hcju/isabelleps huggingface dataset:
one_and_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and one a = modulo a (numeral.numeral one plus (num.Bit0 num.One))"
set_bit_def: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and set_bit :: "'a \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a" shows "class.set_bit plus minus zero times one divide modulo bit set_bit \<equiv> class.semiring_bits plus minus zero times one divide modulo bit \<and> class.set_bit_axioms plus zero times one bit set_bit"
convergence_ext: fixes xs :: "('a \<times> operation) list"   and ys :: "('a \<times> operation) list"   and s :: "int" assumes "set xs = set ys"   and "hb.concurrent_ops_commute xs"   and "hb.concurrent_ops_commute ys"   and "distinct xs"   and "distinct ys"   and "hb.hb_consistent xs"   and "hb.hb_consistent ys" shows "hb.apply_operations xs s = hb.apply_operations ys s"
simps_4: fixes f :: "nat \<Rightarrow> nat"   and tk :: "nat" shows "map_ticket f (REL tk) = REL (f tk)"
coreflexive_comp_top_inf: fixes x :: "'a"   and y :: "'a" assumes "x \<le> top" shows "x \<sqinter> top \<sqinter> y = x \<sqinter> y"
has_lock_upd_locks_implies_has_lock: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and Ls :: "lock_action list"   and t' :: "'c" assumes "has_lock (upd_locks l t Ls) t'"   and "t \<noteq> t'" shows "has_lock l t'"
Diff: fixes a :: "'a set"   and M :: "'a measure"   and b :: "'a set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a - b \<in> sets M"
special_cf_def: fixes S :: "('e, 'f) Ring_scheme" shows "C\<^sub>0 S = (0, \<lambda>j. 1\<^sub>r\<^bsub>S\<^esub>)"
AbstTermP_Ind: fixes v :: "tm"   and w :: "tm"   and t :: "tm" shows "{VarP v, IndP w, OrdP t} \<turnstile> AbstTermP v t w w"
sum_closure: fixes x :: "i"   and y :: "i" shows "\<exists>z. \<forall>v. O v z = (O v x \<or> O v y)"
assms_3: assumes "m < n \<and> x < y"   and "x \<noteq> m"   and "x \<noteq> n"   and "y \<noteq> m"   and "y \<noteq> n" shows "P (arr m) (arr n)"
enc_weakly_respects_barbs_iff_source_target_rel: shows "((\<forall>S x. (\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x) \<longrightarrow> (\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x)) \<and> (\<forall>S x. (\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x) \<longrightarrow> (\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x))) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_weakly_respects_barbs Rel (STCalWB SWB TWB))"
equivalent_objects_reflexive: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.equivalent_objects a a"
sums_integral_count_space_nat: fixes f :: "nat \<Rightarrow> 'c" assumes "integrable (count_space UNIV) f" shows "f sums integral\<^sup>L (count_space UNIV) f"
combSlice_refl: fixes n :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "n \<in> S" shows "n \<in> combine_SDG_slices S"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case right of left \<Rightarrow> f1 | right \<Rightarrow> f2) = f2"
when_neq_zero: fixes a :: "'a"   and P :: "bool" shows "((a when P) \<noteq> (0::'a)) = (P \<and> a \<noteq> (0::'a))"
src_dom: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg (B.dom \<mu>) = trg \<mu>"
list_comb_app: fixes f :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "list_comb f (xs @ ys) = list_comb (list_comb f xs) ys"
carrier_Characters: fixes G :: "('c, 'd) monoid_scheme" shows "carrier (Characters G) = characters G"
done_only_from_recv_marker_trace: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace c t c'"   and "t \<noteq> []"   and "snd (cs c cid) \<noteq> Done"   and "snd (cs c' cid) = Done"   and "channel cid = Some (p, q)" shows "RecvMarker cid q p \<in> set t"
poly_0: fixes x :: "'c" shows "poly 0 x = (0::'c)"
div_smult_left: fixes a :: "'b"   and x :: "'b poly"   and y :: "'b poly" shows "smult a x div y = smult a (x div y)"
measurable_count_space_insert: fixes s :: "'c"   and S :: "'c set"   and A :: "'c set" assumes "s \<in> S"   and "A \<in> sets (count_space S)" shows "insert s A \<in> sets (count_space S)"
Refl_Field_Restr2: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "Refl r"   and "A \<subseteq> Field r" shows "Field (Restr r A) = A"
rep_eq: fixes x :: "nat \<Rightarrow> bool" shows "Rep_uint32 (set_bits x) = set_bits x"
singleton_conv2: fixes a :: "'k" shows "{x. a = x} = {a}"
md2: shows "monomial_decomp (snd (split 0 X S))"
emb_step_arg: fixes t :: "('c, 'd) tm" assumes "is_App t" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b arg t"
shortest_paths_locale_step2_axioms: shows "shortest_paths_locale_step2 G s c num parent_edge dist"
that_2: fixes x :: "'a" assumes "x \<in> U" shows "g x = f x"
tensor_state2: fixes u :: "complex Matrix.mat"   and v :: "complex Matrix.mat" assumes "state 1 u"   and "state 1 v" shows "state 2 (u \<Otimes> v)"
cexpr_typing_map_vars: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and f :: "nat \<Rightarrow> nat"   and e :: "cexpr"   and t :: "pdf_type" assumes "\<Gamma> \<circ> f \<turnstile>\<^sub>c e : t" shows "\<Gamma> \<turnstile>\<^sub>c map_vars f e : t"
tauRemoveTau: fixes P :: "pi"   and P' :: "pi" assumes "\<tau>.P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'"   and "P \<noteq> P'" shows "P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'"
pell_mul_def: shows "pell_mul = (\<lambda>(a, b) (x, y). (x * a + y * b * of_nat D, x * b + y * a))"
lNactGe: fixes c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"   and "\<parallel>c\<parallel>\<^bsub>t n'\<^esub>" shows "n \<le> n'"
prj_chine_assoc_1: shows "\<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>1 \<cdot> \<mu>\<nu>_\<pi>_\<rho>.chine_assoc = \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>1\<^sub>1"
1: shows "II = \<Union> {I \<in> P. I \<subseteq> II}"
split_beta': fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
const_type: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and c :: "'a" assumes "typed_model arity public Ana \<Gamma>"   and "arity c = 0" shows "\<exists>a. \<forall>T. \<Gamma> (Fun c T) = Var a"
current_weight_mono: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and \<Gamma>' :: "('k, 'm) web_scheme" assumes "current \<Gamma> f"   and "edge \<Gamma> = edge \<Gamma>'"   and "A \<Gamma> = A \<Gamma>'"   and "B \<Gamma> = B \<Gamma>'"   and "\<And>x. weight \<Gamma> x \<le> weight \<Gamma>' x" shows "current \<Gamma>' f"
part_equivpE: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "part_equivp R" obtains x :: "'b" where "R x x" and "symp R" and "transp R"
pos: shows "real_of_int \<lfloor>x\<rfloor> \<noteq> 0"
intro: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "OrdinalInduct.strict_mono F" shows "normal_axioms F"
oivf: shows "openin (top_of_set S) (S \<inter> f -` W)"
abs_eq: fixes x :: "int" shows "the_int (Word.Word x) = take_bit LENGTH('k) x"
viewL_correct_empty: shows "FingerTreeStruc.viewL Empty = None"
converse_iff: fixes a :: "'d"   and b :: "'e"   and r :: "('e \<times> 'd) set" shows "((a, b) \<in> r\<inverse>) = ((b, a) \<in> r)"
r_inv_ex: fixes x :: "'c" assumes "x \<in> carrier H" shows "\<exists>y\<in>carrier H. x \<otimes>\<^bsub>H\<^esub> y = \<one>\<^bsub>H\<^esub>"
rd_H1: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R (\<not> (\<lambda>(A, A'). P (A, A'\<lparr>ok := False\<rparr>)) \<turnstile> \<lambda>(A, A'). P (A, A'\<lparr>ok := True\<rparr>)) = R (\<not> H1 (\<lambda>(A, A'). P (A, A'\<lparr>ok := False\<rparr>)) \<turnstile> H1 (\<lambda>(A, A'). P (A, A'\<lparr>ok := True\<rparr>)))"
multi_psub_self: fixes A :: "'b multiset" shows "(A \<subset># A) = False"
AND_def: fixes \<Phi> :: "'b pltl set" shows "AND \<Phi> \<equiv> SOME \<phi>. fold_graph And_ltlp true\<^sub>p \<Phi> \<phi>"
convergent_diff: fixes X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'a" assumes "convergent X"   and "convergent Y" shows "convergent (\<lambda>n. X n - Y n)"
mult_scaleC_right: 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "complex"   and y :: "'b" assumes "class.complex_algebra scaleR scaleC plus zero minus uminus times" shows "times x (scaleC a y) = scaleC a (times x y)"
assms: shows "L\<^sub>1\<^sup>C = L\<^sub>2\<^sup>C"
strict_gL_rel_def: fixes x :: "'a"   and y :: "'a" shows "(x \<prec>\<^sub>L y) = (x \<preceq>\<^sub>L y \<and> \<not> y \<preceq>\<^sub>L x)"
policy_range_comp_def: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'a \<Rightarrow> 'c decision option" shows "f o\<^sub>f p = (\<lambda>x. case p x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>allow y\<rfloor> \<Rightarrow> \<lfloor>allow (f y)\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> \<lfloor>deny (f y)\<rfloor>)"
two_in_limit_iff: fixes a :: "'b"   and b :: "'b"   and x :: "nat \<Rightarrow> 'b" shows "({a, b} \<subseteq> limit x) = ((\<exists>n. x n = a) \<and> (\<forall>n. x n = a \<longrightarrow> (\<exists>m>n. x m = b)) \<and> (\<forall>m. x m = b \<longrightarrow> (\<exists>n>m. x n = a)))"
pp_one: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and conv :: "'b \<Rightarrow> 'b" assumes "class.stone_relation_algebra inf less_eq less sup bot top uminus one times conv" shows "uminus (uminus one) = one"
countable_UN': fixes X :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "countable X"   and "A ` X \<subseteq> M" shows "\<Union> (A ` X) \<in> M"
assms: shows "\<forall>op\<in>set \<pi>. op \<in> set (\<Pi>\<^sub>\<O>)"
E10: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<diamond>F = \<diamond>F"
reachable_blackD: fixes s :: "'mut process_name \<Rightarrow> ('a, 'mut, 'b, 'c) Model.local_state"   and r :: "'c" assumes "no_grey_refs s"   and "reachable_snapshot_inv s"   and "reachable r s" shows "black r s"
finite: fixes P :: "nat \<Rightarrow> bool" assumes "finite {x. P x}" shows "\<not> eventually P \<U>"
foundation9: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x" shows "(\<tau> \<Turnstile> not x) = (\<tau> |\<noteq> x)"
vmc_path_def: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" shows "vmc_path G P v0 p \<sigma> \<equiv> vm_path G P v0 \<and> vmc_path_axioms G P p \<sigma>"
varSortAsSort_inj: shows "varSortAsSort_inj Delta"
th3_5_1: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and d :: "(real, 'a) vec"   and c :: "(real, 'a) vec" assumes "B\<^sub>\<real> a b d"   and "B\<^sub>\<real> b c d" shows "B\<^sub>\<real> a b c"
finite_measure_axioms: shows "finite_measure (restrict_space lborel {0<..<1})"
diff_zero: fixes plus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and minus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and zero :: "'i"   and a :: "'i" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
A4a: fixes x :: "i"   and y :: "i" shows "dom x \<cdot> y \<cong> dom (dom x) \<cdot> y"
rep_eq: fixes x :: "('d, 'e) lgenerator'" shows "lappend_trans x = append_raw (lgenerator x)"
term_poly_list_rel_set_mset: fixes p :: "char list list"   and q :: "char list multiset" assumes "(p, q) \<in> term_poly_list_rel" shows "set p = set_mset q"
get_disconnected_nodes_delete_shadow_root: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr' :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \<noteq> ptr'"   and "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
sorted_list_of_map_inj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "inj_on (linorder.sorted_list_of_map less_eq) (Collect (finite \<circ> dom))"
rep_eq: fixes x :: "float"   and xa :: "float" shows "real_of_float (x * xa) = real_of_float x * real_of_float xa"
1: shows "set b\<^sub>2 \<subseteq> set b"
memClosed: fixes p :: "(name \<times> name) list"   and x :: "'a"   and xs :: "'a list" shows "p \<bullet> x mem p \<bullet> xs = x mem xs"
simps_1: fixes Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "pre SKIP Q = ($ 1 \<and>* Q)"
of_nat_ne_\<omega>: fixes n :: "nat" shows "of_nat n \<noteq> \<omega>"
comp: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "map f \<circ> map g = map (f \<circ> g)"
IH_1: fixes p :: "'a"   and n :: "'a" assumes "p \<notin> snd (pn_atoms F)"   and "n \<notin> fst (pn_atoms F)" shows "(M \<Turnstile> F \<longrightarrow> M(p := True) \<Turnstile> F \<and> M(n := False) \<Turnstile> F) \<and> (\<not> M \<Turnstile> F \<longrightarrow> \<not> M(n := True) \<Turnstile> F \<and> \<not> M(p := False) \<Turnstile> F)"
Gram_Schmidt_matrix_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "Gram_Schmidt_matrix A = Gram_Schmidt_upt_k A (ncols A - 1)"
ipresCons_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'f, 'opSym, 'var, 'd, 'e, 'g) model_scheme" shows "ipresCons h hA MOD \<equiv> ipresVar h MOD \<and> ipresAbs h hA MOD \<and> ipresOp h hA MOD"
inf_zmultiset_def: fixes A :: "'a zmultiset"   and B :: "'a zmultiset" shows "inf A B = (if A < B then A else B)"
Domain_insert: fixes a :: "'c"   and b :: "'d"   and r :: "('c \<times> 'd) set" shows "Domain (insert (a, b) r) = insert a (Domain r)"
SubstTermP_subst: fixes v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm"   and j :: "name"   and w :: "tm" shows "(SubstTermP v i t u)(j::=w) = SubstTermP (subst j w v) (subst j w i) (subst j w t) (subst j w u)"
1: shows "(f \<noteq> Null \<and> Dom f \<in> Obj \<and> Cod f \<in> Obj \<and> Map f \<in> Hom (Dom f) (Cod f)) \<and> Dom f = Cod f"
min_height_size1_if_incomplete: fixes t :: "'b tree" assumes "\<not> complete t" shows "2 ^ min_height t < size1 t"
U'_def: fixes x :: "'s"   and y :: "'s" shows "U' x y = \<integral>\<^sup>+ x. ennreal_of_enat (eSuc (sfirst (HLD {y}) x)) \<partial>T x"
orthonormal_system_eq: fixes S :: "'a set"   and w :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and m :: "'b"   and n :: "'b" assumes "orthonormal_system S w" shows "l2product S (w m) (w n) = (if m = n then 1 else 0)"
finite_image_set2: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "finite {x. P x}"   and "finite {y. Q y}" shows "finite {f x y |x y. P x \<and> Q y}"
a_notin_n_pow1: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "\<not> Unit R a"   and "a \<noteq> \<zero>"   and "0 < n" shows "a \<notin> R \<diamondsuit>\<^sub>p a \<^bsup>\<diamondsuit>R Suc n\<^esup>"
crdt_ops_unique_mid: fixes xs :: "('c \<times> 'd) list"   and oid :: "'c"   and oper :: "'d"   and ys :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set" assumes "crdt_ops (xs @ [(oid, oper)] @ ys) deps" shows "oid \<notin> set (map fst xs) \<and> oid \<notin> set (map fst ys)"
simps_2: fixes \<theta> :: "(name \<times> lam) list"   and e\<^sub>1 :: "lam"   and e\<^sub>2 :: "lam" shows "\<theta><App e\<^sub>1 e\<^sub>2> = App (\<theta><e\<^sub>1>) (\<theta><e\<^sub>2>)"
one_imp_carrier: assumes "\<one> \<in> Idl {\<one>}" shows "Idl {\<one>} = carrier R"
coeff_sign_changes_smult: fixes a :: "'a"   and p :: "'a poly" assumes "(0::'a) < a" shows "sign_changes (coeffs (smult a p)) = sign_changes (coeffs p)"
right_mult_one_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A * 1\<^sub>m nc = A"
simps_7: fixes v1 :: "'b"   and v2 :: "'b" shows "AExp_Lexorder.less_aexp_aux (V v1) (V v2) = (v1 < v2)"
e_hs_def: shows "e_hs = (SOME (e, hs). 0 < e \<and> length hs = k \<and> (\<forall>h\<in>set hs. h \<in> O(\<lambda>x. real x / ln (real x) powr (1 + e))) \<and> (\<forall>t\<in>set ts. \<forall>x\<ge>x\<^sub>1. x\<^sub>0 \<le> t x \<and> t x < x) \<and> (\<forall>i<k. \<forall>x\<ge>x\<^sub>1. bs ! i * real x + (hs ! i) x = real ((ts ! i) x)))"
open_cover_of_open_subset_def: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and U :: "'c set"   and I :: "real set"   and C :: "real \<Rightarrow> 'c set" shows "open_cover_of_open_subset X is_open U I C \<equiv> open_cover_of_subset X is_open U I C \<and> open_cover_of_open_subset_axioms is_open U"
simps_2: fixes v :: "'a"   and l :: "value"   and t :: "value list" shows "fold_In v (l # t) = gOr (Eq (V v) (L l)) (fold_In v t)"
simps_2: fixes x :: "real"   and n :: "nat" shows "sqrtu x (Suc n) = (sqrtu x n + x / sqrtu x n) / 2"
real_sqrt_divide: fixes x :: "real"   and y :: "real" shows "sqrt (x / y) = sqrt x / sqrt y"
705: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "x \<sqinter> (y \<squnion> z) \<squnion> (x \<sqinter> (y \<squnion> z \<sqinter> bot) \<squnion> u) = x \<sqinter> (y \<squnion> z) \<squnion> u"
subgroup_imp_group: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := S\<rparr>)" shows "Group.group (G\<lparr>carrier := S, carrier := H\<rparr>)"
b_e_check_single_type_sound: fixes x1 :: "t list"   and t_in :: "t list"   and t_out :: "t list"   and x2 :: "t list"   and tm :: "t list"   and \<C> :: "t_context"   and e :: "b_e" assumes "type_update (Type x1) (to_ct_list t_in) (Type t_out) = Type x2"   and "c_types_agree (Type x2) tm"   and "\<C> \<turnstile> [e] : t_in _> t_out" shows "\<exists>tn. c_types_agree (Type x1) tn \<and> \<C> \<turnstile> [e] : tn _> tm"
B: fixes x :: "complex" assumes "B \<le> cmod x" shows "cmod (f x) * 2 < cmod (f z)"
b_def: shows "b = {Trn X r |r. Trn X r \<in> rhs}"
not_ev: fixes \<phi> :: "'a stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> ev \<phi> xs) = alw (\<lambda>xs. \<not> \<phi> xs)"
swap_continuous: fixes a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "continuous_on (cbox (a, c) (b, d)) (\<lambda>(x, y). f x y)" shows "continuous_on (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)"
simps_166: fixes x61 :: "('b, 'a, 'c) expr"   and x62 :: "('b, 'a, 'c) expr" shows "LID\<^sub>E (Assign x61 x62) = LID\<^sub>E x61 \<union> LID\<^sub>E x62"
assms: shows "complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>x\<in>k. ball x e))"
empty_filter_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "([] = filter P xs) = (\<forall>x\<in>set xs. \<not> P x)"
cr_mpoly_def: shows "cr_mpoly \<equiv> \<lambda>x y. x = mapping_of y"
induces_meta_adjunction: shows "meta_adjunction (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G F E.\<phi> E.\<psi>"
matrix_to_iarray_Gauss_Jordan_column_k_2: fixes k :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and i :: "nat" assumes "k < ncols A"   and "i \<le> nrows A" shows "matrix_to_iarray (snd (Gauss_Jordan_column_k (i, A) k)) = snd (Gauss_Jordan_column_k_iarrays (i, matrix_to_iarray A) k)"
thinish: fixes C :: "'a set"   and z :: "'a set" assumes "chamber C"   and "z \<lhd> C"   and "\<exists>D\<in>X - {C}. z \<lhd> D" shows "\<exists>!D. D \<in> X - {C} \<and> z \<lhd> D"
Proper_Roots_def: fixes M :: "node list" shows "Proper_Roots M \<equiv> Roots \<noteq> {} \<and> Roots \<subseteq> {i. i < length M}"
basis_gen: fixes w :: "'b list"   and G :: "'b list set" shows "w \<in> G \<longrightarrow> w \<in> \<langle>\<BB> G\<rangle>"
sr: shows "s \<in> reachable (pnet np n) TT"
bal_iff_full: fixes t :: "'b tree234" shows "bal t = (\<exists>n. full n t)"
\<pi>l: shows "\<pi> l = \<pi>l 0"
absorb_iff2: fixes b :: "'c word"   and a :: "'c word" shows "(b \<le>s a) = (signed.min a b = b)"
assms_5: shows "y \<notin> snd ` set txs"
U\<^sub>f_mult_without_empty_summands_even: fixes i :: "nat"   and j :: "nat"   and A :: "complex Matrix.mat" assumes "i < dim_row U\<^sub>f"   and "j < dim_col A"   and "even i"   and "dim_col U\<^sub>f = dim_row A" shows "(U\<^sub>f * A) $$ (i, j) = (\<Sum>k\<in>{i, i + 1}. U\<^sub>f $$ (i, k) * A $$ (k, j))"
order_0I: fixes p :: "'a poly"   and a :: "'a" assumes "poly p a \<noteq> (0::'a)" shows "Polynomial.order a p = 0"
right_commute: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "ac_operator f" shows "f (f a b) c = f (f a c) b"
subsequence_drop: fixes i :: "nat"   and w :: "nat \<Rightarrow> 'a"   and j :: "nat"   and k :: "nat" shows "drop i (w [j \<rightarrow> k]) = w [j + i \<rightarrow> k]"
asymp_equiv_add_right': fixes h :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "h \<in> o[F](g)" shows "(g \<sim>[F] (\<lambda>x. f x + h x)) = (g \<sim>[F] f)"