hcju/isabelleps huggingface dataset:
prems_4: shows "((\<mu> U\<^sub>r \<eta> \<in> old (fst x) \<longrightarrow> \<eta> \<in> old (fst x) \<union> new (fst x) \<or> \<mu> \<in> old (fst x) \<union> new (fst x) \<and> \<mu> U\<^sub>r \<eta> \<in> next (fst x)) \<and> (\<forall>nd\<in>snd x. \<mu> U\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst x) \<longrightarrow> \<eta> \<in> old nd \<or> \<mu> \<in> old nd \<and> \<mu> U\<^sub>r \<eta> \<in> old (fst x) \<union> new (fst x))) \<and> (\<forall>n\<in>snd x. \<forall>nd\<in>snd x. \<mu> U\<^sub>r \<eta> \<in> old n \<and> name n \<in> incoming nd \<longrightarrow> \<eta> \<in> old n \<or> \<mu> \<in> old n \<and> \<mu> U\<^sub>r \<eta> \<in> old nd) \<and> ((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)"
hta_states_correct_1: shows "hs.\<alpha> (hta_states H) = ta_rstates (hta_\<alpha> H)"
val: shows "fpxs_val f = rat_of_int M / rat_of_nat N"
xor_commute: fixes x :: "'b"   and y :: "'b" shows "x \<oplus> y = y \<oplus> x"
tendsto_minus_cancel_left: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and F :: "'a filter" shows "(f \<longlongrightarrow> - y) F = ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
sa_iso_pair: fixes x :: "'c regular"   and y :: "'c dense" shows "(- - x, stone_phi (- x) \<squnion> up_filter y) \<in> stone_phi.pairs"
disc_13: fixes x61 :: "'a OCL_Types.type"   and x62 :: "String.literal"   and x63 :: "'a expr list" shows "\<not> is_Let (StaticOperationCall x61 x62 x63)"
ticks_image_sub': fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat" assumes "dilating f sub r"   and "\<exists>c. hamlet (Rep_run r n c)" shows "\<exists>n\<^sub>0. f n\<^sub>0 = n"
map_comp: fixes g1 :: "'c \<Rightarrow> 'a"   and g2 :: "'d \<Rightarrow> 'b"   and f1 :: "'e \<Rightarrow> 'c"   and f2 :: "'f \<Rightarrow> 'd"   and v :: "('e, 'f) blindable\<^sub>m" shows "map_blindable\<^sub>m g1 g2 (map_blindable\<^sub>m f1 f2 v) = map_blindable\<^sub>m (g1 \<circ> f1) (g2 \<circ> f2) v"
big_unclocked_ignore_3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and st' :: "'b state"   and r3 :: "(v, v) result"   and cnt :: "nat" assumes "evaluate ck env s e (st', r3)"   and "r3 \<noteq> Rerr (Rabort Rtimeout_error)" shows "evaluate False env (update_clock (\<lambda>_. cnt) s) e (update_clock (\<lambda>_. cnt) st', r3)"
worker_wrapper_id: fixes wrap :: "'b \<rightarrow> 'a"   and unwrap :: "'a \<rightarrow> 'b"   and computation :: "'a"   and body :: "'a \<rightarrow> 'a" assumes "wrap oo unwrap = ID"   and "computation = fix\<cdot>body" shows "computation = wrap\<cdot>(fix\<cdot>(unwrap oo body oo wrap))"
rPar1: fixes \<Psi>\<^sub>Q :: "'b"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* C" shows "Prop \<alpha> (P' \<parallel> Q)"
Ccompt: fixes f :: "'b"   and X :: "'a"   and Y :: "'a"   and g :: "'b"   and Z :: "'a" assumes "f maps X to Y"   and "g maps Y to Z" shows "f ;; g maps X to Z"
of_nat_tree: fixes n :: "nat" shows "of_nat n = pure (of_nat n)"
cblinfun_apply: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "cblinfun_apply x \<in> {f. bounded_clinear f}"
simps_2: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and c :: "color"   and lt :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c"   and rt :: "('a, 'c) rbt" shows "RBT_Impl.map f (Branch c lt k v rt) = Branch c (RBT_Impl.map f lt) k (f k v) (RBT_Impl.map f rt)"
typ_of1_subst_bv_gen: fixes Ts' :: "typ list"   and uty :: "typ"   and Ts :: "typ list"   and t :: "Core.term"   and tty :: "typ"   and u :: "Core.term" assumes "typ_of1 (Ts' @ [uty] @ Ts) t = Some tty"   and "typ_of1 Ts u = Some uty" shows "typ_of1 (Ts' @ Ts) (subst_bv1 t (length Ts') u) = Some tty"
history_consistent_append_Prog\<^sub>s\<^sub>b: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and p' :: "'a"   and mis :: "instr list"   and xs :: "'a memref list" assumes "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', mis)"   and "history_consistent \<theta> (hd_prog p xs) xs"   and "last_prog p xs = p" shows "history_consistent \<theta> (hd_prog p' (xs @ [Prog\<^sub>s\<^sub>b p p' mis])) (xs @ [Prog\<^sub>s\<^sub>b p p' mis])"
sees_method_idemp: fixes P :: "'b prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> D sees M, b :  Ts\<rightarrow>T = m in D"
simps_3: fixes c :: "'e" shows "lsumr (Inr (Inr c)) = Inr c"
trans_mult2_ns: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "s O ns \<subseteq> s"   and "refl ns"   and "trans ns" shows "trans (mult2_ns ns s)"
tm_neg_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t :: "taylor_model" assumes "x \<in>\<^sub>i range_tm e t" shows "- x \<in>\<^sub>i range_tm e (tm_neg t)"
weighted_RP_measure_decreasing_N: fixes St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and St' :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and C :: "'a literal multiset"   and l :: "nat" assumes "St \<leadsto>\<^sub>w St'"   and "(C, l) \<in># wN_of_wstate St" shows "(RP_filtered_measure (\<lambda>Di. True) St', RP_filtered_measure (\<lambda>Di. True) St) \<in> natLess <*lex*> natLess <*lex*> natLess"
no_retraction_cball: fixes e :: "real"   and a :: "'a" assumes "0 < e" shows "\<not> frontier (cball a e) retract_of cball a e"
select_convs_1: fixes Leg0 :: "'b"   and Leg1 :: "'b"   and more :: "'c" shows "Leg0 \<lparr>Leg0 = Leg0, Leg1 = Leg1, \<dots> = more\<rparr> = Leg0"
flat_lub_def: fixes b :: "'d"   and A :: "'d set" shows "flat_lub b A = (if A \<subseteq> {b} then b else THE x. x \<in> A - {b})"
assms_2: shows "rt2 \<sqsubset>\<^bsub>dip\<^esub> rt3"
inj_on_image_Int: fixes f :: "'d \<Rightarrow> 'e"   and C :: "'d set"   and A :: "'d set"   and B :: "'d set" assumes "inj_on f C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "f ` (A \<inter> B) = f ` A \<inter> f ` B"
pappus21: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and Q :: "Points"   and R :: "Points" assumes "is_pappus2 A B C A' B' C' P Q R" shows "is_pappus1 A B C A' B' C' P Q R"
bounded: fixes n :: "nat" shows "sum a {..n} \<le> B"
Cut_l_redu_elim: fixes a :: "coname"   and M :: "trm"   and x :: "name"   and N :: "trm"   and R :: "trm" assumes "Cut <a>.M x.N \<longrightarrow>\<^sub>l R" shows "(\<exists>b. R = M[a\<turnstile>c>b]) \<or> (\<exists>y. R = N[x\<turnstile>n>y]) \<or> (\<exists>y M' b N'. M = NotR y.M' a \<and> N = NotL <b>.N' x \<and> R = Cut <b>.N' y.M' \<and> fic M a \<and> fin N x) \<or> (\<exists>b M1 c M2 y N'. M = AndR <b>.M1 <c>.M2 a \<and> N = AndL1 y.N' x \<and> R = Cut <b>.M1 y.N' \<and> fic M a \<and> fin N x) \<or> (\<exists>b M1 c M2 y N'. M = AndR <b>.M1 <c>.M2 a \<and> N = AndL2 y.N' x \<and> R = Cut <c>.M2 y.N' \<and> fic M a \<and> fin N x) \<or> (\<exists>b N' z M1 y M2. M = OrR1 <b>.N' a \<and> N = OrL z.M1 y.M2 x \<and> R = Cut <b>.N' z.M1 \<and> fic M a \<and> fin N x) \<or> (\<exists>b N' z M1 y M2. M = OrR2 <b>.N' a \<and> N = OrL z.M1 y.M2 x \<and> R = Cut <b>.N' y.M2 \<and> fic M a \<and> fin N x) \<or> (\<exists>z b M' c N1 y N2. M = ImpR z.<b>.M' a \<and> N = ImpL <c>.N1 y.N2 x \<and> R = Cut <b>.Cut <c>.N1 z.M' y.N2 \<and> b \<sharp> (c, N1) \<and> fic M a \<and> fin N x)"
mu_commute_fixpoint_1: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "isotone f"   and "has_least_fixpoint (f \<circ> g)"   and "f \<circ> g = g \<circ> f" shows "is_fixpoint f (\<mu> (f \<circ> g))"
intro: fixes A :: "'c set"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> bo x x"   and "\<And>x y z. \<lbrakk>bo x y; bo y z; x \<in> A\<rbrakk> \<Longrightarrow> bo x z"   and "\<And>x y. \<lbrakk>bo x y; bo y x; x \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "blinding_of_on_axioms A bo"
proj_def: fixes r :: "('b \<times> 'a) set"   and x :: "'b" shows "Equiv_Relations.proj r x = r `` \<eta> x"
Imagep_Image: fixes r :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d set" shows "r ``` A = {(a, b). r a b} `` A"
u_def: shows "u = Matrix.vec 4 (\<lambda>i. if i = 0 then v $ 0 * w $ 0 else if i = 3 then v $ 1 * w $ 1 else if i = 1 then v $ 0 * w $ 1 else v $ 1 * w $ 0)"
that: fixes C :: "'a literal set" assumes "cnf f = {C}" shows "thesis"
idinfo_spied: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and X :: "msg" assumes "s\<^sub>0 \<Turnstile> s"   and "\<langle>n, X\<rangle> \<in> parts (used s)" shows "\<langle>n, X\<rangle> \<in> spied s"
obs_postdominate: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and n :: "'b"   and n' :: "'b"   and S :: "'b set" assumes "StandardControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"   and "n \<in> CFG.obs sourcenode targetnode valid_edge n' (StandardControlDependencePDG.PDG_BS_s sourcenode targetnode valid_edge Def Use Exit S)"   and "n \<noteq> Exit" shows "Postdomination.postdominate sourcenode targetnode valid_edge Exit n n'"
abs_eq: shows "local.pmf_of_set = embed_pmf (\<lambda>x. indicat_real S x / real (card S))"
filter_tab_None: fixes t :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and p :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "t k = None" shows "filter_tab p t k = None"
prefixes_nontotal_hyp: fixes t :: "nat"   and j :: "nat"   and b :: "nat"   and i :: "nat" assumes "local.prefixes t j \<down>= b"   and "local.prefixes (Suc t) j \<up>"   and "s b \<down>= i" shows "\<exists>x. \<phi> i x \<up>"
eq: shows "\<forall>s\<in>generator. \<mu> s = \<mu>G s"
intros_7: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and x :: "name" assumes "P \<longmapsto>\<alpha> \<prec> P'"   and "x \<sharp> \<alpha>" shows "\<lparr>\<nu>x\<rparr>P \<longmapsto>\<alpha> \<prec> \<lparr>\<nu>x\<rparr>P'"
first: shows "\<forall>l<n. \<pi> l \<noteq> \<pi> n"
VSLuckyBlocksLimited_def: shows "VSLuckyBlocksLimited = opt_group \<circ> QE_dnf opt_group (step_augment luckyFind' IdentityHeuristic) \<circ> VSLuckiestBlocks \<circ> opt_group"
z_def: shows "z = y + k *\<^sub>R (x - y)"
segment_degen_0: fixes u :: "real"   and a :: "'b"   and b :: "'b" shows "((1 - u) *\<^sub>R a + u *\<^sub>R b = a) = (a = b \<or> u = 0)"
const_is_polynomial: fixes a :: "'a list"   and K :: "'a list set" assumes "a \<in> K - {\<zero>\<^bsub>K [X]\<^esub>}" shows "polynomial\<^bsub>K [X]\<^esub> K [a]"
intros_2: fixes C :: "'a eclause"   and S :: "'a eclause set"   and D :: "'a eclause" assumes "derivable_ecl C S"   and "renaming_cl C D" shows "derivable_ecl D S"
is_class_Object: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_md P" shows "is_class P Object"
well_order_on: fixes A :: "'b set" shows "\<exists>r. well_order_on A r"
P: fixes s :: "'s" assumes "(x, s) \<in> acc" shows "almost_everywhere (T s) P"
abs_eq: fixes y :: "dflt_size word" shows "Abs_uint' y = Abs_uint y"
hyps: fixes as :: "nat"   and am :: "nat list"   and bs :: "nat"   and bm :: "nat list" assumes "abc_steps_l (as, am) ap stp = (bs, bm)" shows "abc_steps_l (as, am) ap (Suc stp) = abc_step_l (bs, bm) (abc_fetch bs ap)"
uniform_complement: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int" assumes "block_design point_set block_collection u_block_size"   and "u_block_size \<le> int (card point_set) - 1" shows "block_design point_set (incidence_system.complement_blocks point_set block_collection) (int (card point_set) - u_block_size)"
map_mmupdE: fixes m :: "'b \<Rightarrow> 'a option"   and K :: "'b set"   and v :: "'a"   and k :: "'b"   and x :: "'a" assumes "map_mmupd m K v k = Some x" obtains "k \<notin> K" and "m k = Some x"   | "k \<in> K" and "x = v"
row_mat_of_row: fixes y :: "'b vec" shows "row (mat_of_row y) 0 = y"
that: fixes u :: "real \<Rightarrow> 'a"   and d :: "real \<Rightarrow> 'a" assumes "arc u"   and "arc d"   and "pathstart u = a"   and "pathfinish u = b"   and "pathstart d = b"   and "pathfinish d = a"   and "path_image u \<inter> path_image d = {a, b}"   and "path_image u \<union> path_image d = path_image g" shows "thesis"
algebraic_div: fixes x :: "'a"   and y :: "'a" assumes "algebraic x"   and "algebraic y" shows "algebraic (x / y)"
alpha_qSwap_preserves_qGood: fixes zs :: "'varSort"   and y :: "'var"   and x :: "'var"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and zs' :: "'varSort"   and y' :: "'var"   and x' :: "'var"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "X #[[y \<and> x]]_zs #= X' #[[y' \<and> x']]_zs'" shows "qGood X = qGood X'"
strategy_attracts_does_not_leave: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and W :: "'a set"   and v :: "'a"   and w :: "'a" assumes "strategy_attracts p \<sigma> A W"   and "strategy p \<sigma>"   and "v \<rightarrow> w"   and "v \<in> A - W"   and "w \<notin> A \<union> W" shows "v \<in> VV p \<and> \<sigma> v \<noteq> w"
both_Nil: assumes "as = []"   and "bs = []" shows "P"
diff_2C: fixes x :: "complex"   and y :: "complex" assumes "x \<in> convex hull {a, b, c}"   and "y \<in> convex hull {a, b, c}" shows "cmod (x - y) \<le> 2 * C"
padic_set_res_closed: fixes f :: "nat \<Rightarrow> int"   and p :: "int"   and m :: "nat" assumes "f \<in> padic_set p" shows "f m \<in> carrier (residue_ring (p ^ m))"
cong: fixes q :: "nat" shows "secure_mult.Ideal2 q \<equiv> secure_mult.Ideal2 q"
coreflexive_comp_top_inf: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_idempotent_semiring.coreflexive x" shows "x \<odot> mtop \<otimes> y = x \<odot> y"
prime_factors_power_subset: fixes x :: "'b"   and n :: "nat" shows "prime_factors (x ^ n) \<subseteq> prime_factors x"
measurable_dist_measure: fixes d :: "pdf_dist" shows "dist_measure d \<in> stock_measure (dist_param_type d) \<rightarrow>\<^sub>M subprob_algebra (stock_measure (dist_result_type d))"
compose_extensional: fixes A :: "'a set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "compose A f g \<in> extensional A"
less_eq_y: fixes x :: "'a" assumes "x \<in> S"   and "P x" shows "x \<le> y"
prime_cong_1_mod_4_gauss_int_norm_exists: fixes p :: "nat" assumes "prime p"   and "[p = 1] (mod 4)" shows "\<exists>z. gauss_int_norm z = p \<and> 0 < ReZ z \<and> 0 < ImZ z"
\<tau>Exec_methr_step': fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "\<tau>Exec_methr ci P ins xt t h s (stk', loc', pc', xcp')"   and "\<tau>exec_meth ci P ins xt t h (stk', loc', pc', xcp') s'" shows "\<tau>Exec_methr ci P ins xt t h s s'"
valid_sops_stmt: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'a"   and \<O> :: "'b" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "valid_sops_stmt t s"
log_add_eq_powr: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "0 < b"   and "b \<noteq> 1"   and "0 < x" shows "log b x + y = log b (x * b powr y)"
minor_eq_row_col: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and j :: "'d" shows "minor M i j = upd_row (upd_col M j (axis i (1::'c))) i (axis j (1::'c))"
gWlsBinp_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Recursion.model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'h option" shows "gWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (gWlsAbs MOD) (barOf delta) binp"
finsert_absorb2: fixes x :: "'a"   and A :: "'a fset" shows "finsert x (finsert x A) = finsert x A"
that: shows "\<forall>\<^sub>F x in F. f x = g x"
foldl_ins_set_tree: fixes ys :: "('a \<times> 'b) list"   and t :: "('a \<times> 'b) tree" assumes "inj_on fst (set ys)"   and "inj_on snd (set ys)"   and "distinct ys"   and "fst ` set ys \<inter> keys t = {}" shows "set_tree (foldl (\<lambda>t' (x, p). ins x p t') t ys) = set ys \<union> set_tree t"
indRelRSTPO_impl_SRel_and_TRel_weakly_preserve_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs (indRelRSTPO SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_preserves_barbs SRel SWB"
no_ub: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.dlo less_eq less" shows "\<exists>u. less x u"
subset_imp: fixes A :: "'t set"   and B :: "'t set" assumes "A \<subseteq> B"   and "A \<noteq> {}"   and "finite B" shows "ord_term_lin.Max A \<preceq>\<^sub>t ord_term_lin.Max B"
arr_mkIde: fixes A :: "'s set" shows "arr (mkIde A) = (A \<subseteq> Univ \<and> |A| <o \<AA>)"
keys_tabulate: fixes ks :: "'b list"   and f :: "'b \<Rightarrow> 'c" shows "Mapping.keys (Mapping.tabulate ks f) = set ks"
aff_dim_convex_Int_nonempty_interior: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "S \<inter> interior T \<noteq> {}" shows "aff_dim (S \<inter> T) = aff_dim S"
l_ewp_def: fixes X :: "'a list set" shows "l_ewp X = ({[]} \<subseteq> X)"
d_wloc: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y = (0::'a)) = (x \<cdot> d y = (0::'a))"
var: fixes \<Gamma> :: "(name, sterm) fmap"   and name :: "name"   and val :: "sterm"   and rs :: "(name \<times> sterm) list" assumes "fmlookup \<Gamma> name = Some val" shows "rs, \<Gamma> \<turnstile>\<^sub>s Svar name \<down> val"
fps_shift_fps_X: fixes n :: "nat" assumes "1 \<le> n" shows "fps_shift n fps_X = (if n = 1 then 1 else 0)"
has_absolute_integral_change_of_variables_compact: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and b :: "(real, 'b) vec" assumes "compact S"   and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "inj_on g S" shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \<and> integral (g ` S) f = b)"
representation_add: fixes basis :: "'b set"   and v :: "'b"   and u :: "'b" assumes "vs1.independent basis"   and "v \<in> vs1.span basis"   and "u \<in> vs1.span basis" shows "vs1.representation basis (u + v) = (\<lambda>b. vs1.representation basis u b + vs1.representation basis v b)"
rbt_filter_def: fixes P :: "'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT.rbt" shows "rbt_filter P t = RBT.fold (\<lambda>k _ A'. if P k then insert k A' else A') t {}"
lift_transform_simps_1: fixes t :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and e :: "exp" shows "lift_transform t \<bottom> e = e"
lr: shows "(c::'a::refinement_lattice)\<^sup>\<infinity> \<sqsubseteq> c\<^sup>\<omega> ; \<top>"
mprefix_Par_Int_distr: fixes B :: "'a set"   and C :: "'a set"   and A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process" assumes "B \<inter> C = {}"   and "A \<subseteq> C" shows "(Mprefix A P \<lbrakk>C\<rbrakk> Mprefix B Q) = Mprefix B (\<lambda>x. Mprefix A P \<lbrakk>C\<rbrakk> Q x)"
semigroup_add_ow_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semigroup_add_ow S pls \<equiv> (\<forall>a b c. a \<in> S \<longrightarrow> b \<in> S \<longrightarrow> c \<in> S \<longrightarrow> pls (pls a b) c = pls a (pls b c)) \<and> (\<forall>a b. a \<in> S \<longrightarrow> b \<in> S \<longrightarrow> pls a b \<in> S)"
right_idem: fixes a :: "'d"   and b :: "'d" shows "max (max a b) b = max a b"
less_restrictive_complete: fixes R1 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and R2 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" assumes "less_restrictive R1 R2"   and "Complete R2" shows "Complete R1"
dim_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c" shows "dim_vec (vec n f) = n"
sig_red_monom_mult: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and c :: "'b"   and s :: "'a" assumes "sig_red sing_reg top_tail F p q"   and "c \<noteq> (0::'b)" shows "sig_red sing_reg top_tail F (monom_mult c s p) (monom_mult c s q)"
not_predI': fixes P :: "unit Predicate.pred" assumes "\<not> pred.eval P ()" shows "pred.eval (Predicate.not_pred P) ()"
function_smult_l_distr: fixes a :: "'a"   and b :: "'a"   and x :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "x \<in> carrier F" shows "(a \<oplus> b) \<odot>\<^bsub>F\<^esub> x = a \<odot>\<^bsub>F\<^esub> x \<oplus>\<^bsub>F\<^esub> b \<odot>\<^bsub>F\<^esub> x"
valid_def: fixes w :: "'person \<Rightarrow> 'color" shows "local.valid w = (\<exists>p. p \<noteq> guru \<and> w p = blue)"
cr_poly_def: shows "cr_poly \<equiv> \<lambda>x y. x = poly.coeff y"
bprv_prv: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c" assumes "Deduct2 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "bprv \<phi>" shows "prv \<phi>"
pn_atoms_atoms: fixes F :: "'a formula" shows "atoms F = fst (pn_atoms F) \<union> snd (pn_atoms F)"
RedCondF: fixes e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>if (false) e\<^sub>1 else e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>e\<^sub>2,s,b\<rangle>"