hcju/isabelleps huggingface dataset:
le_join_disj: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.SL_top le join Top"   and "le x y \<or> le x z" shows "le x (join y z)"
simps_1: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'e" shows "zip_with_option f (Some a) (Some b) = Some (f a b)"
that: assumes "final x"   and "ln = no_wait_locks"   and "wset s t = None" shows "thesis"
mod_diff_cong: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and a' :: "'b"   and b :: "'b"   and b' :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (minus a b) c = modulo (minus a' b') c"
log2_gt: fixes x :: "nat" shows "x < 2 ^ (log2 x + 1)"
addsE: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "plus.adds plus b a" obtains k :: "'a" where "a = plus b k"
fresh_9: fixes a :: "name"   and x1 :: "name"   and x2 :: "pi" shows "a \<sharp> <\<nu>x1>x2 = a \<sharp> [x1].x2"
A: fixes x :: "'a" assumes "cont f"   and "cont g" shows "cont ((|-|) (f x))"
FIXvarFIX: fixes b :: "BExpr" shows "PhiWhile b = (\<lambda>\<Phi> (s, t). (b, \<Phi>, s, t) \<in> var)"
wf_dec2_le: shows "wf abc_dec_2_LE"
TIMES_def: fixes xs :: "'b language list" shows "TIMES xs \<equiv> foldr Times xs One"
ile0_eq: fixes n :: "enat" shows "(n \<le> 0) = (n = 0)"
fmadd_empty_2: fixes m :: "('c, 'd) fmap" shows "m ++\<^sub>f fmempty = m"
env_strengthen: fixes v :: "val"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and \<rho>' :: "(nat \<times> val) list" assumes "v \<in> E e \<rho>"   and "\<forall>x. x \<in> FV e \<longrightarrow> lookup \<rho>' x = lookup \<rho> x" shows "v \<in> E e \<rho>'"
foldr_And: fixes as :: "'a formula list" shows "foldr (\<^bold>\<and>) as (\<^bold>\<not> \<bottom>) = \<^bold>\<And>as"
freshChainPermSimp_1: fixes xvec :: "name list"   and p :: "(name \<times> name) list" assumes "xvec \<sharp>* p" shows "p \<bullet> xvec = xvec"
homotopic_loops_conjugate: fixes p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a"   and s :: "'a set" assumes "path p"   and "path q"   and "path_image p \<subseteq> s"   and "path_image q \<subseteq> s"   and "pathfinish p = pathstart q"   and "pathfinish q = pathstart q" shows "homotopic_loops s (p +++ q +++ reversepath p) q"
cf_h_len: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "PolynRg A B Y"   and "f \<in> rHom S B"   and "pol_coeff S c" shows "fst (cf_h f c) = fst c"
continuous_on_s: fixes T :: "real set" shows "continuous_on T other.s"
euler_trail_conv_connected: fixes a :: "'a \<times> nat \<times> 'a"   and u :: "'a"   and p :: "('a \<times> nat \<times> 'a) list"   and v :: "'a" assumes "Digraph_Component.connected (add_arc a)" shows "H.euler_trail a u p v = (H.trail a u p v \<and> set p = arcs (add_arc a))"
assms_6: shows "insertion (nth_default 0 (L[var := B / C])) c = C"
length_nfresh: fixes fresh :: "'a set \<Rightarrow> 'a"   and S :: "'a set"   and n :: "nat" assumes "freshstuff fresh" shows "length (freshstuff.nfresh fresh S n) = n"
sp_pl: shows "simple_path (p +++ linepath (a - complex_of_real e) (a + complex_of_real e))"
indiff_set_mono: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "indiff_set S A"   and "B \<subseteq> A"   and "B \<noteq> {}" shows "indiff_set S B"
topological_tendstoI: fixes "open" :: "'c set \<Rightarrow> bool"   and l :: "'c"   and f :: "'d \<Rightarrow> 'c"   and F :: "'d filter" assumes "class.topological_space open"   and "\<And>S. \<lbrakk>open S; l \<in> S\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in F. f x \<in> S" shows "filterlim f (topological_space.nhds open l) F"
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'e multiset" shows "image_mset g (image_mset f v) = image_mset (g \<circ> f) v"
power_minus_even: fixes a :: "'c"   and n :: "nat" shows "(- a) ^ (2 * n) = a ^ (2 * n)"
arr_char: fixes f :: "'c \<times> 'c \<times> 'c" shows "C.VVV.arr f = (C.arr (fst f) \<and> C.VV.arr (snd f) \<and> src\<^sub>C (fst f) = trg\<^sub>C (fst (snd f)))"
changes_append: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<noteq> [] \<and> ys \<noteq> [] \<longrightarrow> last xs = hd ys \<and> last xs \<noteq> (0::'b)" shows "changes (xs @ ys) = changes xs + changes ys"
card_of_well_order_on: fixes A :: "'c set" shows "well_order_on A (card_of A)"
alw_ev_imp_ev_alw: fixes P :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" assumes "alw (ev P) \<omega>" shows "ev (\<lambda>xs. P xs \<and> alw (ev P) xs) \<omega>"
div_lt': fixes i :: "'b word"   and k :: "'b word"   and x :: "'b word" assumes "i \<le> k div x" shows "unat i * unat x < 2 ^ LENGTH('b)"
heap_wellformed_induct_rev: fixes h :: "(_) heap"   and P :: "(_) object_ptr \<Rightarrow> bool"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "\<And>child. (\<And>parent child_node. \<lbrakk>cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child_node = child; h \<turnstile> Shadow_DOM.get_parent child_node \<rightarrow>\<^sub>r Some parent\<rbrakk> \<Longrightarrow> P parent) \<Longrightarrow> P child" shows "P ptr"
convex_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.order less_eq less" shows "order.convex less_eq = {A. \<forall>a b. a \<in> A \<and> b \<in> A \<longrightarrow> order.closed_interval less_eq a b \<subseteq> A}"
inj_on_convol_ident: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set" shows "inj_on (\<lambda>x. (x, f x)) X"
infinite_Ioc: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.dense_linorder less_eq less"   and "less a b" shows "infinite (ord.greaterThanAtMost less_eq less a b)"
borel_measurable_sum': fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M" shows "sum f I \<in> borel_measurable M"
mahler_measure_poly_via_monic: fixes p :: "complex poly" shows "mahler_measure_poly p = cmod (lead_coeff p) * mahler_measure_monic p"
T_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tensor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unity :: "'b"   and lunit :: "'b \<Rightarrow> 'b"   and runit :: "'b \<Rightarrow> 'b"   and assoc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "elementary_monoidal_category C tensor unity lunit runit assoc"   and "partial_magma.arr C f"   and "partial_magma.arr C g" shows "elementary_monoidal_category.T C tensor (f, g) = tensor f g"
set_minimize_wrt: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y. Q x y \<Longrightarrow> \<not> Q y x"   and "sorted_wrt Q xs"   and "\<And>x y. \<not> P x y \<Longrightarrow> Q x y"   and "\<And>x. P x x" shows "set (minimize_wrt P xs) = {x \<in> set xs. \<forall>y\<in>set xs. P y x}"
limI: fixes F :: "'a set set"   and x :: "'a" assumes "F \<in> Filters"   and "x \<in> carrier"   and "F \<longlongrightarrow> x" shows "lim F = x"
gt_compat_fun: fixes t' :: "('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "t' >\<^sub>t t" shows "App s t' >\<^sub>t App s t"
merge_guards_simps_8: fixes c :: "'d \<Rightarrow> ('d, 'e, 'f) com" shows "merge_guards (DynCom c) = DynCom (\<lambda>s. merge_guards (c s))"
alphaAbs_all_equal_or_qFresh_imp_qAFresh: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" assumes "alphaAbs_all_equal_or_qFresh xs x X xs' x' X'" shows "alphaAbs_all_equal_or_qAFresh xs x X xs' x' X'"
clop_wtrans: fixes x :: "'a" shows "cl_op (cl_op x) \<le> cl_op x"
disjoint_subheaps_exist: fixes h :: "'a" shows "\<exists>x y. x ## y \<and> h = x + y"
finvimageI2: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet> `\<^sub>\<bullet> A" shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A"
nres_of_nress_impl_def: fixes xi :: "'c nres list" shows "local.nres_of_nress_impl xi \<equiv> rec_list (RETURN []) (\<lambda>x xa xb. x \<bind> (\<lambda>xc. xb \<bind> (\<lambda>xd. RETURN (xc # xd)))) xi"
states_nodes: fixes fP :: "('c \<Rightarrow> nat option) \<times> 'c set"   and A :: "('d, 'c) nba" assumes "fP \<in> nodes (complement A)" shows "snd fP \<subseteq> nodes A"
disj_simps_2: fixes P :: "fm" shows "Ferrack.disj P T = T"
null_uni_uniq_2: fixes v :: "i" shows "[\<^bold>\<exists>!x. TAO_8_Definitions.Universal (x\<^sup>P) in v]"
\<phi>: shows "ring_homomorphism \<phi> st.carrier_stalk st.add_stalk st.mult_stalk (st.zero_stalk V) (st.one_stalk V) pi.carrier_local_ring_at pi.add_local_ring_at pi.mult_local_ring_at pi.zero_local_ring_at pi.one_local_ring_at"
mor_Abs_IF: shows "mor (car_init1 undefined) (car_init2 undefined) (str_init1 undefined) (str_init2 undefined) UNIV UNIV ctor1 ctor2 Abs_IF1 Abs_IF2"
span_sum: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and S :: "'c set" assumes "module scale"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<in> module.span scale S" shows "sum f A \<in> module.span scale S"
assms_1: shows "foldl1 f2 (map (h2 i) js) = Two"
Necessary\<^sub>1_def: shows "Necessary \<equiv> \<lambda>F. \<^bold>\<box>(\<^bold>\<forall>x. \<lparr>F,x\<^sup>P\<rparr>)"
op_out1c: shows "open (outside (path_image c1 \<union> path_image c))"
Q_1: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat" assumes "ProofStepInduct_NonTrivial G v0 v1 paths P_new sep_size" shows "DisjointPaths (DisjointPathsPlusOne.H_x G P_new) v0 v1 (ProofStepInduct_NonTrivial.Q G v0 v1 paths P_new sep_size)"
ge_neg_game: fixes x :: "game"   and y :: "game" shows "ge_game (neg_game x, neg_game y) = ge_game (y, x)"
x1: shows "(\<Union>xa\<in>R `` {x}. {per_union R x y `` {xa}}) = {per_union R x y `` {x}}"
ticonc: fixes xs :: "int list" shows "\<exists>ys. xs = take_incr xs @ ys"
jordan_block_constant_bound2: fixes n :: "nat" shows "\<exists>p. norm a < 1 \<longrightarrow> (\<forall>i j k. i < n \<longrightarrow> j < n \<longrightarrow> norm ((jordan_block n a ^\<^sub>m k) $$ (i, j)) \<le> p)"
HC_intros_9: fixes F :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H (F \<^bold>\<and> G) \<^bold>\<rightarrow> G"
numeral_inc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (Num.inc x) = plus (numeral.numeral one plus x) one"
ipresOp_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'e, 'f, 'opSym, 'g, 'c, 'd, 'h) model_scheme" shows "ipresOp h hA MOD \<equiv> \<forall>delta inp binp. wlsInp delta inp \<and> wlsBinp delta binp \<longrightarrow> h (Op delta inp binp) = igOp MOD delta (lift h inp) (lift hA binp)"
ltl_models_equiv_prop_entailment: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" shows "w \<Turnstile> \<phi> = {\<chi>. w \<Turnstile> \<chi>} \<Turnstile>\<^sub>P \<phi>"
f_def: fixes fs :: "'b list"   and i :: "nat" shows "f fs i = fs ! i"
r: shows "revision_step r' s\<^sub>1 s\<^sub>2'"
disc_iff: fixes a :: "'c" shows "lconst a \<noteq> LNil"
M_alt: fixes \<pi> :: "'b prefix" shows "M \<pi> = {(i, v). i < progress \<pi> \<and> wf_tuple nfv fv v \<and> (\<exists>\<sigma>. prefix_of \<pi> \<sigma> \<and> sat \<sigma> (map the v) i)}"
sop_oneTr: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i (s_set R s A))"   and "assoc_bpp (aug_pm_set z i (s_set R s A)) f"   and "ipp_cond1 (s_set R s A) i"   and "ipp_cond2 z (s_set R s A) i f"   and "ipp_cond3 z i"   and "inv_ipp z i f (s_set R s A)"   and "zeroA z i f (s_set R s A) z"   and "sop_distr2 R s f i (s_set R s A) z"   and "sop_assoc R s (aug_pm_set z i (s_set R s A))"   and "sop_one R s (aug_pm_set z i (s_set R s A))" shows "\<forall>x\<in>add_set f (aug_pm_set z i (s_set R s A)) n. 1\<^sub>r \<^sub>s\<cdot> x = x"
nonsat_push_precond_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat" shows "nonsat_push_precond f l \<equiv> \<lambda>(u, v). (0::'capacity) < excess f u \<and> excess f u < cf_of f (u, v) \<and> (u, v) \<in> cfE_of f \<and> l u = l v + 1"
characteristic_weak_formula_is_characteristic: fixes Q :: "'state"   and P :: "'state" shows "Q \<Turnstile>\<^sub>S S_transform.characteristic_weak_formula P = P \<approx>\<cdot>\<^sub>S Q"
invc2_joinR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invc l"   and "invh l"   and "invc r"   and "invh r"   and "bheight r \<le> bheight l" shows "invc2 (joinR l x r) \<and> (bheight l \<noteq> bheight r \<and> color l = Black \<longrightarrow> invc (joinR l x r))"
annos_ne: fixes C :: "'a acom" shows "annos C \<noteq> []"
powers_10: fixes a :: "'b"   and c :: "'b"   and b :: "'b" shows "(a = c - b) = (a + b = c)"
comparator_list_simps_4: fixes comp\<^sub>'\<^sub>a :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and xa :: "'b list"   and y :: "'b"   and ya :: "'b list" shows "comparator_list comp\<^sub>'\<^sub>a (x # xa) (y # ya) = (case comp\<^sub>'\<^sub>a x y of Eq \<Rightarrow> comparator_list comp\<^sub>'\<^sub>a xa ya | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
uses_in_\<alpha>n: fixes v :: "'a \<times> 'b \<times> Def"   and g :: "('b, 'a, 'c) gen_cfg_wf"   and n :: "'b" assumes "v \<in> gen_wf.uses' g n" shows "n \<in> set (gen_wf_\<alpha>n g)"
mono_det_FD: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process"   and S' :: "'a process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D P'"   and "S \<sqsubseteq>\<^sub>F\<^sub>D S'" shows "(P [+] S) \<sqsubseteq>\<^sub>F\<^sub>D (P' [+] S')"
nonempty_has_size: fixes S :: "'c multiset" shows "(S \<noteq> {#}) = (0 < size S)"
rel_interior_maximal: fixes T :: "'b set"   and S :: "'b set" assumes "T \<subseteq> S"   and "openin (top_of_set (affine hull S)) T" shows "T \<subseteq> rel_interior S"
hasI: fixes xs :: "'b list"   and S :: "'b set" assumes "hasw xs S" shows "has (length xs) S"
trms\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('a, 'b) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p ` set S)"
AE_completion_iff: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" shows "almost_everywhere (completion M) P = almost_everywhere M P"
comp_null_2: fixes f :: "'a" shows "D f D.null = D.null"
conj_overlap: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<and>. q) = (A \<turnstile> p .\<and>. (.\<not> p .\<or>. q))"
IH: assumes "distinct (map fst pairs)"   and "\<And>x y. \<lbrakk>x \<in> set (map fst pairs); y \<in> set (map fst pairs); P x; P y\<rbrakk> \<Longrightarrow> x = y"   and "(x, y) \<in> set pairs" shows "List.find (\<lambda>a. case a of (x, uu_) \<Rightarrow> P x) pairs = Some (x, y)"
rep_plus: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "rep_nat (plus x y) = rep_nat x + rep_nat y"
zip_obtain_same_length: fixes xs :: "'g list"   and ys :: "'h list"   and P :: "('g \<times> 'h) list \<Rightarrow> bool" assumes "\<And>zs ws n. \<lbrakk>length zs = length ws; n = min (length xs) (length ys); zs = take n xs; ws = take n ys\<rbrakk> \<Longrightarrow> P (zip zs ws)" shows "P (zip xs ys)"
"1_2": fixes x :: "'a"   and y :: "'a" assumes "x \<in> set xs'"   and "y \<in> set ys" shows "f x \<noteq> f y"
envObs_def: shows "envObs \<equiv> \<lambda>a s. (envObsC (es s), ODList.lookup (ps s) a)"
ring_times_minusl: fixes a :: "'a" assumes "a \<in> carrier R" shows "-\<^sub>a a = (-\<^sub>a 1\<^sub>r) \<cdot>\<^sub>r a"
assms_2: shows "\<forall>x y. x - y = x \<sqinter> - y"
set_delete: fixes xs :: "'b list"   and t :: "'b trie" shows "Trie_Fun.set (delete xs t) = Trie_Fun.set t - {xs}"
m'_def: shows "Mapping.lookup a2_map (tp - len + 1) = Some m'"
K_block_design_axioms: shows "K_block_design (\<V> \<times> I) (\<Sum>\<^sub># (image_mset f \<B>)) \<K>'"
Greatest_le_nat: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat"   and b :: "nat" assumes "P k"   and "\<And>y. P y \<Longrightarrow> y \<le> b" shows "k \<le> Greatest P"
dbm_le_not_inf: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "a \<preceq> b"   and "b \<noteq> \<infinity>" shows "a \<noteq> \<infinity>"
base_in_hom_2: shows "\<guillemotleft>g \<star> f\<^sup>* : g \<star> f\<^sup>* \<Rightarrow> g \<star> f\<^sup>*\<guillemotright>"
systemOUT_noLOC: fixes x :: "chanID"   and i :: "AbstrLevelsID" assumes "systemOUT x i" shows "\<not> systemLOC x i"
compute_summarize_pdevs_list: fixes p :: "nat"   and I :: "nat \<Rightarrow> 'a list \<Rightarrow> bool"   and d :: "nat"   and xs :: "'a pdevs list" shows "summarize_pdevs_list p I d xs = (let M = pdevs_mapping xs in map2 (summarize_pdevs p (\<lambda>i _. I i (M i))) [d..<d + length xs] xs)"
essential_classI: fixes C :: "'s option set" assumes "C \<in> UNIV // K2.communicating"   and "\<And>x y. \<lbrakk>x \<in> C; (x, y) \<in> K2.acc\<rbrakk> \<Longrightarrow> y \<in> C" shows "K2.essential_class C"
progress_simps_6: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Neg \<phi>) j = Monitor.progress \<sigma> P \<phi> j"
subset_hull: fixes S :: "'a set \<Rightarrow> bool"   and t :: "'a set"   and s :: "'a set" assumes "S t" shows "(S hull s \<subseteq> t) = (s \<subseteq> t)"
enableds_def: fixes enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b" shows "transition_system.enableds enabled p \<equiv> {a. enabled a p}"
CN_FALSE_def: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "CN_FALSE P x \<equiv> False"
unrest_in_rel_var_res: fixes x :: "'b \<Longrightarrow> 'c"   and P :: "'c hrel" assumes "vwb_lens x" shows "$x \<sharp> P \<restriction>\<^sub>\<alpha> x"
assms_2: shows "f \<in> borel_measurable (nat_filtration n)"
simps_2: fixes f :: "'h \<Rightarrow> 'g"   and x :: "'h" shows "(case object_ptr.Ext x of object_ptr.Ext x \<Rightarrow> f x) = f x"
is_nextElem2: fixes vs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "distinct vs"   and "x \<in> set vs"   and "is_nextElem vs x y" shows "nextElem vs (hd vs) x = y"
condCatch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and b :: "'e set"   and R :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c\<^sub>1 Q,(b \<inter> R \<union> - b \<inter> A)"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P condCatch c\<^sub>1 b c\<^sub>2 Q,A"
simps_1: fixes a :: "'a" shows "is_Addr (Addr a) = True"
set_rw2: shows "cols_upt_k (Gram_Schmidt_column_k A j) (Suc k) = insert (column (mod_type_class.from_nat (Suc k)) (Gram_Schmidt_column_k A j)) (cols_upt_k (Gram_Schmidt_column_k A j) k)"
Eq_Eats_I: fixes H :: "fm set"   and i :: "name"   and z :: "tm"   and t :: "tm"   and u :: "tm" assumes "H \<turnstile> SyntaxN.All i (Var i IN z IFF Var i IN t OR Var i EQ u)"   and "atom i \<sharp> (z, t, u)" shows "H \<turnstile> z EQ Eats t u"
simps_10: fixes i :: "int"   and c :: "int"   and e :: "Cooper.num"   and k :: "int" shows "d_\<beta> (NDvd i (CN 0 c e)) k = (c dvd k)"
rev_ImageI: fixes a :: "'a"   and A :: "'a set"   and b :: "'b"   and r :: "('a \<times> 'b) set" assumes "a \<in> A"   and "(a, b) \<in> r" shows "b \<in> r `` A"
c_futures_tr: fixes yps :: "('a \<times> 'b) list"   and Y :: "('a \<times> 'b) set"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>0 :: "'c"   and xps :: "('a \<times> 'b) list" assumes "(yps, Y) \<in> futures (c_process step out s\<^sub>0) xps" shows "yps = c_tr step out (foldl step s\<^sub>0 (map fst xps)) (map fst yps)"
alphaAbs_qAbs_iff_ex_distinct_qAFresh: fixes xs :: "'f"   and x :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
fresh_5: fixes a :: "name"   and x2 :: "('d, 'e, 'f) psi"   and x1 :: "('d, 'e, 'f) psi" shows "a \<sharp> x2 \<parallel> x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
bij_betw_iff_bijections: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" shows "bij_betw f A B = (\<exists>g. (\<forall>x\<in>A. f x \<in> B \<and> g (f x) = x) \<and> (\<forall>y\<in>B. g y \<in> A \<and> f (g y) = y))"
l_cancel_one: fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier G"   and "a \<in> carrier G" shows "(x \<otimes> a = x) = (a = \<one>)"
degree_eq: fixes g :: "graph"   and v :: "nat" assumes "Invariants.inv g"   and "final g"   and "v \<in> \<V> g" shows "degree g v = tri g v + quad g v + except g v"
frechet_derivative_plus_fun: fixes x :: "'c \<Rightarrow> 'd"   and a :: "'c"   and y :: "'c \<Rightarrow> 'd" assumes "x differentiable at a"   and "y differentiable at a" shows "frechet_derivative (x + y) (at a) = frechet_derivative x (at a) + frechet_derivative y (at a)"
is_nan_def: fixes a :: "('a, 'b) IEEE.float" shows "IEEE.is_nan a = (IEEE.exponent a = emax TYPE(('a, 'b) IEEE.float) \<and> fraction a \<noteq> 0)"
subsumees_conv: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "subsumees subs = {v. \<exists>v'. (v, v') \<in> subs}"
dvd_imp_dvdm: fixes a :: "int poly"   and b :: "int poly"   and m :: "int" assumes "a dvd b" shows "poly_mod.dvdm m a b"
stepC_dec: fixes n :: "C"   and c :: "nat" assumes "n \<noteq> C []" shows "Suc (evalC (Suc (Suc c)) (stepC c n)) = evalC (Suc (Suc c)) n"
direct_arc_def: fixes uv :: "'b \<times> 'b" shows "pair_pre_digraph.direct_arc uv \<equiv> SOME e. {fst uv, snd uv} = {fst e, snd e}"
succ_of_enex_def: fixes enex :: "('d \<Rightarrow> 'f set) \<times> ('f \<Rightarrow> 'd \<Rightarrow> 'e)" shows "succ_of_enex enex \<equiv> \<lambda>s. let (en, ex) = enex in {s'. \<exists>a\<in>en s. s' = ex a s}"
chine_simps_3: shows "C.cod r.chine = r.cod.apex"
adj_mat_def: fixes A :: "'b mat" shows "adj_mat A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). cofactor A j i)"