refactor(library): minor reorg
This commit is contained in:
Leonardo de Moura
parent
c816b80855
commit
866eaae8ed
6 changed files with 323 additions and 307 deletions
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@ -9,7 +9,7 @@ This is a work-in-progress, and contains additions to other theories.
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-/
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import data.tuple
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definition bitvec (n : ℕ) := tuple bool n
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def bitvec (n : ℕ) := tuple bool n
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namespace bitvec
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@ -17,15 +17,15 @@ open nat
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open tuple
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-- Create a zero bitvector
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definition zero { n : ℕ } : bitvec n := tuple.repeat ff
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def zero {n : ℕ} : bitvec n := tuple.repeat ff
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-- Create a bitvector with the constant one.
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definition one {n : ℕ} : bitvec (succ n) := tuple.append (tuple.repeat ff : bitvec n) [ tt ]
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def one {n : ℕ} : bitvec (succ n) := tuple.append (tuple.repeat ff : bitvec n) [ tt ]
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section shift
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-- shift left
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definition shl {n:ℕ} : bitvec n → ℕ → bitvec n
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def shl {n : ℕ} : bitvec n → ℕ → bitvec n
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| x i :=
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if le : i ≤ n then
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let eq := nat.sub_add_cancel le in
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@ -34,7 +34,7 @@ section shift
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zero
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-- unsigned shift right
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definition ushr {n:ℕ} : bitvec n → ℕ → bitvec n
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def ushr {n : ℕ} : bitvec n → ℕ → bitvec n
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| x i :=
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if le : i ≤ n then
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let y : bitvec (n-i) := @firstn _ _ _ (sub_le n i) x in
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@ -45,7 +45,7 @@ section shift
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zero
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-- signed shift right
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definition sshr {m:ℕ} : bitvec (succ m) → ℕ → bitvec (succ m)
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def sshr {m : ℕ} : bitvec (succ m) → ℕ → bitvec (succ m)
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| x i :=
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let n := succ m in
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if le : i ≤ n then
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@ -60,48 +60,47 @@ section shift
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end shift
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section bitwise
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variable { n : ℕ }
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variable {n : ℕ}
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definition not : bitvec n → bitvec n := map bnot
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definition and : bitvec n → bitvec n → bitvec n := map₂ band
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definition or : bitvec n → bitvec n → bitvec n := map₂ bor
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definition xor : bitvec n → bitvec n → bitvec n := map₂ bxor
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def not : bitvec n → bitvec n := map bnot
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def and : bitvec n → bitvec n → bitvec n := map₂ band
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def or : bitvec n → bitvec n → bitvec n := map₂ bor
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def xor : bitvec n → bitvec n → bitvec n := map₂ bxor
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end bitwise
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section arith
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variable {n : ℕ}
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variable { n : ℕ }
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protected definition xor3 (x:bool) (y:bool) (c:bool) := bxor (bxor x y) c
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protected definition carry (x:bool) (y:bool) (c:bool) :=
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protected def xor3 (x:bool) (y:bool) (c:bool) := bxor (bxor x y) c
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protected def carry (x:bool) (y:bool) (c:bool) :=
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x && y || x && c || y && c
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definition neg : bitvec n → bitvec n
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def neg : bitvec n → bitvec n
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| x :=
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let f := λy c, (y || c, bxor y c) in
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prod.snd (map_accumr f x ff)
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-- Add with carry (no overflow)
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definition adc : bitvec n → bitvec n → bool → bitvec (n+1)
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def adc : bitvec n → bitvec n → bool → bitvec (n+1)
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| x y c :=
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let f := λx y c, (bitvec.carry x y c, bitvec.xor3 x y c) in
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let z := tuple.map_accumr₂ f x y c in
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prod.fst z :: prod.snd z
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definition add : bitvec n → bitvec n → bitvec n
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def add : bitvec n → bitvec n → bitvec n
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| x y := tail (adc x y ff)
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protected definition borrow (x:bool) (y:bool) (b:bool) :=
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protected def borrow (x:bool) (y:bool) (b:bool) :=
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bnot x && y || bnot x && b || y && b
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-- Subtract with borrow
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definition sbb : bitvec n → bitvec n → bool → bool × bitvec n
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def sbb : bitvec n → bitvec n → bool → bool × bitvec n
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| x y b :=
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let f := λx y c, (bitvec.borrow x y c, bitvec.xor3 x y c) in
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tuple.map_accumr₂ f x y b
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definition sub : bitvec n → bitvec n → bitvec n
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def sub : bitvec n → bitvec n → bitvec n
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| x y := prod.snd (sbb x y ff)
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instance : has_zero (bitvec n) := has_zero.mk zero
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@ -110,7 +109,7 @@ section arith
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instance : has_sub (bitvec n) := has_sub.mk sub
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instance : has_neg (bitvec n) := has_neg.mk neg
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definition mul : bitvec n → bitvec n → bitvec n
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def mul : bitvec n → bitvec n → bitvec n
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| x y :=
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let f := λr b, cond b (r + r + y) (r + r) in
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list.foldl f 0 (to_list x)
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@ -121,26 +120,26 @@ end arith
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section comparison
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variable {n : ℕ}
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definition uborrow : bitvec n → bitvec n → bool := λx y, prod.fst (sbb x y ff)
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def uborrow : bitvec n → bitvec n → bool := λx y, prod.fst (sbb x y ff)
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definition ult (x y : bitvec n) : Prop := uborrow x y = tt
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definition ugt (x y : bitvec n) : Prop := ult y x
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def ult (x y : bitvec n) : Prop := uborrow x y = tt
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def ugt (x y : bitvec n) : Prop := ult y x
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definition ule (x y : bitvec n) : Prop := ¬ (ult y x)
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definition uge : bitvec n → bitvec n → Prop := λx y, ule y x
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def ule (x y : bitvec n) : Prop := ¬ (ult y x)
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def uge : bitvec n → bitvec n → Prop := λx y, ule y x
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definition sborrow : bitvec (succ n) → bitvec (succ n) → bool := λx y,
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def sborrow : bitvec (succ n) → bitvec (succ n) → bool := λx y,
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bool.cases_on
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(head x)
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(bool.cases_on (head y) (uborrow (tail x) (tail y)) tt)
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(bool.cases_on (head y) ff (uborrow (tail x) (tail y)))
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definition slt : bitvec (succ n) → bitvec (succ n) → Prop := λx y,
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def slt : bitvec (succ n) → bitvec (succ n) → Prop := λx y,
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sborrow x y = tt
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definition sgt : bitvec (succ n) → bitvec (succ n) → Prop := λx y, slt y x
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definition sle : bitvec (succ n) → bitvec (succ n) → Prop := λx y, ¬ (slt y x)
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definition sge : bitvec (succ n) → bitvec (succ n) → Prop := λx y, sle y x
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def sgt : bitvec (succ n) → bitvec (succ n) → Prop := λx y, slt y x
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def sle : bitvec (succ n) → bitvec (succ n) → Prop := λx y, ¬ (slt y x)
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def sge : bitvec (succ n) → bitvec (succ n) → Prop := λx y, sle y x
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end comparison
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@ -148,10 +147,9 @@ section from_bitvec
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variable {α : Type}
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-- Convert a bitvector to another number.
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definition from_bitvec [p : has_add α] [q0 : has_zero α] [q1 : has_one α] {n:nat} (v:bitvec n) : α :=
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let f : α → bool → α := λr b, cond b (r + r + 1) (r + r) in
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list.foldl f (0 : α) (to_list v)
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def from_bitvec [has_add α] [has_zero α] [has_one α] {n : nat} (v : bitvec n) : α :=
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let f : α → bool → α := λr b, cond b (r + r + 1) (r + r) in
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list.foldl f (0 : α) (to_list v)
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end from_bitvec
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end bitvec
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@ -12,4 +12,4 @@ import init.monad init.option init.state init.fin init.list init.char init.strin
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import init.monad_combinators init.set
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import init.timeit init.trace init.unsigned init.ordering init.list_classes init.coe
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import init.wf init.nat_div init.meta init.instances init.breakpoint
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import init.sigma_lex init.id_locked init.order init.algebra
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import init.sigma_lex init.id_locked init.order init.algebra init.nat_lemmas
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@ -18,6 +18,12 @@ lemma eq_of_veq : ∀ {i j : fin n}, (val i) = (val j) → i = j
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lemma veq_of_eq : ∀ {i j : fin n}, i = j → (val i) = (val j)
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| ⟨iv, ilt⟩ .⟨iv, ilt⟩ rfl := rfl
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lemma ne_of_vne {i j : fin n} (h : val i ≠ val j) : i ≠ j :=
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λ h', absurd (veq_of_eq h') h
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lemma vne_of_ne {i j : fin n} (h : i ≠ j) : val i ≠ val j :=
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λ h', absurd (eq_of_veq h') h
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end fin
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open fin
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@ -10,77 +10,6 @@ import init.order
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notation `ℕ` := nat
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namespace nat
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protected lemma zero_add : ∀ n : ℕ, 0 + n = n
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| 0 := rfl
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| (n+1) := congr_arg succ (zero_add n)
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lemma succ_add : ∀ n m : ℕ, (succ n) + m = succ (n + m)
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| n 0 := rfl
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| n (m+1) := congr_arg succ (succ_add n m)
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protected lemma add_comm : ∀ n m : ℕ, n + m = m + n
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| n 0 := eq.symm (nat.zero_add n)
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| n (m+1) :=
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suffices succ (n + m) = succ (m + n), from
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eq.symm (succ_add m n) ▸ this,
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congr_arg succ (add_comm n m)
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protected lemma bit0_succ_eq (n : ℕ) : bit0 (succ n) = succ (succ (bit0 n)) :=
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show succ (succ n + n) = succ (succ (n + n)), from
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congr_arg succ (succ_add n n)
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protected lemma bit1_eq_succ_bit0 (n : ℕ) : bit1 n = succ (bit0 n) :=
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rfl
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protected lemma bit1_succ_eq (n : ℕ) : bit1 (succ n) = succ (succ (bit1 n)) :=
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eq.trans (nat.bit1_eq_succ_bit0 (succ n)) (congr_arg succ (nat.bit0_succ_eq n))
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lemma succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
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assume h, nat.no_confusion h
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lemma succ_ne_self : ∀ n : ℕ, succ n ≠ n
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| 0 h := absurd h (nat.succ_ne_zero 0)
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| (n+1) h := succ_ne_self n (nat.no_confusion h (λ h, h))
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protected lemma one_ne_zero : 1 ≠ (0 : ℕ) :=
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assume h, nat.no_confusion h
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protected lemma bit0_ne_zero : ∀ n : ℕ, n ≠ 0 → bit0 n ≠ 0
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| 0 h := absurd rfl h
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| (n+1) h := nat.succ_ne_zero _
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protected lemma bit1_ne_zero (n : ℕ) : bit1 n ≠ 0 :=
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show succ (n + n) ≠ 0, from
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succ_ne_zero (n + n)
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protected lemma bit1_ne_one : ∀ n : ℕ, n ≠ 0 → bit1 n ≠ 1
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| 0 h h1 := absurd rfl h
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| (n+1) h h1 := nat.no_confusion h1 (λ h2, absurd h2 (nat.succ_ne_zero _))
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protected lemma bit0_ne_one : ∀ n : ℕ, bit0 n ≠ 1
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| 0 h := absurd h (ne.symm nat.one_ne_zero)
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| (n+1) h :=
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have h1 : succ (succ (n + n)) = 1, from succ_add n n ▸ h,
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nat.no_confusion h1
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(λ h2, absurd h2 (succ_ne_zero (n + n)))
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protected lemma add_self_ne_one : ∀ (n : ℕ), n + n ≠ 1
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| 0 h := nat.no_confusion h
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| (n+1) h :=
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have h1 : succ (succ (n + n)) = 1, from succ_add n n ▸ h,
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nat.no_confusion h1 (λ h2, absurd h2 (nat.succ_ne_zero (n + n)))
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protected lemma bit1_ne_bit0 : ∀ (n m : ℕ), bit1 n ≠ bit0 m
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| 0 m h := absurd h (ne.symm (nat.add_self_ne_one m))
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| (n+1) 0 h :=
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have h1 : succ (bit0 (succ n)) = 0, from h,
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absurd h1 (nat.succ_ne_zero _)
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| (n+1) (m+1) h :=
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have h1 : succ (succ (bit1 n)) = succ (succ (bit0 m)), from
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nat.bit0_succ_eq m ▸ nat.bit1_succ_eq n ▸ h,
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have h2 : bit1 n = bit0 m, from
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nat.no_confusion h1 (λ h2', nat.no_confusion h2' (λ h2'', h2'')),
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absurd h2 (bit1_ne_bit0 n m)
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inductive le (a : ℕ) : ℕ → Prop
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| nat_refl : le a -- use nat_refl to avoid overloading le.refl
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@ -89,10 +18,6 @@ namespace nat
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instance : has_le ℕ :=
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⟨nat.le⟩
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attribute [refl]
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protected def le_refl : ∀ a : ℕ, a ≤ a :=
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le.nat_refl
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@[reducible] protected def lt (n m : ℕ) := succ n ≤ m
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instance : has_lt ℕ :=
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@ -126,149 +51,42 @@ namespace nat
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| is_false xney := is_false (λ h, nat.no_confusion h (λ xeqy, absurd xeqy xney))
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end
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def {u} repeat {α : Type u} (f : ℕ → α → α) : ℕ → α → α
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| 0 a := a
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| (succ n) a := f n (repeat n a)
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instance : inhabited ℕ :=
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⟨nat.zero⟩
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/- properties of inequality -/
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protected lemma le_of_eq {n m : ℕ} (p : n = m) : n ≤ m :=
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p ▸ le.nat_refl n
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@[refl] protected def le_refl : ∀ a : ℕ, a ≤ a :=
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le.nat_refl
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lemma le_succ (n : ℕ) : n ≤ succ n :=
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le.step (nat.le_refl n)
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lemma pred_le : ∀ (n : ℕ), pred n ≤ n
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| 0 := le.nat_refl 0
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| (succ a) := le.step (le.nat_refl a)
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attribute [simp]
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lemma le_succ_iff_true (n : ℕ) : n ≤ succ n ↔ true :=
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iff_true_intro (le_succ n)
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attribute [simp]
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lemma pred_le_iff_true (n : ℕ) : pred n ≤ n ↔ true :=
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iff_true_intro (pred_le n)
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protected lemma le_trans {n m k : ℕ} (h1 : n ≤ m) : m ≤ k → n ≤ k :=
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le.rec h1 (λ p h2, le.step)
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lemma le_succ_of_le {n m : ℕ} (h : n ≤ m) : n ≤ succ m :=
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nat.le_trans h (le_succ m)
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lemma le_of_succ_le {n m : ℕ} (h : succ n ≤ m) : n ≤ m :=
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nat.le_trans (le_succ n) h
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protected lemma le_of_lt {n m : ℕ} (h : n < m) : n ≤ m :=
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le_of_succ_le h
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lemma succ_le_succ {n m : ℕ} : n ≤ m → succ n ≤ succ m :=
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λ h, le.rec (nat.le_refl (succ n)) (λ a b, le.step) h
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lemma zero_le : ∀ (n : ℕ), 0 ≤ n
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| 0 := nat.le_refl 0
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| (n+1) := le.step (zero_le n)
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lemma zero_lt_succ (n : ℕ) : 0 < succ n :=
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succ_le_succ (zero_le n)
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lemma not_succ_le_zero : ∀ (n : ℕ), succ n ≤ 0 → false
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.
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lemma not_lt_zero (a : ℕ) : ¬ a < 0 := not_succ_le_zero a
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lemma pred_le_pred {n m : ℕ} : n ≤ m → pred n ≤ pred m :=
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λ h, le.rec (nat.le_refl (pred n)) (λ n, nat.rec (λ a b, b) (λ a b c, le.step) n) h
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lemma le_of_succ_le_succ {n m : ℕ} : succ n ≤ succ m → n ≤ m :=
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pred_le_pred
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lemma le_succ_of_pred_le {n m : ℕ} : pred n ≤ m → n ≤ succ m :=
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nat.cases_on n le.step (λ a, succ_le_succ)
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lemma not_succ_le_zero : ∀ (n : ℕ), succ n ≤ 0 → false
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.
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lemma succ_le_zero_iff_false (n : ℕ) : succ n ≤ 0 ↔ false :=
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iff_false_intro (not_succ_le_zero n)
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lemma not_succ_le_self : ∀ n : ℕ, ¬succ n ≤ n :=
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λ n, nat.rec (not_succ_le_zero 0) (λ a b c, b (le_of_succ_le_succ c)) n
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attribute [simp]
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lemma succ_le_self_iff_false (n : ℕ) : succ n ≤ n ↔ false :=
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iff_false_intro (not_succ_le_self n)
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lemma zero_le : ∀ (n : ℕ), 0 ≤ n
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| 0 := nat.le_refl 0
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| (n+1) := le.step (zero_le n)
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attribute [simp]
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lemma zero_le_iff_true (n : ℕ) : 0 ≤ n ↔ true :=
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iff_true_intro (zero_le n)
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protected lemma one_le_bit1 (n : ℕ) : 1 ≤ bit1 n :=
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show 1 ≤ succ (bit0 n), from
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succ_le_succ (zero_le (bit0 n))
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protected lemma one_le_bit0 : ∀ (n : ℕ), n ≠ 0 → 1 ≤ bit0 n
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| 0 h := absurd rfl h
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| (n+1) h :=
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suffices 1 ≤ succ (succ (bit0 n)), from
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eq.symm (nat.bit0_succ_eq n) ▸ this,
|
||||
succ_le_succ (zero_le (succ (bit0 n)))
|
||||
|
||||
def lt.step {n m : ℕ} : n < m → n < succ m := le.step
|
||||
|
||||
lemma zero_lt_succ (n : ℕ) : 0 < succ n :=
|
||||
succ_le_succ (zero_le n)
|
||||
|
||||
attribute [simp]
|
||||
lemma zero_lt_succ_iff_true (n : ℕ) : 0 < succ n ↔ true :=
|
||||
iff_true_intro (zero_lt_succ n)
|
||||
|
||||
protected lemma lt_trans {n m k : ℕ} (h₁ : n < m) : m < k → n < k :=
|
||||
nat.le_trans (le.step h₁)
|
||||
|
||||
protected lemma lt_of_le_of_lt {n m k : ℕ} (h₁ : n ≤ m) : m < k → n < k :=
|
||||
nat.le_trans (succ_le_succ h₁)
|
||||
|
||||
protected lemma lt_of_lt_of_le {n m k : ℕ} : n < m → m ≤ k → n < k := nat.le_trans
|
||||
|
||||
protected lemma lt_irrefl (n : ℕ) : ¬n < n :=
|
||||
not_succ_le_self n
|
||||
|
||||
lemma lt_self_iff_false (n : ℕ) : n < n ↔ false :=
|
||||
iff_false_intro (λ h, absurd h (nat.lt_irrefl n))
|
||||
|
||||
lemma self_lt_succ (n : ℕ) : n < succ n := nat.le_refl (succ n)
|
||||
|
||||
attribute [simp]
|
||||
lemma self_lt_succ_iff_true (n : ℕ) : n < succ n ↔ true :=
|
||||
iff_true_intro (self_lt_succ n)
|
||||
|
||||
def lt.base (n : ℕ) : n < succ n := nat.le_refl (succ n)
|
||||
|
||||
lemma le_lt_antisymm {n m : ℕ} (h₁ : n ≤ m) (h₂ : m < n) : false :=
|
||||
nat.lt_irrefl n (nat.lt_of_le_of_lt h₁ h₂)
|
||||
|
||||
protected lemma le_antisymm {n m : ℕ} (h₁ : n ≤ m) : m ≤ n → n = m :=
|
||||
le.cases_on h₁ (λ a, rfl) (λ a b c, absurd (nat.lt_of_le_of_lt b c) (nat.lt_irrefl n))
|
||||
|
||||
instance : weak_order ℕ :=
|
||||
⟨ @nat.le, @nat.le_refl, @nat.le_trans, @nat.le_antisymm ⟩
|
||||
|
||||
lemma lt_le_antisymm {n m : ℕ} (h₁ : n < m) (h₂ : m ≤ n) : false :=
|
||||
le_lt_antisymm h₂ h₁
|
||||
|
||||
protected lemma nat.lt_asymm {n m : ℕ} (h₁ : n < m) : ¬ m < n :=
|
||||
le_lt_antisymm (nat.le_of_lt h₁)
|
||||
|
||||
lemma not_lt_zero (a : ℕ) : ¬ a < 0 := not_succ_le_zero a
|
||||
|
||||
attribute [simp]
|
||||
lemma lt_zero_iff_false (a : ℕ) : a < 0 ↔ false :=
|
||||
iff_false_intro (not_lt_zero a)
|
||||
|
||||
protected lemma eq_or_lt_of_le {a b : ℕ} (h : a ≤ b) : a = b ∨ a < b :=
|
||||
le.cases_on h (or.inl rfl) (λ n h, or.inr (succ_le_succ h))
|
||||
|
||||
protected lemma le_of_eq_or_lt {a b : ℕ} (h : a = b ∨ a < b) : a ≤ b :=
|
||||
or.elim h nat.le_of_eq nat.le_of_lt
|
||||
|
||||
lemma succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
|
||||
succ_le_succ
|
||||
|
||||
lemma lt_of_succ_lt {a b : ℕ} : succ a < b → a < b :=
|
||||
le_of_succ_le
|
||||
|
||||
lemma lt_of_succ_lt_succ {a b : ℕ} : succ a < succ b → a < b :=
|
||||
le_of_succ_le_succ
|
||||
|
||||
instance decidable_le : ∀ a b : ℕ, decidable (a ≤ b)
|
||||
| 0 b := is_true (zero_le b)
|
||||
| (a+1) 0 := is_false (not_succ_le_zero a)
|
||||
|
|
@ -281,65 +99,33 @@ namespace nat
|
|||
instance decidable_lt : ∀ a b : ℕ, decidable (a < b) :=
|
||||
λ a b, nat.decidable_le (succ a) b
|
||||
|
||||
protected lemma lt_or_ge : ∀ (a b : ℕ), a < b ∨ a ≥ b
|
||||
| a 0 := or.inr (zero_le a)
|
||||
| a (b+1) :=
|
||||
match lt_or_ge a b with
|
||||
| or.inl h := or.inl (le_succ_of_le h)
|
||||
| or.inr h :=
|
||||
match nat.eq_or_lt_of_le h with
|
||||
| or.inl h1 := or.inl (h1 ▸ self_lt_succ b)
|
||||
| or.inr h1 := or.inr h1
|
||||
end
|
||||
end
|
||||
|
||||
protected def {u} lt_ge_by_cases {a b : ℕ} {C : Type u} (h₁ : a < b → C) (h₂ : a ≥ b → C) : C :=
|
||||
decidable.by_cases h₁ (λ h, h₂ (or.elim (nat.lt_or_ge a b) (λ a, absurd a h) (λ a, a)))
|
||||
|
||||
protected def {u} lt_by_cases {a b : ℕ} {C : Type u} (h₁ : a < b → C) (h₂ : a = b → C)
|
||||
(h₃ : b < a → C) : C :=
|
||||
nat.lt_ge_by_cases h₁ (λ h₁,
|
||||
nat.lt_ge_by_cases h₃ (λ h, h₂ (nat.le_antisymm h h₁)))
|
||||
|
||||
protected lemma lt_trichotomy (a b : ℕ) : a < b ∨ a = b ∨ b < a :=
|
||||
nat.lt_by_cases (λ h, or.inl h) (λ h, or.inr (or.inl h)) (λ h, or.inr (or.inr h))
|
||||
|
||||
protected lemma eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ∨ b < a :=
|
||||
or.elim (nat.lt_trichotomy a b)
|
||||
(λ hlt, absurd hlt hnlt)
|
||||
(λ h, h)
|
||||
protected lemma eq_or_lt_of_le {a b : ℕ} (h : a ≤ b) : a = b ∨ a < b :=
|
||||
le.cases_on h (or.inl rfl) (λ n h, or.inr (succ_le_succ h))
|
||||
|
||||
lemma lt_succ_of_le {a b : ℕ} : a ≤ b → a < succ b :=
|
||||
succ_le_succ
|
||||
|
||||
lemma lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
|
||||
|
||||
lemma succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
|
||||
|
||||
attribute [simp]
|
||||
lemma succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
|
||||
@[simp] lemma succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
|
||||
nat.rec_on b
|
||||
(show succ a - succ zero = a - zero, from (eq.refl (succ a - succ zero)))
|
||||
(λ b, congr_arg pred)
|
||||
|
||||
lemma sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
|
||||
eq.symm (succ_sub_succ_eq_sub a b)
|
||||
lemma not_succ_le_self : ∀ n : ℕ, ¬succ n ≤ n :=
|
||||
λ n, nat.rec (not_succ_le_zero 0) (λ a b c, b (le_of_succ_le_succ c)) n
|
||||
|
||||
attribute [simp]
|
||||
lemma zero_sub_eq_zero : ∀ a : ℕ, 0 - a = 0
|
||||
| 0 := rfl
|
||||
| (a+1) := congr_arg pred (zero_sub_eq_zero a)
|
||||
protected lemma lt_irrefl (n : ℕ) : ¬n < n :=
|
||||
not_succ_le_self n
|
||||
|
||||
lemma zero_eq_zero_sub (a : ℕ) : 0 = 0 - a :=
|
||||
eq.symm (zero_sub_eq_zero a)
|
||||
protected lemma le_trans {n m k : ℕ} (h1 : n ≤ m) : m ≤ k → n ≤ k :=
|
||||
le.rec h1 (λ p h2, le.step)
|
||||
|
||||
lemma pred_le : ∀ (n : ℕ), pred n ≤ n
|
||||
| 0 := le.nat_refl 0
|
||||
| (succ a) := le.step (le.nat_refl a)
|
||||
|
||||
lemma sub_le (a b : ℕ) : a - b ≤ a :=
|
||||
nat.rec_on b (nat.le_refl (a - 0)) (λ b₁, nat.le_trans (pred_le (a - b₁)))
|
||||
|
||||
attribute [simp]
|
||||
lemma sub_le_iff_true (a b : ℕ) : a - b ≤ a ↔ true :=
|
||||
iff_true_intro (sub_le a b)
|
||||
|
||||
lemma sub_lt : ∀ {a b : ℕ}, 0 < a → 0 < b → a - b < a
|
||||
| 0 b h1 h2 := absurd h1 (nat.lt_irrefl 0)
|
||||
| (a+1) 0 h1 h2 := absurd h2 (nat.lt_irrefl 0)
|
||||
|
|
@ -348,24 +134,7 @@ namespace nat
|
|||
show a - b < succ a, from
|
||||
lt_succ_of_le (sub_le a b)
|
||||
|
||||
lemma sub_lt_succ (a b : ℕ) : a - b < succ a :=
|
||||
lt_succ_of_le (sub_le a b)
|
||||
protected lemma lt_of_lt_of_le {n m k : ℕ} : n < m → m ≤ k → n < k :=
|
||||
nat.le_trans
|
||||
|
||||
attribute [simp]
|
||||
lemma sub_lt_succ_iff_true (a b : ℕ) : a - b < succ a ↔ true :=
|
||||
iff_true_intro (sub_lt_succ a b)
|
||||
|
||||
lemma le_add_right : ∀ (n k : ℕ), n ≤ n + k
|
||||
| n 0 := nat.le_refl n
|
||||
| n (k+1) := le_succ_of_le (le_add_right n k)
|
||||
|
||||
lemma le_add_left (n m : ℕ): n ≤ m + n :=
|
||||
nat.add_comm n m ▸ le_add_right n m
|
||||
|
||||
def {u} repeat {α : Type u} (f : ℕ → α → α) : ℕ → α → α
|
||||
| 0 a := a
|
||||
| (succ n) a := f n (repeat n a)
|
||||
|
||||
instance : inhabited ℕ :=
|
||||
⟨nat.zero⟩
|
||||
end nat
|
||||
|
|
|
|||
243
library/init/nat_lemmas.lean
Normal file
243
library/init/nat_lemmas.lean
Normal file
|
|
@ -0,0 +1,243 @@
|
|||
/-
|
||||
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Leonardo de Moura
|
||||
-/
|
||||
prelude
|
||||
import init.nat init.meta
|
||||
|
||||
namespace nat
|
||||
|
||||
protected lemma zero_add : ∀ n : ℕ, 0 + n = n
|
||||
| 0 := rfl
|
||||
| (n+1) := congr_arg succ (zero_add n)
|
||||
|
||||
lemma succ_add : ∀ n m : ℕ, (succ n) + m = succ (n + m)
|
||||
| n 0 := rfl
|
||||
| n (m+1) := congr_arg succ (succ_add n m)
|
||||
|
||||
protected lemma add_comm : ∀ n m : ℕ, n + m = m + n
|
||||
| n 0 := eq.symm (nat.zero_add n)
|
||||
| n (m+1) :=
|
||||
suffices succ (n + m) = succ (m + n), from
|
||||
eq.symm (succ_add m n) ▸ this,
|
||||
congr_arg succ (add_comm n m)
|
||||
|
||||
protected lemma bit0_succ_eq (n : ℕ) : bit0 (succ n) = succ (succ (bit0 n)) :=
|
||||
show succ (succ n + n) = succ (succ (n + n)), from
|
||||
congr_arg succ (succ_add n n)
|
||||
|
||||
protected lemma bit1_eq_succ_bit0 (n : ℕ) : bit1 n = succ (bit0 n) :=
|
||||
rfl
|
||||
|
||||
protected lemma bit1_succ_eq (n : ℕ) : bit1 (succ n) = succ (succ (bit1 n)) :=
|
||||
eq.trans (nat.bit1_eq_succ_bit0 (succ n)) (congr_arg succ (nat.bit0_succ_eq n))
|
||||
|
||||
lemma succ_ne_zero (n : ℕ) : succ n ≠ 0 :=
|
||||
assume h, nat.no_confusion h
|
||||
|
||||
lemma succ_ne_self : ∀ n : ℕ, succ n ≠ n
|
||||
| 0 h := absurd h (nat.succ_ne_zero 0)
|
||||
| (n+1) h := succ_ne_self n (nat.no_confusion h (λ h, h))
|
||||
|
||||
protected lemma one_ne_zero : 1 ≠ (0 : ℕ) :=
|
||||
assume h, nat.no_confusion h
|
||||
|
||||
protected lemma bit0_ne_zero : ∀ n : ℕ, n ≠ 0 → bit0 n ≠ 0
|
||||
| 0 h := absurd rfl h
|
||||
| (n+1) h := nat.succ_ne_zero _
|
||||
|
||||
protected lemma bit1_ne_zero (n : ℕ) : bit1 n ≠ 0 :=
|
||||
show succ (n + n) ≠ 0, from
|
||||
succ_ne_zero (n + n)
|
||||
|
||||
protected lemma bit1_ne_one : ∀ n : ℕ, n ≠ 0 → bit1 n ≠ 1
|
||||
| 0 h h1 := absurd rfl h
|
||||
| (n+1) h h1 := nat.no_confusion h1 (λ h2, absurd h2 (nat.succ_ne_zero _))
|
||||
|
||||
protected lemma bit0_ne_one : ∀ n : ℕ, bit0 n ≠ 1
|
||||
| 0 h := absurd h (ne.symm nat.one_ne_zero)
|
||||
| (n+1) h :=
|
||||
have h1 : succ (succ (n + n)) = 1, from succ_add n n ▸ h,
|
||||
nat.no_confusion h1
|
||||
(λ h2, absurd h2 (succ_ne_zero (n + n)))
|
||||
|
||||
protected lemma add_self_ne_one : ∀ (n : ℕ), n + n ≠ 1
|
||||
| 0 h := nat.no_confusion h
|
||||
| (n+1) h :=
|
||||
have h1 : succ (succ (n + n)) = 1, from succ_add n n ▸ h,
|
||||
nat.no_confusion h1 (λ h2, absurd h2 (nat.succ_ne_zero (n + n)))
|
||||
|
||||
protected lemma bit1_ne_bit0 : ∀ (n m : ℕ), bit1 n ≠ bit0 m
|
||||
| 0 m h := absurd h (ne.symm (nat.add_self_ne_one m))
|
||||
| (n+1) 0 h :=
|
||||
have h1 : succ (bit0 (succ n)) = 0, from h,
|
||||
absurd h1 (nat.succ_ne_zero _)
|
||||
| (n+1) (m+1) h :=
|
||||
have h1 : succ (succ (bit1 n)) = succ (succ (bit0 m)), from
|
||||
nat.bit0_succ_eq m ▸ nat.bit1_succ_eq n ▸ h,
|
||||
have h2 : bit1 n = bit0 m, from
|
||||
nat.no_confusion h1 (λ h2', nat.no_confusion h2' (λ h2'', h2'')),
|
||||
absurd h2 (bit1_ne_bit0 n m)
|
||||
|
||||
/- properties of inequality -/
|
||||
|
||||
protected lemma le_of_eq {n m : ℕ} (p : n = m) : n ≤ m :=
|
||||
p ▸ le.nat_refl n
|
||||
|
||||
@[simp] lemma le_succ_iff_true (n : ℕ) : n ≤ succ n ↔ true :=
|
||||
iff_true_intro (le_succ n)
|
||||
|
||||
@[simp] lemma pred_le_iff_true (n : ℕ) : pred n ≤ n ↔ true :=
|
||||
iff_true_intro (pred_le n)
|
||||
|
||||
|
||||
lemma le_succ_of_le {n m : ℕ} (h : n ≤ m) : n ≤ succ m :=
|
||||
nat.le_trans h (le_succ m)
|
||||
|
||||
lemma le_of_succ_le {n m : ℕ} (h : succ n ≤ m) : n ≤ m :=
|
||||
nat.le_trans (le_succ n) h
|
||||
|
||||
protected lemma le_of_lt {n m : ℕ} (h : n < m) : n ≤ m :=
|
||||
le_of_succ_le h
|
||||
|
||||
|
||||
lemma le_succ_of_pred_le {n m : ℕ} : pred n ≤ m → n ≤ succ m :=
|
||||
nat.cases_on n le.step (λ a, succ_le_succ)
|
||||
|
||||
lemma succ_le_zero_iff_false (n : ℕ) : succ n ≤ 0 ↔ false :=
|
||||
iff_false_intro (not_succ_le_zero n)
|
||||
|
||||
|
||||
@[simp] lemma succ_le_self_iff_false (n : ℕ) : succ n ≤ n ↔ false :=
|
||||
iff_false_intro (not_succ_le_self n)
|
||||
|
||||
@[simp] lemma zero_le_iff_true (n : ℕ) : 0 ≤ n ↔ true :=
|
||||
iff_true_intro (zero_le n)
|
||||
|
||||
protected lemma one_le_bit1 (n : ℕ) : 1 ≤ bit1 n :=
|
||||
show 1 ≤ succ (bit0 n), from
|
||||
succ_le_succ (zero_le (bit0 n))
|
||||
|
||||
protected lemma one_le_bit0 : ∀ (n : ℕ), n ≠ 0 → 1 ≤ bit0 n
|
||||
| 0 h := absurd rfl h
|
||||
| (n+1) h :=
|
||||
suffices 1 ≤ succ (succ (bit0 n)), from
|
||||
eq.symm (nat.bit0_succ_eq n) ▸ this,
|
||||
succ_le_succ (zero_le (succ (bit0 n)))
|
||||
|
||||
def lt.step {n m : ℕ} : n < m → n < succ m := le.step
|
||||
|
||||
@[simp] lemma zero_lt_succ_iff_true (n : ℕ) : 0 < succ n ↔ true :=
|
||||
iff_true_intro (zero_lt_succ n)
|
||||
|
||||
protected lemma lt_trans {n m k : ℕ} (h₁ : n < m) : m < k → n < k :=
|
||||
nat.le_trans (le.step h₁)
|
||||
|
||||
protected lemma lt_of_le_of_lt {n m k : ℕ} (h₁ : n ≤ m) : m < k → n < k :=
|
||||
nat.le_trans (succ_le_succ h₁)
|
||||
|
||||
lemma lt_self_iff_false (n : ℕ) : n < n ↔ false :=
|
||||
iff_false_intro (λ h, absurd h (nat.lt_irrefl n))
|
||||
|
||||
lemma self_lt_succ (n : ℕ) : n < succ n := nat.le_refl (succ n)
|
||||
|
||||
@[simp] lemma self_lt_succ_iff_true (n : ℕ) : n < succ n ↔ true :=
|
||||
iff_true_intro (self_lt_succ n)
|
||||
|
||||
def lt.base (n : ℕ) : n < succ n := nat.le_refl (succ n)
|
||||
|
||||
lemma le_lt_antisymm {n m : ℕ} (h₁ : n ≤ m) (h₂ : m < n) : false :=
|
||||
nat.lt_irrefl n (nat.lt_of_le_of_lt h₁ h₂)
|
||||
|
||||
protected lemma le_antisymm {n m : ℕ} (h₁ : n ≤ m) : m ≤ n → n = m :=
|
||||
le.cases_on h₁ (λ a, rfl) (λ a b c, absurd (nat.lt_of_le_of_lt b c) (nat.lt_irrefl n))
|
||||
|
||||
instance : weak_order ℕ :=
|
||||
⟨ @nat.le, @nat.le_refl, @nat.le_trans, @nat.le_antisymm ⟩
|
||||
|
||||
lemma lt_le_antisymm {n m : ℕ} (h₁ : n < m) (h₂ : m ≤ n) : false :=
|
||||
le_lt_antisymm h₂ h₁
|
||||
|
||||
protected lemma nat.lt_asymm {n m : ℕ} (h₁ : n < m) : ¬ m < n :=
|
||||
le_lt_antisymm (nat.le_of_lt h₁)
|
||||
|
||||
attribute [simp]
|
||||
lemma lt_zero_iff_false (a : ℕ) : a < 0 ↔ false :=
|
||||
iff_false_intro (not_lt_zero a)
|
||||
|
||||
protected lemma le_of_eq_or_lt {a b : ℕ} (h : a = b ∨ a < b) : a ≤ b :=
|
||||
or.elim h nat.le_of_eq nat.le_of_lt
|
||||
|
||||
lemma succ_lt_succ {a b : ℕ} : a < b → succ a < succ b :=
|
||||
succ_le_succ
|
||||
|
||||
lemma lt_of_succ_lt {a b : ℕ} : succ a < b → a < b :=
|
||||
le_of_succ_le
|
||||
|
||||
lemma lt_of_succ_lt_succ {a b : ℕ} : succ a < succ b → a < b :=
|
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le_of_succ_le_succ
|
||||
|
||||
|
||||
protected lemma lt_or_ge : ∀ (a b : ℕ), a < b ∨ a ≥ b
|
||||
| a 0 := or.inr (zero_le a)
|
||||
| a (b+1) :=
|
||||
match lt_or_ge a b with
|
||||
| or.inl h := or.inl (le_succ_of_le h)
|
||||
| or.inr h :=
|
||||
match nat.eq_or_lt_of_le h with
|
||||
| or.inl h1 := or.inl (h1 ▸ self_lt_succ b)
|
||||
| or.inr h1 := or.inr h1
|
||||
end
|
||||
end
|
||||
|
||||
protected def {u} lt_ge_by_cases {a b : ℕ} {C : Type u} (h₁ : a < b → C) (h₂ : a ≥ b → C) : C :=
|
||||
decidable.by_cases h₁ (λ h, h₂ (or.elim (nat.lt_or_ge a b) (λ a, absurd a h) (λ a, a)))
|
||||
|
||||
protected def {u} lt_by_cases {a b : ℕ} {C : Type u} (h₁ : a < b → C) (h₂ : a = b → C)
|
||||
(h₃ : b < a → C) : C :=
|
||||
nat.lt_ge_by_cases h₁ (λ h₁,
|
||||
nat.lt_ge_by_cases h₃ (λ h, h₂ (nat.le_antisymm h h₁)))
|
||||
|
||||
protected lemma lt_trichotomy (a b : ℕ) : a < b ∨ a = b ∨ b < a :=
|
||||
nat.lt_by_cases (λ h, or.inl h) (λ h, or.inr (or.inl h)) (λ h, or.inr (or.inr h))
|
||||
|
||||
protected lemma eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ∨ b < a :=
|
||||
or.elim (nat.lt_trichotomy a b)
|
||||
(λ hlt, absurd hlt hnlt)
|
||||
(λ h, h)
|
||||
|
||||
|
||||
lemma lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
|
||||
|
||||
lemma succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
|
||||
|
||||
|
||||
lemma sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
|
||||
eq.symm (succ_sub_succ_eq_sub a b)
|
||||
|
||||
@[simp] lemma zero_sub_eq_zero : ∀ a : ℕ, 0 - a = 0
|
||||
| 0 := rfl
|
||||
| (a+1) := congr_arg pred (zero_sub_eq_zero a)
|
||||
|
||||
lemma zero_eq_zero_sub (a : ℕ) : 0 = 0 - a :=
|
||||
eq.symm (zero_sub_eq_zero a)
|
||||
|
||||
|
||||
@[simp] lemma sub_le_iff_true (a b : ℕ) : a - b ≤ a ↔ true :=
|
||||
iff_true_intro (sub_le a b)
|
||||
|
||||
lemma sub_lt_succ (a b : ℕ) : a - b < succ a :=
|
||||
lt_succ_of_le (sub_le a b)
|
||||
|
||||
@[simp] lemma sub_lt_succ_iff_true (a b : ℕ) : a - b < succ a ↔ true :=
|
||||
iff_true_intro (sub_lt_succ a b)
|
||||
|
||||
lemma le_add_right : ∀ (n k : ℕ), n ≤ n + k
|
||||
| n 0 := nat.le_refl n
|
||||
| n (k+1) := le_succ_of_le (le_add_right n k)
|
||||
|
||||
lemma le_add_left (n m : ℕ): n ≤ m + n :=
|
||||
nat.add_comm n m ▸ le_add_right n m
|
||||
|
||||
end nat
|
||||
|
|
@ -1,4 +1,4 @@
|
|||
The Lean Standard Library
|
||||
dThe Lean Standard Library
|
||||
=========================
|
||||
|
||||
The Lean standard library is contained in the following files and directories:
|
||||
|
|
@ -39,9 +39,9 @@ excluded middle, use `open classical`. The additional axioms are used
|
|||
sparingly. For example:
|
||||
|
||||
* The constructions of finite sets and the rationals use quotients.
|
||||
* The set library uses propext and funext, as well as excluded middle to prove
|
||||
* The set library uses propext and funext, as well as excluded middle to prove
|
||||
some classical identities.
|
||||
* Hilbert choice is used to define the multiplicative inverse on the reals, and
|
||||
* Hilbert choice is used to define the multiplicative inverse on the reals, and
|
||||
also to define function inverses classically.
|
||||
|
||||
You can use `print axioms foo` to see which axioms `foo` depends
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue