hcju/isabelleps huggingface dataset:
simps: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and a :: "'a" shows "SATp \<Gamma> f a = ((\<exists>x. a = x \<and> x \<in> A \<Gamma>) \<or> (\<exists>x. a = x \<and> weight \<Gamma> x \<le> d_IN f x))"
null_sets_subset: fixes B :: "'a set"   and M :: "'a measure"   and A :: "'a set" assumes "B \<in> null_sets M"   and "A \<in> sets M"   and "A \<subseteq> B" shows "A \<in> null_sets M"
lincomb_union: fixes A :: "'c set"   and B :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "finite (A \<union> B)"   and "A \<union> B \<subseteq> carrier M"   and "A \<inter> B = {}"   and "a \<in> A \<union> B \<rightarrow> carrier R" shows "lincomb a (A \<union> B) = lincomb a A \<oplus>\<^bsub>M\<^esub> lincomb a B"
assms_2: shows "\<forall>\<beta>\<in>benv_in_ve ve. Q \<beta>"
map_of_remap_conv: fixes fs :: "('a \<times> 'b) list"   and D :: "'c"   and F :: "'a"   and T :: "'b" assumes "distinct_fst fs"   and "map_of (map (\<lambda>(F, y). ((F, D), y)) fs) (F, D) = \<lfloor>T\<rfloor>" shows "map_of (map (\<lambda>((F, D), T). (F, D, T)) (map (\<lambda>(F, y). ((F, D), y)) fs)) F = \<lfloor>(D, T)\<rfloor>"
hyps: assumes "wf_ll t levellist var"   and "n \<le> length levellist" shows "Nodes n levellist \<subseteq> set_of t"
double_diff: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "B - (C - A) = A"
simps: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "('c, 'd) blindable\<^sub>m"   and a2 :: "('c, 'd) blindable\<^sub>m" shows "blinding_of_blindable h bo a1 a2 = ((\<exists>x y. a1 = Unblinded x \<and> a2 = Unblinded y \<and> bo x y) \<or> (\<exists>t x. a1 = Blinded x \<and> a2 = t \<and> hash_blindable h t = x))"
lappend_lnull2: fixes ys :: "'b llist"   and xs :: "'b llist" assumes "lnull ys" shows "lappend xs ys = xs"
is_singleton_altdef: fixes A :: "'a set" shows "is_singleton A = (card A = 1)"
second_summands_in_sums: fixes y :: "i"   and x :: "i"   and v :: "i"   and z :: "i" shows "P y (x \<oplus> y) \<and> P v (z \<oplus> v)"
fBexE: fixes A :: "'a fset"   and P :: "'a \<Rightarrow> bool" assumes "fBex A P" obtains x :: "'a" where "x |\<in>| A" and "P x"
elims_2: fixes x :: "'a Point" assumes "onAxisT x" obtains u :: "'a Point" where "x = u" and "xval u = (0::'a) \<and> yval u = (0::'a) \<and> zval u = (0::'a)"
assms_2: shows "\<forall>xs x. P xs \<longrightarrow> P (x # xs)"
NO_MATCH_def: fixes pat :: "'k"   and val :: "'l" shows "NO_MATCH pat val \<equiv> True"
typeof_lit_neq_Array: fixes v :: "'a val"   and T :: "ty" shows "typeof v \<noteq> \<lfloor>T\<lfloor>\<rceil>\<rfloor>"
Rel_abs: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>x y. Rel A x y \<Longrightarrow> Rel B (f x) (g y)" shows "Rel (A ===> B) f g"
rev_singleton_conv: fixes xs :: "'a list"   and x :: "'a" shows "(rev xs = [x]) = (xs = [x])"
arr_char: fixes f :: "'a" shows "L.arr f = left g f"
I_minWeightBasisD_4: fixes carrier :: "'c set"   and indep :: "'c set \<Rightarrow> bool"   and weight :: "'c \<Rightarrow> 'd"   and T :: "'c set"   and E :: "'c set"   and x :: "'c"   and y :: "'c" assumes "weighted_matroid carrier indep"   and "weighted_matroid.I_minWeightBasis carrier indep weight (T, E)"   and "x \<in> T"   and "y \<in> E" shows "weight x \<le> weight y"
that: fixes ns' :: "'node SDG_node list" assumes "matched n ns' n'"   and "set ns \<subseteq> set ns'" shows "thesis"
substAbs_preserves_wls: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var" assumes "wlsAbs (xs, s) A"   and "wls (asSort ys) Y" shows "wlsAbs (xs, s) (A $[Y / y]_ys)"
shrK_not_AKcryptSK: fixes evs :: "event list"   and K :: "nat"   and A :: "agent" assumes "evs \<in> kerbV" shows "\<not> AKcryptSK K (shrK A) evs"
induct: fixes P :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a bdd \<Rightarrow> 'b bdd \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a1 :: "'a bdd"   and a2 :: "'b bdd" assumes "\<And>f x y. P f (Leaf x) (Leaf y)"   and "\<And>f l r y. \<lbrakk>P f l (Leaf y); P f r (Leaf y)\<rbrakk> \<Longrightarrow> P f (Branch l r) (Leaf y)"   and "\<And>f x l r. \<lbrakk>P f (Leaf x) l; P f (Leaf x) r\<rbrakk> \<Longrightarrow> P f (Leaf x) (Branch l r)"   and "\<And>f l\<^sub>1 r\<^sub>1 l\<^sub>2 r\<^sub>2. \<lbrakk>P f l\<^sub>1 l\<^sub>2; P f r\<^sub>1 r\<^sub>2\<rbrakk> \<Longrightarrow> P f (Branch l\<^sub>1 r\<^sub>1) (Branch l\<^sub>2 r\<^sub>2)" shows "P a0 a1 a2"
that: fixes i1 :: "complex_homo"   and i2 :: "complex_homo" assumes "i1 \<noteq> i2"   and "ideal_points H = {i1, i2}" shows "thesis"
ImplS: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<psi> in v] = ([\<phi> in v] \<longrightarrow> [\<psi> in v])"
domains_comp: fixes g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "seq g f" shows "domains (local.comp g f) = domains f"
pickBinp_asBinp: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" assumes "qGoodBinp qbinp" shows "(pickBinp (asBinp qbinp), qbinp) \<in> alphaBinp"
igSwapCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "igSwapCls MOD \<equiv> igSwapIGVar MOD \<and> igSwapIGAbs MOD \<and> igSwapIGOp MOD"
l_set_child_nodes_get_tag_name_axioms_def: fixes set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_child_nodes_get_tag_name_axioms set_child_nodes_locs get_tag_name_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
determinant_one_element: fixes A :: "'a mat" assumes "A \<in> carrier_mat 1 1" shows "Determinant.det A = A $$ (0, 0)"
simps_1: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "common_prefix (x # xs) (y # ys) = (if x = y then x # common_prefix xs ys else [])"
simps_1: fixes \<theta> :: "(name \<times> lam) list"   and x :: "name" shows "\<theta><Var x> = lookup \<theta> x"
real_sup_exists: fixes P :: "real \<Rightarrow> bool" assumes "\<exists>x. P x"   and "\<exists>z. \<forall>x. P x \<longrightarrow> x < z" shows "\<exists>s. \<forall>y. (\<exists>x. P x \<and> y < x) = (y < s)"
winning_region_extends_VVpstar: fixes v :: "'a"   and p :: "Player" assumes "v \<in> VV p**"   and "\<And>w. v \<rightarrow> w \<Longrightarrow> w \<in> winning_region p" shows "v \<in> winning_region p"
vec_to_lpoly_poly_of_vec_eq: fixes v :: "rat vec" shows "vec_to_lpoly v = poly_of_vec v"
Rep_hf_cases: fixes y :: "nat" assumes "y \<in> UNIV" obtains x :: "hf" where "y = Rep_hf x"
llist_all2_LCons_LCons: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x21 :: "'a"   and x22 :: "'a llist"   and y21 :: "'b"   and y22 :: "'b llist" shows "llist_all2 R (LCons x21 x22) (LCons y21 y22) = (R x21 y21 \<and> llist_all2 R x22 y22)"
count_node_correct: fixes n :: "('c, 'd) Node" shows "count_node n = length (nodeToList n)"
set_iteratei_I: fixes invar :: "'c \<Rightarrow> bool"   and iti :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e set" assumes "\<And>s. invar s \<Longrightarrow> set_iterator (iti s) (\<alpha> s)" shows "set_iteratei \<alpha> invar iti"
mopup_inv_step: fixes n :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" assumes "n < length lm"   and "mopup_inv (s, l, r) lm n ires" shows "mopup_inv (step0 (s, l, r) (mopup_a n @ shift mopup_b (2 * n))) lm n ires"
is_zero_row_imp_is_zero_row_upt: fixes i :: "'b"   and A :: "(('c, 'd) vec, 'b) vec"   and k :: "nat" assumes "is_zero_row i A" shows "is_zero_row_upt_k i k A"
assms_5: shows "zs \<noteq> ys \<or> z \<noteq> y"
sandwich_funcset: fixes A :: "'o"   and a :: "'a" assumes "A \<in> Ob"   and "a \<in> F \<^bsub>\<o>\<^esub> A" shows "\<sigma>(A,a) \<in> Ob \<rightarrow> Ar\<^bsub>Set\<^esub>"
case_2: fixes f1 :: "nat \<Rightarrow> 'c"   and f2 :: "'c" shows "(case \<infinity> of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f2"
indRelRTPO_impl_TRel_preserves_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred (indRelRTPO TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_preserves_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
prems: shows "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (f # F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)"
singular_relcycle_add: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_relcycle p X S a"   and "singular_relcycle p X S b" shows "singular_relcycle p X S (a + b)"
inf_sup_ord_4: fixes y :: "'b"   and x :: "'b" shows "y \<le> sup x y"
equivp_reflp2: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "equivp R" shows "reflp R"
map_last_prefixes: fixes p :: "'c list" shows "map last (tl (prefixes p)) = p"
pth_b_2: fixes c :: "real"   and x :: "'a"   and z :: "'a"   and d :: "real"   and y :: "'a" shows "c *\<^sub>R x + z + d *\<^sub>R y \<equiv> c *\<^sub>R x + (z + d *\<^sub>R y)"
mset_eq_length: fixes xs :: "'c list"   and ys :: "'c list" assumes "mset xs = mset ys" shows "length xs = length ys"
monad_catch_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and catch :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_catch return bind fail catch" shows "monad_catch (return_env return) (bind_env bind) (fail_env fail) (catch_env catch)"
restrict_star_left_induct: fixes ms :: "'c list"   and l :: "'c"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd" assumes "distinct ms"   and "[l] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<preceq> [l] \<langle>g\<rangle> ms" shows "[l] \<langle>star \<circ> f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<preceq> [l] \<langle>g\<rangle> ms"
le_err_trans: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e1 :: "'b err"   and e2 :: "'b err"   and e3 :: "'b err" assumes "order r"   and "e1 \<sqsubseteq>\<^bsub>le r\<^esub> e2"   and "e2 \<sqsubseteq>\<^bsub>le r\<^esub> e3" shows "e1 \<sqsubseteq>\<^bsub>le r\<^esub> e3"
b_trans: fixes T :: "('a, 'b) TS_scheme"   and s :: "'a"   and b :: "'a list"   and t :: "'a" assumes "behp T (s # b)"   and "(s, t) \<in> TS.trans T" shows "behp T (t # s # b)"
ordered_map_distinct: fixes S :: "('a \<times> 'b) set" assumes "finite S"   and "is_map S" shows "distinct (map fst (ordered_map S))"
i_nonneg: shows "le\<cdot>0\<cdot>i = TT"
dep_set_def: fixes S :: "'a set set" shows "dep_set S \<equiv> \<exists>x S'. S' \<subseteq> S \<and> (\<exists>P\<in>S - S'. dep_path P S' x)"
arr_mset_append: fixes l :: "int"   and r :: "int"   and arr :: "int \<Rightarrow> 'a" assumes "l \<le> r" shows "arr_mset arr l (r + 1) = arr_mset arr l r + {#arr r#}"
is_extensional: fixes f :: "'a" assumes "\<not> A.arr f" shows "\<sigma>' f = \<sigma>.B.null"
config_config: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and init :: "'a list"   and qs :: "'a list" shows "set (fst (config\<^sub>p A init qs)) = set init \<and> distinct (fst (config\<^sub>p A init qs)) = distinct init \<and> length (fst (config\<^sub>p A init qs)) = length init"
eout_know_nonKS_s: fixes m :: "Secrets"   and A :: "specID" assumes "m \<notin> specSecrets A"   and "eout A (sE m)"   and "eoutKnowCorrect A (sKS m)" shows "know A (sKS m)"
vsubstAbs_simp: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and ys :: "'varSort"   and x :: "'var"   and y :: "'var"   and y1 :: "'var" assumes "good X"   and "xs \<noteq> ys \<or> x \<notin> {y, y1}" shows "Abs xs x X $[y1 // y]_ys = Abs xs x (X #[y1 // y]_ys)"
r_unit: fixes a :: "'a" assumes "a \<in> carrier G" shows "a \<cdot> \<one> = a"
step_2_main_dims_main: fixes n :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_row (step_2_main n j A) = dim_row A \<and> dim_col (step_2_main n j A) = dim_col A"
zero_le_imp_eq_int: fixes k :: "int" assumes "0 \<le> k" shows "\<exists>n. k = int n"
fds_zeta_commutes: fixes f :: "'a fds" shows "fds_zeta * f = f * fds_zeta"
f_def: fixes s :: "real"   and m :: "nat" shows "f s m \<equiv> complex_of_real ((x - s) ^ m)"
assms: shows "comp_tfr\<^sub>s\<^sub>s\<^sub>t arity Ana \<Gamma> Pair (SMP0 Ana \<Gamma> (trms_list\<^sub>s\<^sub>s\<^sub>t S @ map pair (setops_list\<^sub>s\<^sub>s\<^sub>t S))) S"
zip_append2: fixes xs :: "'d list"   and ys :: "'e list"   and zs :: "'e list" shows "xs || ys @ zs = (take (length ys) xs || ys) @ (drop (length ys) xs || zs)"
strict_mono_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x < f y) = less x y"
first_countable_topology_ow_axioms: shows "first_countable_topology_ow U\<^sub>2 \<tau>\<^sub>2"
range_lsu_iff: fixes x :: "'b" shows "(x \<in> range \<nu>\<^sup>\<natural>) = (\<nu>\<^sup>\<natural> x = x)"
elims_2: fixes x :: "'e list set"   and xa :: "('e, 'f, 'g) FSM"   and xb :: "('e, 'f, 'g) FSM"   and xc :: "('e \<times> 'f) list"   and xd :: "('e \<times> 'f) list" assumes "minimal_sequence_to_failure_extending x xa xb xc xd" obtains V :: "'e list set" and M1 :: "('e, 'f, 'g) FSM" and M2 :: "('e, 'f, 'g) FSM" and v' :: "('e \<times> 'f) list" and io :: "('e \<times> 'f) list" where "x = V" and "xa = M1" and "xb = M2" and "xc = v'" and "xd = io" and "v' \<in> L\<^sub>i\<^sub>n M1 V \<and> sequence_to_failure M1 M2 (v' @ io) \<and> (\<nexists>io'. \<exists>w'\<in>L\<^sub>i\<^sub>n M1 V. sequence_to_failure M1 M2 (w' @ io') \<and> length io' < length io)"
not_ODord: fixes a :: "'c Order set"   and b :: "'c Order set" assumes "a \<in> ODnums"   and "b \<in> ODnums"   and "a \<sqsubset> b" shows "\<not> b \<sqsubseteq> a"
AbstrLevel1: shows "AbstrLevel level1 = {sA11, sA12, sA21, sA22, sA23, sA31, sA32, sA41, sA42, sA5, sA6, sA71, sA72, sA81, sA82, sA91, sA92, sA93}"
comp_right_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<sqinter> z = x \<sqinter> z \<squnion> y \<sqinter> z"
mat_eq: shows "mat_delete (four_block_mat A B C D) n n = four_block_mat A (Matrix.mat m (n - m) (\<lambda>(i, j). 0::'a)) (Matrix.mat (n - m) m (\<lambda>(i, j). 0::'a)) (1\<^sub>m (n - m))"
wordinterval_empty_set_eq: fixes r :: "'b wordinterval" shows "wordinterval_empty r = (wordinterval_to_set r = {})"
neutr_eq_iff: fixes a :: "'a"   and b :: "'a" shows "(top = inf a b) = (a = top \<and> b = top)"
fun_lub_def: fixes L :: "'g set \<Rightarrow> 'f"   and A :: "('e \<Rightarrow> 'g) set" shows "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
l_one: fixes x :: "nat" assumes "x \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"
fa_intros_3: fixes f :: "'a \<Rightarrow> 'a" shows "Transformers.additive (wp (Apply f))"
add_sum: fixes i :: "nat"   and j :: "nat"   and k :: "nat"   and as :: "'b list" assumes "0 < i"   and "i \<le> j"   and "j < k" shows "SWA.sum as i j + SWA.sum as (Suc j) k = SWA.sum as i k"
cones_map_is_composition: fixes g :: "'a"   and a' :: "'a"   and a :: "'a"   and D :: "'b \<Rightarrow> 'a"   and \<chi> :: "'b \<Rightarrow> 'a" assumes "\<guillemotleft>g : a' \<rightarrow> a\<guillemotright>"   and "Limit.cone J (\<cdot>) D a \<chi>" shows "J_C.MkArr (constant_functor.map J (\<cdot>) a') D ((\<lambda>\<chi>\<in>{\<chi>. Limit.cone J (\<cdot>) D (cod g) \<chi>}. \<lambda>j. if J.arr j then \<chi> j \<cdot> g else null) \<chi>) = J_C.MkArr (constant_functor.map J (\<cdot>) a) D \<chi> \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] \<Delta>.map g"
fun_eq_on_trans: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and h :: "'c \<Rightarrow> 'd" assumes "fun_eq_on f g A"   and "fun_eq_on g h A" shows "fun_eq_on f h A"
utility_less_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "(u x < u y) = x \<prec>[le] y"
axioms_1: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "NFlow c s t f" shows "NPreflow c s t f"
orderE: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" obtains "a = inf a b"
simps_4: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list"   and xs :: "nat list" shows "rec_exec (Cn n f gs) xs = rec_exec f (map (\<lambda>a. rec_exec a xs) gs)"
complete_iff_pointed_semicomplete: fixes A :: "'a set" shows "UNIV-complete A (\<sqsubseteq>) = (semicomplete A (\<sqsubseteq>) \<and> extremed A (\<lambda>x y. y \<sqsubseteq> x))"
Arccos_unique: fixes z :: "complex"   and w :: "complex" assumes "cos z = w"   and "0 < Re z \<and> Re z < pi \<or> Re z = 0 \<and> 0 \<le> Im z \<or> Re z = pi \<and> Im z \<le> 0" shows "Arccos w = z"
eq_resource_on_imp_trace_eq: fixes A :: "'y set"   and res1 :: "('y, 'z) resource"   and res2 :: "('y, 'z) resource" assumes "A \<turnstile>\<^sub>R res1 \<sim> res2" shows "A \<turnstile>\<^sub>R res1 \<approx> res2"
holo2: shows "(\<lambda>z. complex_interval_lebesgue_integral lborel (ereal (exp (- 1))) 1 (f z)) holomorphic_on {z. - 2 < Re z}"
stream_space_def: fixes M :: "'c measure" shows "stream_space M = distr (Pi\<^sub>M UNIV (\<lambda>i. M)) (vimage_algebra (streams (space M)) (!!) (Pi\<^sub>M UNIV (\<lambda>i. M))) to_stream"
insert_eq': shows "insert y (insert x (set (filter (\<lambda>z. z \<noteq> x \<and> z \<noteq> y) vs))) = set vs"
dom_char: fixes f :: "'a" shows "S.dom f = (if S.arr f then C.dom f else C.null)"
lemma2_6_1_list: fixes r :: "('a \<times> 'a) set"   and \<sigma> :: "'a list"   and \<tau> :: "'a list" shows "r \<down>l (\<sigma> @ \<tau>) = r \<down>l \<sigma> \<union> r \<down>l \<tau>"
mult_im_sheaf_def: shows "idim.mult_im_sheaf \<equiv> \<lambda>V. \<cdot>\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
preservation_of_barbs_and_closures_2: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_preserves_binary_pred Rel (\<lambda>a. hasBarb a CWB)" shows "rel_preserves_binary_pred (Rel\<^sup>+) (\<lambda>a. hasBarb a CWB)"
inf_sup_aci_2: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd"   and z :: "'c \<times> 'd" shows "inf_prod (inf_prod x y) z = inf_prod x (inf_prod y z)"
P_k_pre_edges: shows "H.edges_of_walk (P_k_pre @ [y]) \<inter> B = {}"
maps_to_ide: fixes i :: "'i" assumes "i \<in> I" shows "C.ide (D i)"
Diff_iff: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
simps: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and a :: "'e" shows "terminates_onp g a = ((\<exists>s. a = s \<and> g s = Done) \<or> (\<exists>s s'. a = s \<and> g s = Skip s' \<and> terminates_onp g s') \<or> (\<exists>s aa s'. a = s \<and> g s = Yield aa s' \<and> terminates_onp g s'))"
of_nat: fixes n :: "nat" shows "f (of_nat n) = f (0::'a)"
arctan_less_pi4_pos: fixes x :: "real" assumes "x < 1" shows "arctan x < pi / 4"
closest_pair_bf_dist: fixes ps :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "1 < length ps"   and "(c\<^sub>0, c\<^sub>1) = closest_pair_bf ps" shows "sparse (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (set ps)"
rd: shows "\<forall>asset\<in>stocks Mkt. borel_adapt_stoch_proc G (prices Mkt asset)"
simplex_like_def: fixes P :: "'a set" shows "simplex_like P \<equiv> finite P \<and> (\<exists>f A. OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater P f \<and> f ` P = Pow A)"
simps_2: fixes x :: "'b" shows "is_a_constant (Const x) = True"
asym: fixes a :: "nat"   and b :: "nat" assumes "a dvd b \<and> a \<noteq> b"   and "b dvd a \<and> b \<noteq> a" shows "False"
summable_norm: fixes f :: "nat \<Rightarrow> 'a" assumes "summable (\<lambda>n. norm (f n))" shows "norm (suminf f) \<le> (\<Sum>n. norm (f n))"
ring_hom_in_hom_1: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<in> hom R S"
assms_4: shows "G \<psi> \<in> \<G>"
hom_zero: fixes G :: "('e, 'f) ring_scheme"   and H :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "abelian_group_hom G H h" shows "h \<zero>\<^bsub>G\<^esub> = \<zero>\<^bsub>H\<^esub>"
zeroring_def: fixes R :: "('e, 'f) Ring_scheme" shows "zeroring R = (Ring R \<and> carrier R = {\<zero>\<^bsub>R\<^esub>})"
rel_prod_conv: fixes R1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_prod R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
monom_is_polynomial: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and a :: "'c"   and n :: "nat" assumes "ring R"   and "subring K R"   and "a \<in> K - {\<zero>\<^bsub>R\<^esub>}" shows "polynomial\<^bsub>R\<^esub> K (ring.monom R a n)"
finite_fold_deriv: fixes d0 :: "'x \<Rightarrow> 'a \<Rightarrow> ('a, 'k) aformula"   and \<phi> :: "('a, 'k) aformula" assumes "d0 = lderiv0 \<and> lformula \<phi> \<or> d0 = rderiv0" shows "finite {\<langle>fold (deriv d0) xs \<phi>\<rangle> |xs. True}"
update_restr: fixes x :: "'c"   and y :: "'d"   and D :: "'c set"   and al :: "('c \<times> 'd) list" shows "map_of (update x y (restrict D al)) = map_of (update x y (restrict (D - {x}) al))"
apex_def: shows "\<pi>.apex \<equiv> C.dom \<pi>.leg0"
intersection_numbers_def: shows "ins.intersection_numbers \<equiv> {b1 |\<inter>| b2 |b1 b2. b1 \<in># \<A> \<and> b2 \<in># remove1_mset b1 \<A>}"
pred_sexp_t2: fixes M :: "('b, unit) Old_Datatype.node set"   and a :: "('b, unit) Old_Datatype.node set" assumes "M \<in> sexp"   and "a \<in> sexp" shows "(a, Scons M a) \<in> pred_sexp\<^sup>+"
cnj_numeral: fixes x :: "num" shows "Octonions.cnj (numeral x) = numeral x"
ai_def: shows "ai = f2 / (f2 - f2')"
delta_sol: shows "POR.delta = POR_lf (dual POR.delta, POR.delta)"
pmf_of_set: fixes S :: "'b set"   and x :: "'b" assumes "S \<noteq> {}"   and "finite S" shows "pmf (pmf_of_set S) x = indicat_real S x / real (card S)"
structCongWeakBisimulation: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<equiv>\<^sub>s Q" shows "P \<approx> Q"
sfirst_eSuc: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" assumes "\<not> P \<omega>" shows "sfirst P \<omega> = eSuc (sfirst P (stl \<omega>))"
Lower_memD: fixes l :: "'c"   and L :: "('c, 'd) gorder_scheme"   and A :: "'c set"   and x :: "'c" assumes "l \<in> Lower L A"   and "x \<in> A"   and "A \<subseteq> carrier L" shows "l \<sqsubseteq>\<^bsub>L\<^esub> x \<and> l \<in> carrier L"
ww_def: shows "ww \<equiv> \<lambda>j i. if i = 0 \<or> i = Suc j then 1 else 0"
sprod_n_a_def: fixes A :: "('e, 'f) aGroup_scheme"   and n :: "int"   and x :: "'e" shows "n\<triangleright>x\<^bsub>A\<^esub> = (if 0 \<le> n then aSum A (nat n) x else aSum A (nat (- n)) (-\<^sub>a\<^bsub>A\<^esub> x))"