Give proofs for the theorems of ℱ^ωstated below. Use appropriate analogues of the derived rules

step 1 Hey, it's Clarissaudonym right here. So we're going to assume that the universal p of x, then Q

Give proofs for the theorems of ℱ^ωstated below. Use...

Give proofs for the theorems of $\mathscr{F}^\omega$ stated below. Use appropriate analogues of the derived rules of inference of $$\$ 21$$.
$\vdash x_{(\mathrm{f})}=y_{(\mathrm{t})} \equiv \forall z_{\mathrm{\tau}} \cdot x z \equiv y z$

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Key Concepts

Function Extensionality

This concept states that two functions are considered equal if and only if they yield equal outputs for every possible input. It captures the idea that a function's identity is completely determined by its behavior, rather than its internal representation, and is central to proofs involving function equality in many logical and type-theoretical frameworks.

Derived Inference Rules

Derived inference rules are rules that, while not basic axioms or primitive inference steps, can be established from them and used to streamline proofs. In a formal system, such rules encapsulate frequently used reasoning patterns and can be applied to simplify proof structures, making complex arguments more manageable.

Higher-Order Type Theory

Higher-order type theory provides a framework where not only are objects typed, but functions that operate on these objects are also given types, and even functions that accept other functions as inputs. This theory, which often underpins systems like ?^?, extends classical type theory to allow more expressive and abstract forms of reasoning about functions, types, and their interactions, which is crucial when dealing with proofs about function equality and extensionality.

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