HOL/Import: Update HOL4 generated files to current Isabelle.
(* AUTOMATICALLY GENERATED, DO NOT EDIT! *)
theory HOL4Prob imports HOL4Real begin
;setup_theory prob_extra
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: "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: "Suc q * (p div Suc q) <= p"
by (import prob_extra DIV_THEN_MULT)
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: "(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: "(n::nat) = (2::nat) * (n div (2::nat)) + n mod (2::nat)"
by (import prob_extra DIV_TWO)
lemma MOD_TWO: "n mod 2 = (if EVEN n then 0 else 1)"
by (import prob_extra MOD_TWO)
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: "(m::nat) div (2::nat) < (n::nat) div (2::nat) ==> m < n"
by (import prob_extra DIV_TWO_MONO)
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: "2 * n div 2 = n & Suc (2 * n) div 2 = n"
by (import prob_extra DIV_TWO_CANCEL)
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: "EVEN (2 ^ n) = (n ~= 0)"
by (import prob_extra EVEN_EXP_TWO)
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: "prob_extra.inf == %P. - real.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: "prob_extra.inf P = - real.sup (%r. P (- r))"
by (import prob_extra INF_DEF_ALT)
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: "P z & (ALL x. P x --> x <= z) ==> real.sup P = z"
by (import prob_extra REAL_SUP_MAX)
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_CANCEL: "(2::real) * ((1::real) / (2::real)) = (1::real)"
by (import prob_extra HALF_CANCEL)
lemma POW_HALF_POS: "(0::real) < ((1::real) / (2::real)) ^ (n::nat)"
by (import prob_extra POW_HALF_POS)
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: "((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: "(1::real) / (2::real) * (x::real) + (1::real) / (2::real) * x = x"
by (import prob_extra X_HALF_HALF)
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: "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: "((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: "(1::real) - (1::real) / (2::real) = (1::real) / (2::real)"
by (import prob_extra ONE_MINUS_HALF)
lemma HALF_LT_1: "(1::real) / (2::real) < (1::real)"
by (import prob_extra HALF_LT_1)
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: "0 < 1 / real (Suc n)"
by (import prob_extra INV_SUC_POS)
lemma INV_SUC_MAX: "1 / real (Suc x) <= 1"
by (import prob_extra INV_SUC_MAX)
lemma INV_SUC: "0 < 1 / real (Suc n) & 1 / real (Suc n) <= 1"
by (import prob_extra INV_SUC)
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 x. ~ List.member l x) = (l = [])"
by (import prob_extra MEM_NIL)
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: "~ List.member (map (op # x) l) []"
by (import prob_extra MEM_NIL_MAP_CONS)
lemma FILTER_TRUE: "[x<-l. True] = l"
by (import prob_extra FILTER_TRUE)
lemma FILTER_FALSE: "[x<-l. False] = []"
by (import prob_extra FILTER_FALSE)
lemma FILTER_MEM: "List.member (filter P l) x ==> P x"
by (import prob_extra FILTER_MEM)
lemma MEM_FILTER: "List.member (filter P l) x ==> List.member l x"
by (import prob_extra MEM_FILTER)
lemma FILTER_OUT_ELT: "List.member l x | [y<-l. y ~= x] = l"
by (import prob_extra FILTER_OUT_ELT)
lemma IS_PREFIX_NIL: "IS_PREFIX x [] & IS_PREFIX [] x = (x = [])"
by (import prob_extra IS_PREFIX_NIL)
lemma IS_PREFIX_REFL: "IS_PREFIX x x"
by (import prob_extra IS_PREFIX_REFL)
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: "IS_PREFIX x y & IS_PREFIX y z ==> IS_PREFIX x z"
by (import prob_extra IS_PREFIX_TRANS)
lemma IS_PREFIX_BUTLAST: "IS_PREFIX (x # y) (butlast (x # y))"
by (import prob_extra IS_PREFIX_BUTLAST)
lemma IS_PREFIX_LENGTH: "IS_PREFIX y x ==> length x <= length y"
by (import prob_extra IS_PREFIX_LENGTH)
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: "IS_PREFIX (SNOC x y) z = (IS_PREFIX y z | z = SNOC x y)"
by (import prob_extra IS_PREFIX_SNOC)
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: "List.member (h # t) (last (h # t))"
by (import prob_extra LAST_MEM)
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: "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: "pred_set.UNION s t = (%x. s x | t x)"
by (import prob_extra UNION_DEF_ALT)
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: "(x = xa) = (SUBSET x xa & SUBSET xa x)"
by (import prob_extra SUBSET_EQ)
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: "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: "GSPEC (f::'a => 'b * bool) = (%v::'b. EX x::'a. (v, True) = f x)"
by (import prob_extra GSPEC_DEF_ALT)
;end_setup
;setup_theory prob_canon
consts
alg_twin :: "bool list => bool list => bool"
defs
alg_twin_primdef: "alg_twin == %x y. EX l. 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
"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: "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 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 (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 # v7) = True &
alg_order [] [] = True &
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)
consts
alg_sorted :: "bool list list => bool"
defs
alg_sorted_primdef: "alg_sorted ==
WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
(%alg_sorted.
list_case True
(%v2. list_case True
(%v6 v7. alg_order v2 v6 & alg_sorted (v6 # v7))))"
lemma alg_sorted_primitive_def: "alg_sorted =
WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
(%alg_sorted.
list_case True
(%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 (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 # 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
alg_prefixfree :: "bool list list => bool"
defs
alg_prefixfree_primdef: "alg_prefixfree ==
WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
(%alg_prefixfree.
list_case True
(%v2. list_case True
(%v6 v7. ~ IS_PREFIX v6 v2 & alg_prefixfree (v6 # v7))))"
lemma alg_prefixfree_primitive_def: "alg_prefixfree =
WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
(%alg_prefixfree.
list_case True
(%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 (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 # 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
alg_twinfree :: "bool list list => bool"
defs
alg_twinfree_primdef: "alg_twinfree ==
WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
(%alg_twinfree.
list_case True
(%v2. list_case True
(%v6 v7. ~ alg_twin v2 v6 & alg_twinfree (v6 # v7))))"
lemma alg_twinfree_primitive_def: "alg_twinfree =
WFREC (SOME R. WF R & (ALL x z y. R (y # z) (x # y # z)))
(%alg_twinfree.
list_case True
(%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 (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 # 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 t. 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. 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)
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. 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
else alg_canon_prefs l (h # t)))"
by (import prob_canon alg_canon_find_def)
consts
alg_canon1 :: "bool list list => bool list list"
defs
alg_canon1_primdef: "alg_canon1 == FOLDR alg_canon_find []"
lemma alg_canon1_def: "alg_canon1 = FOLDR alg_canon_find []"
by (import prob_canon alg_canon1_def)
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. 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)
consts
alg_canon2 :: "bool list list => bool list list"
defs
alg_canon2_primdef: "alg_canon2 == FOLDR alg_canon_merge []"
lemma alg_canon2_def: "alg_canon2 = FOLDR alg_canon_merge []"
by (import prob_canon alg_canon2_def)
consts
alg_canon :: "bool list list => bool list list"
defs
alg_canon_primdef: "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. 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: "~ alg_twin l [] & ~ alg_twin [] l"
by (import prob_canon ALG_TWIN_NIL)
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: "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: "alg_twin (h # t) (h # t') = alg_twin t t'"
by (import prob_canon ALG_TWIN_REDUCE)
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: "alg_order [] x & alg_order x [] = (x = [])"
by (import prob_canon ALG_ORDER_NIL)
lemma ALG_ORDER_REFL: "alg_order x x"
by (import prob_canon ALG_ORDER_REFL)
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: "alg_order x y & alg_order y z ==> alg_order x z"
by (import prob_canon ALG_ORDER_TRANS)
lemma ALG_ORDER_TOTAL: "alg_order x y | alg_order y x"
by (import prob_canon ALG_ORDER_TOTAL)
lemma ALG_ORDER_PREFIX: "IS_PREFIX y x ==> alg_order x y"
by (import prob_canon ALG_ORDER_PREFIX)
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: "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: "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: "~ alg_order (SNOC x l) l"
by (import prob_canon ALG_ORDER_SNOC)
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: "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: "alg_sorted (h # t) ==> alg_sorted t"
by (import prob_canon ALG_SORTED_TL)
lemma ALG_SORTED_MONO: "alg_sorted (x # y # z) ==> alg_sorted (x # z)"
by (import prob_canon ALG_SORTED_MONO)
lemma ALG_SORTED_TLS: "alg_sorted (map (op # b) l) = alg_sorted l"
by (import prob_canon ALG_SORTED_TLS)
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: "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: "alg_sorted b ==> alg_sorted (filter P b)"
by (import prob_canon ALG_SORTED_FILTER)
lemma ALG_PREFIXFREE_TL: "alg_prefixfree (h # t) ==> alg_prefixfree t"
by (import prob_canon ALG_PREFIXFREE_TL)
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: "[| 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: "alg_prefixfree (map (op # b) l) = alg_prefixfree l"
by (import prob_canon ALG_PREFIXFREE_TLS)
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: "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: "alg_sorted b & alg_prefixfree b ==> alg_prefixfree (filter P b)"
by (import prob_canon ALG_PREFIXFREE_FILTER)
lemma ALG_TWINFREE_TL: "alg_twinfree (h # t) ==> alg_twinfree t"
by (import prob_canon ALG_TWINFREE_TL)
lemma ALG_TWINFREE_TLS: "alg_twinfree (map (op # b) l) = alg_twinfree l"
by (import prob_canon ALG_TWINFREE_TLS)
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: "(~ 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: "~ 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: "length h <= alg_longest (h # t)"
by (import prob_canon ALG_LONGEST_HD)
lemma ALG_LONGEST_TL: "alg_longest t <= alg_longest (h # t)"
by (import prob_canon ALG_LONGEST_TL)
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: "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: "hd (alg_canon_prefs l b) = l"
by (import prob_canon ALG_CANON_PREFS_HD)
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: "alg_sorted (l # b) ==> alg_sorted (alg_canon_prefs l b)"
by (import prob_canon ALG_CANON_PREFS_SORTED)
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: "alg_prefixfree (l # b) ==> alg_canon_prefs l b = l # b"
by (import prob_canon ALG_CANON_PREFS_CONSTANT)
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: "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: "alg_sorted b ==> alg_sorted (alg_canon_find l b)"
by (import prob_canon ALG_CANON_FIND_SORTED)
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: "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: "alg_sorted (alg_canon1 x)"
by (import prob_canon ALG_CANON1_SORTED)
lemma ALG_CANON1_PREFIXFREE: "alg_prefixfree (alg_canon1 l)"
by (import prob_canon ALG_CANON1_PREFIXFREE)
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: "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: "[| !!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: "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: "alg_twinfree (l # b) ==> alg_canon_merge l b = l # b"
by (import prob_canon ALG_CANON_MERGE_CONSTANT)
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: "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: "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: "alg_twinfree l ==> alg_canon2 l = l"
by (import prob_canon ALG_CANON2_CONSTANT)
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: "alg_sorted l & alg_prefixfree l & alg_twinfree l ==> alg_canon l = l"
by (import prob_canon ALG_CANON_CONSTANT)
lemma ALG_CANON_IDEMPOT: "alg_canon (alg_canon l) = alg_canon l"
by (import prob_canon ALG_CANON_IDEMPOT)
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. algebra_canon [x])"
by (import prob_canon ALGEBRA_CANON_BASIC)
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: "algebra_canon (h # t) ==> algebra_canon t"
by (import prob_canon ALGEBRA_CANON_TL)
lemma ALGEBRA_CANON_NIL_MEM: "(algebra_canon l & List.member l []) = (l = [[]])"
by (import prob_canon ALGEBRA_CANON_NIL_MEM)
lemma ALGEBRA_CANON_TLS: "algebra_canon (map (op # b) l) = algebra_canon l"
by (import prob_canon ALGEBRA_CANON_TLS)
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: "(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: "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: "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: "[| P [] &
P [[]] &
(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));
algebra_canon l |]
==> P l"
by (import prob_canon ALGEBRA_CANON_CASES)
lemma ALGEBRA_CANON_INDUCTION: "[| P [] &
P [[]] &
(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));
algebra_canon l |]
==> P l"
by (import prob_canon ALGEBRA_CANON_INDUCTION)
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: "(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 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
;setup_theory boolean_sequence
consts
SHD :: "(nat => bool) => bool"
defs
SHD_primdef: "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 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 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. (SHD s, STL s)"
lemma SDEST_def: "SDEST = (%s. (SHD s, STL s))"
by (import boolean_sequence SDEST_def)
consts
SCONST :: "bool => nat => bool"
defs
SCONST_primdef: "SCONST == K"
lemma SCONST_def: "SCONST = K"
by (import boolean_sequence SCONST_def)
consts
STAKE :: "nat => (nat => bool) => bool list"
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. SDROP (Suc n) = SDROP n o STL)"
by (import boolean_sequence SDROP_def)
lemma SCONS_SURJ: "EX xa t. x = SCONS xa t"
by (import boolean_sequence SCONS_SURJ)
lemma SHD_STL_ISO: "EX x. SHD x = h & STL x = t"
by (import boolean_sequence SHD_STL_ISO)
lemma SHD_SCONS: "SHD (SCONS h t) = h"
by (import boolean_sequence SHD_SCONS)
lemma STL_SCONS: "STL (SCONS h t) = t"
by (import boolean_sequence STL_SCONS)
lemma SHD_SCONST: "SHD (SCONST b) = b"
by (import boolean_sequence SHD_SCONST)
lemma STL_SCONST: "STL (SCONST b) = SCONST b"
by (import boolean_sequence STL_SCONST)
;end_setup
;setup_theory prob_algebra
consts
alg_embed :: "bool list => (nat => bool) => bool"
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 t.
algebra_embed (h # t) = pred_set.UNION (alg_embed h) (algebra_embed t))"
by (import prob_algebra algebra_embed_def)
consts
measurable :: "((nat => bool) => bool) => bool"
defs
measurable_primdef: "measurable == %s. EX b. 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. SHD x = True) (%x. SHD x = False) = EMPTY"
by (import prob_algebra HALVES_INTER)
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: "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 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 (alg_embed c) = (c = [])"
by (import prob_algebra ALG_EMBED_NIL)
lemma ALG_EMBED_POPULATED: "Ex (alg_embed b)"
by (import prob_algebra ALG_EMBED_POPULATED)
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: "SUBSET (alg_embed b) (alg_embed c) = IS_PREFIX b c"
by (import prob_algebra ALG_EMBED_PREFIX_SUBSET)
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. algebra_embed [[b]] = (%s. SHD s = b))"
by (import prob_algebra ALGEBRA_EMBED_BASIC)
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: "algebra_embed (l1 @ l2) =
pred_set.UNION (algebra_embed l1) (algebra_embed l2)"
by (import prob_algebra ALGEBRA_EMBED_APPEND)
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: "algebra_embed (alg_canon_prefs l b) = algebra_embed (l # b)"
by (import prob_algebra ALG_CANON_PREFS_EMBED)
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: "algebra_embed (alg_canon1 x) = algebra_embed x"
by (import prob_algebra ALG_CANON1_EMBED)
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: "algebra_embed (alg_canon2 x) = algebra_embed x"
by (import prob_algebra ALG_CANON2_EMBED)
lemma ALG_CANON_EMBED: "algebra_embed (alg_canon l) = algebra_embed l"
by (import prob_algebra ALG_CANON_EMBED)
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: "(alg_canon b = alg_canon c) = (algebra_embed b = algebra_embed c)"
by (import prob_algebra ALG_CANON_REP)
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: "algebra_canon l ==> All (algebra_embed l) = (l = [[]])"
by (import prob_algebra ALGEBRA_CANON_EMBED_UNIV)
lemma MEASURABLE_ALGEBRA: "measurable (algebra_embed b)"
by (import prob_algebra MEASURABLE_ALGEBRA)
lemma MEASURABLE_BASIC: "measurable EMPTY &
measurable pred_set.UNIV & (ALL b. measurable (%s. SHD s = b))"
by (import prob_algebra MEASURABLE_BASIC)
lemma MEASURABLE_SHD: "measurable (%s. SHD s = b)"
by (import prob_algebra MEASURABLE_SHD)
lemma ALGEBRA_EMBED_COMPL: "EX l'. COMPL (algebra_embed l) = algebra_embed l'"
by (import prob_algebra ALGEBRA_EMBED_COMPL)
lemma MEASURABLE_COMPL: "measurable (COMPL s) = measurable s"
by (import prob_algebra MEASURABLE_COMPL)
lemma MEASURABLE_UNION: "measurable s & measurable t ==> measurable (pred_set.UNION s t)"
by (import prob_algebra MEASURABLE_UNION)
lemma MEASURABLE_INTER: "measurable s & measurable t ==> measurable (pred_set.INTER s t)"
by (import prob_algebra MEASURABLE_INTER)
lemma MEASURABLE_STL: "measurable (p o STL) = measurable p"
by (import prob_algebra MEASURABLE_STL)
lemma MEASURABLE_SDROP: "measurable (p o SDROP n) = measurable p"
by (import prob_algebra MEASURABLE_SDROP)
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: "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: "measurable (pred_set.INTER (%x. SHD x = b) (p o STL)) = measurable p"
by (import prob_algebra MEASURABLE_INTER_SHD)
;end_setup
;setup_theory prob
consts
alg_measure :: "bool list list => real"
specification (alg_measure_primdef: alg_measure) alg_measure_def: "alg_measure [] = 0 &
(ALL l rest. alg_measure (l # rest) = (1 / 2) ^ length l + alg_measure rest)"
by (import prob alg_measure_def)
consts
algebra_measure :: "bool list list => real"
defs
algebra_measure_primdef: "algebra_measure ==
%b. prob_extra.inf
(%r. EX c. 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
prob :: "((nat => bool) => bool) => real"
defs
prob_primdef: "prob ==
%s. real.sup (%r. EX b. 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: "((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. alg_measure [[b]] = 1 / 2)"
by (import prob ALG_MEASURE_BASIC)
lemma ALG_MEASURE_POS: "0 <= alg_measure l"
by (import prob ALG_MEASURE_POS)
lemma ALG_MEASURE_APPEND: "alg_measure (l1 @ l2) = alg_measure l1 + alg_measure l2"
by (import prob ALG_MEASURE_APPEND)
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: "alg_measure (alg_canon_prefs l b) <= alg_measure (l # b)"
by (import prob ALG_CANON_PREFS_MONO)
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: "alg_measure (alg_canon1 x) <= alg_measure x"
by (import prob ALG_CANON1_MONO)
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: "alg_measure (alg_canon2 x) <= alg_measure x"
by (import prob ALG_CANON2_MONO)
lemma ALG_CANON_MONO: "alg_measure (alg_canon l) <= alg_measure l"
by (import prob ALG_CANON_MONO)
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. algebra_measure [[b]] = 1 / 2)"
by (import prob ALGEBRA_MEASURE_BASIC)
lemma ALGEBRA_CANON_MEASURE_MAX: "algebra_canon l ==> alg_measure l <= 1"
by (import prob ALGEBRA_CANON_MEASURE_MAX)
lemma ALGEBRA_MEASURE_MAX: "algebra_measure l <= 1"
by (import prob ALGEBRA_MEASURE_MAX)
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: "[| 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: "[| 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: "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. prob (%s. SHD s = b) = 1 / 2)"
by (import prob PROB_BASIC)
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: "measurable s ==> prob (COMPL s) = 1 - prob s"
by (import prob PROB_COMPL)
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: "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: "(!!s'. measurable s' & SUBSET s' s ==> prob s' <= x) ==> prob s <= x"
by (import prob PROB_LE_X)
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: "SUBSET s t ==> prob s <= prob t"
by (import prob PROB_SUBSET_MONO)
lemma PROB_ALG: "prob (alg_embed x) = (1 / 2) ^ length x"
by (import prob PROB_ALG)
lemma PROB_STL: "measurable p ==> prob (p o STL) = prob p"
by (import prob PROB_STL)
lemma PROB_SDROP: "measurable p ==> prob (p o SDROP n) = prob p"
by (import prob PROB_SDROP)
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: "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: "0 <= algebra_measure l"
by (import prob ALGEBRA_MEASURE_POS)
lemma ALGEBRA_MEASURE_RANGE: "0 <= algebra_measure l & algebra_measure l <= 1"
by (import prob ALGEBRA_MEASURE_RANGE)
lemma PROB_POS: "0 <= prob p"
by (import prob PROB_POS)
lemma PROB_MAX: "prob p <= 1"
by (import prob PROB_MAX)
lemma PROB_RANGE: "0 <= prob p & prob p <= 1"
by (import prob PROB_RANGE)
lemma ABS_PROB: "abs (prob p) = prob p"
by (import prob ABS_PROB)
lemma PROB_SHD: "prob (%s. SHD s = b) = 1 / 2"
by (import prob PROB_SHD)
lemma PROB_COMPL_LE1: "measurable p ==> (prob (COMPL p) <= r) = (1 - r <= prob p)"
by (import prob PROB_COMPL_LE1)
;end_setup
;setup_theory prob_pseudo
consts
pseudo_linear_hd :: "nat => bool"
defs
pseudo_linear_hd_primdef: "pseudo_linear_hd == EVEN"
lemma pseudo_linear_hd_def: "pseudo_linear_hd = EVEN"
by (import prob_pseudo pseudo_linear_hd_def)
consts
pseudo_linear_tl :: "nat => nat => nat => nat => nat"
defs
pseudo_linear_tl_primdef: "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. (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. SHD (pseudo_linear1 x) = pseudo_linear_hd x) &
(ALL x.
STL (pseudo_linear1 x) =
pseudo_linear1
(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)))))))
x))"
by (import prob_pseudo pseudo_linear1_def)
consts
pseudo :: "nat => nat => bool"
defs
pseudo_primdef: "pseudo == pseudo_linear1"
lemma pseudo_def: "pseudo = pseudo_linear1"
by (import prob_pseudo pseudo_def)
;end_setup
;setup_theory prob_indep
consts
indep_set :: "((nat => bool) => bool) => ((nat => bool) => bool) => bool"
defs
indep_set_primdef: "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
alg_cover_set :: "bool list list => bool"
defs
alg_cover_set_primdef: "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 x. SOME b. List.member l b & 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
indep :: "((nat => bool) => 'a * (nat => bool)) => bool"
defs
indep_primdef: "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)))"
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: "measurable p ==> indep_set EMPTY p & indep_set pred_set.UNIV p"
by (import prob_indep INDEP_SET_BASIC)
lemma INDEP_SET_SYM: "indep_set p q = indep_set p q"
by (import prob_indep INDEP_SET_SYM)
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: "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: "~ 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: "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: "[| 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));
alg_cover_set l |]
==> P l"
by (import prob_indep ALG_COVER_SET_CASES)
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));
alg_cover_set l |]
==> P l"
by (import prob_indep ALG_COVER_SET_INDUCTION)
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: "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: "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: "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: "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: "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: "[| 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: "[| 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: "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: "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: "indep (UNIT x)"
by (import prob_indep INDEP_UNIT)
lemma INDEP_SDEST: "indep SDEST"
by (import prob_indep INDEP_SDEST)
lemma BIND_STEP: "BIND SDEST (%k. f o SCONS k) = f"
by (import prob_indep BIND_STEP)
lemma INDEP_BIND_SDEST: "(!!x. indep (f x)) ==> indep (BIND SDEST f)"
by (import prob_indep INDEP_BIND_SDEST)
lemma INDEP_BIND: "indep f & (ALL x. indep (g x)) ==> indep (BIND f g)"
by (import prob_indep INDEP_BIND)
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: "indep f ==> measurable (p o (fst o f))"
by (import prob_indep INDEP_MEASURABLE1)
lemma INDEP_MEASURABLE2: "indep f & measurable q ==> measurable (q o (snd o f))"
by (import prob_indep INDEP_MEASURABLE2)
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
;setup_theory prob_uniform
consts
unif_bound :: "nat => nat"
defs
unif_bound_primdef: "unif_bound ==
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. 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) = Suc (unif_bound (Suc v div 2))"
by (import prob_uniform unif_bound_def)
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
"unif_tupled ==
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 =>
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. 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 =>
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)
consts
unif :: "nat => (nat => bool) => nat * (nat => bool)"
defs
unif_primdef: "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 = (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) (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
"uniform_tupled ==
WFREC
(SOME R.
WF R &
(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 (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.
WF R &
(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 (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 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 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) 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: "(Suc n div 2 = 0) = (n = 0)"
by (import prob_uniform SUC_DIV_TWO_ZERO)
lemma UNIF_BOUND_LOWER: "n < 2 ^ unif_bound n"
by (import prob_uniform UNIF_BOUND_LOWER)
lemma UNIF_BOUND_LOWER_SUC: "Suc n <= 2 ^ unif_bound n"
by (import prob_uniform UNIF_BOUND_LOWER_SUC)
lemma UNIF_BOUND_UPPER: "n ~= 0 ==> 2 ^ unif_bound n <= 2 * n"
by (import prob_uniform UNIF_BOUND_UPPER)
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.
unif (Suc n) =
BIND (unif (Suc n div 2))
(%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. uniform 0 (Suc x) = UNIT 0) &
(ALL x xa.
uniform (Suc 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: "indep (unif n)"
by (import prob_uniform INDEP_UNIF)
lemma INDEP_UNIFORM: "indep (uniform t (Suc n))"
by (import prob_uniform INDEP_UNIFORM)
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: "fst (unif n s) < 2 ^ unif_bound n"
by (import prob_uniform UNIF_RANGE)
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: "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: "1 / 2 <= prob (%s. fst (unif n s) < Suc n)"
by (import prob_uniform PROB_UNIF_GOOD)
lemma UNIFORM_RANGE: "fst (uniform t (Suc n) s) < Suc n"
by (import prob_uniform UNIFORM_RANGE)
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: "[| !!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: "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: "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: "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
end