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