hcju/isabelleps huggingface dataset:
directed_downset_ideals: fixes X :: "'a set" shows "ord.directed (\<lambda>x y. y \<le> x) (ord.downset_set (\<lambda>x y. y \<le> x) X) = (ord.downset_set (\<lambda>x y. y \<le> x) X \<in> ord.ideals (\<lambda>x y. y \<le> x))"
value_minus_flow: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and g :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "flow \<Delta> f"   and "flow \<Delta> g"   and "value_flow \<Delta> g \<le> value_flow \<Delta> f"   and "\<And>y. edge \<Delta> (source \<Delta>) y = (y = x)" shows "value_flow \<Delta> (f \<ominus> g) = value_flow \<Delta> f - value_flow \<Delta> g"
insert_before_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr' :: "(_) object_ptr"   and child :: "(_) node_ptr"   and ref :: "(_) node_ptr option"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'l" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> insert_before ptr' child ref \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_scdom_component ptr'|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
pathint_def: shows "pathint \<equiv> \<lambda>x y. contour_integral (linepath x y) f"
h_real: fixes z :: "complex" assumes "z \<in> \<real>" shows "h z \<in> \<real>"
Enc_Enc: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "(Enc X Y = Enc X' Y') = (X = X' \<and> Y = Y')"
DIM_eq_def: fixes n :: "nat" shows "DIM_precond TYPE('c) n = (DIM('c) = n)"
case_prod_const: fixes c :: "'c" shows "(\<lambda>(i, j). c) = (\<lambda>_. c)"
retain_TCons_conv: fixes A :: "('d, 'b) tllist set" shows "local.retain_TCons A = A \<inter> {xs. \<not> is_TNil xs}"
o_circline_point_0_def: shows "o_circline_point_0 \<equiv> abs_ocircline circline_point_0_clmat"
Dom_comp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "seq g f" shows "(if C1.arr (fst (g \<cdot> f)) \<and> C2.arr (snd (g \<cdot> f)) then (C1.dom (fst (g \<cdot> f)), C2.dom (snd (g \<cdot> f))) else (C1.null, C2.null)) = (if C1.arr (fst f) \<and> C2.arr (snd f) then (C1.dom (fst f), C2.dom (snd f)) else (C1.null, C2.null))"
step_rewrite_pure_1: fixes K :: "('f \<times> 'c) set"   and V :: "('d \<times> 'a) set"   and R :: "('e \<times> 'b) set" shows "pure (p2rel (\<langle>K, V, R\<rangle>pac_step_rel_raw)) = pac_step_rel_assn (pure K) (pure V) (pure R)"
SD_preorderD_1: fixes q :: "'a pmf"   and p :: "'a pmf" assumes "SD le q p" shows "p \<in> lotteries_on carrier"
restrict: fixes a :: "'a" assumes "a \<in> carrier G" shows "((\<lambda>x. the_elem (f ` x)) \<circ> (#>) H) a = f a"
Ostrowski_elem_0: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and x :: "'a" assumes "vals_nonequiv K n vv"   and "x \<in> carrier K"   and "Ostrowski_elem K n vv x" shows "0 < vv 0 (1\<^sub>r \<plusminus> -\<^sub>a x)"
f: shows "f \<in> (\<Pi>\<^sub>E i\<in>I. topspace (X i))"
valid_path_subpath: fixes g :: "real \<Rightarrow> 'a"   and u :: "real"   and v :: "real" assumes "valid_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}" shows "valid_path (subpath u v g)"
get_scdom_component_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" shows "ptr' |\<in>| object_ptr_kinds h"
f6: shows "p \<cdot> (q \<cdot> x) = q \<cdot> p \<cdot> x"
proj_lnil: fixes c :: "'id" shows "\<pi>\<^bsub>c\<^esub>[]\<^sub>l = []\<^sub>l"
min_bot: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.order_bot bot less_eq less" shows "ord.min less_eq bot x = bot"
weight_finite: fixes x :: "'v + 'v" shows "weight (bipartite_web_of \<Gamma>) x \<noteq> \<top>"
free_generator_generator: fixes H :: "'a set" assumes "free_generator R M H" shows "generator R M H"
sum_encode_eq: fixes x :: "nat + nat"   and y :: "nat + nat" shows "(sum_encode x = sum_encode y) = (x = y)"
number_partition1_eq_replicate_mset: fixes n :: "nat" shows "{N. (\<forall>n. n \<in># N \<longrightarrow> n = 1) \<and> number_partition n N} = {replicate_mset n 1}"
abs_linorder_cases: fixes a :: "'a"   and b :: "'a" obtains "\<bar>a\<bar> < \<bar>b\<bar>"   | "\<bar>a\<bar> = \<bar>b\<bar>"   | "\<bar>b\<bar> < \<bar>a\<bar>"
simple_distributed_finite: fixes M :: "'d measure"   and X :: "'d \<Rightarrow> 'e"   and P :: "'e \<Rightarrow> real" assumes "simple_distributed M X P" shows "finite (X ` space M)"
a: shows "(\<lambda>n. f (n - k) / real (n - k + k)) \<longlonglongrightarrow> l"
c1d1_3: shows "c1 \<approx>01 d1"
inf_left_commute: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf y z) = inf y (inf x z)"
relcompE: fixes xz :: "'h \<times> 'i"   and r :: "('h \<times> 'j) set"   and s :: "('j \<times> 'i) set" assumes "xz \<in> r O s" obtains x :: "'h" and y :: "'j" and z :: "'i" where "xz = (x, z)" and "(x, y) \<in> r" and "(y, z) \<in> s"
sel_3: fixes x31 :: "bool"   and x32 :: "'b" shows "fin (Atom2 x31 x32) = x31"
B_Issues_A: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and Nb :: "nat"   and evs :: "event list" assumes "Says B A (Crypt K (Nonce Nb)) \<in> set evs"   and "Key K \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_shared" shows "B Issues A with Crypt K (Nonce Nb) on evs"
compare_refl_Integer: fixes x :: "Integer" assumes "x \<noteq> \<bottom>" shows "compare\<cdot>x\<cdot>x = EQ"
FunctorCompM: fixes f :: "'c"   and X :: "'a"   and Y :: "'a" assumes "f maps\<^bsub>CatDom F\<^esub> X to Y" shows "F ## f maps\<^bsub>CatCod F\<^esub> F @@ X to F @@ Y"
mutual_information_eq_entropy_conditional_entropy: fixes X :: "'key \<Rightarrow> 'a"   and Y :: "'key \<Rightarrow> 'b" assumes "simple_function (point_measure keys (\<lambda>x. ennreal (K x))) X"   and "simple_function (point_measure keys (\<lambda>x. ennreal (K x))) Y" shows "K.mutual_information_Pow X Y = K.entropy_Pow X - K.conditional_entropy_Pow X Y"
r_def: shows "r \<equiv> \<lambda>x i. if i = k then - x i else x i"
closure_subset_eq: fixes S :: "'b set" shows "(closure S \<subseteq> S) = closed S"
Tarjan_invar_eq: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" shows "DFS_invar G (Tarjan_def.tarjan_params G) s = Tarjan_invar G s"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (load_sub3 instr curr_win rd 10 address s1)) \<and> t2 = snd (fst (load_sub3 instr curr_win rd 10 address s2))"
ge_def: fixes c :: "int"   and t :: "MIR.num" shows "ge c t = (if c = 0 then Ge t else if 0 < c then Ge (CN 0 c t) else Le (CN 0 (- c) (Neg t)))"
isom_gch_units_transpTr8_1: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n g"   and "i \<le> n"   and "j \<le> n"   and "g i \<cong> E"   and "\<not> g j \<cong> E" shows "{i. i \<le> n \<and> g i \<cong> E} = {i. i \<le> n \<and> g i \<cong> E} - {j}"
injec_def: fixes F :: "('e, 'f) aGroup_scheme"   and G :: "('g, 'h) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g" shows "injec\<^bsub>F,G\<^esub> f = (f \<in> aHom F G \<and> ker\<^bsub>F,G\<^esub> f = {\<zero>\<^bsub>F\<^esub>})"
derived_is_proper: assumes "\<And>b. b \<in># remove1_mset bl \<B> \<Longrightarrow> 0 < bl |\<inter>| b" shows "proper_design bl \<B>\<^sup>D"
S2_def: fixes y :: "nat"   and s2 :: "nat" shows "S2 y s2 = sample_uniform q \<bind> (\<lambda>b. sample_uniform q \<bind> (\<lambda>e2. let d2 = (e2 * b + (q - s2)) mod q in return_spmf (y, b, d2, e2)))"
map_vec_to_iarray_rw: fixes A :: "(('c, 'd) vec, 'e) vec"   and i :: "'e" shows "map (\<lambda>x. vec_to_iarray (A $ mod_type_class.from_nat x)) [0..<CARD('e)] ! mod_type_class.to_nat i = vec_to_iarray (A $ i)"
rep_eq: fixes x :: "'a mat" shows "($$) x = snd (snd (Rep_mat x))"
fun_pair_subst: fixes d :: "('fun, 'd) Term.term \<times> ('fun, 'd) Term.term"   and \<delta> :: "'d \<Rightarrow> ('fun, 'c) Term.term" shows "pair d \<cdot> \<delta> = pair (d \<cdot>\<^sub>p \<delta>)"
isLub_dual_isGlb: fixes S :: "'b set"   and cl :: "'b potype" shows "isLub S cl = isGlb S (dual cl)"
int_Idl: fixes a :: "int" shows "Idl\<^bsub>\<Z>\<^esub> {a} = {x * a |x. True}"
assms_2: shows "snd (f (k, 0::'b)) = (0::'c)"
semilattice_inf_ow_axioms: shows "semilattice_inf_ow U (\<sqinter>\<^sub>o\<^sub>w) (\<le>\<^sub>o\<^sub>w) (<\<^sub>o\<^sub>w)"
Scons_def: fixes M :: "('a, 'b) node set"   and N :: "('a, 'b) node set" shows "Scons M N \<equiv> Push_Node (Inr 1) ` M \<union> Push_Node (Inr (Suc 1)) ` N"
hyps_1: shows "a \<rightarrow>\<^sub>\<eta>\<^sup>* b"
qFresh_preserves_alpha: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and z :: "'var" assumes "qGood X \<or> qGood X'"   and "X #= X'" shows "qFresh zs z X = qFresh zs z X'"
trans_mult: fixes R :: "('b \<times> 'b) set" shows "trans (mult R)"
has_sgnx_poly_at_bot: fixes p :: "real poly" shows "(poly p has_sgnx sgn_neg_inf p) at_bot"
replicate_add: fixes n :: "nat"   and m :: "nat"   and x :: "'c" shows "replicate (n + m) x = replicate n x @ replicate m x"
Dom_f': shows "\<BB>\<lparr>Dom\<rparr>\<lparr>f'\<rparr> = a'"
init2: fixes G :: "('c \<times> ('c, 'd) RHS) list"   and w :: "'d list"   and k :: "nat" shows "(init G w k = None) = (w = [] \<or> (\<exists>a\<in>set w. init_match G a = []))"
simps_1: fixes i :: "int" shows "numgcdh (C i) = gcd i"
when_mult: fixes a :: "'b"   and P :: "bool"   and b :: "'b" shows "(a when P) * b = (a * b when P)"
ideals_filters_dual: fixes X :: "'a set" shows "idealp X = filterp (\<partial> ` X)"
take_bit_of_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n one = zero_neq_one.of_bool one zero (0 < n)"
span_image: fixes S :: "'b set" shows "span (f ` S) = f ` span S"
intro: fixes D :: "('c, 'd) Order_scheme" assumes "\<And>a b. \<lbrakk>a \<in> carrier D; b \<in> carrier D\<rbrakk> \<Longrightarrow> a \<preceq>\<^bsub>D\<^esub> b \<or> b \<preceq>\<^bsub>D\<^esub> a" shows "Torder_axioms D"
distinct_20: fixes x51 :: "'b gexp"   and x52 :: "'b gexp"   and x41 :: "'b"   and x42 :: "value list" shows "Nor x51 x52 \<noteq> In x41 x42"
axioms: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_delete_shadow_root_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs" shows "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
T_opt_def: fixes step :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'c"   and t :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> nat"   and s :: "'c"   and rs :: "'d list" shows "On_Off.T_opt step t s rs = \<Sqinter> {On_Off.T step t s rs as |as. length as = length rs}"
val1_neq_0: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "v x = 1" shows "x \<noteq> \<zero>"
finals_def: fixes g :: "graph" shows "finals g \<equiv> filter final (faces g)"
on_circline_opposite_ocircline: fixes H :: "ocircline"   and z :: "complex_homo" shows "on_ocircline (opposite_ocircline H) z = on_ocircline H z"
is_cupcake_all_envI: fixes env :: "v sem_env" assumes "is_cupcake_ns (sem_env.v env)"   and "c env = static_cenv" shows "is_cupcake_all_env env"
XC_def: fixes c :: "nat list" shows "XC c = {es \<in> space P. c \<in> ucycles (edge_ugraph es) \<and> uwalk_length c = k}"
all_list_constr_R_def: fixes L :: "(nat list \<times> nat list) list"   and n :: "nat" shows "all_list_constr_R L n \<equiv> \<forall>x. List.member L x \<longrightarrow> list_constr (fst x) n \<and> list_constr (snd x) n"
islin_qf: fixes p :: "fm" assumes "islin p" shows "qfree p"
markedElementsToPrefixElement: fixes p :: "('b \<times> bool) list"   and M :: "('b \<times> bool) list"   and e :: "'b" assumes "isPrefix p M"   and "e \<in> set (elements p)" shows "markedElementsTo e M = markedElementsTo e p"
cont_left_D: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "((\<Squnion>i. Y i) `;` S) = (\<Squnion>i. Y i `;` S)"
fmimage_drop_fset: fixes B :: "'d fset"   and m :: "('d, 'c) fmap"   and A :: "'d fset" shows "fmimage (fmdrop_fset B m) A = fmimage m (A |-| B)"
raise_env: fixes v :: "val"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and \<rho>' :: "(nat \<times> val) list" assumes "v \<in> E e \<rho>"   and "\<rho> \<sqsubseteq> \<rho>'" shows "v \<in> E e \<rho>'"
monoid_mult_isom_axioms: shows "monoid_mult_isom hom"
check_CDPlayer_CTRL: shows "(CDPlayer_CTRL_States, CDPlayer_CTRL_Init, CDPlayer_CTRL_Labels, CDPlayer_CTRL_Delta) \<in> seqauto"
intTI_def: shows "intTI \<equiv> fst3 MI"
locate1_def: fixes f :: "'b \<Rightarrow> nat"   and al :: "'b list"   and i :: "nat" shows "locate1 f al i \<equiv> fst (locate f al i)"
tt_monom_mult: fixes c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "c \<noteq> (0::'c)"   and "p \<noteq> 0" shows "tt (monom_mult c t p) = t \<oplus> tt p"
not_reset_unbounded: fixes i :: "nat"   and c :: "'c"   and xs :: "('c \<Rightarrow> real) stream"   and j :: "nat" assumes "\<forall>j\<ge>i. \<not> zero c ([xs !! j]\<^sub>\<R>)"   and "i \<le> j"   and "c \<in> \<X>"   and "stream_trans xs"   and "\<forall>i. eq_elapsed (xs !! i) (xs !! Suc i)"   and "\<forall>j. xs !! j \<in> V"   and "unbounded c ([xs !! i]\<^sub>\<R>)" shows "unbounded c ([xs !! j]\<^sub>\<R>)"
inc_def: fixes sn :: "nat" shows "inc sn \<equiv> if sn = 0 then sn else sn + 1"
assms_1: shows "z = (x * c + r) mod order \<G>"
infinite_Un: fixes S :: "'e set"   and T :: "'e set" shows "infinite (S \<union> T) = (infinite S \<or> infinite T)"
opti_iterate_var: fixes y :: "'a"   and x :: "'a" shows " |(ad y \<cdot> x)\<^sup>\<star>\<rangle> y =  |x\<^sup>\<star>\<rangle> y"
msubstltpos_nb: fixes t :: "tm"   and a :: "poly"   and r :: "tm"   and c :: "poly" assumes "tmbound0 t"   and "islin (Lt (CNP 0 a r))" shows "bound0 (msubstltpos c t a r)"
total: fixes x :: "'a"   and y :: "'a" assumes "x \<in> alts"   and "y \<in> alts" shows "R x y \<or> R y x"
match_list_concat: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and lss :: "'b match_expr list list"   and a :: "action"   and p :: "'c" shows "match_list \<gamma> (concat lss) a p = (\<exists>ls\<in>set lss. match_list \<gamma> ls a p)"
eq_fmem_trans: fixes a :: "'b"   and b :: "'b"   and A :: "'b fset" assumes "a = b"   and "b |\<in>| A" shows "a |\<in>| A"
unwrapC_def: shows "unwrapC \<equiv> \<Lambda> g e s f. case g\<cdot>e of Nothing \<Rightarrow> f | Just\<cdot>n \<Rightarrow> s\<cdot>n"
sym: fixes S :: "('c, 'd) eq_object_scheme"   and x :: "'c"   and y :: "'c" assumes "equivalence S"   and "x .=\<^bsub>S\<^esub> y"   and "x \<in> carrier S"   and "y \<in> carrier S" shows "y .=\<^bsub>S\<^esub> x"
prime_factorization_Gcd_factorial: fixes A :: "'a set" assumes "\<not> A \<subseteq> {0::'a}" shows "prime_factorization (Gcd_factorial A) = Inf (prime_factorization ` (A - {0::'a}))"
unique_convergence: fixes T :: "'b set set"   and F :: "'b set set"   and x :: "'b"   and y :: "'b" assumes "T2 T"   and "Topology.filter F T"   and "x \<in> carrier\<^bsub>T\<^esub>"   and "y \<in> carrier\<^bsub>T\<^esub>"   and "F \<longlongrightarrow>\<^bsub>T\<^esub> x"   and "F \<longlongrightarrow>\<^bsub>T\<^esub> y" shows "x = y"
linear_injective_imp_surjective: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "linear (*s) (*s) f"   and "inj f"   and "vec.dim UNIV = vec.dim UNIV" shows "surj f"
quotientD: fixes X :: "'b set"   and R :: "('b \<times> 'b) set"   and A :: "'b set"   and x :: "'b" assumes "equiv X R"   and "A \<in> X // R"   and "x \<in> A" shows "A = R `` {x}"
valid_lists_def: shows "valid_lists \<equiv> \<lambda>S G. {x. valid_listsp (\<lambda>x. x \<in> S) G x}"
preorder_one_closed: shows "matrix_idempotent_semiring.preorder mone"
q_lookup: fixes s :: "'t" shows "lookup q s = lookup p s - lookup (monom_mult (lookup p (t \<oplus> lt f) / lc f) t f) s"
Hom_DopxG_map_simp: fixes gf :: "'d \<times> 'c" assumes "DopxC.arr gf" shows "Hom_DopxG.map gf = S.mkArr (HomD.set (D.cod (fst gf), G (C.dom (snd gf)))) (HomD.set (D.dom (fst gf), G (C.cod (snd gf)))) (\<phi>D (D.dom (fst gf), G (C.cod (snd gf))) \<circ> (\<lambda>h. G (snd gf) \<cdot>\<^sub>D h \<cdot>\<^sub>D fst gf) \<circ> \<psi>D (D.cod (fst gf), G (C.dom (snd gf))))"
ln_exp_cf7_upper_bound_nonneg: fixes x :: "real" assumes "0 \<le> x"   and "0 < numer_cf7 (- x)" shows "x \<le> ln (exp_cf7 x)"
d0_def: shows "c.d0 = prod_list c.dims0"
normalize_quot_aux_2: fixes b :: "'c"   and a :: "'c" assumes "b \<noteq> (0::'c)" shows "b = snd (normalize_quot (a, b)) * (gcd a b * unit_factor b)"
mcont_if_bot: fixes lub' :: "'f set \<Rightarrow> 'f"   and ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and bound :: "('g, 'h, 'i) Resumption.resumption"   and bot :: "'f"   and f :: "('g, 'h, 'i) Resumption.resumption \<Rightarrow> 'f"   and g :: "('g, 'h, 'i) Resumption.resumption \<Rightarrow> 'f" assumes "class.ccpo lub' ord (mk_less ord)"   and "\<And>x. \<not> resumption_ord x bound \<Longrightarrow> ord bot (f x)"   and "\<And>x. g x = (if resumption_ord x bound then bot else f x)"   and "\<And>x y. \<lbrakk>resumption_ord x y; \<not> resumption_ord x bound\<rbrakk> \<Longrightarrow> ord (f x) (f y)"   and "\<And>Y. \<lbrakk>Complete_Partial_Order.chain resumption_ord Y; Y \<noteq> {}; \<And>x. x \<in> Y \<Longrightarrow> \<not> resumption_ord x bound\<rbrakk> \<Longrightarrow> f (resumption_lub Y) = lub' (f ` Y)" shows "mcont resumption_lub resumption_ord lub' ord g"
contractible_imp_holomorphic_arccos_bounded: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and a :: "complex" assumes "f holomorphic_on S"   and "contractible S"   and "a \<in> S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 1 \<and> f z \<noteq> - 1" obtains g :: "complex \<Rightarrow> complex" where "g holomorphic_on S" and "cmod (g a) \<le> pi + cmod (f a)" and "\<And>z. z \<in> S \<Longrightarrow> f z = cos (g z)"
norm_pths_2: fixes x :: "'a"   and y :: "'a" shows "(x \<noteq> y) = (\<not> norm (x - y) \<le> 0)"
a_de_morgan: fixes x :: "'a"   and y :: "'a" shows "ad (ad x \<cdot> ad y) = d (x + y)"
assms_5: shows "K1 \<inter> {x. x \<bullet> k \<le> c} = K2 \<inter> {x. x \<bullet> k \<le> c}"
length_list_of_eucl: fixes x :: "'a" shows "length (list_of_eucl x) = DIM('a)"
representation_ne_zero: fixes basis :: "'d set"   and v :: "'d"   and b :: "'d" assumes "representation basis v b \<noteq> 0" shows "b \<in> basis"
cl9: fixes x :: "'a" shows "(x \<sqinter> 1\<^sub>\<sigma>) \<cdot> 1\<^sub>\<pi> \<parallel> 1\<^sub>\<sigma> = x \<sqinter> 1\<^sub>\<sigma>"
Spy_not_see_encrypted_key: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and nb :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Key K \<notin> analz (knows Spy evs)"
simps_1: fixes v1 :: "'a"   and v2 :: "'a" shows "ces v1 [] v2 = (v1 = v2)"
induct: fixes P :: "'a \<Rightarrow> 'a tree \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a tree" assumes "\<And>x. P x \<langle>\<rangle>"   and "\<And>x AB CD. P x \<langle>AB, x, CD\<rangle>"   and "\<And>x b A B CD. x < b \<Longrightarrow> P x \<langle>\<langle>A, x, B\<rangle>, b, CD\<rangle>"   and "\<And>x b CD. x < b \<Longrightarrow> P x \<langle>\<langle>\<rangle>, b, CD\<rangle>"   and "\<And>x a b B CD. \<lbrakk>x < a; x < b\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>\<langle>\<rangle>, a, B\<rangle>, b, CD\<rangle>"   and "\<And>x a b A B CD. \<lbrakk>x < a; x < b; A \<noteq> \<langle>\<rangle>; P x A\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>A, a, B\<rangle>, b, CD\<rangle>"   and "\<And>a x b A CD. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>A, a, \<langle>\<rangle>\<rangle>, b, CD\<rangle>"   and "\<And>a x b B A CD. \<lbrakk>a < x; x < b; B \<noteq> \<langle>\<rangle>; P x B\<rbrakk> \<Longrightarrow> P x \<langle>\<langle>A, a, B\<rangle>, b, CD\<rangle>"   and "\<And>b x AB C D. b < x \<Longrightarrow> P x \<langle>AB, b, \<langle>C, x, D\<rangle>\<rangle>"   and "\<And>b x AB. b < x \<Longrightarrow> P x \<langle>AB, b, \<langle>\<rangle>\<rangle>"   and "\<And>b x c C AB D. \<lbrakk>b < x; x < c; C \<noteq> \<langle>\<rangle>; P x C\<rbrakk> \<Longrightarrow> P x \<langle>AB, b, \<langle>C, c, D\<rangle>\<rangle>"   and "\<And>b x c AB D. \<lbrakk>b < x; x < c\<rbrakk> \<Longrightarrow> P x \<langle>AB, b, \<langle>\<langle>\<rangle>, c, D\<rangle>\<rangle>"   and "\<And>b x c AB C. \<lbrakk>b < x; c < x\<rbrakk> \<Longrightarrow> P x \<langle>AB, b, \<langle>C, c, \<langle>\<rangle>\<rangle>\<rangle>"   and "\<And>a x c D AB C. \<lbrakk>a < x; c < x; D \<noteq> \<langle>\<rangle>; P x D\<rbrakk> \<Longrightarrow> P x \<langle>AB, a, \<langle>C, c, D\<rangle>\<rangle>" shows "P a0 a1"
Cons_listrel1_Cons: fixes x :: "'g"   and xs :: "'g list"   and y :: "'g"   and ys :: "'g list"   and r :: "('g \<times> 'g) set" shows "((x # xs, y # ys) \<in> listrel1 r) = ((x, y) \<in> r \<and> xs = ys \<or> x = y \<and> (xs, ys) \<in> listrel1 r)"
mndet_refine_FD: fixes a :: "'d"   and A :: "'d set"   and P :: "'d \<Rightarrow> 'd process" assumes "a \<in> A" shows "mndet A P \<le> (a \<rightarrow> P a)"
invert_opposite_sides_of_square: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.seq f g"   and "f \<cdot>\<^sub>C g = h \<cdot>\<^sub>C k"   and "C.iso f"   and "C.iso k" shows "C.seq g (C.inv k) \<and> C.seq (C.inv f) h \<and> g \<cdot>\<^sub>C C.inv k = C.inv f \<cdot>\<^sub>C h"
init: fixes p :: "'a" assumes "p \<in> X" shows "P p"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_inverse" shows "a\<inverse>\<^sub>\<rat> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_inverse"
valid_mmuaux_def: fixes args :: "Monitor.args"   and cur :: "nat" shows "valid_mmuaux args cur = valid_mmuaux' args cur cur"
adjuster_wit_carrier_vec: fixes f :: "'a vec"   and n :: "nat"   and gs :: "'a vec list"   and mus :: "'a list" assumes "f \<in> carrier_vec n"   and "set gs \<subseteq> carrier_vec n" shows "snd (gram_schmidt.adjuster_wit n mus f gs) \<in> carrier_vec n"
r_amalgamate_recfn: shows "recfn 2 r_amalgamate"
bell00_index_1: shows "|\<beta>\<^sub>0\<^sub>0\<rangle> $$ (0, 0) = complex_of_real (1 / sqrt 2)"