hcju/isabelleps huggingface dataset:
interior_union_boundary: fixes X :: "'a set"   and Y :: "'a set" shows "(interior (X \<union> Y) = interior X \<union> interior Y) = (boundary X \<inter> boundary Y \<subseteq> boundary (X \<union> Y))"
id: fixes f :: "'b \<Rightarrow> 'c" shows "sum f K = f i + sum f (K - {i})"
simps: fixes a :: "'a"   and S :: "'a set" shows "(a \<in> group_closure S) = ((\<exists>s. a = s \<and> s \<in> insert (0::'a) S) \<or> (\<exists>s t. a = s - t \<and> s \<in> group_closure S \<and> t \<in> group_closure S))"
Axiom_of_Unions: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> a \<in>\<^sub>\<circ> Vset \<alpha>"
\<gamma>_eq_def: fixes a :: "real list \<times> real list" shows "\<gamma>_eq a = (mean (fst a) = mean (snd a) \<and> length (fst a) = length (snd a))"
zero_neq_one: fixes one :: "'a"   and zero :: "'a" assumes "class.zero_neq_one one zero" shows "zero \<noteq> one"
asymp_equiv_at_top_imp_at_right: fixes a :: "real" assumes "(\<lambda>x. f (a + inverse x)) \<sim>[at_top] (\<lambda>x. g (a + inverse x))" shows "f \<sim>[at_right a] g"
Pd'_d'_x: shows "x \<in> Field (Pd' (Xd x))"
flowD_finite_OUT: fixes f :: "'v vertex \<times> 'v vertex \<Rightarrow> ennreal"   and x :: "'v vertex" assumes "flow \<Psi> f"   and "x \<noteq> source \<Psi>"   and "x \<noteq> sink \<Psi>" shows "d_OUT f x \<noteq> \<top>"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "\<lbrace>t \<^bold>\<otimes> u\<rbrace> = \<lbrace>t\<rbrace> \<otimes> \<lbrace>u\<rbrace>"
replacefacesAt2_nth2: fixes k :: "nat"   and F :: "face list list"   and oldf :: "face"   and newfs :: "face list" assumes "k < |F|" shows "replacefacesAt2 [k] oldf newfs F ! k = replace oldf newfs (F ! k)"
bijective_def: fixes \<alpha> :: "'d \<Rightarrow> 'e"   and S :: "'d set"   and T :: "'e set" shows "bijective \<alpha> S T \<equiv> bij_betw \<alpha> S T"
real_cond_exp_intg_2: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M (\<lambda>x. f x * g x)"   and "f \<in> borel_measurable F"   and "g \<in> borel_measurable M" shows "LINT x|M. f x * real_cond_exp M F g x = LINT x|M. f x * g x"
B: shows "offs_set_prev ns xs index key mi ma j \<noteq> {}"
Super: fixes G :: "prog"   and s :: "st" shows "G\<turnstile>Norm s \<midarrow>Super-\<succ>val_this s\<rightarrow> Norm s"
sctn_rel_def: fixes A :: "('a \<times> 'b) set" shows "\<langle>A\<rangle>sctn_rel = {(a, b). rel_sctn (in_rel A) a b}"
GH_def: shows "GH = supp g \<union> supp h"
lim_proc_is_lub3c: fixes S :: "nat \<Rightarrow> 'a process"   and X :: "'a process set" assumes "chain S"   and "X = range S" shows "\<forall>u. X <| u \<longrightarrow> min_elems (D (lim_proc X)) \<subseteq> T u"
borel_Iio: shows "borel = sigma UNIV (range lessThan)"
obtain_subset_with_card_int_n: fixes n :: "int"   and S :: "'b set" assumes "n \<le> int (card S)"   and "0 \<le> n" obtains T :: "'b set" where "T \<subseteq> S" and "int (card T) = n" and "finite T"
simps: fixes uu :: "'c \<Rightarrow> 'd list"   and uv :: "'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set list"   and ALF :: "'c \<times> 'f \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qAbs set list"   and uw :: "('c \<Rightarrow> 'd \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm set option) list"   and xs :: "'c"   and s :: "'f" shows "absOfS (ParS uu uv ALF uw) (xs, s) = set (ALF (xs, s))"
chain_equality: fixes A :: "'f set" shows "Complete_Partial_Order.chain (=) A = (\<forall>x\<in>A. \<forall>y\<in>A. x = y)"
implies_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<squnion> y) \<squnion> z = (- x \<squnion> z) \<sqinter> (- y \<squnion> z)"
Seq_tE: fixes c1 :: "com"   and c2 :: "com"   and s1 :: "char list \<Rightarrow> int"   and p :: "nat"   and s3 :: "char list \<Rightarrow> int" assumes "(c1;; c2, s1) \<Rightarrow> p \<Down> s3" obtains x :: "nat" and s2 :: "char list \<Rightarrow> int" and y :: "nat" where "p = x + y" and "(c1, s1) \<Rightarrow> x \<Down> s2" and "(c2, s2) \<Rightarrow> y \<Down> s3"
cSym: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel" shows "(\<Psi>', T, S) \<in> Rel"
flatten_sigma_2_IH_lifting: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b set" assumes "\<psi> \<in> subfrmlsn \<phi>"   and "suffix i w \<Turnstile>\<^sub>n \<psi>[\<G>\<F> \<psi> (suffix i w)]\<^sub>\<Sigma>\<^sub>2 = suffix i w \<Turnstile>\<^sub>n \<psi>" shows "suffix i w \<Turnstile>\<^sub>n \<psi>[\<G>\<F> \<phi> w]\<^sub>\<Sigma>\<^sub>2 = suffix i w \<Turnstile>\<^sub>n \<psi>"
le_diff_conv2: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b" shows "(c \<le> b - a) = (c + a \<le> b)"
content_cball_conv_ball: fixes c :: "'a"   and r :: "real" shows "content (cball c r) = content (ball c r)"
rank_nullity_theorem_matrices: fixes A :: "(('a, 'b) vec, 'c) vec" shows "ncols A = vec.dim (null_space A) + vec.dim (col_space A)"
AndI: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and b :: "('a, 'b) form" assumes "G \<turnstile> a"   and "G \<turnstile> b" shows "G \<turnstile> And a b"
distinct_sorted_list_of_set: fixes A :: "'a Word.word set" shows "distinct (signed.sorted_list_of_set A)"
rtrancl_eta_subst: fixes s :: "dB"   and t :: "dB"   and u :: "dB"   and i :: "nat" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "u[s/i] \<rightarrow>\<^sub>\<eta>\<^sup>* u[t/i]"
restricted_Dynkin_system: fixes D :: "'a set" assumes "D \<in> Dynkin \<Omega> M" shows "Dynkin_system \<Omega> {Q. Q \<subseteq> \<Omega> \<and> Q \<inter> D \<in> Dynkin \<Omega> M}"
Suc_rw: shows "[0..<Suc (Suc k)] = [0..<Suc k] @ [Suc k]"
g: shows "abc_step_l (as, am) (abc_fetch as ap) = (as', am')"
2136: fixes x :: "'a"   and y :: "'a" shows "- - ((x \<squnion> top) \<sqinter> y) \<sqinter> y = y"
inf_llist_def: fixes xs :: "'b llist"   and ys :: "'b llist" shows "xs \<sqinter> ys = unfold_llist (\<lambda>(xs, ys). xs \<noteq> LNil \<longrightarrow> ys \<noteq> LNil \<longrightarrow> lhd xs \<noteq> lhd ys) (lhd \<circ> snd) (map_prod ltl ltl) (xs, ys)"
Smith_1xn_def: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a mat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op" shows "Smith_Impl.Smith_1xn Smith_1x2 A = (if dim_col A = 0 then (A, 1\<^sub>m (dim_col A)) else Smith_Impl.Smith_1xn_aux Smith_1x2 (dim_col A - 1) A (A, 1\<^sub>m (dim_col A)))"
finite_ChiRel: fixes A :: "('a, 'b, 'c) hierauto" shows "finite (ChiRel A)"
msgrel_imp_eq_freenonces: fixes U :: "freemsg"   and V :: "freemsg" assumes "U \<sim> V" shows "freenonces U = freenonces V"
assms: shows "y \<in> {xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('a) = None}"
NextPropagate'_preserves_prop_invs: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and q :: "'b"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b" assumes "prop_invs (prop_config c0 q)"   and "NextPropagate' c0 c1 p" shows "prop_invs (prop_config c1 q)"
emeasure_subprob_space_less_top: fixes M :: "'b measure"   and A :: "'b set" assumes "subprob_space M" shows "emeasure M A \<noteq> top"
almost_full_on_imp_homogeneous_subseq: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and f :: "nat \<Rightarrow> 'b" assumes "almost_full_on P A"   and "\<forall>i. f i \<in> A" shows "\<exists>\<phi>. \<forall>i j. i < j \<longrightarrow> \<phi> i < \<phi> j \<and> P (f (\<phi> i)) (f (\<phi> j))"
DiaP: fixes a :: "'d"   and ps :: "('c, 'd) fm list"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and p :: "('c, 'd) fm"   and i :: "'d"   and A :: "'d set"   and n :: "nat" assumes "(\<^bold>\<diamond> p) at a in (ps, a) # branch"   and "i \<notin> A \<union> branch_nominals ((ps, a) # branch)"   and "\<nexists>a. p = Nom a"   and "\<not> witnessed p a ((ps, a) # branch)"   and "A, Suc n \<turnstile> ((\<^bold>@ i p) # (\<^bold>\<diamond> Nom i) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
bl2nat_double: fixes xs :: "cell list"   and n :: "nat" shows "bl2nat xs (Suc n) = 2 * bl2nat xs n"
det_row_mul: fixes k :: "'b"   and c :: "'a"   and a :: "'b \<Rightarrow> ('a, 'b) vec"   and b :: "'b \<Rightarrow> ('a, 'b) vec" shows "det (\<chi>i. if i = k then c *s a i else b i) = c * det (\<chi>i. if i = k then a i else b i)"
arrI: fixes f :: "'a" assumes "right (src \<mu>) f" shows "Right.arr f"
lmeasurable_open: fixes S :: "'b set" assumes "bounded S"   and "open S" shows "S \<in> lmeasurable"
"1_3": shows "state_succ (exec_plan s as1) act = s'"
ide_cod: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.ide C (partial_magma.cod C f)"
arsinh_0: shows "arsinh (0::'a) = (0::'a)"
span_eq: fixes Sa :: "('a \<Rightarrow> real) set"   and T :: "('a \<Rightarrow> real) set" assumes "Sa \<subseteq> diff_fun_space"   and "T \<subseteq> diff_fun_space" shows "(sub.diff_fun_space.m1.span Sa = sub.diff_fun_space.m1.span T) = (Sa \<subseteq> sub.diff_fun_space.m1.span T \<and> T \<subseteq> sub.diff_fun_space.m1.span Sa)"
rel_refl_strong: fixes x :: "'c set['d]"   and Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z. z \<in> set_bset x \<Longrightarrow> Ra z z" shows "rel_bset Ra x x"
inf_wf_step: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and order :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'c"   and m :: "'c" assumes "r\<^sup>+\<^sup>+ x y"   and "inf_wf r order n y" shows "inf_wf r order m x"
flip_altsublist_adjacent_in_Q: fixes ss :: "'w list"   and ts :: "'w list" assumes "Abs_freelist ss \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"   and "flip_altsublist_adjacent ss ts" shows "Abs_freelist ts \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
type_definition_def: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "type_definition Rep Abs A \<equiv> (\<forall>x. Rep x \<in> A) \<and> (\<forall>x. Abs (Rep x) = x) \<and> (\<forall>y. y \<in> A \<longrightarrow> Rep (Abs y) = y)"
domainConstant: fixes Y :: "'b set"   and X :: "'c set" shows "Domain (Y \<times> {0}) = Y \<and> Domain (X \<times> {1::'d}) = X"
suffix_def: fixes xs :: "'c list"   and ys :: "'c list" shows "suffix xs ys = (\<exists>zs. ys = zs @ xs)"
indep_sets_finite_index_sets: fixes F :: "'b \<Rightarrow> 'a set set"   and I :: "'b set" shows "indep_sets F I = (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> indep_sets F J)"
that: fixes F' :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and q :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite F'"   and "F' \<subseteq> F"   and "p = (\<Sum>f\<in>F'. q f * f)"   and "\<And>f. homogeneous (q f)"   and "\<And>f. f \<in> F' \<Longrightarrow> poly_deg (q f * f) = poly_deg p"   and "\<And>f. f \<notin> F' \<Longrightarrow> q f = 0" shows "thesis"
validE_def: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool"   and Q :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and E :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>, \<lbrace>E\<rbrace> \<equiv> \<lbrace>P\<rbrace> f \<lbrace>\<lambda>v s. case v of Inl e \<Rightarrow> E e s | Inr r \<Rightarrow> Q r s\<rbrace>"
sumRestrictedToDomainInvariant: fixes f :: "('d \<times> 'e) set"   and X :: "'d set" assumes "runiq (f || X)" shows "sum (toFunction f) (X \<inter> Domain f) = sum snd (f || X)"
subst_of_simps_3: fixes b :: "'c \<times> ('d, 'c) Term.term"   and ss :: "('c \<times> ('d, 'c) Term.term) list" shows "subst_of (b # ss) = subst_of ss \<circ>\<^sub>s subst (fst b) (snd b)"
less_sets_irrefl: fixes A :: "'a set" shows "(A \<lless> A) = (A = {})"
assms: shows "s |\<^sub>s n \<approx> t |\<^sub>s m"
bij_\<alpha>: shows "bij (id(r' := r'', r'' := r'))"
fmpred_foldl: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and init :: "('a, 'b) fmap"   and xs :: "('a, 'b) fmap list" assumes "fmpred P init"   and "\<And>x. x \<in> set xs \<Longrightarrow> fmpred P x" shows "fmpred P (foldl (++\<^sub>f) init xs)"
m\<tau>move_heap: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "m\<tau>move s (t, ta) s'" shows "shr s' = shr s"
constant_on_subset: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "f constant_on A"   and "B \<subseteq> A" shows "f constant_on B"
conts: fixes i :: "'a" shows "continuous_on UNIV (f i)"
l_cancel_one: fixes x :: "nat \<Rightarrow> 'a"   and a :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "a \<in> carrier P" shows "(x \<oplus>\<^bsub>P\<^esub> a = x) = (a = \<zero>\<^bsub>P\<^esub>)"
map_2: fixes f :: "'e \<Rightarrow> 'd"   and x21 :: "'e"   and x22 :: "'e list" shows "map f (x21 # x22) = f x21 # map f x22"
wp_under_wlp_SetPC: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and f :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> supp (P s) \<Longrightarrow> wp_under_wlp (f a)"   and "\<And>s a. a \<in> supp (P s) \<Longrightarrow> 0 \<le> P s a" shows "wp_under_wlp (SetPC f P)"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_stop (s, l, r) lm n ires = (\<exists>ln rn. l = Bk \<up> ln @ Bk # Bk # ires \<and> r = <lm ! n> @ Bk \<up> rn)"
simulation_def: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs" shows "P \<leadsto>[Rel] Q \<equiv> \<forall>a Q'. Q \<longmapsto>a \<prec> Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>a \<prec> P' \<and> (P', Q') \<in> Rel)"
rename_clause_def: fixes A :: "'at set"   and C :: "'at Literal set" shows "rename_clause A C = {L. \<exists>LL. LL \<in> C \<and> L = rename_literal A LL}"
truncate_down_nonneg_iff: fixes p :: "nat"   and x :: "real" shows "(0 \<le> truncate_down p x) = (0 \<le> x)"
eq_SConsD: fixes xs :: "'a stream"   and y :: "'a"   and ys :: "'a stream" assumes "xs = y ## ys" shows "shd xs = y \<and> stl xs = ys"
dyn_accessible_from_commonD: fixes G :: "prog"   and S :: "qtname"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m in C dyn_accessible_from S" shows "G \<turnstile> m member_in C"
pcr_fls_def: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "pcr_fls T \<equiv> rel_fun (=) T OO cr_fls"
exec_1_iff: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and \<sigma>' :: "'addr option \<times> 'heap \<times> 'addr frame list" shows "P,t \<turnstile> \<sigma> -ta-jvm\<rightarrow> \<sigma>' = ((ta, \<sigma>') \<in> exec P t \<sigma>)"
pmf_mix_induct': fixes A :: "'a set"   and p :: "'a pmf"   and P :: "'a pmf \<Rightarrow> bool" assumes "finite A"   and "set_pmf p \<subseteq> A"   and "\<And>x. x \<in> A \<Longrightarrow> P (return_pmf x)"   and "\<And>p q a. \<lbrakk>set_pmf p \<subseteq> A; set_pmf q \<subseteq> A; a \<in> {0<..<1}; P p; P q\<rbrakk> \<Longrightarrow> P (mix_pmf a p q)" shows "P p"
comm_monoid_list_axioms: shows "comm_monoid_list (*) (1::'a)"
gallery_least_length: fixes C :: "'a set"   and D :: "'a set" assumes "folding.chamber C"   and "folding.chamber D"   and "C \<noteq> D" shows "folding.gallery (C # (ARG_MIN length Cs. folding.gallery (C # Cs @ [D])) @ [D])"
normal_set_INTER: fixes A :: "nat \<Rightarrow> ordinal set" assumes "\<forall>n. normal_set (A n)"   and "\<forall>n. A (Suc n) \<subseteq> A n" shows "normal_set (\<Inter> (range A))"
simple_fw_iff_generalized_fw_accept: fixes fw :: "'a simple_rule list"   and p :: "('a, 'b) simple_packet_scheme" shows "(simple_fw fw p = Decision FinalAllow) = (\<exists>r. generalized_sfw (map simple_rule_dtor fw) p = Some (r, Accept))"
3: shows "gfg.Prj\<^sub>1 \<cdot> gfg.chine_assoc \<cdot> chine_hcomp \<eta> g \<cdot> \<langle>g.leg1 \<lbrakk>g.dtrg, g.leg1\<rbrakk> g.chine\<rangle> = gfg.Prj\<^sub>1\<^sub>1 \<cdot> chine_hcomp \<eta> g \<cdot> \<langle>g.leg1 \<lbrakk>g.dtrg, g.leg1\<rbrakk> g.chine\<rangle>"
upper_split_float_interval1: fixes X :: "float interval" shows "upper (fst (split_float_interval X)) = mid X"
ltake_Suc_conv_snoc_lnth: fixes m :: "nat"   and xs :: "'a llist" assumes "enat m < llength xs" shows "ltake (enat (Suc m)) xs = ltake (enat m) xs @\<^sub>l (lnth xs m #\<^sub>l []\<^sub>l)"
pickT_pred_pickT: fixes II :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "II \<in> UNpart cl P" shows "pickT_pred cl P II (pickT cl P II)"
acyclic_5b_def: fixes x :: "'a" shows "acyclic_5b x \<equiv> \<forall>y\<le>x. coreflexive (y\<^sup>\<star> \<sqinter> (x \<sqinter> - y)\<^sup>+)"
connected_component_eq: fixes y :: "'b"   and S :: "'b set"   and x :: "'b" assumes "y \<in> connected_component_set S x" shows "connected_component_set S y = connected_component_set S x"
OutputPres: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "P \<equiv>\<^sub>e Q" shows "a{b}.P \<equiv>\<^sub>e a{b}.Q"
uprodE: fixes c :: "('a, 'b) node set"   and A :: "('a, 'b) node set set"   and B :: "('a, 'b) node set set" assumes "c \<in> uprod A B" obtains x :: "('a, 'b) node set" and y :: "('a, 'b) node set" where "x \<in> A" and "y \<in> B" and "c = Scons x y"
assms_3: shows "C\<^sub>1 \<inter> C\<^sub>2 \<noteq> {}"
norm_zero: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm zero = 0"
abort_Bleast_def: fixes S :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "abort_Bleast S P = (LEAST x. x \<in> S \<and> P x)"
subst_typ_tinstT: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and ty :: "typ" shows "tinstT (subst_typ insts ty) ty"
sps_sub: shows "set sps \<subseteq> set bs \<times> (set gs \<union> set bs)"
26: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases_2: fixes t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (receive\<langle>t\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>) = fv (t \<cdot> \<theta>)"
euler_mascheroni_sequence_nonneg: fixes n :: "nat" assumes "0 < n" shows "0 \<le> harm n - ln (real n)"
elims_2: fixes x :: "'b btree" assumes "nonempty_lasttreebal x" obtains "x = Leaf"   | ts :: "('b btree \<times> 'b) list" and t :: "'b btree" where "x = Node ts t" and "(\<exists>ls tsub tsep. ts = ls @ [(tsub, tsep)] \<and> height tsub = height t) \<and> nonempty_lasttreebal t"
fbasis: shows "set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) vs \<and> R_scalar_mult.R_lin_independent UNIV (\<sharp>\<cdot>) vs"
\<R>\<^sub>2_sym: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option" shows "sym (\<R>\<^sub>2 \<Gamma>)"
length_coeffs_primitive_part: fixes f :: "'a poly" shows "length (coeffs (primitive_part f)) = length (coeffs f)"
eucl_rel_poly_by_0_iff: fixes x :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" shows "eucl_rel_poly x 0 (q, r) = (q = 0 \<and> r = x)"
check_unsat_terminates: fixes s :: "('i, 'a) state" assumes "\<U> s" shows "check_dom s"
monoidal_category_axioms: shows "monoidal_category D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D"
zKs: fixes n :: "nat" shows "(z \<circ> K) n \<in> S"
image2_def: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e" shows "image2 A f g = {(f a, g a) |a. a \<in> A}"
id_arrow: fixes A :: "'a" assumes "A \<in> Ob" shows "Id A \<in> Ar"
sources_aux_subset: fixes U :: "'c set"   and I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "U \<subseteq> sources_aux I D U xs"
n_preserve: fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> !y \<cdot> !x = !y \<cdot> !x"
constVal_eval: fixes e :: "expr"   and c :: "val"   and G :: "prog"   and s0 :: "st"   and v :: "val"   and s :: "state" assumes "constVal e = Some c"   and "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s" shows "v = c \<and> normal s"
chains_un: fixes c :: "'a set set" assumes "c \<in> chains {I. ideal R I \<and> I \<subset> carrier R}"   and "c \<noteq> {}" shows "ideal R (\<Union> c)"
matrix_to_iarray_snd_Gauss_Jordan_in_ij_det_P: fixes A :: "(('c, 'd) vec, 'b) vec"   and j :: "'d"   and i :: "'b" assumes "\<exists>n. A $ n $ j \<noteq> (0::'c) \<and> i \<le> n" shows "matrix_to_iarray (snd (Gauss_Jordan_in_ij_det_P A i j)) = snd (Gauss_Jordan_in_ij_det_P_iarrays (matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j))"
wpt_is_assertion: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "mbt_algebra.wpt one times inf top x \<in> mbt_algebra.assertion one times inf less_eq dual top"
smult_mat_mat_one_element: fixes A :: "'b mat"   and B :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat 1 1"   and "B \<in> carrier_mat 1 n" shows "A * B = A $$ (0, 0) \<cdot>\<^sub>m B"
i_Exec_Stream_Cons: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "i_Exec_Comp_Stream trans_fun ([x] \<frown> input) c = [trans_fun x c] \<frown> i_Exec_Comp_Stream trans_fun input (trans_fun x c)"
Gauss_Jordan_wrapper_def: fixes i :: "'g"   and j :: "'h"   and A :: "(('e, 'f) vec, 'g) vec"   and B :: "(('e, 'h) vec, 'g) vec" shows "Gauss_Jordan_wrapper i j A B = (\<chi>s. if s = i then A $ s else row_add A s i (- B $ s $ j) $ s)"
opposite_ocircline_involution: fixes H :: "ocircline" shows "opposite_ocircline (opposite_ocircline H) = H"
laplace_integrand_def: fixes s :: "complex"   and t :: "real" shows "local.laplace_integrand s t = exp (t *\<^sub>R - s) * f t"
is_poincare_line_iff: fixes H :: "circline" shows "is_poincare_line H = (circline_type H = - 1 \<and> perpendicular H unit_circle)"
check_plan_correct: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi>s :: "char list list" shows "ast_problem.check_plan problem \<pi>s = ast_problem.valid_plan problem \<pi>s"
add_zero_l: fixes x :: "'a" shows "(0::'a) + x = x"