$  and  $(1 + 2432 / 5r)$ -Lipschitz-continuous on  $\mathsf{B}_0$  with respect to  $\| \cdot \|_{\mathfrak{g}_0}$ .
We are now ready to prove the second result of Lemma 22.
Invertibility of  $\operatorname{Jac}[\mathrm{G}_{h,x^{(0)}}]$ . Let  $x \in \mathsf{B}_0$  and  $w \in \mathbb{R}^d$ . We have  $(h/2)\mathrm{G}_{h,x^{(0)}} = \mathrm{M}_0(y)^{-1}\tilde{\phi}_0(y)$  and thus,
$$
\begin{array}{l} \left(h / 2\right) D \mathrm {G} _ {h, x ^ {(0)}} (x) [ w ] = \mathrm {M} _ {0} (x) ^ {- 1} D \tilde {\phi} _ {0} (x) [ w ] + D \mathrm {M} _ {0} (x) ^ {- 1} [ w ] \tilde {\phi} _ {0} (x), \\ D \tilde {\phi} _ {0} (x) [ w ] = (h / 2) \mathrm {M} _ {0} (x) D \mathrm {G} _ {h, x ^ {(0)}} (x) [ w ] - \mathrm {M} _ {0} (x) D \mathrm {M} _ {0} (x) ^ {- 1} [ w ] \tilde {\phi} _ {0} (x). \\ \end{array}
$$
$$
\begin{array}{l} \| D \tilde {\phi} _ {0} (x) [ w ] \| _ {\mathfrak {g} _ {0}} \leq (h / 2) \| \mathfrak {g} _ {0} ^ {1 / 2} \mathrm {M} _ {0} (x) \mathfrak {g} _ {0} ^ {1 / 2} \| _ {2} \| D \mathrm {G} _ {h, x ^ {(0)}} (x) [ w ] \| _ {\mathfrak {g} _ {0} ^ {- 1}} \tag {46} \\ + \| \mathfrak {g} _ {0} ^ {1 / 2} \mathrm {M} _ {0} (x) \mathfrak {g} _ {0} ^ {1 / 2} \| _ {2} \| \mathfrak {g} _ {0} ^ {- 1 / 2} D \mathrm {M} _ {0} (x) ^ {- 1} [ w ] \mathfrak {g} _ {0} ^ {- 1 / 2} \| _ {2} \| \tilde {\phi} _ {0} (x) \| _ {\mathfrak {g} _ {0}} \\ \leq \left\{1 + (1 - r) ^ {- 2} \right\} \left\{\left(h / 2\right) \| D G _ {h, x ^ {(0)}} (x) [ w ] \| _ {\mathfrak {g} _ {0} ^ {- 1}} + \| \mathfrak {g} _ {0} ^ {- 1 / 2} D M _ {0} (x) ^ {- 1} [ w ] \mathfrak {g} _ {0} ^ {- 1 / 2} \| _ {2} \| \tilde {\phi} _ {0} (x) \| _ {\mathfrak {g} _ {0}} \right\}, \\ \end{array}
$$
where  $DM_0(x)^{-1}[w] = -M_0(x)^{-1}DM_0(x)[w]M_0(x)^{-1}$  and
$$
\begin{array}{l} \left\| \mathfrak {g} _ {0} ^ {- 1 / 2} D M _ {0} (x) ^ {- 1} [ w ] \mathfrak {g} _ {0} ^ {- 1 / 2} \right\| _ {2} \leq \left\| \mathfrak {g} _ {0} ^ {- 1 / 2} M _ {0} (x) ^ {- 1} \mathfrak {g} _ {0} ^ {- 1 / 2} \right\| _ {2} ^ {2} \left\| \mathfrak {g} _ {0} ^ {- 1 / 2} D M _ {0} (x) [ w ] \mathfrak {g} _ {0} ^ {- 1 / 2} \right\| _ {2} (47) \\ \leq 2 (1 - r) ^ {- 3} \left(1 + (1 - r) ^ {2}\right) ^ {- 1} \| w \| _ {\mathfrak {g} _ {0}} (see(40)) \\ \leq 2 (1 - r) ^ {- 7} \| w \| _ {\mathfrak {g} _ {0}}. \\ \end{array}
$$
Moreover, using the Lipschitz-continuity of  $\tilde{\phi}_0$  on  $\mathsf{B}_0$  and  $\tilde{\phi}_0(x^{(0)}) = 0$ , we have