hcju/isabelleps huggingface dataset:
nsg_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "G \<triangleright> H = (G \<guillemotright> H  \<and> (\<forall>x\<in>carrier G. H \<bullet>\<^bsub>G\<^esub> x = x \<diamondsuit>\<^bsub>G\<^esub> H))"
integral_iff: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (poly_ring R)"   and "b \<in> carrier (poly_ring R)" shows "(a \<otimes>\<^bsub>poly_ring R\<^esub> b = \<zero>\<^bsub>poly_ring R\<^esub>) = (a = \<zero>\<^bsub>poly_ring R\<^esub> \<or> b = \<zero>\<^bsub>poly_ring R\<^esub>)"
path_component_in_path_components_of: fixes X :: "'c topology"   and a :: "'c" shows "(path_component_of_set X a \<in> path_components_of X) = (a \<in> topspace X)"
exec_step_ind_Load: fixes n :: "nat"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step_ind (StepI (Load n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (loc ! n # stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
ln_prod: fixes I :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> 0 < f i" shows "ln (prod f I) = (\<Sum>x\<in>I. ln (f x))"
witness_1: fixes a :: "'b" assumes "a \<in> S" shows "i (j a) = a"
small_big_antisym: fixes h1 :: "'a \<Rightarrow> real"   and h2 :: "'a \<Rightarrow> real" assumes "h1 \<in> H"   and "h2 \<in> H"   and "h1 \<in> o[F](h2)"   and "h2 \<in> O[F](h1)" shows "False"
replicating_expectation: fixes N :: "'a measure"   and pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and pyf :: "'a \<Rightarrow> real"   and matur :: "nat" assumes "risk_neutral_prob N"   and "filt_equiv F M N"   and "replicating_portfolio pf pyf matur"   and "\<forall>n. \<forall>asset\<in>support_set pf. integrable N (\<lambda>w. prices Mkt asset n w * pf asset (Suc n) w)"   and "\<forall>n. \<forall>asset\<in>support_set pf. integrable N (\<lambda>w. prices Mkt asset (Suc n) w * pf asset (Suc n) w)"   and "viable_market Mkt"   and "sets (F 0) = {{}, space M}"   and "pyf \<in> borel_measurable (F matur)" shows "fair_price Mkt (integral\<^sup>L N (discounted_value r (\<lambda>m. pyf) matur)) pyf matur"
get_add_neq: fixes x :: "'key"   and y :: "'key"   and e :: "'env"   and v :: "'val" assumes "x \<noteq> y" shows "get (add e x v) y = get e y"
gcd_eucl_i: shows "(poly_rel ===> poly_rel ===> poly_rel) poly.gcd_eucl_i normalization_euclidean_semiring_class.gcd"
\<psi>_char_2: shows "D.iso F'.\<psi>"
rel_fundefs_rewrite_both: fixes F1 :: "'fenv_inca"   and F2 :: "'fenv_ubx"   and fd1 :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and fd2 :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and f :: "'fun" assumes "rel_fundefs (Finca_get F1) (Fubx_get F2)"   and "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2" shows "rel_fundefs (Finca_get (Finca_add F1 f fd1)) (Fubx_get (Fubx_add F2 f fd2))"
lemma_2_5_8_a: fixes a :: "'a"   and b :: "'a" shows "a * b \<le> a"
big_small_mult: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> L F g1"   and "f2 \<in> l F g2" shows "(\<lambda>x. f1 x * f2 x) \<in> l F (\<lambda>x. g1 x * g2 x)"
allpolys_split0: fixes p :: "tm" assumes "allpolys isnpoly p" shows "allpolys isnpoly (snd (split0 p))"
bicategory_axioms: shows "bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
elims_2: fixes x :: "(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)" assumes "wf_sig x" obtains ctf :: "String.literal \<Rightarrow> typ option" and arf :: "String.literal \<Rightarrow> nat option" and oss :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" where "x = (ctf, arf, oss)" and "wf_osig oss \<and> dom (tcsigs oss) = dom arf \<and> (\<forall>type\<in>dom (tcsigs oss). \<forall>ars\<in>ran (the (tcsigs oss type)). the (arf type) = length ars) \<and> Ball (ran ctf) (wf_type (ctf, arf, oss))"
zp_z: shows "hom_boundary (int n) (lsphere n) (equator n) zp = z"
Lemma_1: fixes h :: "'b list \<Rightarrow> 'c"   and f :: "'b list \<Rightarrow> 'b list \<Rightarrow> 'b list"   and g :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zs :: "'b list list" assumes "\<And>x y. \<lbrakk>x \<noteq> []; y \<noteq> []\<rbrakk> \<Longrightarrow> h (f x y) = g (h x) (h y)"   and "\<And>x y. \<lbrakk>x \<noteq> []; y \<noteq> []\<rbrakk> \<Longrightarrow> f x y \<noteq> []"   and "\<And>x. x \<in> set zs \<Longrightarrow> x \<noteq> []" shows "map h (candidate f zs) = candidate g (map h zs)"
simps: fixes I :: "'a set"   and v :: "nat \<Rightarrow> 'b"   and BL :: "nat \<Rightarrow> 'b option"   and BU :: "nat \<Rightarrow> 'b option"   and IL :: "nat \<Rightarrow> 'a"   and IU :: "nat \<Rightarrow> 'a" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>b ((BL, BU), IL, IU) = ((\<forall>x c. BL x = Some c \<longrightarrow> IL x \<in> I \<longrightarrow> c \<le> v x) \<and> (\<forall>x c. BU x = Some c \<longrightarrow> IU x \<in> I \<longrightarrow> v x \<le> c))"
\<alpha>nl_def: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and e :: "('c, 'f) label list el_step" shows "\<alpha>nl fg e \<equiv> \<alpha>n fg (le_rem_s e)"
imgTr11: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and H :: "'e set" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "F \<guillemotright> H " shows "G \<guillemotright> f ` H "
doctor_optimal_match_optimal: fixes ds :: "'b set"   and Y :: "'a set"   and x :: "'a"   and X :: "'a set" assumes "doctor_optimal_match ds Y"   and "x \<in> X"   and "stable_on ds X" shows "\<exists>y\<in>Y. (x, y) \<in> Pd (Xd x)"
sfilter_smap: fixes p :: "'r \<rightarrow> tr"   and f :: "'s \<rightarrow> 'r" assumes "p\<cdot>\<bottom> = \<bottom>" shows "sfilter\<cdot>p oo smap\<cdot>f = smap\<cdot>f oo sfilter\<cdot>(p oo f)"
eucl_inf: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "inf x y = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (inf (inner x i) (inner y i)) i) Basis"
AccLVar: fixes vn :: "lname"   and B :: "lname set"   and A :: "assigned"   and Env :: "env" assumes "vn \<in> B"   and "nrm A = B"   and "brk A = (\<lambda>k. UNIV)" shows "Env\<turnstile> B \<guillemotright>\<langle>Acc (LVar vn)\<rangle>\<guillemotright> A"
lParamDefs: shows "lParamDefs \<equiv> lift_ParamDefs ParamDefs"
assms_1: shows "(C \<Rightarrow> S) \<sqsubseteq> F (C \<Rightarrow> S)"
psi_sqrt: fixes n :: "nat" shows "psi (Discrete.sqrt n) = psi_even n"
lc_subst: fixes u :: "expr"   and t :: "expr"   and xT :: "char list \<times> Unary_PCF.type" assumes "lc u"   and "lc t" shows "lc (subst xT t u)"
simps_2: fixes xs :: "'c list" shows "C_merge_all [xs] = 0"
Vs'_3: shows "local.independent K (Vs' @ u # Us)"
scale_cancel_right: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and x :: "'d"   and b :: "'c" assumes "vector_space scale" shows "(scale a x = scale b x) = (a = b \<or> x = (0::'d))"
inner_Basis_mono: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "less_eq a b"   and "c \<in> Basis" shows "inner a c \<le> inner b c"
splits_at_append_prefix: fixes v :: "symbol list"   and i :: "nat"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list"   and u :: "symbol list" assumes "splits_at v i \<alpha> N \<beta>" shows "splits_at (u @ v) (i + length u) (u @ \<alpha>) N \<beta>"
elims_1: fixes \<alpha> :: "real"   and x :: "'b tree"   and y :: "bool" assumes "WBT0 \<alpha>"   and "WBT0.wbt \<alpha> x = y" obtains "x = \<langle>\<rangle>" and "y"   | l :: "'b tree" and uu_ :: "'b" and r :: "'b tree" where "x = \<langle>l, uu_, r\<rangle>" and "y = (WBT0.wbt \<alpha> l \<and> WBT0.wbt \<alpha> r \<and> (let ratio = real (size1 l) / real (size1 l + size1 r) in \<alpha> \<le> ratio \<and> ratio \<le> 1 - \<alpha>))"
insert_not_elem: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<notin> A"   and "A \<noteq> {}" shows "Max (insert x A) = max x (Max A)"
simps: fixes a1 :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and a2 :: "('a, 'b) form"   and a3 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" shows "leftPrincipal a1 a2 a3 = (\<exists>C A Ps R. a1 = (Ps, C) \<and> a2 = A \<and> a3 = R \<and> C = ( \<LM> A  \<RM> \<Rightarrow>* \<Empt>) \<and> A \<noteq> ff \<and> (Ps, C) \<in> R)"
sq_norm_of_int: fixes v :: "int vec" shows "\<parallel>of_int_hom.vec_hom v\<parallel>\<^sup>2 = of_int \<parallel>v\<parallel>\<^sup>2"
rel_funI: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
Iface_single_not_2: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and X :: "iface"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (MatchNot (Match (OIface X))) a p = (\<not> match_iface X (p_oiface p))"
is_arrI: fixes f :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V" assumes "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>"   and "\<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a"   and "\<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b" shows "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
inv_dghm_components_2: fixes \<FF> :: "V" shows "inv_dghm \<FF>\<lparr>ArrMap\<rparr> = (\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ>"
disj: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (MIR.disj p q) = Ifm bs (Or p q)"
f1: shows "\<phi> \<sqinter> \<psi> = fb\<^sub>\<F> (fb\<^sup>-\<^sub>\<F> \<phi>) \<sqinter> fb\<^sub>\<F> (fb\<^sup>-\<^sub>\<F> \<psi>)"
redT_has_locks_inv: fixes ls :: "('l, 't) locks"   and ts :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and m :: "'m"   and ws :: "'t \<Rightarrow> 'w wait_set_status option"   and "is" :: "'t set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and ls' :: "('l, 't) locks"   and ts' :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and m' :: "'m"   and ws' :: "'t \<Rightarrow> 'w wait_set_status option"   and is' :: "'t set"   and t' :: "'t"   and l :: "'l" assumes "\<langle>ls, (ts, m), ws, is\<rangle> -t\<triangleright>ta\<rightarrow> \<langle>ls', (ts', m'), ws', is'\<rangle>"   and "t \<noteq> t'" shows "has_locks (ls $ l) t' = has_locks (ls' $ l) t'"
getNonWatchedUnfalsifiedLiteralNoneCharacterization: fixes clause :: "Literal list"   and w1 :: "Literal"   and w2 :: "Literal"   and M :: "(Literal \<times> bool) list" assumes "getNonWatchedUnfalsifiedLiteral clause w1 w2 M = None" shows "\<forall>l. l el clause \<and> l \<noteq> w1 \<and> l \<noteq> w2 \<longrightarrow> literalFalse l (elements M)"
noStuckE: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>{Stuck}"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Stuck" shows "P"
mkM_simps_1: fixes T :: "'s Trace set" shows "worlds (mkM T) = T"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "\<phi>.A.arr f"   and "\<phi>.A.cod f = b" shows "b \<cdot>\<^sub>A f = f"
\<Theta>_def: fixes i :: "nat" shows "\<Theta> i = (- 1) ^ \<tau> i * pow_int (f i) (int (\<delta> (i - 1)) - 1) * (\<Prod>l\<leftarrow>[3..<Suc i]. (\<beta> l / \<alpha> l) ^ (n (l - 1) - n i) * f (l - 1) ^ (\<delta> (l - 2) + \<delta> (l - 1)))"
pCons_1: shows "a \<noteq> (0::'f) \<or> p \<noteq> 0"
diff_eq: fixes x :: "'a upred"   and y :: "'a upred" shows "diff_upred x y = (x \<and> \<not> y)"
poly_shift_closed: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "poly_shift f \<in> carrier P"
simps_2: fixes v :: "'b"   and va :: "'b list" shows "inter_sorted (v # va) [] = []"
to_ct_list_def: fixes ts :: "t list" shows "to_ct_list ts = map TSome ts"
sw_def: fixes ys :: "'d"   and y1 :: "'c"   and y2 :: "'c"   and xs :: "'d"   and x :: "'c" shows "x @xs[y1 \<and> y2]_ys \<equiv> if ys = xs then if x = y1 then y2 else if x = y2 then y1 else x else x"
size_Conj: fixes A :: "fm"   and B :: "fm" shows "size (A AND B) = size A + size B + 4"
CT_lemma: fixes \<theta> :: "('sf, 'sz) trm"   and \<theta>' :: "('sf, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and var :: "'sf"   and a :: "(real, 'sz) vec"   and b :: "(real, 'sz) vec" assumes "\<forall>I. is_interp I \<longrightarrow> (\<forall>a b. dterm_sem I \<theta> (a, b) = dterm_sem I \<theta>' (a, b))"   and "is_interp I" shows "Functions I var (\<chi>i. dterm_sem I (if i = vid1 then \<theta> else Const 0) (a, b)) = Functions I var (\<chi>i. dterm_sem I (if i = vid1 then \<theta>' else Const 0) (a, b))"
powr_sum: fixes x :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "x \<noteq> (0::'a)"   and "finite A" shows "x powr sum f A = (\<Prod>y\<in>A. x powr f y)"
exec_step_ind_Init_Prepared_nObject: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and sh' :: "sheap"   and D :: "char list"   and P :: "jvm_prog"   and Cs :: "char list list"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list" assumes "sh C = \<lfloor>(sfs, Prepared)\<rfloor>"   and "sh' = sh(C \<mapsto> (fst (the (sh C)), Processing))"   and "C \<noteq> Object"   and "D = fst (the (class P C))" shows "exec_step_ind (StepC C Cs) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Calling D (C # Cs)) # frs, sh')"
AMin: fixes A :: "ant set"   and z :: "int" assumes "A \<subseteq> LBset (ant z)"   and "A \<noteq> {}" shows "AMin A \<in> A \<and> (\<forall>x\<in>A. AMin A \<le> x)"
assms_3: shows "card {z \<in> A. z \<le> x} = card {z \<in> A. z \<le> y}"
comp_funpow: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "(\<circ>) f ^^ n = (\<circ>) (f ^^ n)"
exhaust_sel: fixes option :: "'c option" obtains "option = \<bottom>"   | "option = \<lfloor>the option\<rfloor>"
is_maxsplitter_def: fixes P :: "'c list \<Rightarrow> bool"   and f :: "'c list \<Rightarrow> 'c list \<times> 'c list" shows "is_maxsplitter P f = (\<forall>xs ps qs. (f xs = (ps, qs)) = (xs = ps @ qs \<and> is_maxpref P ps xs))"
ternary_lift2: fixes tv :: "ternaryvalue" shows "(eval_ternary_Not tv \<noteq> TernaryTrue) = (tv = TernaryTrue \<or> tv = TernaryUnknown)"
d_star_sim1: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "\<And>x y z. d z + x \<cdot> y \<le> y \<Longrightarrow> x\<^sup>\<star> \<cdot> d z \<le> y"   and "\<And>x y z. x \<cdot> d y \<cdot> z = x \<cdot> (d y \<cdot> z)"   and "\<And>x y z. d x \<cdot> y \<cdot> z = d x \<cdot> (y \<cdot> z)"   and "x \<cdot> d z \<le> d z \<cdot> y" shows "x\<^sup>\<star> \<cdot> d z \<le> d z \<cdot> y\<^sup>\<star>"
prems_1: shows "I \<in> {..Suc (Suc n)} \<rightarrow> {J. ideal J R}"
i2: shows "closed ((\<lambda>x. x - P *\<^sub>V x) -` {0::'a})"
space_count_space: fixes \<Omega> :: "'d set" shows "space (count_space \<Omega>) = \<Omega>"
some_betw: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "c \<in> Q"   and "a \<noteq> b"   and "a \<noteq> c"   and "b \<noteq> c" shows "betw a b c \<or> betw b c a \<or> betw c a b"
access_matrix_pretty_ipv4_def: fixes c :: "parts_connection"   and rs :: "32 simple_rule list" shows "access_matrix_pretty_ipv4 c rs \<equiv> if \<not> simple_firewall_without_interfaces rs then undefined else let (V, E) = access_matrix c rs; formatted_nodes = map (\<lambda>(v_repr, v_range). (ipv4addr_toString v_repr, ipv4addr_wordinterval_toString v_range)) V in Let (map (\<lambda>(s, d). (ipv4addr_toString s, ipv4addr_toString d)) E) (Pair formatted_nodes)"
UnionP_Mem_E: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and z :: "tm"   and a :: "tm"   and A :: "fm" assumes "H \<turnstile> UnionP x y z"   and "insert (a IN x) H \<turnstile> A"   and "insert (a IN y) H \<turnstile> A" shows "insert (a IN z) H \<turnstile> A"
op_dg_component_intros_1: fixes a :: "V"   and \<CC> :: "V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> op_dg \<CC>\<lparr>Obj\<rparr>"
eovisidifmifiOm: shows "(e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> m\<inverse> \<union> f\<inverse>) O m \<subseteq> m \<union> ov \<union> f\<inverse> \<union> d\<inverse> \<union> s \<union> s\<inverse> \<union> e"
oseqpTEs_6: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and u\<^sub>m\<^sub>s\<^sub>g :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}receive(u\<^sub>m\<^sub>s\<^sub>g) . p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains msg :: "'f" where "a = receive msg" and "\<sigma>' i = u\<^sub>m\<^sub>s\<^sub>g msg (\<sigma> i)" and "q = p"
h2: fixes X :: "atom fm"   and x :: "real list"   and \<Gamma> :: "real list" shows "(\<exists>f\<in>set (dnf_modified X). eval (case f of (L, F, n) \<Rightarrow> ExN n (list_conj (map fm.Atom L @ F))) (x @ \<Gamma>)) = (\<exists>(al, fl, n)\<in>set (dnf_modified X). \<exists>L. length L = n \<and> (\<forall>a\<in>set al. aEval a (L @ x @ \<Gamma>)) \<and> (\<forall>f\<in>set fl. eval f (L @ x @ \<Gamma>)))"
suffix_plus: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" shows "\<sigma> |\<^sub>s n |\<^sub>s m = \<sigma> |\<^sub>s m + n"
mgu_mg: fixes t1 :: "typ"   and t2 :: "typ"   and U :: "nat \<Rightarrow> typ"   and S :: "nat \<Rightarrow> typ" assumes "mgu t1 t2 = Some U"   and "$ S t1 = $ S t2" shows "\<exists>R. S = $ R \<circ> U"
sign_simps_33: fixes am :: "'a"   and bm :: "'a" shows "am * bm = bm * am"
extendID: fixes S :: "'a sequent"   and i :: "nat"   and \<Gamma> :: "'a form multiset"   and \<Delta> :: "'a form" assumes "extend S ( \<LM> At i  \<RM> \<Rightarrow>* At i) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "At i \<in># \<Gamma>"
frestriction_fcomp: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<circ>\<^sub>\<bullet> s) \<restriction>\<^sub>\<bullet> A = r \<restriction>\<^sup>r\<^sub>\<bullet> A \<circ>\<^sub>\<bullet> s \<restriction>\<^sup>l\<^sub>\<bullet> A"
disj_assoc: fixes P :: "'b upred"   and Q :: "'b upred"   and S :: "'b upred" shows "((P \<or> Q) \<or> S) = (P \<or> Q \<or> S)"
ordinal_divI: fixes x :: "ordinal"   and y :: "ordinal"   and q :: "ordinal"   and r :: "ordinal" assumes "x = y * q + r"   and "r < y" shows "x div y = q"
to_fun_zero: fixes a :: "'a" assumes "a \<in> carrier R" shows "to_fun \<zero>\<^bsub>P\<^esub> a = \<zero>"
no_B_in_N1: shows "\<forall>B\<in>Bot. B \<notin> N1"
ext: fixes g :: "('a E.term, 'a) arr"   and f :: "('a E.term, 'a) arr" assumes "g \<cdot> f \<noteq> null" shows "seq g f"
types_preserved_select: fixes v1 :: "v"   and v2 :: "v"   and vn :: "v"   and e :: "b_e"   and v3 :: "v"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<lparr>[$C v1, $C v2, $C vn, $e]\<rparr> \<leadsto> \<lparr>[$C v3]\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> [$C v1, $C v2, $C vn, $e] : ts _> ts'"   and "e = Select" shows "\<S>\<bullet>\<C> \<turnstile> [$C v3] : ts _> ts'"
eqd_equal: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v"   and "\<^bold>|x\<^bold>| = \<^bold>|u\<^bold>|" shows "x = u \<and> y = v"
insert_def: fixes v :: "'c"   and t :: "'c Tree" shows "HeapFunctional.insert v t = merge t (T v E E)"
vareq_def: fixes x0 :: "'c"   and t :: "real" shows "vareq x0 t = ode_d1 (flow0 x0 t)"
B2_2: fixes a :: "'n"   and b :: "'n" assumes "ball (0::'n) B2 \<subseteq> cbox a b" shows "\<exists>z. ((\<lambda>x. if x \<in> i then f x else (0::'a)) has_integral z) (cbox a b) \<and> norm (z - k2) < norm (k1 - k2) / 2"
ccSup_upper2: fixes A :: "'a set"   and u :: "'a"   and v :: "'a" assumes "countable A"   and "u \<in> A"   and "v \<le> u" shows "v \<le> Sup A"
\<kappa>_def: fixes i :: "nat"   and l :: "nat" assumes "i < length fs"   and "l \<le> i" shows "M.sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<l]) = M.sumlist (map (\<lambda>j. \<kappa> i l j \<cdot>\<^sub>v fs ! j) [0..<l])"
thms_excluded_middle: fixes H :: "'b pl set"   and p :: "'b pl"   and q :: "'b pl" shows "H |- (p -> q) -> ((p -> false) -> q) -> q"
wcode_fourtimes_invs_6: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_right_move ires rs (l, r) = (\<exists>ln rn. l = Oc # ires \<and> r = Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
ipresIGAbs_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'varSort, 'sort, 'i, 'j, 'c, 'e, 'k) model_scheme"   and MOD' :: "('l, 'm, 'varSort, 'n, 'o, 'j, 'd, 'f, 'p) model_scheme" shows "ipresIGAbs h hA MOD MOD' \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> hA (igAbs MOD xs x X) = igAbs MOD' xs x (h X)"