hcju/isabelleps huggingface dataset:
intro: fixes is_node :: "'a \<Rightarrow> bool"   and is_alpha :: "'b \<Rightarrow> bool"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'a" assumes "\<And>q a. \<lbrakk>is_node q; is_alpha a\<rbrakk> \<Longrightarrow> is_node (trans q a)" shows "Automaton trans is_node is_alpha"
idempotent_bot_closed: shows "bot \<sqinter> bot = bot"
absolutely_integrable_componentwise: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) absolutely_integrable_on A" shows "f absolutely_integrable_on A"
Cons: fixes xs :: "'a list"   and x :: "'a" assumes "\<forall>y\<in>set xs. x \<le> y"   and "Specialisation_Examples.sorted xs" shows "Specialisation_Examples.sorted (x \<Colon> xs)"
intros_1: fixes c :: "i32"   and iop :: "unop_i" shows "\<lparr>[$C ConstInt32 c, $Unop_i T_i32 iop]\<rparr> \<leadsto> \<lparr>[$C ConstInt32 (app_unop_i iop c)]\<rparr>"
map_transfer: fixes R1b :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S1d :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R2b :: "'h \<Rightarrow> 'l \<Rightarrow> bool"   and S2d :: "'i \<Rightarrow> 'm \<Rightarrow> bool"   and R3b :: "'j \<Rightarrow> 'n \<Rightarrow> bool"   and S3d :: "'k \<Rightarrow> 'o \<Rightarrow> bool" shows "rel_fun (rel_fun R1b S1d) (rel_fun (rel_fun R2b S2d) (rel_fun (rel_fun R3b S3d) (rel_fun (F2rel R1b R2b R3b) (F2rel S1d S2d S3d)))) F2map F2map"
termdiffs_sums_strong: fixes K :: "real"   and c :: "nat \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and f' :: "'b"   and z :: "'b" assumes "\<And>z. norm z < K \<Longrightarrow> (\<lambda>n. c n * z ^ n) sums f z"   and "(f has_field_derivative f') (at z)"   and "norm z < K" shows "(\<lambda>n. diffs c n * z ^ n) sums f'"
assms_8: shows "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"
unlessI: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A - B co A \<union> B" shows "F \<in> A unless B"
word_mult_rbl: fixes v :: "'b word"   and vbl :: "bool list"   and w :: "'b word"   and wbl :: "bool list" assumes "to_bl v = vbl"   and "to_bl w = wbl" shows "to_bl (v * w) = rev (rbl_mult (rev vbl) (rev wbl))"
routing_ipassmt: fixes rt :: "'b routing_rule list"   and p :: "('b, 'c) simple_packet_scheme" assumes "valid_prefixes rt"   and "output_iface (routing_table_semantics rt (p_dst p)) = p_oiface p" shows "\<exists>p_ips. map_of (routing_ipassmt rt) (Iface (p_oiface p)) = Some p_ips \<and> p_dst p \<in> ipcidr_union_set (set p_ips)"
homeomorphism_restrict: fixes a :: "'a set"   and b :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'a"   and s :: "'a set" assumes "homeomorphism a b f f'" shows "homeomorphism (a \<inter> s) (b \<inter> f' -` s) f f'"
that: fixes b :: "complex" assumes "ball b r \<subseteq> f ` S" shows "thesis"
mC_C_mC_ZOC: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c :: "('d, 'c, 'a) cmd"   and d :: "('d, 'c, 'a) cmd"   and s :: "'b"   and t :: "'b"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" assumes "PL_Indis cval indis"   and "PL_Indis.mC_C aval tval cval indis theta c d s t P F" shows "PL_Indis.mC_ZOC aval tval cval indis theta c d s t {} (P \<union> {{}}) (\<lambda>I. if I \<in> P then F I else {})"
elims_3: fixes x :: "nat"   and xa :: "'b btree" assumes "\<not> order x xa" obtains k :: "nat" and ts :: "('b btree \<times> 'b) list" and t :: "'b btree" where "x = k" and "xa = Node ts t" and "\<not> (k \<le> length ts \<and> length ts \<le> 2 * k \<and> Ball (set (subtrees ts)) (order k) \<and> order k t)"
possible_steps_finsert: fixes s :: "nat"   and s' :: "nat"   and t :: "transition"   and e :: "((nat \<times> nat) \<times> transition) fset"   and ss :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" shows "possible_steps (finsert ((s, s'), t) e) ss r l i = (if s = ss \<and> Label t = l \<and> length i = Arity t \<and> apply_guards (Guards t) (join_ir i r) then finsert (s', t) (possible_steps e s r l i) else possible_steps e ss r l i)"
sp_spec_subj_subj_def: fixes configuration_subj_obj :: "'a \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "abstract_policy_derivation.sp_spec_subj_subj configuration_subj_obj a b \<equiv> \<exists>f m1 m2. abstract_policy_derivation.sp_spec_subj_obj configuration_subj_obj a (FILEP f) m1 \<and> abstract_policy_derivation.sp_spec_subj_obj configuration_subj_obj b (FILEP f) m2"
lset_ltake: fixes n :: "enat"   and xs :: "'b llist" shows "lset (ltake n xs) \<subseteq> lset xs"
Ana_keys_wf: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and t :: "('a, 'b) Term.term"   and K :: "('a, 'b) Term.term list"   and R :: "('a, 'b) Term.term list"   and k :: "('a, 'b) Term.term"   and f :: "'a"   and T :: "('a, 'b) Term.term list" assumes "intruder_model arity Ana"   and "Ana t = (K, R)"   and "\<And>g S. Fun g S \<sqsubseteq> t \<Longrightarrow> length S = arity g"   and "k \<in> set K"   and "Fun f T \<sqsubseteq> k" shows "length T = arity f"
sup_ge2: fixes y :: "'b"   and x :: "'b" shows "y \<le> x \<sqinter> y"
ln_upper_3_def: shows "ln_upper_3 \<equiv> \<lambda>x. (x + 5) * (x - 1) / (2 * (2 * x + 1))"
mem_size_def: fixes m :: "mem" shows "mem_size m = length (Rep_mem m)"
simps_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "monoidal_language C" shows "monoidal_language.Diagonalize C \<^bold>\<langle>f\<^bold>\<rangle> = \<^bold>\<langle>f\<^bold>\<rangle>"
n_kat_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<cdot> y = !x \<cdot> y \<cdot> !z) = \<lparr>!x\<rparr>y\<lparr>y \<cdot> !z\<rparr>"
Form_def: fixes y :: "hf" shows "Form y \<equiv> \<exists>s k. SeqForm s k y"
lcm_list_dvd_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and xs :: "'a list"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (semiring_gcd.Lcm_fin one zero lcm (set xs)) b = (\<forall>a\<in>set xs. dvd.dvd times a b)"
conc_abs_swap: fixes R :: "('a \<times> 'b) set"   and m' :: "'a nres"   and m :: "'b nres" assumes "single_valued R" shows "(m' \<le> \<Down> R m) = (\<Up> R m' \<le> m)"
exhaust: fixes y :: "ct" obtains "y = TAny"   | x2 :: "t" where "y = TSome x2"
analz_insert_freshK: fixes evs :: "event list"   and KAB :: "nat"   and K :: "nat" assumes "evs \<in> otway"   and "KAB \<notin> range shrK" shows "(Key K \<in> analz (insert (Key KAB) (knows Spy evs))) = (K = KAB \<or> Key K \<in> analz (knows Spy evs))"
prim_impl_correct_1: fixes \<pi>i :: "'m" assumes "prim_impl = \<pi>i" shows "invar_MST \<pi>i"
cod_in_codomains: fixes f :: "'a \<times> 'a" assumes "VV.codomains f \<noteq> {}" shows "VV.cod f \<in> VV.codomains f"
len_right: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "right (len v ts c) \<le> right (ext v)"
complex_Im_fact: fixes n :: "nat" shows "Im (fact n) = 0"
l2_inv4_init: shows "init l2 \<subseteq> l2_inv4"
simps_2: fixes m :: "nat"   and n :: "nat" shows "m - Suc n = (case m - n of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> k)"
the_inverseD_1: fixes g :: "'a" assumes "g \<in> G" shows "\<ii> g \<in> G"
floor_less_iff: fixes x :: "'b"   and z :: "int" shows "(\<lfloor>x\<rfloor> < z) = (x < of_int z)"
add_eq_0_iff_both_eq_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
simps: fixes p :: "real" shows "bernoulli p = coin_spmf \<bind> (\<lambda>b. if b then return_spmf (1 / 2 \<le> p) else if p < 1 / 2 then bernoulli (2 * p) else bernoulli (2 * p - 1))"
prems_1: shows "\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)"
fun_upd_smaller_oexp: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e"   and y :: "'d" assumes "wo_rel2 r s"   and "f \<in> FinFunc r s"   and "x \<in> Field s"   and "y \<in> Field r"   and "(y, f x) \<in> r" shows "(f(x := y), f) \<in> r ^o s"
of_int_mask_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_1.of_int one plus zero minus (mask n) = mask n"
assms_3: shows "v \<in> local.writes (path \<sigma> l)"
simps_1: fixes p :: "real \<times> real \<times> real" shows "substNegInfinityUni (EqUni p) = allZero' p"
simple_action_to_decision_def: fixes a :: "simple_action" shows "simple_action_to_decision a \<equiv> case a of Accept \<Rightarrow> Decision FinalAllow | Drop \<Rightarrow> Decision FinalDeny"
upper: fixes z :: "'a" assumes "z \<in> (\<squnion>) x ` Y" shows "z \<le> y"
trimono_def: fixes B :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" shows "trimono B \<equiv> monotone flatf_ge flatf_ge B \<and> mono B"
samet_imp_iso_subseteq: fixes gsopt :: "(nat, nat list list list) trie option"   and arch :: "nat list list list" assumes "\<And>gs g. \<lbrakk>gsopt = Some gs; g \<in> set_tries gs\<rbrakk> \<Longrightarrow> pre_iso_test g"   and "\<And>g. g \<in> set arch \<Longrightarrow> pre_iso_test g"   and "\<And>gs. gsopt = Some gs \<Longrightarrow> invar_trie gs"   and "samet gsopt arch" shows "\<exists>gs. gsopt = Some gs \<and> set_tries gs \<subseteq>\<^sub>\<simeq> set arch"
pw_bind_leI: fixes S :: "'c nres"   and M :: "'d nres"   and f :: "'d \<Rightarrow> 'c nres" assumes "nofail S \<Longrightarrow> nofail M"   and "\<And>x. \<lbrakk>nofail M; inres M x\<rbrakk> \<Longrightarrow> f x \<le> S" shows "M \<bind> f \<le> S"
mono_Increasing_o: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'b" assumes "mono g" shows "Increasing f \<subseteq> Increasing (g \<circ> f)"
rel_inject_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "'a"   and y2 :: "'b" shows "rel_withERR R (OK x2) (OK y2) = R x2 y2"
det_permute_rows: fixes p :: "'b \<Rightarrow> 'b"   and A :: "(('c, 'b) vec, 'b) vec" assumes "p permutes UNIV" shows "det (\<chi>i. A $ p i) = of_int (sign p) * det A"
eq_fmem_trans: fixes a :: "'a"   and b :: "'a"   and A :: "'a fset" assumes "a = b"   and "b |\<in>| A" shows "a |\<in>| A"
extf_map: fixes A :: "('s, 'v) tm set"   and ys :: "('s, 'v) tm list"   and xs :: "('s, 'v) tm list"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and f :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm"   and fa :: "'s" assumes "finite A"   and "ys \<in> lists A"   and "xs \<in> lists A"   and "\<forall>x\<in>A. \<not> gt (f x) (f x)"   and "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt (f z) (f y) \<longrightarrow> gt (f y) (f x) \<longrightarrow> gt (f z) (f x)"   and "\<forall>y\<in>A. \<forall>x\<in>A. gt y x \<longrightarrow> gt (f y) (f x)"   and "extf fa gt ys xs" shows "extf fa gt (map f ys) (map f xs)"
length_list_of_mset: fixes A :: "'b multiset" shows "length (list_of_mset A) = size A"
natural_isomorphism_\<ll>: shows "natural_isomorphism (\<cdot>) (\<cdot>) L local.map \<ll>"
Abs_multiset_inverse: fixes y :: "'a \<Rightarrow> nat" assumes "y \<in> multiset" shows "count (Abs_multiset y) = y"
convex_hull_caratheodory_aff_dim: fixes p :: "'c set" shows "convex hull p = {y. \<exists>S u. finite S \<and> S \<subseteq> p \<and> int (card S) \<le> aff_dim p + 1 \<and> (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = y}"
is_lub_thelub_ex: fixes S :: "'a set"   and x :: "'a" assumes "\<exists>u. S <<| u"   and "S <| x" shows "lub S \<sqsubseteq> x"
emeasure_pair_measure_Times: fixes A :: "'d set"   and N :: "'d measure"   and B :: "'e set"   and Xa :: "'e set"   and Aa :: "'e set set" assumes "A \<in> sets N"   and "B \<in> sets (sigma Xa Aa)" shows "emeasure (N \<Otimes>\<^sub>M sigma Xa Aa) (A \<times> B) = emeasure N A * emeasure (sigma Xa Aa) B"
simps: fixes a :: "'b literal multiset"   and \<Gamma> :: "'b inference set"   and CC :: "'b literal multiset set" shows "(a \<in> inference_system.saturate \<Gamma> CC) = ((\<exists>C. a = C \<and> C \<in> CC) \<or> (\<exists>CC' D E. a = E \<and> Infer CC' D E \<in> \<Gamma> \<and> (\<forall>x. x \<in># CC' \<longrightarrow> x \<in> inference_system.saturate \<Gamma> CC) \<and> D \<in> inference_system.saturate \<Gamma> CC))"
getMinTree_min_prio: fixes bq :: "('c, 'd) BinomialTree list"   and y :: "'c \<times> 'd" assumes "queue_invar bq"   and "y \<in># queue_to_multiset bq" shows "prio (getMinTree bq) \<le> snd y"
dghm_tdghm_comp_def: fixes \<HH> :: "V"   and \<NN> :: "V" shows "\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN> = [\<lambda>a\<in>\<^sub>\<circ>\<NN>\<lparr>NTDGDom\<rparr>\<lparr>Obj\<rparr>. \<HH>\<lparr>ArrMap\<rparr>\<lparr>\<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr>\<rparr>, \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN>\<lparr>NTDom\<rparr>, \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN>\<lparr>NTCod\<rparr>, \<NN>\<lparr>NTDGDom\<rparr>, \<HH>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
int_ins_loop_term: fixes w :: "('c, 'd option) insert_message list"   and p :: "'c extended"   and l :: "nat"   and s :: "'c extended"   and u :: "nat"   and d :: "'c extended list list"   and m :: "('c, 'e) insert_message"   and r :: "'c extended \<times> 'c extended" assumes "idx w p = Inr l"   and "idx w s = Inr u"   and "mapM (concurrent w l u) (substr w l u) = Inr d"   and "concat d \<noteq> []"   and "find ((\<lambda>x. \<lbrakk>I m\<rbrakk> < x \<or> x = s) \<circ> snd) (zip (p # concat d) (concat d @ [s])) = Some r" shows "int_ins_measure (m, w, r) < u - l"
fund_sol_minimal: fixes a :: "nat"   and b :: "nat" assumes "nontriv_solution (a, b)" shows "pell_valuation (case fund_sol of (x, y) \<Rightarrow> (int x, int y)) \<le> pell_valuation (int a, int b)"
axioms_1: fixes R :: "('e, 'f) ring_scheme" assumes "domain R" shows "cring R"
measurable_marg_dens: fixes x :: "nat"   and \<rho> :: "nat \<Rightarrow> val" assumes "x \<in> V"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "marg_dens \<Y> x \<rho> \<in> borel_measurable (stock_measure (\<Gamma> x))"
A1a: shows "\<lfloor>(\<lambda>w. \<forall>x. (\<P> (\<^bold>\<rightharpoondown>x) \<^bold>\<rightarrow> \<^bold>\<not>\<P> x) w)\<rfloor>"
Abs_coreflexive_cases: fixes x :: "'b coreflexive" obtains y :: "'b" where "x = Abs_coreflexive y" and "y \<in> \<down>(1::'b)"
leD: fixes y :: "'e"   and x :: "'e" assumes "y \<le> x" shows "\<not> x < y"
1: shows "\<guillemotleft>assoc\<mu>\<nu>\<pi>.chine : HH\<mu>\<nu>\<pi>.chine \<rightarrow>\<^sub>C H\<mu>H\<nu>\<pi>.chine\<guillemotright>"
m_impl: shows "get_impl ((m_impl, m_spec) # M) = m_impl # get_impl M"
deg_monom_le: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "deg R (monom P a n) \<le> n"
Pause: fixes r :: "('a, 'b, 'c) Resumption.resumption"   and r' :: "('a, 'b, 'c) Resumption.resumption" assumes "X r r'"   and "\<not> Resumption.resumption.is_Done r"   and "\<not> Resumption.resumption.is_Done r'" shows "Resumption.resumption.output r = Resumption.resumption.output r' \<and> ((=) ===> (\<lambda>r r'. X r r' \<or> resumption_ord r r')) (Resumption.resumption.resume r) (Resumption.resumption.resume r')"
integral_distr: fixes g :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "g \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> borel_measurable N" shows "integral\<^sup>L (distr M N g) f = LINT x|M. f (g x)"
refl: fixes x :: "'a" assumes "x \<in> P" shows "x \<sqsubseteq> x"
child_ancestors: shows "h \<turnstile> get_ancestors_si (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root) \<rightarrow>\<^sub>r child_ancestors"
wfp_on_restrict_to: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "wfp_on (restrict_to P A) A = wfp_on P A"
divide_nonneg_pos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less zero y" shows "less_eq zero (divide x y)"
height_bal_list: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "height (bal_list n xs) = nat \<lceil>log 2 (real (n + 1))\<rceil>"
hoare_sequential: fixes S :: "'a \<Rightarrow> 'b"   and p :: "'b"   and T :: "'c \<Rightarrow> 'a"   and r :: "'c" assumes "mono S" shows "\<Turnstile> p{| S \<circ> T |}r = (\<exists>q. \<Turnstile> p{| S |}q \<and> \<Turnstile> q{| T |}r)"
a: shows "x:M \<in> ANDLEFT1 (B AND C) (\<parallel>(B)\<parallel>)"
runit_char_3: fixes f :: "'a" assumes "ide f" shows "\<r>[f] \<star> src f = (f \<star> \<i>[src f]) \<cdot> \<a>[f, src f, src f]"
seqSubstSimps_2: fixes M :: "'a"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and \<sigma> :: "(name list \<times> 'a list) list" shows "M\<langle>N\<rangle>.P[<\<sigma>>] = (M[<\<sigma>>])\<langle>(N[<\<sigma>>])\<rangle>.P[<\<sigma>>]"
llist_le_trans: fixes r :: "'b llist"   and s :: "'b llist"   and t :: "'b llist" assumes "r \<le> s"   and "s \<le> t" shows "r \<le> t"
sunflower_card_subset_lift: fixes k :: "nat"   and c :: "nat"   and r :: "nat"   and F :: "'b set set" assumes "\<And>G. \<lbrakk>\<forall>A\<in>G. finite A \<and> card A = k; c < card G\<rbrakk> \<Longrightarrow> \<exists>S\<subseteq>G. sunflower S \<and> card S = r"   and "\<forall>A\<in>F. finite A \<and> card A \<le> k"   and "c < card F" shows "\<exists>S\<subseteq>F. sunflower S \<and> card S = r"
better_Cut_substn1: fixes y :: "name"   and x :: "name"   and N :: "trm"   and a :: "coname"   and b :: "coname"   and P :: "trm"   and M :: "trm" assumes "y \<sharp> (x, N)"   and "a \<sharp> (b, P)"   and "M \<noteq> Ax y a" shows "Cut <a>.M x.N{y:=<b>.P} = Cut <a>.M{y:=<b>.P} x.N"
mangoldt_nonneg: fixes d :: "nat" shows "0 \<le> mangoldt d"
fps_of_ratfps_is_unit: fixes a :: "'b ratfps" shows "(fps_of_ratfps a $ 0 \<noteq> (0::'b)) = (ratfps_nth a 0 \<noteq> (0::'b))"
iMODb_iMOD_append_union': fixes r :: "nat"   and m :: "nat"   and r' :: "nat"   and c :: "nat" assumes "r mod m = r' mod m"   and "r' \<le> r + m * Suc c" shows "[ r, mod m, c ] \<union> [ r', mod m ] = [ min r r', mod m ]"
simps: fixes a :: "'v"   and ps :: "'v \<times> 'inPort" shows "(a \<in> scope ps) = (\<exists>v. a = v \<and> v |\<in>| vertices \<and> (\<forall>x xa. path v xa x \<longrightarrow> terminal_vertex xa \<longrightarrow> ps \<in> snd ` set x))"
list_to_poly_def: fixes ts :: "'c list"   and cs :: "'d list" shows "list_to_poly ts cs = Abs_poly_mapping (list_to_fun ts cs)"
a_comm_var: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> ad y \<le> ad y \<cdot> ad x"
S: shows "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p S"
counit_simps_4: shows "src \<i>[trg \<mu>] = trg (trg \<mu>)"
simps_11: fixes v :: "'a acom"   and va :: "'a acom"   and vb :: "char list"   and vc :: "aexp"   and vd :: "'a" shows "(v;; va \<le> vb ::= vc {vd}) = False"
assms: shows "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>process_ret (add_match m rs), Undecided\<rangle> \<Rightarrow> t"
ev_2: shows "\<forall>x'\<in>{b<..<x}. 0 \<le> f x'"
optimal_tree_def: fixes F :: "('c, 'd) graph"   and G :: "('c, 'd) graph" shows "optimal_tree F G \<equiv> \<forall>F'. spanning_tree F' G \<longrightarrow> edge_weight F \<le> edge_weight F'"
Normal_pre_star: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cfg\<^sub>1 :: "('d, 'e, 'f) com \<times> ('d, 'f) xstate"   and p' :: "('d, 'e, 'f) com"   and t :: "'d" assumes "\<Gamma> \<turnstile> cfg\<^sub>1 \<rightarrow>\<^sup>* (p', Normal t)" shows "\<exists>p s. cfg\<^sub>1 = (p, Normal s)"
assms: shows "insert_ops (xs @ [(i1, r1)] @ ys @ [(i2, r2)])"
mapping_of_inject: fixes x :: "'q mpoly"   and y :: "'q mpoly" shows "(mapping_of x = mapping_of y) = (x = y)"
inj_on_finite: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "finite B" shows "finite A"
states_Accessible_dfa: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.states (dfa.Accessible_dfa M) = dfa.accessible M"
mono2_step: fixes c1 :: "(char list \<Rightarrow> int) set acom"   and c2 :: "(char list \<Rightarrow> int) set acom"   and S1 :: "(char list \<Rightarrow> int) set"   and S2 :: "(char list \<Rightarrow> int) set" assumes "c1 \<le> c2"   and "S1 \<subseteq> S2" shows "step S1 c1 \<le> step S2 c2"
widens_refl: fixes 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 xs :: "ty list" shows "P \<turnstile> xs [\<le>] xs"
HG: shows "bij_betw (inv_into (carrier G) h) (carrier H) (carrier G)"
insert_into_member_partition1: fixes elem :: "'b"   and P :: "'b set set"   and set :: "'b set" shows "\<Union> (insert_into_member elem P set) = \<Union> (insert (set \<union> {elem}) (P - {set}))"
path_append_elim: fixes A :: "('c, 'd) nba"   and r :: "('c \<times> 'd) list"   and s :: "('c \<times> 'd) list"   and p :: "'d" assumes "path A (r @ s) p" obtains "path A r p" and "path A s (target r p)"
phi_phiDefs: fixes g :: "'g"   and v :: "'val"   and vs :: "'val list" assumes "phi g v = Some vs" shows "v \<in> phiDefs g (defNode g v)"
c1: fixes e2 :: "nat"   and x :: "nat"   and c1 :: "nat" assumes "e2 = (x + c1) mod q"   and "x < q"   and "c1 < q" shows "c1 = (e2 + q - x) mod q"
has_derivative_scaleC: fixes f :: "'a \<Rightarrow> complex"   and f' :: "'a \<Rightarrow> complex"   and x :: "'a"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. f x *\<^sub>C g x) has_derivative (\<lambda>h. f x *\<^sub>C g' h + f' h *\<^sub>C g x)) (at x within s)"