hcju/isabelleps huggingface dataset:
yields_isomorphic_representation_2: shows "local.iso (T0.trnr\<^sub>\<epsilon> r \<rho>)"
h1_is_pr: shows "(\<lambda>x y z. h x y) \<in> PrimRec3"
TC_equiv_tranclp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "TC R a b = R\<^sup>+\<^sup>+ a b"
v_in_some_subst: shows "\<theta> v \<noteq> Var v \<or> \<sigma> v \<noteq> Var v"
distinct_ordered_keys: fixes m :: "('a, 'b) mapping" shows "distinct (Mapping.ordered_keys m)"
path_cons: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and v :: "'c" assumes "Graph G"   and "Graph.walk G xs \<and> distinct xs"   and "xs \<noteq> []"   and "v \<rightarrow>\<^bsub>G\<^esub> hd xs"   and "v \<notin> set xs" shows "Graph.walk G (v # xs) \<and> distinct (v # xs)"
not_outside_3: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and x :: "'b"   and y :: "'b" assumes "pref_profile_wf agents alts R"   and "R i x y" shows "y \<in> alts"
intro: fixes preprocess :: "('a \<times> 'b ns_constraint) list \<Rightarrow> (nat \<times> linear_poly) list \<times> ('a \<times> 'b atom) list \<times> ((nat, 'b) mapping \<Rightarrow> (nat, 'b) mapping) \<times> 'a list"   and assert_all :: "(nat \<times> linear_poly) list \<Rightarrow> ('a \<times> 'b atom) list \<Rightarrow> 'a list + (nat, 'b) mapping" assumes "Preprocess preprocess"   and "AssertAll assert_all" shows "Solve_exec_ns' preprocess assert_all"
foldr_Cons: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "foldr f (x # xs) = (f x \<circ>\<circ>\<circ> foldr) f xs"
unilateral_substitutes_on_doctor_separable_on: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" assumes "unilateral_substitutes_on A f"   and "irc_on A f"   and "\<forall>B\<subseteq>A. inj_on Xd (f B)"   and "f_range_on A f" shows "doctor_separable_on A f"
grounding_of_state_Liminf_state_subseteq: shows "grounding_of_clss (N_of_state (Liminf_state Sts) \<union> P_of_state (Liminf_state Sts) \<union> Q_of_state (Liminf_state Sts)) \<subseteq> Liminf_llist (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts)"
Eq\<kappa>S: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" shows "[x \<^bold>=\<^sub>\<kappa> y in v] = (\<exists>o\<^sub>1 o\<^sub>2. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2)"
oZero_least: fixes x :: "ordinal" shows "oZero \<le> x"
assms: assumes "c \<noteq> (0::'b)" shows "f abs_summable_on A"
attach_parallel2: fixes conv1 :: "('i, 'k, 'm, 'o) converter"   and conv2 :: "('j, 'l, 'n, 'p) converter"   and res1 :: "('m, 'o) resource"   and res2 :: "('n, 'p) resource" shows "conv1 |\<^sub>= conv2 \<rhd> res1 \<parallel> res2 = (conv1 \<rhd> res1) \<parallel> (conv2 \<rhd> res2)"
Sigma_mset_plus_distrib2: fixes A :: "'a multiset"   and B :: "'a \<Rightarrow> 'b multiset"   and C :: "'a \<Rightarrow> 'b multiset" shows "(SIGMAMSET i\<in>#A. B i + C i) = Sigma_mset A B + Sigma_mset A C"
assms_3: fixes p :: "trm"   and q :: "trm"   and k :: "stack" assumes "\<And>p'. p \<mapsto> p' \<Longrightarrow> P p' q k"   and "\<And>q' k. q \<mapsto> q' \<Longrightarrow> P p q' k"   and "\<And>k'. |k'| < |k| \<Longrightarrow> P p q k'" shows "P p q k"
take_nemp_len: fixes u :: "'b list"   and r :: "'c list" assumes "u \<noteq> \<epsilon>"   and "r \<noteq> \<epsilon>" shows "take \<^bold>|r\<^bold>| u \<noteq> \<epsilon>"
ladder_stepdown_j: fixes L :: "(nat \<times> nat \<times> nat) list"   and L' :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" assumes "1 < length L"   and "L' = ladder_stepdown L"   and "index < length L'" shows "ladder_j L' index = ladder_j L (Suc index)"
to_tree_order_ptrs_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to" shows "ptr' |\<in>| object_ptr_kinds h"
map_respect_cancel1: shows "((=) ===> cancel1 ===> equivclp cancel1) map map"
bmh: shows "h \<in> borel_measurable (lebesgue_on S)"
ffb: fixes x :: "'a" assumes "x \<in> B" shows "f x = fb x"
truncate_down_log2_eqI: fixes x :: "real"   and y :: "real"   and p :: "nat" assumes "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"   and "\<lfloor>x * 2 powr real_of_int (int p - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>)\<rfloor> = \<lfloor>y * 2 powr real_of_int (int p - \<lfloor>log 2 \<bar>x\<bar>\<rfloor>)\<rfloor>" shows "truncate_down p x = truncate_down p y"
iso_tuple_update_accessor_eq_assist_def: fixes upd :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and ac :: "'b \<Rightarrow> 'a"   and v :: "'b"   and f :: "'a \<Rightarrow> 'a"   and v' :: "'b"   and x :: "'a" shows "iso_tuple_update_accessor_eq_assist upd ac v f v' x = (upd f v = v' \<and> ac v = x \<and> iso_tuple_update_accessor_cong_assist upd ac)"
sorted_tl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (tl xs)"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> state option) \<times> 'a" shows "initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> toy_openproc_par_qmsg.someinit i | Some s \<Rightarrow> s, snd \<sigma>)"
surj_fun_eq: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set"   and g1 :: "'a \<Rightarrow> 'c"   and g2 :: "'a \<Rightarrow> 'c" assumes "f ` X = UNIV"   and "\<forall>x\<in>X. (g1 \<circ> f) x = (g2 \<circ> f) x" shows "g1 = g2"
group_l_invI: fixes G :: "('b, 'c) monoid_scheme" assumes "Group.monoid G"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "Group.group G"
ffUnion_subset_elem: fixes x :: "'b fset"   and X :: "'b fset fset" assumes "x |\<in>| X" shows "x |\<subseteq>| ffUnion X"
nat_div_eq_Suc_0_iff: fixes n :: "nat"   and m :: "nat" shows "(n div m = Suc 0) = (m \<le> n \<and> n < 2 * m)"
mC_C_trans: fixes theta1 :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set"   and theta2 :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and e :: "('test, 'atom, 'choice) cmd"   and u :: "'state"   and G :: "nat set \<Rightarrow> nat set" assumes "mC_C theta1 c d s t P F"   and "mC_C theta2 d e t u (F ` P) G" shows "mC_C (theta1 O theta2) c e s u P (G \<circ> F)"
PP_Pinp: fixes t :: "'trm" assumes "t \<in> trm" shows "PP t = instInp Pinp t"
distinct_127: fixes x6 :: "char list"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "Var x6 \<noteq> Call x101 x102 x103 x104"
not_primepow_Suc_0_nat: shows "\<not> primepow (Suc 0)"
constructive_security_aux_obsf: fixes real_resource :: "('m + 'n, 'o + 'p) resource"   and ideal_resource :: "('q + 'n, 'r + 'p) resource"   and sim :: "('m, 'o, 'q, 'r) converter"   and \<I>_real :: "('m, 'o) \<I>"   and \<I>_ideal :: "('q, 'r) \<I>"   and \<I>_common :: "('n, 'p) \<I>"   and \<A> :: "('m + 'n, 'o + 'p) distinguisher_obsf"   and adv :: "real" assumes "constructive_security_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common \<A> adv" shows "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv"
cUNION_cempty: fixes f :: "'e \<Rightarrow> 'd cset" shows "cUnion (cimage f cempty) = cempty"
o_id: fixes f :: "'c \<Rightarrow> 'd" shows "f \<circ> id = f"
assms: shows "\<forall>i\<le>n. s ! (x + i) = []"
lossless_I: fixes I :: "('a \<times> 'b) spmf"   and domain :: "'a \<Rightarrow> 'c set"   and range :: "'a \<Rightarrow> 'c set"   and F :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and F\<^sub>i\<^sub>n\<^sub>v :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "etp I domain range F F\<^sub>i\<^sub>n\<^sub>v" shows "lossless_spmf I"
af_subsequence_R: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and n :: "nat" shows "af (\<phi> R\<^sub>n \<psi>) (prefix (Suc n) w) = af \<psi> (prefix (Suc n) w) and\<^sub>n af \<phi> (prefix (Suc n) w) or\<^sub>n af (\<phi> R\<^sub>n \<psi>) (w [1 \<rightarrow> Suc n])"
characteriseLast: fixes C :: "sequent"   and m :: "nat"   and R :: "(sequent list \<times> sequent) set" assumes "(C, m + 1) \<in> derivable R" shows "\<exists>Ps. Ps \<noteq> [] \<and> (Ps, C) \<in> R \<and> (\<forall>p\<in>set Ps. \<exists>n\<le>m. (p, n) \<in> derivable R)"
special_soundness_def: shows "\<Sigma>0.special_soundness \<equiv> \<forall>h e e' a z z'. h \<in> valid_pub0 \<longrightarrow> e \<in> carrier L \<longrightarrow> e' \<in> carrier L \<longrightarrow> e \<noteq> e' \<longrightarrow> check0 h a e z \<longrightarrow> check0 h a e' z' \<longrightarrow> lossless_spmf (\<A>ss0 h (a, e, z) (a, e', z')) \<and> (\<forall>w'\<in>set_spmf (\<A>ss0 h (a, e, z) (a, e', z')). (h, w') \<in> Rel0)"
minus_divide_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "uminus (divide a b) = divide a (uminus b)"
mult_one_left: fixes P :: "real mpoly" shows "Const 1 * P = P"
slow_median_correct: fixes xs :: "'a list" shows "slow_median xs = median xs"
Extend_def: fixes A :: "'c set"   and S :: "(('b, 'c) fm list \<times> 'c) set"   and f :: "nat \<Rightarrow> ('b, 'c) fm list \<times> 'c" shows "Extend A S f \<equiv> \<Union> (range (extend A S f))"
not_le: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "(\<not> less_eq x y) = less y x"
assms_5: shows "A\<^sub>G \<sharp>* \<Psi>\<^sub>F"
Z': fixes n :: "nat" shows "Z' n \<subseteq> Z n"
fbb_ifbb_galois: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and f :: "'c \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "(\<phi> = bb\<^sub>\<F> f) = (f = bb\<^sup>-\<^sub>\<F> \<phi>)"
CCfix_def: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "CoCallArityAnalysis.CCfix cccExp \<Gamma> = (\<Lambda> aeG. \<mu> x. CoCallAnalysis.ccBindsExtra (CoCallArityAnalysis.CCexp cccExp) \<Gamma>\<cdot>(fst aeG, x) \<squnion> snd aeG)"
a'_def: shows "a' = (\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i)"
prems_1: shows "\<forall>x\<in>set (xs @ [a]). dim_vec x = n"
enc_alt: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and x :: "('a \<times> bool list) list" assumes "wf_interp w I" shows "(x \<in> enc (w, I)) = (x @- sconst (any, replicate (length I) False) = stream_enc (w, I))"
big_by_small: fixes 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"   and 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)" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"
cSup_upper: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> X"   and "bdd_above X" shows "x \<le> Sup X"
a_r_coset_def: fixes G :: "('c, 'd) ring_scheme" shows "(+>\<^bsub>G\<^esub>) = (#>\<^bsub>add_monoid G\<^esub>)"
solvable_img_imp_solvable: fixes K :: "'a set" assumes "subgroup K G"   and "inj_on h K"   and "solvable_seq H (h ` K)" shows "solvable_seq G K"
wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_code: fixes M :: "('fun, 'a) Term.term set" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M = wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s' arity M"
iT_Minus_singleton: fixes k :: "nat"   and a :: "nat" shows "k \<ominus> {a} = (if a \<le> k then {k - a} else {})"
Vr_one: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "1\<^sub>r \<in> carrier (Vr K v)"
diff_eq_0_ennreal: fixes a :: "ennreal"   and b :: "ennreal" assumes "a < \<top>"   and "a \<le> b" shows "a - b = 0"
prems_4: shows "\<forall>k k'. cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k' \<and> l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k \<longrightarrow> \<pi> (Suc k) = \<pi>' (Suc k')"
Gramian_matrix_def: fixes G :: "'c vec list"   and k :: "nat" shows "Gramian_matrix G k = (let M = mat k n (\<lambda>(i, j). G ! i $ j) in M * M\<^sup>T)"
parts_insert_subset_Un: fixes X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "X \<in> G" shows "parts (insert X H) \<subseteq> parts G \<union> parts H"
Sbis_coind: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and theta :: "(('c, 'd, 'a) cmd \<times> ('c, 'd, 'a) cmd) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "theta \<subseteq> PL_Indis.Sretr aval tval cval indis (theta \<union> PL_Indis.Sbis aval tval cval indis)" shows "theta \<subseteq> PL_Indis.Sbis aval tval cval indis"
EE: fixes l :: "'b"   and r :: "'b" assumes "l \<in> S_Min_Le"   and "r \<in> S_Max_Le" shows "l \<le> r"
complex_interval_tendsto_def: fixes C :: "nat \<Rightarrow> complex_interval"   and c :: "complex" shows "C \<longlonglongrightarrow>\<^sub>c c \<equiv> (Re_interval \<circ> C) \<longlonglongrightarrow>\<^sub>i Re c \<and> (Im_interval \<circ> C) \<longlonglongrightarrow>\<^sub>i Im c"
multrow_i_def: fixes ops :: "'b arith_ops_record" shows "multrow_i ops \<equiv> mat_multrow_gen (arith_ops_record.times ops)"
sup_commute: fixes sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
nxt_Collapse_resp: fixes x :: "'a" shows "(\<lambda>q. eq_right_lang `` {nxt M q x}) respects eq_right_lang"
assms_7: shows "qf1 x2 y2 = 0"
assms_4: shows "R = Ax \<union> R1 \<union> R2 \<union> R3"
insort_wrt_def: fixes R :: "('b \<times> 'b) set"   and xs :: "'b list" shows "insort_wrt R xs = foldr (insert_wrt R) xs []"
lookup_pair_map_val_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and k :: "'a" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "f k (0::'b) = (0::'c)" shows "comparator.lookup_pair comp (map_val_pair f xs) k = f k (comparator.lookup_pair comp xs k)"
fresh_star_eqvt: fixes p :: "perm"   and as :: "atom set"   and x :: "'e" shows "p \<bullet> as \<sharp>* x = (p \<bullet> as) \<sharp>* (p \<bullet> x)"
kcomp_idr: fixes f :: "'a \<Rightarrow> 'b set" shows "f \<circ>\<^sub>K \<eta> = f"
rat_plus_code: fixes p :: "rat"   and q :: "rat" shows "quotient_of (p + q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q in Rat.normalize (a * d + b * c, c * d))"
ln_msetprod: fixes I :: "nat multiset" assumes "\<And>x. x \<in># I \<Longrightarrow> 0 < x" shows "(\<Sum>p\<in>#I. ln (real p)) = ln (\<Prod>\<^sub># (image_mset real I))"
Plus: fixes U :: "freeExp"   and V :: "freeExp" shows "Plus (Abs_Exp (exprel `` {U})) (Abs_Exp (exprel `` {V})) = Abs_Exp (exprel `` {PLUS U V})"
keys_lift_poly_syz: fixes n :: "nat"   and b :: "'t \<Rightarrow>\<^sub>0 'c"   and i :: "nat" shows "keys (lift_poly_syz n b i) = insert (term_of_pair (0::'a, i)) (map_component (\<lambda>k. k + n) ` keys b)"
is_ring: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "ring R"
proved_terms_well_formed_1: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and p :: "Core.term" assumes "\<Theta>,\<Gamma> \<turnstile> p" shows "typ_of p = Some propT"
tendsto_const_iff: fixes "open" :: "'a set \<Rightarrow> bool"   and F :: "'b filter"   and a :: "'a"   and b :: "'a" assumes "class.t2_space open"   and "F \<noteq> bot" shows "(LIM x F. a :> topological_space.nhds open b) = (a = b)"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_ring_hom hom"   and "zero_hom_0 hom" shows "inj_comm_ring_hom hom"
large: shows "\<parallel>factor\<parallel>\<^sup>2 ^ degree local.g * \<parallel>local.g\<parallel>\<^sup>2 ^ degree factor < (p ^ local.ll)\<^sup>2"
restrict_inj: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and A1 :: "'e set" assumes "inj_on f A"   and "A1 \<subseteq> A" shows "inj_on f A1"
hdvd_by_all: shows "\<forall>M. \<exists>N>0. \<forall>m. 0 < m \<and> m \<le> M \<longrightarrow> m dvd N"
le_iff_diff_le_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a b = less_eq (minus a b) zero"
a: shows " App t s \<longrightarrow>\<^sub>1 r"
c: fixes r :: "'d rexp"   and s :: "'d rexp" shows "Alt r s ~ Alt s r"
prems: fixes p :: "pat"   and e :: "exp0"   and s' :: "'a state"   and env :: "v sem_env" assumes "(p, e) \<in> set (pe # pes)"   and "clock s' \<le> clock s" shows "\<exists>s'' r. evaluate True env s' e (s'', r)"
UN_extend_simps_2: fixes A :: "'e \<Rightarrow> 'd set"   and C :: "'e set"   and B :: "'d set" shows "\<Union> (A ` C) \<union> B = (if C = {} then B else \<Union>x\<in>C. A x \<union> B)"
HT_partial_conseq: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT_partial \<pi> P c Q"   and "\<And>s. P' s \<Longrightarrow> P s"   and "\<And>s\<^sub>0 s. \<lbrakk>P s\<^sub>0; P' s\<^sub>0; Q s\<^sub>0 s\<rbrakk> \<Longrightarrow> Q' s\<^sub>0 s" shows "HT_partial \<pi> P' c Q'"
vcomp_vid_on: fixes A :: "V"   and B :: "V" shows "vid_on A \<circ>\<^sub>\<circ> vid_on B = vid_on (A \<inter>\<^sub>\<circ> B)"
proj2_Col_iff: fixes a :: "proj2"   and r :: "proj2"   and t :: "proj2" assumes "a \<noteq> r" shows "proj2_Col a r t = (t = a \<or> (\<exists>i. t = proj2_abs (i *\<^sub>R proj2_rep a + proj2_rep r)))"
bisim: shows "(bisim12 \<circ>\<^sub>B bisim23) s1 s3"
r_prepend_total: fixes r :: "recf"   and vs :: "nat list"   and x :: "nat" assumes "recfn 1 r"   and "Partial_Recursive.total r" shows "eval (r_prepend vs r) [x] \<down>= (if x < length vs then vs ! x else the (eval r [x - length vs]))"
PCF_consts_rel_PE: shows "PCF_consts_rel (unlr PE.delta)"
discr0_BisT: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "discr0 c1"   and "discr0 c2" shows "c1 \<approx>T c2"
InterferenceWithFWB: shows "\<forall>c w x. (\<^bold>\<diamond>\<^sub>aFWB x) c w = (\<forall>xa. (\<^bold>\<not>InterferesWith xa (FWB x)) c w)"
circ_sup_sub_sup_one_1: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y \<le> x\<^sup>\<circ> * ((1::'a) \<squnion> y)"
assms_3: shows "\<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright>"
hg: fixes x :: "'b" assumes "x \<in> S'" shows "h x = g x"
power_0_left: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero n = (if n = 0 then one else zero)"
not_eq_order_implies_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<in> U"   and "b \<in> U"   and "a \<noteq> b"   and "a \<le>\<^sub>o\<^sub>w b" shows "a <\<^sub>o\<^sub>w b"
dg_Arr_vsubset_Vset: shows "\<BB>\<lparr>Arr\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
inversion_id_iff_on_unit_circle: fixes a :: "complex_homo" shows "(inversion a = a) = on_circline unit_circle a"
least: shows "(LEAST n. Hermite_of A ass res bezout $ i $ n \<noteq> (0::'a)) = (LEAST n. echelon_form_of A bezout $ i $ n \<noteq> (0::'a))"
plus_left_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "class.ordered_ab_semigroup plus zero less_eq"   and "less_eq y x" shows "less_eq (plus y z) (plus x z)"
S_def: fixes h :: "'g"   and e :: "'e" shows "\<Sigma>0.S h e = map_spmf (\<lambda>(a, z). (a, e, z)) (S0_raw h e)"
simps_1: fixes i :: "nat"   and j :: "nat" shows "\<sigma>s 0 i j = \<mu>' i 0 * \<mu>' j 0"
ESem_ignores_fresh_restr: fixes S :: "'b set"   and e :: "'a"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "atom ` S \<sharp>* e" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho> f|` (- S)\<^esub>"
processed_def: fixes p :: "'b \<times> 'b"   and xs :: "'b list"   and ps :: "('b \<times> 'b) list" shows "processed p xs ps = (p \<in> set xs \<times> set xs \<and> p \<notin>\<^sub>p set ps)"
sum_infinity_imp_summand_infinity: fixes a :: "eint"   and b :: "eint" assumes "a + b = \<infinity>" shows "a = \<infinity> \<or> b = \<infinity>"
rbt_bulkload_is_rbt: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "('c \<times> 'd) list" assumes "class.linorder less_eq less" shows "ord.is_rbt less (ord.rbt_bulkload less xs)"
delay_bisimulation_diverge_axioms: shows "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim (\<lambda>s tl s'. False) (\<lambda>s tl s'. False)"
hyps_9: shows "A\<^sub>P \<sharp>* P"
fls_regpart_integral: fixes f :: "'a fls" shows "fls_regpart (fls_integral f) = fps_integral0 (fls_regpart f)"
simps_1: fixes t' :: "transition"   and t :: "transition"   and r :: "nat" shows "input_stored_in_reg_aux t' t 0 r = input_i_stored_in_reg t' t 0 r"
elims_2: fixes x :: "'b dres"   and xa :: "'b dres" assumes "x \<le> xa" obtains uu_ :: "'b dres" where "x = dSUCCEEDi" and "xa = uu_"   | "x = dFAILi" and "xa = dFAILi"   | v :: "'b" where "x = dRETURN v" and "xa = dFAILi"   | a :: "'b" and b :: "'b" where "x = dRETURN a" and "xa = dRETURN b" and "a = b"
openin_subtopology_refl: fixes U :: "'b topology"   and V :: "'b set" shows "openin (subtopology U V) V = (V \<subseteq> topspace U)"
ereal_minus_7: fixes x :: "ereal" shows "x - 0 = x"
less_\<tau>D: fixes \<sigma> :: "'a trace"   and i :: "nat"   and j :: "nat" assumes "\<tau> \<sigma> i < \<tau> \<sigma> j" shows "i < j"