cons 1 (cons 2 (cons 3 nil))

cons 1 (cons 2 nil)+cons 3 nil

cons nil (cons 1 (cons 2 (cons 3)))

cons (1,2,3) nil

Anything can be proven from the assumption that P is false.

Anything can be proven from the assumption that P is true.

It is possible to prove false.

Anything can be proven from the assumption false.

It automates the injection tactic.

It ensures that different constructors always produce distinct values.

It prevents the creation of infinite lists.

It allows lists of different types to be compared.

exact h,

apply h,

constructor,

cases h with p q,

9

4

1

2

Reflexivity, asymmetry, transitivity

Irreflexivity, symmetry, transitivity

Reflexivity, symmetry, transitivity

Reflexivity, symmetry, anti-transitivity

It rewrites the assumption eq to prove reflexivity.

It rewrites the assumption p using eq.

It rewrites the goal PP y to PP x using eq.

It rewrites the goal PP x to PP y using eq.

⊢ PP x

⊢ QQ x

⊢ PP a

⊢ PP a → QQ a

It applies the assumption pq to the goal.

It decomposes the assumption p of ∃ x : A, PP x into an element a : A and an assumption pa : PP a, making a and pa available for use in the proof.

It introduces a specific element a of type A and an assumption pa : PP a, effectively proving the existential quantifier.

It constructs a term of type ∃ z : A, QQ z using p directly.

A proposition is a statement that can be true or false, while a predicate is a function-like construct that takes inputs and returns a proposition.

Predicates have a fixed truth value, while propositions can have varying truth values.

There is no difference between predicates and propositions; they are interchangeable terms.

A predicate is a statement that can be true or false, while a proposition is a function that returns a truth value.