calc.md

How to use calc

calc is an environment -- so a "mode" like tactic mode, term mode and conv mode. Documentation and basic examples for how to use it are in Theorem Proving In Lean, in section 4.3.

Basic example usage:

example (a b c : ℕ) (H1 : a = b + 1) (H2 : b = c) : a = c + 1 :=
calc a = b + 1 : H1
...    = c + 1 : by rw H2

Error messages, and how to avoid them.

Note that the error messages can be quite obscure when things aren't quite right, and often the red squiggles end up under a random .... A tip to avoid these problems with calc usage is to first populate a skeleton proof such as

example : A = D :=
calc A = B : sorry
...    = C : sorry
...    = D : sorry

(in tactic mode this might look more like

have H : A = D,
  exact calc A = B : sorry
  ...          = C : sorry
  ...          = D : sorry,

with a comma at the end), and then to start filling in the sorries after that. (Idle thought: could one write a VS Code snippet to write this skeleton?)

Using operators other than equality.

Many of the basic examples in TPIL use equality for most or all of the operators, but actually calc will work with any relation for which the corresponding transitivity statement is tagged [trans]:

definition r : ℕ → ℕ → Prop := sorry 
@[trans] theorem r_trans (a b c : ℕ) : r a b → r b c → r a c := sorry
infix `***`: 50 := r

example (a b c : ℕ) (H1 : a *** b) (H2 : b *** c) : a *** c :=
calc a *** b : H1
...    *** c : H2

Using more than one operator.

This is possible; TPIL has the following example:

theorem T2 (a b c d : ℕ)
  (h1 : a = b) (h2 : b ≤ c) (h3 : c + 1 < d) : a < d :=
calc
  a     = b     : h1
    ... < b + 1 : nat.lt_succ_self b
    ... ≤ c + 1 : nat.succ_le_succ h2
    ... < d     : h3

What is actually going on here? The proofs themselves are not a mystery, for example nat.succ_le_succ h2 is a proof of b + 1 ≤ c + 1. The clever part is that lean can put all of these together to correctly deduce that if U = V < W ≤ X < Y then U < Y. The way this is done, Kevin thinks (can someone verify this?) is that Lean continually tries to amalgamate the first two operators in the list, until there is only one left. In other words, Lean will attempt to reduce the equations thus:

U = V < W ≤ X < Y
U < W ≤ X < Y
U < X < Y
U < Y

Note the following subtlety: given U op1 V and V op2 W Lean has to conclude U op3 W for some operator, which might be op1 or op2 (or even, as we shall see, a new operator). How is Lean doing this? The easiest case is when one of op1 and op2 is =. Lean knows

#check @trans_rel_right -- ∀ {α : Sort u_1} {a b c : α} (r : α → α → Prop), a = b → r b c → r a c
#check @trans_rel_left -- ∀ {α : Sort u_1} {a b c : α} (r : α → α → Prop), r a b → b = c → r a c

and (Kevin believes) uses them if one of the operators is an equality operator. If however neither operator is the equality operator, Lean looks through the theorems in its database which are tagged [trans] and applies these instead. For example Lean has the following definitions:

@[trans] lemma lt_of_lt_of_le [preorder α] : ∀ {a b c : α}, a < b → b ≤ c → a < c
@[trans] lemma lt_trans [preorder α] : ∀ {a b c : α}, a < b → b < c → a < c

and it is easily seen that these lemmas can be used to justify the example in the manual.

Using user-defined operators

It is as simple as tagging the relevant results with trans. For example

definition r : ℕ → ℕ → Prop := sorry 
definition s : ℕ → ℕ → Prop := sorry 
definition t : ℕ → ℕ → Prop := sorry 
@[trans] theorem rst_trans (a b c : ℕ) : r a b → s b c → t a c := sorry
infix `***`: 50 := r
infix `&&&` : 50 := s
infix `%%%` : 50 := t

example (a b c : ℕ) (H1 : a *** b) (H2 : b &&& c) : a %%% c :=
calc a *** b : H1
...    &&& c : H2

This example shows us that the third operator op3 can be different to both op1 and op2.