59 lines
2.6 KiB
Text
59 lines
2.6 KiB
Text
(**
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import("tactic.lua")
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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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**)
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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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The (show [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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*)
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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 (show B /\ A by auto)
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Show Environment 1. (* Show proof for the previous theorem *)
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(*
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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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*)
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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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(*
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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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*)
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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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apply Conj. exact. done. (* tactic command sequence for the third hole *)
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(*
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We can also mix the two styles (hints and command sequences)
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*)
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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 (show B /\ A by auto).
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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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