hcju/isabelleps huggingface dataset:
term_ok_mk_eqD: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and s :: "Core.term"   and t :: "Core.term" assumes "term_ok \<Theta> (mk_eq s t)" shows "term_ok \<Theta> s \<and> term_ok \<Theta> t"
prj_simps_17: shows "C.cod f_src_src.Prj\<^sub>1\<^sub>0 = src.apex"
5: shows "local.comp (eval b c) (tuple \<Lambda>_pr1 b_pr0) = local.comp (eval b c) (tuple \<Lambda>_pr1' b_pr0')"
finite_measure_dist: fixes C :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "AE x in M. x \<notin> C \<longrightarrow> (x \<in> A) = (x \<in> B)"   and "A \<in> events"   and "B \<in> events"   and "C \<in> events" shows "dist (prob A) (prob B) \<le> prob C"
verit_sko_ex_indirect: fixes x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "x = (SOME x. P x)" shows "(\<exists>x. P x) = P x"
simps_3: shows "OUTfromV data3 = {stA4}"
unifiable_def: fixes E :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) set" shows "unifiable E = (\<exists>\<sigma>. \<sigma> \<in> unifiers E)"
not_0_cCons_eq: fixes p :: "'b poly"   and a :: "'b" assumes "p \<noteq> 0" shows "cCons a (coeffs p) = a # coeffs p"
proj_arcs_G_def: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d"   and HM :: "'d pre_map"   and x :: "'d" assumes "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM" shows "subdiv1_contr.proj_arcs_G rev_G uv uw vw x \<equiv> if x = uv then uw else if x = rev_G uv then vw else x"
NotInDom_Contra: fixes H :: "fm set"   and d :: "tm"   and r :: "tm"   and x :: "tm"   and y :: "tm"   and A :: "fm" assumes "H \<turnstile> NotInDom d r"   and "H \<turnstile> HPair x y IN r" shows "insert (x EQ d) H \<turnstile> A"
right_adjoint_functor_axioms: shows "right_adjoint_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G"
converse_relcompp: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'b \<Rightarrow> bool" shows "(r OO s)\<inverse>\<inverse> = s\<inverse>\<inverse> OO r\<inverse>\<inverse>"
sync_orderE: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and a :: "nat"   and a' :: "nat" assumes "P,E \<turnstile> a \<le>so a'" obtains "a \<in> sactions P E" and "a' \<in> sactions P E" and "E \<turnstile> a \<le>a a'"
unforwarded_not_written: fixes a :: "nat"   and sb :: "'b memref list"   and W :: "nat set" assumes "a \<in> unforwarded_non_volatile_reads sb W" shows "a \<notin> W"
authKeys_not_insert: fixes ev :: "event"   and evs :: "event list" assumes "\<forall>A Ta akey Peer. ev \<noteq> Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>akey, Agent Peer, Ta\<rbrace>, Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, akey, Ta\<rbrace>\<rbrace>" shows "authKeys (ev # evs) = authKeys evs"
elims_3: fixes x :: "('a \<times> 'a) list" assumes "\<not> exe_antisym x" obtains xa :: "'a" and y :: "'a" and r :: "('a \<times> 'a) list" where "x = (xa, y) # r" and "\<not> (((y, xa) \<in> set r \<longrightarrow> xa = y) \<and> exe_antisym r)"
assms_1: shows "f maps\<^bsub>SET\<^esub> X to Y"
FL_bisimilar_is_L_bisimulation: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.is_L_bisimulation satisfies transition effect_apply L (effect_nominal_ts.FL_bisimilar satisfies transition effect_apply L)"
real_is_rat_def: fixes x :: "real" shows "is_rat x = (x \<in> \<rat>)"
similar_mat_witI: fixes P :: "'b mat"   and Q :: "'b mat"   and n :: "nat"   and A :: "'b mat"   and B :: "'b mat" assumes "P * Q = 1\<^sub>m n"   and "Q * P = 1\<^sub>m n"   and "A = P * B * Q"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "P \<in> carrier_mat n n"   and "Q \<in> carrier_mat n n" shows "similar_mat_wit A B P Q"
valof_eq: fixes x :: "('a, 'b) IEEE.float" shows "valof x = (if IEEE.exponent x = 0 then (- 1) ^ sign x * (2 / 2 ^ bias TYPE(('a, 'b) IEEE.float)) * (real (fraction x) / 2 ^ LENGTH('b)) else (- 1) ^ sign x * (2 ^ IEEE.exponent x / 2 ^ bias TYPE(('a, 'b) IEEE.float)) * (1 + real (fraction x) / 2 ^ LENGTH('b)))"
has_derivative_vec: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and f' :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and F :: "'e filter" assumes "\<And>i. (f i has_derivative f' i) F" shows "((\<lambda>x. \<chi>i. f i x) has_derivative (\<lambda>h. \<chi>i. f' i h)) F"
invar: shows "2 \<cdot> g \<cdot> x = 2 \<cdot> g \<cdot> h + v \<cdot> v"
rCase: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> Rs"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P Rs A\<^sub>P \<Psi>\<^sub>P"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P"   and "\<Psi>\<^sub>P \<simeq> \<one>"   and "supp \<Psi>\<^sub>P = {}"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Rs"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (Cases Cs) Rs [] (\<one>)"
list_def: fixes n :: "nat"   and A :: "'b set" shows "list n A = {xs. length xs = n \<and> set xs \<subseteq> A}"
S: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and C :: "fmla"   and H' :: "fmla set" assumes "H \<turnstile> A IMP B IMP C"   and "H' \<turnstile> A IMP B" shows "H \<union> H' \<turnstile> A IMP C"
one_add_floor: fixes x :: "'b" shows "\<lfloor>x\<rfloor> + 1 = \<lfloor>x + (1::'b)\<rfloor>"
h_def: shows "h = trdsp (map fst bs) (p, q)"
eval_next: fixes t :: "nat"   and v :: "nat"   and j :: "nat" shows "eval r_next [t, v, j] \<down>= e_snoc (e_append_zeros v (the (eval r_find_k [v]))) 1"
Rtrans': fixes P :: "'prog"   and \<sigma> :: "'state"   and \<sigma>1 :: "'state"   and coll1 :: "'coll"   and \<sigma>' :: "'state"   and coll2 :: "'coll" assumes "R' P \<sigma> \<sigma>1 coll1 \<and> R' P \<sigma>1 \<sigma>' coll2" shows "R' P \<sigma> \<sigma>' (combine coll1 coll2)"
arr_char: fixes f :: "cospan_shape.Arr" shows "partial_magma.arr cospan_shape.comp f = (f \<noteq> cospan_shape.Null)"
assms_2: shows "\<forall>M w. M, w \<Turnstile>\<^sub>! p"
mulex_on_add_mset: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset"   and m :: "'b" assumes "mulex_on P A M N"   and "m \<in> A" shows "mulex_on P A (add_mset m M) (add_mset m N)"
ide_cod: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.ide (hom'.cod f)"
is_discrete: fixes f :: "(bool, unit) J.arr" shows "J.ide f = J.arr f"
dvdE: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "dvd.dvd times b a" obtains k :: "'c" where "a = times b k"
that: fixes r :: "'a list" assumes "u \<cdot> r = r \<cdot> v" shows "thesis"
is_ground_cls_list_def: fixes CC :: "'a literal multiset list" shows "is_ground_cls_list CC = Ball (set CC) is_ground_cls"
R'_walk_not_Nil: shows "R'_walk \<noteq> []"
subsetD: fixes A :: "'k set"   and B :: "'k set"   and c :: "'k" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
subtype_refl: fixes P :: "'b prog"   and t :: "ty" shows "subtype P t t"
Un_insert_right: fixes A :: "'c set"   and a :: "'c"   and B :: "'c set" shows "A \<union> Set.insert a B = Set.insert a (A \<union> B)"
pred_envirD: fixes p :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" assumes "pred_envir p f" shows "p (f x)"
AT_def: fixes s :: "('h, 'g, 'i, 'j, 'k) system_state"   and p :: "'i" shows "AT s p = atCs (cPGM (GST s p))"
times_fun_var2: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f \<oplus> g = (\<lambda>x. \<Squnion>{f y \<oplus> g z |y z. D y z \<and> x = y \<oplus> z})"
HT_conseq: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT \<pi> P c Q"   and "\<And>s. P' s \<Longrightarrow> P s"   and "\<And>s\<^sub>0 s. \<lbrakk>P s\<^sub>0; P' s\<^sub>0; Q s\<^sub>0 s\<rbrakk> \<Longrightarrow> Q' s\<^sub>0 s" shows "HT \<pi> P' c Q'"
list_update_beyond: fixes xs :: "'a list"   and i :: "nat"   and x :: "'a" assumes "length xs \<le> i" shows "xs[i := x] = xs"
in_mono: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "(A::'b set) \<subseteq> (B::'b set)" shows "(x::'b) \<in> (A::'b set) \<longrightarrow> x \<in> (B::'b set)"
edge_succ_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(edge_succ HM x = edge_succ HM y) = (x = y)"
dvd_mult_unit_iff: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "is_unit b" shows "(a dvd c * b) = (a dvd c)"
cases: fixes a :: "('exp, 'id) MWLfCom list" assumes "\<turnstile>\<^bsub>\<V>\<^esub> a" obtains V :: "('exp, 'id) MWLfCom list" where "a = V" and "\<forall>i<length V. \<turnstile>\<^bsub>\<C>\<^esub> V ! i"
flow_unlift: fixes f :: "'v node \<times> 'v node \<Rightarrow> ennreal" assumes "flow \<Delta>' f" shows "flow \<Delta> (\<lambda>(x, y). f (Inner x, Inner y))"
cong_diff_iff_cong_0_nat: fixes b :: "nat"   and a :: "nat"   and m :: "nat" assumes "b \<le> a" shows "[a - b = 0] (mod m) = [a = b] (mod m)"
is_pot_ord_def: shows "is_pot_ord = (\<forall>u v. component_of_term u < component_of_term v \<longrightarrow> u \<prec>\<^sub>t v)"
atomicity: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.atomic_boolean_algebra minus uminus inf less_eq less sup bot top"   and "x \<noteq> bot" shows "\<exists>y. bounded_lattice.atom less bot y \<and> less_eq y x"
max_x'_def: fixes y :: "nat list"   and i :: "nat" shows "max_x' y i = (if i < m \<and> Di' i y \<noteq> {} then Min (Di' i y) else Max (set b))"
iadd_le_enat_iff: fixes x :: "enat"   and y :: "enat"   and n :: "nat" shows "(x + y \<le> enat n) = (\<exists>y' x'. x = enat x' \<and> y = enat y' \<and> x' + y' \<le> n)"
degree_prod_list_eq: fixes ps :: "'b poly list" assumes "\<And>p. p \<in> set ps \<Longrightarrow> p \<noteq> 0" shows "degree (prod_list ps) = sum_list (map degree ps)"
dom_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "C.dom (src\<^sub>C \<mu>) = src\<^sub>C \<mu>"
simps_6: fixes cl :: "('a, 'b, 'c) cmd list" shows "noWhile (Par cl) = Ball (set cl) noWhile"
ext_inject: fixes rel :: "('c \<times> 'c) set"   and more :: "'d"   and rel' :: "('c \<times> 'c) set"   and more' :: "'d" shows "(\<lparr>rel = rel, \<dots> = more\<rparr> = \<lparr>rel = rel', \<dots> = more'\<rparr>) = (rel = rel' \<and> more = more')"
leadsTo_Trans: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set"   and C :: "'d set" assumes "F \<in> A \<longmapsto> B"   and "F \<in> B \<longmapsto> C" shows "F \<in> A \<longmapsto> C"
conucleus_def: fixes f :: "'b \<Rightarrow> 'b" shows "conucleus f = (coclop f \<and> (\<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y)))"
head_of_item_\<beta>_is_next_symbol: fixes x :: "item"   and t :: "symbol"   and \<delta> :: "symbol list" assumes "wellformed_item x"   and "item_\<beta> x = t # \<delta>" shows "next_symbol x = Some t"
MtransC_Refl: fixes cf :: "('test, 'atom) com \<times> 'state" shows "cf \<rightarrow>*c cf"
card_eq: shows "card {a. a < m \<and> a \<in> f ` {0..<n}} = n"
commute: fixes a :: "'g multiset"   and b :: "'g multiset" shows "a \<union># b = b \<union># a"
semicontract: fixes x :: "'a"   and y :: "'a" shows "dist (T x) (T y) \<le> dist x y"
last_message_append: fixes xs :: "'d message_af list"   and ys :: "'d message_af list" shows "last_message (xs @ ys) = (if last_message ys = NoMsg then last_message xs else last_message ys)"
real_of_float_eq: fixes f1 :: "Float.float"   and f2 :: "Float.float" shows "(f1 = f2) = (real_of_float f1 = real_of_float f2)"
random_serial_dictatorship_nonempty: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite agents"   and "agents \<noteq> {}" shows "random_serial_dictatorship agents alts R = pmf_of_set agents \<bind> (\<lambda>i. random_serial_dictatorship (agents - {i}) (Max_wrt_among (R i) alts) R)"
mircfrqe_def: fixes p :: "fm" shows "mircfrqe p = qelim (prep p) mircfr"
lc_times: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "lcf (p * q) = lcf p * lcf q"
sum_case_Inl: fixes f :: "V \<Rightarrow> 'a"   and g :: "V \<Rightarrow> 'a"   and x :: "V" shows "sum_case f g (ZFC_Cardinals.Inl x) = f x"
bet_suma__sams: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Bet G H I" shows "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F"
split_beta: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and prod :: "'h \<times> 'i" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
empty_dilated_prefix': fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and c :: "clock" assumes "dilating f sub r" shows "{i. f 0 \<le> i \<and> i \<le> f n \<and> hamlet (Rep_run r i c)} = {i. i \<le> f n \<and> hamlet (Rep_run r i c)}"
mem_antichain_nonempty: fixes s :: "'a"   and A :: "'a antichain" assumes "s \<in>\<^sub>A A" shows "A \<noteq> {}\<^sub>A"
mon1_closed: shows "monom P \<one> 1 \<in> carrier P"
hseqE: fixes \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.hseq \<nu> \<mu>" obtains "S.arr \<mu>" and "S.arr \<nu>" and "S.src \<nu> = S.trg \<mu>"
be_def: shows "be = real_of_float (upper bivl - lower bivl) / 2"
shift_inj: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I1 :: "nat set"   and I2 :: "nat set" shows "(local.shift cl n I1 = local.shift cl n I2) = (I1 = I2)"
ltl_step_alt: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and t :: "String.literal \<times> value list" shows "ltl_step e (Some s) r t = (let possibilities = possible_steps e s r (fst t) (snd t) in if possibilities = {||} then (None, [], r) else let (s', t') = SOME x. x |\<in>| possibilities in (Some s', evaluate_outputs t' (snd t) r, evaluate_updates t' (snd t) r))"
fold_map: fixes g :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "fold g (map f xs) = fold (g \<circ> f) xs"
rsp: shows "eq_onp overrider (\<lambda>x y. y) (\<lambda>x y. y)"
r_min_alt_def: fixes t :: "('b, unit) RBT.rbt" shows "r_min t = fold1_keys min t"
that: fixes zs' :: "'a llist" assumes "zs = lappend xs zs'" shows "thesis"
surj_to_el: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "surj_to f A B" shows "\<forall>b\<in>B. \<exists>a\<in>A. f a = b"
iterable_set: shows "set_iteratei \<alpha> invar iteratei"
is_class_xcpt: fixes C :: "char list"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list" assumes "C \<in> sys_xcpts"   and "wf_prog wf_md P" shows "is_class P C"
cs': shows "\<Phi> \<turnstile> t:(xcp, h', frs) \<surd>"
FEI: fixes y :: "('a, 'b) BsSkewElem" assumes "y \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map bsmapt q))" shows "elem_invar y"
nth_nodes: fixes i :: "nat"   and xs :: "'b list"   and ls :: "'b tree list"   and rs :: "'b tree list" assumes "i < length xs" shows "nodes ls xs rs ! i = \<langle>if i < length ls then ls ! i else \<langle>\<rangle>, xs ! i, if i < length rs then rs ! i else \<langle>\<rangle>\<rangle>"
image_zmset_single: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "Auxiliary.image_zmset f {#x#}\<^sub>z = {#f x#}\<^sub>z"
lemma2: fixes r :: "nat \<Rightarrow> nat \<Rightarrow> b"   and p :: "nat \<times> nat list \<times> nat"   and j :: "nat"   and k :: "nat" assumes "is_path r p 0 j k" shows "r j k = T"
emeasure_lborel_box_eq: fixes l :: "'c"   and u :: "'c" shows "emeasure lborel (box l u) = ennreal (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then prod ((\<bullet>) (u - l)) Basis else 0)"
2: shows "set (map snd (dbproj i D)) \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I> \<subseteq> set (map snd D) \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>"
roofedD: fixes x :: "'k"   and G :: "('k, 'l) graph_scheme"   and B :: "'k set"   and S :: "'k set"   and p :: "'k list"   and y :: "'k" assumes "x \<in> roofed_gen G B S"   and "path G x p y"   and "y \<in> B" shows "(\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S"
vector_N_pp: fixes x :: "'a" assumes "vector x" shows "N x = - x '"
IH_2: fixes \<rho> :: "Vars.var \<Rightarrow> Value" shows "eq (fst (a, Aheap \<Gamma> e\<cdot>a)) (\<lbrakk> \<T>\<^bsub>fst (a, Aheap \<Gamma> e\<cdot>a)\<^esub> e \<rbrakk>\<^bsub>\<rho>\<^esub>) (\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)"
zero_less_binomial: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "0 < n choose k"
hyps: fixes ps :: "'a list" assumes "\<not> suffix ps ys" shows "P ps ys"
simps_1: fixes n :: "'a" shows "parent_node (CFG_node n) = n"
matrix_vector_mult_def: fixes m :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and x :: "('a, 'c) Finite_Cartesian_Product.vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. m $h i $h j * x $h j"
br_table_length: fixes "is" :: "nat list"   and c :: "i32"   and i :: "nat" assumes "length is \<le> nat_of_int c" shows "\<lparr>[$C ConstInt32 c, $Br_table is i]\<rparr> \<leadsto> \<lparr>[$Br i]\<rparr>"
defloat_float_zerosign_round_finite: fixes x :: "real"   and s :: "nat" assumes "\<bar>x\<bar> < threshold TYPE(('a, 'b) IEEE.float)" shows "is_finite (zerosign s (IEEE.round To_nearest x))"
path_connected_convex_diff_countable: fixes U :: "'b set"   and S :: "'b set" assumes "convex U"   and "\<not> collinear U"   and "countable S" shows "path_connected (U - S)"
snd_INF: fixes f :: "'e \<Rightarrow> 'd \<times> 'c"   and A :: "'e set" shows "snd (Inf (f ` A)) = (INF x\<in>A. snd (f x))"
subadditive_def: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> 'c" shows "subadditive M f = (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> f (x \<union> y) \<le> f x + f y)"
Rep_inverse: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'b" assumes "type_definition Rep Abs A" shows "Abs (Rep x) = x"
oa_contingent_4: fixes v :: "i" shows "[Contingent O! in v]"
cross3_simps_16: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a = c - b) = (a + b = c)"
atMost_nat_numeral: fixes k :: "num" shows "{..numeral k} = insert (numeral k) {..pred_numeral k}"
sums_divide: fixes f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b" assumes "f sums a" shows "(\<lambda>n. f n / c) sums (a / c)"
Q_impl_struct: fixes t\<^sub>1 :: "value"   and t\<^sub>2 :: "value" assumes "Q t\<^sub>1 t\<^sub>2" shows "veq_structure t\<^sub>1 t\<^sub>2"
p01: fixes x :: "'a" assumes "x \<in> S" shows "p x \<in> {0..1}"
less_imp_le: fixes x :: "'a"   and y :: "'a" assumes "lt x y" shows "le x y"
irc_def: fixes f :: "'a set \<Rightarrow> 'a set" shows "irc f = (\<forall>B a. a \<notin> f (insert a B) \<longrightarrow> f (insert a B) = f B)"
r_2: fixes n :: "nat" shows "real (2 * n) \<le> u (r n)"
pow_zero: fixes u :: "'b list" shows "u \<^sup>@ 0 = \<epsilon>"
Init_temp: fixes F :: "behavior \<Rightarrow> bool" shows "\<turnstile> Init F = F"
diff_fun_compose: fixes k :: "enat"   and M1 :: "('c, 'd) chart set"   and M2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'g" assumes "diff k M1 M2 f"   and "diff_fun k M2 g" shows "diff_fun k M1 (g \<circ> f)"
update_pre_def: fixes v :: "'V"   and \<sigma> :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)" shows "update_pre v \<sigma> \<equiv> let (wl, res) = \<sigma> in v \<in> V \<and> (\<forall>v'\<in>V - wl. v' \<noteq> v \<longrightarrow> (\<forall>p. is_path v0 p v' \<longrightarrow> path_weight' (res v') \<le> path_weight' (Some p))) \<and> (\<forall>v'\<in>V. \<forall>p. res v' = Some p \<longrightarrow> is_path v0 p v')"
to_fun_Fun_add: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "to_fun (x \<oplus>\<^bsub>P\<^esub> y) = to_fun x \<oplus>\<^bsub>function_ring (carrier R) R\<^esub> to_fun y"
snd_scaleR: fixes r :: "real"   and A :: "'e \<times> 'd" shows "snd (r *\<^sub>R A) = r *\<^sub>R snd A"
simps_46: fixes x51 :: "CTypeId"   and x52 :: "nat" shows "nullV \<noteq> objV x51 x52"
elims_2: fixes x :: "('a, 'b) lit" assumes "M.wtL x" obtains a :: "('a, 'b) atm" where "x = Pos a__" and "M.wtA a__"   | a :: "('a, 'b) atm" where "x = Neg a" and "M.wtA a"
assert_all_state_code_def: shows "assert_all_state_code = AssertAllState''.assert_all_state init_state assert_bound_code check_code"
that_2: assumes "\<not> is_exn \<sigma>'"   and "\<sigma>' = \<tau>"   and "r = s" shows "thesis"
chop_subst_emb: fixes s :: "('a, 'b) tm"   and \<rho> :: "'b \<Rightarrow> ('a, 'b) tm" assumes "is_App s" shows "chop (subst \<rho> s) \<unrhd>\<^sub>e\<^sub>m\<^sub>b subst \<rho> (chop s)"
sum: fixes g :: "'b \<Rightarrow> real"   and A :: "'b set"   and x :: "'a" shows "sum g A *\<^sub>R x = (\<Sum>xa\<in>A. g xa *\<^sub>R x)"
StrongEq_L_subst2_rev: fixes \<tau> :: "'a state \<times> 'a state"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and P :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> y \<triangleq> x"   and "cp P"   and "\<tau> \<Turnstile> P x" shows "\<tau> \<Turnstile> P y"
d_gt_0: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real" assumes "circle i j d" shows "0 < d"
entire_continuation_imp_abs_conv_abscissa_MInfty: fixes f :: "complex fds"   and c :: "real"   and g :: "complex \<Rightarrow> complex" assumes "nonneg_dirichlet_series f"   and "abs_conv_abscissa f \<le> ereal c"   and "\<And>s. c < Re s \<Longrightarrow> g s = eval_fds f s"   and "g holomorphic_on UNIV" shows "abs_conv_abscissa f = - \<infinity>"
prems: shows "p \<noteq> 0 \<and> count_roots_below p a = (if poly p a = 0 then 1 else 0)"