hcju/isabelleps huggingface dataset:
obsf_comp_converter: fixes \<I> :: "('y, 'z) \<I>"   and \<I>' :: "('aa, 'ab) \<I>"   and conv1 :: "('y, 'z, 'aa, 'ab) converter"   and \<I>'' :: "('ac, 'ad) \<I>"   and conv2 :: "('aa, 'ab, 'ac, 'ad) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv1 \<surd>"   and "\<I>', \<I>'' \<turnstile>\<^sub>C conv2 \<surd>"   and "pfinite_converter \<I> \<I>' conv1" shows "exception_\<I> \<I>,exception_\<I> \<I>'' \<turnstile>\<^sub>C obsf_converter (conv1 \<odot> conv2) \<sim> obsf_converter conv1 \<odot> obsf_converter conv2"
r_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>add_monoid G\<^esub> x \<subseteq> carrier G"
context_le_def: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "\<Gamma> \<sqsubseteq>\<^sub>c \<Gamma>' \<equiv> dom \<Gamma> = dom \<Gamma>' \<and> (\<forall>x\<in>dom \<Gamma>. the (\<Gamma> x) \<sqsubseteq> the (\<Gamma>' x))"
t_def: shows "t \<equiv> \<lambda>cfg'. THE sa. sa \<in> set_pmf (rept s (action cfg)) \<and> abss sa = state cfg'"
AsThrow: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and p :: "'e set" shows "assertionsR \<Gamma> \<Theta> Q A (AnnExpr p) Throw p"
dsr1: fixes x :: "'a" shows "x + ad (ad x) \<cdot> x = ad (ad x) \<cdot> x"
fref_def: fixes P :: "'g \<Rightarrow> bool"   and R :: "('e \<times> 'g) set"   and S :: "('f \<times> 'h) set" shows "[P]\<^sub>f R \<rightarrow> S \<equiv> {(f, g). \<forall>x y. P y \<and> (x, y) \<in> R \<longrightarrow> (f x, g y) \<in> S}"
f1: shows "!x \<cdot> !y = t (!x \<cdot> !y)"
simps_1: fixes f :: "'c" shows "Arr \<^bold>\<langle>f\<^bold>\<rangle> = C.arr f"
equivalence: fixes G :: "('b, 'c) labeled_graph"   and I :: "'b" assumes "graph G"   and "maintainedA {reflexivity_rule I, transitive_rule I, symmetry_rule I} G" shows "equiv (vertices G) (getRel I G)"
assms: shows "(\<sigma>', coll') \<in> csmall_nstep P \<sigma> (Suc n)"
VPG_MST: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" assumes "VP_G G" shows "MST_valid"
ide_cod: fixes f :: "'a \<times> 'a \<times> 'a" assumes "CCC.arr f" shows "CCC.ide (CCC.cod f)"
decides_subset: fixes \<F> :: "nat set set"   and S :: "nat set"   and M :: "nat set"   and N :: "nat set" assumes "decides \<F> S M"   and "N \<subseteq> M" shows "decides \<F> S N"
cf: shows "P,\<Phi> |- (None, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]"
emeasure_eq_ennreal_measure: fixes M :: "'b measure"   and A :: "'b set" assumes "emeasure M A \<noteq> \<top>" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
simple_extension_in_carrier: fixes K :: "'a set"   and x :: "'a" assumes "K \<subseteq> carrier R"   and "x \<in> carrier R" shows "simple_extension K x \<subseteq> carrier R"
induct: fixes x1 :: "BExpr"   and x2 :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and x3 :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and x4 :: "Var \<Rightarrow> Val"   and x5 :: "Var \<Rightarrow> Val"   and P :: "BExpr \<Rightarrow> ((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> ((Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "(x1, x2, x3, x4, x5) \<in> var"   and "\<And>b t A s \<Phi>. \<lbrakk>\<not> evalB b t; A s t\<rbrakk> \<Longrightarrow> P b A \<Phi> s t"   and "\<And>b t \<Phi> r A s. \<lbrakk>evalB b t; \<Phi> (r, t); \<forall>w. A r w \<longrightarrow> (b, A, \<Phi>, s, w) \<in> var \<and> P b A \<Phi> s w\<rbrakk> \<Longrightarrow> P b A \<Phi> s t" shows "P x1 x2 x3 x4 x5"
strand_sem_eq_defs_1: fixes M :: "('fun, 'var) Term.term set"   and \<A> :: "('fun, 'var) strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "\<lbrakk>M; \<A>\<rbrakk>\<^sub>c' \<I> = \<lbrakk>M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>; \<A>\<rbrakk>\<^sub>c \<I>"
desargues_config_2D_non_collinear_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "rk {A, B, B'} = 3"
Least_le: fixes P :: "'b \<Rightarrow> bool"   and k :: "'b" assumes "P k" shows "(LEAST x. P x) \<le> k"
rel_envirI: fixes p :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" assumes "\<And>x. p (f x) (g x)" shows "rel_envir p f g"
inject_2: fixes x31 :: "'a bro"   and x32 :: "'a"   and x33 :: "'a bro"   and y31 :: "'a bro"   and y32 :: "'a"   and y33 :: "'a bro" shows "(N2 x31 x32 x33 = N2 y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
sel_1: fixes x :: "complex" shows "Re (- x) = - Re x"
lists_accD: fixes xs :: "'a list"   and r :: "('a \<times> 'a) set" assumes "xs \<in> lists (Wellfounded.acc r)" shows "xs \<in> Wellfounded.acc (listrel1 r)"
bilinear_rmul: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a"   and c :: "real"   and y :: "'b" assumes "bilinear h" shows "h x (c *\<^sub>R y) = c *\<^sub>R h x y"
nonneg_Reals_subset_Reals: shows "\<real>\<^sub>\<ge>\<^sub>0 \<subseteq> \<real>"
Lcm_0_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "finite A" shows "(Lcm A = zero) = (zero \<in> A)"
Skip: fixes s :: "char list \<Rightarrow> int" shows "(SKIP, s) \<Rightarrow> s"
pos_imp_zdiv_nonneg_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(0 \<le> a div b) = (0 \<le> a)"
bij_to_real_Gromov_completion: shows "bij to_real_Gromov_completion"
simps_1: fixes i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and Q :: "(('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "diagonal_to_Smith_i_PQ [] i bezout (P, A, Q) = (P, A, Q)"
gt_in_args_imp: fixes ti :: "('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "ti \<in> set (args t)"   and "ti >\<^sub>t s" shows "t >\<^sub>t s"
a_alt_def: fixes l :: "nat"   and i :: "nat" assumes "l < length (map of_int_hom.vec_hom fs)" shows "gs.a i (Suc l) = (let v = gs.\<mu>' l l \<cdot>\<^sub>v gs.a i l - gs.\<mu>' i l \<cdot>\<^sub>v gs.a l l in if l = 0 then v else 1 / gs.\<mu>' (l - 1) (l - 1) \<cdot>\<^sub>v v)"
lcp_mismatch: fixes u :: "'b list"   and v :: "'b list" assumes "\<^bold>|u \<and>\<^sub>p v\<^bold>| < \<^bold>|u\<^bold>|"   and "\<^bold>|u \<and>\<^sub>p v\<^bold>| < \<^bold>|v\<^bold>|" shows "u ! \<^bold>|u \<and>\<^sub>p v\<^bold>| \<noteq> v ! \<^bold>|u \<and>\<^sub>p v\<^bold>|"
sel_1: fixes z :: "gauss_int" shows "Re (gauss2complex z) = real_of_int (ReZ z)"
blubb: fixes e :: "'a ifex"   and v :: "'a"   and vr :: "'a"   and vl :: "bool" assumes "e \<in> set [i, t, ea__]"   and "ifex_ordered e"   and "v \<in> ifex_var_set (restrict_top e vr vl)" shows "a_ \<le> v"
det_fold_set_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and \<sigma> :: "'d"   and result :: "'c set \<Rightarrow> 'd" shows "det_fold_set R c f \<sigma> result \<equiv> \<forall>l. distinct l \<and> sorted_wrt R l \<longrightarrow> foldli l c f \<sigma> = result (set l)"
assms: shows "Complete_Partial_Order.chain (fun_ord le) Y"
a: shows "d z +  |x\<rangle> y \<le> d y"
create_factor_tree_balanced: fixes xs :: "(int poly \<times> nat) list" assumes "xs \<noteq> []" shows "factors_of_factor_tree (create_factor_tree_balanced xs) = mset (map fst xs)"
axis_eq_axis: fixes i :: "'e"   and x :: "'d"   and j :: "'e"   and y :: "'d" shows "(axis i x = axis j y) = (x = y \<and> i = j \<or> x = (0::'d) \<and> y = (0::'d))"
id_ord_isom: shows "ord_isom D D (idmap (carrier D))"
succ_in_diff: fixes r :: "('a \<times> 'a) set"   and i :: "'a" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "(i, succ r i) \<in> r \<and> succ r i \<noteq> i"
rep_eq: fixes x :: "'b fset"   and xa :: "'b fset" shows "(x |\<subseteq>| xa) = (\<langle>x\<rangle>\<^sub>f \<subseteq> \<langle>xa\<rangle>\<^sub>f)"
rep_nat_term_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.nat_pp_term plus zero rep_nat_term splus" shows "rep_nat_term zero = (0, 0)"
simps_4: fixes b :: "bexp"   and P\<^sub>1 :: "'b"   and C\<^sub>1 :: "'b acom"   and P\<^sub>2 :: "'b"   and C\<^sub>2 :: "'b acom"   and P :: "'b" shows "strip (IF b THEN {P\<^sub>1} C\<^sub>1 ELSE {P\<^sub>2} C\<^sub>2 {P}) = IF b THEN strip C\<^sub>1 ELSE strip C\<^sub>2"
esing_eqvt: fixes \<pi> :: "perm"   and x :: "'b" shows "\<pi> \<bullet> esing x = esing (\<pi> \<bullet> x)"
assms_2: shows "ring_isomorphism f A addA multA zeroA oneA B addB multB zeroB oneB"
that_2: fixes s :: "'a set" assumes "s \<in> S" shows "open s"
alw_ev_HLD_cycle: fixes xs :: "'a stream"   and as :: "'a set list"   and a :: "'a set" assumes "stream_all2 (\<in>) xs (cycle as)"   and "a \<in> set as" shows "infs a xs"
class_add_method: fixes P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "jvm_method"   and D :: "char list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> C\<^sub>0 sees M\<^sub>0, b :  Ts\<rightarrow>T = m in D"   and "\<not> is_class P C" shows "method P C\<^sub>0 M\<^sub>0 = method (class_add P (C, cdec)) C\<^sub>0 M\<^sub>0"
W: fixes I :: "nat set" assumes "I \<in> P" shows "sum (wt (fst cf1) (snd cf1)) I = sum (wt (fst cf2) (snd cf2)) (F I)"
F: shows "\<forall>x1 x2. \<exists>y1 y2. f x1 x2 = (y1, y2) \<and> P x1 x2 (fst (f x1 x2)) (snd (f x1 x2))"
eo: shows "\<forall>\<^sub>F t in at_top. norm (f t) \<le> K + M * exp (c * t)"
abs_eq: fixes x :: "real" shows "bernoulli_pmf x = embed_pmf (\<lambda>b. if b then min 1 (max 0 x) else 1 - min 1 (max 0 x))"
peek_res_def2: fixes P :: "vals \<Rightarrow> 'b assn"   and Y :: "vals" shows "peek_res P Y = P Y Y"
iLast_induct_rule: fixes P :: "nat \<Rightarrow> bool"   and I :: "nat set"   and t :: "nat" assumes "P (Max I)"   and "\<box> t I. P t \<longrightarrow> iLast t I P"   and "finite I"   and "t \<in> I" shows "P t"
topological_basis_imp_subbasis: fixes B :: "'c set set" assumes "topological_basis B" shows "open = generate_topology B"
opeB: fixes C :: "'a set" assumes "C \<in> \<B>" shows "openin (top_of_set S) C"
hcomplex_hcnj_cancel_iff: fixes x :: "complex star"   and y :: "complex star" shows "(hcnj x = hcnj y) = (x = y)"
select_convs_1: fixes igWls :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and igWlsAbs :: "'e \<times> 'a \<Rightarrow> 'h \<Rightarrow> bool"   and igVar :: "'e \<Rightarrow> 'g \<Rightarrow> 'b"   and igAbs :: "'e \<Rightarrow> 'g \<Rightarrow> 'b \<Rightarrow> 'h"   and igOp :: "'f \<Rightarrow> ('c \<Rightarrow> 'b option) \<Rightarrow> ('d \<Rightarrow> 'h option) \<Rightarrow> 'b"   and igFresh :: "'e \<Rightarrow> 'g \<Rightarrow> 'b \<Rightarrow> bool"   and igFreshAbs :: "'e \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> bool"   and igSwap :: "'e \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'b \<Rightarrow> 'b"   and igSwapAbs :: "'e \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'h"   and igSubst :: "'e \<Rightarrow> 'b \<Rightarrow> 'g \<Rightarrow> 'b \<Rightarrow> 'b"   and igSubstAbs :: "'e \<Rightarrow> 'b \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'h"   and more :: "'i" shows "igWls \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igWls"
dot_norm_neg: fixes x :: "'b"   and y :: "'b" shows "x \<bullet> y = ((norm x)\<^sup>2 + (norm y)\<^sup>2 - (norm (x - y))\<^sup>2) / 2"
DOM_in_dom: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.DOM f \<in> \<F>C.dom f"
vpair_def: fixes a :: "V"   and b :: "V" shows "\<langle>a, b\<rangle> = set {set {a}, set {a, b}}"
vpeq_obj_def: fixes u :: "partition_id_t"   and s :: "state_t"   and t :: "state_t" shows "vpeq_obj u s t \<equiv> \<forall>obj_id. Policy.sp_spec_subj_obj u obj_id READ \<longrightarrow> obj s obj_id = obj t obj_id"
max_bag_card_lower_bound_1: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set" assumes "TreeDecomposition G T bag"   and "V\<^bsub>G\<^esub> \<noteq> {}" shows "0 < TreeDecomposition.max_bag_card T bag"
zero_leastIII: fixes x :: "'a" shows "(0::'a) \<le> x"
election_def: fixes agents :: "'a set"   and alts :: "'b set" shows "election agents alts \<equiv> (finite agents \<and> finite alts) \<and> agents \<noteq> {} \<and> alts \<noteq> {}"
filtermap_le_iff_le_filtercomap: fixes f :: "'b \<Rightarrow> 'a"   and F :: "'b filter"   and G :: "'a filter" shows "(filtermap f F \<le> G) = (F \<le> filtercomap f G)"
maxR_absorb2: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "MaxR r"   and "(x, y) \<in> r" shows "MaxR.maxR r x y = y"
has_fields_types: fixes P :: "'b prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and FD :: "char list \<times> char list"   and b :: "staticb"   and T :: "ty"   and wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool" assumes "P \<turnstile> C has_fields FDTs"   and "(FD, b, T) \<in> set FDTs"   and "wf_prog wf_md P" shows "is_type P T"
exhaust: fixes y :: "('c, 'd) operation" obtains x11 :: "'c" and x12 :: "'d" where "y = Create x11 x12"   | x21 :: "'c set" and x22 :: "'d" where "y = Delete x21 x22"   | x31 :: "'c" and x32 :: "'d" where "y = Append x31 x32"   | x41 :: "'d" and x42 :: "'c" and x43 :: "'c" where "y = Expunge x41 x42 x43"   | x51 :: "'d" and x52 :: "'c" and x53 :: "'c" where "y = Store x51 x52 x53"
B: shows "\<forall>u\<in>sinks_aux_less I D U (ys @ [y]). \<forall>z\<in>Z \<union> set zs. (u, D z) \<notin> I"
cycle_rotate_2': fixes xs :: "nat list"   and i :: "nat"   and j :: "nat"   and a :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c" assumes "xs \<noteq> []"   and "(i, j) \<in> set (arcs a a xs)" shows "\<exists>ys. len M a a xs = len M i i (j # ys) \<and> set (i # j # ys) = set (a # xs) \<and> 1 + length ys = length xs \<and> set (arcs a a xs) = set (arcs i i (j # ys))"
compatible_refl: fixes cmds :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "list_all com_sifum_secure cmds"   and "mem\<^sub>1 =\<^sup>l mem\<^sub>2" shows "makes_compatible (cmds, mem\<^sub>1) (cmds, mem\<^sub>2) (replicate (length cmds) (mem\<^sub>1, mem\<^sub>2))"
assms_5: shows "\<forall>i<length hs. f ((1::'f) \<delta>\<delta> hs ! i) = f' ((1::'f) \<delta>\<delta> hs ! i)"
hyps: assumes "Lyndon_mono xs" shows "xs = LynFac (concat xs)"
assms_2: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "P cfg" shows "state cfg \<in> \<S>"
dghm_id_ArrMap_vsv: fixes \<CC> :: "V" shows "vsv (dghm_id \<CC>\<lparr>ArrMap\<rparr>)"
g_group_sort: fixes xs :: "(primfun \<times> real) list" shows "eval_primfuns (groupsort_primfun.group_sort xs) = eval_primfuns xs"
second_countable_borel_measurable: fixes X :: "'c set set" assumes "open = generate_topology X" shows "borel = sigma UNIV X"
semiring_0_cancel_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.semiring_0_cancel plus minus zero times \<equiv> class.cancel_comm_monoid_add plus minus zero \<and> class.semiring plus times"
steps_Fault_prop: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and s :: "('d, 'f) xstate"   and c' :: "('d, 'e, 'f) com"   and s' :: "('d, 'f) xstate"   and f :: "'f" assumes "\<Gamma> \<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s')"   and "s = Fault f" shows "s' = Fault f"
change_eval_eq: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real"   and y :: "real" assumes "aEvalUni (EqUni (a, b, c)) x \<noteq> aEvalUni (EqUni (a, b, c)) y \<and> x < y" shows "\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0"
parallel_converter2_map1_out: fixes f :: "'h \<Rightarrow> 'p"   and g :: "'q \<Rightarrow> 'j"   and h :: "'r \<Rightarrow> 'l"   and k :: "'n \<Rightarrow> 's"   and conv1 :: "('p, 'q, 'r, 's) converter"   and conv2 :: "('i, 'k, 'm, 'o) converter" shows "map_converter f g h k conv1 |\<^sub>= conv2 = map_converter (map_sum f id) (map_sum g id) (map_sum h id) (map_sum k id) (conv1 |\<^sub>= conv2)"
admissibleD: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and A :: "'c set" assumes "ccpo.admissible lub ord P"   and "Complete_Partial_Order.chain ord A"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> P x" shows "P (lub A)"
simps: fixes p :: "int poly"   and ri :: "root_info"   and cr :: "rat \<Rightarrow> rat \<Rightarrow> nat"   and lrs :: "(rat \<times> rat) list"   and rais :: "real_alg_2 list" shows "roots_of_2_main p ri cr lrs rais = (case lrs of [] \<Rightarrow> rais | (l, r) # lrs \<Rightarrow> let c = cr l r in if c = 0 then roots_of_2_main p ri cr lrs rais else if c = 1 then roots_of_2_main p ri cr lrs (real_alg_2'' ri p l r # rais) else let m = (l + r) / 2 in roots_of_2_main p ri cr ((m, r) # (l, m) # lrs) rais)"
warp_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "warp_on A f = (\<forall>(x, y)\<in>rsp_on A f. (y, x) \<notin> rwp_on A f)"
eqI: fixes a :: "'a"   and b :: "'a" assumes "f a = f b" shows "a = b"
assms_2: shows "linear_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 (*s\<^sub>1) (*s\<^sub>2) f"
all_red_crit: shows "\<forall>q\<in>Q. calculus Bot_F Inf_F (\<lambda>N1 N2. entails_q q (\<G>_Fset_q q N1) (\<G>_Fset_q q N2)) (Red_I_\<G>_q q) (Red_F_\<G>_empty_q q)"
1: shows "\<forall>l. g l < ack kg (sum_list l)"
isZFfun_def: fixes drf :: "ZF" shows "isZFfun drf \<equiv> let f = Snd drf in isOpair drf \<and> isOpair (Fst drf) \<and> isFun f \<and> f |\<subseteq>| HOLZF.Domain f |\<times>| HOLZF.Range f \<and> HOLZF.Domain f = |dom|drf \<and> HOLZF.Range f |\<subseteq>| |cod|drf"
eq: shows "p - (\<Sum>t\<in>keys q - {0}. monomial (lookup q t) 0 * subst_pp (\<lambda>x. monomial (1::'a) (monomial 1 x) - monomial (a x) 0) t) = monomial (lookup q 0) 0"
not_Interp_imp_general: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and D' :: "'b literal multiset"   and D :: "'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset"   and CC :: "'b literal multiset set" assumes "ground_resolution_with_selection S"   and "D' \<le> D"   and "D' \<in> N \<and> S D' = {#} \<or> Max (atms_of D') < Max (atms_of D)"   and "\<not> ground_resolution_with_selection.Interp S N D \<Turnstile> D'"   and "D < C"   and "ground_resolution_with_selection.interp S N C \<subseteq> \<Union> (ground_resolution_with_selection.production S N ` CC)" shows "\<not> \<Union> (ground_resolution_with_selection.production S N ` CC) \<Turnstile> D'"
g: shows "retraction_maps (nsphere p) (nsphere p) g f"
map_of_eq_Some_iff: fixes xys :: "('c \<times> 'd) list"   and x :: "'c"   and y :: "'d" assumes "distinct (map fst xys)" shows "(map_of xys x = Some y) = ((x, y) \<in> set xys)"
resolution_class_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and S :: "'b set set" assumes "incidence_system point_set block_collection" shows "incidence_system.resolution_class point_set block_collection S = (partition_on point_set S \<and> (\<forall>bl\<in>S. bl \<in># block_collection))"
distinct_2: fixes b' :: "'c"   and a :: "'b" shows "Inr b' \<noteq> Inl a"
zero_left: fixes b :: "'a" shows "0 $ b = (0::'a)"
lfp_lockstep_induct: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "mono f"   and "mono g"   and "\<And>A B. \<lbrakk>A \<le> lfp f; B \<le> lfp g; R A B\<rbrakk> \<Longrightarrow> R (f A) (g B)"   and "\<And>M. \<forall>(A, B)\<in>M. R A B \<Longrightarrow> R (Sup (fst_proj M)) (Sup (snd_proj M))" shows "R (lfp f) (lfp g)"
connected_component_eq_eq: fixes S :: "'c set"   and x :: "'c"   and y :: "'c" shows "(connected_component_set S x = connected_component_set S y) = (x \<notin> S \<and> y \<notin> S \<or> x \<in> S \<and> y \<in> S \<and> connected_component S x y)"
alist_meld: fixes xs :: "('d, 'c) bintree option list"   and ys :: "('d, 'c) bintree option list" assumes "distinct (vals xs)"   and "distinct (vals ys)"   and "set (vals xs) \<inter> set (vals ys) = {}" shows "set (elements (meld xs ys)) = set (elements xs) \<union> set (elements ys)"
s: fixes p :: "'b"   and q :: "'b" shows "((p, q) \<in> s) = (\<exists>k u v. k \<parallel> p \<and> p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> q \<and> q \<parallel> v)"
remdups_eq_member_eq: fixes xa :: "'b list"   and ya :: "'b list" assumes "remdups xa = remdups ya" shows "List.member xa = List.member ya"
M_subset_calM_prep: fixes M' :: "'a set set" assumes "M' \<in> calM // RelM" shows "M' \<subseteq> calM"
any_Err: fixes 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 T :: "ty err" shows "P \<turnstile> T \<le>\<^sub>\<top> Err"
adv_P1_def: fixes n :: "nat"   and m1 :: "bool \<times> bool"   and m2 :: "bool \<times> bool"   and D :: "((bool \<times> bool) \<times> bool \<times> 'v_14_OT1) \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "and_secret_sharing.adv_P1 n m1 m2 D \<equiv> \<bar>spmf (and_R1 n m1 m2 \<bind> D) True - spmf (and_funct m1 m2 \<bind> (\<lambda>(o1, o2). and_secret_sharing.Ideal1 n m1 m2 o1 \<bind> D)) True\<bar>"
distinct_1: fixes x1 :: "('b ann_com option \<times> 'b set) list"   and x2 :: "'b \<Rightarrow> 'b" shows "Parallel x1 \<noteq> Basic x2"
get_pending_nofail: fixes s :: "'s" assumes "pre_get_pending s" shows "nofail (gds_get_pending gds s)"
singletonI: fixes a :: "'c" shows "a \<in> {a}"