hcju/isabelleps huggingface dataset:
a_is_strongly_dom_component_safe_def: fixes S\<^sub>a\<^sub>r\<^sub>g :: "(_) object_ptr set"   and S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t :: "(_) object_ptr set"   and h :: "(_) heap"   and h' :: "(_) heap" shows "a_is_strongly_dom_component_safe S\<^sub>a\<^sub>r\<^sub>g S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t h h' = (let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h'); added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h); arg_components = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> a_get_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> a_get_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> added_pointers \<subseteq> arg_components' \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<and> (\<forall>outside_ptr\<in>fset (object_ptr_kinds h) \<inter> fset (object_ptr_kinds h') - (\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
pairwise_alt: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set" shows "pairwise R S = (\<forall>x\<in>S. \<forall>y\<in>S - {x}. R x y)"
NewInitRed: fixes sh :: "sheap"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and l :: "sfields" assumes "\<nexists>sfs. sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "is_class P C" shows "P \<turnstile> \<langle>new C,(h, l, sh),False\<rangle> \<rightarrow> \<langle>INIT C ([C],False) \<leftarrow> new C,(h, l, sh),False\<rangle>"
uniform: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "int (card bl) = \<k>"
WT_gpvD_1: fixes \<Gamma> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and out :: "'a"   and c :: "('c, 'a, 'b) rpv" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)" shows "out \<in> outs_\<I> \<Gamma>"
sum_rel_simp_3: fixes a :: "'a"   and a' :: "'d"   and Rl :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set" shows "(Inl a, Inr a') \<notin> \<langle>Rl, Rr\<rangle>sum_rel"
fixed_length_sublist_def: fixes xs :: "'b list"   and l :: "nat"   and i :: "nat" shows "fixed_length_sublist xs l i = take l (drop (l * i) xs)"
simps_13: fixes e\<^sub>1 :: "('f, 'g, 'h) exp"   and F :: "String.literal"   and D :: "String.literal"   and e\<^sub>2 :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (e\<^sub>1\<bullet>F{D} := e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
unity_root_periodic_arithmetic_mult_minus: fixes k :: "nat"   and m :: "nat" shows "periodic_arithmetic (\<lambda>i. unity_root k (- int i * int m)) k"
n_3: shows "lset (ltake (enat n) P) \<subseteq> A"
countably_additive_iff_continuous_from_below: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f"   and "Measure_Space.additive M f" shows "countably_additive M f = (\<forall>A. range A \<subseteq> M \<longrightarrow> incseq A \<longrightarrow> \<Union> (range A) \<in> M \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union> (range A)))"
nsum_mem: fixes A :: "('c, 'd) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "aGroup A"   and "\<forall>j\<le>n. f j \<in> carrier A" shows "\<Sigma>\<^sub>e A f n \<in> carrier A"
lbound_ofD1: fixes x :: "'a"   and y :: "'a"   and b :: "'a" assumes "lbound_of x y b" shows "x \<succeq> b"
finite_Set: fixes x :: "('b, unit) RBT.rbt" shows "finite (Set x)"
node\<^sub>i_inorder: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" shows "inorder_up\<^sub>i (node\<^sub>i k ts t) = inorder (Node ts t)"
INV: shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
relpowp_Suc_E: fixes n :: "nat"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and z :: "'c" assumes "(P ^^ Suc n) x z" obtains y :: "'c" where "(P ^^ n) x y" and "P y z"
inf_sup_aci_5: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "sup_prod x y = sup_prod y x"
convergent_add: fixes X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'a" assumes "convergent X"   and "convergent Y" shows "convergent (\<lambda>n. X n + Y n)"
map_pre_GF\<^sub>M_def: shows "map_pre_GF\<^sub>M \<equiv> \<lambda>f1 f2 f3. BNF_Composition.id_bnf \<circ> map_F0 f2 f3 \<circ> BNF_Composition.id_bnf"
map_ide: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "local.map (b, a) = S.mkIde (local.set (b, a))"
distinct_7: fixes x2 :: "fresh_t"   and x3 :: "key" shows "aNon x2 \<noteq> aKey x3"
mpt_factor_def: fixes proj :: "(int \<Rightarrow> 'a) \<Rightarrow> 'b"   and M2 :: "'b measure"   and T2 :: "'b \<Rightarrow> 'b" shows "I.mpt_factor proj M2 T2 = (proj \<in> measure_preserving natural_extension_measure M2 \<and> (AE x in natural_extension_measure. proj (natural_extension_map x) = T2 (proj x)) \<and> mpt M2 T2)"
stops\<^sub>C_def: fixes c :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)" shows "stops\<^sub>C c \<equiv> \<forall>c'. (c, c') \<notin> C.eval\<^sub>w"
bisimulation_final_axioms: shows "bisimulation_final trsys1 trsys2 bisim tlsim final1 final2"
ag_pOp_closed: fixes A :: "('d, 'e) aGroup_scheme"   and x :: "'d"   and y :: "'d" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y \<in> carrier A"
simps_34: fixes vd :: "'a"   and v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a" shows "(SKIP {vd} \<sqsubseteq> {v} WHILE va DO vb {vc}) = False"
cobounded2: fixes b :: "'b word"   and a :: "'b word" shows "b \<le>s signed.max a b"
axioms_1: fixes x :: "'a" shows "[] \<bullet> x = x"
seq_comp_traces: fixes P :: "'a option process"   and Q :: "'a option process" assumes "weakly_sequential P" shows "traces (P ; Q) = Domain (seq_comp_failures P Q)"
param_rec_array: fixes Ra :: "('a \<times> 'c) set"   and Rb :: "('b \<times> 'd) set" shows "(rec_array, rec_array) \<in> (\<langle>Ra\<rangle>list_rel \<rightarrow> Rb) \<rightarrow> \<langle>Ra\<rangle>array_rel \<rightarrow> Rb"
select_convs_2: fixes asig :: "'f signature"   and start :: "'e set"   and trans :: "('e \<times> 'f \<times> 'e) set"   and more :: "'g" shows "ioa.start \<lparr>ioa.asig = asig, start = start, trans = trans, \<dots> = more\<rparr> = start"
map_prod_simp: fixes f :: "'h \<Rightarrow> 'f"   and g :: "'i \<Rightarrow> 'g"   and a :: "'h"   and b :: "'i" shows "map_prod f g (a, b) = (f a, g b)"
deriv: fixes x\<^sub>0 :: "real" assumes "poly p' x\<^sub>0 = 0" shows "\<forall>\<^sub>F x in at x\<^sub>0. sgn (poly (p' * sturm_squarefree p ! 1) x) = (if x\<^sub>0 < x then 1 else - 1)"
that: fixes p :: "'a"   and n' :: "'a"   and k :: "nat" assumes "P p"   and "prime p"   and "p dvd n"   and "\<not> p dvd n'"   and "0 < k"   and "n = p ^ k * n'" shows "thesis"
Option_bind_eq_None_conv: fixes x :: "'b option"   and y :: "'b \<Rightarrow> 'a option" shows "(x \<bind> y = None) = (x = None \<or> (\<exists>x'. x = \<lfloor>x'\<rfloor> \<and> y x' = None))"
arr_Rel_def: shows "T = [T\<lparr>ArrVal\<rparr>, T\<lparr>ArrDom\<rparr>, T\<lparr>ArrCod\<rparr>]\<^sub>\<circ>"
LANDAU_PROD_bigo_iff: fixes c1 :: "real"   and c2 :: "real"   and fs :: "(primfun \<times> real) list" shows "LANDAU_PROD bigo_at_top c1 c2 fs = (c1 = 0 \<or> c2 \<noteq> 0 \<and> nonneg_primfun_list (group_primfuns fs))"
chamberD_maximal: fixes X :: "'b set set"   and x :: "'b set"   and z :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X x"   and "z \<in> X"   and "order.greater_eq z x" shows "z = x"
finite_ik\<^sub>s\<^sub>t: fixes S :: "('a, 'b) strand_step list" shows "finite (ik\<^sub>s\<^sub>t S)"
pseudo_rationalizable_on_def: fixes A :: "'b set"   and Rs :: "('b \<times> 'b) set set"   and f :: "'b set \<Rightarrow> 'b set" shows "pseudo_rationalizable_on A Rs f = ((\<forall>r\<in>Rs. Linear_order r) \<and> (\<forall>B\<subseteq>A. f B = (\<Union>r\<in>Rs. greatest r (B \<inter> Field r))))"
simps_1: fixes uu :: "'b" shows "is_literal_formula (Atom uu) = True"
fds_truncate_exp: fixes m :: "nat"   and f :: "'a fds" shows "fds_truncate m (fds_exp (fds_truncate m f)) = fds_truncate m (fds_exp f)"
elims_3: fixes x :: "arctic"   and xa :: "arctic" assumes "\<not> x < xa" obtains uu_ :: "int" where "x = Num_arc uu_" and "xa = MinInfty"   | y :: "int" and xb :: "int" where "x = Num_arc y" and "xa = Num_arc xb" and "\<not> y < xb"
le_boolD: fixes P :: "bool"   and Q :: "bool" assumes "P \<le> Q" shows "P \<longrightarrow> Q"
ProblemIk_axioms: shows "ProblemIk wtFsym wtPsym arOf resOf parOf \<Phi> infTp"
continuous_at_setdist: fixes x :: "'b"   and S :: "'b set" shows "isCont (\<lambda>y. setdist {y} S) x"
degs_less: shows "\<forall>n>0. degree (F' $ n) < degree (F' $ 0)"
reflp_ge_eq: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflp R" shows "(=) \<le> R"
ag_mOp_closed: fixes x :: "'a" assumes "x \<in> carrier A" shows "-\<^sub>a x \<in> carrier A"
closure_eq: fixes S :: "'c set" shows "(closure S = S) = closed S"
mult_div_mod_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (times b (divide a b)) (modulo a b) = a"
hom_deduct_if_deduct: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and M :: "('fun, 'var) Term.term set"   and m :: "('fun, 'var) Term.term" assumes "par_comp \<A> Sec"   and "\<forall>m\<in>M. homogeneous\<^sub>l\<^sub>s\<^sub>t m \<A> Sec \<and> m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"   and "M \<turnstile> m"   and "m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)" shows "\<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m m"
lu_props_3: fixes p :: "real poly"   and x :: "real" assumes "p \<in> set ps"   and "u \<le> x" shows "sgn (poly p x) = poly_inf p"
add_diff_inverse: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<not> less a b" shows "plus b (minus a b) = a"
asAbs_def: fixes qA :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "asAbs qA \<equiv> proj alphaAbsGood qA"
cases: fixes a1 :: "((nat \<times> nat) \<times> transition) fset"   and a2 :: "nat"   and a3 :: "nat \<Rightarrow>f value option"   and a4 :: "(String.literal \<times> value list \<times> value list) list" assumes "accepts_trace a1 a2 a3 a4" obtains e :: "((nat \<times> nat) \<times> transition) fset" and s :: "nat" and r :: "nat \<Rightarrow>f value option" where "a1 = e__" and "a2 = s__" and "a3 = r__" and "a4 = []"   | e :: "((nat \<times> nat) \<times> transition) fset" and s :: "nat" and r :: "nat \<Rightarrow>f value option" and l :: "String.literal" and i :: "value list" and p :: "value list" and t :: "(String.literal \<times> value list \<times> value list) list" where "a1 = e" and "a2 = s" and "a3 = r" and "a4 = (l, i, p) # t" and "\<exists>(s', T)|\<in>|possible_steps e s r l i. evaluate_outputs T i r = map Some p \<and> accepts_trace e s' (evaluate_updates T i r) t"
empty_Mapping: shows "Mapping.empty = AList_Mapping.Mapping []"
1: shows "singular_chain (p - Suc 0) (subtopology X S) (chain_boundary p (chain_boundary (Suc p) d))"
normalize_cnf_Cons: fixes x :: "'a"   and xs :: "'a list" shows "normalize_cnf (x # xs) = (case xs of [] \<Rightarrow> [x] | y # ys \<Rightarrow> (if x < y then [] else [x]) @ normalize_cnf xs)"
lfinite_lmap: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist" shows "lfinite (lmap f xs) = lfinite xs"
C: shows "Min (offs_set_next (offs (enum xs index key n mi ma) k) xs index key mi ma i) \<in> offs_set_next (offs (enum xs index key n mi ma) k) xs index key mi ma i"
algo_def: shows "algo \<equiv> ({1, 2, 3} ::# rel_set1, {1, 2, 3} ::# rel_set2)"
Suc_le_length_iff: fixes n :: "nat"   and xs :: "'d list" shows "(Suc n \<le> length xs) = (\<exists>x ys. xs = x # ys \<and> n \<le> length ys)"
e'_def: fixes i :: "nat" shows "e' i = (if i < d then e i else e2 (i - d))"
less_induct: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "class.wellorder less_eq less"   and "\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
simps_1: fixes S :: "'node SDG_node set"   and cs :: "'edge list" shows "slice_edges S cs [] = []"
read_only_reads_append: fixes \<O> :: "nat set"   and xs :: "'b memref list"   and ys :: "'b memref list" shows "read_only_reads \<O> (xs @ ys) = read_only_reads \<O> xs \<union> read_only_reads (acquired True xs \<O>) ys"
ennsqrt_mult: fixes a :: "ennreal"   and b :: "ennreal" shows "ennsqrt (a * b) = ennsqrt a * ennsqrt b"
add_succ_right: fixes zero :: "'a"   and succ :: "'a \<Rightarrow> 'a"   and m :: "'a"   and n :: "'a" assumes "peano zero succ" shows "peano.add zero succ m (succ n) = succ (peano.add zero succ m n)"
graph_plan_not_eq_last_diff_paths: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'c) fmap \<times> ('a, 'd) fmap) set"   and vs :: "'a set" assumes "s \<in> valid_states PROB" shows "fmrestrict_set vs s \<in> valid_states (prob_proj PROB vs)"
Quotient_alt_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd"   and T :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)))"
absolutely_integrable_on_image_real: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> real" assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) absolutely_integrable_on S" shows "f absolutely_integrable_on g ` S"
pdivmod2_rel: shows "(MP_Rel ===> MP_Rel ===> MP_Rel_Pair) pdivmod2 pdivmod"
\<gamma>3: shows "\<gamma>3 \<in> conv ars"
multiple_block_in_original: fixes b :: "'a set"   and n :: "nat"   and na :: "nat" assumes "b \<in># mult_sys.multiple_blocks na n" shows "b \<in># multiple_blocks na"
simps_49: fixes x1 :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>t\<^sub>p (send\<langle>x1\<rangle>\<^sub>s\<^sub>t) = fv x1"
af_unfold: shows "\<up>af = (\<lambda>\<phi>. \<up>step (\<up>Unf \<phi>))"
initial_of_def: shows "initial_of \<equiv> id \<circ> Record.iso_tuple_fst Record.tuple_iso_tuple \<circ> Record.iso_tuple_snd Record.tuple_iso_tuple \<circ> Record.iso_tuple_fst sas_plus_problem_ext_Tuple_Iso"
diff_delta_arctan_upper_32_def: shows "diff_delta_arctan_upper_32 \<equiv> \<lambda>x. - 256 * x ^ 12 / ((5 * x ^ 6 + 105 * x ^ 4 + 315 * x\<^sup>2 + 231)\<^sup>2 * (1 + x\<^sup>2))"
pow2_list: fixes u :: "'b list" shows "CoWBasic.power2 u = u \<cdot> u"
simps_11: fixes var :: "nat"   and a :: "atom fm" shows "freeIn var (AllQ a) = freeIn (var + 1) a"
nonzero_neg_divide_eq_eq2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "(c = uminus (divide a b)) = (times c b = uminus a)"
\<psi>_def: shows "\<psi> \<equiv> \<lambda>a. if a \<in> GRepHomSet (\<star>) W then THE S. \<psi>_condition a S else 0"
bpspec: fixes A :: "('a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "\<forall>(x, y)\<in>A. P x y"   and "(x, y) \<in> A" shows "P x y"
mono: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<le> c"   and "b \<le> d" shows "a \<sqinter> b \<le> c \<sqinter> d"
elims_2: fixes x :: "nat"   and xa :: "'b btree" assumes "almost_order x xa" obtains k :: "nat" where "x = ka__" and "xa = Leaf"   | k :: "nat" and ts :: "('b btree \<times> 'b) list" and t :: "'b btree" where "x = k" and "xa = Node ts t" and "length ts \<le> 2 * k \<and> Ball (set (subtrees ts)) (order k) \<and> order k t"
heap_hupd_same: fixes a :: "loc"   and obj :: "obj"   and s :: "state" shows "heap (hupd(a\<mapsto>obj) s) a = Some obj"
natUnion_eq: fixes f :: "nat \<Rightarrow> 'b set"   and g :: "nat \<Rightarrow> 'b set" assumes "\<And>n. \<exists>m. f n \<subseteq> g m"   and "\<And>n. \<exists>m. g n \<subseteq> f m" shows "natUnion f = natUnion g"
ll: shows "measure_pmf.prob (bernoulli_pmf p \<bind> (\<lambda>b. if b then map_pmf Inl Da else map_pmf Inr Db)) {f. \<exists>e. Inl e = f} = p"
w_simps_5: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c" assumes "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta>" shows "partial_magma.dom V w = w"
qty_single_val_process: fixes Mkt :: "('c, 'd) discrete_market"   and asset :: "'d"   and qty :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and n :: "nat"   and w :: "'c" shows "val_process Mkt (qty_single asset qty) n w = prices Mkt asset n w * qty (Suc n) w"
simps: fixes f :: "'a \<times> 'd \<Rightarrow> 'a \<times> 'b"   and xs :: "('a \<times> 'd) list"   and ko :: "'c" shows "map_raw f (xs, ko) = (map_pair f xs, ko)"
simps_2: fixes x :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" shows "BsSkewBinomialHeapStruc.bs_to_mset (Inr x) = BsSkewBinomialHeapStruc.elem_to_mset x"
upt_map_Suc: fixes k :: "nat" assumes "0 < k" shows "[0..<k + 1] = 0 # map Suc [0..<k]"
diff_numeral_special_9: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "minus one one = zero"
the_sorted_list_of_set: fixes l :: "'a list" assumes "distinct l"   and "sorted l" shows "sorted_list_of_set (set l) = l"
assms_5: shows "(red G)\<^sup>*\<^sup>* p r"
case_1: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'e \<Rightarrow> 'd"   and x1 :: "'e" shows "(case allow x1 of allow x \<Rightarrow> f1 x | deny x \<Rightarrow> f2 x) = f1 x1"
senior_states_card_le: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and p :: "'d"   and n :: "nat"   and q :: "'d" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "(card (semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w p n) < card (semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n)) = (semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w p n \<subset> semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n)"
ctr_2: fixes x :: "('f, 'g, 'h) Resumption.resumption"   and f :: "'f \<Rightarrow> ('i, 'g, 'h) Resumption.resumption" assumes "\<not> Resumption.resumption.is_Done x \<or> \<not> (Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x))))" shows "x \<bind> f = Resumption.resumption.Pause (if Resumption.resumption.is_Done x then Resumption.resumption.output (f (the (Resumption.resumption.result x))) else Resumption.resumption.output x) (\<lambda>inp. if Resumption.resumption.is_Done x then Resumption.resumption.resume (f (the (Resumption.resumption.result x))) inp else Resumption.resumption.resume x inp \<bind> f)"
image_mset_union: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset"   and N :: "'b multiset" shows "f `# (M + N) = f `# M + f `# N"
assms_2: shows "(\<box>(\<lambda>s. P s \<longrightarrow> (Q \<U> R) s)) \<sigma>"
intros_2: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Q :: "ccs" assumes "P \<longmapsto>\<alpha> \<prec> P'" shows "P \<oplus> Q \<longmapsto>\<alpha> \<prec> P'"
oreachable_local': fixes \<sigma> :: "'d"   and p :: "'e"   and A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'f"   and \<sigma>' :: "'d"   and p' :: "'e" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "(\<sigma>', p') \<in> oreachable A S U"
exhaust: fixes y :: "st" obtains x1 :: "oref \<Rightarrow> obj option" and x2 :: "lname \<Rightarrow> val option" where "y = st x1 x2"
linear_singular_into_hyperplane: fixes f :: "'a \<Rightarrow> 'a" assumes "linear f" shows "(\<not> inj f) = (\<exists>a. a \<noteq> (0::'a) \<and> (\<forall>x. a \<bullet> f x = 0))"
resOutputOutputTrans: fixes x :: "name"   and a :: "name"   and P :: "pi"   and b :: "name"   and y :: "name"   and P' :: "pi" assumes "<\<nu>x>a{x}.P \<longmapsto> b[y] \<prec> P'" shows "False"
not_in_iff_zmset: fixes x :: "'b"   and M :: "'b zmultiset" shows "(x \<notin>#\<^sub>z M) = (zcount M x = 0)"
arcs_H: shows "arcs H = {uw, rev_H uw, vw, rev_H vw} \<union> arcs I - {uv, rev_I uv}"
design_defs_1: fixes P :: "'b alpha_d_scheme \<times> 'b alpha_d_scheme \<Rightarrow> bool"   and Q :: "'b alpha_d_scheme \<times> 'b alpha_d_scheme \<Rightarrow> bool" shows "(P \<turnstile> Q) \<equiv> \<lambda>(A, A'). ok A \<and> P (A, A') \<longrightarrow> ok A' \<and> Q (A, A')"
dual_equivalence: shows "D.internal_equivalence (map\<^sub>0 a) (\<tau>\<^sub>0 a) (D.inv (counit a)) (D.inv (unit a))"
lower_zero: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and v :: "'c" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.lower ord_term 0 v = 0"
anycast_simps_4: fixes P :: "'a \<Rightarrow> bool"   and ip :: "nat" shows "anycast P (\<not>unicast ip) = True"
2: fixes x :: "'n"   and d :: "real" assumes "x \<in> S \<and> 0 < d" shows "\<exists>i. i \<in> K' \<and> x \<in> cball (a i) (r i) \<and> r i < d"
real_restrict_merge: fixes I :: "nat set"   and J :: "nat set"   and n :: "nat"   and tI :: "'b option list"   and tJ :: "'b option list" assumes "I \<inter> J = {}"   and "wf_tuple n I tI"   and "wf_tuple n J tJ" shows "restrict I (New_max.merge tI tJ) = restrict I tI \<and> restrict J (New_max.merge tI tJ) = restrict J tJ"
val_Zp_dist_infty: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "val_Zp_dist a b = \<infinity>" shows "a = b"