hcju/isabelleps huggingface dataset:
Union_A: shows "(concat, \<Union>) \<in> \<langle>A\<rangle>list_wset_rel \<rightarrow> A"
assms: fixes y :: "'a"   and z :: "'a" assumes "y \<in> set ys"   and "z \<in> set zs" shows "y \<le> z"
replicate_0_2: fixes xs :: "'b" shows "replicate\<cdot>(MkI\<cdot>0)\<cdot>xs = []"
adv: assumes "\<And>\<eta>. exception_\<I> (\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) \<turnstile>g \<A> \<eta> \<surd>" shows "Negligible.negligible (\<lambda>\<eta>. advantage (\<A> \<eta>) (obsf_resource (sim_outer \<eta> |\<^sub>= 1\<^sub>C \<rhd> middle \<eta>)) (obsf_resource (real \<eta>)))"
insert_subset: fixes x :: "'d"   and A :: "'d set"   and B :: "'d set" shows "(insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
C: shows "(Asset n, Token n (Auth_PriK n) B C SK) \<in> s"
FH: fixes H :: "'a set"   and G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and rrsmult :: "('b, 'a) aezfun \<Rightarrow> (('b, 'a) aezfun \<Rightarrow> 'c) \<Rightarrow> ('b, 'a) aezfun \<Rightarrow> 'c" assumes "InducedFHModule H G smult V rrsmult" shows "aezfun_setspan H \<equiv> aezfun_setspan H"
vdisjnt_vunion_left: fixes A :: "V"   and B :: "V"   and C :: "V" shows "vdisjnt (A \<union>\<^sub>\<circ> B) C = (vdisjnt A C \<and> vdisjnt B C)"
IH: fixes y :: "nat"   and k :: "nat" assumes "y < l"   and "0 < y"   and "0 < k" shows "\<exists>a. vdw k y a"
simps_3: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "Unf (\<phi> M\<^sub>n \<psi>) = (\<phi> M\<^sub>n \<psi>) or\<^sub>n Unf \<phi> and\<^sub>n Unf \<psi>"
is_non_volatile_Write\<^sub>s\<^sub>b_simps_3: fixes p :: "'b"   and p' :: "'b"   and "is" :: "instr list" shows "is_non_volatile_Write\<^sub>s\<^sub>b (Prog\<^sub>s\<^sub>b p p' is) = False"
sets_UNIV: fixes A :: "'d set" shows "A \<in> sets (count_space UNIV)"
nn_integral_snd: fixes f :: "'a \<times> 'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (lborel \<Otimes>\<^sub>M lborel)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>lborel \<partial>lborel = integral\<^sup>N (lborel \<Otimes>\<^sub>M lborel) f"
iMODb_0_subset_conv: fixes r' :: "nat"   and m' :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" shows "([ r', mod m', 0 ] \<subseteq> [ r, mod m, c ]) = (r' \<in> [ r, mod m, c ])"
pac_ideal_idemp: fixes A :: "int mpoly set" shows "pac_ideal (pac_ideal A) = pac_ideal A"
size_height: fixes t :: "'b tree" shows "size t \<le> 2 ^ height t - 1"
trg_trg: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>" shows "trg (trg \<mu>) = trg \<mu>"
reachNT_PairI: fixes s :: "'state"   and a :: "'act"   and ou :: "'out"   and s' :: "'state" assumes "reachNT s"   and "step s a = (ou, s')"   and "\<not> T (Trans s a ou s')" shows "reachNT s'"
current_mono_web: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "current \<Gamma>' f" shows "current \<Gamma> f"
ereal_add_left_cancel_less: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "0 \<le> a"   and "0 \<le> b" shows "(a + b < a + c) = (a \<noteq> \<infinity> \<and> b < c)"
empty_subset_add_mset: fixes x :: "'k"   and M :: "'k multiset" shows "{#} \<subset># add_mset x M"
zero_le: fixes x :: "'a" shows "(0::'a) \<le> x"
prim_rootD: fixes x :: "'b list"   and r :: "'b list" assumes "x \<in>\<^sub>p r *" shows "x \<in> r*"
pickQFreshEnv: fixes V :: "'var set"   and XS :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Rho :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option) set"   and xs :: "'varSort" assumes "|V| <o |UNIV| \<or> finite V"   and "|XS| <o |UNIV| \<or> finite XS"   and "Ball XS qGood"   and "|Rho| <o |UNIV| \<or> finite Rho"   and "Ball Rho qGoodEnv" shows "pickQFreshEnv xs V XS Rho \<notin> V \<and> (\<forall>X\<in>XS. qFresh xs (pickQFreshEnv xs V XS Rho) X) \<and> (\<forall>rho\<in>Rho. qFreshEnv xs (pickQFreshEnv xs V XS Rho) rho)"
Mf: shows "f \<in> stock_measure t \<rightarrow>\<^sub>M stock_measure t'"
hyps: fixes Us :: "LambdaType.type list" assumes "e\<langle>i:T\<rangle> \<tturnstile> bs : Us"   and "listall (\<lambda>t. \<forall>e T' u i. e\<langle>i:T\<rangle> \<turnstile> t : T' \<longrightarrow> NF u \<longrightarrow> e \<turnstile> u : T \<longrightarrow> (\<exists>t'. t[u/i] \<rightarrow>\<^sub>\<beta>\<^sup>* t' \<and> NF t')) bs" shows "\<exists>as'. \<forall>j. Var j \<degree>\<degree> map (\<lambda>t. f (t[u/i])) bs \<rightarrow>\<^sub>\<beta>\<^sup>* Var j \<degree>\<degree> map f as' \<and> NF (Var j \<degree>\<degree> map f as')"
biB: fixes i :: "'b" assumes "i \<in> I" shows "b i \<in> B"
scene_override_commute_indep: fixes X :: "'a scene"   and Y :: "'a scene"   and S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a"   and S\<^sub>3 :: "'a" assumes "X \<bowtie>\<^sub>S Y" shows "S\<^sub>1 \<oplus>\<^sub>S S\<^sub>2 on X \<oplus>\<^sub>S S\<^sub>3 on Y = S\<^sub>1 \<oplus>\<^sub>S S\<^sub>3 on Y \<oplus>\<^sub>S S\<^sub>2 on X"
lexord_sufE: fixes xs :: "'a list"   and zs :: "'a list"   and ys :: "'a list"   and qs :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(xs @ zs, ys @ qs) \<in> lexord r"   and "xs \<noteq> ys"   and "length xs = length ys"   and "length zs = length qs" shows "(xs, ys) \<in> lexord r"
IH: fixes pre :: "'a list" assumes "xs = pre @ ref # suf"   and "distinct xs"   and "\<And>i. i \<in> set xs \<Longrightarrow> i < e" shows "insert_rga xs (e, Some ref) = pre @ ref # e # suf"
simps_2: fixes i :: "Integer" assumes "i \<noteq> \<bottom>" shows "stake\<cdot>i\<cdot>[::] = [::]"
check_no_overlapI: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "('c, 'e) flow_entry_match list" assumes "no_overlaps \<gamma> t" shows "check_no_overlap \<gamma> t"
finite_image_set2: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite {x. P x}"   and "finite {y. Q y}" shows "finite {f x y |x y. P x \<and> Q y}"
lossless_S2_14_OT: fixes c0 :: "bool"   and c1 :: "bool"   and b :: "bool" shows "lossless_spmf (S2_14_OT (c0, c1) b)"
eq_zero_str: fixes U :: "'b set" assumes "U \<subseteq> X" shows "ind_zero_str U = zero_str U"
vec_component: fixes x :: "'b"   and i :: "'c" shows "Finite_Cartesian_Product.vec x $h i = x"
simps_1: fixes p :: "nat"   and vs :: "nat list" shows "fv (PAtom p vs) = vs"
min_elems_Collect_ftF_is_Nil: shows "min_elems (Collect front_tickFree) = {[]}"
env_restr_perm: fixes p :: "perm"   and S :: "'a set"   and \<rho> :: "'a \<Rightarrow> 'b" assumes "supp p \<sharp>* S"   and "finite S" shows "(p \<bullet> \<rho>) f|` S = \<rho> f|` S"
owned_simp: fixes p :: "'c"   and instrs :: "'d"   and \<theta> :: "'e"   and sb :: "'f"   and \<D> :: "'g"   and \<O> :: "'b"   and \<R> :: "'h" shows "owned (p, instrs, \<theta>, sb, \<D>, \<O>, \<R>) = \<O>"
subspace_scale: fixes S :: "'b set"   and x :: "'b"   and c :: "'a" assumes "m1.subspace S"   and "x \<in> S" shows "c *a x \<in> S"
distinct_21: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "Skip \<noteq> While x61 x62"
refl_on_gState_progress_rel: fixes prog :: "program" shows "refl_on (Collect (gState_inv prog)) (gState_progress_rel prog)"
continuous_on_compose2: fixes t :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and s :: "'e set"   and f :: "'e \<Rightarrow> 'c" assumes "continuous_on t g"   and "continuous_on s f"   and "f ` s \<subseteq> t" shows "continuous_on s (\<lambda>x. g (f x))"
dg_prod_2_Cod_app: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and g :: "V"   and f :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "[g, f]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Arr\<rparr>" shows "(\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Cod\<rparr> \<lparr>g, f\<rparr>\<^sub>\<bullet> = [\<AA>\<lparr>Cod\<rparr>\<lparr>g\<rparr>, \<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr>]\<^sub>\<circ>"
cmod_Im_le_iff: fixes x :: "complex"   and y :: "complex" assumes "Re x = Re y" shows "(cmod x \<le> cmod y) = (\<bar>Im x\<bar> \<le> \<bar>Im y\<bar>)"
steps_def: fixes P :: "'b"   and Cal :: "'b processCalculus"   and Q :: "'b" shows "P \<longmapsto>Cal* Q \<equiv> \<exists>n. P \<longmapsto>Cal\<^bsup>n\<^esup> Q"
source: fixes S1 :: "'procS"   and S2 :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "(S1, S2) \<in> SRel" shows "SourceTerm S1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> SourceTerm S2"
fps_dvd1_left_trivial_unit_factor: fixes f :: "'a fps" assumes "\<exists>g. 1 = f * g" shows "unit_factor f = f"
ran_map_add: fixes m1 :: "'d \<Rightarrow> 'e option"   and m2 :: "'d \<Rightarrow> 'e option" assumes "dom m1 \<inter> dom m2 = {}" shows "ran (m1 ++ m2) = ran m1 \<union> ran m2"
spmf_ub_tight: fixes p :: "'a spmf"   and f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "\<And>x. spmf p x \<le> f x"   and "(\<Sum>\<^sup>+ x. ennreal (f x)) = ennreal (weight_spmf p)" shows "spmf p x = f x"
strict_trans: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subset># b"   and "b \<subset># c" shows "a \<subset># c"
linear_linear: fixes f :: "'c \<Rightarrow> 'd" shows "bounded_linear f = linear f"
A4bRedundant: fixes x :: "i"   and y :: "i" assumes "AxiomsSet6" shows "cod x \<cdot> y \<cong> cod x \<cdot> cod y"
padd_def: fixes a :: "int"   and p\<^sub>1 :: "int \<times> int \<times> int"   and p\<^sub>2 :: "int \<times> int \<times> int" shows "padd a p\<^sub>1 p\<^sub>2 = (let (x\<^sub>1, y\<^sub>1, z\<^sub>1) = p\<^sub>1; (x\<^sub>2, y\<^sub>2, z\<^sub>2) = p\<^sub>2 in if z\<^sub>1 = \<zero> then p\<^sub>2 else if z\<^sub>2 = \<zero> then p\<^sub>1 else let d\<^sub>1 = x\<^sub>2 \<otimes> z\<^sub>1; d\<^sub>2 = x\<^sub>1 \<otimes> z\<^sub>2; l = d\<^sub>1 \<ominus> d\<^sub>2; m = y\<^sub>2 \<otimes> z\<^sub>1 \<ominus> y\<^sub>1 \<otimes> z\<^sub>2 in if l = \<zero> then if m = \<zero> then pdouble a p\<^sub>1 else (\<zero>, \<zero>, \<zero>) else let h = m [^] 2 \<otimes> z\<^sub>1 \<otimes> z\<^sub>2 \<ominus> (d\<^sub>1 \<oplus> d\<^sub>2) \<otimes> l [^] 2 in (l \<otimes> h, (d\<^sub>2 \<otimes> l [^] 2 \<ominus> h) \<otimes> m \<ominus> l [^] 3 \<otimes> y\<^sub>1 \<otimes> z\<^sub>2, l [^] 3 \<otimes> z\<^sub>1 \<otimes> z\<^sub>2))"
Liminf_const: fixes F :: "'a filter"   and c :: "'b" assumes "F \<noteq> bot" shows "Liminf F (\<lambda>x. c) = c"
rel_interior_nonempty_interior: fixes S :: "'a set" assumes "interior S \<noteq> {}" shows "rel_interior S = interior S"
F_2: fixes n :: "nat" shows "F n \<in> sets M"
simps_2: fixes t :: "'a bro" shows "inorder (N1 t) = inorder t"
eq0: shows "\<lbrace>e\<rbrace> ^ 3 + a * \<lbrace>e\<rbrace>\<^sup>2 + b * \<lbrace>e\<rbrace> + c = 0"
add_less_le_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less a b"   and "less_eq c d" shows "less (plus a c) (plus b d)"
spmf_fast_uniform_ub: fixes x :: "nat" shows "spmf local.fast_uniform x \<le> (if x < n then 1 / real n else 0)"
bdd_below_mono: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and A :: "'b set" assumes "class.preorder less_eq less"   and "preorder.bdd_below less_eq B"   and "A \<subseteq> B" shows "preorder.bdd_below less_eq A"
assms_2: assumes "\<And>x y. x \<triangleleft>\<triangleright> y\<cdot>C\<^sup>\<infinity> \<Longrightarrow> f\<cdot>x \<triangleleft>\<triangleright> g\<cdot>y\<cdot>C\<^sup>\<infinity>" shows "P"
fix_strict: fixes F :: "'a \<rightarrow> 'a" assumes "F\<cdot>\<bottom> = \<bottom>" shows "fix\<cdot>F = \<bottom>"
vimage2p_comp: fixes f1 :: "'h \<Rightarrow> 'c"   and f2 :: "'f \<Rightarrow> 'h"   and g1 :: "'i \<Rightarrow> 'd"   and g2 :: "'g \<Rightarrow> 'i" shows "vimage2p (f1 \<circ> f2) (g1 \<circ> g2) = vimage2p f2 g2 \<circ> vimage2p f1 g1"
subgroup_of_subgroup_generated: fixes H :: "'a set"   and B :: "'a set" assumes "H \<subseteq> B"   and "subgroup H (G\<lparr>carrier := H'\<rparr>)" shows "subgroup H (subgroup_generated (G\<lparr>carrier := H'\<rparr>) B)"
rel_spmfE: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf" assumes "rel_spmf P p q" obtains pq :: "('a \<times> 'b) spmf" where "\<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> P x y" and "p = map_spmf fst pq" and "q = map_spmf snd pq"
expand: fixes foo :: "('f, 'g) foo"   and foo' :: "('f, 'g) foo" assumes "foo.field_1 foo = foo.field_1 foo' \<and> foo.field_2 foo = foo.field_2 foo'" shows "foo = foo'"
wf_trm_subst_compose: fixes \<theta> :: "'c \<Rightarrow> ('fun, 'b) Term.term"   and x :: "'c"   and \<delta> :: "'b \<Rightarrow> ('fun, 'b) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m (\<theta> x)"   and "\<And>x. wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)" shows "wf\<^sub>t\<^sub>r\<^sub>m ((\<theta> \<circ>\<^sub>s \<delta>) x)"
e_e'_mod: shows "e' mod order \<G> < e mod order \<G>"
bind_assoc_pmf: fixes A :: "'g pmf"   and B :: "'g \<Rightarrow> 'f pmf"   and C :: "'f \<Rightarrow> 'e pmf" shows "A \<bind> B \<bind> C = A \<bind> (\<lambda>x. B x \<bind> C)"
of_circline_x_axis: shows "of_circline x_axis = o_x_axis"
UINF_def: shows "UINF \<equiv> map_fun id (map_fun id mk\<^sub>e) (\<lambda>P F b. \<Sqinter>{uu_. \<exists>x. uu_ = \<lbrakk>F x\<rbrakk>\<^sub>e b \<and> \<lbrakk>P x\<rbrakk>\<^sub>e b})"
edge_end_tup: fixes x :: "('d \<times> 'c) \<times> 'd \<times> 'b" shows "edge_end x = fst (snd x)"
admissible_targets_clocks_1: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "(l, g, \<mu>) \<in> PTA.trans_of A"   and "(X, l') \<in> set_pmf \<mu>" shows "X \<subseteq> \<X>"
equal: fixes equal :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "class.equal equal" shows "equal = (=)"
hyps: fixes ll :: "('b, 'c) label list"   and u :: "'a"   and c :: "'a list multiset" assumes "length ll \<le> length w"   and "isolated_ret fg p"   and "proc_of fg u = p"   and "u \<noteq> return fg p"   and "(([u], c), ll, [return fg p'], c') \<in> trcl (trss fg)" shows "False"
arcs_del_vert: fixes u :: "'a" shows "arcs (del_vert u) = {a \<in> arcs G. tail G a \<noteq> u \<and> head G a \<noteq> u}"
pred_of_succ_def: fixes s :: "'d \<Rightarrow> 'e set" shows "pred_of_succ s \<equiv> \<lambda>a b. b \<in> s a"
the_\<theta>_props_2: shows "local.iso w.the_\<theta>"
c_def: shows "U \<cdot> (0::'a) = 1\<^sub>\<pi>"
open_imp_subset: fixes U :: "'b set" assumes "is_open U" shows "U \<subseteq> X"
cnf_semantics_def: fixes \<A> :: "'b \<Rightarrow> bool"   and S :: "'b literal set set" shows "cnf_semantics \<A> S \<equiv> \<forall>C\<in>S. clause_semantics \<A> C"
simplex_map_def: fixes p :: "nat"   and g :: "'e \<Rightarrow> 'd"   and c :: "(nat \<Rightarrow> real) \<Rightarrow> 'e" shows "simplex_map p g c \<equiv> restrict (g \<circ> c) (standard_simplex p)"
IH: fixes ts1 :: "'t \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option"   and ts2 :: "'t \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option"   and tas1 :: "('t, 'x1, 'm1) new_thread_action list" assumes "\<forall>t. (ts1 t = None) = (ts2 t = None)"   and "list_all2 (nta_bisim bisim) tas1 TAS2" shows "thread_oks ts1 tas1 = thread_oks ts2 TAS2"
prems_4: shows "x \<in> rvars (\<T> (l ! i))"
sum_constant_scale: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c"   and A :: "'d set" assumes "module scale" shows "(\<Sum>x\<in>A. y) = scale (of_nat (card A)) y"
ex_conv_ex_member: fixes P :: "'b \<Rightarrow> bool"   and t :: "'b set_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Set2.ex P t = (\<exists>x. RBT_Set2.member t x \<and> P x)"
reachableNodes_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat" shows "Graph.reachableNodes c u \<equiv> {v. Graph.connected c u v}"
IH: shows "fsafe \<phi> \<longrightarrow> ssafe \<sigma> \<longrightarrow> (\<forall>\<nu>. (\<nu> \<in> fml_sem I (Fsubst \<phi> \<sigma>)) = (\<nu> \<in> fml_sem (local.adjoint I \<sigma> \<nu>) \<phi>))"
index_swap_if_distinct: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat"   and x :: "'a" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "index (xs[i := xs ! j, j := xs ! i]) x = (if x = xs ! i then j else if x = xs ! j then i else index xs x)"
supp_weak_tau_modality: fixes x :: "('idx, 'pred, 'act) formula" shows "supp (weak_tau_modality x) = supp x"
shorten_position: fixes t :: "('a, 'b) tm"   and p :: "dir list"   and q :: "dir list" assumes "position_of t (p @ q)" shows "position_of t p"
terminal_char: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "terminal a = (\<exists>x. a = MkIde {x})"
noS: shows "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> (\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` set (uset p)"
Nml_HcompD_1: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml (t \<^bold>\<star> u)" shows "\<^bold>\<langle>un_Prim t\<^bold>\<rangle> = t"
simps_2: fixes x :: "'k"   and y :: "'k"   and xs :: "'k list" shows "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
fill_index: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and ns :: "nat list"   and ub :: "nat"   and i :: "nat"   and j :: "nat"   and x :: "'c" assumes "index_less index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "offs_pred ns ub xs index key mi ma"   and "i < length ns"   and "0 < offs_num (length ns) xs index key mi ma i"   and "j \<in> {ns ! i..<offs_next ns ub xs index key mi ma i}"   and "fill xs ns index key ub mi ma ! j = Some x" shows "index key x (length ns) mi ma = i"
divide_nonneg_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less y zero" shows "less_eq (divide x y) zero"
Solutions_def: fixes a :: "nat list"   and b :: "nat list" shows "Linear_Diophantine_Equations.Solutions a b = {(x, y). a \<bullet> x = b \<bullet> y \<and> length x = length a \<and> length y = length b}"
eval_and_def: fixes \<phi> :: "tvl"   and \<psi> :: "tvl" shows "eval_and \<phi> \<psi> = (case (\<phi>, \<psi>) of (Yes, Yes) \<Rightarrow> Yes | (Yes, No) \<Rightarrow> No | (Yes, Maybe) \<Rightarrow> Maybe | (No, xa) \<Rightarrow> No | (Maybe, No) \<Rightarrow> No | (Maybe, _) \<Rightarrow> Maybe)"
exec_bind_SE_failure: fixes A :: "'b \<Rightarrow> ('a \<times> 'b) option"   and \<sigma> :: "'b"   and M :: "'a \<Rightarrow> 'b \<Rightarrow> (bool \<times> 'b) option" assumes "A \<sigma> = None" shows "\<not> (\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E A M)"
\<epsilon>_uh_def: shows "\<epsilon>_uh = (\<Squnion>w w'. spmf (game_hash w w') True)"
multiplicity_gcd: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and p :: "'b" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm"   and "x \<noteq> zero"   and "y \<noteq> zero"   and "normalization_semidom.prime zero times one normalize p" shows "factorial_semiring.multiplicity times one p (gcd x y) = min (factorial_semiring.multiplicity times one p x) (factorial_semiring.multiplicity times one p y)"
in_br_conv: fixes c :: "'c"   and a :: "'d"   and \<alpha> :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool" shows "((c, a) \<in> br \<alpha> I) = (a = \<alpha> c \<and> I c)"
llist_gen_induct: fixes P :: "'b llist \<Rightarrow> bool"   and xs :: "'b llist" assumes "ccpo.admissible lSup lprefix P"   and "\<exists>zs. lprefix zs xs \<and> lfinite zs \<and> (\<forall>ys. lprefix zs ys \<longrightarrow> lprefix ys xs \<longrightarrow> lfinite ys \<longrightarrow> P ys)" shows "P xs"
if_False: fixes x :: "'e"   and y :: "'e" shows "(if False then x else y) = y"
mr_cos_sprod_distrib1: fixes H :: "'a set"   and a :: "'b"   and b :: "'b"   and X :: "'a set" assumes "submodule R M H"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "X \<in> set_mr_cos M H" shows "mr_cos_sprod M H (a \<plusminus>\<^bsub>R\<^esub> b) X = mr_cospOp M H (mr_cos_sprod M H a X) (mr_cos_sprod M H b X)"
l0_f2f2': shows "(f2 + 1) / (f2 - f2') \<le> real l0"
lvl_Suc_iff: fixes t :: "('b \<times> nat) tree"   and n :: "nat" shows "(lvl t = Suc n) = (\<exists>l a r. t = \<langle>l, (a, Suc n), r\<rangle>)"
assms_1: shows "\<S>\<bullet>\<C> \<turnstile> [$Get_local i] : ts _> ts'"
trace_def: fixes sig :: "'j signature" shows "trace sig \<equiv> filter (\<lambda>a. a \<in> externals sig) \<circ> schedule"
bet__ts: fixes A :: "'p"   and Y :: "'p"   and B :: "'p"   and X :: "'p" assumes "A \<noteq> Y"   and "\<not> Col A B X"   and "Bet X A Y" shows "A B TS X Y"
strict_implies_order: fixes b :: "'a word"   and a :: "'a word" assumes "b <s a" shows "b \<le>s a"
relative_homology_group_def: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set" shows "relative_homology_group p X S \<equiv> if p < 0 then singleton_group undefined else relcycle_group (nat p) X S Mod singular_relboundary_set (nat p) X S"
onorm_def: fixes f :: "'c \<Rightarrow> 'd" shows "onorm f = (SUP x. norm (f x) / norm x)"
intro: fixes a :: "char list \<Longrightarrow> 'a" assumes "vwb_lens a" shows "myss' a"
simps_1: fixes R :: "('c, 'd) ring_scheme"   and p2 :: "'c list" assumes "ring R" shows "ring.poly_mult R [] p2 = []"
Im_interval_def: fixes complex_interval :: "complex_interval" shows "Im_interval complex_interval = (case complex_interval of Complex_Interval x1 x2 \<Rightarrow> x2)"
assms_3: shows "n\<^sub>2 \<in> s"
uniform_limit_iff: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and S :: "'e set"   and l :: "'e \<Rightarrow> 'f"   and F :: "'d filter" shows "uniform_limit S f l F = (\<forall>e>0. \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e)"
smooth_on_add: fixes k :: "enat"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x + g x)"
R1: fixes p :: "'a"   and q :: "'a" shows "H p (Ref p q) q"
repeInt_simps: fixes w :: "32 Word.word" shows "(w \<equiv>\<^sub>E real_of_int (sint w)) = (w = 0x7FFFFFFF \<and> 2147483647 \<le> signed_real_of_word w \<or> w = 0x80000001 \<and> signed_real_of_word w \<le> - 2147483647 \<or> sint w < 2147483647 \<and> - 2147483647 < sint w)"
r_pdec222_prim: shows "prim_recfn 1 r_pdec222"
V0_in_FIN_minus_NUM: shows "V\<^sub>0 \<in> FIN - NUM"
set_to_nat_of_interval: fixes m :: "nat" shows "set_to_nat {i. i < m} = 2 ^ m - 1"
carr_dsumag_def: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme" shows "carr_dsumag I A = {f. finiteHom I A f}"