This is a guide for coq users getting into writing lean proofs.
Please PR your own or request additions via issue tracker or lean gitter.
n/a doesn't mean it doesn't exist, only that I don't know about it.
The main table is for the Lean release v3.3.0.
Coq Tactic | Lean Tactic | Notes |
---|---|---|
; |
; |
|
. |
, |
|
assumption |
assumption |
|
admit |
admit |
|
apply |
apply |
lean apply is coq eapply |
apply H in A |
have A2 := H A |
Will create a new hypothesis A2, and A will persist, see coq-tactic-substitutes.lean for a closer approximation |
assert |
have |
|
auto |
n/a | |
autorewrite |
n/a | simp using <attribute> approximates |
change |
change |
|
change with |
n/a | |
clear |
clear |
no clear - |
constructor |
constructor |
|
destruct |
cases |
|
destruct x eqn:? |
destruct x |
|
eapply |
apply |
|
exact |
exact |
|
exfalso |
exfalso |
|
exists |
existsi |
use is the same, but takes the expected type into account. use requires mathlib |
eexists |
existsi _ |
|
f_equal |
congr' 1 |
requires mathlib |
fail |
fail_if_success {skip} |
|
first [A | B |.. | X] |
first [A B .. X] |
|
generalize t |
generalize : t = y |
y is name of the new variable, the name must be provided |
generalize dependent |
revert |
|
idtac |
skip |
skip does not print, succeeds trivially |
induction |
induction |
|
intro |
intro |
|
intros |
intros |
|
intuition |
n/a | |
inversion |
cases |
Cases should be applied to dependent arguments first |
left |
left |
|
omega |
omega |
requires mathlib. Might not have the same features as Coq's omega |
pose |
let |
|
pose proof |
have |
|
progress |
n/a | lean tactics by convention should fail if they don't progress |
remember x as y eqn:h |
generalize h : x = y |
names must be provided |
revert |
n/a | always dependent |
revert dependent |
revert |
|
rewrite |
rewrite , rw |
|
rewrite <- |
rewrite <- , rw |
|
right |
right |
|
simpl |
dsimp |
to some approximation at least. simp only might be closer |
simpl in |
dsimp at |
|
simpl in * |
dsimp at * |
|
solve |
solve1 |
|
specialize (H e) |
specialize (H e) |
|
split |
split |
|
subst x |
subst x |
|
subst |
n/a | |
symmetry |
symmetry |
|
transitivity |
transitivity |
|
trivial |
trivial |
|
try T |
try {T} |
curly braces required |
typeclasses eauto |
apply_instance |
|
unfold |
unfold |
|
unfold in |
unfold at |
|
unshelve eapply |
fapply |
Similar options from Coq exist in Lean as well. Whereas Coq options are set and unset with the Vernacular Set option
and Unset option
, Lean options toggled with set_option <option> true
and set_option <option> false
. Here is a list of Coq options and their Lean versions:
Coq option | Lean option | Notes |
---|---|---|
Printing Implicit |
pp.implicit |
default false |
Printing Universes |
pp.universes |
default false |
Printing Notations |
pp.notation |
default true |
And other Vernacular:
Coq Vernacular | Lean directive | Notes |
---|---|---|
Opaque ident |
attribute [irreducible] ident |
|
Transparent ident |
attribute [reducible] ident |
|
Check term |
#check term |
|
Print term |
#print term |
Can also be used to print structures, inductive types, notation |
Proof using P |
include P |
include P makes P available to all proofs, as well as typeclass resolution if it is an instance. omit P un-includes P |
Variables and sections
Coq construct | Lean construct | Notes |
---|---|---|
Variable |
parameter |
only within a section. Lean variable does not automatically apply arguments within the section |
Universes |
universes |
use universe variables if you want to declare the universes, but not fix them for the file |