hcju/isabelleps huggingface dataset:
assms: shows "exec_step P h stk loc C M pc ics frs sh = (xp', h', frs', sh')"
subst_introduces_no_rids: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and e :: "('a, 'b, 'c) expr"   and x :: "'c"   and e' :: "('a, 'b, 'c) expr" assumes "substitution subst" shows "RID\<^sub>E (subst e x e') \<subseteq> RID\<^sub>E e \<union> RID\<^sub>E e'"
closure_affine_hull: fixes S :: "'a set" shows "closure S \<subseteq> affine hull S"
that: fixes lambda :: "nat"   and mu :: "nat" assumes "properties lambda mu" shows "thesis"
a: shows "nonzero_positions (Rep_matrix A) \<noteq> {}"
otherassms_4: shows "((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A"
iso_counit: fixes a :: "'a" assumes "C.obj a" shows "D.iso (counit a)"
cmp_inj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C"   and "inj_on f A"   and "inj_on g B" shows "inj_on (cmp g f) A"
induct: fixes x1 :: "BExpr"   and x2 :: "TP"   and P :: "BExpr \<Rightarrow> TP \<Rightarrow> bool" assumes "(x1, x2) \<in> VS_Bexpr"   and "\<And>e1 t e2 f. \<lbrakk>(e1, t) \<in> VS_expr; (e2, t) \<in> VS_expr; compBGood f\<rbrakk> \<Longrightarrow> P (compB f e1 e2) t"   and "\<And>e. P e high" shows "P x1 x2"
is_process3: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "is_process P" shows "\<forall>s t. (s @ t, {}) \<in> FAILURES P \<longrightarrow> (s, {}) \<in> FAILURES P"
sinpoly_minus: fixes n :: "nat"   and x :: "real" shows "sinpoly n (- x) = - sinpoly n x"
zero_left: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and b :: "'e" assumes "bounded_cbilinear prod" shows "prod (0::'d) b = (0::'f)"
fundfoldpairs_induced_autormorph_order2: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "(f, g) \<in> fundfoldpairs" shows "induced_automorph f g \<circ> induced_automorph f g = id"
inverse_group_isomorphism: fixes \<eta> :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and G' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d" assumes "group_isomorphism \<eta> G composition unit G' composition' unit'" shows "group_isomorphism (restrict (inv_into G \<eta>) G') G' composition' unit' G composition unit"
code: fixes gpv :: "('a, 'b, 'd) gpv" shows "left_gpv gpv = GPV (map_spmf (map_generat id Inl left_rpv) (the_gpv gpv))"
invar2_finish: fixes Q :: "'v \<Rightarrow> enat"   and \<pi> :: "'v \<Rightarrow> 'v option" assumes "prim_invar2 Q \<pi>"   and "Q = (\<lambda>_. \<infinity>)" shows "is_MST (G_\<alpha>w g) rg (graph {r} {(u, v). \<pi> u = Some v})"
is_equalized_by_cone: fixes a :: "'c"   and \<chi> :: "(bool, bool list) concrete_category.arr \<Rightarrow> 'c" assumes "local.cone a \<chi>" shows "is_equalized_by (\<chi> J.Zero)"
IncreasingD: fixes F :: "'b program"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "F \<in> Increasing f" shows "F \<in> Stable {s. x \<le> f s}"
less_eqE: fixes plus :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and zero :: "'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and a :: "'a::type"   and b :: "'a::type" assumes "class.canonically_ordered_monoid_add (plus::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (zero::'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)"   and "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (a::'a::type) (b::'a::type)" obtains c :: "'a::type" where "(b::'a::type) = (plus::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (a::'a::type) (c::'a::type)"
size_psubset: fixes M :: "'b multiset"   and M' :: "'b multiset" assumes "M \<subseteq># M'"   and "size M < size M'" shows "M \<subset># M'"
cod_cod: fixes f :: "'d" assumes "Dop.arr f" shows "Dop.cod (Dop.cod f) = Dop.cod f"
elims_2: fixes x :: "nat \<Rightarrow> 'a"   and xa :: "(nat \<Rightarrow> 'a option) \<times> (nat \<Rightarrow> 'a option)" assumes "x \<Turnstile>\<^sub>b xa" obtains v :: "nat \<Rightarrow> 'a" and b :: "(nat \<Rightarrow> 'a option) \<times> (nat \<Rightarrow> 'a option)" where "x = v" and "xa = b" and "\<forall>x. in_bounds x v b"
legacy_Complex_simps_25: fixes r :: "real" shows "complex_of_real r = Complex r 0"
winning_strategy_only_for_one_player: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v :: "'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "strategy p \<sigma>"   and "winning_strategy p \<sigma> v"   and "strategy p** \<sigma>'"   and "winning_strategy p** \<sigma>' v"   and "v \<in> V" shows "False"
eq_\<I>_converterD_WT1: fixes \<I> :: "('y, 'z) \<I>"   and \<I>' :: "('aa, 'ab) \<I>"   and conv1 :: "('y, 'z, 'aa, 'ab) converter"   and conv2 :: "('y, 'z, 'aa, 'ab) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2"   and "\<I>, \<I>' \<turnstile>\<^sub>C conv1 \<surd>" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv2 \<surd>"
taylor_deg_one_expansion_subring: fixes f :: "nat \<Rightarrow> 'a"   and S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "f \<in> carrier (UP R)"   and "subring S R"   and "\<And>i. f i \<in> S"   and "a \<in> S"   and "b \<in> S" shows "\<exists>c\<in>S. to_fun f b = to_fun f a \<oplus> deriv f a \<otimes> (b \<ominus> a) \<oplus> c \<otimes> (b \<ominus> a) [^] 2"
lateEarlyOutput: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'" shows "P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'"
dvd_gcd_list_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and b :: "'a"   and xs :: "'a list" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times b (semiring_gcd.Gcd_fin one zero gcd (set xs)) = (\<forall>a\<in>set xs. dvd.dvd times b a)"
Union_Zero1: fixes H :: "fm set"   and x :: "tm" shows "H \<turnstile> UnionP Zero x x"
lookup_map_values: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and m :: "('c, 'd) mapping"   and k :: "'c" shows "lookup (map_values f m) k = map_option (f k) (lookup m k)"
nz: fixes x :: "rat" shows "poly (real_of_int_poly p) (real_of_rat x) \<noteq> 0"
shows_prec_vec_def: fixes p :: "nat"   and v :: "'a vec" shows "shows_prec p v \<equiv> shows_vec v"
GroupByPresentation_S_P: shows "GroupByPresentation S (map (charpair S) ` R)"
Q_good_def: shows "Q_good \<equiv> \<lambda>Q. DisjointPaths H_x v0 v1 Q \<and> card Q = sep_size \<and> (\<forall>Q'. DisjointPaths H_x v0 v1 Q' \<and> card Q' = sep_size \<longrightarrow> Q_weight Q \<le> Q_weight Q')"
Pgv: fixes g :: "'g"   and v :: "'v" assumes "g \<in> G"   and "v \<in> V" shows "P (g *\<cdot> v) \<in> V"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b" shows "map_literal f (Pos x1) = Pos (f x1)"
spaceN_contains_zero: fixes N :: "'b quasinorm" shows "(0::'b) \<in> space\<^sub>N N"
fps_mult_right_inverse_unit_factor_divring: fixes f :: "'a fps" assumes "f \<noteq> 0" shows "f * inverse (unit_factor f) = fps_X ^ subdegree f"
coerce_WriterT: fixes k :: "'h\<cdot>'j writer\<cdot>'i" shows "COERCE('h\<cdot>('i, 'j) writerT, 'e\<cdot>('f, 'g) writerT)\<cdot>(WriterT\<cdot>k) = WriterT\<cdot>(COERCE('h\<cdot>'j writer\<cdot>'i, 'e\<cdot>'g writer\<cdot>'f)\<cdot>k)"
intros_2: fixes P :: "'b prog"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'b mdecl list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "class P Object = \<lfloor>(D, fs, ms)\<rfloor>"   and "FDTs = map (\<lambda>(F, b, T). ((F, Object), b, T)) fs" shows "P \<turnstile> Object has_fields FDTs"
tj_stack_incr_disc: fixes k :: "nat"   and j :: "nat" assumes "k < length (tj_stack s)"   and "j < k" shows "\<delta> s (tj_stack s ! k) < \<delta> s (tj_stack s ! j)"
lift_def: fixes return :: "'f \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'f"   and x :: "'e" shows "monad_base.lift return bind f x = bind x (\<lambda>x. return (f x))"
proj_arcs_H_simps_2: shows "proj_arcs_H (rev_H vw) = uv"
path_evalF': fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and A :: "formula" assumes "branch subs (pseq fs) f"   and "\<forall>n. \<not> proofTree (tree subs (f n))" shows "(\<exists>i. EV (contains f (i, A))) \<longrightarrow> \<not> evalF (counterModel f) counterAssign A"
points_index_single_rep_num: fixes B :: "'b set multiset"   and x :: "'b" shows "int (B index {x}) = B rep x"
Times_mset_assoc: fixes A :: "'a multiset"   and B :: "'b multiset"   and C :: "'c multiset" shows "(A \<times># B) \<times># C = {#((a, b), c). (a, b, c) \<in># A \<times># B \<times># C#}"
scale_zero_left: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" assumes "x \<in> sub.tangent_space p" shows "0 *\<^sub>R x = 0"
permute_exp: fixes X :: "msg"   and Y :: "msg"   and Z :: "msg" shows "Exp (Exp X Y) Z = Exp (Exp X Z) Y"
ae_1: shows "AE x in M. pf x = ennreal (g x)"
AB_fv_disj: shows "fv\<^sub>s\<^sub>e\<^sub>t (set A) \<inter> fv\<^sub>s\<^sub>e\<^sub>t (set (B \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>)) = {}"
cos_one_2pi: fixes x :: "real" shows "(cos x = 1) = ((\<exists>xa. x = real xa * 2 * pi) \<or> (\<exists>xa. x = - (real xa * 2 * pi)))"
admissible\<^sub>2_def: fixes w :: "face \<Rightarrow> nat"   and g :: "graph" shows "admissible\<^sub>2 w g \<equiv> \<forall>v\<in>\<V> g. except g v = 0 \<longrightarrow> \<b> (tri g v) (quad g v) \<le> ListSum (facesAt g v) w"
north_def: shows "north = proj2_abs (vector [0, 1, 1])"
v11_app_vconverse_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_inverse" shows "(vreal_inverse\<inverse>\<^sub>\<circ>\<lparr>a\<rparr>)\<inverse>\<^sub>\<real> = a"
subgraph_trans: fixes G :: "('b, 'c) pre_digraph"   and H :: "('b, 'c) pre_digraph"   and I :: "('b, 'c) pre_digraph" assumes "subgraph G H"   and "subgraph H I" shows "subgraph G I"
slice_complete: fixes xs :: "'c list" shows "slice 0 (length xs) xs = xs"
no_marker_if_no_snapshot: fixes t :: "('a, 'b, 'c) event list"   and cid :: "nat"   and p :: "'a"   and q :: "'a"   and i :: "nat" assumes "trace init t final"   and "channel cid = Some (p, q)"   and "\<not> ps (S t i) p \<noteq> None" shows "Marker \<notin> set (msgs (S t i) cid)"
assms_1: shows "kKS k \<in> LocalSecrets PQ"
atMost_def: fixes u :: "'a word" shows "signed.atMost u \<equiv> {x. x \<le>s u}"
notin_alphabet_imp_splitLeaf_id: fixes a :: "'a"   and t :: "'a tree"   and w\<^sub>a :: "nat"   and w\<^sub>b :: "nat"   and b :: "'a" assumes "a \<notin> alphabet t" shows "splitLeaf t w\<^sub>a a w\<^sub>b b = t"
insert: fixes G :: "'b triple set"   and t :: "'b triple"   and ts :: "'b triple set" assumes "G|-t"   and "G||-ts" shows "G||-insert t ts"
assms: shows "K\<^bsub>5\<^esub> (with_proj G)"
MkArr_in_hom: fixes A :: "'j \<Rightarrow> 'c"   and B :: "'j \<Rightarrow> 'c"   and f :: "'j \<Rightarrow> 'c" assumes "A \<in> Collect (functor (\<cdot>\<^sub>J) (\<cdot>))"   and "B \<in> Collect (functor (\<cdot>\<^sub>J) (\<cdot>))"   and "f \<in> Collect (natural_transformation (\<cdot>\<^sub>J) (\<cdot>) A B)" shows "\<guillemotleft>J_C.MkArr A B f : J_C.MkIde A \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] J_C.MkIde B\<guillemotright>"
chainE: fixes ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f set"   and x :: "'f"   and y :: "'f" assumes "Complete_Partial_Order.chain ord S"   and "x \<in> S"   and "y \<in> S" obtains "ord x y"   | "ord y x"
continuous_on_blinfun_componentwise: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'd" assumes "\<And>i. i \<in> Basis \<Longrightarrow> continuous_on s (\<lambda>x. blinfun_apply (f x) i)" shows "continuous_on s f"
simps_23: fixes r :: "'b rexp"   and s :: "'b rexp" shows "le_rexp (Star r) (Star s) = le_rexp r s"
eq_\<I>_gpv_mono: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and \<I>' :: "('c, 'd) \<I>" assumes "A \<le> A'"   and "\<I> \<le> \<I>'" shows "eq_\<I>_gpv A \<I> \<le> eq_\<I>_gpv A' \<I>'"
is_offending_flows_min_set_def: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and f :: "('c \<times> 'c) set"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" shows "SecurityInvariant_withOffendingFlows.is_offending_flows_min_set sinvar f G nP \<equiv> SecurityInvariant_withOffendingFlows.is_offending_flows sinvar f G nP \<and> (\<forall>(e1, e2)\<in>f. \<not> sinvar (add_edge e1 e2 (delete_edges G f)) nP)"
assms_2: shows "\<forall>i<dim_vec y. (0::'a) \<le> y $ i"
assms_1: shows "P \<leadsto>\<^sup>^<Rel> Q"
elims_3: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list" assumes "class.linorder less_eq less"   and "\<not> linorder.sorted less_eq x" obtains xa :: "'a" and ys :: "'a list" where "x = xa # ys" and "\<not> (Ball (set ys) (less_eq xa) \<and> linorder.sorted less_eq ys)"
ide_img: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "ide (img f)"
mat0_leastIII: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A \<ge>\<^sub>m 0\<^sub>m nr nc"
Cast_step_Cast_or_fin: fixes C :: "char list"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   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 E :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "final e' \<or> (\<exists>e''. e' = \<lparr>C\<rparr>e'')"
Inf_finite_4_def: fixes A :: "finite_4 set" shows "\<Sqinter> A = (if a\<^sub>1 \<in> A \<or> a\<^sub>2 \<in> A \<and> a\<^sub>3 \<in> A then a\<^sub>1 else if a\<^sub>2 \<in> A then a\<^sub>2 else if a\<^sub>3 \<in> A then a\<^sub>3 else a\<^sub>4)"
neg_equal_zero: fixes 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 a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "(uminus a = a) = (a = zero)"
simps_9: fixes C :: "char list"   and M :: "char list"   and n :: "nat" shows "is_relevant_class (Invokestatic C M n) = (\<lambda>P C. True)"
reduction_prf_def: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv" shows "reduction_prf \<A> = lift_spmf upf_key_gen \<bind> (\<lambda>k. lift_spmf coin_spmf \<bind> (\<lambda>b. inline (intercept_prf k b) \<A> () \<bind> (\<lambda>(b', uu_). Generative_Probabilistic_Value.Done (b' = b))))"
polynomial_pow_division: fixes p :: "'a list"   and n :: "nat"   and m :: "nat" assumes "p \<in> carrier (poly_ring R)"   and "n \<le> m" shows "p [^]\<^bsub>poly_ring R\<^esub> n pdivides p [^]\<^bsub>poly_ring R\<^esub> m"
rep_eq: fixes x :: "'a"   and xa :: "('a, 'b) mapping" shows "mapping.rep (Mapping.delete x xa) = (mapping.rep xa)(x := None)"
V01_in_CONS: shows "V\<^sub>0\<^sub>1 \<in> CONS"
funion_finsert_left: fixes a :: "'a"   and B :: "'a fset"   and C :: "'a fset" shows "finsert a B |\<union>| C = finsert a (B |\<union>| C)"
that_3: assumes "1 < LENGTH('a)"   and "sint 1 = 1" shows "P"
error_returns_result: fixes h :: "'a"   and e :: "'b"   and y :: "'c" shows "\<not> h \<turnstile> error e \<rightarrow>\<^sub>r y"
strong_exhaust: fixes y :: "trm" obtains "y = zer"   | x2 :: "name" where "y = Var x2"   | x3 :: "trm" where "y = suc x3"   | x41 :: "trm" and x42 :: "trm" where "y = pls x41 x42"   | x51 :: "trm" and x52 :: "trm" where "y = tms x51 x52"
prime_elem_dvd_power: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a"   and x :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p"   and "dvd.dvd times p (power.power one times x n)" shows "dvd.dvd times p x"
s_delayed_has_real_derivative': fixes t :: "real" assumes "\<delta> \<le> t" shows "(ego2.s \<circ> \<tau> has_real_derivative (ego2.s' \<circ> \<tau>) t) (at t within {\<delta>..})"
l_set_mode_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs \<equiv> l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs \<and> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
measurable_pmf_measure2: fixes N :: "'a measure"   and M :: "'b pmf" shows "N \<rightarrow>\<^sub>M measure_pmf M = N \<rightarrow>\<^sub>M count_space UNIV"
P: fixes i :: "nat" shows "P (lnth (ldrop (enat k) Sts) i)"
iso_bij_betwn_block_sets: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map" shows "bij_betw ((`) bij_map) (set_mset \<B>) (set_mset \<B>')"
check: shows "check_OR (x0, x1) (a0, a1) s ((e0, z0), e1, z1)"
with_proj_simps_1: fixes G :: "'c pair_pre_digraph" shows "verts (with_proj G) = pverts G"
R12_R14: shows "pmf (sds R14) a \<le> pmf (sds R12) a"
complete_isometric_image: fixes e :: "real"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "0 < e"   and "subspace s"   and "bounded_linear f"   and "\<forall>x\<in>s. e * norm x \<le> norm (f x)"   and "complete s" shows "complete (f ` s)"
Inf_set_def: fixes A :: "'c set set" shows "\<Inter> A = {x. \<Sqinter>B\<in>A. s2p B x}"
iMin_le_Max: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "iMin A \<le> Max A"
iso_char: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.iso \<mu> = (Span.arr \<mu> \<and> Maps.iso (Chn \<mu>))"
select_convs_8: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'a" shows "ConvSchema.more \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = more"
rel_gpv_conv_rel_gpv'': fixes A :: "'k \<Rightarrow> 'n \<Rightarrow> bool"   and C :: "'l \<Rightarrow> 'o \<Rightarrow> bool" shows "rel_gpv A C = rel_gpv'' A C (=)"
simps_1: fixes x1 :: "('a, 'b) form multiset"   and x2 :: "('a, 'b) form multiset"   and y1 :: "('a, 'b) form multiset"   and y2 :: "('a, 'b) form multiset" shows "(( x1 \<Rightarrow>* x2) = ( y1 \<Rightarrow>* y2)) = (x1 = y1 \<and> x2 = y2)"
degree_smult_eq: fixes a :: "'d"   and p :: "'d poly" shows "degree (smult a p) = (if a = (0::'d) then 0 else degree p)"
homeomorphic_convex_sets_le: fixes S :: "'a set"   and T :: "'b set" assumes "convex S"   and "S homeomorphic T" shows "aff_dim S \<le> aff_dim T"
prems: shows "interval_lebesgue_integrable lborel a b f"
t2: shows "(1 - t2) * (a \<bullet> c) + t2 * (b \<bullet> c) = d"
oalist_eq_alt: fixes xs :: "('b, 'c) oalist_ntm"   and ys :: "('b, 'c) oalist_ntm" shows "OAlist_eq_ntm xs ys = (OAlist_lookup_ntm xs = OAlist_lookup_ntm ys)"
l_inv_ex: fixes x :: "'c" assumes "x \<in> carrier M" shows "\<exists>y\<in>carrier M. y \<oplus>\<^bsub>M\<^esub> x = \<zero>\<^bsub>M\<^esub>"
subgroup_is_submonoid: shows "submonoid {\<one>} G"
starset_n_Int: fixes A :: "nat \<Rightarrow> 'a set"   and B :: "nat \<Rightarrow> 'a set" shows "*sn* (\<lambda>n. A n \<inter> B n) = *sn* A \<inter> *sn* B"
bi_unique: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique T" shows "bi_unique (pcr_nonzero T)"
independent_aforms_def: fixes x :: "'a \<times> 'a pdevs"   and env :: "('a \<times> 'a pdevs) list" shows "independent_aforms x env = msum_aform (fresh_index_aforms env) (0::'a, zero_pdevs) x # env"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>D :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>D :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>D :: "'d"   and V :: "'b \<Rightarrow> 'd" assumes "category C"   and "monoidal_category D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D"   and "evaluation_map C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V"   and "free_monoidal_category C" shows "FreeMonoidalCategory.evaluation_functor C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V"
surjective: fixes r :: "('b, 'c) routing_action_scheme" shows "r = \<lparr>output_iface = output_iface r, next_hop = next_hop r, \<dots> = routing_action.more r\<rparr>"
restrict_compose_right: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e"   and S :: "'c set" shows "restrict (g \<circ> restrict f S) S = restrict (g \<circ> f) S"
simps_3: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a" shows "ltl (greedy_conforming_path p \<sigma> \<sigma>' v0) = (if deadend v0 then LNil else if v0 \<in> VV p then greedy_conforming_path p \<sigma> \<sigma>' (\<sigma> v0) else greedy_conforming_path p \<sigma> \<sigma>' (\<sigma>' v0))"
induced_play_infinite: fixes \<sigma> :: "'d list \<Rightarrow> 'd"   and p :: "'d list" shows "\<not> lfinite (strategy_progression \<sigma> p)"
sectionE: fixes f :: "'s" assumes "section f" obtains g :: "'s" where "ide (g \<cdot> f)"
abs_eq_infinity_cases: fixes x :: "ereal" assumes "\<bar>x\<bar> = \<infinity>" obtains "x = \<infinity>"   | "x = - \<infinity>"
Quotient3_Tree\<^sub>p: shows "Quotient3 hull_relp abs_Tree\<^sub>p rep_Tree\<^sub>p"
poly_set_rev_iterateoi_axioms: shows "poly_set_rev_iterateoi \<alpha> invar list_rev_it"