hcju/isabelleps huggingface dataset:
landau_pair_axioms: shows "landau_pair L l L' l' Lr lr R"
subprob_space_return: fixes x :: "'c"   and M :: "'c measure" assumes "x \<in> space M" shows "subprob_space (return M x)"
pickInp_def: fixes inp :: "'d \<Rightarrow> 'e set option" shows "pickInp inp \<equiv> lift pick inp"
mcont_ldropn_aux: shows "mcont lSup (\<sqsubseteq>) llist.lub_fun llist.le_fun (\<lambda>xs n. ldropn n xs)"
g_2: fixes x :: "'a metric_completion" assumes "x \<in> range to_metric_completion" shows "f (I x) = g x"
execn_to_execn_strip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "\<not> isFault t" shows "strip F \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
delete_def: fixes x :: "'b"   and t :: "('b \<times> bal) tree" shows "delete x t = tree (del x t)"
ABinds_restr: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "Vars.var set"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "domA \<Gamma> \<subseteq> S" shows "ABinds \<Gamma>\<cdot>ae f|` S = ABinds \<Gamma>\<cdot>(ae f|` S) f|` S"
simps_4: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "flatten (Seq c\<^sub>1 c\<^sub>2) = flatten c\<^sub>1 @ flatten c\<^sub>2"
y_Triple: assumes "y = Triple p a x" shows "(\<forall>q\<in>#prime_factorization x. [a ^ ((p - 1) div q) \<noteq> 1] (mod p)) \<and> 0 < x"
IH_1: fixes Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" assumes "qGood Y \<and> qSkel Y = qSkel qX" shows "\<forall>ys y1 y. qAFresh ys y1 Y \<longrightarrow> qInit MOD (Y #[[y1 \<and> y]]_ys) = igSubst MOD ys (igVar MOD ys y1) y (qInit MOD Y)"
real_sqrt_eq_iff: fixes x :: "real"   and y :: "real" shows "(sqrt x = sqrt y) = (x = y)"
bij_swap_iff: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "bij (Fun.swap a b f) = bij f"
uniform_limit_at_within_subset: fixes x :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and S :: "'b set"   and l :: "'b \<Rightarrow> 'c"   and t :: "'a"   and T :: "'a set"   and U :: "'a set" assumes "uniform_limit S x l (at t within T)"   and "U \<subseteq> T" shows "uniform_limit S x l (at t within U)"
Aw_def: shows "Aw = A(decode_Var v::=Var (decode_Var w))"
embed_bool_bounded_by_1: fixes P :: "'b \<Rightarrow> bool" shows "bounded_by 1 \<guillemotleft> P \<guillemotright>"
nat_term_order_of_le_def: shows "nat_term_order_of_le = Abs_nat_term_order comparator_of"
fls_regpart_uminus: fixes f :: "'a fls" shows "fls_regpart (- f) = - fls_regpart f"
fpr_nullI: fixes P :: "cld list"   and ctx :: "ctx"   and fqn :: "fqn"   and ctxclds :: "(ctx \<times> cld) list" assumes "\<not> acyclic_clds P \<or> find_cld P ctx fqn None" shows "find_path_rec P ctx (cl_fqn fqn) ctxclds None"
aux2: fixes P :: "('a, 'b, 'c) psi" shows "bangPred P (P \<parallel> !P)"
primerec_rec_pred_1: shows "primerec UF.rec_pred (Suc 0)"
eq: shows "{x \<in> topspace X. g (f x) \<in> U''} = {x \<in> topspace X. f x \<in> {y \<in> topspace X'. g y \<in> U''}}"
downset_onI: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "\<And>x y. \<lbrakk>(x, y) \<in> r; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A" shows "downset_on A r"
convex_bound_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and a :: "'c"   and y :: "'c"   and u :: "'c"   and v :: "'c" assumes "class.linordered_semiring_1 plus minus less_eq less zero times one"   and "less_eq x a"   and "less_eq y a"   and "less_eq zero u"   and "less_eq zero v"   and "plus u v = one" shows "less_eq (plus (times u x) (times v y)) a"
bottomI: fixes z :: "'a"   and P :: "'a set" assumes "z \<in> P"   and "\<And>y. y \<in> P \<Longrightarrow> y \<succeq> z" shows "z = bottom P"
updVal_preserves_eqBut: fixes val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and val' :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and ys :: "'varSort"   and y :: "'var"   and x :: "'var"   and sX :: "'b"   and xs :: "'varSort" assumes "eqBut val val' ys y" shows "eqBut (val (x := sX)_xs) (val' (x := sX)_xs) ys y"
bdd_below_Icc: fixes a :: "'a multiset"   and b :: "'a multiset" shows "subset_mset.bdd_below (subset_mset.atLeastAtMost a b)"
Quotient3_abs_rep: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and a :: "'d" assumes "Quotient3 R Abs Rep" shows "Abs (Rep a) = a"
liftrT_setD: fixes a :: "char list"   and T :: "POPLmarkRecord.type"   and n :: "nat"   and k :: "nat"   and fs :: "(char list \<times> POPLmarkRecord.type) list" assumes "(a, T) \<in> set (\<up>\<^sub>r\<^sub>\<tau> n k fs)" shows "\<exists>T'. (a, T') \<in> set fs \<and> T = \<up>\<^sub>\<tau> n k T'"
that: assumes "v11 f"   and "n \<in>\<^sub>\<circ> \<omega>"   and "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"   and "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A" shows "thesis"
concat_eq_append_conv: fixes xss :: "'b list list"   and ys :: "'b list"   and zs :: "'b list" shows "(concat xss = ys @ zs) = (if xss = [] then ys = [] \<and> zs = [] else \<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2)"
G_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and p :: "'b \<Rightarrow> 'a"   and I :: "'b set" assumes "comm_monoid_set f z" shows "comm_monoid_set.G f z p I \<equiv> if finite {x \<in> I. p x \<noteq> z} then comm_monoid_set.F f z p {x \<in> I. p x \<noteq> z} else z"
eq: shows "(\<Sum>i\<le>n. coeff p i * real_of_int a ^ i * real_of_int b ^ (n - i)) = real_of_int b ^ n * poly p (real_of_int a / real_of_int b)"
sat_atom_sat_eq_sat_constraint_non_monom: fixes v :: "nat \<Rightarrow> QDelta"   and h :: "QDelta ns_constraint"   and var :: "nat" assumes "v \<Turnstile>\<^sub>a qdelta_constraint_to_atom h var"   and "v \<Turnstile>\<^sub>e linear_poly_to_eq (poly h) var"   and "\<not> is_monom (poly h)" shows "v \<Turnstile>\<^sub>n\<^sub>s h"
take_eq_map_nth: fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "take n xs = map ((!) xs) [0..<n]"
add_block_def: fixes bl :: "'a set"   and b :: "'a set" shows "del_block_sys.add_block bl b \<equiv> add_point_sys.del_block bl + {#b#}"
coeff_poly_from_roots: fixes A :: "'a set"   and k :: "nat"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "k \<le> card A" shows "poly.coeff (\<Prod>x\<in>A. [:- f x, 1::'b:]) k = (- (1::'b)) ^ (card A - k) * (\<Sum>X | X \<subseteq> A \<and> card X = card A - k. prod f X)"
l: fixes x :: "nat \<Rightarrow> real" assumes "x \<in> standard_simplex p" shows "oriented_simplex p l x \<in> S"
Comm2: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<longmapsto> a[b] \<prec> P'"   and "Q \<longmapsto> a<b> \<prec> Q'" shows "P \<parallel> Q \<longmapsto> \<tau> \<prec> P' \<parallel> Q'"
add_less_le_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a < b"   and "c \<le> d" shows "a + c < b + d"
rest_compl: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "A - {x \<in> A. P x} = {x \<in> A. \<not> P x}"
locked_in_vf_def: fixes Quorum :: "process set set"   and v_f :: "process \<Rightarrow> val option"   and v :: "val" assumes "quorum_process Quorum" shows "quorum_process.locked_in_vf Quorum v_f v \<equiv> \<exists>Q. quorum_process.quorum_for Quorum Q v v_f"
simps_3: fixes rel' :: "('c \<times> 'c) set \<Rightarrow> ('c \<times> 'c) set"   and carrier :: "'c set"   and rel :: "('c \<times> 'c) set"   and more :: "'d" shows "rel_update rel' \<lparr>carrier = carrier, rel = rel, \<dots> = more\<rparr> = \<lparr>carrier = carrier, rel = rel' rel, \<dots> = more\<rparr>"
simps_2: fixes f :: "'b match_expr \<Rightarrow> 'b match_expr option"   and fs :: "('b match_expr \<Rightarrow> 'b match_expr option) list"   and m :: "'b match_expr" shows "compress_normalize_primitive_monad (f # fs) m = (case f m of None \<Rightarrow> None | Some x \<Rightarrow> compress_normalize_primitive_monad fs x)"
is_Skip_def: fixes step :: "('c, 'd) step" shows "is_Skip step = (\<exists>x2. step = Skip x2)"
unique_row_col_1: fixes nr1 :: "nat"   and nc1 :: "nat"   and M :: "'b list list"   and nr2 :: "nat"   and nc2 :: "nat" assumes "mat nr1 nc1 M"   and "mat nr2 nc2 M"   and "M \<noteq> []" shows "nr1 = nr2"
hom_boundary_hom: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set" shows "hom_boundary p X S \<in> hom (relative_homology_group p X S) (homology_group (p - 1) (subtopology X S))"
asm: shows "good_context s \<and> s' = s \<and> get_trap_set s' \<noteq> {} \<and> reset_trap_val s' = False \<and> get_ET (cpu_reg_val PSR s') = 0"
vsv_axioms: shows "vsv (\<AA>\<lparr>Dom\<rparr>)"
bi_total_rel: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "bi_total R" shows "bi_total (rel_mset R)"
fresh_def: fixes a :: "'a"   and x :: "'b" shows "a \<sharp> x = (a \<notin> supp x)"
polynomial_decomp_alien_var_2: fixes q :: "int mpoly"   and A :: "int mpoly"   and x' :: "nat"   and b :: "int mpoly" assumes "q = A * MPoly_Type.monom (monomial (Suc 0) x') 1 + b"   and "x' \<notin> vars q"   and "x' \<notin> vars b" shows "q = b"
disc_doc_disc_node_dual: fixes h :: "(_) heap"   and disc_node :: "(_) node_ptr"   and disc_doc :: "(_) document_ptr" assumes "h \<turnstile> get_disconnected_document disc_node \<rightarrow>\<^sub>r disc_doc" obtains disc_nodes :: "(_) node_ptr list" where "h \<turnstile> get_disconnected_nodes disc_doc \<rightarrow>\<^sub>r disc_nodes" and "disc_node \<in> set disc_nodes"
lemma_3_7_ii: fixes F :: "'a set" assumes "F \<in> filters" shows "F = {a. cong_l F a (1::'a)}"
segment_to_rel_frontier: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "convex S"   and "bounded S"   and "x \<in> rel_interior S"   and "y \<in> S"   and "\<not> (x = y \<and> S = {x})" obtains z :: "'b" where "z \<in> rel_frontier S" and "y \<in> closed_segment x z" and "open_segment x z \<subseteq> rel_interior S"
streams_stl: fixes s :: "'c stream"   and A :: "'c set" assumes "s \<in> streams A" shows "stl s \<in> streams A"
memo_nat_def: fixes low :: "nat"   and up :: "nat"   and f :: "nat \<Rightarrow> 'c" shows "memo_nat low up f \<equiv> let m = Mapping.tabulate [low..<up] f in (\<lambda>x. if low \<le> x \<and> x < up then the (Mapping.lookup m x) else f x)"
image_image: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and A :: "'f set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
map\<^sub>0_simp: fixes a :: "'c" assumes "C'.obj a" shows "map\<^sub>0 a = F.map\<^sub>0 a"
add_mono_thms_linordered_semiring_1: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i \<le>\<^sub>\<K> j \<and> k \<le>\<^sub>\<K> l" shows "i + k \<le>\<^sub>\<K> j + l"
f_aggregate_take_mult: fixes n :: "nat"   and k :: "nat"   and xs :: "'d list"   and ag :: "'d list \<Rightarrow> 'd" shows "f_aggregate (xs \<down> (n * k)) k ag = f_aggregate xs k ag \<down> n"
order_finite_chain2: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and X :: "'b set"   and i :: "nat"   and j :: "nat"   and l :: "nat" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "MinkowskiBetweenness.long_ch_by_ord2 betw f X"   and "finite X"   and "0 \<le> i \<and> i < j \<and> j < l \<and> l < card X" shows "betw (f i) (f j) (f l)"
extreme_bound_singleton: fixes x :: "'a" assumes "x \<in> A" shows "extreme_bound A (\<lambda>x y. y \<sqsubseteq> x) {x} x"
simps_3: fixes m :: "int"   and d :: "int"   and i :: "int"   and k :: "int"   and ks :: "int list" shows "hd_coeff1 m (NDvd d i (k # ks)) = (if k = 0 then NDvd d i (k # ks) else let m' = m div k in NDvd (m' * d) (m' * i) (1 # m' *\<^sub>s ks))"
p1mod22k: fixes b :: "int"   and n :: "nat" shows "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
AS_prec_first_action: shows "\<forall>s d aseq. invariant s \<and> aseq \<in> AS_set \<and> aseq \<noteq> [] \<longrightarrow> AS_precondition s d (hd aseq)"
ext_wf_bounded_def: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" shows "ext_wf_bounded ext \<equiv> ext_irrefl_before_trans ext \<and> ext_hd_or_tl ext"
to_list_repr: shows "list_repr i f (map ((!) fs) [0..<m])"
ShuffleLR_ZeroL: fixes r :: "'a language" shows "ShuffleLR Zero r = Zero"
rel_funI: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
uniformly_convergent_uniform_limit_iff: fixes X :: "'c set"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" shows "uniformly_convergent_on X f = uniform_limit X f (\<lambda>x. lim (\<lambda>n. f n x)) sequentially"
hd_append2: fixes xs :: "'f list"   and ys :: "'f list" assumes "xs \<noteq> []" shows "hd (xs @ ys) = hd xs"
injfun_elim_image: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and x :: "'e" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "x \<in> A" shows "f ` (A - {x}) = f ` A - {f x}"
group_axioms_def: fixes G :: "('b, 'c) monoid_scheme" shows "Group.group_axioms G \<equiv> carrier G \<subseteq> Units G"
prems_2: fixes e :: "exp" assumes "e \<in> snd ` set as" shows "eqvt_at ESem_sumC e"
proper_implies_distinct: fixes x :: "i"   and y :: "i" assumes "PP x y" shows "x \<noteq> y"
leg0_commutes: shows "\<mu>.cod.leg0 \<cdot> \<mu>.chine = \<mu>.dom.leg0"
finite_converse: fixes r :: "('b \<times> 'a) set" shows "finite (r\<inverse>) = finite r"
cmp_simps_5: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.cod (cmp (f, g)) = local.map (f \<star>\<^sub>C g)"
linorder_on_axioms: shows "linorder_on D (cmp_img f cmp)"
bi_unique_mod_ring_rel_1: shows "bi_unique mod_ring_rel"
preserves_local_guarantee_compliance_def2: fixes \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set" shows "preserves_local_guarantee_compliance \<R> \<equiv> \<forall>c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C. A.respects_own_guarantees (c\<^sub>A, mds\<^sub>A) \<and> (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> conc.respects_own_guarantees (c\<^sub>C, mds\<^sub>C)"
span_explicit: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b set" assumes "module scale" shows "module.span scale b = {\<Sum>a\<in>t. scale (r a) a |t r. finite t \<and> t \<subseteq> b}"
prv_scnj_imp_in: fixes F :: "'fmla set"   and \<phi> :: "'fmla" assumes "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> F" shows "prv (imp (scnj F) \<phi>)"
cases: fixes x :: "('a\<^sub>m, 'a\<^sub>h) blindable\<^sub>m \<times> ('a\<^sub>m, 'a\<^sub>h) blindable\<^sub>m" obtains xa :: "'a\<^sub>m" and y :: "'a\<^sub>m" where "x = (Unblinded xa__, Unblinded y__)"   | xa :: "'a\<^sub>h blindable\<^sub>h" and y :: "'a\<^sub>m" where "x = (Blinded xaa__, Unblinded ya__)"   | y :: "'a\<^sub>m" and xa :: "'a\<^sub>h blindable\<^sub>h" where "x = (Unblinded y, Blinded xa)"   | t :: "'a\<^sub>h blindable\<^sub>h" and u :: "'a\<^sub>h blindable\<^sub>h" where "x = (Blinded t, Blinded u)"
Id_on_subset_Times: fixes A :: "'a set" shows "Id_on A \<subseteq> A \<times> A"
it_to_it_correct: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list"   and S :: "'a set" assumes "set_iterator it S" shows "set_iterator (it_to_it it) S"
iNextStrong_False: fixes t0 :: "nat"   and I :: "nat set" shows "\<not> (\<circle>\<^sub>S t t0 I. False)"
find_sub_False_proj_list_3_same_right: fixes xs :: "'b set list"   and ns :: "'b set list"   and rs :: "'b set list"   and w :: "'b set"   and ws :: "'b set list"   and ys :: "'b set list"   and z :: "'b set"   and zs :: "'b set list"   and ns' :: "'c list"   and ys' :: "'c list"   and z' :: "'c"   and zs' :: "'c list" assumes "find_sub_False xs ns = Some ((rs, w, ws), ys, z, zs)"   and "length ns = length ns'"   and "proj_list_3 ns' (ys, z, zs) = (ys', z', zs')" shows "ns' = ys' @ z' # zs'"
permute_pure: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "perm"   and x :: "'a" assumes "class.pure permute" shows "permute p x = x"
not_isAbrD: fixes s :: "('e, 'f) xstate" assumes "\<not> isAbr s" shows "(\<exists>s'. s = Normal s') \<or> s = Stuck \<or> (\<exists>f. s = Fault f)"
rbt_lookup_rbt_delete: fixes t :: "('a, 'c) rbt"   and k :: "'a" assumes "is_rbt t" shows "rbt_lookup (rbt_delete k t) = rbt_lookup t |` (- {k})"
monoI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f x \<le> f y" shows "order.mono less_eq f"
cases: fixes a1 :: "(status \<times> 'x) \<times> 'm"   and a2 :: "('l, 't, status \<times> 'x, 'm, 'w, 'o action) thread_action"   and a3 :: "(status \<times> 'x) \<times> 'm" assumes "init_fin_\<tau>move a1 a2 a3" obtains x :: "'x" and m :: "'m" and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" where "a1 = ((Running, x), m)" and "a2 = convert_TA_initial (convert_obs_initial ta)" and "a3 = ((Running, x'), m')" and "\<tau>move (x, m) ta (x', m')"
ltln_weak_to_strong_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi>"   and "w \<Turnstile>\<^sub>n F\<^sub>n \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi>"
cases: fixes x :: "exp" obtains xa :: "Vars.var" where "x = Var xa__"   | xa :: "Vars.var" and e :: "exp" where "x = Lam [xaa__]. e__"   | e :: "exp" and xa :: "Vars.var" where "x = App ea__ xa"   | as :: "(Vars.var \<times> exp) list" and e :: "exp" where "x = Terms.Let as e"   | b :: "bool" where "x = Bool b"   | scrut :: "exp" and e1 :: "exp" and e2 :: "exp" where "x = (scrut ? e1 : e2)"
real_poly_real_coeff: fixes p :: "'a poly"   and x :: "nat" assumes "set (coeffs p) \<subseteq> \<real>" shows "coeff p x \<in> \<real>"
continuous_interval_bij: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and u :: "'a"   and v :: "'a" shows "isCont (interval_bij (a, b) (u, v)) x"
weak: shows "\<forall>x xs. P xs \<longrightarrow> P (x # xs)"
hyps_3: assumes "s1 \<approx> s2" shows "s1' \<approx> s2 \<and> \<mu>1\<^sup>+\<^sup>+ s1' s1 \<or> (\<exists>s2'. s2 -\<tau>2\<rightarrow>+ s2' \<and> s1' \<approx> s2')"
eq_\<I>_converterD_WT1: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and conv1 :: "('i, 'j, 'k, 'l) converter"   and conv2 :: "('i, 'j, 'k, 'l) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2"   and "\<I>, \<I>' \<turnstile>\<^sub>C conv1 \<surd>" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv2 \<surd>"
pwd_owner_iv_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_owner_iv n of (s, S, A, B, C, D) \<Rightarrow> pwd_owner_iv_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
lim_def: fixes X :: "nat \<Rightarrow> 'e" shows "lim X = (THE L. X \<longlonglongrightarrow> L)"
ZFfunCompApp: fixes f :: "ZF"   and g :: "ZF" assumes "isZFfun f"   and "isZFfun g"   and "|dom|g = |cod|f" shows "f |o| g = ZFfun (|dom|f) (|cod|g) (\<lambda>x. g |@| (f |@| x))"
abs_eq: shows "dflt_size_integer = integer_of_int (int dflt_size)"
A: shows "JMPND (size (ccomp c\<^sub>1) + 1) # ccomp c\<^sub>1 @ JMP (size (ccomp c\<^sub>2)) # ccomp c\<^sub>2 \<Turnstile> cfs\<box>"
action_tid_change_prefix: fixes n :: "enat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat" assumes "ltake n E [\<approx>] ltake n E'"   and "enat a < n" shows "action_tid E a = action_tid E' a"
covering_space_locally_connected_eq: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "covering_space C p S" shows "locally connected S = locally connected C"
verit_minus_simplify_3: fixes b :: "'b" shows "(0::'b) - b = - b"
X_distr: fixes a :: "real" assumes "0 \<le> a" shows "emeasure M {x \<in> space M. X x \<le> a} = ennreal (erlang_CDF k l a)"
wf_acyclic: fixes r :: "('c \<times> 'c) set" assumes "wf r" shows "acyclic r"
is_pole_hurwitz_zeta: fixes a :: "real" assumes "0 < a" shows "is_pole (hurwitz_zeta a) 1"
are_operator_effects_consistent_def: fixes op :: "('c, 'd) sas_plus_operator"   and op' :: "('c, 'd) sas_plus_operator" shows "SAS_Plus_Semantics.are_operator_effects_consistent op op' \<equiv> let effect = effect_of op; effect' = effect_of op' in list_all (\<lambda>(v, a). list_all (\<lambda>(v', a'). v \<noteq> v' \<or> a = a') effect') effect"
elims_2: fixes x :: "(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 xa :: "char list \<Rightarrow> ty option"   and xb :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and xc :: "expr list"   and xd :: "ty list" assumes "x,xa,xb \<turnstile> xc [:]\<^bsub>NT\<^esub> xd" obtains 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" where "x = Pa__" and "xa = Ea__" and "xb = h__" and "xc = []" and "xd = []"   | 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and e :: "expr" and es :: "expr list" and T :: "ty" and Ts :: "ty list" where "x = P" and "xa = E" and "xb = h" and "xc = e # es" and "xd = T # Ts" and "P,E,h \<turnstile> e :\<^bsub>NT\<^esub> T \<and> P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> Ts"
elims_3: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list" assumes "class.linorder less_eq less"   and "\<not> linorder.sorted less_eq x" obtains x :: "'b" and ys :: "'b list" where "xa__ = x # ys" and "\<not> (Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
find_crossing_edge_on_path: fixes g :: "'a ugraph"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "path g u p v"   and "\<not> u \<notin> S"   and "v \<notin> S" obtains u' :: "'a" and v' :: "'a" where "(u', v') \<in> set p" and "\<not> u' \<notin> S" and "v' \<notin> S"
semIntAbs_prFreshAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prFreshAbs (semIntAbs SEM) SEM"
leg0_composite: shows "src_src.leg0 = src.leg0 \<cdot> src_src.prj\<^sub>0"
rreq_oip: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:3, PRreq-:4, PRreq-:13, PRreq-:21} \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>) \<and> (osn \<xi> < sqn (rt \<xi>) (oip \<xi>) \<or> sqn (rt \<xi>) (oip \<xi>) = osn \<xi> \<and> the (dhops (rt \<xi>) (oip \<xi>)) \<le> Suc (hops \<xi>) \<and> the (flag (rt \<xi>) (oip \<xi>)) = val))"