--- a/src/HOL/Import/HOL/HOL4Prob.thy Wed Sep 07 00:08:09 2011 +0200
+++ b/src/HOL/Import/HOL/HOL4Prob.thy Wed Sep 07 07:59:45 2011 +0900
@@ -4,357 +4,211 @@
;setup_theory prob_extra
-lemma BOOL_BOOL_CASES_THM: "ALL f::bool => bool.
- f = (%b::bool. False) |
- f = (%b::bool. True) | f = (%b::bool. b) | f = Not"
+lemma BOOL_BOOL_CASES_THM: "f = (%b. False) | f = (%b. True) | f = (%b. b) | f = Not"
by (import prob_extra BOOL_BOOL_CASES_THM)
lemma EVEN_ODD_BASIC: "EVEN 0 & ~ EVEN 1 & EVEN 2 & ~ ODD 0 & ODD 1 & ~ ODD 2"
by (import prob_extra EVEN_ODD_BASIC)
-lemma EVEN_ODD_EXISTS_EQ: "ALL n::nat.
- EVEN n = (EX m::nat. n = 2 * m) & ODD n = (EX m::nat. n = Suc (2 * m))"
+lemma EVEN_ODD_EXISTS_EQ: "EVEN n = (EX m. n = 2 * m) & ODD n = (EX m. n = Suc (2 * m))"
by (import prob_extra EVEN_ODD_EXISTS_EQ)
-lemma DIV_THEN_MULT: "ALL (p::nat) q::nat. Suc q * (p div Suc q) <= p"
+lemma DIV_THEN_MULT: "Suc q * (p div Suc q) <= p"
by (import prob_extra DIV_THEN_MULT)
-lemma DIV_TWO_UNIQUE: "ALL (n::nat) (q::nat) r::nat.
- n = 2 * q + r & (r = 0 | r = 1) --> q = n div 2 & r = n mod 2"
+lemma DIV_TWO_UNIQUE: "(n::nat) = (2::nat) * (q::nat) + (r::nat) & (r = (0::nat) | r = (1::nat))
+==> q = n div (2::nat) & r = n mod (2::nat)"
by (import prob_extra DIV_TWO_UNIQUE)
-lemma DIVISION_TWO: "ALL n::nat. n = 2 * (n div 2) + n mod 2 & (n mod 2 = 0 | n mod 2 = 1)"
+lemma DIVISION_TWO: "(n::nat) = (2::nat) * (n div (2::nat)) + n mod (2::nat) &
+(n mod (2::nat) = (0::nat) | n mod (2::nat) = (1::nat))"
by (import prob_extra DIVISION_TWO)
-lemma DIV_TWO: "ALL n::nat. n = 2 * (n div 2) + n mod 2"
+lemma DIV_TWO: "(n::nat) = (2::nat) * (n div (2::nat)) + n mod (2::nat)"
by (import prob_extra DIV_TWO)
-lemma MOD_TWO: "ALL n::nat. n mod 2 = (if EVEN n then 0 else 1)"
+lemma MOD_TWO: "n mod 2 = (if EVEN n then 0 else 1)"
by (import prob_extra MOD_TWO)
-lemma DIV_TWO_BASIC: "(op &::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((op div::nat => nat => nat) (0::nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- (0::nat))
- ((op &::bool => bool => bool)
- ((op =::nat => nat => bool)
- ((op div::nat => nat => nat) (1::nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- (0::nat))
- ((op =::nat => nat => bool)
- ((op div::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- (1::nat)))"
+lemma DIV_TWO_BASIC: "(0::nat) div (2::nat) = (0::nat) &
+(1::nat) div (2::nat) = (0::nat) & (2::nat) div (2::nat) = (1::nat)"
by (import prob_extra DIV_TWO_BASIC)
-lemma DIV_TWO_MONO: "ALL (m::nat) n::nat. m div 2 < n div 2 --> m < n"
+lemma DIV_TWO_MONO: "(m::nat) div (2::nat) < (n::nat) div (2::nat) ==> m < n"
by (import prob_extra DIV_TWO_MONO)
-lemma DIV_TWO_MONO_EVEN: "ALL (m::nat) n::nat. EVEN n --> (m div 2 < n div 2) = (m < n)"
+lemma DIV_TWO_MONO_EVEN: "EVEN n ==> (m div 2 < n div 2) = (m < n)"
by (import prob_extra DIV_TWO_MONO_EVEN)
-lemma DIV_TWO_CANCEL: "ALL n::nat. 2 * n div 2 = n & Suc (2 * n) div 2 = n"
+lemma DIV_TWO_CANCEL: "2 * n div 2 = n & Suc (2 * n) div 2 = n"
by (import prob_extra DIV_TWO_CANCEL)
-lemma EXP_DIV_TWO: "(All::(nat => bool) => bool)
- (%n::nat.
- (op =::nat => nat => bool)
- ((op div::nat => nat => nat)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((Suc::nat => nat) n))
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- n))"
+lemma EXP_DIV_TWO: "(2::nat) ^ Suc (n::nat) div (2::nat) = (2::nat) ^ n"
by (import prob_extra EXP_DIV_TWO)
-lemma EVEN_EXP_TWO: "ALL n::nat. EVEN (2 ^ n) = (n ~= 0)"
+lemma EVEN_EXP_TWO: "EVEN (2 ^ n) = (n ~= 0)"
by (import prob_extra EVEN_EXP_TWO)
-lemma DIV_TWO_EXP: "ALL (n::nat) k::nat. (k div 2 < 2 ^ n) = (k < 2 ^ Suc n)"
+lemma DIV_TWO_EXP: "((k::nat) div (2::nat) < (2::nat) ^ (n::nat)) = (k < (2::nat) ^ Suc n)"
by (import prob_extra DIV_TWO_EXP)
consts
inf :: "(real => bool) => real"
defs
- inf_primdef: "inf == %P::real => bool. - sup (IMAGE uminus P)"
+ inf_primdef: "prob_extra.inf == %P. - real.sup (IMAGE uminus P)"
-lemma inf_def: "ALL P::real => bool. inf P = - sup (IMAGE uminus P)"
+lemma inf_def: "prob_extra.inf P = - real.sup (IMAGE uminus P)"
by (import prob_extra inf_def)
-lemma INF_DEF_ALT: "ALL P::real => bool. inf P = - sup (%r::real. P (- r))"
+lemma INF_DEF_ALT: "prob_extra.inf P = - real.sup (%r. P (- r))"
by (import prob_extra INF_DEF_ALT)
-lemma REAL_SUP_EXISTS_UNIQUE: "ALL P::real => bool.
- Ex P & (EX z::real. ALL x::real. P x --> x <= z) -->
- (EX! s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s))"
+lemma REAL_SUP_EXISTS_UNIQUE: "Ex (P::real => bool) & (EX z::real. ALL x::real. P x --> x <= z)
+==> EX! s::real. ALL y::real. (EX x::real. P x & y < x) = (y < s)"
by (import prob_extra REAL_SUP_EXISTS_UNIQUE)
-lemma REAL_SUP_MAX: "ALL (P::real => bool) z::real.
- P z & (ALL x::real. P x --> x <= z) --> sup P = z"
+lemma REAL_SUP_MAX: "P z & (ALL x. P x --> x <= z) ==> real.sup P = z"
by (import prob_extra REAL_SUP_MAX)
-lemma REAL_INF_MIN: "ALL (P::real => bool) z::real.
- P z & (ALL x::real. P x --> z <= x) --> inf P = z"
+lemma REAL_INF_MIN: "P z & (ALL x. P x --> z <= x) ==> prob_extra.inf P = z"
by (import prob_extra REAL_INF_MIN)
-lemma HALF_POS: "(op <::real => real => bool) (0::real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))"
- by (import prob_extra HALF_POS)
-
-lemma HALF_CANCEL: "(op =::real => real => bool)
- ((op *::real => real => real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))))
- (1::real)"
+lemma HALF_CANCEL: "(2::real) * ((1::real) / (2::real)) = (1::real)"
by (import prob_extra HALF_CANCEL)
-lemma POW_HALF_POS: "(All::(nat => bool) => bool)
- (%n::nat.
- (op <::real => real => bool) (0::real)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- n))"
+lemma POW_HALF_POS: "(0::real) < ((1::real) / (2::real)) ^ (n::nat)"
by (import prob_extra POW_HALF_POS)
-lemma POW_HALF_MONO: "(All::(nat => bool) => bool)
- (%m::nat.
- (All::(nat => bool) => bool)
- (%n::nat.
- (op -->::bool => bool => bool) ((op <=::nat => nat => bool) m n)
- ((op <=::real => real => bool)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- n)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- m))))"
+lemma POW_HALF_MONO: "(m::nat) <= (n::nat)
+==> ((1::real) / (2::real)) ^ n <= ((1::real) / (2::real)) ^ m"
by (import prob_extra POW_HALF_MONO)
-lemma POW_HALF_TWICE: "(All::(nat => bool) => bool)
- (%n::nat.
- (op =::real => real => bool)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- n)
- ((op *::real => real => real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((Suc::nat => nat) n))))"
+lemma POW_HALF_TWICE: "((1::real) / (2::real)) ^ (n::nat) =
+(2::real) * ((1::real) / (2::real)) ^ Suc n"
by (import prob_extra POW_HALF_TWICE)
-lemma X_HALF_HALF: "ALL x::real. 1 / 2 * x + 1 / 2 * x = x"
+lemma X_HALF_HALF: "(1::real) / (2::real) * (x::real) + (1::real) / (2::real) * x = x"
by (import prob_extra X_HALF_HALF)
-lemma REAL_SUP_LE_X: "ALL (P::real => bool) x::real.
- Ex P & (ALL r::real. P r --> r <= x) --> sup P <= x"
+lemma REAL_SUP_LE_X: "Ex P & (ALL r. P r --> r <= x) ==> real.sup P <= x"
by (import prob_extra REAL_SUP_LE_X)
-lemma REAL_X_LE_SUP: "ALL (P::real => bool) x::real.
- Ex P &
- (EX z::real. ALL r::real. P r --> r <= z) &
- (EX r::real. P r & x <= r) -->
- x <= sup P"
+lemma REAL_X_LE_SUP: "Ex P & (EX z. ALL r. P r --> r <= z) & (EX r. P r & x <= r)
+==> x <= real.sup P"
by (import prob_extra REAL_X_LE_SUP)
-lemma ABS_BETWEEN_LE: "ALL (x::real) (y::real) d::real.
- (0 <= d & x - d <= y & y <= x + d) = (abs (y - x) <= d)"
+lemma ABS_BETWEEN_LE: "((0::real) <= (d::real) & (x::real) - d <= (y::real) & y <= x + d) =
+(abs (y - x) <= d)"
by (import prob_extra ABS_BETWEEN_LE)
-lemma ONE_MINUS_HALF: "(op =::real => real => bool)
- ((op -::real => real => real) (1::real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))))
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))"
+lemma ONE_MINUS_HALF: "(1::real) - (1::real) / (2::real) = (1::real) / (2::real)"
by (import prob_extra ONE_MINUS_HALF)
-lemma HALF_LT_1: "(op <::real => real => bool)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- (1::real)"
+lemma HALF_LT_1: "(1::real) / (2::real) < (1::real)"
by (import prob_extra HALF_LT_1)
-lemma POW_HALF_EXP: "(All::(nat => bool) => bool)
- (%n::nat.
- (op =::real => real => bool)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- n)
- ((inverse::real => real)
- ((real::nat => real)
- ((op ^::nat => nat => nat)
- ((number_of \<Colon> int => nat)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int))))
- n))))"
+lemma POW_HALF_EXP: "((1::real) / (2::real)) ^ (n::nat) = inverse (real ((2::nat) ^ n))"
by (import prob_extra POW_HALF_EXP)
-lemma INV_SUC_POS: "ALL n::nat. 0 < 1 / real (Suc n)"
+lemma INV_SUC_POS: "0 < 1 / real (Suc n)"
by (import prob_extra INV_SUC_POS)
-lemma INV_SUC_MAX: "ALL x::nat. 1 / real (Suc x) <= 1"
+lemma INV_SUC_MAX: "1 / real (Suc x) <= 1"
by (import prob_extra INV_SUC_MAX)
-lemma INV_SUC: "ALL n::nat. 0 < 1 / real (Suc n) & 1 / real (Suc n) <= 1"
+lemma INV_SUC: "0 < 1 / real (Suc n) & 1 / real (Suc n) <= 1"
by (import prob_extra INV_SUC)
-lemma ABS_UNIT_INTERVAL: "ALL (x::real) y::real.
- 0 <= x & x <= 1 & 0 <= y & y <= 1 --> abs (x - y) <= 1"
+lemma ABS_UNIT_INTERVAL: "(0::real) <= (x::real) &
+x <= (1::real) & (0::real) <= (y::real) & y <= (1::real)
+==> abs (x - y) <= (1::real)"
by (import prob_extra ABS_UNIT_INTERVAL)
-lemma MEM_NIL: "ALL l::'a::type list. (ALL x::'a::type. ~ x mem l) = (l = [])"
+lemma MEM_NIL: "(ALL x. ~ List.member l x) = (l = [])"
by (import prob_extra MEM_NIL)
-lemma MAP_MEM: "ALL (f::'a::type => 'b::type) (l::'a::type list) x::'b::type.
- x mem map f l = (EX y::'a::type. y mem l & x = f y)"
+lemma MAP_MEM: "List.member (map (f::'a => 'b) (l::'a list)) (x::'b) =
+(EX y::'a. List.member l y & x = f y)"
by (import prob_extra MAP_MEM)
-lemma MEM_NIL_MAP_CONS: "ALL (x::'a::type) l::'a::type list list. ~ [] mem map (op # x) l"
+lemma MEM_NIL_MAP_CONS: "~ List.member (map (op # x) l) []"
by (import prob_extra MEM_NIL_MAP_CONS)
-lemma FILTER_TRUE: "ALL l::'a::type list. [x::'a::type<-l. True] = l"
+lemma FILTER_TRUE: "[x<-l. True] = l"
by (import prob_extra FILTER_TRUE)
-lemma FILTER_FALSE: "ALL l::'a::type list. [x::'a::type<-l. False] = []"
+lemma FILTER_FALSE: "[x<-l. False] = []"
by (import prob_extra FILTER_FALSE)
-lemma FILTER_MEM: "ALL (P::'a::type => bool) (x::'a::type) l::'a::type list.
- x mem filter P l --> P x"
+lemma FILTER_MEM: "List.member (filter P l) x ==> P x"
by (import prob_extra FILTER_MEM)
-lemma MEM_FILTER: "ALL (P::'a::type => bool) (l::'a::type list) x::'a::type.
- x mem filter P l --> x mem l"
+lemma MEM_FILTER: "List.member (filter P l) x ==> List.member l x"
by (import prob_extra MEM_FILTER)
-lemma FILTER_OUT_ELT: "ALL (x::'a::type) l::'a::type list. x mem l | [y::'a::type<-l. y ~= x] = l"
+lemma FILTER_OUT_ELT: "List.member l x | [y<-l. y ~= x] = l"
by (import prob_extra FILTER_OUT_ELT)
-lemma IS_PREFIX_NIL: "ALL x::'a::type list. IS_PREFIX x [] & IS_PREFIX [] x = (x = [])"
+lemma IS_PREFIX_NIL: "IS_PREFIX x [] & IS_PREFIX [] x = (x = [])"
by (import prob_extra IS_PREFIX_NIL)
-lemma IS_PREFIX_REFL: "ALL x::'a::type list. IS_PREFIX x x"
+lemma IS_PREFIX_REFL: "IS_PREFIX x x"
by (import prob_extra IS_PREFIX_REFL)
-lemma IS_PREFIX_ANTISYM: "ALL (x::'a::type list) y::'a::type list.
- IS_PREFIX y x & IS_PREFIX x y --> x = y"
+lemma IS_PREFIX_ANTISYM: "IS_PREFIX y x & IS_PREFIX x y ==> x = y"
by (import prob_extra IS_PREFIX_ANTISYM)
-lemma IS_PREFIX_TRANS: "ALL (x::'a::type list) (y::'a::type list) z::'a::type list.
- IS_PREFIX x y & IS_PREFIX y z --> IS_PREFIX x z"
+lemma IS_PREFIX_TRANS: "IS_PREFIX x y & IS_PREFIX y z ==> IS_PREFIX x z"
by (import prob_extra IS_PREFIX_TRANS)
-lemma IS_PREFIX_BUTLAST: "ALL (x::'a::type) y::'a::type list. IS_PREFIX (x # y) (butlast (x # y))"
+lemma IS_PREFIX_BUTLAST: "IS_PREFIX (x # y) (butlast (x # y))"
by (import prob_extra IS_PREFIX_BUTLAST)
-lemma IS_PREFIX_LENGTH: "ALL (x::'a::type list) y::'a::type list.
- IS_PREFIX y x --> length x <= length y"
+lemma IS_PREFIX_LENGTH: "IS_PREFIX y x ==> length x <= length y"
by (import prob_extra IS_PREFIX_LENGTH)
-lemma IS_PREFIX_LENGTH_ANTI: "ALL (x::'a::type list) y::'a::type list.
- IS_PREFIX y x & length x = length y --> x = y"
+lemma IS_PREFIX_LENGTH_ANTI: "IS_PREFIX y x & length x = length y ==> x = y"
by (import prob_extra IS_PREFIX_LENGTH_ANTI)
-lemma IS_PREFIX_SNOC: "ALL (x::'a::type) (y::'a::type list) z::'a::type list.
- IS_PREFIX (SNOC x y) z = (IS_PREFIX y z | z = SNOC x y)"
+lemma IS_PREFIX_SNOC: "IS_PREFIX (SNOC x y) z = (IS_PREFIX y z | z = SNOC x y)"
by (import prob_extra IS_PREFIX_SNOC)
-lemma FOLDR_MAP: "ALL (f::'b::type => 'c::type => 'c::type) (e::'c::type)
- (g::'a::type => 'b::type) l::'a::type list.
- foldr f (map g l) e = foldr (%x::'a::type. f (g x)) l e"
+lemma FOLDR_MAP: "foldr (f::'b => 'c => 'c) (map (g::'a => 'b) (l::'a list)) (e::'c) =
+foldr (%x::'a. f (g x)) l e"
by (import prob_extra FOLDR_MAP)
-lemma LAST_MEM: "ALL (h::'a::type) t::'a::type list. last (h # t) mem h # t"
+lemma LAST_MEM: "List.member (h # t) (last (h # t))"
by (import prob_extra LAST_MEM)
-lemma LAST_MAP_CONS: "ALL (b::bool) (h::bool list) t::bool list list.
- EX x::bool list. last (map (op # b) (h # t)) = b # x"
+lemma LAST_MAP_CONS: "EX x::bool list.
+ last (map (op # (b::bool)) ((h::bool list) # (t::bool list list))) =
+ b # x"
by (import prob_extra LAST_MAP_CONS)
-lemma EXISTS_LONGEST: "ALL (x::'a::type list) y::'a::type list list.
- EX z::'a::type list.
- z mem x # y &
- (ALL w::'a::type list. w mem x # y --> length w <= length z)"
+lemma EXISTS_LONGEST: "EX z. List.member (x # y) z &
+ (ALL w. List.member (x # y) w --> length w <= length z)"
by (import prob_extra EXISTS_LONGEST)
-lemma UNION_DEF_ALT: "ALL (s::'a::type => bool) t::'a::type => bool.
- pred_set.UNION s t = (%x::'a::type. s x | t x)"
+lemma UNION_DEF_ALT: "pred_set.UNION s t = (%x. s x | t x)"
by (import prob_extra UNION_DEF_ALT)
-lemma INTER_UNION_RDISTRIB: "ALL (p::'a::type => bool) (q::'a::type => bool) r::'a::type => bool.
- pred_set.INTER (pred_set.UNION p q) r =
- pred_set.UNION (pred_set.INTER p r) (pred_set.INTER q r)"
+lemma INTER_UNION_RDISTRIB: "pred_set.INTER (pred_set.UNION p q) r =
+pred_set.UNION (pred_set.INTER p r) (pred_set.INTER q r)"
by (import prob_extra INTER_UNION_RDISTRIB)
-lemma SUBSET_EQ: "ALL (x::'a::type => bool) xa::'a::type => bool.
- (x = xa) = (SUBSET x xa & SUBSET xa x)"
+lemma SUBSET_EQ: "(x = xa) = (SUBSET x xa & SUBSET xa x)"
by (import prob_extra SUBSET_EQ)
-lemma INTER_IS_EMPTY: "ALL (s::'a::type => bool) t::'a::type => bool.
- (pred_set.INTER s t = EMPTY) = (ALL x::'a::type. ~ s x | ~ t x)"
+lemma INTER_IS_EMPTY: "(pred_set.INTER s t = EMPTY) = (ALL x. ~ s x | ~ t x)"
by (import prob_extra INTER_IS_EMPTY)
-lemma UNION_DISJOINT_SPLIT: "ALL (s::'a::type => bool) (t::'a::type => bool) u::'a::type => bool.
- pred_set.UNION s t = pred_set.UNION s u &
- pred_set.INTER s t = EMPTY & pred_set.INTER s u = EMPTY -->
- t = u"
+lemma UNION_DISJOINT_SPLIT: "pred_set.UNION s t = pred_set.UNION s u &
+pred_set.INTER s t = EMPTY & pred_set.INTER s u = EMPTY
+==> t = u"
by (import prob_extra UNION_DISJOINT_SPLIT)
-lemma GSPEC_DEF_ALT: "ALL f::'a::type => 'b::type * bool.
- GSPEC f = (%v::'b::type. EX x::'a::type. (v, True) = f x)"
+lemma GSPEC_DEF_ALT: "GSPEC (f::'a => 'b * bool) = (%v::'b. EX x::'a. (v, True) = f x)"
by (import prob_extra GSPEC_DEF_ALT)
;end_setup
@@ -365,158 +219,56 @@
alg_twin :: "bool list => bool list => bool"
defs
- alg_twin_primdef: "alg_twin ==
-%(x::bool list) y::bool list.
- EX l::bool list. x = SNOC True l & y = SNOC False l"
+ alg_twin_primdef: "alg_twin == %x y. EX l. x = SNOC True l & y = SNOC False l"
-lemma alg_twin_def: "ALL (x::bool list) y::bool list.
- alg_twin x y = (EX l::bool list. x = SNOC True l & y = SNOC False l)"
+lemma alg_twin_def: "alg_twin x y = (EX l. x = SNOC True l & y = SNOC False l)"
by (import prob_canon alg_twin_def)
-definition alg_order_tupled :: "bool list * bool list => bool" where
- "(op ==::(bool list * bool list => bool)
- => (bool list * bool list => bool) => prop)
- (alg_order_tupled::bool list * bool list => bool)
- ((WFREC::(bool list * bool list => bool list * bool list => bool)
- => ((bool list * bool list => bool)
- => bool list * bool list => bool)
- => bool list * bool list => bool)
- ((Eps::((bool list * bool list => bool list * bool list => bool) => bool)
- => bool list * bool list => bool list * bool list => bool)
- (%R::bool list * bool list => bool list * bool list => bool.
- (op &::bool => bool => bool)
- ((WF::(bool list * bool list => bool list * bool list => bool)
- => bool)
- R)
- ((All::(bool => bool) => bool)
- (%h'::bool.
- (All::(bool => bool) => bool)
- (%h::bool.
- (All::(bool list => bool) => bool)
- (%t'::bool list.
- (All::(bool list => bool) => bool)
- (%t::bool list.
- R ((Pair::bool list
- => bool list => bool list * bool list)
- t t')
- ((Pair::bool list
- => bool list => bool list * bool list)
- ((op #::bool => bool list => bool list) h
- t)
- ((op #::bool => bool list => bool list) h'
- t')))))))))
- (%alg_order_tupled::bool list * bool list => bool.
- (split::(bool list => bool list => bool)
- => bool list * bool list => bool)
- (%(v::bool list) v1::bool list.
- (list_case::bool
- => (bool => bool list => bool) => bool list => bool)
- ((list_case::bool
- => (bool => bool list => bool)
- => bool list => bool)
- (True::bool) (%(v8::bool) v9::bool list. True::bool) v1)
- (%(v4::bool) v5::bool list.
- (list_case::bool
- => (bool => bool list => bool)
- => bool list => bool)
- (False::bool)
- (%(v10::bool) v11::bool list.
- (op |::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((op =::bool => bool => bool) v4 (True::bool))
- ((op =::bool => bool => bool) v10 (False::bool)))
- ((op &::bool => bool => bool)
- ((op =::bool => bool => bool) v4 v10)
- (alg_order_tupled
- ((Pair::bool list
- => bool list => bool list * bool list)
- v5 v11))))
- v1)
- v)))"
+definition
+ alg_order_tupled :: "bool list * bool list => bool" where
+ "alg_order_tupled ==
+WFREC (SOME R. WF R & (ALL h' h t' t. R (t, t') (h # t, h' # t')))
+ (%alg_order_tupled (v, v1).
+ case v of [] => case v1 of [] => True | _ => True
+ | v4 # v5 =>
+ case v1 of [] => False
+ | v10 # v11 =>
+ v4 = True & v10 = False |
+ v4 = v10 & alg_order_tupled (v5, v11))"
-lemma alg_order_tupled_primitive_def: "(op =::(bool list * bool list => bool)
- => (bool list * bool list => bool) => bool)
- (alg_order_tupled::bool list * bool list => bool)
- ((WFREC::(bool list * bool list => bool list * bool list => bool)
- => ((bool list * bool list => bool)
- => bool list * bool list => bool)
- => bool list * bool list => bool)
- ((Eps::((bool list * bool list => bool list * bool list => bool) => bool)
- => bool list * bool list => bool list * bool list => bool)
- (%R::bool list * bool list => bool list * bool list => bool.
- (op &::bool => bool => bool)
- ((WF::(bool list * bool list => bool list * bool list => bool)
- => bool)
- R)
- ((All::(bool => bool) => bool)
- (%h'::bool.
- (All::(bool => bool) => bool)
- (%h::bool.
- (All::(bool list => bool) => bool)
- (%t'::bool list.
- (All::(bool list => bool) => bool)
- (%t::bool list.
- R ((Pair::bool list
- => bool list => bool list * bool list)
- t t')
- ((Pair::bool list
- => bool list => bool list * bool list)
- ((op #::bool => bool list => bool list) h
- t)
- ((op #::bool => bool list => bool list) h'
- t')))))))))
- (%alg_order_tupled::bool list * bool list => bool.
- (split::(bool list => bool list => bool)
- => bool list * bool list => bool)
- (%(v::bool list) v1::bool list.
- (list_case::bool
- => (bool => bool list => bool) => bool list => bool)
- ((list_case::bool
- => (bool => bool list => bool)
- => bool list => bool)
- (True::bool) (%(v8::bool) v9::bool list. True::bool) v1)
- (%(v4::bool) v5::bool list.
- (list_case::bool
- => (bool => bool list => bool)
- => bool list => bool)
- (False::bool)
- (%(v10::bool) v11::bool list.
- (op |::bool => bool => bool)
- ((op &::bool => bool => bool)
- ((op =::bool => bool => bool) v4 (True::bool))
- ((op =::bool => bool => bool) v10 (False::bool)))
- ((op &::bool => bool => bool)
- ((op =::bool => bool => bool) v4 v10)
- (alg_order_tupled
- ((Pair::bool list
- => bool list => bool list * bool list)
- v5 v11))))
- v1)
- v)))"
+lemma alg_order_tupled_primitive_def: "alg_order_tupled =
+WFREC (SOME R. WF R & (ALL h' h t' t. R (t, t') (h # t, h' # t')))
+ (%alg_order_tupled (v, v1).
+ case v of [] => case v1 of [] => True | _ => True
+ | v4 # v5 =>
+ case v1 of [] => False
+ | v10 # v11 =>
+ v4 = True & v10 = False |
+ v4 = v10 & alg_order_tupled (v5, v11))"
by (import prob_canon alg_order_tupled_primitive_def)
consts
alg_order :: "bool list => bool list => bool"
defs
- alg_order_primdef: "alg_order == %(x::bool list) x1::bool list. alg_order_tupled (x, x1)"
+ alg_order_primdef: "alg_order == %x x1. alg_order_tupled (x, x1)"
-lemma alg_order_curried_def: "ALL (x::bool list) x1::bool list. alg_order x x1 = alg_order_tupled (x, x1)"
+lemma alg_order_curried_def: "alg_order x x1 = alg_order_tupled (x, x1)"
by (import prob_canon alg_order_curried_def)
-lemma alg_order_ind: "ALL P::bool list => bool list => bool.
- (ALL (x::bool) xa::bool list. P [] (x # xa)) &
- P [] [] &
- (ALL (x::bool) xa::bool list. P (x # xa) []) &
- (ALL (x::bool) (xa::bool list) (xb::bool) xc::bool list.
- P xa xc --> P (x # xa) (xb # xc)) -->
- (ALL x::bool list. All (P x))"
+lemma alg_order_ind: "(ALL (x::bool) xa::bool list.
+ (P::bool list => bool list => bool) [] (x # xa)) &
+P [] [] &
+(ALL (x::bool) xa::bool list. P (x # xa) []) &
+(ALL (x::bool) (xa::bool list) (xb::bool) xc::bool list.
+ P xa xc --> P (x # xa) (xb # xc))
+==> P (x::bool list) (xa::bool list)"
by (import prob_canon alg_order_ind)
-lemma alg_order_def: "alg_order [] ((v6::bool) # (v7::bool list)) = True &
+lemma alg_order_def: "alg_order [] (v6 # v7) = True &
alg_order [] [] = True &
-alg_order ((v2::bool) # (v3::bool list)) [] = False &
-alg_order ((h::bool) # (t::bool list)) ((h'::bool) # (t'::bool list)) =
+alg_order (v2 # v3) [] = False &
+alg_order (h # t) (h' # t') =
(h = True & h' = False | h = h' & alg_order t t')"
by (import prob_canon alg_order_def)
@@ -525,42 +277,28 @@
defs
alg_sorted_primdef: "alg_sorted ==
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_sorted::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_sorted.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- alg_order v2 v6 & alg_sorted (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. alg_order v2 v6 & alg_sorted (v6 # v7))))"
lemma alg_sorted_primitive_def: "alg_sorted =
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_sorted::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_sorted.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- alg_order v2 v6 & alg_sorted (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. alg_order v2 v6 & alg_sorted (v6 # v7))))"
by (import prob_canon alg_sorted_primitive_def)
-lemma alg_sorted_ind: "ALL P::bool list list => bool.
- (ALL (x::bool list) (y::bool list) z::bool list list.
- P (y # z) --> P (x # y # z)) &
- (ALL v::bool list. P [v]) & P [] -->
- All P"
+lemma alg_sorted_ind: "(ALL (x::bool list) (y::bool list) z::bool list list.
+ (P::bool list list => bool) (y # z) --> P (x # y # z)) &
+(ALL v::bool list. P [v]) & P []
+==> P (x::bool list list)"
by (import prob_canon alg_sorted_ind)
-lemma alg_sorted_def: "alg_sorted ((x::bool list) # (y::bool list) # (z::bool list list)) =
-(alg_order x y & alg_sorted (y # z)) &
-alg_sorted [v::bool list] = True & alg_sorted [] = True"
+lemma alg_sorted_def: "alg_sorted (x # y # z) = (alg_order x y & alg_sorted (y # z)) &
+alg_sorted [v] = True & alg_sorted [] = True"
by (import prob_canon alg_sorted_def)
consts
@@ -568,42 +306,28 @@
defs
alg_prefixfree_primdef: "alg_prefixfree ==
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_prefixfree::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_prefixfree.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- ~ IS_PREFIX v6 v2 & alg_prefixfree (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. ~ IS_PREFIX v6 v2 & alg_prefixfree (v6 # v7))))"
lemma alg_prefixfree_primitive_def: "alg_prefixfree =
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_prefixfree::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_prefixfree.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- ~ IS_PREFIX v6 v2 & alg_prefixfree (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. ~ IS_PREFIX v6 v2 & alg_prefixfree (v6 # v7))))"
by (import prob_canon alg_prefixfree_primitive_def)
-lemma alg_prefixfree_ind: "ALL P::bool list list => bool.
- (ALL (x::bool list) (y::bool list) z::bool list list.
- P (y # z) --> P (x # y # z)) &
- (ALL v::bool list. P [v]) & P [] -->
- All P"
+lemma alg_prefixfree_ind: "(ALL (x::bool list) (y::bool list) z::bool list list.
+ (P::bool list list => bool) (y # z) --> P (x # y # z)) &
+(ALL v::bool list. P [v]) & P []
+==> P (x::bool list list)"
by (import prob_canon alg_prefixfree_ind)
-lemma alg_prefixfree_def: "alg_prefixfree ((x::bool list) # (y::bool list) # (z::bool list list)) =
-(~ IS_PREFIX y x & alg_prefixfree (y # z)) &
-alg_prefixfree [v::bool list] = True & alg_prefixfree [] = True"
+lemma alg_prefixfree_def: "alg_prefixfree (x # y # z) = (~ IS_PREFIX y x & alg_prefixfree (y # z)) &
+alg_prefixfree [v] = True & alg_prefixfree [] = True"
by (import prob_canon alg_prefixfree_def)
consts
@@ -611,60 +335,44 @@
defs
alg_twinfree_primdef: "alg_twinfree ==
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_twinfree::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_twinfree.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- ~ alg_twin v2 v6 & alg_twinfree (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. ~ alg_twin v2 v6 & alg_twinfree (v6 # v7))))"
lemma alg_twinfree_primitive_def: "alg_twinfree =
-WFREC
- (SOME R::bool list list => bool list list => bool.
- WF R &
- (ALL (x::bool list) (z::bool list list) y::bool list.
- R (y # z) (x # y # z)))
- (%alg_twinfree::bool list list => bool.
+WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
+ (%alg_twinfree.
list_case True
- (%v2::bool list.
- list_case True
- (%(v6::bool list) v7::bool list list.
- ~ alg_twin v2 v6 & alg_twinfree (v6 # v7))))"
+ (%v2. list_case True
+ (%v6 v7. ~ alg_twin v2 v6 & alg_twinfree (v6 # v7))))"
by (import prob_canon alg_twinfree_primitive_def)
-lemma alg_twinfree_ind: "ALL P::bool list list => bool.
- (ALL (x::bool list) (y::bool list) z::bool list list.
- P (y # z) --> P (x # y # z)) &
- (ALL v::bool list. P [v]) & P [] -->
- All P"
+lemma alg_twinfree_ind: "(ALL (x::bool list) (y::bool list) z::bool list list.
+ (P::bool list list => bool) (y # z) --> P (x # y # z)) &
+(ALL v::bool list. P [v]) & P []
+==> P (x::bool list list)"
by (import prob_canon alg_twinfree_ind)
-lemma alg_twinfree_def: "alg_twinfree ((x::bool list) # (y::bool list) # (z::bool list list)) =
-(~ alg_twin x y & alg_twinfree (y # z)) &
-alg_twinfree [v::bool list] = True & alg_twinfree [] = True"
+lemma alg_twinfree_def: "alg_twinfree (x # y # z) = (~ alg_twin x y & alg_twinfree (y # z)) &
+alg_twinfree [v] = True & alg_twinfree [] = True"
by (import prob_canon alg_twinfree_def)
consts
alg_longest :: "bool list list => nat"
defs
- alg_longest_primdef: "alg_longest ==
-FOLDR (%(h::bool list) t::nat. if t <= length h then length h else t) 0"
+ alg_longest_primdef: "alg_longest == FOLDR (%h t. if t <= length h then length h else t) 0"
-lemma alg_longest_def: "alg_longest =
-FOLDR (%(h::bool list) t::nat. if t <= length h then length h else t) 0"
+lemma alg_longest_def: "alg_longest = FOLDR (%h t. if t <= length h then length h else t) 0"
by (import prob_canon alg_longest_def)
consts
alg_canon_prefs :: "bool list => bool list list => bool list list"
-specification (alg_canon_prefs_primdef: alg_canon_prefs) alg_canon_prefs_def: "(ALL l::bool list. alg_canon_prefs l [] = [l]) &
-(ALL (l::bool list) (h::bool list) t::bool list list.
+specification (alg_canon_prefs_primdef: alg_canon_prefs) alg_canon_prefs_def: "(ALL l. alg_canon_prefs l [] = [l]) &
+(ALL l h t.
alg_canon_prefs l (h # t) =
(if IS_PREFIX h l then alg_canon_prefs l t else l # h # t))"
by (import prob_canon alg_canon_prefs_def)
@@ -672,8 +380,8 @@
consts
alg_canon_find :: "bool list => bool list list => bool list list"
-specification (alg_canon_find_primdef: alg_canon_find) alg_canon_find_def: "(ALL l::bool list. alg_canon_find l [] = [l]) &
-(ALL (l::bool list) (h::bool list) t::bool list list.
+specification (alg_canon_find_primdef: alg_canon_find) alg_canon_find_def: "(ALL l. alg_canon_find l [] = [l]) &
+(ALL l h t.
alg_canon_find l (h # t) =
(if alg_order h l
then if IS_PREFIX l h then h # t else h # alg_canon_find l t
@@ -692,8 +400,8 @@
consts
alg_canon_merge :: "bool list => bool list list => bool list list"
-specification (alg_canon_merge_primdef: alg_canon_merge) alg_canon_merge_def: "(ALL l::bool list. alg_canon_merge l [] = [l]) &
-(ALL (l::bool list) (h::bool list) t::bool list list.
+specification (alg_canon_merge_primdef: alg_canon_merge) alg_canon_merge_def: "(ALL l. alg_canon_merge l [] = [l]) &
+(ALL l h t.
alg_canon_merge l (h # t) =
(if alg_twin l h then alg_canon_merge (butlast h) t else l # h # t))"
by (import prob_canon alg_canon_merge_def)
@@ -711,365 +419,308 @@
alg_canon :: "bool list list => bool list list"
defs
- alg_canon_primdef: "alg_canon == %l::bool list list. alg_canon2 (alg_canon1 l)"
+ alg_canon_primdef: "alg_canon == %l. alg_canon2 (alg_canon1 l)"
-lemma alg_canon_def: "ALL l::bool list list. alg_canon l = alg_canon2 (alg_canon1 l)"
+lemma alg_canon_def: "alg_canon l = alg_canon2 (alg_canon1 l)"
by (import prob_canon alg_canon_def)
consts
algebra_canon :: "bool list list => bool"
defs
- algebra_canon_primdef: "algebra_canon == %l::bool list list. alg_canon l = l"
+ algebra_canon_primdef: "algebra_canon == %l. alg_canon l = l"
-lemma algebra_canon_def: "ALL l::bool list list. algebra_canon l = (alg_canon l = l)"
+lemma algebra_canon_def: "algebra_canon l = (alg_canon l = l)"
by (import prob_canon algebra_canon_def)
-lemma ALG_TWIN_NIL: "ALL l::bool list. ~ alg_twin l [] & ~ alg_twin [] l"
+lemma ALG_TWIN_NIL: "~ alg_twin l [] & ~ alg_twin [] l"
by (import prob_canon ALG_TWIN_NIL)
-lemma ALG_TWIN_SING: "ALL (x::bool) l::bool list.
- alg_twin [x] l = (x = True & l = [False]) &
- alg_twin l [x] = (l = [True] & x = False)"
+lemma ALG_TWIN_SING: "alg_twin [x] l = (x = True & l = [False]) &
+alg_twin l [x] = (l = [True] & x = False)"
by (import prob_canon ALG_TWIN_SING)
-lemma ALG_TWIN_CONS: "ALL (x::bool) (y::bool) (z::bool list) (h::bool) t::bool list.
- alg_twin (x # y # z) (h # t) = (x = h & alg_twin (y # z) t) &
- alg_twin (h # t) (x # y # z) = (x = h & alg_twin t (y # z))"
+lemma ALG_TWIN_CONS: "alg_twin (x # y # z) (h # t) = (x = h & alg_twin (y # z) t) &
+alg_twin (h # t) (x # y # z) = (x = h & alg_twin t (y # z))"
by (import prob_canon ALG_TWIN_CONS)
-lemma ALG_TWIN_REDUCE: "ALL (h::bool) (t::bool list) t'::bool list.
- alg_twin (h # t) (h # t') = alg_twin t t'"
+lemma ALG_TWIN_REDUCE: "alg_twin (h # t) (h # t') = alg_twin t t'"
by (import prob_canon ALG_TWIN_REDUCE)
-lemma ALG_TWINS_PREFIX: "ALL (x::bool list) l::bool list.
- IS_PREFIX x l -->
- x = l | IS_PREFIX x (SNOC True l) | IS_PREFIX x (SNOC False l)"
+lemma ALG_TWINS_PREFIX: "IS_PREFIX x l
+==> x = l | IS_PREFIX x (SNOC True l) | IS_PREFIX x (SNOC False l)"
by (import prob_canon ALG_TWINS_PREFIX)
-lemma ALG_ORDER_NIL: "ALL x::bool list. alg_order [] x & alg_order x [] = (x = [])"
+lemma ALG_ORDER_NIL: "alg_order [] x & alg_order x [] = (x = [])"
by (import prob_canon ALG_ORDER_NIL)
-lemma ALG_ORDER_REFL: "ALL x::bool list. alg_order x x"
+lemma ALG_ORDER_REFL: "alg_order x x"
by (import prob_canon ALG_ORDER_REFL)
-lemma ALG_ORDER_ANTISYM: "ALL (x::bool list) y::bool list. alg_order x y & alg_order y x --> x = y"
+lemma ALG_ORDER_ANTISYM: "alg_order x y & alg_order y x ==> x = y"
by (import prob_canon ALG_ORDER_ANTISYM)
-lemma ALG_ORDER_TRANS: "ALL (x::bool list) (y::bool list) z::bool list.
- alg_order x y & alg_order y z --> alg_order x z"
+lemma ALG_ORDER_TRANS: "alg_order x y & alg_order y z ==> alg_order x z"
by (import prob_canon ALG_ORDER_TRANS)
-lemma ALG_ORDER_TOTAL: "ALL (x::bool list) y::bool list. alg_order x y | alg_order y x"
+lemma ALG_ORDER_TOTAL: "alg_order x y | alg_order y x"
by (import prob_canon ALG_ORDER_TOTAL)
-lemma ALG_ORDER_PREFIX: "ALL (x::bool list) y::bool list. IS_PREFIX y x --> alg_order x y"
+lemma ALG_ORDER_PREFIX: "IS_PREFIX y x ==> alg_order x y"
by (import prob_canon ALG_ORDER_PREFIX)
-lemma ALG_ORDER_PREFIX_ANTI: "ALL (x::bool list) y::bool list. alg_order x y & IS_PREFIX x y --> x = y"
+lemma ALG_ORDER_PREFIX_ANTI: "alg_order x y & IS_PREFIX x y ==> x = y"
by (import prob_canon ALG_ORDER_PREFIX_ANTI)
-lemma ALG_ORDER_PREFIX_MONO: "ALL (x::bool list) (y::bool list) z::bool list.
- alg_order x y & alg_order y z & IS_PREFIX z x --> IS_PREFIX y x"
+lemma ALG_ORDER_PREFIX_MONO: "alg_order x y & alg_order y z & IS_PREFIX z x ==> IS_PREFIX y x"
by (import prob_canon ALG_ORDER_PREFIX_MONO)
-lemma ALG_ORDER_PREFIX_TRANS: "ALL (x::bool list) (y::bool list) z::bool list.
- alg_order x y & IS_PREFIX y z --> alg_order x z | IS_PREFIX x z"
+lemma ALG_ORDER_PREFIX_TRANS: "alg_order x y & IS_PREFIX y z ==> alg_order x z | IS_PREFIX x z"
by (import prob_canon ALG_ORDER_PREFIX_TRANS)
-lemma ALG_ORDER_SNOC: "ALL (x::bool) l::bool list. ~ alg_order (SNOC x l) l"
+lemma ALG_ORDER_SNOC: "~ alg_order (SNOC x l) l"
by (import prob_canon ALG_ORDER_SNOC)
-lemma ALG_SORTED_MIN: "ALL (h::bool list) t::bool list list.
- alg_sorted (h # t) --> (ALL x::bool list. x mem t --> alg_order h x)"
+lemma ALG_SORTED_MIN: "[| alg_sorted (h # t); List.member t x |] ==> alg_order h x"
by (import prob_canon ALG_SORTED_MIN)
-lemma ALG_SORTED_DEF_ALT: "ALL (h::bool list) t::bool list list.
- alg_sorted (h # t) =
- ((ALL x::bool list. x mem t --> alg_order h x) & alg_sorted t)"
+lemma ALG_SORTED_DEF_ALT: "alg_sorted (h # t) =
+((ALL x. List.member t x --> alg_order h x) & alg_sorted t)"
by (import prob_canon ALG_SORTED_DEF_ALT)
-lemma ALG_SORTED_TL: "ALL (h::bool list) t::bool list list. alg_sorted (h # t) --> alg_sorted t"
+lemma ALG_SORTED_TL: "alg_sorted (h # t) ==> alg_sorted t"
by (import prob_canon ALG_SORTED_TL)
-lemma ALG_SORTED_MONO: "ALL (x::bool list) (y::bool list) z::bool list list.
- alg_sorted (x # y # z) --> alg_sorted (x # z)"
+lemma ALG_SORTED_MONO: "alg_sorted (x # y # z) ==> alg_sorted (x # z)"
by (import prob_canon ALG_SORTED_MONO)
-lemma ALG_SORTED_TLS: "ALL (l::bool list list) b::bool. alg_sorted (map (op # b) l) = alg_sorted l"
+lemma ALG_SORTED_TLS: "alg_sorted (map (op # b) l) = alg_sorted l"
by (import prob_canon ALG_SORTED_TLS)
-lemma ALG_SORTED_STEP: "ALL (l1::bool list list) l2::bool list list.
- alg_sorted (map (op # True) l1 @ map (op # False) l2) =
- (alg_sorted l1 & alg_sorted l2)"
+lemma ALG_SORTED_STEP: "alg_sorted (map (op # True) l1 @ map (op # False) l2) =
+(alg_sorted l1 & alg_sorted l2)"
by (import prob_canon ALG_SORTED_STEP)
-lemma ALG_SORTED_APPEND: "ALL (h::bool list) (h'::bool list) (t::bool list list) t'::bool list list.
- alg_sorted ((h # t) @ h' # t') =
- (alg_sorted (h # t) & alg_sorted (h' # t') & alg_order (last (h # t)) h')"
+lemma ALG_SORTED_APPEND: "alg_sorted ((h # t) @ h' # t') =
+(alg_sorted (h # t) & alg_sorted (h' # t') & alg_order (last (h # t)) h')"
by (import prob_canon ALG_SORTED_APPEND)
-lemma ALG_SORTED_FILTER: "ALL (P::bool list => bool) b::bool list list.
- alg_sorted b --> alg_sorted (filter P b)"
+lemma ALG_SORTED_FILTER: "alg_sorted b ==> alg_sorted (filter P b)"
by (import prob_canon ALG_SORTED_FILTER)
-lemma ALG_PREFIXFREE_TL: "ALL (h::bool list) t::bool list list.
- alg_prefixfree (h # t) --> alg_prefixfree t"
+lemma ALG_PREFIXFREE_TL: "alg_prefixfree (h # t) ==> alg_prefixfree t"
by (import prob_canon ALG_PREFIXFREE_TL)
-lemma ALG_PREFIXFREE_MONO: "ALL (x::bool list) (y::bool list) z::bool list list.
- alg_sorted (x # y # z) & alg_prefixfree (x # y # z) -->
- alg_prefixfree (x # z)"
+lemma ALG_PREFIXFREE_MONO: "alg_sorted (x # y # z) & alg_prefixfree (x # y # z)
+==> alg_prefixfree (x # z)"
by (import prob_canon ALG_PREFIXFREE_MONO)
-lemma ALG_PREFIXFREE_ELT: "ALL (h::bool list) t::bool list list.
- alg_sorted (h # t) & alg_prefixfree (h # t) -->
- (ALL x::bool list. x mem t --> ~ IS_PREFIX x h & ~ IS_PREFIX h x)"
+lemma ALG_PREFIXFREE_ELT: "[| alg_sorted (h # t) & alg_prefixfree (h # t); List.member t x |]
+==> ~ IS_PREFIX x h & ~ IS_PREFIX h x"
by (import prob_canon ALG_PREFIXFREE_ELT)
-lemma ALG_PREFIXFREE_TLS: "ALL (l::bool list list) b::bool.
- alg_prefixfree (map (op # b) l) = alg_prefixfree l"
+lemma ALG_PREFIXFREE_TLS: "alg_prefixfree (map (op # b) l) = alg_prefixfree l"
by (import prob_canon ALG_PREFIXFREE_TLS)
-lemma ALG_PREFIXFREE_STEP: "ALL (l1::bool list list) l2::bool list list.
- alg_prefixfree (map (op # True) l1 @ map (op # False) l2) =
- (alg_prefixfree l1 & alg_prefixfree l2)"
+lemma ALG_PREFIXFREE_STEP: "alg_prefixfree (map (op # True) l1 @ map (op # False) l2) =
+(alg_prefixfree l1 & alg_prefixfree l2)"
by (import prob_canon ALG_PREFIXFREE_STEP)
-lemma ALG_PREFIXFREE_APPEND: "ALL (h::bool list) (h'::bool list) (t::bool list list) t'::bool list list.
- alg_prefixfree ((h # t) @ h' # t') =
- (alg_prefixfree (h # t) &
- alg_prefixfree (h' # t') & ~ IS_PREFIX h' (last (h # t)))"
+lemma ALG_PREFIXFREE_APPEND: "alg_prefixfree ((h # t) @ h' # t') =
+(alg_prefixfree (h # t) &
+ alg_prefixfree (h' # t') & ~ IS_PREFIX h' (last (h # t)))"
by (import prob_canon ALG_PREFIXFREE_APPEND)
-lemma ALG_PREFIXFREE_FILTER: "ALL (P::bool list => bool) b::bool list list.
- alg_sorted b & alg_prefixfree b --> alg_prefixfree (filter P b)"
+lemma ALG_PREFIXFREE_FILTER: "alg_sorted b & alg_prefixfree b ==> alg_prefixfree (filter P b)"
by (import prob_canon ALG_PREFIXFREE_FILTER)
-lemma ALG_TWINFREE_TL: "ALL (h::bool list) t::bool list list.
- alg_twinfree (h # t) --> alg_twinfree t"
+lemma ALG_TWINFREE_TL: "alg_twinfree (h # t) ==> alg_twinfree t"
by (import prob_canon ALG_TWINFREE_TL)
-lemma ALG_TWINFREE_TLS: "ALL (l::bool list list) b::bool.
- alg_twinfree (map (op # b) l) = alg_twinfree l"
+lemma ALG_TWINFREE_TLS: "alg_twinfree (map (op # b) l) = alg_twinfree l"
by (import prob_canon ALG_TWINFREE_TLS)
-lemma ALG_TWINFREE_STEP1: "ALL (l1::bool list list) l2::bool list list.
- alg_twinfree (map (op # True) l1 @ map (op # False) l2) -->
- alg_twinfree l1 & alg_twinfree l2"
+lemma ALG_TWINFREE_STEP1: "alg_twinfree (map (op # True) l1 @ map (op # False) l2)
+==> alg_twinfree l1 & alg_twinfree l2"
by (import prob_canon ALG_TWINFREE_STEP1)
-lemma ALG_TWINFREE_STEP2: "ALL (l1::bool list list) l2::bool list list.
- (~ [] mem l1 | ~ [] mem l2) & alg_twinfree l1 & alg_twinfree l2 -->
- alg_twinfree (map (op # True) l1 @ map (op # False) l2)"
+lemma ALG_TWINFREE_STEP2: "(~ List.member l1 [] | ~ List.member l2 []) &
+alg_twinfree l1 & alg_twinfree l2
+==> alg_twinfree (map (op # True) l1 @ map (op # False) l2)"
by (import prob_canon ALG_TWINFREE_STEP2)
-lemma ALG_TWINFREE_STEP: "ALL (l1::bool list list) l2::bool list list.
- ~ [] mem l1 | ~ [] mem l2 -->
- alg_twinfree (map (op # True) l1 @ map (op # False) l2) =
- (alg_twinfree l1 & alg_twinfree l2)"
+lemma ALG_TWINFREE_STEP: "~ List.member l1 [] | ~ List.member l2 []
+==> alg_twinfree (map (op # True) l1 @ map (op # False) l2) =
+ (alg_twinfree l1 & alg_twinfree l2)"
by (import prob_canon ALG_TWINFREE_STEP)
-lemma ALG_LONGEST_HD: "ALL (h::bool list) t::bool list list. length h <= alg_longest (h # t)"
+lemma ALG_LONGEST_HD: "length h <= alg_longest (h # t)"
by (import prob_canon ALG_LONGEST_HD)
-lemma ALG_LONGEST_TL: "ALL (h::bool list) t::bool list list. alg_longest t <= alg_longest (h # t)"
+lemma ALG_LONGEST_TL: "alg_longest t <= alg_longest (h # t)"
by (import prob_canon ALG_LONGEST_TL)
-lemma ALG_LONGEST_TLS: "ALL (h::bool list) (t::bool list list) b::bool.
- alg_longest (map (op # b) (h # t)) = Suc (alg_longest (h # t))"
+lemma ALG_LONGEST_TLS: "alg_longest (map (op # b) (h # t)) = Suc (alg_longest (h # t))"
by (import prob_canon ALG_LONGEST_TLS)
-lemma ALG_LONGEST_APPEND: "ALL (l1::bool list list) l2::bool list list.
- alg_longest l1 <= alg_longest (l1 @ l2) &
- alg_longest l2 <= alg_longest (l1 @ l2)"
+lemma ALG_LONGEST_APPEND: "alg_longest l1 <= alg_longest (l1 @ l2) &
+alg_longest l2 <= alg_longest (l1 @ l2)"
by (import prob_canon ALG_LONGEST_APPEND)
-lemma ALG_CANON_PREFS_HD: "ALL (l::bool list) b::bool list list. hd (alg_canon_prefs l b) = l"
+lemma ALG_CANON_PREFS_HD: "hd (alg_canon_prefs l b) = l"
by (import prob_canon ALG_CANON_PREFS_HD)
-lemma ALG_CANON_PREFS_DELETES: "ALL (l::bool list) (b::bool list list) x::bool list.
- x mem alg_canon_prefs l b --> x mem l # b"
+lemma ALG_CANON_PREFS_DELETES: "List.member (alg_canon_prefs l b) x ==> List.member (l # b) x"
by (import prob_canon ALG_CANON_PREFS_DELETES)
-lemma ALG_CANON_PREFS_SORTED: "ALL (l::bool list) b::bool list list.
- alg_sorted (l # b) --> alg_sorted (alg_canon_prefs l b)"
+lemma ALG_CANON_PREFS_SORTED: "alg_sorted (l # b) ==> alg_sorted (alg_canon_prefs l b)"
by (import prob_canon ALG_CANON_PREFS_SORTED)
-lemma ALG_CANON_PREFS_PREFIXFREE: "ALL (l::bool list) b::bool list list.
- alg_sorted b & alg_prefixfree b --> alg_prefixfree (alg_canon_prefs l b)"
+lemma ALG_CANON_PREFS_PREFIXFREE: "alg_sorted b & alg_prefixfree b ==> alg_prefixfree (alg_canon_prefs l b)"
by (import prob_canon ALG_CANON_PREFS_PREFIXFREE)
-lemma ALG_CANON_PREFS_CONSTANT: "ALL (l::bool list) b::bool list list.
- alg_prefixfree (l # b) --> alg_canon_prefs l b = l # b"
+lemma ALG_CANON_PREFS_CONSTANT: "alg_prefixfree (l # b) ==> alg_canon_prefs l b = l # b"
by (import prob_canon ALG_CANON_PREFS_CONSTANT)
-lemma ALG_CANON_FIND_HD: "ALL (l::bool list) (h::bool list) t::bool list list.
- hd (alg_canon_find l (h # t)) = l | hd (alg_canon_find l (h # t)) = h"
+lemma ALG_CANON_FIND_HD: "hd (alg_canon_find l (h # t)) = l | hd (alg_canon_find l (h # t)) = h"
by (import prob_canon ALG_CANON_FIND_HD)
-lemma ALG_CANON_FIND_DELETES: "ALL (l::bool list) (b::bool list list) x::bool list.
- x mem alg_canon_find l b --> x mem l # b"
+lemma ALG_CANON_FIND_DELETES: "List.member (alg_canon_find l b) x ==> List.member (l # b) x"
by (import prob_canon ALG_CANON_FIND_DELETES)
-lemma ALG_CANON_FIND_SORTED: "ALL (l::bool list) b::bool list list.
- alg_sorted b --> alg_sorted (alg_canon_find l b)"
+lemma ALG_CANON_FIND_SORTED: "alg_sorted b ==> alg_sorted (alg_canon_find l b)"
by (import prob_canon ALG_CANON_FIND_SORTED)
-lemma ALG_CANON_FIND_PREFIXFREE: "ALL (l::bool list) b::bool list list.
- alg_sorted b & alg_prefixfree b --> alg_prefixfree (alg_canon_find l b)"
+lemma ALG_CANON_FIND_PREFIXFREE: "alg_sorted b & alg_prefixfree b ==> alg_prefixfree (alg_canon_find l b)"
by (import prob_canon ALG_CANON_FIND_PREFIXFREE)
-lemma ALG_CANON_FIND_CONSTANT: "ALL (l::bool list) b::bool list list.
- alg_sorted (l # b) & alg_prefixfree (l # b) -->
- alg_canon_find l b = l # b"
+lemma ALG_CANON_FIND_CONSTANT: "alg_sorted (l # b) & alg_prefixfree (l # b) ==> alg_canon_find l b = l # b"
by (import prob_canon ALG_CANON_FIND_CONSTANT)
-lemma ALG_CANON1_SORTED: "ALL x::bool list list. alg_sorted (alg_canon1 x)"
+lemma ALG_CANON1_SORTED: "alg_sorted (alg_canon1 x)"
by (import prob_canon ALG_CANON1_SORTED)
-lemma ALG_CANON1_PREFIXFREE: "ALL l::bool list list. alg_prefixfree (alg_canon1 l)"
+lemma ALG_CANON1_PREFIXFREE: "alg_prefixfree (alg_canon1 l)"
by (import prob_canon ALG_CANON1_PREFIXFREE)
-lemma ALG_CANON1_CONSTANT: "ALL l::bool list list. alg_sorted l & alg_prefixfree l --> alg_canon1 l = l"
+lemma ALG_CANON1_CONSTANT: "alg_sorted l & alg_prefixfree l ==> alg_canon1 l = l"
by (import prob_canon ALG_CANON1_CONSTANT)
-lemma ALG_CANON_MERGE_SORTED_PREFIXFREE_TWINFREE: "ALL (l::bool list) b::bool list list.
- alg_sorted (l # b) & alg_prefixfree (l # b) & alg_twinfree b -->
- alg_sorted (alg_canon_merge l b) &
- alg_prefixfree (alg_canon_merge l b) & alg_twinfree (alg_canon_merge l b)"
+lemma ALG_CANON_MERGE_SORTED_PREFIXFREE_TWINFREE: "alg_sorted (l # b) & alg_prefixfree (l # b) & alg_twinfree b
+==> alg_sorted (alg_canon_merge l b) &
+ alg_prefixfree (alg_canon_merge l b) &
+ alg_twinfree (alg_canon_merge l b)"
by (import prob_canon ALG_CANON_MERGE_SORTED_PREFIXFREE_TWINFREE)
-lemma ALG_CANON_MERGE_PREFIXFREE_PRESERVE: "ALL (l::bool list) (b::bool list list) h::bool list.
- (ALL x::bool list. x mem l # b --> ~ IS_PREFIX h x & ~ IS_PREFIX x h) -->
- (ALL x::bool list.
- x mem alg_canon_merge l b --> ~ IS_PREFIX h x & ~ IS_PREFIX x h)"
+lemma ALG_CANON_MERGE_PREFIXFREE_PRESERVE: "[| !!x. List.member (l # b) x ==> ~ IS_PREFIX h x & ~ IS_PREFIX x h;
+ List.member (alg_canon_merge l b) x |]
+==> ~ IS_PREFIX h x & ~ IS_PREFIX x h"
by (import prob_canon ALG_CANON_MERGE_PREFIXFREE_PRESERVE)
-lemma ALG_CANON_MERGE_SHORTENS: "ALL (l::bool list) (b::bool list list) x::bool list.
- x mem alg_canon_merge l b -->
- (EX y::bool list. y mem l # b & IS_PREFIX y x)"
+lemma ALG_CANON_MERGE_SHORTENS: "List.member (alg_canon_merge l b) x
+==> EX y. List.member (l # b) y & IS_PREFIX y x"
by (import prob_canon ALG_CANON_MERGE_SHORTENS)
-lemma ALG_CANON_MERGE_CONSTANT: "ALL (l::bool list) b::bool list list.
- alg_twinfree (l # b) --> alg_canon_merge l b = l # b"
+lemma ALG_CANON_MERGE_CONSTANT: "alg_twinfree (l # b) ==> alg_canon_merge l b = l # b"
by (import prob_canon ALG_CANON_MERGE_CONSTANT)
-lemma ALG_CANON2_PREFIXFREE_PRESERVE: "ALL (x::bool list list) xa::bool list.
- (ALL xb::bool list.
- xb mem x --> ~ IS_PREFIX xa xb & ~ IS_PREFIX xb xa) -->
- (ALL xb::bool list.
- xb mem alg_canon2 x --> ~ IS_PREFIX xa xb & ~ IS_PREFIX xb xa)"
+lemma ALG_CANON2_PREFIXFREE_PRESERVE: "[| !!xb. List.member x xb ==> ~ IS_PREFIX xa xb & ~ IS_PREFIX xb xa;
+ List.member (alg_canon2 x) xb |]
+==> ~ IS_PREFIX xa xb & ~ IS_PREFIX xb xa"
by (import prob_canon ALG_CANON2_PREFIXFREE_PRESERVE)
-lemma ALG_CANON2_SHORTENS: "ALL (x::bool list list) xa::bool list.
- xa mem alg_canon2 x --> (EX y::bool list. y mem x & IS_PREFIX y xa)"
+lemma ALG_CANON2_SHORTENS: "List.member (alg_canon2 x) xa ==> EX y. List.member x y & IS_PREFIX y xa"
by (import prob_canon ALG_CANON2_SHORTENS)
-lemma ALG_CANON2_SORTED_PREFIXFREE_TWINFREE: "ALL x::bool list list.
- alg_sorted x & alg_prefixfree x -->
- alg_sorted (alg_canon2 x) &
- alg_prefixfree (alg_canon2 x) & alg_twinfree (alg_canon2 x)"
+lemma ALG_CANON2_SORTED_PREFIXFREE_TWINFREE: "alg_sorted x & alg_prefixfree x
+==> alg_sorted (alg_canon2 x) &
+ alg_prefixfree (alg_canon2 x) & alg_twinfree (alg_canon2 x)"
by (import prob_canon ALG_CANON2_SORTED_PREFIXFREE_TWINFREE)
-lemma ALG_CANON2_CONSTANT: "ALL l::bool list list. alg_twinfree l --> alg_canon2 l = l"
+lemma ALG_CANON2_CONSTANT: "alg_twinfree l ==> alg_canon2 l = l"
by (import prob_canon ALG_CANON2_CONSTANT)
-lemma ALG_CANON_SORTED_PREFIXFREE_TWINFREE: "ALL l::bool list list.
- alg_sorted (alg_canon l) &
- alg_prefixfree (alg_canon l) & alg_twinfree (alg_canon l)"
+lemma ALG_CANON_SORTED_PREFIXFREE_TWINFREE: "alg_sorted (alg_canon l) &
+alg_prefixfree (alg_canon l) & alg_twinfree (alg_canon l)"
by (import prob_canon ALG_CANON_SORTED_PREFIXFREE_TWINFREE)
-lemma ALG_CANON_CONSTANT: "ALL l::bool list list.
- alg_sorted l & alg_prefixfree l & alg_twinfree l --> alg_canon l = l"
+lemma ALG_CANON_CONSTANT: "alg_sorted l & alg_prefixfree l & alg_twinfree l ==> alg_canon l = l"
by (import prob_canon ALG_CANON_CONSTANT)
-lemma ALG_CANON_IDEMPOT: "ALL l::bool list list. alg_canon (alg_canon l) = alg_canon l"
+lemma ALG_CANON_IDEMPOT: "alg_canon (alg_canon l) = alg_canon l"
by (import prob_canon ALG_CANON_IDEMPOT)
-lemma ALGEBRA_CANON_DEF_ALT: "ALL l::bool list list.
- algebra_canon l = (alg_sorted l & alg_prefixfree l & alg_twinfree l)"
+lemma ALGEBRA_CANON_DEF_ALT: "algebra_canon l = (alg_sorted l & alg_prefixfree l & alg_twinfree l)"
by (import prob_canon ALGEBRA_CANON_DEF_ALT)
-lemma ALGEBRA_CANON_BASIC: "algebra_canon [] &
-algebra_canon [[]] & (ALL x::bool list. algebra_canon [x])"
+lemma ALGEBRA_CANON_BASIC: "algebra_canon [] & algebra_canon [[]] & (ALL x. algebra_canon [x])"
by (import prob_canon ALGEBRA_CANON_BASIC)
-lemma ALG_CANON_BASIC: "alg_canon [] = [] &
-alg_canon [[]] = [[]] & (ALL x::bool list. alg_canon [x] = [x])"
+lemma ALG_CANON_BASIC: "alg_canon [] = [] & alg_canon [[]] = [[]] & (ALL x. alg_canon [x] = [x])"
by (import prob_canon ALG_CANON_BASIC)
-lemma ALGEBRA_CANON_TL: "ALL (h::bool list) t::bool list list.
- algebra_canon (h # t) --> algebra_canon t"
+lemma ALGEBRA_CANON_TL: "algebra_canon (h # t) ==> algebra_canon t"
by (import prob_canon ALGEBRA_CANON_TL)
-lemma ALGEBRA_CANON_NIL_MEM: "ALL l::bool list list. (algebra_canon l & [] mem l) = (l = [[]])"
+lemma ALGEBRA_CANON_NIL_MEM: "(algebra_canon l & List.member l []) = (l = [[]])"
by (import prob_canon ALGEBRA_CANON_NIL_MEM)
-lemma ALGEBRA_CANON_TLS: "ALL (l::bool list list) b::bool.
- algebra_canon (map (op # b) l) = algebra_canon l"
+lemma ALGEBRA_CANON_TLS: "algebra_canon (map (op # b) l) = algebra_canon l"
by (import prob_canon ALGEBRA_CANON_TLS)
-lemma ALGEBRA_CANON_STEP1: "ALL (l1::bool list list) l2::bool list list.
- algebra_canon (map (op # True) l1 @ map (op # False) l2) -->
- algebra_canon l1 & algebra_canon l2"
+lemma ALGEBRA_CANON_STEP1: "algebra_canon (map (op # True) l1 @ map (op # False) l2)
+==> algebra_canon l1 & algebra_canon l2"
by (import prob_canon ALGEBRA_CANON_STEP1)
-lemma ALGEBRA_CANON_STEP2: "ALL (l1::bool list list) l2::bool list list.
- (l1 ~= [[]] | l2 ~= [[]]) & algebra_canon l1 & algebra_canon l2 -->
- algebra_canon (map (op # True) l1 @ map (op # False) l2)"
+lemma ALGEBRA_CANON_STEP2: "(l1 ~= [[]] | l2 ~= [[]]) & algebra_canon l1 & algebra_canon l2
+==> algebra_canon (map (op # True) l1 @ map (op # False) l2)"
by (import prob_canon ALGEBRA_CANON_STEP2)
-lemma ALGEBRA_CANON_STEP: "ALL (l1::bool list list) l2::bool list list.
- l1 ~= [[]] | l2 ~= [[]] -->
- algebra_canon (map (op # True) l1 @ map (op # False) l2) =
- (algebra_canon l1 & algebra_canon l2)"
+lemma ALGEBRA_CANON_STEP: "l1 ~= [[]] | l2 ~= [[]]
+==> algebra_canon (map (op # True) l1 @ map (op # False) l2) =
+ (algebra_canon l1 & algebra_canon l2)"
by (import prob_canon ALGEBRA_CANON_STEP)
-lemma ALGEBRA_CANON_CASES_THM: "ALL l::bool list list.
- algebra_canon l -->
- l = [] |
- l = [[]] |
- (EX (l1::bool list list) l2::bool list list.
- algebra_canon l1 &
- algebra_canon l2 & l = map (op # True) l1 @ map (op # False) l2)"
+lemma ALGEBRA_CANON_CASES_THM: "algebra_canon l
+==> l = [] |
+ l = [[]] |
+ (EX l1 l2.
+ algebra_canon l1 &
+ algebra_canon l2 & l = map (op # True) l1 @ map (op # False) l2)"
by (import prob_canon ALGEBRA_CANON_CASES_THM)
-lemma ALGEBRA_CANON_CASES: "ALL P::bool list list => bool.
- P [] &
+lemma ALGEBRA_CANON_CASES: "[| P [] &
P [[]] &
- (ALL (l1::bool list list) l2::bool list list.
+ (ALL l1 l2.
algebra_canon l1 &
algebra_canon l2 &
algebra_canon (map (op # True) l1 @ map (op # False) l2) -->
- P (map (op # True) l1 @ map (op # False) l2)) -->
- (ALL l::bool list list. algebra_canon l --> P l)"
+ P (map (op # True) l1 @ map (op # False) l2));
+ algebra_canon l |]
+==> P l"
by (import prob_canon ALGEBRA_CANON_CASES)
-lemma ALGEBRA_CANON_INDUCTION: "ALL P::bool list list => bool.
- P [] &
+lemma ALGEBRA_CANON_INDUCTION: "[| P [] &
P [[]] &
- (ALL (l1::bool list list) l2::bool list list.
+ (ALL l1 l2.
algebra_canon l1 &
algebra_canon l2 &
P l1 &
P l2 & algebra_canon (map (op # True) l1 @ map (op # False) l2) -->
- P (map (op # True) l1 @ map (op # False) l2)) -->
- (ALL l::bool list list. algebra_canon l --> P l)"
+ P (map (op # True) l1 @ map (op # False) l2));
+ algebra_canon l |]
+==> P l"
by (import prob_canon ALGEBRA_CANON_INDUCTION)
-lemma MEM_NIL_STEP: "ALL (l1::bool list list) l2::bool list list.
- ~ [] mem map (op # True) l1 @ map (op # False) l2"
+lemma MEM_NIL_STEP: "~ List.member (map (op # True) l1 @ map (op # False) l2) []"
by (import prob_canon MEM_NIL_STEP)
-lemma ALG_SORTED_PREFIXFREE_MEM_NIL: "ALL l::bool list list.
- (alg_sorted l & alg_prefixfree l & [] mem l) = (l = [[]])"
+lemma ALG_SORTED_PREFIXFREE_MEM_NIL: "(alg_sorted l & alg_prefixfree l & List.member l []) = (l = [[]])"
by (import prob_canon ALG_SORTED_PREFIXFREE_MEM_NIL)
-lemma ALG_SORTED_PREFIXFREE_EQUALITY: "ALL (l::bool list list) l'::bool list list.
- (ALL x::bool list. x mem l = x mem l') &
- alg_sorted l & alg_sorted l' & alg_prefixfree l & alg_prefixfree l' -->
- l = l'"
+lemma ALG_SORTED_PREFIXFREE_EQUALITY: "(ALL x. List.member l x = List.member l' x) &
+alg_sorted l & alg_sorted l' & alg_prefixfree l & alg_prefixfree l'
+==> l = l'"
by (import prob_canon ALG_SORTED_PREFIXFREE_EQUALITY)
;end_setup
@@ -1080,34 +731,33 @@
SHD :: "(nat => bool) => bool"
defs
- SHD_primdef: "SHD == %f::nat => bool. f 0"
+ SHD_primdef: "SHD == %f. f 0"
-lemma SHD_def: "ALL f::nat => bool. SHD f = f 0"
+lemma SHD_def: "SHD f = f 0"
by (import boolean_sequence SHD_def)
consts
STL :: "(nat => bool) => nat => bool"
defs
- STL_primdef: "STL == %(f::nat => bool) n::nat. f (Suc n)"
+ STL_primdef: "STL == %f n. f (Suc n)"
-lemma STL_def: "ALL (f::nat => bool) n::nat. STL f n = f (Suc n)"
+lemma STL_def: "STL f n = f (Suc n)"
by (import boolean_sequence STL_def)
consts
SCONS :: "bool => (nat => bool) => nat => bool"
-specification (SCONS_primdef: SCONS) SCONS_def: "(ALL (h::bool) t::nat => bool. SCONS h t 0 = h) &
-(ALL (h::bool) (t::nat => bool) n::nat. SCONS h t (Suc n) = t n)"
+specification (SCONS_primdef: SCONS) SCONS_def: "(ALL h t. SCONS h t 0 = h) & (ALL h t n. SCONS h t (Suc n) = t n)"
by (import boolean_sequence SCONS_def)
consts
SDEST :: "(nat => bool) => bool * (nat => bool)"
defs
- SDEST_primdef: "SDEST == %s::nat => bool. (SHD s, STL s)"
+ SDEST_primdef: "SDEST == %s. (SHD s, STL s)"
-lemma SDEST_def: "SDEST = (%s::nat => bool. (SHD s, STL s))"
+lemma SDEST_def: "SDEST = (%s. (SHD s, STL s))"
by (import boolean_sequence SDEST_def)
consts
@@ -1122,32 +772,32 @@
consts
STAKE :: "nat => (nat => bool) => bool list"
-specification (STAKE_primdef: STAKE) STAKE_def: "(ALL s::nat => bool. STAKE 0 s = []) &
-(ALL (n::nat) s::nat => bool. STAKE (Suc n) s = SHD s # STAKE n (STL s))"
+specification (STAKE_primdef: STAKE) STAKE_def: "(ALL s. STAKE 0 s = []) &
+(ALL n s. STAKE (Suc n) s = SHD s # STAKE n (STL s))"
by (import boolean_sequence STAKE_def)
consts
SDROP :: "nat => (nat => bool) => nat => bool"
-specification (SDROP_primdef: SDROP) SDROP_def: "SDROP 0 = I & (ALL n::nat. SDROP (Suc n) = SDROP n o STL)"
+specification (SDROP_primdef: SDROP) SDROP_def: "SDROP 0 = I & (ALL n. SDROP (Suc n) = SDROP n o STL)"
by (import boolean_sequence SDROP_def)
-lemma SCONS_SURJ: "ALL x::nat => bool. EX (xa::bool) t::nat => bool. x = SCONS xa t"
+lemma SCONS_SURJ: "EX xa t. x = SCONS xa t"
by (import boolean_sequence SCONS_SURJ)
-lemma SHD_STL_ISO: "ALL (h::bool) t::nat => bool. EX x::nat => bool. SHD x = h & STL x = t"
+lemma SHD_STL_ISO: "EX x. SHD x = h & STL x = t"
by (import boolean_sequence SHD_STL_ISO)
-lemma SHD_SCONS: "ALL (h::bool) t::nat => bool. SHD (SCONS h t) = h"
+lemma SHD_SCONS: "SHD (SCONS h t) = h"
by (import boolean_sequence SHD_SCONS)
-lemma STL_SCONS: "ALL (h::bool) t::nat => bool. STL (SCONS h t) = t"
+lemma STL_SCONS: "STL (SCONS h t) = t"
by (import boolean_sequence STL_SCONS)
-lemma SHD_SCONST: "ALL b::bool. SHD (SCONST b) = b"
+lemma SHD_SCONST: "SHD (SCONST b) = b"
by (import boolean_sequence SHD_SCONST)
-lemma STL_SCONST: "ALL b::bool. STL (SCONST b) = SCONST b"
+lemma STL_SCONST: "STL (SCONST b) = SCONST b"
by (import boolean_sequence STL_SCONST)
;end_setup
@@ -1157,16 +807,15 @@
consts
alg_embed :: "bool list => (nat => bool) => bool"
-specification (alg_embed_primdef: alg_embed) alg_embed_def: "(ALL s::nat => bool. alg_embed [] s = True) &
-(ALL (h::bool) (t::bool list) s::nat => bool.
- alg_embed (h # t) s = (h = SHD s & alg_embed t (STL s)))"
+specification (alg_embed_primdef: alg_embed) alg_embed_def: "(ALL s. alg_embed [] s = True) &
+(ALL h t s. alg_embed (h # t) s = (h = SHD s & alg_embed t (STL s)))"
by (import prob_algebra alg_embed_def)
consts
algebra_embed :: "bool list list => (nat => bool) => bool"
specification (algebra_embed_primdef: algebra_embed) algebra_embed_def: "algebra_embed [] = EMPTY &
-(ALL (h::bool list) t::bool list list.
+(ALL h t.
algebra_embed (h # t) = pred_set.UNION (alg_embed h) (algebra_embed t))"
by (import prob_algebra algebra_embed_def)
@@ -1174,157 +823,127 @@
measurable :: "((nat => bool) => bool) => bool"
defs
- measurable_primdef: "measurable ==
-%s::(nat => bool) => bool. EX b::bool list list. s = algebra_embed b"
+ measurable_primdef: "measurable == %s. EX b. s = algebra_embed b"
-lemma measurable_def: "ALL s::(nat => bool) => bool.
- measurable s = (EX b::bool list list. s = algebra_embed b)"
+lemma measurable_def: "measurable s = (EX b. s = algebra_embed b)"
by (import prob_algebra measurable_def)
-lemma HALVES_INTER: "pred_set.INTER (%x::nat => bool. SHD x = True)
- (%x::nat => bool. SHD x = False) =
-EMPTY"
+lemma HALVES_INTER: "pred_set.INTER (%x. SHD x = True) (%x. SHD x = False) = EMPTY"
by (import prob_algebra HALVES_INTER)
-lemma INTER_STL: "ALL (p::(nat => bool) => bool) q::(nat => bool) => bool.
- pred_set.INTER p q o STL = pred_set.INTER (p o STL) (q o STL)"
+lemma INTER_STL: "pred_set.INTER p q o STL = pred_set.INTER (p o STL) (q o STL)"
by (import prob_algebra INTER_STL)
-lemma COMPL_SHD: "ALL b::bool.
- COMPL (%x::nat => bool. SHD x = b) = (%x::nat => bool. SHD x = (~ b))"
+lemma COMPL_SHD: "COMPL (%x. SHD x = b) = (%x. SHD x = (~ b))"
by (import prob_algebra COMPL_SHD)
lemma ALG_EMBED_BASIC: "alg_embed [] = pred_set.UNIV &
-(ALL (h::bool) t::bool list.
- alg_embed (h # t) =
- pred_set.INTER (%x::nat => bool. SHD x = h) (alg_embed t o STL))"
+(ALL h t.
+ alg_embed (h # t) = pred_set.INTER (%x. SHD x = h) (alg_embed t o STL))"
by (import prob_algebra ALG_EMBED_BASIC)
-lemma ALG_EMBED_NIL: "ALL c::bool list. All (alg_embed c) = (c = [])"
+lemma ALG_EMBED_NIL: "All (alg_embed c) = (c = [])"
by (import prob_algebra ALG_EMBED_NIL)
-lemma ALG_EMBED_POPULATED: "ALL b::bool list. Ex (alg_embed b)"
+lemma ALG_EMBED_POPULATED: "Ex (alg_embed b)"
by (import prob_algebra ALG_EMBED_POPULATED)
-lemma ALG_EMBED_PREFIX: "ALL (b::bool list) (c::bool list) s::nat => bool.
- alg_embed b s & alg_embed c s --> IS_PREFIX b c | IS_PREFIX c b"
+lemma ALG_EMBED_PREFIX: "alg_embed b s & alg_embed c s ==> IS_PREFIX b c | IS_PREFIX c b"
by (import prob_algebra ALG_EMBED_PREFIX)
-lemma ALG_EMBED_PREFIX_SUBSET: "ALL (b::bool list) c::bool list.
- SUBSET (alg_embed b) (alg_embed c) = IS_PREFIX b c"
+lemma ALG_EMBED_PREFIX_SUBSET: "SUBSET (alg_embed b) (alg_embed c) = IS_PREFIX b c"
by (import prob_algebra ALG_EMBED_PREFIX_SUBSET)
-lemma ALG_EMBED_TWINS: "ALL l::bool list.
- pred_set.UNION (alg_embed (SNOC True l)) (alg_embed (SNOC False l)) =
- alg_embed l"
+lemma ALG_EMBED_TWINS: "pred_set.UNION (alg_embed (SNOC True l)) (alg_embed (SNOC False l)) =
+alg_embed l"
by (import prob_algebra ALG_EMBED_TWINS)
lemma ALGEBRA_EMBED_BASIC: "algebra_embed [] = EMPTY &
algebra_embed [[]] = pred_set.UNIV &
-(ALL b::bool. algebra_embed [[b]] = (%s::nat => bool. SHD s = b))"
+(ALL b. algebra_embed [[b]] = (%s. SHD s = b))"
by (import prob_algebra ALGEBRA_EMBED_BASIC)
-lemma ALGEBRA_EMBED_MEM: "ALL (b::bool list list) x::nat => bool.
- algebra_embed b x --> (EX l::bool list. l mem b & alg_embed l x)"
+lemma ALGEBRA_EMBED_MEM: "algebra_embed b x ==> EX l. List.member b l & alg_embed l x"
by (import prob_algebra ALGEBRA_EMBED_MEM)
-lemma ALGEBRA_EMBED_APPEND: "ALL (l1::bool list list) l2::bool list list.
- algebra_embed (l1 @ l2) =
- pred_set.UNION (algebra_embed l1) (algebra_embed l2)"
+lemma ALGEBRA_EMBED_APPEND: "algebra_embed (l1 @ l2) =
+pred_set.UNION (algebra_embed l1) (algebra_embed l2)"
by (import prob_algebra ALGEBRA_EMBED_APPEND)
-lemma ALGEBRA_EMBED_TLS: "ALL (l::bool list list) b::bool.
- algebra_embed (map (op # b) l) (SCONS (h::bool) (t::nat => bool)) =
- (h = b & algebra_embed l t)"
+lemma ALGEBRA_EMBED_TLS: "algebra_embed (map (op # b) l) (SCONS h t) = (h = b & algebra_embed l t)"
by (import prob_algebra ALGEBRA_EMBED_TLS)
-lemma ALG_CANON_PREFS_EMBED: "ALL (l::bool list) b::bool list list.
- algebra_embed (alg_canon_prefs l b) = algebra_embed (l # b)"
+lemma ALG_CANON_PREFS_EMBED: "algebra_embed (alg_canon_prefs l b) = algebra_embed (l # b)"
by (import prob_algebra ALG_CANON_PREFS_EMBED)
-lemma ALG_CANON_FIND_EMBED: "ALL (l::bool list) b::bool list list.
- algebra_embed (alg_canon_find l b) = algebra_embed (l # b)"
+lemma ALG_CANON_FIND_EMBED: "algebra_embed (alg_canon_find l b) = algebra_embed (l # b)"
by (import prob_algebra ALG_CANON_FIND_EMBED)
-lemma ALG_CANON1_EMBED: "ALL x::bool list list. algebra_embed (alg_canon1 x) = algebra_embed x"
+lemma ALG_CANON1_EMBED: "algebra_embed (alg_canon1 x) = algebra_embed x"
by (import prob_algebra ALG_CANON1_EMBED)
-lemma ALG_CANON_MERGE_EMBED: "ALL (l::bool list) b::bool list list.
- algebra_embed (alg_canon_merge l b) = algebra_embed (l # b)"
+lemma ALG_CANON_MERGE_EMBED: "algebra_embed (alg_canon_merge l b) = algebra_embed (l # b)"
by (import prob_algebra ALG_CANON_MERGE_EMBED)
-lemma ALG_CANON2_EMBED: "ALL x::bool list list. algebra_embed (alg_canon2 x) = algebra_embed x"
+lemma ALG_CANON2_EMBED: "algebra_embed (alg_canon2 x) = algebra_embed x"
by (import prob_algebra ALG_CANON2_EMBED)
-lemma ALG_CANON_EMBED: "ALL l::bool list list. algebra_embed (alg_canon l) = algebra_embed l"
+lemma ALG_CANON_EMBED: "algebra_embed (alg_canon l) = algebra_embed l"
by (import prob_algebra ALG_CANON_EMBED)
-lemma ALGEBRA_CANON_UNIV: "ALL l::bool list list.
- algebra_canon l --> algebra_embed l = pred_set.UNIV --> l = [[]]"
+lemma ALGEBRA_CANON_UNIV: "[| algebra_canon l; algebra_embed l = pred_set.UNIV |] ==> l = [[]]"
by (import prob_algebra ALGEBRA_CANON_UNIV)
-lemma ALG_CANON_REP: "ALL (b::bool list list) c::bool list list.
- (alg_canon b = alg_canon c) = (algebra_embed b = algebra_embed c)"
+lemma ALG_CANON_REP: "(alg_canon b = alg_canon c) = (algebra_embed b = algebra_embed c)"
by (import prob_algebra ALG_CANON_REP)
-lemma ALGEBRA_CANON_EMBED_EMPTY: "ALL l::bool list list.
- algebra_canon l --> (ALL v::nat => bool. ~ algebra_embed l v) = (l = [])"
+lemma ALGEBRA_CANON_EMBED_EMPTY: "algebra_canon l ==> (ALL v. ~ algebra_embed l v) = (l = [])"
by (import prob_algebra ALGEBRA_CANON_EMBED_EMPTY)
-lemma ALGEBRA_CANON_EMBED_UNIV: "ALL l::bool list list.
- algebra_canon l --> All (algebra_embed l) = (l = [[]])"
+lemma ALGEBRA_CANON_EMBED_UNIV: "algebra_canon l ==> All (algebra_embed l) = (l = [[]])"
by (import prob_algebra ALGEBRA_CANON_EMBED_UNIV)
-lemma MEASURABLE_ALGEBRA: "ALL b::bool list list. measurable (algebra_embed b)"
+lemma MEASURABLE_ALGEBRA: "measurable (algebra_embed b)"
by (import prob_algebra MEASURABLE_ALGEBRA)
lemma MEASURABLE_BASIC: "measurable EMPTY &
-measurable pred_set.UNIV &
-(ALL b::bool. measurable (%s::nat => bool. SHD s = b))"
+measurable pred_set.UNIV & (ALL b. measurable (%s. SHD s = b))"
by (import prob_algebra MEASURABLE_BASIC)
-lemma MEASURABLE_SHD: "ALL b::bool. measurable (%s::nat => bool. SHD s = b)"
+lemma MEASURABLE_SHD: "measurable (%s. SHD s = b)"
by (import prob_algebra MEASURABLE_SHD)
-lemma ALGEBRA_EMBED_COMPL: "ALL l::bool list list.
- EX l'::bool list list. COMPL (algebra_embed l) = algebra_embed l'"
+lemma ALGEBRA_EMBED_COMPL: "EX l'. COMPL (algebra_embed l) = algebra_embed l'"
by (import prob_algebra ALGEBRA_EMBED_COMPL)
-lemma MEASURABLE_COMPL: "ALL s::(nat => bool) => bool. measurable (COMPL s) = measurable s"
+lemma MEASURABLE_COMPL: "measurable (COMPL s) = measurable s"
by (import prob_algebra MEASURABLE_COMPL)
-lemma MEASURABLE_UNION: "ALL (s::(nat => bool) => bool) t::(nat => bool) => bool.
- measurable s & measurable t --> measurable (pred_set.UNION s t)"
+lemma MEASURABLE_UNION: "measurable s & measurable t ==> measurable (pred_set.UNION s t)"
by (import prob_algebra MEASURABLE_UNION)
-lemma MEASURABLE_INTER: "ALL (s::(nat => bool) => bool) t::(nat => bool) => bool.
- measurable s & measurable t --> measurable (pred_set.INTER s t)"
+lemma MEASURABLE_INTER: "measurable s & measurable t ==> measurable (pred_set.INTER s t)"
by (import prob_algebra MEASURABLE_INTER)
-lemma MEASURABLE_STL: "ALL p::(nat => bool) => bool. measurable (p o STL) = measurable p"
+lemma MEASURABLE_STL: "measurable (p o STL) = measurable p"
by (import prob_algebra MEASURABLE_STL)
-lemma MEASURABLE_SDROP: "ALL (n::nat) p::(nat => bool) => bool.
- measurable (p o SDROP n) = measurable p"
+lemma MEASURABLE_SDROP: "measurable (p o SDROP n) = measurable p"
by (import prob_algebra MEASURABLE_SDROP)
-lemma MEASURABLE_INTER_HALVES: "ALL p::(nat => bool) => bool.
- (measurable (pred_set.INTER (%x::nat => bool. SHD x = True) p) &
- measurable (pred_set.INTER (%x::nat => bool. SHD x = False) p)) =
- measurable p"
+lemma MEASURABLE_INTER_HALVES: "(measurable (pred_set.INTER (%x. SHD x = True) p) &
+ measurable (pred_set.INTER (%x. SHD x = False) p)) =
+measurable p"
by (import prob_algebra MEASURABLE_INTER_HALVES)
-lemma MEASURABLE_HALVES: "ALL (p::(nat => bool) => bool) q::(nat => bool) => bool.
- measurable
- (pred_set.UNION (pred_set.INTER (%x::nat => bool. SHD x = True) p)
- (pred_set.INTER (%x::nat => bool. SHD x = False) q)) =
- (measurable (pred_set.INTER (%x::nat => bool. SHD x = True) p) &
- measurable (pred_set.INTER (%x::nat => bool. SHD x = False) q))"
+lemma MEASURABLE_HALVES: "measurable
+ (pred_set.UNION (pred_set.INTER (%x. SHD x = True) p)
+ (pred_set.INTER (%x. SHD x = False) q)) =
+(measurable (pred_set.INTER (%x. SHD x = True) p) &
+ measurable (pred_set.INTER (%x. SHD x = False) q))"
by (import prob_algebra MEASURABLE_HALVES)
-lemma MEASURABLE_INTER_SHD: "ALL (b::bool) p::(nat => bool) => bool.
- measurable (pred_set.INTER (%x::nat => bool. SHD x = b) (p o STL)) =
- measurable p"
+lemma MEASURABLE_INTER_SHD: "measurable (pred_set.INTER (%x. SHD x = b) (p o STL)) = measurable p"
by (import prob_algebra MEASURABLE_INTER_SHD)
;end_setup
@@ -1335,8 +954,7 @@
alg_measure :: "bool list list => real"
specification (alg_measure_primdef: alg_measure) alg_measure_def: "alg_measure [] = 0 &
-(ALL (l::bool list) rest::bool list list.
- alg_measure (l # rest) = (1 / 2) ^ length l + alg_measure rest)"
+(ALL l rest. alg_measure (l # rest) = (1 / 2) ^ length l + alg_measure rest)"
by (import prob alg_measure_def)
consts
@@ -1344,16 +962,12 @@
defs
algebra_measure_primdef: "algebra_measure ==
-%b::bool list list.
- inf (%r::real.
- EX c::bool list list.
- algebra_embed b = algebra_embed c & alg_measure c = r)"
+%b. prob_extra.inf
+ (%r. EX c. algebra_embed b = algebra_embed c & alg_measure c = r)"
-lemma algebra_measure_def: "ALL b::bool list list.
- algebra_measure b =
- inf (%r::real.
- EX c::bool list list.
- algebra_embed b = algebra_embed c & alg_measure c = r)"
+lemma algebra_measure_def: "algebra_measure b =
+prob_extra.inf
+ (%r. EX c. algebra_embed b = algebra_embed c & alg_measure c = r)"
by (import prob algebra_measure_def)
consts
@@ -1361,209 +975,148 @@
defs
prob_primdef: "prob ==
-%s::(nat => bool) => bool.
- sup (%r::real.
- EX b::bool list list.
- algebra_measure b = r & SUBSET (algebra_embed b) s)"
+%s. real.sup (%r. EX b. algebra_measure b = r & SUBSET (algebra_embed b) s)"
-lemma prob_def: "ALL s::(nat => bool) => bool.
- prob s =
- sup (%r::real.
- EX b::bool list list.
- algebra_measure b = r & SUBSET (algebra_embed b) s)"
+lemma prob_def: "prob s =
+real.sup (%r. EX b. algebra_measure b = r & SUBSET (algebra_embed b) s)"
by (import prob prob_def)
-lemma ALG_TWINS_MEASURE: "(All::(bool list => bool) => bool)
- (%l::bool list.
- (op =::real => real => bool)
- ((op +::real => real => real)
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((size::bool list => nat)
- ((SNOC::bool => bool list => bool list) (True::bool) l)))
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((size::bool list => nat)
- ((SNOC::bool => bool list => bool list) (False::bool) l))))
- ((op ^::real => nat => real)
- ((op /::real => real => real) (1::real)
- ((number_of \<Colon> int => real)
- ((Int.Bit0 \<Colon> int => int)
- ((Int.Bit1 \<Colon> int => int) (Int.Pls \<Colon> int)))))
- ((size::bool list => nat) l)))"
+lemma ALG_TWINS_MEASURE: "((1::real) / (2::real)) ^ length (SNOC True (l::bool list)) +
+((1::real) / (2::real)) ^ length (SNOC False l) =
+((1::real) / (2::real)) ^ length l"
by (import prob ALG_TWINS_MEASURE)
lemma ALG_MEASURE_BASIC: "alg_measure [] = 0 &
-alg_measure [[]] = 1 & (ALL b::bool. alg_measure [[b]] = 1 / 2)"
+alg_measure [[]] = 1 & (ALL b. alg_measure [[b]] = 1 / 2)"
by (import prob ALG_MEASURE_BASIC)
-lemma ALG_MEASURE_POS: "ALL l::bool list list. 0 <= alg_measure l"
+lemma ALG_MEASURE_POS: "0 <= alg_measure l"
by (import prob ALG_MEASURE_POS)
-lemma ALG_MEASURE_APPEND: "ALL (l1::bool list list) l2::bool list list.
- alg_measure (l1 @ l2) = alg_measure l1 + alg_measure l2"
+lemma ALG_MEASURE_APPEND: "alg_measure (l1 @ l2) = alg_measure l1 + alg_measure l2"
by (import prob ALG_MEASURE_APPEND)
-lemma ALG_MEASURE_TLS: "ALL (l::bool list list) b::bool.
- 2 * alg_measure (map (op # b) l) = alg_measure l"
+lemma ALG_MEASURE_TLS: "2 * alg_measure (map (op # b) l) = alg_measure l"
by (import prob ALG_MEASURE_TLS)
-lemma ALG_CANON_PREFS_MONO: "ALL (l::bool list) b::bool list list.
- alg_measure (alg_canon_prefs l b) <= alg_measure (l # b)"
+lemma ALG_CANON_PREFS_MONO: "alg_measure (alg_canon_prefs l b) <= alg_measure (l # b)"
by (import prob ALG_CANON_PREFS_MONO)
-lemma ALG_CANON_FIND_MONO: "ALL (l::bool list) b::bool list list.
- alg_measure (alg_canon_find l b) <= alg_measure (l # b)"
+lemma ALG_CANON_FIND_MONO: "alg_measure (alg_canon_find l b) <= alg_measure (l # b)"
by (import prob ALG_CANON_FIND_MONO)
-lemma ALG_CANON1_MONO: "ALL x::bool list list. alg_measure (alg_canon1 x) <= alg_measure x"
+lemma ALG_CANON1_MONO: "alg_measure (alg_canon1 x) <= alg_measure x"
by (import prob ALG_CANON1_MONO)
-lemma ALG_CANON_MERGE_MONO: "ALL (l::bool list) b::bool list list.
- alg_measure (alg_canon_merge l b) <= alg_measure (l # b)"
+lemma ALG_CANON_MERGE_MONO: "alg_measure (alg_canon_merge l b) <= alg_measure (l # b)"
by (import prob ALG_CANON_MERGE_MONO)
-lemma ALG_CANON2_MONO: "ALL x::bool list list. alg_measure (alg_canon2 x) <= alg_measure x"
+lemma ALG_CANON2_MONO: "alg_measure (alg_canon2 x) <= alg_measure x"
by (import prob ALG_CANON2_MONO)
-lemma ALG_CANON_MONO: "ALL l::bool list list. alg_measure (alg_canon l) <= alg_measure l"
+lemma ALG_CANON_MONO: "alg_measure (alg_canon l) <= alg_measure l"
by (import prob ALG_CANON_MONO)
-lemma ALGEBRA_MEASURE_DEF_ALT: "ALL l::bool list list. algebra_measure l = alg_measure (alg_canon l)"
+lemma ALGEBRA_MEASURE_DEF_ALT: "algebra_measure l = alg_measure (alg_canon l)"
by (import prob ALGEBRA_MEASURE_DEF_ALT)
lemma ALGEBRA_MEASURE_BASIC: "algebra_measure [] = 0 &
-algebra_measure [[]] = 1 & (ALL b::bool. algebra_measure [[b]] = 1 / 2)"
+algebra_measure [[]] = 1 & (ALL b. algebra_measure [[b]] = 1 / 2)"
by (import prob ALGEBRA_MEASURE_BASIC)
-lemma ALGEBRA_CANON_MEASURE_MAX: "ALL l::bool list list. algebra_canon l --> alg_measure l <= 1"
+lemma ALGEBRA_CANON_MEASURE_MAX: "algebra_canon l ==> alg_measure l <= 1"
by (import prob ALGEBRA_CANON_MEASURE_MAX)
-lemma ALGEBRA_MEASURE_MAX: "ALL l::bool list list. algebra_measure l <= 1"
+lemma ALGEBRA_MEASURE_MAX: "algebra_measure l <= 1"
by (import prob ALGEBRA_MEASURE_MAX)
-lemma ALGEBRA_MEASURE_MONO_EMBED: "ALL (x::bool list list) xa::bool list list.
- SUBSET (algebra_embed x) (algebra_embed xa) -->
- algebra_measure x <= algebra_measure xa"
+lemma ALGEBRA_MEASURE_MONO_EMBED: "SUBSET (algebra_embed x) (algebra_embed xa)
+==> algebra_measure x <= algebra_measure xa"
by (import prob ALGEBRA_MEASURE_MONO_EMBED)
-lemma ALG_MEASURE_COMPL: "ALL l::bool list list.
- algebra_canon l -->
- (ALL c::bool list list.
- algebra_canon c -->
- COMPL (algebra_embed l) = algebra_embed c -->
- alg_measure l + alg_measure c = 1)"
+lemma ALG_MEASURE_COMPL: "[| algebra_canon l; algebra_canon c;
+ COMPL (algebra_embed l) = algebra_embed c |]
+==> alg_measure l + alg_measure c = 1"
by (import prob ALG_MEASURE_COMPL)
-lemma ALG_MEASURE_ADDITIVE: "ALL l::bool list list.
- algebra_canon l -->
- (ALL c::bool list list.
- algebra_canon c -->
- (ALL d::bool list list.
- algebra_canon d -->
- pred_set.INTER (algebra_embed c) (algebra_embed d) = EMPTY &
- algebra_embed l =
- pred_set.UNION (algebra_embed c) (algebra_embed d) -->
- alg_measure l = alg_measure c + alg_measure d))"
+lemma ALG_MEASURE_ADDITIVE: "[| algebra_canon l; algebra_canon c; algebra_canon d;
+ pred_set.INTER (algebra_embed c) (algebra_embed d) = EMPTY &
+ algebra_embed l = pred_set.UNION (algebra_embed c) (algebra_embed d) |]
+==> alg_measure l = alg_measure c + alg_measure d"
by (import prob ALG_MEASURE_ADDITIVE)
-lemma PROB_ALGEBRA: "ALL l::bool list list. prob (algebra_embed l) = algebra_measure l"
+lemma PROB_ALGEBRA: "prob (algebra_embed l) = algebra_measure l"
by (import prob PROB_ALGEBRA)
lemma PROB_BASIC: "prob EMPTY = 0 &
-prob pred_set.UNIV = 1 &
-(ALL b::bool. prob (%s::nat => bool. SHD s = b) = 1 / 2)"
+prob pred_set.UNIV = 1 & (ALL b. prob (%s. SHD s = b) = 1 / 2)"
by (import prob PROB_BASIC)
-lemma PROB_ADDITIVE: "ALL (s::(nat => bool) => bool) t::(nat => bool) => bool.
- measurable s & measurable t & pred_set.INTER s t = EMPTY -->
- prob (pred_set.UNION s t) = prob s + prob t"
+lemma PROB_ADDITIVE: "measurable s & measurable t & pred_set.INTER s t = EMPTY
+==> prob (pred_set.UNION s t) = prob s + prob t"
by (import prob PROB_ADDITIVE)
-lemma PROB_COMPL: "ALL s::(nat => bool) => bool. measurable s --> prob (COMPL s) = 1 - prob s"
+lemma PROB_COMPL: "measurable s ==> prob (COMPL s) = 1 - prob s"
by (import prob PROB_COMPL)
-lemma PROB_SUP_EXISTS1: "ALL s::(nat => bool) => bool.
- EX (x::real) b::bool list list.
- algebra_measure b = x & SUBSET (algebra_embed b) s"
+lemma PROB_SUP_EXISTS1: "EX x b. algebra_measure b = x & SUBSET (algebra_embed b) s"
by (import prob PROB_SUP_EXISTS1)
-lemma PROB_SUP_EXISTS2: "ALL s::(nat => bool) => bool.
- EX x::real.
- ALL r::real.
- (EX b::bool list list.
- algebra_measure b = r & SUBSET (algebra_embed b) s) -->
+lemma PROB_SUP_EXISTS2: "EX x. ALL r.
+ (EX b. algebra_measure b = r & SUBSET (algebra_embed b) s) -->
r <= x"
by (import prob PROB_SUP_EXISTS2)
-lemma PROB_LE_X: "ALL (s::(nat => bool) => bool) x::real.
- (ALL s'::(nat => bool) => bool.
- measurable s' & SUBSET s' s --> prob s' <= x) -->
- prob s <= x"
+lemma PROB_LE_X: "(!!s'. measurable s' & SUBSET s' s ==> prob s' <= x) ==> prob s <= x"
by (import prob PROB_LE_X)
-lemma X_LE_PROB: "ALL (s::(nat => bool) => bool) x::real.
- (EX s'::(nat => bool) => bool.
- measurable s' & SUBSET s' s & x <= prob s') -->
- x <= prob s"
+lemma X_LE_PROB: "EX s'. measurable s' & SUBSET s' s & x <= prob s' ==> x <= prob s"
by (import prob X_LE_PROB)
-lemma PROB_SUBSET_MONO: "ALL (s::(nat => bool) => bool) t::(nat => bool) => bool.
- SUBSET s t --> prob s <= prob t"
+lemma PROB_SUBSET_MONO: "SUBSET s t ==> prob s <= prob t"
by (import prob PROB_SUBSET_MONO)
-lemma PROB_ALG: "ALL x::bool list. prob (alg_embed x) = (1 / 2) ^ length x"
+lemma PROB_ALG: "prob (alg_embed x) = (1 / 2) ^ length x"
by (import prob PROB_ALG)
-lemma PROB_STL: "ALL p::(nat => bool) => bool. measurable p --> prob (p o STL) = prob p"
+lemma PROB_STL: "measurable p ==> prob (p o STL) = prob p"
by (import prob PROB_STL)
-lemma PROB_SDROP: "ALL (n::nat) p::(nat => bool) => bool.
- measurable p --> prob (p o SDROP n) = prob p"
+lemma PROB_SDROP: "measurable p ==> prob (p o SDROP n) = prob p"
by (import prob PROB_SDROP)
-lemma PROB_INTER_HALVES: "ALL p::(nat => bool) => bool.
- measurable p -->
- prob (pred_set.INTER (%x::nat => bool. SHD x = True) p) +
- prob (pred_set.INTER (%x::nat => bool. SHD x = False) p) =
- prob p"
+lemma PROB_INTER_HALVES: "measurable p
+==> prob (pred_set.INTER (%x. SHD x = True) p) +
+ prob (pred_set.INTER (%x. SHD x = False) p) =
+ prob p"
by (import prob PROB_INTER_HALVES)
-lemma PROB_INTER_SHD: "ALL (b::bool) p::(nat => bool) => bool.
- measurable p -->
- prob (pred_set.INTER (%x::nat => bool. SHD x = b) (p o STL)) =
- 1 / 2 * prob p"
+lemma PROB_INTER_SHD: "measurable p
+==> prob (pred_set.INTER (%x. SHD x = b) (p o STL)) = 1 / 2 * prob p"
by (import prob PROB_INTER_SHD)
-lemma ALGEBRA_MEASURE_POS: "ALL l::bool list list. 0 <= algebra_measure l"
+lemma ALGEBRA_MEASURE_POS: "0 <= algebra_measure l"
by (import prob ALGEBRA_MEASURE_POS)
-lemma ALGEBRA_MEASURE_RANGE: "ALL l::bool list list. 0 <= algebra_measure l & algebra_measure l <= 1"
+lemma ALGEBRA_MEASURE_RANGE: "0 <= algebra_measure l & algebra_measure l <= 1"
by (import prob ALGEBRA_MEASURE_RANGE)
-lemma PROB_POS: "ALL p::(nat => bool) => bool. 0 <= prob p"
+lemma PROB_POS: "0 <= prob p"
by (import prob PROB_POS)
-lemma PROB_MAX: "ALL p::(nat => bool) => bool. prob p <= 1"
+lemma PROB_MAX: "prob p <= 1"
by (import prob PROB_MAX)
-lemma PROB_RANGE: "ALL p::(nat => bool) => bool. 0 <= prob p & prob p <= 1"
+lemma PROB_RANGE: "0 <= prob p & prob p <= 1"
by (import prob PROB_RANGE)
-lemma ABS_PROB: "ALL p::(nat => bool) => bool. abs (prob p) = prob p"
+lemma ABS_PROB: "abs (prob p) = prob p"
by (import prob ABS_PROB)
-lemma PROB_SHD: "ALL b::bool. prob (%s::nat => bool. SHD s = b) = 1 / 2"
+lemma PROB_SHD: "prob (%s. SHD s = b) = 1 / 2"
by (import prob PROB_SHD)
-lemma PROB_COMPL_LE1: "ALL (p::(nat => bool) => bool) r::real.
- measurable p --> (prob (COMPL p) <= r) = (1 - r <= prob p)"
+lemma PROB_COMPL_LE1: "measurable p ==> (prob (COMPL p) <= r) = (1 - r <= prob p)"
by (import prob PROB_COMPL_LE1)
;end_setup
@@ -1583,41 +1136,41 @@
pseudo_linear_tl :: "nat => nat => nat => nat => nat"
defs
- pseudo_linear_tl_primdef: "pseudo_linear_tl ==
-%(a::nat) (b::nat) (n::nat) x::nat. (a * x + b) mod (2 * n + 1)"
+ pseudo_linear_tl_primdef: "pseudo_linear_tl == %a b n x. (a * x + b) mod (2 * n + 1)"
-lemma pseudo_linear_tl_def: "ALL (a::nat) (b::nat) (n::nat) x::nat.
- pseudo_linear_tl a b n x = (a * x + b) mod (2 * n + 1)"
+lemma pseudo_linear_tl_def: "pseudo_linear_tl a b n x = (a * x + b) mod (2 * n + 1)"
by (import prob_pseudo pseudo_linear_tl_def)
-lemma PSEUDO_LINEAR1_EXECUTE: "EX x::nat => nat => bool.
- (ALL xa::nat. SHD (x xa) = pseudo_linear_hd xa) &
- (ALL xa::nat.
- STL (x xa) =
- x (pseudo_linear_tl
- (NUMERAL
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT2 (NUMERAL_BIT1 (NUMERAL_BIT2 ALT_ZERO)))))))
- (NUMERAL
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT1 (NUMERAL_BIT2 ALT_ZERO)))))))
- (NUMERAL
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1
- (NUMERAL_BIT1 (NUMERAL_BIT2 (NUMERAL_BIT1 ALT_ZERO)))))))
- xa))"
+lemma PSEUDO_LINEAR1_EXECUTE: "EX x. (ALL xa. SHD (x xa) = pseudo_linear_hd xa) &
+ (ALL xa.
+ STL (x xa) =
+ x (pseudo_linear_tl
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT2
+ (NUMERAL_BIT1 (NUMERAL_BIT2 ALT_ZERO)))))))
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1 (NUMERAL_BIT2 ALT_ZERO)))))))
+ (NUMERAL
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT1
+ (NUMERAL_BIT2 (NUMERAL_BIT1 ALT_ZERO)))))))
+ xa))"
by (import prob_pseudo PSEUDO_LINEAR1_EXECUTE)
consts
pseudo_linear1 :: "nat => nat => bool"
-specification (pseudo_linear1_primdef: pseudo_linear1) pseudo_linear1_def: "(ALL x::nat. SHD (pseudo_linear1 x) = pseudo_linear_hd x) &
-(ALL x::nat.
+specification (pseudo_linear1_primdef: pseudo_linear1) pseudo_linear1_def: "(ALL x. SHD (pseudo_linear1 x) = pseudo_linear_hd x) &
+(ALL x.
STL (pseudo_linear1 x) =
pseudo_linear1
(pseudo_linear_tl
@@ -1657,13 +1210,11 @@
defs
indep_set_primdef: "indep_set ==
-%(p::(nat => bool) => bool) q::(nat => bool) => bool.
- measurable p & measurable q & prob (pred_set.INTER p q) = prob p * prob q"
+%p q. measurable p &
+ measurable q & prob (pred_set.INTER p q) = prob p * prob q"
-lemma indep_set_def: "ALL (p::(nat => bool) => bool) q::(nat => bool) => bool.
- indep_set p q =
- (measurable p &
- measurable q & prob (pred_set.INTER p q) = prob p * prob q)"
+lemma indep_set_def: "indep_set p q =
+(measurable p & measurable q & prob (pred_set.INTER p q) = prob p * prob q)"
by (import prob_indep indep_set_def)
consts
@@ -1671,24 +1222,19 @@
defs
alg_cover_set_primdef: "alg_cover_set ==
-%l::bool list list.
- alg_sorted l & alg_prefixfree l & algebra_embed l = pred_set.UNIV"
+%l. alg_sorted l & alg_prefixfree l & algebra_embed l = pred_set.UNIV"
-lemma alg_cover_set_def: "ALL l::bool list list.
- alg_cover_set l =
- (alg_sorted l & alg_prefixfree l & algebra_embed l = pred_set.UNIV)"
+lemma alg_cover_set_def: "alg_cover_set l =
+(alg_sorted l & alg_prefixfree l & algebra_embed l = pred_set.UNIV)"
by (import prob_indep alg_cover_set_def)
consts
alg_cover :: "bool list list => (nat => bool) => bool list"
defs
- alg_cover_primdef: "alg_cover ==
-%(l::bool list list) x::nat => bool.
- SOME b::bool list. b mem l & alg_embed b x"
+ alg_cover_primdef: "alg_cover == %l x. SOME b. List.member l b & alg_embed b x"
-lemma alg_cover_def: "ALL (l::bool list list) x::nat => bool.
- alg_cover l x = (SOME b::bool list. b mem l & alg_embed b x)"
+lemma alg_cover_def: "alg_cover l x = (SOME b. List.member l b & alg_embed b x)"
by (import prob_indep alg_cover_def)
consts
@@ -1696,195 +1242,148 @@
defs
indep_primdef: "indep ==
-%f::(nat => bool) => 'a::type * (nat => bool).
- EX (l::bool list list) r::bool list => 'a::type.
- alg_cover_set l &
- (ALL s::nat => bool.
- f s =
- (let c::bool list = alg_cover l s in (r c, SDROP (length c) s)))"
+%f. EX l r.
+ alg_cover_set l &
+ (ALL s. f s = (let c = alg_cover l s in (r c, SDROP (length c) s)))"
-lemma indep_def: "ALL f::(nat => bool) => 'a::type * (nat => bool).
- indep f =
- (EX (l::bool list list) r::bool list => 'a::type.
- alg_cover_set l &
- (ALL s::nat => bool.
- f s =
- (let c::bool list = alg_cover l s in (r c, SDROP (length c) s))))"
+lemma indep_def: "indep f =
+(EX l r.
+ alg_cover_set l &
+ (ALL s. f s = (let c = alg_cover l s in (r c, SDROP (length c) s))))"
by (import prob_indep indep_def)
-lemma INDEP_SET_BASIC: "ALL p::(nat => bool) => bool.
- measurable p --> indep_set EMPTY p & indep_set pred_set.UNIV p"
+lemma INDEP_SET_BASIC: "measurable p ==> indep_set EMPTY p & indep_set pred_set.UNIV p"
by (import prob_indep INDEP_SET_BASIC)
-lemma INDEP_SET_SYM: "ALL (p::(nat => bool) => bool) q::(nat => bool) => bool.
- indep_set p q = indep_set p q"
+lemma INDEP_SET_SYM: "indep_set p q = indep_set p q"
by (import prob_indep INDEP_SET_SYM)
-lemma INDEP_SET_DISJOINT_DECOMP: "ALL (p::(nat => bool) => bool) (q::(nat => bool) => bool)
- r::(nat => bool) => bool.
- indep_set p r & indep_set q r & pred_set.INTER p q = EMPTY -->
- indep_set (pred_set.UNION p q) r"
+lemma INDEP_SET_DISJOINT_DECOMP: "indep_set p r & indep_set q r & pred_set.INTER p q = EMPTY
+==> indep_set (pred_set.UNION p q) r"
by (import prob_indep INDEP_SET_DISJOINT_DECOMP)
lemma ALG_COVER_SET_BASIC: "~ alg_cover_set [] & alg_cover_set [[]] & alg_cover_set [[True], [False]]"
by (import prob_indep ALG_COVER_SET_BASIC)
-lemma ALG_COVER_WELL_DEFINED: "ALL (l::bool list list) x::nat => bool.
- alg_cover_set l --> alg_cover l x mem l & alg_embed (alg_cover l x) x"
+lemma ALG_COVER_WELL_DEFINED: "alg_cover_set l
+==> List.member l (alg_cover l x) & alg_embed (alg_cover l x) x"
by (import prob_indep ALG_COVER_WELL_DEFINED)
lemma ALG_COVER_UNIV: "alg_cover [[]] = K []"
by (import prob_indep ALG_COVER_UNIV)
-lemma MAP_CONS_TL_FILTER: "ALL (l::bool list list) b::bool.
- ~ [] mem l -->
- map (op # b) (map tl [x::bool list<-l. hd x = b]) =
- [x::bool list<-l. hd x = b]"
+lemma MAP_CONS_TL_FILTER: "~ List.member (l::bool list list) []
+==> map (op # (b::bool)) (map tl [x::bool list<-l. hd x = b]) =
+ [x::bool list<-l. hd x = b]"
by (import prob_indep MAP_CONS_TL_FILTER)
-lemma ALG_COVER_SET_CASES_THM: "ALL l::bool list list.
- alg_cover_set l =
- (l = [[]] |
- (EX (x::bool list list) xa::bool list list.
- alg_cover_set x &
- alg_cover_set xa & l = map (op # True) x @ map (op # False) xa))"
+lemma ALG_COVER_SET_CASES_THM: "alg_cover_set l =
+(l = [[]] |
+ (EX x xa.
+ alg_cover_set x &
+ alg_cover_set xa & l = map (op # True) x @ map (op # False) xa))"
by (import prob_indep ALG_COVER_SET_CASES_THM)
-lemma ALG_COVER_SET_CASES: "ALL P::bool list list => bool.
- P [[]] &
- (ALL (l1::bool list list) l2::bool list list.
+lemma ALG_COVER_SET_CASES: "[| P [[]] &
+ (ALL l1 l2.
alg_cover_set l1 &
alg_cover_set l2 &
alg_cover_set (map (op # True) l1 @ map (op # False) l2) -->
- P (map (op # True) l1 @ map (op # False) l2)) -->
- (ALL l::bool list list. alg_cover_set l --> P l)"
+ P (map (op # True) l1 @ map (op # False) l2));
+ alg_cover_set l |]
+==> P l"
by (import prob_indep ALG_COVER_SET_CASES)
-lemma ALG_COVER_SET_INDUCTION: "ALL P::bool list list => bool.
- P [[]] &
- (ALL (l1::bool list list) l2::bool list list.
+lemma ALG_COVER_SET_INDUCTION: "[| P [[]] &
+ (ALL l1 l2.
alg_cover_set l1 &
alg_cover_set l2 &
P l1 &
P l2 & alg_cover_set (map (op # True) l1 @ map (op # False) l2) -->
- P (map (op # True) l1 @ map (op # False) l2)) -->
- (ALL l::bool list list. alg_cover_set l --> P l)"
+ P (map (op # True) l1 @ map (op # False) l2));
+ alg_cover_set l |]
+==> P l"
by (import prob_indep ALG_COVER_SET_INDUCTION)
-lemma ALG_COVER_EXISTS_UNIQUE: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL s::nat => bool. EX! x::bool list. x mem l & alg_embed x s)"
+lemma ALG_COVER_EXISTS_UNIQUE: "alg_cover_set l ==> EX! x. List.member l x & alg_embed x s"
by (import prob_indep ALG_COVER_EXISTS_UNIQUE)
-lemma ALG_COVER_UNIQUE: "ALL (l::bool list list) (x::bool list) s::nat => bool.
- alg_cover_set l & x mem l & alg_embed x s --> alg_cover l s = x"
+lemma ALG_COVER_UNIQUE: "alg_cover_set l & List.member l x & alg_embed x s ==> alg_cover l s = x"
by (import prob_indep ALG_COVER_UNIQUE)
-lemma ALG_COVER_STEP: "ALL (l1::bool list list) (l2::bool list list) (h::bool) t::nat => bool.
- alg_cover_set l1 & alg_cover_set l2 -->
- alg_cover (map (op # True) l1 @ map (op # False) l2) (SCONS h t) =
- (if h then True # alg_cover l1 t else False # alg_cover l2 t)"
+lemma ALG_COVER_STEP: "alg_cover_set l1 & alg_cover_set l2
+==> alg_cover (map (op # True) l1 @ map (op # False) l2) (SCONS h t) =
+ (if h then True # alg_cover l1 t else False # alg_cover l2 t)"
by (import prob_indep ALG_COVER_STEP)
-lemma ALG_COVER_HEAD: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL f::bool list => bool. f o alg_cover l = algebra_embed (filter f l))"
+lemma ALG_COVER_HEAD: "alg_cover_set l ==> f o alg_cover l = algebra_embed (filter f l)"
by (import prob_indep ALG_COVER_HEAD)
-lemma ALG_COVER_TAIL_STEP: "ALL (l1::bool list list) (l2::bool list list) q::(nat => bool) => bool.
- alg_cover_set l1 & alg_cover_set l2 -->
- q o
- (%x::nat => bool.
- SDROP
- (length (alg_cover (map (op # True) l1 @ map (op # False) l2) x))
- x) =
- pred_set.UNION
- (pred_set.INTER (%x::nat => bool. SHD x = True)
- (q o ((%x::nat => bool. SDROP (length (alg_cover l1 x)) x) o STL)))
- (pred_set.INTER (%x::nat => bool. SHD x = False)
- (q o ((%x::nat => bool. SDROP (length (alg_cover l2 x)) x) o STL)))"
+lemma ALG_COVER_TAIL_STEP: "alg_cover_set l1 & alg_cover_set l2
+==> q o
+ (%x. SDROP
+ (length (alg_cover (map (op # True) l1 @ map (op # False) l2) x))
+ x) =
+ pred_set.UNION
+ (pred_set.INTER (%x. SHD x = True)
+ (q o ((%x. SDROP (length (alg_cover l1 x)) x) o STL)))
+ (pred_set.INTER (%x. SHD x = False)
+ (q o ((%x. SDROP (length (alg_cover l2 x)) x) o STL)))"
by (import prob_indep ALG_COVER_TAIL_STEP)
-lemma ALG_COVER_TAIL_MEASURABLE: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL q::(nat => bool) => bool.
- measurable
- (q o (%x::nat => bool. SDROP (length (alg_cover l x)) x)) =
- measurable q)"
+lemma ALG_COVER_TAIL_MEASURABLE: "alg_cover_set l
+==> measurable (q o (%x. SDROP (length (alg_cover l x)) x)) = measurable q"
by (import prob_indep ALG_COVER_TAIL_MEASURABLE)
-lemma ALG_COVER_TAIL_PROB: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL q::(nat => bool) => bool.
- measurable q -->
- prob (q o (%x::nat => bool. SDROP (length (alg_cover l x)) x)) =
- prob q)"
+lemma ALG_COVER_TAIL_PROB: "[| alg_cover_set l; measurable q |]
+==> prob (q o (%x. SDROP (length (alg_cover l x)) x)) = prob q"
by (import prob_indep ALG_COVER_TAIL_PROB)
-lemma INDEP_INDEP_SET_LEMMA: "ALL l::bool list list.
- alg_cover_set l -->
- (ALL q::(nat => bool) => bool.
- measurable q -->
- (ALL x::bool list.
- x mem l -->
- prob
- (pred_set.INTER (alg_embed x)
- (q o (%x::nat => bool. SDROP (length (alg_cover l x)) x))) =
- (1 / 2) ^ length x * prob q))"
+lemma INDEP_INDEP_SET_LEMMA: "[| alg_cover_set l; measurable q; List.member l x |]
+==> prob
+ (pred_set.INTER (alg_embed x)
+ (q o (%x. SDROP (length (alg_cover l x)) x))) =
+ (1 / 2) ^ length x * prob q"
by (import prob_indep INDEP_INDEP_SET_LEMMA)
-lemma INDEP_SET_LIST: "ALL (q::(nat => bool) => bool) l::bool list list.
- alg_sorted l &
- alg_prefixfree l &
- measurable q &
- (ALL x::bool list. x mem l --> indep_set (alg_embed x) q) -->
- indep_set (algebra_embed l) q"
+lemma INDEP_SET_LIST: "alg_sorted l &
+alg_prefixfree l &
+measurable q & (ALL x. List.member l x --> indep_set (alg_embed x) q)
+==> indep_set (algebra_embed l) q"
by (import prob_indep INDEP_SET_LIST)
-lemma INDEP_INDEP_SET: "ALL (f::(nat => bool) => 'a::type * (nat => bool)) (p::'a::type => bool)
- q::(nat => bool) => bool.
- indep f & measurable q --> indep_set (p o (fst o f)) (q o (snd o f))"
+lemma INDEP_INDEP_SET: "indep f & measurable q ==> indep_set (p o (fst o f)) (q o (snd o f))"
by (import prob_indep INDEP_INDEP_SET)
-lemma INDEP_UNIT: "ALL x::'a::type. indep (UNIT x)"
+lemma INDEP_UNIT: "indep (UNIT x)"
by (import prob_indep INDEP_UNIT)
lemma INDEP_SDEST: "indep SDEST"
by (import prob_indep INDEP_SDEST)
-lemma BIND_STEP: "ALL f::(nat => bool) => 'a::type * (nat => bool).
- BIND SDEST (%k::bool. f o SCONS k) = f"
+lemma BIND_STEP: "BIND SDEST (%k. f o SCONS k) = f"
by (import prob_indep BIND_STEP)
-lemma INDEP_BIND_SDEST: "ALL f::bool => (nat => bool) => 'a::type * (nat => bool).
- (ALL x::bool. indep (f x)) --> indep (BIND SDEST f)"
+lemma INDEP_BIND_SDEST: "(!!x. indep (f x)) ==> indep (BIND SDEST f)"
by (import prob_indep INDEP_BIND_SDEST)
-lemma INDEP_BIND: "ALL (f::(nat => bool) => 'a::type * (nat => bool))
- g::'a::type => (nat => bool) => 'b::type * (nat => bool).
- indep f & (ALL x::'a::type. indep (g x)) --> indep (BIND f g)"
+lemma INDEP_BIND: "indep f & (ALL x. indep (g x)) ==> indep (BIND f g)"
by (import prob_indep INDEP_BIND)
-lemma INDEP_PROB: "ALL (f::(nat => bool) => 'a::type * (nat => bool)) (p::'a::type => bool)
- q::(nat => bool) => bool.
- indep f & measurable q -->
- prob (pred_set.INTER (p o (fst o f)) (q o (snd o f))) =
- prob (p o (fst o f)) * prob q"
+lemma INDEP_PROB: "indep f & measurable q
+==> prob (pred_set.INTER (p o (fst o f)) (q o (snd o f))) =
+ prob (p o (fst o f)) * prob q"
by (import prob_indep INDEP_PROB)
-lemma INDEP_MEASURABLE1: "ALL (f::(nat => bool) => 'a::type * (nat => bool)) p::'a::type => bool.
- indep f --> measurable (p o (fst o f))"
+lemma INDEP_MEASURABLE1: "indep f ==> measurable (p o (fst o f))"
by (import prob_indep INDEP_MEASURABLE1)
-lemma INDEP_MEASURABLE2: "ALL (f::(nat => bool) => 'a::type * (nat => bool)) q::(nat => bool) => bool.
- indep f & measurable q --> measurable (q o (snd o f))"
+lemma INDEP_MEASURABLE2: "indep f & measurable q ==> measurable (q o (snd o f))"
by (import prob_indep INDEP_MEASURABLE2)
-lemma PROB_INDEP_BOUND: "ALL (f::(nat => bool) => nat * (nat => bool)) n::nat.
- indep f -->
- prob (%s::nat => bool. fst (f s) < Suc n) =
- prob (%s::nat => bool. fst (f s) < n) +
- prob (%s::nat => bool. fst (f s) = n)"
+lemma PROB_INDEP_BOUND: "indep f
+==> prob (%s. fst (f s) < Suc n) =
+ prob (%s. fst (f s) < n) + prob (%s. fst (f s) = n)"
by (import prob_indep PROB_INDEP_BOUND)
;end_setup
@@ -1896,47 +1395,36 @@
defs
unif_bound_primdef: "unif_bound ==
-WFREC
- (SOME R::nat => nat => bool. WF R & (ALL v::nat. R (Suc v div 2) (Suc v)))
- (%unif_bound::nat => nat.
- nat_case 0 (%v1::nat. Suc (unif_bound (Suc v1 div 2))))"
+WFREC (SOME R. WF R & (ALL v. R (Suc v div 2) (Suc v)))
+ (%unif_bound. nat_case 0 (%v1. Suc (unif_bound (Suc v1 div 2))))"
lemma unif_bound_primitive_def: "unif_bound =
-WFREC
- (SOME R::nat => nat => bool. WF R & (ALL v::nat. R (Suc v div 2) (Suc v)))
- (%unif_bound::nat => nat.
- nat_case 0 (%v1::nat. Suc (unif_bound (Suc v1 div 2))))"
+WFREC (SOME R. WF R & (ALL v. R (Suc v div 2) (Suc v)))
+ (%unif_bound. nat_case 0 (%v1. Suc (unif_bound (Suc v1 div 2))))"
by (import prob_uniform unif_bound_primitive_def)
-lemma unif_bound_def: "unif_bound 0 = 0 &
-unif_bound (Suc (v::nat)) = Suc (unif_bound (Suc v div 2))"
+lemma unif_bound_def: "unif_bound 0 = 0 & unif_bound (Suc v) = Suc (unif_bound (Suc v div 2))"
by (import prob_uniform unif_bound_def)
-lemma unif_bound_ind: "ALL P::nat => bool.
- P 0 & (ALL v::nat. P (Suc v div 2) --> P (Suc v)) --> All P"
+lemma unif_bound_ind: "P 0 & (ALL v. P (Suc v div 2) --> P (Suc v)) ==> P x"
by (import prob_uniform unif_bound_ind)
-definition unif_tupled :: "nat * (nat => bool) => nat * (nat => bool)" where
+definition
+ unif_tupled :: "nat * (nat => bool) => nat * (nat => bool)" where
"unif_tupled ==
-WFREC
- (SOME R::nat * (nat => bool) => nat * (nat => bool) => bool.
- WF R & (ALL (s::nat => bool) v2::nat. R (Suc v2 div 2, s) (Suc v2, s)))
- (%(unif_tupled::nat * (nat => bool) => nat * (nat => bool)) (v::nat,
- v1::nat => bool).
+WFREC (SOME R. WF R & (ALL s v2. R (Suc v2 div 2, s) (Suc v2, s)))
+ (%unif_tupled (v, v1).
case v of 0 => (0, v1)
- | Suc (v3::nat) =>
- let (m::nat, s'::nat => bool) = unif_tupled (Suc v3 div 2, v1)
+ | Suc v3 =>
+ let (m, s') = unif_tupled (Suc v3 div 2, v1)
in (if SHD s' then 2 * m + 1 else 2 * m, STL s'))"
lemma unif_tupled_primitive_def: "unif_tupled =
-WFREC
- (SOME R::nat * (nat => bool) => nat * (nat => bool) => bool.
- WF R & (ALL (s::nat => bool) v2::nat. R (Suc v2 div 2, s) (Suc v2, s)))
- (%(unif_tupled::nat * (nat => bool) => nat * (nat => bool)) (v::nat,
- v1::nat => bool).
+WFREC (SOME R. WF R & (ALL s v2. R (Suc v2 div 2, s) (Suc v2, s)))
+ (%unif_tupled (v, v1).
case v of 0 => (0, v1)
- | Suc (v3::nat) =>
- let (m::nat, s'::nat => bool) = unif_tupled (Suc v3 div 2, v1)
+ | Suc v3 =>
+ let (m, s') = unif_tupled (Suc v3 div 2, v1)
in (if SHD s' then 2 * m + 1 else 2 * m, STL s'))"
by (import prob_uniform unif_tupled_primitive_def)
@@ -1944,247 +1432,160 @@
unif :: "nat => (nat => bool) => nat * (nat => bool)"
defs
- unif_primdef: "unif == %(x::nat) x1::nat => bool. unif_tupled (x, x1)"
+ unif_primdef: "unif == %x x1. unif_tupled (x, x1)"
-lemma unif_curried_def: "ALL (x::nat) x1::nat => bool. unif x x1 = unif_tupled (x, x1)"
+lemma unif_curried_def: "unif x x1 = unif_tupled (x, x1)"
by (import prob_uniform unif_curried_def)
-lemma unif_def: "unif 0 (s::nat => bool) = (0, s) &
-unif (Suc (v2::nat)) s =
-(let (m::nat, s'::nat => bool) = unif (Suc v2 div 2) s
+lemma unif_def: "unif 0 s = (0, s) &
+unif (Suc v2) s =
+(let (m, s') = unif (Suc v2 div 2) s
in (if SHD s' then 2 * m + 1 else 2 * m, STL s'))"
by (import prob_uniform unif_def)
-lemma unif_ind: "ALL P::nat => (nat => bool) => bool.
- All (P 0) &
- (ALL (v2::nat) s::nat => bool. P (Suc v2 div 2) s --> P (Suc v2) s) -->
- (ALL v::nat. All (P v))"
+lemma unif_ind: "All ((P::nat => (nat => bool) => bool) (0::nat)) &
+(ALL (v2::nat) s::nat => bool. P (Suc v2 div (2::nat)) s --> P (Suc v2) s)
+==> P (v::nat) (x::nat => bool)"
by (import prob_uniform unif_ind)
-definition uniform_tupled :: "nat * nat * (nat => bool) => nat * (nat => bool)" where
+definition
+ uniform_tupled :: "nat * nat * (nat => bool) => nat * (nat => bool)" where
"uniform_tupled ==
WFREC
- (SOME R::nat * nat * (nat => bool) => nat * nat * (nat => bool) => bool.
+ (SOME R.
WF R &
- (ALL (t::nat) (s::nat => bool) (n::nat) (res::nat) s'::nat => bool.
+ (ALL t s n res s'.
(res, s') = unif n s & ~ res < Suc n -->
R (t, Suc n, s') (Suc t, Suc n, s)))
- (%(uniform_tupled::nat * nat * (nat => bool) => nat * (nat => bool))
- (v::nat, v1::nat * (nat => bool)).
- case v of
- 0 => (%(v3::nat, v4::nat => bool).
- case v3 of 0 => ARB | Suc (v5::nat) => (0, v4))
- v1
- | Suc (v2::nat) =>
- (%(v7::nat, v8::nat => bool).
- case v7 of 0 => ARB
- | Suc (v9::nat) =>
- let (res::nat, s'::nat => bool) = unif v9 v8
- in if res < Suc v9 then (res, s')
- else uniform_tupled (v2, Suc v9, s'))
- v1)"
+ (%uniform_tupled (v, v1).
+ case v of 0 => case v1 of (0, v4) => ARB | (Suc v5, v4) => (0, v4)
+ | Suc v2 =>
+ case v1 of (0, v8) => ARB
+ | (Suc v9, v8) =>
+ let (res, s') = unif v9 v8
+ in if res < Suc v9 then (res, s')
+ else uniform_tupled (v2, Suc v9, s'))"
lemma uniform_tupled_primitive_def: "uniform_tupled =
WFREC
- (SOME R::nat * nat * (nat => bool) => nat * nat * (nat => bool) => bool.
+ (SOME R.
WF R &
- (ALL (t::nat) (s::nat => bool) (n::nat) (res::nat) s'::nat => bool.
+ (ALL t s n res s'.
(res, s') = unif n s & ~ res < Suc n -->
R (t, Suc n, s') (Suc t, Suc n, s)))
- (%(uniform_tupled::nat * nat * (nat => bool) => nat * (nat => bool))
- (v::nat, v1::nat * (nat => bool)).
- case v of
- 0 => (%(v3::nat, v4::nat => bool).
- case v3 of 0 => ARB | Suc (v5::nat) => (0, v4))
- v1
- | Suc (v2::nat) =>
- (%(v7::nat, v8::nat => bool).
- case v7 of 0 => ARB
- | Suc (v9::nat) =>
- let (res::nat, s'::nat => bool) = unif v9 v8
- in if res < Suc v9 then (res, s')
- else uniform_tupled (v2, Suc v9, s'))
- v1)"
+ (%uniform_tupled (v, v1).
+ case v of 0 => case v1 of (0, v4) => ARB | (Suc v5, v4) => (0, v4)
+ | Suc v2 =>
+ case v1 of (0, v8) => ARB
+ | (Suc v9, v8) =>
+ let (res, s') = unif v9 v8
+ in if res < Suc v9 then (res, s')
+ else uniform_tupled (v2, Suc v9, s'))"
by (import prob_uniform uniform_tupled_primitive_def)
consts
uniform :: "nat => nat => (nat => bool) => nat * (nat => bool)"
defs
- uniform_primdef: "uniform == %(x::nat) (x1::nat) x2::nat => bool. uniform_tupled (x, x1, x2)"
+ uniform_primdef: "uniform == %x x1 x2. uniform_tupled (x, x1, x2)"
-lemma uniform_curried_def: "ALL (x::nat) (x1::nat) x2::nat => bool.
- uniform x x1 x2 = uniform_tupled (x, x1, x2)"
+lemma uniform_curried_def: "uniform x x1 x2 = uniform_tupled (x, x1, x2)"
by (import prob_uniform uniform_curried_def)
-lemma uniform_ind: "ALL P::nat => nat => (nat => bool) => bool.
- (ALL x::nat. All (P (Suc x) 0)) &
- All (P 0 0) &
- (ALL x::nat. All (P 0 (Suc x))) &
- (ALL (x::nat) (xa::nat) xb::nat => bool.
- (ALL (xc::nat) xd::nat => bool.
- (xc, xd) = unif xa xb & ~ xc < Suc xa --> P x (Suc xa) xd) -->
- P (Suc x) (Suc xa) xb) -->
- (ALL (x::nat) xa::nat. All (P x xa))"
+lemma uniform_ind: "(ALL x. All (P (Suc x) 0)) &
+All (P 0 0) &
+(ALL x. All (P 0 (Suc x))) &
+(ALL x xa xb.
+ (ALL xc xd.
+ (xc, xd) = unif xa xb & ~ xc < Suc xa --> P x (Suc xa) xd) -->
+ P (Suc x) (Suc xa) xb)
+==> P x xa xb"
by (import prob_uniform uniform_ind)
-lemma uniform_def: "uniform 0 (Suc (n::nat)) (s::nat => bool) = (0, s) &
-uniform (Suc (t::nat)) (Suc n) s =
-(let (xa::nat, x::nat => bool) = unif n s
+lemma uniform_def: "uniform 0 (Suc n) s = (0, s) &
+uniform (Suc t) (Suc n) s =
+(let (xa, x) = unif n s
in if xa < Suc n then (xa, x) else uniform t (Suc n) x)"
by (import prob_uniform uniform_def)
-lemma SUC_DIV_TWO_ZERO: "ALL n::nat. (Suc n div 2 = 0) = (n = 0)"
+lemma SUC_DIV_TWO_ZERO: "(Suc n div 2 = 0) = (n = 0)"
by (import prob_uniform SUC_DIV_TWO_ZERO)
-lemma UNIF_BOUND_LOWER: "ALL n::nat. n < 2 ^ unif_bound n"
+lemma UNIF_BOUND_LOWER: "n < 2 ^ unif_bound n"
by (import prob_uniform UNIF_BOUND_LOWER)
-lemma UNIF_BOUND_LOWER_SUC: "ALL n::nat. Suc n <= 2 ^ unif_bound n"
+lemma UNIF_BOUND_LOWER_SUC: "Suc n <= 2 ^ unif_bound n"
by (import prob_uniform UNIF_BOUND_LOWER_SUC)
-lemma UNIF_BOUND_UPPER: "ALL n::nat. n ~= 0 --> 2 ^ unif_bound n <= 2 * n"
+lemma UNIF_BOUND_UPPER: "n ~= 0 ==> 2 ^ unif_bound n <= 2 * n"
by (import prob_uniform UNIF_BOUND_UPPER)
-lemma UNIF_BOUND_UPPER_SUC: "ALL n::nat. 2 ^ unif_bound n <= Suc (2 * n)"
+lemma UNIF_BOUND_UPPER_SUC: "2 ^ unif_bound n <= Suc (2 * n)"
by (import prob_uniform UNIF_BOUND_UPPER_SUC)
lemma UNIF_DEF_MONAD: "unif 0 = UNIT 0 &
-(ALL n::nat.
+(ALL n.
unif (Suc n) =
BIND (unif (Suc n div 2))
- (%m::nat.
- BIND SDEST (%b::bool. UNIT (if b then 2 * m + 1 else 2 * m))))"
+ (%m. BIND SDEST (%b. UNIT (if b then 2 * m + 1 else 2 * m))))"
by (import prob_uniform UNIF_DEF_MONAD)
-lemma UNIFORM_DEF_MONAD: "(ALL x::nat. uniform 0 (Suc x) = UNIT 0) &
-(ALL (x::nat) xa::nat.
+lemma UNIFORM_DEF_MONAD: "(ALL x. uniform 0 (Suc x) = UNIT 0) &
+(ALL x xa.
uniform (Suc x) (Suc xa) =
- BIND (unif xa)
- (%m::nat. if m < Suc xa then UNIT m else uniform x (Suc xa)))"
+ BIND (unif xa) (%m. if m < Suc xa then UNIT m else uniform x (Suc xa)))"
by (import prob_uniform UNIFORM_DEF_MONAD)
-lemma INDEP_UNIF: "ALL n::nat. indep (unif n)"
+lemma INDEP_UNIF: "indep (unif n)"
by (import prob_uniform INDEP_UNIF)
-lemma INDEP_UNIFORM: "ALL (t::nat) n::nat. indep (uniform t (Suc n))"
+lemma INDEP_UNIFORM: "indep (uniform t (Suc n))"
by (import prob_uniform INDEP_UNIFORM)
-lemma PROB_UNIF: "ALL (n::nat) k::nat.
- prob (%s::nat => bool. fst (unif n s) = k) =
- (if k < 2 ^ unif_bound n then (1 / 2) ^ unif_bound n else 0)"
+lemma PROB_UNIF: "prob (%s. fst (unif n s) = k) =
+(if k < 2 ^ unif_bound n then (1 / 2) ^ unif_bound n else 0)"
by (import prob_uniform PROB_UNIF)
-lemma UNIF_RANGE: "ALL (n::nat) s::nat => bool. fst (unif n s) < 2 ^ unif_bound n"
+lemma UNIF_RANGE: "fst (unif n s) < 2 ^ unif_bound n"
by (import prob_uniform UNIF_RANGE)
-lemma PROB_UNIF_PAIR: "ALL (n::nat) (k::nat) k'::nat.
- (prob (%s::nat => bool. fst (unif n s) = k) =
- prob (%s::nat => bool. fst (unif n s) = k')) =
- ((k < 2 ^ unif_bound n) = (k' < 2 ^ unif_bound n))"
+lemma PROB_UNIF_PAIR: "(prob (%s. fst (unif n s) = k) = prob (%s. fst (unif n s) = k')) =
+((k < 2 ^ unif_bound n) = (k' < 2 ^ unif_bound n))"
by (import prob_uniform PROB_UNIF_PAIR)
-lemma PROB_UNIF_BOUND: "ALL (n::nat) k::nat.
- k <= 2 ^ unif_bound n -->
- prob (%s::nat => bool. fst (unif n s) < k) =
- real k * (1 / 2) ^ unif_bound n"
+lemma PROB_UNIF_BOUND: "k <= 2 ^ unif_bound n
+==> prob (%s. fst (unif n s) < k) = real k * (1 / 2) ^ unif_bound n"
by (import prob_uniform PROB_UNIF_BOUND)
-lemma PROB_UNIF_GOOD: "ALL n::nat. 1 / 2 <= prob (%s::nat => bool. fst (unif n s) < Suc n)"
+lemma PROB_UNIF_GOOD: "1 / 2 <= prob (%s. fst (unif n s) < Suc n)"
by (import prob_uniform PROB_UNIF_GOOD)
-lemma UNIFORM_RANGE: "ALL (t::nat) (n::nat) s::nat => bool. fst (uniform t (Suc n) s) < Suc n"
+lemma UNIFORM_RANGE: "fst (uniform t (Suc n) s) < Suc n"
by (import prob_uniform UNIFORM_RANGE)
-lemma PROB_UNIFORM_LOWER_BOUND: "(All::(real => bool) => bool)
- (%b::real.
- (op -->::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k ((Suc::nat => nat) (n::nat)))
- ((op <::real => real => bool)
- ((prob::((nat => bool) => bool) => real)
- (%s::nat => bool.
- (op =::nat => nat => bool)
- ((fst::nat * (nat => bool) => nat)
- ((uniform::nat
- => nat
-=> (nat => bool) => nat * (nat => bool))
- (t::nat) ((Suc::nat => nat) n) s))
- k))
- b)))
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) m ((Suc::nat => nat) n))
- ((op <::real => real => bool)
- ((prob::((nat => bool) => bool) => real)
- (%s::nat => bool.
- (op <::nat => nat => bool)
- ((fst::nat * (nat => bool) => nat)
- ((uniform::nat
- => nat
-=> (nat => bool) => nat * (nat => bool))
- t ((Suc::nat => nat) n) s))
- ((Suc::nat => nat) m)))
- ((op *::real => real => real)
- ((real::nat => real) ((Suc::nat => nat) m)) b)))))"
+lemma PROB_UNIFORM_LOWER_BOUND: "[| !!k. k < Suc n ==> prob (%s. fst (uniform t (Suc n) s) = k) < b;
+ m < Suc n |]
+==> prob (%s. fst (uniform t (Suc n) s) < Suc m) < real (Suc m) * b"
by (import prob_uniform PROB_UNIFORM_LOWER_BOUND)
-lemma PROB_UNIFORM_UPPER_BOUND: "(All::(real => bool) => bool)
- (%b::real.
- (op -->::bool => bool => bool)
- ((All::(nat => bool) => bool)
- (%k::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) k ((Suc::nat => nat) (n::nat)))
- ((op <::real => real => bool) b
- ((prob::((nat => bool) => bool) => real)
- (%s::nat => bool.
- (op =::nat => nat => bool)
- ((fst::nat * (nat => bool) => nat)
- ((uniform::nat
- => nat
-=> (nat => bool) => nat * (nat => bool))
- (t::nat) ((Suc::nat => nat) n) s))
- k)))))
- ((All::(nat => bool) => bool)
- (%m::nat.
- (op -->::bool => bool => bool)
- ((op <::nat => nat => bool) m ((Suc::nat => nat) n))
- ((op <::real => real => bool)
- ((op *::real => real => real)
- ((real::nat => real) ((Suc::nat => nat) m)) b)
- ((prob::((nat => bool) => bool) => real)
- (%s::nat => bool.
- (op <::nat => nat => bool)
- ((fst::nat * (nat => bool) => nat)
- ((uniform::nat
- => nat
-=> (nat => bool) => nat * (nat => bool))
- t ((Suc::nat => nat) n) s))
- ((Suc::nat => nat) m)))))))"
+lemma PROB_UNIFORM_UPPER_BOUND: "[| !!k. k < Suc n ==> b < prob (%s. fst (uniform t (Suc n) s) = k);
+ m < Suc n |]
+==> real (Suc m) * b < prob (%s. fst (uniform t (Suc n) s) < Suc m)"
by (import prob_uniform PROB_UNIFORM_UPPER_BOUND)
-lemma PROB_UNIFORM_PAIR_SUC: "ALL (t::nat) (n::nat) (k::nat) k'::nat.
- k < Suc n & k' < Suc n -->
- abs (prob (%s::nat => bool. fst (uniform t (Suc n) s) = k) -
- prob (%s::nat => bool. fst (uniform t (Suc n) s) = k'))
- <= (1 / 2) ^ t"
+lemma PROB_UNIFORM_PAIR_SUC: "k < Suc n & k' < Suc n
+==> abs (prob (%s. fst (uniform t (Suc n) s) = k) -
+ prob (%s. fst (uniform t (Suc n) s) = k'))
+ <= (1 / 2) ^ t"
by (import prob_uniform PROB_UNIFORM_PAIR_SUC)
-lemma PROB_UNIFORM_SUC: "ALL (t::nat) (n::nat) k::nat.
- k < Suc n -->
- abs (prob (%s::nat => bool. fst (uniform t (Suc n) s) = k) -
- 1 / real (Suc n))
- <= (1 / 2) ^ t"
+lemma PROB_UNIFORM_SUC: "k < Suc n
+==> abs (prob (%s. fst (uniform t (Suc n) s) = k) - 1 / real (Suc n))
+ <= (1 / 2) ^ t"
by (import prob_uniform PROB_UNIFORM_SUC)
-lemma PROB_UNIFORM: "ALL (t::nat) (n::nat) k::nat.
- k < n -->
- abs (prob (%s::nat => bool. fst (uniform t n s) = k) - 1 / real n)
- <= (1 / 2) ^ t"
+lemma PROB_UNIFORM: "k < n
+==> abs (prob (%s. fst (uniform t n s) = k) - 1 / real n) <= (1 / 2) ^ t"
by (import prob_uniform PROB_UNIFORM)
;end_setup