hcju/isabelleps huggingface dataset:
inv_baldR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invh l"   and "invh r"   and "bheight l = bheight r + 1"   and "invc l"   and "invc2 r" shows "invh (baldR l a r) \<and> bheight (baldR l a r) = bheight l \<and> invc2 (baldR l a r) \<and> (color l = Black \<longrightarrow> invc (baldR l a r))"
inorder_del: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and x :: "'c" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (RBT_Map.del x t) = AList_Upd_Del.del_list x (Tree2.inorder t)"
summable'': shows "(\<lambda>(x, m, n). f m * g n / nat_power x s) abs_summable_on (SIGMA x:UNIV. {(m, n). m * n = x})"
dependent_explicit: fixes s :: "'b set" shows "dependent s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v *s v) = (0::'b) \<and> (\<exists>v\<in>t. u v \<noteq> (0::'a)))"
select_convs_1: fixes init :: "'a"   and prod :: "'a \<Rightarrow> ('b + 'a) list fset"   and more :: "'c" shows "init \<lparr>init = init, prod = prod, \<dots> = more\<rparr> = init"
zero_closed: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "FinGroupRepresentation G smult V" shows "(0::'c) \<in> V"
leadsTo_Union_Int: fixes S :: "'c set set"   and F :: "'c program"   and C :: "'c set"   and B :: "'c set" assumes "\<And>A. A \<in> S \<Longrightarrow> F \<in> A \<inter> C \<longmapsto> B" shows "F \<in> \<Union> S \<inter> C \<longmapsto> B"
bres_canc2: fixes y :: "'b \<times> 'c"   and x :: "'b \<times> 'c" shows "less_eq_prod y (prod.bres x (times_prod' x y))"
obj_split: fixes obj :: "obj" shows "\<exists>t vs. obj = \<lparr>tag = t, values = vs\<rparr>"
nth_find_first: fixes a :: "'b"   and xs :: "'b list" assumes "a \<in> set xs" shows "xs ! find_first a xs = a"
is_aligned_AND_less_0: fixes u :: "'b word"   and n :: "nat"   and v :: "'b word" assumes "u AND mask n = 0"   and "v < 2 ^ n" shows "u AND v = 0"
max_ge_y: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a" assumes "class.non_strict_order less_eq" shows "less_eq y (ord.max less_eq x y)"
simps_1: fixes u :: "'a" shows "awalk_verts u [] = [u]"
bind_altc1: fixes C :: "'c cset"   and g :: "'c \<Rightarrow> 'm"   and f :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind (altc C g) f = altc C (\<lambda>c. bind (g c) f)"
neg_img: shows "\<forall>p\<in>path_image \<gamma>. Re p \<le> Re z"
arity_head_ge: fixes s :: "('s, 'v) tm" shows "arity s \<le> arity_hd (head s)"
fds_truncate_truncate: fixes m :: "nat"   and n :: "nat"   and f :: "'b fds" shows "fds_truncate m (fds_truncate n f) = fds_truncate (min m n) f"
EF_lem2d: fixes x :: "'a"   and f :: "'a set" assumes "x \<notin> EF f" shows "x \<notin> f"
implmt_dynimethd_access: fixes G :: "prog"   and I :: "qtname"   and sig :: "sig"   and dynC :: "qtname"   and accC :: "qtname" assumes "DeclConcepts.imethds G I sig \<noteq> {}"   and "G\<turnstile>dynC\<leadsto>I"   and "is_iface G I"   and "wf_prog G" shows "\<exists>dynM. dynimethd G I dynC sig = Some dynM \<and> G \<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
Image_subset_eq: fixes r :: "('f \<times> 'e) set"   and A :: "'f set"   and B :: "'e set" shows "(r `` A \<subseteq> B) = (A \<subseteq> - r\<inverse> `` (- B))"
vars2': shows "free_vars (\<langle>e2\<rangle>\<^sub>c) \<subseteq> V \<union> V'"
\<psi>_asymptotics: shows "\<psi> \<sim>[at_top] (\<lambda>x. x)"
adjuster_carrier: fixes w :: "'a vec"   and n :: "nat"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "adjuster n w us \<in> carrier_vec n"
Theorem_11_Pd'_tax: shows "\<exists>y\<in>TruePref_tax.cop ds. (x, y) \<in> Pd'_tax (Xd x)"
Seq: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C\<^sub>1 :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C\<^sub>2 :: "com"   and R :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>' {P} C\<^sub>1 { Q}"   and "\<turnstile>\<^sub>3\<^sub>' {Q} C\<^sub>2 { R}" shows "\<turnstile>\<^sub>3\<^sub>' {P} C\<^sub>1;; C\<^sub>2 { R}"
aff_dim_affine_hull2: fixes S :: "'a set"   and T :: "'a set" assumes "affine hull S = affine hull T" shows "aff_dim S = aff_dim T"
assms_1: shows "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"
concls_of_inferences_between_eq_Bin_ord_resolve_rename: fixes Q :: "'a literal multiset set"   and C :: "'a literal multiset" shows "concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) Q C) = Bin_ord_resolve_rename C C \<union> (\<Union>D\<in>Q. Bin_ord_resolve_rename C D \<union> Bin_ord_resolve_rename D C)"
SUP_set_fold: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` set xs) = fold (sup \<circ> f) xs bot"
MATCH1_matchrel_conv: fixes x :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)"   and y :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)" shows "MATCH1 x y = ((x, y) \<in> matchrel)"
finite_dom_fmap: fixes F :: "'g \<Rightarrow> 'h option" shows "finite (dom F)"
le_cases: fixes x :: "'a word"   and y :: "'a word" obtains "x \<le>s y"   | "y \<le>s x"
snga_prec: shows "precise (\<lambda>x p. p \<mapsto>\<^sub>a x)"
find_slot_def: shows "find_slot \<equiv> l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_find_slot Shadow_DOM.get_parent get_shadow_root get_mode get_attribute get_tag_name Shadow_DOM.first_in_tree_order"
bintr_lt2p: fixes n :: "nat"   and w :: "int" shows "take_bit n w < 2 ^ n"
2: fixes y :: "'b" assumes "y \<in> indets p" shows "f (g y) = y"
f_Exec_eq_f_Exec_Stream_last2: fixes xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "xs \<noteq> []" shows "f_Exec_Comp trans_fun xs c = last (f_Exec_Comp_Stream trans_fun xs c)"
scaleR1_nonzero_eq: fixes e :: "real"   and a :: "'a"   and b :: "'a" assumes "e \<noteq> 0" shows "ccw (0::'a) (e *\<^sub>R a) b = (if 0 < e then ccw (0::'a) a b else ccw (0::'a) b a)"
AE_bind: fixes N :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and B :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra B"   and "Measurable.pred B P" shows "almost_everywhere (M \<bind> N) P = (AE x in M. almost_everywhere (N x) P)"
mantissa_neg_iff: fixes x :: "Float.float" shows "(mantissa x < 0) = (x < 0)"
pderiv_exp_prod: shows "pderiv p = (\<Prod>(a, i)\<in>as. a ^ i) * (\<Sum>(a, i)\<in>as. (\<Prod>(b, j)\<in>as - {(a, i)}. b) * smult (of_nat (Suc i)) (pderiv a))"
minus_add_minus_vec: fixes u :: "'b vec"   and n :: "nat"   and v :: "'b vec"   and w :: "'b vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "u - (v + w) = u - v - w"
assms_1: shows "\<forall>t2<t. a t2 = []"
takeWhile_foo: fixes x :: "'b"   and ys :: "'b list"   and zs :: "'b list" assumes "x \<notin> set ys" shows "ys = takeWhile (\<lambda>y. y \<noteq> x) (ys @ x # zs)"
ra0: shows "r (Suc k) (a $ 0) ^ Suc k = a $ 0"
division_of_closed: fixes s :: "'c set set"   and i :: "'c set" assumes "s division_of i" shows "closed i"
rb_spp_aux_Cons: fixes bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and ss :: "'t list"   and p :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat) list"   and z :: "nat" shows "rb_spp_aux ((bs, ss, p # ps), z) = rb_spp_aux (rb_spp_body ((bs, ss, p # ps), z))"
inconsist_converg_eq: fixes i :: "nat"   and e :: "nat" assumes "inconsist i e \<down>= e_length e" shows "\<forall>x<e_length e. \<phi> i x \<down>= e_nth e x"
invar_imp_correct: fixes G :: "('a, 'b) graph"   and v0 :: "'a"   and res :: "'a \<Rightarrow> ('a \<times> 'b \<times> 'a) list option" assumes "Dijkstra G v0"   and "Dijkstra.dinvar G v0 ({}, res)" shows "weighted_graph.is_shortest_path_map G v0 res"
Remove_def: shows "FIG.Remove = (\<lambda>(A, A'). list_select FIG_alphabet.x A \<in> idS A \<and> idS A' = idS A - {list_select FIG_alphabet.x A})"
lappend_code_2: fixes xa :: "'a"   and x :: "'a llist"   and ys :: "'a llist" shows "xa #\<^sub>l x @\<^sub>l ys = xa #\<^sub>l (x @\<^sub>l ys)"
comp_in_homI': fixes f :: "(bool, bool list) concrete_category.arr"   and g :: "(bool, bool list) concrete_category.arr"   and a :: "(bool, bool list) concrete_category.arr"   and c :: "(bool, bool list) concrete_category.arr" assumes "J.arr f"   and "J.arr g"   and "J.dom f = a"   and "J.cod g = c"   and "J.dom g = J.cod f" shows "\<guillemotleft>g \<cdot>\<^sub>J f : a \<rightarrow>\<^sub>J c\<guillemotright>"
bounded: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
inv_inv: fixes x :: "'a" assumes "x \<in> carrier H'" shows "inv\<^bsub>H'\<^esub> (inv\<^bsub>H'\<^esub> x) = x"
linear_scale_self: fixes c :: "complex" shows "clinear ((*\<^sub>C) c)"
cardSumCommute: fixes XX :: "'c set set"   and X :: "'c set" assumes "XX partitions X"   and "finite X"   and "finite XX" shows "card (\<Union> XX) = sum card XX"
1: shows "bprv (NN \<langle>\<phi>\<rangle> \<langle>neg \<phi>\<rangle>)"
in_insert_fset_iff: fixes y :: "'b"   and S :: "'b fset"   and x :: "'b" shows "(x |\<in>| insert_fset y S) = (x = y \<or> x |\<in>| S)"
qml_4: shows "[[\<^bold>\<diamond>(\<^bold>\<exists>x. \<lparr>E!,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>) \<^bold>& \<^bold>\<diamond>\<^bold>\<not>(\<^bold>\<exists>x. \<lparr>E!,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<diamond>\<^bold>\<not>\<lparr>E!,x\<^sup>P\<rparr>)]]"
zin: shows "z \<in> inside (path_image ((linepath (a + complex_of_real kde) (a + complex_of_real e) +++ p +++ linepath (a - complex_of_real d) (a - complex_of_real kde)) +++ linepath (a - complex_of_real kde) (a + complex_of_real kde)))"
K_def: shows "K = {i \<in> {1..}. (T ^^ i) x \<in> A}"
2: shows "common_sudiv_exists (two_chain_vertical_boundary {rot_diamond_cube}) (boundary diamond_cube)"
measurable_count_space: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "f \<in> count_space A \<rightarrow>\<^sub>M count_space UNIV"
solves_ode_congI: fixes x :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set"   and y :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "real set"   and Y :: "'b set" assumes "(x solves_ode f) T X"   and "\<And>t. t \<in> T \<Longrightarrow> x t = y t"   and "\<And>t. t \<in> T \<Longrightarrow> f t (x t) = g t (x t)"   and "T = S"   and "X = Y" shows "(y solves_ode g) S Y"
nchotomy: fixes y :: "ValD" shows "y = \<bottom> \<or> (\<exists>cfun. y = ValF\<cdot>cfun) \<or> y = ValTT \<or> y = ValFF \<or> (\<exists>nat. y = ValN\<cdot>nat)"
i_Exec_Stream_expand_aggregate_map_drop: fixes k :: "nat"   and n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k" shows "i_aggregate (f \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) k ag \<Up> n = i_aggregate (f \<circ> i_Exec_Comp_Stream trans_fun (input \<Up> n \<odot> k) (f_Exec_Comp trans_fun (input \<Down> n \<odot> k) c)) k ag"
backward_sim_execs: fixes f :: "'b \<Rightarrow> 'c set"   and B :: "('b, 'd) ioa"   and A :: "('c, 'd) ioa"   and e_B :: "'b \<times> ('d \<times> 'b) list" assumes "is_backward_sim f B A"   and "is_exec_of B e_B" shows "\<exists>e_A. is_exec_of A e_A \<and> trace (ioa.asig A) e_A = trace (ioa.asig A) e_B"
nop_vdomain: shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"
set_mode_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_mode_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.type_wf h = ShadowRootClass.type_wf h'"
mask_out_first_mask_some: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word"   and m :: "nat" assumes "x AND NOT (mask n) = y"   and "n \<le> m" shows "x AND NOT (mask m) = y AND NOT (mask m)"
sup_left_commute: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
Pxy: shows "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy"
wlsAbs_freshAbs_nchotomy: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and x :: "'var" assumes "wlsAbs (xs, s) A"   and "freshAbs xs x A" shows "\<exists>X. wls s X \<and> A = Abs xs x X"
span_li_not_depend_1: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule R N M" shows "module.span R (md N) S = span S"
union_single_eq_diff: fixes x :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "add_mset x M = N" shows "M = remove1_mset x N"
add_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.add divide times one plus minus uminus a p\<^sub>1 p\<^sub>2 = (case p\<^sub>1 of Infinity \<Rightarrow> p\<^sub>2 | Point x\<^sub>1 y\<^sub>1 \<Rightarrow> case p\<^sub>2 of Infinity \<Rightarrow> p\<^sub>1 | Point x\<^sub>2 y\<^sub>2 \<Rightarrow> if x\<^sub>1 = x\<^sub>2 then if y\<^sub>1 = uminus y\<^sub>2 then Infinity else let l = divide (plus (times (numeral.numeral one plus (num.Bit1 num.One)) (power.power one times x\<^sub>1 2)) a) (times (numeral.numeral one plus (num.Bit0 num.One)) y\<^sub>1); x\<^sub>3 = minus (power.power one times l 2) (times (numeral.numeral one plus (num.Bit0 num.One)) x\<^sub>1) in Point x\<^sub>3 (minus (uminus y\<^sub>1) (times l (minus x\<^sub>3 x\<^sub>1))) else let l = divide (minus y\<^sub>2 y\<^sub>1) (minus x\<^sub>2 x\<^sub>1); x\<^sub>3 = minus (minus (power.power one times l 2) x\<^sub>1) x\<^sub>2 in Point x\<^sub>3 (minus (uminus y\<^sub>1) (times l (minus x\<^sub>3 x\<^sub>1))))"
mask_AND_NOT_mask: fixes w :: "'a word"   and n :: "nat" shows "(w AND NOT (mask n)) AND mask n = 0"
A_def: shows "A = \<Union> (arith_prog_fb 0 ` {p. prime p})"
simps_10: fixes i :: "int"   and a :: "Cooper.num" shows "bound0 (NDvd i a) = numbound0 a"
IH: assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>2"   and "subst_domain \<sigma>1 \<inter> subst_domain \<theta>2 = {}"   and "subst_domain \<sigma>1 \<inter> range_vars \<theta>2 = {}" shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (\<sigma>1 \<circ>\<^sub>s \<theta>2)"
strict_coboundedI1: fixes a :: "'c DBMEntry"   and c :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<prec> c" shows "linordered_monoid.min a b \<prec> c"
top_on_Step: fixes X :: "char list set"   and f :: "char list \<Rightarrow> aexp \<Rightarrow> (char list \<Rightarrow> 'av) option \<Rightarrow> (char list \<Rightarrow> 'av) option"   and g :: "bexp \<Rightarrow> (char list \<Rightarrow> 'av) option \<Rightarrow> (char list \<Rightarrow> 'av) option"   and C :: "(char list \<Rightarrow> 'av) option acom"   and S :: "(char list \<Rightarrow> 'av) option" assumes "\<And>x e S. \<lbrakk>top_on\<^sub>o S X; x \<notin> X; vars e \<subseteq> - X\<rbrakk> \<Longrightarrow> top_on\<^sub>o (f x e S) X"   and "\<And>b S. \<lbrakk>top_on\<^sub>o S X; vars b \<subseteq> - X\<rbrakk> \<Longrightarrow> top_on\<^sub>o (g b S) X"   and "vars C \<subseteq> - X"   and "top_on\<^sub>o S X"   and "top_on\<^sub>c C X" shows "top_on\<^sub>c (Step f g S C) X"
main: shows "complex_roots_complex [:- a, 1:] = [a]"
assms: shows "\<R>\<^sub>\<circ> s \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
lift_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'m1" shows "m1.lift f x = bind1 x (\<lambda>x. return1 (f x))"
valid_graph_def: fixes G :: "('a, 'b) graph" shows "valid_graph G \<equiv> fst ` edges G \<subseteq> nodes G \<and> snd ` snd ` edges G \<subseteq> nodes G"
axioms_2: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" assumes "Height_Bounded_Labeling c s t f l" shows "Height_Bounded_Labeling_axioms c l"
simps_6: shows "opt_MatchAny_match_expr_once (MatchNot MatchAny) = MatchNot (opt_MatchAny_match_expr_once MatchAny)"
proots_card_sphere_axis_eq: fixes p :: "complex poly" assumes "p \<noteq> 0" shows "card (proots_within p (sphere 0 1 - {- 1})) = card (proots_within (fcompose p [:\<i>, - 1:] [:\<i>, 1:]) {x. 0 = Im x})"
Inf_utrans_greatest: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "\<forall>u\<in>S. le_utrans t u" shows "le_utrans t (Inf_utrans S)"
distinct_93: fixes x4 :: "val"   and x121 :: "expr"   and x122 :: "expr" shows "Val x4 \<noteq> x121;; x122"
axioms_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "functor A B F" shows "category B"
mf: assumes "A $$ (k, l) \<noteq> (0::'a)" shows "A $$ (l, l) = q' * g \<and> A $$ (k, l) * q' = A $$ (l, l) * p'"
ceiling_of_nat: fixes n :: "nat" shows "\<lceil>of_nat n\<rceil> = int n"
induct: fixes x1 :: "('fun, 'var) strand_step list"   and x2 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and x3 :: "('fun, 'var) strand_step list"   and x4 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and P :: "('fun, 'var) strand_step list \<Rightarrow> ('var \<Rightarrow> ('fun, 'var) Term.term) \<Rightarrow> ('fun, 'var) strand_step list \<Rightarrow> ('var \<Rightarrow> ('fun, 'var) Term.term) \<Rightarrow> bool" assumes "(x1, x2) \<leadsto> (x3, x4)"   and "\<And>S T f S' \<theta>. \<lbrakk>simple S; length T = arity f; public f\<rbrakk> \<Longrightarrow> P (S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S') \<theta> (S @ map Send T @ S') \<theta>"   and "\<And>S f T' \<delta> T S' \<theta>. \<lbrakk>simple S; Fun f T' \<in> ik\<^sub>s\<^sub>t S; Some \<delta> = mgu (Fun f T) (Fun f T')\<rbrakk> \<Longrightarrow> P (S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S') \<theta> (S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>) (\<theta> \<circ>\<^sub>s \<delta>)"   and "\<And>S \<delta> t t' uu_ S' \<theta>. \<lbrakk>simple S; Some \<delta> = mgu t t'\<rbrakk> \<Longrightarrow> P (S @ \<langle>uu_: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S') \<theta> (S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>) (\<theta> \<circ>\<^sub>s \<delta>)" shows "P x1 x2 x3 x4"
indep_sets_sigma: fixes F :: "'b \<Rightarrow> 'a set set"   and I :: "'b set" assumes "indep_sets F I"   and "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I"
regions_closed'_spec: fixes R :: "('c \<Rightarrow> real) set"   and v :: "'c \<Rightarrow> real"   and t :: "real" assumes "R \<in> \<R>"   and "v \<in> R"   and "0 \<le> t" shows "v \<oplus> t \<in> [v \<oplus> t]\<^sub>\<R>"
upper: shows "space M \<subseteq> \<Union> (f ` s)"
of_real_poly_map_mult: fixes p :: "real poly"   and q :: "real poly" shows "map_poly of_real (p * q) = map_poly of_real p * map_poly of_real q"
abs_eq: fixes xa :: "((real, 'a) vec, 'a) vec"   and x :: "((real, 'a) vec, 'a) vec" shows "to_mtx xa - to_mtx x = to_mtx (xa - x)"
exhaust: fixes y :: "'b trie" obtains x1 :: "bool" and x2 :: "'b \<Rightarrow> 'b trie option" where "y = Nd x1 x2"
Suc: shows "(\<forall>n. (\<forall>i<n. inpt_st (\<lambda>a. case a of (u, v) \<Rightarrow> \<lambda>a. case a of (u', v') \<Rightarrow> r u u' OO r' v v') (u i) (u (Suc i)) (x i)) \<longrightarrow> inpt_st r (fst (u n)) (fst (u (Suc n))) (x n) \<and> (\<forall>y. r (fst (u n)) (fst (u (Suc n))) (x n) y \<longrightarrow> inpt_st r' (snd (u n)) (snd (u (Suc n))) y)) \<longrightarrow> (\<forall>i<n. inpt_st r ((fst \<circ> u) i) ((fst \<circ> u) (Suc i)) (x i) \<and> (\<forall>y. r (fst (u i)) (fst (u (Suc i))) (x i) y \<longrightarrow> inpt_st r' (snd (u i)) (snd (u (Suc i))) y))"
prems: shows "a \<in> read_only_reads \<O> (x # xs)"
step: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto>\<^sup>^<(X \<union> weakBisim)> Q \<and> ((Q, P) \<in> X \<or> Q \<approx> P)"
abs_eq: fixes x :: "complex \<times> complex \<times> complex \<times> complex" assumes "eq_onp (\<lambda>M. mat_det M \<noteq> 0) x x" shows "unit_disc_fix_mmat (Abs_moebius_mat x) = unit_disc_fix_cmat x"
left_idem: fixes a :: "'a"   and b :: "'a" shows "a AND a AND b = a AND b"
finfun_rec_const: fixes g :: "'c \<Rightarrow> 'b"   and c :: "'c" shows "finfun_rec (\<lambda>b. finfun_const (g b)) (\<lambda>a b c. finfun_update c a (g b)) (finfun_const c) = finfun_const (g c)"
fbox_arden_whilei: fixes t :: "'a"   and x :: "'a"   and i :: "'a"   and q :: "'a" assumes "\<nabla>(d t \<cdot> x) = (zero_class.zero::'a)"   and "d i = (d t + d q) \<cdot>  |d t \<cdot> x] i" shows "d i =  |while t inv i do x od] q"
sinks_idem: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list" shows "sinks I D u (ipurge_tr I D u xs) = {}"
in_arcs_eq: fixes w :: "'a" shows "in_arcs (del_vert u) w = edge_rev M ` out_arcs (del_vert u) w"
pt_prod_inst: assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)" shows "pt TYPE('a \<times> 'c) TYPE('b)"
post_meas1_index_0_alice_state: fixes \<phi> :: "complex Matrix.mat" assumes "state 1 \<phi>" shows "state 3 (post_meas1 3 (alice \<phi>) 0)"
top_right_mult_increasing: fixes x :: "'a" shows "x \<le> x \<sqinter> top"
bij_inv_2: fixes x :: "'a" assumes "is_bij x" shows "x\<^sup>\<smile> ; x = 1'"
rtsem_Max_simps: fixes \<theta>\<^sub>1 :: "trm"   and \<theta>\<^sub>2 :: "trm"   and \<nu> :: "fin_string \<Rightarrow> real"   and r :: "real" shows "([trm.Max \<theta>\<^sub>1 \<theta>\<^sub>2]\<nu> \<down> r) = (\<exists>r1 r2. r = max r1 r2 \<and> ([\<theta>\<^sub>1]\<nu> \<down> r1) \<and> ([\<theta>\<^sub>2]\<nu> \<down> r2))"
c_def: shows "c = sgnx (poly p) (at_right a)"
the_NF_def: fixes A :: "('b \<times> 'b) set"   and a :: "'b" shows "the_NF A a = (THE b. (a, b) \<in> A\<^sup>!)"
hsubsetCE: fixes A :: "hf"   and B :: "hf"   and c :: "hf" assumes "A \<le> B" obtains "c \<^bold>\<notin> A"   | "c \<^bold>\<in> B"
not_cd_impl_ipd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and i :: "nat" assumes "is_path \<pi>"   and "k < i"   and "\<not> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> i \<noteq> local.return" obtains j :: "nat" where "j \<in> {k..i}" and "\<pi> j = ipd (\<pi> k)"
\<phi>_def: shows "\<phi> \<equiv> \<lambda>(l, i) x. max 0 (1 - \<bar>x * 2 ^ (l + 1) - real_of_int i\<bar>)"
nb\<^sub>1: shows "cnf (\<Phi>\<^sub>I \<Pi>) = \<Union> {cnf (encode_state_variable 0 (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>I) v)) |v. v \<in> set (\<Pi>\<^sub>\<V>) \<and> (\<Pi>\<^sub>I) v \<noteq> None}"
scaleC_left: fixes r :: "complex"   and a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b" shows "cblinfun_apply (r *\<^sub>C a) b = r *\<^sub>C cblinfun_apply a b"
word_of_int_0: shows "word_of_int 0 = 0"