2.7 KiB
Calculational Proofs
A calculational proof is just a chain of intermediate results that are
meant to be composed by basic principles such as the transitivity of
=. In Lean, a calculation proof starts with the keyword calc, and has
the following syntax
calc <expr>_0 'op_1' <expr>_1 ':' <proof>_1
'...' 'op_2' <expr>_2 ':' <proof>_2
...
'...' 'op_n' <expr>_n ':' <proof>_n
Each <proof>_i is a proof for <expr>_{i-1} op_i <expr>_i.
Recall that proofs are also expressions in Lean. The <proof>_i
may also be of the form { <pr> }, where <pr> is a proof
for some equality a = b. The form { <pr> } is just syntax sugar
for subst <pr> (refl <expr>_{i-1})
That is, we are claiming we can obtain <expr>_i by replacing a with b
in <expr>_{i-1}.
Here is an example
import nat
using nat
variables a b c d e : nat.
axiom Ax1 : a = b.
axiom Ax2 : b = c + 1.
axiom Ax3 : c = d.
axiom Ax4 : e = 1 + d.
theorem T : a = e
:= calc a = b : Ax1
... = c + 1 : Ax2
... = d + 1 : { Ax3 }
... = 1 + d : add_comm d 1
... = e : symm Ax4.
The proof expressions <proof>_i do not need to be explicitly provided.
We can use by <tactic> or by <solver> to (try to) automatically construct the
proof expression using the given tactic or solver.
Even when tactics and solvers are not used, we can still use the elaboration engine to fill
gaps in our calculational proofs. In the previous examples, we can use _ as arguments for the
add_comm theorem. The Lean elaboration engine will infer d and 1 for us.
Here is the same example using placeholders.
import nat
using nat
variables a b c d e : nat.
axiom Ax1 : a = b.
axiom Ax2 : b = c + 1.
axiom Ax3 : c = d.
axiom Ax4 : e = 1 + d.
theorem T : a = e
:= calc a = b : Ax1
... = c + 1 : Ax2
... = d + 1 : { Ax3 }
... = 1 + d : add_comm _ _
... = e : symm Ax4.
The calc command can be configured for any relation that supports
some form of transitivity. It can even combine different relations.
import nat
using nat
theorem T2 (a b c : nat) (H1 : a = b) (H2 : b = c + 1) : a ≠ 0
:= calc a = b : H1
... = c + 1 : H2
... = succ c : add_one _
... ≠ 0 : succ_ne_zero _The Lean simplifier can be used to reduce the size of calculational proofs. In the following example, we create a rewrite rule set with basic theorems from the Natural number library, and some of the axioms declared above.