Proof mode is used to prove theorems.
Coq enters proof mode when you begin a proof,
such as with the
Theorem command. It exits proof mode when
you complete a proof, such as with the
Qed command. Tactics,
which are available only in proof mode, incrementally transform incomplete
proofs to eventually generate a complete proof.
When you run Coq interactively, such as through CoqIDE, Proof General or
coqtop, Coq shows the current proof state (the incomplete proof) as you
enter tactics. This information isn't shown when you run Coq in batch
The proof state consists of one or more unproven goals. Each goal has a conclusion (the statement that is to be proven) and a local context, which contains named hypotheses (which are propositions), variables and local definitions that can be used in proving the conclusion. The proof may also use constants from the global environment such as definitions and proven theorems.
(Note that conclusion is also used to refer to the last part of an implication.
For example, in
A -> B -> C,
B are premises and
is the conclusion.)
The term "goal" may refer to an entire goal or to the conclusion of a goal, depending on the context.
The conclusion appears below a line and the local context appears above the line.
The conclusion is a type. Each item in the local context begins with a name
and ends, after a colon, with an associated type.
Local definitions are shown in the form
n := 0 : nat, for example, in which
nat is the
The local context of a goal contains items specific to the goal as well as section-local variables and hypotheses (see Assumptions) defined in the current section. The latter are included in the initial proof state. Items in the local context are ordered; an item can only refer to items that appear before it. (A more mathematical description of the local context is here.)
The global environment has definitions and proven theorems that are global in scope. (A more mathematical description of the global environment is here.)
When you begin proving a theorem, the proof state shows the statement of the theorem below the line and often nothing in the local context:
- Parameter P: nat -> Prop.
- P is declared
- Goal forall n m: nat, n > m -> P 1 /\ P 2.
- 1 goal ============================ forall n m : nat, n > m -> P 1 /\ P 2
- 1 goal n, m : nat H : n > m ============================ P 1 /\ P 2
Some tactics, such as
split, create new goals, which may
be referred to as subgoals for clarity.
Goals are numbered from 1 to N at each step of the proof to permit applying a
tactic to specific goals. The local context is only shown for the first goal.
- 2 goals n, m : nat H : n > m ============================ P 1 goal 2 is: P 2
"Variables" may refer specifically to local context items for which the type of their type
Type, and "hypotheses" refers to items that are
for which the type of their type is
but these terms are also used interchangeably.
- let t_n := type of n in idtac "type of n :" t_n; let tt_n := type of t_n in idtac "type of" t_n ":" tt_n.
- type of n : nat type of nat : Set
- let t_H := type of H in idtac "type of H :" t_H; let tt_H := type of t_H in idtac "type of" t_H ":" tt_H.
- type of H : (n > m) type of (n > m) : Prop
A proof script, consisting of the tactics that are applied to prove a theorem, is often informally referred to as a "proof". The real proof, whether complete or incomplete, is a term, the proof term, which users may occasionally want to examine. (This is based on the Curry-Howard isomorphism [How80][Bar81][GLT89][Hue89], which is a correspondence between between proofs and terms and between propositions and types of λ-calculus. The isomorphism is also sometimes called the "propositions-as-types correspondence".)
- Show Proof.
- (fun (n m : nat) (H : n > m) => conj ?Goal ?Goal0)
The incomplete parts, the goals, are represented by
with names that begin with
Show Existentials command
shows each existential with the hypotheses and conclusion for the associated goal.
- Show Existentials.
- Existential 1 = ?Goal : [n : nat m : nat H : n > m |- P 1] Existential 2 = ?Goal0 : [n : nat m : nat H : n > m |- P 2]
Coq's kernel verifies the correctness of proof terms when it exits proof mode by checking that the proof term is well-typed and that its type is the same as the theorem statement.
After a proof is completed,
shows the proof term and its type. The type appears after
the colon (
forall ...), as for this theorem from Coq's standard library:
- Print proj1.
- Fetching opaque proofs from disk for Coq.Init.Logic proj1 = fun (A B : Prop) (H : A /\ B) => match H with | conj x x0 => (fun (H0 : A) (_ : B) => H0) x x0 end : forall A B : Prop, A /\ B -> A Arguments proj1 [A B]%type_scope _
Entering and exiting proof mode¶
Tactics are available only in proof mode (currently they give syntax errors outside of proof mode). Most commands can be used both in and out of proof mode, but some commands only work in or outside of proof mode.
Asserts an unnamed proposition. This is intended for quick tests that a proposition is provable. If the proof is eventually completed and validated, you can assign a name with the
Definedcommands. If no name is given, the name will be
Unnamed_thm(or, if that name is already defined, a variant of that).
Passes a completed proof term to Coq's kernel to check that the proof term is well-typed and to verify that its type matches the theorem statement. If it's verified, the proof term is added to the global environment as an opaque constant using the declared name from the original goal.
It's very rare for a proof term to fail verification. Generally this indicates a bug in a tactic you used or that you misused some unsafe tactics.
Attempt to save an incomplete proof.¶
No focused proof (No proof-editing in progress).¶
You tried to use a proof mode command such as
Qedoutside of proof mode.
Sometimes an error occurs when building the proof term, because tactics do not enforce completely the term construction constraints.
The user should also be aware of the fact that since the proof term is completely rechecked at this point, one may have to wait a while when the proof is large. In some exceptional cases one may even incur a memory overflow.
Save, except the proof is made transparent, which means that its content can be explicitly used for type checking and that it can be unfolded in conversion tactics (see Applying conversion rules,
identis specified, the proof is defined with the given name, which overrides any name provided by the
Theoremcommand or its variants.
This command is available in proof mode to give up the current proof and declare the initial goal as an axiom.
Cancels the current proof development, switching back to the previous proof development, or to the Coq toplevel if no other proof was being edited.
Aborts all current proofs.
No focused proof (No proof-editing in progress).¶
Is a no-op which is useful to delimit the sequence of tactic commands which start a proof, after a
Theoremcommand. It is a good practice to use
Proofas an opening parenthesis, closed in the script with a closing
Proof using section_var_expr with ltac_expr?¶
::=section_var_expr0 - section_var_expr0
|section_var_expr0 + section_var_expr0
|( section_var_expr ) *?
Opens proof mode, declaring the set of section variables (see Assumptions) used by the proof. At
Qedtime, the system verifies that the set of section variables used in the proof is a subset of the declared one.
The set of declared variables is closed under type dependency. For example, if
Tis a variable and
ais a variable of type
T, then the commands
Proof using aand
Proof using T aare equivalent.
The set of declared variables always includes the variables used by the statement. In other words
Proof using eis equivalent to
Proof using Type + efor any declaration expression
Use all section variables except those specified by
section_var_expr0 + section_var_expr0
Use section variables from the union of both collections. See Name a set of section hypotheses for Proof using to see how to form a named collection.
section_var_expr0 - section_var_expr0
Use section variables which are in the first collection but not in the second one.
Use the transitive closure of the specified collection.
Use only section variables occurring in the statement. Specifying
*uses the forward transitive closure of all the section variables occurring in the statement. For example, if the variable
p < 5then
poccurs in the type of
Use all section variables.
This attribute can be applied to the
CoFixpointcommands as well as to
Lemmaand its variants. It takes a
section_var_expr, in quotes, as its value. This is equivalent to specifying the same
- Section Test.
- Variable n : nat.
- n is declared
- Hypothesis Hn : n <> 0.
- Hn is declared
- #[using="Hn"] Lemma example : 0 < n.
- 1 goal n : nat Hn : n <> 0 ============================ 0 < n
- End Test.
Proof using options¶
The following options modify the behavior of
Default Proof Using "section_var_expr"¶
Name a set of section hypotheses for
Collection ident := section_var_expr¶
This can be used to name a set of section hypotheses, with the purpose of making
Proof usingannotations more compact.
Define the collection named
Collection Some := x y z.
Define the collection named
Collection Fewer := Some - z
Define the collection named
Manycontaining the set union or set difference of
Collection Many := Fewer + Some Collection Many := Fewer - Some
Define the collection named
Manycontaining the set difference of
Fewerand the unnamed collection
Collection Many := Fewer - (x y)
Existential natural : type? := term¶
This command is intended to be used to instantiate existential variables when the proof is completed but some uninstantiated existential variables remain. To instantiate existential variables during proof edition, you should use the tactic
Deprecated since version 8.13.
When entering proof mode through commands such as
Coq picks by default the
Ltac mode. Nonetheless, there exist other proof modes
shipped in the standard Coq installation, and furthermore some plugins define
their own proof modes. The default proof mode used when opening a proof can
be changed using the following option.
Default Proof Mode string¶
This option selects the proof mode to use when starting a proof. Depending on the proof mode, various syntactic constructs are allowed when writing a proof. All proof modes support commands; the proof mode determines which tactic language and set of tactic definitions are available. The possible option values are:
Ltac language and the tactics with the syntax documented in this manual. Some tactics are not available until the associated plugin is loaded, such as
micromega. This proof mode is set when the prelude is loaded.
No tactic language is activated at all. This is the default when the prelude is not loaded, e.g. through the
Some external plugins also define their own proof mode, which can be activated with this command.
Displays the current goals.
Display only the
Displays the named goal
ident. This is useful in particular to display a shelved goal but only works if the corresponding existential variable has been named by the user (see Existential variables) as in the following example.
- Goal exists n, n = 0.
- 1 goal ============================ exists n : nat, n = 0
- eexists ?[n].
- 1 focused goal (shelved: 1) ============================ ?n = 0
- Show n.
- goal n is: ============================ nat
No focused proof.¶
No such goal.¶
Show Proof Diffs removed??¶
Displays the proof term generated by the tactics that have been applied so far. If the proof is incomplete, the term will contain holes, which correspond to subterms which are still to be constructed. Each hole is an existential variable, which appears as a question mark followed by an identifier.
Specifying “Diffs” highlights the difference between the current and previous proof step. By default, the command shows the output once with additions highlighted. Including “removed” shows the output twice: once showing removals and once showing additions. It does not examine the
Diffsoption. See "Show Proof" differences.
Prints the names of all the theorems that are currently being proved. As it is possible to start proving a previous lemma during the proof of a theorem, there may be multiple names.
If the current goal begins by at least one product, prints the name of the first product as it would be generated by an anonymous
intro. The aim of this command is to ease the writing of more robust scripts. For example, with an appropriate Proof General macro, it is possible to transform any anonymous
introinto a qualified one such as
intro y13. In the case of a non-product goal, it prints nothing.
Displays all open goals / existential variables in the current proof along with the context and type of each variable.
Show Match qualid¶
- Show Match nat.
- match # with | O => | S x => end
Unknown inductive type.¶
Displays the set of all universe constraints and its normalized form at the current stage of the proof, useful for debugging universe inconsistencies.
Show Goal natural at natural¶
Available in coqtop. Displays a goal at a proof state using the goal ID number and the proof state ID number. It is primarily for use by tools such as Prooftree that need to fetch goal history in this way. Prooftree is a tool for visualizing a proof as a tree that runs in Proof General.
Some tactics (e.g.
refine) allow to build proofs using fixpoint or co-fixpoint constructions. Due to the incremental nature of proof construction, the check of the termination (or guardedness) of the recursive calls in the fixpoint or cofixpoint constructions is postponed to the time of the completion of the proof.
Guardedallows checking if the guard condition for fixpoint and cofixpoint is violated at some time of the construction of the proof without having to wait the completion of the proof.
Showing differences between proof steps¶
Coq can automatically highlight the differences between successive proof steps
and between values in some error messages. Coq can also highlight differences
in the proof term.
For example, the following screenshots of CoqIDE and coqtop show the application
of the same
intros tactic. The tactic creates two new hypotheses, highlighted in green.
The conclusion is entirely in pale green because although it’s changed, no tokens were added
to it. The second screenshot uses the "removed" option, so it shows the conclusion a
second time with the old text, with deletions marked in red. Also, since the hypotheses are
new, no line of old text is shown for them.
This image shows an error message with diff highlighting in CoqIDE:
How to enable diffs¶
This option is used to enable diffs. The “on” setting highlights added tokens in green, while the “removed” setting additionally reprints items with removed tokens in red. Unchanged tokens in modified items are shown with pale green or red. Diffs in error messages use red and green for the compared values; they appear regardless of the setting. (Colors are user-configurable.)
For coqtop, showing diffs can be enabled when starting coqtop with the
-diffs on|off|removed command-line option or by setting the
within Coq. You will need to provide the
-color on|auto command-line option when
you start coqtop in either case.
Colors for coqtop can be configured by setting the
variable. See section By environment variables. Diffs
use the tags
In CoqIDE, diffs should be enabled from the
View menu. Don’t use the
command in CoqIDE. You can change the background colors shown for diffs from the
Edit | Preferences | Tags panel by changing the settings for the
diff.removed.bg tags. This panel also
lets you control other attributes of the highlights, such as the foreground
color, bold, italic, underline and strikeout.
Proof General can also display Coq-generated proof diffs automatically. Please see the PG documentation section "Showing Proof Diffs") for details.
How diffs are calculated¶
Diffs are calculated as follows:
Select the old proof state to compare to, which is the proof state before the last tactic that changed the proof. Changes that only affect the view of the proof, such as
all: swap 1 2, are ignored.
For each goal in the new proof state, determine what old goal to compare it to—the one it is derived from or is the same as. Match the hypotheses by name (order is ignored), handling compacted items specially.
For each hypothesis and conclusion (the “items”) in each goal, pass them as strings to the lexer to break them into tokens. Then apply the Myers diff algorithm [Mye86] on the tokens and add appropriate highlighting.
Aside from the highlights, output for the "on" option should be identical to the undiffed output.
Goals completed in the last proof step will not be shown even with the "removed" setting.
This screen shot shows the result of applying a
split tactic that replaces one goal
with 2 goals. Notice that the goal
P 1 is not highlighted at all after
the split because it has not changed.
Diffs may appear like this after applying a
intro tactic that results
in a compacted hypotheses:
"Show Proof" differences¶
To show differences in the proof term:
In coqtop and Proof General, use the
In CoqIDE, position the cursor on or just after a tactic to compare the proof term after the tactic with the proof term before the tactic, then select
View / Show Prooffrom the menu or enter the associated key binding. Differences will be shown applying the current
Show Diffssetting from the
Viewmenu. If the current setting is
Don't show diffs, diffs will not be shown.
Output with the "added and removed" option looks like this:
Controlling proof mode¶
Hyps Limit natural¶
This option controls the maximum number of hypotheses displayed in goals after the application of a tactic. All the hypotheses remain usable in the proof development. When unset, it goes back to the default mode which is to print all available hypotheses.
Nested Proofs Allowed¶
When turned on (it is off by default), this flag enables support for nested proofs: a new assertion command can be inserted before the current proof is finished, in which case Coq will temporarily switch to the proof of this nested lemma. When the proof of the nested lemma is finished (with
Defined), its statement will be made available (as if it had been proved before starting the previous proof) and Coq will switch back to the proof of the previous assertion.
Controlling memory usage¶
Print Debug GC¶
Prints heap usage statistics, which are values from the
stattype of the
Gcmodule described here in the OCaml documentation. The
top_heap_wordsvalues give the basic information. Words are 8 bytes or 4 bytes, respectively, for 64- and 32-bit executables.
When experiencing high memory usage the following commands can be used to force Coq to optimize some of its internal data structures.
Shrink the data structure used to represent the current proof.
Memory usage parameters can be set through the OCAMLRUNPARAM environment variable.