Import kernel. Variable Nat : Type. Alias ℕ : Nat. Namespace Nat. Builtin numeral. Builtin add : Nat → Nat → Nat. Infixl 65 + : add. Builtin mul : Nat → Nat → Nat. Infixl 70 * : mul. Builtin le : Nat → Nat → Bool. Infix 50 <= : le. Infix 50 ≤ : le. Definition ge (a b : Nat) := b ≤ a. Infix 50 >= : ge. Infix 50 ≥ : ge. Definition lt (a b : Nat) := ¬ (a ≥ b). Infix 50 < : lt. Definition gt (a b : Nat) := ¬ (a ≤ b). Infix 50 > : gt. Definition id (a : Nat) := a. Notation 55 | _ | : id. Axiom SuccInj {a b : Nat} (H : a + 1 = b + 1) : a = b Axiom PlusZero (a : Nat) : a + 0 = a. Axiom PlusSucc (a b : Nat) : a + (b + 1) = (a + b) + 1. Axiom MulZero (a : Nat) : a * 0 = 0. Axiom MulSucc (a b : Nat) : a * (b + 1) = a * b + a. Axiom Induction {P : Nat → Bool} (Hb : P 0) (Hi : Π (n : Nat) (H : P n), P (n + 1)) (a : Nat) : P a. Theorem ZeroNeOne : 0 ≠ 1 := Trivial. Theorem ZeroPlus (a : Nat) : 0 + a = a := Induction (show 0 + 0 = 0, Trivial) (λ (n : Nat) (Hi : 0 + n = n), let L1 : 0 + (n + 1) = (0 + n) + 1 := PlusSucc 0 n in show 0 + (n + 1) = n + 1, Subst L1 Hi) a. Theorem SuccPlus (a b : Nat) : (a + 1) + b = (a + b) + 1 := Induction (show (a + 1) + 0 = (a + 0) + 1, (Subst (PlusZero (a + 1)) (Symm (PlusZero a)))) (λ (n : Nat) (Hi : (a + 1) + n = (a + n) + 1), let L1 : (a + 1) + (n + 1) = ((a + 1) + n) + 1 := PlusSucc (a + 1) n, L2 : (a + 1) + (n + 1) = ((a + n) + 1) + 1 := Subst L1 Hi, L3 : (a + n) + 1 = a + (n + 1) := Symm (PlusSucc a n) in show (a + 1) + (n + 1) = (a + (n + 1)) + 1, Subst L2 L3) b. Theorem PlusComm (a b : Nat) : a + b = b + a := Induction (show a + 0 = 0 + a, let L1 : a + 0 = a := PlusZero a, L2 : a = 0 + a := Symm (ZeroPlus a) in Trans L1 L2) (λ (n : Nat) (Hi : a + n = n + a), let L1 : a + (n + 1) = (a + n) + 1 := PlusSucc a n, L2 : a + (n + 1) = (n + a) + 1 := Subst L1 Hi, L3 : (n + a) + 1 = (n + 1) + a := Symm (SuccPlus n a) in show a + (n + 1) = (n + 1) + a, Trans L2 L3) b. Theorem PlusAssoc (a b c : Nat) : a + (b + c) = (a + b) + c := Induction (show 0 + (b + c) = (0 + b) + c, Subst (ZeroPlus (b + c)) (Symm (ZeroPlus b))) (λ (n : Nat) (Hi : n + (b + c) = (n + b) + c), let L1 : (n + 1) + (b + c) = (n + (b + c)) + 1 := SuccPlus n (b + c), L2 : (n + 1) + (b + c) = ((n + b) + c) + 1 := Subst L1 Hi, L3 : ((n + b) + 1) + c = ((n + b) + c) + 1 := SuccPlus (n + b) c, L4 : (n + b) + 1 = (n + 1) + b := Symm (SuccPlus n b), L5 : ((n + 1) + b) + c = ((n + b) + c) + 1 := Subst L3 L4, L6 : ((n + b) + c) + 1 = ((n + 1) + b) + c := Symm L5 in show (n + 1) + (b + c) = ((n + 1) + b) + c, Trans L2 L6) a. Theorem ZeroMul (a : Nat) : 0 * a = 0 := Induction (show 0 * 0 = 0, Trivial) (λ (n : Nat) (Hi : 0 * n = 0), let L1 : 0 * (n + 1) = (0 * n) + 0 := MulSucc 0 n, L2 : 0 * (n + 1) = 0 + 0 := Subst L1 Hi in show 0 * (n + 1) = 0, L2) a. Theorem SuccMul (a b : Nat) : (a + 1) * b = a * b + b := Induction (show (a + 1) * 0 = a * 0 + 0, Trans (MulZero (a + 1)) (Symm (Subst (PlusZero (a * 0)) (MulZero a)))) (λ (n : Nat) (Hi : (a + 1) * n = a * n + n), let L1 : (a + 1) * (n + 1) = (a + 1) * n + (a + 1) := MulSucc (a + 1) n, L2 : (a + 1) * (n + 1) = a * n + n + (a + 1) := Subst L1 Hi, L3 : a * n + n + (a + 1) = a * n + n + a + 1 := PlusAssoc (a * n + n) a 1, L4 : a * n + n + a = a * n + (n + a) := Symm (PlusAssoc (a * n) n a), L5 : a * n + n + (a + 1) = a * n + (n + a) + 1 := Subst L3 L4, L6 : a * n + n + (a + 1) = a * n + (a + n) + 1 := Subst L5 (PlusComm n a), L7 : a * n + (a + n) = a * n + a + n := PlusAssoc (a * n) a n, L8 : a * n + n + (a + 1) = a * n + a + n + 1 := Subst L6 L7, L9 : a * n + a = a * (n + 1) := Symm (MulSucc a n), L10 : a * n + n + (a + 1) = a * (n + 1) + n + 1 := Subst L8 L9, L11 : a * (n + 1) + n + 1 = a * (n + 1) + (n + 1) := Symm (PlusAssoc (a * (n + 1)) n 1) in show (a + 1) * (n + 1) = a * (n + 1) + (n + 1), Trans (Trans L2 L10) L11) b. Theorem OneMul (a : Nat) : 1 * a = a := Induction (show 1 * 0 = 0, Trivial) (λ (n : Nat) (Hi : 1 * n = n), let L1 : 1 * (n + 1) = 1 * n + 1 := MulSucc 1 n in show 1 * (n + 1) = n + 1, Subst L1 Hi) a. Theorem MulOne (a : Nat) : a * 1 = a := Induction (show 0 * 1 = 0, Trivial) (λ (n : Nat) (Hi : n * 1 = n), let L1 : (n + 1) * 1 = n * 1 + 1 := SuccMul n 1 in show (n + 1) * 1 = n + 1, Subst L1 Hi) a. Theorem MulComm (a b : Nat) : a * b = b * a := Induction (show a * 0 = 0 * a, Trans (MulZero a) (Symm (ZeroMul a))) (λ (n : Nat) (Hi : a * n = n * a), let L1 : a * (n + 1) = a * n + a := MulSucc a n, L2 : (n + 1) * a = n * a + a := SuccMul n a, L3 : (n + 1) * a = a * n + a := Subst L2 (Symm Hi) in show a * (n + 1) = (n + 1) * a, Trans L1 (Symm L3)) b. Theorem Distribute (a b c : Nat) : a * (b + c) = a * b + a * c := Induction (let L1 : 0 * (b + c) = 0 := ZeroMul (b + c), L2 : 0 * b + 0 * c = 0 + 0 := Subst (Subst (Refl (0 * b + 0 * c)) (ZeroMul b)) (ZeroMul c), L3 : 0 + 0 = 0 := Trivial in show 0 * (b + c) = 0 * b + 0 * c, Trans L1 (Symm (Trans L2 L3))) (λ (n : Nat) (Hi : n * (b + c) = n * b + n * c), let L1 : (n + 1) * (b + c) = n * (b + c) + (b + c) := SuccMul n (b + c), L2 : (n + 1) * (b + c) = n * b + n * c + (b + c) := Subst L1 Hi, L3 : n * b + n * c + (b + c) = n * b + n * c + b + c := PlusAssoc (n * b + n * c) b c, L4 : n * b + n * c + b = n * b + (n * c + b) := Symm (PlusAssoc (n * b) (n * c) b), L5 : n * b + n * c + b = n * b + (b + n * c) := Subst L4 (PlusComm (n * c) b), L6 : n * b + (b + n * c) = n * b + b + n * c := PlusAssoc (n * b) b (n * c), L7 : n * b + (b + n * c) = (n + 1) * b + n * c := Subst L6 (Symm (SuccMul n b)), L8 : n * b + n * c + b = (n + 1) * b + n * c := Trans L5 L7, L9 : n * b + n * c + (b + c) = (n + 1) * b + n * c + c := Subst L3 L8, L10 : (n + 1) * b + n * c + c = (n + 1) * b + (n * c + c) := Symm (PlusAssoc ((n + 1) * b) (n * c) c), L11 : (n + 1) * b + n * c + c = (n + 1) * b + (n + 1) * c := Subst L10 (Symm (SuccMul n c)), L12 : n * b + n * c + (b + c) = (n + 1) * b + (n + 1) * c := Trans L9 L11 in show (n + 1) * (b + c) = (n + 1) * b + (n + 1) * c, Trans L2 L12) a. Theorem Distribute2 (a b c : Nat) : (a + b) * c = a * c + b * c := let L1 : (a + b) * c = c * (a + b) := MulComm (a + b) c, L2 : c * (a + b) = c * a + c * b := Distribute c a b, L3 : (a + b) * c = c * a + c * b := Trans L1 L2 in Subst (Subst L3 (MulComm c a)) (MulComm c b). Theorem MulAssoc (a b c : Nat) : a * (b * c) = a * b * c := Induction (let L1 : 0 * (b * c) = 0 := ZeroMul (b * c), L2 : 0 * b * c = 0 * c := Subst (Refl (0 * b * c)) (ZeroMul b), L3 : 0 * c = 0 := ZeroMul c in show 0 * (b * c) = 0 * b * c, Trans L1 (Symm (Trans L2 L3))) (λ (n : Nat) (Hi : n * (b * c) = n * b * c), let L1 : (n + 1) * (b * c) = n * (b * c) + (b * c) := SuccMul n (b * c), L2 : (n + 1) * (b * c) = n * b * c + (b * c) := Subst L1 Hi, L3 : n * b * c + (b * c) = (n * b + b) * c := Symm (Distribute2 (n * b) b c), L4 : n * b * c + (b * c) = (n + 1) * b * c := Subst L3 (Symm (SuccMul n b)) in show (n + 1) * (b * c) = (n + 1) * b * c, Trans L2 L4) a. SetOpaque ge true. SetOpaque lt true. SetOpaque gt true. SetOpaque id true. EndNamespace.