hcju/isabelleps huggingface dataset:
pt_empty_supp_fun_subset: fixes f :: "'a \<Rightarrow> 'c"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "supp f = {}" shows "supp (f x) \<subseteq> supp x"
HSem_def: fixes \<Gamma> :: "('var \<times> 'exp) list" shows "HSem \<Gamma> = (\<Lambda> \<rho>. \<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>x\<^esub>)"
E_Let: fixes t :: "trm"   and t' :: "trm"   and p :: "pat"   and u :: "trm" assumes "t \<longmapsto> t'" shows "(LET p = t IN u) \<longmapsto> (LET p = t' IN u)"
k12_2: shows "\<forall>x. k2 (0, x) = g x"
fmdom_fmupd: fixes a :: "'c"   and b :: "'d"   and m :: "('c, 'd) fmap" shows "fmdom (fmupd a b m) = finsert a (fmdom m)"
Frame'_def: fixes P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a \<Rightarrow> bool" shows "Frame' P P' Q F \<equiv> \<forall>s. (P' \<and>* P) s \<longrightarrow> (Q \<and>* F) s"
map_le_iff_map_add_commute: fixes f :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" shows "(f \<subseteq>\<^sub>m f ++ g) = (f ++ g = g ++ f)"
af_G_prefix_prop: fixes \<phi> :: "'b ltln"   and w' :: "'b set list"   and w :: "'b set list" shows "af (G\<^sub>n \<phi>) (w' @ w) \<longrightarrow>\<^sub>P af (G\<^sub>n \<phi>) w"
All_rev_wrap_def: fixes P :: "'b list \<Rightarrow> bool" shows "All_rev_wrap P = (\<forall>x. P (rev x))"
distinct_2: fixes x2 :: "nat"   and x1 :: "nat" shows "R x2 \<noteq> I x1"
lselect_lproject': fixes s :: "nat set"   and w :: "'b llist"   and A :: "'b set" assumes "\<And>i. i \<in> s \<Longrightarrow> enat i < llength w" shows "lproject A (lselect s w) = lselect (s \<inter> liset A w) w"
norm_power: fixes x :: "'c"   and n :: "nat" shows "norm (x ^ n) = norm x ^ n"
simps_2: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "bexp"   and c :: "com"   and n :: "nat" shows "get_back P b c (Suc n) = (\<lambda>t t'. \<exists>s. (c, s) \<Rightarrow> t \<and> bval b s \<and> get_back P b c n s t')"
cfcomp2: fixes f :: "'f \<rightarrow> 'd"   and g :: "'e \<rightarrow> 'f"   and x :: "'e" shows "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
rank_link: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.rank t = SkewBinomialHeapStruc.rank t'" shows "SkewBinomialHeapStruc.rank (SkewBinomialHeapStruc.link t t') = SkewBinomialHeapStruc.rank t + 1"
limit_jordan_matrix: fixes i :: "nat"   and j :: "nat"   and off :: "nat" assumes "i < n"   and "j < n" shows "(\<lambda>N. (J ^\<^sub>m K off N) $$ (i, j) * complex_of_real (C off N)) \<longlonglongrightarrow> (if j \<in> I \<and> i = j - (m - 1) then Lam j ^ off else 0)"
ODE_unbound_zero: fixes i :: "'d"   and ODE :: "('e, 'd) ODE"   and I :: "('e, 'f, 'd) interp"   and x :: "(real, 'd) vec" assumes "Inl i \<notin> BVO ODE" shows "ODE_sem I ODE x $ i = 0"
elims_3: fixes x :: "real_min"   and xa :: "real_min" assumes "\<not> x < xa" obtains x :: "real" and y :: "real" where "xa__ = Rmin x" and "xa = Rmin y" and "\<not> x < y"
assms_3: fixes x :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "h \<turnstile> g x \<rightarrow>\<^sub>r y"
that: fixes A2 :: "V set" assumes "tp A2 = \<alpha>"   and "A2 \<subseteq> A1"   and "{x} \<lless> A2" shows "thesis"
seq_def: shows "seq \<equiv> \<lambda>ars. {(xa, x). seqp (\<lambda>x xa xb. (x, xa, xb) \<in> ars) xa x}"
fps_unit_factor_mult: fixes f :: "'a fps"   and g :: "'a fps" shows "unit_factor (f * g) = unit_factor f * unit_factor g"
f_Exec_Stream_Init_not_empty: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream_Init trans_fun xs c \<noteq> []"
finite_list: fixes A :: "'a set" assumes "finite A" shows "\<exists>xs. set xs = A"
cdleswap: shows "\<forall>j<k. k + n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<longrightarrow> (\<exists>j'<k'. k' + n' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> j' \<and> cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j')"
le_iff_inf: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type) = ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) x y = x)"
generate_eq_finprod_PiE_image: fixes gs :: "'a set" assumes "finite gs"   and "gs \<subseteq> carrier G" shows "generate G gs = (\<lambda>x. finprod G x gs) ` (\<Pi>\<^sub>E a\<in>gs. generate G {a})"
cod_eqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.ide C b"   and "partial_magma.arr C (C b f)" shows "partial_magma.cod C f = b"
is_constant_semantics: fixes \<phi> :: "'a ltln" assumes "is_constant \<phi>" shows "(\<forall>w. w \<Turnstile>\<^sub>n \<phi>) \<or> (\<nexists>w. w \<Turnstile>\<^sub>n \<phi>)"
IH: fixes es' :: "char list exp list" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"   and "finals es" shows "es = es' \<and> s = s'"
intersect_poincare_circles_conjugate_centers: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and r :: "real"   and u :: "complex_homo" assumes "z1 \<in> unit_disc"   and "z2 \<in> unit_disc"   and "z1 \<noteq> z2"   and "z1 = conjugate z2"   and "0 < r"   and "u \<in> poincare_circle z1 r \<inter> poincare_circle z2 r" shows "is_real (to_complex u)"
indep_sets_def: fixes F :: "'c \<Rightarrow> ('a \<Rightarrow> 'b) set set"   and I :: "'c set" shows "indep_sets F I = ((\<forall>i\<in>I. F i \<subseteq> events) \<and> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> (\<forall>A\<in>Pi J F. prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j)))))"
divide_pos_pos: fixes x :: "'d"   and y :: "'d" assumes "(0::'d) < x"   and "(0::'d) < y" shows "(0::'d) < x / y"
induct: fixes P :: "(event_data option list set list \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> event_data option list set list list \<Rightarrow> bool"   and a0 :: "event_data option list set list \<Rightarrow> 'a \<Rightarrow> 'a"   and a1 :: "'a"   and a2 :: "event_data option list set list list" assumes "\<And>f z buf. (\<not> (buf = [] \<or> [] \<in> set buf) \<Longrightarrow> P f (f (map hd buf) z) (map tl buf)) \<Longrightarrow> P f z buf" shows "P a0 a1 a2"
fin: shows "finite {i. n cd\<^bsup>\<pi> \<guillemotleft> k\<^esup>\<rightarrow> i}"
subresultant_mat_def: fixes J :: "nat"   and F :: "'b poly"   and G :: "'b poly" shows "subresultant_mat J F G = (let dg = degree G; df = degree F; f = coeff_int F; g = coeff_int G; n = df - J + (dg - J) in mat n n (\<lambda>(i, j). if j < dg - J then if i = n - 1 then monom (1::'b) (dg - J - 1 - j) * F else [:f (int df - int i + int j):] else let jj = j - (dg - J) in if i = n - 1 then monom (1::'b) (df - J - 1 - jj) * G else [:g (int dg - int i + int jj):]))"
rep_eq: fixes x :: "('d, 'c) sq_matrix" shows "Square_Matrix.row x = (\<lambda>i. vec_lambda (to_fun x i))"
path_finite_leq: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" assumes "wf_prog wf_md P" shows "finite {Cs. Subobjs P C Cs \<and> length Cs \<le> length P}"
LIMSEQ_le_zero: fixes g :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real" assumes "g \<longlonglongrightarrow> 0"   and "\<forall>\<^sup>\<infinity>n. 0 \<le> f n"   and "\<forall>\<^sup>\<infinity>n. f n \<le> g n" shows "f \<longlonglongrightarrow> 0"
disjoint_Aux2: fixes c :: "('a, 'b) sequent"   and A :: "('a, 'b) form"   and B :: "('a, 'b) form"   and S :: "('a, 'b) sequent" assumes "ModalSequents.mset c = \<LM> A  \<RM>"   and "A \<noteq> B"   and "ModalSequents.mset (extend S c) = \<LM> B  \<RM>" shows "False"
eq_refl: fixes x :: "'a list"   and y :: "'a list" assumes "x = y" shows "prefix x y"
mset_neg_empty_iff: fixes M :: "'a zmultiset" shows "(mset_neg M = {#}) = (\<forall>t. 0 \<le> zcount M t)"
uncurry_def: fixes \<tau> :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f :: "'j \<times> 'a" shows "Curry.uncurry \<tau> f \<equiv> if JxA.arr f then E.map (\<tau> (fst f), snd f) else B.null"
ainvf_def: fixes F :: "('d, 'e) aGroup_scheme"   and G :: "('c, 'f) aGroup_scheme"   and f :: "'d \<Rightarrow> 'c" shows "ainvf\<^bsub>F,G\<^esub> f = f\<inverse>\<^bsub>carrier G,carrier F\<^esub>"
Right_Coset_Class_unit: fixes g :: "'a" assumes "g \<in> G" shows "Normal |\<cdot> g = Class g"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>k Ks u Us. P Ks Us \<Longrightarrow> P (k # Ks) (u # Us)"   and "\<And>Us. P [] Us"   and "\<And>Ks. P Ks []" shows "P a0 a1"
case_eq_if: fixes f :: "'e \<Rightarrow> 'd"   and nondetT :: "('f, 'e) nondetT" shows "(case nondetT of NondetT x \<Rightarrow> f x) = f (run_nondet nondetT)"
path_weight_conv_path: fixes s :: "'sum"   and l1 :: "'loc"   and l2 :: "'loc" assumes "s \<in>\<^sub>A path_summary l1 l2" shows "\<exists>xs. path l1 l2 xs \<and> s = sum_weights (map (\<lambda>(s, l, t). l) xs) \<and> (\<forall>ys. path l1 l2 ys \<longrightarrow> \<not> sum_weights (map (\<lambda>(s, l, t). l) ys) < sum_weights (map (\<lambda>(s, l, t). l) xs))"
cases: fixes x :: "wall" obtains xa :: "brick list" where "x = basic xa__"   | xa :: "brick list" and ys :: "wall" where "x = xa * ys"
inject_4: fixes x4 :: "nat"   and y4 :: "nat" shows "(Pan x4 = Pan y4) = (x4 = y4)"
simps_1: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and project :: "'c \<Rightarrow> 'c"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "project \<Sigma> project" shows "project.lang \<Sigma> project lookup n Zero = {}"
simps_7: shows "eval_ternary_And TernaryUnknown TernaryFalse = TernaryFalse"
PiE_mono: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and C :: "'b \<Rightarrow> 'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C x" shows "Pi\<^sub>E A B \<subseteq> Pi\<^sub>E A C"
map2_cong: fixes xs1 :: "'a list"   and xs2 :: "'a list"   and ys1 :: "'b list"   and ys2 :: "'b list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "xs1 = xs2"   and "ys1 = ys2"   and "\<And>x y. \<lbrakk>x \<in> set xs1; y \<in> set ys1\<rbrakk> \<Longrightarrow> f x y = g x y" shows "Term_Utils.map2 f xs1 ys1 = Term_Utils.map2 g xs2 ys2"
div_mult_swap: 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 c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c b" shows "times a (divide b c) = divide (times a b) c"
f'_def: shows "f' = f - fds_const (fds_nth f 1)"
LI_equality_measure_lt: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list"   and S' :: "('b, 'a) strand_step list"   and \<theta>\<^sub>1 :: "'a \<Rightarrow> ('b, 'a) Term.term"   and a :: "poscheckvariant"   and \<theta>\<^sub>2 :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "Some \<delta> = mgu t t'" shows "((S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>, \<theta>\<^sub>1), S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S', \<theta>\<^sub>2) \<in> measure\<^sub>s\<^sub>t"
stutter_extend_simps_2: fixes G :: "('a, 'b) graph_rec_scheme" shows "g_E (stutter_extend G) = stutter_extend_edges (g_V G) (g_E G)"
simps: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'c Monomorphic_Monad.id"   and a2 :: "'d pmf" shows "cr_id_prob A a1 a2 = (\<exists>x y. a1 = return_id x \<and> a2 = return_pmf y \<and> A x y)"
eq: shows "opp p\<^sub>3 = local.add a p\<^sub>1 p\<^sub>2"
subtype_refl': fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile> T <: T"
subspace_def: fixes Sa :: "('a \<Rightarrow> real) set" assumes "Sa \<subseteq> diff_fun_space" shows "diff_fun_space.m1.subspace Sa = (0 \<in> Sa \<and> (\<forall>x\<in>Sa. \<forall>y\<in>Sa. x + y \<in> Sa) \<and> (\<forall>c. \<forall>x\<in>Sa. c *\<^sub>R x \<in> Sa))"
intros_1: fixes a :: "'c"   and r :: "('c \<times> 'c) set" shows "(a, a) \<in> r\<^sup>*"
Spec: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and r :: "('b \<times> 'b) set"   and s :: "'b" shows "\<Gamma>\<turnstile>Spec r \<down> Normal s"
quotient_map_def: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "quotient_map X X' f = (f ` topspace X = topspace X' \<and> (\<forall>U\<subseteq>topspace X'. openin X {x \<in> topspace X. f x \<in> U} = openin X' U))"
eq_fin_le_inf_concat_end: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w :: "'item stream" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "w\<^sub>1 @- w \<preceq>\<^sub>I w\<^sub>2 @- w"
eq_iff_4: fixes fm :: "fm"   and fma :: "fm" shows "(Neg fm = Neg fma) = (fm = fma)"
simps_1: fixes q :: "hf" shows "Accessible.nextl q [] = q"
single_valued_has_lubs: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and u :: "'b"   and y :: "'b" assumes "single_valued r"   and "(x, u) \<in> r\<^sup>*"   and "(y, u) \<in> r\<^sup>*" shows "\<exists>z. is_lub (r\<^sup>*) x y z"
right_neutral: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and z :: "'g"   and a :: "'g" assumes "monoid f z" shows "f a z = a"
intro: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat" assumes "wf_dfa A n" shows "aut_dfa A n"
prems_3: shows "well_shaped (Node i j a t1 t2)"
induct: fixes x1 :: "com"   and x2 :: "state"   and x3 :: "nat"   and x4 :: "state"   and P :: "com \<Rightarrow> state \<Rightarrow> nat \<Rightarrow> state \<Rightarrow> bool" assumes "<x1,x2> -x3-> x4"   and "\<And>s n. P SKIP s n s"   and "\<And>X a s n. P (X:==a) s n (s[X::=a s])"   and "\<And>c s0 Y a n s1. \<lbrakk><c,s0[Loc Y::=a s0]> -n-> s1; P c (s0[Loc Y::=a s0]) n s1\<rbrakk> \<Longrightarrow> P (LOCAL Y:=a IN c) s0 n (s1[Loc Y::=s0<Y>])"   and "\<And>c0 s0 n s1 c1 s2. \<lbrakk><c0,s0> -n-> s1; P c0 s0 n s1; <c1,s1> -n-> s2; P c1 s1 n s2\<rbrakk> \<Longrightarrow> P (c0;; c1) s0 n s2"   and "\<And>b s c0 n s1 c1. \<lbrakk>b s; <c0,s> -n-> s1; P c0 s n s1\<rbrakk> \<Longrightarrow> P (IF b THEN c0 ELSE c1) s n s1"   and "\<And>b s c1 n s1 c0. \<lbrakk>\<not> b s; <c1,s> -n-> s1; P c1 s n s1\<rbrakk> \<Longrightarrow> P (IF b THEN c0 ELSE c1) s n s1"   and "\<And>b s c n. \<not> b s \<Longrightarrow> P (WHILE b DO c) s n s"   and "\<And>b s0 c n s1 s2. \<lbrakk>b s0; <c,s0> -n-> s1; P c s0 n s1; <WHILE b DO c,s1> -n-> s2; P (WHILE b DO c) s1 n s2\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s0 n s2"   and "\<And>pn s0 n s1. \<lbrakk><the (body pn),s0> -n-> s1; P (the (body pn)) s0 n s1\<rbrakk> \<Longrightarrow> P (BODY pn) s0 (Suc n) s1"   and "\<And>pn s0 a n s1 X. \<lbrakk><BODY pn,setlocs s0 newlocs[Loc Arg::=a s0]> -n-> s1; P (BODY pn) (setlocs s0 newlocs[Loc Arg::=a s0]) n s1\<rbrakk> \<Longrightarrow> P (X:=CALL pn(a)) s0 n (setlocs s1 (loc s0)[X::=s1<Res>])" shows "P x1 x2 x3 x4"
minus_equality: fixes y :: "nat \<Rightarrow> 'a"   and x :: "nat \<Rightarrow> 'a" assumes "y \<oplus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>"   and "x \<in> carrier P"   and "y \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> x = y"
set_child_nodes_get_child_nodes_is_l_set_child_nodes_get_child_nodes: shows "l_set_child_nodes_get_child_nodes ShadowRootClass.type_wf ShadowRootClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs"
add: shows "\<forall>i<d. \<forall>j<d. (\<lambda>n. f n $$ (i, j) + g n $$ (i, j)) \<longlonglongrightarrow> matrix_seq.lowner_lub (\<lambda>n. f n + g n) $$ (i, j)"
add_mono_thms_linordered_field_3: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i < j \<and> k \<le> l" shows "i + k < j + l"
run_def: fixes sys :: "('f, 'g, 'h, 'i, 'j) system" shows "run sys = (\<lambda>s. prerun sys s \<and> FAIR sys s)"
aux114: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" assumes "x \<noteq> DenyAll" shows "removeShadowRules1_alternative_rev (x # xs) = x # removeShadowRules1_alternative_rev xs"
bind_eq_Some_conv: fixes f :: "'f option"   and g :: "'f \<Rightarrow> 'e option"   and x :: "'e" shows "(f \<bind> g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
simps_2: shows "eval_ternary_And TernaryTrue TernaryFalse = TernaryFalse"
memory_write_asi_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and addr :: "word32"   and bm :: "word4"   and data :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> Some t1 = memory_write_asi 10 addr bm data s1 \<and> Some t2 = memory_write_asi 10 addr bm data s2" shows "low_equal t1 t2"
insertion_sort_alt_def: fixes a :: "'b array" shows "insertion_sort a = Array.len a \<bind> sort_upto a"
IntE: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A \<inter> B" obtains "c \<in> A" and "c \<in> B"
has_locks_has_locks_t_eq: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and n :: "nat"   and t' :: "'b"   and n' :: "nat" assumes "has_locks l t = Suc n"   and "has_locks l t' = Suc n'" shows "t = t'"
elims: fixes key :: "'a \<Rightarrow> 'b"   and x :: "'a list"   and y :: "'a list" assumes "msort_key key x = y" obtains xs :: "'a list" where "x = xs" and "y = merge_all key (sequences key xs)"
on_triple_dest: fixes R :: "('d \<times> 'c) set"   and E :: "('b \<times> 'd \<times> 'd) set"   and G :: "('b \<times> 'c \<times> 'c) set"   and l :: "'b"   and x :: "'d"   and y :: "'d"   and xx :: "'c"   and yy :: "'c" assumes "on_triple R `` E \<subseteq> G"   and "(l, x, y) \<in> E"   and "(x, xx) \<in> R"   and "(y, yy) \<in> R" shows "(l, xx, yy) \<in> G"
card_gt_0_iff: fixes A :: "'b set" shows "(0 < card A) = (A \<noteq> {} \<and> finite A)"
carrier_subgroup_generated_subgroup: shows "carrier (subgroup_generated G B) = B"
converse_pseudonatural_equivalence_def: fixes V\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>C :: "'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and V\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>D :: "'s \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and \<tau>\<^sub>0 :: "'r \<Rightarrow> 's"   and \<tau>\<^sub>1 :: "'r \<Rightarrow> 's" shows "converse_pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1 \<equiv> (bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D) \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G \<and> pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
starset_diff: fixes A :: "'b set"   and B :: "'b set" shows "*s* (A - B) = *s* A - *s* B"
ntrp_mon_env_e_no_ctx: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), ENV e, s', c') \<in> ntrp fg" shows "mon_w fg e \<inter> mon_s fg s = {}"
cone_iff_finite_cone: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec m"   and "finite A" shows "m.cone A = m.finite_cone A"
ltrm_of_sub_in_ltrm: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "deg R f = n"   and "0 < deg R g" shows "monom P (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g (deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g))) (deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g)) = f (deg R f) \<odot>\<^bsub>P\<^esub> monom P (g (deg R g)) (deg R g) [^]\<^bsub>P\<^esub> n"
ring_primeI: fixes p :: "'a" assumes "p \<noteq> \<zero>"   and "prime R p" shows "ring_prime p"
exhaust: fixes y :: "order_tag" obtains "y = Lex"   | "y = Mul"
asymp_equiv_cong: fixes f1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f1 x = f2 x"   and "\<forall>\<^sub>F x in F. g1 x = g2 x" shows "(f1 \<sim>[F] g1) = (f2 \<sim>[F] g2)"
atLeastatMost_empty_iff2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.preorder less_eq less" shows "({} = ord.atLeastAtMost less_eq a b) = (\<not> less_eq a b)"
sup_less_eq_cases: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "- z \<le> - x \<squnion> - y"   and "- z \<le> - - x \<squnion> - y" shows "- z \<le> - y"
X0_3: shows "X0 \<times> C \<subseteq> W"
x: shows "x \<in> vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (s \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>)"
Arg_unique_lemma: fixes z :: "complex"   and t :: "real"   and t' :: "real" assumes "is_Arg z t"   and "is_Arg z t'"   and "- pi < t"   and "t \<le> pi"   and "- pi < t'"   and "t' \<le> pi"   and "z \<noteq> 0" shows "t' = t"
bind_writerT_def: shows "Monad_Overloading.bind = bind_writer Monad_Overloading.return Monad_Overloading.bind"
second_countable_topology_ow_def: fixes U :: "'c set"   and \<tau> :: "'c set \<Rightarrow> bool" shows "second_countable_topology_ow U \<tau> \<equiv> topological_space_ow U \<tau> \<and> second_countable_topology_ow_axioms U \<tau>"