New simpler representation of numerals, using Bit0 and Bit1 instead of BIT, B0, and B1
(* AUTOMATICALLY GENERATED, DO NOT EDIT! *)
theory HOL4Prob imports HOL4Real begin
;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"
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))"
by (import prob_extra EVEN_ODD_EXISTS_EQ)
lemma DIV_THEN_MULT: "ALL (p::nat) q::nat. 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"
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)"
by (import prob_extra DIVISION_TWO)
lemma DIV_TWO: "ALL n::nat. n = 2 * (n div 2) + n mod 2"
by (import prob_extra DIV_TWO)
lemma MOD_TWO: "ALL n::nat. 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)))"
by (import prob_extra DIV_TWO_BASIC)
lemma DIV_TWO_MONO: "ALL (m::nat) n::nat. m div 2 < n div 2 --> 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)"
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"
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))"
by (import prob_extra EXP_DIV_TWO)
lemma EVEN_EXP_TWO: "ALL n::nat. 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)"
by (import prob_extra DIV_TWO_EXP)
consts
inf :: "(real => bool) => real"
defs
inf_primdef: "inf == %P::real => bool. - sup (IMAGE uminus P)"
lemma inf_def: "ALL P::real => bool. inf P = - 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))"
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))"
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"
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"
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)"
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))"
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))))"
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))))"
by (import prob_extra POW_HALF_TWICE)
lemma X_HALF_HALF: "ALL x::real. 1 / 2 * x + 1 / 2 * 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"
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"
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)"
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)))))"
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)"
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))))"
by (import prob_extra POW_HALF_EXP)
lemma INV_SUC_POS: "ALL n::nat. 0 < 1 / real (Suc n)"
by (import prob_extra INV_SUC_POS)
lemma INV_SUC_MAX: "ALL x::nat. 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"
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"
by (import prob_extra ABS_UNIT_INTERVAL)
lemma MEM_NIL: "ALL l::'a::type list. (ALL x::'a::type. ~ x mem l) = (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)"
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"
by (import prob_extra MEM_NIL_MAP_CONS)
lemma FILTER_TRUE: "ALL l::'a::type list. [x::'a::type<-l. True] = l"
by (import prob_extra FILTER_TRUE)
lemma FILTER_FALSE: "ALL l::'a::type list. [x::'a::type<-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"
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"
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"
by (import prob_extra FILTER_OUT_ELT)
lemma IS_PREFIX_NIL: "ALL x::'a::type list. 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"
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"
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"
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))"
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"
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"
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)"
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"
by (import prob_extra FOLDR_MAP)
lemma LAST_MEM: "ALL (h::'a::type) t::'a::type list. last (h # t) mem 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"
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)"
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)"
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)"
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)"
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)"
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"
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)"
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::bool list) y::bool list.
EX l::bool list. 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)"
by (import prob_canon alg_twin_def)
constdefs
alg_order_tupled :: "bool list * bool list => bool"
"(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)))"
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)))"
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)"
lemma alg_order_curried_def: "ALL (x::bool list) x1::bool list. 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))"
by (import prob_canon alg_order_ind)
lemma alg_order_def: "alg_order [] ((v6::bool) # (v7::bool list)) = True &
alg_order [] [] = True &
alg_order ((v2::bool) # (v3::bool list)) [] = False &
alg_order ((h::bool) # (t::bool list)) ((h'::bool) # (t'::bool list)) =
(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::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.
list_case True
(%v2::bool list.
list_case True
(%(v6::bool list) v7::bool list list.
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.
list_case True
(%v2::bool list.
list_case True
(%(v6::bool list) v7::bool list list.
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"
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"
by (import prob_canon alg_sorted_def)
consts
alg_prefixfree :: "bool list list => bool"
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.
list_case True
(%v2::bool list.
list_case True
(%(v6::bool list) v7::bool list list.
~ 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.
list_case True
(%v2::bool list.
list_case True
(%(v6::bool list) v7::bool list list.
~ 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"
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"
by (import prob_canon alg_prefixfree_def)
consts
alg_twinfree :: "bool list list => bool"
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.
list_case True
(%v2::bool list.
list_case True
(%(v6::bool list) v7::bool list list.
~ 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.
list_case True
(%v2::bool list.
list_case True
(%(v6::bool list) v7::bool list list.
~ 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"
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"
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"
lemma alg_longest_def: "alg_longest =
FOLDR (%(h::bool list) t::nat. 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.
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::bool list. alg_canon_find l [] = [l]) &
(ALL (l::bool list) (h::bool list) t::bool list list.
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::bool list. alg_canon_merge l [] = [l]) &
(ALL (l::bool list) (h::bool list) t::bool list list.
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::bool list list. alg_canon2 (alg_canon1 l)"
lemma alg_canon_def: "ALL l::bool list list. 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"
lemma algebra_canon_def: "ALL l::bool list list. 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"
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)"
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))"
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'"
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)"
by (import prob_canon ALG_TWINS_PREFIX)
lemma ALG_ORDER_NIL: "ALL x::bool list. 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"
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"
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"
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"
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"
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"
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"
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"
by (import prob_canon ALG_ORDER_PREFIX_TRANS)
lemma ALG_ORDER_SNOC: "ALL (x::bool) l::bool list. ~ 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)"
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)"
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"
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)"
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"
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)"
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')"
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)"
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"
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)"
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)"
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"
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)"
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)))"
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)"
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"
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"
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"
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)"
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)"
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)"
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)"
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))"
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)"
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"
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"
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)"
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)"
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"
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"
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"
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)"
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)"
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"
by (import prob_canon ALG_CANON_FIND_CONSTANT)
lemma ALG_CANON1_SORTED: "ALL x::bool list list. 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)"
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"
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)"
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)"
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)"
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"
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)"
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)"
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)"
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"
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)"
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"
by (import prob_canon ALG_CANON_CONSTANT)
lemma ALG_CANON_IDEMPOT: "ALL l::bool list list. 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)"
by (import prob_canon ALGEBRA_CANON_DEF_ALT)
lemma ALGEBRA_CANON_BASIC: "algebra_canon [] &
algebra_canon [[]] & (ALL x::bool list. 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])"
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"
by (import prob_canon ALGEBRA_CANON_TL)
lemma ALGEBRA_CANON_NIL_MEM: "ALL l::bool list list. (algebra_canon l & [] mem 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"
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"
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)"
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)"
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)"
by (import prob_canon ALGEBRA_CANON_CASES_THM)
lemma ALGEBRA_CANON_CASES: "ALL P::bool list list => bool.
P [] &
P [[]] &
(ALL (l1::bool list list) l2::bool list list.
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)"
by (import prob_canon ALGEBRA_CANON_CASES)
lemma ALGEBRA_CANON_INDUCTION: "ALL P::bool list list => bool.
P [] &
P [[]] &
(ALL (l1::bool list list) l2::bool list list.
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)"
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"
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 = [[]])"
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'"
by (import prob_canon ALG_SORTED_PREFIXFREE_EQUALITY)
;end_setup
;setup_theory boolean_sequence
consts
SHD :: "(nat => bool) => bool"
defs
SHD_primdef: "SHD == %f::nat => bool. f 0"
lemma SHD_def: "ALL f::nat => bool. 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)"
lemma STL_def: "ALL (f::nat => bool) n::nat. 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)"
by (import boolean_sequence SCONS_def)
consts
SDEST :: "(nat => bool) => bool * (nat => bool)"
defs
SDEST_primdef: "SDEST == %s::nat => bool. (SHD s, STL s)"
lemma SDEST_def: "SDEST = (%s::nat => bool. (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::nat => bool. STAKE 0 s = []) &
(ALL (n::nat) s::nat => bool. 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)"
by (import boolean_sequence SDROP_def)
lemma SCONS_SURJ: "ALL x::nat => bool. EX (xa::bool) t::nat => bool. 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"
by (import boolean_sequence SHD_STL_ISO)
lemma SHD_SCONS: "ALL (h::bool) t::nat => bool. 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"
by (import boolean_sequence STL_SCONS)
lemma SHD_SCONST: "ALL b::bool. SHD (SCONST b) = b"
by (import boolean_sequence SHD_SCONST)
lemma STL_SCONST: "ALL b::bool. 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::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)))"
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.
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::(nat => bool) => bool. EX b::bool list list. s = algebra_embed b"
lemma measurable_def: "ALL s::(nat => bool) => bool.
measurable s = (EX b::bool list list. 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"
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)"
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))"
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))"
by (import prob_algebra ALG_EMBED_BASIC)
lemma ALG_EMBED_NIL: "ALL c::bool list. All (alg_embed c) = (c = [])"
by (import prob_algebra ALG_EMBED_NIL)
lemma ALG_EMBED_POPULATED: "ALL b::bool list. 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"
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"
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"
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))"
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)"
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)"
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)"
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)"
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)"
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"
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)"
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"
by (import prob_algebra ALG_CANON2_EMBED)
lemma ALG_CANON_EMBED: "ALL l::bool list list. 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 = [[]]"
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)"
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 = [])"
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 = [[]])"
by (import prob_algebra ALGEBRA_CANON_EMBED_UNIV)
lemma MEASURABLE_ALGEBRA: "ALL b::bool list list. 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))"
by (import prob_algebra MEASURABLE_BASIC)
lemma MEASURABLE_SHD: "ALL b::bool. measurable (%s::nat => bool. 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'"
by (import prob_algebra ALGEBRA_EMBED_COMPL)
lemma MEASURABLE_COMPL: "ALL s::(nat => bool) => bool. 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)"
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)"
by (import prob_algebra MEASURABLE_INTER)
lemma MEASURABLE_STL: "ALL p::(nat => bool) => bool. 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"
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"
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))"
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"
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::bool list) rest::bool list list.
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::bool list list.
inf (%r::real.
EX c::bool list list.
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)"
by (import prob algebra_measure_def)
consts
prob :: "((nat => bool) => bool) => real"
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)"
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)"
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)))"
by (import prob ALG_TWINS_MEASURE)
lemma ALG_MEASURE_BASIC: "alg_measure [] = 0 &
alg_measure [[]] = 1 & (ALL b::bool. alg_measure [[b]] = 1 / 2)"
by (import prob ALG_MEASURE_BASIC)
lemma ALG_MEASURE_POS: "ALL l::bool list list. 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"
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"
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)"
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)"
by (import prob ALG_CANON_FIND_MONO)
lemma ALG_CANON1_MONO: "ALL x::bool list list. 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)"
by (import prob ALG_CANON_MERGE_MONO)
lemma ALG_CANON2_MONO: "ALL x::bool list list. 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"
by (import prob ALG_CANON_MONO)
lemma ALGEBRA_MEASURE_DEF_ALT: "ALL l::bool list list. 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)"
by (import prob ALGEBRA_MEASURE_BASIC)
lemma ALGEBRA_CANON_MEASURE_MAX: "ALL l::bool list list. 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"
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"
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)"
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))"
by (import prob ALG_MEASURE_ADDITIVE)
lemma PROB_ALGEBRA: "ALL l::bool list list. 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)"
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"
by (import prob PROB_ADDITIVE)
lemma PROB_COMPL: "ALL s::(nat => bool) => bool. 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"
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) -->
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"
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"
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"
by (import prob PROB_SUBSET_MONO)
lemma PROB_ALG: "ALL x::bool list. 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"
by (import prob PROB_STL)
lemma PROB_SDROP: "ALL (n::nat) p::(nat => bool) => bool.
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"
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"
by (import prob PROB_INTER_SHD)
lemma ALGEBRA_MEASURE_POS: "ALL l::bool list list. 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"
by (import prob ALGEBRA_MEASURE_RANGE)
lemma PROB_POS: "ALL p::(nat => bool) => bool. 0 <= prob p"
by (import prob PROB_POS)
lemma PROB_MAX: "ALL p::(nat => bool) => bool. prob p <= 1"
by (import prob PROB_MAX)
lemma PROB_RANGE: "ALL p::(nat => bool) => bool. 0 <= prob p & prob p <= 1"
by (import prob PROB_RANGE)
lemma ABS_PROB: "ALL p::(nat => bool) => bool. 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"
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)"
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::nat) (b::nat) (n::nat) x::nat. (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)"
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))"
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.
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::(nat => bool) => bool) q::(nat => bool) => bool.
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)"
by (import prob_indep indep_set_def)
consts
alg_cover_set :: "bool list list => bool"
defs
alg_cover_set_primdef: "alg_cover_set ==
%l::bool list list.
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)"
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"
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)"
by (import prob_indep alg_cover_def)
consts
indep :: "((nat => bool) => 'a * (nat => bool)) => bool"
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)))"
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))))"
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"
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"
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"
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"
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]"
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))"
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.
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)"
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.
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)"
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)"
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"
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)"
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))"
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)))"
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)"
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)"
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))"
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"
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))"
by (import prob_indep INDEP_INDEP_SET)
lemma INDEP_UNIT: "ALL x::'a::type. 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"
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)"
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)"
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"
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))"
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))"
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)"
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::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))))"
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))))"
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))"
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"
by (import prob_uniform unif_bound_ind)
constdefs
unif_tupled :: "nat * (nat => bool) => nat * (nat => bool)"
"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).
case v of 0 => (0, v1)
| Suc (v3::nat) =>
let (m::nat, s'::nat => bool) = 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).
case v of 0 => (0, v1)
| Suc (v3::nat) =>
let (m::nat, s'::nat => bool) = 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::nat) x1::nat => bool. unif_tupled (x, x1)"
lemma unif_curried_def: "ALL (x::nat) x1::nat => bool. 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
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))"
by (import prob_uniform unif_ind)
constdefs
uniform_tupled :: "nat * nat * (nat => bool) => nat * (nat => bool)"
"uniform_tupled ==
WFREC
(SOME R::nat * nat * (nat => bool) => nat * nat * (nat => bool) => bool.
WF R &
(ALL (t::nat) (s::nat => bool) (n::nat) (res::nat) s'::nat => bool.
(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)"
lemma uniform_tupled_primitive_def: "uniform_tupled =
WFREC
(SOME R::nat * nat * (nat => bool) => nat * nat * (nat => bool) => bool.
WF R &
(ALL (t::nat) (s::nat => bool) (n::nat) (res::nat) s'::nat => bool.
(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)"
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)"
lemma uniform_curried_def: "ALL (x::nat) (x1::nat) x2::nat => bool.
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))"
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
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)"
by (import prob_uniform SUC_DIV_TWO_ZERO)
lemma UNIF_BOUND_LOWER: "ALL n::nat. 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"
by (import prob_uniform UNIF_BOUND_LOWER_SUC)
lemma UNIF_BOUND_UPPER: "ALL n::nat. 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)"
by (import prob_uniform UNIF_BOUND_UPPER_SUC)
lemma UNIF_DEF_MONAD: "unif 0 = UNIT 0 &
(ALL n::nat.
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))))"
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.
uniform (Suc x) (Suc xa) =
BIND (unif xa)
(%m::nat. 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)"
by (import prob_uniform INDEP_UNIF)
lemma INDEP_UNIFORM: "ALL (t::nat) n::nat. 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)"
by (import prob_uniform PROB_UNIF)
lemma UNIF_RANGE: "ALL (n::nat) s::nat => bool. 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))"
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"
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)"
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"
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)))))"
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)))))))"
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"
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"
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"
by (import prob_uniform PROB_UNIFORM)
;end_setup
end