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