hcju/isabelleps huggingface dataset:
plus_period: fixes x :: "real" shows "fds_perzeta (x + 1) = fds_perzeta x"
hull_closed: fixes w1 :: "'b list"   and G :: "'b list set"   and w2 :: "'b list" assumes "w1 \<in> \<langle>G\<rangle>"   and "w2 \<in> \<langle>G\<rangle>" shows "w1 \<cdot> w2 \<in> \<langle>G\<rangle>"
pick_le: fixes n :: "nat"   and i :: "nat"   and S :: "nat set" assumes "n < card {a. a < i \<and> a \<in> S}" shows "pick S n < i"
additionTr0: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "0 < n"   and "\<forall>j\<le>n. f j \<in> carrier A" shows "\<Sigma>\<^sub>e A cmp f (transpos (n - 1) n) n = \<Sigma>\<^sub>e A f n"
delivery_has_a_cause: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m :: "'c"   and i :: "nat" assumes "network history msg_id"   and "Deliver m \<in> set (history i)" shows "\<exists>j. Broadcast m \<in> set (history j)"
word_overflow_unat: fixes x :: "'b word" shows "unat (x + 1) = unat x + 1 \<or> x + 1 = 0"
arith: fixes x :: "int" assumes "x \<le> 2 ^ 31 - 1" shows "x + - 2147483647 < 2147483647"
2: shows "(x \<oplus>\<^bsub>F\<^esub> (y \<oplus>\<^bsub>F\<^esub> z)) a = x a \<oplus> y a \<oplus> z a"
zariski_is_topological_space: shows "topological_space local.local.spectrum local.local.is_zariski_open"
ActualForallI: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>" assumes "[\<^bold>\<forall>x. \<^bold>\<A>\<phi> x in v]" shows "[\<^bold>\<A>(\<^bold>\<forall>x. \<phi> x) in v]"
mono: fixes c :: "'c"   and a :: "'c"   and d :: "'c"   and b :: "'c" assumes "c \<le> a"   and "d \<le> b" shows "sup c d \<le> sup a b"
lfp_lowerbound: fixes a :: "'a set acom"   and i :: "com"   and f :: "'a set acom \<Rightarrow> 'a set acom" assumes "a \<in> {c'. strip c' = i}"   and "f a \<le> a" shows "Collecting.lfp f i \<le> a"
rep_eq: fixes x :: "Vars.var set"   and xa :: "Vars.var set" shows "Rep_CoCall (x G\<times> xa) = x \<times> xa \<union> xa \<times> x"
FunctionalExpr: fixes A :: "'t"   and e :: "('t, 'f) Expression"   and B :: "'t"   and i :: "('o, 'm) IType"   and j :: "('o, 'm) IType" assumes "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> i"   and "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> j" shows "i = j"
zimage_iff: fixes y :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c zet" shows "zin y (zimage f A) = (\<exists>x. zin x A \<and> y = f x)"
add_diff_cancel_left: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and c :: "'k"   and a :: "'k"   and b :: "'k" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus c a) (plus c b) = minus a b"
le_Limsup: fixes F :: "'b filter"   and l :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "F \<noteq> bot"   and "\<forall>\<^sub>F x in F. l \<le> f x" shows "l \<le> Limsup F f"
functions_of_def: fixes A :: "'c set"   and B :: "'d set" shows "Twelvefold_Way_Entry5.functions_of A B = {f \<in> A \<rightarrow>\<^sub>E B. f ` A = B}"
n_mult_lb1': fixes x :: "'a"   and y :: "'a" shows "\<lparr>!x\<rparr>!y\<lparr>!x\<rparr>"
Basis_list: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Basis_list :: "'b list"   and eucl_down :: "int \<Rightarrow> 'b \<Rightarrow> 'b"   and eucl_truncate_down :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and eucl_truncate_up :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "set Basis_list = Basis"
strong_fusion_idempotence: fixes z :: "i" shows "(\<sigma>x. z = x) = z"
distinct_17: fixes x31 :: "com"   and x32 :: "com"   and x51 :: "bexp"   and x52 :: "com" shows "x31;; x32 \<noteq> WHILE x51 DO x52"
consPerm: fixes x :: "name"   and y :: "name"   and p :: "(name \<times> name) list"   and C :: "'d" shows "((x, y) # p) \<bullet> C = [(x, y)] \<bullet> p \<bullet> C"
l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors_di :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_di_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_host get_host_locs get_disconnected_document get_disconnected_document_locs get_ancestors_di get_ancestors_di_locs \<equiv> (l_get_parent type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs \<and> l_get_host get_host get_host_locs) \<and> l_get_disconnected_document get_disconnected_document get_disconnected_document_locs \<and> l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_host get_host_locs get_disconnected_document get_disconnected_document_locs get_ancestors_di get_ancestors_di_locs"
supt_ctxtE: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "s \<rhd> t" obtains C :: "('a, 'b) ctxt" where "C \<noteq> \<box>" and "s = C\<langle>t\<rangle>"
drop_Cons: fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" shows "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
eq_divide_eq_numeral1_1: fixes a :: "'a"   and b :: "'a"   and w :: "num" shows "(a = b / numeral w) = (if numeral w \<noteq> (0::'a) then a * numeral w = b else a = (0::'a))"
span_alt: fixes B :: "'b set" shows "m1.span B = {\<Sum>x | f x \<noteq> (0::'a). f x *a x |f. {x. f x \<noteq> (0::'a)} \<subseteq> B \<and> finite {x. f x \<noteq> (0::'a)}}"
simps_5: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "nnf (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = Or (nnf \<phi>\<^sub>1) (nnf \<phi>\<^sub>2)"
path_2: fixes xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and ys :: "('vtx \<times> 'lbl \<times> 'vtx) list" assumes "xsa__ = xs @ ys" shows "\<exists>l2a. path l2a l2 ys \<and> path l1 l2a xs"
open_halfspace_component_lt: fixes i :: "'b"   and a :: "real" shows "open {x. x \<bullet> i < a}"
simps_3: fixes ao :: "'a option"   and x :: "'a"   and xs :: "'a list" shows "set_less_aux_Compl ao (x # xs) [] = (proper_interval ao (Some x) \<or> set_less_aux_Compl (Some x) xs [])"
simps_2: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'b"   and y :: "'d"   and l :: "('b \<times> 'd) list" shows "map_subst f ((x, y) # l) = (x, f y) # map_subst f l"
simps_1: fixes r :: "'a language"   and s :: "'a language" shows "\<oo> (TimesLR r s) = (\<oo> r \<and> \<oo> s)"
invar_reachable: fixes I :: "'a list \<times> (nat \<Rightarrow> 'a option) \<times> 'b \<Rightarrow> bool"   and s :: "'a list \<times> (nat \<Rightarrow> 'a option) \<times> 'b" assumes "is_invar I"   and "reachable s" shows "I s"
restr_to_subalg_def: fixes M :: "'c measure"   and F :: "'c measure" shows "restr_to_subalg M F = measure_of (space M) (sets F) (emeasure M)"
Im_image_box: fixes a :: "complex"   and b :: "complex" assumes "Re a < Re b"   and "Im a < Im b" shows "Im ` box a b = {Im a<..<Im b}"
assms_2: shows "foldr (\<otimes>) fs \<one> \<sim> a"
Un_absorb2: fixes B :: "'b set"   and A :: "'b set" assumes "B \<subseteq> A" shows "A \<union> B = A"
select_convs_8: fixes red :: "('e \<times> nat) rgraph"   and black :: "('e, 'f, 'g) lts"   and subs :: "(('e \<times> nat) \<times> 'e \<times> nat) set"   and init_conf :: "('f, 'g) conf"   and confs :: "'e \<times> nat \<Rightarrow> ('f, 'g) conf"   and marked :: "'e \<times> nat \<Rightarrow> bool"   and strengthenings :: "'e \<times> nat \<Rightarrow> ('f \<Rightarrow> 'g) \<Rightarrow> bool"   and more :: "'d" shows "pre_RedBlack.more \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = more"
rem: shows "(\<Prod>(a, i)\<in>as - UNIV \<times> {0..<n} - {(a, i)}. a) \<noteq> 0"
assms: shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s"
expand_prop_all: fixes n_ns :: "'b node \<times> 'b node set"   and \<xi> :: "nat \<Rightarrow> 'b set" assumes "((\<forall>nm\<in>incoming (fst n_ns). nm < name (fst n_ns)) \<and> 0 < name (fst n_ns) \<and> (\<forall>q\<in>snd n_ns. name q < name (fst n_ns) \<and> (\<forall>nm\<in>incoming q. nm < name (fst n_ns)))) \<and> expand_name_ident (snd n_ns)" shows "expand n_ns \<le> SPEC (expand_rslt_all \<xi> n_ns)"
S_mono_one: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and f :: "'f"   and ss1 :: "('f, 'b) Term.term list"   and ss2 :: "('f, 'b) Term.term list" assumes "S s t" shows "S (Fun f (ss1 @ s # ss2)) (Fun f (ss1 @ t # ss2))"
fds_nth_shift: fixes c :: "'b"   and f :: "'b fds"   and n :: "nat" shows "fds_nth (fds_shift c f) n = fds_nth f n * nat_power n c"
nfoldli_append: fixes l1 :: "'b list"   and l2 :: "'b list"   and ctd :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a nres"   and s :: "'a" shows "nfoldli (l1 @ l2) ctd f s = nfoldli l1 ctd f s \<bind> nfoldli l2 ctd f"
rewrite_negated_primitives_not_has_disc: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and disc2 :: "'b \<Rightarrow> bool"   and negate_f :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<not> has_disc disc2 m"   and "has_disc_negated disc False m \<Longrightarrow> \<forall>pts. \<not> has_disc disc2 (negate_f C pts)"   and "\<forall>a. \<not> disc2 (C a)" shows "\<not> has_disc disc2 (rewrite_negated_primitives (disc, sel) C negate_f m)"
sum_pair_less_iff: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a"   and c :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "(\<Sum>(k, m)\<in>{(k, m). k + m \<le> n}. a k * b m * c (k + m)) = (\<Sum>s = 0..n. \<Sum>i = 0..s. a i * b (s - i) * c s)"
rev_notE: fixes P :: "bool"   and R :: "bool" assumes "P"   and "\<not> R \<Longrightarrow> \<not> P" shows "R"
rphil_transitions_def: fixes i :: "nat"   and s :: "nat" shows "Tr\<^sub>r\<^sub>p i s = (if s = 0 then {picks i i} else if s = 1 then {picks i (i - 1)} else if s = 2 then {putsdown i (i - 1)} else if s = 3 then {putsdown i i} else {})"
case_prod_app: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'b"   and x :: "'c \<times> 'd"   and y :: "'e" shows "(case x of (x, xa) \<Rightarrow> f x xa) y = (case x of (l, r) \<Rightarrow> f l r y)"
smult_r_null: fixes a :: "'a" shows "a \<cdot>\<^sub>v 0\<^sub>v n = 0\<^sub>v n"
build_termination: fixes ps :: "(real, 'a) vec list"   and ks :: "'a list" assumes "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)" shows "build_dom (ks, ps)"
static_to_dynamic_accessible_from: fixes G :: "prog"   and accC :: "qtname"   and m :: "qtname \<times> memberdecl"   and statC :: "qtname"   and dynC :: "qtname" assumes "G \<turnstile> m of statC accessible_from accC"   and "G\<turnstile>dynC\<preceq>\<^sub>C statC"   and "wf_prog G" shows "G \<turnstile> m in dynC dyn_accessible_from accC"
test_inf_closed_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y"   and "\<forall>u v. test u \<and> test v \<longrightarrow> test (u \<sqinter> v)" shows "test (x \<squnion> y)"
replication_numbers_def: shows "source.replication_numbers \<equiv> {\<B> rep x |x. x \<in> \<V>}"
is_tame13a_def: fixes g :: "graph" shows "is_tame13a g \<equiv> squanderLowerBound g < squanderTarget"
lang_nderiv: fixes a :: "nat"   and S :: "nat set"   and r :: "nat rexp" assumes "a \<in> S" shows "lang S (nderiv a r) = Deriv a (lang S r)"
cos_int_times_real: fixes m :: "int"   and x :: "real" shows "cos (of_int m * of_real x) = of_real (cos (real_of_int m * x))"
sup_aci_3: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x + (y + z) = y + (x + z)"
fls_deriv_fps_to_fls: fixes f :: "'a fps" shows "fls_deriv (fps_to_fls f) = fps_to_fls (fps_deriv f)"
coprime_prod2: fixes A :: "'c set"   and a :: "'c" assumes "\<forall>x\<in>A. comm_monoid_mult_class.coprime a x"   and "finite A" shows "comm_monoid_mult_class.coprime a (\<Prod>A)"
IH: fixes I :: "'a \<Rightarrow> bool option" assumes "length (trace_parallel_plan_strips I \<pi>) < length \<pi> + 1" shows "\<not> are_all_operators_applicable (execute_parallel_plan I (take (length (trace_parallel_plan_strips I \<pi>) - 1) \<pi>)) (\<pi> ! (length (trace_parallel_plan_strips I \<pi>) - 1)) \<or> \<not> are_all_operator_effects_consistent (\<pi> ! (length (trace_parallel_plan_strips I \<pi>) - 1))"
config'_rand_induct: fixes init :: "('state \<times> 'b) pmf"   and P :: "'state \<Rightarrow> bool"   and A :: "('state \<Rightarrow> 'b pmf) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> ('answer \<times> 'b) pmf)"   and qs :: "'request list" assumes "\<forall>x\<in>set_pmf init. P (fst x)"   and "\<And>s q a. P s \<Longrightarrow> P (step s q a)" shows "\<forall>x\<in>set_pmf (config'_rand A init qs). P (fst x)"
order_zmset_exists_foundation: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a zmultiset"   and t :: "'a" assumes "class.order less_eq less"   and "0 < zcount M t" shows "\<exists>s. less_eq s t \<and> 0 < zcount M s \<and> (\<forall>u. 0 < zcount M u \<longrightarrow> \<not> less u s)"
xor_commute: fixes x :: "'g"   and y :: "'g" shows "x \<oplus> y = y \<oplus> x"
simps_3: fixes x2 :: "'a" shows "Msg x2 \<noteq> Marker"
closure_finite_cspan: fixes T :: "'a set" assumes "finite T" shows "closure (cspan T) = cspan T"
equal_pair_def: fixes p :: "'b \<times> 'b" shows "equal_pair p \<equiv> fst p = snd p"
induct: fixes x1 :: "trm"   and x2 :: "trm"   and P :: "trm \<Rightarrow> trm \<Rightarrow> bool" assumes "x1 \<mapsto> x2"   and "\<And>s s' t. \<lbrakk>s \<mapsto> s'; P s s'\<rbrakk> \<Longrightarrow> P (App s t) (App s' t)"   and "\<And>t t' s. \<lbrakk>t \<mapsto> t'; P t t'\<rbrakk> \<Longrightarrow> P (App s t) (App s t')"   and "\<And>x s t. x \<sharp> s \<Longrightarrow> P (App (\<Lambda> x . t) s) (t[x::=s])"   and "\<And>t t' x. \<lbrakk>t \<mapsto> t'; P t t'\<rbrakk> \<Longrightarrow> P (\<Lambda> x . t) (\<Lambda> x . t')"   and "\<And>x t. x \<sharp> t \<Longrightarrow> P (\<Lambda> x . App t (Var x)) t"   and "\<And>x s s' t. \<lbrakk>x \<sharp> (s, s'); s \<mapsto> s'; P s s'\<rbrakk> \<Longrightarrow> P (s to x in t) (s' to x in t)"   and "\<And>x s t t'. \<lbrakk>x \<sharp> s; t \<mapsto> t'; P t t'\<rbrakk> \<Longrightarrow> P (s to x in t) (s to x in t')"   and "\<And>x s t. x \<sharp> s \<Longrightarrow> P ([s] to x in t) (t[x::=s])"   and "\<And>x s. x \<sharp> s \<Longrightarrow> P (s to x in [Var x]) s"   and "\<And>x y s u t. \<lbrakk>x \<sharp> (y, s, u); y \<sharp> (s, t)\<rbrakk> \<Longrightarrow> P ((s to x in t) to y in u) (s to x in t to y in u)"   and "\<And>s s'. \<lbrakk>s \<mapsto> s'; P s s'\<rbrakk> \<Longrightarrow> P [s] [s']" shows "P x1 x2"
h_contour: shows "((\<lambda>x. deriv h x / h x) has_contour_integral 0) \<gamma>"
one_less_HNatInfinite: fixes x :: "nat star" assumes "x \<in> HNatInfinite" shows "1 < x"
unstream_zipWithS: fixes a :: "('b, 'c) Stream"   and b :: "('d, 'e) Stream"   and f :: "'b \<rightarrow> 'd \<rightarrow> 'f" assumes "a \<noteq> \<bottom>"   and "b \<noteq> \<bottom>" shows "unstream\<cdot>(zipWithS\<cdot>f\<cdot>a\<cdot>b) = zipWithL\<cdot>f\<cdot>(unstream\<cdot>a)\<cdot>(unstream\<cdot>b)"
implAuth_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> valid \<union> range LtK \<union> Tags"   and "implAuth A B M \<in> synth (analz H)" shows "implAuth A B M \<in> H \<or> M \<in> synth (analz H) \<and> (A, B) \<in> broken H"
mono_fun: fixes S1 :: "'a st"   and S2 :: "'a st"   and x :: "char list" assumes "S1 \<le> S2" shows "fun S1 x \<le> fun S2 x"
simps: fixes \<delta> :: "real \<Rightarrow> real"   and a1 :: "real \<times> real"   and a2 :: "(real \<times> real \<times> real) list" shows "fine \<delta> a1 a2 = ((\<exists>a. a1 = (a, a) \<and> a2 = []) \<or> (\<exists>b c D a x. a1 = (a, c) \<and> a2 = (a, x, b) # D \<and> fine \<delta> (b, c) D \<and> a < b \<and> a \<le> x \<and> x \<le> b \<and> b - a < \<delta> x))"
Even_ok_N: shows "{}|-{Z=Arg+0}. BODY Even .{Res_ok}"
mkarr_Assoc: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Arr v" shows "\<F>C.mkarr \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = \<F>C.assoc (\<F>C.mkarr t) (\<F>C.mkarr u) (\<F>C.mkarr v)"
filter_mset_is_empty_iff: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b multiset" shows "(filter_mset P A = {#}) = (\<forall>x. x \<in># A \<longrightarrow> \<not> P x)"
simptm_tmbound0: fixes t :: "tm" assumes "tmbound0 t" shows "tmbound0 (simptm t)"
fps_dvd_iff: fixes f :: "'a fps"   and g :: "'a fps" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "(f dvd g) = (subdegree f \<le> subdegree g)"
klift_prop: fixes f :: "'b \<Rightarrow> 'a set"   and X :: "'b set" shows "bd\<^sub>\<F> f X = \<mu> (\<P> f X)"
l7_13_R1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and P' :: "'a"   and Q' :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> P"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A P' P"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A Q' Q" shows "Cong P Q P' Q'"
s1'_e1_s2': shows "s1' e1\<longrightarrow>\<^bsub>SES\<^esub> s2'"
bell01_def: shows "|\<beta>\<^sub>0\<^sub>1\<rangle> \<equiv> complex_of_real (1 / sqrt 2) \<cdot>\<^sub>m |vec 4 (\<lambda>i. if i = 1 \<or> i = 2 then 1 else 0)\<rangle>"
degree_synthetic_div: fixes p :: "'a poly"   and c :: "'a" shows "degree (synthetic_div p c) = degree p - 1"
map_poly_inj_zero_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "map_poly_inj_zero_hom hom \<equiv> inj_zero_hom hom"
eWlsInp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'a option" shows "eWlsInp MOD delta (OKI inp) = igWlsInp MOD delta inp"
expect_prob_comp: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and f :: "bool stream \<Rightarrow> real"   and n :: "nat" assumes "infinite_cts_filtration p M F"   and "f \<in> borel_measurable (F n)" shows "integral\<^sup>L M f = (\<Sum>w\<in>range (pseudo_proj_True n). prod (prob_component p w) {0..<n} * f w)"
convolution_delta_right: fixes f :: "'a \<Rightarrow> 'b"   and b :: "'b"   and a :: "'a" shows "convolution f (b \<delta> a) = (\<lambda>x. f (x - a) * b)"
pendingD: fixes u :: "'v"   and v :: "'v" assumes "(u, v) \<in> pending s" shows "(u, v) \<in> E \<and> u \<in> dom (discovered s)"
not_eq_order_implies_strict: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" assumes "a \<noteq> b"   and "a \<preceq> b" shows "a \<prec> b"
isUnitClauseIsReason: fixes uClause :: "Literal list"   and uLiteral :: "Literal"   and valuation :: "Literal list"   and valuation' :: "Literal list" assumes "isUnitClause uClause uLiteral valuation"   and "uLiteral el valuation'" shows "isReason uClause uLiteral (valuation @ valuation')"
nat_to_bv0: shows "nat_to_bv 0 = []"
inf_continuous_const: fixes c :: "'e" shows "inf_continuous (\<lambda>x. c)"
indep_card_eq_dim_span: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and Basis :: "'e set"   and B :: "'e set" assumes "finite_dimensional_vector_space scale Basis"   and "\<not> module.dependent scale B" shows "finite B \<and> card B = vector_space.dim scale (module.span scale B)"
Cons_1: fixes accu :: "('a \<times> 'a) list" assumes "\<forall>a\<in>set M. case a of (a, b) \<Rightarrow> SecurityInvariant_complies_formal_def a b"   and "wf_list_graph G"   and "set Es \<subseteq> set (edgesL G)"   and "set accu \<subseteq> set (edgesL G)"   and "distinct (Es @ accu)" shows "TopoS_Stateful_Policy_impl.filter_IFS_no_violations_accu G (get_impl M) accu Es = TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations_accu (list_graph_to_graph G) (get_spec M) accu Es"
monotone_comp_is_utility: fixes f :: "real \<Rightarrow> real" assumes "monotone (\<lambda>x y. y < x) (\<lambda>x y. y < x) f" shows "ordinal_utility carrier relation (f \<circ> u)"
u_3: shows "csupport_on carrier u \<inter> carrier \<subseteq> U"
preimage_of_max_lideal: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and B :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d"   and \<ww>\<^sub>A :: "'c set"   and \<ww>\<^sub>B :: "'d set" assumes "local_ring_morphism f A addition multiplication zero unit B addition' multiplication' zero' unit'"   and "max_lideal \<ww>\<^sub>A A addition multiplication zero unit"   and "max_lideal \<ww>\<^sub>B B addition' multiplication' zero' unit'" shows "f \<^sup>\<inverse> A \<ww>\<^sub>B = \<ww>\<^sub>A"
two_step_def: fixes r :: "nat" shows "two_step r \<equiv> r mod 2"
aligned_mask_step: fixes n' :: "nat"   and n :: "nat"   and p' :: "'b word"   and p :: "'b word" assumes "n' \<le> n"   and "p' \<le> p + mask n"   and "is_aligned p n"   and "is_aligned p' n'" shows "p' + mask n' \<le> p + mask n"
d1: fixes p :: "(real \<times> real set) set" assumes "p tagged_division_of {a..c}"   and "d1 fine p" shows "norm ((\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) - integral {a..c} f) < e / 3"
lookup_inverse: fixes x :: "'a \<Rightarrow>\<^sub>0 'b" shows "Abs_poly_mapping (lookup x) = x"
exhaust: fixes y :: "('e, 'f, 'g) var0" obtains x1 :: "('e \<times> 'f) pmf" where "y = PMF x1"   | x2 :: "('e \<times> 'f) set['g]" where "y = BPS x2"
c_process_def: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" shows "c_process step out s\<^sub>0 \<equiv> Abs_process (c_failures step out s\<^sub>0, {})"
mismatchIdLeft: fixes Rel :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name"   and P :: "pi" assumes "Id \<subseteq> Rel"   and "a \<noteq> b" shows "[a\<noteq>b]P \<leadsto>[Rel] P"
summable_R: shows "summable (\<lambda>i. R (f i))"