hcju/isabelleps huggingface dataset:
CASRed2: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and e :: "'addr expr"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and v :: "'addr val"   and D :: "String.literal"   and F :: "String.literal"   and e3 :: "'addr expr" assumes "extTA,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "extTA,P,t \<turnstile> \<langle>Val v\<bullet>compareAndSwap(D\<bullet>F, e, e3),s\<rangle> -ta\<rightarrow> \<langle>Val v\<bullet>compareAndSwap(D\<bullet>F, e', e3),s'\<rangle>"
mod_eq_imp_eq_aux: fixes b :: "nat"   and a :: "nat" assumes "b mod N = a mod N"   and "a \<le> b"   and "b < a + N" shows "b = a"
sel_13: fixes x41 :: "'b mrexp2"   and x42 :: "'b mrexp2"   and x43 :: "bool"   and x44 :: "bool" shows "nul (Plus2 x41 x42 x43 x44) = x44"
finter_commute: fixes A :: "'c fset"   and B :: "'c fset" shows "A |\<inter>| B = B |\<inter>| A"
simps_1: fixes x :: "'a"   and v :: "'b" shows "ins x v Leaf = Node Leaf R x v Leaf"
simps_2: fixes x :: "'g"   and xs :: "'g list"   and ys :: "'g list" shows "(x # xs) @ ys = x # xs @ ys"
lift_gi_def: fixes f :: "'e \<Rightarrow> 'a"   and gi :: "bool \<times> 'e" shows "G.lift_gi f gi = (if fst gi then inv\<^bsub>G\<^esub> f (snd gi) else f (snd gi))"
accessible_on_tranclp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "accessible_on P A x" shows "accessible_on (restrict_to P A)\<^sup>+\<^sup>+ A x"
widen_asym: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "ty"   and D :: "ty" assumes "wf_prog wf_md P"   and "P \<turnstile> C \<le> D"   and "C \<noteq> D" shows "\<not> P \<turnstile> D \<le> C"
fps_nth_inject: fixes x :: "'b fps"   and y :: "'b fps" shows "(($) x = ($) y) = (x = y)"
normalise_decomp: fixes y :: "'a"   and x :: "'a"   and n :: "nat" assumes "is_unit (lift y)" shows "lift (fst (normalise_decomp x y n)) + (\<Sum>i<n. from_decomp (snd (normalise_decomp x y n) ! i) y (n - i)) = lift x div lift y ^ n"
assert_all_state_tableau_equiv: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and s' :: "('i, 'a) state"   and v :: "nat \<Rightarrow> 'a" assumes "\<triangle> t"   and "assert_all_state t as = s'" shows "v \<Turnstile>\<^sub>t t = v \<Turnstile>\<^sub>t \<T> s'"
assms: fixes i :: "'a"   and v :: "'b" assumes "inp i = Some v" shows "h v = h' v"
map\<^sub>0_def: fixes a :: "'b" shows "E.G.map\<^sub>0 a \<equiv> src\<^sub>C (F.right_map a)"
can_take_transition_def: fixes t :: "'b transition_scheme"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "can_take_transition t i r = can_take (Arity t) (Guards t) i r"
minus: fixes x :: "real"   and xa :: "'b" shows "- x *\<^sub>R xa = - (x *\<^sub>R xa)"
r_cancel_one: fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier M"   and "a \<in> carrier M" shows "(a \<oplus>\<^bsub>M\<^esub> x = x) = (a = \<zero>\<^bsub>M\<^esub>)"
int: shows "integrable M (\<lambda>x. - subcocycle_lim u x)"
l_neg: fixes G :: "('h, 'i) ring_scheme"   and x :: "'h" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x \<oplus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"
oghoare_Cond: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and b :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Cond b c\<^sub>1 c\<^sub>2) Q, A" shows "\<exists>P' P\<^sub>1 P\<^sub>2 Q' A'. P = AnnBin P' P\<^sub>1 P\<^sub>2 \<and> P' \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> pre P\<^sub>1) \<and> (s \<notin> b \<longrightarrow> s \<in> pre P\<^sub>2)} \<and> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 Q', A' \<and> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 Q', A' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A"
compat_pair_def: fixes S :: "('b \<times> 'b) set"   and NS :: "('b \<times> 'b) set" shows "compat_pair S NS \<equiv> NS O S \<subseteq> S \<and> S O NS \<subseteq> S"
ladder_j_bound: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list"   and index :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "LocalLexing.LeftDerivationLadder \<NN> \<TT> \<RR> \<alpha> D L \<gamma>"   and "index < length L" shows "ladder_j L index < length (LocalLexing.ladder_\<gamma> \<NN> \<TT> \<RR> \<alpha> D L index)"
simps_5: shows "dist_param_type UniformReal = PRODUCT REAL REAL"
CConcEnd: fixes a :: "((int \<times> int) set set, int) Combinators"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and y :: "unit decision"   and xs :: "((int \<times> int) set set, int) Combinators list" assumes "C a x = \<lfloor>y\<rfloor>" shows "C (list2FWpolicy (xs @ [a])) x = \<lfloor>y\<rfloor>"
add_le_imp_le_diff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and i :: "'d"   and k :: "'d"   and n :: "'d" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (plus i k) n" shows "less_eq i (minus n k)"
iter_count: fixes b :: "bool"   and T' :: "bool list list"   and bd :: "nat bdd list"   and as :: "bool list"   and T :: "bool list list" assumes "(b, T') = iter (bd, as) T"   and "b" shows "count_tr T' < count_tr T"
mapping_one_closed: shows "coreflexive ((1::'a)[[(1::'a)]]) \<and> times_top_class.total (1::'a)"
simps_12: fixes e :: "'a exp"   and e\<^sub>1 :: "'a exp"   and e\<^sub>2 :: "'a exp"   and A :: "'a set option" shows "\<D> (if (e) e\<^sub>1 else e\<^sub>2) A = (\<D> e A \<and> \<D> e\<^sub>1 (A \<squnion> \<A> e) \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e))"
simps_5: fixes b :: "bexp"   and c :: "com"   and X :: "char list set" shows "L (WHILE b DO c) X = lfp (\<lambda>Y. vars b \<union> X \<union> L c Y)"
long_division_zero_2: fixes K :: "'a list set"   and q :: "'a list list" assumes "subfield K (K [X])"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "UP.pmod [] q = []"
attach_map_converter: fixes f :: "'a \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'b"   and h :: "'g \<Rightarrow> 'c"   and k :: "'d \<Rightarrow> 'h"   and conv :: "('e, 'f, 'g, 'h) converter"   and res :: "('c, 'd) resource" shows "map_converter f g h k conv \<rhd> res = map_resource f g (conv \<rhd> map_resource h k res)"
succ_smallest: fixes r :: "('c \<times> 'c) set"   and i :: "'c"   and j :: "'c" assumes "wo_rel r"   and "(i, j) \<in> r"   and "i \<noteq> j" shows "(succ r i, j) \<in> r"
action_dom_def: fixes s1 :: "('d, 'e) fmap"   and s2 :: "('d, 'f) fmap" shows "action_dom s1 s2 \<equiv> fmdom' s1 \<union> fmdom' s2"
nn_integral_diff: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and g :: "'c \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integral\<^sup>N M g \<noteq> \<infinity>"   and "AE x in M. g x \<le> f x" shows "\<integral>\<^sup>+ x. f x - g x \<partial>M = integral\<^sup>N M f - integral\<^sup>N M g"
collapse_3: fixes formula :: "'b formula" assumes "is_Neg formula" shows "Neg (un_Neg formula) = formula"
compile_env_entry_def: fixes \<A> :: "'fun \<Rightarrow> nat option"   and \<O> :: "'k \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and x :: "'k \<times> ('b, 'l, 'fun, 'd, 'e) Inca.instr fundef" shows "compile_env_entry \<A> \<O> x = map_result id (Pair (fst x)) (compile_fundef \<A> (\<O> (fst x)) (snd x))"
fold_plus_prod_list_rev: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'a list" assumes "class.monoid_mult one times" shows "fold times xs = times (monoid_mult.prod_list one times (rev xs))"
vrange_vsingleton: fixes a :: "V"   and b :: "V" shows "\<R>\<^sub>\<circ> (set {\<langle>a, b\<rangle>}) = set {b}"
elims: fixes x :: "('b, 'sz) trm"   and xa :: "'sz"   and y :: "('b, 'sz) trm" assumes "singleton x xa = y" obtains t :: "('b, 'sz) trm" and i :: "'sz" where "x = t" and "xa = i" and "y = (if i = vid1 then t else Const 0)"
left_adjoint_prod: fixes b :: "'a" assumes "ide b" shows "left_adjoint_functor (\<cdot>) (\<cdot>) (\<lambda>x. x \<otimes> b)"
omega_one: shows "(1::'a) ^ \<omega> = \<bottom>"
mset_un_cases: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" assumes "a \<in># A + B" obtains "a \<in># A"   | "a \<in># B"
assms_5: shows "poincare_distance u v = poincare_distance u' v'"
subgroup_is_group: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "subgroup H G"   and "Group.group G" shows "Group.group (G\<lparr>carrier := H\<rparr>)"
B_def: shows "B = (\<lambda>i. (next_good_set ^^ i) A)"
trss_mon_w_no_ctx: fixes s :: "'n list"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" shows "mon_w fg w \<inter> mon_c fg c = {}"
maxsimpD_maximal: fixes x :: "'a set"   and z :: "'a set" assumes "maxsimp x"   and "z \<in> X"   and "order.greater_eq z x" shows "z = x"
u_simps_4: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c" assumes "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta>" shows "partial_magma.dom V u = u"
llength_LCons: fixes x :: "'a"   and xs :: "'a llist" shows "llength (LCons x xs) = eSuc (llength xs)"
skip: fixes \<iota> :: "nat"   and m :: "'id \<Rightarrow> 'val" shows "SRabbr (skip\<^bsub>\<iota>\<^esub>) m [] None m"
IH: fixes xs :: "'a list"   and ys :: "'a list" shows "(replicate n x = xs @ ys) = (\<exists>m\<le>n. xs = replicate m x \<and> ys = replicate (n - m) x)"
h: assumes "freeIn 0 \<phi>'" shows "eval (lowerFm 0 1 \<phi>') xs = eval (ExQ \<phi>') xs"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>C.map f : C.map a \<rightarrow> C.map b\<guillemotright>"
Dynkin_systemI: fixes M :: "'c set set"   and \<Omega> :: "'c set" assumes "\<And>A. A \<in> M \<Longrightarrow> A \<subseteq> \<Omega>"   and "\<Omega> \<in> M"   and "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"   and "\<And>A. \<lbrakk>disjoint_family A; range A \<subseteq> M\<rbrakk> \<Longrightarrow> \<Union> (range A) \<in> M" shows "Dynkin_system \<Omega> M"
invariant_steps: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and as :: "'c list" assumes "Graph_Invariant E P"   and "Graph_Defs.steps E (a # as)"   and "P a" shows "list_all P as"
g_def: shows "g \<equiv> \<lambda>n. (ln (real n) + c * f n * ln (ln (real n))) / ln (f n)"
weight_sample_uniform_gt_0: fixes n :: "nat" assumes "0 < n" shows "weight_spmf (sample_uniform n) = 1"
eventually_mem_existence_ivl: fixes b :: "real"   and t0 :: "real"   and x0 :: "real" assumes "b \<in> existence_ivl t0 x0" shows "\<forall>\<^sub>F x in at x0. b \<in> existence_ivl t0 x"
if_not_P: fixes P :: "bool"   and x :: "'a"   and y :: "'a" assumes "\<not> P" shows "(if P then x else y) = y"
simps_1: fixes C :: "String.literal" shows "ty_of_htype (Class_type C) = Class C"
lexord_cons_cons: fixes a :: "'b"   and x :: "'b list"   and b :: "'b"   and y :: "'b list"   and r :: "('b \<times> 'b) set" shows "((a # x, b # y) \<in> lexord r) = ((a, b) \<in> r \<or> a = b \<and> (x, y) \<in> lexord r)"
Pair_Rep_def: fixes a :: "'b"   and b :: "'c" shows "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
sum_le_2: fixes f :: "nat \<Rightarrow> complex" shows "sum f {0..<2} = f 0 + f 1"
integrable_abs_iff: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "integrable M (\<lambda>x. \<bar>f x\<bar>) = integrable M f"
sum_strict_mono: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset"   and g :: "'c \<Rightarrow> 'd multiset" assumes "finite A"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subset># g x" shows "comm_monoid_add.sum (+) {#} f A \<subset># comm_monoid_add.sum (+) {#} g A"
empty_iff: fixes c :: "'c" shows "s2p {} c = False"
dvd_Gcd_fin_iff: fixes A :: "'a set"   and b :: "'a" assumes "finite A" shows "(b dvd Gcd\<^sub>f\<^sub>i\<^sub>n A) = (\<forall>a\<in>A. b dvd a)"
tab_in_hom_1: shows "\<guillemotleft>D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho> : src\<^sub>D (F f) \<rightarrow>\<^sub>D trg\<^sub>D (F r)\<guillemotright>"
int_eq: fixes X :: "'b \<Rightarrow> 'c"   and Y :: "'b \<Rightarrow> 'c" assumes "\<turnstile> X = Y" shows "X = Y"
derive_interval_1: fixes i :: "'b"   and x :: "'b sort_key extended \<times> 'b sort_key extended" assumes "\<lbrakk>Final i\<rbrakk> \<le> fst x"   and "is_interval x" shows "is_interval (derive_left x)"
rep_eq: fixes x :: "'c program"   and xa :: "'c program" shows "(x < xa) = (rep_program x \<sqsubseteq> rep_program xa \<and> \<not> rep_program xa \<sqsubseteq> rep_program x)"
fres_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> y \<le> z) = (x \<le> z \<leftarrow> y)"
VDMSkip: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" shows "G \<rhd> Skip : (\<lambda>s t. t = s)"
Qg_iop: fixes N :: "'a set" assumes "G \<triangleright> N" shows "c_iop G N \<in> set_rcs G N \<rightarrow> set_rcs G N"
neg1: fixes A :: "'c"   and k :: "'c"   and a :: "'c" assumes "A \<equiv> k + a" shows "- A \<equiv> - k + - a"
iso_arcs_eq_iff: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and e1 :: "'d"   and e2 :: "'d" assumes "iG.digraph_isomorphism hom"   and "e1 \<in> arcs (app_iso hom G)"   and "e2 \<in> arcs (app_iso hom G)" shows "(iso_arcs hom e1 = iso_arcs hom e2) = (e1 = e2)"
simps: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" shows "prognosis ae as a (\<Gamma>, e, S) = pathsCard (paths (substitute (FBinds \<Gamma>\<cdot>ae) (thunks \<Gamma>) (Texp e\<cdot>a \<otimes>\<otimes> Fstack as S)))"
Pd_range: shows "\<forall>d. Field (Pd_shuffle_to_top ds' Y d) \<subseteq> {x. Xd x = d}"
Suc: fixes r :: "nat" shows "(\<Sum>k\<le>r. (m choose k) * (n choose (r - k))) = m + n choose r"
distrib_right': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y) \<cdot> z = x \<cdot> z + y \<cdot> z"
arg_max_list_element: fixes l :: "'b list"   and f :: "'b \<Rightarrow> nat" assumes "1 \<le> length l" shows "arg_max_list f l \<in> set l"
sum_support_set: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" shows "support_set (qty_sum pf1 pf2) \<subseteq> support_set pf1 \<union> support_set pf2"
equiv_up_to_if_True: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c1 :: "com"   and c2 :: "com" assumes "\<And>s. P s \<Longrightarrow> bval b s" shows "P \<Turnstile> IF b THEN c1 ELSE c2 \<sim> c1"
X_deX_connection: fixes n :: "nat"   and A :: "vbl set" shows "(X n \<in> A) = (n \<in> deX ` A)"
induct: fixes x1 :: "cmd"   and x2 :: "cmd"   and P :: "cmd \<Rightarrow> cmd \<Rightarrow> bool" assumes "x1 \<^sub>C\<longlonglongrightarrow>\<^sup>* x2"   and "\<And>c. P c c"   and "\<And>c d e. \<lbrakk>c \<^sub>C\<longlonglongrightarrow> d; d \<^sub>C\<longlonglongrightarrow>\<^sup>* e; P d e\<rbrakk> \<Longrightarrow> P c e" shows "P x1 x2"
basic_cqe_conv1_1: fixes p :: "'b poly" shows "(\<exists>x. poly p x = (0::'b) \<and> poly 0 x \<noteq> (0::'b)) = False"
bis_Gr: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B1' :: "'g set"   and B2' :: "'h set"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and f1 :: "'d \<Rightarrow> 'g"   and f2 :: "'e \<Rightarrow> 'h" assumes "coalg B1 B2 s1 s2"   and "mor B1 B2 s1 s2 B1' B2' s1' s2' f1 f2" shows "bis B1 B2 s1 s2 B1' B2' s1' s2' (BNF_Def.Gr B1 f1) (BNF_Def.Gr B2 f2)"
inverse_mod_coprime_exp: fixes p :: "int"   and n :: "nat"   and x :: "int" assumes "m = p ^ n"   and "prime p"   and "n \<noteq> 0"   and "comm_monoid_mult_class.coprime x p" shows "M (inverse_mod x m * x) = 1"
subnet_hosts_def: shows "subnet_hosts \<equiv> [''v11'', ''v12'', ''v13'', ''v1b'', ''v21'', ''v22'', ''v23'', ''v2b'', ''v3b'', ''vo'']"
scale_flow: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and c :: "ennreal" assumes "flow \<Delta> f"   and "c \<le> 1" shows "flow \<Delta> (\<lambda>e. c * f e)"
m_def: shows "m = root (traverse_tree pref (stern_brocot_iterate_aux I))"
simps_2: fixes x :: "'b equation" shows "mk_lit neg x = Neg x"
chain_mono: fixes A :: "'d set"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'d set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "pred_on.chain A P C" shows "pred_on.chain A Q C"
delete_append_commute: fixes i :: "'c"   and "is" :: "'c set"   and e1 :: "'d"   and e2 :: "'d" assumes "i \<notin> is" shows "\<langle>Delete is e1\<rangle> \<rhd> \<langle>Append i e2\<rangle> = \<langle>Append i e2\<rangle> \<rhd> \<langle>Delete is e1\<rangle>"
tl_hd_empty: fixes l :: "'a list"   and a :: "'a" assumes "tl (l @ [a]) = []" shows "l = []"
exhaust_sel: fixes sum :: "'a + 'b" obtains "sum = Inl (projl sum)"   | "sum = Inr (projr sum)"
sig_trd_aux_domI: fixes args0 :: "'a \<times> ('t \<Rightarrow>\<^sub>0 'b)"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "fst args0 \<in> keys (rep_list (snd args0))" shows "sig_trd_aux_dom bs args0"
singular_relboundary_cmul: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and a :: "int" assumes "singular_relboundary p X S c" shows "singular_relboundary p X S (frag_cmul a c)"
subgroup_Units: fixes H :: "'a set" assumes "subgroup H G" shows "H \<subseteq> Units (G\<lparr>carrier := H\<rparr>)"
unfold: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and s :: "'a" assumes "fst g s"   and "terminates_onp g (snd (snd g s))" shows "terminates_onp g s"
levy_continuity: fixes M :: "nat \<Rightarrow> real measure"   and M' :: "real measure" assumes "\<And>n. real_distribution (M n)"   and "real_distribution M'"   and "\<And>t. (\<lambda>n. char (M n) t) \<longlonglongrightarrow> char M' t" shows "weak_conv_m M M'"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (Cones.map f) = Cop.arr f"
measure_times: fixes A :: "'a \<Rightarrow> 'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M.events i" shows "emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
equal_intr_rule: fixes phi :: "prop"   and psi :: "prop" assumes "PROP phi \<Longrightarrow> PROP psi"   and "PROP psi \<Longrightarrow> PROP phi" shows "PROP phi \<equiv> PROP psi"
strategyproofness_1: shows "pmf (sds R13) b + pmf (sds R13) a \<le> pmf (sds R27) c + pmf (sds R27) a"
elementLevelLtLevelImpliesMemberPrefixToLevel: fixes e :: "'b"   and M :: "('b \<times> bool) list"   and level :: "nat" assumes "e \<in> set (elements M)"   and "elementLevel e M \<le> level" shows "e \<in> set (elements (prefixToLevel level M))"
Call: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and ps :: "char list exp list"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and l\<^sub>2' :: "sfields"   and e' :: "char list exp"   and h\<^sub>3 :: "heap"   and l\<^sub>3 :: "sfields"   and sh\<^sub>3 :: "sheap" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = (pns, body) in D"   and "length vs = length pns"   and "l\<^sub>2' = [this \<mapsto> Addr a, pns [\<mapsto>] vs]"   and "P \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2', sh\<^sub>2)\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>3, sh\<^sub>3)\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>2, sh\<^sub>3)\<rangle>"
simps_9: shows "isrlfm T = (isatom T \<and> bound0 T)"
assms: shows "(f, g) \<in> [P]\<^sub>a R \<rightarrow> S"
coreflexive_comp_inf_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "coreflexive x"   and "coreflexive y" shows "x * z \<sqinter> y * z = (x \<sqinter> y) * z"
igSwapIGVar_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSwapIGVar MOD \<equiv> \<forall>zs z1 z2 xs x. igSwap MOD zs z1 z2 (igVar MOD xs x) = igVar MOD xs (x @xs[z1 \<and> z2]_zs)"
cases: fixes G :: "prog"   and accclass :: "qtname"   and a1 :: "qtname \<times> memberdecl"   and a2 :: "qtname" assumes "G \<turnstile> a1 in a2 dyn_accessible_from accclass" obtains membr :: "qtname \<times> memberdecl" and "class" :: "qtname" where "a1 = membr__" and "a2 = class__" and "G \<turnstile> membr__ member_in class__" and "G \<turnstile> membr__ in class__ permits_acc_from accclass"   | membr :: "qtname \<times> memberdecl" and C :: "qtname" and new :: "mdecl" and old :: "qtname \<times> mdecl" and supr :: "qtname" and "class" :: "qtname" where "a1 = membr" and "a2 = class" and "G \<turnstile> membr member_in class" and "membr = (C, mdecl new)" and "G \<turnstile> (C, new) overrides old" and "G\<turnstile>class\<prec>\<^sub>C supr" and "G \<turnstile>Method old in supr dyn_accessible_from accclass"
assms_3: fixes x :: "nat" assumes "x < longest_prefix j" shows "f x = \<psi> j x"
negligible_poly_times: fixes f :: "nat \<Rightarrow> real"   and n :: "real"   and g :: "nat \<Rightarrow> real" assumes "f \<in> O(\<lambda>x. real x powr n)"   and "Negligible.negligible g" shows "Negligible.negligible (\<lambda>x. f x * g x)"
matched_same_level_CFG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "matched n ns n'" obtains as :: "'edge list" where "parent_node n -as\<rightarrow>\<^bsub>sl\<^esub>* parent_node n'"
inj_on_fst_map_to_set: fixes m :: "'a \<Rightarrow> 'b option" shows "inj_on fst (map_to_set m)"
B: shows "xs @ ipurge_tr I D (D y) ys \<in> traces P"