hcju/isabelleps huggingface dataset:
D_IFF: fixes b :: "BExpr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c1 :: "OBJ"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and c2 :: "OBJ"   and \<Psi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes "BExpr_low b"   and "(G, c1, Sec \<Phi>) \<in> Deriv"   and "(G, c2, Sec \<Psi>) \<in> Deriv" shows "(G, Iff b c1 c2, Sec (\<lambda>(s, t, \<beta>). (evalB b (fst t) \<longrightarrow> \<Phi> (s, t, \<beta>)) \<and> (\<not> evalB b (fst t) \<longrightarrow> \<Psi> (s, t, \<beta>)))) \<in> Deriv"
gray_alt_def: shows "sievert \<cong>\<^sub>Q joule \<^bold>\<cdot> kilogram\<^sup>-\<^sup>\<one>"
extending_increasing: fixes F :: "'a program"   and func :: "'a \<Rightarrow> 'e" shows "extending (\<lambda>G. UNIV) h F (increasing (func \<circ> f)) (increasing func)"
CastRedsNull: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>" shows "P \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>"
cancellation_1: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "is_conjugation f g" shows "f (- g x) \<le> - x"
independent_if_scalars_zero: fixes A :: "'b set" assumes "finite A"   and "\<And>f x. \<lbrakk>(\<Sum>x\<in>A. f x *s x) = (0::'b); x \<in> A\<rbrakk> \<Longrightarrow> f x = (0::'a)" shows "independent A"
\<phi>: shows "\<guillemotleft>\<phi> : a \<star> a \<Rightarrow> a\<guillemotright> \<and> local.iso \<phi>"
exec_move_CASI1: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "'a J1_prog"   and t :: "'b"   and e :: "'a expr1"   and h :: "'c"   and s :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and ta :: "('a, 'b, 'c) jvm_thread_action"   and h' :: "'c"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and D :: "String.literal"   and F :: "String.literal"   and e' :: "'a expr1"   and e'' :: "'a expr1" assumes "JVM_heap_base.exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t e h s ta h' s'" shows "JVM_heap_base.exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t (e\<bullet>compareAndSwap(D\<bullet>F, e', e'')) h s ta h' s'"
rel_G_neg_distr: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra1' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Contra2' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and uu :: "('ab \<times> 'ac \<times> 'ad \<times> 'ae \<times> 'af \<times> 'ag \<times> 'ah) itself"   and L1 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and L1' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and L2 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and L2' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu" shows "rel_G (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') \<le> rel_G L1 L2 Co1 Co2 Contra1 Contra2 OO rel_G L1' L2' Co1' Co2' Contra1' Contra2'"
diff_col_ex3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "\<exists>D. A \<noteq> D \<and> B \<noteq> D \<and> C \<noteq> D \<and> Tarski_neutral_dimensionless.Col Bet A B D"
w2: shows "real (Esigma h) = real h / real g * real (Esigma g)"
elims: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "'fsym trm"   and y :: "univ" assumes "Ik.int x xa = y" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and xb :: "Preliminaries.var" where "x = \<xi>'__" and "xa = Var xb" and "y = \<xi>'__ xb"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and f :: "'fsym" and Tl :: "'fsym trm list" where "x = \<xi>" and "xa = Fn f Tl" and "y = intF f (map (Ik.int \<xi>) Tl)"
WT_auth: fixes \<eta> :: "nat"   and s :: "'a list cstate" assumes "pred_cstate (\<lambda>x. length x = \<eta>) s" shows "\<I>_uniform UNIV (insert None (Some ` nlists UNIV \<eta>)) \<turnstile>c A_CHAN.auth_oracle s \<surd>"
alls2_nth0: fixes A :: "nat"   and B :: "nat"   and as :: "nat list"   and bs :: "nat list" shows "alls2 A B as bs ! 0 = ((zeroes (length as), 0), zeroes (length bs), 0)"
nb\<^sub>1: shows "\<forall>(v, a)\<in>dom (\<phi>\<^sub>S \<Psi> s). \<forall>(v, a')\<in>dom (\<phi>\<^sub>S \<Psi> s). (\<phi>\<^sub>S \<Psi> s) (v, a) = Some True \<and> (\<phi>\<^sub>S \<Psi> s) (v, a') = Some True \<longrightarrow> (v, a) = (v, a')"
icard_singleton_enat: fixes a :: "'a" shows "icard {a} = eSuc (enat 0)"
constrains_UNIV2: fixes F :: "'c program"   and A :: "'c set" shows "F \<in> A co UNIV"
present_alloc_neq: fixes h :: "heap"   and r :: "'a ref"   and v :: "'b" assumes "Ref.present h r" shows "r =!= fst (Ref.alloc v h)"
vsubst_eq_swap: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and y1 :: "'var"   and y2 :: "'var"   and ys :: "'varSort" assumes "good X"   and "y1 = y2 \<or> fresh ys y1 X" shows "X #[y1 // y2]_ys = X #[y1 \<and> y2]_ys"
induct: fixes x :: "'c"   and G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> generate G H"   and "P \<one>\<^bsub>G\<^esub>"   and "\<And>h. h \<in> H \<Longrightarrow> P h"   and "\<And>h. h \<in> H \<Longrightarrow> P (inv\<^bsub>G\<^esub> h)"   and "\<And>h1 h2. \<lbrakk>h1 \<in> generate G H; P h1; h2 \<in> generate G H; P h2\<rbrakk> \<Longrightarrow> P (h1 \<otimes>\<^bsub>G\<^esub> h2)" shows "P x"
coeff_poly_of_pm: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and k :: "nat" shows "poly.coeff (poly_of_pm x p) k = lookup p (monomial k x)"
sup_commute: fixes x :: "'a"   and y :: "'a" shows "sup x y = sup y x"
BINDINGc_elim: fixes a :: "coname"   and M :: "trm"   and B :: "ty" assumes "<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>)" shows "\<forall>x P. x:P \<in> \<parallel>(B)\<parallel> \<longrightarrow> SNa (M{a:=x.P})"
asize_nz: fixes c :: "com" shows "asize c \<noteq> 0"
sup_set_def: fixes A :: "'d set"   and B :: "'d set" shows "A \<union> B = Collect (sup (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
map_prod_def: shows "HOLCF_ROOT.map_prod = (\<Lambda> f g (x, y). (f\<cdot>x, g\<cdot>y))"
filterlim_real_of_int_at_bot: shows "filterlim real_of_int at_bot at_bot"
SUP_cong: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g \<Rightarrow> 'h"   and D :: "'g \<Rightarrow> 'h"   and Sup :: "'h set \<Rightarrow> 'h" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
set_iterator_foldri_correct: fixes xs :: "'d list" assumes "distinct xs" shows "set_iterator (foldri xs) (set xs)"
summable_hurwitz_zeta: fixes s :: "complex"   and a :: "real" assumes "1 < Re s"   and "0 < a" shows "summable (\<lambda>n. (of_nat n + complex_of_real a) powr - s)"
AE_I2: fixes M :: "'a measure"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in> space M \<Longrightarrow> P x" shows "almost_everywhere M P"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "('a, 'b) lit list set" shows "U.satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. U.satC \<xi> c"
empty_not_insert: fixes a :: "'e"   and A :: "'e set" shows "{} \<noteq> insert a A"
vector_scalar_mult_def: fixes c :: "'b"   and x :: "('b, 'c) vec" shows "c *s x = (\<chi>i. c * x $ i)"
simps_1: shows "B 0 = p div gcd p (pderiv p)"
extendFalsum: fixes S :: "'a sequent"   and \<Gamma> :: "'a form multiset"   and \<Delta> :: "'a form multiset" assumes "extend S ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "ff \<in># \<Gamma>"
axioms_2: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "Group_Theory.group G composition unit" shows "Group_Theory.group_axioms G composition unit"
update_def: fixes F :: "'b Abs_State.st"   and x :: "char list"   and y :: "'b" shows "update F x y = FunDom ((fun F)(x := y)) (if x \<in> set (Abs_State.dom F) then Abs_State.dom F else x # Abs_State.dom F)"
stake_szip: fixes n :: "nat"   and s1 :: "'b stream"   and s2 :: "'c stream" shows "stake n (s1 ||| s2) = stake n s1 || stake n s2"
ik_snd_empty: fixes X :: "('c, 'd) Term.term list" shows "ik\<^sub>s\<^sub>t (map Send X) = {}"
multi_self_add_other_not_self: fixes M :: "'d multiset"   and x :: "'d" shows "(M = add_mset x M) = False"
filterlim_filtercomap: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'b filter" shows "filterlim f F (filtercomap f F)"
del_cl: fixes t :: "('a, 'b) rbt"   and x :: "'a" assumes "cl_inv t" shows "if cl t = R then cl_inv (del x t) else cl_inv' (del x t)"
simps_2: fixes x :: "'b"   and t :: "'b tree" shows "cost (Splay x) [t] = T_splay x t"
Collect_cong: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "_Collect x (P x) = _Collect x (Q x)"
of_int_power_eq_of_int_cancel_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "int"   and b :: "int"   and w :: "nat" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus x = power.power one times (ring_1.of_int one plus zero minus b) w) = (x = b ^ w)"
find_meth_def_f_def: fixes P :: "cld list"   and ty :: "ty"   and m :: "char list" shows "find_meth_def_f P ty m = (case find_path_ty_f P ty of None \<Rightarrow> None | Some ctxclds \<Rightarrow> find_meth_def_in_path_f ctxclds m)"
of_real_exp: fixes x :: "real" shows "of_real (exp x) = exp (of_real x)"
Class_def: shows "Class = (\<lambda>a\<in>G. {b \<in> G. (b, a) \<in> Congruence})"
cons_conv_snoc: fixes x :: "'b"   and xs :: "'b list" shows "\<exists>ys y. x # xs = ys @ [y]"
cadjoint_univ_prop': fixes G :: "'c \<Rightarrow> 'd" assumes "bounded_clinear G" shows "\<forall>x y. x \<bullet>\<^sub>C (G\<^sup>\<dagger>) y = G x \<bullet>\<^sub>C y"
in_tset_conv_tnth: fixes x :: "'a"   and xs :: "('a, 'b) tllist" shows "(x \<in> tset xs) = (\<exists>n. enat n < tlength xs \<and> tnth xs n = x)"
lock_actions_ok_Lock_may_lock: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and Ls :: "lock_action list" assumes "lock_actions_ok l t Ls"   and "Lock \<in> set Ls" shows "may_lock l t"
intensional_simps_2: shows "\<not> #True \<equiv> #False"
plus_prod_def: fixes m1 :: "'e \<times> 'f"   and m2 :: "'e \<times> 'f" shows "m1 + m2 \<equiv> (fst m1 + fst m2, snd m1 + snd m2)"
disjoint_add_mset_2: fixes A :: "'k multiset"   and b :: "'k"   and B :: "'k multiset" shows "({#} = A \<inter># add_mset b B) = (b \<notin># A \<and> {#} = A \<inter># B)"
1: shows "x (Suc n) \<noteq> \<zero>\<^bsub>residue_ring (p ^ Suc n)\<^esub>"
simps_2: fixes uu :: "('b ivl \<times> 'b) tree"   and uv :: "'b ivl"   and m :: "'b"   and uw :: "('b ivl \<times> 'b) tree" shows "max_hi \<langle>uu, (uv, m), uw\<rangle> = m"
lspasl_p_eq: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a" shows "((h1,h2\<triangleright>h0) \<and> (h1,h2\<triangleright>h3)) = ((h1,h2\<triangleright>h0) \<and> h0 = h3)"
simps_1: fixes ops :: "'b arith_ops_record"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "minus_poly_rev_list_i ops (x # xs) (y # ys) = arith_ops_record.minus ops x y # minus_poly_rev_list_i ops xs ys"
iterates_above_Field: fixes x :: "'a"   and a :: "'a" assumes "x \<in> iterates_above a"   and "a \<in> Field leq" shows "x \<in> Field leq"
index_take: fixes i :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "i \<le> index xs x" shows "x \<notin> set (take i xs)"
isCont_ln_Gamma_complex: fixes z :: "'a"   and f :: "'a \<Rightarrow> complex" assumes "isCont f z"   and "f z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "isCont (\<lambda>z. ln_Gamma (f z)) z"
simple_bochner_integral_nonneg: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "\<And>x. 0 \<le> f x" shows "0 \<le> simple_bochner_integral M f"
Fun_mapsto: fixes f :: "'s" assumes "arr f" shows "Fun f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f)"
cod_char: fixes f :: "('o, 'a) arr" shows "cod f = (if arr f then MkIde (Cod f) else null)"
qmodule_carr: fixes H :: "'a set" assumes "submodule R M H" shows "carrier (M /\<^sub>m H) = set_mr_cos M H"
log_rec: fixes n :: "nat" assumes "2 \<le> n" shows "local.log n = Suc (local.log (n div 2))"
sset_cycle: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "sset (cycle xs) = set xs"
indefinite_integral_continuous_right: fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and c :: "real"   and e :: "real" assumes "f integrable_on {a..b}"   and "a \<le> c"   and "c < b"   and "0 < e" obtains d :: "real" where "0 < d" and "\<forall>t. c \<le> t \<and> t < c + d \<longrightarrow> norm (integral {a..c} f - integral {a..t} f) < e"
e_append_def: fixes xs :: "nat"   and ys :: "nat" shows "e_append xs ys \<equiv> Partial_Recursive.list_encode (Partial_Recursive.list_decode xs @ Partial_Recursive.list_decode ys)"
mem_CH_Ch: fixes x :: "'x"   and X :: "'x set" shows "(x \<in> CH X) = (x \<in> Ch (Xh x) X)"
nsum_splitTr: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>Suc (n + m). f j \<in> carrier A) \<longrightarrow> \<Sigma>\<^sub>e A f Suc (n + m) = \<Sigma>\<^sub>e A f n \<plusminus> \<Sigma>\<^sub>e A cmp f (slide (Suc n)) m"
prems: shows "\<forall>c. c \<in> set (b # w) \<longrightarrow> c = a"
method_typing_aux_def: fixes CT :: "nat \<Rightarrow> classDef option"   and m :: "nat"   and D :: "nat"   and Cs :: "nat list"   and C :: "nat" shows "method_typing_aux CT m D Cs C = (\<not> (\<forall>Ds D0. mtype(CT,m,D) = Ds \<rightarrow> D0 \<longrightarrow> Cs = Ds \<and> C = D0))"
q: shows "q = p \<otimes>\<^bsub>poly_ring R\<^esub> b"
LLL_d_bound: fixes i :: "nat" assumes "i \<le> m" shows "\<bar>d fs i\<bar> \<le> int (N ^ i) \<and> \<bar>d fs i\<bar> \<le> int (N ^ m)"
assms: fixes n :: "nat"   and e :: "CValue"   and a :: "C \<rightarrow> CValue"   and c :: "C" shows "\<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>((e \<down>CFn a)\<cdot>c) = (\<psi>\<^sup>E\<^bsub>Suc n\<^esub>\<cdot>e \<down>CFn \<psi>\<^sup>A\<^bsub>n\<^esub>\<cdot>a)\<cdot>c"
inverse_functors_axioms: shows "inverse_functors C' (\<cdot>) \<psi> (\<lambda>i. if C'.arr i then \<phi> i else null)"
Trans: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and z :: "tm" assumes "H \<turnstile> x EQ y"   and "H \<turnstile> y EQ z" shows "H \<turnstile> x EQ z"
euclid_ext_aux_i: shows "(R ===> R ===> R ===> R ===> R ===> R ===> rel_prod (rel_prod R R) R) euclid_ext_aux_i euclid_ext_aux"
nbth: shows "bound0 (Ferrack.disj (minusinf (rlfm (simpfm p))) (Ferrack.disj (plusinf (rlfm (simpfm p))) (evaldjf (simpfm \<circ> usubst (rlfm (simpfm p))) (remdups (map simp_num_pair (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs (uset (rlfm (simpfm p))))))))))"
eq_implies: fixes x :: "'a"   and y :: "'a" assumes "x = y"   and "x \<in> A" shows "x \<sqsubseteq> y"
simps_2: fixes s :: "('c, 'd) fmap"   and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "state_list s (a # as) = s # state_list (state_succ s a) as"
det_mult: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "det (A * B) = det A * det B"
pmf_of_set_split_inter_diff: fixes A :: "'a set"   and B :: "'a set"   and p :: "real" assumes "finite A"   and "finite B"   and "A \<noteq> {}"   and "B \<noteq> {}"   and "p = real (card (A \<inter> B)) / real (card B)" shows "bernoulli_pmf p \<bind> (\<lambda>b. if b then pmf_of_set (A \<inter> B) else pmf_of_set (B - A)) = pmf_of_set B"
nneg_const: fixes c :: "'c" shows "nneg (\<lambda>x. c) = ((0::'c) \<le> c)"
not_less: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.linorder less_eq less" shows "(\<not> less x y) = less_eq y x"
inject_14: fixes x14 :: "'a exp"   and y14 :: "'a exp" shows "(throw x14 = throw y14) = (x14 = y14)"
map_cong0: fixes x :: "('l, 'k, 'm) F2"   and f1 :: "'k \<Rightarrow> 'n"   and g1 :: "'k \<Rightarrow> 'n"   and f2 :: "'m \<Rightarrow> 'o"   and g2 :: "'m \<Rightarrow> 'o" assumes "\<And>z1. z1 \<in> F2set1 x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> F2set2 x \<Longrightarrow> f2 z2 = g2 z2" shows "F2map f1 f2 x = F2map g1 g2 x"
genby_uminus_add_closed: fixes g :: "'a"   and S :: "'a set"   and h :: "'a" assumes "g \<in> \<langle>S\<rangle>"   and "h \<in> \<langle>S\<rangle>" shows "- g + h \<in> \<langle>S\<rangle>"
natural_transformation_axioms: shows "natural_transformation CC.comp (\<cdot>) local.Prod local.Prod local.Prod"
length_abc_comp: fixes A :: "abc_inst list"   and B :: "abc_inst list" shows "length (A [+] B) = length A + length B"
wf_preallocatedD: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and h :: "'heap"   and C :: "String.literal" assumes "wf_prog wf_md P"   and "preallocated h"   and "C \<in> sys_xcpts" shows "typeof_addr h (addr_of_sys_xcpt C) = \<lfloor>Class_type C\<rfloor> \<and> P \<turnstile> C \<preceq>\<^sup>* Throwable"
DiamondOne: fixes cfg :: "('p, 'v, 's) configuration"   and m :: "('p, 'v) message"   and cfg1 :: "('p, 'v, 's) configuration"   and p :: "'p"   and q :: "'p"   and m' :: "('p, 'v) message"   and cfg2 :: "('p, 'v, 's) configuration" assumes "cfg \<turnstile> m \<mapsto> cfg1"   and "p \<noteq> q"   and "isReceiverOf p m"   and "\<not> isReceiverOf p m'"   and "isReceiverOf q m'"   and "\<not> isReceiverOf q m"   and "cfg \<turnstile> m' \<mapsto> cfg2" shows "\<exists>cfg'. cfg1 \<turnstile> m' \<mapsto> cfg' \<and> cfg2 \<turnstile> m \<mapsto> cfg'"
387: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> y) \<sqinter> - x = - x"
none_in_empty_fset: fixes S :: "'b fset" shows "(\<forall>x. x |\<notin>| S) = (S = {||})"
swap_atom_def: fixes ab :: "atom \<times> atom"   and c :: "atom" shows "swap ab c \<equiv> swap_atom ab c"
cp\<^sub>P\<^sub>R\<^sub>E: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "PRE self a1 a2 \<tau> = PRE (\<lambda>_. self \<tau>) (\<lambda>_. a1 \<tau>) (\<lambda>_. a2 \<tau>) \<tau>"
closed_def: fixes S :: "'c set" shows "closed S = open (- S)"
VLambda_vintersection: fixes A :: "V"   and B :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda (A \<inter>\<^sub>\<circ> B) f = VLambda A f \<inter>\<^sub>\<circ> VLambda B f"
fixed_points_def: shows "fixed_points = {m \<in> M. carrier G \<subseteq> stabilizer m}"
icard_Suc_Diff1: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "eSuc (icard (A - {x})) = icard A"
assn_aci_3: fixes x :: "assn"   and y :: "assn"   and z :: "assn" shows "x \<and>\<^sub>A y \<and>\<^sub>A z = y \<and>\<^sub>A x \<and>\<^sub>A z"
SN_s_mul_ext: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "order_pair s ns"   and "SN s" shows "SN (s_mul_ext ns s)"
A: fixes y :: "real" assumes "y \<in> I" shows "psi y = phi y"
ipurge_rec_ext_def: fixes Pol :: "('c \<times> 'c) set"   and Map :: "'b \<Rightarrow> 'c"   and Doms :: "'c set"   and List :: "'b list"   and ListOp :: "'b list option"   and Set :: "'b set"   and more :: "'d" shows "\<lparr>Pol = Pol, Map = Map, Doms = Doms, List = List, ListOp = ListOp, Set = Set, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons ipurge_rec_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple (Record.iso_tuple_cons Record.tuple_iso_tuple Pol (Record.iso_tuple_cons Record.tuple_iso_tuple Map Doms)) (Record.iso_tuple_cons Record.tuple_iso_tuple List (Record.iso_tuple_cons Record.tuple_iso_tuple ListOp Set))) more"
assms_2: shows "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"
removeKey_subset: fixes a :: "'c"   and ps :: "('c \<times> 'd) list" shows "set (removeKey a ps) \<subseteq> set ps"
sum_vimage: fixes S :: "'c set"   and f :: "'c \<Rightarrow> nat"   and bd :: "nat"   and g :: "nat \<Rightarrow> real" assumes "\<And>w. w \<in> S \<Longrightarrow> f w < bd"   and "0 < bd"   and "finite S" shows "(\<Sum>w\<in>S. g (f w)) = (\<Sum>m = 0..<bd. real (card (f -` {m} \<inter> S)) * g m)"
is_faithful: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and f :: "'a"   and f' :: "'a" assumes "faithful_functor A B F"   and "partial_magma.arr A f \<and> partial_magma.arr A f' \<and> partial_magma.dom A f = partial_magma.dom A f' \<and> partial_magma.cod A f = partial_magma.cod A f'"   and "F f = F f'" shows "f = f'"
induct: fixes P :: "'a ifex \<Rightarrow> 'a ifex \<Rightarrow> 'a ifex \<Rightarrow> bool"   and a0 :: "'a ifex"   and a1 :: "'a ifex"   and a2 :: "'a ifex" assumes "\<And>i t e. \<lbrakk>\<And>x2. \<lbrakk>param_opt i t e = None; lowest_tops [i, t, e] = Some x2\<rbrakk> \<Longrightarrow> P (restrict_top i x2 True) (restrict_top t x2 True) (restrict_top e x2 True); \<And>x2. \<lbrakk>param_opt i t e = None; lowest_tops [i, t, e] = Some x2\<rbrakk> \<Longrightarrow> P (restrict_top i x2 False) (restrict_top t x2 False) (restrict_top e x2 False)\<rbrakk> \<Longrightarrow> P i t e" shows "P a0 a1 a2"
ce: fixes EXP :: "(real \<Rightarrow> bool) measure" assumes "\<And>P. ((\<lambda>(x, y). P x y) \<in> POW \<Otimes>\<^sub>M COCOUNT \<rightarrow>\<^sub>M BOOL) = (P \<in> POW \<rightarrow>\<^sub>M EXP)" shows "False"