* 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

#+BEGIN_SRC
  calc  <expr>_0  'op_1'  <expr>_1  ':'  <proof>_1
          '...'   'op_2'  <expr>_2  ':'  <proof>_2
           ...
          '...'   'op_n'  <expr>_n  ':'  <proof>_n
#+END_SRC

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

#+BEGIN_SRC lean
  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.

#+END_SRC

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.

#+BEGIN_SRC lean
  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.

#+END_SRC

The =calc= command can be configured for any relation that supports
some form of transitivity. It can even combine different relations.

#+BEGIN_SRC lean
  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 _
#+END_SRC

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.