$$ \begin{aligned} F = (P, Q): \mathbb{C}^2 \to \mathbb{C}^2...

Prompt

$$ \begin{aligned} F = (P, Q): \mathbb{C}^2 \to \mathbb{C}^2 \quad : \quad & \begin{cases} P(x, y) = x + A(x, y) + x^{16} y^{16} & (\deg P = 32) \\ Q(x, y) = y + B(x, y) + x^{56} y^{56} & (\deg Q = 112) \end{cases} \\ \wedge\quad & \gcd(\deg P, \deg Q) = 16 \;\wedge\; \neg\operatorname{Prime}(\gcd(\deg P, \deg Q)) \;\wedge\; \max(\deg P, \deg Q) = 112 > 100 \\ \wedge\quad & \deg P \nmid \deg Q \;\wedge\; \deg Q \nmid \deg P \\ \wedge\quad & \frac{\partial P}{\partial x}\frac{\partial Q}{\partial y} - \frac{\partial P}{\partial y}\frac{\partial Q}{\partial x} \equiv 1 \\ \wedge\quad & P(0,0) = Q(0,0) = 0 \;\wedge\; \begin{pmatrix} \frac{\partial P}{\partial x}(0,0) & \frac{\partial P}{\partial y}(0,0) \\ \frac{\partial Q}{\partial x}(0,0) & \frac{\partial Q}{\partial y}(0,0) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \\ \wedge\quad & 2\left(\frac{\partial B}{\partial x}\frac{\partial H}{\partial y} - \frac{\partial B}{\partial y}\frac{\partial H}{\partial x}\right) = 7 H^5 \left(\frac{\partial A}{\partial x}\frac{\partial H}{\partial y} - \frac{\partial A}{\partial y}\frac{\partial H}{\partial x}\right) \quad \text{ahol} \quad H = x^8 y^8 \\ \wedge\quad & \det \begin{pmatrix} \frac{\partial}{\partial x}\left(\frac{P+Q}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{P+Q}{\sqrt{2}}\right) \\ \frac{\partial}{\partial x}\left(\frac{Q-P}{\sqrt{2}}\right) & \frac{\partial}{\partial y}\left(\frac{Q-P}{\sqrt{2}}\right) \end{pmatrix} \equiv 1 \\ \wedge\quad & S_{P,Q} \supseteq \big\{ (u, v) \in \mathbb{C}^2 \;\big|\; v^2 - u^7 = 0 \big\} \ne \emptyset \implies \neg\operatorname{Proper}(F) \\ \wedge\quad & [\mathbb{C}(x,y) : \mathbb{C}(P, Q)] = d \ge 5 \implies \exists z_1 \ne z_2 \in \mathbb{C}^2 : F(z_1) = F(z_2) \implies \neg\operatorname{Inj}(P, Q) \\ \wedge\quad & \neg\operatorname{Inv}(P, Q) \end{aligned} $$ 1. `A(x,y)` and `B(x,y)` are never defined. Their type, degree and the requirement that they have no constant or linear terms are all missing, yet they are used in P, Q and the bracket identity. 2. `S_{P,Q}` is never defined, so its type is unknown. Because of that, `S ⊇ {…}` is not checkably well-typed. 3. `d` is introduced inside `[ℂ(x,y):ℂ(P,Q)] = d ≥ 5` without a declaration or quantifier. The chained `= d ≥ 5` mixes an equality and an inequality in one relation. 4. `S_{P,Q} ⊇ {…} ≠ ∅` is a chained relation that mixes ⊇ and ≠. It is ambiguous whether `≠ ∅` applies to the set or to the whole expression. 5. `Proper(F)` takes a map, but `Inj(P,Q)` and `Inv(P,Q)` take two polynomials instead of `F=(P,Q)`. The predicate signatures are inconsistent. 6. The chain `A ⇒ ∃ z₁≠z₂ : F(z₁)=F(z₂) ⇒ ¬Inj(P,Q)` has ambiguous implication associativity. The ∃ quantifier's scope is also unclear, and a quantified statement is used as the antecedent of another implication. 7. The two implications are joined to the other conditions by `∧` with no parentheses. Precedence is ambiguous, so the conjunction is not well-formed. 8. The word `ahol` ("where", Hungarian) is plain text inside a formula. It also introduces H only after H is used, and the bracket expression `∂B/∂x ∂H/∂y − …` is not labelled as a Poisson bracket. 9. The `cases` block is treated as the first conjunct of a logical formula. It is a definition of P and Q, not a proposition, so definitions and propositions are mixed. 10. `F = (P,Q): ℂ²→ℂ² : {…}` uses `:` for both "has type" and "such that". 11. `Inv(P,Q)`, `Prime(·)`, `Proper(·)` and `Inj(·)` are undeclared predicates. Send back the complete code with all the fixes. Fix each of the listed errors one by one, making sure to actually correct them so that there are 0 errors remaining. Keep the original imports, since the files exist. Write out every single character; do not abbreviate anything. Fix every error. There must be exactly one file. Do not write anything else; just output the complete code, and it must not contain any comments. Never, under any circumstances, use simplified, substitute, dummy, simulated, or fake code. Write the entire file as complete, unabridged, production-ready code in a single code block. It must be 100% error-free, a complete, error-free file, and must be submitted as a downloadable file. These requirements are mandatory and must be strictly adhered to. If no list of errors is provided, you must find all the errors and fix them. If there were comments in the original code, delete them. And most importantly: YOU MUST NEVER SIMPLIFY!