hcju/isabelleps huggingface dataset:
sum_mset_replicate_mset: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and n :: "nat"   and a :: "'k" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum_mset plus zero (replicate_mset n a) = times (semiring_1.of_nat one plus zero n) a"
sphere_params: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "x * x + y * y + z * z = 1" shows "x = cos (arcsin z) * cos (atan2 y x) \<and> y = cos (arcsin z) * sin (atan2 y x) \<and> z = sin (arcsin z)"
fps_conv_radius_uminus: fixes f :: "'a fps" shows "fps_conv_radius (- f) = fps_conv_radius f"
elims: fixes x :: "nat"   and y :: "nat" assumes "fib x = y" obtains "x = 0" and "y = 0"   | "x = Suc 0" and "y = 1"   | xa :: "nat" where "x = Suc (Suc xa)" and "y = fib xa + fib (Suc xa)"
le_Inf_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<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 b :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq b (Inf A) = (\<forall>a\<in>A. less_eq b a)"
assms_5: fixes x :: "'b"   and y :: "'a" assumes "a = Inr x"   and "b = Inl y" shows "thesis"
to_bl_rotl: fixes n :: "nat"   and w :: "'b word" shows "to_bl (word_rotl n w) = rotate n (to_bl w)"
take_prefixeq_le_conv: fixes a :: "nat"   and xs :: "'b list"   and b :: "nat" assumes "a \<le> length xs" shows "prefix (xs \<down> a) (xs \<down> b) = (a \<le> b)"
simps: fixes c :: "'a \<times> (Mode \<Rightarrow> 'b set)"   and mem :: "Eg1.var \<Rightarrow> nat"   and x :: "Eg1.var"   and v :: "nat" shows "A.lc_set_var (c, mem) x v = (c, mem(x := v))"
cod_inv: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c" assumes "category C"   and "category.iso C f" shows "partial_magma.cod C (category.inv C f) = partial_magma.dom C f"
splitFace_face_face_op2: fixes g :: "graph"   and u :: "nat"   and v :: "nat"   and f :: "face"   and vs :: "nat list" assumes "minGraphProps g"   and "pre_splitFace g u v f vs" shows "face_face_op (snd (snd (splitFace g u v f vs)))"
x0'_def: shows "x0' \<equiv> einterpret x0 vs"
circline_line': fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> z2" shows "circline 0 (\<i> * (z2 - z1)) (Re (- cnj_mix (\<i> * (z2 - z1)) z1)) = line z1 z2"
smap_union_mono: fixes ve1 :: "'c \<Rightarrow> 'd set"   and ve1' :: "'c \<Rightarrow> 'd set"   and ve2 :: "'c \<Rightarrow> 'd set"   and ve2' :: "'c \<Rightarrow> 'd set" assumes "ve1 \<subseteq>. ve1'"   and "ve2 \<subseteq>. ve2'" shows "ve1 \<union>. ve2 \<subseteq>. ve1' \<union>. ve2'"
add_aform'_float_1: fixes p :: "nat"   and x :: "(real \<times> real pdevs) \<times> real"   and y :: "(real \<times> real pdevs) \<times> real"   and a :: "real"   and b :: "real pdevs"   and ba :: "real" assumes "add_aform' p x y = ((a, b), ba)" shows "a \<in> float"
m: shows "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> m"
simps_1: fixes s :: "('a, 'b) fmap" shows "exec_plan s [] = s"
delete_point_p_not_in_bl_blocks: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and p :: "'b" assumes "incidence_system point_set block_collection"   and "\<And>bl. bl \<in># block_collection \<Longrightarrow> p \<notin> bl" shows "incidence_system.del_point_blocks block_collection p = block_collection"
is_TComp_var_instance_closed_def: fixes \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('c, 'd) Term.term"   and M :: "('a, 'b) Term.term list" shows "is_TComp_var_instance_closed \<Gamma> M \<equiv> \<forall>x\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). is_Fun (\<Gamma> (Var x)) \<longrightarrow> list_ex (\<lambda>t. is_Fun t \<and> \<Gamma> t = \<Gamma> (Var x) \<and> list_all is_Var (args t) \<and> distinct (args t)) M"
exhaust_sel: fixes BinomialTree :: "('a, 'b) BinomialHeapStruc.BinomialTree" obtains "BinomialTree = BinomialHeapStruc.Node (BinomialHeapStruc.val BinomialTree) (BinomialHeapStruc.prio BinomialTree) (BinomialHeapStruc.rank BinomialTree) (BinomialHeapStruc.children BinomialTree)"
quasi_measure_preserving_def: fixes M :: "'c measure"   and N :: "'d measure" shows "quasi_measure_preserving M N = {f \<in> M \<rightarrow>\<^sub>M N. \<forall>A\<in>sets N. (f -` A \<inter> space M \<in> null_sets M) = (A \<in> null_sets N)}"
\<a>_def: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "finite X" shows "\<a> ps = (LEAST k. standard_decomp k ps)"
inf_sup_aci_3: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.lattice inf less_eq less sup" shows "inf x (inf y z) = inf y (inf x z)"
step_diff: fixes i :: "nat"   and x :: "real" assumes "i < k"   and "x\<^sub>1 \<le> x" shows "bs ! i * x + (hs ! i) x + 1 < x"
eq2: shows "lt ` set ((a # xs) @ ys) = lt ` set (xs @ trd (xs @ ys) a # ys)"
cnf_of_encode_all_operator_preconditions_structure: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_all_operator_preconditions \<Pi> (\<Pi>\<^sub>\<O>) t) = (\<Union>(t, op)\<in>{..<t} \<times> set (\<Pi>\<^sub>\<O>). \<Union>v\<in>set (precondition_of op). {{(Operator t (index (\<Pi>\<^sub>\<O>) op))\<inverse>, (State t (index (\<Pi>\<^sub>\<V>) v))\<^sup>+}})"
singular_subdivision_add: fixes p :: "nat"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_subdivision p (a + b) = singular_subdivision p a + singular_subdivision p b"
rt_strictly_fresher_def: shows "rt_strictly_fresher \<equiv> \<lambda>dip rt1 rt2. rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2 \<and> \<not> rt1 \<approx>\<^bsub>dip\<^esub> rt2"
reducecoeffh_numbound0: fixes t :: "Ferrack.num"   and g :: "int" assumes "numbound0 t" shows "numbound0 (reducecoeffh t g)"
read_only_unowned: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<O> \<inter> read_only \<S> = {}"
nat_pow_eone: fixes x :: "int vec" assumes "x \<in> carrier_vec n" shows "[1] \<cdot>\<^bsub>V\<^esub> x = x"
le_infI: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a"   and "x \<le> b" shows "x \<le> a \<cdot> b"
multi_member_last: fixes x :: "'c" shows "x \<in># {#x#}"
subst_app: fixes x :: "nat"   and v :: "exp"   and e1 :: "exp"   and e2 :: "exp" shows "subst x v (EApp e1 e2) = EApp (subst x v e1) (subst x v e2)"
assms_7: shows "occurs_on ev \<noteq> occurs_on ev'"
excess_eq1: fixes g :: "graph"   and v :: "nat" assumes "Invariants.inv g"   and "final g"   and "tame g"   and "except g v = 0"   and "v \<in> \<V> g" shows "ExcessAt g v + tri g v * \<d> 3 + quad g v * \<d> 4 = \<b> (tri g v) (quad g v)"
pth_7_2: fixes x :: "'b" shows "x + (0::'b) \<equiv> x"
last_prog_append_Ghost\<^sub>s\<^sub>b: fixes p :: "'b"   and sb :: "'b memref list"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "last_prog p (sb @ [Ghost\<^sub>s\<^sub>b A L R W]) = last_prog p sb"
integral_mult_right: fixes c :: "'d"   and M :: "'e measure"   and f :: "'e \<Rightarrow> 'd" assumes "c \<noteq> (0::'d) \<Longrightarrow> integrable M f" shows "LINT x|M. c * f x = c * integral\<^sup>L M f"
seq_extract_append: fixes A :: "nat set"   and xs :: "'b list"   and ys :: "'b list" shows "A \<upharpoonleft>\<^sub>l (xs @ ys) = A \<upharpoonleft>\<^sub>l xs @ {j. j + length xs \<in> A} \<upharpoonleft>\<^sub>l ys"
dghm_ObjMap_vrange: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
lprefix_lappend_same: fixes xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" shows "lprefix (lappend xs ys) (lappend xs zs) = (lfinite xs \<longrightarrow> lprefix ys zs)"
PPc_in_PPV: fixes c :: "('a, 'b) MWLsCom"   and V :: "('a, 'b) MWLsCom list" assumes "c \<in> set V" shows "set (PPc c) \<subseteq> set (PPV V)"
coeffs_0_eq_Nil: shows "Polynomial.coeffs 0 = []"
GroupIso_axioms_def: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and H :: "'b set" shows "GroupIso_axioms G T H \<equiv> bij_betw T G H"
pid_gc_proc_eq: fixes pid :: "nat"   and gc :: "('b, 'd, 'e) pid_global_config" assumes "pid_valid gc pid" shows "global_config.processes (pidgc_\<alpha> gc) = {#lc_of_pid gc pid#} + (global_config.processes (pidgc_\<alpha> gc) - {#lc_of_pid gc pid#})"
qty_sum_def: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" shows "qty_sum pf1 pf2 = (\<lambda>x n w. pf1 x n w + pf2 x n w)"
cd'ca: shows "c \<otimes> somegcd G a b divides c \<otimes> a"
mset_heap_empty: fixes h :: "'b hp option" shows "(mset_heap h = {#}) = (h = None)"
bound_of_greater: fixes P :: "'b \<Rightarrow> real"   and x :: "'b" assumes "Expectations.bounded P" shows "P x \<le> bound_of P"
mono: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "x \<le> y" shows "f y \<le> f x"
eq_iff: fixes a :: "'c"   and b :: "'c" shows "(a = b) = (b \<le> a \<and> a \<le> b)"
rep_eq: fixes x :: "('a, 'b) ffun"   and xa :: "'a"   and xb :: "'b" shows "pfun_of (x(xa \<mapsto> xb)\<^sub>f) = pfun_of x(xa \<mapsto> xb)\<^sub>p"
assms: shows "xs \<noteq> ys \<or> x \<noteq> y"
permutes_inv_o_2: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S" shows "inv p \<circ> p = id"
\<pi>_def_on_S: fixes u :: "'v" assumes "u \<in> S (A Q \<pi>)"   and "u \<noteq> r" shows "\<pi> u \<noteq> None"
exhaust_sel: fixes llist :: "'c llist" obtains "llist = LNil"   | "llist = LCons (un_LCons1 llist) (un_LCons2 llist)"
rep_eq: fixes x :: "'a Filters.filter"   and xa :: "'a Filters.filter" shows "Filters.filter.Rep_filter (x \<sqinter> xa) = Filters.filter.Rep_filter x \<inter> Filters.filter.Rep_filter xa"
FUren: fixes I :: "('sf, 'sc, 'sz) interp"   and \<phi> :: "('sf, 'sc, 'sz) formula"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and x :: "'sz"   and y :: "'sz" assumes "is_interp I"   and "FRadmit \<phi>"   and "fsafe \<phi>" shows "(\<nu> \<in> fml_sem I (FUrename x y \<phi>)) = (Radj x y \<nu> \<in> fml_sem I \<phi>)"
chine_simps_2: shows "C.dom HHfgh.chine = HHfgh.dom.apex"
Normal_pre: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and p' :: "('e, 'd, 'f) com"   and u :: "'e" assumes "\<Gamma>\<turnstile> (p, s) \<rightarrow> (p', Normal u)" shows "\<exists>v. s = Normal v"
advantage\<^sub>n_def: fixes \<A> :: "nat \<Rightarrow> ('b \<times> 'message \<times> 'signature, 'b \<times> (unit + 'message), 'vkey + 'signature) gpv"   and \<eta> :: "nat" shows "advantage\<^sub>n \<A> \<eta> = spmf (suf_cma\<^sub>n \<A> \<eta>) True"
kop_def: shows "op\<^sub>K = \<F> \<circ> \<smile> \<circ> \<R>"
meld_xlate: fixes q :: "('c, 'd) BsSkewBinomialTree list"   and q' :: "('c, 'd) BsSkewBinomialTree list" shows "map bsmapt (meld q q') = SkewBinomialHeapStruc.meld (map bsmapt q) (map bsmapt q')"
trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_fv_subst_subset': fixes t :: "('c, 'd) Term.term"   and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F" shows "fv (t \<cdot> \<theta>) \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>)"
cases: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c list"   and a2 :: "'c list" assumes "list_emb P a1 a2" obtains ys :: "'c list" where "a1 = []" and "a2 = ys__"   | xs :: "'c list" and ys :: "'c list" and y :: "'c" where "a1 = xs__" and "a2 = y__ # ysa__" and "list_emb P xs__ ysa__"   | x :: "'c" and y :: "'c" and xs :: "'c list" and ys :: "'c list" where "a1 = x # xs" and "a2 = y # ys" and "P x y" and "list_emb P xs ys"
fold_insert2: fixes A :: "'a set"   and x :: "'a"   and z :: "'b" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold f z (insert x A) = Finite_Set.fold f (f x z) A"
lang_lang_trad: shows "subst \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>], [\<bb>], [\<aa>, S, \<aa>], [\<bb>, S, \<bb>]|}\<rparr> {|[Inr (init \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>], [\<bb>], [\<aa>, S, \<aa>], [\<bb>, S, \<bb>]|}\<rparr>)]|} = to_language (lang_trad \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>], [\<bb>], [\<aa>, S, \<aa>], [\<bb>, S, \<bb>]|}\<rparr>)"
induced_function_integral_nonergodic_1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets MA"   and "integrable MA f" shows "integrable MA (A.induced_function A f)"
sorted01: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less"   and "length xs \<le> 1" shows "linorder.sorted less_eq xs"
quotient_of_int_div: fixes x :: "int"   and y :: "int"   and a :: "int"   and b :: "int" assumes "quotient_of (rat_of_int x / rat_of_int y) = (a, b)"   and "y \<noteq> 0" shows "\<exists>z. z \<noteq> 0 \<and> x = a * z \<and> y = b * z"
prefixes_at_0: fixes j :: "nat" shows "local.prefixes 0 j \<down>= singleton_encode j"
simps: fixes a :: "'d tree"   and S :: "'d set" shows "(a \<in> trees S) = (a = \<langle>\<rangle> \<or> (\<exists>l r v. a = \<langle>l, v, r\<rangle> \<and> l \<in> trees S \<and> r \<in> trees S \<and> v \<in> S))"
nnfis_integral_1: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "a \<in> nnfis f M"   and "measure_space M" shows "integrable f M"
simps_7: fixes D :: "ast_domain"   and ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and v :: "'a atom formula"   and va :: "'a atom formula" shows "ast_domain.wf_fmla_atom D ty_ent (v \<^bold>\<rightarrow> va) = False"
assigns_stmt_simp: fixes s :: "stmt" shows "assigns (In1r s) = {}"
Sup_atMost: fixes y :: "'b \<times> 'c" shows "Sup_prod (prod.atMost y) = y"
product_topology_empty_discrete: fixes T :: "'c \<Rightarrow> 'd topology" shows "product_topology T {} = discrete_topology {\<lambda>x. undefined}"
c1_3: shows "fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) = lst \<tau>3'"
center: shows "center x y z \<in> {x--y} \<inter> {x--z} \<inter> {y--z}"
lincomb_isolate_2: fixes A :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "finite A"   and "A \<subseteq> carrier V"   and "a \<in> A \<rightarrow> carrier K"   and "v \<in> A"   and "a v \<noteq> \<zero>\<^bsub>K\<^esub>"   and "lincomb a A = \<zero>\<^bsub>V\<^esub>" shows "v \<in> span (A - {v})"
C4_b: fixes x :: "'a" shows "-l (-r x) = x"
cospan_\<nu>\<pi>: shows "C.cospan src.leg0 src.leg1"
F1set3_Lev1: fixes xf :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xe :: "'d"   and xc :: "'d"   and x :: "'e" assumes "xf \<in> Lev1 s1 s2 n xe"   and "rv1 s1 s2 xf xe = Inl xc"   and "x \<in> F1set3 (s1 xc)" shows "xf @ [Inr (tobd_F13 s1 xc x)] \<in> Lev1 s1 s2 (Suc n) xe"
abs_plus: fixes x :: "'a"   and y :: "'a" assumes "sgn x = sgn y" shows "\<bar>x + y\<bar> = \<bar>x\<bar> + \<bar>y\<bar>"
events_paths: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<E>"   and "b \<in> \<E>"   and "a \<noteq> b" shows "\<exists>R\<in>\<P>. \<exists>S\<in>\<P>. a \<in> R \<and> b \<in> S \<and> R \<inter> S \<noteq> {}"
pcr_cr_eq: shows "pcr_mini_alg_unique = cr_mini_alg_unique"
prv_neg_neg: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "prv \<phi>" shows "prv (neg (neg \<phi>))"
inj_uni_samp_plus: fixes y :: "nat"   and q :: "nat" shows "inj_on (\<lambda>b. (y + b) mod q) {..<q}"
collinear_sym1: fixes z1 :: "complex"   and z2 :: "complex"   and z3 :: "complex" shows "collinear z1 z2 z3 = collinear z1 z3 z2"
cond_action_oks_Join: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and cas :: "'b conditional_action list"   and T :: "'b"   and x :: "'x"   and ln :: "'a \<Rightarrow>f nat" assumes "cond_action_oks s t cas"   and "Join T \<in> set cas"   and "thr s T = \<lfloor>(x, ln)\<rfloor>" shows "final x \<and> ln = no_wait_locks \<and> wset s T = None \<and> t \<noteq> T"
inf_import_p: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> - (x \<squnion> y) = x \<squnion> - y"
less_eq_quasinorm_def: fixes N1 :: "'b quasinorm"   and N2 :: "'b quasinorm" shows "(N1 \<subseteq>\<^sub>N N2) = (\<exists>C\<ge>0. \<forall>f. eNorm N2 f \<le> ennreal C * eNorm N1 f)"
simps_1: fixes c :: "'c"   and bs :: "'c list" shows "splitAtRec c bs [] = (bs, [])"
NonUniformExecutionBase: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and cfg :: "('a, 'c, 'b) configuration" assumes "flpPseudoConsensus trans sends start"   and "asynchronousSystem.initial start cfg"   and "asynchronousSystem.initReachable trans sends start cfg \<and> \<not> flpSystem.vUniform trans sends start False cfg \<and> \<not> flpSystem.vUniform trans sends start True cfg" shows "execution trans sends start [cfg] [] \<and> (asynchronousSystem.initReachable trans sends start (last [cfg]) \<and> \<not> flpSystem.vUniform trans sends start False (last [cfg]) \<and> \<not> flpSystem.vUniform trans sends start True (last [cfg])) \<and> (\<exists>cfgList' msgList'. (asynchronousSystem.initReachable trans sends start (last cfgList') \<and> \<not> flpSystem.vUniform trans sends start False (last cfgList') \<and> \<not> flpSystem.vUniform trans sends start True (last cfgList')) \<and> prefixList [cfg] cfgList' \<and> prefixList [] msgList' \<and> execution trans sends start cfgList' msgList' \<and> (\<exists>msg'. execution.minimalEnabled [cfg] [] msg' \<and> msg' \<in> set msgList'))"
power_strict_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less a b"   and "less_eq zero a" shows "0 < n \<longrightarrow> less (power.power one times a n) (power.power one times b n)"
ex_\<sigma>: shows "\<exists>\<sigma>. tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (g \<star> w) \<sigma> m (g \<star> n)"
samequot_wf: fixes n :: "nat"   and r :: "'b rexp"   and as :: "'a"   and b :: "bool"   and rs :: "'b rexp list" assumes "local.wf n r"   and "while_option fst (rderiv_and_add as) (True, [\<guillemotleft>r\<guillemotright>]) = Some (b, rs)" shows "local.wf n (PLUS rs)"
ceiling_zero: shows "\<lceil>0::'a\<rceil> = 0"
51: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (x \<sqinter> y) = bot"
derivs_Plus: fixes xs :: "'b list"   and r :: "'b rexp"   and s :: "'b rexp" shows "derivs xs (Plus r s) = Plus (derivs xs r) (derivs xs s)"
truncate_up_nonneg_iff: fixes p :: "nat"   and x :: "real" shows "(0 \<le> truncate_up p x) = (0 \<le> x)"
arctan_upper_34_def: shows "arctan_upper_34 \<equiv> \<lambda>x. pi / 2 - (33 + 170 * x\<^sup>2 + 165 * x ^ 4) * 7 * x / (5 * (5 + 105 * x\<^sup>2 + 315 * x ^ 4 + 231 * x ^ 6))"
that: fixes u :: "'a set"   and v :: "'a set" assumes "openin (top_of_set S) u"   and "P v"   and "x \<in> u"   and "u \<subseteq> v"   and "v \<subseteq> w" shows "thesis"
x': fixes s :: "real" assumes "s \<in> {t..u}" shows "(x has_vector_derivative f (s, x s)) (at s within {t..u})"
assms_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M"
upd_nbors_def: fixes n :: "nat"   and p :: "(real, 'b) vec"   and q :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list" shows "upd_nbors n p q ps = take n (insort_key (\<lambda>q. dist q p) q ps)"
simps: fixes ly :: "nat list"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and inres :: "cell list" shows "crsp ly (as, lm) (s, l, r) inres = (s = start_of ly as \<and> (\<exists>x. r = <lm> @ Bk \<up> x) \<and> l = Bk # Bk # inres)"
hyps_1: fixes xs :: "'a list"   and w :: "'a" assumes "n = length xs"   and "v0 \<leadsto>xs\<leadsto> w"   and "tl xs \<noteq> []"   and "v1 \<notin> set xs"   and "new_last \<notin> set xs"   and "hd (tl xs) = hd (tl P)"   and "edges_of_walk xs \<subseteq> \<Union> (edges_of_walk ` paths_with_new)" shows "edges_of_walk xs \<subseteq> edges_of_walk P"
cases: fixes a :: "(name \<times> ty) list" assumes "valid a" obtains "a = []"   | \<Gamma> :: "(name \<times> ty) list" and x :: "name" and T :: "ty" where "a = (x, T) # \<Gamma>" and "valid \<Gamma>" and "x \<sharp> \<Gamma>"
dvdm_smult: fixes m :: "int"   and f :: "int poly"   and g :: "int poly"   and c :: "int" assumes "poly_mod.dvdm m f g" shows "poly_mod.dvdm m f (smult c g)"
le: fixes x__ :: "real list"   and x'__ :: "'c"   and x :: "real list"   and x' :: "'c" assumes "(x__, x'__) \<in> lv_rel"   and "(x, x') \<in> lv_rel" shows "(list_all2 (\<le>) x__ x, x'__ \<le> x') \<in> bool_rel"
fact_mono: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "fact m \<le> fact n"
proj2_abs_cart2_append1: fixes p :: "proj2" assumes "z_non_zero p" shows "proj2_abs (cart2_append1 p) = p"
mk_cycles_path_awalk: fixes u :: "'a"   and c :: "'b list"   and n :: "nat" assumes "awalk u c u" shows "awalk u (mk_cycles_path n c) u"
assms_2: shows "(g, h) \<in> A\<^sub>2 \<rightarrow> B\<^sub>2"
And_Not_def: fixes \<phi> :: "'c MFOTL.formula"   and \<psi> :: "'c MFOTL.formula" shows "MFOTL.And_Not \<phi> \<psi> = formula.Neg (formula.Or (formula.Neg \<phi>) \<psi>)"
snoc_2: fixes r :: "'state"   and a :: "'action"   and b :: "'action" assumes "r \<in> nodes"   and "a \<in> psen p s - {a. en a s}"   and "b \<in> {a. en a r} - pac p"   and "en a (ex b r)" shows "en a r"
simps_4: fixes v :: "'c"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree" shows "siftDown (T v E (T va vb vc)) = (if val (T va vb vc) \<le> v then T v E (T va vb vc) else T (val (T va vb vc)) E (siftDown (set_val (T va vb vc) v)))"
bound_of_sound: fixes P :: "'b \<Rightarrow> real" assumes "sound P" shows "0 \<le> bound_of P"
simps_1: fixes h :: "nat \<Rightarrow> nat option"   and x :: "nat"   and y :: "nat" shows "Path h x [] y = (x = y)"
sup_left_commute: fixes x :: "'k multiset"   and y :: "'k multiset"   and z :: "'k multiset" shows "x \<union># (y \<union># z) = y \<union># (x \<union># z)"
prems_3: fixes y :: "'t \<Rightarrow>\<^sub>0 'b" assumes "y \<in> set ys" shows "\<not> is_red (set ((a # xs) @ ys) - {y}) y"
last_drop: fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "last (drop n xs) = last xs"
neg_implies_prv_N: fixes \<phi> :: "'fmla" shows "let NN = \<lambda>t1 t2. psubst N [(t1, xx), (t2, yy)] in \<phi> \<in> fmla \<longrightarrow> Fvars \<phi> = {} \<longrightarrow> bprv (NN \<langle>\<phi>\<rangle> \<langle>neg \<phi>\<rangle>)"