hcju/isabelleps huggingface dataset:
interrupt_a_conv: fixes ls :: "'h lock_actions"   and nts :: "('g, 'i, 'j) new_thread_action list"   and js :: "'g conditional_action list"   and ws :: "('g, 'k) wait_set_action list"   and "is" :: "'g interrupt_action list"   and obs :: "'l list" shows "\<lbrace>(ls, nts, js, ws, is, obs)\<rbrace>\<^bsub>i\<^esub> = is"
assms_1: shows "vs @ xs \<in> C M2 M1 \<Omega> V m (Suc (Suc i))"
tensor_mat_hermitian: fixes m1 :: "'a mat"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat ps_P.d1 ps_P.d1"   and "m2 \<in> carrier_mat ps_P.d2 ps_P.d2"   and "hermitian m1"   and "hermitian m2" shows "hermitian (ps_P.tensor_mat m1 m2)"
return_def: fixes a :: "'d" shows "return a \<equiv> State (Pair a)"
rw: shows "(A $ from_nat i $ from_nat i - v * (p * A $ from_nat i $ from_nat i) - v * A $ from_nat k $ from_nat k * q) * u = (0::'a)"
simps_11: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "merge_guards (Catch c\<^sub>1 c\<^sub>2) = Catch (merge_guards c\<^sub>1) (merge_guards c\<^sub>2)"
norm_dist_def: fixes x :: "'b"   and y :: "'b" shows "norm_dist x y \<equiv> norm (x - y)"
has_derivative_norm_compose: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" assumes "(f has_derivative f') (at x within s)"   and "f x \<noteq> (0::'d)" shows "((\<lambda>x. norm (f x)) has_derivative (\<lambda>xa. f' xa \<bullet> sgn (f x))) (at x within s)"
less_le: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.order less_eq less" shows "less x y = (less_eq x y \<and> x \<noteq> y)"
lexord_linear: assumes "\<forall>a b. (a, b) \<in> less_than_char \<or> a = b \<or> (b, a) \<in> less_than_char" shows "(xs, ys) \<in> lexord less_than_char \<or> xs = ys \<or> (ys, xs) \<in> lexord less_than_char"
lookup_monom_mult: fixes c :: "'c"   and t :: "'d"   and p :: "'d \<Rightarrow>\<^sub>0 'c"   and v :: "'d" shows "lookup (MPoly_Type_Class.punit.monom_mult c t p) v = (if MPoly_Type_Class.punit.adds_pp t v then c * lookup p (MPoly_Type_Class.punit.sminus v t) else (0::'c))"
fdisjnt_subset_right: fixes N' :: "'a fset"   and N :: "'a fset"   and M :: "'a fset" assumes "N' |\<subseteq>| N"   and "fdisjnt M N" shows "fdisjnt M N'"
qo_on_imp_transp_on: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "qo_on P A" shows "transp_on P A"
retract_of_locally_path_connected: fixes T :: "'a set"   and S :: "'a set" assumes "locally path_connected T"   and "S retract_of T" shows "locally path_connected S"
fstI: fixes x :: "'a \<times> 'c"   and y :: "'a"   and z :: "'c" assumes "x = (y, z)" shows "fst x = y"
less_imp_le: fixes x :: "'a list"   and y :: "'a list" assumes "strict_prefix x y" shows "prefix x y"
bounded_iff: fixes a :: "'d upred"   and b :: "'d upred"   and c :: "'d upred" shows "((b \<and> c) \<sqsubseteq> a) = (b \<sqsubseteq> a \<and> c \<sqsubseteq> a)"
S_def: shows "S = (*) f ` normal_form L ` P[X]"
no_shared_edges_have_similar_orientations: shows "\<forall>twoCube1\<in>two_chain. \<forall>twoCube2\<in>two_chain. twoCube1 \<noteq> twoCube2 \<longrightarrow> boundary twoCube1 \<inter> boundary twoCube2 = {}"
excess_notame: fixes g :: "graph"   and g' :: "graph"   and p :: "nat" assumes "Invariants.inv g"   and "g' \<in> set (next_plane\<^bsub>p\<^esub> g)"   and "g' \<notin> set (next_tame0\<^bsub>p\<^esub> g)" shows "notame7 g'"
vsv_vcard_vdomain: shows "vcard (\<D>\<^sub>\<circ> xs') = vcard xs'"
tendsto_ln_powr_over_ln_powr'': fixes a :: "real"   and d :: "real" assumes "0 < a" shows "((\<lambda>x. ln (a * x) powr d / ln x powr d) \<longlongrightarrow> 1) at_top"
d_IN_Sup: fixes Y :: "('a \<times> 'a \<Rightarrow> ennreal) set"   and y :: "'a" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "countable (support_flow (\<Squnion> Y))" shows "d_IN (\<Squnion> Y) y = (\<Squnion>f\<in>Y. d_IN f y)"
not_treach_ref_impl_not_reach: fixes x :: "Value"   and l :: "Location"   and s :: "Store" assumes "\<not> treach \<tau> x \<tau> ref l" shows "\<not> s\<turnstile> l reachable_from x"
fls_lr_inverse_fps_to_fls_2: fixes f :: "'a fps"   and x :: "'a" assumes "subdegree f = 0" shows "fls_right_inverse (fps_to_fls f) x = fps_to_fls (fps_right_inverse f x)"
proves_eq_elim: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term"   and B :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "term_ok \<Theta> B"   and "typ_of B = Some propT"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq A B \<longmapsto> A \<longmapsto> B"
msb_succ: fixes w :: "32 Word.word" assumes "uint w \<noteq> 4294967295"   and "uint w \<noteq> 2147483647" shows "msb (w + 1) = msb w"
eulerian_poly: fixes k :: "nat" shows "fps_of_poly (eulerian_poly k) = Abs_fps (\<lambda>n. of_nat (n + 1) ^ k) * (1 - fps_X) ^ (k + 1)"
diff_open_coverI: fixes U :: "'a set set"   and f :: "'a \<Rightarrow> 'c" assumes "\<And>u. u \<in> U \<Longrightarrow> diff k (src.charts_submanifold u) charts2 f"   and "\<And>u. u \<in> U \<Longrightarrow> open u"   and "src.carrier \<subseteq> \<Union> U" shows "diff k charts1 charts2 f"
Cauchy_product_sums: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a" assumes "summable (\<lambda>k. norm (a k))"   and "summable (\<lambda>k. norm (b k))" shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
prems_1: shows "[a] \<preceq>\<^sub>F w"
next_InvInfoAtBetaUpright: fixes s :: "('c, 'd) configuration stream" assumes "next s"   and "InvInfoAtBetaUpright (shd s)"   and "InvIncomingInfoUpright (shd s)"   and "InvTempUpright (shd s)" shows "InvInfoAtBetaUpright (shd (stl s))"
\<psi>_mapsto: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "HomC.\<psi> (b, a) \<in> HomC.set (b, a) \<rightarrow> C.hom b a"
of_rat_numeral_eq: fixes w :: "num" shows "of_rat (numeral w) = numeral w"
lookup_index: fixes l1 :: "'a list"   and f :: "'a \<Rightarrow> bool"   and e :: "'a"   and l2 :: "'b list" assumes "lookup l1 f = Some e" shows "\<exists>i<length l1. e = l1 ! i \<and> (length l1 = length l2 \<longrightarrow> lookup2 l1 l2 f = Some (l2 ! i))"
orderI: fixes a :: "'a word"   and b :: "'a word" assumes "a = signed.min a b" shows "a \<le>s b"
basis_subset_carrier: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set" assumes "indep_system carrier indep"   and "indep_system.basis carrier indep X" shows "X \<subseteq> carrier"
next_step_of_def: fixes l :: "'b"   and i :: "nat" shows "next_step_of l i \<equiv> LEAST j. i \<le> j \<and> (\<not> can_step l (s j) \<or> rstep l s j)"
induct: fixes P :: "'g list \<Rightarrow> 'g list \<Rightarrow> bool"   and a0 :: "'g list"   and a1 :: "'g list" assumes "\<And>ys. P [] ys"   and "\<And>xs. P xs []"   and "\<And>x xs y ys. \<lbrakk>P xs (y # ys); P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
simps_11: fixes a :: "floatarith" shows "fold_const_fa (Arctan a) = Arctan (fold_const_fa a)"
rerename_def: fixes V :: "'var set"   and "from" :: "nat"   and to :: "nat"   and f :: "'var \<Rightarrow> 'var"   and x :: "'var" shows "rerename V from to f x = (if x \<in> freshenLC from ` V then freshenLC to (inv (freshenLC from) x) else f x)"
antecedent: shows "\<forall>w. P w z = (\<not> O w x)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R S.map \<rr>.map"
safe_distance0_def: fixes v\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and \<delta> :: "real" shows "safe_distance0 v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = v\<^sub>e * \<delta> - v\<^sub>o * \<delta> - a\<^sub>o * \<delta>\<^sup>2 / 2"
hom_equality: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b" assumes "hom_graph f S x a" shows "hom f x = a"
pcr_residue_ring_def: shows "pcr_residue_ring \<equiv> (=) OO cr_residue_ring"
HPair_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> HPair t u EQ HPair t' u'"
symmetric_reachable': fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w" shows "w \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
cbelow_on_def: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "cbelow_on A ord z rel a b \<equiv> a = b \<and> b \<in> A \<and> ord b z \<or> cbelow_on_1 A ord z rel a b"
galleryD_chamber: fixes X :: "'b set set"   and xs :: "'b set list"   and x :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimpchain X xs"   and "x \<in> set xs" shows "SimplicialComplex.maxsimp X x"
assms: shows "\<forall>w\<in>space M. 0 \<le> X w"
ilist_slice_nth_nth: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b"   and m :: "nat" assumes "n < k" shows "ilist_slice f k m ! n = f (m * k + n)"
isomorphisms_UNIV_UNIV: assumes "DIM('c) = DIM('d)" obtains f :: "'c \<Rightarrow> 'd" and g :: "'d \<Rightarrow> 'c" where "linear f" and "linear g" and "\<And>x. norm (f x) = norm x" and "\<And>y. norm (g y) = norm y" and "\<And>x. g (f x) = x" and "\<And>y. f (g y) = y"
assms_1: shows "rell_FGf L1 L2 x y"
simps_10: fixes f1 :: "'n \<Rightarrow> 'l"   and f2 :: "'o \<Rightarrow> 'm"   and x2 :: "'o" shows "map_result f1 f2 (Ok x2) = Ok (f2 x2)"
IH: fixes t :: "pdf_type" assumes "\<Gamma> \<turnstile> e : t"   and "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"   and "free_vars e \<subseteq> set vs \<union> set vs'" shows "dens_ctxt_\<alpha> (vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> (\<lambda>x xa. ennreal (eval_cexpr f x xa)) \<and> is_density_expr (vs, vs', \<Gamma>, \<delta>) t f"
simps_1: fixes n :: "nat"   and p :: "int poly"   and l :: "rat"   and r :: "rat" shows "get_itvl_2 (Irrational n (p, l, r)) = Interval (real_of_rat l) (real_of_rat r)"
\<phi>''': fixes j :: "nat" shows "Monitor.progress \<sigma> \<phi>''' j = Monitor.progress \<sigma> \<phi>'' j"
check_plan_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and \<pi>s :: "char list list" shows "ast_problem.check_plan problem \<pi>s = (case ast_problem.simulate_plan problem \<pi>s (ast_problem.I problem) of None \<Rightarrow> False | Some s' \<Rightarrow> s' \<in> ast_problem.G problem)"
min_is_min: fixes xs :: "('c, 'd) bintree option list"   and a :: "'c" assumes "normalized xs"   and "xs \<noteq> []"   and "Binomial_Queue.min xs = Some a" shows "\<forall>x\<in>set xs. x = None \<or> a \<le> priority (the x)"
mult_1_right: fixes a :: "'a" shows "a \<odot> (1::'a) = a"
deg_pm_zero: shows "deg_pm 0 = (0::'b)"
ext_hd_or_tl_axioms: shows "ext_hd_or_tl lexext_rev"
d_def: shows "d = Cn 1 S [Cn 1 psi [recf.Id 1 0, recf.Id 1 0]]"
sign_gt_0_conv: fixes x :: "('a, 'b) IEEE.float" shows "(0 < sign x) = (sign x = 1)"
pc: fixes k :: "real" shows "piecewise_continuous_on 0 k D f"
surjective: fixes r :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme" shows "r = \<lparr>CatDom = CatDom r, CatCod = CatCod r, MapM = (##) r, \<dots> = Functor.more r\<rparr>"
insert_DiffM2: fixes x :: "'d"   and M :: "'d multiset" assumes "x \<in># M" shows "M - {#x#} + {#x#} = M"
indspace0: fixes FH :: "('f, 'g) aezfun set" shows "indspace G FH 0 = 0"
p: fixes ps :: "sequent list"   and c :: "sequent"   and R :: "(sequent list \<times> sequent) set"   and F :: "char list"   and x :: "NominalSequents.var"   and A :: "form"   and S :: "sequent" assumes "(ps, c) \<in> R"   and "(ps, c) \<in> provRules"   and "NominalSequents.mset c = \<LM> F \<nabla> [x].A  \<RM>"   and "x \<sharp> NominalSequents.set_of_seq S" shows "extendRule S (ps, c) \<in>  R*"
getLastAssertedLiteral_def: fixes clause :: "Literal list"   and valuation :: "Literal list" shows "getLastAssertedLiteral clause valuation \<equiv> last (filter (\<lambda>l. l el clause) valuation)"
s_Tr_3: shows "\<lless>\<pi>\<^sub>2, eP\<^sub>2\<ggreater> P\<rightarrow> \<lless>\<pi>\<^sub>1 @ \<pi>, eP\<^sub>3\<ggreater>"
assms: shows "\<exists>o\<^sub>1 o\<^sub>2. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2"
add_mono: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "a \<le> b"   and "c \<le> d" shows "a + c \<le> b + d"
strict_order_iff: fixes b :: "'b word"   and a :: "'b word" shows "(b <s a) = (a = signed.max a b \<and> a \<noteq> b)"
simps_2: fixes x :: "constraint"   and xs :: "constraint list" shows "transf_constraints (x # xs) = transf_constraint x @ transf_constraints xs"
qring_def: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" shows "R /\<^sub>r I = \<lparr>carrier = set_rcs (b_ag R) I, pop = c_top (b_ag R) I, mop = c_iop (b_ag R) I, zero = I, tp = rcostOp R I, un = 1\<^sub>r\<^bsub>R\<^esub> \<uplus>\<^bsub>R\<^esub> I\<rparr>"
direct_lim_axioms: shows "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"
iso_iff: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" shows "(h \<in> Group.iso G H) = (h \<in> hom G H \<and> h ` carrier G = carrier H \<and> inj_on h (carrier G))"
onorm: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "bounded_linear f" shows "norm (f x) \<le> onorm f * norm x"
that_5: fixes CTa :: "nat \<Rightarrow> classDef option"   and e0 :: "exp"   and m :: "nat"   and esa :: "exp list"   and e0' :: "exp" assumes "x = CTa"   and "xa = MethodInvk e0 m esa"   and "xb = MethodInvk e0' m esa"   and "CTa \<turnstile> e0 \<rightarrow> e0'" shows "thesis"
simps_11: fixes v :: "nat"   and va :: "agg_type \<times> event_data"   and vb :: "nat"   and vc :: "trm"   and vd :: "formula" shows "get_and_list (Agg v va vb vc vd) = [Agg v va vb vc vd]"
distinct_41: fixes x4 :: "('a, 'b, 'c) expr"   and x7 :: "('a, 'b, 'c) expr" shows "Ref x4 \<noteq> Rfork x7"
strange_Un_eq: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> (B \<union> A) = B \<union> A"
lexordp_irreflexive: fixes xs :: "'a list" assumes "\<forall>x. \<not> x \<prec> x" shows "\<not> local.lexordp xs xs"
that: fixes b :: "complex" assumes "ball b r' \<subseteq> f ` ball a r" shows "thesis"
homotopic_paths_imp_pathfinish: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "homotopic_paths s p q" shows "pathfinish p = pathfinish q"
d_equal_def: fixes DA :: "'a \<Rightarrow> 'b"   and d :: "'b"   and m :: "'a \<Rightarrow> 'c"   and m' :: "'a \<Rightarrow> 'c" shows "Strong_Security.d_equal DA d m m' \<equiv> \<forall>x. DA x \<le> d \<longrightarrow> m x = m' x"
P: shows "Measurable.pred (stream_space (count_space UNIV)) P"
reusables_lchild_neq_Nil_under_some_conditions: fixes t :: "'c tree"   and w :: "nat \<times> nat" assumes "well_shaped t"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "fst w \<noteq> l t"   and "fst w \<le> r t"   and "fst w < l (rchild t)" shows "reusables (lchild t) w \<noteq> []"
assms: fixes k :: "'a"   and v :: "'b" assumes "fmlookup m k = Some v" shows "\<exists>v'. f v' = v"
inner_commute_zero: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and xa :: "'b"   and x :: "'b" assumes "inner_product_space scale inner" shows "(inner xa x = (0::'a)) = (inner x xa = (0::'a))"
o_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and x :: "'e" shows "(f \<circ> g) x = f (g x)"
combine_infinity_invariant_2: fixes i :: "enat"   and x :: "'a ltln"   and j :: "enat"   and y :: "'a ltln"   and z :: "'a ltln"   and k :: "enat" assumes "(i = \<infinity>) = is_constant x"   and "(j = \<infinity>) = is_constant y"   and "combine mk_or (x, i) (y, j) = (z, k)" shows "(k = \<infinity>) = is_constant z"
mult_scaleC_left: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "complex"   and x :: "'a"   and y :: "'a" assumes "class.complex_algebra scaleR scaleC plus zero minus uminus times" shows "times (scaleC a x) y = scaleC a (times x y)"
q2: shows "f \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> n}"
reduce_basis_cost_def: fixes fs :: "int vec list" shows "reduce_basis_cost fs = (case fs of [] \<Rightarrow> (fs, 0) | f # x \<Rightarrow> case basis_reduction_cost fs of (state, c) \<Rightarrow> (fs_state state, c))"
inc: shows "pseudo_proj_True n -` {pseudo_proj_True n w} \<subseteq> A"
termi_merge_c2: shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>merge_guards c1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>merge_guards c2 \<down> Normal s'"
coeff_in_ring: fixes a :: "'d \<Rightarrow> 'a"   and A :: "'d set"   and x :: "'d" assumes "a \<in> A \<rightarrow> carrier R"   and "x \<in> A" shows "a x \<in> carrier R"
Un_open: fixes T :: "'b set set"   and A :: "'b set"   and B :: "'b set" assumes "topology T"   and "A open\<^bsub>T\<^esub>"   and "B open\<^bsub>T\<^esub>" shows "A \<union> B open\<^bsub>T\<^esub>"
le_infI: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x a"   and "less_eq x b" shows "less_eq x (inf a b)"
sdrop_0_1: fixes xs :: "[:'r:]" shows "sdrop\<cdot>0\<cdot>xs = xs"
IH: fixes r :: "'trm"   and ts :: "'trm list"   and us :: "'var list"   and vs :: "'var list" assumes "r \<in> atrm"   and "set ts \<subseteq> atrm"   and "set xs \<subseteq> var"   and "distinct xs"   and "set us \<subseteq> var"   and "distinct us"   and "set us \<inter> FvarsT r = {}"   and "set us \<inter> \<Union> (FvarsT ` set ts) = {}"   and "set us \<inter> set xs = {}"   and "set vs \<subseteq> var"   and "distinct vs"   and "set vs \<inter> FvarsT r = {}"   and "set vs \<inter> \<Union> (FvarsT ` set ts) = {}"   and "set vs \<inter> set xs = {}"   and "length us = length xs"   and "length vs = length xs"   and "length ts = length xs"   and "set us \<inter> set vs = {}" shows "rawpsubstT (rawpsubstT r (zip (map Var us) xs)) (zip ts us) = rawpsubstT (rawpsubstT r (zip (map Var vs) xs)) (zip ts vs)"
encode2_lt: fixes i :: "nat" assumes "i < d" shows "encode2 i < d2"
empty_storebuffer_step_def: fixes c :: "(nat \<Rightarrow> nat) \<times> 'b \<times> 'c \<times> 'd \<times> 'e"   and c' :: "(nat \<Rightarrow> nat) \<times> 'b \<times> 'c \<times> 'd \<times> 'e" shows "empty_storebuffer_step c c' = False"
DomainI: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" assumes "r a b" shows "Domainp r a"
fcomp_def: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" shows "fcomp f g = (\<lambda>x. g (f x))"
ordLeq_ordIso_trans: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and r'' :: "('d \<times> 'd) set" assumes "r \<le>o r'"   and "r' =o r''" shows "r \<le>o r''"
list_rel_split_right_iff: fixes x :: "'d"   and xs :: "'d list"   and l :: "'e list"   and R :: "('d \<times> 'e) set" shows "((x # xs, l) \<in> \<langle>R\<rangle>list_rel) = (\<exists>y ys. l = y # ys \<and> (x, y) \<in> R \<and> (xs, ys) \<in> \<langle>R\<rangle>list_rel)"
remove_key_keys: fixes f :: "'a \<Rightarrow>\<^sub>0 'b"   and k :: "'a" shows "keys f - {k} = keys (remove_key k f)"
ale_linear: fixes z :: "ant"   and w :: "ant" shows "z \<le> w \<or> w \<le> z"
pred_inject_1: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" shows "bdd_all P (Leaf a) = P a"
leq: shows "\<forall>i. f (Suc i) \<le> f i"
gen_binomial_real: fixes z :: "real"   and a :: "real" assumes "norm z < 1" shows "(\<lambda>n. (a gchoose n) * z ^ n) sums (1 + z) powr a"
connectedin_discrete_topology: fixes U :: "'b set"   and S :: "'b set" shows "connectedin (discrete_topology U) S = (S \<subseteq> U \<and> (\<exists>a. S \<subseteq> {a}))"
det_lower_triangular: fixes n :: "nat"   and A :: "'b mat" assumes "\<And>i j. \<lbrakk>i < j; j < n\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'b)"   and "A \<in> carrier_mat n n" shows "Determinant.det A = prod_list (diag_mat A)"
wf_bs_def: fixes bs :: "'b list"   and p :: "'b poly" shows "wf_bs bs p = (maxindex p \<le> length bs)"
direct_decompD_3: fixes A :: "'b set"   and ss :: "'b set list" assumes "direct_decomp A ss" shows "sum_list ` listset ss = A"
strategy_attracts_def: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and W :: "'a set" shows "strategy_attracts p \<sigma> A W \<equiv> \<forall>v0\<in>A. strategy_attracts_via p \<sigma> v0 A W"
antimonoD: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "antimono f"   and "x \<le> y" shows "f y \<le> f x"
card_of_leq: fixes A :: "'s set" assumes "A \<subseteq> Univ" shows "|A| <o cardSuc (cmax |UNIV| natLeq)"
intro: fixes S :: "('b \<times> 'b) set" assumes "SN S" shows "SN_ars S"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<Phi>'\<Psi>.map f) = Cop.arr f"
cod_dom: fixes f :: "'s" assumes "arr f" shows "cod (local.dom f) = local.dom f"
v': shows "(vars_term_ms (SCF (Fun g ts)) \<subseteq># vars_term_ms (SCF (Fun f (bef @ s # aft)))) = True"
mat_of_cols_cols: fixes A :: "'b mat" shows "mat_of_cols (dim_row A) (cols A) = A"
stable_on_Xd: fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "Pds'.stable_on ds X"   and "x \<in> X" shows "Xd x \<in> ds"
exec_gpv_restrict_gpv_invariant: fixes gpv :: "('d, 'a, 'b) gpv"   and s :: "'s2" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I2 s" shows "exec_gpv oracle2 (restrict_gpv \<I> gpv) s = exec_gpv oracle2 gpv s"
mat_mult_sub_lattice: fixes fs :: "rat vec list"   and gs :: "rat vec list"   and A :: "int mat" assumes "set fs \<subseteq> Rn"   and "set gs \<subseteq> Rn"   and "A \<in> carrier_mat (length fs) (length gs)"   and "mat_of_rows n fs = of_int_hom.mat_hom A * mat_of_rows n gs" shows "gs.lattice_of fs \<subseteq> gs.lattice_of gs"
the_inv_into_onto: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "the_inv_into A f ` f ` A = A"
that: fixes cs :: "(Term_Class.term \<times> sterm) list" assumes "rhs = Sabs cs" shows "thesis"
axioms_1: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor" assumes "Functor F : AA \<longrightarrow> BB" shows "two_cats AA BB"
map_pmf_compose: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
Hset1_alt: shows "Hset1 = \<Union> \<circ> BNF_Def.collect {(`) F1set1 \<circ> Gset1, (`) F2set1 \<circ> Gset2, (`) F3set1 \<circ> Gset3}"
continuous_within_compose: fixes x :: "'c"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "continuous (at x within s) f"   and "continuous (at (f x) within f ` s) g" shows "continuous (at x within s) (g \<circ> f)"
proj_arcs_eq_swap: fixes a :: "'b" assumes "a \<notin> {uv, rev_G uv, rev_H uw, rev_H vw}" shows "proj_arcs_H a = (uw \<rightleftharpoons>\<^sub>F uv \<circ> vw \<rightleftharpoons>\<^sub>F rev_G uv) a"