hcju/isabelleps huggingface dataset:
dvd_prod_eqI: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and A :: "'d set"   and a :: "'d"   and b :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "a \<in> A"   and "b = f a" shows "dvd.dvd times b (comm_monoid_mult.prod times one f A)"
pwd_spy_sesk_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_sesk n)"
finite_insert: fixes a :: "'d"   and A :: "'d set" shows "finite (insert a A) = finite A"
prune_decreasing: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "prune A w V \<subseteq> V"
Cond: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and b :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P \<inter> b) c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P \<inter> - b) c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Cond b c\<^sub>1 c\<^sub>2 Q,A"
lt_minus_higher: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p" shows "punit.lt (p - punit.higher p v) = v"
less_prod_def': fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "(x < y) = (fst x < fst y \<or> fst x = fst y \<and> snd x < snd y)"
simps_2: fixes p :: "'b fm"   and ps :: "'b fm list"   and q :: "'b fm" shows "conjoin (p # ps) q = (p \<^bold>\<and> conjoin ps q)"
IH: fixes Y :: "'a set" assumes "X \<subset> Y"   and "Y \<subseteq> carrier"   and "indep Y" shows "\<exists>B. basis B \<and> Y \<subseteq> B"
Cod_in_Obj: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "Cod f \<in> IDE"
deMorgan2: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "not (X or Y) = (not X and not Y)"
is_exec_of_def: fixes A :: "('b, 'c) ioa"   and e :: "'b \<times> ('c \<times> 'b) list" shows "is_exec_of A e \<equiv> fst e \<in> start A \<and> is_exec_frag_of A e"
quot_simps_5: fixes A :: "fm"   and B :: "fm" shows "\<guillemotleft>A OR B\<guillemotright> = Q_Disj \<guillemotleft>A\<guillemotright> \<guillemotleft>B\<guillemotright>"
extend_2: shows "w =\<^sub>F w\<^sub>1 @ w\<^sub>2"
Form_elim_upair: fixes l :: "nat"   and U :: "nat list set" assumes "Form l U" obtains xs :: "nat list" and ys :: "nat list" where "xs \<noteq> ys" and "U = {xs, ys}" and "length xs \<le> length ys"
ladder_shift_n_length: fixes N :: "nat"   and L :: "(nat \<times> nat \<times> nat) list" shows "length (ladder_shift_n N L) = length L"
decomps\<^sub>e\<^sub>s\<^sub>t_finite_ik_append: fixes M :: "('fun, 'var) extstrand_step list set"   and A :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "finite M"   and "M \<subseteq> decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>" shows "\<exists>D\<in>decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>. ik\<^sub>e\<^sub>s\<^sub>t D = \<Union> (ik\<^sub>e\<^sub>s\<^sub>t ` M)"
bpE_3: fixes P :: "'a set set" assumes "bp P" shows "\<Union> P = U"
ins_height_2: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat"   and d :: "nat"   and x :: "'b"   and t :: "'b tree"   and t' :: "'b tree" assumes "RBTi1 TYPE('b) bal_i"   and "RBTi1.ins bal_i n d x t = Unbal t'" shows "height t \<le> height t' \<and> height t' \<le> height t + 1"
assms_2: fixes \<eta> :: "nat" shows "interaction_any_bounded_converter (conv \<eta>) (bound_conv \<eta>)"
runFw_def: fixes s :: "'a word"   and d :: "'a word"   and c :: "'b parts_connection_scheme"   and rs :: "'a simple_rule list" shows "runFw s d c rs = simple_fw rs \<lparr>p_iiface = pc_iiface c, p_oiface = pc_oiface c, p_src = s, p_dst = d, p_proto = pc_proto c, p_sport = pc_sport c, p_dport = pc_dport c, p_tcp_flags = {TCP_SYN}, p_payload = []\<rparr>"
MonotProblem_axioms_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set" shows "MonotProblem_axioms wtFsym wtPsym arOf resOf parOf \<Phi> \<equiv> Problem.monot wtFsym wtPsym arOf resOf parOf \<Phi>"
bij_iff: fixes f :: "'c \<Rightarrow> 'd" shows "bij f = (\<forall>x. \<exists>!y. f y = x)"
InitDmd: fixes F :: "behavior \<Rightarrow> bool" shows "\<turnstile> F \<longrightarrow> \<diamond>F"
gridgen_dim_restrict: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds' :: "nat set"   and ds :: "nat set" assumes "p \<in> grid b (ds' \<union> ds)"   and "\<forall>d\<in>ds'. length b \<le> d" shows "p \<in> grid b ds"
path_image_subpath_gen: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'a" shows "path_image (subpath u v g) = g ` closed_segment u v"
mult_rat: fixes a :: "int"   and b :: "int"   and c :: "int"   and d :: "int" shows "Fract a b * Fract c d = Fract (a * c) (b * d)"
a1: shows "s' = snd (fst (write_state_reg_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
nuc_lax: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "nucleus f" shows "f x \<cdot> f y \<le> f (x \<cdot> y)"
rcs_Unit2: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and h :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "h \<in> H" shows "H \<bullet>\<^bsub>G\<^esub> h = H"
comp_reduce: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and k :: "'b"   and h :: "'b" assumes "category C"   and "C f g = k"   and "partial_magma.arr C (C f g)"   and "partial_magma.arr C (C g h)" shows "C f (C g h) = C k h"
A: shows "\<forall>\<^sub>F n in sequentially. 3 * alpha < Sigma_Algebra.measure (distr P borel (\<lambda>x. (g x + birkhoff_sum f n x) / B n)) {x. 2 * a < \<bar>x\<bar>}"
RedFAccNull: fixes F :: "char list"   and D :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>null\<bullet>F{D},s,b\<rangle> \<rightarrow> \<langle>THROW NullPointer,s,b\<rangle>"
bot_entails_all: fixes B :: "'g"   and N1 :: "'g set" assumes "B \<in> Bot_G" shows "{B} \<Turnstile>G N1"
DFS_invar_step: fixes s :: "('v, 'es) state_scheme" assumes "nofail it_dfs"   and "DFS_invar G param s"   and "cond s" shows "step s \<le> SPEC (DFS_invar G param)"
simps_3: fixes uv :: "'b"   and a :: "('a, 'b) Node"   and b :: "('a, 'b) Node"   and c :: "('a, 'b) Node" shows "FingerTreeStruc.nodeToDigit (Node3 uv a b c) = Three a b c"
to_fun_to_poly: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "UP_cring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "UP_cring.to_fun R (to_polynomial R a) b = a"
simps_20: fixes traps' :: "Trap set \<Rightarrow> Trap set"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "traps_update traps' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps' traps, undef = undef, \<dots> = more\<rparr>"
simps_13: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm" shows "push_forall (AllQ (Or \<phi> \<psi>)) = (if freeIn 0 \<phi> then if freeIn 0 \<psi> then or (lowerFm 0 1 \<phi>) (lowerFm 0 1 \<psi>) else or (lowerFm 0 1 \<phi>) (AllQ \<psi>) else if freeIn 0 \<psi> then or (AllQ \<phi>) (lowerFm 0 1 \<psi>) else AllQ (or \<phi> \<psi>))"
has_overlap_at_k_equiv2: fixes rects :: "'a rectangle list"   and ops :: "'a operation list" assumes "is_rect_list rects"   and "ops = all_ops rects"   and "has_rect_overlap rects" shows "\<exists>k<length ops. has_overlap_at_k rects k"
col_trivial_2: 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" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Col Bet A B B"
continuous_divide: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous F f"   and "continuous F g"   and "g (Lim F (\<lambda>x. x)) \<noteq> (0::'b)" shows "continuous F (\<lambda>x. f x / g x)"
cut1: fixes A :: "fmla"   and B :: "fmla"   and H :: "fmla set" assumes "{A} \<turnstile> B"   and "H \<turnstile> A" shows "H \<turnstile> B"
funionM_fUnionM1: fixes A :: "'m fset"   and x :: "'m" shows "funionM (fUnionM A) x = ffold funionM x A"
simps: fixes a1 :: "lam"   and a2 :: "lam" shows "a1 \<longrightarrow>x a2 = (\<exists>E x t t'. a1 = E\<lbrakk>App (Lam [x].t) t'\<rbrakk> \<and> a2 = E\<lbrakk>t[x::=t']\<rbrakk>)"
n_kat_1_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(t x \<cdot> y \<cdot> t z = y \<cdot> t z) = (!x \<cdot> y \<cdot> t z = (0::'a))"
flat_interpretation: fixes b :: "'c" shows "partial_function_definitions (flat_ord b) (flat_lub b)"
1: shows "insertion_fun f (\<lambda>m. a when Poly_Mapping.single v n = m) = (\<Sum>m. (a when Poly_Mapping.single v n = m) * (\<Prod>v. f v ^ lookup m v))"
in_KeysI: fixes s :: "'d"   and f :: "'d \<Rightarrow>\<^sub>0 'e"   and F :: "('d \<Rightarrow>\<^sub>0 'e) set" assumes "s \<in> keys f"   and "f \<in> F" shows "s \<in> Keys F"
hyps_2: shows "While_CFG.path (prog;; c\<^sub>2) n'' as (_ l _)"
assms_2: shows "path_image g2 \<subseteq> s"
cgenideal_ideal: fixes a :: "int" assumes "a \<in> UNIV" shows "ideal (PIdl\<^bsub>\<Z>\<^esub> a) \<Z>"
spr_jview_tObsCI: fixes t :: "('a, 'es, 'as) BEState Trace"   and t' :: "('a, 'es, 'as) BEState Trace"   and a :: "'a" assumes "tObsC t = tObsC t'"   and "envObs a (tFirst t) = envObs a (tFirst t')"   and "tMap (\<lambda>s. ODList.lookup (ps s) a) t = tMap (\<lambda>s. ODList.lookup (ps s) a) t'" shows "spr_jview a t = spr_jview a t'"
hyps: fixes ds :: "nat list"   and a :: "nat" shows "take n (digit_encode ds a) = digit_encode (take n ds) a"
flowsto_subset: fixes X0 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and T :: "real set"   and CX :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and Y :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and Z :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and S :: "real set"   and CZ :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and W :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "flowsto X0 T CX Y"   and "Z \<subseteq> X0"   and "T \<subseteq> S"   and "CX \<subseteq> CZ"   and "Y \<subseteq> W" shows "flowsto Z S CZ W"
longer_falsifies\<^sub>c: fixes C :: "fterm literal set"   and ds :: "bool list"   and d :: "bool list" assumes "falsifies\<^sub>c ds C" shows "falsifies\<^sub>c (ds @ d) C"
Subobjs_Subobjs: 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 C :: "char list"   and Cs :: "char list list"   and C' :: "char list"   and Cs' :: "char list list" assumes "Subobjs P C (Cs @ C' # Cs')" shows "Subobjs P C' (C' # Cs')"
step: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and \<R> :: "('e \<Rightarrow> real) set set"   and l :: "'d"   and R :: "('e \<Rightarrow> real) set"   and l' :: "'d"   and R' :: "('e \<Rightarrow> real) set"   and l'' :: "'d"   and R'' :: "('e \<Rightarrow> real) set" assumes "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle>"   and "A,\<R> \<turnstile> \<langle>l', R'\<rangle> \<leadsto> \<langle>l'', R''\<rangle>" shows "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l'', R''\<rangle>"
cases: fixes x :: "intv" obtains d :: "nat" where "x = Const d__"   | d :: "nat" where "x = Intv da__"   | d :: "nat" where "x = Greater d"
subcocycle_avg_add_1: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and v :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u"   and "subcocycle v"   and "- \<infinity> < subcocycle_avg_ereal u"   and "- \<infinity> < subcocycle_avg_ereal v" shows "- \<infinity> < subcocycle_avg_ereal (\<lambda>n x. u n x + v n x)"
induct: fixes P :: "'a \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "('a \<times> 'b) list" assumes "\<And>k. P k []"   and "\<And>k l w ls. (k \<noteq> l \<Longrightarrow> P k ls) \<Longrightarrow> P k ((l, w) # ls)" shows "P a0 a1"
fds_converges_norm_iff': fixes f :: "'a fds"   and s :: "real" shows "fds_converges (fds_norm f) s = fds_abs_converges f (of_real s)"
seq_char: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "('a, 'b) concrete_category.arr"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp" shows "partial_magma.arr (concrete_category.COMP Obj Hom Comp) (concrete_category.COMP Obj Hom Comp g f) = (partial_magma.arr (concrete_category.COMP Obj Hom Comp) f \<and> partial_magma.arr (concrete_category.COMP Obj Hom Comp) g \<and> concrete_category.Dom g = concrete_category.Cod f)"
assms_2: fixes x :: "real" assumes "a < x"   and "x < b" shows "\<exists>y. (f has_real_derivative y) (at x) \<and> 0 < y"
one_poly_i_def: fixes ops :: "'b arith_ops_record" shows "one_poly_i ops = [arith_ops_record.one ops]"
C_def: shows "C \<equiv> mk_finite_char (close {set G |G. \<not> \<turnstile> imply G \<^bold>\<bottom>})"
allNeededINChannels_def: fixes i :: "AbstrLevelsID"   and chSet :: "chanID set" shows "allNeededINChannels i chSet \<equiv> \<forall>Z\<in>minSetOfComponents i chSet. \<exists>x\<in>IN Z. systemIN x i \<longrightarrow> x \<in> chSet"
subst_subset_mono: fixes D :: "'a literal multiset"   and C :: "'a literal multiset"   and \<sigma> :: "'s" assumes "D \<subset># C" shows "D \<cdot> \<sigma> \<subset># C \<cdot> \<sigma>"
WCFG_WhileFalse: fixes b :: "expr"   and c' :: "cmd" shows "while (b) c' \<turnstile> (_ 0 _) -(\<lambda>s. interpret b s = Some false)\<^sub>\<surd>\<rightarrow> (_ 1 _)"
init_config_def: fixes c :: "('d, 'e, 'f) configuration_scheme" shows "init_config c \<equiv> \<forall>loc. c_imp c loc = {#}\<^sub>z \<and> c_work c loc = zmset_of (mset_set (set_antichain (frontier (c_pts c loc))))"
finite_imp_compactin: fixes S :: "'b set"   and X :: "'b topology" assumes "S \<subseteq> topspace X"   and "finite S" shows "compactin X S"
Nil_eqvt: fixes p :: "perm" shows "p \<bullet> [] = []"
i_last_message_hold_nth: fixes f :: "nat \<Rightarrow> 'b message_af"   and k :: "nat"   and n :: "nat" shows "(f \<longmapsto> k) n = last_message (f \<Up> (n - n mod k) \<Down> Suc (n mod k))"
simps: fixes r :: "Vars.var list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "to_gc_conf r (\<Gamma>, e, S) = (restrictA (- set r) \<Gamma>, e, restr_stack (- set r) S @ map Dummy (rev r))"
asIMOD_igSubstAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c) semDom"   and ys :: "'varSort"   and Y :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'c) \<Rightarrow> 'c"   and y :: "'var"   and A :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'c) \<Rightarrow> ('varSort, 'c) sAbs" shows "igSubstAbs (asIMOD SEM) ys Y y A = (\<lambda>val. if sWlsVal SEM val then A (val (y := Y val)_ys) else undefined)"
finset: shows "finite {y. a * y\<^sup>2 + b * y + c = 0}"
replicate_count: fixes n :: "nat"   and x :: "'c" shows "count (mset (replicate n x)) x = n"
sym_iff: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "sympartp r x y = sympartp r y x"
clo: fixes X :: "'a set" assumes "X \<in> \<F>" shows "closed X"
strict_boundedE: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "sup b c < a" obtains "b < a" and "c < a"
simps_2: fixes uu :: "ct" shows "ct_eq TAny uu = True"
echelon_form_of_upt_k_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" shows "echelon_form_of_upt_k A k bezout = fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])"
dom_rhs: shows "\<D>\<^sub>\<circ> [T\<lparr>ArrVal\<rparr>, T\<lparr>ArrDom\<rparr>, T\<lparr>ArrCod\<rparr>]\<^sub>\<circ> = 3\<^sub>\<nat>"
retraction: fixes S :: "'a set"   and T :: "'a set"   and r :: "'a \<Rightarrow> 'a" shows "retraction S T r = (T \<subseteq> S \<and> continuous_on S r \<and> r ` S = T \<and> (\<forall>x\<in>T. r x = x))"
inf_bot_left: fixes x :: "'a" shows "0 \<cdot> x = 0"
Above_Pd_Xd: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> Above (Pd d) X" shows "Xd x = d"
nn_integral_cmult_indicator: fixes A :: "'b set"   and M :: "'b measure"   and c :: "ennreal" assumes "A \<in> sets M" shows "\<integral>\<^sup>+ x. c * indicator A x \<partial>M = c * emeasure M A"
F_nonneg: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "f \<in> Field leq" shows "0 \<le> f e"
Ord_LeastI2: fixes a :: "V"   and P :: "V \<Rightarrow> bool"   and Q :: "V \<Rightarrow> bool" assumes "Ord a"   and "P a"   and "\<And>x. \<lbrakk>Ord x; P x\<rbrakk> \<Longrightarrow> Q x" shows "Q (LEAST i. Ord i \<and> P i)"
non_wildcard_ifaces_def: fixes n :: "nat" shows "non_wildcard_ifaces n \<equiv> filter (\<lambda>i. \<not> iface_name_is_wildcard i) (List.n_lists n all_chars)"
cblinfun_eq_0_on_UNIV_span: fixes basis :: "'a set"   and F :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "cspan basis = UNIV"   and "\<And>s. s \<in> basis \<Longrightarrow> F *\<^sub>V s = (0::'b)" shows "F = 0"
ctr_transfer_1: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "list_all2 R [] []"
segment_2: shows "\<forall>\<^sub>F x' in at x within {x. \<forall>j\<in>Basis. x \<bullet> j \<in> X j}. x' \<noteq> x"
inf_par_distrib2: fixes par :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and skip :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice"   and c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice" assumes "par_distrib (par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (skip::'b::refinement_lattice)" shows "(par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (d::'b::refinement_lattice) ((c\<^sub>0::'b::refinement_lattice) \<sqinter> (c\<^sub>1::'b::refinement_lattice)) = par d c\<^sub>0 \<sqinter> par d c\<^sub>1"
padic_no_zero_divisors: fixes p :: "int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "prime p"   and "a \<in> carrier (padic_int p)"   and "b \<in> carrier (padic_int p)"   and "a \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"   and "b \<noteq> \<zero>\<^bsub>padic_int p\<^esub>" shows "a \<otimes>\<^bsub>padic_int p\<^esub> b \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"
R1: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and X :: "'b list \<times> 'b list \<times> 'b list" assumes "(x # xs, ys, []) \<in> l_sort_set X" shows "(xs, [], ys @ [x]) \<in> l_sort_set X"
comparator_proj_comp: fixes cmp :: "(('a, 'b) pp \<times> nat) nat_term_order" shows "comparator (proj_comp cmp)"
simps: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d list"   and a2 :: "'d list" shows "ord.lexordp_eq less a1 a2 = ((\<exists>ys. a1 = [] \<and> a2 = ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> less x y) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> \<not> less x y \<and> \<not> less y x \<and> ord.lexordp_eq less xs ys))"
assms: shows "f\<cdot>C\<^sup>\<infinity> \<noteq> \<bottom>"
otherwith_def: fixes Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "'e set"   and P :: "('e \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> bool"   and \<sigma> :: "'e \<Rightarrow> 'd"   and \<sigma>' :: "'e \<Rightarrow> 'd"   and a :: "'f" shows "otherwith Q I P \<sigma> \<sigma>' a \<equiv> (\<forall>i. i \<notin> I \<longrightarrow> Q (\<sigma> i) (\<sigma>' i)) \<and> P \<sigma> a"
simps_1: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "dfa_of_pf' n (Eq ks l) = eq_dfa n ks l"
ipassmt_inj_p: shows "\<forall>ips'. p_dst p \<in> ipcidr_union_set (set ips') \<and> (\<exists>k. ipassmt k = Some ips') \<longrightarrow> ips' = i_ips"
map_ident: fixes t :: "'d stream" shows "smap (\<lambda>x. x) t = t"
countable_bipartite_web_of: shows "countable_bipartite_web (bipartite_web_of (bipartite_web_of \<Gamma>))"
that: fixes S :: "'a spmf \<Rightarrow> 'g spmf \<Rightarrow> bool" assumes "S p q"   and "\<And>p q e. \<lbrakk>S p q; e \<in> E\<rbrakk> \<Longrightarrow> weight_spmf (p \<bind> (\<lambda>s. cpoke core1 s e)) = weight_spmf (q \<bind> (\<lambda>s. cpoke core2 s e))"   and "\<And>p q e. \<lbrakk>S p q; e \<in> E\<rbrakk> \<Longrightarrow> S (mk_lossless (p \<bind> (\<lambda>s. cpoke core1 s e))) (mk_lossless (q \<bind> (\<lambda>s. cpoke core2 s e)))"   and "\<And>p q ia. \<lbrakk>S p q; ia \<in> IA\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (cfunc_adv core1 s1 ia)) = q \<bind> (\<lambda>s2. map_spmf fst (cfunc_adv core2 s2 ia))"   and "\<And>p q ia oa. \<lbrakk>S p q; ia \<in> IA\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_adv core2 s2 ia)) oa)"   and "\<And>p q iu. \<lbrakk>S p q; iu \<in> IU\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (cfunc_usr core1 s1 iu)) = q \<bind> (\<lambda>s2. map_spmf fst (cfunc_usr core2 s2 iu))"   and "\<And>p q iu ou. \<lbrakk>S p q; iu \<in> IU\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_usr core2 s2 iu)) ou)" shows "thesis"
smult_power: fixes a :: "'b"   and p :: "'b poly"   and n :: "nat" shows "Polynomial.smult a p ^ n = Polynomial.smult (a ^ n) (p ^ n)"
fps_exp_add_mult: fixes a :: "'a"   and b :: "'a" shows "fps_exp (a + b) = fps_exp a * fps_exp b"
is_tangle_diagram_weak_matrix_match: fixes w1 :: "brick list"   and ws1 :: "wall"   and w2 :: "brick list" assumes "is_tangle_diagram (w1 * ws1)"   and "codomain_block w2 = 0" shows "weak_matrix_match (blockmat w1) (kauff_mat ws1) (blockmat w2)"
expl_cond_expect_prop_sets: fixes Y :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and D :: "'c set"   and P :: "real \<Rightarrow> bool"   and X :: "'c \<Rightarrow> real" assumes "disc_fct Y"   and "point_measurable M (space N) Y"   and "D = {w \<in> space M. Y w \<in> space N \<and> P (expl_cond_expect M Y X w)}" shows "D \<in> sets M"
Fake_parts_insert_in_Un: fixes Z :: "msg"   and X :: "msg"   and H :: "msg set" assumes "Z \<in> parts (insert X H)"   and "X \<in> synth (analz H)" shows "Z \<in> synth (analz H) \<union> parts H"
strand_id_subst: fixes S :: "('a, 'b) strand_step list" shows "S \<cdot>\<^sub>s\<^sub>t Var = S"
map_sum_sel_1: fixes s :: "'h + 'i"   and f :: "'h \<Rightarrow> 'j"   and g :: "'i \<Rightarrow> 'k" assumes "isl s" shows "projl (map_sum f g s) = f (projl s)"