139 lines
2.9 KiB
Text
139 lines
2.9 KiB
Text
import Lean
|
||
open Lean Meta Elab Tactic
|
||
|
||
theorem bv0_eq (x : BitVec 0) : x = 0 := BitVec.of_length_zero
|
||
|
||
set_option warn.sorry false
|
||
|
||
elab "sym_simp" "[" declNames:ident,* "]" : tactic => do
|
||
let declNames ← declNames.getElems.mapM fun s => realizeGlobalConstNoOverload s.raw
|
||
liftMetaTactic1 <| Sym.simpGoalUsing declNames
|
||
|
||
theorem heq_self : (x ≍ x) = True := by simp
|
||
theorem forall_true {α : Sort u} : (∀ _ : α, True) = True := by simp
|
||
|
||
example : x + 0 ≍ x := by
|
||
fail_if_success sym_simp []
|
||
sym_simp [Nat.add_zero, heq_self]
|
||
|
||
example : 0 + x + 0 = x := by
|
||
sym_simp [Nat.add_zero, Nat.zero_add, eq_self]
|
||
|
||
example : x = x := by
|
||
sym_simp [bv0_eq, eq_self]
|
||
|
||
example (x y : BitVec 0) : x = y := by
|
||
sym_simp [bv0_eq, eq_self]
|
||
|
||
example : ∀ x, 0 + x + 0 = x := by
|
||
sym_simp [Nat.add_zero, Nat.zero_add, eq_self]
|
||
sym_simp [forall_true]
|
||
|
||
example : ∀ x, 0 + x + 0 = x := by
|
||
sym_simp [Nat.add_zero, Nat.zero_add, eq_self, forall_true]
|
||
|
||
example (p q : Prop) (hp : p) : if x + 0 = x then p else q := by
|
||
sym_simp [Nat.add_zero, eq_self, if_true]
|
||
exact hp
|
||
|
||
example (as : Array Int) (i : Nat) (h : 0 + i < as.size) : as[0 + i] = as[i] := by
|
||
sym_simp [Nat.zero_add, eq_self]
|
||
|
||
/-- trace: ⊢ Nat.add 0 = id -/
|
||
#guard_msgs in
|
||
example : Nat.add (0 + 0) = id := by
|
||
sym_simp [Nat.zero_add]
|
||
trace_state
|
||
funext
|
||
simp
|
||
|
||
/--
|
||
trace: a : Nat
|
||
β✝ : Type
|
||
f : β✝ → Prop
|
||
h : HEq a = f
|
||
⊢ HEq a = f
|
||
-/
|
||
#guard_msgs in
|
||
example (h : HEq a = f) : HEq (α := Nat) (0 + a) = f := by
|
||
sym_simp [Nat.zero_add]
|
||
trace_state
|
||
exact h
|
||
|
||
/--
|
||
trace: a b : Nat
|
||
f : Nat → Nat
|
||
h : f a = b
|
||
⊢ id f a = b
|
||
-/
|
||
#guard_msgs in
|
||
example (f : Nat → Nat) (h : f a = b) : id f (0 + a) = b := by
|
||
sym_simp [Nat.zero_add]
|
||
trace_state
|
||
exact h
|
||
|
||
def f (_ : α) {β : Type} (b : β) : β := b
|
||
|
||
/--
|
||
trace: a : Nat
|
||
g : Nat → Nat
|
||
⊢ f 0 g a = g a
|
||
-/
|
||
#guard_msgs in
|
||
example (g : Nat → Nat) : f (0 + 0) g (0 + a) = g a := by
|
||
sym_simp [Nat.zero_add]
|
||
trace_state
|
||
rfl
|
||
|
||
def f' (_ : α) (b : β) := b
|
||
|
||
/--
|
||
trace: a : Nat
|
||
g : Nat → Nat
|
||
⊢ f' 0 g a = g a
|
||
-/
|
||
#guard_msgs in
|
||
example (g : Nat → Nat) : f' (0 + 0) g (0 + a) = g a := by
|
||
sym_simp [Nat.zero_add]
|
||
trace_state
|
||
rfl
|
||
|
||
/--
|
||
trace: a b : Nat
|
||
as : Array (Nat → Nat)
|
||
i : Nat
|
||
x✝ : i < as.size
|
||
h : as[i] a = b
|
||
⊢ as[i] a = b
|
||
-/
|
||
#guard_msgs in
|
||
example (as : Array (Nat → Nat)) (i : Nat) (_ : i < as.size) (h : as[i] a = b) : as[0 + i] (0 + a) = b := by
|
||
sym_simp [Nat.zero_add]
|
||
trace_state
|
||
exact h
|
||
|
||
/--
|
||
trace: c a : Nat
|
||
g : Nat → Nat
|
||
h : ite (0 < c) a = g
|
||
⊢ ite (0 < c) a = g
|
||
-/
|
||
#guard_msgs in
|
||
example (h : ite (c > 0) a = g) : ite (c > 0) (0 + a) = g := by
|
||
sym_simp [Nat.zero_add]
|
||
trace_state
|
||
exact h
|
||
|
||
example (f : Nat → Nat) : id f a = f a := by
|
||
sym_simp [id_eq, eq_self]
|
||
|
||
example (f : Nat → Nat) : id f (0 + a) = f a := by
|
||
sym_simp [id_eq, eq_self, Nat.zero_add]
|
||
|
||
def foo (x : Nat) :=
|
||
match x with
|
||
| 0 => true
|
||
| _+1 => false
|
||
|
||
example : foo 0 = true := by
|
||
sym_simp [foo.eq_def, foo.match_1.eq_1, eq_self]
|