hcju/isabelleps huggingface dataset:
dup_boxE: fixes F :: "'a \<Rightarrow> bool"   and sigma :: "behavior" assumes "sigma \<Turnstile> \<box>F" obtains "sigma \<Turnstile> \<box>\<box>F"
subgraph_def: fixes G\<^sub>1 :: "('d, 'e) labeled_graph"   and G\<^sub>2 :: "('d, 'e) labeled_graph" shows "subgraph G\<^sub>1 G\<^sub>2 = (graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2)"
abelian_group_def: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "abelian_group G composition unit \<equiv> Group_Theory.group G composition unit \<and> commutative_monoid G composition unit"
alt_group_is_group: fixes n :: "nat" shows "Group.group (alt_group n)"
filter_tab_all_True: fixes t :: "'a \<Rightarrow> 'b option"   and p :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>k y. t k = Some y \<longrightarrow> p k y" shows "filter_tab p t = t"
simps_2: fixes x :: "bit"   and l :: "bit list"   and n :: "nat" shows "selecthelp (x # l) n = (if n \<le> 0 then [x] else x # selecthelp l (n - 1))"
zero_less_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (numeral.numeral one plus n)"
step: fixes P :: "'a llist"   and v0 :: "'a" assumes "vmc_path_no_deadend G P v0 p \<sigma>"   and "v0 \<in> S" shows "vmc_path_no_deadend.w0 P \<in> S"
cones_map_mapsto: fixes f :: "'b" assumes "D.C.arr f" shows "D'.cones_map f \<in> extensional (D'.cones (D.C.cod f)) \<inter> (D'.cones (D.C.cod f) \<rightarrow> D'.cones (D.C.dom f))"
add_point_wf: fixes p :: "'a" shows "incidence_system (add_point p) \<B>"
upset_faithful: fixes dual :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.preorder_with_dual dual less_eq less"   and "ord.upset less_eq x \<subseteq> ord.upset less_eq y" shows "less_eq y x"
ex_complex_to_complex: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> 0"   and "z2 \<noteq> 0" shows "\<exists>k. k \<noteq> 0 \<and> z2 = k * z1"
card: shows "card ((\<lambda>i. cis (real i * 2 * pi / real n)) ` {0..<n}) = card {x. x ^ n = 1}"
n_part: shows "N = {#y \<in># N. y = x#} + {#y \<in># N. y \<noteq> x#}"
successive_ends_1': fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and ys :: "'b list" assumes "successive P xs"   and "\<not> P x"   and "P y"   and "\<not> P z"   and "successive P ys" shows "successive P (xs @ x # y # z # ys)"
option_Max_by_def: fixes cmp_f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "option_Max_by cmp_f A \<equiv> if A = {} then None else Some (Max_by cmp_f A)"
f_1_def: fixes x :: "int" shows "f_1 x = (x mod int p, x mod int q)"
norm_trunc: fixes S :: "'a set" assumes "finite S" shows "norm (trunc_ell2 S \<psi>) = sqrt (sum f S)"
simps_3: fixes m :: "'node"   and m' :: "'node"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" shows "slice_kind_aux m m' S (Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs) = (if m \<in> \<lfloor>S\<rfloor>\<^bsub>CFG\<^esub> then Q:r\<hookrightarrow>\<^bsub>p\<^esub>cspp m' S fs else \<lambda>cf. False:r\<hookrightarrow>\<^bsub>p\<^esub>fs)"
nonzero_mult_divide_mult_cancel_right2: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times a c) (times c b) = divide a b"
syntactical_implies_correct: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "\<phi> \<turnstile>\<^sub>s \<psi>"   and "w \<Turnstile>\<^sub>n \<phi>" shows "w \<Turnstile>\<^sub>n \<psi>"
\<R>_regions_distinct: fixes R :: "('a \<Rightarrow> real) set"   and v :: "'a \<Rightarrow> real"   and R' :: "('a \<Rightarrow> real) set" assumes "R \<in> \<R>"   and "v \<in> R"   and "R' \<in> \<R>"   and "R \<noteq> R'" shows "v \<notin> R'"
dickson_lessD1: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "gd_powerprod ord ord_strict"   and "gd_powerprod.dickson_less ord_strict d m s t" shows "d s \<le> m"
floor_le_iff: fixes x :: "'b"   and z :: "int" shows "(\<lfloor>x\<rfloor> \<le> z) = (x < of_int z + (1::'b))"
diff_le_mono2: fixes m :: "nat"   and n :: "nat"   and l :: "nat" assumes "m \<le> n" shows "l - n \<le> l - m"
monoI: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> f x \<le> f y" shows "order.mono less_eq f"
lcons_correct_2: fixes t :: "('a, 'b) FingerTreeStruc"   and a :: "'a \<times> 'b" assumes "FingerTreeStruc.ft_invar t" shows "FingerTreeStruc.toList (FingerTreeStruc.lcons a t) = a # FingerTreeStruc.toList t"
\<R>_mds: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds', mem\<^sub>2\<rangle>" shows "mds = mds'"
flow_attainability_axioms: shows "flow_attainability (residual_network f)"
exec_step_ind_Init_Done: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and Cs :: "char list list"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>" shows "exec_step_ind (StepC C Cs) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Called Cs) # frs, sh)"
b_least_le_arg: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" shows "b_least f x \<le> x"
gen_length_def: fixes n :: "nat"   and xs :: "'c list" shows "List.gen_length n xs = n + length xs"
U1: shows "(\<Union>(p, n, s)\<in>{(p, n, s). (p, n, s) \<in> set (rsplit0 a) \<and> n = 0}. set (case (p, n, s) of (p, n, s) \<Rightarrow> if n = 0 then [(p, 0, Floor s)] else map (case (p, n, s) of (p, n, s) \<Rightarrow> \<lambda>j. ((case (p, n, s) of (p, n, s) \<Rightarrow> fp p n s) j, 0, Add (Floor s) (C j))) (if 0 < n then [0..n] else [n..0]))) = (\<Union>(p, n, s)\<in>{(p, n, s). (p, n, s) \<in> set (rsplit0 a) \<and> n = 0}. set [(p, 0, Floor s)])"
worecSL_isLim: fixes L :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c"   and i :: "'a"   and S :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "adm_woL L"   and "isLim i" shows "worecSL S L i = L (worecSL S L) i"
fps_inv_right: fixes a :: "'a fps" assumes "fps_nth a 0 = (0::'a)"   and "fps_nth a 1 \<noteq> (0::'a)" shows "a oo fps_inv a = fps_X"
pff_def: shows "pff \<equiv> \<lambda>x. \<Prod>w\<in>subA. ff w x"
simps_15: fixes x21 :: "'b"   and x22 :: "'b list" shows "set (x21 # x22) = Set.insert x21 (set x22)"
face_of_polyhedron: fixes S :: "'b set"   and C :: "'b set" assumes "polyhedron S"   and "C face_of S"   and "C \<noteq> {}"   and "C \<noteq> S" shows "C = \<Inter> {F. F facet_of S \<and> C \<subseteq> F}"
SeqAbstFormP_Mem: fixes s :: "name"   and k :: "name"   and x :: "tm"   and y :: "tm"   and x' :: "tm"   and y' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, x, y, x', y', v, i)"   and "atom k \<sharp> (x, y, x', y', v, i)" shows "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqAbstFormP v i (Q_Mem x y) (Q_Mem x' y') (Var s) (Var k)))"
assms: shows "pair \<in> N \<times> (Pow G - {{}})"
left_unit: fixes a :: "'a" assumes "a \<in> G" shows "\<one> \<cdot> a = a"
assms: shows "\<langle>\<langle>r, A\<rangle>, s\<rangle> \<in>\<^sub>\<circ> invimage D"
connected_simple_path_image: fixes g :: "real \<Rightarrow> 'b" assumes "simple_path g" shows "connected (path_image g)"
updates_2: fixes i' :: "nat"   and j' :: "nat"   and c :: "int" assumes "i' < m"   and "j' < i'" shows "d\<mu> (fs[i := fs ! i - c \<cdot>\<^sub>v fs ! j]) i' j' = (if i' = i \<and> j' < j then d\<mu> fs i' j' - c * d\<mu> fs j j' else if i' = i \<and> j' = j then d\<mu> fs i' j' - c * d fs (Suc j) else d\<mu> fs i' j')"
eventually_conj_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "(\<forall>\<^sub>F x in F. P x \<and> Q x) = (eventually P F \<and> eventually Q F)"
pi_it_to_it: fixes I :: "('c list \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'c list \<Rightarrow> 'c list) \<Rightarrow> 'c list \<Rightarrow> 'c list" shows "proper_it (it_to_it I) (it_to_it I)"
prems: fixes kv :: "'b \<times> 'c" assumes "kv \<in> set (x # xs)" shows "f kv = g kv"
Sup_prod_def: fixes X :: "('c \<times> 'd) set" shows "Sup_prod X = (\<Squnion>{fst x |x. x \<in> X}, \<Squnion>{snd x |x. x \<in> X})"
intro: fixes vars1 :: "nat set"   and vars2 :: "nat set" assumes "vars1 \<inter> vars2 = {}" shows "partial_state2 vars1 vars2"
generate_valid_stateful_policy_IFSACS_def: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and edgesList :: "('b \<times> 'b) list" shows "generate_valid_stateful_policy_IFSACS G M edgesList \<equiv> let filterIFS = filter_IFS_no_violations G M edgesList; filterACS = filter_compliant_stateful_ACS G M filterIFS in \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = set filterACS\<rparr>"
restrict_map_sub_disj: fixes h :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "h |` S \<bottom> h `- S"
oarrivemsg_simps_7: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a" shows "oarrivemsg P \<sigma> \<tau>\<^sub>n = True"
c2: shows "0 < Re \<langle>v|v\<rangle>"
continuous_map_def: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "continuous_map X Y f \<equiv> (\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> (\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U})"
dom_mkArr: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and B :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and F :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr (mkArr A B F)" shows "local.dom (mkArr A B F) = mkIde A"
sel_1: fixes x1 :: "'d" shows "un_Link (Link x1) = x1"
simps_2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and delta :: "'c"   and inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs option" shows "qOp delta inp binp #[[rho]] = qOp delta (lift (qPsubst rho) inp) (lift (qPsubstAbs rho) binp)"
F_def: shows "F \<equiv> \<lambda>n z. interpret_floatarith ((DERIV_floatarith x ^^ n) f) (xs[x := z])"
sum_split: fixes x :: "'a"   and y :: "'a" shows "(ad x \<oplus> ad y) \<cdot> (ad x \<oplus> d y) = ad x"
\<gamma>_2: fixes \<D>1 :: "('n \<times> 'n set) set"   and m :: "nat"   and \<D>2 :: "('n \<times> 'n set) set" assumes "\<D>1 tagged_division_of cbox a b"   and "\<gamma> m fine \<D>1"   and "\<D>2 tagged_division_of cbox a b"   and "\<gamma> m fine \<D>2" shows "norm ((\<Sum>(x, K)\<in>\<D>1. content K *\<^sub>R f x) - (\<Sum>(x, K)\<in>\<D>2. content K *\<^sub>R f x)) < 1 / (real m + 1)"
map_plus_restrict_dist: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option"   and A :: "'c set" shows "(f ++ g) |` A = f |` A ++ g |` A"
left_adjoint_determines_right_up_to_iso: fixes f :: "'b"   and g :: "'b"   and g' :: "'b" assumes "D.adjoint_pair f g"   and "D.adjoint_pair f g'" shows "g \<cong>\<^sub>D g'"
USS: shows "U - S \<subseteq> \<Union> \<B>"
vpeq_s_nt: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and control :: "'a \<Rightarrow> 'd \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "'a"   and execs2 :: "'d \<Rightarrow> 'b list list"   and u :: "'d"   and s :: "'a" assumes "Separation_Kernel kstep output_f current cswitch kprecondition control kinvolved ifp vpeq"   and "Kernel.precondition kprecondition (Kernel.next_state current control t execs2) (Kernel.next_action current control t execs2)"   and "\<not> ifp\<^sup>*\<^sup>* (current t) u"   and "\<forall>v. ifp\<^sup>*\<^sup>* v u \<longrightarrow> vpeq v s t" shows "\<forall>v. ifp\<^sup>*\<^sup>* v u \<longrightarrow> vpeq v s (Kernel.step kstep (Kernel.next_state current control t execs2) (Kernel.next_action current control t execs2))"
simps: fixes A :: "'b ring"   and B :: "'b ring" shows "A \<lesssim> B = (id \<in> ring_hom A B)"
assms_2: shows "\<forall>x y. eval r_psi [x, y] = \<psi> x y"
common_part_in_product: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "CM"   and "P z x \<and> P z y" shows "P z (x \<otimes> y)"
not_add_mem_right: fixes x :: "V"   and y :: "V" shows "x + y \<notin>\<^sub>\<circ> x"
sdnets_in_subnets: fixes p :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and x :: "((int \<times> int) set set, int) Combinators" assumes "p \<in> dom (C x)"   and "\<not> member DenyAll x" shows "\<exists>(a, b)\<in>sdnets x. a \<in> subnetsOfAdr (src p) \<and> b \<in> subnetsOfAdr (dest p)"
trans: fixes a :: "'e multiset"   and b :: "'e multiset"   and c :: "'e multiset" assumes "a \<subseteq># b"   and "b \<subseteq># c" shows "a \<subseteq># c"
equivalence_in_bicategory_axioms: shows "internal_equivalence (f \<star> f') (g' \<star> g) gg'.\<eta> gg'.\<epsilon>"
list_it_correct: fixes m :: "'s" assumes "invar m" shows "map_iterator (list_it m) (\<alpha> m)"
order_trans: fixes x :: "'loc \<times> 't"   and y :: "'loc \<times> 't"   and z :: "'loc \<times> 't" assumes "x\<le>\<^sub>py"   and "y\<le>\<^sub>pz" shows "x\<le>\<^sub>pz"
ex_bij_betw: fixes A :: "'d set"   and r :: "('e \<times> 'e) set" assumes "|A| \<le>o r" shows "\<exists>f B. bij_betw f B A"
subgroups_card_coprime_imp_compl: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G"   and "coprime (card H) (card J)" shows "complementary H J"
invertibleMtx1: shows "invertible_mat (M_mat signs1 subsets1)"
roots_over_subfield: fixes P :: "'a list" assumes "P \<in> carrier (K [X])" shows "splitted P"
dim_def: fixes V :: "('d, 'e) vec set" shows "vec.dim V = (if \<exists>b. vec.independent b \<and> vec.span b = vec.span V then card (SOME b. vec.independent b \<and> vec.span b = vec.span V) else 0)"
preserves_cod: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "cod (\<alpha>.map f) = HoVH (VVV.cod f)"
divide_less_cancel: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide a c) (divide b c) = ((less zero c \<longrightarrow> less a b) \<and> (less c zero \<longrightarrow> less b a) \<and> c \<noteq> zero)"
order_iff: fixes a :: "'f multiset"   and b :: "'f multiset" shows "(a \<subseteq># b) = (a = a \<inter># b)"
poly_lift_hom_is_hom: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e" assumes "cring S"   and "\<phi> \<in> ring_hom R S" shows "poly_lift_hom R S \<phi> \<in> ring_hom (UP R) (UP S)"
LAss: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and l' :: "sfields"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h, l, sh)\<rangle>"   and "l' = l(V \<mapsto> v)" shows "P \<turnstile> \<langle>V:=e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h, l', sh)\<rangle>"
quantum_machine_def: fixes n :: "nat"   and s :: "complex Matrix.vec"   and U :: "complex Matrix.mat" shows "quantum_machine n s U \<equiv> (dim_vec s = 2 ^ n \<and> dim_col U = 2 ^ n * 2 ^ n) \<and> square_mat U \<and> unitary U"
ALL_MOST: fixes P :: "'a \<Rightarrow> bool" assumes "\<forall>x. P x" shows "\<forall>\<^sub>\<infinity>x. P x"
iia_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "iia_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. \<forall>a\<in>C. a \<in> f B \<longrightarrow> a \<in> f C)"
uf: shows "poly_mod.unique_factorization_m (p ^ n) f (lead_coeff f, mset (berlekamp_hensel p n f))"
indets_map_indets_subset: fixes f :: "'f \<Rightarrow> 'd"   and p :: "('f \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'e" shows "indets (map_indets f p) \<subseteq> f ` indets p"
left_total_rel_resumption: fixes R1 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "left_total R1"   and "left_total R2" shows "left_total (Resumption.resumption.rel_resumption R1 R2)"
fold_insert: fixes B :: "'a set"   and x :: "'a"   and z :: "'a" assumes "finite B"   and "x \<notin> B"   and "insert x B \<subseteq> A"   and "z \<in> A" shows "Finite_Set.fold (\<lambda>y. f y ^^ count (add_mset x M) y) z (insert x B) = (f x ^^ count (add_mset x M) x) (Finite_Set.fold (\<lambda>y. f y ^^ count (add_mset x M) y) z B)"
card_set_mset': fixes M :: "'b multiset" assumes "\<forall>x. count M x \<le> 1" shows "card (set_mset M) = size M"
line_is_pair_img: shows "{p. \<exists>x. x \<in> cbox a b \<and> p = (g2 x, x)} = (\<lambda>x. (g2 x, x)) ` cbox a b"
point_rep_num_inv_non_empty: fixes x :: "'a" assumes "str_del_point_blocks p rep x < int (size (str_del_point_blocks p))"   and "str_del_point_blocks p \<noteq> {#}" shows "filter_mset ((\<notin>) x) (str_del_point_blocks p) \<noteq> {#}"
strict_iff_order: fixes b :: "'b list"   and a :: "'b list" shows "strict_prefix b a = (prefix b a \<and> a \<noteq> b)"
f_tendsto_2: shows "(f \<longlongrightarrow> f x) (at_right x)"
odiff_def: fixes y :: "V"   and x :: "V" shows "odiff y x \<equiv> THE z. x + z = y \<or> z = 0 \<and> \<not> x \<unlhd> y"
bs: shows "bs \<equiv> THE ds. length ds = length vs \<and> v' = ds \<bullet>\<cdot> vs"
interchange: fixes \<nu> :: "'b"   and \<mu> :: "'b"   and \<tau> :: "'b"   and \<sigma> :: "'b" assumes "seq \<nu> \<mu>"   and "seq \<tau> \<sigma>" shows "\<nu> \<cdot> \<mu> \<star> \<tau> \<cdot> \<sigma> = (\<nu> \<star> \<tau>) \<cdot> (\<mu> \<star> \<sigma>)"
trss_c_fmt_s: fixes s :: "'f list"   and c :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "((s, c), e, s', c') \<in> trss fg" shows "\<exists>csp. c' = csp + c \<and> (csp = {#} \<or> (\<exists>p. e = LSpawn p \<and> csp = {#[entry fg p]#}))"
insertI1: fixes a :: "'g"   and B :: "'g set" shows "a \<in> insert a B"
while_unfold: fixes b :: "bexp"   and c :: "com" shows "WHILE b DO c \<sim> IF b THEN c;; WHILE b DO c ELSE SKIP"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Accept of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2) = f1"
that: fixes f :: "real \<Rightarrow> real" assumes "continuous_on {a..b} f"   and "f a = d"   and "f b = 0"   and "l2norm {a..b} f < e" shows "thesis"
assms: shows "Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> f' \<alpha>' t\<^sub>\<alpha>'"
min_zero: fixes x :: "nat"   and k :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "x \<in> {1..k}"   and "T x = 0" shows "T (min\<^sub>k T k) = 0"
swap_psubst: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "good X"   and "goodEnv rho" shows "X #[rho] #[z1 \<and> z2]_zs = X #[z1 \<and> z2]_zs #[rho &[z1 \<and> z2]_zs]"
Edges_disj: fixes vs :: "nat list"   and x :: "nat"   and z :: "nat"   and y :: "nat" assumes "distinct vs"   and "x \<in> set vs"   and "z \<in> set vs"   and "x \<noteq> y"   and "y \<noteq> z"   and "y \<in> set (between vs x z)" shows "Edges (x # between vs x y @ [y]) \<inter> Edges (y # between vs y z @ [z]) = {}"
f0: shows "U\<^sub>f $$ (i, i + 1) * U\<^sub>f $$ (i + 1, j) = of_nat (f (i div 2)) * U\<^sub>f $$ (i + 1, j)"
igWlsBinp_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'h option" shows "igWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (igWlsAbs MOD) (barOf delta) binp"
document_ptr_kinds_M_def: shows "document_ptr_kinds_M \<equiv> l_ptr_kinds_M.a_ptr_kinds_M document_ptr_kinds"
leg1_commutes: shows "src_f.cod.leg1 \<cdot> src_f.chine = src_f.dom.leg1"
intros_1: fixes s :: "char list \<Rightarrow> int" shows "(SKIP, s) \<Rightarrow> s"
group_by_def: fixes f :: "'f \<Rightarrow> 'd \<times> 'e"   and s :: "'f set" shows "group_by f s = {(fst (f a), {snd (f a') |a'. a' \<in> s \<and> fst (f a) = fst (f a')}) |a. a \<in> s}"
currentD_outside': fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and e :: "'k \<times> 'k" assumes "current \<Gamma> f"   and "e \<notin> \<^bold>E\<^bsub>\<Gamma>\<^esub>" shows "f e = 0"
less_le_trans: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b"   and "b \<preceq>\<^bsub>D\<^esub> c" shows "a \<prec>\<^bsub>D\<^esub> c"
crypts_mpair_2: fixes X :: "msg"   and Y :: "msg" shows "insert \<lbrace>X, Y\<rbrace> (crypts {X} \<union> crypts {Y}) \<subseteq> crypts {\<lbrace>X, Y\<rbrace>}"
closed_add: fixes x :: "'b"   and y :: "'b" assumes "x \<in> U\<^sub>M"   and "y \<in> U\<^sub>M" shows "x + y \<in> U\<^sub>M"
kD_is_vD_and_iD: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "kD rt = vD rt \<union> iD rt"
interior_of_Int: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "X interior_of (S \<inter> T) = X interior_of S \<inter> X interior_of T"
restrict_out: fixes x :: "'d"   and A :: "'d set"   and m :: "'d \<Rightarrow> 'e option" assumes "x \<notin> A" shows "(m|\<^bsub>A\<^esub>) x = None"