as_aux_lemma => tac
does the same as tac
, except that it wraps the resulting expression
into an auxiliary lemma. In some cases, this significantly reduces the size of expressions
because the proof term is not duplicated.
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with_annotate_state stx t
annotates the lexical range of stx : Syntax
with
the initial and final state of running tactic t
.
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Introduces one or more hypotheses, optionally naming and/or pattern-matching them.
For each hypothesis to be introduced, the remaining main goal's target type must
be a let
or function type.
intro
by itself introduces one anonymous hypothesis, which can be accessed by e.g.assumption
.intro x y
introduces two hypotheses and names them. Individual hypotheses can be anonymized via_
, or matched against a pattern:-- ... ⊢ α × β → ... intro (a, b) -- ..., a : α, b : β ⊢ ...
- Alternatively,
intro
can be combined with pattern matching much likefun
:intro | n + 1, 0 => tac | ...
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Introduces zero or more hypotheses, optionally naming them.
intros
is equivalent to repeatedly applyingintro
until the goal is not an obvious candidate forintro
, which is to say that so long as the goal is alet
or a pi type (e.g. an implication, function, or universal quantifier), theintros
tactic will introduce an anonymous hypothesis. This tactic does not unfold definitions.intros x y ...
is equivalent tointro x y ...
, introducing hypotheses for each supplied argument and unfolding definitions as necessary. Each argument can be either an identifier or a_
. An identifier indicates a name to use for the corresponding introduced hypothesis, and a_
indicates that the hypotheses should be introduced anonymously.
Examples #
Basic properties:
def AllEven (f : Nat → Nat) := ∀ n, f n % 2 = 0
-- Introduces the two obvious hypotheses automatically
example : ∀ (f : Nat → Nat), AllEven f → AllEven (fun k => f (k + 1)) := by
intros
/- Tactic state
f✝ : Nat → Nat
a✝ : AllEven f✝
⊢ AllEven fun k => f✝ (k + 1) -/
sorry
-- Introduces exactly two hypotheses, naming only the first
example : ∀ (f : Nat → Nat), AllEven f → AllEven (fun k => f (k + 1)) := by
intros g _
/- Tactic state
g : Nat → Nat
a✝ : AllEven g
⊢ AllEven fun k => g (k + 1) -/
sorry
-- Introduces exactly three hypotheses, which requires unfolding `AllEven`
example : ∀ (f : Nat → Nat), AllEven f → AllEven (fun k => f (k + 1)) := by
intros f h n
/- Tactic state
f : Nat → Nat
h : AllEven f
n : Nat
⊢ (fun k => f (k + 1)) n % 2 = 0 -/
apply h
Implications:
example (p q : Prop) : p → q → p := by
intros
/- Tactic state
a✝¹ : p
a✝ : q
⊢ p -/
assumption
Let bindings:
example : let n := 1; let k := 2; n + k = 3 := by
intros
/- n✝ : Nat := 1
k✝ : Nat := 2
⊢ n✝ + k✝ = 3 -/
rfl
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rename t => x
renames the most recent hypothesis whose type matches t
(which may contain placeholders) to x
, or fails if no such hypothesis could be found.
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clear x...
removes the given hypotheses, or fails if there are remaining
references to a hypothesis.
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Syntax for trying to clear the values of all local definitions.
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Syntax for creating a hypothesis before clearing values.
In (hx : x = _)
, the value of x
is unified with _
.
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Argument for the clear_value
tactic.
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clear_value x...
clears the values of the given local definitions. A local definitionx : α := v
becomes a hypothesisx : α
.clear_value (h : x = _)
adds a hypothesish : x = v
before clearing the value ofx
. This is short forhave h : x = v := rfl; clear_value x
. Any value definitionally equal tov
can be used in place of_
.clear_value *
clears values of all hypotheses that can be cleared. Fails if none can be cleared.
These syntaxes can be combined. For example, clear_value x y *
ensures that x
and y
are cleared
while trying to clear all other local definitions,
and clear_value (hx : x = _) y * with hx
does the same while first adding the hx : x = v
hypothesis.
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subst x...
substitutes each hypothesis x
with a definition found in the local context,
then eliminates the hypothesis.
- If
x
is a local definition, then its definition is used. - Otherwise, if there is a hypothesis of the form
x = e
ore = x
, thene
is used for the definition ofx
.
If h : a = b
, then subst h
may be used if either a
or b
unfolds to a local hypothesis.
This is similar to the cases h
tactic.
See also: subst_vars
for substituting all local hypotheses that have a defining equation.
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assumption
tries to solve the main goal using a hypothesis of compatible type, or else fails.
Note also the ‹t›
term notation, which is a shorthand for show t by assumption
.
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contradiction
closes the main goal if its hypotheses are "trivially contradictory".
- Inductive type/family with no applicable constructors
example (h : False) : p := by contradiction
- Injectivity of constructors
example (h : none = some true) : p := by contradiction --
- Decidable false proposition
example (h : 2 + 2 = 3) : p := by contradiction
- Contradictory hypotheses
example (h : p) (h' : ¬ p) : q := by contradiction
- Other simple contradictions such as
example (x : Nat) (h : x ≠ x) : p := by contradiction
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Changes the goal to False
, retaining as much information as possible:
- If the goal is
False
, do nothing. - If the goal is an implication or a function type, introduce the argument and restart.
(In particular, if the goal is
x ≠ y
, introducex = y
.) - Otherwise, for a propositional goal
P
, replace it with¬ ¬ P
(attempting to find aDecidable
instance, but otherwise falling back to working classically) and introduce¬ P
. - For a non-propositional goal use
False.elim
.
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apply e
tries to match the current goal against the conclusion of e
's type.
If it succeeds, then the tactic returns as many subgoals as the number of premises that
have not been fixed by type inference or type class resolution.
Non-dependent premises are added before dependent ones.
The apply
tactic uses higher-order pattern matching, type class resolution,
and first-order unification with dependent types.
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If the main goal's target type is an inductive type, constructor
solves it with
the first matching constructor, or else fails.
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case tag => tac
focuses on the goal with case nametag
and solves it usingtac
, or else fails.case tag x₁ ... xₙ => tac
additionally renames then
most recent hypotheses with inaccessible names to the given names.case tag₁ | tag₂ => tac
is equivalent to(case tag₁ => tac); (case tag₂ => tac)
.
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next => tac
focuses on the next goal and solves it using tac
, or else fails.
next x₁ ... xₙ => tac
additionally renames the n
most recent hypotheses with
inaccessible names to the given names.
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all_goals tac
runs tac
on each goal, concatenating the resulting goals.
If the tactic fails on any goal, the entire all_goals
tactic fails.
See also any_goals tac
.
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any_goals tac
applies the tactic tac
to every goal,
concatenating the resulting goals for successful tactic applications.
If the tactic fails on all of the goals, the entire any_goals
tactic fails.
This tactic is like all_goals try tac
except that it fails if none of the applications of tac
succeeds.
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focus tac
focuses on the main goal, suppressing all other goals, and runs tac
on it.
Usually · tac
, which enforces that the goal is closed by tac
, should be preferred.
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trace_state
displays the current state in the info view.
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trace msg
displays msg
in the info view.
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fail_if_success t
fails if the tactic t
succeeds.
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(tacs)
executes a list of tactics in sequence, without requiring that
the goal be closed at the end like · tacs
. Like by
itself, the tactics
can be either separated by newlines or ;
.
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with_reducible tacs
executes tacs
using the reducible transparency setting.
In this setting only definitions tagged as [reducible]
are unfolded.
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with_reducible_and_instances tacs
executes tacs
using the .instances
transparency setting.
In this setting only definitions tagged as [reducible]
or type class instances are unfolded.
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with_unfolding_all tacs
executes tacs
using the .all
transparency setting.
In this setting all definitions that are not opaque are unfolded.
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rotate_left n
rotates goals to the left by n
. That is, rotate_left 1
takes the main goal and puts it to the back of the subgoal list.
If n
is omitted, it defaults to 1
.
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Rotate the goals to the right by n
. That is, take the goal at the back
and push it to the front n
times. If n
is omitted, it defaults to 1
.
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try tac
runs tac
and succeeds even if tac
failed.
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tac <;> tac'
runs tac
on the main goal and tac'
on each produced goal,
concatenating all goals produced by tac'
.
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fail msg
is a tactic that always fails, and produces an error using the given message.
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This tactic applies to a goal whose target has the form x ~ x
,
where ~
is equality, heterogeneous equality or any relation that
has a reflexivity lemma tagged with the attribute @[refl].
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rfl'
is similar to rfl
, but disables smart unfolding and unfolds all kinds of definitions,
theorems included (relevant for declarations defined by well-founded recursion).
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ac_rfl
proves equalities up to application of an associative and commutative operator.
instance : Associative (α := Nat) (.+.) := ⟨Nat.add_assoc⟩
instance : Commutative (α := Nat) (.+.) := ⟨Nat.add_comm⟩
example (a b c d : Nat) : a + b + c + d = d + (b + c) + a := by ac_rfl
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The sorry
tactic is a temporary placeholder for an incomplete tactic proof,
closing the main goal using exact sorry
.
This is intended for stubbing-out incomplete parts of a proof while still having a syntactically correct proof skeleton.
Lean will give a warning whenever a proof uses sorry
, so you aren't likely to miss it,
but you can double check if a theorem depends on sorry
by looking for sorryAx
in the output
of the #print axioms my_thm
command, the axiom used by the implementation of sorry
.
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admit
is a synonym for sorry
.
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infer_instance
is an abbreviation for exact inferInstance
.
It synthesizes a value of any target type by typeclass inference.
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+opt
is short for (opt := true)
. It sets the opt
configuration option to true
.
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-opt
is short for (opt := false)
. It sets the opt
configuration option to false
.
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(opt := val)
sets the opt
configuration option to val
.
As a special case, (config := ...)
sets the entire configuration.
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A configuration item for a tactic configuration.
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Configuration options for tactics.
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Optional configuration option for tactics. (Deprecated. Replace (config)?
with optConfig
.)
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The *
location refers to all hypotheses and the goal.
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The ⊢
location refers to the current goal.
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A sequence of one or more locations at which a tactic should operate. These can include local
hypotheses and ⊢
, which denotes the goal.
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Location specifications are used by many tactics that can operate on either the hypotheses or the goal. It can have one of the forms:
- 'empty' is not actually present in this syntax, but most tactics use
(location)?
matchers. It means to target the goal only. at h₁ ... hₙ
: target the hypothesesh₁
, ...,hₙ
at h₁ h₂ ⊢
: target the hypothesesh₁
andh₂
, and the goalat *
: target all hypotheses and the goal
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Extracts let
and let_fun
expressions from within the target or a local hypothesis,
introducing new local definitions.
extract_lets
extracts all the lets from the target.extract_lets x y z
extracts all the lets from the target and usesx
,y
, andz
for the first names. Using_
for a name leaves it unnamed.extract_lets x y z at h
operates on the local hypothesish
instead of the target.
For example, given a local hypotheses if the form h : let x := v; b x
, then extract_lets z at h
introduces a new local definition z := v
and changes h
to be h : b z
.
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Lifts let
and let_fun
expressions within a term as far out as possible.
It is like extract_lets +lift
, but the top-level lets at the end of the procedure
are not extracted as local hypotheses.
lift_lets
lifts let expressions in the target.lift_lets at h
lifts let expressions at the given local hypothesis.
For example,
example : (let x := 1; x) = 1 := by
lift_lets
-- ⊢ let x := 1; x = 1
...
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If thm
is a theorem a = b
, then as a rewrite rule,
thm
means to replacea
withb
, and← thm
means to replaceb
witha
.
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rewrite [e]
applies identity e
as a rewrite rule to the target of the main goal.
If e
is preceded by left arrow (←
or <-
), the rewrite is applied in the reverse direction.
If e
is a defined constant, then the equational theorems associated with e
are used.
This provides a convenient way to unfold e
.
rewrite [e₁, ..., eₙ]
applies the given rules sequentially.rewrite [e] at l
rewritese
at location(s)l
, wherel
is either*
or a list of hypotheses in the local context. In the latter case, a turnstile⊢
or|-
can also be used, to signify the target of the goal.
Using rw (occs := .pos L) [e]
,
where L : List Nat
, you can control which "occurrences" are rewritten.
(This option applies to each rule, so usually this will only be used with a single rule.)
Occurrences count from 1
.
At each allowed occurrence, arguments of the rewrite rule e
may be instantiated,
restricting which later rewrites can be found.
(Disallowed occurrences do not result in instantiation.)
(occs := .neg L)
allows skipping specified occurrences.
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rw
is like rewrite
, but also tries to close the goal by "cheap" (reducible) rfl
afterwards.
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The injection
tactic is based on the fact that constructors of inductive data
types are injections.
That means that if c
is a constructor of an inductive datatype, and if (c t₁)
and (c t₂)
are two terms that are equal then t₁
and t₂
are equal too.
If q
is a proof of a statement of conclusion t₁ = t₂
, then injection applies
injectivity to derive the equality of all arguments of t₁
and t₂
placed in
the same positions. For example, from (a::b) = (c::d)
we derive a=c
and b=d
.
To use this tactic t₁
and t₂
should be constructor applications of the same constructor.
Given h : a::b = c::d
, the tactic injection h
adds two new hypothesis with types
a = c
and b = d
to the main goal.
The tactic injection h with h₁ h₂
uses the names h₁
and h₂
to name the new hypotheses.
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injections
applies injection
to all hypotheses recursively
(since injection
can produce new hypotheses). Useful for destructing nested
constructor equalities like (a::b::c) = (d::e::f)
.
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The discharger clause of simp
and related tactics.
This is a tactic used to discharge the side conditions on conditional rewrite rules.
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Use this rewrite rule before entering the subterms
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Use this rewrite rule after entering the subterms
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A simp lemma specification is:
- optional
↑
or↓
to specify use before or after entering the subterm - optional
←
to use the lemma backward thm
for the theorem to rewrite with
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An erasure specification -thm
says to remove thm
from the simp set
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The simp lemma specification *
means to rewrite with all hypotheses
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The simp
tactic uses lemmas and hypotheses to simplify the main goal target or
non-dependent hypotheses. It has many variants:
simp
simplifies the main goal target using lemmas tagged with the attribute[simp]
.simp [h₁, h₂, ..., hₙ]
simplifies the main goal target using the lemmas tagged with the attribute[simp]
and the givenhᵢ
's, where thehᵢ
's are expressions.-- If an
hᵢ
is a defined constantf
, thenf
is unfolded. Iff
has equational lemmas associated with it (and is not a projection or areducible
definition), these are used to rewrite withf
. simp [*]
simplifies the main goal target using the lemmas tagged with the attribute[simp]
and all hypotheses.simp only [h₁, h₂, ..., hₙ]
is likesimp [h₁, h₂, ..., hₙ]
but does not use[simp]
lemmas.simp [-id₁, ..., -idₙ]
simplifies the main goal target using the lemmas tagged with the attribute[simp]
, but removes the ones namedidᵢ
.simp at h₁ h₂ ... hₙ
simplifies the hypothesesh₁ : T₁
...hₙ : Tₙ
. If the target or another hypothesis depends onhᵢ
, a new simplified hypothesishᵢ
is introduced, but the old one remains in the local context.simp at *
simplifies all the hypotheses and the target.simp [*] at *
simplifies target and all (propositional) hypotheses using the other hypotheses.
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simp_all
is a stronger version of simp [*] at *
where the hypotheses and target
are simplified multiple times until no simplification is applicable.
Only non-dependent propositional hypotheses are considered.
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A simpArg
is either a *
, -lemma
or a simp lemma specification
(which includes the ↑
↓
←
specifications for pre, post, reverse rewriting).
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The common arguments of simp?
and simp?!
.
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simp?
takes the same arguments as simp
, but reports an equivalent call to simp only
that would be sufficient to close the goal. This is useful for reducing the size of the simp
set in a local invocation to speed up processing.
example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
simp? -- prints "Try this: simp only [ite_true]"
This command can also be used in simp_all
and dsimp
.
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simp?
takes the same arguments as simp
, but reports an equivalent call to simp only
that would be sufficient to close the goal. This is useful for reducing the size of the simp
set in a local invocation to speed up processing.
example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
simp? -- prints "Try this: simp only [ite_true]"
This command can also be used in simp_all
and dsimp
.
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The common arguments of simp_all?
and simp_all?!
.
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simp?
takes the same arguments as simp
, but reports an equivalent call to simp only
that would be sufficient to close the goal. This is useful for reducing the size of the simp
set in a local invocation to speed up processing.
example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
simp? -- prints "Try this: simp only [ite_true]"
This command can also be used in simp_all
and dsimp
.
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simp?
takes the same arguments as simp
, but reports an equivalent call to simp only
that would be sufficient to close the goal. This is useful for reducing the size of the simp
set in a local invocation to speed up processing.
example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
simp? -- prints "Try this: simp only [ite_true]"
This command can also be used in simp_all
and dsimp
.
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The common arguments of dsimp?
and dsimp?!
.
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simp?
takes the same arguments as simp
, but reports an equivalent call to simp only
that would be sufficient to close the goal. This is useful for reducing the size of the simp
set in a local invocation to speed up processing.
example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
simp? -- prints "Try this: simp only [ite_true]"
This command can also be used in simp_all
and dsimp
.
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simp?
takes the same arguments as simp
, but reports an equivalent call to simp only
that would be sufficient to close the goal. This is useful for reducing the size of the simp
set in a local invocation to speed up processing.
example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
simp? -- prints "Try this: simp only [ite_true]"
This command can also be used in simp_all
and dsimp
.
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This is a "finishing" tactic modification of simp
. It has two forms.
simpa [rules, ⋯] using e
will simplify the goal and the type ofe
usingrules
, then try to close the goal usinge
.
Simplifying the type of e
makes it more likely to match the goal
(which has also been simplified). This construction also tends to be
more robust under changes to the simp lemma set.
simpa [rules, ⋯]
will simplify the goal and the type of a hypothesisthis
if present in the context, then try to close the goal using theassumption
tactic.
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This is a "finishing" tactic modification of simp
. It has two forms.
simpa [rules, ⋯] using e
will simplify the goal and the type ofe
usingrules
, then try to close the goal usinge
.
Simplifying the type of e
makes it more likely to match the goal
(which has also been simplified). This construction also tends to be
more robust under changes to the simp lemma set.
simpa [rules, ⋯]
will simplify the goal and the type of a hypothesisthis
if present in the context, then try to close the goal using theassumption
tactic.
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This is a "finishing" tactic modification of simp
. It has two forms.
simpa [rules, ⋯] using e
will simplify the goal and the type ofe
usingrules
, then try to close the goal usinge
.
Simplifying the type of e
makes it more likely to match the goal
(which has also been simplified). This construction also tends to be
more robust under changes to the simp lemma set.
simpa [rules, ⋯]
will simplify the goal and the type of a hypothesisthis
if present in the context, then try to close the goal using theassumption
tactic.
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This is a "finishing" tactic modification of simp
. It has two forms.
simpa [rules, ⋯] using e
will simplify the goal and the type ofe
usingrules
, then try to close the goal usinge
.
Simplifying the type of e
makes it more likely to match the goal
(which has also been simplified). This construction also tends to be
more robust under changes to the simp lemma set.
simpa [rules, ⋯]
will simplify the goal and the type of a hypothesisthis
if present in the context, then try to close the goal using theassumption
tactic.
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delta id1 id2 ...
delta-expands the definitions id1
, id2
, ....
This is a low-level tactic, it will expose how recursive definitions have been
compiled by Lean.
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unfold id
unfolds all occurrences of definitionid
in the target.unfold id1 id2 ...
is equivalent tounfold id1; unfold id2; ...
.unfold id at h
unfolds at the hypothesish
.
Definitions can be either global or local definitions.
For non-recursive global definitions, this tactic is identical to delta
.
For recursive global definitions, it uses the "unfolding lemma" id.eq_def
,
which is generated for each recursive definition, to unfold according to the recursive definition given by the user.
Only one level of unfolding is performed, in contrast to simp only [id]
, which unfolds definition id
recursively.
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Auxiliary macro for lifting have/suffices/let/...
It makes sure the "continuation" ?_
is the main goal after refining.
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The have
tactic is for adding hypotheses to the local context of the main goal.
have h : t := e
adds the hypothesish : t
ife
is a term of typet
.have h := e
uses the type ofe
fort
.have : t := e
andhave := e
usethis
for the name of the hypothesis.have pat := e
for a patternpat
is equivalent tomatch e with | pat => _
, where_
stands for the tactics that follow this one. It is convenient for types that have only one applicable constructor. For example, givenh : p ∧ q ∧ r
,have ⟨h₁, h₂, h₃⟩ := h
produces the hypothesesh₁ : p
,h₂ : q
, andh₃ : r
.
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Given a main goal ctx ⊢ t
, suffices h : t' from e
replaces the main goal with ctx ⊢ t'
,
e
must have type t
in the context ctx, h : t'
.
The variant suffices h : t' by tac
is a shorthand for suffices h : t' from by tac
.
If h :
is omitted, the name this
is used.
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The let
tactic is for adding definitions to the local context of the main goal.
let x : t := e
adds the definitionx : t := e
ife
is a term of typet
.let x := e
uses the type ofe
fort
.let : t := e
andlet := e
usethis
for the name of the hypothesis.let pat := e
for a patternpat
is equivalent tomatch e with | pat => _
, where_
stands for the tactics that follow this one. It is convenient for types that let only one applicable constructor. For example, givenp : α × β × γ
,let ⟨x, y, z⟩ := p
produces the local variablesx : α
,y : β
, andz : γ
.
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show t
finds the first goal whose target unifies with t
. It makes that the main goal,
performs the unification, and replaces the target with the unified version of t
.
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Instances For
let rec f : t := e
adds a recursive definition f
to the current goal.
The syntax is the same as term-mode let rec
.
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The left hand side of an induction arm, | foo a b c
or | @foo a b c
where foo
is a constructor of the inductive type and a b c
are the arguments
to the constructor.
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In induction alternative, which can have 1 or more cases on the left
and _
, ?_
, or a tactic sequence after the =>
.
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After with
, there is an optional tactic that runs on all branches, and
then a list of alternatives.
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Assuming x
is a variable in the local context with an inductive type,
induction x
applies induction on x
to the main goal,
producing one goal for each constructor of the inductive type,
in which the target is replaced by a general instance of that constructor
and an inductive hypothesis is added for each recursive argument to the constructor.
If the type of an element in the local context depends on x
,
that element is reverted and reintroduced afterward,
so that the inductive hypothesis incorporates that hypothesis as well.
For example, given n : Nat
and a goal with a hypothesis h : P n
and target Q n
,
induction n
produces one goal with hypothesis h : P 0
and target Q 0
,
and one goal with hypotheses h : P (Nat.succ a)
and ih₁ : P a → Q a
and target Q (Nat.succ a)
.
Here the names a
and ih₁
are chosen automatically and are not accessible.
You can use with
to provide the variables names for each constructor.
induction e
, wheree
is an expression instead of a variable, generalizese
in the goal, and then performs induction on the resulting variable.induction e using r
allows the user to specify the principle of induction that should be used. Herer
should be a term whose result type must be of the formC t
, whereC
is a bound variable andt
is a (possibly empty) sequence of bound variablesinduction e generalizing z₁ ... zₙ
, wherez₁ ... zₙ
are variables in the local context, generalizes overz₁ ... zₙ
before applying the induction but then introduces them in each goal. In other words, the net effect is that each inductive hypothesis is generalized.- Given
x : Nat
,induction x with | zero => tac₁ | succ x' ih => tac₂
uses tactictac₁
for thezero
case, andtac₂
for thesucc
case.
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generalize ([h :] e = x),+
replaces all occurrencese
s in the main goal with a fresh hypothesisx
s. Ifh
is given,h : e = x
is introduced as well.generalize e = x at h₁ ... hₙ
also generalizes occurrences ofe
insideh₁
, ...,hₙ
.generalize e = x at *
will generalize occurrences ofe
everywhere.
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Assuming x
is a variable in the local context with an inductive type,
cases x
splits the main goal, producing one goal for each constructor of the
inductive type, in which the target is replaced by a general instance of that constructor.
If the type of an element in the local context depends on x
,
that element is reverted and reintroduced afterward,
so that the case split affects that hypothesis as well.
cases
detects unreachable cases and closes them automatically.
For example, given n : Nat
and a goal with a hypothesis h : P n
and target Q n
,
cases n
produces one goal with hypothesis h : P 0
and target Q 0
,
and one goal with hypothesis h : P (Nat.succ a)
and target Q (Nat.succ a)
.
Here the name a
is chosen automatically and is not accessible.
You can use with
to provide the variables names for each constructor.
cases e
, wheree
is an expression instead of a variable, generalizese
in the goal, and then cases on the resulting variable.- Given
as : List α
,cases as with | nil => tac₁ | cons a as' => tac₂
, uses tactictac₁
for thenil
case, andtac₂
for thecons
case, anda
andas'
are used as names for the new variables introduced. cases h : e
, wheree
is a variable or an expression, performs cases one
as above, but also adds a hypothesish : e = ...
to each hypothesis, where...
is the constructor instance for that particular case.
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The fun_induction
tactic is a convenience wrapper around the induction
tactic to use the the
functional induction principle.
The tactic invocation
fun_induction f x₁ ... xₙ y₁ ... yₘ
where f
is a function defined by non-mutual structural or well-founded recursion, is equivalent to
induction y₁, ... yₘ using f.induct_unfolding x₁ ... xₙ
where the arguments of f
are used as arguments to f.induct_unfolding
or targets of the
induction, as appropriate.
The form
fun_induction f
(with no arguments to f
) searches the goal for an unique eligible application of f
, and uses
these arguments. An application of f
is eligible if it is saturated and the arguments that will
become targets are free variables.
The forms fun_induction f x y generalizing z₁ ... zₙ
and
fun_induction f x y with | case1 => tac₁ | case2 x' ih => tac₂
work like with induction.
Under set_option tactic.fun_induction.unfolding true
(the default), fun_induction
uses the
f.induct_unfolding
induction principle, which will try to automatically unfold the call to f
in
the goal. With set_option tactic.fun_induction.unfolding false
, it uses f.induct
instead.
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The fun_cases
tactic is a convenience wrapper of the cases
tactic when using a functional
cases principle.
The tactic invocation
fun_cases f x ... y ...`
is equivalent to
cases y, ... using f.fun_cases_unfolding x ...
where the arguments of f
are used as arguments to f.fun_cases_unfolding
or targets of the case
analysis, as appropriate.
The form
fun_cases f
(with no arguments to f
) searches the goal for an unique eligible application of f
, and uses
these arguments. An application of f
is eligible if it is saturated and the arguments that will
become targets are free variables.
The form fun_cases f x y with | case1 => tac₁ | case2 x' ih => tac₂
works like with cases
.
Under set_option tactic.fun_induction.unfolding true
(the default), fun_induction
uses the
f.fun_cases_unfolding
theorem, which will try to automatically unfold the call to f
in
the goal. With set_option tactic.fun_induction.unfolding false
, it uses f.fun_cases
instead.
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rename_i x_1 ... x_n
renames the last n
inaccessible names using the given names.
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repeat tac
repeatedly applies tac
so long as it succeeds.
The tactic tac
may be a tactic sequence, and if tac
fails at any point in its execution,
repeat
will revert any partial changes that tac
made to the tactic state.
The tactic tac
should eventually fail, otherwise repeat tac
will run indefinitely.
See also:
try tac
is likerepeat tac
but will applytac
at most once.repeat' tac
recursively appliestac
to each goal.first | tac1 | tac2
implements the backtracking used byrepeat
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repeat' tac
recursively applies tac
on all of the goals so long as it succeeds.
That is to say, if tac
produces multiple subgoals, then repeat' tac
is applied to each of them.
See also:
repeat tac
simply repeatedly appliestac
.repeat1' tac
isrepeat' tac
but requires thattac
succeed for some goal at least once.
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repeat1' tac
recursively applies to tac
on all of the goals so long as it succeeds,
but repeat1' tac
fails if tac
succeeds on none of the initial goals.
See also:
repeat tac
simply appliestac
repeatedly.repeat' tac
is likerepeat1' tac
but it does not require thattac
succeed at least once.
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trivial
tries different simple tactics (e.g., rfl
, contradiction
, ...)
to close the current goal.
You can use the command macro_rules
to extend the set of tactics used. Example:
macro_rules | `(tactic| trivial) => `(tactic| simp)
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classical tacs
runs tacs
in a scope where Classical.propDecidable
is a low priority
local instance.
Note that classical
is a scoping tactic: it adds the instance only within the
scope of the tactic.
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The split
tactic is useful for breaking nested if-then-else and match
expressions into separate cases.
For a match
expression with n
cases, the split
tactic generates at most n
subgoals.
For example, given n : Nat
, and a target if n = 0 then Q else R
, split
will generate
one goal with hypothesis n = 0
and target Q
, and a second goal with hypothesis
¬n = 0
and target R
. Note that the introduced hypothesis is unnamed, and is commonly
renamed used the case
or next
tactics.
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stop
is a helper tactic for "discarding" the rest of a proof:
it is defined as repeat sorry
.
It is useful when working on the middle of a complex proofs,
and less messy than commenting the remainder of the proof.
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The tactic specialize h a₁ ... aₙ
works on local hypothesis h
.
The premises of this hypothesis, either universal quantifications or
non-dependent implications, are instantiated by concrete terms coming
from arguments a₁
... aₙ
.
The tactic adds a new hypothesis with the same name h := h a₁ ... aₙ
and tries to clear the previous one.
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unhygienic tacs
runs tacs
with name hygiene disabled.
This means that tactics that would normally create inaccessible names will instead
make regular variables. Warning: Tactics may change their variable naming
strategies at any time, so code that depends on autogenerated names is brittle.
Users should try not to use unhygienic
if possible.
example : ∀ x : Nat, x = x := by unhygienic
intro -- x would normally be intro'd as inaccessible
exact Eq.refl x -- refer to x
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The tactic sleep ms
sleeps for ms
milliseconds and does nothing.
It is used for debugging purposes only.
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Apply congruence (recursively) to goals of the form ⊢ f as = f bs
and ⊢ HEq (f as) (f bs)
.
The optional parameter is the depth of the recursive applications.
This is useful when congr
is too aggressive in breaking down the goal.
For example, given ⊢ f (g (x + y)) = f (g (y + x))
,
congr
produces the goals ⊢ x = y
and ⊢ y = x
,
while congr 2
produces the intended ⊢ x + y = y + x
.
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In tactic mode, if h : t then tac1 else tac2
can be used as alternative syntax for:
by_cases h : t
· tac1
· tac2
It performs case distinction on h : t
or h : ¬t
and tac1
and tac2
are the subproofs.
You can use ?_
or _
for either subproof to delay the goal to after the tactic, but
if a tactic sequence is provided for tac1
or tac2
then it will require the goal to be closed
by the end of the block.
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In tactic mode, if t then tac1 else tac2
is alternative syntax for:
by_cases t
· tac1
· tac2
It performs case distinction on h† : t
or h† : ¬t
, where h†
is an anonymous
hypothesis, and tac1
and tac2
are the subproofs. (It doesn't actually use
nondependent if
, since this wouldn't add anything to the context and hence would be
useless for proving theorems. To actually insert an ite
application use
refine if t then ?_ else ?_
.)
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The tactic nofun
is shorthand for exact nofun
: it introduces the assumptions, then performs an
empty pattern match, closing the goal if the introduced pattern is impossible.
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Acts like have
, but removes a hypothesis with the same name as
this one if possible. For example, if the state is:
f : α → β
h : α
⊢ goal
Then after replace h := f h
the state will be:
f : α → β
h : β
⊢ goal
whereas have h := f h
would result in:
f : α → β
h† : α
h : β
⊢ goal
This can be used to simulate the specialize
and apply at
tactics of Coq.
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subst_eq
repeatedly substitutes according to the equality proof hypotheses in the context,
replacing the left side of the equality with the right, until no more progress can be made.
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haveI
behaves like have
, but inlines the value instead of producing a let_fun
term.
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letI
behaves like let
, but inlines the value instead of producing a let_fun
term.
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Configuration for the decide
tactic family.
- kernel : Bool
If true (default: false), then use only kernel reduction when reducing the
Decidable
instance. This is more efficient, since the default mode reduces twice (once in the elaborator and again in the kernel), however kernel reduction ignores transparency settings. - native : Bool
If true (default: false), then uses the native code compiler to evaluate the
Decidable
instance, admitting the result via the axiomLean.ofReduceBool
. This can be significantly more efficient, but it is at the cost of increasing the trusted code base, namely the Lean compiler and all definitions with an@[implemented_by]
attribute. The instance is only evaluated once. Thenative_decide
tactic is a synonym fordecide +native
. - zetaReduce : Bool
If true (default: true), then when preprocessing the goal, do zeta reduction to attempt to eliminate free variables.
- revert : Bool
If true (default: false), then when preprocessing, removes irrelevant variables and reverts the local context. A variable is relevant if it appears in the target, if it appears in a relevant variable, or if it is a proposition that refers to a relevant variable.
Instances For
decide
attempts to prove the main goal (with target type p
) by synthesizing an instance of Decidable p
and then reducing that instance to evaluate the truth value of p
.
If it reduces to isTrue h
, then h
is a proof of p
that closes the goal.
The target is not allowed to contain local variables or metavariables.
If there are local variables, you can first try using the revert
tactic with these local variables to move them into the target,
or you can use the +revert
option, described below.
Options:
decide +revert
begins by reverting local variables that the target depends on, after cleaning up the local context of irrelevant variables. A variable is relevant if it appears in the target, if it appears in a relevant variable, or if it is a proposition that refers to a relevant variable.decide +kernel
uses kernel for reduction instead of the elaborator. It has two key properties: (1) since it uses the kernel, it ignores transparency and can unfold everything, and (2) it reduces theDecidable
instance only once instead of twice.decide +native
uses the native code compiler (#eval
) to evaluate theDecidable
instance, admitting the result via theLean.ofReduceBool
axiom. This can be significantly more efficient than using reduction, but it is at the cost of increasing the size of the trusted code base. Namely, it depends on the correctness of the Lean compiler and all definitions with an@[implemented_by]
attribute. Like with+kernel
, theDecidable
instance is evaluated only once.
Limitation: In the default mode or +kernel
mode, since decide
uses reduction to evaluate the term,
Decidable
instances defined by well-founded recursion might not work because evaluating them requires reducing proofs.
Reduction can also get stuck on Decidable
instances with Eq.rec
terms.
These can appear in instances defined using tactics (such as rw
and simp
).
To avoid this, create such instances using definitions such as decidable_of_iff
instead.
Examples #
Proving inequalities:
example : 2 + 2 ≠ 5 := by decide
Trying to prove a false proposition:
example : 1 ≠ 1 := by decide
/-
tactic 'decide' proved that the proposition
1 ≠ 1
is false
-/
Trying to prove a proposition whose Decidable
instance fails to reduce
opaque unknownProp : Prop
open scoped Classical in
example : unknownProp := by decide
/-
tactic 'decide' failed for proposition
unknownProp
since its 'Decidable' instance reduced to
Classical.choice ⋯
rather than to the 'isTrue' constructor.
-/
Properties and relations #
For equality goals for types with decidable equality, usually rfl
can be used in place of decide
.
example : 1 + 1 = 2 := by decide
example : 1 + 1 = 2 := by rfl
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native_decide
is a synonym for decide +native
.
It will attempt to prove a goal of type p
by synthesizing an instance
of Decidable p
and then evaluating it to isTrue ..
. Unlike decide
, this
uses #eval
to evaluate the decidability instance.
This should be used with care because it adds the entire lean compiler to the trusted
part, and the axiom Lean.ofReduceBool
will show up in #print axioms
for theorems using
this method or anything that transitively depends on them. Nevertheless, because it is
compiled, this can be significantly more efficient than using decide
, and for very
large computations this is one way to run external programs and trust the result.
example : (List.range 1000).length = 1000 := by native_decide
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The omega
tactic, for resolving integer and natural linear arithmetic problems.
It is not yet a full decision procedure (no "dark" or "grey" shadows), but should be effective on many problems.
We handle hypotheses of the form x = y
, x < y
, x ≤ y
, and k ∣ x
for x y
in Nat
or Int
(and k
a literal), along with negations of these statements.
We decompose the sides of the inequalities as linear combinations of atoms.
If we encounter x / k
or x % k
for literal integers k
we introduce new auxiliary variables
and the relevant inequalities.
On the first pass, we do not perform case splits on natural subtraction.
If omega
fails, we recursively perform a case split on
a natural subtraction appearing in a hypothesis, and try again.
The options
omega +splitDisjunctions +splitNatSub +splitNatAbs +splitMinMax
can be used to:
splitDisjunctions
: split any disjunctions found in the context, if the problem is not otherwise solvable.splitNatSub
: for each appearance of((a - b : Nat) : Int)
, split ona ≤ b
if necessary.splitNatAbs
: for each appearance ofInt.natAbs a
, split on0 ≤ a
if necessary.splitMinMax
: for each occurrence ofmin a b
, split onmin a b = a ∨ min a b = b
Currently, all of these are on by default.
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assumption_mod_cast
is a variant of assumption
that solves the goal
using a hypothesis. Unlike assumption
, it first pre-processes the goal and
each hypothesis to move casts as far outwards as possible, so it can be used
in more situations.
Concretely, it runs norm_cast
on the goal. For each local hypothesis h
, it also
normalizes h
with norm_cast
and tries to use that to close the goal.
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The norm_cast
family of tactics is used to normalize certain coercions (casts) in expressions.
The tactic is basically a version of simp
with a specific set of lemmas to move casts
upwards in the expression.
Therefore even in situations where non-terminal simp
calls are discouraged (because of fragility),
norm_cast
is considered to be safe.
It also has special handling of numerals.
For instance, given an assumption
a b : ℤ
h : ↑a + ↑b < (10 : ℚ)
writing norm_cast at h
will turn h
into
h : a + b < 10
There are also variants of basic tactics that use norm_cast
to normalize expressions during
their operation, to make them more flexible about the expressions they accept
(we say that it is a tactic modulo the effects of norm_cast
):
exact_mod_cast
forexact
andapply_mod_cast
forapply
. Writingexact_mod_cast h
andapply_mod_cast h
will normalize casts in the goal andh
before usingexact h
orapply h
.rw_mod_cast
forrw
. It appliesnorm_cast
between rewrites.assumption_mod_cast
forassumption
. This is effectivelynorm_cast at *; assumption
, but more efficient. It normalizes casts in the goal and, for every hypothesish
in the context, it will try to normalize casts inh
and useexact h
.
See also push_cast
, which moves casts inwards rather than lifting them outwards.
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push_cast
rewrites the goal to move certain coercions (casts) inward, toward the leaf nodes.
This uses norm_cast
lemmas in the forward direction.
For example, ↑(a + b)
will be written to ↑a + ↑b
.
push_cast
moves casts inward in the goal.push_cast at h
moves casts inward in the hypothesish
. It can be used with extra simp lemmas with, for example,push_cast [Int.add_zero]
.
Example:
example (a b : Nat)
(h1 : ((a + b : Nat) : Int) = 10)
(h2 : ((a + b + 0 : Nat) : Int) = 10) :
((a + b : Nat) : Int) = 10 := by
/-
h1 : ↑(a + b) = 10
h2 : ↑(a + b + 0) = 10
⊢ ↑(a + b) = 10
-/
push_cast
/- Now
⊢ ↑a + ↑b = 10
-/
push_cast at h1
push_cast [Int.add_zero] at h2
/- Now
h1 h2 : ↑a + ↑b = 10
-/
exact h1
See also norm_cast
.
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ac_nf
normalizes equalities up to application of an associative and commutative operator.
ac_nf
normalizes all hypotheses and the goal target of the goal.ac_nf at l
normalizes at location(s)l
, wherel
is either*
or a list of hypotheses in the local context. In the latter case, a turnstile⊢
or|-
can also be used, to signify the target of the goal.
instance : Associative (α := Nat) (.+.) := ⟨Nat.add_assoc⟩
instance : Commutative (α := Nat) (.+.) := ⟨Nat.add_comm⟩
example (a b c d : Nat) : a + b + c + d = d + (b + c) + a := by
ac_nf
-- goal: a + (b + (c + d)) = a + (b + (c + d))
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For every hypothesis h : a ~ b
where a @[symm]
lemma is available,
add a hypothesis h_symm : b ~ a
.
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Syntax for omitting a local hypothesis in solve_by_elim
.
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Syntax for including all local hypotheses in solve_by_elim
.
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Syntax for adding or removing a term, or *
, in solve_by_elim
.
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Syntax for adding and removing terms in solve_by_elim
.
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Syntax for using all lemmas labelled with an attribute in solve_by_elim
.
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solve_by_elim
calls apply
on the main goal to find an assumption whose head matches
and then repeatedly calls apply
on the generated subgoals until no subgoals remain,
performing at most maxDepth
(defaults to 6) recursive steps.
solve_by_elim
discharges the current goal or fails.
solve_by_elim
performs backtracking if subgoals can not be solved.
By default, the assumptions passed to apply
are the local context, rfl
, trivial
,
congrFun
and congrArg
.
The assumptions can be modified with similar syntax as for simp
:
solve_by_elim [h₁, h₂, ..., hᵣ]
also applies the given expressions.solve_by_elim only [h₁, h₂, ..., hᵣ]
does not include the local context,rfl
,trivial
,congrFun
, orcongrArg
unless they are explicitly included.solve_by_elim [-h₁, ... -hₙ]
removes the given local hypotheses.solve_by_elim using [a₁, ...]
uses all lemmas which have been labelled with the attributesaᵢ
(these attributes must be created usingregister_label_attr
).
solve_by_elim*
tries to solve all goals together, using backtracking if a solution for one goal
makes other goals impossible.
(Adding or removing local hypotheses may not be well-behaved when starting with multiple goals.)
Optional arguments passed via a configuration argument as solve_by_elim (config := { ... })
maxDepth
: number of attempts at discharging generated subgoalssymm
: adds all hypotheses derived bysymm
(defaults totrue
).exfalso
: allow callingexfalso
and trying again ifsolve_by_elim
fails (defaults totrue
).transparency
: change the transparency mode when callingapply
. Defaults to.default
, but it is often useful to change to.reducible
, so semireducible definitions will not be unfolded when trying to apply a lemma.
See also the doc-comment for Lean.Meta.Tactic.Backtrack.BacktrackConfig
for the options
proc
, suspend
, and discharge
which allow further customization of solve_by_elim
.
Both apply_assumption
and apply_rules
are implemented via these hooks.
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apply_assumption
looks for an assumption of the form ... → ∀ _, ... → head
where head
matches the current goal.
You can specify additional rules to apply using apply_assumption [...]
.
By default apply_assumption
will also try rfl
, trivial
, congrFun
, and congrArg
.
If you don't want these, or don't want to use all hypotheses, use apply_assumption only [...]
.
You can use apply_assumption [-h]
to omit a local hypothesis.
You can use apply_assumption using [a₁, ...]
to use all lemmas which have been labelled
with the attributes aᵢ
(these attributes must be created using register_label_attr
).
apply_assumption
will use consequences of local hypotheses obtained via symm
.
If apply_assumption
fails, it will call exfalso
and try again.
Thus if there is an assumption of the form P → ¬ Q
, the new tactic state
will have two goals, P
and Q
.
You can pass a further configuration via the syntax apply_rules (config := {...}) lemmas
.
The options supported are the same as for solve_by_elim
(and include all the options for apply
).
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apply_rules [l₁, l₂, ...]
tries to solve the main goal by iteratively
applying the list of lemmas [l₁, l₂, ...]
or by applying a local hypothesis.
If apply
generates new goals, apply_rules
iteratively tries to solve those goals.
You can use apply_rules [-h]
to omit a local hypothesis.
apply_rules
will also use rfl
, trivial
, congrFun
and congrArg
.
These can be disabled, as can local hypotheses, by using apply_rules only [...]
.
You can use apply_rules using [a₁, ...]
to use all lemmas which have been labelled
with the attributes aᵢ
(these attributes must be created using register_label_attr
).
You can pass a further configuration via the syntax apply_rules (config := {...})
.
The options supported are the same as for solve_by_elim
(and include all the options for apply
).
apply_rules
will try calling symm
on hypotheses and exfalso
on the goal as needed.
This can be disabled with apply_rules (config := {symm := false, exfalso := false})
.
You can bound the iteration depth using the syntax apply_rules (config := {maxDepth := n})
.
Unlike solve_by_elim
, apply_rules
does not perform backtracking, and greedily applies
a lemma from the list until it gets stuck.
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Searches environment for definitions or theorems that can solve the goal using exact
with conditions resolved by solve_by_elim
.
The optional using
clause provides identifiers in the local context that must be
used by exact?
when closing the goal. This is most useful if there are multiple
ways to resolve the goal, and one wants to guide which lemma is used.
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Searches environment for definitions or theorems that can refine the goal using apply
with conditions resolved when possible with solve_by_elim
.
The optional using
clause provides identifiers in the local context that must be
used when closing the goal.
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Syntax for excluding some names, e.g. [-my_lemma, -my_theorem]
.
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rw?
tries to find a lemma which can rewrite the goal.
rw?
should not be left in proofs; it is a search tool, like apply?
.
Suggestions are printed as rw [h]
or rw [← h]
.
You can use rw? [-my_lemma, -my_theorem]
to prevent rw?
using the named lemmas.
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show_term tac
runs tac
, then prints the generated term in the form
"exact X Y Z" or "refine X ?_ Z" (prefixed by expose_names
if necessary)
if there are remaining subgoals.
(For some tactics, the printed term will not be human readable.)
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show_term e
elaborates e
, then prints the generated term.
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The command by?
will print a suggestion for replacing the proof block with a proof term
using show_term
.
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expose_names
renames all inaccessible variables with accessible names, making them available
for reference in generated tactics. However, this renaming introduces machine-generated names
that are not fully under user control. expose_names
is primarily intended as a preamble for
auto-generated end-game tactic scripts. It is also useful as an alternative to
set_option tactic.hygienic false
. If explicit control over renaming is needed in the
middle of a tactic script, consider using structured tactic scripts with
match .. with
, induction .. with
, or intro
with explicit user-defined names,
as well as tactics such as next
, case
, and rename_i
.
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#suggest_premises
will suggest premises for the current goal, using the currently registered premise selector.
The suggestions are printed in the order of their confidence, from highest to lowest.
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Close fixed-width BitVec
and Bool
goals by obtaining a proof from an external SAT solver and
verifying it inside Lean. The solvable goals are currently limited to
- the Lean equivalent of
QF_BV
- automatically splitting up
structure
s that contain information aboutBitVec
orBool
example : ∀ (a b : BitVec 64), (a &&& b) + (a ^^^ b) = a ||| b := by
intros
bv_decide
If bv_decide
encounters an unknown definition it will be treated like an unconstrained BitVec
variable. Sometimes this enables solving goals despite not understanding the definition because
the precise properties of the definition do not matter in the specific proof.
If bv_decide
fails to close a goal it provides a counter-example, containing assignments for all
terms that were considered as variables.
In order to avoid calling a SAT solver every time, the proof can be cached with bv_decide?
.
If solving your problem relies inherently on using associativity or commutativity, consider enabling
the bv.ac_nf
option.
Note: bv_decide
uses ofReduceBool
and thus trusts the correctness of the code generator.
Note: include import Std.Tactic.BVDecide
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Suggest a proof script for a bv_decide
tactic call. Useful for caching LRAT proofs.
Note: include import Std.Tactic.BVDecide
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Run the normalization procedure of bv_decide
only. Sometimes this is enough to solve basic
BitVec
goals already.
Note: include import Std.Tactic.BVDecide
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Theorems tagged with the simp
attribute are used by the simplifier
(i.e., the simp
tactic, and its variants) to simplify expressions occurring in your goals.
We call theorems tagged with the simp
attribute "simp theorems" or "simp lemmas".
Lean maintains a database/index containing all active simp theorems.
Here is an example of a simp theorem.
@[simp] theorem ne_eq (a b : α) : (a ≠ b) = Not (a = b) := rfl
This simp theorem instructs the simplifier to replace instances of the term
a ≠ b
(e.g. x + 0 ≠ y
) with Not (a = b)
(e.g., Not (x + 0 = y)
).
The simplifier applies simp theorems in one direction only:
if A = B
is a simp theorem, then simp
replaces A
s with B
s,
but it doesn't replace B
s with A
s. Hence a simp theorem should have the
property that its right-hand side is "simpler" than its left-hand side.
In particular, =
and ↔
should not be viewed as symmetric operators in this situation.
The following would be a terrible simp theorem (if it were even allowed):
@[simp] lemma mul_right_inv_bad (a : G) : 1 = a * a⁻¹ := ...
Replacing 1 with a * a⁻¹ is not a sensible default direction to travel. Even worse would be a theorem that causes expressions to grow without bound, causing simp to loop forever.
By default the simplifier applies simp
theorems to an expression e
after its sub-expressions have been simplified.
We say it performs a bottom-up simplification.
You can instruct the simplifier to apply a theorem before its sub-expressions
have been simplified by using the modifier ↓
. Here is an example
@[simp↓] theorem not_and_eq (p q : Prop) : (¬ (p ∧ q)) = (¬p ∨ ¬q) :=
You can instruct the simplifier to rewrite the lemma from right-to-left:
attribute @[simp ←] and_assoc
When multiple simp theorems are applicable, the simplifier uses the one with highest priority. The equational theorems of functions are applied at very low priority (100 and below). If there are several with the same priority, it is uses the "most recent one". Example:
@[simp high] theorem cond_true (a b : α) : cond true a b = a := rfl
@[simp low+1] theorem or_true (p : Prop) : (p ∨ True) = True :=
propext <| Iff.intro (fun _ => trivial) (fun _ => Or.inr trivial)
@[simp 100] theorem ite_self {d : Decidable c} (a : α) : ite c a a = a := by
cases d <;> rfl
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Theorems tagged with the wf_preprocess
attribute are used during the processing of functions defined
by well-founded recursion. They are applied to the function's body to add additional hypotheses,
such as replacing if c then _ else _
with if h : c then _ else _
or xs.map
with
xs.attach.map
. Also see wfParam
.
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The norm_cast
attribute should be given to lemmas that describe the
behaviour of a coercion with respect to an operator, a relation, or a particular
function.
It only concerns equality or iff lemmas involving ↑
, ⇑
and ↥
, describing the behavior of
the coercion functions.
It does not apply to the explicit functions that define the coercions.
Examples:
@[norm_cast] theorem coe_nat_inj' {m n : ℕ} : (↑m : ℤ) = ↑n ↔ m = n
@[norm_cast] theorem coe_int_denom (n : ℤ) : (n : ℚ).denom = 1
@[norm_cast] theorem cast_id : ∀ n : ℚ, ↑n = n
@[norm_cast] theorem coe_nat_add (m n : ℕ) : (↑(m + n) : ℤ) = ↑m + ↑n
@[norm_cast] theorem cast_coe_nat (n : ℕ) : ((n : ℤ) : α) = n
@[norm_cast] theorem cast_one : ((1 : ℚ) : α) = 1
Lemmas tagged with @[norm_cast]
are classified into three categories: move
, elim
, and
squash
. They are classified roughly as follows:
- elim lemma: LHS has 0 head coes and ≥ 1 internal coe
- move lemma: LHS has 1 head coe and 0 internal coes, RHS has 0 head coes and ≥ 1 internal coes
- squash lemma: LHS has ≥ 1 head coes and 0 internal coes, RHS has fewer head coes
norm_cast
uses move
and elim
lemmas to factor coercions toward the root of an expression
and to cancel them from both sides of an equation or relation. It uses squash
lemmas to clean
up the result.
It is typically not necessary to specify these categories, as norm_cast
lemmas are
automatically classified by default. The automatic classification can be overridden by
giving an optional elim
, move
, or squash
parameter to the attribute.
@[simp, norm_cast elim] lemma nat_cast_re (n : ℕ) : (n : ℂ).re = n := by
rw [← of_real_nat_cast, of_real_re]
Don't do this unless you understand what you are doing.
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‹t›
resolves to an (arbitrary) hypothesis of type t
.
It is useful for referring to hypotheses without accessible names.
t
may contain holes that are solved by unification with the expected type;
in particular, ‹_›
is a shortcut for by assumption
.
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get_elem_tactic_trivial
is an extensible tactic automatically called
by the notation arr[i]
to prove any side conditions that arise when
constructing the term (e.g. the index is in bounds of the array).
The default behavior is to just try trivial
(which handles the case
where i < arr.size
is in the context) and simp +arith
and omega
(for doing linear arithmetic in the index).
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get_elem_tactic
is the tactic automatically called by the notation arr[i]
to prove any side conditions that arise when constructing the term
(e.g. the index is in bounds of the array). It just delegates to
get_elem_tactic_trivial
and gives a diagnostic error message otherwise;
users are encouraged to extend get_elem_tactic_trivial
instead of this tactic.
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Searches environment for definitions or theorems that can be substituted in
for exact?%
to solve the goal.