66 lines
2.8 KiB
Text
66 lines
2.8 KiB
Text
|
(*
|
||
|
|
This example demonstrates how to specify a proof skeleton that contains
|
||
|
|
"holes" that must be filled using user-defined tactics.
|
||
|
|
*)
|
||
|
|
|
||
|
|
(**
|
||
|
|
-- Define a simple tactic using Lua
|
||
|
|
auto = REPEAT(ORELSE(assumption_tac, conj_tac, conj_hyp_tac))
|
||
|
|
**)
|
||
|
|
|
||
|
|
(*
|
||
|
|
The (by [tactic]) expression is essentially creating a "hole" and associating a "hint" to it.
|
||
|
|
The "hint" is a tactic that should be used to fill the "hole".
|
||
|
|
In the following example, we use the tactic "auto" defined by the Lua code above.
|
||
|
|
|
||
|
|
The (show [expr] by [tactic]) expression is also creating a "hole" and associating a "hint" to it.
|
||
|
|
The expression [expr] after the shows is fixing the type of the "hole"
|
||
|
|
*)
|
||
|
|
Theorem T1 (A B : Bool) : A /\ B -> B /\ A :=
|
||
|
|
fun assumption : A /\ B,
|
||
|
|
let lemma1 : A := (by auto),
|
||
|
|
lemma2 : B := (by auto)
|
||
|
|
in (show B /\ A by auto)
|
||
|
|
|
||
|
|
Show Environment 1. (* Show proof for the previous theorem *)
|
||
|
|
|
||
|
|
(*
|
||
|
|
When hints are not provided, the user must fill the (remaining) holes using tactic command sequences.
|
||
|
|
Each hole must be filled with a tactic command sequence that terminates with the command 'done' and
|
||
|
|
successfully produces a proof term for filling the hole. Here is the same example without hints
|
||
|
|
This style is more convenient for interactive proofs
|
||
|
|
*)
|
||
|
|
Theorem T2 (A B : Bool) : A /\ B -> B /\ A :=
|
||
|
|
fun assumption : A /\ B,
|
||
|
|
let lemma1 : A := _, (* first hole *)
|
||
|
|
lemma2 : B := _ (* second hole *)
|
||
|
|
in _. (* third hole *)
|
||
|
|
apply auto. done. (* tactic command sequence for the first hole *)
|
||
|
|
apply auto. done. (* tactic command sequence for the second hole *)
|
||
|
|
apply auto. done. (* tactic command sequence for the third hole *)
|
||
|
|
|
||
|
|
(*
|
||
|
|
In the following example, instead of using the "auto" tactic, we apply a sequence of even simpler tactics.
|
||
|
|
*)
|
||
|
|
Theorem T3 (A B : Bool) : A /\ B -> B /\ A :=
|
||
|
|
fun assumption : A /\ B,
|
||
|
|
let lemma1 : A := _, (* first hole *)
|
||
|
|
lemma2 : B := _ (* second hole *)
|
||
|
|
in _. (* third hole *)
|
||
|
|
apply conj_hyp_tac. apply assumption_tac. done. (* tactic command sequence for the first hole *)
|
||
|
|
apply conj_hyp_tac. apply assumption_tac. done. (* tactic command sequence for the second hole *)
|
||
|
|
apply conj_tac. apply assumption_tac. done. (* tactic command sequence for the third hole *)
|
||
|
|
|
||
|
|
(*
|
||
|
|
We can also mix the two styles (hints and command sequences)
|
||
|
|
*)
|
||
|
|
Theorem T4 (A B : Bool) : A /\ B -> B /\ A :=
|
||
|
|
fun assumption : A /\ B,
|
||
|
|
let lemma1 : A := _, (* first hole *)
|
||
|
|
lemma2 : B := _ (* second hole *)
|
||
|
|
in (show B /\ A by auto).
|
||
|
|
apply conj_hyp_tac. apply assumption_tac. done. (* tactic command sequence for the first hole *)
|
||
|
|
apply conj_hyp_tac. apply assumption_tac. done. (* tactic command sequence for the second hole *)
|
||
|
|
|
||
|
|
|