hcju/isabelleps huggingface dataset:
random_variable_component: fixes i :: "nat" shows "random_variable (count_space UNIV) (\<lambda>f. f i)"
dense_list_of_pdevs_def: fixes x :: "'a pdevs" shows "dense_list_of_pdevs x = map (pdevs_apply x) [0..<degree x]"
sel: fixes x :: "'a \<Rightarrow> 'b + 'c \<times> 'a" shows "the_prog (Prog x) = x"
dX_def: fixes X :: "'a set"   and d :: "'b" shows "Pdds'.dX X d = {x \<in> X. Xd x = d}"
fix_index: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and n :: "nat"   and v :: "'d \<Rightarrow> nat" assumes "\<one> < M 0 i + M i 0"   and "cycle_free M n"   and "canonical M n"   and "i \<le> n"   and "i \<noteq> 0" shows "\<exists>M'. ((\<exists>u. u \<turnstile>\<^bsub>v,n\<^esub> M') \<longrightarrow> (\<exists>u. u \<turnstile>\<^bsub>v,n\<^esub> M)) \<and> M' 0 i + M' i 0 = \<one> \<and> cycle_free M' n \<and> (\<forall>j. i \<noteq> j \<and> M 0 j + M j 0 = \<one> \<longrightarrow> M' 0 j + M' j 0 = \<one>) \<and> (\<forall>j. i \<noteq> j \<and> \<one> < M 0 j + M j 0 \<longrightarrow> \<one> < M' 0 j + M' j 0)"
5: shows "- Rep_regular x \<le> Rep_dense (stone_phi.rho_pair (x, y))"
vfinite_vunionI: fixes A :: "V"   and B :: "V" assumes "vfinite A"   and "vfinite B" shows "vfinite (A \<union>\<^sub>\<circ> B)"
succ_\<omega>_iff: fixes n :: "V" shows "(ZFC_in_HOL.succ n \<in> elts \<omega>) = (n \<in> elts \<omega>)"
insert_DiffM: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "add_mset x (remove1_mset x M) = M"
star_trans: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
\<beta>_simps_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and f :: "'b"   and g :: "'b"   and u :: "'b"   and w :: "'b"   and \<theta> :: "'b"   and w' :: "'b"   and \<theta>' :: "'b"   and \<beta> :: "'b" assumes "uw\<theta>w'\<theta>'\<beta> V H \<a> \<iota> src trg r \<rho> f g u w \<theta> w' \<theta>' \<beta>" shows "partial_magma.arr V \<beta>"
strict_mono_mono: fixes f :: "'a list \<Rightarrow> 'b" assumes "border_order.strict_mono f" shows "border_order.mono f"
digraph_map_del_vert: shows "digraph_map (del_vert u) M"
act: shows "\<exists>i. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"
Synchronized1Throw2Fail: fixes uf :: "bool"   and xs :: "'addr val list"   and V :: "nat"   and a' :: "'addr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and ad :: "'addr"   and h :: "'heap" assumes "uf"   and "xs ! V = Addr a'"   and "V < length xs" shows "uf,P,t \<turnstile>1 \<langle>insync\<^bsub>V\<^esub> (a) Throw ad,(h, xs)\<rangle> -\<lbrace>(UnlockFail, a')\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt IllegalMonitorState),(h, xs)\<rangle>"
openin_subopen: fixes U :: "'b topology"   and S :: "'b set" shows "openin U S = (\<forall>x\<in>S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
sup_ge2: fixes y :: "'d multiset"   and x :: "'d multiset" shows "y \<subseteq># x \<union># y"
P: shows "fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma ! k = Some y"
nvars_Extend: fixes k :: "'k"   and i :: "nat"   and \<AA> :: "'i"   and P :: "'v" shows "#\<^sub>V (Extend k i \<AA> P) = SUC k (#\<^sub>V \<AA>)"
poly_add: fixes p :: "'b poly"   and q :: "'b poly"   and x :: "'b" shows "poly (p + q) x = poly p x + poly q x"
div_add: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
sin_ge_zero: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> pi" shows "0 \<le> sin x"
intros_1: fixes \<tau> :: "'a basic_type"   and \<sigma> :: "'a basic_type" assumes "\<tau> \<sqsubset>\<^sub>B \<sigma>" shows "\<tau>[1] \<sqsubset> \<sigma>[1]"
\<phi>_natural: fixes f :: "'c"   and x :: "'c"   and x' :: "'c"   and g :: "'d"   and y' :: "'d"   and y :: "'d"   and h :: "'c" assumes "\<guillemotleft>f : x \<rightarrow>\<^sub>C x'\<guillemotright>"   and "\<guillemotleft>g : y' \<rightarrow>\<^sub>D y\<guillemotright>"   and "\<guillemotleft>h : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<phi> y' (f \<cdot>\<^sub>C h \<cdot>\<^sub>C F g) = (G f \<cdot>\<^sub>D \<phi> y h) \<cdot>\<^sub>D g"
lprod_3_3: fixes xr :: "'a"   and x :: "'a"   and R :: "('a \<times> 'a) set"   and y :: "'a"   and z :: "'a" assumes "(xr, x) \<in> R" shows "([xr, y, z], [x, y, z]) \<in> lprod R"
carrier_subgroup_generated: fixes G :: "('a, 'b) monoid_scheme"   and S :: "'a set" shows "carrier (subgroup_generated G S) = generate G (carrier G \<inter> S)"
K_absorb_HK: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "H \<subseteq> K" shows "H \<diamondop> K = K"
continuous_ordinal_rec: fixes s :: "ordinal \<Rightarrow> ordinal \<Rightarrow> ordinal"   and z :: "ordinal" assumes "\<forall>p x. x \<le> s p x" shows "continuous (ordinal_rec z s)"
ns_of_s_def: fixes s :: "form list" shows "ns_of_s s = map (Pair 0) s"
cyclic_cong: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "\<And>s. s \<in> S \<Longrightarrow> f s = g s" shows "cyclic_on f S = cyclic_on g S"
ss_p_subset: shows "\<forall>p<length ss. set (ss ! p) \<subseteq> set nodes"
inverse_renaming: fixes atom_ordering :: "('a \<times> 'a) set"   and A :: "'a set"   and L :: "'a Literal" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.rename_literal A (propositional_atoms.rename_literal A L) = L"
pinf_3: fixes t :: "'c" shows "\<exists>z. \<forall>x>z. (x = t) = False"
correct_def: shows "abstract_com.correct \<equiv> \<forall>m. valid_msg m \<longrightarrow> spmf (abstract_com.correct_game m) True = 1"
polish_projective_def: fixes I :: "'b set"   and P :: "'b set \<Rightarrow> ('b \<Rightarrow> 'c) measure" shows "polish_projective I P \<equiv> projective_family I P (\<lambda>_. borel)"
in_apartments_2: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "D \<in> A \<inter> B'"
j: shows "j < dim_col (A $$ (0, 0) \<cdot>\<^sub>m B)"
at_ds3: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "at TYPE('a)"   and "distinct [a, b, c]" shows " [(a, c), (b, c), (a, c)] \<triangleq> [(a, b)] "
transpE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "transp r"   and "r x y"   and "r y z" obtains "r x z"
cofinite_def: shows "cofinite = Abs_filter (\<lambda>P. finite {x. \<not> P x})"
abc_list_crsp_simp2: fixes lma :: "nat list"   and lmb :: "nat list"   and m :: "nat"   and n :: "nat" assumes "abc_list_crsp lma lmb"   and "m < length lma"   and "\<not> m < length lmb" shows "abc_list_crsp (lma[m := n]) (lmb @ 0 \<up> (m - length lmb) @ [n])"
inter_sctn_specs_def: fixes d :: "nat"   and X :: "real list set"   and sctn :: "real list sctn" shows "inter_sctn_specs d X sctn = SPEC (\<lambda>R. env_len R d \<and> X \<inter> plane_ofs sctn \<subseteq> R \<and> R \<subseteq> plane_ofs sctn)"
qe_eps\<^sub>1_def: fixes f :: "atom fm" shows "qe_eps\<^sub>1 f = (let as = atoms\<^sub>0 f; lbs = lbounds as; ebs = ebounds as in list_disj (inf\<^sub>- f # map (subst\<^sub>+ f) lbs @ map (subst f) ebs))"
inv_in_homI: fixes f :: "('a, 'a set) Maps.arr"   and a :: "('a, 'a set) Maps.arr"   and b :: "('a, 'a set) Maps.arr" assumes "Maps.iso f"   and "Maps.in_hom f a b" shows "Maps.in_hom (Maps.inv f) b a"
indep_system_subset: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> carrier" shows "indep_system \<E> (\<E>.indep_in \<E>)"
alphaAbs_all_equal_or_qFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_all_equal_or_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. (y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
cut_ge_Int_right: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>\<ge> t = A \<inter> B \<down>\<ge> t"
Tw_func: fixes T :: "('e, 'f) Order_scheme" assumes "Worder T"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier T. ord_equiv (Iod D (segment D a)) (Iod T (segment T b))" shows "Tw\<^bsub>D,T\<^esub> \<in> carrier D \<rightarrow> carrier T"
assms_9: shows "NQ_pos = newQuery J Q_J_pos (I, t)"
strategy_attractsE: fixes A :: "'a set"   and W :: "'a set" assumes "strategy_attracts p** \<sigma>' A W"   and "v \<in> A" shows "visits_via P A W"
trans: fixes x :: "'a\<^sub>m"   and y :: "'a\<^sub>m"   and z :: "'a\<^sub>m" assumes "bo x y"   and "bo y z"   and "x \<in> UNIV" shows "bo x z"
simps_5: fixes \<pi> :: "'b prm"   and A :: "'b ptrm"   and B :: "'b ptrm" shows "\<pi> \<bullet> PPair A B = PPair (\<pi> \<bullet> A) (\<pi> \<bullet> B)"
key_channel_recv_def: fixes s :: "bool list option \<times> bool list cstate"   and m :: "'a" shows "local.key_channel_recv s m \<equiv> \<dagger>channel.recv_oracle s () \<bind> (\<lambda>(c, s). case c of None \<Rightarrow> return_spmf (None, s) | Some c' \<Rightarrow> (key.key_oracle (key \<eta>))\<dagger> s () \<bind> (\<lambda>(k, s). return_spmf (dec \<eta> k c', s)))"
word_ao_nth: fixes x :: "'a word"   and y :: "'a word"   and n :: "nat" shows "(x OR y) !! n = (x !! n \<or> y !! n) \<and> (x AND y) !! n = (x !! n \<and> y !! n)"
dimension_pair: shows "DIM('a \<times> 'b) = DIM('a) + DIM('b)"
border_length_le: fixes xs :: "'b list"   and ys :: "'b list" assumes "border xs ys" shows "length xs \<le> length ys"
scene_indep_copy: fixes X :: "'a scene"   and Y :: "'a scene" shows "(X \<bowtie>\<^sub>S Y) = (\<forall>s\<^sub>1 s\<^sub>2. (\<lambda>s'. s' \<oplus>\<^sub>S s\<^sub>1 on X) \<circ> (\<lambda>s'. s' \<oplus>\<^sub>S s\<^sub>2 on Y) = (\<lambda>s'. s' \<oplus>\<^sub>S s\<^sub>2 on Y) \<circ> (\<lambda>s'. s' \<oplus>\<^sub>S s\<^sub>1 on X))"
lcoeff_gbinomial_series_aux: fixes abort :: "bool"   and x :: "real"   and m :: "nat"   and n :: "nat" shows "lcoeff (gbinomial_series_aux abort x (real m) (x gchoose m)) n = x gchoose (n + m)"
axioms_5: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'" shows "monoid_homomorphism \<eta> R multiplication unit R' multiplication' unit'"
hyps_1: shows "\<Gamma>\<turnstile> cfg\<^sub>1 \<rightarrow> cfg''"
cod_eqI: fixes b :: "'c"   and f :: "'c" assumes "C.ide b"   and "C.seq b f" shows "C.cod f = b"
assms: shows "\<forall>e f g. e,f,g,z \<Turnstile> p"
intros_2: fixes l :: "nat"   and ts :: "('c, 'd) bintree list"   and t :: "('c, 'd) bintree" assumes "is_bintree_list l ts"   and "is_bintree_list l (children t)" shows "is_bintree_list (Suc l) (t # ts)"
keysFor_parts_insert: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "K \<in> keysFor (parts (insert X H))"   and "X \<in> synth (analz H)" shows "K \<in> keysFor (parts H) \<or> Key (invKey K) \<in> parts H"
fps_mult_right_inverse_unit_factor: fixes f :: "'a fps"   and y :: "'a" assumes "fps_nth f (subdegree f) * y = (1::'a)" shows "f * fps_right_inverse (unit_factor f) y = fps_X ^ subdegree f"
set_times_intro2: fixes b :: "'c"   and C :: "'c set"   and a :: "'c" assumes "b \<in> C" shows "a * b \<in> a *o C"
igSubstAbsIPresIGWlsAbs_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstAbsIPresIGWlsAbs MOD \<equiv> \<forall>ys Y y us s A. isInBar (us, s) \<and> igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSubstAbs MOD ys Y y A)"
word_bw_comms_1: fixes x :: "'a word"   and y :: "'a word" shows "x && y = y && x"
x_def: shows "x = \<langle>a, c\<rangle>"
adjunction_data_in_bicategory_axioms: shows "adjunction_data_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
bl2bin_drop: fixes k :: "nat"   and bl :: "bool list" shows "bl_to_bin (drop k bl) = take_bit (length bl - k) (bl_to_bin bl)"
snd_foldl_Gauss_Jordan_column_k_eq: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "snd (foldl Gauss_Jordan_column_k_PA (mat (1::'b), 0, A) [0..<k]) = foldl Gauss_Jordan_column_k (0, A) [0..<k]"
m_o_h: fixes X :: "char list set"   and opt :: "(char list \<Rightarrow> 'av) option" assumes "finite X" shows "m\<^sub>o opt X \<le> h * card X + 1"
det_interchange_rows: fixes A :: "(('b, 'c) vec, 'c) vec"   and i :: "'c"   and j :: "'c" shows "det (interchange_rows A i j) = of_int (if i = j then 1 else - 1) * det A"
poly_add_minus_mult_eq: fixes p :: "'a list"   and q :: "'a list"   and r :: "'a list" shows "poly (p *** q +++ -- (p *** r)) = poly (p *** (q +++ -- r))"
suffix_def: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'b" shows "suffix k x \<equiv> \<lambda>n. x (k + n)"
distinct_9: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c"   and x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c" shows "Send x21 x22 x23 x24 x25 x26 \<noteq> Recv x31 x32 x33 x34 x35 x36"
disjointPartitioning: fixes ss :: "'b set list" shows "disjoint (partitioning ss {})"
cSup_insert_If: fixes X :: "'a multiset set"   and a :: "'a multiset" assumes "subset_mset.bdd_above X" shows "Sup (insert a X) = (if X = {} then a else a \<union># Sup X)"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) (trg f)"
singletonI: fixes a :: "'f" shows "a \<in> {a}"
fpxs_supp_of_fls: fixes f :: "'b fls" shows "fpxs_supp (fpxs_of_fls f) = rat_of_int ` supp (fls_nth f)"
leg_simps_3: shows "C.arr trg_\<mu>.cod.leg1"
CastNull\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and C :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"
matrix_to_iarray_Gram_Schmidt_upt_k: fixes k :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "k < ncols A" shows "matrix_to_iarray (Gram_Schmidt_upt_k A k) = Gram_Schmidt_upt_k_iarrays (matrix_to_iarray A) k"
try_spmf_def: fixes p :: "'b spmf"   and q :: "'b spmf" shows "TRY p ELSE q = p \<bind> (\<lambda>x. case x of None \<Rightarrow> q | Some y \<Rightarrow> return_spmf y)"
hensel_step_def: fixes p :: "int"   and q :: "int"   and S1 :: "int poly"   and T1 :: "int poly"   and D1 :: "int poly"   and H1 :: "int poly"   and S :: "int poly"   and T :: "int poly"   and D :: "int poly"   and H :: "int poly" shows "local.hensel_step p q S1 T1 D1 H1 S T D H = (let U = poly_mod.Mp p (sdiv_poly (C - D * H) q); (A, B) = dupe_monic_dynamic p D1 H1 S1 T1 U; D' = D + Polynomial.smult q B; H' = H + Polynomial.smult q A; U' = poly_mod.Mp q (sdiv_poly (S * D' + T * H' - 1) p); (A', B') = dupe_monic_dynamic q D H S T U'; q' = p * q; S' = poly_mod.Mp q' (S - Polynomial.smult p A'); T' = poly_mod.Mp q' (T - Polynomial.smult p B') in (S', T', D', H'))"
of_nat_less_of_nat_power_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "nat"   and w :: "nat"   and x :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less (power.power one times (semiring_1.of_nat one plus zero b) w) (semiring_1.of_nat one plus zero x) = (b ^ w < x)"
simps: fixes n :: "nat" shows "string_of_nat n = (if n < 10 then [char_of (48 + n)] else string_of_nat (n div 10) @ [char_of (48 + n mod 10)])"
correctCompositionDiffLevelsA2: shows "correctCompositionDiffLevels sA2"
pwt_eq: shows "\<forall>x\<in>vars_hd \<zeta>. A (PWt x) = B (PWt x)"
step_thread_eq_None_conv: fixes update_state :: "('l, 't, 'x, 'm, 'w, 'o) thread_action \<Rightarrow> 'a"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" shows "(\<alpha>.step_thread update_state s t = None) = (t \<notin> \<alpha>.active_threads s)"
fresh_at_base_2: fixes a :: "atom"   and b :: "'a" shows "a \<sharp> b = (a \<noteq> atom b)"
dimX: shows "\<forall>n. X n \<in> carrier_mat m m"
fp_cop_F_def: fixes ds :: "'b set" shows "fp_cop_F ds = bourbaki_witt_fixpoint.fixp_above \<Union> {(x, y). x \<subseteq> y} (cop_F ds) {}"
distinct_3: fixes x1 :: "'a"   and x3 :: "'b" shows "Normal x1 \<noteq> Fault x3"
analz_image_priK: fixes evs :: "event list"   and B :: "agent"   and KK :: "nat set" assumes "evs \<in> tls" shows "(Key (priEK B) \<in> analz (Key ` KK \<union> knows Spy evs)) = (priEK B \<in> KK \<or> B \<in> bad)"
assms_2: shows "S \<Turnstile>\<^sub>P af\<^sub>G \<phi> ys"
fgh: shows "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>x. f (g (fst x)) (h (snd x)))"
\<alpha>nz: shows "\<alpha> i ^ \<delta> (i - 1) \<noteq> 0"
prime_power_cancel_less: fixes p :: "'c"   and m :: "'c"   and k :: "nat"   and m' :: "'c"   and k' :: "nat" assumes "prime p"   and "m * p ^ k = m' * p ^ k'"   and "k < k'"   and "\<not> p dvd m" shows "False"
prj'_joint_monic: fixes h :: "'a"   and a :: "'a"   and h' :: "'a" assumes "\<guillemotleft>h : a \<rightarrow>\<^sub>C \<mu>\<nu>\<pi>.chine\<guillemotright>"   and "\<guillemotleft>h' : a \<rightarrow>\<^sub>C \<mu>\<nu>\<pi>.chine\<guillemotright>"   and "Prj\<^sub>1 \<cdot> h = Prj\<^sub>1 \<cdot> h'"   and "Prj\<^sub>1\<^sub>0 \<cdot> h = Prj\<^sub>1\<^sub>0 \<cdot> h'"   and "Prj\<^sub>0\<^sub>0 \<cdot> h = Prj\<^sub>0\<^sub>0 \<cdot> h'" shows "h = h'"
extends_not_cons: fixes x :: "'b"   and S :: "'b list" shows "\<not> x # S \<lesssim> S"
tm_union_decomp: shows "tm_union prec I a t1 t2 = TaylorModel p e"
unit_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>C \<i>\<^sub>C[a] = a"
sum_upto_def: fixes f :: "nat \<Rightarrow> 'b"   and x :: "real" shows "sum_upto f x = (\<Sum>i | 0 < i \<and> real i \<le> x. f i)"
subgraph_def: fixes G\<^sub>1 :: "('c, 'd) labeled_graph"   and G\<^sub>2 :: "('c, 'd) labeled_graph" shows "subgraph G\<^sub>1 G\<^sub>2 = (graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2)"
reach_def: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and "is" :: "'b set" shows "reach A H is \<equiv> {uu_. \<exists>i j. uu_ = assign A H j \<and> i \<in> is \<and> (\<^bold>\<diamond> Nom j) at i in' H}"
univ_for_pr_def: shows "univ_for_pr = pr_conv_2_to_1 nat_to_pr"
serial_sas_plus_equivalent_to_serial_strips: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<psi> :: "('c, 'd) sas_plus_operator list" assumes "is_valid_problem_sas_plus \<Psi>"   and "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi>" shows "STRIPS_Semantics.is_serial_solution_for_problem (\<phi> \<Psi> ) (map (sasp_op_to_strips \<Psi>) \<psi>)"
disc_eq_case_2: fixes option :: "'d option" shows "(option \<noteq> None) = (case option of None \<Rightarrow> False | Some uu_ \<Rightarrow> True)"
wfp_on_imp_has_min_elt: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wfp_on P A"   and "A \<noteq> {}" shows "\<exists>x\<in>A. \<forall>y\<in>A. \<not> P y x"
ctr_1: fixes n :: "enat"   and xs :: "'c llist" assumes "n = 0 \<or> lnull xs" shows "ltake n xs = LNil"
mult: fixes u :: "'a \<Rightarrow> ennreal"   and c :: "ennreal" assumes "P u" shows "P (\<lambda>x. c * u x)"
ex_inverse_of_nat_less: fixes x :: "'c" assumes "(0::'c) < x" shows "\<exists>n>0. inverse (of_nat n) < x"
ide_is_iso: fixes a :: "'a" assumes "L'.C.ide a" shows "L'.C.iso a"
simps_2: fixes n :: "nat"   and m :: "nat" shows "pr_index_enumerator n (Suc m) = comp_by_index index_of_id (pr_index_enumerator n m)"
dist_norm: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c" assumes "class.dist_norm minus dist norm" shows "dist x y = norm (minus x y)"
ANR_imp_absolute_closed_neighbourhood_retract: fixes S :: "'a set"   and S' :: "'b set"   and U :: "'b set" assumes "ANR S"   and "S homeomorphic S'"   and "closedin (top_of_set U) S'" obtains V :: "'b set" and W :: "'b set" where "openin (top_of_set U) V" and "closedin (top_of_set U) W" and "S' \<subseteq> V" and "V \<subseteq> W" and "S' retract_of W"
find_ce_refine: shows "gfind_ce \<le> \<Down> Id find_ce"
type_definition_Quotient_not_empty: fixes P :: "'d \<Rightarrow> bool"   and Abs :: "'d \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd"   and T :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "Quotient (eq_onp P) Abs Rep T" shows "\<exists>x. P x"
algebraic_imp_non_trivial_ker: fixes K :: "'a list set"   and x :: "'a list" assumes "subring K (K [X])"   and "x \<in> carrier (K [X])"   and "(UP.algebraic over K) x" shows "a_kernel (K [X]\<^bsub>K [X]\<^esub>) (K [X]) (\<lambda>p. UP.eval p x) \<noteq> {[]}"
T1_1: fixes w :: "i"   and F :: "\<Pi>\<^sub>1"   and x :: "\<kappa>" shows "[\<lparr>F,x\<rparr> in w] = (\<exists>r o\<^sub>1. Some r = d\<^sub>1 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> ex1 r w)"
fst_eqvt: fixes pi :: "('b \<times> 'b) list"   and x :: "'a \<times> 'c" shows "pi \<bullet> fst x = fst (pi \<bullet> x)"
prc_refl: fixes f :: "'f" shows "prc f f = (False, True)"
asymp_equiv_tendsto_transfer: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "f \<sim>[F] g"   and "(f \<longlongrightarrow> c) F" shows "(g \<longlongrightarrow> c) F"
list_comb_snoc: fixes f :: "'a"   and xs :: "'a list"   and y :: "'a" shows "app (list_comb f xs) y = list_comb f (xs @ [y])"
notempty_nextnotzero: fixes s :: "nat \<Rightarrow> 'b" assumes "notemptyseq s" shows "next (Suc 0) s 0 \<noteq> s 0"
sub: fixes x :: "'d" assumes "x \<in> S" shows "closed_segment (f x) (g x) \<subseteq> t"
IH_2: assumes "A \<noteq> 0"   and "\<And>i. i \<^bold>\<in> A \<Longrightarrow> Ord i" shows "\<Squnion>A \<^bold>\<in> A"
app_def: fixes s :: "'b"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "s ## \<sigma> \<equiv> \<lambda>n. if n = 0 then s else \<sigma> (n - 1)"
errMOD_igSwapIGVarSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igVarIPresIGWls MOD"   and "igSwapIGVar MOD" shows "igSwapIGVar (errMOD MOD)"
TRANS: fixes r :: "('c \<times> 'c) set" assumes "wo_rel r" shows "trans r"
rep_eq: fixes x :: "('a, 'b) lifted_pair"   and xa :: "('a, 'b) lifted_pair" shows "(x < xa) = ((\<forall>f. triple.pairs_less_eq (Rep_lifted_pair x f) (Rep_lifted_pair xa f)) \<and> \<not> (\<forall>f. triple.pairs_less_eq (Rep_lifted_pair xa f) (Rep_lifted_pair x f)))"