hcju/isabelleps huggingface dataset:
subfield_long_division_theorem_shell: fixes K :: "'a list set"   and p :: "'a list list"   and b :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "b \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "b \<noteq> \<zero>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub>" shows "\<exists>q r. q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>) \<and> r \<in> carrier (K [X]\<^bsub>K [X]\<^esub>) \<and> p = b \<otimes>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> q \<oplus>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> r \<and> (r = \<zero>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> \<or> degree r < degree b)"
AddSA_CompFun_the2: fixes S' :: "'d"   and SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "S' \<in> States SA"   and "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A" shows "the (CompFun (A [++] (S, SA)) S') = {}"
simps: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" shows "P \<turnstile> (E, ws) \<surd> = (is_write_seen P E ws \<and> thread_start_actions_ok E)"
power2_norm_eq_cinner: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(complex_of_real (norm x))\<^sup>2 = cinner x x"
subst_mono: fixes t :: "'b trm"   and u :: "'b trm"   and s :: "('b \<times> 'b trm) list" assumes "t \<prec> u" shows "t \<lhd> s \<prec> u \<lhd> s"
assms: shows "J \<subseteq>\<^sub>\<circ> I"
avar_def: fixes G :: "prog"   and i' :: "val"   and a' :: "val"   and s :: "state" shows "avar G i' a' s = (let oref = Inl (the_Addr a'); i = the_Intg i'; n = Inr i; (T, k, cs) = the_Arr (globs (snd s) oref); f = \<lambda>v (x, s). (raise_if (\<not> G,s\<turnstile>v fits T) ArrStore x, upd_gobj oref n v s) in ((the (cs n), f), abupd (raise_if (\<not> i in_bounds k) IndOutBound \<circ> np a') s))"
uniformly_continuous_on_extension_on_closure: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "uniformly_continuous_on X f" obtains g :: "'b \<Rightarrow> 'c" where "uniformly_continuous_on (closure X) g" and "\<And>x. x \<in> X \<Longrightarrow> f x = g x" and "\<And>Y h x. \<lbrakk>X \<subseteq> Y; Y \<subseteq> closure X; continuous_on Y h; \<And>x. x \<in> X \<Longrightarrow> f x = h x; x \<in> Y\<rbrakk> \<Longrightarrow> h x = g x"
BOresChainSimps_4: fixes xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput"   and B' :: "('d, 'e, 'f) boundOutput" shows "(\<lparr>\<nu>*xvec\<rparr>B = \<lparr>\<nu>*xvec\<rparr>B') = (B = B')"
tendsto_const: fixes "open" :: "'c set \<Rightarrow> bool"   and k :: "'c"   and F :: "'d filter" assumes "class.topological_space open" shows "LIM x F. k :> topological_space.nhds open k"
abc_mopup_measure_def: shows "abc_mopup_measure = measures [\<lambda>(c, n). abc_mopup_stage1 c n, \<lambda>(c, n). abc_mopup_stage2 c n, \<lambda>(c, n). abc_mopup_stage3 c n]"
foldr_cons_def: fixes s :: "'s" shows "local.foldr_cons s = foldr f (unstream g s) z"
subM: fixes t :: "'b set"   and M :: "'b set set"   and T :: "'b set set" assumes "t \<in> M"   and "M \<subseteq> T" shows "t open\<^bsub>T\<^esub>"
append_def: fixes v1 :: "name"   and v2 :: "name" shows "local.append v1 v2 = Name (as_string v1 @ as_string v2)"
mult_min_right: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "0 \<le> a" shows "a * min b c = min (a * b) (a * c)"
term_subst_eq: fixes t :: "('d, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('d, 'e) Term.term"   and \<tau> :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "\<And>x. x \<in> fv t \<Longrightarrow> \<sigma> x = \<tau> x" shows "t \<cdot> \<sigma> = t \<cdot> \<tau>"
qSwap_preserves_qGood: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm" shows "qGood (X #[[x \<and> y]]_zs) = qGood X"
bij_betw_restrict_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" shows "bij_betw (restrict f A) A B = bij_betw f A B"
simps: fixes key_gen :: "('a \<times> 'b) spmf"   and commit :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) spmf"   and valid_msg :: "'c \<Rightarrow> bool"   and \<A>1 :: "'b \<Rightarrow> (('c \<times> 'c) \<times> 'f) spmf"   and \<A>2 :: "'d \<Rightarrow> 'f \<Rightarrow> bool spmf" shows "abstract_commitment.hiding_game_ind_cpa key_gen commit valid_msg (\<A>1, \<A>2) = TRY key_gen \<bind> (\<lambda>(ck, vk). \<A>1 vk \<bind> (\<lambda>((m0, m1), \<sigma>). assert_spmf (valid_msg m0 \<and> valid_msg m1) \<bind> (\<lambda>_. coin_spmf \<bind> (\<lambda>b. commit ck (if b then m0 else m1) \<bind> (\<lambda>(c, d). \<A>2 c \<sigma> \<bind> (\<lambda>b'. return_spmf (b' = b))))))) ELSE coin_spmf"
r_one: fixes x :: "'a set" assumes "x \<in> carrier (ideals_set R)" shows "x \<otimes>\<^bsub>ideals_set R\<^esub> \<one>\<^bsub>ideals_set R\<^esub> = x"
graph_union_vertices: fixes G\<^sub>1 :: "('c, 'b) labeled_graph"   and G\<^sub>2 :: "('c, 'b) labeled_graph" shows "vertices (graph_union G\<^sub>1 G\<^sub>2) = vertices G\<^sub>1 \<union> vertices G\<^sub>2"
weakBisimCoinduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> X \<Longrightarrow> \<Psi>' \<rhd> R \<lessapprox><(X \<union> weakBisim)> S"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> X \<Longrightarrow> \<Psi>' \<rhd> R \<leadsto><(X \<union> weakBisim)> S"   and "\<And>\<Psi>' R S \<Psi>''. (\<Psi>', R, S) \<in> X \<Longrightarrow> (\<Psi>' \<otimes> \<Psi>'', R, S) \<in> X \<or> \<Psi>' \<otimes> \<Psi>'' \<rhd> R \<approx> S"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> X \<Longrightarrow> (\<Psi>', S, R) \<in> X \<or> \<Psi>' \<rhd> S \<approx> R" shows "\<Psi> \<rhd> P \<approx> Q"
awalk_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "awalk u p v \<equiv> u \<in> pverts G \<and> set p \<subseteq> parcs G \<and> pcas u p v"
eventually_def: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" shows "Filter.eventually P F = Rep_filter F P"
elims: fixes x :: "'a ext0"   and y :: "'a ext0" assumes "- x = y" obtains "x = ext0.Bot" and "y = ext0.Top"   | uu_ :: "'a" where "x = ext0.Val uu_" and "y = ext0.Bot"   | "x = ext0.Top" and "y = ext0.Bot"
rel_sel_conj_overlap: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A" shows "\<gamma>\<^sub>T\<^sub>R K (\<phi> .\<and>. \<psi>) \<subseteq> \<gamma>\<^sub>T\<^sub>R K \<phi> \<union> \<gamma>\<^sub>T\<^sub>R K \<psi>"
bet_cop__cop: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and E :: "'p"   and D :: "'p" assumes "Coplanar A B C E"   and "Bet C D E" shows "Coplanar A B C D"
simps_16: shows "OUTfromV data16 = {stA6}"
rep_eq: fixes x :: "'b ratfps" shows "Rep_ratfps (- x) = - Rep_ratfps x"
galleryD_adj: fixes xs :: "'a set list" assumes "gallery xs" shows "adjacentchain xs"
length_insort: fixes x :: "'c"   and xs :: "'c list" shows "length (insort x xs) = length xs + 1"
delta_le_one: fixes l :: "(QDelta \<times> QDelta) list" shows "\<delta>_min l \<le> 1"
sum_list_mult_const: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and c :: "'a"   and xs :: "'b list" assumes "class.semiring_0 plus zero times" shows "monoid_add.sum_list plus zero (map (\<lambda>x. times (f x) c) xs) = times (monoid_add.sum_list plus zero (map f xs)) c"
2: shows "\<langle>z\<rangle>\<^sup>2 * \<langle>w\<rangle>\<^sup>2 = Re ((z1 * cnj z1 + z2 * cnj z2) * (w1 * cnj w1 + w2 * cnj w2))"
cring_simprules_3: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "cring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<in> carrier R"
valid_intv_distinct: fixes c :: "nat"   and I :: "Regions.intv"   and I' :: "Regions.intv"   and x :: "'a"   and u :: "'a \<Rightarrow> real" assumes "Regions.valid_intv c I"   and "Regions.valid_intv c I'"   and "Regions.intv_elem x u I"   and "Regions.intv_elem x u I'" shows "I = I'"
faces: fixes S :: "'b set"   and x :: "'b set set"   and y :: "'b set set" assumes "CoxeterComplex S"   and "x \<in> CoxeterComplex.TheComplex S"   and "order.greater_eq x y" shows "y \<in> CoxeterComplex.TheComplex S"
load_def: fixes m :: "mem"   and n :: "nat"   and off :: "nat"   and l :: "nat" shows "load m n off l = (if n + off + l \<le> mem_size m then Some (read_bytes m (n + off) l) else None)"
proj_preserves: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "P x" shows "(P /// \<phi>) (proj \<phi> x)"
parent_def: fixes G :: "('a, 'b) Graph_scheme"   and root :: "'a"   and v :: "'a" assumes "RootedTree G root" shows "RootedTree.parent G root v \<equiv> hd (tl (v \<leadsto>\<^bsub>G\<^esub> root))"
isomorphic_implies_ide_1: fixes f :: "'c"   and g :: "'c" assumes "C.isomorphic f g" shows "C.ide f"
finsertI1: fixes a :: "'c"   and B :: "'c fset" shows "a |\<in>| finsert a B"
Nil_in_tl_Lm_if_final: fixes r :: "(bool \<times> 'b) rexp" assumes "final r" shows "[] \<in> tl ` Lm r"
pmf_expectation_bind_pmf_of_set: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b pmf"   and h :: "'b \<Rightarrow> 'c" assumes "A \<noteq> {}"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))" shows "measure_pmf.expectation (pmf_of_set A \<bind> f) h = (\<Sum>a\<in>A. measure_pmf.expectation (f a) h /\<^sub>R real (card A))"
hyps_2: shows "y \<Rightarrow>\<^sub>d z"
prime_factorization_empty_iff: fixes x :: "'a" shows "(prime_factorization x = {#}) = (x = (0::'a) \<or> is_unit x)"
Sup_comp_fun: fixes M :: "('d \<Rightarrow> 'c) set"   and f :: "'b \<Rightarrow> 'd" shows "\<Squnion>M \<circ> f = (\<Squnion>m\<in>M. m \<circ> f)"
a0n_inv1_niagree_def: shows "a0n_inv1_niagree \<equiv> {s. \<forall>h d. set h \<subseteq> good \<longrightarrow> d \<notin> corrupted s \<longrightarrow> 0 < signals s (Commit h d) \<longrightarrow> 0 < signals s (Running h d)}"
Sup_insert: fixes a :: "'e"   and A :: "'e set" shows "Sup (insert a A) = sup a (Sup A)"
primerec_all_iff: fixes rt :: "recf"   and n :: "nat"   and rf :: "recf" assumes "primerec rt n"   and "primerec rf (Suc n)"   and "0 < n" shows "primerec (rec_all rt rf) n"
inv_solve_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier \<G>"   and "b \<in> carrier \<G>"   and "c \<in> carrier \<G>" shows "(a = b \<otimes> inv c) = (b = a \<otimes> c)"
tinres_steps1: fixes l :: "cell list"   and l' :: "cell list"   and ss :: "nat"   and r :: "cell list"   and t :: "(action \<times> nat) list"   and stp :: "nat"   and sa :: "nat"   and la :: "cell list"   and ra :: "cell list"   and sb :: "nat"   and lb :: "cell list"   and rb :: "cell list" assumes "tinres l l'"   and "steps0 (ss, l, r) t stp = (sa, la, ra)"   and "steps0 (ss, l', r) t stp = (sb, lb, rb)" shows "tinres la lb \<and> ra = rb \<and> sa = sb"
length_trace: fixes d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and i :: "nat"   and xs :: "'b list" shows "length (trace d i xs) = length xs"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier H'"   and "y \<in> carrier H'" shows "x \<otimes>\<^bsub>H'\<^esub> y \<in> carrier H'"
Step1_4_4a: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> Read rmCh mm ires p \<and> $S4 rmhist p \<and> S4 rmhist p$ \<and> unchanged (e p, c p, r p, rmhist!p) \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> Read memCh mm (resbar rmhist) p"
assms_1: shows "borel_adapt_stoch_proc F (prices Mkt asset)"
b_ne: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j \<le> k" shows "b k (j, i) \<noteq> {}"
project_set_iff: fixes x :: "'d"   and h :: "'d \<times> 'e \<Rightarrow> 'f"   and C :: "'f set" shows "(x \<in> project_set h C) = (\<exists>y. h (x, y) \<in> C)"
et_1: fixes v :: "'a"   and e :: "'a"   and t :: "'a" assumes "is_vector v"   and "e \<le> v ; - v\<^sup>T"   and "t \<le> v ; v\<^sup>T" shows "e ; t = 0"
pdens: shows "has_subprob_density M (stock_measure REAL) \<delta>"
conj_galois_1: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "(f x \<le> y) = (x \<le> - g (- y))"
l_to_tree_order_wf_axioms: shows "l_to_tree_order_wf heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_parent get_child_nodes"
fds_abs_converges_add: fixes f :: "'a fds"   and s :: "'a"   and g :: "'a fds" assumes "fds_abs_converges f s"   and "fds_abs_converges g s" shows "fds_abs_converges (f + g) s"
transpose_mult: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * B\<^sup>t = B\<^sup>t * A\<^sup>t"
comp_red_basis_nonzero: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> set (comp_red_basis xs)" shows "p \<noteq> 0"
div_less_imp_less_mult: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "0 < m"   and "n div m < k" shows "n < k * m"
simps_3: fixes v :: "'b"   and va :: "'b list" shows "gcp (v # va) [] = []"
le_add_diff_inverse: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and b :: "'e"   and a :: "'e" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus b (minus a b) = a"
image_vimage_eq: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'c set" shows "f ` f -` A = A \<inter> range f"
contg: fixes i :: "'a" assumes "i \<in> Basis" shows "continuous_on UNIV (g i)"
expands_to_hd: fixes f :: "real \<Rightarrow> real"   and ys :: "('a \<times> real) msllist"   and h :: "real \<Rightarrow> real"   and e :: "real"   and xs :: "('a ms \<times> real) msllist"   and g :: "real \<Rightarrow> real"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "(f expands_to MS (MSLCons (MS ys h, e) xs) g) (b # basis)" shows "(h expands_to MS ys h) basis"
simps_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Acute of Acute \<Rightarrow> f1 | Grave \<Rightarrow> f2 | Macron \<Rightarrow> f3) = f1"
inject_2: fixes x2 :: "nat"   and y2 :: "nat" shows "(R x2 = R y2) = (x2 = y2)"
alphaAbs_qAbs_imp_all_equal_or_qFresh: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "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)"
HPhase0Read_HInv4d: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "HInv4d s q" shows "HInv4d s' q"
min_def: fixes q :: "('c, 'd) pq" shows "PQ.min q = fst (hd (alist_of q))"
rem_condless_valid_4: fixes as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and A :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" assumes "set as \<subseteq> A" shows "set (rem_condless_act s [] as) \<subseteq> A"
init_def: fixes users :: "nat set" shows "init users = Env \<lparr>owner = 0, others = {Readable}\<rparr> (\<lambda>u. if u \<in> users then Some (Env \<lparr>owner = u, others = {Readable}\<rparr> Map.empty) else None)"
mo_valid_ref_locs_def: shows "mut_m.mo_valid_ref_locs = prefixed ''store_del'' \<union> prefixed ''store_ins'' \<union> {mut_m.deref_del, mut_m.lop_store_ins}"
mapCollect_map_upd: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and m :: "'f \<Rightarrow> 'g option"   and k :: "'f"   and v :: "'g" shows "mapCollect f (m(k \<mapsto> v)) = insert (f k v) (mapCollect f (m(k := None)))"
program_inv_length_states: fixes prog :: "'a program_scheme"   and n :: "nat" assumes "program_inv prog"   and "n < IArray.length (states prog)" shows "0 < IArray.length (states prog !! n)"
map\<^sub>1_simps_1: fixes f :: "'a" assumes "C.ide f" shows "D.seq \<l>\<^sub>D\<^sup>-\<^sup>1[G f] \<r>\<^sub>D[G f]"
iMOD_sub: fixes k :: "nat"   and r :: "nat"   and m :: "nat" shows "k \<ominus> [ r, mod m ] = (if r \<le> k then [ (k - r) mod m, mod m, (k - r) div m ] else {})"
assms_5: fixes it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> \<alpha> m"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k\<in>it. \<forall>j\<in>\<alpha> m - it. j \<le> k" shows "P \<sigma>"
tape_normalize: fixes tp :: "(action \<times> nat) list"   and lm :: "'a"   and m :: "nat"   and n :: "nat" assumes "\<forall>stp. \<not> is_final (steps0 (Suc 0, [Bk, Bk], <[code tp, bl2wc (<lm>)]>) t_utm stp)" shows "\<forall>stp. \<not> is_final (steps0 (Suc 0, Bk \<up> m, <[code tp, bl2wc (<lm>)]> @ Bk \<up> n) t_utm stp)"
equiv_class_eq_iff: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "equiv A r" shows "((x, y) \<in> r) = (r `` {x} = r `` {y} \<and> x \<in> A \<and> y \<in> A)"
f_def: fixes ka :: "'b" shows "f ka = (if a = i \<and> ka = i then p else if a = i \<and> ka = j then q else if a = j \<and> ka = i then u else if a = j \<and> ka = j then v else if a = ka then 1::'a else (0::'a)) * A $ ka $ b"
signed_lconjactionD: fixes s :: "'a"   and x :: "'a \<times> bool" shows "signed_lconjaction s x \<equiv> map_prod (lconjby s) (\<lambda>b. b \<noteq> (fst x = s)) x"
induct: fixes x1 :: "char list"   and x2 :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and Pa :: "char list \<Rightarrow> char list \<Rightarrow> bool" assumes "P \<turnstile> x1 \<prec>\<^sup>1 x2"   and "\<And>C D rest. \<lbrakk>class P C = \<lfloor>(D, rest)\<rfloor>; C \<noteq> Object\<rbrakk> \<Longrightarrow> Pa C D" shows "Pa x1 x2"
span_in_category_axioms: shows "span_in_category (\<cdot>) \<lparr>Leg0 = k0, Leg1 = r1 \<cdot> k1\<rparr>"
BObjOp: shows "B \<in> obj\<^bsub>Op C\<^esub>"
vcard_vsingleton: fixes a :: "V" shows "vcard (ZFC_in_HOL.set {a}) = 1\<^sub>\<nat>"
compose_axiom_def: shows "compose_axiom \<equiv> [[$\<alpha> vid1 ;; $\<alpha> vid2]]Pc pid1 \<leftrightarrow> ([[$\<alpha> vid1]][[$\<alpha> vid2]]Pc pid1)"
normal_density_affine: fixes X :: "'a \<Rightarrow> real"   and \<mu> :: "real"   and \<sigma> :: "real"   and \<alpha> :: "real"   and \<beta> :: "real" assumes "distributed M lborel X (\<lambda>x. ennreal (normal_density \<mu> \<sigma> x))"   and "0 < \<sigma>"   and "\<alpha> \<noteq> 0" shows "distributed M lborel (\<lambda>x. \<beta> + \<alpha> * X x) (\<lambda>x. ennreal (normal_density (\<beta> + \<alpha> * \<mu>) (\<bar>\<alpha>\<bar> * \<sigma>) x))"
odd_mirror_pure: fixes x :: "'a" shows "odd_mirror (pure x) = pure x"
rminusinf_ex: fixes p :: "fm"   and a :: "real"   and bs :: "real list" assumes "isrlfm p"   and "Ifm (a # bs) (minusinf p)" shows "\<exists>x. Ifm (x # bs) p"
Ker0_imp_inj_on: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T"   and "ker T \<inter> V \<subseteq> 0" shows "inj_on T V"
cosrt: shows "Re (croot n x) / cmod y = cos (arg (croot n x))"
2: fixes i :: "nat" assumes "i < n" shows "map of_int_hom.vec_hom (factorization_lattice u (n - d) k) ! i $ i \<noteq> 0"
stable_pair_onE: fixes ds :: "D2 set"   and XD_XH :: "X4 set \<times> X4 set" assumes "StableNoDecomp.stable_pair_on ds XD_XH" obtains "- Rf StableNoDecomp.CH (snd XD_XH) = fst XD_XH" and "- Rf (StableNoDecomp.CD_on ds) (fst XD_XH) = snd XD_XH"
set_category_given_img_axioms: shows "set_category_given_img (\<cdot>) img \<AA>"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>C c\<guillemotright>" shows "C.seq g f"
induct: fixes P :: "'b poly \<Rightarrow> bool"   and a0 :: "'b poly" assumes "\<And>c n p. P p \<Longrightarrow> P (CN c n p)"   and "\<And>n. P (n)\<^sub>p"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
reduced_GB_is_reduced_GB_Polys: fixes X :: "'a option set"   and F :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "extended_ord.punit.is_reduced_GB (extended_ord.punit.reduced_GB F)"
it_dfs_correct: shows "it_dfs \<le> SPEC (\<lambda>s. DFS_invar G param s \<and> \<not> cond s)"
order_divides: fixes a :: "'a"   and n :: "nat"   and p :: "'a poly" shows "([:- a, 1::'a:] ^ n dvd p) = (p = 0 \<or> n \<le> order a p)"
Lyndon_suf_less: fixes w :: "'a list"   and s :: "'a list" assumes "Lyndon w"   and "s \<le>ns w"   and "s \<noteq> w" shows "w <lex s"
indspaceI: fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set"   and G :: "'g set" assumes "f \<in> addfunset FH V"   and "f \<in> smultfunset G FH" shows "f \<in> indspace G FH V"
fps_of_ratfps_unit_factor: fixes x :: "'a ratfps" shows "fps_of_ratfps (unit_factor x) = unit_factor (fps_of_ratfps x)"
mult_commute: fixes U :: "'a set"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "ab_semigroup_mult_ow U times"   and "a \<in> U"   and "b \<in> U" shows "times a b = times b a"
lm067: fixes a :: "('d \<times> 'e set) set"   and G :: "'e set"   and N :: "'d set" assumes "a \<in> allAllocations N G"   and "finite G" shows "\<forall>x\<in>omega ` a. finite x"
update_since_2: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and aux :: "(nat \<times> 'a option list set) list"   and ne :: "nat"   and rel1 :: "'a option list set"   and rel2 :: "'a option list set"   and rel :: "'a option list set"   and aux' :: "(nat \<times> 'a option list set) list" assumes "wf_since_aux \<sigma> n R pos \<phi> I \<psi> aux ne"   and "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) ne \<phi>) rel1"   and "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) ne \<psi>) rel2"   and "(rel, aux') = update_since I pos rel1 rel2 (\<tau> \<sigma> ne) aux"   and "fv \<phi> \<subseteq> fv \<psi>" shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) ne (Sincep pos \<phi> I \<psi>)) rel"
C_prism_def: shows "C\<^sub>\<triangle> = \<lparr>prism_match = fromC, prism_build = C\<rparr>"
Diag_Diagonalize_2: fixes t :: "'a local.term" assumes "Arr t" shows "Dom \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Dom t\<^bold>\<rfloor>"
dagger_refl: fixes x :: "'b \<times> 'c" shows "less_eq_prod one_prod (prod.qstar x)"
omega_le_Limit: fixes \<mu> :: "V" assumes "Limit \<mu>" shows "\<omega> \<subseteq>\<^sub>\<circ> \<mu>"
prems_3: shows "v \<in> wfrestrictedvars\<^sub>s\<^sub>t (S @ [x])"
SComplex_minus_iff: fixes x :: "complex star" shows "(- x \<in> SComplex) = (x \<in> SComplex)"