hcju/isabelleps huggingface dataset:
SBHGhost: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'c memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Ghost A L R W # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>, sb @ [Ghost\<^sub>s\<^sub>b A L R W], m, \<D>, \<O>, \<R>, \<S>)"
HPhase0Read_blocksOf: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d" shows "blocksOf s' q \<subseteq> blocksOf s q"
fcomp_assoc: fixes f :: "'c \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> 'e"   and h :: "'e \<Rightarrow> 'd" shows "fcomp (fcomp f g) h = fcomp f (fcomp g h)"
l: shows "l V' = \<lfloor>v\<rfloor>"
E: shows "\<forall>n. (Owner n, SesKey SK) \<notin> s'"
leadsTo_imp_wens_set: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" assumes "F \<in> A \<longmapsto> B" shows "\<exists>C\<in>wens_set F B. A \<subseteq> C"
distinct_5: fixes x2 :: "agent"   and x3 :: "agent" shows "publK x2 \<noteq> privK x3"
last_op_greatest: fixes op_list :: "('a \<times> 'a option) list"   and oid :: "'a"   and oper :: "'a option"   and x :: "'a" assumes "insert_ops (op_list @ [(oid, oper)])"   and "x \<in> set (map fst op_list)" shows "x < oid"
borel_measurable_diff: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
gamma_def: shows "\<gamma> = (\<lambda>x. (y2 + y1 * x, b))"
run_state_def: fixes state :: "('b, 'c) state" shows "run_state state = (case state of State x \<Rightarrow> x)"
increasing_def: fixes f :: "'c \<Rightarrow> 'd" shows "increasing f \<equiv> \<Inter>z. stable {s. z \<le> f s}"
fset_of_fmap_iff': fixes a :: "'c"   and b :: "'d"   and m :: "('c, 'd) fmap" shows "((a, b) \<in> fset (fset_of_fmap m)) = (fmlookup m a = Some b)"
vector_conv_compl: fixes v :: "'a" assumes "vector v" shows "covector (- v\<^sup>T)"
proots_pcompose_bij_eq: fixes q :: "'b poly"   and A :: "'b set"   and B :: "'b set"   and p :: "'b poly" assumes "bij_betw (poly q) A B"   and "p \<noteq> 0"   and "degree q = 1" shows "proots_count p B = proots_count (p \<circ>\<^sub>p q) A"
surjective: shows "induced.kernel.Class ` kernel.Partition = induced.kernel.Partition"
aext_bop: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'c"   and u :: "('f, 'e) uexpr"   and v :: "('g, 'e) uexpr"   and a :: "'e \<Longrightarrow> 'd" shows "bop f u v \<oplus>\<^sub>p a = bop f (u \<oplus>\<^sub>p a) (v \<oplus>\<^sub>p a)"
simps: fixes P :: "'d \<Rightarrow> bool"   and s :: "'d stream" shows "sdrop_while P s = (if P (shd s) then sdrop_while P (stl s) else s)"
integrable_abs_iff: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" assumes "f \<in> borel_measurable M" shows "integrable M (\<lambda>x. \<bar>f x\<bar>) = integrable M f"
is_isoI: fixes h :: "'f \<Rightarrow> 'g"   and G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme" assumes "h \<in> Group.iso G H" shows "G \<cong> H"
order_trans_rules_21: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y"   and "y < z" shows "x < z"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "P T"   and "P F"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Le v)"   and "\<And>v. P (Gt v)"   and "\<And>v. P (Ge v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)"   and "\<And>v va. P (Dvd v va)"   and "\<And>v va. P (NDvd v va)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Or v va)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
rcliff_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and xs :: "(accent \<times> 'a) list" shows "rcliff r a b xs = ((\<exists>as b' cs a' bs. snd ` set as \<subseteq> under r a \<inter> under r b \<and> b' \<in> {[(Grave, b)], []} \<and> snd ` set cs \<subseteq> under r b \<and> a' = [(Macron, a)] \<and> snd ` set bs \<subseteq> under r b \<and> xs = as @ b' @ cs @ a' @ bs) \<or> (\<exists>as b' cs. snd ` set as \<subseteq> under r a \<and> b' \<in> {[(Grave, b)], []} \<and> snd ` set cs \<subseteq> under r a \<union> under r b \<and> xs = as @ b' @ cs))"
separating_of_bipartite: fixes \<Gamma> :: "('a, 'b) web_scheme"   and S :: "('a + 'a) set" assumes "web \<Gamma>"   and "separating (bipartite_web_of \<Gamma>) S" shows "separating \<Gamma> (web.separating_of_bipartite \<Gamma> S)"
openin_subset_trans: fixes U :: "'c set"   and S :: "'c set"   and T :: "'c set" assumes "openin (top_of_set U) S"   and "S \<subseteq> T"   and "T \<subseteq> U" shows "openin (top_of_set T) S"
assms_1: shows "a \<rightarrow>\<^sup>* b"
differentiable_chain_within: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and S :: "'a set"   and g :: "'b \<Rightarrow> 'c" assumes "f differentiable at x within S"   and "g differentiable at (f x) within f ` S" shows "g \<circ> f differentiable at x within S"
HPT_Target_HAStates2: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d" assumes "TS \<in> HPT ST"   and "S \<in> Target TS" shows "S \<in> HAStates (HA ST)"
elims: fixes x :: "'c match_expr list"   and y :: "'c rule list \<Rightarrow> 'c rule list" assumes "add_match_MatchAnd_foldr x = y" obtains "x = []" and "y = id"   | v :: "'c match_expr" and va :: "'c match_expr list" where "x = v # va" and "y = add_match (MatchAnd_foldr (v # va))"
sat_solve_sasp_axioms: shows "sat_solve_sasp \<Pi>"
assms_1: shows "is_set_to_sorted_list ordR Rk Rs tsl"
AB: shows "\<forall>a\<in>A. \<forall>b\<in>B. \<not> b \<sqsubset> a"
basis_orthogonal: fixes B :: "'a set" assumes "finite B" shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
prime_power_inj'': fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q" shows "(power.power one times p m = power.power one times q n) = (m = 0 \<and> n = 0 \<or> p = q \<and> m = n)"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>) (\<cdot>) R"
blinding_of_blindable_simps_3: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "('c, 'd) blindable\<^sub>m"   and x :: "'c" shows "blinding_of_blindable h bo z (Unblinded x) = ((\<exists>x. z = Unblinded x \<and> bo x x) \<or> z = Blinded (Content (h x)))"
mk_vec_def: fixes f :: "'e \<Rightarrow> 'd" shows "local.mk_vec f = (\<chi>i. f i)"
simps_3: fixes c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd" shows "brn (c1 ;; c2) = brn c1"
bind_option_cong: fixes x :: "'a option"   and y :: "'a option"   and f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "x = y"   and "\<And>z. z \<in> set_option y \<Longrightarrow> f z = g z" shows "Option.bind x f = Option.bind y g"
invertible_inverse_closed: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> N" shows "sub.inverse u \<in> N"
convex_hull_linear_image: fixes f :: "'a \<Rightarrow> 'b"   and s :: "'a set" assumes "linear f" shows "f ` (convex hull s) = convex hull f ` s"
simps_2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" shows "power.power one times a (Suc n) = times a (power.power one times a n)"
dom_op_def: fixes x :: "('a \<times> 'a) set" shows "rel_aka.dom_op x = rel_ad (rel_ad x)"
H1: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> \<box>[P]_v"   and "\<turnstile> \<box>[P \<longrightarrow> Q]_v" shows "\<turnstile> \<box>[Q]_v"
neg_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and \<phi> :: "'c" assumes "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls" shows "Syntax_with_Connectives_False.neg imp fls \<phi> = imp \<phi> fls"
ln_fact: fixes n :: "nat" shows "ln (fact n) = (\<Sum>d = 1..n. ln (real d))"
step_fold'_def: fixes g :: "'f \<Rightarrow> (String.literal, nat) assoc_list \<times> 'g \<times> nat \<times> edgeIndex \<times> edgeIndex option \<times> 'h \<Rightarrow> 'e list \<times> edgeIndex \<times> (String.literal, nat) assoc_list"   and steps :: "'f list"   and lbls :: "(String.literal, nat) assoc_list"   and pri :: "'g"   and pos :: "nat"   and nxt :: "edgeIndex"   and onxt :: "edgeIndex option"   and iB :: "'h" shows "step_fold' g steps lbls pri pos nxt onxt iB = foldr (\<lambda>step (pos, nxt, lbls, es). let (e, enxt, lbls) = g step (lbls, pri, pos, nxt, onxt, iB) in (pos + length e, enxt, lbls, es @ e)) steps (pos, nxt, lbls, [])"
cases: fixes a1 :: "'a \<rightharpoonup>\<^sub>f 'b"   and a2 :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f 'd"   and a3 :: "'a"   and a4 :: "'c"   and a5 :: "'d" assumes "unique_association_class_end a1 a2 a3 a4 a5" obtains association_classes :: "'a \<rightharpoonup>\<^sub>f 'b" and associations :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f 'd" and \<A> :: "'a" and role :: "'c" and "end" :: "'d" where "a1 = association_classes" and "a2 = associations" and "a3 = \<A>" and "a4 = role" and "a5 = end" and "association_class_end' association_classes associations \<A> role end" and "\<not> association_class_end_not_unique association_classes associations \<A> role end"
fst_proj_prop: fixes P :: "'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "\<forall>x\<in>fst_proj {(A, B) |A B. P A \<and> R A B}. P x"
chain_prepend_coinduct: fixes X :: "'b llist \<Rightarrow> bool"   and x :: "'b llist"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> (\<exists>z. x = LCons z LNil) \<or> (\<exists>y xs zs. x = LCons y (prepend zs xs) \<and> (X xs \<or> chain R xs) \<and> chain R (llist_of (y # zs @ [lhd xs])))" shows "chain R x"
hinter_hinsert_left: fixes x :: "hf"   and A :: "hf"   and B :: "hf" shows "(A \<triangleleft> x) \<sqinter> B = (if x \<^bold>\<in> B then A \<sqinter> B \<triangleleft> x else A \<sqinter> B)"
not_empty: shows "set_pmf p \<inter> s \<noteq> {}"
gamma_o_Top: shows "local.\<gamma>\<^sub>o \<top> = UNIV"
interaction_bounded_by_left_gpv: fixes "consider" :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and n :: "enat"   and consider' :: "'a + 'd \<Rightarrow> bool" assumes "interaction_bounded_by consider gpv n"   and "\<And>x. consider' (Inl x) \<Longrightarrow> consider x" shows "interaction_bounded_by consider' (left_gpv gpv) n"
Af: shows "A = mat dim dim (\<lambda>(i, j). lim (\<lambda>n. f n $$ (i, j)))"
bijective_eq_bij: fixes f :: "'a \<Rightarrow> 'b" shows "bijective f = bij f"
dim_col_transfer_rule: fixes A :: "'b mat"   and A' :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" assumes "HMA_M A A'" shows "dim_col A = CARD('c)"
domain_inter_vimage: shows "carrier \<inter> f -` carrier\<^bsub>T\<^esub> = carrier"
final_def: shows "final \<equiv> \<lambda>(ts, ws, r). ts = [] \<and> dom ws \<inter> {0..<n} = {}"
emb_step_is_emb: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
gcd_eq_1_imp_coprime: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "gcd a b = one" shows "algebraic_semidom.coprime times one a b"
drop_all: fixes xs :: "'b list"   and n :: "nat" assumes "length xs \<le> n" shows "xs \<up> n = []"
independent_extend_basis: fixes B :: "'b set" assumes "vs1.independent B" shows "vs1.independent (vs1.extend_basis B)"
block_unique: fixes a :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "a \<in> A"   and "A \<in> Partition"   and "a \<in> B"   and "B \<in> Partition" shows "A = B"
affine_hyperplane_sums_eq_UNIV: fixes S :: "'a set"   and a :: "'a"   and b :: "real" assumes "affine S"   and "S \<inter> {v. a \<bullet> v = b} \<noteq> {}"   and "S - {v. a \<bullet> v = b} \<noteq> {}" shows "{x + y |x y. x \<in> S \<and> a \<bullet> y = b} = UNIV"
set_list_of_dlist_Abs_dlist: fixes xs :: "'b list" shows "set (list_of_dlist (Abs_dlist xs)) = set xs"
sas_plus_operator_inverse_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)" shows "\<phi>\<^sub>O\<inverse> \<Psi> \<phi>\<^sub>O \<Psi> op = op"
intros: fixes p :: "perm"   and x :: "'e" shows "(p \<bullet> x, x) \<in> hull_rel"
d: shows "diagonal_step_PQ_JNF A i j bezout = (P, Q)"
v0_v1_Digraph_axioms_def: fixes G :: "('a, 'b) Graph_scheme"   and v0 :: "'a"   and v1 :: "'a" shows "v0_v1_Digraph_axioms G v0 v1 \<equiv> (v0 \<in> V\<^bsub>G\<^esub> \<and> v1 \<in> V\<^bsub>G\<^esub>) \<and> (v0, v1) \<notin> E\<^bsub>G\<^esub> \<and> v0 \<noteq> v1"
inv_inv_f_eq: shows "inv (inv (inv (inv (inv (inv hom))))) = inv (inv (inv (inv hom)))"
simps_2: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a" shows "heap_base.typeof_h typeof_addr h Unit = typeof Unit"
assms: shows "SimplicialComplex.maxsimp \<Sigma> x"
simps_15: fixes \<C> :: "t_context"   and tn :: "t list"   and tm :: "t list"   and es :: "b_e list"   and ts :: "checker_type" shows "check_single \<C> (Block (tn _> tm) es) ts = (if b_e_type_checker (\<C>\<lparr>label := [tm] @ label \<C>\<rparr>) es (tn _> tm) then type_update ts (to_ct_list tn) (Type tm) else Bot)"
simps: fixes v :: "'c"   and l :: "'c Tree"   and r :: "'c Tree" shows "hs_remove_max (T v l r) = (v, merge l r)"
ord_spmf_return_spmf2: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c spmf"   and y :: "'c" shows "ord_spmf R p (return_spmf y) = (\<forall>x\<in>set_spmf p. R x y)"
pred_of_set_set_fold_sup: fixes xs :: "'a list" shows "pred_of_set (set xs) = fold (\<squnion>) (list.map single xs) \<bottom>"
simps_1: fixes r :: "'b llist \<rightarrow> 'b llist \<rightarrow> 'b llist" shows "lrev_body1\<cdot>r\<cdot>lnil = list2H\<cdot>lnil"
less_one: fixes n :: "nat" shows "order.greater 1 n = (n = 0)"
forget_eval_fmla: fixes k :: "name"   and A :: "fmla"   and e :: "(name, nat) finfun"   and x :: "nat" assumes "atom k \<sharp> A" shows "eval_fmla (finfun_update e k x) A = eval_fmla e A"
finsum_insert: fixes G :: "('b, 'c) ring_scheme"   and F :: "'d set"   and a :: "'d"   and f :: "'d \<Rightarrow> 'b" assumes "abelian_monoid G"   and "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier G"   and "f a \<in> carrier G" shows "finsum G f (insert a F) = f a \<oplus>\<^bsub>G\<^esub> finsum G f F"
subset: shows "X \<subseteq> {x} \<union> \<C>_vars x"
surjective: fixes r :: "'a prog_scheme" shows "r = \<lparr>vars = vars r, body = body r, \<dots> = prog.more r\<rparr>"
multLessState_def: shows "multLessState = {(state1, state2). getM state1 = getM state2 \<and> (getC state1, getC state2) \<in> multLess (getM state1)}"
id_act_prop: fixes v :: "i"   and \<alpha> :: "'b"   and \<beta> :: "'b" assumes "[\<^bold>\<A>(\<alpha> \<^bold>= \<beta>) in v]" shows "[\<alpha> \<^bold>= \<beta> in v]"
remdups_concat_map: fixes xss :: "'a list list" shows "remdups (concat (map remdups xss)) = remdups (concat xss)"
double_dual_iso_one: fixes G :: "('c, 'd) monoid_scheme" assumes "finite_comm_group G" shows "finite_comm_group.double_dual_iso G \<one>\<^bsub>G\<^esub> = principal_char (Characters G)"
strategyproofness_1: shows "(pmf (sds R19) d + pmf (sds R19) c < pmf (sds R27) d + pmf (sds R27) b \<or> pmf (sds R19) d + (pmf (sds R19) c + pmf (sds R19) a) < pmf (sds R27) d + (pmf (sds R27) b + pmf (sds R27) a)) \<or> pmf (sds R19) d + pmf (sds R19) c = pmf (sds R27) d + pmf (sds R27) b \<and> pmf (sds R19) d + (pmf (sds R19) c + pmf (sds R19) a) = pmf (sds R27) d + (pmf (sds R27) b + pmf (sds R27) a)"
cases: fixes a1 :: "('a, 'b, 'c) configuration"   and a2 :: "('a, 'b, 'c) event list"   and a3 :: "('a, 'b, 'c) configuration" assumes "trace a1 a2 a3" obtains c :: "('a, 'b, 'c) configuration" where "a1 = c__" and "a2 = []" and "a3 = c__"   | c :: "('a, 'b, 'c) configuration" and ev :: "('a, 'b, 'c) event" and c' :: "('a, 'b, 'c) configuration" and t :: "('a, 'b, 'c) event list" and c'' :: "('a, 'b, 'c) configuration" where "a1 = c" and "a2 = ev # t" and "a3 = c''" and "c \<turnstile> ev \<mapsto> c'" and "trace c' t c''"
Gramian_determinant_def: fixes G :: "'b vec list"   and k :: "nat" shows "Gramian_determinant G k = det (Gramian_matrix G k)"
C2: fixes n :: "nat" assumes "0 < n" shows "B n \<le> C * Max {B i |i. i < n}"
Sigma_Un_distrib1: fixes I :: "'b set"   and J :: "'b set"   and C :: "'b \<Rightarrow> 'c set" shows "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C"
to_total_def: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option"   and v :: "'b" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B" shows "sifum_types.to_total dma \<Gamma> v \<equiv> if v \<in> dom \<Gamma> then the (\<Gamma> v) else dma v"
Mertens_convergent: shows "convergent (\<lambda>x. \<MM> (real x) - ln (real x))"
intro: fixes A :: "('a, 'b, 'c) sa_rec_scheme"   and B :: "('d, 'b, 'e) sa_rec_scheme"   and R :: "('a \<times> 'd) set" assumes "sa A"   and "sa B"   and "lsimulation R A B"   and "lsimulation (R\<inverse>) B A"   and "bisimulation R A B" shows "lbisimulation R A B"
mset_ssort': fixes l :: "'b"   and sl :: "'a list" assumes "inv l" shows "mset (ssort' l sl) = multiset l + mset sl"
vseq_implies_hpar_1: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.seq \<nu> \<mu>" shows "src\<^sub>D \<nu> = src\<^sub>D \<mu>"
more_temp_simps1_1: fixes F :: "'a \<Rightarrow> bool" shows "\<box>\<box>F \<equiv> \<box>F"
bind_UNION: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'b set" shows "A \<bind> f = \<Union> (f ` A)"
disj_zero_right: fixes x :: "'a" shows "x OR (0::'a) = x"
associatedD1: fixes a :: "'a"   and b :: "'a" assumes "normalize a = normalize b" shows "a dvd b"
lesub_def: fixes x :: "'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> y) = r x y"
ring_simprules_9: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
Weierstrass_prod_ineq: fixes A :: "'e set"   and f :: "'e \<Rightarrow> real" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {0..1}" shows "1 - sum f A \<le> (\<Prod>x\<in>A. 1 - f x)"
diff_fun_space_add: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "f \<in> diff_fun_space"   and "g \<in> diff_fun_space" shows "f + g \<in> diff_fun_space"
f11: shows "\<one> \<otimes> (s \<otimes> s'' \<otimes> (s' \<otimes> s'') \<otimes> ((s' \<otimes> r \<oplus> s \<otimes> r') \<otimes> r'') \<ominus> s \<otimes> s' \<otimes> s'' \<otimes> (s' \<otimes> s'' \<otimes> (r \<otimes> r'') \<oplus> s \<otimes> s'' \<otimes> (r' \<otimes> r''))) = \<zero>"
exhaust: fixes y :: "freeRes_Rep" obtains x11 :: "name" and x12 :: "name" where "y = OutputR_Rep x11 x12"   | "y = TauR_Rep"
treezise_delete: fixes p :: "bool list"   and T :: "tree" assumes "internal p T" shows "treesize (delete p T) < treesize T"
Keys_hom_components: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "Keys (hom_components p) = keys p"
UnitsI_2: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "a \<otimes> b = \<one>" shows "b \<in> Units Zp"
image_insert: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and B :: "'b set" shows "\<P> f (insert a B) = insert (f a) (\<P> f B)"
inf_par_distrib: fixes c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "((c\<^sub>0::'a::refinement_lattice) \<sqinter> (c\<^sub>1::'a::refinement_lattice)) \<parallel> (d::'a::refinement_lattice) = c\<^sub>0 \<parallel> d \<sqinter> c\<^sub>1 \<parallel> d"
change_def: fixes f :: "'a \<Rightarrow> 'a"   and r :: "'a ref" shows "Ref.change f r = !r \<bind> (\<lambda>x. let y = f x in r := y \<bind> (\<lambda>_. return y))"
lupd_def: fixes vn :: "lname"   and v :: "val" shows "lupd(vn\<mapsto>v) = case_st (\<lambda>g l. st g (l(vn \<mapsto> v)))"
sorted_spvec_addmult_spvec_helper: fixes y :: "'a"   and a :: "nat"   and b :: "'a"   and arr :: "(nat \<times> 'a) list"   and brr :: "(nat \<times> 'a) list"   and aa :: "nat"   and ba :: "'a" assumes "sorted_spvec (addmult_spvec y ((a, b) # arr) brr)"   and "aa < a"   and "sorted_spvec ((a, b) # arr)"   and "sorted_spvec ((aa, ba) # brr)" shows "sorted_spvec ((aa, y * ba) # addmult_spvec y ((a, b) # arr) brr)"
elem_out_vp_unit: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v)"   and "x \<notin> vp K v" shows "v x = 0"
Inf_to_Sup_var: fixes X :: "'a set" shows "\<Sqinter> X = \<partial> (\<Squnion> (\<partial> ` X))"
simps_1: fixes t\<^sub>1 :: "term"   and t\<^sub>2 :: "term"   and u :: "'a" shows "match (t\<^sub>1 $ t\<^sub>2) u = unapp u \<bind> (\<lambda>(u\<^sub>1, u\<^sub>2). match t\<^sub>1 u\<^sub>1 \<bind> (\<lambda>env\<^sub>1. match t\<^sub>2 u\<^sub>2 \<bind> (\<lambda>env\<^sub>2. Some (env\<^sub>1 ++\<^sub>f env\<^sub>2))))"
noS: shows "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> (\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` set (\<Upsilon> p)"
bind_is_OK_E: fixes h :: "'ap"   and f :: "('ap, 'aq, 'as) prog"   and g :: "'as \<Rightarrow> ('ap, 'aq, 'ar) prog" assumes "h \<turnstile> ok Heap_Error_Monad.bind f g" obtains x :: "'as" and h' :: "'ap" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> ok g x"