hcju/isabelleps huggingface dataset:
is_float_ivl_err_1: fixes x :: "real interval"   and a :: "real"   and b :: "real pdevs"   and c :: "real" assumes "ivl_err x = ((a, b), c)"   and "is_float (lower x)"   and "is_float (upper x)" shows "is_float a"
borel_measurable_continuous_on_restrict: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on A f" shows "f \<in> borel_measurable (restrict_space borel A)"
fps_nth_induct: fixes y :: "nat \<Rightarrow> 'a"   and P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "y \<in> {f. True}"   and "\<And>x. P (fps_nth x)" shows "P y"
comm_semiring_0_cancel_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "comm_semiring_0_cancel_ow U plus minus zero times \<equiv> comm_semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero"
arr_iff_in_hom: fixes f :: "'a" shows "V.arr f = V.in_hom f (V.dom f) (V.cod f)"
constructive_security: shows "constructive_security real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common bound lossless w"
t_4: assumes "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> set TI"
ideal_prod_ideal: fixes I :: "'a set"   and J :: "'a set" assumes "ideal R I"   and "ideal R J" shows "ideal R (I \<diamondsuit>\<^sub>r J)"
dvd_Lcm_fin: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "a dvd Lcm\<^sub>f\<^sub>i\<^sub>n A"
trnl\<^sub>\<eta>_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b"   and v :: "'b"   and \<mu> :: "'b" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "adjunction_in_bicategory.trnl\<^sub>\<eta> V H \<a> \<i> src trg f g \<eta> v \<mu> \<equiv> V (H g \<mu>) (V (\<a> g f v) (V (H \<eta> v) (category.inv V (bicategory.lunit V H \<a> \<i> src trg v))))"
pw: shows "int (\<Prod>i\<in>#ps + {#p#}. i) = int p * int (\<Prod>i\<in>#ps. i) \<and> odd (int p) \<and> odd (int (\<Prod>i\<in>#ps. i))"
zero_closed: shows "\<zero>\<^bsub>K\<^esub> \<in> carrier K"
fst_set_before: shows "\<forall>t\<in>fst ` set (linearize data_in). \<forall>t'\<in>fst ` set move. t \<le> t'"
cartesian_product_margin1: fixes a :: "'a"   and A :: "'a set"   and B :: "'b set" assumes "a \<in> A" shows "{p \<in> A \<times> B. fst p = a} = {a} \<times> B"
distrib_1: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x \<union># (y \<inter># z) = x \<union># y \<inter># (x \<union># z)"
star_unfold_left_Nil: fixes A :: "'a list set" shows "star A = (A - {[]}) @@ star A \<union> {[]}"
assms_2: fixes cf :: "('test, 'atom) com \<times> 'state" shows "phi cf cf"
cmd: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set" assumes "class.arelations meets \<I> e m b ov d s f" shows "m O d \<subseteq> s \<union> ov \<union> d"
topspace_prod_topology: fixes X :: "'d topology"   and Y :: "'e topology" shows "topspace (prod_topology X Y) = topspace X \<times> topspace Y"
assms: shows "a * 2 = 1 + (2 * b - 1)"
simps_1: fixes x :: "value"   and s :: "'a \<Rightarrow> value option" shows "aval (L x) s = Some x"
strict: fixes a :: "'a"   and b :: "'a" shows "(a < b) = (a \<le> b \<and> \<not> b \<le> a)"
lstrict_prefix_code_4: fixes x :: "'b"   and xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" shows "lstrict_prefix (LCons x xs) (LCons y ys) = (x = y \<and> lstrict_prefix xs ys)"
UP_img_hfset_ide_to_hf: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "UP ` hfset (ide_to_hf a) = local.set a"
permutation_permutes: fixes p :: "'e \<Rightarrow> 'e" shows "permutation p = (\<exists>S. finite S \<and> p permutes S)"
Quotient3_uprod: shows "Quotient3 eq_upair abs_uprod rep_uprod"
prems: shows "\<lless>[], e\<^sub>1\<ggreater> I\<rightarrow>i + 1 \<lless>[], u'\<ggreater>"
open_UNIV: fixes "open" :: "'e set \<Rightarrow> bool" assumes "class.topological_space open" shows "open UNIV"
is_expansion_ms_def': fixes F :: "'a ms" shows "is_expansion F = (case F of MS xs f \<Rightarrow> is_expansion_aux xs f)"
AssumpProof_2: shows "\<forall>index<length xa. xa ! index = xb ! index"
noFault_startD': fixes t :: "('a, 'b) xstate"   and f :: "'b"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'c, 'b) com option"   and c :: "('a, 'c, 'b) com"   and s :: "('a, 'b) xstate" assumes "t \<noteq> Fault f"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "s \<noteq> Fault f"
dvd_ideal_generated_singleton: fixes a :: "'a"   and b :: "'a" assumes "ideal_generated {a} \<subseteq> ideal_generated {b}" shows "b dvd a"
1: shows "dom\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>"
dedekind_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y \<sqinter> z \<le> x * (y \<sqinter> x\<^sup>T * z)"
orderE: fixes b :: "'b upred"   and a :: "'b upred" assumes "a \<sqsubseteq> b" obtains "a = (a \<or> b)"
rep_eq: fixes x :: "Arity" shows "Rep_Arity (inc_Arity x) = Suc (Rep_Arity x)"
inputPermSubject: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and Xs :: "name set"   and Ys :: "name set" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "set p \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* P"   and "Ys \<sharp>* P"   and "Xs \<sharp>* \<Psi>"   and "Ys \<sharp>* \<Psi>" shows "\<Psi> \<rhd> P \<longmapsto> (p \<bullet> M)\<lparr>N\<rparr> \<prec> P'"
eq_on_trans': fixes r :: "'b \<Rightarrow> 'c"   and s :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and t :: "'b \<Rightarrow> 'c"   and X' :: "'b set" assumes "r = s on X"   and "s = t on X'" shows "r = t on X \<inter> X'"
deepL_list: fixes n :: "nat"   and m :: "('a, 'b) FingerTreeStruc"   and sf :: "('a, 'b) Digit"   and pr :: "('a, 'b) Node list" assumes "FingerTreeStruc.is_leveln_ftree (Suc n) m \<and> FingerTreeStruc.is_measured_ftree m"   and "FingerTreeStruc.is_leveln_digit n sf \<and> FingerTreeStruc.is_measured_digit sf"   and "\<forall>x\<in>set pr. (FingerTreeStruc.is_measured_node x \<and> FingerTreeStruc.is_leveln_node n x) \<and> length pr \<le> 4" shows "FingerTreeStruc.toList (FingerTreeStruc.deepL pr m sf) = FingerTreeStruc.nlistToList pr @ FingerTreeStruc.toList m @ FingerTreeStruc.digitToList sf"
xeq1_refl: fixes x :: "Value"   and s :: "Store" shows "s \<equiv>[x] s"
prv_eqv_eqv_neg_prv_fls2: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv (eqv \<phi> \<chi>)"   and "bprv (eqv \<chi> (neg \<phi>))" shows "bprv fls"
prod_ballI: fixes A :: "('b \<times> 'c) set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. (a, b) \<in> A \<Longrightarrow> P a b" shows "\<forall>(a, b)\<in>A. P a b"
strict_trans1: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "order.greater_eq a b"   and "order.greater b c" shows "order.greater a c"
min_elems6: fixes s :: "'c event list"   and P :: "'c process"   and c :: "'c event"   and S :: "'c process" assumes "s \<notin> D P"   and "s @ [c] \<in> D P"   and "P \<sqsubseteq> S" shows "(s @ [c], {}) \<in> F S"
simps_3: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "Formula.sat \<sigma> V v i (formula.Eq t1 t2) = (Formula.eval_trm v t1 = Formula.eval_trm v t2)"
fimg: shows "f ` {0..<n} = {0..<Suc n} - {j}"
permute_finite: fixes p :: "perm"   and X :: "'a set" shows "finite (p \<bullet> X) = finite X"
OICI: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and p :: "'c" assumes "(\<sigma>, p) \<in> OI" shows "(\<sigma> i, p) \<in> CI"
eqvt_def: fixes f :: "'d" shows "eqvt f \<equiv> \<forall>p. p \<bullet> f = f"
nres_monad1: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b nres" shows "RETURN x \<bind> f = f x"
l4_16R3: 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"   and D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.FSC Bet Cong A B C D A' B' C' D'"   and "A \<noteq> B"   and "Bet C A B" shows "Cong C D C' D'"
measurable_sets: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and A :: "'b measure"   and S :: "'b set" assumes "f \<in> M \<rightarrow>\<^sub>M A"   and "S \<in> sets A" shows "f -` S \<inter> space M \<in> sets M"
app_vimage_def: fixes r :: "V"   and A :: "V" shows "r `\<^sub>\<circ> A = \<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
H: shows "list_all2 (eintP \<circ> Guard) (arOf f) (map singl al)"
has_integral_implies_lebesgue_measurable_real: fixes f :: "'a \<Rightarrow> real"   and I :: "real"   and \<Omega> :: "'a set" assumes "(f has_integral I) \<Omega>" shows "(\<lambda>x. f x * indicat_real \<Omega> x) \<in> borel_measurable lebesgue"
drop_bit_rec: 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"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n a = (if n = 0 then a else drop_bit (n - 1) (divide a (numeral.numeral one plus (num.Bit0 num.One))))"
equivp_reflp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "equivp R" shows "R x x"
less_convert1: fixes a :: "'c"   and c :: "'c"   and b :: "'c" assumes "a = c"   and "a < b" shows "c < b"
div_free_DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P: fixes A :: "'a set" shows "D (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) = {}"
Gromov_extension_onto': fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd" assumes "lambda C -quasi_isometry_between UNIV UNIV f" shows "Gromov_extension f ` Gromov_boundary = Gromov_boundary"
assms_2: shows "((Suc 0, vs), 0, vs') \<in> (step M)\<^sup>*"
prems_2: shows "rtrancl_path r x ((a # as) @ y # ys) z"
cont_forall: shows "\<forall>b\<in>basis. continuous_on (\<Union>(k, \<gamma>)\<in>one_chain. path_image \<gamma>) (\<lambda>x. F x \<bullet> b)"
subst_typ'_incr_bv: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and inc :: "nat"   and lev :: "nat"   and t :: "Core.term" shows "subst_typ' insts (incr_bv inc lev t) = incr_bv inc lev (subst_typ' insts t)"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "min b (min a c) = min a (min b c)"
HInfinite_inverse_Infinitesimal: fixes x :: "'a star" assumes "x \<in> HInfinite" shows "inverse x \<in> Infinitesimal"
when_def: fixes a :: "'a"   and P :: "bool" shows "(a when P) = (if P then a else (0::'a))"
mono_on_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "Fun.mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
num_denom_bound: shows "\<bar>num\<bar> \<le> int (N ^ (2 * m) * 2 ^ m * m) \<and> \<bar>denom\<bar> \<le> int (N ^ m)"
is_SNF_HMA_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and R :: "(('b, 'd) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec \<times> (('b, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec \<times> (('b, 'f) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "is_SNF_HMA A R = (case R of (P, S, Q) \<Rightarrow> invertible P \<and> invertible Q \<and> Smith_normal_form S \<and> S = P ** A ** Q)"
f_shrink_take: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" shows "xs \<down> n \<div> k = xs \<div> k \<down> (n div k)"
directedI: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. r x z \<and> r y z" shows "directed A r"
transfer_prover_start: fixes x :: "'d"   and x' :: "'d"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e" assumes "x = x'"   and "Transfer.Rel R x' y" shows "Transfer.Rel R x y"
\<Sigma>_protocol: shows "chaum_ped_sigma.\<Sigma>_protocol"
BIT_competitive2: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list" assumes "BIT_Off acts qs init"   and "init \<noteq> []"   and "\<forall>i<length qs. qs ! i \<in> set init" shows "T_on_rand' BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs \<le> (7 / 4 - 3 / real (4 * length init)) * real (T init qs acts)"
act_def: fixes I :: "'e \<Rightarrow> bool" shows "act I \<equiv> \<lambda>_ _. I"
maximal_imp_preferred: fixes R :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and x :: "'alt" assumes "total_preorder_on alts R"   and "x \<in> alts" shows "Max_wrt R \<subseteq> preferred_alts R x"
M: shows "index key x n mi ma = Max {i. i < n \<and> offs (enum xs index key n mi ma) 0 ! i \<le> j \<and> 0 < offs_num n xs index key mi ma i}"
a_create_element_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and document_ptr :: "(_) document_ptr"   and tag :: "char list" shows "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_element get_disconnected_nodes set_disconnected_nodes set_tag_name document_ptr tag = Heap_Error_Monad.bind new_element (\<lambda>new_element_ptr. Heap_Error_Monad.bind (set_tag_name new_element_ptr tag) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes document_ptr (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr # disc_nodes)) (\<lambda>_. return new_element_ptr))))"
map_of_eq_None_iff: fixes xys :: "('e \<times> 'd) list"   and x :: "'e" shows "(map_of xys x = None) = (x \<notin> fst ` set xys)"
parts_crypt_key: fixes s :: "(agent \<times> msg) set"   and K :: "key"   and K' :: "key" assumes "s\<^sub>0 \<Turnstile> s"   and "Crypt K (Key K') \<in> parts (used s)"   and "K' \<notin> range PriK \<union> range PubK" shows "{Key K', Key K} \<subseteq> spied s"
equivalent_objects_def: fixes a :: "'b"   and b :: "'b" shows "D.equivalent_objects a b \<equiv> \<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright> \<and> D.equivalence_map f"
distinct_5: fixes a :: "'d"   and boundOutput :: "('d, 'e, 'f) boundOutput"   and psi' :: "('d, 'e, 'f) psi" shows "ROut a boundOutput \<noteq> RTau psi'"
transfer_operations_rl_11: shows "((RETURN \<circ>\<circ>\<circ> width_appr) ops optns, width_spec) \<in> appr_rell \<rightarrow> \<langle>rnv_rel\<rangle>nres_rel"
assms_4: assumes "p \<in> dterms \<Gamma> p2" shows "P"
nn_integral_indicator_neq_top: fixes f :: "'b \<Rightarrow> 'd"   and A :: "'d set"   and p :: "'b spmf" assumes "f -` A \<in> sets (measure_spmf p)" shows "\<integral>\<^sup>+ x. indicator A (f x) \<partial>measure_spmf p \<noteq> \<top>"
eq1: shows "F x * of_real (ln x) = dirichlet_prod' (\<lambda>n. \<Sum>d | d dvd n. moebius_mu d) (\<lambda>x. F x * of_real (ln x)) x"
r_cancel_one': fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier H"   and "a \<in> carrier H" shows "(x = a \<oplus>\<^bsub>H\<^esub> x) = (a = \<zero>\<^bsub>H\<^esub>)"
Sbis_coind: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "theta \<subseteq> Sretr (theta \<union> Sbis)" shows "theta \<subseteq> Sbis"
joinableI: fixes x :: "'b"   and z :: "'c"   and R :: "('b \<times> 'c) set"   and y :: "'b" assumes "(x, z) \<in> R"   and "(y, z) \<in> R" shows "(x, y) \<in> joinable R"
Max_eq_image_if: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "finite S"   and "finite T"   and "\<forall>x\<in>S. \<exists>y\<in>T. f x \<le> g y"   and "\<forall>x\<in>T. \<exists>y\<in>S. g x \<le> f y" shows "Max (f ` S) = Max (g ` T)"
simps_1: fixes shared :: "bool"   and M :: "nat" shows "count_weights shared (Input M) = 0"
conj_cancel_right: fixes x :: "'c" shows "x AND NOT x = (0::'c)"
ord_inj_def: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e" shows "ord_inj D E f = (f \<in> extensional (carrier D) \<and> f \<in> carrier D \<rightarrow> carrier E \<and> inj_on f (carrier D) \<and> (\<forall>a\<in>carrier D. \<forall>b\<in>carrier D. a \<prec>\<^bsub>D\<^esub> b = f a \<prec>\<^bsub>E\<^esub> f b))"
a_meet_d: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<sqinter> d_aux a b c = a \<sqinter> b \<squnion> c \<sqinter> a"
ThrowRed: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>throw e,s\<rangle> \<rightarrow> \<langle>throw e',s'\<rangle>"
independent_rotate1_aux: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and u :: "'a"   and Us :: "'a list"   and Vs :: "'a list" assumes "ring R"   and "subfield K R"   and "ring.independent R K (u # Us @ Vs)" shows "ring.independent R K ((Us @ [u]) @ Vs)"
AAss_\<tau>red1r_xt1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr expr1"   and xs :: "'addr val list"   and a' :: "'addr expr1"   and xs' :: "'addr val list"   and i :: "'addr expr1"   and e :: "'addr expr1" assumes "\<tau>red1gr uf P t h (a, xs) (a', xs')" shows "\<tau>red1gr uf P t h (a\<lfloor>i\<rceil> := e, xs) (a'\<lfloor>i\<rceil> := e, xs')"
num: shows "rat_of_int \<bar>num\<bar> \<le> rat_of_int (d fs i * int N)"
absorb_iff2: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (sup a b = b)"
no_volatile_Read\<^sub>s\<^sub>bs_conv: fixes sb :: "'b memref list" shows "(outstanding_refs is_volatile_Read\<^sub>s\<^sub>b sb = {}) = (\<forall>r\<in>set sb. \<forall>v' t' a'. r \<noteq> Read\<^sub>s\<^sub>b True a' t' v')"
scale_eq_0_iff: fixes a :: "'a"   and x :: "'a poly" shows "(smult a x = 0) = (a = (0::'a) \<or> x = 0)"
m1_trans_def: shows "m1_trans \<equiv> \<Union>A B Ra Rb Rs Na Kab Ts Ta T. (m1_step1 Ra A B Na \<union> m1_step2 Rb A B \<union> m1_step3 Rs A B Kab Na Ts \<union> m1_step4 Ra A B Na Kab Ts Ta \<union> m1_step5 Rb A B Kab Ts Ta \<union> m1_step6 Ra A B Na Kab Ts Ta \<union> m1_leak Rs A B Na Ts \<union> m1_tick T \<union> m1_purge A)\<^sup>="
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and sample :: "'c pmf \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_prob return bind sample" shows "monad return bind"
IS: fixes i :: "nat"   and \<sigma> :: "'a" assumes "lb \<le> i"   and "i < ub"   and "I i \<sigma>"   and "c \<sigma>" shows "f i \<sigma> \<le> SPEC (I (i + 1))"
ttree_restr_tree_restr: fixes S :: "'b set"   and S' :: "'b set"   and t :: "'b ttree" shows "ttree_restr S (ttree_restr S' t) = ttree_restr (S' \<inter> S) t"
sigma_finite_measure_axioms: shows "sigma_finite_measure (density M h)"
enumerate_eq_zip: fixes n :: "nat"   and xs :: "'a list" shows "List.enumerate n xs = zip [n..<n + length xs] xs"
zero_eq_add_iff_both_eq_0: fixes x :: "'a multiset"   and y :: "'a multiset" shows "(\<Empt> = x + y) = (x = \<Empt> \<and> y = \<Empt>)"
uf_graph_rel_def: shows "uf_graph_rel \<equiv> br snd uf_graph_invar"
rep_eq: shows "pmf local.binomial_pmf = (\<lambda>k. real (n choose k) * p ^ k * (1 - p) ^ (n - k))"
group_disjoint_sum_def: fixes G :: "('e, 'f) monoid_scheme"   and A :: "'e set"   and B :: "'e set" shows "group_disjoint_sum G A B \<equiv> Group.group G \<and> subgroup A G \<and> subgroup B G"
addSubsetSetCom: fixes a :: "'b set"   and b :: "'b set"   and As :: "'b set set" shows "addSubsetSet a (addSubsetSet b As) = addSubsetSet b (addSubsetSet a As)"
vars_\<C>_def: fixes X :: "'Var set" shows "vars_\<C> X \<equiv> \<Union> (\<C>_vars ` X)"
graph_axioms: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and m :: "'a" shows "graph (b_graph.to_gbg_ext (degeneralize_ext ma) m)"
subset_of_functions_of: fixes A :: "'b set"   and X :: "'c set" assumes "finite A"   and "finite X"   and "card A = card X" shows "subset_of A (functions_of A X) = X"
comp_ide_self: fixes a :: "'b" assumes "B.ide a" shows "B a a = a"
simps_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "page_t \<Rightarrow> 'a" shows "(case WAIT of PREP \<Rightarrow> f1 | WAIT \<Rightarrow> f2 | BUF x \<Rightarrow> f3 x) = f2"
handshake_phase_invD: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state"   and m :: "'e" assumes "handshake_phase_inv s" shows "(sys_ghost_hs_in_sync m s, sys_hs_type s, sys_ghost_hs_phase s, ghost_hs_phase (s (mutator m))) \<in> hp_step_rel \<and> (sys_hs_pending m s \<longrightarrow> \<not> sys_ghost_hs_in_sync m s)"
lNactLe: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n"
pp_def: shows "pp = (\<lambda>x. poly p x / x ^ degree p)"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "seq (trg g) (trg f)"
fmapU_llist_def: shows "fmapU \<equiv> llist_map"
msg_fresh_inc_sn: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg" assumes "msg_fresh \<sigma> m" shows "rreq_rrep_sn m"
axioms: fixes A :: "V"   and f :: "V" assumes "binop A f" shows "nop A (2\<^sub>\<nat>) f"
max_tstpE: fixes tstp :: "nat + nat"   and tstp' :: "nat + nat" assumes "isl tstp = isl tstp'" obtains "max_tstp tstp tstp' = tstp"   | "max_tstp tstp tstp' = tstp'"
mult_1s_1: fixes a :: "'b" shows "Numeral1 * a = a"
sorted_upper: shows "sorted_sublist_map R h xs' (Suc p) hi"
simps: fixes a1 :: "('c, 'd) tm"   and a2 :: "('c, 'd) tm" shows "(a1 \<rightarrow>\<^sub>e\<^sub>m\<^sub>b a2) = ((\<exists>t1 t2. a1 = App t1 t2 \<and> a2 = t1) \<or> (\<exists>t1 t2. a1 = App t1 t2 \<and> a2 = t2) \<or> (\<exists>t s u. a1 = App t u \<and> a2 = App s u \<and> t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s) \<or> (\<exists>t s u. a1 = App u t \<and> a2 = App u s \<and> t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s))"
add_ideals: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "ideal (I <+> J) R"
map_id0: shows "map_rose_tree\<^sub>h id = id"
assms_2: shows "inverse (q $ 0) \<noteq> (0::'a)"