propositions_and_proofs.md

Propositions and Proofs

By now, you have seen some ways of defining objects and functions in Lean. In this chapter, we will begin to explain how to write mathematical assertions and proofs in the language of dependent type theory as well.

Propositions as Types

One strategy for proving assertions about objects defined in the language of dependent type theory is to layer an assertion language and a proof language on top of the definition language. But there is no reason to multiply languages in this way: dependent type theory is flexible and expressive, and there is no reason we cannot represent assertions and proofs in the same general framework.

For example, we could introduce a new type, Prop, to represent propositions, and introduce constructors to build new propositions from others.

# def Implies (p q : Prop) : Prop := p → q
#check And     -- Prop → Prop → Prop
#check Or      -- Prop → Prop → Prop
#check Not     -- Prop → Prop
#check Implies -- Prop → Prop → Prop

variable (p q r : Prop)
#check And p q                      -- Prop
#check Or (And p q) r               -- Prop
#check Implies (And p q) (And q p)  -- Prop

We could then introduce, for each element p : Prop, another type Proof p, for the type of proofs of p. An "axiom" would be a constant of such a type.

# def Implies (p q : Prop) : Prop := p → q
# structure Proof (p : Prop) : Type where
#   proof : p
#check Proof   -- Proof : Prop → Type

axiom and_comm (p q : Prop) : Proof (Implies (And p q) (And q p))

variable (p q : Prop)
#check and_comm p q     -- Proof (Implies (And p q) (And q p))

In addition to axioms, however, we would also need rules to build new proofs from old ones. For example, in many proof systems for propositional logic, we have the rule of modus ponens:

From a proof of Implies p q and a proof of p, we obtain a proof of q.

We could represent this as follows:

# def Implies (p q : Prop) : Prop := p → q
# structure Proof (p : Prop) : Type where
#   proof : p
axiom modus_ponens : (p q : Prop) → Proof (Implies p q) → Proof p → Proof q

Systems of natural deduction for propositional logic also typically rely on the following rule:

Suppose that, assuming p as a hypothesis, we have a proof of q. Then we can "cancel" the hypothesis and obtain a proof of Implies p q.

We could render this as follows:

# def Implies (p q : Prop) : Prop := p → q
# structure Proof (p : Prop) : Type where
#   proof : p
axiom implies_intro : (p q : Prop) → (Proof p → Proof q) → Proof (Implies p q)

This approach would provide us with a reasonable way of building assertions and proofs. Determining that an expression t is a correct proof of assertion p would then simply be a matter of checking that t has type Proof p.

Some simplifications are possible, however. To start with, we can avoid writing the term Proof repeatedly by conflating Proof p with p itself. In other words, whenever we have p : Prop, we can interpret p as a type, namely, the type of its proofs. We can then read t : p as the assertion that t is a proof of p.

Moreover, once we make this identification, the rules for implication show that we can pass back and forth between Implies p q and p → q. In other words, implication between propositions p and q corresponds to having a function that takes any element of p to an element of q. As a result, the introduction of the connective Implies is entirely redundant: we can use the usual function space constructor p → q from dependent type theory as our notion of implication.

This is the approach followed in the Calculus of Constructions, and hence in Lean as well. The fact that the rules for implication in a proof system for natural deduction correspond exactly to the rules governing abstraction and application for functions is an instance of the Curry-Howard isomorphism, sometimes known as the propositions-as-types paradigm. In fact, the type Prop is syntactic sugar for Sort 0, the very bottom of the type hierarchy described in the last chapter. Moreover, Type u is also just syntactic sugar for Sort (u+1). Prop has some special features, but like the other type universes, it is closed under the arrow constructor: if we have p q : Prop, then p → q : Prop.

There are at least two ways of thinking about propositions as types. To some who take a constructive view of logic and mathematics, this is a faithful rendering of what it means to be a proposition: a proposition p represents a sort of data type, namely, a specification of the type of data that constitutes a proof. A proof of p is then simply an object t : p of the right type.

Those not inclined to this ideology can view it, rather, as a simple coding trick. To each proposition p we associate a type that is empty if p is false and has a single element, say *, if p is true. In the latter case, let us say that (the type associated with) p is inhabited. It just so happens that the rules for function application and abstraction can conveniently help us keep track of which elements of Prop are inhabited. So constructing an element t : p tells us that p is indeed true. You can think of the inhabitant of p as being the "fact that p is true." A proof of p → q uses "the fact that p is true" to obtain "the fact that q is true."

Indeed, if p : Prop is any proposition, Lean's kernel treats any two elements t1 t2 : p as being definitionally equal, much the same way as it treats (fun x => t) s and t[s/x] as definitionally equal. This is known as proof irrelevance, and is consistent with the interpretation in the last paragraph. It means that even though we can treat proofs t : p as ordinary objects in the language of dependent type theory, they carry no information beyond the fact that p is true.

The two ways we have suggested thinking about the propositions-as-types paradigm differ in a fundamental way. From the constructive point of view, proofs are abstract mathematical objects that are denoted by suitable expressions in dependent type theory. In contrast, if we think in terms of the coding trick described above, then the expressions themselves do not denote anything interesting. Rather, it is the fact that we can write them down and check that they are well-typed that ensures that the proposition in question is true. In other words, the expressions themselves are the proofs.

In the exposition below, we will slip back and forth between these two ways of talking, at times saying that an expression "constructs" or "produces" or "returns" a proof of a proposition, and at other times simply saying that it "is" such a proof. This is similar to the way that computer scientists occasionally blur the distinction between syntax and semantics by saying, at times, that a program "computes" a certain function, and at other times speaking as though the program "is" the function in question.

In any case, all that really matters is the bottom line. To formally express a mathematical assertion in the language of dependent type theory, we need to exhibit a term p : Prop. To prove that assertion, we need to exhibit a term t : p. Lean's task, as a proof assistant, is to help us to construct such a term, t, and to verify that it is well-formed and has the correct type.

Working with Propositions as Types

In the propositions-as-types paradigm, theorems involving only → can be proved using lambda abstraction and application. In Lean, the theorem command introduces a new theorem:

variable {p : Prop}
variable {q : Prop}

theorem t1 : p → q → p := fun hp : p => fun hq : q => hp

Compare this proof to the expression fun x : α => fun y : β => x of type α → β → α, where α and β are data types. This describes the function that takes arguments x and y of type α and β, respectively, and returns x. The proof of t1 has the same form, the only difference being that p and q are elements of Prop rather than Type. Intuitively, our proof of p → q → p assumes p and q are true, and uses the first hypothesis (trivially) to establish that the conclusion, p, is true.

Note that the theorem command is really a version of the def command: under the propositions and types correspondence, proving the theorem p → q → p is really the same as defining an element of the associated type. To the kernel type checker, there is no difference between the two.

There are a few pragmatic differences between definitions and theorems, however. In normal circumstances, it is never necessary to unfold the "definition" of a theorem; by proof irrelevance, any two proofs of that theorem are definitionally equal. Once the proof of a theorem is complete, typically we only need to know that the proof exists; it doesn't matter what the proof is. In light of that fact, Lean tags proofs as irreducible, which serves as a hint to the parser (more precisely, the elaborator) that there is generally no need to unfold them when processing a file. In fact, Lean is generally able to process and check proofs in parallel, since assessing the correctness of one proof does not require knowing the details of another.

As with definitions, the #print command will show you the proof of a theorem:

# variable {p : Prop}
# variable {q : Prop}
theorem t1 : p → q → p := fun hp : p => fun hq : q => hp

#print t1

Notice that the lambda abstractions hp : p and hq : q can be viewed as temporary assumptions in the proof of t1. Lean also allows us to specify the type of the final term hp, explicitly, with a show statement:

# variable {p : Prop}
# variable {q : Prop}
theorem t1 : p → q → p :=
  fun hp : p =>
  fun hq : q =>
  show p from hp

Adding such extra information can improve the clarity of a proof and help detect errors when writing a proof. The show command does nothing more than annotate the type, and, internally, all the presentations of t1 that we have seen produce the same term.

As with ordinary definitions, we can move the lambda-abstracted variables to the left of the colon:

# variable {p : Prop}
# variable {q : Prop}
theorem t1 (hp : p) (hq : q) : p := hp

#print t1    -- p → q → p

We can use the theorem t1 just as a function application:

# variable {p : Prop}
# variable {q : Prop}
theorem t1 (hp : p) (hq : q) : p := hp

axiom hp : p

theorem t2 : q → p := t1 hp

The axiom declaration postulates the existence of an element of the given type and may compromise logical consistency. For example, we can use it to postulate that the empty type False has an element:

axiom unsound : False
-- Everything follows from false
theorem ex : 1 = 0 :=
  False.elim unsound

Declaring an "axiom" hp : p is tantamount to declaring that p is true, as witnessed by hp. Applying the theorem t1 : p → q → p to the fact hp : p that p is true yields the theorem t1 hp : q → p.

Recall that we can also write theorem t1 as follows:

theorem t1 {p q : Prop} (hp : p) (hq : q) : p := hp

#print t1

The type of t1 is now ∀ {p q : Prop}, p → q → p. We can read this as the assertion "for every pair of propositions p q, we have p → q → p." For example, we can move all parameters to the right of the colon:

theorem t1 : ∀ {p q : Prop}, p → q → p :=
  fun {p q : Prop} (hp : p) (hq : q) => hp

If p and q have been declared as variables, Lean will generalize them for us automatically:

variable {p q : Prop}

theorem t1 : p → q → p := fun (hp : p) (hq : q) => hp

In fact, by the propositions-as-types correspondence, we can declare the assumption hp that p holds, as another variable:

variable {p q : Prop}
variable (hp : p)

theorem t1 : q → p := fun (hq : q) => hp

Lean detects that the proof uses hp and automatically adds hp : p as a premise. In all cases, the command #print t1 still yields ∀ p q : Prop, p → q → p. Remember that this type can just as well be written ∀ (p q : Prop) (hp : p) (hq : q), p, since the arrow denotes nothing more than an arrow type in which the target does not depend on the bound variable.

When we generalize t1 in such a way, we can then apply it to different pairs of propositions, to obtain different instances of the general theorem.

theorem t1 (p q : Prop) (hp : p) (hq : q) : p := hp

variable (p q r s : Prop)

#check t1 p q                -- p → q → p
#check t1 r s                -- r → s → r
#check t1 (r → s) (s → r)    -- (r → s) → (s → r) → r → s

variable (h : r → s)
#check t1 (r → s) (s → r) h  -- (s → r) → r → s

Once again, using the propositions-as-types correspondence, the variable h of type r → s can be viewed as the hypothesis, or premise, that r → s holds.

As another example, let us consider the composition function discussed in the last chapter, now with propositions instead of types.

variable (p q r s : Prop)

theorem t2 (h₁ : q → r) (h₂ : p → q) : p → r :=
  fun h₃ : p =>
  show r from h₁ (h₂ h₃)

As a theorem of propositional logic, what does t2 say?

Note that it is often useful to use numeric unicode subscripts, entered as \0, \1, \2, ..., for hypotheses, as we did in this example.

Propositional Logic

Lean defines all the standard logical connectives and notation. The propositional connectives come with the following notation:

Ascii	Unicode	Editor shortcut	Definition
True			True
False			False
Not	¬	\not, \neg	Not
/\	∧	\and	And
\/	∨	\or	Or
->	→	\to, \r, \imp
<->	↔	\iff, \lr	Iff

They all take values in Prop.

variable (p q : Prop)

#check p → q → p ∧ q
#check ¬p → p ↔ False
#check p ∨ q → q ∨ p

The order of operations is as follows: unary negation ¬ binds most strongly, then ∧, then ∨, then →, and finally ↔. For example, a ∧ b → c ∨ d ∧ e means (a ∧ b) → (c ∨ (d ∧ e)). Remember that → associates to the right (nothing changes now that the arguments are elements of Prop, instead of some other Type), as do the other binary connectives. So if we have p q r : Prop, the expression p → q → r reads "if p, then if q, then r." This is just the "curried" form of p ∧ q → r.

In the last chapter we observed that lambda abstraction can be viewed as an "introduction rule" for →. In the current setting, it shows how to "introduce" or establish an implication. Application can be viewed as an "elimination rule," showing how to "eliminate" or use an implication in a proof. The other propositional connectives are defined in Lean's library in the file Prelude.core (see importing files for more information on the library hierarchy), and each connective comes with its canonical introduction and elimination rules.

Conjunction

The expression And.intro h1 h2 builds a proof of p ∧ q using proofs h1 : p and h2 : q. It is common to describe And.intro as the and-introduction rule. In the next example we use And.intro to create a proof of p → q → p ∧ q.

variable (p q : Prop)

example (hp : p) (hq : q) : p ∧ q := And.intro hp hq

#check fun (hp : p) (hq : q) => And.intro hp hq

The example command states a theorem without naming it or storing it in the permanent context. Essentially, it just checks that the given term has the indicated type. It is convenient for illustration, and we will use it often.

The expression And.left h creates a proof of p from a proof h : p ∧ q. Similarly, And.right h is a proof of q. They are commonly known as the left and right and-elimination rules.

variable (p q : Prop)

example (h : p ∧ q) : p := And.left h
example (h : p ∧ q) : q := And.right h

We can now prove p ∧ q → q ∧ p with the following proof term.

variable (p q : Prop)

example (h : p ∧ q) : q ∧ p :=
  And.intro (And.right h) (And.left h)

Notice that and-introduction and and-elimination are similar to the pairing and projection operations for the Cartesian product. The difference is that given hp : p and hq : q, And.intro hp hq has type p ∧ q : Prop, while Prod hp hq has type p × q : Type. The similarity between ∧ and × is another instance of the Curry-Howard isomorphism, but in contrast to implication and the function space constructor, ∧ and × are treated separately in Lean. With the analogy, however, the proof we have just constructed is similar to a function that swaps the elements of a pair.

We will see in Chapter Structures and Records that certain types in Lean are structures, which is to say, the type is defined with a single canonical constructor which builds an element of the type from a sequence of suitable arguments. For every p q : Prop, p ∧ q is an example: the canonical way to construct an element is to apply And.intro to suitable arguments hp : p and hq : q. Lean allows us to use anonymous constructor notation ⟨arg1, arg2, ...⟩ in situations like these, when the relevant type is an inductive type and can be inferred from the context. In particular, we can often write ⟨hp, hq⟩ instead of And.intro hp hq:

variable (p q : Prop)
variable (hp : p) (hq : q)

#check (⟨hp, hq⟩ : p ∧ q)

These angle brackets are obtained by typing \< and \>, respectively.

Lean provides another useful syntactic gadget. Given an expression e of an inductive type Foo (possibly applied to some arguments), the notation e.bar is shorthand for Foo.bar e. This provides a convenient way of accessing functions without opening a namespace. For example, the following two expressions mean the same thing:

variable (xs : List Nat)

#check List.length xs
#check xs.length

As a result, given h : p ∧ q, we can write h.left for And.left h and h.right for And.right h. We can therefore rewrite the sample proof above conveniently as follows:

variable (p q : Prop)

example (h : p ∧ q) : q ∧ p :=
  ⟨h.right, h.left⟩

There is a fine line between brevity and obfuscation, and omitting information in this way can sometimes make a proof harder to read. But for straightforward constructions like the one above, when the type of h and the goal of the construction are salient, the notation is clean and effective.

It is common to iterate constructions like "And." Lean also allows you to flatten nested constructors that associate to the right, so that these two proofs are equivalent:

variable (p q : Prop)

example (h : p ∧ q) : q ∧ p ∧ q :=
  ⟨h.right, ⟨h.left, h.right⟩⟩

example (h : p ∧ q) : q ∧ p ∧ q :=
  ⟨h.right, h.left, h.right⟩

This is often useful as well.

Disjunction

The expression Or.intro_left q hp creates a proof of p ∨ q from a proof hp : p. Similarly, Or.intro_right p hq creates a proof for p ∨ q using a proof hq : q. These are the left and right or-introduction rules.

variable (p q : Prop)
example (hp : p) : p ∨ q := Or.intro_left q hp
example (hq : q) : p ∨ q := Or.intro_right p hq

The or-elimination rule is slightly more complicated. The idea is that we can prove r from p ∨ q, by showing that r follows from p and that r follows from q. In other words, it is a proof by cases. In the expression Or.elim hpq hpr hqr, Or.elim takes three arguments, hpq : p ∨ q, hpr : p → r and hqr : q → r, and produces a proof of r. In the following example, we use Or.elim to prove p ∨ q → q ∨ p.

variable (p q r : Prop)

example (h : p ∨ q) : q ∨ p :=
  Or.elim h
    (fun hp : p =>
      show q ∨ p from Or.intro_right q hp)
    (fun hq : q =>
      show q ∨ p from Or.intro_left p hq)

In most cases, the first argument of Or.intro_right and Or.intro_left can be inferred automatically by Lean. Lean therefore provides Or.inr and Or.inl which can be viewed as shorthand for Or.intro_right _ and Or.intro_left _. Thus the proof term above could be written more concisely:

variable (p q r : Prop)

example (h : p ∨ q) : q ∨ p :=
  Or.elim h (fun hp => Or.inr hp) (fun hq => Or.inl hq)

Notice that there is enough information in the full expression for Lean to infer the types of hp and hq as well. But using the type annotations in the longer version makes the proof more readable, and can help catch and debug errors.

Because Or has two constructors, we cannot use anonymous constructor notation. But we can still write h.elim instead of Or.elim h:

variable (p q r : Prop)

example (h : p ∨ q) : q ∨ p :=
  h.elim (fun hp => Or.inr hp) (fun hq => Or.inl hq)

Once again, you should exercise judgment as to whether such abbreviations enhance or diminish readability.

Negation and Falsity

Negation, ¬p, is actually defined to be p → False, so we obtain ¬p by deriving a contradiction from p. Similarly, the expression hnp hp produces a proof of False from hp : p and hnp : ¬p. The next example uses both these rules to produce a proof of (p → q) → ¬q → ¬p. (The symbol ¬ is produced by typing \not or \neg.)

variable (p q : Prop)

example (hpq : p → q) (hnq : ¬q) : ¬p :=
  fun hp : p =>
  show False from hnq (hpq hp)

The connective False has a single elimination rule, False.elim, which expresses the fact that anything follows from a contradiction. This rule is sometimes called ex falso (short for ex falso sequitur quodlibet), or the principle of explosion.

variable (p q : Prop)

example (hp : p) (hnp : ¬p) : q := False.elim (hnp hp)

The arbitrary fact, q, that follows from falsity is an implicit argument in False.elim and is inferred automatically. This pattern, deriving an arbitrary fact from contradictory hypotheses, is quite common, and is represented by absurd.

variable (p q : Prop)

example (hp : p) (hnp : ¬p) : q := absurd hp hnp

Here, for example, is a proof of ¬p → q → (q → p) → r:

variable (p q r : Prop)

example (hnp : ¬p) (hq : q) (hqp : q → p) : r :=
  absurd (hqp hq) hnp

Incidentally, just as False has only an elimination rule, True has only an introduction rule, True.intro : true. In other words, True is simply true, and has a canonical proof, True.intro.

Logical Equivalence

The expression Iff.intro h1 h2 produces a proof of p ↔ q from h1 : p → q and h2 : q → p. The expression Iff.mp h produces a proof of p → q from h : p ↔ q. Similarly, Iff.mpr h produces a proof of q → p from h : p ↔ q. Here is a proof of p ∧ q ↔ q ∧ p:

variable (p q : Prop)

theorem and_swap : p ∧ q ↔ q ∧ p :=
  Iff.intro
    (fun h : p ∧ q =>
     show q ∧ p from And.intro (And.right h) (And.left h))
    (fun h : q ∧ p =>
     show p ∧ q from And.intro (And.right h) (And.left h))

#check and_swap p q    -- p ∧ q ↔ q ∧ p

variable (h : p ∧ q)
example : q ∧ p := Iff.mp (and_swap p q) h

We can use the anonymous constructor notation to construct a proof of p ↔ q from proofs of the forward and backward directions, and we can also use . notation with mp and mpr. The previous examples can therefore be written concisely as follows:

variable (p q : Prop)

theorem and_swap : p ∧ q ↔ q ∧ p :=
  ⟨ fun h => ⟨h.right, h.left⟩, fun h => ⟨h.right, h.left⟩ ⟩

example (h : p ∧ q) : q ∧ p := (and_swap p q).mp h

Introducing Auxiliary Subgoals

This is a good place to introduce another device Lean offers to help structure long proofs, namely, the have construct, which introduces an auxiliary subgoal in a proof. Here is a small example, adapted from the last section:

variable (p q : Prop)

example (h : p ∧ q) : q ∧ p :=
  have hp : p := h.left
  have hq : q := h.right
  show q ∧ p from And.intro hq hp

Internally, the expression have h : p := s; t produces the term (fun (h : p) => t) s. In other words, s is a proof of p, t is a proof of the desired conclusion assuming h : p, and the two are combined by a lambda abstraction and application. This simple device is extremely useful when it comes to structuring long proofs, since we can use intermediate have's as stepping stones leading to the final goal.

Lean also supports a structured way of reasoning backwards from a goal, which models the "suffices to show" construction in ordinary mathematics. The next example simply permutes the last two lines in the previous proof.

variable (p q : Prop)

example (h : p ∧ q) : q ∧ p :=
  have hp : p := h.left
  suffices hq : q from And.intro hq hp
  show q from And.right h

Writing suffices hq : q leaves us with two goals. First, we have to show that it indeed suffices to show q, by proving the original goal of q ∧ p with the additional hypothesis hq : q. Finally, we have to show q.

Classical Logic

The introduction and elimination rules we have seen so far are all constructive, which is to say, they reflect a computational understanding of the logical connectives based on the propositions-as-types correspondence. Ordinary classical logic adds to this the law of the excluded middle, p ∨ ¬p. To use this principle, you have to open the classical namespace.

open Classical

variable (p : Prop)
#check em p

Intuitively, the constructive "Or" is very strong: asserting p ∨ q amounts to knowing which is the case. If RH represents the Riemann hypothesis, a classical mathematician is willing to assert RH ∨ ¬RH, even though we cannot yet assert either disjunct.

One consequence of the law of the excluded middle is the principle of double-negation elimination:

open Classical

theorem dne {p : Prop} (h : ¬¬p) : p :=
  Or.elim (em p)
    (fun hp : p => hp)
    (fun hnp : ¬p => absurd hnp h)

Double-negation elimination allows one to prove any proposition, p, by assuming ¬p and deriving false, because that amounts to proving ¬¬p. In other words, double-negation elimination allows one to carry out a proof by contradiction, something which is not generally possible in constructive logic. As an exercise, you might try proving the converse, that is, showing that em can be proved from dne.

The classical axioms also give you access to additional patterns of proof that can be justified by appeal to em. For example, one can carry out a proof by cases:

open Classical
variable (p : Prop)

example (h : ¬¬p) : p :=
  byCases
    (fun h1 : p => h1)
    (fun h1 : ¬p => absurd h1 h)

Or you can carry out a proof by contradiction:

open Classical
variable (p : Prop)

example (h : ¬¬p) : p :=
  byContradiction
    (fun h1 : ¬p =>
     show False from h h1)

If you are not used to thinking constructively, it may take some time for you to get a sense of where classical reasoning is used. It is needed in the following example because, from a constructive standpoint, knowing that p and q are not both true does not necessarily tell you which one is false:

# open Classical
# variable (p q : Prop)
example (h : ¬(p ∧ q)) : ¬p ∨ ¬q :=
  Or.elim (em p)
    (fun hp : p =>
      Or.inr
        (show ¬q from
          fun hq : q =>
          h ⟨hp, hq⟩))
    (fun hp : ¬p =>
      Or.inl hp)

We will see later that there are situations in constructive logic where principles like excluded middle and double-negation elimination are permissible, and Lean supports the use of classical reasoning in such contexts without relying on excluded middle.

The full list of axioms that are used in Lean to support classical reasoning are discussed in Axioms and Computation.

Examples of Propositional Validities

Lean's standard library contains proofs of many valid statements of propositional logic, all of which you are free to use in proofs of your own. The following list includes a number of common identities.

Commutativity:

1. p ∧ q ↔ q ∧ p
2. p ∨ q ↔ q ∨ p

Associativity:

3. (p ∧ q) ∧ r ↔ p ∧ (q ∧ r)
4. (p ∨ q) ∨ r ↔ p ∨ (q ∨ r)

Distributivity:

5. p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r)
6. p ∨ (q ∧ r) ↔ (p ∨ q) ∧ (p ∨ r)

Other properties:

7. (p → (q → r)) ↔ (p ∧ q → r)
8. ((p ∨ q) → r) ↔ (p → r) ∧ (q → r)
9. ¬(p ∨ q) ↔ ¬p ∧ ¬q
10. ¬p ∨ ¬q → ¬(p ∧ q)
11. ¬(p ∧ ¬p)
12. p ∧ ¬q → ¬(p → q)
13. ¬p → (p → q)
14. (¬p ∨ q) → (p → q)
15. p ∨ False ↔ p
16. p ∧ False ↔ False
17. ¬(p ↔ ¬p)
18. (p → q) → (¬q → ¬p)

These require classical reasoning:

19. (p → r ∨ s) → ((p → r) ∨ (p → s))
20. ¬(p ∧ q) → ¬p ∨ ¬q
21. ¬(p → q) → p ∧ ¬q
22. (p → q) → (¬p ∨ q)
23. (¬q → ¬p) → (p → q)
24. p ∨ ¬p
25. (((p → q) → p) → p)

The sorry identifier magically produces a proof of anything, or provides an object of any data type at all. Of course, it is unsound as a proof method -- for example, you can use it to prove False -- and Lean produces severe warnings when files use or import theorems which depend on it. But it is very useful for building long proofs incrementally. Start writing the proof from the top down, using sorry to fill in subproofs. Make sure Lean accepts the term with all the sorry's; if not, there are errors that you need to correct. Then go back and replace each sorry with an actual proof, until no more remain.

Here is another useful trick. Instead of using sorry, you can use an underscore _ as a placeholder. Recall this tells Lean that the argument is implicit, and should be filled in automatically. If Lean tries to do so and fails, it returns with an error message "don't know how to synthesize placeholder," followed by the type of the term it is expecting, and all the objects and hypotheses available in the context. In other words, for each unresolved placeholder, Lean reports the subgoal that needs to be filled at that point. You can then construct a proof by incrementally filling in these placeholders.

For reference, here are two sample proofs of validities taken from the list above.

open Classical

-- distributivity
example (p q r : Prop) : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) :=
  Iff.intro
    (fun h : p ∧ (q ∨ r) =>
      have hp : p := h.left
      Or.elim (h.right)
        (fun hq : q =>
          show (p ∧ q) ∨ (p ∧ r) from Or.inl ⟨hp, hq⟩)
        (fun hr : r =>
          show (p ∧ q) ∨ (p ∧ r) from Or.inr ⟨hp, hr⟩))
    (fun h : (p ∧ q) ∨ (p ∧ r) =>
      Or.elim h
        (fun hpq : p ∧ q =>
          have hp : p := hpq.left
          have hq : q := hpq.right
          show p ∧ (q ∨ r) from ⟨hp, Or.inl hq⟩)
        (fun hpr : p ∧ r =>
          have hp : p := hpr.left
          have hr : r := hpr.right
          show p ∧ (q ∨ r) from ⟨hp, Or.inr hr⟩))

-- an example that requires classical reasoning
example (p q : Prop) : ¬(p ∧ ¬q) → (p → q) :=
  fun h : ¬(p ∧ ¬q) =>
  fun hp : p =>
  show q from
    Or.elim (em q)
      (fun hq : q => hq)
      (fun hnq : ¬q => absurd (And.intro hp hnq) h)

Exercises

Prove the following identities, replacing the "sorry" placeholders with actual proofs.

variable (p q r : Prop)

-- commutativity of ∧ and ∨
example : p ∧ q ↔ q ∧ p := sorry
example : p ∨ q ↔ q ∨ p := sorry

-- associativity of ∧ and ∨
example : (p ∧ q) ∧ r ↔ p ∧ (q ∧ r) := sorry
example : (p ∨ q) ∨ r ↔ p ∨ (q ∨ r) := sorry

-- distributivity
example : p ∧ (q ∨ r) ↔ (p ∧ q) ∨ (p ∧ r) := sorry
example : p ∨ (q ∧ r) ↔ (p ∨ q) ∧ (p ∨ r) := sorry

-- other properties
example : (p → (q → r)) ↔ (p ∧ q → r) := sorry
example : ((p ∨ q) → r) ↔ (p → r) ∧ (q → r) := sorry
example : ¬(p ∨ q) ↔ ¬p ∧ ¬q := sorry
example : ¬p ∨ ¬q → ¬(p ∧ q) := sorry
example : ¬(p ∧ ¬p) := sorry
example : p ∧ ¬q → ¬(p → q) := sorry
example : ¬p → (p → q) := sorry
example : (¬p ∨ q) → (p → q) := sorry
example : p ∨ False ↔ p := sorry
example : p ∧ False ↔ False := sorry
example : (p → q) → (¬q → ¬p) := sorry

Prove the following identities, replacing the "sorry" placeholders with actual proofs. These require classical reasoning.

open Classical

variable (p q r : Prop)

example : (p → q ∨ r) → ((p → q) ∨ (p → r)) := sorry
example : ¬(p ∧ q) → ¬p ∨ ¬q := sorry
example : ¬(p → q) → p ∧ ¬q := sorry
example : (p → q) → (¬p ∨ q) := sorry
example : (¬q → ¬p) → (p → q) := sorry
example : p ∨ ¬p := sorry
example : (((p → q) → p) → p) := sorry

Prove ¬(p ↔ ¬p) without using classical logic.