hcju/isabelleps huggingface dataset:
can_lock_devreserp: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and wf_state :: "(('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set) set"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t' :: "'b"   and ta' :: "('d, 'b, 'a, 'c, 'e, 'f) thread_action"   and s' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'a"   and L :: "('d + 'b + 'b) set" assumes "preserve_deadlocked final r convert_RA wf_state"   and "s \<in> wf_state"   and "multithreaded_base.redT final r convert_RA s (t', ta') s'"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "multithreaded_base.can_sync r t x (shr s') L" shows "\<exists>L'\<subseteq>L. multithreaded_base.can_sync r t x (shr s) L'"
ev: shows "\<forall>\<^sub>F x in F. f x ` T \<subseteq> S"
conditional_entropy_def: fixes b :: "real"   and S :: "'b measure"   and T :: "'e measure"   and X :: "'a \<Rightarrow> 'b"   and Y :: "'a \<Rightarrow> 'e" shows "conditional_entropy b S T X Y = - (LINT (x, y)|distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)). log b (enn2real (RN_deriv (S \<Otimes>\<^sub>M T) (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x))) (x, y)) / enn2real (RN_deriv T (distr M T Y) y)))"
compatAtmSyntactic: fixes expr :: "exp"   and v :: "Concrete.var" assumes "exprSec expr = Lo \<or> sec v = Hi" shows "Example_PL.compatAtm (Assign v expr)"
commute: fixes a :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool" shows "(a \<and>* b) = (b \<and>* a)"
prv_tms_zer: fixes t :: "'trm" assumes "t \<in> atrm" shows "prv (eql (tms t zer) zer)"
conn_tail: shows "tail G a \<notin> H.isolated_verts"
Collect_cong: fixes P :: "'k \<Rightarrow> bool"   and Q :: "'k \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
sup_inf_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<sqinter> z) = x \<squnion> y \<sqinter> (x \<squnion> z)"
s2: fixes x :: "real" assumes "x \<in> (\<lambda>t. r *\<^sub>R \<parallel>f *\<^sub>v t\<parallel>) ` ball (0::'a) 1" shows "x \<le> r * (SUP t\<in>ball (0::'a) 1. \<parallel>f *\<^sub>v t\<parallel>)"
simps_8: fixes t :: "'c local.term" shows "Inv \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = \<^bold>\<r>\<^bold>[Inv t\<^bold>]"
r: shows "D_HMA = P_HMA ** Mod_Type_Connect.to_hma\<^sub>m A ** Q_HMA \<and> invertible P_HMA \<and> invertible Q_HMA \<and> isDiagonal D_HMA"
card_UNIV_bit1_def: shows "card_UNIV_class.card_UNIV = Phantom('a bit1) (1 + 2 * of_phantom card_UNIV_class.card_UNIV)"
sig_crit'I_sig_crit: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "rb_aux_inv (bs, ss, p # ps)"   and "sig_crit bs (new_syz_sigs ss bs p) p" shows "sig_crit' bs p"
index_distinct: fixes G1 :: "'a set"   and G2 :: "'a set"   and x :: "'a"   and y :: "'a" assumes "G1 \<in> \<G>"   and "G2 \<in> \<G>"   and "G1 \<noteq> G2"   and "x \<in> G1"   and "y \<in> G2" shows "int (\<B> index {x, y}) = \<Lambda>"
simps: fixes y :: "int"   and x :: "int" shows "enumFromToStep\<cdot>(up\<cdot>y)\<cdot>(up\<cdot>(up\<cdot>x)) = (if x \<le> y then Yield\<cdot>(up\<cdot>x)\<cdot>(up\<cdot>(up\<cdot>(x + 1))) else Done)"
singleton_insert_inj_eq: fixes b :: "'b"   and a :: "'b"   and A :: "'b set" shows "({b} = insert a A) = (a = b \<and> A \<subseteq> {b})"
unop_type_det: fixes op :: "any_unop + boolean_unop + numeric_unop + string_unop + collection_unop"   and k :: "call_kind"   and \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'a OCL_Types.type"   and \<sigma>\<^sub>2 :: "'a OCL_Types.type" assumes "unop_type op k \<tau> \<sigma>\<^sub>1"   and "unop_type op k \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
measurableI: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<in> space N"   and "\<And>A. A \<in> sets N \<Longrightarrow> f -` A \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M N"
surjective: fixes r :: "'b sparc_state_var_scheme" shows "r = \<lparr>annul = annul r, resett = resett r, exe = exe r, reset = sparc_state_var.reset r, err = err r, ticc = ticc r, itrpt_lvl = itrpt_lvl r, st_bar = st_bar r, atm_ldst_byte = atm_ldst_byte r, atm_ldst_word = atm_ldst_word r, \<dots> = sparc_state_var.more r\<rparr>"
filternew_flows_state_alt2: fixes \<T> :: "'b stateful_policy" shows "filternew_flows_state \<T> = {e \<in> flows_state \<T>. e \<notin> backflows (flows_fix \<T>)}"
T_def: fixes s :: "'b cfg" shows "MC.T s = distr (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K_cfg x)))) (stream_space (count_space UNIV)) (MC.walk s)"
jkbpCn_step_inv: fixes t :: "'s Trace"   and s :: "'s"   and n :: "nat" assumes "t \<leadsto> s \<in> jkbpC\<^bsub>Suc n\<^esub>" shows "t \<in> jkbpC\<^bsub>n\<^esub>"
Ok_in_err: fixes x :: "'b"   and A :: "'b set" shows "(OK x \<in> err A) = (x \<in> A)"
lm010: fixes a :: "('c set \<times> 'd) set"   and G :: "'c set"   and N :: "'d set" assumes "runiq a"   and "runiq (a\<inverse>)"   and "Domain a partitions G"   and "Range a \<subseteq> N" shows "a \<in> possible_allocations_rel G N"
crename_swap: fixes a :: "coname"   and M :: "trm"   and b :: "coname" assumes "a \<sharp> M" shows "[(a, b)] \<bullet> M = M[b\<turnstile>c>a]"
quorum_non_empty: fixes Q :: "'a set" assumes "Q \<in> Quorum" shows "Q \<noteq> {}"
cSup_least: fixes X :: "'b multiset set"   and z :: "'b multiset" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq># z" shows "Sup X \<subseteq># z"
vbrelation_vconverse: shows "vbrelation (r\<inverse>\<^sub>\<circ>)"
pderiv_prod: fixes f :: "'d \<Rightarrow> 'c poly"   and as :: "'d set" shows "pderiv (prod f as) = (\<Sum>a\<in>as. prod f (as - {a}) * pderiv (f a))"
induct: fixes P :: "(('c, 'd, 'b) prot_fun, 'a) Term.term \<Rightarrow> (nat \<times> bool \<Rightarrow> 'b set) \<Rightarrow> bool"   and a0 :: "(('c, 'd, 'b) prot_fun, 'a) Term.term"   and a1 :: "nat \<times> bool \<Rightarrow> 'b set" assumes "\<And>x \<alpha>. P (Var x) \<alpha>"   and "\<And>n T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (Val n) T) \<alpha>"   and "\<And>v T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (Fu v) T) \<alpha>"   and "\<And>v T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (Set v) T) \<alpha>"   and "\<And>v T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (Abs v) T) \<alpha>"   and "\<And>T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun prot_fun.Pair T) \<alpha>"   and "\<And>v T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (Attack v) T) \<alpha>"   and "\<And>v va T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (PubConstAtom v va) T) \<alpha>"   and "\<And>v T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (PubConstSetType v) T) \<alpha>"   and "\<And>v T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (PubConstAttackType v) T) \<alpha>"   and "\<And>v T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (PubConstBottom v) T) \<alpha>"   and "\<And>v T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun (PubConstOccursSecType v) T) \<alpha>"   and "\<And>T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun OccursFact T) \<alpha>"   and "\<And>T \<alpha>. (\<And>x. x \<in> set T \<Longrightarrow> P x \<alpha>) \<Longrightarrow> P (Fun OccursSec T) \<alpha>" shows "P a0 a1"
div_right_strict_mono_on: fixes k :: "nat"   and I :: "nat set" assumes "0 < k"   and "\<forall>x\<in>I. \<forall>y\<in>I. x < y \<longrightarrow> x + k \<le> y" shows "SetInterval2.strict_mono_on (\<lambda>x. x div k) I"
r_consistent_def: fixes r_psi :: "recf" shows "r_consistent r_psi \<equiv> Cn 2 (r_consist_upto r_psi) [Cn 2 r_length [recf.Id 2 1], recf.Id 2 0, recf.Id 2 1]"
const_map_def: fixes v :: "'d"   and S :: "'c set" shows "const_map v S \<equiv> (\<lambda>_. Some v) |` S"
UnI1: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "c \<in> A" shows "c \<in> A \<union> B"
hyps_2: fixes f :: "'a \<Rightarrow> 'a" shows "map (psubst f) z \<turnstile> psubst f (p[App c []/0])"
that: fixes p :: "int poly" assumes "p \<noteq> 0"   and "poly (of_int_poly p) x = (0::'a)" shows "thesis"
rel_cong: fixes x :: "'c pmf"   and ya :: "'c pmf"   and y :: "'d pmf"   and xa :: "'d pmf"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set_pmf ya; yb \<in> set_pmf xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "rel_pmf R x y = rel_pmf Ra ya xa"
K_FFI: fixes p :: "'b fm" shows "\<turnstile> (p \<^bold>\<longrightarrow> \<^bold>\<not> \<^bold>\<not> p)"
strict_iff_order: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" shows "(b \<prec> a) = (b \<preceq> a \<and> a \<noteq> b)"
recurrent_iffI_communicating: fixes x :: "'s"   and y :: "'s" assumes "(x, y) \<in> communicating" shows "recurrent x = recurrent y"
except_min_def: fixes ko :: "'o"   and xs :: "'x" shows "except_min ko xs = local.tl (reorder ko xs)"
Pi_pmf_union: fixes A :: "'b set"   and B :: "'b set"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "Pi_pmf (A \<union> B) dflt p = map_pmf (\<lambda>(f, g) x. if x \<in> A then f x else g x) (pair_pmf (Pi_pmf A dflt p) (Pi_pmf B dflt p))"
lspasl_magicl_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" assumes "class.heap_sep_algebra plus zero sep_disj"   and "heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0"   and "sep_algebra.sep_impl plus sep_disj A B h2" shows "heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> \<not> A h1 \<and> sep_algebra.sep_impl plus sep_disj A B h2 \<and> heap_sep_algebra.magicl_applied plus sep_disj h1 h2 h0 (sep_algebra.sep_impl plus sep_disj A B) \<or> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> B h0 \<and> sep_algebra.sep_impl plus sep_disj A B h2 \<and> heap_sep_algebra.magicl_applied plus sep_disj h1 h2 h0 (sep_algebra.sep_impl plus sep_disj A B)"
correctCompositionKS_exprChannel_s_Q: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and secret :: "Secrets"   and ch :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionKS PQ"   and "sKS secret \<notin> LocalSecrets PQ"   and "ch \<in> ins Q"   and "exprChannel ch (sE secret)"   and "sKS secret \<notin> specKeysSecrets PQ"   and "correctCompositionIn PQ" shows "ch \<in> ins PQ \<and> exprChannel ch (sE secret)"
finite_field_mult_group_has_gen: assumes "finite (carrier R)" shows "\<exists>a\<in>carrier (Multiplicative_Group.mult_of R). carrier (Multiplicative_Group.mult_of R) = {a [^]\<^bsub>R\<^esub> i |i. i \<in> UNIV}"
simps_4: fixes f1 :: "nat \<Rightarrow> 'c"   and f2 :: "'c"   and nat :: "nat" shows "(case enat nat of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f1 nat"
deg_le_sectD_2: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and X :: "'b set"   and d :: "nat" assumes "t \<in> deg_le_sect X d" shows "deg_pm t \<le> d"
next_commutes_4: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<circle>(F = G) \<equiv> \<circle>F = \<circle>G"
fimage_def: shows "(|`|) \<equiv> map_fun id (map_fun fset Abs_fset) (`)"
well_formed_path_io_containment: fixes M :: "('d, 'e, 'f) FSM"   and p :: "(('d \<times> 'e) \<times> 'f) list"   and q :: "'f" assumes "well_formed M"   and "path M p q" shows "set (map fst p) \<subseteq> inputs M \<times> outputs M"
Ass: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and va :: "Term.var"   and Q :: "'b assn"   and e :: "expr"   and R :: "'b assn" assumes "G,A\<turnstile>{Normal P} va=\<succ> {Q}"   and "\<forall>vf. G,A\<turnstile>{Q\<leftarrow>In2 vf} e-\<succ> {\<lambda>Val:v:. assign (snd vf) v .; R}" shows "G,A\<turnstile>{Normal P} va:=e-\<succ> {R}"
uminus_prod_def: fixes x :: "'c \<times> 'd" shows "- x = (- fst x, - snd x)"
prems_1: shows "markedElements (a # M') \<noteq> []"
fps_conv_radius_1: shows "fps_conv_radius 1 = \<infinity>"
u: shows "u \<turnstile>\<^bsub>v,n\<^esub> \<lambda>i' j'. if i' = 0 \<and> j' = v c2 then Le r else if i' = v c2 \<and> j' = 0 then Le (- r) else M i' j'"
D_gchain_def: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "D_gchain G n g = (if n = 0 then G \<guillemotright> g 0  else d_gchain G n g \<and> (\<forall>l\<le>n - Suc 0. g (Suc l) \<subset> g l))"
iptables_bigstepD_7: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr"   and a :: "action" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m a]"   and "s = Undecided"   and "\<not> matches \<gamma> m p" shows "t = Undecided"
least_power_of_permutation_1: fixes p :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "permutation p" shows "(p ^^ least_power p a) a = a"
isTrue_all: fixes x :: "name"   and \<phi> :: "fm" assumes "x \<in> UNIV"   and "\<phi> \<in> UNIV"   and "Fvars \<phi> = {x}"   and "\<forall>n\<in>{t. ground t}. eval_fm e0 (\<phi>(x::=n))" shows "eval_fm e0 (SyntaxN.All x \<phi>)"
surj_simplex_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexAutomorphism X f" shows "f \<turnstile> X = X"
polar_def: fixes p :: "proj2" shows "polar p \<equiv> proj2_line_abs (M *v proj2_rep p)"
simps_3: fixes CatDom :: "('c, 'a, 'e) Category_scheme"   and CatCod :: "('d, 'b, 'f) Category_scheme"   and MapM :: "'a \<Rightarrow> 'b"   and more :: "'g" shows "(##) \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = MapM"
disc_6: fixes x61 :: "'a OCL_Types.type"   and x62 :: "String.literal"   and x63 :: "'a expr list" shows "\<not> is_Literal (StaticOperationCall x61 x62 x63)"
simple_set_image_id: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "\<forall>x\<in>X. f x = x" shows "f ` X = X"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "group_add_hom hom"   and "zero_hom_0 hom" shows "inj_group_add_hom hom"
cod_simps_2: shows "f_src.cod.leg0 = f_src.leg0"
almost_all_eventually_provable_monotonI: fixes \<phi> :: "'b ltl"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<G>' :: "'b ltl set" assumes "\<PP>\<^sub>\<infinity> \<phi> \<G> w"   and "\<G> \<subseteq> \<G>'" shows "\<PP>\<^sub>\<infinity> \<phi> \<G>' w"
cExt': fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', S, T) \<in> Rel'" shows "(\<Psi>' \<otimes> \<Psi>'', S, T) \<in> Rel'"
mat_of_rows_rows: fixes A :: "'b mat" shows "mat_of_rows (dim_col A) (rows A) = A"
ennreal_mult_strict_right_mono: fixes a :: "ennreal"   and c :: "ennreal"   and b :: "ennreal" assumes "a < c"   and "0 < b"   and "b < \<top>" shows "a * b < c * b"
fminus_iff: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset" shows "(c |\<in>| A |-| B) = (c |\<in>| A \<and> c |\<notin>| B)"
in_proc_cdep_SDG_path: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'"   and "n \<noteq> n'"   and "n' \<noteq> (_Exit_)"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "targetnode a = n" obtains ns :: "'node SDG_node list" where "CFG_node n cd-ns\<rightarrow>\<^sub>d* CFG_node n'" and "ns \<noteq> []" and "\<forall>n''\<in>set ns. parent_node n'' \<in> set (sourcenodes as)"
function_smult_closed: fixes a :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier F" shows "a \<odot>\<^bsub>F\<^esub> f \<in> carrier F"
HEndPhase0_HInv4a2_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv1 s"   and "Inv2c_inner s p" shows "HInv4a2 s' p"
decs_unclocked_ignore: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec list"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_decs ck mn env s d res" shows "\<forall>s' r count. res = (s', r) \<and> r \<noteq> Rerr (Rabort Rtimeout_error) \<longrightarrow> evaluate_decs False mn env (update_clock (\<lambda>_. count) s) d (update_clock (\<lambda>_. count) s', r)"
per_union_same: fixes R :: "('a \<times> 'a) set"   and l :: "'a" assumes "part_equiv R" shows "per_union R l l = R"
assms_2: shows "m \<le> \<Down> R (f' x')"
multi_self_add_other_not_self: fixes M :: "'a multiset"   and x :: "'a" shows "(M = add_mset x M) = False"
path2_split_first_last: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and x :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "x \<in> set ns" obtains ns\<^sub>1 :: "'node list" and ns\<^sub>3 :: "'node list" and ns\<^sub>2 :: "'node list" where "ns = ns\<^sub>1 @ ns\<^sub>3 @ ns\<^sub>2" and "prefix (ns\<^sub>1 @ [x]) ns" and "suffix (x # ns\<^sub>2) ns" and "g \<turnstile> n-ns\<^sub>1 @ [x]\<rightarrow>x" and "x \<notin> set ns\<^sub>1" and "g \<turnstile> x-ns\<^sub>3\<rightarrow>x" and "g \<turnstile> x-x # ns\<^sub>2\<rightarrow>m" and "x \<notin> set ns\<^sub>2"
IO: fixes out :: "'a"   and \<I> :: "('a, 'b) \<I>"   and D :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'c"   and c' :: "'b \<Rightarrow> 'd"   and A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "out \<in> outs_\<I> \<I>"   and "\<And>input. input \<in> responses_\<I> \<I> out \<Longrightarrow> D (c input) (c' input)" shows "eq_\<I>_generat A \<I> D (IO out c) (IO out c')"
closedin_Int: fixes U :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "closedin U S"   and "closedin U T" shows "closedin U (S \<inter> T)"
c_def: shows "c = ennreal (real u * q / (real u * q + real l * p))"
columnvector_def: fixes v :: "('d, 'f) vec" shows "columnvector v = (\<chi>i j. v $ i)"
fun_eq_on_bij_betw: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "fun_eq_on f g A" shows "bij_betw f A B = bij_betw g A B"
list_comb: fixes n :: "nat"   and f :: "Term_Class.term"   and xs :: "Term_Class.term list" shows "wellformed' n (list_comb f xs) = (wellformed' n f \<and> list_all (wellformed' n) xs)"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_inI_4: fixes t :: "('a, 'b) Term.term"   and s :: "('c, 'b) Term.term"   and F :: "(('a, 'b) Term.term \<times> ('c, 'b) Term.term) list"   and x :: "'b" assumes "(t, s) \<in> set F"   and "x \<in> fv t" shows "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F"
member_lookup: fixes t :: "'b set_rbt"   and x :: "'b" shows "member t x = (RBT_Mapping2.lookup t x = Some ())"
set_less_eq_aux_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" shows "ord.set_less_eq_aux less_eq A B = (A \<in> infinite_complement_partition \<and> A = B \<or> ord.set_less_aux less_eq A B)"
special_P: shows "\<exists>U. \<forall>x. \<not> Elem x U \<longrightarrow> P x"
sel: fixes b :: "'b"   and a :: "'a"   and c :: "'b" assumes "set_of b \<noteq> {}"   and "sel b = (a, c)" shows "set_of b = insert a (set_of c) \<and> size c < size b"
col_transpose_is_row: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and i :: "nat" assumes "mat nr nc m"   and "i < nr" shows "col (Matrix_Legacy.transpose nr m) i = row m i"
embedS_comp_embed: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and r'' :: "('e \<times> 'e) set"   and f' :: "'d \<Rightarrow> 'e" assumes "Well_order r"   and "Well_order r'"   and "embedS r r' f"   and "embed r' r'' f'" shows "embedS r r'' (f' \<circ> f)"
valid_weak_action_modality: fixes P :: "'state"   and \<alpha> :: "'act"   and x :: "('idx, 'pred, 'act) formula" shows "P \<Turnstile> \<langle>\<langle>\<alpha>\<rangle>\<rangle>x = (\<exists>\<alpha>' x' P'. Act \<alpha> x = Act \<alpha>' x' \<and> P \<Rightarrow>\<langle>\<alpha>'\<rangle> P' \<and> P' \<Turnstile> x')"
lotteries_on_subset: fixes A :: "'a set"   and B :: "'a set"   and p :: "'a pmf" assumes "A \<subseteq> B"   and "p \<in> lotteries_on A" shows "p \<in> lotteries_on B"
im_sheaf_is_presheaf: shows "presheaf_of_rings S' is_open' local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf"
assms_2: shows "(v \<longlongrightarrow> ereal m) F"
of_bool_eq_2: shows "of_bool True = (1::'a)"
frag_extend_of: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and a :: "'d" shows "frag_extend f (frag_of a) = f a"
elims_2: fixes x :: "'at Literal" assumes "positive x" obtains A :: "'at" where "x = Pos A"
simps_1: fixes P :: "'a"   and Cal :: "'a processCalculus"   and Q :: "'a" shows "P \<longmapsto>Cal\<^bsup>0\<^esup> Q = (P = Q)"
CARD_mod_ring: shows "CARD('a mod_ring) = CARD('a)"
prems_2: shows "\<forall>a\<in>set fs. case a of (l, t) \<Rightarrow> t \<in> value"
simps_1: fixes y :: "'c arctic_delta" shows "MinInfty_delta * y = MinInfty_delta"
det_linear_rows_sum_lemma: fixes S :: "'a set"   and T :: "'a set"   and a :: "'a \<Rightarrow> 'a \<Rightarrow> ('b, 'a) vec"   and c :: "'a \<Rightarrow> ('b, 'a) vec" assumes "finite S"   and "finite T" shows "det (\<chi>i. if i \<in> T then sum (a i) S else c i) = (\<Sum>f | (\<forall>i\<in>T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i). det (\<chi>i. if i \<in> T then a i (f i) else c i))"
integrate_insert_result_helper: fixes pm :: "'a extended"   and sm :: "'a extended"   and m' :: "('a, 'b) insert_message"   and s' :: "('a, 'b option) insert_message list" assumes "invariant pm sm"   and "m' = m"   and "s' = s" shows "is_certified_associated_string' (integrate_insert m' s' pm sm)"
TS_c2'': fixes x :: "nat"   and y :: "nat"   and x0 :: "nat"   and y0 :: "nat"   and s :: "(nat list \<times> nat list) pmf"   and qs :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "TS_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom x, Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])" shows "TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h0)) s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS h0)) s qs = real (length qs - 3)"
preserves_arr: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "functor A B F"   and "partial_magma.arr A f" shows "partial_magma.arr B (F f)"