# Commands¶

Imandra has a number of powerful verification commands:

• verify <upto> <func>: takes a function representing a goal and attempts to prove it. If the proof attempt fails, Imandra will try to synthesize a concrete counterexample illustrating the failure. Found counterexamples are installed by Imandra in the CX module. When verifying a formula that doesn't depend on function parameters, verify (<expr>) is a shorthand for verify (fun () -> <expr>). If <upto> is provided as one of ~upto:<n> or ~upto_bound:<n>, verification will be bound by unrolling limits.

• instance <upto> <func>: takes a function representing a goal and attempts to synthesize an instance (i.e., a concrete value) that satisfies it. It is useful for answering the question "What is an example value that satisfies this particular property?". Found instances are installed by Imandra in the CX module. If <upto> is provided as one of ~upto:<n> or ~upto_bound:<n>, instance search will be bound by unrolling limits.

• theorem <name> <vars> = <body>: takes a name, variables and a function of the variables representing a goal to be proved. If Imandra proves the goal, the named theorem is installed and may be used in subsequent proofs. Theorems can be tagged with attributes instructing Imandra how the theorem should be (automatically) applied to prove other theorems in the future. Found counterexamples are installed by Imandra in the CX module.

• lemma <name> <vars> = <body>: synonym of theorem, but idiomatically often used for "smaller" subsidiary results as one is working up to a larger theorem.

• axiom <name> <vars> = <body>: declares an axiom, effectively the same as theorem but forcing Imandra to assume the truth of the conjecture, rather than verifying it. This is of course dangerous and should be used with extreme care.

## Examples¶

In [1]:
verify (fun x -> x + 1 > x)

Out[1]:
- : Z.t -> bool = <fun>

Proved
proof
ground_instances:0
definitions:0
inductions:0
search_time:
0.016s
details:
Expand
smt_stats:
 rlimit count: 14 num allocs: 646488 time: 0.006 memory: 5.16 max memory: 5.16
Expand
• start[0.016s] (( :var_0: ) + 1) > ( :var_0: )
• #### simplify

 into: true expansions: [] rewrite_steps: forward_chaining:
• Unsat
In [2]:
instance (fun x y -> x < 0 && x + y = 4)

Out[2]:
- : Z.t -> Z.t -> bool = <fun>
module CX : sig val x : Z.t val y : Z.t end

Instance (after 0 steps, 0.016s):
let x : int = (-1)
let y : int = 5

Instance
proof attempt
ground_instances:0
definitions:0
inductions:0
search_time:
0.016s
details:
Expand
smt_stats:
 num checks: 1 eliminated vars: 1 rlimit count: 154 mk bool var: 1 eliminated applications: 2 num allocs: 3.81216e+06 final checks: 1 time: 0.007 memory: 5.32 max memory: 5.75
Expand
• start[0.016s] ( :var_0: ) < 0 && ( :var_0: ) + ( :var_1: ) = 4
• #### simplify

 into: not (0 <= ( :var_0: )) && ( :var_0: ) + ( :var_1: ) = 4 expansions: [] rewrite_steps: forward_chaining:
• Sat (Some let x : int = (Z.of_nativeint (-1n)) let y : int = (Z.of_nativeint (5n)) )
In [3]:
theorem succ_mono n m = succ n > succ m <==> n > m

Out[3]:
val succ_mono : Z.t -> Z.t -> bool = <fun>

Proved
proof
ground_instances:0
definitions:0
inductions:0
search_time:
0.016s
details:
Expand
smt_stats:
 rlimit count: 24 num allocs: 8.49023e+06 time: 0.007 memory: 5.71 max memory: 5.75
Expand
• start[0.016s]
(( :var_0: ) + 1) > (( :var_1: ) + 1) = ( :var_0: ) > ( :var_1: )
• #### simplify

 into: true expansions: [] rewrite_steps: forward_chaining:
• Unsat
In [4]:
verify (fun n -> succ n <> 100)

Out[4]:
- : Z.t -> bool = <fun>
module CX : sig val n : Z.t end

Counterexample (after 0 steps, 0.015s):
let n : int = 99

Refuted
proof attempt
ground_instances:0
definitions:0
inductions:0
search_time:
0.015s
details:
Expand
smt_stats:
 eliminated vars: 1 rlimit count: 40 num allocs: 1.48267e+07 time: 0.006 memory: 5.89 max memory: 5.89
Expand
• start[0.015s] not (( :var_0: ) + 1 = 100)
• #### simplify

 into: not (( :var_0: ) = 99) expansions: [] rewrite_steps: forward_chaining:
• Sat (Some let n : int = (Z.of_nativeint (99n)) )