hcju/isabelleps huggingface dataset:
l_new_document_get_child_nodes_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_document_get_child_nodes_axioms get_child_nodes get_child_nodes_locs \<equiv> (\<forall>ptr' new_document_ptr h h' r. ptr' \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr \<longrightarrow> h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr \<longrightarrow> h \<turnstile> new_document \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_child_nodes_locs ptr' \<longrightarrow> r h h') \<and> (\<forall>h new_document_ptr h'. h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr \<longrightarrow> h \<turnstile> new_document \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_child_nodes (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr) \<rightarrow>\<^sub>r [])"
pfsubset_imp_ex_fmem: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<subset>| B" shows "\<exists>b. b |\<in>| B |-| A"
min_list_param_in: fixes xs :: "'a list"   and rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "xs \<noteq> []" shows "min_list_param rel xs \<in> set xs"
line_integral_on_path_eq_line_integral_on_equiv_chain_1: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "chain_subdiv_path \<gamma> one_chain"   and "boundary_chain one_chain"   and "\<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>"   and "finite basis" shows "one_chain_line_integral F basis one_chain = line_integral F basis \<gamma>"
flift1_strict: fixes f :: "'c \<Rightarrow> 'b" shows "flift1 f\<cdot>\<bottom> = \<bottom>"
nat_filtration_subalgebra: fixes n :: "nat" shows "subalgebra M (nat_filtration n)"
is_interval_cbox: fixes a :: "'b"   and b :: "'b" shows "is_interval (cbox a b)"
measure_Int_set_pmf: fixes p :: "'c pmf"   and A :: "'c set" shows "measure_pmf.prob p (A \<inter> set_pmf p) = measure_pmf.prob p A"
lcond_def: fixes lp :: "'c set"   and lp' :: "'c set"   and b :: "'d \<Rightarrow> bool"   and s :: "'d" shows "lcond lp lp' b s = (if b s then lp else lp')"
dvd_div_eq_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 b :: "'a"   and a :: "'a" assumes "class.semidom_divide divide plus minus zero times one"   and "dvd.dvd times b a" shows "(divide a b = zero) = (a = zero)"
p_antitone_sup: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) \<le> - x"
rec_if_def: shows "rec_if = CN rec_ifz [CN rec_not [recf.Id 3 0], recf.Id 3 1, recf.Id 3 2]"
UP_ring_subring_2: fixes S :: "'a set" assumes "subring S R" shows "UP_ring (R\<lparr>carrier := S\<rparr>)"
simps_4: fixes \<E> :: "char list \<Rightarrow> nat option"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" shows "eval \<E> (Add e\<^sub>1 e\<^sub>2) = eval \<E> e\<^sub>1 \<oplus> eval \<E> e\<^sub>2"
a_ro_share: shows "a \<in> read_only (share sb \<S>)"
collapse: fixes dual :: "'b dual" shows "\<partial> (\<partial>\<^sup>- dual) = dual"
subdag_NodeD: fixes t :: "dag"   and lt :: "dag"   and a :: "ref"   and rt :: "dag" assumes "subdag t (Node lt a rt)" shows "subdag t lt \<and> subdag t rt"
assms_2: shows "aEvalUni At x \<noteq> aEvalUni At y"
mk_cert_def: fixes P :: "jvm_prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and et :: "ex_table"   and bs :: "bytecode"   and phi :: "(ty list \<times> ty err list) option list" shows "mk_cert P mxs T\<^sub>r et bs phi \<equiv> make_cert (exec P mxs T\<^sub>r et bs) (map OK phi) (OK None)"
Form_Body_length: fixes ka :: "nat"   and kb :: "nat"   and xs :: "nat list"   and ys :: "nat list"   and zs :: "nat list" assumes "Form_Body ka kb xs ys zs" shows "length xs < length ys"
msed_map_invR: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset"   and b :: "'c"   and N :: "'c multiset" assumes "f `# M = add_mset b N" shows "\<exists>M1 a. M = add_mset a M1 \<and> f a = b \<and> f `# M1 = N"
progress_regex_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option" shows "progress_regex \<sigma> P = min_regex_default (Monitor.progress \<sigma> P)"
list_of_eucl_nth_if: fixes x :: "'b"   and n :: "nat" shows "list_of_eucl x ! n = (if n < DIM('b) then x \<bullet> Basis_list ! n else [] ! (n - DIM('b)))"
qTerm_templateInduct: fixes rel :: "(('a, 'b, 'c, 'd, 'e) qTerm \<times> ('a, 'b, 'c, 'd, 'e) qTerm) set"   and phi :: "('a, 'b, 'c, 'd, 'e) qTerm \<Rightarrow> bool"   and phiAbs :: "('a, 'b, 'c, 'd, 'e) qAbs \<Rightarrow> bool"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and A :: "('a, 'b, 'c, 'd, 'e) qAbs" assumes "\<And>X Y. (X, Y) \<in> rel \<Longrightarrow> qSkel Y = qSkel X"   and "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. (\<And>Y. (X, Y) \<in> rel \<Longrightarrow> phi Y) \<Longrightarrow> phiAbs (qAbs xs x X)" shows "phi X \<and> phiAbs A"
subprob_space_dist_measure: fixes v :: "val"   and dst :: "pdf_dist" assumes "v \<in> type_universe (dist_param_type dst)" shows "subprob_space (dist_measure dst v)"
Iff_def: fixes A :: "fm"   and B :: "fm" shows "A IFF B = (A IMP B) AND (B IMP A)"
atoms_subset: fixes f :: "atom fm" assumes "qfree f" shows "set (atoms\<^sub>0 f) \<subseteq> atoms f"
integrableD_1: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "integrable M f" shows "f \<in> borel_measurable M"
zero_Measurement_System_ext_def: shows "0 = \<lparr>unit_sys = unit, \<dots> = 0::'b\<rparr>"
cone_decomp_closed_minus: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and b :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "cone_decomp T ps"   and "a \<in> T"   and "b \<in> T" shows "a - b \<in> T"
f'_nonneg: fixes x :: "nat" shows "0 \<le> f' x"
set_decode_def: fixes x :: "nat" shows "set_decode x = {n. odd (x div 2 ^ n)}"
the_paramL: fixes l :: "'a option"   and r :: "'b option"   and R :: "('a \<times> 'b) set" assumes "l \<noteq> None"   and "(l, r) \<in> \<langle>R\<rangle>option_rel" shows "(the l, the r) \<in> R"
P_Gauss_Jordan_def: fixes A :: "(('b, 'd) vec, 'c) vec" shows "P_Gauss_Jordan A = fst (Gauss_Jordan_PA A)"
equalitySubsetI: fixes A :: "'b set"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B"   and "\<And>x. x \<in> B \<Longrightarrow> x \<in> A" shows "A = B"
mix_pmf_def: fixes \<alpha> :: "real"   and p :: "'a pmf"   and q :: "'a pmf" shows "mix_pmf \<alpha> p q = bernoulli_pmf \<alpha> \<bind> (\<lambda>X. if X then p else q)"
sat_lit_alt: fixes \<sigma> :: "nat \<Rightarrow> bool"   and p :: "bool"   and v :: "nat" shows "sat_lit \<sigma> (Lit p v) = (\<sigma> v = p)"
enum_DF_less_iff: fixes j :: "nat"   and k :: "nat"   and i :: "nat"   and i' :: "nat" assumes "j \<le> k" shows "(enum (DF k i) j < enum (DF k i') j) = (i < i')"
iff_2: fixes x :: "nat"   and y :: "nat" shows "(f x < f y) = (x < y)"
surjective: fixes r :: "('a, 'b) simple_match_scheme" shows "r = \<lparr>iiface = iiface r, oiface = oiface r, src = src r, dst = dst r, proto = proto r, sports = sports r, dports = dports r, \<dots> = simple_match.more r\<rparr>"
vrange_VLambda_vsubset: fixes A :: "V"   and f :: "V \<Rightarrow> V"   and B :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> f x \<in>\<^sub>\<circ> B" shows "\<R>\<^sub>\<circ> (VLambda A f) \<subseteq>\<^sub>\<circ> B"
chine_simps_3: shows "C.cod \<eta>.chine = \<eta>.cod.apex"
simps_2: fixes c :: "'a"   and s :: "'a list"   and r :: "'a rexp" shows "pderivs (c # s) r = pderivs_set s (pderiv c r)"
lepoll_antisym: fixes A :: "'b set"   and B :: "'c set" assumes "A \<lesssim> B"   and "B \<lesssim> A" shows "A \<approx> B"
Collect_conv_if2: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "{x. a = x \<and> P x} = (if P a then {a} else {})"
ennreal_mult_top: fixes a :: "ennreal" shows "a * \<top> = (if a = 0 then 0 else \<top>)"
agsum_add1p: fixes a :: "'a"   and n :: "int"   and m :: "int" assumes "a \<in> carrier A"   and "0 \<le> n"   and "0 \<le> m" shows "(n + m)\<triangleright>a\<^bsub>A\<^esub> = n\<triangleright>a\<^bsub>A\<^esub> \<plusminus> m\<triangleright>a\<^bsub>A\<^esub>"
times_fun_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "(f * g) x = f x * g x"
that_2: fixes p :: "'a"   and q :: "'b" assumes "x = Some (p, q)" shows "thesis"
that: fixes ns :: "'node list" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "\<forall>n\<in>set ns. v \<notin> defs g n"   and "length (predecessors g n) \<noteq> 1"   and "unvisitedPath un ns" shows "thesis"
respects_own_guarantees_def: fixes cm :: "'Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)" shows "abs.respects_own_guarantees cm \<equiv> \<forall>x. (x \<in> snd cm GuarNoReadOrWrite \<longrightarrow> abs.doesnt_read_or_modify (fst cm) x) \<and> (x \<in> snd cm GuarNoWrite \<longrightarrow> abs.doesnt_modify (fst cm) x)"
allVars_def: fixes g :: "'g" shows "uninst_code.ssa.allVars uses phis g \<equiv> \<Union>n\<in>set (\<alpha>n g). uninst_code.ssa.allDefs phis g n \<union> uninst_code.ssa.allUses uses phis g n"
monoid_axioms: shows "Group_Theory.monoid M (\<cdot>) \<one>"
cltn2_transpose_compose: fixes A :: "cltn2"   and B :: "cltn2" shows "cltn2_transpose (cltn2_compose A B) = cltn2_compose (cltn2_transpose B) (cltn2_transpose A)"
union: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a set"   and B :: "'a set" assumes "semilattice_set f"   and "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "semilattice_set.F f (A \<union> B) = f (semilattice_set.F f A) (semilattice_set.F f B)"
filter_sup_left_isotone: fixes G :: "'a set"   and H :: "'a set"   and F :: "'a set" assumes "G \<subseteq> H" shows "filter_sup G F \<subseteq> filter_sup H F"
I_notcarr: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "maximalideal I R" shows "carrier R \<noteq> I"
dvd_div_unit_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times a (divide c b) = dvd.dvd times a c"
matrix_scalar_ac: fixes A :: "(('a, 'd) vec, 'c) vec"   and k :: "real"   and B :: "(('a, 'b) vec, 'd) vec" shows "A ** k *\<^sub>R B = k *\<^sub>R A ** B"
action_closed: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and m :: "'e"   and g :: "'c" assumes "group_action G \<phi> M"   and "m \<in> M"   and "g \<in> carrier G" shows "\<phi> g m \<in> M"
sigma_le_sets: fixes A :: "'b set set"   and X :: "'b set"   and N :: "'b measure" assumes "A \<subseteq> Pow X" shows "(sets (sigma X A) \<subseteq> sets N) = (X \<in> sets N \<and> A \<subseteq> sets N)"
size_index_conv: fixes xs :: "'c list"   and x :: "'c" shows "(length xs = index xs x) = (x \<notin> set xs)"
eq_iff: fixes a :: "'b list"   and b :: "'b list" shows "(a = b) = (subseq a b \<and> subseq b a)"
is_L_bisimulation_def: fixes R :: "'effect fs_set \<Rightarrow> 'state \<Rightarrow> 'state \<Rightarrow> bool" shows "is_L_bisimulation R \<equiv> \<forall>F. symp (R F) \<and> (\<forall>P Q. R F P Q \<longrightarrow> (\<forall>f. f \<in>\<^sub>f\<^sub>s F \<longrightarrow> (\<forall>\<phi>. \<langle>f\<rangle>P \<turnstile> \<phi> \<longrightarrow> \<langle>f\<rangle>Q \<turnstile> \<phi>))) \<and> (\<forall>P Q. R F P Q \<longrightarrow> (\<forall>f. f \<in>\<^sub>f\<^sub>s F \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* (\<langle>f\<rangle>Q, F, f) \<longrightarrow> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. \<langle>f\<rangle>Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle> \<and> R (L (\<alpha>, F, f)) P' Q'))))"
counit_in_vhom: shows "\<guillemotleft>S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g) : S.UP f \<star>\<^sub>S S.UP g \<Rightarrow>\<^sub>S S.src (S.UP g)\<guillemotright>"
that: fixes B :: "real"   and Z :: "complex set" assumes "0 < B"   and "open Z"   and "w \<in> Z"   and "Z \<subseteq> S"   and "\<And>h z. \<lbrakk>h \<in> Y; z \<in> Z\<rbrakk> \<Longrightarrow> cmod (h z) \<le> B" shows "thesis"
DynCast: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and C :: "char list"   and Cs' :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs),(h, l)\<rangle>"   and "h a = \<lfloor>(D, S)\<rfloor>"   and "P \<turnstile> Path D to C via Cs' "   and "P \<turnstile> Path D to C unique" shows "P,E \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs'),(h, l)\<rangle>"
atms_def: fixes r :: "Formula.formula Regex.regex" shows "atms r = (\<Union>\<phi>\<in>regex.atms r. if safe_formula \<phi> then {\<phi>} else case \<phi> of formula.Neg \<phi>' \<Rightarrow> {\<phi>'} | _ \<Rightarrow> {})"
finite: fixes a :: "'a" assumes "a \<in> A" shows "emeasure M {a} < \<infinity>"
lunit_simps_2: fixes f :: "'b" assumes "D.ide f" shows "src\<^sub>D \<l>\<^sub>D[f] = src\<^sub>D f"
eq_onp_to_eq: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "eq_onp P x y" shows "x = y"
Cons_1: assumes "set as \<subseteq> set bs" shows "\<delta>_min bs \<le> \<delta>_min as"
not_icard_seteq: shows "\<exists>A B. A \<subseteq> B \<and> icard B \<le> icard A \<and> A \<noteq> B"
fold_map: fixes g :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "fold g (map f xs) = fold (g \<circ> f) xs"
C: shows "offs_set_prev ns (x # xs) index key mi ma i \<noteq> {}"
div_mult_mod_eq: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (times (divide a b) b) (modulo a b) = a"
substInp_preserves_wls: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var" assumes "wlsInp delta inp"   and "wls (asSort ys) Y" shows "wlsInp delta (inp %[Y / y]_ys)"
d_2: fixes t :: "'a" assumes "t \<in> T"   and "dist t a < d" shows "dist (f t b) (f a b) < e / 2"
fps_compose_1: fixes a :: "'a fps" shows "1 oo a = 1"
deadlockedp_deadlocked_eq: fixes final :: "'b \<Rightarrow> bool"   and r :: "('d, 'a, 'b, 'c, 'e, 'f) semantics"   and s :: "('d, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set" shows "multithreaded_base.deadlockedp final r s = (\<lambda>x. x \<in> multithreaded_base.deadlocked final r s)"
EndPhase0_HInv3_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase0 s s' p"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
less_Suc_eq_0_disj: fixes m :: "nat"   and n :: "nat" shows "order.greater (Suc n) m = (m = 0 \<or> (\<exists>j. m = Suc j \<and> order.greater n j))"
wits: shows "wits = map (\<mu> i) [j..<i]"
zero_below_ord: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp"   and "int n \<le> ord_Zp x" shows "x n = 0"
cong_solve_unique_nontrivial: fixes p :: "nat"   and a :: "nat"   and x :: "nat" assumes "prime p"   and "coprime p a"   and "0 < x"   and "x < p" shows "\<exists>!y. 0 < y \<and> y < p \<and> [x * y = a] (mod p)"
subset_singleton_conv: fixes A :: "'a set"   and a :: "'a" shows "(A \<subseteq> {a}) = (A = {} \<or> A = {a})"
mult_1: fixes a :: "'g" shows "(1::'g) * a = a"
swap_cs_Send_Snapshot: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"
sWlsDisj_def: fixes SEM :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) semDom_scheme" shows "sWlsDisj SEM \<equiv> \<forall>s s' sX. sWls SEM s sX \<and> sWls SEM s' sX \<longrightarrow> s = s'"
convex_hull_list_def: fixes Vs :: "'a vec list" shows "convex_hull_list Vs = {x. \<exists>c. convex_lincomb_list c Vs x}"
hnorm_mult_ineq: fixes x :: "'a star"   and y :: "'a star" shows "hnorm (x * y) \<le> hnorm x * hnorm y"
nq: shows "(Ree q)\<^sup>2 + (Im1 q)\<^sup>2 + (Im2 q)\<^sup>2 + (Im3 q)\<^sup>2 + (Im4 q)\<^sup>2 + (Im5 q)\<^sup>2 + (Im6 q)\<^sup>2 + (Im7 q)\<^sup>2 = 1"
pcr_SomeE: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'b"   and y :: "'d option" assumes "pcr_Some R x y" obtains z :: "'d" where "y = Some z" and "R x z"
toList_def: fixes t :: "('d, 'e) FingerTree" shows "toList t \<equiv> FingerTreeStruc.toList (Rep_FingerTree t)"
less_eq_term_def: shows "(\<le>) = in_rel leq_term"
eqvt: fixes x1 :: "lam"   and x2 :: "lam"   and pi :: "(name \<times> name) list" assumes " x1 \<longrightarrow>\<^sub>d x2" shows " pi \<bullet> x1 \<longrightarrow>\<^sub>d pi \<bullet> x2"
reachableNodes_E_closed: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and s :: "nat" shows "Graph.E c `` Graph.reachableNodes c s \<subseteq> Graph.reachableNodes c s"
left_idem: fixes a :: "'a"   and b :: "'a" shows "min a (min a b) = min a b"
hyp: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> xs" shows "x \<in>\<^sub>\<circ> A"
races: fixes a :: "nat"   and ad :: "'addr"   and al :: "addr_loc"   and v :: "'addr val" assumes "enat a < llength E"   and "action_obs E a = NormalAction (ReadMem ad al v)"   and "\<not> P,E \<turnstile> ws a \<le>hb a" shows "\<exists>T. P \<turnstile> ad@al : T \<and> P \<turnstile> v :\<le> T"
dbm_not_lt_eq: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" assumes "\<not> a \<prec> b"   and "\<not> b \<prec> a" shows "a = b"
icard_inj_on_le: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "inj_on f A"   and "f ` A \<subseteq> B" shows "icard A \<le> icard B"
l2_inv8_step4: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and gnx :: "msg" shows "{l2_inv8 \<inter> l2_inv1 \<inter> l2_inv3 \<inter> l2_inv4 \<inter> l2_inv4'} l2_step4 Rb A B gnx {> l2_inv8}"
nat_div_distrib': fixes y :: "int"   and x :: "int" assumes "0 \<le> y" shows "nat (x div y) = nat x div nat y"
simps_1: shows "Cp DenyAll = D\<^sub>U"
bounded_le_Sup: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "bounded (f ` S)" shows "\<forall>x\<in>S. norm (f x) \<le> Sup (norm ` f ` S)"
someI: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" assumes "P x" shows "P (Eps P)"
hom_mult: fixes x :: "int"   and y :: "int" shows "of_int (x * y) = of_int x * of_int y"
cases: fixes a1 :: "real \<Rightarrow> real \<times> real"   and a2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "chain_subdiv_path a1 a2" obtains l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" and \<gamma> :: "real \<Rightarrow> real \<times> real" where "a1 = \<gamma>" and "a2 = set l" and "distinct l" and "rec_join l = \<gamma>" and "valid_chain_list l"
IV_axiom: shows "universal (\<lambda>x y. \<lfloor>\<lambda>z. x r y \<and> y r z \<longrightarrow> x r z\<rfloor>)"
inf_absorb1: fixes x :: "'a multiset"   and y :: "'a multiset" assumes "x \<subseteq># y" shows "x \<inter># y = x"
a_rcosI: fixes h :: "'a"   and H :: "'a set"   and x :: "'a" assumes "h \<in> H"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "h \<oplus> x \<in> H +> x"
tr_s_field_writeI: fixes x :: "x"   and T :: "x \<Rightarrow> x option"   and x' :: "x"   and y :: "x"   and y' :: "x"   and f :: "char list" assumes "(case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'"   and "(case T y of None \<Rightarrow> y | Some x' \<Rightarrow> x') = y'" shows "tr_s T (s_write x f y) (s_write x' f y')"
jozsa_transform_is_unitary_index_odd: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row U\<^sub>f"   and "j < dim_col U\<^sub>f"   and "odd i" shows "(U\<^sub>f * U\<^sub>f) $$ (i, j) = 1\<^sub>m (dim_col U\<^sub>f) $$ (i, j)"
functions_of_is_inj_on: fixes A :: "'c set"   and B :: "'d set"   and N :: "nat multiset" assumes "finite A"   and "finite B"   and "\<forall>n. n \<in># N \<longrightarrow> n = 1"   and "number_partition (card A) N"   and "size N \<le> card B" shows "univ (\<lambda>f. inj_on f A) (functions_of A B N)"
xcpt_subcls_Throwable: fixes C :: "String.literal"   and wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" assumes "C \<in> sys_xcpts"   and "wf_prog wf_md P" shows "P \<turnstile> C \<preceq>\<^sup>* Throwable"
valid_bool_split: fixes \<tau> :: "'b state \<times> 'b state"   and A :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> \<upsilon> A) = (\<tau> \<Turnstile> A \<triangleq> null \<or> \<tau> \<Turnstile> A \<or> \<tau> \<Turnstile> not A)"
inject_2: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and y21 :: "nat"   and y22 :: "nat"   and y23 :: "nat" shows "(Triple x21 x22 x23 = Triple y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
rep_eq: fixes x :: "'a Filters.filter"   and xa :: "'a Filters.filter" shows "(x \<le> xa) = (Filters.filter.Rep_filter x \<subseteq> Filters.filter.Rep_filter xa)"
1: fixes y :: "'b" shows "monotone orda ordc (\<lambda>x. f x y)"
token_run_step': fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "\<MM>.token_run x (Suc n) = \<up>af\<^sub>G (\<MM>.token_run x n) (w n)"
subst_Bool_def: fixes P :: "'b assn"   and b :: "bool" shows "P\<leftarrow>=b = (\<lambda>Y s Z. \<exists>v. P (In1 v) s Z \<and> (normal s \<longrightarrow> the_Bool v = b))"