hcju/isabelleps huggingface dataset:
iMODb_leD: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "x \<in> [ r, mod m, c ]" shows "x \<le> r + m * c"
fmap_fmap: fixes f :: "'g \<rightarrow> 'e"   and g :: "'h \<rightarrow> 'g"   and xs :: "'h\<cdot>'f" shows "fmap\<cdot>f\<cdot>(fmap\<cdot>g\<cdot>xs) = fmap\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"
is_extensional: fixes f :: "'c" assumes "\<not> C.arr f" shows "G\<epsilon>'_\<eta>G.map f = D.null"
inv_translationD_6: fixes r :: "nat set"   and x :: "nat" assumes "inv_translation r" shows "(x < card r) = (x \<in> r)"
orbit_size: fixes m :: "'c" assumes "finite (carrier G)"   and "m \<in> M" shows "order G = card (orbit m) * card (stabilizer m)"
fs_deriv: fixes k :: "nat"   and x :: "real" assumes "k \<le> 2 * N"   and "x \<in> {real a..} - Y" shows "(fs k has_vector_derivative fs (Suc k) x) (at x)"
inf_left_commute: fixes x :: "'b upred"   and y :: "'b upred"   and z :: "'b upred" shows "(x \<and> y \<and> z) = (y \<and> x \<and> z)"
degree_mult_Var': fixes x' :: "nat"   and A :: "int mpoly" shows "MPoly_Type.degree (Var x' * A) x' = (if A = 0 then 0 else Suc (MPoly_Type.degree A x'))"
closed_segment_commute: fixes a :: "'a"   and b :: "'a" shows "closed_segment a b = closed_segment b a"
lemma_4_5_i: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b) \<squnion>1 (b l\<rightarrow> a) = (1::'a)"
stepResultsinR_def: fixes p :: "('exp, 'id) MWLsCom option"   and p' :: "('exp, 'id) MWLsCom option"   and R :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set" shows "stepResultsinR p p' R \<equiv> p = None \<and> p' = None \<or> (\<exists>c c'. p = Some c \<and> p' = Some c' \<and> ([c], [c']) \<in> R)"
hyps: assumes "trace c 0 k" shows "(\<forall>l\<le>k. fst (c l) ! i = fst (c k) ! i) \<or> (\<exists>last<k. fst (c last) ! i \<noteq> fst (c (Suc last)) ! i \<and> (\<forall>l>last. l \<le> k \<longrightarrow> fst (c l) ! i = fst (c k) ! i))"
STT: shows "closedin (top_of_set (S \<union> T)) T"
dghm_tdghm_comp_is_tm_tdghm: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<HH> :: "V"   and \<AA> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN> : \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<CC>"
poly_comp_expansion: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "deg R p \<le> n" shows "Cring_Poly.compose R p q = (\<Oplus>\<^bsub>P\<^esub>i\<in>{..n}. p i \<odot>\<^bsub>P\<^esub> q [^]\<^bsub>P\<^esub> i)"
primes_M_nonneg: fixes x :: "real" shows "0 \<le> \<MM> x"
wary_cases_apps\<^sub>h: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and t :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "arity.wary arity_sym arity_var t" obtains \<zeta> :: "('b, 'a) hd" and ss :: "('b, 'a) tm list" where "t = apps (Hd \<zeta>) ss" and "\<And>sa. sa \<in> set ss \<Longrightarrow> arity.wary arity_sym arity_var sa" and "of_nat (length ss) \<le> hmset_of_enat (arity.arity_hd arity_sym arity_var \<zeta>)"
assms_1: shows "mbufnt_take f z buf nts = (z', buf', nts')"
subtype_x_Optional: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a basic_type" assumes "\<tau> \<sqsubset> \<sigma>[?]" obtains \<tau>' :: "'a basic_type" where "\<tau> = \<tau>'[?]" and "\<tau>' \<sqsubset>\<^sub>B \<sigma>"   | "\<tau> = \<sigma>[1]"
cSim: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel" shows "\<Psi>' \<rhd> S \<leadsto><Rel> T"
simps_4: fixes x :: "nat"   and v :: "exp"   and e1 :: "exp"   and e2 :: "exp" shows "subst x v (EApp e1 e2) = EApp (subst x v e1) (subst x v e2)"
simps_1: fixes \<theta> :: "(name \<times> trm) list"   and x :: "name" shows "\<theta><Var x> = lookup \<theta> x"
GFP_fixed_point: fixes f :: "'a \<Rightarrow> 'a" assumes "Mono\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "GFP\<^bsub>L\<^esub> f \<in> fps L f"
deriv_cmult_right: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a"   and c :: "'a" assumes "f field_differentiable at z" shows "deriv (\<lambda>w. f w * c) z = deriv f z * c"
box_ne_empty_1: fixes a :: "'a"   and b :: "'a" shows "(cbox a b \<noteq> {}) = (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
sees_field_fun: fixes P :: "'a prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal"   and T' :: "ty"   and fm' :: "fmod"   and D' :: "String.literal" assumes "P \<turnstile> C sees F:T (fm) in D"   and "P \<turnstile> C sees F:T' (fm') in D'" shows "T' = T \<and> D' = D \<and> fm = fm'"
fv_at_base: fixes a :: "'a" shows "fv a = {a}"
assms_1: fixes x :: "'a" assumes "x \<in> A" shows "\<exists>y\<in>B. dist x y \<le> D"
rank_def: fixes x :: "hf" shows "rank x = (LEAST n. Ord n \<and> x \<^bold>\<in> Vset (succ n))"
sfunction_case: fixes I :: "('b, 'c, 'a) interp"   and f :: "'b"   and args :: "'a \<Rightarrow> ('b, 'a) trm" shows "sterm_sem I ($f f args) = (\<lambda>v. Functions I f (\<chi>i. sterm_sem I (args i) v))"
Inf_UNIV: shows "Inf_prod UNIV = bot_prod"
simps: fixes x :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop5_loop x (l, r) = (\<exists>i j k t. i + j = Suc x \<and> 0 < i \<and> 0 < j \<and> k + t = j \<and> 0 < t \<and> (l, r) = (Oc \<up> k @ Bk \<up> j @ Oc \<up> i, Oc \<up> t))"
cases: fixes x :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" obtains \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and S :: "stack_elem list" where "x = (\<Gamma>, e, S)"
equivalence_map_preserved_by_iso: fixes f :: "'a"   and f' :: "'a" assumes "equivalence_map f"   and "f \<cong> f'" shows "equivalence_map f'"
HPT_def: fixes ST :: "('d, 'e, 'f) status" shows "HPT ST \<equiv> {T. MaxNonConflict ST T}"
smult_diff_right: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "Polynomial.smult a (p - q) = Polynomial.smult a p - Polynomial.smult a q"
Substable_intro_not: fixes cond :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<psi> :: "'a \<Rightarrow> \<o>" assumes "Substable cond \<psi>" shows "Substable cond (\<lambda>\<phi>. \<^bold>\<not>\<psi> \<phi>)"
mult_less_le_imp_less: fixes a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "a < b"   and "c \<le> d"   and "(0::'d) \<le> a"   and "(0::'d) < c" shows "a * c < b * d"
scast_eq: fixes w :: "'c word" shows "scast w = word_of_int (sint w)"
degree_gt: fixes x :: "'b pdevs"   and j :: "nat" assumes "pdevs_apply x j \<noteq> (0::'b)" shows "j < degree x"
Expr: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and v :: "val"   and s1 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>Expr e\<rightarrow> s1"
partition_filter2: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "snd (partition P xs) = filter (Not \<circ> P) xs"
iT_Minus_iUntil_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<U> t2 k \<ominus> I. Q t2) = (P (k - t1). t1 \<S> t2 I \<down>\<le> k. Q (k - t2))"
Inv_def: fixes xs :: "'c list"   and ys :: "'c list" shows "Inv xs ys = {(x, y). x < y in xs \<and> y < x in ys}"
sing_word: fixes us :: "'b list" assumes "\<^bold>|us\<^bold>| = 1" shows "[hd us] = us"
monad_fail_writerT: fixes return :: "'a \<times> 'b list \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b list \<Rightarrow> 'c) \<Rightarrow> 'c"   and fail :: "'c" assumes "monad_fail return bind fail" shows "monad_fail (return_writer return) (bind_writer return bind) (fail_writer fail)"
cinfinite_def: fixes r :: "('b \<times> 'b) set" shows "cinfinite r = infinite (Field r)"
list_rel_append1: fixes as :: "'a list"   and bs :: "'a list"   and l :: "'b list"   and R :: "('a \<times> 'b) set" shows "((as @ bs, l) \<in> \<langle>R\<rangle>list_rel) = (\<exists>cs ds. l = cs @ ds \<and> (as, cs) \<in> \<langle>R\<rangle>list_rel \<and> (bs, ds) \<in> \<langle>R\<rangle>list_rel)"
c: fixes x :: "real" shows "norm (pbernpoly n x) \<le> c"
NAe_DA_equiv: fixes A :: "'b \<times> ('c option \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and w :: "'c list" shows "DA.accepts (nae2da A) w = NAe.accepts A w"
fst_crit_pair_below_lcs: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "fst (crit_pair p q) \<prec>\<^sub>p monomial (1::'c) (term_of_pair (lcs (lp p) (lp q), component_of_term (lt p)))"
anno_type_stable_def: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and upd :: "'Var ModeUpd" shows "anno_type_stable \<Gamma> \<S> upd \<equiv> case upd of v +=\<^sub>m m \<Rightarrow> v \<notin> \<C> \<and> v \<in> add_anno_dom \<Gamma> \<S> upd - dom \<Gamma> \<longrightarrow> (\<forall>x\<in>\<C>_vars v. stable \<S> x) | v -=\<^sub>m m \<Rightarrow> v \<in> \<C> \<and> v \<notin> add_anno_dom \<Gamma> \<S> upd \<longrightarrow> (\<forall>x\<in>dom \<Gamma>. v \<notin> vars_of_type (the (\<Gamma> x)))"
function_factors_left_gen: fixes P :: "'c \<Rightarrow> bool"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "(\<forall>x y. P x \<and> P y \<and> g x = g y \<longrightarrow> f x = f y) = (\<exists>h. \<forall>x. P x \<longrightarrow> f x = h (g x))"
vunion_vempty_right: fixes a :: "V" shows "a \<union>\<^sub>\<circ> []\<^sub>\<circ> = a"
assumption_L: fixes \<phi> :: "'a"   and A :: "'a set" assumes "\<phi> \<in> A" shows "A \<turnstile> \<phi>"
emeasure_stream_space: fixes X :: "'a stream stream set" assumes "X \<in> sets (stream_space (stream_space M))" shows "emeasure (stream_space (stream_space M)) X = \<integral>\<^sup>+ t. emeasure (stream_space (stream_space M)) {x \<in> space (stream_space (stream_space M)). t ## x \<in> X} \<partial>stream_space M"
Abs_swap_fresh: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and xs :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "good X"   and "fresh xs x' X" shows "Abs xs x X = Abs xs x' (X #[x' \<and> x]_xs)"
isSubobj_Subobjs_rev: fixes 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"   and C' :: "char list"   and Cs' :: "char list list"   and 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" assumes "is_subobj P (C, C' # rev Cs')"   and "wf_prog wf_md P" shows "Subobjs P C (C' # rev Cs')"
C_not_in: shows "C \<notin> local.Red_F_\<G> N"
rep_eq: fixes x :: "'a interval" shows "lower x = fst (bounds_of_interval x)"
VAsub: fixes a :: "('sf + 'sz, 'sz) trm" assumes "SFunctions \<sigma> x1a = Some a" shows "FVT a \<subseteq> (\<Union>i\<in>SIGT ($f x1a x2a). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"
reflection: shows "rel_reflects_binary_pred (indRelLTPO TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))"
eq: fixes x :: "real" shows "indicat_real {a..b} (- x) *\<^sub>R f (- x) = indicat_real {- b..- a} x *\<^sub>R f (- x)"
assms: shows "continuous_on (A \<times> B) (\<lambda>(x, y). d x y)"
sdp_loop: fixes body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and G :: "'b \<Rightarrow> bool" assumes "sub_distrib_pconj body"   and "nearly_healthy (wlp body)"   and "healthy (wp body)" shows "sub_distrib_pconj (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)"
assms_1: shows "\<Psi> \<rhd> \<alpha>\<cdot>P \<longmapsto> \<tau> \<prec> P'"
simple_term_inverse: fixes y :: "('a, 'b) trm" assumes "y \<in> {\<theta>. dfree \<theta>}" shows "raw_term (simple_term y) = y"
image_stretch_interval: fixes m :: "'a \<Rightarrow> real"   and a :: "'a"   and b :: "'a" shows "(\<lambda>x. \<Sum>k\<in>Basis. (m k * (x \<bullet> k)) *\<^sub>R k) ` cbox a b = (if cbox a b = {} then {} else cbox (\<Sum>k\<in>Basis. min (m k * (a \<bullet> k)) (m k * (b \<bullet> k)) *\<^sub>R k) (\<Sum>k\<in>Basis. max (m k * (a \<bullet> k)) (m k * (b \<bullet> k)) *\<^sub>R k))"
holo3: shows "(\<lambda>z. u ^ n * (deriv ^^ n) f (u * z)) holomorphic_on S"
Basic_sound: fixes pre :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and post :: "'b set"   and guar :: "('b \<times> 'b) set"   and rely :: "('b \<times> 'b) set" assumes "pre \<subseteq> {s. f s \<in> post}"   and "{(s, t). s \<in> pre \<and> t = f s} \<subseteq> guar"   and "stable pre rely"   and "stable post rely" shows "\<Turnstile> Basic f sat [pre, rely, guar, post]"
gauss2complex_diff: fixes x :: "gauss_int"   and y :: "gauss_int" shows "gauss2complex (x - y) = gauss2complex x - gauss2complex y"
Proj_inj: fixes X :: "'a ccsubspace"   and Y :: "'a ccsubspace" assumes "Proj X = Proj Y" shows "X = Y"
map_simp: fixes f :: "'j" assumes "J.arr f" shows "A'.G.map f = C.cod a'"
root_len: fixes u :: "'b list"   and q :: "'b list" assumes "u \<in> q*" shows "\<exists>k. \<^bold>|u\<^bold>| = k * \<^bold>|q\<^bold>|"
hyps_4: shows "t\<^sub>i \<in> set T'"
real_is_int_def: fixes x :: "real" shows "real_is_int x = (\<exists>u. x = real_of_int u)"
lookup_multimap_def: fixes m :: "('b, 'a set) mapping"   and k :: "'b" shows "lookup_multimap m k \<equiv> case Mapping.lookup m k of None \<Rightarrow> {} | Some x \<Rightarrow> id x"
well\<^sub>h_Ma: shows "local.well_base\<^sub>h Ma"
share_append_commute: fixes xs :: "'b memref list"   and ys :: "'c memref list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "(all_shared xs \<union> all_unshared xs \<union> all_acquired xs) \<inter> (all_shared ys \<union> all_unshared ys \<union> all_acquired ys) = {}" shows "share xs (share ys \<S>) = share ys (share xs \<S>)"
distinct_1: fixes x2 :: "enat" shows "0 \<noteq> eSuc x2"
ptensor_mat_def: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and dims :: "nat list"   and m1 :: "'a mat"   and m2 :: "'a mat" assumes "partial_state2 vars1 vars2" shows "partial_state2.ptensor_mat dims vars1 vars2 m1 m2 = tensor_mat (partial_state2.dims0 dims vars1 vars2) (partial_state2.vars1' vars1 vars2) m1 m2"
mult_rel_class_of: fixes U :: "'a set"   and x :: "'b"   and V :: "'a set"   and y :: "'b"   and W :: "'a set" assumes "(U, x) \<in> Sigma {U. is_open\<^sub>X U \<and> x \<in> U} \<O>\<^sub>X"   and "(V, y) \<in> Sigma {U. is_open\<^sub>X U \<and> x \<in> U} \<O>\<^sub>X"   and "W \<in> {U. is_open\<^sub>X U \<and> x \<in> U}"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "stx.mult_rel (stx.class_of U x) (stx.class_of V y) = stx.class_of W (mult_str\<^sub>X W (\<rho>\<^sub>X U W x) (\<rho>\<^sub>X V W y))"
quat_inner_j_right: fixes x :: "quat" shows "x \<bullet> \<j> = Im2 x"
path_def: fixes g :: "real \<Rightarrow> 'b" shows "path g = continuous_on {0..1} g"
assms: shows "x \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<bullet> A"
map_le_antisym: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "f \<subseteq>\<^sub>m g"   and "g \<subseteq>\<^sub>m f" shows "f = g"
vwb_lens_def: fixes x :: "'b \<Longrightarrow> 'c" shows "vwb_lens x \<equiv> wb_lens x \<and> mwb_lens x"
less_cINF_D: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'a"   and i :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_below less_eq (f ` A)"   and "less y (Inf (f ` A))"   and "i \<in> A" shows "less y (f i)"
final: fixes \<sigma> :: "'\<sigma>" assumes "I \<sigma> []" shows "P \<sigma>"
list2FWpolicy_eq: fixes zs :: "'b list"   and x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and y :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and z :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and p :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "zs \<noteq> []" shows "Cp (list2FWpolicy (x \<oplus> y # z)) p = Cp (x \<oplus> list2FWpolicy (y # z)) p"
nres_inequalities_1: fixes X :: "'b set" shows "FAIL \<noteq> RES X"
delta0: fixes qd1 :: "QDelta"   and qd2 :: "QDelta" assumes "qd1 \<le> qd2" shows "\<forall>\<epsilon>. 0 < \<epsilon> \<and> \<epsilon> \<le> \<delta>0 qd1 qd2 \<longrightarrow> val qd1 \<epsilon> \<le> val qd2 \<epsilon>"
rel_cfg_eq: fixes cfg1 :: "'a cfg"   and cfg2 :: "'a cfg" shows "rel_cfg (=) cfg1 cfg2 = (cfg1 = cfg2)"
intro: fixes get_ancestors_di :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_ancestors_di_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "get_ancestors_di = l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_di get_parent get_host get_disconnected_document"   and "get_ancestors_di_locs = l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_di_locs get_parent_locs get_host_locs get_disconnected_document_locs" shows "l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_host get_host_locs get_disconnected_document get_disconnected_document_locs get_ancestors_di get_ancestors_di_locs"
meet_connection: fixes x :: "int"   and y :: "int" shows "(x \<le> y) = (int.meet x y = x)"
CatIdInMor: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X \<in> mor\<^bsub>C\<^esub>"
fully_connected_def: fixes G :: "'b graph" shows "fully_connected G \<equiv> \<lparr>nodes = nodes G, edges = nodes G \<times> nodes G\<rparr>"
subgroup_is_submonoid: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "subgroup H G" shows "submonoid H G"
simps_2: fixes l :: "'a list"   and i :: "int" shows "Ipolex l (Const i) = \<guillemotleft>i\<guillemotright>"
fold_Mapping_filter_Some_Some: fixes ts :: "('b, 'a) mapping"   and as :: "'b"   and t :: "'a"   and ds :: "('a \<times> 'b set) list" assumes "Mapping.lookup ts as = Some t"   and "\<And>X. (t, X) \<in> set ds \<Longrightarrow> as \<notin> X" shows "Mapping.lookup (fold (\<lambda>(t, X) ts. Mapping.filter (filter_cond X ts t) ts) ds ts) as = Some t"
weakening: fixes \<Gamma> :: "binding list"   and S :: "ty"   and T :: "ty"   and \<Delta> :: "binding list" assumes "\<Gamma>\<turnstile>S<:T"   and "\<turnstile> \<Delta> ok"   and "\<Delta> extends \<Gamma>" shows "\<Delta>\<turnstile>S<:T"
T_vrestr_composed_3: fixes A :: "'a set"   and m :: "nat" assumes "A \<in> sets M" shows "(T ^^ m) --` T --` A = (T ^^ (m + 1)) --` A"
cinner_eq_flip: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<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 dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and w :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(cinner x y = cinner z w) = (cinner y x = cinner w z)"
explosion_def: fixes \<omega> :: "(real \<times> 'a) stream" shows "explosion \<omega> = (\<Squnion>i. ereal (fst (\<omega> !! i)))"
sorted_less_sorted_list_of_set: fixes S :: "'b set" shows "Sorted_Less.sorted (sorted_list_of_set S)"
even_pow: fixes n :: "nat"   and P :: "pol" assumes "even n" shows "Commutative_Ring.pow n P = Commutative_Ring.pow (n div 2) (sqr P)"
col_def: fixes A :: "'a mat"   and j :: "nat" shows "col A j = vec (dim_row A) (\<lambda>i. A $$ (i, j))"
l4_6: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C'" shows "Bet A' B' C'"
a'_def: shows "a' \<equiv> \<Sum>j\<in>Basis. (if j = i then c else a \<bullet> j) *\<^sub>R j"
sets: fixes x :: "'a" assumes "x \<in> X" shows "{x} \<in> sets M"
stable_translation_length_inv: fixes f :: "'b \<Rightarrow> 'b" assumes "isometry f" shows "stable_translation_length (inv f) = stable_translation_length f"
ran_def: fixes m :: "'c \<Rightarrow> 'b option" shows "ran m = {b. \<exists>a. m a = \<lfloor>b\<rfloor>}"
one_strongly_connected_iff_5_eq: fixes x :: "'a" shows "one_strongly_connected x = (x ; 1 ; x\<^sup>T = x\<^sup>+)"
BVG_def: fixes \<alpha> :: "game" shows "BVG \<alpha> = {x. \<exists>I \<omega> X. \<omega> \<in> game_sem I \<alpha> X \<and> \<omega> \<notin> game_sem I \<alpha> (selectlike X \<omega> {x})}"
degree_zero_imp_splitted: fixes p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "p \<in> carrier (poly_ring M)"   and "degree p = 0" shows "M.splitted p"
ball_empty: fixes e :: "real"   and x :: "'b" assumes "e \<le> 0" shows "ball x e = {}"
has_integral_reflect_lemma_real: fixes f :: "real \<Rightarrow> 'a"   and i :: "'a"   and a :: "real"   and b :: "real" assumes "(f has_integral i) {a..b}" shows "((\<lambda>x. f (- x)) has_integral i) {- b..- a}"
mpt_factorI: fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "proj \<in> measure_preserving M M2"   and "AE x in M. proj (T x) = T2 (proj x)"   and "mpt M2 T2" shows "mpt_factor proj M2 T2"
hyps: shows "four_block_mat A (0\<^sub>m n m) (0\<^sub>m m n) B ^\<^sub>m k = four_block_mat (A ^\<^sub>m k) (0\<^sub>m n m) (0\<^sub>m m n) (B ^\<^sub>m k)"