diff --git a/examples/lean/even.lean b/examples/lean/even.lean index a6134a68aa..15081fa686 100644 --- a/examples/lean/even.lean +++ b/examples/lean/even.lean @@ -54,7 +54,7 @@ theorem OddPlusOne {a : Nat} (H : odd a) : even (a + 1) := obtain (w : Nat) (Hw : a = 2*w + 1), from H, exists_intro (w + 1) (calc a + 1 = 2*w + 1 + 1 : { Hw } - ... = 2*w + (1 + 1) : symm (add_assoc _ _ _) + ... = 2*w + (1 + 1) : add_assoc _ _ _ ... = 2*w + 2*1 : refl _ ... = 2*(w + 1) : symm (distributer _ _ _)) diff --git a/src/builtin/Nat.lean b/src/builtin/Nat.lean index f1c042d0cd..92c94d0395 100644 --- a/src/builtin/Nat.lean +++ b/src/builtin/Nat.lean @@ -83,15 +83,15 @@ theorem add_comm (a b : Nat) : a + b = b + a ... = (n + a) + 1 : { iH } ... = (n + 1) + a : symm (add_succl n a)) -theorem add_assoc (a b c : Nat) : a + (b + c) = (a + b) + c -:= induction_on a +theorem add_assoc (a b c : Nat) : (a + b) + c = a + (b + c) +:= symm (induction_on a (calc 0 + (b + c) = b + c : add_zerol (b + c) ... = (0 + b) + c : { symm (add_zerol b) }) (λ (n : Nat) (iH : n + (b + c) = (n + b) + c), calc (n + 1) + (b + c) = (n + (b + c)) + 1 : add_succl n (b + c) ... = ((n + b) + c) + 1 : { iH } ... = ((n + b) + 1) + c : symm (add_succl (n + b) c) - ... = ((n + 1) + b) + c : { have (n + b) + 1 = (n + 1) + b : symm (add_succl n b) }) + ... = ((n + 1) + b) + c : { have (n + b) + 1 = (n + 1) + b : symm (add_succl n b) })) theorem mul_zerol (a : Nat) : 0 * a = 0 := induction_on a @@ -109,12 +109,12 @@ theorem mul_succl (a b : Nat) : (a + 1) * b = a * b + b (λ (n : Nat) (iH : (a + 1) * n = a * n + n), calc (a + 1) * (n + 1) = (a + 1) * n + (a + 1) : mul_succr (a + 1) n ... = a * n + n + (a + 1) : { iH } - ... = a * n + n + a + 1 : add_assoc (a * n + n) a 1 - ... = a * n + (n + a) + 1 : { have a * n + n + a = a * n + (n + a) : symm (add_assoc (a * n) n a) } + ... = a * n + n + a + 1 : symm (add_assoc (a * n + n) a 1) + ... = a * n + (n + a) + 1 : { have a * n + n + a = a * n + (n + a) : add_assoc (a * n) n a } ... = a * n + (a + n) + 1 : { add_comm n a } - ... = a * n + a + n + 1 : { add_assoc (a * n) a n } + ... = a * n + a + n + 1 : { symm (add_assoc (a * n) a n) } ... = a * (n + 1) + n + 1 : { symm (mul_succr a n) } - ... = a * (n + 1) + (n + 1) : symm (add_assoc (a * (n + 1)) n 1)) + ... = a * (n + 1) + (n + 1) : add_assoc (a * (n + 1)) n 1) theorem mul_onel (a : Nat) : 1 * a = a := induction_on a @@ -148,12 +148,12 @@ theorem distributer (a b c : Nat) : a * (b + c) = a * b + a * c (λ (n : Nat) (iH : n * (b + c) = n * b + n * c), calc (n + 1) * (b + c) = n * (b + c) + (b + c) : mul_succl n (b + c) ... = n * b + n * c + (b + c) : { iH } - ... = n * b + n * c + b + c : add_assoc (n * b + n * c) b c - ... = n * b + (n * c + b) + c : { symm (add_assoc (n * b) (n * c) b) } + ... = n * b + n * c + b + c : symm (add_assoc (n * b + n * c) b c) + ... = n * b + (n * c + b) + c : { add_assoc (n * b) (n * c) b } ... = n * b + (b + n * c) + c : { add_comm (n * c) b } - ... = n * b + b + n * c + c : { add_assoc (n * b) b (n * c) } + ... = n * b + b + n * c + c : { symm (add_assoc (n * b) b (n * c)) } ... = (n + 1) * b + n * c + c : { symm (mul_succl n b) } - ... = (n + 1) * b + (n * c + c) : symm (add_assoc ((n + 1) * b) (n * c) c) + ... = (n + 1) * b + (n * c + c) : add_assoc ((n + 1) * b) (n * c) c ... = (n + 1) * b + (n + 1) * c : { symm (mul_succl n c) }) theorem distributel (a b c : Nat) : (a + b) * c = a * c + b * c @@ -162,8 +162,8 @@ theorem distributel (a b c : Nat) : (a + b) * c = a * c + b * c ... = a * c + c * b : { mul_comm c a } ... = a * c + b * c : { mul_comm c b } -theorem mul_assoc (a b c : Nat) : a * (b * c) = a * b * c -:= induction_on a +theorem mul_assoc (a b c : Nat) : (a * b) * c = a * (b * c) +:= symm (induction_on a (calc 0 * (b * c) = 0 : mul_zerol (b * c) ... = 0 * c : symm (mul_zerol c) ... = (0 * b) * c : { symm (mul_zerol b) }) @@ -171,7 +171,13 @@ theorem mul_assoc (a b c : Nat) : a * (b * c) = a * b * c calc (n + 1) * (b * c) = n * (b * c) + (b * c) : mul_succl n (b * c) ... = n * b * c + (b * c) : { iH } ... = (n * b + b) * c : symm (distributel (n * b) b c) - ... = (n + 1) * b * c : { symm (mul_succl n b) }) + ... = (n + 1) * b * c : { symm (mul_succl n b) })) + +theorem add_left_comm (a b c : Nat) : a + (b + c) = b + (a + c) +:= left_comm add_comm add_assoc a b c + +theorem mul_left_comm (a b c : Nat) : a * (b * c) = b * (a * c) +:= left_comm mul_comm mul_assoc a b c theorem add_injr {a b c : Nat} : a + b = a + c → b = c := induction_on a @@ -181,13 +187,13 @@ theorem add_injr {a b c : Nat} : a + b = a + c → b = c ... = c : add_zerol c) (λ (n : Nat) (iH : n + b = n + c → b = c) (H : n + 1 + b = n + 1 + c), let L1 : n + b + 1 = n + c + 1 - := (calc n + b + 1 = n + (b + 1) : symm (add_assoc n b 1) + := (calc n + b + 1 = n + (b + 1) : add_assoc n b 1 ... = n + (1 + b) : { add_comm b 1 } - ... = n + 1 + b : add_assoc n 1 b + ... = n + 1 + b : symm (add_assoc n 1 b) ... = n + 1 + c : H - ... = n + (1 + c) : symm (add_assoc n 1 c) + ... = n + (1 + c) : add_assoc n 1 c ... = n + (c + 1) : { add_comm 1 c } - ... = n + c + 1 : add_assoc n c 1), + ... = n + c + 1 : symm (add_assoc n c 1)), L2 : n + b = n + c := succ_inj L1 in iH L2) @@ -202,9 +208,9 @@ theorem add_eqz {a b : Nat} (H : a + b = 0) : a = 0 (λ (n : Nat) (H1 : a = n + 1), absurd_elim (a = 0) H (calc a + b = n + 1 + b : { H1 } - ... = n + (1 + b) : symm (add_assoc n 1 b) + ... = n + (1 + b) : add_assoc n 1 b ... = n + (b + 1) : { add_comm 1 b } - ... = n + b + 1 : add_assoc n b 1 + ... = n + b + 1 : symm (add_assoc n b 1) ... ≠ 0 : succ_nz (n + b))) theorem le_intro {a b c : Nat} (H : a + c = b) : a ≤ b @@ -221,22 +227,22 @@ theorem le_zero (a : Nat) : 0 ≤ a := le_intro (add_zerol a) theorem le_trans {a b c : Nat} (H1 : a ≤ b) (H2 : b ≤ c) : a ≤ c := obtain (w1 : Nat) (Hw1 : a + w1 = b), from (le_elim H1), obtain (w2 : Nat) (Hw2 : b + w2 = c), from (le_elim H2), - le_intro (calc a + (w1 + w2) = a + w1 + w2 : add_assoc a w1 w2 + le_intro (calc a + (w1 + w2) = a + w1 + w2 : symm (add_assoc a w1 w2) ... = b + w2 : { Hw1 } ... = c : Hw2) theorem le_add {a b : Nat} (H : a ≤ b) (c : Nat) : a + c ≤ b + c := obtain (w : Nat) (Hw : a + w = b), from (le_elim H), - le_intro (calc a + c + w = a + (c + w) : symm (add_assoc a c w) + le_intro (calc a + c + w = a + (c + w) : add_assoc a c w ... = a + (w + c) : { add_comm c w } - ... = a + w + c : add_assoc a w c + ... = a + w + c : symm (add_assoc a w c) ... = b + c : { Hw }) theorem le_antisym {a b : Nat} (H1 : a ≤ b) (H2 : b ≤ a) : a = b := obtain (w1 : Nat) (Hw1 : a + w1 = b), from (le_elim H1), obtain (w2 : Nat) (Hw2 : b + w2 = a), from (le_elim H2), let L1 : w1 + w2 = 0 - := add_injr (calc a + (w1 + w2) = a + w1 + w2 : { add_assoc a w1 w2 } + := add_injr (calc a + (w1 + w2) = a + w1 + w2 : { symm (add_assoc a w1 w2) } ... = b + w2 : { Hw1 } ... = a : Hw2 ... = a + 0 : symm (add_zeror a)), @@ -249,9 +255,9 @@ theorem not_lt_0 (a : Nat) : ¬ a < 0 := not_intro (λ H : a + 1 ≤ 0, obtain (w : Nat) (Hw1 : a + 1 + w = 0), from (le_elim H), absurd - (calc a + w + 1 = a + (w + 1) : symm (add_assoc _ _ _) + (calc a + w + 1 = a + (w + 1) : add_assoc _ _ _ ... = a + (1 + w) : { add_comm _ _ } - ... = a + 1 + w : add_assoc _ _ _ + ... = a + 1 + w : symm (add_assoc _ _ _) ... = 0 : Hw1) (succ_nz (a + w))) @@ -263,7 +269,7 @@ theorem lt_elim {a b : Nat} (H : a < b) : ∃ x, a + 1 + x = b theorem lt_le {a b : Nat} (H : a < b) : a ≤ b := obtain (w : Nat) (Hw : a + 1 + w = b), from (le_elim H), - le_intro (calc a + (1 + w) = a + 1 + w : add_assoc _ _ _ + le_intro (calc a + (1 + w) = a + 1 + w : symm (add_assoc _ _ _) ... = b : Hw) theorem lt_ne {a b : Nat} (H : a < b) : a ≠ b @@ -271,7 +277,7 @@ theorem lt_ne {a b : Nat} (H : a < b) : a ≠ b obtain (w : Nat) (Hw : a + 1 + w = b), from (lt_elim H), absurd (calc w + 1 = 1 + w : add_comm _ _ ... = 0 : - add_injr (calc b + (1 + w) = b + 1 + w : add_assoc b 1 w + add_injr (calc b + (1 + w) = b + 1 + w : symm (add_assoc b 1 w) ... = a + 1 + w : { symm H1 } ... = b : Hw ... = b + 0 : symm (add_zeror b))) @@ -284,40 +290,40 @@ theorem lt_nrefl (a : Nat) : ¬ a < a theorem lt_trans {a b c : Nat} (H1 : a < b) (H2 : b < c) : a < c := obtain (w1 : Nat) (Hw1 : a + 1 + w1 = b), from (lt_elim H1), obtain (w2 : Nat) (Hw2 : b + 1 + w2 = c), from (lt_elim H2), - lt_intro (calc a + 1 + (w1 + 1 + w2) = a + 1 + (w1 + (1 + w2)) : { symm (add_assoc w1 1 w2) } - ... = (a + 1 + w1) + (1 + w2) : add_assoc _ _ _ + lt_intro (calc a + 1 + (w1 + 1 + w2) = a + 1 + (w1 + (1 + w2)) : { add_assoc w1 1 w2 } + ... = (a + 1 + w1) + (1 + w2) : symm (add_assoc _ _ _) ... = b + (1 + w2) : { Hw1 } - ... = b + 1 + w2 : add_assoc _ _ _ + ... = b + 1 + w2 : symm (add_assoc _ _ _) ... = c : Hw2) theorem lt_le_trans {a b c : Nat} (H1 : a < b) (H2 : b ≤ c) : a < c := obtain (w1 : Nat) (Hw1 : a + 1 + w1 = b), from (lt_elim H1), obtain (w2 : Nat) (Hw2 : b + w2 = c), from (le_elim H2), - lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : add_assoc _ _ _ + lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : symm (add_assoc _ _ _) ... = b + w2 : { Hw1 } ... = c : Hw2) theorem le_lt_trans {a b c : Nat} (H1 : a ≤ b) (H2 : b < c) : a < c := obtain (w1 : Nat) (Hw1 : a + w1 = b), from (le_elim H1), obtain (w2 : Nat) (Hw2 : b + 1 + w2 = c), from (lt_elim H2), - lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : add_assoc _ _ _ - ... = a + (1 + w1) + w2 : { symm (add_assoc a 1 w1) } + lt_intro (calc a + 1 + (w1 + w2) = a + 1 + w1 + w2 : symm (add_assoc _ _ _) + ... = a + (1 + w1) + w2 : { add_assoc a 1 w1 } ... = a + (w1 + 1) + w2 : { add_comm 1 w1 } - ... = a + w1 + 1 + w2 : { add_assoc a w1 1 } + ... = a + w1 + 1 + w2 : { symm (add_assoc a w1 1) } ... = b + 1 + w2 : { Hw1 } ... = c : Hw2) theorem ne_lt_succ {a b : Nat} (H1 : a ≠ b) (H2 : a < b + 1) : a < b := obtain (w : Nat) (Hw : a + 1 + w = b + 1), from (lt_elim H2), - let L : a + w = b := add_injl (calc a + w + 1 = a + (w + 1) : symm (add_assoc _ _ _) + let L : a + w = b := add_injl (calc a + w + 1 = a + (w + 1) : add_assoc _ _ _ ... = a + (1 + w) : { add_comm _ _ } - ... = a + 1 + w : add_assoc _ _ _ + ... = a + 1 + w : symm (add_assoc _ _ _) ... = b + 1 : Hw) in discriminate (λ Hz : w = 0, absurd_elim (a < b) (calc a = a + 0 : symm (add_zeror _) ... = a + w : { symm Hz } ... = b : L) H1) - (λ (p : Nat) (Hp : w = p + 1), lt_intro (calc a + 1 + p = a + (1 + p) : symm (add_assoc _ _ _) + (λ (p : Nat) (Hp : w = p + 1), lt_intro (calc a + 1 + p = a + (1 + p) : add_assoc _ _ _ ... = a + (p + 1) : { add_comm _ _ } ... = a + w : { symm Hp } ... = b : L)) diff --git a/src/builtin/obj/Nat.olean b/src/builtin/obj/Nat.olean index d8d5b4879a..79f555851e 100644 Binary files a/src/builtin/obj/Nat.olean and b/src/builtin/obj/Nat.olean differ diff --git a/src/library/arith/Nat_decls.cpp b/src/library/arith/Nat_decls.cpp index bce59a39f9..dc343031ff 100644 --- a/src/library/arith/Nat_decls.cpp +++ b/src/library/arith/Nat_decls.cpp @@ -34,6 +34,8 @@ MK_CONSTANT(Nat_mul_comm_fn, name({"Nat", "mul_comm"})); MK_CONSTANT(Nat_distributer_fn, name({"Nat", "distributer"})); MK_CONSTANT(Nat_distributel_fn, name({"Nat", "distributel"})); MK_CONSTANT(Nat_mul_assoc_fn, name({"Nat", "mul_assoc"})); +MK_CONSTANT(Nat_add_left_comm_fn, name({"Nat", "add_left_comm"})); +MK_CONSTANT(Nat_mul_left_comm_fn, name({"Nat", "mul_left_comm"})); MK_CONSTANT(Nat_add_injr_fn, name({"Nat", "add_injr"})); MK_CONSTANT(Nat_add_injl_fn, name({"Nat", "add_injl"})); MK_CONSTANT(Nat_add_eqz_fn, name({"Nat", "add_eqz"})); diff --git a/src/library/arith/Nat_decls.h b/src/library/arith/Nat_decls.h index d7ed8f6d90..1df8d744a2 100644 --- a/src/library/arith/Nat_decls.h +++ b/src/library/arith/Nat_decls.h @@ -92,6 +92,12 @@ inline expr mk_Nat_distributel_th(expr const & e1, expr const & e2, expr const & expr mk_Nat_mul_assoc_fn(); bool is_Nat_mul_assoc_fn(expr const & e); inline expr mk_Nat_mul_assoc_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_mul_assoc_fn(), e1, e2, e3}); } +expr mk_Nat_add_left_comm_fn(); +bool is_Nat_add_left_comm_fn(expr const & e); +inline expr mk_Nat_add_left_comm_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_add_left_comm_fn(), e1, e2, e3}); } +expr mk_Nat_mul_left_comm_fn(); +bool is_Nat_mul_left_comm_fn(expr const & e); +inline expr mk_Nat_mul_left_comm_th(expr const & e1, expr const & e2, expr const & e3) { return mk_app({mk_Nat_mul_left_comm_fn(), e1, e2, e3}); } expr mk_Nat_add_injr_fn(); bool is_Nat_add_injr_fn(expr const & e); inline expr mk_Nat_add_injr_th(expr const & e1, expr const & e2, expr const & e3, expr const & e4) { return mk_app({mk_Nat_add_injr_fn(), e1, e2, e3, e4}); } diff --git a/tests/lean/simp3.lean.expected.out b/tests/lean/simp3.lean.expected.out index bab4db32d8..2da8ef33db 100644 --- a/tests/lean/simp3.lean.expected.out +++ b/tests/lean/simp3.lean.expected.out @@ -6,16 +6,16 @@ 9 ⊥ 2 + 2 + (2 + 2) + 1 ≥ 3 -3 ≤ 2 * 2 + 2 * 2 + 2 * 2 + 2 * 2 + 1 +3 ≤ 2 * 2 + (2 * 2 + (2 * 2 + (2 * 2 + 1))) Assumed: a Assumed: b Assumed: c Assumed: d Imported 'if_then_else' -a * c + a * d + b * c + b * d +a * c + (a * d + (b * c + b * d)) trans (Nat::distributel a b (c + d)) (trans (congr (congr2 Nat::add (Nat::distributer a c d)) (Nat::distributer b c d)) - (Nat::add_assoc (a * c + a * d) (b * c) (b * d))) + (Nat::add_assoc (a * c) (a * d) (b * c + b * d))) Proved: congr2_congr1 Proved: congr2_congr2 Proved: congr1_congr2 @@ -28,7 +28,7 @@ trans (congr (congr2 eq let κ::1 := congr2 (λ x : ℕ, eq ((λ x : ℕ, x + 10) x)) (trans (congr2 (ite (a > 0) b) (Nat::add_zeror b)) (if_a_a (a > 0) b)) in trans (congr κ::1 (congr2 (λ x : ℕ, x + 10) (if_a_a (a > 0) b))) (eq_id (b + 10)) -a * a + a * b + b * a + b * b +a * a + (a * b + (b * a + b * b)) ⊤ → ⊥ refl (⊤ → ⊥) ⊤ → ⊤ refl (⊤ → ⊤) ⊥ → ⊥ refl (⊥ → ⊥) diff --git a/tests/lean/simp8.lean b/tests/lean/simp8.lean new file mode 100644 index 0000000000..6088279c7d --- /dev/null +++ b/tests/lean/simp8.lean @@ -0,0 +1,10 @@ +variables a b c d e f : Nat +rewrite_set simple +add_rewrite Nat::add_assoc Nat::add_comm Nat::add_left_comm Nat::distributer Nat::distributel : simple +(* +local t = parse_lean("f + (c + f + d) + (e * (a + c) + (d + a))") +local t2, pr = simplify(t, "simple") +print(t) +print("====>") +print(t2) +*) diff --git a/tests/lean/simp8.lean.expected.out b/tests/lean/simp8.lean.expected.out new file mode 100644 index 0000000000..f88a808740 --- /dev/null +++ b/tests/lean/simp8.lean.expected.out @@ -0,0 +1,11 @@ + Set: pp::colors + Set: pp::unicode + Assumed: a + Assumed: b + Assumed: c + Assumed: d + Assumed: e + Assumed: f +f + (c + f + d) + (e * (a + c) + (d + a)) +====> +a + (c + (d + (d + (f + (f + (e * a + e * c)))))) diff --git a/tests/lean/using.lean.expected.out b/tests/lean/using.lean.expected.out index eb09767d43..70c5ef22bb 100644 --- a/tests/lean/using.lean.expected.out +++ b/tests/lean/using.lean.expected.out @@ -2,7 +2,7 @@ Set: pp::unicode Using: Nat 0 + 1 -Nat::add_assoc : ∀ a b c : ℕ, a + (b + c) = a + b + c +Nat::add_assoc : ∀ a b c : ℕ, a + b + c = a + (b + c) using.lean:7:6: error: unknown identifier 'add' Using: Nat 0 + 1