hcju/isabelleps huggingface dataset:
ge_induct: fixes i :: "nat"   and j :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "i \<le> j"   and "\<And>n. \<lbrakk>i \<le> n; \<forall>m\<ge>i. m < n \<longrightarrow> P m\<rbrakk> \<Longrightarrow> P n" shows "P j"
monad_resumption_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and pause :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" shows "monad_resumption return bind pause \<equiv> monad return bind \<and> monad_resumption_axioms bind pause"
simps_2: fixes g :: "char list \<Rightarrow> bool"   and uw :: "'b fm \<Rightarrow> bool"   and i :: "char list" shows "eval g uw (Pro i) = g i"
float_add_l0: fixes e :: "int"   and x :: "real" shows "float (0, e) + x = x"
hcard_hdiff1_less: fixes x :: "hf"   and z :: "hf" assumes "x \<^bold>\<in> z" shows "hcard (z - \<lbrace>x\<rbrace>) < hcard z"
exec_move_FAssI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e2 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e1 :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and v :: "'addr val" assumes "exec_move ci P t e2 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (e1\<bullet>F{D} := e2) h (stk @ [v], loc, length (compE2 e1) + pc, xcp) ta h' (stk' @ [v], loc', length (compE2 e1) + pc', xcp')"
divide_self: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "divide a a = one"
adds_trans: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.comm_monoid_add plus zero"   and "plus.adds plus a b"   and "plus.adds plus b c" shows "plus.adds plus a c"
grid_invariant: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and p :: "(nat \<times> int) list" assumes "d < length b"   and "d \<notin> ds"   and "p \<in> grid b ds" shows "p ! d = b ! d"
length_append: fixes xs :: "'g list"   and ys :: "'g list" shows "length (xs @ ys) = length xs + length ys"
dickson_less_pI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<in> dgrad_p_set d m"   and "q \<in> dgrad_p_set d m"   and "p \<prec>\<^sub>p q" shows "dickson_less_p d m p q"
subst_refl: fixes s :: "('b \<times> 'b trm) list" shows "s \<doteq> s"
pow_un1: shows "\<mu> \<circ> \<P> \<eta> = id"
wf_\<Delta>'': shows "\<Delta>''.wf_residual_network"
ccw'_half_segments_lowest_last: fixes c :: "real \<times> real"   and d :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "(c, d) \<in> set (butlast (half_segments_of_aform X))"   and "inl (snd X) \<noteq> []" shows "ccw' (lowest_vertex (fst X, nlex_pdevs (snd X))) d (snd (last (half_segments_of_aform X)))"
greachable_subset: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set"   and v :: "nat \<times> 'a" shows "greachable A w V v \<subseteq> insert v V"
only_blue_eyes_leave: fixes blue :: "'a"   and brown :: "'a"   and green :: "'a"   and red :: "'a"   and guru :: "'b"   and n :: "nat"   and p :: "'b"   and w :: "'b \<Rightarrow> 'a" assumes "distinct [blue, brown, green, red]"   and "finite UNIV"   and "Blue_Eyes.leaves blue guru n p w"   and "Blue_Eyes.valid blue guru w" shows "w p = blue"
inverse_of_nat_le: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m"   and "n \<noteq> 0" shows "(1::'c) / of_nat m \<le> (1::'c) / of_nat n"
merge_as_tailrec_inefficient: fixes as :: "'a list"   and bs :: "'a list" shows "merge as bs = merge_tailrec_inefficient as bs []"
P2: shows "C B J' CongA D E F \<and> \<not> B C OS A J' \<and> Coplanar A B C J' \<and> A B J' CongA G' H' I'"
EphK_Enc: fixes X :: "ephkey"   and X' :: "msg"   and Y' :: "msg" shows "EphK X \<noteq> Enc X' Y'"
real_div_sqrt: fixes x :: "real" assumes "0 \<le> x" shows "x / sqrt x = sqrt x"
less_mult_imp_div_less: fixes n :: "nat"   and k :: "nat"   and m :: "nat" assumes "n < k * m" shows "n div m < k"
fiter_fun_exp: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'd" shows "fiter_fun f g h = f \<sqinter> (g \<circ> h)"
y2_prop: shows "q < y2 \<and> (case z of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y2}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
prj_tuple_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = k"
simps_2: fixes x :: "nat"   and y :: "nat"   and ps :: "(nat \<times> nat) list" shows "ExcessNotAtRecList ((x, y) # ps) = (\<lambda>g. let l1 = ExcessNotAtRecList ps g; l2 = ExcessNotAtRecList (deleteAround g x ps) g in if ExcessNotAtRec ps g \<le> y + ExcessNotAtRec (deleteAround g x ps) g then x # l2 else l1)"
abs_eq: fixes y :: "'d \<Rightarrow> 'd \<Rightarrow> 'c" shows "Square_Matrix.transpose (of_fun (\<lambda>j i. y i j)) = of_fun y"
uI_int: fixes n :: "nat" shows "integrable MI (uI n)"
FactRing_iso: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "ring_hom_ring R S h"   and "h ` carrier R = carrier S" shows "R Quot a_kernel R S h \<simeq> S"
echelon_form_of_column_k_efficient_def: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A' :: "(('a, 'b) vec, 'c) vec \<times> nat"   and k :: "nat" shows "echelon_form_of_column_k_efficient bezout A' k = (let (A, i) = A'; from_nat_k = mod_type_class.from_nat k; from_nat_i = mod_type_class.from_nat i; all_zero_below_i = \<forall>m>from_nat_i. A $ m $ from_nat_k = (0::'a) in if i = nrows A \<or> A $ from_nat_i $ from_nat_k = (0::'a) \<and> all_zero_below_i then (A, i) else if all_zero_below_i then (A, i + 1) else let n = LEAST n. A $ n $ from_nat_k \<noteq> (0::'a) \<and> from_nat_i \<le> n; interchange_A = interchange_rows A from_nat_i n in (bezout_iterate interchange_A (nrows A - 1) from_nat_i from_nat_k bezout, i + 1))"
perp_not_eq_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D" shows "A \<noteq> B"
prems: shows "(l, a) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (c # A))"
measure_Ioi: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'b pmf"   and x :: "'b" assumes "rel_pmf R p q"   and "reflp R"   and "transp R" shows "measure_pmf.prob p {y. R x y \<and> \<not> R y x} \<le> measure_pmf.prob q {y. R x y \<and> \<not> R y x}"
exec_gpv_oracle_bisim_bad': fixes bad2 :: "'b \<Rightarrow> bool"   and s2 :: "'b"   and X_bad :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and s1 :: "'d"   and X :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and bad1 :: "'d \<Rightarrow> bool"   and \<I> :: "('e, 'f) \<I>"   and oracle1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and oracle2 :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b) spmf"   and gpv :: "('g, 'e, 'f) gpv" assumes "if bad2 s2 then X_bad s1 s2 else X s1 s2"   and "bad1 s1 = bad2 s2"   and "\<And>s1 s2 x. \<lbrakk>X s1 s2; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"   and "\<And>s2. bad2 s2 \<Longrightarrow> callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"   and "\<And>s1. bad1 s1 \<Longrightarrow> callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"   and "\<And>s1 x. \<lbrakk>bad1 s1; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> lossless_spmf (oracle1 s1 x)"   and "\<And>s2 x. \<lbrakk>bad2 s2; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> lossless_spmf (oracle2 s2 x)"   and "lossless_gpv \<I> gpv"   and "\<And>s1. \<I> \<turnstile>c oracle1 s1 \<surd>"   and "\<And>s2. \<I> \<turnstile>c oracle2 s2 \<surd>"   and "\<I> \<turnstile>g gpv \<surd>" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
Units_m_closed: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.monoid G"   and "x \<in> Units G"   and "y \<in> Units G" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> Units G"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "Right_a.in_hom f a b"   and "Right_a.in_hom g b c" shows "Right_a.seq g f"
almost_full_on_imp_af: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "almost_full_on P A" shows "af A P"
all_is_list_nominals: fixes p :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and xs :: "nat set" assumes "all_is p ps xs" shows "nominals p \<union> list_nominals (omit xs ps) = nominals p \<union> list_nominals ps"
HigherPriority_def: fixes A :: "('d, 'e, 'f) hierauto" shows "HigherPriority A \<equiv> \<lambda>(t, t')\<in>HADelta A \<times> HADelta A. (source t', source t) \<in> ChiPlus A"
eq_set_not_inc: fixes a :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "a \<notin> A"   and "A = B" shows "a \<notin> B"
f_t: shows "eval (Cn 3 r_not [r_result]) [t, i, x] \<down>= 0"
REF0: shows "((S, \<sigma>0), \<alpha> ` S, \<sigma>0') \<in> R"
non_empty_v1_v2: shows "v=v1\<parallel>v2 \<and> 0 < \<parallel>ext v1\<parallel> \<and> 0 < \<parallel>ext v2\<parallel>"
idem: fixes a :: "'a multiset" shows "a \<inter># a = a"
assms_2: fixes f :: "'b \<Rightarrow> 'a" assumes "mcont \<Or> (\<sqsubseteq>) Sup (\<le>) f" shows "mcont \<Or> (\<sqsubseteq>) Sup (\<le>) (F f)"
assms_2: shows "\<mu> U\<^sub>r \<eta> \<in> old (\<sigma> 0)"
fst_tmp: fixes tp' :: "nat" assumes "tp' \<in> fst ` tmp" shows "tp - len \<le> tp' \<and> tp' < tp + 1"
isometryD_2: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "isometry f" shows "dist (f x) (f y) = dist x y"
component_antisym: fixes F :: "'b program"   and G :: "'b program" assumes "F \<le> G"   and "G \<le> F" shows "F = G"
twos: fixes r :: "rat"   and y :: "rat" assumes "0 < r" shows "\<exists>n. y / 2 ^ n < r"
mset_subset_size: fixes A :: "'k multiset"   and B :: "'k multiset" assumes "A \<subset># B" shows "size A < size B"
bdd_aboveI: fixes A :: "'a multiset set"   and M :: "'a multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<subseteq># M" shows "subset_mset.bdd_above A"
inf_le2: fixes x :: "'b upred"   and y :: "'b upred" shows "y \<sqsubseteq> (x \<and> y)"
DisjointPaths_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set" shows "DisjointPaths G v0 v1 paths \<equiv> v0_v1_Digraph G v0 v1 \<and> DisjointPaths_axioms G v0 v1 paths"
runit'_in_hom: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'c \<times> 'c \<Rightarrow> 'c"   and \<alpha> :: "'c \<times> 'c \<times> 'c \<Rightarrow> 'c"   and \<iota> :: "'c"   and a :: "'c" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.in_hom C (category.inv C (monoidal_category.runit C T \<alpha> \<iota> a)) a (T (a, monoidal_category.unity C \<iota>))"
cond_abs: fixes s :: "'C" assumes "s \<in> wa_invar WAC"   and "s \<in> wa_cond WAC" shows "\<alpha> s \<in> wa_cond WAA"
SEND_def: shows "SEND \<equiv> \<mu> x. left`?`xa \<rightarrow> mid`!`xa \<rightarrow> ack \<rightarrow> x"
bisim1While3: fixes P :: "'a prog"   and h :: "'heap"   and c :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e :: "'addr expr1" assumes "P,c,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,while (c) e,h \<turnstile> (if (e') (e;; while (c) e) else unit, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
Loop: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and E :: "assigned"   and c :: "stmt"   and C :: "assigned"   and A :: "assigned"   and l :: "label" assumes "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E"   and "Env\<turnstile> B \<union> assigns_if True e \<guillemotright>\<langle>c\<rangle>\<guillemotright> C"   and "nrm A = nrm C \<inter> (B \<union> assigns_if False e)"   and "brk A = brk C" shows "Env\<turnstile> B \<guillemotright>\<langle>l\<bullet> While(e) c\<rangle>\<guillemotright> A"
domain_eq: shows "Domainp (pcr_fract (=)) = (\<lambda>x. fractrel x x)"
aHom_mem: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and a :: "'c" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G"   and "a \<in> carrier F" shows "f a \<in> carrier G"
arbitrary_union_of_inc: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "P S" shows "(arbitrary union_of P) S"
hom_closed: fixes G :: "('e, 'f) ring_scheme"   and H :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and x :: "'e" assumes "abelian_group_hom G H h"   and "x \<in> carrier G" shows "h x \<in> carrier H"
vrange_small: fixes r :: "V" shows "small {b. \<exists>a. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
integer_less_eq_iff: fixes k :: "integer"   and l :: "integer" shows "(k \<le> l) = (int_of_integer k \<le> int_of_integer l)"
semiring_normalization_rules_29: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "times x x = power.power one times x 2"
siso0_transC_indis: fixes c :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "siso0 c"   and "mustT c s"   and "mustT c t"   and "(c, s) \<rightarrow>c (c', s')"   and "s \<approx> t" shows "\<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>c (c', t')"
derivation_times_eq_zeroI: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "c_manifold charts k"   and "X \<in> c_manifold.tangent_space charts k p"   and "f \<in> c_manifold.diff_fun_space charts k"   and "g \<in> c_manifold.diff_fun_space charts k"   and "f p = 0"   and "g p = 0" shows "X (f * g) = 0"
R23s_def: shows "R23s \<equiv> {(s, s'). s = med23s s'}"
assms: shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr>"
ID_distr_global_2: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r :: "'b"   and ls :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr" shows "LID\<^sub>G (s(r \<mapsto> ls)) = LID\<^sub>G (s(r := None)) \<union> LID\<^sub>L ls"
mbody_functional: fixes CT :: "nat \<Rightarrow> classDef option"   and m :: "nat"   and C :: "nat"   and xs :: "nat list"   and e0 :: "exp"   and ys :: "nat list"   and d0 :: "exp" assumes "mbody(CT,m,C) = xs . e0"   and "mbody(CT,m,C) = ys . d0" shows "xs = ys \<and> e0 = d0"
abs_zero: fixes abs :: "'y \<Rightarrow> 'y"   and plus :: "'y \<Rightarrow> 'y \<Rightarrow> 'y"   and zero :: "'y"   and minus :: "'y \<Rightarrow> 'y \<Rightarrow> 'y"   and uminus :: "'y \<Rightarrow> 'y"   and less_eq :: "'y \<Rightarrow> 'y \<Rightarrow> bool"   and less :: "'y \<Rightarrow> 'y \<Rightarrow> bool" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs zero = zero"
extend_set_inverse: fixes C :: "'a set" shows "project_set h (extend_set h C) = C"
bet_col: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C" shows "Col A B C"
inf_le2: fixes x :: "'c multiset"   and y :: "'c multiset" shows "x \<inter># y \<subseteq># y"
subst_typ_typ_ok: fixes \<Sigma> :: "(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)"   and \<tau> :: "typ"   and insts :: "((variable \<times> String.literal set) \<times> typ) list" assumes "typ_ok_sig \<Sigma> \<tau>"   and "list_all (typ_ok_sig \<Sigma>) (map snd insts)" shows "typ_ok_sig \<Sigma> (subst_typ insts \<tau>)"
power_gt1_lemma: fixes a :: "'a"   and n :: "nat" assumes "(1::'a) < a" shows "(1::'a) < a * a ^ n"
map_us_def: fixes f :: "'d \<Rightarrow> 'e" shows "map_us f = us \<circ> f \<circ> IInf"
not_int_zless_negative: fixes n :: "nat"   and m :: "nat" shows "\<not> int n < - int m"
arr_cod_iff_arr: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and f :: "'ag" assumes "category C" shows "partial_magma.arr C (partial_magma.cod C f) = partial_magma.arr C f"
integral_norm_eq_0_iff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" shows "(LINT x|M. norm (f x) = 0) = (emeasure M {x \<in> space M. f x \<noteq> (0::'d)} = 0)"
eventually_top: fixes P :: "'a \<Rightarrow> bool" shows "eventually P top = (\<forall>x. P x)"
nsphere_def: fixes n :: "nat" shows "nsphere n \<equiv> subtopology (Euclidean_space (Suc n)) {x. (\<Sum>i\<le>n. (x i)\<^sup>2) = 1}"
in_hom_def: fixes f :: "('obj, 'obj) concrete_category.arr"   and a :: "('obj, 'obj) concrete_category.arr"   and b :: "('obj, 'obj) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<equiv> a \<in> domains f \<and> b \<in> codomains f"
delta: fixes S :: "'c set"   and a :: "'c"   and b :: "'c \<Rightarrow> 'a" assumes "finite S" shows "F (\<lambda>k. if k = a then b k else \<^bold>1) S = (if a \<in> S then b a \<^bold>* \<^bold>1 else \<^bold>1)"
below_antisym: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.po below"   and "below x y"   and "below y x" shows "x = y"
inner_Pair_0_1: fixes x :: "'a \<times> 'b"   and b :: "'b" shows "x \<bullet> (0::'a, b) = snd x \<bullet> b"
acrrs1: fixes rs\<^sub>1 :: "'a rule list"   and rs\<^sub>2 :: "'a rule list" assumes "rrs = rs\<^sub>1 @ rs\<^sub>2" shows "all_chains (no_call_to c) \<Gamma> rs\<^sub>1"
productF_language: fixes A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM"   and io :: "('d \<times> 'e) list" assumes "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "io \<in> L A \<inter> L B" shows "io \<in> L AB"
assms: shows "r\<^sub>q \<in> pre_ranks r \<nu> q - {max_rank}"
project_Stable: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set" shows "(F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> Stable A) = (extend h F \<squnion> G \<in> Stable (extend_set h A))"
not_null_ang_ang: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "QCongAnNull a" shows "QCongA a"
\<C>_vars_subset_\<C>: fixes x :: "'Var\<^sub>C" shows "\<C>_vars\<^sub>C x \<subseteq> \<C>\<^sub>C"
elims_2: fixes x :: "('c, 'd) tree" assumes "tree_sorted x" obtains "x = Tip"   | l :: "('c, 'd) tree" and k :: "'c" and v :: "'d" and r :: "('c, 'd) tree" where "x = Node l k v r" and "(\<forall>x\<in>tree_set l. x < k) \<and> Ball (tree_set r) ((<) k) \<and> tree_sorted l \<and> tree_sorted r"
set_eq_iff_mset_remdups_eq: fixes x :: "'a list"   and y :: "'a list" shows "(set x = set y) = (mset (remdups x) = mset (remdups y))"
igAbsRen_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gAbsRen MOD" shows "igAbsRen (fromMOD MOD)"
det_eq: shows "det3 b c p / det3 a b c + det3 c a p / det3 a b c + det3 a b p / det3 a b c = 1"
intros_1: fixes A :: "'b set" shows "[] \<in> lists A"
intro: fixes F :: "('d, 'b, 'e, 'c) functor"   and AA :: "('d, 'b, 'f) category_scheme"   and BB :: "('e, 'c, 'g) category_scheme" assumes "F \<^bsub>\<a>\<^esub> \<in> Ar\<^bsub>AA\<^esub> \<rightarrow> Ar\<^bsub>BB\<^esub>"   and "F \<^bsub>\<o>\<^esub> \<in> Ob\<^bsub>AA\<^esub> \<rightarrow> Ob\<^bsub>BB\<^esub>"   and "\<forall>f\<in>Ar\<^bsub>AA\<^esub>. F \<^bsub>\<o>\<^esub> (Dom\<^bsub>AA\<^esub> f) = Dom\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f"   and "\<forall>f\<in>Ar\<^bsub>AA\<^esub>. F \<^bsub>\<o>\<^esub> (Cod\<^bsub>AA\<^esub> f) = Cod\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f"   and "\<forall>A\<in>Ob\<^bsub>AA\<^esub>. F \<^bsub>\<a>\<^esub> (Id\<^bsub>AA\<^esub> A) = Id\<^bsub>BB\<^esub> F \<^bsub>\<o>\<^esub> A"   and "\<forall>f\<in>Ar\<^bsub>AA\<^esub>. \<forall>g\<in>Ar\<^bsub>AA\<^esub>. Cod\<^bsub>AA\<^esub> f = Dom\<^bsub>AA\<^esub> g \<longrightarrow> F \<^bsub>\<a>\<^esub> (g \<bullet>\<^bsub>AA\<^esub> f) = F \<^bsub>\<a>\<^esub> g \<bullet>\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f" shows "functor_axioms AA BB F"
fds_shift_1: fixes a :: "'b" shows "fds_shift a 1 = 1"
neq1: shows "close ([q] @ \<Gamma>) (p \<leftrightarrow> q) \<noteq> \<Gamma>"
hs_it_has_next_def: shows "hs_it_has_next \<equiv> hm_it_has_next"
agree_def: fixes s1 :: "('d, 'e) fmap"   and s2 :: "('d, 'e) fmap" shows "agree s1 s2 \<equiv> \<forall>v. v \<in> fmdom' s1 \<and> v \<in> fmdom' s2 \<longrightarrow> fmlookup s1 v = fmlookup s2 v"
atoms_formula_vars: fixes f :: "'a formula" shows "atoms f = set (formula_vars f)"
and_less_eq: fixes l :: "int"   and k :: "int" assumes "l < 0" shows "k AND l \<le> k"
IH: fixes a :: "'a list"   and b :: "'a list" assumes "l \<in> a \<otimes> b" shows "l @ [x] \<in> a @ [x] \<otimes> b"
ex: shows "\<exists>c. \<forall>w\<in>space (sigma (space M) {{}, space M}). disc_rfr_proc r n w = c"
assms_4: shows "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"
invoke_def: fixes xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> 'aa \<Rightarrow> ((_) heap, exception, 'z) prog)) list"   and ptr :: "(_) object_ptr"   and args :: "'aa" shows "invoke xs ptr args = Heap_Error_Monad.bind (check_in_heap ptr) (\<lambda>_. invoke_rec xs ptr args)"
prod_zero: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "\<exists>a\<in>A. f a = zero" shows "comm_monoid_mult.prod times one f A = zero"
bintr_bin_set_ge: fixes n :: "nat"   and w :: "int"   and m :: "nat" shows "take_bit n w \<le> take_bit n (bin_sc m True w)"
Times_mset_image_mset2: fixes A :: "'a multiset"   and f :: "'c \<Rightarrow> 'b"   and B :: "'c multiset" shows "A \<times># image_mset f B = {#(a, f b). (a, b) \<in># A \<times># B#}"
lexordp_antisym: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list" assumes "class.order less_eq less"   and "ord.lexordp less xs ys"   and "ord.lexordp less ys xs" shows "False"