hcju/isabelleps huggingface dataset:
types_preserved_loop: fixes vs :: "e list"   and t1s :: "t list"   and t2s :: "t list"   and es :: "b_e list"   and n :: "nat"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and m :: "nat" assumes "\<lparr>vs @ [$Loop (t1s _> t2s) es]\<rparr> \<leadsto> \<lparr>[Label n [$Loop (t1s _> t2s) es] (vs @ ($* es))]\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> vs @ [$Loop (t1s _> t2s) es] : ts _> ts'"   and "const_list vs"   and "length vs = n"   and "length t1s = n"   and "length t2s = m" shows "\<S>\<bullet>\<C> \<turnstile> [Label n [$Loop (t1s _> t2s) es] (vs @ ($* es))] : ts _> ts'"
l_distr: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
ar_cos_mem: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> carrier R" shows "a \<uplus>\<^bsub>R\<^esub> I \<in> set_rcs (b_ag R) I"
T_ls2: shows "c_assoc_value (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_pair (c_snd (c_snd n)) x) = univ_for_pr (c_pair (c_snd (c_snd n)) x)"
fmapU_maybe_simps_2: fixes f :: "udom \<rightarrow> udom" shows "fmapU\<cdot>f\<cdot>Nothing = Nothing"
neg_closed: fixes x :: "'a" assumes "x \<in> V" shows "- x \<in> V"
rec_1: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a" shows "rec_option f1 f2 None = f1"
IF_PARALLEL: fixes G :: "Var \<Rightarrow> L"   and p :: "L"   and b :: "BExpr"   and H :: "Var \<Rightarrow> L"   and q :: "L"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c1 :: "IMP"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c2 :: "IMP"   and \<Psi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "\<forall>s ss. EQ G p s ss \<longrightarrow> evalB b s = evalB b ss"   and "\<forall>x. LEQ (G x) (H x) \<or> LEQ p (H x)"   and "\<exists>x. LEQ p (H x) \<and> LEQ (H x) q"   and "X \<rhd> c1 : Sec (Q p H) (EQ G q) (EQ H q) \<Phi>"   and "X \<rhd> c2 : Sec (Q p H) (EQ G q) (EQ H q) \<Psi>" shows "X \<rhd> Iff b c1 c2 : Sec (Q p H) (EQ G q) (EQ H q) (\<lambda>(r, u). (evalB b u \<longrightarrow> \<Phi> (r, u)) \<and> (\<not> evalB b u \<longrightarrow> \<Psi> (r, u)))"
simps: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'b" shows "vec_map f (x, y) = (f x, f y)"
variety_of_ideal_of_variety: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "\<V> (\<I> (\<V> F)) = \<V> F"
that: fixes q :: "'a set set" assumes "p \<subseteq> q"   and "q division_of cbox a b" shows "thesis"
polynomial_function_cmul: fixes f :: "'c \<Rightarrow> 'd"   and c :: "real" assumes "polynomial_function f" shows "polynomial_function (\<lambda>x. c *\<^sub>R f x)"
rec: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "rec_id f (return_id x) = f x"
goodInp_def: fixes inp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option" shows "goodInp inp \<equiv> liftAll good inp \<and> |{i. inp i \<noteq> None}| <o |UNIV|"
set_objectE: fixes A :: "'b set"   and U :: "'b set" assumes "A \<in> Ob\<^bsub>set_cat U\<^esub>" shows "A \<subseteq> U"
complex_mat_has_unitary_schur_decomposition: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "\<exists>B P es. similar_mat_wit A B P (adjoint P) \<and> unitary P \<and> char_poly A = (\<Prod>e\<leftarrow>es. [:- e, 1:]) \<and> diag_mat B = es"
simps_3: fixes y :: "'a"   and ys :: "[:'a:]" assumes "y \<noteq> \<bottom>"   and "ys \<noteq> \<bottom>" shows "eq\<cdot>[::]\<cdot>(y :# ys) = FF"
monom_sub: fixes a :: "'a"   and g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "a \<in> carrier R"   and "g \<in> carrier P" shows "Cring_Poly.compose R (monom P a n) g = a \<odot>\<^bsub>P\<^esub> g [^]\<^bsub>P\<^esub> n"
inj_ab_group_add_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "inj_ab_group_add_hom hom \<equiv> ab_group_add_hom hom \<and> inj_group_add_hom hom"
of_int_0_eq_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(zero = ring_1.of_int one plus zero minus z) = (z = 0)"
independent_bound_general: fixes S :: "('a, 'b) vec set" assumes "vec.independent S" shows "finite S \<and> card S \<le> vec.dim S"
carrier_vec_dim_vec: fixes v :: "'c vec" shows "v \<in> carrier_vec (dim_vec v)"
bex_Infinitesimal_iff2: fixes x :: "'a star"   and z :: "'a star" shows "(\<exists>y\<in>Infinitesimal. x = z + y) = (x \<approx> z)"
set_eq1: shows "{a [^] n |n. n \<in> {1..d}} = {x \<in> carrier (mult_of R). x [^] d = \<one>}"
subgroup_generated_subset_carrier_subset: fixes S :: "'a set" assumes "S \<subseteq> carrier G" shows "S \<subseteq> carrier (subgroup_generated G S)"
simps_1: fixes n :: "nat" shows "0 * n = 0"
assms: shows "q2 \<in> io_targets M q1 [xy]"
start: shows "OK first \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! 0"
ordering_axioms: shows "ordering (\<sqsubseteq>) (\<lambda>x y. x \<sqsubseteq> y \<and> x \<noteq> y)"
induct: fixes P :: "Syntax.term \<Rightarrow> Syntax.term \<Rightarrow> Syntax.var \<Rightarrow> bool"   and a0 :: "Syntax.term"   and a1 :: "Syntax.term"   and a2 :: "Syntax.var" assumes "\<And>t' x. P Unit t' x"   and "\<And>y t' x. P (Var y) t' x"   and "\<And>y x t' t. \<lbrakk>atom y \<sharp> (x, t'); P t t' x\<rbrakk> \<Longrightarrow> P (Lam y t) t' x"   and "\<And>y x t' t. \<lbrakk>atom y \<sharp> (x, t'); P t t' x\<rbrakk> \<Longrightarrow> P (Rec y t) t' x"   and "\<And>t t' x. P t t' x \<Longrightarrow> P (Inj1 t) t' x"   and "\<And>t t' x. P t t' x \<Longrightarrow> P (Inj2 t) t' x"   and "\<And>t1 t2 t' x. \<lbrakk>P t1 t' x; P t2 t' x\<rbrakk> \<Longrightarrow> P (Syntax.Pair t1 t2) t' x"   and "\<And>t t' x. P t t' x \<Longrightarrow> P (Roll t) t' x"   and "\<And>y x t' t1 t2. \<lbrakk>atom y \<sharp> (x, t'); P t1 t' x; P t2 t' x\<rbrakk> \<Longrightarrow> P (Syntax.Let t1 y t2) t' x"   and "\<And>t1 t2 t' x. \<lbrakk>P t1 t' x; P t2 t' x\<rbrakk> \<Longrightarrow> P (App t1 t2) t' x"   and "\<And>t1 t2 t3 t' x. \<lbrakk>P t1 t' x; P t2 t' x; P t3 t' x\<rbrakk> \<Longrightarrow> P (Case t1 t2 t3) t' x"   and "\<And>t t' x. P t t' x \<Longrightarrow> P (Prj1 t) t' x"   and "\<And>t t' x. P t t' x \<Longrightarrow> P (Prj2 t) t' x"   and "\<And>t t' x. P t t' x \<Longrightarrow> P (Unroll t) t' x"   and "\<And>t t' x. P t t' x \<Longrightarrow> P (Auth t) t' x"   and "\<And>t t' x. P t t' x \<Longrightarrow> P (Unauth t) t' x"   and "\<And>h t' x. P (Hash h) t' x"   and "\<And>h t t' x. P t t' x \<Longrightarrow> P (Hashed h t) t' x" shows "P a0 a1 a2"
is_natural_2: fixes f :: "'b \<times> 'b" assumes "B.VV.arr f" shows "cmp (B.VV.cod f) \<cdot>\<^sub>B cmp f = cmp f"
prod_zero: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "\<exists>a\<in>A. f a = zero" shows "comm_monoid_mult.prod times one f A = zero"
dvd_prodI: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and A :: "'d set"   and a :: "'d"   and f :: "'d \<Rightarrow> 'c" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "a \<in> A" shows "dvd.dvd times (f a) (comm_monoid_mult.prod times one f A)"
SMP_Cons: fixes x :: "('fun, 'var) strand_step"   and S :: "('fun, 'var) strand_step list" shows "SMP (trms\<^sub>s\<^sub>t (x # S)) = SMP (trms\<^sub>s\<^sub>t [x]) \<union> SMP (trms\<^sub>s\<^sub>t S)"
inter_scheme_injective: fixes l :: "nat"   and U :: "nat list set"   and U' :: "nat list set" assumes "Form l U"   and "Form l U'"   and "0 < l"   and "inter_scheme l U' = inter_scheme l U" shows "U' = U"
continuous_at_Arctan: fixes z :: "complex" assumes "Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1" shows "isCont Arctan z"
SDG_return_or_param_out_edge_unique_CFG_return_edge: fixes n :: "'node SDG_node"   and Vopt :: "'var option"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "SDG_edge n Vopt (Some (p, False)) n'" shows "\<exists>!a. valid_edge a \<and> sourcenode a = parent_node n \<and> targetnode a = parent_node n' \<and> (\<exists>Q f. kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f)"
exhaust: fixes y :: "('a, 'b) antecedent" obtains x1 :: "'a fset" and x2 :: "'a" and x3 :: "'b set" where "y = Antecedent x1 x2 x3"
lookup_join: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "('a, 'b) mapping_rbt"   and t2 :: "('a, 'b) mapping_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Mapping2.lookup (RBT_Mapping2.join f t1 t2) = (\<lambda>k. case RBT_Mapping2.lookup t1 k of None \<Rightarrow> RBT_Mapping2.lookup t2 k | Some v1 \<Rightarrow> Some (case RBT_Mapping2.lookup t2 k of None \<Rightarrow> v1 | Some v2 \<Rightarrow> f k v1 v2))"
standard_control_dependence_def: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g"   and n :: "'g"   and n' :: "'g" assumes "Postdomination sourcenode targetnode kind valid_edge Entry Exit" shows "Postdomination.standard_control_dependence sourcenode targetnode valid_edge Exit n n' = (\<exists>a a' as. n' \<notin> set (CFG.sourcenodes sourcenode (a # as)) \<and> CFG.path sourcenode targetnode valid_edge n (a # as) n' \<and> Postdomination.postdominate sourcenode targetnode valid_edge Exit n' (targetnode a) \<and> valid_edge a' \<and> sourcenode a' = n \<and> \<not> Postdomination.postdominate sourcenode targetnode valid_edge Exit n' (targetnode a'))"
bounded_linear_mult_const: fixes g :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "bounded_linear g" shows "bounded_linear (\<lambda>x. g x * y)"
hinter_hempty_left: fixes A :: "hf" shows "0 \<sqinter> A = 0"
seq_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "B.VxVxV.seq g f = (B.seq (fst g) (fst f) \<and> B.VxV.seq (snd g) (snd f))"
ok_JN_iff1: fixes F :: "'b program"   and I :: "'c set"   and G :: "'c \<Rightarrow> 'b program" shows "F ok JOIN I G = (\<forall>i\<in>I. F ok G i)"
surj_iff: fixes f :: "'c \<Rightarrow> 'b" shows "top_pres ((`) f) = (f \<circ> inv f = id)"
CAND_ImpR_elim: fixes a :: "coname"   and x :: "name"   and b :: "coname"   and M :: "trm"   and B :: "ty" assumes "<a>:ImpR x.<b>.M a \<in> \<parallel><B>\<parallel>"   and "<a>:ImpR x.<b>.M a \<notin> BINDINGc B (\<parallel>(B)\<parallel>)" shows "\<exists>B1 B2. B = B1 IMP B2 \<and> (\<forall>z P. x \<sharp> (z, P) \<and> z:P \<in> \<parallel>(B2)\<parallel> \<longrightarrow> x:M{b:=z.P} \<in> \<parallel>(B1)\<parallel>) \<and> (\<forall>c Q. b \<sharp> (c, Q) \<and> <c>:Q \<in> \<parallel><B1>\<parallel> \<longrightarrow> <b>:M{x:=<c>.Q} \<in> \<parallel><B2>\<parallel>)"
spec2: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and j :: "'b"   and i :: "'c" assumes "\<forall>j i. P j i" shows "P j i"
that: fixes ek :: "real" assumes "a * aform_val e X + b * aform_val e Y + c + u = aform_val (e(k := ek)) (affine_binop' p X Y a b c d k)"   and "ek \<in> {- 1..1}" shows "thesis"
intros_5: fixes Ts :: "typ list"   and u :: "Core.term"   and U :: "typ"   and t :: "Core.term"   and T :: "typ" assumes "Ts \<turnstile>\<^sub>\<tau> u : U"   and "Ts \<turnstile>\<^sub>\<tau> t : U \<rightarrow> T" shows "Ts \<turnstile>\<^sub>\<tau> t $ u : T"
le_less_trans: fixes x :: "'g"   and y :: "'g"   and z :: "'g" assumes "x \<le> y"   and "y < z" shows "x < z"
deg_sect_Suc: fixes X :: "'b set"   and d :: "nat" shows "deg_sect X (Suc d) = (\<Union>x\<in>X. (+) (monomial 1 x) ` deg_sect X d)"
compact': fixes \<epsilon> :: "real" shows "compact (B' \<epsilon>)"
ltakeWhile_lmap: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" shows "ltakeWhile P (lmap f xs) = lmap f (ltakeWhile (P \<circ> f) xs)"
abs_ge_self: fixes a :: "'a" shows "a \<le> \<bar>a\<bar>"
rcosets_inv_mult_group_eq: fixes M :: "'a set" assumes "M \<in> rcosets H" shows "set_inv M <#> M = H"
antisym: fixes x :: "'b"   and y :: "'b" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "x = y"
has_vector_derivative_eq_rhs: fixes f :: "real \<Rightarrow> 'b"   and X :: "'b"   and F :: "real filter"   and Y :: "'b" assumes "(f has_vector_derivative X) F"   and "X = Y" shows "(f has_vector_derivative Y) F"
that_1: fixes xs :: "'a list" assumes "xa = []"   and "xb = xs"   and "xc = xs" shows "thesis"
1: shows "val_Zp a \<le> val_Zp c"
cmp_simps_5: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.cod (\<Phi>\<^sub>G (f, g)) = G (f \<star>\<^sub>C g)"
pt_name3: fixes pi1 :: "(name \<times> name) list"   and pi2 :: "(name \<times> name) list"   and x :: "'a" assumes " pi1 \<triangleq> pi2 " shows "pi1 \<bullet> x = pi2 \<bullet> x"
landau_flip_2: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "(f \<in> o[F](g)) = (g \<in> \<omega>[F](f))"
stirling_same: fixes n :: "nat" shows "stirling n n = 1"
split_llist_first: fixes x :: "'a"   and xs :: "'a llist" assumes "x \<in> lset xs" shows "\<exists>ys zs. xs = lappend ys (LCons x zs) \<and> lfinite ys \<and> x \<notin> lset ys"
mcont_ldrop: shows "mcont (prod_lub the_Sup lSup) (rel_prod (=) (\<sqsubseteq>)) lSup (\<sqsubseteq>) (\<lambda>(x, y). ldrop x y)"
fusion_part_character: fixes F :: "i \<Rightarrow> bool" assumes "\<exists>x. F x" shows "\<forall>y. P y (\<sigma>v. F v) = (\<forall>w. P w y \<longrightarrow> (\<exists>v. F v \<and> O w v))"
mk_lossless_return: fixes x :: "'c option" shows "mk_lossless (return_pmf x) = return_pmf x"
cn_unhalt_case: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list"   and ap :: "abc_inst list"   and ar :: "nat"   and ft :: "nat"   and args :: "nat list"   and i :: "nat"   and gap :: "abc_inst list"   and gar :: "nat"   and gft :: "nat"   and anything :: "nat list" assumes "rec_ci (Cn n f gs) = (ap, ar, ft) \<and> length args = ar"   and "i < length gs"   and "rec_ci (gs ! i) = (gap, gar, gft) \<and> gar = length args"   and "\<And>anything. {\<lambda>nl. nl = args @ 0 \<up> (gft - gar) @ anything} gap \<up>"   and "\<And>apj arj ftj j anything. \<lbrakk>j < i; rec_ci (gs ! j) = (apj, arj, ftj)\<rbrakk> \<Longrightarrow> {\<lambda>nl. nl = args @ 0 \<up> (ftj - arj) @ anything} apj {\<lambda>nl. nl = args @ rec_exec (gs ! j) args # 0 \<up> (ftj - Suc arj) @ anything}"   and "\<forall>j<i. terminate (gs ! j) args" shows "{\<lambda>nl. nl = args @ 0 \<up> (ft - ar) @ anything} ap \<up>"
cancel_separate_eq: fixes x :: "'a"   and y :: "'a" assumes "coreflexive (x * y)" shows "x\<^sup>\<star> * y\<^sup>\<star> = x\<^sup>\<star> \<squnion> y\<^sup>\<star>"
line_line1: fixes a :: "'b"   and b :: "'b"   and c :: "real"   and x :: "real" shows "{{a -- b}\<^bsub>c\<^esub> -- b}\<^bsub>x\<^esub> = {a -- b}\<^bsub>c + x - c * x\<^esub>"
fds_subseries_diff: fixes P :: "nat \<Rightarrow> bool"   and f :: "'b fds"   and g :: "'b fds" shows "fds_subseries P (f - g) = fds_subseries P f - fds_subseries P g"
RInitReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and Cs :: "char list list"   and e\<^sub>0 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>RI (C,e) ; Cs \<leftarrow> e\<^sub>0,s,b\<rangle> \<rightarrow>* \<langle>RI (C,e') ; Cs \<leftarrow> e\<^sub>0,s',b'\<rangle>"
map_add_find_right: fixes n :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and xx :: "'c"   and m :: "'d \<Rightarrow> 'c option" assumes "n k = Some xx" shows "(m ++ n) k = Some xx"
iT_Minus_iAll_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t k \<ominus> I. P t) = (\<box> t I \<down>\<le> k. P (k - t))"
polyfun_linear_factor_root: fixes c :: "nat \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "(\<Sum>i\<le>n. c i * a ^ i) = (0::'b)" shows "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z ^ i) = (z - a) * (\<Sum>i<n. b i * z ^ i)"
cone_mono_1: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and h' :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and U :: "'c set"   and h :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "pm_powerprod ord ord_strict"   and "h' \<in> P[U]" shows "pm_powerprod.cone (h' * h, U) \<subseteq> pm_powerprod.cone (h, U)"
le_acom_subset_same_annos: shows "(strict {(c, c'). c \<sqsubseteq> c'})\<inverse> \<subseteq> (strict (inv_image (listrel {(a, a'). a \<sqsubseteq> a'} - {([], [])}) annos))\<inverse>"
ax1: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> F"
ap_sum_comp: fixes h :: "('g \<Rightarrow> 'e) + bool"   and g :: "('f \<Rightarrow> 'g) + bool"   and x :: "'f + bool" shows "ap_sum (\<and>) (ap_sum (\<and>) (ap_sum (\<and>) (pure (\<circ>)) h) g) x = ap_sum (\<and>) h (ap_sum (\<and>) g x)"
\<phi>_in_terms_of_\<eta>: fixes y :: "'b"   and f :: "'a"   and x :: "'a" assumes "Adj.D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<phi> y f = G f \<cdot>\<^sub>D Adj.\<eta> y"
abs_fun_def: fixes R :: "('b \<times> 'a) set"   and m :: "'b nres" shows "\<Up> R m \<equiv> case m of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> if X \<subseteq> Domain R then RES (R `` X) else FAIL"
last_map: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'e" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
\<pi>_upper_bound_strong: fixes n :: "nat"   and \<alpha> :: "real" assumes "2 \<le> n"   and "\<alpha> \<in> {0<..<1}" shows "\<pi> (real n) < (1 / ((1 - \<alpha>) * exp 1) + ln 4 / \<alpha>) * real n / ln (real n)"
affine_ineq: fixes x :: "'b"   and v :: "'b"   and u :: "'b" assumes "x \<le> (1::'b)"   and "v \<le> u" shows "v + x * u \<le> u + x * v"
matcher_agree_on_exact_matches_def: fixes exact :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and approx :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue" shows "matcher_agree_on_exact_matches exact approx \<equiv> \<forall>p m. approx m p \<noteq> TernaryUnknown \<longrightarrow> exact m p = the (ternary_to_bool (approx m p))"
prems_5: shows "insert e S \<union> X \<union> Y = {0..<m}"
assms_5: shows "\<forall>q\<in>set (p # ps). real_of_int l - \<delta> < real_of_int (fst q) \<and> real_of_int (fst q) < real_of_int l + \<delta>"
length_removeAll_less_eq: fixes x :: "'b"   and xs :: "'b list" shows "length (removeAll x xs) \<le> length xs"
nat_pow_inv: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and i :: "nat" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> i = inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> i)"
P_Nil: shows "\<turnstile> [] [:] [] \<Rightarrow> []"
that: fixes p :: "'state" assumes "p \<in> nodes"   and "enabled a p" shows "thesis"
simps: fixes B :: "ty"   and C :: "ty"   and X :: "ctrm set"   and Y :: "ntrm set" shows "IMPLEFT (B IMP C) X Y = {y:ImpL <a>.M x.N y |x a y M N. fin (ImpL <a>.M x.N y) y \<and> <a>:M \<in> X \<and> x:N \<in> Y}"
supset_lbound_ofI: fixes x :: "'a set"   and b :: "'a set"   and y :: "'a set" assumes "order.greater_eq b x"   and "order.greater_eq b y" shows "dual_order.lbound_of x y b"
eval_simps_5: fixes f :: "'c local.term" assumes "Arr f" shows "\<lbrace>Cod f\<rbrace> = cod \<lbrace>f\<rbrace>"
simps_6: fixes v :: "'a formula"   and va :: "'a formula" shows "is_mini_formula (v \<^bold>\<or> va) = False"
Poly_eq_0: fixes as :: "'a list" shows "(Poly as = 0) = (\<exists>n. as = replicate n (0::'a))"
mset_zero_iff_right: fixes x :: "'a list" shows "({#} = mset x) = (x = [])"
prime_factorsI: fixes x :: "'a"   and p :: "'a" assumes "x \<noteq> (0::'a)"   and "prime p"   and "p dvd x" shows "p \<in> prime_factors x"
termMOD_igOpIPresIGWls: shows "igOpIPresIGWls termMOD"
rel_sel: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a option"   and b :: "'b option" shows "rel_option R a b = ((a = None) = (b = None) \<and> (a \<noteq> None \<longrightarrow> b \<noteq> None \<longrightarrow> R (the a) (the b)))"
indis_sym: fixes s :: "'state"   and s' :: "'state" assumes "s \<approx> s'" shows "s' \<approx> s"
canE_associator_1: fixes f :: "'a E.term"   and g :: "'a E.term"   and h :: "'a E.term" assumes "E.Ide f"   and "E.Ide g"   and "E.Ide h"   and "E.Src f = E.Trg g"   and "E.Src g = E.Trg h" shows "B.can (f \<^bold>\<star> g \<^bold>\<star> h) ((f \<^bold>\<star> g) \<^bold>\<star> h) = \<a>\<^sub>B[EVAL f, EVAL g, EVAL h]"
fls_X_shift_to_one: shows "fls_shift 1 fls_X = 1"
rep_eq: fixes x :: "perm"   and xa :: "('a, 'b, 'c) formula" shows "Rep_formula (x \<bullet> xa) = x \<bullet> Rep_formula xa"
that: fixes a :: "V" assumes "F = ZFC_in_HOL.set {\<langle>j, a\<rangle>}"   and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "thesis"
App: fixes as1 :: "nat list list"   and bs1 :: "nat list list"   and as2 :: "nat list list"   and bs2 :: "nat list list"   and as :: "nat list list"   and bs :: "nat list list" assumes "as1 \<noteq> []"   and "bs1 \<noteq> []"   and "concat as1 < concat bs1"   and "concat bs1 < concat as2"   and "merge as2 bs2 as bs" shows "merge (as1 @ as2) (bs1 @ bs2) (concat as1 # as) (concat bs1 # bs)"
subspace_span: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set" assumes "module scale" shows "module.subspace scale (module.span scale S)"
prems: shows "\<forall>a\<in>set (z # neq). case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0"
intros_4: fixes r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and seq :: "('a, 'b) sequent"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" assumes "r \<in> R"   and "case r of (x, xa) \<Rightarrow> (x, xa) \<in> modRules2" shows "extendRule seq r \<in> ext R R' M N"
seen_def: shows "seen = dds_of (\<lambda>S X. (S |\<inter>| X, S |\<union>| X)) {||}"
lex_polychain_of_center: fixes d :: "(real \<times> real) \<times> real \<times> real"   and x0 :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "d \<in> set (polychain_of x0 xs)"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs" shows "Counterclockwise_2D_Arbitrary.lex (snd d) x0"
bal_stackD: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "c \<Rightarrow>\<^sup>b\<^sup>*\<^bsub>T\<^esub> c'" shows "stack c' = stack c"
singular_cycle: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_relcycle p X {} c = (singular_chain p X c \<and> chain_boundary p c = 0)"
lookup_the_value: fixes xs :: "('c \<times> 'b) list"   and k :: "'c" shows "lookup (the_value xs) k = (case map_of xs k of None \<Rightarrow> 0::'b | Some v \<Rightarrow> v)"