hcju/isabelleps huggingface dataset:
preserves_seq: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and g :: "'a"   and f :: "'a" assumes "functor A B F"   and "partial_magma.arr A (A g f)" shows "partial_magma.arr B (B (F g) (F f))"
all_frees_app: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and wellformed :: "'b \<Rightarrow> bool"   and all_frees :: "'b \<Rightarrow> name fset"   and u\<^sub>1 :: "'b"   and u\<^sub>2 :: "'b" assumes "class.pre_strong_term size frees subst consts app unapp const unconst free unfree abs_pred wellformed all_frees" shows "all_frees (app u\<^sub>1 u\<^sub>2) = all_frees u\<^sub>1 |\<union>| all_frees u\<^sub>2"
sigma_finite_measure_count_space_countable: fixes A :: "'d set" assumes "countable A" shows "sigma_finite_measure (count_space A)"
eval_ternaryD_1: fixes tv :: "ternaryvalue" assumes "eval_ternary_Not tv = TernaryTrue" shows "tv = TernaryFalse"
select_convs_2: fixes precondition_of :: "('a \<times> 'b) list"   and effect_of :: "('a \<times> 'b) list"   and more :: "'c" shows "effect_of \<lparr>precondition_of = precondition_of, effect_of = effect_of, \<dots> = more\<rparr> = effect_of"
subgroup_of_subset: fixes K :: "'a set" assumes "Group.group G"   and "H \<subseteq> K"   and "subgroup K G" shows "subgroup H (G\<lparr>carrier := K\<rparr>)"
map_comp: fixes g1 :: "'k \<Rightarrow> 'h"   and g2 :: "'l \<Rightarrow> 'i"   and f1 :: "'m \<Rightarrow> 'k"   and f2 :: "'n \<Rightarrow> 'l"   and v :: "('m, 'n, 'j) gpv" shows "map_gpv g1 g2 (map_gpv f1 f2 v) = map_gpv (g1 \<circ> f1) (g2 \<circ> f2) v"
assms: fixes i :: "'a"   and j :: "'b" assumes "i \<in> CBasis"   and "j \<in> CBasis" shows "continuous_on S (\<lambda>s. g s i j)"
adds_term_triv: fixes v :: "'t"   and t :: "'a" shows "v adds\<^sub>t t \<oplus> v"
\<zeta>_def: shows "\<zeta> = Re (zeta 2)"
prems_12: shows "length as' \<le> Suc (length bs')"
mat_max_id: fixes A :: "'a mat"   and B :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<ge>\<^sub>m B"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "max\<^sub>m A B = A"
vdisjntI: fixes A :: "V"   and B :: "V" assumes "\<And>x y. \<lbrakk>x \<in>\<^sub>\<circ> A; y \<in>\<^sub>\<circ> B\<rbrakk> \<Longrightarrow> x \<noteq> y" shows "A \<inter>\<^sub>\<circ> B = []\<^sub>\<circ>"
equivalent_bicategories_def: fixes V\<^sub>D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>D :: "'b \<Rightarrow> 'b"   and src\<^sub>D :: "'b \<Rightarrow> 'b"   and trg\<^sub>D :: "'b \<Rightarrow> 'b"   and V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c" shows "equivalent_bicategories V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<equiv> \<exists>F \<Phi>. equivalence_pseudofunctor V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>"
simps_7: fixes uv :: "nat \<Rightarrow> val"   and uw :: "pdf_dist"   and ux :: "expr" shows "expr_sem_rf uv (Random uw ux) = undefined"
sinh_less_cosh_real: fixes x :: "real" shows "sinh x < cosh x"
Kab_authentic: fixes A :: "agent"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> bankerb_gets" shows "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"
abs_eq: fixes x :: "'c list"   and xa :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" assumes "eq_onp (\<lambda>xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None) x x" shows "DList_Set.fold xa (Abs_dlist x) = List.fold xa x"
P_mod_add: fixes I :: "'a set"   and p :: "'a"   and q :: "'a" assumes "ideal S I"   and "p \<in> carrier R"   and "q \<in> carrier R"   and "P_mod R S X I p"   and "P_mod R S X I q" shows "P_mod R S X I (p \<plusminus> q)"
matches_add_match_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
Dflow_def: fixes x0 :: "'a" shows "Dflow x0 = mvar.flow x0 0 1\<^sub>L"
ffact_eq_fact_mult_binomial: fixes k :: "nat"   and n :: "nat" shows "ffact k n = fact k * (n choose k)"
group_int_pow_integer_group: fixes x :: "int"   and n :: "int" shows "x [^]\<^bsub>integer_group\<^esub> n = n * x"
shEx_pbm_left: fixes P :: "'h \<Rightarrow> ('d, 'f) urel"   and M :: "(('d, 'f, 'g) mrg, 'e) urel"   and Q :: "('d, 'g) urel" shows "ushEx P \<parallel>\<^bsub>M\<^esub> Q = (\<^bold>\<exists> x \<bullet> P x \<parallel>\<^bsub>M\<^esub> Q)"
config'_snoc: fixes step :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and A :: "('a \<Rightarrow> 'b) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b)"   and s :: "'a \<times> 'b"   and rs :: "'c list"   and r :: "'c" shows "On_Off.config' step A s (rs @ [r]) = On_Off.Step step A (On_Off.config' step A s rs) r"
substitutes_fvs: fixes A :: "'a trm"   and x :: "'a"   and M :: "'a trm"   and A' :: "'a trm" assumes "substitutes A x M A'" shows "fvs A' \<subseteq> fvs A - {x} \<union> fvs M"
has_densityD_2: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "has_density M N f" shows "M = density N f"
assms_2: fixes \<sigma> :: "nat \<Rightarrow> ('a, 'b, 'c, 'd, 'e) system_state"   and i :: "nat" assumes "prerun sys \<sigma>" shows "P (\<sigma> i)"
eqInt: fixes S :: "'a set"   and P :: "'a \<Rightarrow> bool"   and T :: "'a set"   and Q :: "'a \<Rightarrow> bool" shows "S \<inter> Collect P \<inter> (T \<inter> Collect Q) = S \<inter> T \<inter> (Collect P \<inter> Collect Q)"
induced_pow_fun_surj: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "f \<turnstile> Pow A = Pow (f ` A)"
2: shows "(\<exists>\<alpha>. finished_item \<alpha> \<in> \<II>) = Bex \<II> is_finished"
numeral_neq_neg_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and m :: "num"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "numeral.numeral one plus m \<noteq> uminus (numeral.numeral one plus n)"
dvd_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b"   and "b dvd c" shows "a dvd c"
eqfelem_imp_iff: fixes x :: "'b"   and y :: "'b"   and A :: "'b fset" assumes "x = y" shows "(x |\<in>| A) = (y |\<in>| A)"
not_eq_extremum: fixes a :: "'c" shows "(a \<noteq> bot) = (bot < a)"
c3: fixes i :: "nat"   and x :: "real" assumes "i < k"   and "x\<^sub>1 \<le> x" shows "c3 \<le> c1 * ((1 - bs ! i) / 2) / max 1 ((bs ! i + (hs ! i) x / x) powr (p + 1))"
curryI: fixes f :: "'e \<times> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f" assumes "f (a, b)" shows "curry f a b"
bitlen_eq_zero_iff: fixes x :: "int" shows "(bitlen x = 0) = (x \<le> 0)"
simps: fixes a1 :: "'c \<Rightarrow> 'd"   and a2 :: "('c, 'd) label"   and a3 :: "'c \<Rightarrow> 'd" shows "ce a1 a2 a3 = ((\<exists>\<sigma>. a1 = \<sigma> \<and> a2 = Skip \<and> a3 = \<sigma>) \<or> (\<exists>e \<sigma>. a1 = \<sigma> \<and> a2 = Assume e \<and> a3 = \<sigma> \<and> e \<sigma>) \<or> (\<exists>\<sigma> v e. a1 = \<sigma> \<and> a2 = Assign v e \<and> a3 = \<sigma>(v := e \<sigma>)))"
assms: shows "\<not> Option.is_none (devBC t n nid n')"
phis'_finite: fixes g :: "('a, 'b, 'c) gen_cfg_wf" shows "finite (dom (gen_wf.phis' g))"
factorization_m_smult: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset"   and d :: "int" assumes "factorization_m f (c, fs)" shows "factorization_m (Polynomial.smult d f) (c * d, fs)"
simps_3: fixes c :: "int"   and s :: "MIR.num"   and t :: "MIR.num" shows "numgcdh (CF c s t) = (\<lambda>g. gcd c (numgcdh t g))"
det30_minus_scaled2: fixes a :: "real \<times> real"   and x :: "real"   and b :: "real \<times> real" shows "coll 0 (a - x *\<^sub>R a) b = (if x = 1 then True else coll 0 a b)"
outs_gpv_def: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" shows "outs_gpv \<I> gpv \<equiv> {x. Generative_Probabilistic_Value.outsp_gpv \<I> x gpv}"
bisim1: shows "P,e,h \<turnstile> (e, xs) \<leftrightarrow> ([], xs, 0, None)"
ileq_interleave_alt: fixes l' :: "'b list"   and l :: "'b list" shows "l' \<preceq> l \<equiv> \<exists>lb. l \<in> lb \<otimes> l'"
restrict_untouched_id: fixes x :: "'a"   and t :: "'a ifex"   and val :: "bool" assumes "x \<notin> ifex_var_set t" shows "restrict t x val = t"
UniqueAux: fixes v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> \<o>"   and \<alpha> :: "\<nu>" assumes "[\<^bold>\<A>\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= \<alpha>) in v]" shows "[\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<equiv> z \<^bold>= \<alpha> in v]"
l15: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> y) = - - x \<squnion> - - y"
set_shadow_root_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_shadow_root_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
sup_left_idem: fixes x :: "'b"   and y :: "'b" shows "x + (x + y) = x + y"
assms: fixes i :: "nat" assumes "i < length ts" shows "0 < f i"
min_nat: fixes a :: "nat"   and b :: "nat" shows "min a (a + b) = a"
SAT_h': shows "B \<Gamma> \<inter> \<^bold>V \<subseteq> SAT \<Gamma> h'"
a_expr: fixes m :: "nat" shows "a k m = (if gcd m k = 1 then 1 else 0)"
Pi_def: fixes A :: "'h set"   and B :: "'h \<Rightarrow> 'i set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
transversal_segment_def: fixes a :: "'a"   and b :: "'a" shows "rev.transversal_segment a b = (a \<noteq> b \<and> {a--b} \<subseteq> X \<and> (\<forall>z\<in>{a--b}. (- f) z \<bullet> rev.rot (a - b) \<noteq> 0))"
list_all2_eq: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs = ys) = list_all2 (=) xs ys"
left_move_to_left: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq (plus (uminus c) a) b" shows "less_eq a (plus c b)"
C: fixes self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "\<lambda>_. self \<tau>.boss@pre.contents@pre()->including\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. self \<tau>.salary@pre) \<tau> = self.boss@pre.contents@pre()->including\<^sub>S\<^sub>e\<^sub>t(self.salary@pre) \<tau>"
inf_absorb2: fixes y :: "'a upred"   and x :: "'a upred" assumes "x \<sqsubseteq> y" shows "(x \<and> y) = y"
assms_2: fixes c1' :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>c (c1', s')"   and "c' = c1' ;; c2" shows "phi"
assms: shows "\<Gamma>' \<turnstile> t is s over \<Gamma>"
Class_widen_Class: fixes P :: "'a prog"   and C :: "char list"   and D :: "char list" shows "P \<turnstile> Class C \<le> Class D = P \<turnstile> C \<preceq>\<^sup>* D"
K4_imp_K2: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "\<exists>Ta. Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace> \<in> set evs"
distinct_57: fixes x3 :: "nonce_t"   and x11 :: "tag" shows "cNonce x3 \<noteq> cTag x11"
sporadic_on_e2: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<tau> :: "'b tag_const"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>"
locally_faithful_iff_faithful: shows "UP.locally_faithful = faithful_functor (\<cdot>\<^sub>B) (\<cdot>) UP"
simps_980: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x261 :: "t"   and x262 :: "binop_i" shows "(case Binop_i x261 x262 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f26 x261 x262"
d_exp_ax: fixes x :: "'a"   and y :: "'a" shows "d (d x \<cdot> y) = d x \<cdot> d y"
complete_exhaustive': fixes j :: "int"   and k :: "int"   and f :: "int \<Rightarrow> (bool \<times> term list) option" shows "(\<exists>i\<ge>j. i \<le> k \<and> is_some (f i)) = is_some (Quickcheck_Exhaustive.exhaustive_int' f k j)"
neither_endpoint: fixes I :: "'b set"   and a :: "'b"   and b :: "'b"   and x :: "'b" assumes "Q I a b"   and "x \<in> I"   and "x \<noteq> a \<and> x \<noteq> b" shows "P x I"
reaches_def: fixes x :: "'g"   and y :: "'g" shows "x reaches y = (\<lambda>s. (\<lambda>x y. (x points_to y) s)\<^sup>*\<^sup>* x y)"
assms_1: shows "negligible {x \<in> S - T. f x \<noteq> (0::'b)}"
Seq_succ_iff: fixes s :: "hf"   and k :: "hf" shows "Seq s (succ k) = (Seq s k \<and> (\<exists>y. \<langle>k, y\<rangle> \<^bold>\<in> s))"
fml_sem_not: fixes I :: "interp"   and \<phi> :: "fml" shows "fml_sem I (! \<phi>) = - fml_sem I \<phi>"
vimage_sets_compl_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and M :: "'c measure" shows "(f -` A \<inter> space M \<in> sets M) = (f -` (- A) \<inter> space M \<in> sets M)"
of_nat_simps_1: shows "of_nat 0 = (0::'d)"
divide_le_0_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide one a) zero = less_eq a zero"
AC1: shows "ac_Zp x 1 * p ^ m = x (m + 1)"
kernel_is_csubspace: fixes f :: "'d \<Rightarrow> 'e" assumes "clinear f" shows "csubspace (f -` {0::'e})"
of_nat_ge_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat" assumes "class.ordered_semiring_1 one times plus zero less_eq less" shows "less_eq zero (semiring_1.of_nat one plus zero n)"
mult_cancel2_gr0: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < k" shows "(m * k = n * k) = (m = n)"
subsetI: fixes A :: "'g set"   and B :: "'g set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
inj_on_strict_subset: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'c set" assumes "inj_on f B"   and "A \<subset> B" shows "f ` A \<subset> f ` B"
nth_append_drop_is_nth_conv: fixes j :: "nat"   and i :: "nat"   and xs :: "'c list"   and y :: "'c" assumes "j < i"   and "j \<le> length xs"   and "i \<le> length xs" shows "(take j xs @ y # drop (Suc j) xs) ! i = xs ! i"
llast_singleton: fixes x :: "'b" shows "Coinductive_List.llast (x ## LNil) = x"
M1: fixes s :: "'a" assumes "s \<in> M" shows "s \<in> S \<union> N"
ordinal_div_greater: fixes x :: "ordinal"   and y :: "ordinal" assumes "x < y" shows "x div y = 0"
single_eq_single: fixes a :: "'d"   and b :: "'d" shows "({#a#} = {#b#}) = (a = b)"
am_d_def: fixes x :: "'a" shows "antidomain_left_monoid.am_d antirange_op x = ar ar x"
Const_def: shows "Const \<equiv> map_fun id MPoly Const\<^sub>0"
simps_1: fixes M :: "('fun, 'var) Term.term set" shows "\<lbrakk>M; []\<rbrakk>\<^sub>d = (\<lambda>\<I>. True)"
knows1s: fixes A :: "specID"   and m :: "Secrets" shows "know A (sKS m) = knows A [sE m]"
conforms_to_another_strategy: fixes \<sigma>' :: "'a \<Rightarrow> 'a" assumes "path_conforms_with_strategy p P' \<sigma>'" shows "vmc_path G P' (P $ n) p \<sigma>'"
assms_2: shows "length xs = card {a \<in> A. a < length xs + length ys}"
loc_upb_lm: fixes p :: "nat \<times> nat" shows "c_assoc_have_key (pr_gr (loc_upb (fst p) (snd p))) (c_pair (fst p) (snd p)) = 0"
conjl1_sup: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<lhd> x \<squnion> y = (z \<lhd> x) \<squnion> (z \<lhd> y)"
simps_5: fixes t :: "'c local.term" shows "Arr \<^bold>\<l>\<^bold>[t\<^bold>] = Arr t"
synthetic_div_correct': fixes c :: "'a"   and p :: "'a poly" shows "[:- c, 1::'a:] * synthetic_div p c + [:poly p c:] = p"
Wbis_matchC_M_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>w t" shows "matchC_M Wbis t s"
REV_PLUS: fixes xs :: "'c rexp list" shows "REV (PLUS xs) = PLUS (map REV xs)"
ltl_to_rabin_correct: fixes \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>" shows "w \<Turnstile> F G \<phi> = (\<exists>\<G>\<subseteq>\<^bold>G (G \<phi>). G \<phi> \<in> \<G> \<and> (\<forall>\<psi>. G \<psi> \<in> \<G> \<longrightarrow> accept\<^sub>R (ltl_to_rabin \<Sigma> \<psi> \<G>) w))"
case4: assumes "x \<le> 0"   and "y \<le> 0" shows "- 4611686018427387904 \<le> x * y"
all_nos_inc: fixes i :: "nat"   and S :: "(nat \<Rightarrow> state) \<Rightarrow> msg seq_action \<Rightarrow> bool" shows "optoy i \<Turnstile>\<^sub>A (otherwith nos_inc {i} S, other nos_inc {i} \<rightarrow>) onll \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>((\<sigma>, uu_), a, \<sigma>', uu_). \<forall>j. nos_inc (\<sigma> j) (\<sigma>' j))"
validCE_cont: fixes cid :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and t' :: "nat \<Rightarrow> 'cmp"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i. \<parallel>cid\<parallel>\<^bsub>t i\<^esub>"   and "\<not> (\<exists>i'\<ge>n. \<parallel>cid\<parallel>\<^bsub>t i'\<^esub>)"   and "eval cid t t' n \<gamma>" shows "\<gamma> (lnth (\<pi>\<^bsub>cid\<^esub>inf_llist t @\<^sub>l inf_llist t')) (\<^bsub>cid\<^esub>\<down>\<^bsub>t\<^esub>n)"
chain_lprefix_lSup: fixes A :: "'b llist set"   and xs :: "'b llist" assumes "Complete_Partial_Order.chain lprefix A"   and "xs \<in> A" shows "lprefix xs (lSup A)"
CSP4_def: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "CSP4 P \<equiv> P ;; action.relation_of Skip"
simps_1: fixes \<phi> :: "Formula.formula" shows "convert_multiway (formula.Neg \<phi>) = formula.Neg (convert_multiway \<phi>)"
select_convs_1: fixes om :: "'d \<Rightarrow> 'e"   and am :: "'f \<Rightarrow> 'g"   and more :: "'h" shows "\<lparr>om = om, am = am, \<dots> = more\<rparr> \<^bsub>\<o>\<^esub> = om"
Init''_def: shows "Init'' \<equiv> {(p, t, lnk, lbl, mrk). lnk = link0 \<and> lbl = label0}"
list_all2_appendI: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g list"   and b :: "'h list"   and c :: "'g list"   and d :: "'h list" assumes "list_all2 P a b"   and "list_all2 P c d" shows "list_all2 P (a @ c) (b @ d)"
fmeasurableI_null_sets: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> null_sets M" shows "A \<in> fmeasurable M"
generate_inv_eq: fixes a :: "'a" assumes "a \<in> carrier G" shows "generate G {a} = generate G {inv a}"
finite_bot_prod: shows "bot = \<Sqinter>\<^sub>f\<^sub>i\<^sub>n UNIV"
oo: fixes T :: "'a set" assumes "openin (top_of_set S) T" shows "openin (top_of_set (f ` S)) (f ` T)"
map_of_entries: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "map_of (RBT_Impl.entries t) = ord.rbt_lookup less t"