hcju/isabelleps huggingface dataset:
tensor_eq: fixes B :: "wall"   and D :: "wall"   and A :: "wall"   and C :: "wall" assumes "B ~ D \<and> A ~ C \<and> is_tangle_diagram A \<and> is_tangle_diagram B \<and> is_tangle_diagram C \<and> is_tangle_diagram D" shows "A \<otimes> B ~ C \<otimes> D"
prime_int_iff: fixes p :: "int" shows "prime p = (0 < p \<and> prime_elem p)"
CutAux1: fixes H :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "H \<rhd> c : A" shows "CutAuxProp H c A"
has_derivative: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and g' :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "bounded_linear f"   and "(g has_derivative g') F" shows "((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
dvd_by_all: shows "\<forall>M. \<exists>N>0. \<forall>m. 0 < m \<and> m \<le> M \<longrightarrow> m dvd N"
rq_emp: fixes u :: "'b list" shows "u\<^sup><\<inverse>\<epsilon>  = u"
poly_power_zero_iff: fixes p :: "'c poly"   and n :: "nat"   and x :: "'c" shows "(poly (p ^ n) x = (0::'c)) = (n \<noteq> 0 \<and> poly p x = (0::'c))"
eq_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun A (rel_fun A (=)) (=) (=)"
set_plus_image: fixes S :: "'c set"   and T :: "'c set" shows "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
right_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "f (f a b) c = f a (f b c)"
1: shows "\<mu> \<star> local.dom \<mu> \<noteq> null"
xml_tree_anything: fixes x :: "xml_tree"   and y :: "xml_tree" shows "x = y"
conjunctive_simp: fixes S :: "'d \<Rightarrow> 'e"   and Q :: "'d set" assumes "conjunctive S" shows "S (Inf Q) = Inf (S ` Q)"
m_o_def: fixes opt :: "'av st option"   and X :: "char list set" shows "m\<^sub>o opt X = (case opt of None \<Rightarrow> h * card X + 1 | Some S \<Rightarrow> m\<^sub>s S X)"
ideD_1: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "VV.arr a"
monoI: fixes g :: "'a" assumes "arr g"   and "\<And>f f'. seq g f \<and> seq g f' \<and> g \<cdot> f = g \<cdot> f' \<Longrightarrow> f = f'" shows "local.mono g"
"1_1": fixes j :: "nat"   and l :: "nat"   and k :: "nat" assumes "local.neville_aitken_merge (e1 # e2 # es) = map (\<lambda>i. (x i, x (i + j), local.neville_aitken_main i (i + j))) [l..<Suc (l + k)]" shows "local.neville_aitken_impl_main (local.neville_aitken_merge (e1 # e2 # es)) = local.neville_aitken_main l (l + j + k)"
simps_1: fixes x1 :: "label"   and y1 :: "label" shows "(Break x1 = Break y1) = (x1 = y1)"
dynlookup_declC: fixes G :: "prog"   and statT :: "ref_ty"   and dynC :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "dynlookup G statT dynC sig = Some m"   and "wf_prog G"   and "is_class G dynC"   and "isrtype G statT" shows "G\<turnstile>dynC\<preceq>\<^sub>C declclass m \<and> is_class G (declclass m)"
isPath_ex_edge2: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat"   and u1 :: "nat"   and v1 :: "nat" assumes "cf.isPath u p v"   and "(u1, v1) \<in> set p"   and "v1 \<noteq> v" shows "\<exists>v2. (v1, v2) \<in> set p"
uf_def: shows "uf \<equiv> fls_base_factor f"
nfs: shows "\<not> (fin_sat (F \<triangleright> S) \<or> fin_sat (\<^bold>\<not> F \<triangleright> S))"
res_neg_eq: fixes x :: "int" shows "\<ominus>\<^bsub>R\<^esub> x = - x mod int p"
arrE: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f" obtains "f \<noteq> concrete_category.Null" and "concrete_category.Dom f \<in> Obj" and "concrete_category.Cod f \<in> Obj" and "concrete_category.Map f \<in> Hom (concrete_category.Dom f) (concrete_category.Cod f)"
euclidean_eqI: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and x :: "'a"   and y :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "\<And>b. b \<in> Basis \<Longrightarrow> inner x b = inner y b" shows "x = y"
in_idealE_rep_list_dgrad_sig_set_take: fixes d :: "'a \<Rightarrow> nat"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "hom_grading d"   and "p \<in> punit_dgrad_max_set d"   and "p \<in> ideal (set (take j fs))" obtains r :: "'t \<Rightarrow>\<^sub>0 'b" where "r \<in> dgrad_sig_set' (length fs) d" and "r \<in> dgrad_sig_set' j d" and "p = rep_list r"
minWeightBasis_def: shows "minWeightBasis \<equiv> obtain_sorted_carrier \<bind> (\<lambda>l. ASSERT (set l = carrier) \<bind> (\<lambda>_. nfoldli l (\<lambda>_. True) (\<lambda>e T. ASSERT (indep T \<and> e \<in> carrier \<and> T \<subseteq> carrier) \<bind> (\<lambda>_. if indep (insert e T) then RETURN (insert e T) else RETURN T)) {} \<bind> RETURN))"
distinct_10: fixes x4 :: "e list"   and x21 :: "nat"   and x22 :: "v list" shows "RSNormal x4 \<noteq> RSBreak x21 x22"
order_sym_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<sim> x"   and "sympartp (\<sim>) y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sim> x"
R'_simps_3: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "((x, Suc n) \<in> R' n) = (\<exists>y'>n. (x, y') \<in> R)"
wf_config_def: 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 E :: "char list \<Rightarrow> ty option"   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 T :: "ty" shows "P,E,s \<turnstile> e : T \<surd> \<equiv> P,E \<turnstile> s \<surd> \<and> P,E,hp s \<turnstile> e : T"
piecewise_C1_differentiable_neg: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set" assumes "f piecewise_C1_differentiable_on S" shows "(\<lambda>x. - f x) piecewise_C1_differentiable_on S"
AC_mono: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and d :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<sqsubseteq> b"   and "c \<sqsubseteq> d" shows "(a \<Squnion> c) \<sqsubseteq> (b \<Squnion> d)"
tensor_mat_add2: fixes m1 :: "'a mat"   and dims :: "nat list"   and vars :: "nat set"   and m2 :: "'a mat"   and m3 :: "'a mat" assumes "m1 \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "m2 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)"   and "m3 \<in> carrier_mat (partial_state.d2 dims vars) (partial_state.d2 dims vars)" shows "tensor_mat dims vars m1 (m2 + m3) = tensor_mat dims vars m1 m2 + tensor_mat dims vars m1 m3"
brc_iq_def: fixes \<delta> :: "('a, 'b) ta_rule dlist" shows "brc_iq \<delta> \<equiv> lh_set_xy.g_image_filter (\<lambda>r. if rhsq r = [] then Some (lhs r) else None) \<delta>"
assms_2: fixes X :: "real \<Rightarrow> 'a" assumes " D X = (\<lambda>\<tau>. f \<tau> (X \<tau>)) on U (X t\<^sub>0)" shows "\<forall>\<tau>\<in>U (X t\<^sub>0). t\<^sub>0 < \<tau> \<longrightarrow> G (X \<tau>) \<longrightarrow> \<nu>' (X \<tau>) \<le> \<mu>' (X \<tau>)"
rep_def: shows "rep \<equiv> map_fun eval\<kappa> id the"
not_less_zero_hmset: fixes M :: "hmultiset" shows "\<not> M < 0"
Phi_Psi: fixes n :: "nat"   and l :: "nat" shows "\<Phi> (n, l) = \<Psi> (int l0 \<le> \<lfloor>real l / c\<rfloor>) ai ad (f1'' * real l) (f2'' * real l) n"
1: shows "castAbs xs s (Abs xs x X) = (x', X')"
inject: fixes x :: "'b \<Rightarrow> bool"   and ya :: "'b \<Rightarrow> bool" shows "(pred.Pred x = pred.Pred ya) = (x = ya)"
assms: shows "\<forall>c\<in>set cs. u c = d"
ordered_finite_map_def: fixes \<alpha> :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and invar :: "'c \<Rightarrow> bool" shows "ordered_finite_map \<alpha> invar \<equiv> finite_map \<alpha> invar"
iptables_bigstep_process_ret_DecisionD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and X :: "final_decision" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_ret rs, s\<rangle> \<Rightarrow> Decision X" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> Decision X"
ensures: fixes P :: "state \<Rightarrow> bool"   and N :: "state \<times> state \<Rightarrow> bool"   and Q :: "state \<Rightarrow> bool"   and A :: "state \<times> state \<Rightarrow> bool" assumes "\<turnstile> $P \<and> N \<longrightarrow> P$ \<or> Q$"   and "\<turnstile> ($P \<and> N) \<and> A \<longrightarrow> Q$" shows "\<turnstile> \<box>N \<and> \<box>(\<box>P \<longrightarrow> \<diamond>A) \<longrightarrow> (P \<leadsto> Q)"
inject_4: fixes x41 :: "aexp"   and x42 :: "aexp"   and y41 :: "aexp"   and y42 :: "aexp" shows "(Less x41 x42 = Less y41 y42) = (x41 = y41 \<and> x42 = y42)"
bij_comp: fixes f :: "'h \<Rightarrow> 'i"   and g :: "'i \<Rightarrow> 'j" assumes "bij f"   and "bij g" shows "bij (g \<circ> f)"
concurrent_comm: fixes s1 :: "'a"   and s2 :: "'a" shows "(s1 \<parallel> s2) = (s2 \<parallel> s1)"
prefD: fixes A :: "'a \<Rightarrow> bool"   and w :: "'a" assumes "|~ A" shows "A w"
connected_convex_1_gen: fixes S :: "'a set" assumes "DIM('a) = 1" shows "connected S = convex S"
simTrans: shows "\<forall>a ec t. t \<in> jkbpC \<and> simAbs ec = sim_equiv_class a t \<longrightarrow> simAbs ` set (KBP.k_succs a ec) = {sim_equiv_class a (t' \<leadsto> s) |t' s. t' \<leadsto> s \<in> jkbpC \<and> jview a t' = jview a t}"
rep_eq: fixes x :: "nat_int"   and xa :: "nat_int" shows "(x \<le> xa) = (Rep_nat_int x \<subseteq> Rep_nat_int xa)"
equal_3: fixes x :: "real_alg_3"   and y :: "real_alg_3" shows "equal_3 x y = (real_of_3 x = real_of_3 y)"
space_PiM: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" shows "space (Pi\<^sub>M I M) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
bind_eq_join_pmf: fixes M :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf" shows "M \<bind> f = join_pmf (map_pmf f M)"
holf': shows "f holomorphic_on (\<lambda>z. a + complex_of_real r * z) ` ball 0 1"
bin_to_bl_aux_append: fixes n :: "nat"   and w :: "int"   and bs :: "bool list"   and cs :: "bool list" shows "bin_to_bl_aux n w bs @ cs = bin_to_bl_aux n w (bs @ cs)"
raw_match'_map_defined: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and p :: "variable \<times> String.literal set" assumes "raw_match' T U subs = Some subs'"   and "p \<in> tvsT T" shows "subs' p \<noteq> None"
continuous_map_id_subt: fixes X :: "'a topology"   and S :: "'a set" shows "continuous_map (subtopology X S) X id"
restrict_indets_plus: fixes p :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "restrict_indets (p + q) = restrict_indets p + restrict_indets q"
Args_min_on_def: fixes f :: "int \<Rightarrow> 'a"   and I :: "int set" shows "Args_min_on f I = {k. is_arg_min_on f I k}"
encode_problem_parallel_correct_vii: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t" shows "length (map (decode_state_at \<Pi> \<A>) [0..<Suc (length (\<Phi>\<inverse> \<Pi> \<A> t))]) = length (trace_parallel_plan_strips (\<Pi>\<^sub>I) (\<Phi>\<inverse> \<Pi> \<A> t))"
setdist_eq_0_closed: fixes S :: "'a set"   and x :: "'a" assumes "closed S" shows "(setdist {x} S = 0) = (S = {} \<or> x \<in> S)"
eta_reds_into_beta_eta_reds: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* t" shows "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t"
cases: fixes x :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>"   and a :: "('d, 'e, 'f) gpv" assumes "x \<in> sub_gpvs \<I> a" obtains out :: "'e" and c :: "('d, 'e, 'f) rpv" and gpv :: "('d, 'e, 'f) gpv" and input :: "'f" where "a = gpva__" and "IO out__ c__ \<in> set_spmf (the_gpv gpva__)" and "input__ \<in> responses_\<I> \<I> out__" and "x = c__ input__"   | out :: "'e" and c :: "('d, 'e, 'f) rpv" and gpv :: "('d, 'e, 'f) gpv" and input :: "'f" where "a = gpv" and "IO out c \<in> set_spmf (the_gpv gpv)" and "input \<in> responses_\<I> \<I> out" and "x \<in> sub_gpvs \<I> (c input)"
leg1_iso: shows "\<exists>\<nu>. \<guillemotleft>\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> \<mu>.chine\<guillemotright> \<and> local.iso \<nu>"
g'_def: fixes \<G> :: "'a cyclic_group"   and x :: "nat" assumes "chaum_ped_\<Sigma>_base \<G>" shows "chaum_ped_\<Sigma>_base.g' \<G> x = \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> x"
a_repr_independenceD: fixes y :: "int"   and x :: "int" assumes "y \<in> UNIV"   and "Idl\<^bsub>\<Z>\<^esub> {m} +>\<^bsub>\<Z>\<^esub> x = Idl\<^bsub>\<Z>\<^esub> {m} +>\<^bsub>\<Z>\<^esub> y" shows "y \<in> Idl\<^bsub>\<Z>\<^esub> {m} +>\<^bsub>\<Z>\<^esub> x"
star2: fixes x :: "('a \<times> ('b \<times> 'a) list) set" shows "t_star (t_one \<union> x) = t_star x"
intro: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "('b, 'c) epsym \<Rightarrow> univ list \<Rightarrow> bool" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl"   and "CM.Model (ProblemIkTpartG.GE_wtFsym wtFsym resOf protCl) (ProblemIkTpartG.GE_wtPsym wtPsym prot protFw) (ProblemIkTpartG.GE_arOf arOf) (ProblemIkTpartG.GE_resOf resOf) (ProblemIkTpartG.GE_parOf parOf) (ProblemIkTpartG.gPB wtFsym arOf resOf \<Phi> prot protFw protCl) eintT eintF eintP" shows "ProblemIkTpartG_GEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl eintT eintF eintP"
lead_coeff_poly_mult: fixes p :: "'a poly"   and q :: "'a poly" assumes "monic p"   and "monic q"   and "0 < degree p"   and "0 < degree q"   and "coeff q 0 \<noteq> (0::'a)" shows "monic (poly_mult p q)"
declclass_method_simp: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "declclass (method sig m) = declclass m"
sums_def_le: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a" shows "f sums s = (\<lambda>n. sum f {..n}) \<longlonglongrightarrow> s"
stcons_def: shows "stcons \<equiv> \<Lambda> a Stream. Stream_abs\<cdot>(:up\<cdot>a, up\<cdot>Stream:)"
fms_sb: fixes m :: "nat" shows "(f ^^ m) ` s \<subseteq> s"
PRelQ: fixes Tvec :: "'a list"   and \<Psi>' :: "'b" assumes "length xvec = length Tvec" shows "(\<Psi> \<otimes> \<Psi>', P[xvec::=Tvec], Q[xvec::=Tvec]) \<in> Rel"
single_zero: fixes k :: "'b" shows "single k (0::'c) = 0"
assms: fixes x :: "nat" assumes "x \<in># I" shows "0 < x"
hcomp_isomorphic_ide: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "horizontal_composition V H src trg"   and "category.isomorphic V f g"   and "partial_magma.ide V h"   and "src f = trg h" shows "category.isomorphic V (H f h) (H g h)"
vector_scaleR_component: fixes r :: "real"   and x :: "('b, 'c) Finite_Cartesian_Product.vec"   and i :: "'c" shows "(r *\<^sub>R x) $h i = r *\<^sub>R x $h i"
map_scale_distrib_left: fixes k :: "'c"   and s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "k \<cdot> (s + t) = k \<cdot> s + k \<cdot> t"
Gamma_nonzero: fixes z :: "'a" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "Gamma z \<noteq> (0::'a)"
main_part_null_part_Un: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets (completion M)" shows "main_part M S \<union> null_part M S = S"
axioms_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b"   and \<eta> :: "'b \<Rightarrow> 'b"   and \<epsilon> :: "'a \<Rightarrow> 'a" assumes "equivalence_of_categories C D F G \<eta> \<epsilon>" shows "category C"
assms: shows "xvec \<sharp>* (\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>)"
compatible_awalk_verts: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "compatible G H" shows "pre_digraph.awalk_verts G = pre_digraph.awalk_verts H"
mono_bal_i: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat"   and h :: "nat"   and n' :: "nat"   and h' :: "nat" assumes "RBTi1 TYPE('b) bal_i"   and "bal_i n h"   and "n \<le> n'"   and "h' \<le> h" shows "bal_i n' h'"
prems: shows "n is-nx # nsx\<rightarrow>\<^sub>d* n'"
push_down_rank_tokens: fixes x :: "nat"   and n :: "nat"   and y :: "nat"   and i :: "nat" assumes "rank x n = rank y n"   and "rank x n = Some i" shows "\<exists>q. x \<in> configuration q n \<and> y \<in> configuration q n"
tick_count_strict_suc: fixes r :: "'b run"   and c :: "clock"   and n :: "nat" shows "tick_count_strict r c (Suc n) = tick_count r c n"
map_graph_countable: fixes f :: "'c \<Rightarrow> 'd option" assumes "countable (dom f)" shows "countable (map_graph f)"
fls_subdegree_fls_X_inv_pow: fixes n :: "nat" shows "fls_subdegree (fls_X_inv ^ n) = - int n"
simps_2: fixes uu :: "'a"   and uv :: "bool" shows "restrict_top Trueif uu uv = Trueif"
lengths: shows "(\<forall>w\<in>set m. length w = nr) \<and> length v = nr \<and> length m = ncc"
tid_def: fixes v :: "Value" shows "tid v = (case v of objV C a \<Rightarrow> C)"
MP_null: fixes A :: "fm"   and B :: "fm"   and H :: "fm set" assumes "{} \<turnstile> A IMP B"   and "H \<turnstile> A" shows "H \<turnstile> B"
mset_left_cancel_union: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "a \<in># A + B"   and "a \<notin># A" shows "a \<in># B"
trm_is_terminal_arr: fixes a :: "'a" assumes "ide a" shows "terminal_arr \<t>[a]"
tab_def: shows "tab \<equiv> tab\<^sub>0.trnr\<^sub>\<eta> (tab\<^sub>1 r) (rep r)"
strict_implies_order: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subset># b" shows "a \<subseteq># b"
inputChainEqLength: fixes xvec :: "name list"   and M :: "'a"   and P :: "('a, 'b, 'c) psi"   and yvec :: "name list"   and N :: "'a"   and Q :: "('a, 'b, 'c) psi" assumes "inputChain xvec M P = inputChain yvec N Q" shows "length xvec = length yvec"
aNDSubset: fixes a :: "('b set set, 'c) Combinators list"   and b :: "('b set set, 'c) Combinators list" assumes "singleCombinators a"   and "set a \<subseteq> set b"   and "allNetsDistinct b" shows "allNetsDistinct a"
count_eq_zero_iff: fixes M :: "'a multiset"   and x :: "'a" shows "(count M x = 0) = (x \<notin># M)"
topD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b"   and x :: "'b" assumes "Semilat.top r T" shows "x \<sqsubseteq>\<^bsub>r\<^esub> T"
degree_of_nat: fixes n :: "nat" shows "Polynomial.degree (of_nat n) = 0"
max_id: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.non_strict_order less_eq"   and "less_eq y x" shows "ord.max less_eq x y = x"
equivalence_map_is_ide: fixes f :: "'a" assumes "equivalence_map f" shows "ide f"
isSPath_sg_outgoing: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat"   and u1 :: "nat"   and v1 :: "nat"   and v2 :: "nat" assumes "cf.isSimplePath u p v"   and "(u1, v1) \<in> set p"   and "v1 \<noteq> v2" shows "(u1, v2) \<notin> set p"
vc_mono: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C :: "acom" assumes "\<forall>s. P s \<longrightarrow> Q s"   and "vc C P" shows "vc C Q"
scale_minus_left: fixes a :: "complex"   and x :: "'a" shows "- a *\<^sub>C x = - (a *\<^sub>C x)"
free_tv_subst: fixes S :: "nat \<Rightarrow> typ" shows "free_tv S = Type.dom S \<union> cod S"
simpnum_numbound0: fixes t :: "MIR.num" assumes "numbound0 t" shows "numbound0 (simpnum t)"
VectorSpace_VectorSpaceHomSet: fixes smult' :: "'f \<Rightarrow> 'a \<Rightarrow> 'a"   and W :: "'a set" assumes "VectorSpace smult' W" shows "VectorSpace (\<lambda>a T v. smult' a (T v)) (VectorSpaceHomSet (\<cdot>) V smult' W)"
tab_in_hom_2: shows "\<guillemotleft>\<rho> : g \<Rightarrow>\<^sub>C r \<star>\<^sub>C f\<guillemotright>"
loop_invar: shows "loop \<in> dwa_invar (brec_det_algo (build_rqrm \<delta>) Qi \<delta>)"
open_segment_image_interval: fixes a :: "'b"   and b :: "'b" shows "open_segment a b = (if a = b then {} else (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0<..<1})"
sublist_rev: fixes xs :: "'b list"   and ys :: "'b list" shows "rev xs \<le>f rev ys = xs \<le>f ys"
b': shows "\<forall>r'\<in>R. leftPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"
language_state_for_inputs_in_language_state: fixes M :: "('e, 'f, 'g) FSM"   and q :: "'g"   and T :: "'e list set" shows "LS\<^sub>i\<^sub>n M q T \<subseteq> LS M q"
cp_dot\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<S>\<A>\<L>\<A>\<R>\<Y>: fixes X :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "(X.salary) \<tau> = (\<lambda>_. X \<tau>.salary) \<tau>"
mcont_lset: shows "mcont lSup (\<sqsubseteq>) \<Union> (\<subseteq>) lset"
lexer_correct_Some: fixes s :: "'b list"   and r :: "'b rexp" shows "(s \<in> lang r) = (\<exists>v. lexer r s = Some v \<and> s \<in> r \<rightarrow> v)"
bind_btree_simps_1: fixes k :: "'b \<rightarrow> 'a\<cdot>btree" shows "\<bottom> \<bind> k = \<bottom>"
poly_mod_type_axioms: shows "poly_mod_type TYPE('a) m"
aseval_ext: fixes P :: "assn_raw"   and P' :: "assn_raw" assumes "\<forall>h. aseval P h = aseval P' h" shows "P = P'"
simps_8: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b aexp"   and va :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (Eq e1 e2) (Gt v va) = True"
2: shows "\<nu>.leg0 \<cdot> Prj\<^sub>1\<^sub>0 = \<pi>.leg1 \<cdot> Prj\<^sub>0\<^sub>0"
sylow_thm: shows "\<exists>H. subgroup H (G\<lparr>carrier := P\<rparr>) \<and> card H = p ^ a"
older_seniors_card_mono: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and A :: "nat set"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "A \<subseteq> semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n" shows "card A \<le> card (semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n)"
echelon_form_condition1: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "echelon_form A" shows "\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)"
Quotient3_ivl: shows "Quotient3 eq_ivl abs_ivl rep_ivl"