hcju/isabelleps huggingface dataset:
slice_kind_def: 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 Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and S :: "'b SDG_node set"   and a :: "'a" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.slice_kind sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S a \<equiv> SDG.slice_kind_aux sourcenode targetnode kind valid_edge get_proc procs Exit ParamDefs (sourcenode a) (targetnode a) (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S) (kind a)"
lconjby_inj: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.group_add minus plus zero uminus" shows "inj (\<lambda>y. minus (plus x y) x)"
choice_alt_def: fixes t :: "transition"   and t' :: "transition" shows "choice_alt t t' = (\<exists>i r. apply_guards (Guards t @ Guards t') (join_ir i r))"
atoms_formula_union: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set" shows "atoms_formula (S1 \<union> S2) = atoms_formula S1 \<union> atoms_formula S2"
strict_order_order: fixes x :: "'a" assumes "x \<le> - top \<and> x \<sqinter> x \<le> x" shows "(top \<le> x \<squnion> top \<and> (x \<squnion> top) \<sqinter> (x \<squnion> top) \<le> x \<squnion> top) \<and> (x \<squnion> top) \<sqinter> id (x \<squnion> top) \<le> top"
simps_4: shows "sch_to_nat Base_snd = c_pair 3 0"
finite_ordLess_infinite: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "Well_order r"   and "Well_order r'"   and "finite (Field r)"   and "infinite (Field r')" shows "r <o r'"
l9_41_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and R :: "'p"   and Q :: "'p" assumes "A B CTSP P R"   and "A B CTSP Q R" shows "A B C OSP P Q"
elims: fixes x :: "'b list"   and y :: "int" assumes "changes x = y" obtains "x = []" and "y = 0"   | uu_ :: "'b" where "x = [uu_]" and "y = 0"   | x1 :: "'b" and x2 :: "'b" and xs :: "'b list" where "x = x1 # x2 # xs" and "y = (if x1 * x2 < (0::'b) then 1 + changes (x2 # xs) else if x2 = (0::'b) then changes (x1 # xs) else changes (x2 # xs))"
that: fixes D' :: "('fun, 'var) extstrand_step list" assumes "D @ D' \<in> decomps\<^sub>e\<^sub>s\<^sub>t M N \<I>"   and "M \<union> ik\<^sub>e\<^sub>s\<^sub>t (D @ D') \<turnstile>\<^sub>c t"   and "M \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<subset> M \<union> ik\<^sub>e\<^sub>s\<^sub>t (D @ D')" shows "thesis"
ardenable_all_def: fixes ES :: "('b \<times> 'c trm set) set" shows "ardenable_all ES \<equiv> \<forall>(X, rhs)\<in>ES. ardenable rhs"
in_idem: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and x :: "'b" assumes "semilattice_set f"   and "finite A"   and "x \<in> A" shows "f x (semilattice_set.F f A) = semilattice_set.F f A"
dehomogenize_dehomogenize: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "dehomogenize x (dehomogenize x p) = dehomogenize x p"
op_norm_diag_mat_eq: fixes f :: "'a \<Rightarrow> real" shows "\<parallel>diag_mat f\<parallel>\<^sub>o\<^sub>p = Max {\<bar>f i\<bar> |i. i \<in> UNIV}"
ddh_1_def: fixes \<G> :: "'d cyclic_group"   and \<A> :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool spmf" shows "Diffie_Hellman.ddh.ddh_1 \<G> \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. sample_uniform (order \<G>) \<bind> (\<lambda>z. \<A> (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> x) (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> y) (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> z))))"
induct: fixes P :: "nat \<Rightarrow> bool list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "bool list" assumes "\<And>n. P n []"   and "\<And>n bs. P (Suc n) bs \<Longrightarrow> P n (False # bs)"   and "\<And>n bs. P (Suc n) (mark_out n bs) \<Longrightarrow> P n (True # bs)" shows "P a0 a1"
test_preserves_equation: fixes p :: "'a"   and x :: "'a" assumes "dense_rel p"   and "coreflexive p" shows "(p * x \<le> x * p) = (p * x = p * x * p)"
is_Fun_SetE: fixes t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term" assumes "is_Fun_Set t" obtains s :: "'g" where "t = Fun (Set s) []"
join_below: fixes x :: "'a"   and y :: "'a"   and a :: "'a" assumes "compatible x y"   and "x \<sqsubseteq> a"   and "y \<sqsubseteq> a" shows "x \<squnion> y \<sqsubseteq> a"
m_sup': shows "concl_of ` F.Inf_between (fst ` FL.active_subset N) {C} \<subseteq> fst ` M"
lhs_def: shows "lhs \<equiv> \<lambda>plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1. module_pair_ow (Collect (Domainp B\<^sub>1)) plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 (Collect (Domainp B\<^sub>2))"
norm_blinop_def: shows "norm \<equiv> map_fun Rep_blinop id norm"
arr_iff_in_hom: fixes f :: "'a" shows "B.arr f = \<guillemotleft>f : B.dom f \<Rightarrow>\<^sub>B B.cod f\<guillemotright>"
R2_xor_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "R2_xor A B = return_spmf B"
aact: shows "\<forall>a. aact a \<in> set (jAction (SPR.MCn (tLength t)) t a)"
bij_betw_id: fixes A :: "'c set" shows "bij_betw id A A"
filter_min_append_minimal: fixes xs :: "'c list"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ys :: "'c list"   and x :: "'c"   and y :: "'c" assumes "\<And>x' y'. \<lbrakk>x' \<in> set xs; y' \<in> set xs; rel x' y'\<rbrakk> \<Longrightarrow> x' = y'"   and "\<And>x' y'. \<lbrakk>x' \<in> set ys; y' \<in> set ys; rel x' y'\<rbrakk> \<Longrightarrow> x' = y'"   and "x \<in> set (filter_min_append rel xs ys)"   and "y \<in> set (filter_min_append rel xs ys)"   and "rel x y" shows "x = y"
assms: fixes param :: "'k" shows "Rel (crel_vs (=)) (dp param) (dp\<^sub>T param)"
conjugate_zero: shows "conjugate 0\<^sub>h = 0\<^sub>h"
seqSubsNil: fixes P :: "('a, 'b, 'c) psi" shows "P[<[]>] = P"
f_eq: fixes x :: "'b"   and i :: "'a" assumes "x \<in> space M"   and "P i x" shows "f x = i"
eqD_15: fixes y :: "cmsg"   and N :: "nonce_t" assumes "eq y (cNonce N)" shows "y = cNonce N"
A_fin: shows "finite {k. k < length xs \<and> xs ! k = i}"
UP_def: fixes R :: "('a, 'b) ring_scheme" shows "UP R = \<lparr>carrier = up R, mult = \<lambda>p\<in>up R. \<lambda>q\<in>up R. \<lambda>n. \<Oplus>\<^bsub>R\<^esub>i\<in>{..n}. p i \<otimes>\<^bsub>R\<^esub> q (n - i), one = \<lambda>i. if i = 0 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>, zero = \<lambda>i. \<zero>\<^bsub>R\<^esub>, add = \<lambda>p\<in>up R. \<lambda>q\<in>up R. \<lambda>i. p i \<oplus>\<^bsub>R\<^esub> q i, smult = \<lambda>a\<in>carrier R. \<lambda>p\<in>up R. \<lambda>i. a \<otimes>\<^bsub>R\<^esub> p i, monom = \<lambda>a\<in>carrier R. \<lambda>n i. if i = n then a else \<zero>\<^bsub>R\<^esub>, coeff = \<lambda>p\<in>up R. p\<rparr>"
C_merge_ub: fixes xs :: "'c list"   and ys :: "'c list" shows "C_merge xs ys \<le> length xs + length ys"
simps: fixes i :: "nat"   and j :: "nat" shows "fs'.gs.\<mu> i j = (if j < i then map of_int_hom.vec_hom fs' ! i \<bullet> fs'.gs.gso j / \<parallel>fs'.gs.gso j\<parallel>\<^sup>2 else if i = j then 1 else 0)"
simps: fixes a :: "'b"   and s :: "'b + 'c" shows "(a \<in> Basic_BNFs.setl s) = (\<exists>x. a = x \<and> s = Inl x)"
\<sigma>\<^sub>T\<^sub>O\<^sub>Y_def: fixes i :: "nat" shows "\<sigma>\<^sub>T\<^sub>O\<^sub>Y i \<equiv> {(toy_init i, \<Gamma>\<^sub>T\<^sub>O\<^sub>Y PToy)}"
simps_6: fixes r :: "'b mrexp2"   and f :: "bool" shows "ok2 (Star2 r f) = (ok2 r \<and> f = final (strip2 r))"
unfolding_monotonic: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set" assumes "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<nu>" shows "w \<Turnstile>\<^sub>n Unf \<phi>[X]\<^sub>\<nu>"
simps: fixes A :: "rat mat"   and signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list"   and lhs_vec :: "rat vec" shows "reduction_step_R A signs subsets lhs_vec = (let reduce_cols_A = reduce_mat_cols A lhs_vec; rows_keep = rows_to_keep reduce_cols_A in (take_rows_from_matrix reduce_cols_A rows_keep, take_indices subsets rows_keep, take_indices signs (find_nonzeros_from_input_vec lhs_vec)))"
ideal_view_1_def: fixes funct :: "'f \<Rightarrow> 'g \<Rightarrow> ('h \<times> 'e) spmf"   and x :: "'f"   and y :: "'g"   and z :: "'i"   and S :: "('j \<Rightarrow> 'f \<Rightarrow> 'i \<Rightarrow> ('f \<times> 'k) spmf) \<times> ('j \<Rightarrow> 'f \<Rightarrow> 'i \<Rightarrow> 'h \<Rightarrow> 'k \<Rightarrow> 'd spmf)"   and \<A> :: "'j" shows "malicious_base.ideal_view_1 funct x y z S \<A> = (let (S1, S2) = S in malicious_base.ideal_game_1 funct x y z (S1 \<A>, S2 \<A>))"
index_distinct_alt_ss: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2"   and "\<And>G. G \<in> \<G> \<Longrightarrow> \<not> ps \<subseteq> G" shows "int (\<B> index ps) = \<Lambda>"
lconjby_Abs_freelist_relator_freeword: fixes S :: "'b set"   and R :: "'b list set"   and rs :: "'b list"   and xs :: "'b list" assumes "GroupWithGeneratorsRelators S R"   and "rs \<in> R"   and "xs \<in> lists S" shows "lconjby (Abs_freelist xs) (Abs_freeword (map (charpair S) rs)) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
assoc_is_natural_1: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and H :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<a> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<i> :: "'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a> \<mu> \<nu> \<tau> = V (H \<mu> (H \<nu> \<tau>)) (\<a> (partial_magma.dom V \<mu>) (partial_magma.dom V \<nu>) (partial_magma.dom V \<tau>))"
cap_objects_def: fixes cap :: "cdl_cap" shows "cap_objects cap \<equiv> case cap of EndpointCap x xa \<Rightarrow> {x} | CNodeCap x \<Rightarrow> {x} | TcbCap x \<Rightarrow> {x}"
abs_eq: fixes x :: "circline_mat" shows "of_circline (abs_circline x) = abs_ocircline (of_circline_clmat x)"
SeqWRP_Succ: fixes s :: "name"   and s1 :: "tm"   and k1 :: "tm"   and y :: "tm" assumes "atom s \<sharp> (s1, k1, y)" shows "{SeqWRP s1 k1 y} \<turnstile> SyntaxN.Ex s (SeqWRP (Var s) (SUCC k1) (Q_Succ y))"
aug_ipp_closed: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set"   and x :: "'e" assumes "commute_bpp f (aug_pm_set z i A)"   and "assoc_bpp (aug_pm_set z i A) f"   and "ipp_cond1 A i"   and "ipp_cond2 z A i f"   and "ipp_cond3 z i"   and "x \<in> addition_set f (aug_pm_set z i A)" shows "\<^sub>i- x \<in> addition_set f (aug_pm_set z i A)"
thin2: fixes A1 :: "fm"   and H :: "fm set"   and B :: "fm"   and A2 :: "fm" assumes "insert A1 H \<turnstile> B" shows "insert A1 (insert A2 H) \<turnstile> B"
in_fst_imageE: fixes x :: "'e"   and S :: "('e \<times> 'f) set" assumes "x \<in> fst ` S" obtains y :: "'f" where "(x, y) \<in> S"
fact: shows "dterm_sem I (f1 fid1 vid1) (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) = snd (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) $ vid1"
d_OUT_lift_SOURCE: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow_attainability \<Delta>" shows "d_OUT (flow_attainability.lift \<Delta> f) SOURCE = value_flow \<Delta> f"
assms_2: fixes n :: "nat" shows "0 \<le> a n"
castmsg_simps_5: fixes P :: "'d \<Rightarrow> bool"   and i1 :: "nat"   and i2 :: "nat" shows "castmsg P (disconnect(i1, i2)) = True"
OrdP_Trans: fixes H :: "fm set"   and k :: "tm"   and i :: "tm"   and j :: "tm" assumes "H \<turnstile> OrdP k"   and "H \<turnstile> i IN j"   and "H \<turnstile> j IN k" shows "H \<turnstile> i IN k"
induct: fixes x1 :: "('a, 'b, 'c) expr"   and x2 :: "('a, 'b, 'c) cntxt"   and x3 :: "('a, 'b, 'c) expr"   and P :: "('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) cntxt \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> bool" assumes "decompose x1 x2 x3"   and "\<And>e. redex e \<Longrightarrow> P e \<box> e"   and "\<And>e\<^sub>1 e\<^sub>2 \<E> r. \<lbrakk>\<not> redex (Apply e\<^sub>1 e\<^sub>2); decompose e\<^sub>1 \<E> r; P e\<^sub>1 \<E> r\<rbrakk> \<Longrightarrow> P (Apply e\<^sub>1 e\<^sub>2) (ApplyL\<^sub>\<E> \<E> e\<^sub>2) r"   and "\<And>v e \<E> r. \<lbrakk>\<not> redex (Apply (VE v) e); decompose e \<E> r; P e \<E> r\<rbrakk> \<Longrightarrow> P (Apply (VE v) e) (ApplyR\<^sub>\<E> v \<E>) r"   and "\<And>e\<^sub>1 e\<^sub>2 e\<^sub>3 \<E> r. \<lbrakk>\<not> redex (Ite e\<^sub>1 e\<^sub>2 e\<^sub>3); decompose e\<^sub>1 \<E> r; P e\<^sub>1 \<E> r\<rbrakk> \<Longrightarrow> P (Ite e\<^sub>1 e\<^sub>2 e\<^sub>3) (Ite\<^sub>\<E> \<E> e\<^sub>2 e\<^sub>3) r"   and "\<And>e \<E> r. \<lbrakk>\<not> redex (Ref e); decompose e \<E> r; P e \<E> r\<rbrakk> \<Longrightarrow> P (Ref e) (Ref\<^sub>\<E> \<E>) r"   and "\<And>e \<E> r. \<lbrakk>\<not> redex (Read e); decompose e \<E> r; P e \<E> r\<rbrakk> \<Longrightarrow> P (Read e) (Read\<^sub>\<E> \<E>) r"   and "\<And>e\<^sub>1 e\<^sub>2 \<E> r. \<lbrakk>\<not> redex (Assign e\<^sub>1 e\<^sub>2); decompose e\<^sub>1 \<E> r; P e\<^sub>1 \<E> r\<rbrakk> \<Longrightarrow> P (Assign e\<^sub>1 e\<^sub>2) (AssignL\<^sub>\<E> \<E> e\<^sub>2) r"   and "\<And>l e\<^sub>2 \<E> r. \<lbrakk>\<not> redex (Assign (VE (Loc l)) e\<^sub>2); decompose e\<^sub>2 \<E> r; P e\<^sub>2 \<E> r\<rbrakk> \<Longrightarrow> P (Assign (VE (Loc l)) e\<^sub>2) (AssignR\<^sub>\<E> l \<E>) r"   and "\<And>e \<E> r. \<lbrakk>\<not> redex (Rjoin e); decompose e \<E> r; P e \<E> r\<rbrakk> \<Longrightarrow> P (Rjoin e) (Rjoin\<^sub>\<E> \<E>) r" shows "P x1 x2 x3"
False: shows "\<not> (x = 0 \<or> y = 0 \<or> z = 0)"
unfold_ruleset_CHAIN_safe_def: shows "unfold_ruleset_CHAIN_safe = unfold_optimize_ruleset_CHAIN optimize_primitive_univ"
IntD1: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A \<inter> B" shows "c \<in> A"
mod_ring_inverse_integer: shows "(mod_ring_rel_integer ===> mod_ring_rel_integer) (inverse_p_integer pp) inverse"
simps_1: shows "min_height \<langle>\<rangle> = 0"
insert_stays_forest_means_not_connected: fixes u :: "'b"   and v :: "'b"   and E :: "'b uprod set" assumes "forest (insert (Upair u v) E)"   and "Upair u v \<notin> E"   and "u \<noteq> v" shows "(u, v) \<notin> uconnected E"
assms_3: shows "fds_nth f 1 \<noteq> (0::'a)"
nn_integral_diff: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and g :: "'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integral\<^sup>N M g \<noteq> \<infinity>"   and "AE x in M. g x \<le> f x" shows "\<integral>\<^sup>+ x. f x - g x \<partial>M = integral\<^sup>N M f - integral\<^sup>N M g"
bot_similar'_base: shows "similar'_base \<bottom> \<bottom>"
RS: shows "\<forall>i. (r i, r (Suc i)) \<in> E"
mono_mndet_D_process: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process" assumes "\<forall>x\<in>A. P x \<sqsubseteq>\<^sub>D Q x" shows "mndet A P \<sqsubseteq>\<^sub>D mndet A Q"
vfr_1: shows "vfp holomorphic_on cball z vfr"
invariantD: fixes A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "A \<TTurnstile> (I \<rightarrow>) P"   and "s \<in> reachable A I" shows "P s"
insertCI: fixes a :: "'g"   and B :: "'g set"   and b :: "'g" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
Inf_insert: fixes Inf :: "'f set \<Rightarrow> 'f"   and Sup :: "'f set \<Rightarrow> 'f"   and inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and bot :: "'f"   and top :: "'f"   and a :: "'f"   and A :: "'f set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (insert a A) = inf a (Inf A)"
ramanujan_sum_dirichlet_form: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "c k n = (\<Sum>d | d dvd gcd n k. of_nat d * moebius_mu (k div d))"
inj: shows "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
funpow_add: fixes m :: "nat"   and n :: "nat"   and f :: "'a \<Rightarrow> 'a" shows "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
abs_ocircline_def: shows "abs_ocircline \<equiv> quot_type.abs ocircline_eq_clmat Abs_ocircline"
singleCombinatorsAppend: fixes xs :: "('b, 'c) Combinators list"   and ys :: "('b, 'c) Combinators list" assumes "singleCombinators xs"   and "singleCombinators ys" shows "singleCombinators (xs @ ys)"
sim_sym: fixes c :: "com"   and c' :: "com" shows "(c \<sim> c') = (c' \<sim> c)"
15: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<sqinter> z = x \<sqinter> z \<squnion> y \<sqinter> z"
M_2: shows "ord_of_nat ` A \<subseteq> ON"
\<sigma>3's: shows "set (labels \<sigma>3') \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step}"
distinct_7: fixes x21 :: "'a"   and x22 :: "('a, 'b) form list"   and x31 :: "'b"   and x32 :: "('a, 'b) form list" shows "Compound x21 x22 \<noteq> Modal x31 x32"
card_UNIV_sdim_def: shows "card_UNIV_class.card_UNIV = Phantom(sdim) 7"
invariantCEntailedAfterConflict: fixes conflictFlag :: "bool"   and F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and clause :: "Literal list"   and C' :: "Literal list"   and conflictFlag' :: "bool" assumes "conflictFlag = False"   and "formulaFalse F (elements M)"   and "clause el F"   and "clauseFalse clause (elements M)"   and "C' = clause"   and "conflictFlag' = True" shows "InvariantCEntailed conflictFlag' F C'"
Q_weight_def: shows "Q_weight \<equiv> \<lambda>Q. card (\<Union> (H.edges_of_walk ` Q) \<inter> B)"
bij_apply: fixes f :: "'a perm" shows "bij ((\<langle>$\<rangle>) f)"
bound_of_mono: fixes P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "P \<tturnstile> Q"   and "Expectations.bounded P"   and "Expectations.bounded Q" shows "bound_of P \<le> bound_of Q"
polyfun_extremal_lemma: fixes e :: "real"   and c :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "0 < e" shows "\<exists>M. \<forall>z. M \<le> norm z \<longrightarrow> norm (\<Sum>i\<le>n. c i * z ^ i) \<le> e * norm z ^ Suc n"
has_codomain_iff_arr: fixes f :: "'a \<times> 'a" shows "(VoV.codomains f \<noteq> {}) = VoV.arr f"
tighten_bounds_3_def: shows "tighten_bounds_3 \<equiv> (rep_real_alg_3 ---> Real_Alg_Invariant) tighten_bounds_2"
no_zeros: fixes r :: "nat" assumes "r \<in> set [10, 10, 10]" shows "0 < r"
even_sum_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "finite A" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (comm_monoid_add.sum plus zero f A) = even (card {a \<in> A. \<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (f a)})"
cases: fixes x :: "nat list" obtains "xa__ = []"   | x :: "nat" and xs :: "nat list" where "xa__ = x # xs"
in_diffD: fixes a :: "'k"   and M :: "'k multiset"   and N :: "'k multiset" assumes "a \<in># M - N" shows "a \<in># M"
\<A>_def: fixes \<alpha> :: "'index"   and \<sigma> :: "bool"   and b\<^sub>\<sigma> :: "bool"   and D2 :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf"   and x :: "'range" shows "\<A> \<alpha> \<sigma> b\<^sub>\<sigma> D2 x = coin_spmf \<bind> (\<lambda>\<beta>\<^sub>\<sigma>'. local.etp.S \<alpha> \<bind> (\<lambda>x\<^sub>\<sigma>. let \<beta>\<^sub>\<sigma> = B \<alpha> x\<^sub>\<sigma> \<oplus> b\<^sub>\<sigma> in D2 (\<sigma>, \<alpha>, \<beta>\<^sub>\<sigma>, \<beta>\<^sub>\<sigma>') \<bind> (\<lambda>d. return_spmf (if d then \<beta>\<^sub>\<sigma>' else \<not> \<beta>\<^sub>\<sigma>'))))"
run_axioms: shows "DF_System.run as\<^sub>0 alstep s"
monom_mult_in_pmdl: fixes b :: "'a \<Rightarrow>\<^sub>0 'b"   and B :: "('a \<Rightarrow>\<^sub>0 'b) set"   and c :: "'b"   and t :: "'a" assumes "b \<in> B" shows "MPoly_Type_Class.punit.monom_mult c t b \<in> MPoly_Type_Class.punit.pmdl B"
parallelE: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<parallel> ys" obtains "\<not> xs \<le>p ys \<and> \<not> ys \<le>p xs"
simps_3: fixes c :: "int"   and a :: "MIR.num"   and b :: "MIR.num" shows "split_int (CF c a b) = (let (bv, bi) = split_int b in (bv, CF c a bi))"
rep_eq: fixes x :: "('a, 'b) chart" shows "apply_chart x = (case Rep_chart x of (d, d', f, f') \<Rightarrow> f)"
Un_singleton_iff: fixes A :: "'d set"   and B :: "'d set"   and x :: "'d" shows "(A \<union> B = {x}) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
Un_UNIV_right: fixes A :: "'c set" shows "A \<union> UNIV = UNIV"
intro: fixes ord_term :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>v w t. ord_term v w \<Longrightarrow> ord_term (term_powerprod.splus pair_of_term term_of_pair t v) (term_powerprod.splus pair_of_term term_of_pair t w)"   and "\<And>v w. \<lbrakk>ord (term_powerprod.pp_of_term pair_of_term v) (term_powerprod.pp_of_term pair_of_term w); term_powerprod.component_of_term pair_of_term v \<le> term_powerprod.component_of_term pair_of_term w\<rbrakk> \<Longrightarrow> ord_term v w" shows "ordered_term_axioms pair_of_term term_of_pair ord ord_term"
linear_span_inc_0: fixes A :: "'b set"   and H :: "'a set" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "\<zero> \<in> linear_span R M A H"