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theory HOL4Compat(* Title: HOL/Import/HOL4Compat.thy ID: $Id: HOL4Compat.thy,v 1.9 2007/06/14 16:33:32 wenzelm Exp $ Author: Sebastian Skalberg (TU Muenchen) *) theory HOL4Compat imports HOL4Setup Divides Primes Real begin lemma EXISTS_UNIQUE_DEF: "(Ex1 P) = (Ex P & (ALL x y. P x & P y --> (x = y)))" by auto lemma COND_DEF:"(If b t f) = (@x. ((b = True) --> (x = t)) & ((b = False) --> (x = f)))" by auto constdefs LET :: "['a => 'b,'a] => 'b" "LET f s == f s" lemma [hol4rew]: "LET f s = Let s f" by (simp add: LET_def Let_def) lemmas [hol4rew] = ONE_ONE_rew lemma bool_case_DEF: "(bool_case x y b) = (if b then x else y)" by simp; lemma INR_INL_11: "(ALL y x. (Inl x = Inl y) = (x = y)) & (ALL y x. (Inr x = Inr y) = (x = y))" by safe (*lemma INL_neq_INR: "ALL v1 v2. Sum_Type.Inr v2 ~= Sum_Type.Inl v1" by simp*) consts ISL :: "'a + 'b => bool" ISR :: "'a + 'b => bool" primrec ISL_def: "ISL (Inl x) = True" "ISL (Inr x) = False" primrec ISR_def: "ISR (Inl x) = False" "ISR (Inr x) = True" lemma ISL: "(ALL x. ISL (Inl x)) & (ALL y. ~ISL (Inr y))" by simp lemma ISR: "(ALL x. ISR (Inr x)) & (ALL y. ~ISR (Inl y))" by simp consts OUTL :: "'a + 'b => 'a" OUTR :: "'a + 'b => 'b" primrec OUTL_def: "OUTL (Inl x) = x" primrec OUTR_def: "OUTR (Inr x) = x" lemma OUTL: "OUTL (Inl x) = x" by simp lemma OUTR: "OUTR (Inr x) = x" by simp lemma sum_case_def: "(ALL f g x. sum_case f g (Inl x) = f x) & (ALL f g y. sum_case f g (Inr y) = g y)" by simp; lemma one: "ALL v. v = ()" by simp; lemma option_case_def: "(!u f. option_case u f None = u) & (!u f x. option_case u f (Some x) = f x)" by simp lemma OPTION_MAP_DEF: "(!f x. option_map f (Some x) = Some (f x)) & (!f. option_map f None = None)" by simp consts IS_SOME :: "'a option => bool" IS_NONE :: "'a option => bool" primrec IS_SOME_def: "IS_SOME (Some x) = True" "IS_SOME None = False" primrec IS_NONE_def: "IS_NONE (Some x) = False" "IS_NONE None = True" lemma IS_NONE_DEF: "(!x. IS_NONE (Some x) = False) & (IS_NONE None = True)" by simp lemma IS_SOME_DEF: "(!x. IS_SOME (Some x) = True) & (IS_SOME None = False)" by simp consts OPTION_JOIN :: "'a option option => 'a option" primrec OPTION_JOIN_def: "OPTION_JOIN None = None" "OPTION_JOIN (Some x) = x" lemma OPTION_JOIN_DEF: "(OPTION_JOIN None = None) & (ALL x. OPTION_JOIN (Some x) = x)" by simp; lemma PAIR: "(fst x,snd x) = x" by simp lemma PAIR_MAP: "prod_fun f g p = (f (fst p),g (snd p))" by (simp add: prod_fun_def split_def) lemma pair_case_def: "split = split" ..; lemma LESS_OR_EQ: "m <= (n::nat) = (m < n | m = n)" by auto constdefs nat_gt :: "nat => nat => bool" "nat_gt == %m n. n < m" nat_ge :: "nat => nat => bool" "nat_ge == %m n. nat_gt m n | m = n" lemma [hol4rew]: "nat_gt m n = (n < m)" by (simp add: nat_gt_def) lemma [hol4rew]: "nat_ge m n = (n <= m)" by (auto simp add: nat_ge_def nat_gt_def) lemma GREATER_DEF: "ALL m n. (n < m) = (n < m)" by simp lemma GREATER_OR_EQ: "ALL m n. n <= (m::nat) = (n < m | m = n)" by auto lemma LESS_DEF: "m < n = (? P. (!n. P (Suc n) --> P n) & P m & ~P n)" proof safe assume "m < n" def P == "%n. n <= m" have "(!n. P (Suc n) --> P n) & P m & ~P n" proof (auto simp add: P_def) assume "n <= m" from prems show False by auto qed thus "? P. (!n. P (Suc n) --> P n) & P m & ~P n" by auto next fix P assume alln: "!n. P (Suc n) --> P n" assume pm: "P m" assume npn: "~P n" have "!k q. q + k = m --> P q" proof fix k show "!q. q + k = m --> P q" proof (induct k,simp_all) show "P m" by fact next fix k assume ind: "!q. q + k = m --> P q" show "!q. Suc (q + k) = m --> P q" proof (rule+) fix q assume "Suc (q + k) = m" hence "(Suc q) + k = m" by simp with ind have psq: "P (Suc q)" by simp from alln have "P (Suc q) --> P q" .. with psq show "P q" by simp qed qed qed hence "!q. q + (m - n) = m --> P q" .. hence hehe: "n + (m - n) = m --> P n" .. show "m < n" proof (rule classical) assume "~(m<n)" hence "n <= m" by simp with hehe have "P n" by simp with npn show "m < n" .. qed qed; constdefs FUNPOW :: "('a => 'a) => nat => 'a => 'a" "FUNPOW f n == f ^ n" lemma FUNPOW: "(ALL f x. (f ^ 0) x = x) & (ALL f n x. (f ^ Suc n) x = (f ^ n) (f x))" proof auto fix f n x have "ALL x. f ((f ^ n) x) = (f ^ n) (f x)" by (induct n,auto) thus "f ((f ^ n) x) = (f ^ n) (f x)" .. qed lemma [hol4rew]: "FUNPOW f n = f ^ n" by (simp add: FUNPOW_def) lemma ADD: "(!n. (0::nat) + n = n) & (!m n. Suc m + n = Suc (m + n))" by simp lemma MULT: "(!n. (0::nat) * n = 0) & (!m n. Suc m * n = m * n + n)" by simp lemma SUB: "(!m. (0::nat) - m = 0) & (!m n. (Suc m) - n = (if m < n then 0 else Suc (m - n)))" by (simp, arith) lemma MAX_DEF: "max (m::nat) n = (if m < n then n else m)" by (simp add: max_def) lemma MIN_DEF: "min (m::nat) n = (if m < n then m else n)" by (simp add: min_def) lemma DIVISION: "(0::nat) < n --> (!k. (k = k div n * n + k mod n) & k mod n < n)" by simp constdefs ALT_ZERO :: nat "ALT_ZERO == 0" NUMERAL_BIT1 :: "nat => nat" "NUMERAL_BIT1 n == n + (n + Suc 0)" NUMERAL_BIT2 :: "nat => nat" "NUMERAL_BIT2 n == n + (n + Suc (Suc 0))" NUMERAL :: "nat => nat" "NUMERAL x == x" lemma [hol4rew]: "NUMERAL ALT_ZERO = 0" by (simp add: ALT_ZERO_def NUMERAL_def) lemma [hol4rew]: "NUMERAL (NUMERAL_BIT1 ALT_ZERO) = 1" by (simp add: ALT_ZERO_def NUMERAL_BIT1_def NUMERAL_def) lemma [hol4rew]: "NUMERAL (NUMERAL_BIT2 ALT_ZERO) = 2" by (simp add: ALT_ZERO_def NUMERAL_BIT2_def NUMERAL_def) lemma EXP: "(!m. m ^ 0 = (1::nat)) & (!m n. m ^ Suc n = m * (m::nat) ^ n)" by auto lemma num_case_def: "(!b f. nat_case b f 0 = b) & (!b f n. nat_case b f (Suc n) = f n)" by simp; lemma divides_def: "(a::nat) dvd b = (? q. b = q * a)" by (auto simp add: dvd_def); lemma list_case_def: "(!v f. list_case v f [] = v) & (!v f a0 a1. list_case v f (a0#a1) = f a0 a1)" by simp consts list_size :: "('a => nat) => 'a list => nat" primrec "list_size f [] = 0" "list_size f (a0#a1) = 1 + (f a0 + list_size f a1)" lemma list_size_def: "(!f. list_size f [] = 0) & (!f a0 a1. list_size f (a0#a1) = 1 + (f a0 + list_size f a1))" by simp lemma list_case_cong: "! M M' v f. M = M' & (M' = [] --> v = v') & (!a0 a1. (M' = a0#a1) --> (f a0 a1 = f' a0 a1)) --> (list_case v f M = list_case v' f' M')" proof clarify fix M M' v f assume "M' = [] --> v = v'" and "!a0 a1. M' = a0 # a1 --> f a0 a1 = f' a0 a1" show "list_case v f M' = list_case v' f' M'" proof (rule List.list.case_cong) show "M' = M'" .. next assume "M' = []" with prems show "v = v'" by auto next fix a0 a1 assume "M' = a0 # a1" with prems show "f a0 a1 = f' a0 a1" by auto qed qed lemma list_Axiom: "ALL f0 f1. EX fn. (fn [] = f0) & (ALL a0 a1. fn (a0#a1) = f1 a0 a1 (fn a1))" proof safe fix f0 f1 def fn == "list_rec f0 f1" have "fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))" by (simp add: fn_def) thus "EX fn. fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))" by auto qed lemma list_Axiom_old: "EX! fn. (fn [] = x) & (ALL h t. fn (h#t) = f (fn t) h t)" proof safe def fn == "list_rec x (%h t r. f r h t)" have "fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)" by (simp add: fn_def) thus "EX fn. fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)" by auto next fix fn1 fn2 assume "ALL h t. fn1 (h # t) = f (fn1 t) h t" assume "ALL h t. fn2 (h # t) = f (fn2 t) h t" assume "fn2 [] = fn1 []" show "fn1 = fn2" proof fix xs show "fn1 xs = fn2 xs" by (induct xs,simp_all add: prems) qed qed lemma NULL_DEF: "(null [] = True) & (!h t. null (h # t) = False)" by simp constdefs sum :: "nat list => nat" "sum l == foldr (op +) l 0" lemma SUM: "(sum [] = 0) & (!h t. sum (h#t) = h + sum t)" by (simp add: sum_def) lemma APPEND: "(!l. [] @ l = l) & (!l1 l2 h. (h#l1) @ l2 = h# l1 @ l2)" by simp lemma FLAT: "(concat [] = []) & (!h t. concat (h#t) = h @ (concat t))" by simp lemma LENGTH: "(length [] = 0) & (!h t. length (h#t) = Suc (length t))" by simp lemma MAP: "(!f. map f [] = []) & (!f h t. map f (h#t) = f h#map f t)" by simp lemma MEM: "(!x. x mem [] = False) & (!x h t. x mem (h#t) = ((x = h) | x mem t))" by auto lemma FILTER: "(!P. filter P [] = []) & (!P h t. filter P (h#t) = (if P h then h#filter P t else filter P t))" by simp lemma REPLICATE: "(ALL x. replicate 0 x = []) & (ALL n x. replicate (Suc n) x = x # replicate n x)" by simp constdefs FOLDR :: "[['a,'b]=>'b,'b,'a list] => 'b" "FOLDR f e l == foldr f l e" lemma [hol4rew]: "FOLDR f e l = foldr f l e" by (simp add: FOLDR_def) lemma FOLDR: "(!f e. foldr f [] e = e) & (!f e x l. foldr f (x#l) e = f x (foldr f l e))" by simp lemma FOLDL: "(!f e. foldl f e [] = e) & (!f e x l. foldl f e (x#l) = foldl f (f e x) l)" by simp lemma EVERY_DEF: "(!P. list_all P [] = True) & (!P h t. list_all P (h#t) = (P h & list_all P t))" by simp consts list_exists :: "['a => bool,'a list] => bool" primrec list_exists_Nil: "list_exists P Nil = False" list_exists_Cons: "list_exists P (x#xs) = (if P x then True else list_exists P xs)" lemma list_exists_DEF: "(!P. list_exists P [] = False) & (!P h t. list_exists P (h#t) = (P h | list_exists P t))" by simp consts map2 :: "[['a,'b]=>'c,'a list,'b list] => 'c list" primrec map2_Nil: "map2 f [] l2 = []" map2_Cons: "map2 f (x#xs) l2 = f x (hd l2) # map2 f xs (tl l2)" lemma MAP2: "(!f. map2 f [] [] = []) & (!f h1 t1 h2 t2. map2 f (h1#t1) (h2#t2) = f h1 h2#map2 f t1 t2)" by simp lemma list_INDUCT: "[| P [] ; !t. P t --> (!h. P (h#t)) |] ==> !l. P l" proof fix l assume "P []" assume allt: "!t. P t --> (!h. P (h # t))" show "P l" proof (induct l) show "P []" by fact next fix h t assume "P t" with allt have "!h. P (h # t)" by auto thus "P (h # t)" .. qed qed lemma list_CASES: "(l = []) | (? t h. l = h#t)" by (induct l,auto) constdefs ZIP :: "'a list * 'b list => ('a * 'b) list" "ZIP == %(a,b). zip a b" lemma ZIP: "(zip [] [] = []) & (!x1 l1 x2 l2. zip (x1#l1) (x2#l2) = (x1,x2)#zip l1 l2)" by simp lemma [hol4rew]: "ZIP (a,b) = zip a b" by (simp add: ZIP_def) consts unzip :: "('a * 'b) list => 'a list * 'b list" primrec unzip_Nil: "unzip [] = ([],[])" unzip_Cons: "unzip (xy#xys) = (let zs = unzip xys in (fst xy # fst zs,snd xy # snd zs))" lemma UNZIP: "(unzip [] = ([],[])) & (!x l. unzip (x#l) = (fst x#fst (unzip l),snd x#snd (unzip l)))" by (simp add: Let_def) lemma REVERSE: "(rev [] = []) & (!h t. rev (h#t) = (rev t) @ [h])" by simp; lemma REAL_SUP_ALLPOS: "[| ALL x. P (x::real) --> 0 < x ; EX x. P x; EX z. ALL x. P x --> x < z |] ==> EX s. ALL y. (EX x. P x & y < x) = (y < s)" proof safe fix x z assume allx: "ALL x. P x --> 0 < x" assume px: "P x" assume allx': "ALL x. P x --> x < z" have "EX s. ALL y. (EX x : Collect P. y < x) = (y < s)" proof (rule posreal_complete) show "ALL x : Collect P. 0 < x" proof safe fix x assume "P x" from allx have "P x --> 0 < x" .. thus "0 < x" by (simp add: prems) qed next from px show "EX x. x : Collect P" by auto next from allx' show "EX y. ALL x : Collect P. x < y" apply simp .. qed thus "EX s. ALL y. (EX x. P x & y < x) = (y < s)" by simp qed lemma REAL_10: "~((1::real) = 0)" by simp lemma REAL_ADD_ASSOC: "(x::real) + (y + z) = x + y + z" by simp lemma REAL_MUL_ASSOC: "(x::real) * (y * z) = x * y * z" by simp lemma REAL_ADD_LINV: "-x + x = (0::real)" by simp lemma REAL_MUL_LINV: "x ~= (0::real) ==> inverse x * x = 1" by simp lemma REAL_LT_TOTAL: "((x::real) = y) | x < y | y < x" by auto; lemma [hol4rew]: "real (0::nat) = 0" by simp lemma [hol4rew]: "real (1::nat) = 1" by simp lemma [hol4rew]: "real (2::nat) = 2" by simp lemma real_lte: "((x::real) <= y) = (~(y < x))" by auto lemma real_of_num: "((0::real) = 0) & (!n. real (Suc n) = real n + 1)" by (simp add: real_of_nat_Suc) lemma abs: "abs (x::real) = (if 0 <= x then x else -x)" by (simp add: abs_if) lemma pow: "(!x::real. x ^ 0 = 1) & (!x::real. ALL n. x ^ (Suc n) = x * x ^ n)" by simp constdefs real_gt :: "real => real => bool" "real_gt == %x y. y < x" lemma [hol4rew]: "real_gt x y = (y < x)" by (simp add: real_gt_def) lemma real_gt: "ALL x (y::real). (y < x) = (y < x)" by simp constdefs real_ge :: "real => real => bool" "real_ge x y == y <= x" lemma [hol4rew]: "real_ge x y = (y <= x)" by (simp add: real_ge_def) lemma real_ge: "ALL x y. (y <= x) = (y <= x)" by simp end
lemma EXISTS_UNIQUE_DEF:
Ex1 P = (Ex P ∧ (∀x y. P x ∧ P y --> x = y))
lemma COND_DEF:
(if b then t else f) = (SOME x. (b = True --> x = t) ∧ (b = False --> x = f))
lemma
LET f1 s1 == Let s1 f1
lemma
ONE_ONE f1 == inj f1
lemma bool_case_DEF:
(case b of True => x | False => y) = (if b then x else y)
lemma INR_INL_11:
(∀y x. (Inl x = Inl y) = (x = y)) ∧ (∀y x. (Inr x = Inr y) = (x = y))
lemma ISL:
(∀x. ISL (Inl x)) ∧ (∀y. ¬ ISL (Inr y))
lemma ISR:
(∀x. ISR (Inr x)) ∧ (∀y. ¬ ISR (Inl y))
lemma OUTL:
OUTL (Inl x) = x
lemma OUTR:
OUTR (Inr x) = x
lemma sum_case_def:
(∀f g x. sum.sum_case f g (Inl x) = f x) ∧
(∀f g y. sum.sum_case f g (Inr y) = g y)
lemma one:
∀v. v = ()
lemma option_case_def:
(∀u f. option_case u f None = u) ∧ (∀u f x. option_case u f (Some x) = f x)
lemma OPTION_MAP_DEF:
(∀f x. option_map f (Some x) = Some (f x)) ∧ (∀f. option_map f None = None)
lemma IS_NONE_DEF:
(∀x. IS_NONE (Some x) = False) ∧ IS_NONE None = True
lemma IS_SOME_DEF:
(∀x. IS_SOME (Some x) = True) ∧ IS_SOME None = False
lemma OPTION_JOIN_DEF:
OPTION_JOIN None = None ∧ (∀x. OPTION_JOIN (Some x) = x)
lemma PAIR:
(fst x, snd x) = x
lemma PAIR_MAP:
prod_fun f g p = (f (fst p), g (snd p))
lemma pair_case_def:
split = split
lemma LESS_OR_EQ:
(m ≤ n) = (m < n ∨ m = n)
lemma
nat_gt m1 n1 == n1 < m1
lemma
nat_ge m1 n1 == n1 ≤ m1
lemma GREATER_DEF:
∀m n. (n < m) = (n < m)
lemma GREATER_OR_EQ:
∀m n. (n ≤ m) = (n < m ∨ m = n)
lemma LESS_DEF:
(m < n) = (∃P. (∀n. P (Suc n) --> P n) ∧ P m ∧ ¬ P n)
lemma FUNPOW:
(∀f x. (f ^ 0) x = x) ∧ (∀f n x. (f ^ Suc n) x = (f ^ n) (f x))
lemma
FUNPOW f1 n1 == f1 ^ n1
lemma ADD:
(∀n. 0 + n = n) ∧ (∀m n. Suc m + n = Suc (m + n))
lemma MULT:
(∀n. 0 * n = 0) ∧ (∀m n. Suc m * n = m * n + n)
lemma SUB:
(∀m. 0 - m = 0) ∧ (∀m n. Suc m - n = (if m < n then 0 else Suc (m - n)))
lemma MAX_DEF:
max m n = (if m < n then n else m)
lemma MIN_DEF:
min m n = (if m < n then m else n)
lemma DIVISION:
0 < n --> (∀k. k = k div n * n + k mod n ∧ k mod n < n)
lemma
NUMERAL ALT_ZERO == 0
lemma
NUMERAL (NUMERAL_BIT1 ALT_ZERO) == 1
lemma
NUMERAL (NUMERAL_BIT2 ALT_ZERO) == 2
lemma EXP:
(∀m. m ^ 0 = 1) ∧ (∀m n. m ^ Suc n = m * m ^ n)
lemma num_case_def:
(∀b f. nat_case b f 0 = b) ∧ (∀b f n. nat_case b f (Suc n) = f n)
lemma divides_def:
(a dvd b) = (∃q. b = q * a)
lemma list_case_def:
(∀v f. list_case v f [] = v) ∧ (∀v f a0 a1. list_case v f (a0 # a1) = f a0 a1)
lemma list_size_def:
(∀f. list_size f [] = 0) ∧
(∀f a0 a1. list_size f (a0 # a1) = 1 + (f a0 + list_size f a1))
lemma list_case_cong:
∀M M' v f.
M = M' ∧
(M' = [] --> v = v') ∧ (∀a0 a1. M' = a0 # a1 --> f a0 a1 = f' a0 a1) -->
list_case v f M = list_case v' f' M'
lemma list_Axiom:
∀f0 f1. ∃fn. fn [] = f0 ∧ (∀a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))
lemma list_Axiom_old:
∃!fn. fn [] = x ∧ (∀h t. fn (h # t) = f (fn t) h t)
lemma NULL_DEF:
null [] = True ∧ (∀h t. null (h # t) = False)
lemma SUM:
sum [] = 0 ∧ (∀h t. sum (h # t) = h + sum t)
lemma APPEND:
(∀l. [] @ l = l) ∧ (∀l1 l2 h. (h # l1) @ l2 = h # l1 @ l2)
lemma FLAT:
concat [] = [] ∧ (∀h t. concat (h # t) = h @ concat t)
lemma LENGTH:
length [] = 0 ∧ (∀h t. length (h # t) = Suc (length t))
lemma MAP:
(∀f. map f [] = []) ∧ (∀f h t. map f (h # t) = f h # map f t)
lemma MEM:
(∀x. x mem [] = False) ∧ (∀x h t. x mem h # t = (x = h ∨ x mem t))
lemma FILTER:
(∀P. filter P [] = []) ∧
(∀P h t. filter P (h # t) = (if P h then h # filter P t else filter P t))
lemma REPLICATE:
(∀x. replicate 0 x = []) ∧ (∀n x. replicate (Suc n) x = x # replicate n x)
lemma
FOLDR f1 e1 l1 == foldr f1 l1 e1
lemma FOLDR:
(∀f e. foldr f [] e = e) ∧ (∀f e x l. foldr f (x # l) e = f x (foldr f l e))
lemma FOLDL:
(∀f e. foldl f e [] = e) ∧ (∀f e x l. foldl f e (x # l) = foldl f (f e x) l)
lemma EVERY_DEF:
(∀P. list_all P [] = True) ∧ (∀P h t. list_all P (h # t) = (P h ∧ list_all P t))
lemma list_exists_DEF:
(∀P. list_exists P [] = False) ∧
(∀P h t. list_exists P (h # t) = (P h ∨ list_exists P t))
lemma MAP2:
(∀f. map2 f [] [] = []) ∧
(∀f h1 t1 h2 t2. map2 f (h1 # t1) (h2 # t2) = f h1 h2 # map2 f t1 t2)
lemma list_INDUCT:
[| P []; ∀t. P t --> (∀h. P (h # t)) |] ==> ∀l. P l
lemma list_CASES:
l = [] ∨ (∃t h. l = h # t)
lemma ZIP:
zip [] [] = [] ∧ (∀x1 l1 x2 l2. zip (x1 # l1) (x2 # l2) = (x1, x2) # zip l1 l2)
lemma
ZIP (a1, b1) == zip a1 b1
lemma UNZIP:
unzip [] = ([], []) ∧
(∀x l. unzip (x # l) = (fst x # fst (unzip l), snd x # snd (unzip l)))
lemma REVERSE:
rev [] = [] ∧ (∀h t. rev (h # t) = rev t @ [h])
lemma REAL_SUP_ALLPOS:
[| ∀x. P x --> 0 < x; ∃x. P x; ∃z. ∀x. P x --> x < z |]
==> ∃s. ∀y. (∃x. P x ∧ y < x) = (y < s)
lemma REAL_10:
1 ≠ 0
lemma REAL_ADD_ASSOC:
x + (y + z) = x + y + z
lemma REAL_MUL_ASSOC:
x * (y * z) = x * y * z
lemma REAL_ADD_LINV:
- x + x = 0
lemma REAL_MUL_LINV:
x ≠ 0 ==> inverse x * x = 1
lemma REAL_LT_TOTAL:
x = y ∨ x < y ∨ y < x
lemma
real 0 == 0
lemma
real 1 == 1
lemma
real 2 == 2
lemma real_lte:
(x ≤ y) = (¬ y < x)
lemma real_of_num:
0 = 0 ∧ (∀n. real (Suc n) = real n + 1)
lemma abs:
¦x¦ = (if 0 ≤ x then x else - x)
lemma pow:
(∀x. x ^ 0 = 1) ∧ (∀x n. x ^ Suc n = x * x ^ n)
lemma
real_gt x1 y1 == y1 < x1
lemma real_gt:
∀x y. (y < x) = (y < x)
lemma
real_ge x1 y1 == y1 ≤ x1
lemma real_ge:
∀x y. (y ≤ x) = (y ≤ x)