hcju/isabelleps huggingface dataset:
atLeastAtMost_subseteq_greaterThanLessThan_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.dense_linorder less_eq less" shows "(ord.atLeastAtMost less_eq a b \<subseteq> ord.greaterThanLessThan less c d) = (less_eq a b \<longrightarrow> less c a \<and> less b d)"
uminus_INF: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 bot :: "'b"   and top :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "uminus (Inf (B ` A)) = Sup ((\<lambda>x. uminus (B x)) ` A)"
vfsequence_axioms_def: fixes xs :: "V" shows "vfsequence_axioms xs \<equiv> \<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>"
infinite_nxtN: fixes N :: "nat set"   and n :: "nat" assumes "infinite N" shows "infinite (nxt N n)"
image_mono: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "order.greater_eq B A" shows "order.greater_eq (f ` B) (f ` A)"
distinct_53: fixes x5 :: "char list" shows "Call x5 \<noteq> Return"
finsum_vec: fixes n :: "nat" shows "finsum_vec TYPE('b) n = finsum (module_vec TYPE('b) n)"
distinct_31: fixes x3 :: "nat"   and x6 :: "msg" shows "Nonce x3 \<noteq> Hash x6"
mult_less_cancel_left_disj: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times c a) (times c b) = (less zero c \<and> less a b \<or> less c zero \<and> less b a)"
geral_U_is_expected_value_of_ocU: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and p :: "'a pmf" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "continuous_vnm (\<P> outcomes) \<R>"   and "set_pmf p \<subseteq> outcomes" shows "general_U outcomes \<R> p = measure_pmf.expectation p (ocU outcomes \<R>)"
map_endofun_def: shows "map_endofun \<equiv> (id ---> id ---> rep_endofun ---> abs_endofun) map_endofun'"
loc_imps_fw_M_nonneg: fixes c :: "('loc, 't) configuration"   and loc1 :: "'loc"   and loc2 :: "'loc"   and M :: "'t zmultiset"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and t :: "'t" assumes "loc_imps_fw c loc1 loc2 M xs"   and "inv_implications_nonneg c" shows "0 \<le> zcount M t"
ag_eq_sol1: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<in> carrier M"   and "x \<in> carrier M"   and "b \<in> carrier M"   and "a \<plusminus> x = b" shows "x = -\<^sub>a a \<plusminus> b"
cri_less_def: fixes x :: "'loc \<times> 't"   and y :: "'loc \<times> 't" shows "(x<\<^sub>py) = (x\<le>\<^sub>py \<and> x \<noteq> y)"
dsg1: fixes x :: "'a" shows "x \<cdot> antidomain_semiringl.ads_d antirange_op x = x"
rtrancl_subset: fixes R :: "('g \<times> 'g) set"   and S :: "('g \<times> 'g) set" assumes "R \<subseteq> S"   and "S \<subseteq> R\<^sup>*" shows "S\<^sup>* = R\<^sup>*"
homeomorphic_maps_iff_homeomorphism: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" shows "homeomorphic_maps (top_of_set S) (top_of_set T) f g = homeomorphism S T f g"
prj_simps_9: shows "C.cod f_src_src.Prj\<^sub>0 = src.apex"
pcr_cr_eq: shows "pcr_moebius_mat = cr_moebius_mat"
IH: fixes ss :: "'a list" assumes "ss \<in> list n A"   and "p < length ss" shows "merges f ps' ss ! p = map snd (filter (\<lambda>a. case a of (p', t') \<Rightarrow> p' = p) ps') \<Squnion>\<^bsub>f\<^esub> ss ! p"
vint_uminus_def: shows "vint_uminus = (\<lambda>x\<in>\<^sub>\<circ>\<int>\<^sub>\<circ>. (- int_of_vint x)\<^sub>\<int>)"
get_M_Element_preserved8: fixes element_ptr :: "(_) element_ptr"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and setter :: "('ay \<Rightarrow> 'ay) \<Rightarrow> ('ar, 'ap, 'aq, 'as, 'av option) RElement_scheme \<Rightarrow> ('ar, 'ap, 'aq, 'as, 'av option) RElement_scheme"   and v :: "'ay"   and h' :: "(_) heap"   and getter :: "(_) RObject_ext \<Rightarrow> 'az" assumes "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr \<noteq> object_ptr"   and "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
simps: fixes ant :: "('a, 'b) form multiset"   and suc :: "('a, 'b) form multiset" shows "antec ( ant \<Rightarrow>* suc) = ant"
min_key_val_raw_in: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ko :: "'o" assumes "fst xs \<noteq> []" shows "min_key_val_raw ko xs \<in> set (fst xs)"
simps_2: fixes f :: "('fsym, 'tp) efsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "TE.wt (Fn f Tl) = (TE_wtFsym f \<and> list_all TE.wt Tl \<and> TE_arOf f = map TE.tpOf Tl)"
wf_subst_apply: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "wf\<^sub>s\<^sub>t V S" shows "wf\<^sub>s\<^sub>t (fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)) (S \<cdot>\<^sub>s\<^sub>t \<delta>)"
th3: shows "(\<lambda>p. p \<circ> q) ` {p. p permutes S} = {p. p permutes S}"
ipresWlsAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'sort, 'h, 'i, 'd, 'e, 'j) model_scheme" shows "ipresWlsAll h hA MOD \<equiv> ipresWls h MOD \<and> ipresWlsAbs hA MOD"
mset_map_split_orig_le: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b multiset"   and M1 :: "'a multiset"   and M2 :: "'a multiset" assumes "f `# P \<subseteq># M1 + M2" obtains P1 :: "'b multiset" and P2 :: "'b multiset" where "P = P1 + P2" and "f `# P1 \<subseteq># M1" and "f `# P2 \<subseteq># M2"
map_graph_fn_id_2: fixes X :: "('e, 'f) labeled_graph" shows "map_graph (Id_on (vertices X)) X = restrict X"
combine_options_cases: fixes x :: "'a option"   and P :: "'a option \<Rightarrow> 'b option \<Rightarrow> bool"   and y :: "'b option" assumes "x = None \<Longrightarrow> P x y"   and "y = None \<Longrightarrow> P x y"   and "\<And>a b. \<lbrakk>x = Some a; y = Some b\<rbrakk> \<Longrightarrow> P x y" shows "P x y"
locally_sound_mode_use_def2: fixes lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "locally_sound_mode_use lc \<equiv> \<forall>lc'\<in>loc_reach lc. respects_own_guarantees (fst lc')"
t_A'_def: fixes n :: "nat" shows "t_A' n = c_A' n + p_A' n"
assms_1: shows "pd_kleene_allegory_class.acyclic w"
nat_cut_ge_infinite_not_empty: fixes I :: "nat set"   and t :: "nat" assumes "infinite I" shows "I \<down>\<ge> t \<noteq> {}"
mono_gamma_c: fixes C1 :: "'av st option acom"   and C2 :: "'av st option acom" assumes "C1 \<le> C2" shows "\<gamma>\<^sub>c C1 \<le> \<gamma>\<^sub>c C2"
consequence_graphI: fixes Rs :: "(('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph) set"   and G :: "('b, 'd) labeled_graph" assumes "\<And>R. R \<in> Rs \<Longrightarrow> maintained R G"   and "\<And>R. R \<in> Rs \<Longrightarrow> subgraph (fst R) (snd R)"   and "graph G" shows "consequence_graph Rs G"
simps: fixes a :: "'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set" shows "(a \<in> derivable' R) = ((\<exists>C. a = C \<and> ([], C) \<in> R) \<or> (\<exists>r. a = snd r \<and> r \<in> R \<and> fst r \<noteq> [] \<and> (\<forall>p\<in>set (fst r). p \<in> derivable' R)))"
resultant_def: fixes p :: "'b poly"   and q :: "'b poly" shows "resultant p q = det (sylvester_mat p q)"
idemp_embedding_maintained_preserved: fixes G :: "('b, 'c) labeled_graph"   and f :: "'c \<Rightarrow> 'c"   and r :: "('b, 'd) labeled_graph \<times> ('b, 'd) labeled_graph" assumes "subgraph (map_graph_fn G f) G"   and "\<And>x. x \<in> vertices G \<Longrightarrow> (f \<circ> f) x = f x"   and "maintained r G" shows "maintained r (map_graph_fn G f)"
subalg: fixes M :: "'b measure"   and F :: "'b measure" assumes "sigma_finite_subalgebra M F" shows "subalgebra M F"
IdNatTrans_defs_3: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "MakeNT \<eta> \<equiv> \<lparr>NTDom = NTDom \<eta>, NTCod = NTCod \<eta>, NatTransMap = restrict (($$) \<eta>) (obj\<^bsub>NTCatDom \<eta>\<^esub>)\<rparr>"
new_document_is_l_new_document: shows "l_new_document ShadowRootClass.type_wf"
has_ell2_norm_finite: fixes x :: "'c \<Rightarrow> complex" shows "has_ell2_norm x"
red2_Nml_Src: fixes t :: "'c local.term" assumes "Ide t"   and "Nml t" shows "\<lbrace>t \<^bold>\<Down> Src t\<rbrace> = \<r>[\<lbrace>t\<rbrace>]"
trans_trancl: fixes r :: "('b \<times> 'b) set" shows "trans (r\<^sup>+)"
Compl_subset_Compl_iff: fixes A :: "'b set"   and B :: "'b set" shows "(- A \<subseteq> - B) = (B \<subseteq> A)"
neq_mono: fixes x :: "real"   and bs :: "real list" shows "\<forall>a n s. Inum (x # bs) a = Inum (x # bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x # bs) (neq n s) = Ifm (x # bs) (NEq a)"
assms: shows "P\<^sub>1 \<sqsubseteq> P\<^sub>2"
positive_zero: fixes n :: "nat" shows "positive (0\<^sub>m n n)"
funpow_mult: fixes n :: "nat"   and m :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "(f ^^ m) ^^ n = f ^^ (m * n)"
span_clauses_3: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "x \<in> ideal S"   and "y \<in> ideal S" shows "x + y \<in> ideal S"
alpha_Tree_eqvt_aux: fixes t :: "('a, 'b, 'c) Formula.Tree"   and p :: "perm" assumes "\<And>a b. ((a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> t) = (p \<bullet> (a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> p \<bullet> t)" shows "p \<bullet> supp_rel (=\<^sub>\<alpha>) t = supp_rel (=\<^sub>\<alpha>) (p \<bullet> t)"
Polygamma_approx_2_real: fixes x :: "real"   and m :: "nat" assumes "0 < x"   and "0 < m" shows "Polygamma_approx (Suc (Suc 0)) m x = stirling_sum' 1 m x"
fin_B: shows "finite {x. f x \<noteq> (0::'c)}"
I_list_disj: fixes fs :: "atom fm list"   and xs :: "real list" shows "R.I (list_disj fs) xs = (\<exists>\<phi>\<in>set fs. R.I \<phi> xs)"
G_reach_1: shows "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> u"
Abs_freeword_Cons: fixes x :: "'a \<times> bool"   and xs :: "('a \<times> bool) list" assumes "proper_signed_list (x # xs)" shows "Abs_freeword (x # xs) = Abs_freeword [x] + Abs_freeword xs"
insert_not_empty: fixes a :: "'b"   and A :: "'b set" shows "Set.insert a A \<noteq> {}"
partition_on_mset_elems: fixes A :: "'a multiset"   and P :: "'a multiset multiset"   and p1 :: "'a multiset"   and x :: "'a" assumes "partition_on_mset A P"   and "p1 \<in># P"   and "x \<in># p1" shows "x \<in># A"
restrict_space_measurable: fixes X :: "'c set"   and K :: "'c measure"   and N :: "'d \<Rightarrow> 'c measure"   and M :: "'d measure" assumes "X \<noteq> {}"   and "X \<in> sets K"   and "N \<in> M \<rightarrow>\<^sub>M subprob_algebra K" shows "(\<lambda>x. restrict_space (N x) X) \<in> M \<rightarrow>\<^sub>M subprob_algebra (restrict_space K X)"
subo: shows "Subobjs\<^sub>R P (hd (C' # Cs')) ((C' # Cs') @ [D])"
Cons_1: assumes "sorted_wrt r (insort_wrt s x xs)" shows "sorted_wrt r xs"
a: shows "a \<in> Class \<one> |\<cdot> g"
distinct_35: fixes x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm"   and x7 :: "'b ptrm" shows "PFn x41 x42 x43 \<noteq> PSnd x7"
rotater_map2: fixes xs :: "'b list"   and ys :: "'c list"   and n :: "nat"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "length xs = length ys" shows "rotater n (map2 f xs ys) = map2 f (rotater n xs) (rotater n ys)"
left_neutral: fixes a :: "'b" shows "(zero_class.zero::'b) + a = a"
valid_path_aux_callstack_prefix: fixes cs :: "'edge list"   and cs' :: "'edge list"   and as :: "'edge list" assumes "valid_path_aux (cs @ cs') as" shows "valid_path_aux cs as"
combine_rbt_sorted: fixes l :: "('a, 'c) rbt"   and r :: "('a, 'c) rbt"   and k :: "'a" assumes "rbt_sorted l"   and "rbt_sorted r"   and "l |\<guillemotleft> k"   and "k \<guillemotleft>| r" shows "rbt_sorted (combine l r)"
map_carrier_mat: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b mat"   and nr :: "nat"   and nc :: "nat" shows "(map_mat f A \<in> carrier_mat nr nc) = (A \<in> carrier_mat nr nc)"
quotient_of_def: fixes x :: "rat" shows "quotient_of x = (THE pair. x = Rat.Fract (fst pair) (snd pair) \<and> 0 < snd pair \<and> coprime (fst pair) (snd pair))"
span_is_submodule: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and S :: "'g set" assumes "module R M"   and "S \<subseteq> carrier M" shows "LinearCombinations.submodule R (module.span R M S) M"
scale_shift_smooth: fixes a :: "real"   and b :: "real"   and s :: "real set" shows "(\<lambda>x. a + b * x) C1_differentiable_on s"
emeasure_real: fixes A :: "('a \<Rightarrow> 'b) set" shows "\<exists>r\<ge>0. emeasure (Pi\<^sub>M I M) A = ennreal r"
internal_case_prod_conv: fixes c :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'c" shows "??.Product_Type.internal_case_prod c (a, b) = c a b"
Max_by_in: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "A \<noteq> {}" shows "Max_by f A \<in> A"
code: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" shows "derivation_from St = LCons St (if is_final_dstate St then LNil else derivation_from (deterministic_RP_step St))"
shiftr1_bl: fixes w :: "'b word" shows "shiftr1 w = of_bl (butlast (to_bl w))"
asym: fixes a :: "'g"   and b :: "'g" assumes "a < b"   and "b < a" shows "False"
castmsg_simps_1: fixes P :: "'d \<Rightarrow> bool"   and R :: "nat set"   and m :: "'d" shows "castmsg P (R:*cast(m)) = P m"
image_f_inv_f: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'a set" assumes "surj f" shows "f ` inv f ` A = A"
not_equiv_trans2: fixes eqv :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.equiv eqv"   and "eqv x y"   and "\<not> eqv y z" shows "\<not> eqv x z"
assms_1: shows "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
NoRW\<^sub>A_implies_NoRW\<^sub>C: fixes x :: "Eg1.var"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set" assumes "x \<in> mds\<^sub>A_of mds\<^sub>C AsmNoReadOrWrite" shows "Eg2_var\<^sub>C_of_Eg1 x \<in> mds\<^sub>C AsmNoReadOrWrite"
trading_strategy_init: fixes p :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "trading_strategy p"   and "support_adapt Mkt p" shows "\<exists>c. \<forall>w\<in>space M. val_process Mkt p 0 w = c"
shunt2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> '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 bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (inf x (uminus y)) z = less_eq x (sup y z)"
idAA: shows "id \<noteq> rotate_A \<circ> rotate_A"
rtrancl_on_into_rtrancl_on: fixes a :: "'a"   and b :: "'a"   and F :: "'a set"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> rtrancl_on F r"   and "(b, c) \<in> r"   and "c \<in> F" shows "(a, c) \<in> rtrancl_on F r"
length_drop: fixes n :: "nat"   and xs :: "'b list" shows "\<^bold>|drop n xs\<^bold>| = \<^bold>|xs\<^bold>| - n"
scene_compat_copy: fixes X :: "'a scene"   and Y :: "'a scene" shows "(X ##\<^sub>S Y) = (\<forall>s. (\<lambda>s'. s' \<oplus>\<^sub>S s on X) \<circ> (\<lambda>s'. s' \<oplus>\<^sub>S s on Y) = (\<lambda>s'. s' \<oplus>\<^sub>S s on Y) \<circ> (\<lambda>s'. s' \<oplus>\<^sub>S s on X))"
irreducible_not_unit: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.irreducible times one zero p" shows "\<not> dvd.dvd times p one"
induct: fixes x1 :: "'c"   and x2 :: "'c"   and U :: "'c set"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "(x1, x2) \<in> on U r\<^sup>+"   and "\<And>a b. \<lbrakk>a \<in> U; b \<in> U; (a, b) \<in> r\<rbrakk> \<Longrightarrow> P a b"   and "\<And>a b c. \<lbrakk>a \<in> U; b \<in> U; c \<in> U; (a, b) \<in> on U r\<^sup>+; P a b; (b, c) \<in> r\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
to_nat_from_nat: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and j :: "'a"   and k :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.to_nat Rep j = k" shows "mod_type.from_nat Abs k = j"
not_in_isovarspar: fixes p :: "real mpoly"   and var :: "nat"   and x :: "nat"   and q :: "real mpoly" assumes "isolate_variable_sparse p var x = q" shows "var \<notin> vars q"
connected_space_closedin_eq: fixes X :: "'b topology" shows "connected_space X = (\<nexists>E1 E2. closedin X E1 \<and> closedin X E2 \<and> E1 \<union> E2 = topspace X \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
npowf_minus: fixes x :: "'a"   and j :: "int" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "(x\<^bsub>K\<^esub>\<^bsup>j\<^esup>)\<^bsup>\<hyphen> K\<^esup> = x\<^bsub>K\<^esub>\<^bsup>- j\<^esup>"
residuated_conj2: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "conjugation_pair (conjugate f) f"
NSLIM_NSCRLIM_Re_Im_iff: fixes f :: "'a \<Rightarrow> complex"   and a :: "'a"   and L :: "complex" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L = ((\<lambda>x. Re (f x)) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S Re L \<and> (\<lambda>x. Im (f x)) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S Im L)"
gcsort_subset: fixes t' :: "nat \<times> nat list \<times> 'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and t :: "nat \<times> nat list \<times> 'c list" assumes "t' \<in> gcsort_set index key p t" shows "gcsort_set index key p t' \<subseteq> gcsort_set index key p t"
stable: fixes y :: "'a" assumes "y \<in> set ys" shows "filter (\<lambda>ya. compare cmp y ya = Equiv) ys = filter (\<lambda>ya. compare cmp y ya = Equiv) xs"
assms_4: shows "\<forall>x. insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var 2) \<noteq> 0 \<or> insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var 1) \<noteq> 0 \<or> insertion (nth_default 0 (xs @ x # \<Gamma>)) (isolate_variable_sparse p var 0) \<noteq> 0"
cut_greater_CommuteWithIntervalCut4: fixes t :: "'a" shows "CommuteWithIntervalCut4 (\<lambda>I. I \<down>> t)"
sameDom_sym: fixes inp :: "'d \<Rightarrow> 'e option"   and inp' :: "'d \<Rightarrow> 'f option" shows "sameDom inp inp' = sameDom inp' inp"
eq: shows "More_Modules.ideal (insert p (set_mset A \<union> polynomial_bool)) = More_Modules.ideal (insert p (set_mset A \<union> (\<lambda>c. (Var c)\<^sup>2 - Var c) ` {c. c \<noteq> x'}))"
pop_pow_one: fixes m :: "nat"   and r :: "'b list" assumes "m \<noteq> 0" shows "r \<^sup>@ m = r \<cdot> r \<^sup>@ (m - 1)"
linear_surjective_right_inverse: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "surj f" shows "\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> f \<circ> g = id"
count_list_eq_0_iff: fixes xs :: "'c list"   and x :: "'c" shows "(count_list xs x = 0) = (x \<notin> set xs)"
igSubstInpIPresIGWlsInp_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstInpIPresIGWlsInp MOD \<equiv> \<forall>ys y Y delta inp. igWls MOD (asSort ys) Y \<and> igWlsInp MOD delta inp \<longrightarrow> igWlsInp MOD delta (igSubstInp MOD ys Y y inp)"
imult_ile_mono1: fixes i :: "enat"   and j :: "enat"   and k :: "enat" assumes "i \<le> j" shows "i * k \<le> j * k"
lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S_def: fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>" shows "formula.lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S \<Sigma> n \<phi> = \<Union> {formula.enc \<Sigma> (w, I) |w I. length I = n \<and> formula.wf_interp_for_formula \<Sigma> (w, I) \<phi> \<and> formula.satisfies \<Sigma> (w, I) \<phi>}"
rolling_rule: fixes g :: "'b \<rightarrow> 'a"   and f :: "'a \<rightarrow> 'b" shows "fix\<cdot>(g oo f) = g\<cdot>(fix\<cdot>(f oo g))"
all_security_requirements_fulfilled_IFS_D: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" assumes "all_security_requirements_fulfilled M G" shows "all_security_requirements_fulfilled (get_IFS M) G"
graph_symmetric: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^bsub>del_vert u\<^esub> v" shows "v \<rightarrow>\<^bsub>del_vert u\<^esub> u"
card1: fixes S :: "nat set"   and X :: "nat set"   and Y :: "nat set"   and m :: "nat" assumes "finite S"   and "finite X"   and "finite Y"   and "X \<inter> Y = {}"   and "S \<inter> (X \<union> Y) = {}"   and "S \<union> X \<union> Y = {0..<m}" shows "card {xs. (\<forall>i\<in>X. xs ! i) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m} = 2 ^ (m - card X - card Y)"
singleton_insert_inj_eq': fixes a :: "'a"   and A :: "'a set"   and b :: "'a" shows "(set_ins a A = {b}) = (a = b \<and> A \<subseteq> {b})"
fl: shows "VFun f \<in> E (ELam x e) \<rho>"
assms_2: shows "A = P\<^sup>-\<^sup>1 * (\<d>\<i>\<a>\<g> i. f i) * P"
elims_2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and x :: "'a list"   and xa :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.valid_path_aux kind get_return_edges x xa" obtains cs :: "'a list" where "x = cs__" and "xa = []"   | cs :: "'a list" and a :: "'a" and as :: "'a list" where "x = cs" and "xa = a # as" and "case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> CFG.valid_path_aux kind get_return_edges (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> CFG.valid_path_aux kind get_return_edges [] as | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> CFG.valid_path_aux kind get_return_edges cs' as | _ \<Rightarrow> CFG.valid_path_aux kind get_return_edges cs as"
eq_scons: fixes w :: "'c stream"   and a :: "'c"   and v :: "'c stream" shows "(w = a ## v) = (a = shd w \<and> v = stl w)"
precond: shows "pre_splitFace g' v a f' [countVertices g'..<countVertices g' + n]"