hcju/isabelleps huggingface dataset:
awalk_verts_rev_path: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u p v" shows "pawalk_verts v (rev_path p) = rev (pawalk_verts u p)"
RETURN_inter_aform_plane_ortho_annot: fixes p :: "nat"   and Z :: "'b \<times> 'b pdevs"   and n :: "'b"   and g :: "real" shows "RETURN (inter_aform_plane_ortho p Z n g) = (case those (map (\<lambda>b. bound_intersect_2d_ud p (inner2_aform Z n b) g) Basis_list) of None \<Rightarrow> RETURN None | Some mMs \<Rightarrow> ASSERT (length mMs = DIM('b)) \<bind> (\<lambda>_. let l = sum_list (map2 (\<lambda>b m. m *\<^sub>R b) Basis_list (map fst mMs)); u = sum_list (map2 (\<lambda>b M. M *\<^sub>R b) Basis_list (map snd mMs)) in RETURN (Some (aform_of_ivl l u))))"
empt: fixes x :: "nat \<times> nat" assumes "prod_is_node A B x" shows "\<not> prod_memb x (prod_empt A B)"
induct: fixes P :: "X3 \<Rightarrow> bool"   and X3 :: "X3" assumes "P X3.Xd1"   and "P X3.Xd1'"   and "P X3.Xd2" shows "P X3"
\<mu>_def: fixes f :: "'b \<Rightarrow> real"   and u :: "'a"   and v :: "'a" shows "\<mu> f u v \<equiv> INF p\<in>{p. awalk u p v}. ereal (awalk_cost f p)"
fds_converges: fixes f :: "'a fds"   and s :: "'a" assumes "conv_abscissa f < ereal (s \<bullet> (1::'a))" shows "fds_converges f s"
combine_pairs_prop: fixes \<P> :: "(('a \<times> 'c \<times> 'a) set \<times> ('a \<times> 'c \<times> 'a) set) set"   and \<delta> :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and q\<^sub>0 :: "'a"   and w :: "nat \<Rightarrow> 'c" shows "(\<forall>P\<in>\<P>. accepting_pair\<^sub>R \<delta> q\<^sub>0 P w) = accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (local.combine_pairs \<P>) w"
semiring_normalization_rules_18: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and lx :: "'b"   and rx :: "'b"   and ry :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "times lx (times rx ry) = times (times lx rx) ry"
th2: shows "(\<Sum>f\<in>{p. p permutes UNIV}. det (\<chi>i. A $ i $ f i *s B $ f i)) = det A * det B"
sbintrunc_sbintrunc_min: fixes m :: "nat"   and n :: "nat"   and w :: "int" shows "signed_take_bit m (signed_take_bit n w) = signed_take_bit (min m n) w"
univ_poly_infinite_dimension: fixes K :: "'a list set" assumes "subfield K (A [X])" shows "\<not> ring.finite_dimension (K [X]\<^bsub>A [X]\<^esub>) (UP_A.poly_of_const ` K) (carrier (K [X]\<^bsub>A [X]\<^esub>))"
poly_y_x_fixed_deg: fixes p :: "'c poly poly" shows "poly_y_x p = (\<Sum>i\<le>degree p. \<Sum>j\<le>Max {degree (coeff p i) |i. i \<le> degree p}. monom (monom (coeff (coeff p i) j) i) j)"
axioms_2: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FGModule G smult V" shows "RModule (aezfun_setspan G) smult V"
aa_bb_pos_1: fixes k :: "nat" shows "0 < aa k"
Interp_imp_interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and D' :: "'a literal multiset" assumes "C \<le> D"   and "D < D'"   and "Interp D \<Turnstile> C" shows "interp D' \<Turnstile> C"
tensor_smult0: fixes A :: "'b tensor" shows "(0::'b) \<cdot> A = tensor0 (dims A)"
simps_1: fixes xs :: "'a msstream" shows "mssnth xs 0 = (case xs of MSSCons x xs \<Rightarrow> x)"
wellformed_items_\<I>: fixes k :: "nat" shows "wellformed_items (\<I> k)"
upower_Suc_Healthy: fixes P :: "'a hrel"   and n :: "nat" assumes "P is \<H>" shows "power.power II (;;) P (Suc n) is \<H>"
outgoing_def: fixes G :: "('a, 'b) graph_scheme"   and x :: "'a" shows "\<^bold>O\<^bold>U\<^bold>T\<^bsub>G\<^esub> x = {y. (x, y) \<in> \<^bold>E\<^bsub>G\<^esub>}"
\<alpha>n_\<alpha>nl: fixes fg :: "('d, 'a, 'e, 'c, 'f) flowgraph_rec_scheme" shows "\<alpha>n fg \<circ> le_rem_s = \<alpha>nl fg"
count_diff: fixes M :: "'c multiset"   and N :: "'c multiset"   and a :: "'c" shows "count (M - N) a = count M a - count N a"
fnth_def: fixes s :: "'b fset"   and n :: "nat" shows "s |!| n = list_of_fset s ! n"
a3: shows "\<forall>S'. S' supports x \<longrightarrow> S \<subseteq> S'"
remove_def: fixes x :: "'x"   and A :: "'x set" shows "Set.remove x A = A - {x}"
vec_of_ell2_inverse: fixes B :: "'a ell2" shows "ell2_of_vec (vec_of_ell2 B) = B"
slp_of_fas_def: fixes fas :: "floatarith list" shows "slp_of_fas fas = (let (M, slp) = slp_of_fas' fas Mapping.empty []; fasi = map (the \<circ> Mapping.lookup M) fas in Let (map2 (\<lambda>a b. Var (length slp + a - Suc b)) [0..<length fasi] (rev fasi)) ((@) slp))"
ennreal_suminf_SUP_eq_directed: fixes I :: "'a set"   and f :: "'a \<Rightarrow> nat \<Rightarrow> ennreal" assumes "\<And>N i j. \<lbrakk>i \<in> I; j \<in> I; finite N\<rbrakk> \<Longrightarrow> \<exists>k\<in>I. \<forall>n\<in>N. f i n \<le> f k n \<and> f j n \<le> f k n" shows "(\<Sum>n. SUP i\<in>I. f i n) = (SUP i\<in>I. \<Sum>n. f i n)"
sym_preproc_def: fixes gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "sym_preproc gs fs = sym_preproc_aux gs [] (Keys_to_list fs, fs)"
cosh_dist_positive: fixes a :: "proj2"   and b :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2" shows "0 < cosh_dist a b"
uniform_complement_block_size: fixes bl :: "'a set" assumes "bl \<in># \<B>\<^sup>C" shows "int (card bl) = des.\<v> - \<k>"
distinct_set_conv_list: fixes xs :: "'c list"   and g :: "'c \<Rightarrow> 'd" assumes "distinct xs" shows "prod g (set xs) = prod_list (map g xs)"
I_intP_Fext_not_intT: fixes p :: "'psym"   and al :: "univ list" assumes "polC p = Fext"   and "\<not> list_all2 Ik_intT (parOf p) al" shows "\<not> I_intP p al"
step_elim_cases_11: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and u :: "('e, 'd, 'f) com \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Catch c1 c2, s) \<rightarrow> u" obtains c\<^sub>1' :: "('e, 'd, 'f) com" and s' :: "('e, 'f) xstate" where "u = (Catch c\<^sub>1' c2, s')" and "\<Gamma>\<turnstile> (c1, s) \<rightarrow> (c\<^sub>1', s')"   | "c1 = Skip" and "u = (Skip, s)"   | sa :: "'e" where "u = (c2, Normal sa)" and "c1 = Throw" and "s = Normal sa"   | f :: "'f" where "s = Fault f" and "u = (Skip, Fault f)" and "redex c1 = Catch c1 c2"   | "s = Stuck" and "u = (Skip, Stuck)" and "redex c1 = Catch c1 c2"
put\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_def: fixes element_ptr :: "(_) element_ptr"   and element :: "('o, 'm, 'n, 'p, 's option) RElement_scheme" shows "put element_ptr element = put (cast element_ptr) (cast element)"
head_is_vert: fixes G :: "nat graph_rec"   and u :: "nat"   and v :: "nat" assumes "cfg_doms G"   and "(u, v) \<in> g_E G" shows "v \<in> set (g_V G)"
that: fixes k :: "nat" assumes "u <p r \<^sup>@ k" shows "thesis"
compiler_axioms_def: fixes load1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and load2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and match :: "'h \<Rightarrow> 'e \<Rightarrow> 'g \<Rightarrow> bool"   and compile :: "'d \<Rightarrow> 'f option" shows "compiler_axioms load1 load2 match compile \<equiv> \<forall>p1 p2 s2. compile p1 = Some p2 \<longrightarrow> load2 p2 s2 \<longrightarrow> (\<exists>s1 i. load1 p1 s1 \<and> match i s1 s2)"
into_itrm_cong: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and x :: "'a itrm"   and y :: "'a itrm" assumes "R x y" shows "itrm_cong R x y"
Form_Body_imp_inter_scheme: fixes ka :: "nat"   and kb :: "nat"   and xs :: "nat list"   and ys :: "nat list"   and zs :: "nat list" assumes "Form_Body ka kb xs ys zs"   and "0 < kb"   and "kb \<le> ka"   and "ka \<le> Suc kb" shows "zs = inter_scheme (ka + kb - Suc 0) {xs, ys}"
mcont2mcont_insert: fixes lub :: "'f set \<Rightarrow> 'f"   and ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and t :: "'f \<Rightarrow> 'g set"   and x :: "'g" assumes "mcont lub ord \<Union> (\<subseteq>) t" shows "mcont lub ord \<Union> (\<subseteq>) (\<lambda>xa. insert x (t xa))"
correct2: fixes \<D> :: "nat \<Rightarrow> ('k + 'h, 'l + 'j) distinguisher" assumes "\<And>\<eta>. \<I>_inner2 \<eta> \<oplus>\<^sub>\<I> \<I>_common2 \<eta> \<turnstile>g \<D> \<eta> \<surd>"   and "\<And>\<eta>. interaction_any_bounded_by (\<D> \<eta>) (bound2 \<eta>)"   and "\<And>\<eta>. lossless2 \<Longrightarrow> plossless_gpv (\<I>_inner2 \<eta> \<oplus>\<^sub>\<I> \<I>_common2 \<eta>) (\<D> \<eta>)" shows "(\<forall>\<eta>. wiring (\<I>_inner2 \<eta>) (\<I>_real2 \<eta>) (cnv2 \<eta>) (w2 \<eta>)) \<and> Negligible.negligible (\<lambda>\<eta>. advantage (\<D> \<eta>) (ideal2 \<eta>) (cnv2 \<eta> |\<^sub>= 1\<^sub>C \<rhd> real2 \<eta>))"
that: fixes x :: "(real, 'n) vec"   and y :: "(real, 'n) vec" assumes "x \<noteq> y"   and "x \<in> S"   and "y \<in> S"   and "\<And>i. (x - y) $ i \<in> \<int>" shows "thesis"
p: shows "p = (\<Prod>(a, i)\<in>as. a ^ Suc i)"
funpow_dist_step: fixes x :: "'b"   and y :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "x \<noteq> y"   and "y \<in> orbit f x" shows "funpow_dist f x y = funpow_dist1 f x y"
hyps: fixes y :: "'a list"   and sl :: "'a list" assumes "length y < length xs" shows "quicksort_by_rel R sl y = quicksort_by_rel R [] y @ sl"
unit_less: fixes i :: "nat"   and n :: "nat"   and x :: "nat list"   and b :: "nat" assumes "i < n"   and "x <\<^sub>v (zeroes n)[i := b]" shows "x ! i < b \<and> (\<forall>j<n. j \<noteq> i \<longrightarrow> x ! j = 0)"
real_sin_eq: fixes x :: "real" shows "Re (sin (complex_of_real x)) = sin x"
cosh_real_at_top: shows "filterlim cosh at_top at_top"
intro: fixes V :: "'b set"   and f :: "'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real" assumes "\<exists>c. \<forall>x\<in>V. \<bar>f x\<bar> \<le> c * norm x" shows "continuous_axioms V f norm"
CartProd_def: fixes A :: "ZF"   and B :: "ZF" shows "CartProd A B \<equiv> HOLZF.Sum (Repl A (\<lambda>a. Repl B (Opair a)))"
"case": fixes f :: "'h \<Rightarrow> 'g"   and x :: "'h" shows "(case object_ptr.Ext x of object_ptr.Ext x \<Rightarrow> f x) = f x"
may_acquire_allI: fixes ln :: "'c \<Rightarrow>f nat"   and ls :: "('c, 'd) locks"   and t :: "'d" assumes "\<And>l. 0 < ln $ l \<Longrightarrow> may_lock (ls $ l) t" shows "may_acquire_all ls t ln"
uses_in_\<alpha>n: fixes v :: "'var"   and g :: "'g"   and n :: "'node" assumes "v \<in> uses g n" shows "n \<in> set (\<alpha>n g)"
u_servTicket_authentic: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "B \<noteq> Tgs"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "\<exists>Ta authK. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs \<and> Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace>) \<in> set evs \<and> servKlife + Ts \<le> authKlife + Ta"
Basis_zero: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "zero \<notin> Basis"
frame_hoare_r: fixes a :: "'a \<Longrightarrow> 'b"   and r :: "'b upred"   and q :: "'b upred"   and p :: "'b upred"   and P :: "'b hrel" assumes "vwb_lens a"   and "a \<sharp> r"   and "a \<natural> q"   and "\<lbrace>p\<rbrace> P \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p \<and> r\<rbrace> {a}:[P] \<lbrace>q \<and> r\<rbrace>\<^sub>u"
inverse_im_exists': fixes T :: "'v \<Rightarrow> 'w" assumes "T \<in> GRepHomSet (\<star>) W" shows "\<exists>!S. VectorSpaceHom (\<currency>\<currency>) indV (\<sharp>\<star>) S \<and> map S (concat indVfbasis) = concat (negHorbit_homVfbasis T)"
oth_class_taut_4_b: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<phi> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<not>\<phi> in v]"
Int_Diff: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and B :: "'c set" assumes "finite A" shows "sum g A = sum g (A \<inter> B) + sum g (A - B)"
sup_left_commute: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
R1_def: shows "R1 = {(V, V'). (V, V') \<in> R1' \<and> set (PPV V) \<subseteq> set (PPc c1) \<and> set (PPV V') \<subseteq> set (PPc c1)}"
simps_1: fixes all :: "String.literal"   and S :: "typ"   and T :: "typ"   and t :: "Core.term" shows "strip_all_vars (Ct all S $ Abs T t) = (if all = STR ''Pure.all'' \<and> S = (T \<rightarrow> propT) \<rightarrow> propT then T # strip_all_vars t else [])"
is_SNF_Smith_2xn: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat 2 n" shows "is_SNF A (Smith_2xn A)"
ccBind_strict: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and v :: "Vars.var"   and e :: "exp" shows "CoCallAnalysis.ccBind ccExp v e\<cdot>\<bottom> = \<bottom>"
is_nextElem_b: fixes vs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "is_nextElem vs a b" shows "b \<in> set vs"
domI: fixes x :: "'a" assumes "\<not> Q x \<Longrightarrow> local.dom (h x)" shows "local.dom x"
less_natural_def: shows "(<) \<equiv> map_fun nat_of_natural (map_fun nat_of_natural id) (<)"
Bit_B1: fixes k :: "int" shows "k BIT True = k + k + 1"
normFaces_makeFaceFinalFaceList: fixes f :: "face"   and fs :: "face list" shows "normFaces (makeFaceFinalFaceList f fs) = normFaces fs"
cod_simps_2: shows "src_src.cod.leg0 = src_src.leg0"
\<gamma>1''': shows "\<gamma>1''' \<in> seq ars"
closedin_closed_subtopology: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set" assumes "closedin X S" shows "closedin (subtopology X S) T = (closedin X T \<and> T \<subseteq> S)"
dist_fs_sym: fixes z1 :: "complex_homo"   and z2 :: "complex_homo" shows "dist_fs z1 z2 = dist_fs z2 z1"
axioms_2: 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 r :: "'b"   and \<rho> :: "'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s :: "'b"   and \<sigma> :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and \<mu> :: "'b" assumes "arrow_of_tabulations_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 \<mu>" shows "tabulation_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1"
L_init: fixes r :: "'b rexp" assumes "local.wf n r" shows "L (init r) = lang n r"
fmpredI: fixes m :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. m $$ x = Some y \<Longrightarrow> P x y" shows "fmpred P m"
dom_simp: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "VVV.dom f = VxVxV.dom f"
find_consistent_signs_at_roots: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "\<And>q. q \<in> set qs \<Longrightarrow> algebraic_semidom_class.coprime p q" shows "set (find_consistent_signs_at_roots p qs) = set (characterize_consistent_signs_at_roots p qs)"
Mismatch: fixes P :: "pi"   and Rs :: "residual"   and a :: "name"   and b :: "name" assumes "P \<longmapsto> Rs"   and "a \<noteq> b" shows "[a\<noteq>b]P \<longmapsto> Rs"
atoms_biimp: fixes F :: "'a formula"   and G :: "'a formula" shows "atoms (F \<^bold>\<leftrightarrow> G) = atoms F \<union> atoms G"
rat_number_collapse_5: shows "Fract (- 1) 1 = - 1"
assms: shows "\<forall>x\<in>fv t. (\<exists>f. \<theta> x = Fun f []) \<or> (\<exists>y. \<theta> x = Var y)"
cases: fixes a1 :: "nat"   and a2 :: "com" assumes "a1 \<turnstile> a2" obtains l :: "nat" where "a1 = la__" and "a2 = SKIP"   | x :: "char list" and a :: "aexp" and l :: "nat" where "a1 = lb__" and "a2 = x ::= a" and "sec a \<le> sec x" and "lb__ \<le> sec x"   | l :: "nat" and c\<^sub>1 :: "com" and c\<^sub>2 :: "com" where "a1 = lc__" and "a2 = c\<^sub>1__;; c\<^sub>2__" and "lc__ \<turnstile> c\<^sub>1__" and "lc__ \<turnstile> c\<^sub>2__"   | b :: "bexp" and l :: "nat" and c\<^sub>1 :: "com" and c\<^sub>2 :: "com" where "a1 = l" and "a2 = IF b__ THEN c\<^sub>1 ELSE c\<^sub>2" and "max (sec b__) l \<turnstile> c\<^sub>1" and "max (sec b__) l \<turnstile> c\<^sub>2"   | b :: "bexp" and c :: "com" where "a1 = 0" and "a2 = WHILE b DO c" and "sec b = 0" and "0 \<turnstile> c"
less_sqmatrix_def: fixes A :: "('a, 'b) sqmatrix"   and B :: "('a, 'b) sqmatrix" shows "A < B \<equiv> A \<le> B \<and> A \<noteq> B"
chg_map_upd: fixes m :: "'c \<Rightarrow> 'b option"   and a :: "'c"   and b :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "m a = Some b" shows "chg_map f a m = m(a \<mapsto> f b)"
rScope: fixes P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> x\<alpha> \<prec> P'"   and "xP' = \<lparr>\<nu>x\<rparr>P'" shows "Prop"
fcomp_comp: fixes f :: "'g \<Rightarrow> 'i"   and g :: "'i \<Rightarrow> 'h" shows "fcomp f g = g \<circ> f"
prv_imp_trans1: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla" shows "prv (imp (imp \<chi> \<psi>) (imp (imp \<phi> \<chi>) (imp \<phi> \<psi>)))"
Fst: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> parts H" shows "X \<in> parts H"
simps_6: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case Delete x2 of Insert x \<Rightarrow> f1 x | Delete x \<Rightarrow> f2 x) = f2 x2"
eqvt: shows "eqvt Strong_Early_Bisim.bisim"
verticesFrom_def: fixes f :: "face" shows "verticesFrom f \<equiv> rotate_to (vertices f)"
of_int_less_numeral_iff: fixes z :: "int"   and n :: "num" shows "(of_int z < numeral n) = (z < numeral n)"
pres: shows "\<forall>(q, \<tau>)\<in>set (step (hd b) (a ! hd b)). \<tau> \<in> A"
size_mset_union: fixes M :: "('c \<times> 'd) multiset"   and N :: "('c \<times> 'd) multiset" shows "size_mset (M + N) = size_mset N + size_mset M"
union_frontiers_nonneg: fixes c :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t" shows "0 \<le> zcount (\<Sum>loc'\<in>UNIV. after_summary (zmset_of (mset_set (set_antichain (frontier (c_imp c loc'))))) (summary loc' loc)) t"
intros: fixes \<delta> :: "('d, 'e) ta_rule set"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option" assumes "\<And>r. r \<in> \<delta> \<Longrightarrow> rcm r = Some (card (set (rhsq r)))" shows "(br_iq \<delta>, br_iq \<delta>, rcm) \<in> br'_initial \<delta>"
fst_set_full_gb: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "fst ` set (full_gb bs) = (\<lambda>v. monomial (1::'d) (term_of_pair (0::'a, component_of_term v))) ` Keys (fst ` set bs)"
contg: shows "continuous_on ({x. b \<bullet> x = 0} \<inter> T) g"
inj_on_Plus: fixes a :: "nat"   and A :: "nat set" shows "inj_on ((+) a) A"
simps_8: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm"   and q :: "fm" shows "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)"
lm008: fixes X :: "'c set"   and Y :: "'d set" shows "injections X Y = totalRels X Y \<inter> injectionsUniverse"
assms: shows "v \<Turnstile>\<^sub>c\<^sub>s set cs"
T_completeness: 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"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP" shows "CM.Model wtFsym wtPsym arOf resOf parOf \<Phi> (ProblemIkTpart_TEModel.intT prot protFw eintT eintF) (ProblemIkTpart_TEModel.intF arOf resOf prot protFw eintT eintF) (ProblemIkTpart_TEModel.intP parOf prot protFw eintT eintF eintP)"
finprod_infinite: fixes G :: "('c, 'd) monoid_scheme"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'c" assumes "Group.comm_monoid G"   and "infinite A" shows "finprod G f A = \<one>\<^bsub>G\<^esub>"
append_prefixD: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "prefix (xs @ ys) zs" shows "prefix xs zs"
t18_18: 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 D :: "'a"   and E :: "'a"   and C :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B D E"   and "Cong A C D F"   and "Tarski_neutral_dimensionless.LtA Bet Cong F D E C A B" shows "Tarski_neutral_dimensionless.Lt Bet Cong E F B C"
hyps: shows "(\<forall>u\<in>set u. fst u \<noteq> r) = no_illegal_updates_code u r"
inv: shows "FGModuleIso G (\<star>) W (\<cdot>) invT V"
lm063: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "a \<in> allAllocations N G" shows "sum (summedBidVector bids N G) a = sum (sum bids \<circ> omega) a"
assms_1: shows "S \<union> T retract_of UNIV"
ncf5: fixes y :: "real" assumes "x \<le> y"   and "y \<le> 0" shows "0 < numer_cf5 y"
prognosis_Var_lam: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and u :: "Arity"   and as :: "Arity list"   and S :: "stack_elem list"   and a :: "Arity" assumes "CardinalityPrognosisVar prognosis"   and "map_of \<Gamma> x = Some e"   and "ae x = up\<cdot>u"   and "isVal e" shows "prognosis ae as u (\<Gamma>, e, S) \<sqsubseteq> record_call x\<cdot>(prognosis ae as a (\<Gamma>, Var x, S))"
add: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and b1 :: "'e"   and b2 :: "'e" assumes "module_hom s1 s2 f" shows "f (b1 + b2) = f b1 + f b2"
reflp_list_all2: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflp R" shows "reflp (list_all2 R)"
face_cycle_set_succ: fixes a :: "'b" shows "G.face_cycle_set (G.face_cycle_succ a) = G.face_cycle_set a"
a_in_rcs: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and a :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G" shows "a \<in> H \<bullet>\<^bsub>G\<^esub> a"
ideal_inc_one1: fixes I :: "'a set" assumes "ideal R I" shows "(1\<^sub>r \<in> I) = (I = carrier R)"
cSUP_upper2: fixes f :: "'d \<Rightarrow> 'c multiset"   and A :: "'d set"   and x :: "'d"   and u :: "'c multiset" assumes "subset_mset.bdd_above (f ` A)"   and "x \<in> A"   and "u \<subseteq># f x" shows "u \<subseteq># \<Squnion> (f ` A)"
closed_interval_Int_compl: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "x < y"   and "y < z" shows "- {x..y} \<inter> - {y..z} = - {x..z}"
mult_assoc: fixes a :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and c :: "('b \<times> 'b) set" shows "(a O b) O c = a O b O c"
infer_varE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and x :: "'a"   and \<tau> :: "PreSimplyTyped.type" assumes "infer \<Gamma> (Var x) = Some \<tau>" shows "\<Gamma> x = Some \<tau>"