72 lines
3 KiB
Text
72 lines
3 KiB
Text
(**
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-- This example demonstrates how to create a new tactic using Lua.
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-- The basic idea is to reimplement the tactic conj_tactic in Lua.
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-- Tactic for splitting goals of the form
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-- n : Hs |- A /\ B
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-- into
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-- n::1 : Hs |- A
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-- n::2 : Hs |- B
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function conj_fn(env, ios, s)
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local gs = s:goals()
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-- We store the information needed by the proof_builder in the
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-- array proof_info.
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-- proof_info has the format {{name_1, expr_1}, ... {name_k, expr_k}}
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-- where name_i is a goal splitted by this tactic, and expr_i
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-- is the conclusion of the theorem, that is, an expression of the form
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-- A /\ B
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local proof_info = {}
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-- We store the new goals into the Lua array new_gs.
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-- new_gs has the format {{name_1, goal_1}, ..., {name_n, goal_n}}
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local new_gs = {}
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local found = false
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for n, g in gs:pairs() do
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yield() -- Give a chance to other tactics to run
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local c = g:conclusion()
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if c:is_and() then
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-- Goal g is of the form Hs |- A /\ B
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found = true -- The tactic managed to split at least one goal
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local Hs = g:hypotheses()
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local A = c:arg(1)
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local B = c:arg(2)
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proof_info[#proof_info + 1] = {n, c} -- Save information for implementing the proof builder
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new_gs[#new_gs + 1] = {name(n, 1), goal(Hs, A)} -- Add goal n::1 : Hs |- A
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new_gs[#new_gs + 1] = {name(n, 2), goal(Hs, B)} -- Add goal n::1 : Hs |- B
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else
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new_gs[#new_gs + 1] = {n, g} -- Keep the goal
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end
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end
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if not found then
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return nil -- Tactic is not applicable
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end
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local pb = s:proof_builder()
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local new_pb =
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function(m, a)
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local Conj = Const("Conj")
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local new_m = proof_map(m) -- Copy proof map m
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for _, p in ipairs(proof_info) do
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local n = p[1] -- n is the name of the goal splitted by this tactic
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local c = p[2] -- c is the conclusion of the goal splitted by this tactic
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assert(c:is_and()) -- c is of the form A /\ B
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-- The proof for goal n is
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-- Conj(A, B, H1, H2)
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-- where H1 and H2 are the proofs for goals n::1 and n::2
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new_m:insert(n, Conj(c:arg(1), c:arg(2), m:find(name(n, 1)), m:find(name(n, 2))))
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-- We don't need the proofs for n::1 and n::2 anymore
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new_m:erase(name(n, 1))
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new_m:erase(name(n, 2))
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end
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return pb(new_m, a) -- Apply the proof builder for the original state
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end
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return proof_state(s, goals(new_gs), proof_builder(new_pb))
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end
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conj_in_lua = tactic(conj_fn) -- Create a new tactic object using the Lua function conj_fn
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-- Now, the tactic conj_in_lua can be used when proving theorems in Lean.
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**)
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Theorem T (a b : Bool) : a => b => a /\ b := _.
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apply (** REPEAT(ORELSE(imp_tac, conj_in_lua)) .. assumption_tac **)
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done
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(* Show proof created using our script *)
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Show Environment 1.
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