hcju/isabelleps huggingface dataset:
fixp_above_inside: fixes a :: "'a" assumes "a \<in> Field leq" shows "fixp_above a = lub (iterates_above a)"
in_Keys: fixes s :: "'a"   and F :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "(s \<in> Keys F) = (\<exists>f\<in>F. s \<in> keys f)"
one_compl: fixes x :: "'a" shows "- (x ; 1) ; 1 = - (x ; 1)"
smc_dg_def: fixes \<CC> :: "V" shows "smc_dg \<CC> = [\<CC>\<lparr>Obj\<rparr>, \<CC>\<lparr>Arr\<rparr>, \<CC>\<lparr>Dom\<rparr>, \<CC>\<lparr>Cod\<rparr>]\<^sub>\<circ>"
euler_mascheroni_less_13_over_22: shows "euler_mascheroni < 13 / 22"
span_is_submodule: fixes S :: "complex vec set" assumes "S \<subseteq> carrier_vec d" shows "LinearCombinations.submodule class_ring (local.span S) V"
local_lipschitz_compact_implies_lipschitz: fixes T :: "'b set"   and X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "local_lipschitz T X f"   and "compact X"   and "compact T"   and "\<And>x. x \<in> X \<Longrightarrow> continuous_on T (\<lambda>t. f t x)" obtains L :: "real" where "\<And>t. t \<in> T \<Longrightarrow> L-lipschitz_on X (f t)"
modify_alt_def: fixes f :: "'a \<Rightarrow> 'a" shows "modify x f = State_Monad.update (update f)"
pmf_pos: fixes p :: "'e pmf"   and x :: "'e" assumes "pmf p x \<noteq> 0" shows "0 < pmf p x"
exhaust: fixes y :: "alphabet" obtains "y = a"   | "y = b"
e: shows "E = (\<Sum>\<^sub># (mset Cs) + D) \<cdot> \<sigma>"
full_abstraction_impl_SRel_iff_TRel_is_trans: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)" shows "trans SRel = trans (_Collect (T1, T2) (\<exists>S1 S2. T1 = \<lbrakk>S1\<rbrakk> \<and> T2 = \<lbrakk>S2\<rbrakk> \<and> (T1, T2) \<in> TRel))"
dvd_pos_nat: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "m dvd n" shows "0 < m"
block_complement_size: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "finite_incidence_system point_set block_collection"   and "b \<subseteq> point_set" shows "card (incidence_system.block_complement point_set b) = card point_set - card b"
polychain_half_segments_of_aform: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" shows "polychain (half_segments_of_aform X)"
simps_5: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c llist" shows "(case LCons x21 x22 of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
lexord_mismatch: fixes u :: "'b list"   and v :: "'b list"   and r :: "('b \<times> 'b) set" assumes "\<not> u \<bowtie> v"   and "irrefl r" shows "(mismatch_pair u v \<in> r) = ((u, v) \<in> lexord r)"
module_vec_simps_1: fixes ty :: "'b itself"   and n :: "nat" shows "(\<oplus>\<^bsub>module_vec ty n\<^esub>) = (+)"
zero_sqmatrix_def: shows "0 \<equiv> SqMatrix (\<lambda>i j. 0::'c)"
disj: shows "(\<forall>K\<in>\<D>. K \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}) \<or> (\<forall>K\<in>\<D>. K \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {})"
BseqD: fixes X :: "nat \<Rightarrow> 'a" assumes "Bseq X" shows "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
sep_impl_sep_true_P: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "(sep_true \<longrightarrow>\<^sup>* P) s" shows "P s"
diff_closed: fixes U :: "'c set"   and V :: "'c set"   and x :: "'c"   and y :: "'c" assumes "Subspace.subspace U V"   and "vectorspace V"   and "x \<in> U"   and "y \<in> U" shows "x - y \<in> U"
assms: fixes a :: "'a poly"   and i :: "nat" assumes "(a, i) \<in> as" shows "i < n"
exists_fresh_1: fixes x :: "'a" assumes "finite (supp x)" obtains c :: "tyvrs" where "c \<sharp> x"
bot_assn_def: shows "bot \<equiv> Abs_assn (\<lambda>_. False)"
wiring: fixes \<I> :: "('m, 'n) \<I>"   and \<I>' :: "('o, 'p) \<I>"   and cnv :: "('m, 'n, 'o, 'p) converter"   and f :: "'m \<Rightarrow> 'o"   and g :: "'p \<Rightarrow> 'n" assumes "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C"   and "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>" shows "wiring \<I> \<I>' cnv (f, g)"
isomorphic_implies_hpar_2: fixes f :: "'a"   and f' :: "'a" assumes "f \<cong>\<^sub>B f'" shows "B.ide f'"
simps_4: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and X :: "'b ltln set" shows "(\<psi>\<^sub>1 W\<^sub>n \<psi>\<^sub>2)[X]\<^sub>\<nu> = \<psi>\<^sub>1[X]\<^sub>\<nu> W\<^sub>n \<psi>\<^sub>2[X]\<^sub>\<nu>"
fut_terminate: fixes \<phi>\<Phi> :: "('a, 'k) aformula \<times> ('a, 'k) aformula list" assumes "fut_invariant \<phi>\<Phi>"   and "\<not> (case \<phi>\<Phi> of (\<phi>, \<Phi>) \<Rightarrow> \<phi> \<notin> set \<Phi>)" shows "fut_spec \<phi>\<Phi>"
identity_rules_mono: fixes C\<^sub>1 :: "'b Standard_Constant set"   and C\<^sub>2 :: "'b Standard_Constant set" assumes "C\<^sub>1 \<subseteq> C\<^sub>2" shows "identity_rules C\<^sub>1 \<subseteq> identity_rules C\<^sub>2"
nn_integral_indicator_singleton': fixes y :: "'c"   and M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "{y} \<in> sets M" shows "\<integral>\<^sup>+ x. ennreal (f x * indicat_real {y} x) \<partial>M = ennreal (f y) * emeasure M {y}"
A: fixes nr :: "nat"   and nc :: "nat"   and A :: "'c mat" assumes "Matrix_Kernel.kernel nr nc A" shows "A \<in> carrier_mat nr nc"
plus_fun_def: fixes m1 :: "'a \<Rightarrow> 'b"   and m2 :: "'a \<Rightarrow> 'b" shows "m1 + m2 \<equiv> \<lambda>x. if m2 x = none then m1 x else m2 x"
get_trms_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b literal set"   and E :: "'b trm set"   and t :: "inferences" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.get_trms filter_trms C E t = (if t = FirstOrder then filter_trms C E else E)"
impr_ded: fixes a :: "'a N"   and b :: "'a N"   and c :: "'a N" shows "a * b r\<rightarrow> c = b r\<rightarrow> a r\<rightarrow> c"
S_def: shows "S = (\<lambda>b. \<Sum>k\<le>b. (of_nat b + a gchoose k) * x ^ k * y ^ (b - k))"
bind_pure_returns_result_I2: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog"   and y :: "'d" assumes "pure f h"   and "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
poincare_line_cvec_cmat_hermitean: fixes z :: "complex \<times> complex"   and w :: "complex \<times> complex" shows "hermitean (poincare_line_cvec_cmat z w)"
characteristic_formula_is_characteristic': fixes Q :: "'state"   and P :: "'state" shows "Q \<Turnstile> characteristic_formula P = (P =\<cdot> Q)"
positive_imp_inverse_positive: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less zero (inverse a)"
right_idem: fixes a :: "'d upred"   and b :: "'d upred" shows "((a \<and> b) \<and> b) = (a \<and> b)"
1: shows "is_left_adjoint f \<and> is_left_adjoint g \<and> g \<star> f\<^sup>* \<cong> r"
supt_supteq_conv: fixes s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" shows "s \<rhd> t = (s \<unrhd> t \<and> s \<noteq> t)"
bind_eq_join_pmf: fixes M :: "'f pmf"   and f :: "'f \<Rightarrow> 'e pmf" shows "M \<bind> f = join_pmf (map_pmf f M)"
Act\<^sub>\<alpha>_def: shows "Act\<^sub>\<alpha> \<equiv> (id ---> rep_Tree\<^sub>\<alpha> ---> abs_Tree\<^sub>\<alpha>) tAct"
dom_mult_closed: fixes x :: "'a"   and y :: "'a" shows "ad (ad (ad (ad x) \<cdot> ad (ad y))) = ad (ad x) \<cdot> ad (ad y)"
prod_list_prod_nth: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'b list" assumes "class.monoid_add plus zero" shows "prod_list xs = prod ((!) xs) {0..<length xs}"
isomorphic_categories_axioms_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "isomorphic_categories_axioms A B \<equiv> \<exists>F G. inverse_functors A B F G"
natural_transformation_axioms: shows "natural_transformation A C (K \<circ> F) (K \<circ> G) (K \<circ> \<tau>)"
adj_in_verts_1: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^bsub>G\<^esub> v" shows "u \<in> verts G"
bottom_weak_eq: fixes b :: "'a" assumes "b \<in> carrier L"   and "\<And>x. x \<in> carrier L \<Longrightarrow> b \<sqsubseteq> x" shows "b .= \<bottom>"
\<tau>external'_red_external_heap_unchanged: fixes P :: "'a prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap" assumes "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "\<tau>external' P h a M" shows "h' = h"
rtrancl_beta_App: fixes s :: "Core.term"   and s' :: "Core.term"   and t :: "Core.term"   and t' :: "Core.term" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'"   and "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'" shows "s $ t \<rightarrow>\<^sub>\<beta>\<^sup>* s' $ t'"
simps_2: fixes t :: "('a, 'c, 'b) Tree" shows "S_transform_Tree (tNot t) = Not\<^sub>\<alpha> (S_transform_Tree t)"
discI_2: fixes llist :: "'b llist"   and x21 :: "'b"   and x22 :: "'b llist" assumes "llist = x21 ## x22" shows "\<not> lnull llist"
res_lim_def: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" shows "res_lim s = (\<lambda>k. THE r. \<exists>N. \<forall>m>N. s m k = r)"
RP_saturated_if_fair: assumes "chain (\<leadsto>) Sts"   and "fair_state_seq Sts"   and "Q_of_state (lhd Sts) = {}" shows "sr.saturated_upto (Liminf_llist (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts))"
partition_groups_on_size: shows "partition_on \<G> {{G \<in> \<G>. card G = g} |g. g \<in> group_sizes}"
tyenv_wellformed_sub: fixes P\<^sub>1 :: "'BExp set"   and P\<^sub>2 :: "'BExp set"   and \<Gamma>\<^sub>2 :: "'Var \<Rightarrow> 'BExp set option"   and \<Gamma>\<^sub>1 :: "'Var \<Rightarrow> 'BExp set option"   and mds :: "Mode \<Rightarrow> 'Var set"   and \<S> :: "'Var set \<times> 'Var set" assumes "P\<^sub>1 \<subseteq> P\<^sub>2"   and "\<Gamma>\<^sub>2 = \<Gamma>\<^sub>1"   and "tyenv_wellformed mds \<Gamma>\<^sub>2 \<S> P\<^sub>2" shows "tyenv_wellformed mds \<Gamma>\<^sub>1 \<S> P\<^sub>1"
2: shows "strip C = c \<and> local.step' \<top> C \<le> C"
nth_i: shows "x = dropWhile (\<lambda>x. t < f x) xs ! i"
simps_1: fixes c :: "poly"   and n :: "nat"   and p :: "poly" shows "headn (CN c n p) = (\<lambda>m. if n \<le> m then headn p m else CN c n p)"
ts_prop_2: fixes k :: "cnf" shows "\<parallel>the_bb\<parallel>\<^bsub>k\<^esub>"
fds_nth_deriv: fixes f :: "'a fds"   and n :: "nat" shows "fds_nth (fds_deriv f) n = - ln (real n) *\<^sub>R fds_nth f n"
igFreshInp_def: fixes MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme"   and ys :: "'d"   and y :: "'g"   and inp :: "'k \<Rightarrow> 'h option" shows "igFreshInp MOD ys y inp \<equiv> liftAll (igFresh MOD ys y) inp"
lucas_lehmer_add_cong_2: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat" shows "[snd (lucas_lehmer_add m x y) = snd x + snd y] (mod m)"
lemVecPlusAssoc: fixes u :: "'a Vector"   and v :: "'a Vector"   and w :: "'a Vector" shows "u \<oplus> v \<oplus> w = (u \<oplus> v) \<oplus> w"
bdd_aboveI2: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ('b, 'c) Term.term"   and M :: "('b, 'c) Term.term" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<sqsubseteq> M" shows "term.bdd_above (f ` A)"
bij_betw_number_partition_of: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "bij_betw (number_partition_of A B) ((A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B) {N. number_partition (card A) N \<and> size N \<le> card B}"
c: shows "bv_to_nat (nat_to_bv a) = 0"
ffb_kcomp_pres: fixes f :: "'d \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'c set" shows "fb\<^sub>\<F> (f \<circ>\<^sub>K g) = fb\<^sub>\<F> f \<circ> fb\<^sub>\<F> g"
der': fixes x :: "(real, 1) vec" assumes "x \<in> vec ` S" shows "(vec \<circ> g \<circ> (\<lambda>x. x $ 1) has_derivative (*\<^sub>R) (g' (x $ 1))) (at x within vec ` S)"
UN_extend_simps_8: fixes B :: "'d \<Rightarrow> 'c set"   and A :: "'d set set" shows "\<mu> (\<P> (\<lambda>y. \<mu> (\<P> B y)) A) = \<mu> (\<P> B (\<mu> A))"
independent_AbGroups_pairwise_int0_double: fixes G :: "'a set"   and G' :: "'a set" assumes "AbGroup G"   and "AbGroup G'"   and "add_independentS [G, G']" shows "G \<inter> G' = 0"
reds: shows "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e'',s'',b''\<rangle>"
simps_3: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "vwalk_arcs (x # y # xs) = (x, y) # vwalk_arcs (y # xs)"
is_interval_io: fixes r :: "real" shows "is_interval {..<r}"
subset_imageE_inj: fixes B :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "B \<subseteq> f ` A" obtains C :: "'e set" where "C \<subseteq> A" and "B = f ` C" and "inj_on f C"
st_proj1_inv_inv: fixes x :: "'b" shows "st_proj1 (st_proj1_inv x) = x"
\<Delta>0: shows "\<Delta>0 \<in> \<Delta>s"
DF_all_tickfree_traces1: shows "T (DF UNIV) = {s. set s \<subseteq> range ev}"
map_merge_def: fixes f :: "'var \<Rightarrow> 'val option"   and g :: "'var \<Rightarrow> 'val option"   and Q :: "nat \<Rightarrow> bool"   and xs :: "'var list" shows "map_merge f g Q xs \<equiv> \<lambda>V. if \<exists>i<length xs. xs ! i = V \<and> Q i then g V else f V"
cond_var_subst_left: fixes x :: "bool \<Longrightarrow> 'a"   and P :: "('b, 'a) uexpr"   and Q :: "('b, 'a) uexpr" assumes "vwb_lens x" shows "P\<lbrakk>true/x\<rbrakk> \<triangleleft> x \<triangleright> Q = P \<triangleleft> x \<triangleright> Q"
total: fixes x :: "'alt"   and y :: "'alt" assumes "x \<in> alts"   and "y \<in> alts" shows "R x y \<or> R y x"
c_in: shows "C \<in> {C \<in> N. \<not> I \<Turnstile> {C}}"
lexord_transI: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "trans (lexord r)"
cases: fixes x :: "'a list \<times> 'a list" obtains ys :: "'a list" where "x = ([], ys__)"   | xa :: "'a" and xs :: "'a list" and ys :: "'a list" where "x = (xa # xs, ys)"
assms: fixes n :: "nat" shows "\<exists>i\<in>I. of_nat n \<le> f i"
prems_1: fixes a :: "'a" assumes "a \<in> insert a__ A"   and "\<And>b. rel b a \<Longrightarrow> b \<notin> insert a__ A" shows "thesis"
npow_suc: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and n :: "nat" shows "x^\<^bsup>R Suc n\<^esup> = x^\<^bsup>R n\<^esup> \<cdot>\<^sub>r\<^bsub>R\<^esub> x"
homeomorphic_spheres': fixes \<delta> :: "real"   and a :: "'c"   and b :: "'d" assumes "0 < \<delta>"   and "DIM('c) = DIM('d)" shows "sphere a \<delta> homeomorphic sphere b \<delta>"
triangle_strict_inequality: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B D"   and "Cong B C B D"   and "\<not> Bet A B C" shows "Tarski_neutral_dimensionless.Lt Bet Cong A C A D"
fold_insert_idem: fixes A :: "'a set"   and z :: "'b"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z A)"
alw_elim: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "alw \<phi> \<pi>l" obtains "\<phi> \<pi>l" and "alw \<phi> (map stl \<pi>l)"
lim: shows "(\<lambda>n. f n x0) \<longlonglongrightarrow> l"
inverse_inverse: fixes a :: "'a perm" shows "inverse (inverse a) = a"
wf_trm_param_idx: fixes f :: "'fun"   and T :: "('fun, 'b) Term.term list"   and i :: "nat" assumes "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)"   and "i < length T" shows "wf\<^sub>t\<^sub>r\<^sub>m (T ! i)"
orderE: fixes b :: "'a"   and a :: "'a" assumes "b \<le> a" obtains "a = a + b"
real_of_int_div: fixes d :: "int"   and n :: "int" assumes "d dvd n" shows "real_of_int (n div d) = real_of_int n / real_of_int d"
bset_cases: fixes X :: "'d bset" obtains "X = bempty"   | x :: "'d" where "X = bsingleton x__"   | x :: "'d" and y :: "'d" where "x \<noteq> y" and "X = bdoubleton x y"
r_unit: fixes G :: "('e, 'f) Group_scheme"   and a :: "'e" assumes "Group G"   and "a \<in> carrier G" shows "a \<cdot>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = a"
face_distance_eq_0: fixes X :: "'b set set"   and C :: "'b set"   and F :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "order.greater_eq C F" shows "ChamberComplex.face_distance X F C = 0"
one_circ_circ_split: shows "(1::'a)\<^sup>\<circ>\<^sup>\<circ> = L \<squnion> (1::'a)"
ag_l_zero: fixes A :: "('h, 'i) aGroup_scheme"   and x :: "'h" assumes "aGroup A"   and "x \<in> carrier A" shows "\<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> x = x"
nat_wf_ind: fixes P :: "nat \<Rightarrow> bool"   and z :: "nat" assumes "\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P z"
UNIV_lit: shows "UNIV = range (\<lambda>(x, pos). Lit x pos)"
retraction_map_def: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" shows "retraction_map X Y f \<equiv> \<exists>g. retraction_maps X Y f g"
simps_2: fixes i :: "nat"   and j :: "nat"   and xs :: "'b list" shows "I\<^sub>d\<^sub>l\<^sub>o (Less i j) xs = (xs ! i < xs ! j)"
WT_resource_of_oracle: fixes callee :: "'h \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'h) spmf"   and I :: "'h \<Rightarrow> bool"   and \<I> :: "('i, 'j) \<I>"   and s :: "'h" assumes "callee_invariant_on callee I \<I>"   and "I s" shows "\<I> \<turnstile>res RES callee s \<surd>"
surjective: fixes A :: "'a" obtains B :: "'b" where "(A, B) \<in> F"
rel_optionTE: fixes M :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "('f, 'd) optionT"   and m' :: "('g, 'e) optionT" assumes "rel_optionT M m m'" obtains x :: "'d" and y :: "'e" where "m = OptionT x" and "m' = OptionT y" and "M x y"
double_orthogonal_complement_id: fixes M :: "'b set" assumes "closed_csubspace M" shows "orthogonal_complement (orthogonal_complement M) = M"
ltl_logical_characterization: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" shows "w \<Turnstile> \<phi> = (\<exists>\<G>\<subseteq>\<^bold>G \<phi>. accept\<^sub>M \<phi> \<G> w \<and> closed \<G> w)"
3135: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (- y \<sqinter> - x) = - y \<sqinter> - x"
convex: fixes S :: "'b set" shows "convex S = (\<forall>k u x. (\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R x i) \<in> S)"
run_reachable: fixes i :: "nat" shows "C.reachable (cs i)"
pCons2: fixes a :: "'a"   and p :: "'a poly"   and b :: "'b"   and q :: "'b poly" assumes "a \<noteq> (0::'a) \<or> p \<noteq> 0"   and "b \<noteq> (0::'b) \<or> q \<noteq> 0"   and "P p q" shows "P (pCons a p) (pCons b q)"
length_induced_play: fixes s :: "'b list \<Rightarrow> 'b"   and p :: "'b list" shows "llength (induced_play s p) = 2 * N"
assms_2: shows "1\<^sub>h \<in> circline_set H"
bi_unique_rel_set_lemma: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and X :: "'d set"   and Y :: "'e set" assumes "bi_unique R"   and "rel_set R X Y" obtains f :: "'d \<Rightarrow> 'e" where "Y = f ` X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
split_intervalD: fixes X :: "'a interval"   and x :: "'a"   and A :: "'a interval"   and B :: "'a interval" assumes "split_interval X x = (A, B)" shows "set_of X \<subseteq> set_of A \<union> set_of B"
xy2: shows "det2 a11 a13 a21 a23 * x2 + det2 a12 a13 a22 a23 * y2 = (0::'a)"
sorted_wrt_true: fixes xs :: "'d list" shows "sorted_wrt (\<lambda>_ _. True) xs"
countable_lfp: fixes F :: "'c set \<Rightarrow> 'c set" assumes "\<And>Y. countable Y \<Longrightarrow> countable (F Y)"   and "sup_continuous F" shows "countable (lfp F)"
finite_tree: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent"   and pth :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) list" assumes "valid_in_port (v, p)"   and "terminal_vertex v" shows "tfinite (tree v p pth)"
inf_of_two_greatest: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (division_rel G)"   and "y \<in> carrier (division_rel G)" shows "is_glb (division_rel G) (\<Sqinter>\<^bsub>division_rel G\<^esub>{x, y}) {x, y}"
rep_eq: fixes x :: "natural"   and xa :: "natural" shows "(x \<le> xa) = (nat_of_natural x \<le> nat_of_natural xa)"
leibniz_rule_field_differentiable: fixes U :: "'a set"   and a :: "'b"   and b :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and fx :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and x0 :: "'a" assumes "\<And>x t. \<lbrakk>x \<in> U; t \<in> cbox a b\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"   and "\<And>x. x \<in> U \<Longrightarrow> f x integrable_on cbox a b"   and "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). fx x t)"   and "x0 \<in> U"   and "convex U" shows "(\<lambda>x. integral (cbox a b) (f x)) field_differentiable at x0 within U"
assms_2: shows "f ` {..<q} = {..<q}"
base_def: fixes ds :: "nat set"   and p :: "(nat \<times> int) list" shows "base ds p = (THE b. b \<in> grid (start (length p)) ({0..<length p} - ds) \<and> p \<in> grid b ds)"
finite_emeasure_space: shows "emeasure P (space P) \<noteq> \<top>"
hS: shows "h ` ({0..1} \<times> S) \<subseteq> S"
edgesinvertices: shows "\<Union> (vertices ` E) \<subseteq> V"
power_increasing_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 :: "'a"   and x :: "nat"   and y :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one b" shows "less_eq (power.power one times b x) (power.power one times b y) = (x \<le> y)"
add_block_alt: fixes b :: "'a set" shows "add_point_sys.add_block b = add_mset b \<B>"
simps_2: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and m :: "'b common_primitive match_expr"   and rs :: "'b common_primitive rule list"   and allowed :: "'b word set"   and denied1 :: "'b word set" shows "no_spoofing_algorithm iface ipassmt (Rule m Accept # rs) allowed denied1 = no_spoofing_algorithm iface ipassmt rs (allowed \<union> get_exists_matching_src_ips iface m) denied1"
graph_union_edges: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph" shows "edges (graph_union G\<^sub>1 G\<^sub>2) = edges G\<^sub>1 \<union> edges G\<^sub>2"
size_eq_0_iff_empty: fixes M :: "'a multiset" shows "(size M = 0) = (M = {#})"
strips_equivalent_to_sas_plus_i_a_III: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops :: "('c, 'd) sas_plus_operator list"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)" shows "execute_parallel_operator (\<phi>\<^sub>S \<Psi> s) (map (sasp_op_to_strips \<Psi>) ops) = \<phi>\<^sub>S \<Psi> execute_parallel_operator_sas_plus s ops"