hcju/isabelleps huggingface dataset:
circ_simulate_right_plus: fixes z :: "'a"   and x :: "'a"   and y :: "'a"   and w :: "'a" shows "z * x \<le> y * y\<^sup>\<circ> * z \<squnion> w \<longrightarrow> z * x\<^sup>\<circ> \<le> y\<^sup>\<circ> * (z \<squnion> w * x\<^sup>\<circ>)"
independent_columns_Gram_Schmidt_matrix: fixes A :: "((real, 'b) vec, 'a) vec" assumes "vec.independent (columns A)"   and "card (columns A) = ncols A" shows "vec.independent (columns (Gram_Schmidt_matrix A)) \<and> card (columns (Gram_Schmidt_matrix A)) = ncols A"
mult_comp_cls_val_process0: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and qty :: "nat \<Rightarrow> 'd \<Rightarrow> real" assumes "portfolio pf1" shows "\<forall>w. cls_val_process Mkt (qty_mult_comp pf1 qty) 0 w = cls_val_process Mkt pf1 0 w * qty (Suc 0) w"
case_1: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> 'd"   and f5 :: "'e \<Rightarrow> 'd"   and f6 :: "nat \<Rightarrow> 'd"   and f7 :: "'e \<Rightarrow> 'd"   and f8 :: "'d"   and x1 :: "'e" shows "(case broadcast x1 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f1 x1"
finfun_const_inject: fixes b :: "'b"   and b' :: "'b" shows "(K$ b) = (K$ b') \<equiv> b = b'"
countable_image_inj_on: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "countable (f ` A)"   and "inj_on f A" shows "countable A"
natural_transformation_axioms: shows "natural_transformation A A \<tau>.A.map \<tau>.A.map \<tau>.A.map"
ABind_below_ABinds: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and v :: "Vars.var"   and e :: "exp" assumes "map_of \<Gamma> v = Some e" shows "ABind v e \<sqsubseteq> ABinds \<Gamma>"
snd_eqD: fixes x :: "'c"   and y :: "'b"   and a :: "'b" assumes "snd (x, y) = a" shows "y = a"
s: shows "{w \<in> space N. w !! n} \<in> sets N"
union_mset_add_mset_right: fixes A :: "'k multiset"   and a :: "'k"   and B :: "'k multiset" shows "A + add_mset a B = add_mset a (A + B)"
ff': fixes x :: "'a"   and \<sigma> :: "'b" assumes "x \<in> set l'"   and "c' \<sigma>" shows "f x \<sigma> = f' x \<sigma>"
fcomp_def: fixes r :: "V"   and s :: "V" shows "r \<circ>\<^sub>\<bullet> s = set {[a, c]\<^sub>\<circ> |a c. \<exists>b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> s \<and> [b, c]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
not_inf_on_unit_circle'': shows "\<infinity>\<^sub>h \<notin> circline_set unit_circle"
path_decomp_2: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and ys :: "'c list" assumes "Digraph G"   and "Digraph.walk G (xs @ ys) \<and> distinct (xs @ ys)" shows "Digraph.walk G ys \<and> distinct ys"
eq: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "(key_compare ko x y = Eq) = (x = y)"
divisor_count_upper_bound: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "\<forall>\<^sub>F n in sequentially. real (divisor_count n) < 2 powr ((1 + \<epsilon>) * ln (real n) / ln (ln (real n)))"
Tinv_measure_preserving: assumes "invertible_qmpt" shows "Tinv \<in> measure_preserving M M"
rel_fun_mono: fixes X :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and Y :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "rel_fun X A f g"   and "\<And>x y. Y x y \<longrightarrow> X x y"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_fun Y B f g"
IH: fixes y :: "('a \<times> ('b \<times> 'a) list) \<times> 'a \<times> ('b \<times> 'a) list" assumes "(y, (s, [\<beta>_step] @ \<upsilon>_step), s, [\<alpha>_step] @ \<rho>_step) \<in> pex r"   and "peak ars y" shows "\<exists>\<sigma>' \<tau>'. DD ars r (fst y, snd y, \<sigma>', \<tau>')"
that: fixes U :: "'x set" assumes "(g, U) \<in> set ps" shows "thesis"
Un_empty_right: fixes A :: "'b set" shows "A \<union> {} = A"
all_after_eq_1: fixes c :: "'b" shows "#c$ \<equiv> #c"
out321__inangle: fixes C :: "'p"   and B :: "'p"   and A :: "'p"   and P :: "'p" assumes "C \<noteq> B"   and "B Out A P" shows "P InAngle A B C"
finite_rtrancl_Image: fixes R :: "('d \<times> 'd) set"   and A :: "'d set" assumes "finite R"   and "finite A" shows "finite (R\<^sup>* `` A)"
dom_def: fixes S :: "nat \<Rightarrow> typ" shows "Type.dom S = {n. S n \<noteq> TVar n}"
inner_eq_zero_abs_BasisI: fixes y :: "'e"   and b :: "'e" assumes "\<bar>y\<bar> \<in> Basis"   and "b \<in> Basis"   and "\<bar>y\<bar> \<noteq> b" shows "y \<bullet> b = 0"
subst_notIn: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "x \<notin> Fvars \<phi>" shows "subst \<phi> t x = \<phi>"
is_rbt_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT_Impl.rbt" shows "ord.is_rbt less t = (inv1 t \<and> inv2 t \<and> color_of t = color.B \<and> ord.rbt_sorted less t)"
f_Exec_Stream_State_Idle_append_replicate_NoMsg_gr0_output: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'g message_af list"   and c :: "'d"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun xs c))" shows "map output_fun (f_Exec_Comp_Stream trans_fun (xs @ NoMsg\<^bsup>n\<^esup>) c) = map output_fun (f_Exec_Comp_Stream trans_fun xs c) @ NoMsg\<^bsup>n\<^esup>"
refl: fixes x :: "'a" assumes "x \<in> X" shows "x \<sqsubseteq> x"
that: fixes t :: "'a list"   and k :: "nat"   and m :: "nat" assumes "x = t \<^sup>@ k"   and "y = t \<^sup>@ m" shows "thesis"
X: fixes x :: "loc"   and s :: "loc \<Rightarrow> nat" shows "(X x, s) -|-> (s x, s)"
dimension_def: shows "finite_dimensional_vector_space.dimension {1::'a} = card {1::'a}"
lexordp_linorder: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "class.linorder (ord.lexordp_eq less) (ord.lexordp less)"
exp_boundaries: fixes x :: "float"   and prec :: "nat" shows "exp (real_of_float x) \<in> {real_of_float (lb_exp prec x)..real_of_float (ub_exp prec x)}"
Base_foo_member_of_Base: shows "tprg \<turnstile> (Base, mdecl Base_foo) member_of Base"
perm_set_def: fixes pi :: "('e \<times> 'e) list"   and X :: "'d set" shows "pi \<bullet> X = {pi \<bullet> x |x. x \<in> X}"
order_eq: fixes t :: "real" shows "order t g = order (complex_of_real t) pc"
seqE: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.seq g f" obtains "A_B.arr f" and "A_B.arr g" and "A_B.dom g = A_B.cod f"
assms_1: shows "(\<lambda>x. sum_upto (\<lambda>n. f n * sum_upto g (x / real n)) (sqrt x) - R x) \<in> O(b)"
the1_equality: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "\<exists>!x. P x"   and "P a" shows "(THE x. P x) = a"
rt: shows "returns_to {x \<in> S. x \<bullet> i - c = 0} x"
complete2: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure" assumes "A \<subseteq> B"   and "B \<in> null_sets (completion M)" shows "A \<in> null_sets (completion M)"
abs_eq: fixes xa :: "nat"   and x :: "nat \<Rightarrow> 'b" shows "Matrix.vec xa x = Abs_vec (xa, Matrix.mk_vec xa x)"
oInv_less: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal"   and z :: "ordinal" assumes "normal F"   and "F 0 \<le> a"   and "a < F z" shows "oInv F a < z"
extendFalsum: fixes S :: "'b sequent"   and \<Gamma> :: "'b form multiset"   and \<Delta> :: "'b form multiset" assumes "extend S ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "ff \<in># \<Gamma>"
poly_add_normalize_1: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "poly_add p1 p2 = poly_add (local.normalize p1) p2"
is_Write\<^sub>s\<^sub>b_simps_2: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "is_Write\<^sub>s\<^sub>b (Read\<^sub>s\<^sub>b volatile a t v) = False"
single_diff: fixes k :: "'c"   and a :: "'d"   and b :: "'d" shows "monomial (a - b) k = monomial a k - monomial b k"
trancl_Int_subset: fixes R :: "('b \<times> 'b) set"   and A :: "'b set" shows "(Restr R A)\<^sup>+ \<subseteq> Restr (R\<^sup>+) A"
neighbourhood_base_at_def: fixes x :: "'b"   and P :: "'b set \<Rightarrow> bool"   and X :: "'b topology" shows "neighbourhood_base_at x P X \<equiv> \<forall>W. openin X W \<and> x \<in> W \<longrightarrow> (\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W)"
nu_zero: shows "\<nu> (0::'a) = (0::'a)"
swap_cs_Trans_Snapshot: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isTrans ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"
j: shows "j < length (map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs)))"
comp_closed: fixes f :: "'a"   and g :: "'a" assumes "Arr f"   and "Arr g"   and "C.cod f = C.dom g" shows "Arr (g \<cdot>\<^sub>C f)"
measurable_If: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and M' :: "'e measure"   and g :: "'d \<Rightarrow> 'e"   and P :: "'d \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "g \<in> M \<rightarrow>\<^sub>M M'"   and "{x \<in> space M. P x} \<in> sets M" shows "(\<lambda>x. if P x then f x else g x) \<in> M \<rightarrow>\<^sub>M M'"
bin_rest_numeral_simps_5: fixes w :: "num" shows "numeral (num.Bit0 w) div 2 = numeral w"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "P TBase"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 \<rightarrow> ty2)"   and "\<And>ty. P ty \<Longrightarrow> P (T ty)" shows "P ty"
viterbi_ix_def: fixes \<O> :: "'c \<Rightarrow> 'd pmf"   and \<K> :: "'c \<Rightarrow> 'c pmf"   and state_list :: "'c list"   and os :: "'d list" shows "HMM3_defs.viterbi_ix \<O> \<K> state_list os = HMM3_defs.viterbi_ix_rec \<O> \<K> state_list (IArray os)"
eq_iff_5: fixes name :: "name"   and fm :: "fm"   and namea :: "name"   and fma :: "fm" shows "(SyntaxN.Ex name fm = SyntaxN.Ex namea fma) = ([[atom name]]lst. fm = [[atom namea]]lst. fma)"
Chn_triangle_eq_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and \<eta> :: "'b arrow_of_spans_data"   and \<epsilon> :: "'b arrow_of_spans_data" assumes "adjunction_data_in_span_bicategory C prj0 prj1 f g \<eta> \<epsilon>" shows "Chn (span_vertical_category.vcomp C (bicategory.runit (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_bicategory.unit C prj0 prj1) (span_vertical_category.src C) g) (span_vertical_category.vcomp C (span_bicategory.hcomp C prj0 prj1 g \<epsilon>) (span_vertical_category.vcomp C (span_bicategory.assoc C prj0 prj1 g f g) (span_vertical_category.vcomp C (span_bicategory.hcomp C prj0 prj1 \<eta> g) (category.inv (span_vertical_category.vcomp C) (bicategory.lunit (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_bicategory.unit C prj0 prj1) (span_vertical_category.src C) (span_vertical_category.trg C) g)))))) = C (prj1 (Leg0 (Dom g)) (Leg1 (Dom f))) (C (Chn \<eta>) (Leg1 (Dom g)))"
n_val_surj: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "\<exists>x\<in>carrier K. n_val K v x = 1"
is_derivation_step: fixes u :: "symbol list"   and N :: "symbol"   and v :: "symbol list"   and \<alpha> :: "symbol list" assumes "is_derivation (u @ [N] @ v)"   and "(N, \<alpha>) \<in> \<RR>" shows "is_derivation (u @ \<alpha> @ v)"
unicoherentD: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "unicoherent U"   and "connected S"   and "connected T"   and "U = S \<union> T"   and "closedin (top_of_set U) S"   and "closedin (top_of_set U) T" shows "connected (S \<inter> T)"
get_root_node_def: fixes ptr :: "(_) object_ptr" shows "get_root_node ptr = Heap_Error_Monad.bind (get_ancestors ptr) (\<lambda>ancestors. return (last ancestors))"
l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root_safe get_shadow_root_safe_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs"
length_transpose_sorted: fixes xs :: "'g list list" assumes "sorted (rev (map length xs))" shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
content_primitive_part: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "content (primitive_part p) = (1::'b)"
not_sublist_len: fixes ys :: "'a list"   and xs :: "'a list" assumes "|ys| < |xs|" shows "\<not> is_sublist xs ys"
ov: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and p :: "'b"   and q :: "'b" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> ov) = (\<exists>k l u v t. (meets k p \<and> meets p u \<and> meets u v) \<and> (meets k l \<and> meets l q \<and> meets q v) \<and> meets l t \<and> meets t u)"
card_pick_le: fixes n :: "nat"   and S :: "nat set" assumes "n < card S" shows "card {a \<in> S. a < pick S n} = n"
cos_of_real_pi: shows "cos (of_real pi) = - (1::'a)"
decomp_assignment_rhs_empty: fixes t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term list" assumes "Ana t = (K, M)" shows "assignment_rhs\<^sub>s\<^sub>t (decomp t) = {}"
prob_1_bell_fst: fixes v :: "complex mat" assumes "v = |\<beta>\<^sub>0\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>0\<^sub>1\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>1\<rangle>" shows "prob1 2 v 0 = 1 / 2"
res_to_cong_simps_2: fixes m :: "int"   and x :: "int"   and y :: "int" assumes "residues m" shows "x mod m \<otimes>\<^bsub>residue_ring m\<^esub> (y mod m) = x * y mod m"
taut_iff_eq: fixes P :: "'a upred"   and Q :: "'a upred" shows "`P \<Leftrightarrow> Q` = (P = Q)"
submod: fixes K :: "('e, 'f) ring_scheme"   and W :: "'g set"   and V :: "('e, 'g, 'h) module_scheme" assumes "subspace K W V" shows "LinearCombinations.submodule K W V"
EXJ: shows "\<exists>j\<ge>i. n \<in> acc (fst (r j))"
As0'_As: shows "As0' \<cdot>al \<eta>0' = As"
SUP_apply: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and A :: "'c set"   and x :: "'b" shows "Sup (f ` A) x = (SUP y\<in>A. f y x)"
seqI: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and g :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.arr f"   and "Cop_S.arr g"   and "Cop_S.dom g = Cop_S.cod f" shows "Cop_S.seq g f"
transpose_matrix_vector: fixes A :: "(('d, 'e) vec, 'f) vec"   and x :: "('d, 'f) vec" shows "Finite_Cartesian_Product.transpose A *v x = x v* A"
r_result1_bivalent_phi: fixes i :: "nat"   and x :: "nat"   and v :: "nat"   and t :: "nat" assumes "\<phi> i x \<down>= v" shows "eval r_result1 [t, i, x] \<down>= Suc v \<or> eval r_result1 [t, i, x] \<down>= 0"
assms: fixes h :: "heap \<times> nat set" assumes "h \<Turnstile> P" shows "<P> c <Q>"
assms_1: shows "(_Entry_) -as\<rightarrow>* n"
fref_param1: fixes R :: "('a \<times> 'c) set"   and S :: "('b \<times> 'd) set" shows "R \<rightarrow> S = R \<rightarrow>\<^sub>f S"
map_id_on_UNIV: shows "map_id_on UNIV = Some"
refl_onI: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" assumes "r \<subseteq> A \<times> A"   and "\<And>x. x \<in> A \<Longrightarrow> (x, x) \<in> r" shows "refl_on A r"
rep_props_3: fixes r :: "'a" assumes "ide r" shows "r \<cong> Span.VVV.iso_class_rep \<lbrakk>r\<rbrakk>"
simps_13: fixes b :: "bool"   and C :: "qtname"   and sig :: "sig" shows "assigns_if b (Methd C sig) = {}"
min_set_finite: fixes d :: "'a \<Rightarrow> nat"   and seq :: "nat \<Rightarrow> 't \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "range seq \<subseteq> dgrad_sig_set' (length fs) d"   and "0 \<notin> rep_list ` range seq"   and "\<And>i j. i < j \<Longrightarrow> lt (seq i) \<prec>\<^sub>t lt (seq j)" shows "finite {i. \<not> (\<exists>j<i. lt (seq j) adds\<^sub>t lt (seq i) \<and> punit.lt (rep_list (seq j)) adds punit.lt (rep_list (seq i)))}"
cmim: shows "m\<inverse> O m \<subseteq> s \<union> s\<inverse> \<union> e"
order_refl: fixes x :: "'b \<times> 'c" shows "less_eq_prod x x"
iT_Minus_iprev: fixes k :: "nat"   and I :: "nat set"   and n :: "nat" assumes "k \<ominus> I \<noteq> {}"   and "n \<le> k" shows "iprev (k - n) (k \<ominus> I) = k - inext n (I \<down>\<le> k)"
is_interval_existence_ivl: fixes x0 :: "'a" shows "is_interval (existence_ivl t0 x0)"
intro: fixes r1 :: "('d, 'a, 'b, 'c, 'e, 'f) semantics"   and r2 :: "('d, 'a, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'g \<Rightarrow> bool" assumes "\<And>t. bisimulation (r1 t) (r2 t) (bisim t) (ta_bisim bisim)"   and "\<And>t x1 m1 x2 m2. bisim t (x1, m1) (x2, m2) \<Longrightarrow> final1 x1 = final2 x2"   and "\<And>t' x m1 xx m2 t x1 x2 ta1 x1' m1' ta2 x2' m2'. \<lbrakk>bisim t' (x, m1) (xx, m2); bisim t (x1, m1) (x2, m2); r1 t (x1, m1) ta1 (x1', m1'); r2 t (x2, m2) ta2 (x2', m2'); bisim t (x1', m1') (x2', m2'); ta_bisim bisim ta1 ta2\<rbrakk> \<Longrightarrow> bisim t' (x, m1') (xx, m2')"   and "(\<exists>x1. final1 x1) = (\<exists>x2. final2 x2)" shows "FWbisimulation_axioms final1 r1 final2 r2 bisim"
sep_conj_def: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" shows "P \<and>* Q \<equiv> \<lambda>h. \<exists>x y. x ## y \<and> h = x + y \<and> P x \<and> Q y"
lt_eq_min_term_monomial: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "lpp p = punit.min_term" shows "monomial (lcf p) punit.min_term = p"
finprod_comp: fixes g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and f :: "'d \<Rightarrow> 'a" assumes "inj_on g A"   and "(f \<circ> g) ` A \<subseteq> carrier G" shows "finprod G f (g ` A) = finprod G (f \<circ> g) A"
rep_comp: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>C.seq g f" shows "\<F>C.rep (g \<cdot> f) = \<^bold>\<parallel>\<F>C.rep g \<^bold>\<cdot> \<F>C.rep f\<^bold>\<parallel>"
disjoints_spec_def: fixes X :: "'c set"   and Y :: "'c set" shows "disjoints_spec X Y = ivls_of_sets X \<bind> (\<lambda>Xi. inter_overappr (Xi ::: clw_rel lvivl_rel) (Y ::: clw_rel lvivl_rel) \<bind> (\<lambda>IS. RETURN (is_empty IS)))"
ex_in_conv: fixes A :: "'g set" shows "(\<exists>x. x \<in> A) = (A \<noteq> {})"
domA_def: fixes h :: "('b \<times> 'c) list" shows "domA h = fst ` set h"
resChainFreshSimps_3: fixes xvec :: "name list"   and P :: "('d, 'e, 'f) psi" shows "xvec \<sharp>* (\<lparr>\<nu>*xvec\<rparr>P)"
hyps_3: fixes ss :: "lam list" assumes "ys [\<rightarrow>\<^sub>\<beta>]\<^sub>1 ss" shows "xs [\<rightarrow>\<^sub>s] ss"
trace_callee_eqI: fixes A :: "'d set"   and callee1 :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'f) spmf"   and p :: "'f spmf"   and callee2 :: "'g \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'g) spmf"   and q :: "'g spmf" assumes "\<And>xs x. \<lbrakk>set xs \<subseteq> A \<times> UNIV; x \<in> A\<rbrakk> \<Longrightarrow> trace_callee callee1 p xs x = trace_callee callee2 q xs x" shows "trace_callee_eq callee1 callee2 A p q"
empty_eq_map_add_iff: fixes f :: "'g \<Rightarrow> 'h option"   and g :: "'g \<Rightarrow> 'h option" shows "(Map.empty = f ++ g) = (f = Map.empty \<and> g = Map.empty)"
disc_1: fixes x11 :: "'a"   and x12 :: "'a process" shows "isAction (Action x11 x12)"
hyp: fixes cs :: "('a, 'b, 'c) com list" shows "\<Gamma>\<turnstile>cs,css \<Down> Stuck"
surj_pair: fixes p :: "'c \<times> 'd" shows "\<exists>x y. p = (x, y)"
subseq_reduce: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and s :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "Diagonal_Subsequence.subseqs P"   and "strict_mono s" shows "strict_mono (subseqs.reduce P s n)"
omega_sum_length_leD: fixes ms :: "nat list"   and ns :: "nat list" assumes "omega_sum ms \<le> omega_sum ns"   and "normal ms" shows "length ms \<le> length ns"
N_def: fixes q :: "real"   and n :: "int" assumes "fbnorm q" shows "fbnorm.N q n = (\<Sum>k. if k = 0 \<or> int k dvd n then 0 else 1 / q ^ k)"
enc_atom_dec: fixes s :: "('a \<times> bool list) stream"   and n :: "nat"   and FO :: "nat set"   and a :: "'a"   and p :: "nat" assumes "\<forall>p. length (snd (s !! p)) = n"   and "\<forall>i\<in>FO. i < n \<longrightarrow> (\<exists>!p. snd (s !! p) ! i)"   and "a = fst (s !! p)" shows "enc_atom (stream_dec n FO s) p a = s !! p"
test_domain: fixes x :: "'a" shows "(x = d x) = (x \<le> 1')"
a_r_coset_def: fixes G :: "('d, 'e) ring_scheme" shows "(+>\<^bsub>G\<^esub>) = (#>\<^bsub>add_monoid G\<^esub>)"
seq_lexord_irreflexive: fixes R :: "('b \<times> 'b) set"   and xs :: "'b seq" assumes "\<forall>x. (x, x) \<notin> R" shows "(xs, xs) \<notin> seq_lexord R"
min_ot_def: shows "min_ot \<equiv> Min (pre_oldest_tokens q n)"
ntr_mon_increasing_s: fixes c :: "'n list multiset"   and ee :: "('p, 'ba) label list"   and c' :: "'n list multiset" assumes "(c, ee, c') \<in> ntr fg" shows "mon_c fg c \<subseteq> mon_c fg c'"
distinct_map: fixes f :: "'g \<Rightarrow> 'f"   and xs :: "'g list" shows "distinct (map f xs) = (distinct xs \<and> inj_on f (set xs))"
elims_2: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b tree" assumes "bst_wrt x xa" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" where "x = P__" and "xa = \<langle>\<rangle>"   | P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and l :: "'b tree" and a :: "'b" and r :: "'b tree" where "x = P" and "xa = \<langle>l, a, r\<rangle>" and "(\<forall>x\<in>set_tree l. P x a) \<and> Ball (set_tree r) (P a) \<and> bst_wrt P l \<and> bst_wrt P r"
disjE_realizer: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and x :: "'a + 'b"   and R :: "'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "case x of Inl p \<Rightarrow> P p | Inr q \<Rightarrow> Q q"   and "\<And>p. P p \<Longrightarrow> R (f p)"   and "\<And>q. Q q \<Longrightarrow> R (g q)" shows "R (case x of Inl p \<Rightarrow> f p | Inr q \<Rightarrow> g q)"
zero_in_cone: fixes hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set" shows "0 \<in> cone hU"
reduced_GB_is_auto_reduced_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "is_auto_reduced (reduced_GB F)"
OrdI: fixes x :: "\<kappa>"   and v :: "i" assumes "\<exists>o\<^sub>1 y. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<omega>\<nu> y" shows "[\<lparr>O!,x\<rparr> in v]"
non_null_1: fixes i :: "'b" assumes "i \<in> S" shows "0 \<le> a i"