c73398a0b8
The idea is to support conditional equations where the left-hand-side does not contain all theorem arguments, but the missing arguments can be inferred using type inference.
For example, we will be able to have the eta theorem as rewrite rule:
theorem eta {A : TypeU} {B : A → TypeU} (f : ∀ x : A, B x) : (λ x : A, f x) = f
:= funext (λ x : A, refl (f x))
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
89 lines
2.5 KiB
Lua
89 lines
2.5 KiB
Lua
local env = get_environment()
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assert(env:find_object("neq_elim"))
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function show_ceqs(ceqs)
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for i = 1, #ceqs do
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print(ceqs[i][1], ceqs[i][2])
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env:type_check(ceqs[i][2])
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assert(is_ceq(env, ceqs[i][1]))
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end
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end
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function test_ceq(name, expected)
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local obj = env:find_object(name)
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local r = to_ceqs(env, obj:get_type(), Const(name))
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show_ceqs(r)
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assert(#r == expected)
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end
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parse_lean_cmds([[
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import if_then_else
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variable f : Nat -> Nat
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axiom Ax1 : forall x : Nat, x > 0 -> f x < 0 /\ not (f x = 1)
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axiom Ax2 : forall x : Nat, x < 0 -> f (f x) = x
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variable g : Nat -> Nat -> Nat
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axiom Ax3 : forall x : Nat, not (x = 1) -> if (x < 0) then (g x x = 0) else (g x x < 0 /\ g x 0 = 1 /\ g 0 x = 2)
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axiom Ax4 : forall x y : Nat, f x = x
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axiom Ax5 : forall x y : Nat, f x = y /\ g x y = x
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axiom Ax6 : forall x : Nat, f x ≠ 0
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]])
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test_ceq("Ax1", 2)
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test_ceq("Ax2", 1)
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test_ceq("Ax3", 4)
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test_ceq("Ax4", 0)
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test_ceq("Ax5", 1)
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test_ceq("Ax6", 1)
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test_ceq("eta", 1)
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test_ceq("not_not_eq", 1)
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test_ceq("or_comm", 1)
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test_ceq("or_assoc", 1)
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test_ceq("or_id", 1)
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test_ceq("or_falsel", 1)
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test_ceq("or_falser", 1)
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test_ceq("or_truel", 1)
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test_ceq("or_truer", 1)
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test_ceq("or_tauto", 1)
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test_ceq("and_comm", 1)
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test_ceq("and_assoc", 1)
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test_ceq("and_id", 1)
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test_ceq("and_falsel", 1)
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test_ceq("and_falser", 1)
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test_ceq("and_truel", 1)
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test_ceq("and_truer", 1)
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test_ceq("and_absurd", 1)
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test_ceq("imp_truer", 1)
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test_ceq("imp_truel", 1)
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test_ceq("imp_falser", 1)
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test_ceq("imp_falsel", 1)
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test_ceq("not_and", 1)
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test_ceq("not_or", 1)
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test_ceq("not_iff", 1)
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test_ceq("not_implies", 1)
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test_ceq("or_left_comm", 1)
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test_ceq("and_left_comm", 1)
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test_ceq("forall_or_distributer", 1)
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test_ceq("forall_or_distributel", 1)
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test_ceq("forall_and_distribute", 1)
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test_ceq("exists_and_distributer", 1)
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test_ceq("exists_and_distributel", 1)
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test_ceq("exists_or_distribute", 1)
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test_ceq({"Nat", "add_zeror"}, 1)
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test_ceq({"Nat", "add_zerol"}, 1)
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test_ceq({"Nat", "add_comm"}, 1)
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test_ceq({"Nat", "add_assoc"}, 1)
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test_ceq({"Nat", "mul_zerol"}, 1)
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test_ceq({"Nat", "mul_zeror"}, 1)
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test_ceq({"Nat", "mul_onel"}, 1)
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test_ceq({"Nat", "mul_oner"}, 1)
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test_ceq({"Nat", "mul_comm"}, 1)
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test_ceq({"Nat", "mul_assoc"}, 1)
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test_ceq({"Nat", "add_comm"}, 1)
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test_ceq({"Nat", "add_assoc"}, 1)
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test_ceq({"Nat", "distributer"}, 1)
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test_ceq({"Nat", "distributel"}, 1)
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test_ceq({"Nat", "le_refl"}, 1)
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test_ceq({"Nat", "le_zero"}, 1)
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test_ceq({"Nat", "not_lt_0"}, 1)
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test_ceq({"Nat", "lt_nrefl"}, 1)
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