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doc(examples/lean): new tactic example

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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
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Leonardo de Moura 2013-12-06 16:34:59 -08:00
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(*
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 *)