diff --git a/src/builtin/Nat.lean b/src/builtin/Nat.lean index 3509b2abf..577019274 100644 --- a/src/builtin/Nat.lean +++ b/src/builtin/Nat.lean @@ -41,135 +41,117 @@ 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) + calc 0 + (n + 1) = (0 + n) + 1 : PlusSucc 0 n + ... = n + 1 : { 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)))) +:= Induction (calc (a + 1) + 0 = a + 1 : PlusZero (a + 1) + ... = (a + 0) + 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) + calc (a + 1) + (n + 1) = ((a + 1) + n) + 1 : PlusSucc (a + 1) n + ... = ((a + n) + 1) + 1 : { Hi } + ... = (a + (n + 1)) + 1 : { show (a + n) + 1 = a + (n + 1), Symm (PlusSucc a n) }) 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) +:= Induction (calc a + 0 = a : PlusZero a + ... = 0 + a : Symm (ZeroPlus a)) (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) + calc a + (n + 1) = (a + n) + 1 : PlusSucc a n + ... = (n + a) + 1 : { Hi } + ... = (n + 1) + a : Symm (SuccPlus n a)) 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))) +:= Induction (calc 0 + (b + c) = b + c : ZeroPlus (b + c) + ... = (0 + 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) + calc (n + 1) + (b + c) = (n + (b + c)) + 1 : SuccPlus n (b + c) + ... = ((n + b) + c) + 1 : { Hi } + ... = ((n + b) + 1) + c : Symm (SuccPlus (n + b) c) + ... = ((n + 1) + b) + c : { show (n + b) + 1 = (n + 1) + b, Symm (SuccPlus n b) }) 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) + calc 0 * (n + 1) = (0 * n) + 0 : MulSucc 0 n + ... = 0 + 0 : { Hi } + ... = 0 : Trivial) 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)))) +:= Induction (calc (a + 1) * 0 = 0 : MulZero (a + 1) + ... = a * 0 : Symm (MulZero a) + ... = a * 0 + 0 : Symm (PlusZero (a * 0))) (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) + calc (a + 1) * (n + 1) = (a + 1) * n + (a + 1) : MulSucc (a + 1) n + ... = a * n + n + (a + 1) : { Hi } + ... = a * n + n + a + 1 : PlusAssoc (a * n + n) a 1 + ... = a * n + (n + a) + 1 : { show a * n + n + a = a * n + (n + a), Symm (PlusAssoc (a * n) n a) } + ... = a * n + (a + n) + 1 : { PlusComm n a } + ... = a * n + a + n + 1 : { PlusAssoc (a * n) a n } + ... = a * (n + 1) + n + 1 : { Symm (MulSucc a n) } + ... = a * (n + 1) + (n + 1) : Symm (PlusAssoc (a * (n + 1)) n 1)) 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) + calc 1 * (n + 1) = 1 * n + 1 : MulSucc 1 n + ... = n + 1 : { 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) + calc (n + 1) * 1 = n * 1 + 1 : SuccMul n 1 + ... = n + 1 : { Hi }) a. Theorem MulComm (a b : Nat) : a * b = b * a -:= Induction (show a * 0 = 0 * a, Trans (MulZero a) (Symm (ZeroMul a))) +:= Induction (calc a * 0 = 0 : MulZero a + ... = 0 * 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)) + calc a * (n + 1) = a * n + a : MulSucc a n + ... = n * a + a : { Hi } + ... = (n + 1) * a : Symm (SuccMul n a)) 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))) +:= Induction (calc 0 * (b + c) = 0 : ZeroMul (b + c) + ... = 0 + 0 : Trivial + ... = 0 * b + 0 : { Symm (ZeroMul b) } + ... = 0 * b + 0 * c : { Symm (ZeroMul c) }) (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) + calc (n + 1) * (b + c) = n * (b + c) + (b + c) : SuccMul n (b + c) + ... = n * b + n * c + (b + c) : { Hi } + ... = n * b + n * c + b + c : PlusAssoc (n * b + n * c) b c + ... = n * b + (n * c + b) + c : { Symm (PlusAssoc (n * b) (n * c) b) } + ... = n * b + (b + n * c) + c : { PlusComm (n * c) b } + ... = n * b + b + n * c + c : { PlusAssoc (n * b) b (n * c) } + ... = (n + 1) * b + n * c + c : { Symm (SuccMul n b) } + ... = (n + 1) * b + (n * c + c) : Symm (PlusAssoc ((n + 1) * b) (n * c) c) + ... = (n + 1) * b + (n + 1) * c : { Symm (SuccMul n c) }) 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). +:= calc (a + b) * c = c * (a + b) : MulComm (a + b) c + ... = c * a + c * b : Distribute c a b + ... = a * c + c * b : { MulComm c a } + ... = a * c + b * c : { 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))) +:= Induction (calc 0 * (b * c) = 0 : ZeroMul (b * c) + ... = 0 * c : Symm (ZeroMul c) + ... = (0 * b) * c : { Symm (ZeroMul b) }) (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) + calc (n + 1) * (b * c) = n * (b * c) + (b * c) : SuccMul n (b * c) + ... = n * b * c + (b * c) : { Hi } + ... = (n * b + b) * c : Symm (Distribute2 (n * b) b c) + ... = (n + 1) * b * c : { Symm (SuccMul n b) }) a. SetOpaque ge true. diff --git a/src/builtin/obj/Nat.olean b/src/builtin/obj/Nat.olean index b5e571a5f..44c6d2416 100644 Binary files a/src/builtin/obj/Nat.olean and b/src/builtin/obj/Nat.olean differ