hcju/isabelleps huggingface dataset:
real_cond_exp_F_meas: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f"   and "f \<in> borel_measurable F" shows "AE x in M. real_cond_exp M F f x = f x"
psi: shows "next_free \<psi> \<and> atoms_pltl \<psi> \<subseteq> A \<and> (\<forall>\<sigma>. stutter_free \<sigma> \<and> canonical_sequence A \<sigma> \<longrightarrow> \<sigma> \<Turnstile>\<^sub>p \<psi> = \<sigma> \<Turnstile>\<^sub>p \<phi>)"
volume_finite_additive: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal"   and I :: "'c set"   and A :: "'c \<Rightarrow> 'b set" assumes "volume M f"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M"   and "disjoint_family_on A I"   and "finite I"   and "\<Union> (A ` I) \<in> M" shows "f (\<Union> (A ` I)) = (\<Sum>i\<in>I. f (A i))"
S_edges_def: fixes p :: "real"   and n :: "nat" assumes "edge_space p" shows "edge_space.S_edges n = all_edges (edge_space.S_verts n)"
lossless_res: fixes \<eta> :: "nat" assumes "lossless" shows "lossless_resource (\<I>_res \<eta>) (res \<eta>)"
k: fixes e :: "real"   and n :: "nat" assumes "0 < e"   and "0 < n" shows "\<exists>k. Sigma_Algebra.measure M (space M) - e * 2 powr - real n \<le> Sigma_Algebra.measure M (\<Union>i\<in>{0..k}. cball (from_nat_into X i) (1 / real n))"
strict_border_example: shows "{b. strict_border b ''aabaabaa''} = {[], ''a'', ''aa'', ''aabaa''}"
simps_1: fixes uu :: "'a ext" shows "(Bot \<le> uu) = True"
invariant: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and s :: "'a"   and D :: "'a pmf" assumes "Markov_Decision_Process_Invariant K S"   and "s \<in> S"   and "D \<in> K s" shows "\<forall>s'\<in>set_pmf D. s' \<in> S"
assoc_in_homs_4: shows "\<guillemotleft>\<lparr>Chn = \<mu>_\<nu>_\<pi>\<rho>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi> \<star> \<rho>)\<rparr> : (\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho> \<Rightarrow> \<mu> \<star> \<nu> \<star> \<pi> \<star> \<rho>\<guillemotright>"
stabiliser_def: fixes x :: "Vertex" shows "tetrahedral.stabiliser x = {g \<in> carrier tetrahedral_group. apply_rotation g x = x}"
fvi_plus: fixes x :: "nat"   and b :: "nat"   and c :: "nat"   and \<phi> :: "Formula.formula" shows "(x \<in> Formula.fvi (b + c) \<phi>) = (x + c \<in> Formula.fvi b \<phi>)"
uminus_of_complex: fixes z :: "complex" shows "~\<^sub>h (of_complex z) = of_complex (- z)"
PutHigh: fixes f :: "Field"   and e :: "Expr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var" assumes "GAMMA f = high"   and "\<forall>s. noLowDPs s \<longrightarrow> Expr_good e s" shows "G \<rhd> Put x f e : HighSec"
binop_G_comp_binop_\<ii>G: fixes g :: "'a"   and x :: "'a" assumes "g \<in> G"   and "x \<in> G" shows "binop g (binop (\<ii> g) x) = x"
act_rews_3: fixes v :: "state \<Rightarrow> 'c"   and s :: "state"   and t :: "state" shows "(s, t) \<Turnstile> unchanged v \<equiv> v t = v s"
order_le_less_trans: fixes x :: "'d"   and y :: "'d"   and z :: "'d" assumes "x \<le> y"   and "y < z" shows "x < z"
list_nonempty_induct: fixes xs :: "'e list"   and P :: "'e list \<Rightarrow> bool" assumes "xs \<noteq> []"   and "\<And>x. P [x]"   and "\<And>x xs. \<lbrakk>xs \<noteq> []; P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P xs"
Conj_eqvt: fixes xset :: "('a, 'b, 'c) formula set['a]"   and p :: "perm" assumes "finite (supp xset)" shows "p \<bullet> Conj xset = Conj (p \<bullet> xset)"
tjs_union: assumes "u \<in> set tjs" shows "set tjs = set (tl (dropWhile ((\<noteq>) u) tjs)) \<union> insert u (set (takeWhile ((\<noteq>) u) tjs))"
submonoid_def: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "submonoid N M composition unit \<equiv> monoid M composition unit \<and> submonoid_axioms N M composition unit"
min_dnf_iff_prop_assignment_subset: fixes \<A> :: "'b ltln set"   and \<phi> :: "'b ltln" shows "\<A> \<Turnstile>\<^sub>P \<phi> = (\<exists>B. fset B \<subseteq> \<A> \<and> B \<in> min_dnf \<phi>)"
intvs_lower_incr: fixes l :: "int"   and h :: "int" assumes "l < h" shows "{l..<h} = insert l {l + 1..<h}"
dominated_convergence_integrable: fixes S :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>k. f k absolutely_integrable_on S"   and "h integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (g x) \<le> h x"   and "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x" shows "g integrable_on S"
is_node_ptr_kind_obtains: fixes ptr :: "(_) object_ptr" assumes "is_node_ptr_kind ptr" obtains node_ptr :: "(_) node_ptr" where "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some node_ptr"
N: shows "KBP.k_isNode a (runJP k_mkAlgAuto t a)"
depthTransition_1: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "hnf P"   and "P \<longmapsto> a[b] \<prec> P'" shows "depth P' < depth P"
Unit_Left: fixes P :: "'a hrel" assumes "P is \<H>" shows "\<I>\<I> ;; P = P"
fim1_state: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "state = (f, mu, ds)"   and "i < m"   and "i \<noteq> 0" shows "fim1_state state = fs ! (i - 1)"
disj_split: fixes P :: "bool \<Rightarrow> bool"   and a :: "bool"   and b :: "bool" shows "P (a \<or> b) = ((a \<longrightarrow> P True) \<and> (b \<longrightarrow> P True) \<and> (\<not> a \<and> \<not> b \<longrightarrow> P False))"
assms_2: shows "v \<in> dom (\<Pi>\<^sub>G)"
i_extension_ideal: fixes f :: "'a \<Rightarrow> 'c"   and I :: "'a set" assumes "f \<in> rHom R R'"   and "ideal R I" shows "ideal R' (i_extension f R R' I)"
varDefs_types_def: shows "varDefs_types = map vdType"
dvd_prod: fixes A :: "'a set"   and a :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "\<exists>b\<in>A. a dvd f b"   and "finite A" shows "a dvd prod f A"
Gcd_empty: 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 Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<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" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Gcd {} = zero"
inv_Mp2_def: fixes m2 :: "int" shows "inv_Mp2 m2 = map_poly (inv_M2 m2)"
param_append: fixes R :: "('g \<times> 'h) set" shows "((@), (@)) \<in> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel"
Red_F_of_subset: fixes Inf :: "'a inference set"   and Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N :: "'a set"   and N' :: "'a set" assumes "calculus_with_standard_redundancy Inf Bot entails"   and "N \<subseteq> N'" shows "calculus_with_standard_redundancy.Red_F entails N \<subseteq> calculus_with_standard_redundancy.Red_F entails N'"
has_derivative_norm: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "(norm has_derivative (\<lambda>h. h \<bullet> sgn x)) (at x)"
alist_insert: fixes xs :: "('d, 'c) bintree option list"   and v :: "'c"   and a :: "'d" assumes "distinct (vals xs)"   and "v \<notin> set (vals xs)" shows "set (elements (Binomial_Queue.insert a v xs)) = set (elements xs) \<union> {(v, a)}"
prime_elem_imp_irreducible: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and p :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.prime_elem times one zero p" shows "comm_semiring_1.irreducible times one zero p"
simplex_map_id: fixes p :: "nat" shows "simplex_map p id = (\<lambda>c. restrict c (standard_simplex p))"
enforce_option_alt_def: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option" shows "enforce_option P x = x \<bind> (\<lambda>a. assert_option (P a) \<bind> (\<lambda>_. Some a))"
subst_cls_lists_def: fixes Cs :: "'a literal multiset list"   and \<sigma>s :: "'s list" shows "Cs \<cdot>\<cdot>cl \<sigma>s = Map2.map2 (\<cdot>) Cs \<sigma>s"
interleaving_Nil_iff1: fixes ys :: "'a list"   and zs :: "'a list" shows "interleavings [] ys zs = (ys = zs)"
cut: fixes G' :: "'b triple set"   and ts :: "'b triple set"   and G :: "'b triple set" assumes "G'||-ts"   and "G||-G'" shows "G||-ts"
perm_1: fixes pi :: "(vrs \<times> vrs) list"   and x2 :: "vrs"   and x1 :: "ty" shows "pi \<bullet> VarB x2 x1 = VarB (pi \<bullet> x2) (pi \<bullet> x1)"
intro: fixes x :: "'c \<Longrightarrow> 'b" assumes "\<And>\<sigma>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>" shows "wb_lens_axioms x"
perm_remdups_iff_eq_set: fixes x :: "'b list"   and y :: "'b list" shows "(remdups x <~~> remdups y) = (set x = set y)"
pseudo_proj_True_suc_img: fixes n :: "nat"   and w :: "bool stream" shows "pseudo_proj_True (Suc n) w \<in> {pseudo_proj_True n w, pseudo_proj_False n w}"
closure_is_csubspace: fixes A :: "'a set" assumes "csubspace A" shows "csubspace (closure A)"
fx: fixes x :: "'a"   and t :: "'b" assumes "x \<in> U"   and "t \<in> cbox a b" shows "((\<lambda>x. f x t) has_derivative blinfun_apply (fx x t)) (at x within U)"
inv_renaming_cancel_r: fixes r :: "'s" assumes "is_renaming r" shows "r \<odot> inv_renaming r = id_subst"
neighborhoods_eq: shows "target.neighborhoods (f x) = {U. is_open\<^sub>Y U \<and> f x \<in> U}"
gallery_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and Cs :: "'b set list" assumes "ChamberComplexRetraction X f"   and "SimplicialComplex.maxsimpchain X Cs" shows "SimplicialComplex.maxsimpchain X (f \<Turnstile> Cs)"
neg_dsj_I: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> neg A"   and "H \<turnstile> neg B" shows "H \<turnstile> neg (A OR B)"
the_def: fixes option :: "'f option" shows "the option = (case option of \<lfloor>x2\<rfloor> \<Rightarrow> x2)"
AE_T': fixes P :: "'s stream \<Rightarrow> bool"   and I :: "'s pmf" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "almost_everywhere (T' I) P = (\<forall>s\<in>set_pmf I. AE x in T s. P (s ## x))"
permits_acc_static_declC: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname"   and accC :: "qtname" assumes "G \<turnstile> m in C permits_acc_from accC"   and "G \<turnstile> m member_in C"   and "is_static m" shows "G \<turnstile> m in declclass m permits_acc_from accC"
axioms_2: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "tolerance A equiv" shows "reflexive A equiv"
goedel_first_classic: assumes "consistent" shows "\<not> prv \<phi>G \<and> \<not> prv (neg \<phi>G)"
child_ix_left: fixes d :: "nat"   and p :: "(nat \<times> int) list" assumes "d < length p" shows "ix (child p left d) d = 2 * ix p d - 1"
has_subprob_densityD_3: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "M = density N f"
axiom_or_def: fixes p :: "fol fm"   and q :: "fol fm" shows "axiom_or p q \<equiv> Thm (Iff (Or p q) (fm.Not (And (fm.Not p) (fm.Not q))))"
sba2_sub_sup_demorgan: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) = - (- - x \<squnion> - - y)"
approx_mult_HFinite: fixes a :: "'a star"   and b :: "'a star"   and c :: "'a star"   and d :: "'a star" assumes "a \<approx> b"   and "c \<approx> d"   and "b \<in> HFinite"   and "d \<in> HFinite" shows "a * c \<approx> b * d"
sha1blocklen: fixes b :: "bit list"   and x :: "bit list"   and A :: "bit list"   and B :: "bit list"   and C :: "bit list"   and D :: "bit list"   and E :: "bit list" shows "length (sha1block b x A B C D E) = 160"
find_closest_pair_dist_mono: fixes C\<^sub>0 :: "int \<times> int"   and C\<^sub>1 :: "int \<times> int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and ps :: "(int \<times> int) list" assumes "(C\<^sub>0, C\<^sub>1) = find_closest_pair (c\<^sub>0, c\<^sub>1) ps" shows "dist (case C\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))"
pochhammer_Suc_prod: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a (Suc n) = comm_monoid_mult.prod times one (\<lambda>i. plus a (semiring_1.of_nat one plus zero i)) {0..n}"
islimpt_diff_finite: fixes t :: "'a set"   and x :: "'a"   and s :: "'a set" assumes "finite t" shows "x islimpt s - t = x islimpt s"
assms: shows "x \<in> {exp (- 1)..exp 1}"
Fvars_fmla_simps_1: fixes x :: "trm"   and y :: "trm" shows "Fvars (x EQ y) = Fvars x \<union> Fvars y"
bconf_Call: fixes P :: "'a prog"   and sh :: "sheap"   and e :: "'b exp"   and M :: "char list"   and es :: "'b exp list"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (e\<bullet>M(es),b) \<surd>) = (case val_of e of None \<Rightarrow> P,sh \<turnstile>\<^sub>b (e,b) \<surd> | \<lfloor>v\<rfloor> \<Rightarrow> P,sh \<turnstile>\<^sub>b (es,b) \<surd>)"
iso_cycle_group_sum: fixes \<U> :: "'b set set"   and X :: "'b topology"   and p :: "nat" assumes "disjoint \<U>"   and "\<Union> \<U> = topspace X"   and "\<And>C T. \<lbrakk>compactin X C; path_connectedin X C; T \<in> \<U>; \<not> disjnt C T\<rbrakk> \<Longrightarrow> C \<subseteq> T" shows "(\<lambda>f. sum' f \<U>) \<in> Group.iso (sum_group \<U> (\<lambda>T. relcycle_group p (subtopology X T) {})) (relcycle_group p X {})"
Bs_4: shows "B4 \<in> carrier_mat (n - k) (n - k)"
le_of_nat_term_order_alt: fixes to :: "'a nat_term_order" shows "le_of_nat_term_order to = (\<lambda>u v. ko.le (key_order_of_nat_term_order_inv to) v u)"
power: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "completely_multiplicative_function f" shows "f (n ^ m) = f n ^ m"
parallel_cases: fixes V :: "('exp, 'id) MWLsCom list" assumes "\<turnstile>\<^bsub>\<V>\<^esub> V" obtains "\<forall>i<length V. \<turnstile>\<^bsub>\<C>\<^esub> V ! i"
fs0_gso0: fixes n :: "nat"   and fs :: "'a vec list" assumes "gram_schmidt_fs_Rn n fs"   and "0 < length fs" shows "fs ! 0 = gram_schmidt_fs.gso n fs 0"
env_restr_split: fixes m :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "m = m f|` S \<squnion> m f|` (- S)"
sub: shows "subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>"
lcons_correct_1: fixes t :: "('a, 'b) FingerTreeStruc"   and a :: "'a \<times> 'b" assumes "FingerTreeStruc.ft_invar t" shows "FingerTreeStruc.ft_invar (FingerTreeStruc.lcons a t)"
infinite_insert_eqpoll: fixes A :: "'a set"   and a :: "'a" assumes "infinite A" shows "insert a A \<approx> A"
\<rho>: shows "\<rho> \<in> space (state_measure (set vs') \<Gamma>)"
gbm_from_list_def: shows "gbm_from_list \<equiv> gga_from_list gbm_empty gbm_add_node gbm_add_edge"
Min_le: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set"   and x :: "'k" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq (linorder.Min less_eq A) x"
simps_2: fixes x :: "name"   and xs :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "inputChain (x # xs) N P = \<nu>xinputChain xs N P"
trans_inv: shows "\<forall>p f x. 0 \<le> x \<longrightarrow> cfn p (\<lambda>y. f y + x) = cfn p f + x"
circ_mult_upper_bound: fixes x :: "'a"   and y :: "'a" assumes "x \<le> top"   and "y \<le> top" shows "x \<sqinter> y \<le> top"
1: fixes d :: "'a"   and f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "d \<in> S"   and "f d \<in> S"   and "homeomorphism T T f g"   and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"   and "bounded {x. \<not> (f x = x \<and> g x = x)}" shows "\<exists>h k. homeomorphism T T h k \<and> h (f d) = d \<and> {x. \<not> (h x = x \<and> k x = x)} \<subseteq> S \<and> bounded {x. \<not> (h x = x \<and> k x = x)}"
distinct_11: fixes x2 :: "bool" shows "Bool x2 \<noteq> Null"
a_asymptotics: shows "(\<lambda>x. dirichlet_prod' (\<lambda>n. ind prime n * ln (real n)) (\<lambda>x. real_of_int \<lfloor>x\<rfloor>) x - x * ln x) \<in> O(\<lambda>x. x)"
ObjCg: shows "cod\<^bsub>C\<^esub> g \<in> obj\<^bsub>C\<^esub>"
expand: fixes step :: "('c, 'd) step"   and step' :: "('c, 'd) step" assumes "(step = Done) = (step' = Done)"   and "is_Skip step = is_Skip step'"   and "\<lbrakk>is_Skip step; is_Skip step'\<rbrakk> \<Longrightarrow> un_Skip step = un_Skip step'"   and "\<lbrakk>is_Yield step; is_Yield step'\<rbrakk> \<Longrightarrow> un_Yield1 step = un_Yield1 step' \<and> un_Yield2 step = un_Yield2 step'" shows "step = step'"
sets_PiM_I: fixes J :: "'d set"   and I :: "'d set"   and E :: "'d \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure" assumes "finite J"   and "J \<subseteq> I"   and "\<forall>i\<in>J. E i \<in> sets (M i)" shows "prod_emb I M J (Pi\<^sub>E J E) \<in> sets (Pi\<^sub>M I M)"
rep_eq: fixes x :: "real_alg_3" shows "sgn_3 x = sgn_2 (rep_real_alg_3 x)"
append_Nil2: fixes xs :: "'f list" shows "xs @ [] = xs"
finter_finsert_left_if1: fixes a :: "'c"   and C :: "'c fset"   and B :: "'c fset" assumes "a |\<in>| C" shows "finsert a B |\<inter>| C = finsert a (B |\<inter>| C)"
idempotent_conv_closed: fixes x :: "'a" assumes "idempotent x" shows "x[[x\<^sup>T]] = x\<^sup>T"
biholomorphic_to_disc: assumes "connected S"   and "S \<noteq> {}"   and "S \<noteq> UNIV"   and "\<And>f. \<lbrakk>f holomorphic_on S; \<forall>z\<in>S. f z \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2)" shows "\<exists>f g. f holomorphic_on S \<and> g holomorphic_on ball 0 1 \<and> (\<forall>z\<in>S. f z \<in> ball 0 1 \<and> g (f z) = z) \<and> (\<forall>z\<in>ball 0 1. g z \<in> S \<and> f (g z) = z)"
tangent_not_meet: fixes x1 :: "complex_homo"   and x2 :: "complex_homo"   and i :: "complex_homo"   and a :: "complex_homo" assumes "x1 \<in> unit_disc"   and "x2 \<in> unit_disc"   and "x1 \<noteq> x2"   and "\<not> poincare_collinear {0\<^sub>h, x1, x2}"   and "i \<in> ideal_points (poincare_line x1 x2)"   and "a \<in> unit_disc"   and "a \<noteq> 0\<^sub>h"   and "poincare_collinear {0\<^sub>h, a, i}" shows "\<not> poincare_ray_meets_line 0\<^sub>h a x1 x2"
UniverseUPair: fixes u :: "ZF"   and v :: "ZF" assumes "Elem u U"   and "Elem v U" shows "Elem (Upair u v) U"
total_order_axioms_def: fixes L :: "('a, 'b) gorder_scheme" shows "total_order_axioms L \<equiv> \<forall>x y. x \<in> carrier L \<longrightarrow> y \<in> carrier L \<longrightarrow> x \<sqsubseteq>\<^bsub>L\<^esub> y \<or> y \<sqsubseteq>\<^bsub>L\<^esub> x"
nonsing_concat_len: fixes us :: "'b list list" assumes "\<^bold>|us\<^bold>| \<noteq> 1"   and "concat us \<noteq> \<epsilon>" shows "1 < \<^bold>|us\<^bold>|"
no_grey_refs_not_reachable: fixes s :: "'mut process_name \<Rightarrow> ('d, 'mut, 'e, 'f) Model.local_state"   and r :: "'f" assumes "no_grey_refs s"   and "reachable_snapshot_inv s"   and "white r s" shows "\<not> reachable r s"
wp_abort: fixes r :: "'b upred" assumes "\<not> r = true" shows "true wp r = false"
lm010: fixes N :: "'d set"   and G :: "'e set" assumes "N \<noteq> {}"   and "G \<noteq> {}" shows "allAllocations N G \<noteq> {}"
simps_2: fixes t :: "('a, 'b, 'c, 'd) FL_Formula.Tree" shows "L_transform_Tree (FL_Formula.Tree.tNot t) = Formula.Not\<^sub>\<alpha> (L_transform_Tree t)"
not_eq_order_implies_strict: fixes a :: "'c list"   and b :: "'c list" assumes "a \<noteq> b"   and "prefix a b" shows "strict_prefix a b"
le: shows "\<forall>x\<in>A. f x \<le> g x"
zero_prod_def: shows "0 \<equiv> (0::'c, 0::'d)"
p_def: fixes x :: "nat \<Rightarrow> complex"   and n :: "nat"   and k :: "nat" shows "power_sum_puzzle.p x n k = (\<Sum>i<n. x i ^ k)"
Suc: shows "\<exists>cs. length cs = m \<and> list_all (\<lambda>c. new c p) cs \<and> distinct cs"
create_delete_commute: fixes i :: "'c"   and "is" :: "'c set"   and e1 :: "'d"   and e2 :: "'d" assumes "i \<notin> is" shows "\<langle>Create i e1\<rangle> \<rhd> \<langle>Delete is e2\<rangle> = \<langle>Delete is e2\<rangle> \<rhd> \<langle>Create i e1\<rangle>"
act_act: fixes g :: "cltn2"   and h :: "cltn2"   and b :: "hyp2" assumes "g \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"   and "h \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "hyp2_cltn2 (hyp2_cltn2 b g) h = hyp2_cltn2 b (g \<otimes>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> h)"
less_cSupD: 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 X :: "'a set"   and z :: "'a" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "less z (Sup X)" shows "\<exists>x\<in>X. less z x"
nondet_exp_def: fixes return :: "int \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "('c \<Rightarrow> int option) \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "int cset \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b" shows "nondet_exp return bind get put altc \<equiv> monad_state_altc return bind get put altc \<and> memoization return bind get put"
reads_subset: fixes S :: "('x \<Rightarrow> 'x \<Rightarrow> bool) set"   and f :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x"   and S' :: "('x \<Rightarrow> 'x \<Rightarrow> bool) set" assumes "reads S f h h'"   and "\<forall>P\<in>S' - S. reflp P \<and> transp P"   and "S \<subseteq> S'" shows "reads S' f h h'"
step: fixes l1 :: "'a list"   and x :: "'a"   and l2 :: "'a list"   and \<sigma> :: "'\<sigma>" assumes "l0 = l1 @ x # l2"   and "I \<sigma> (x # l2)" shows "I (f \<sigma> x) l2"
ae3: shows "AE x in S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P. Pxz (fst x, snd (snd x)) = 0 \<longrightarrow> Pxyz x = 0"
WeakConjunctive_def: fixes H :: "'b upred \<Rightarrow> 'b upred" shows "WeakConjunctive H = (\<forall>P. \<exists>Q. H P = (P \<and> Q))"
compl_sup: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "\<^bold>- (x \<squnion> y) = \<^bold>- x \<sqinter> \<^bold>- y"
env_restr_cont: fixes S :: "'c set" shows "cont (env_restr S)"
is_invar_def: fixes I :: "('v, 'es) state_scheme \<Rightarrow> bool" shows "is_invar I \<equiv> is_rwof_invar init cond step I"
invariant: fixes a :: "'a"   and b :: "'a" assumes "P a"   and "a \<rightarrow> b" shows "P b"
T2_def: shows "T2 = (*\<^sub>C) (- \<i>) ` (T \<inter> B')"
append: fixes xs :: "'b list"   and ys :: "'b list" shows "prod_list (xs @ ys) = prod_list xs * prod_list ys"
shortest_path_pos_cost_axioms: shows "shortest_path_pos_cost G dist c s num"