hcju/isabelleps huggingface dataset:
berlekamp_zassenhaus_factorization_irreducible\<^sub>d: fixes f :: "int poly"   and fs :: "int poly list" assumes "berlekamp_zassenhaus_factorization f = fs"   and "square_free f"   and "0 < degree f" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"
simps_3: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and as :: "'c list"   and b :: "'d"   and bs :: "'d list" shows "filter2 P (a # as) (b # bs) = (if P a b then (a # fst (filter2 P as bs), b # snd (filter2 P as bs)) else filter2 P as bs)"
bia: shows "bef @ i # aft @ [a] = bia @ [a]"
square_integrable_neg_eq: fixes f :: "real \<Rightarrow> real"   and S :: "real set" shows "((\<lambda>x. - f x) square_integrable S) = (f square_integrable S)"
mset_image_comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and E :: "'d multiset" shows "image_mset f (image_mset g E) = {#f (g x). x \<in># E#}"
ex_norm_eq_1: shows "\<exists>x. norm x = 1"
vwb_lens_mwb: fixes x :: "'d \<Longrightarrow> 'e" assumes "vwb_lens x" shows "mwb_lens x"
hyps_3: assumes "is_word w"   and "is_prefix w a" shows "derives (drop (length w) a) (drop (length w) y)"
ext: fixes g :: "'a"   and f :: "'a" assumes "g \<cdot> f \<noteq> null" shows "seq g f"
IH: assumes "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds" shows "dims (listsum ds As) = ds"
sublist_all_positions: fixes xs :: "'a list"   and ys :: "'a list"   and i :: "nat" assumes "sublist_at xs ys i" shows "\<forall>jj<length xs. ys ! (i + jj) = xs ! jj"
narrow_acom_def: shows "(\<triangle>) = map2_acom (\<triangle>)"
Subring1D1: fixes S :: "'r set" assumes "Subring1 S" shows "Ring1 S"
in_space_prob_algebra: fixes x :: "'b measure"   and M :: "'b measure" assumes "x \<in> space (prob_algebra M)" shows "emeasure x (space M) = 1"
mssnth_sin_series_stream: fixes n :: "nat" shows "mssnth sin_series_stream n = (- 1) ^ n / fact (2 * n + 1)"
nop_onto_n: shows "2\<^sub>\<nat> \<in>\<^sub>\<circ> \<omega>"
modulo_int_def: fixes k :: "int"   and l :: "int" shows "k mod l = (if l = 0 then k else if sgn k = sgn l then sgn l * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>) else sgn l * (\<bar>l\<bar> * of_bool (\<not> l dvd k) - int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)))"
combine_singleton: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and j :: "nat"   and i :: "nat"   and a :: "'c"   and b :: "'d" assumes "f (0::'c) (0::'d) = (0::'b)" shows "combine_matrix f (singleton_matrix j i a) (singleton_matrix j i b) = singleton_matrix j i (f a b)"
list_induct3: fixes xs :: "'a list"   and ys :: "'b list"   and zs :: "'c list"   and P :: "'a list \<Rightarrow> 'b list \<Rightarrow> 'c list \<Rightarrow> bool" assumes "length xs = length ys"   and "length ys = length zs"   and "P [] [] []"   and "\<And>x xs y ys z zs. \<lbrakk>length xs = length ys; length ys = length zs; P xs ys zs\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys) (z # zs)" shows "P xs ys zs"
monitor_def: fixes \<phi> :: "Formula.formula"   and \<pi> :: "(String.literal \<times> event_data list) prefix" shows "local.monitor \<phi> \<pi> = msteps_stateless \<pi> (minit_safe \<phi>)"
sub2: shows "G \<subseteq> dgrad_p_set d m"
fract_content_def: fixes p :: "'a fract poly" shows "fract_content p = (let d = Lcm (snd ` quot_of_fract ` set (coeffs p)) in Fraction_Field.Fract (content (map_poly (fst \<circ> quot_of_fract) (smult (to_fract d) p))) d)"
evalHeap_cong: fixes heap1 :: "('d \<times> 'e) list"   and heap2 :: "('d \<times> 'e) list"   and eval1 :: "'e \<Rightarrow> 'f"   and eval2 :: "'e \<Rightarrow> 'f" assumes "heap1 = heap2"   and "\<And>e. e \<in> snd ` set heap2 \<Longrightarrow> eval1 e = eval2 e" shows "evalHeap heap1 eval1 = evalHeap heap2 eval2"
top_unique: fixes x :: "'b"   and y :: "'b" shows "- x \<squnion> - - x = - y \<squnion> - - y"
interesting_subst1': fixes x :: "name"   and y :: "name"   and P :: "trm"   and N :: "trm"   and c :: "coname"   and a :: "coname" assumes "x \<noteq> y"   and "x \<sharp> P"   and "y \<sharp> P" shows "N{y:=<c>.P}{x:=<c>.P} = N{x:=<a>.Ax y a}{y:=<c>.P}"
le_cases: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" obtains "a \<preceq>\<^bsub>D\<^esub> b"   | "b \<preceq>\<^bsub>D\<^esub> a"
delta_remove: fixes S :: "'a set"   and a :: "'a"   and b :: "'a \<Rightarrow> 'b"   and c :: "'a \<Rightarrow> 'b" assumes "finite S" shows "(\<Sum>k\<in>S. if k = a then b k else c k) = (if a \<in> S then b a + sum c (S - {a}) else sum c (S - {a}))"
Fvar_pbeta: fixes x :: "char list"   and t :: "sterm" shows "(Fvar x => t) = (t = Fvar x)"
disjoint_eq_subset_Compl: fixes A :: "'d set"   and B :: "'d set" shows "(A \<inter> B = {}) = (A \<subseteq> - B)"
compl_sup_top: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<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 x :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "sup (uminus x) x = top"
t_BIT_ub: fixes n :: "nat" assumes "qs ! n \<in> set init" shows "t_BIT n \<le> real (length init)"
dghm_dghm_comp_dghm_id_left: shows "dghm_id \<CC> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> = \<GG>"
carrier_subgroup_generated_subgroup: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "subgroup H G" shows "carrier (subgroup_generated G H) = H"
KBasic_6: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<psi>) in v]"
nop_bij_vrange: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_bij A n f" shows "\<R>\<^sub>\<circ> f = A"
g_def: shows "g = (\<lambda>x. f x - u x + u (T x))"
rank_eq_dim_image: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A = vec.dim (range ((*v) A))"
SeqAppendP_graph_aux_def: shows "SeqAppendP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqAppendP_aux g1 g2 f1 k1 f2 k2 g. x1 = (f1, k1, f2, k2, g) \<and> x2 = SyntaxN.Ex g1 (SyntaxN.Ex g2 (RestrictedP f1 k1 (Var g1) AND ShiftP f2 k2 k1 (Var g2) AND UnionP (Var g1) (Var g2) g)) \<and> atom g1 \<sharp> (g2, f1, k1, f2, k2, g) \<and> atom g2 \<sharp> (f1, k1, f2, k2, g))"
assms_7: shows "mod_type_class.to_nat i \<le> n"
ns_mul_ext_bottom: fixes A :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "(A, {#}) \<in> ns_mul_ext ns s"
semilattice_fset_axioms: shows "semilattice_fset (\<^bold>*)"
gr_res: shows "ground_resolution_with_selection.ord_resolve S_G CAs DA AAs As E"
s: shows "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<Suc k] bezout) k"
power_one_right: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.monoid_mult one times" shows "power.power one times a 1 = a"
g_rw: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) = mod_type_class.from_nat (mod_type_class.to_nat ((GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows)))"
lunit'_naturality: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "(trg\<^sub>D \<mu> \<star>\<^sub>D \<mu>) \<cdot>\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[D.dom \<mu>] = \<l>\<^sub>D\<^sup>-\<^sup>1[D.cod \<mu>] \<cdot>\<^sub>D \<mu>"
Group_indspace: fixes FH :: "('f, 'g) aezfun set" assumes "Ring1 FH" shows "Group (BaseFGMod.indspace G FH V)"
monad_alt_nondetT': assumes "monad_commute Monad_Overloading.return Monad_Overloading.bind" shows "monad_alt Monad_Overloading.return Monad_Overloading.bind alt"
s3: shows "s3 = sum_weights (map (\<lambda>(s, l, t). l) (path1 @ path2))"
typeof: shows "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor>"
ih_1: assumes "n \<noteq> 0" shows "local.well_base\<^sub>h (local.encode 0 e)"
replace_bound_eq: fixes i :: "nat"   and t :: "Term_Class.term"   and t' :: "Term_Class.term" assumes "i |\<notin>| bounds t" shows "replace_bound i t t' = incr_bounds (- 1) (i + 1) t"
insert_def: fixes x :: "'b"   and t :: "('b \<times> bal) tree" shows "AVL_Bal2_Set.insert x t = tree (ins x t)"
DmdImpldup: fixes A :: "'a \<Rightarrow> bool"   and sigma :: "behavior"   and F :: "behavior \<Rightarrow> bool"   and G :: "behavior \<Rightarrow> bool" assumes "sigma \<Turnstile> \<box>A"   and "sigma \<Turnstile> \<diamond>F"   and "\<turnstile> \<box>A \<and> F \<longrightarrow> G" shows "sigma \<Turnstile> \<diamond>G"
WT_plus_oracleI: fixes \<I>l :: "('a, 'b) \<I>"   and s :: "'s"   and \<I>r :: "('c, 'd) \<I>" assumes "\<I>l \<turnstile>c left s \<surd>"   and "\<I>r \<turnstile>c right s \<surd>" shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c (left \<oplus>\<^sub>O right) s \<surd>"
a_def': shows "a = insertion (nth_default 0 (xs' @ 0 # xs)) (isolate_variable_sparse p var 2)"
eVar_simp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and xs :: "'varSort"   and x :: "'var" shows "eVar MOD xs x = OK (igVar MOD xs x)"
T_p: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "d p \<cdot> (p \<leadsto> x \<leadsto> q) = d p \<cdot>  |x] d q"
d2: fixes k :: "nat" assumes "k < n" shows "adjoint (M k) * M k * A \<in> carrier_mat d d"
is_processT2_TR: fixes P :: "'b process" shows "\<forall>s. s \<in> T P \<longrightarrow> front_tickFree s"
Zfun: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "bounded_linear f"   and "Zfun g F" shows "Zfun (\<lambda>x. f (g x)) F"
pa: shows "nmu (sports m) \<or> nmu (dports m) \<longrightarrow> php m"
2: shows "val_Zp (f \<bullet> \<one>) = eint 3 + eint 4 * val_Zp a"
ball_UN: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>z\<in>\<Union> (B ` A). P z) = (\<forall>x\<in>A. \<forall>z\<in>B x. P z)"
TransLam_subst: fixes v :: "Vars.var"   and x :: "Vars.var"   and y :: "Vars.var"   and a :: "'a"   and e :: "exp" assumes "atom v \<sharp> (x, y)" shows "(TransLam a v e)[x::=y] = TransLam a v[x::v=y] e[x::=y]"
limit_o_inv: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'c"   and w :: "nat \<Rightarrow> 'b" assumes "finite (f -` {x})"   and "x \<in> limit (f \<circ> w)" shows "\<exists>a\<in>f -` {x}. a \<in> limit w"
row_add_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and a :: "'d"   and b :: "'d"   and q :: "'b" shows "row_add A a b q = (\<chi>i j. if i = a then A $h a $h j + q * A $h b $h j else A $h i $h j)"
in_span_in_range_construct: fixes B :: "'b set"   and x :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "vs1.independent B"   and "x \<in> vs2.span (f ` B)" shows "x \<in> range (construct B f)"
WStandardControlDependence_aux: fixes prog :: "cmd" shows "StandardControlDependencePDG sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (_Exit_)"
restrict_def: fixes A :: "nat set"   and v :: "'a option list" shows "restrict A v = map (\<lambda>i. if i \<in> A then v ! i else None) [0..<length v]"
IV: shows "set_of_bv (nfa_steps (quantify_nfa v N) q xs) = (\<Union>x\<in>{x. length x = length xs}. set_of_bv (nfa_steps N q (insertll v x xs)))"
simps_2: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f"   and ct :: "'f conditional_action"   and cts :: "'f conditional_action list" shows "cond_action_oks s t (ct # cts) = (cond_action_ok s t ct \<and> cond_action_oks s t cts)"
subgroupE_1: fixes H :: "'c set" assumes "subgroup H H" shows "H \<subseteq> carrier H"
i_cut_defs_3: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<ge> t \<equiv> {x \<in> I. t \<le> x}"
path_component_path_component: fixes S :: "'a set"   and x :: "'a" shows "path_component_set (path_component_set S x) x = path_component_set S x"
bounded_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a \<le> inf b c) = (a \<le> b \<and> a \<le> c)"
transfer_sum: fixes \<alpha> :: "'c \<Rightarrow> 'b"   and fl :: "'a \<Rightarrow> 'c"   and Fl :: "'a \<Rightarrow> 'b"   and fr :: "'d \<Rightarrow> 'c"   and Fr :: "'d \<Rightarrow> 'b"   and x :: "'a + 'd" assumes "\<And>l. \<alpha> (fl l) \<le> Fl l"   and "\<And>r. \<alpha> (fr r) \<le> Fr r" shows "\<alpha> (case x of Inl x \<Rightarrow> fl x | Inr x \<Rightarrow> fr x) \<le> (case x of Inl x \<Rightarrow> Fl x | Inr x \<Rightarrow> Fr x)"
iso_is_arr: fixes f :: "'d" assumes "D'.iso f" shows "D'.arr f"
sup_eq_bot_iff: fixes x :: "'a"   and y :: "'a" shows "(x + y = (zero_class.zero::'a)) = (x = (zero_class.zero::'a) \<and> y = (zero_class.zero::'a))"
nres_order_simps_5: fixes X :: "'c set"   and Y :: "'c set" shows "(RES X \<le> RES Y) = (X \<subseteq> Y)"
edom_mono: fixes x :: "'c \<Rightarrow> 'd"   and y :: "'c \<Rightarrow> 'd" assumes "x \<sqsubseteq> y" shows "edom x \<subseteq> edom y"
cil_assoc1: fixes w :: "'d list"   and wl :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and w3 :: "'d list"   and w1 :: "'d list"   and w2 :: "'d list" assumes "w \<in> wl \<otimes>\<^bsub>\<alpha>\<^esub> w3"   and "wl \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "\<exists>wr. w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> wr \<and> wr \<in> w2 \<otimes>\<^bsub>\<alpha>\<^esub> w3"
csrp_fn_pol_coeff: fixes P :: "'a set"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and n :: "nat"   and c' :: "nat \<Rightarrow> 'a set" assumes "ideal S P"   and "PolynRg R' (S /\<^sub>r P) Y"   and "pol_coeff (S /\<^sub>r P) (n, c')" shows "pol_coeff S (n, cmp (csrp_fn S P) c')"
comp_sufs_comp: fixes z :: "'b list"   and u :: "'b list"   and w :: "'b list"   and v :: "'b list" assumes "z \<cdot> u \<bowtie>\<^sub>s w \<cdot> v" shows "u \<bowtie>\<^sub>s v"
flip_bit_def: fixes n :: "nat"   and a :: "'a" shows "flip_bit n a = a XOR push_bit n (1::'a)"
lfp_def: fixes f :: "'a \<Rightarrow> 'a" shows "local.lfp f = Glb {a \<in> L. f a \<le> a}"
simps_3: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>c' = \<lbrakk>insert t M; S\<rbrakk>\<^sub>c'"
irrefl: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "irreflexive A less"   and "x \<in> A" shows "\<not> less x x"
elims_3: fixes x :: "'at Literal" assumes "\<not> positive x" obtains A :: "'at" where "x = Neg A"
measure_ge_1_iff: fixes M :: "'b pmf"   and A :: "'b set" shows "(1 \<le> measure_pmf.prob M A) = (measure_pmf.prob M A = 1)"
simps_1: fixes x :: "num" shows "Num.pow x num.One = x"
ring_hom_in_hom_1: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<in> hom R S"
col_four_block_mat_1: fixes A :: "'a mat"   and nr1 :: "nat"   and nc1 :: "nat"   and B :: "'a mat"   and nc2 :: "nat"   and C :: "'a mat"   and nr2 :: "nat"   and D :: "'a mat"   and j :: "nat" assumes "A \<in> carrier_mat nr1 nc1"   and "B \<in> carrier_mat nr1 nc2"   and "C \<in> carrier_mat nr2 nc1"   and "D \<in> carrier_mat nr2 nc2"   and "j < nc1" shows "col (four_block_mat A B C D) j = col A j @\<^sub>v col C j"
sel'E: fixes s :: "'s"   and x :: "'x"   and P :: "'x \<Rightarrow> bool" assumes "invar s"   and "x \<in> \<alpha> s"   and "P x" obtains x :: "'x" where "sel' s P = Some x" and "x \<in> \<alpha> s" and "P x"
T: fixes r :: "int"   and t :: "int" assumes "r \<in> T"   and "t \<in> T" shows "r + t \<in> T"
ivl_rel_br: fixes a :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool" shows "\<langle>br a I\<rangle>ivl_rel = br (\<lambda>(x, y). set_of_ivl (a x, a y)) (\<lambda>(x, y). I x \<and> I y)"
1: shows "homeomorphism (cbox a c) (cbox a u) f1 g1"
walk_edges_vertices_1: fixes v :: "'a"   and w :: "'a"   and xs :: "'a list" assumes "(v, w) \<in> edges_of_walk xs" shows "v \<in> set xs"
fst_gb_schema_dummy_pmdl: fixes ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set" assumes "struct_spec sel ap ab compl"   and "compl_pmdl compl"   and "is_Groebner_basis (fst ` set gs)"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "unique_idx (gs @ bs) (snd data)"   and "rem_comps_spec (gs @ bs) data" shows "pmdl (fst ` set (fst (gb_schema_dummy data D bs ps))) = pmdl (fst ` set (gs @ bs))"
r: fixes x :: "'a" assumes "x \<in> cball a r \<and> x \<noteq> a" shows "f x = g x"
cd_ipd_is_cd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and k :: "nat"   and m :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and j :: "nat" assumes "IFC prob"   and "k < m"   and "\<pi> m = IFC_def.ipd prob (\<pi> k)"   and "\<forall>n\<in>{k..<m}. \<pi> n \<noteq> IFC_def.ipd prob (\<pi> k)"   and "IFC_def.is_cdi prob m \<pi> j" shows "IFC_def.is_cdi prob k \<pi> j"
iso_assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c" shows "D.iso \<a>\<^sub>D[a, b, c]"
tuple_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "hfsetcat.tuple f g = mkArr (Dom f) (local.set (hfsetcat.prod (cod f) (cod g))) (\<lambda>x. UP \<langle>DOWN (Fun f x), DOWN (Fun g x)\<rangle>)"
assms_3: shows "m \<in> space\<^sub>N (\<LL> (ennreal p) M)"
smooth_on_basis_comp_inv: fixes a :: "'b"   and b :: "'b" assumes "a \<in> Basis"   and "b \<in> Basis" shows "smooth_on {x. (x + (1 - x \<bullet> a) *\<^sub>R a) \<bullet> b \<noteq> 0} (chart_basis b \<circ> chart_basis_inv a)"
safe_distance_1r_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "safe_distance_normal.safe_distance_1r a\<^sub>e v\<^sub>e \<delta> = v\<^sub>e * \<delta> - v\<^sub>e\<^sup>2 / a\<^sub>e / 2"
cct: fixes y :: "'b" assumes "y \<in> t" shows "connected_component_set t y = {y}"
add_ac_3: fixes b :: "'d"   and a :: "'d"   and c :: "'d" shows "b + (a + c) = a + (b + c)"
outpt_Inv6: fixes s :: "state"   and s' :: "state" assumes "outpt s = outpt s'"   and "\<forall>p. outpt s p \<in> {chosen s, NotAnInput}"   and "Inv2c s"   and "HNextPart s s'" shows "\<forall>p. outpt s' p \<in> {chosen s', NotAnInput}"
bounded_byD: fixes b :: "real"   and P :: "'c \<Rightarrow> real"   and x :: "'c" assumes "bounded_by b P" shows "P x \<le> b"
Abs_eq_iff_1: fixes bs :: "atom set"   and x :: "'b"   and bs' :: "atom set"   and y :: "'b" shows "([bs]set. x = [bs']set. y) = (\<exists>p. (bs, x) \<approx>set (=) supp p (bs', y))"
vec_eq_iff: fixes x :: "'a vec"   and y :: "'a vec" shows "(x = y) = (dim_vec x = dim_vec y \<and> (\<forall>i<dim_vec y. x $ i = y $ i))"
Janiszewski_connected: fixes S :: "complex set"   and T :: "complex set" assumes "compact S"   and "closed T"   and "connected (S \<inter> T)"   and "connected (- S)"   and "connected (- T)" shows "connected (- (S \<union> T))"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "ide (a \<otimes> cod \<iota>)"
prob0_jozsa_algo_of_const_1: assumes "const 1" shows "prob0_fst_qubits n jozsa_algo = 1"
replicate_in_nlists: fixes x :: "'b"   and A :: "'b set"   and n :: "nat" assumes "x \<in> A" shows "replicate n x \<in> nlists A n"
mult_pCons_right: fixes p :: "'b poly"   and a :: "'b"   and q :: "'b poly" shows "p * pCons a q = smult a p + pCons (0::'b) (p * q)"
main: fixes N :: "nat"   and K :: "real"   and e :: "real" assumes "1 < N"   and "0 < K"   and "0 < e" shows "AE x in M. limsup (\<lambda>xa. ereal (u xa x / real xa)) \<le> ereal (F K e x) + ereal \<bar>F K e x\<bar> * ereal (real_cond_exp M Invariants (I N K e) x)"
eSec_def: fixes c :: "('test, 'atom, 'choice) cmd" shows "eSec c = (\<forall>s1 s2 t. s1 \<approx> s2 \<longrightarrow> T.prob (c, s1) {bT \<in> space (T (c, s1)). \<exists>n. qsend ((c, s1) ## bT) = enat n \<and> snd (((c, s1) ## bT) !! n) \<approx> t} = T.prob (c, s2) {bT \<in> space (T (c, s2)). \<exists>n. qsend ((c, s2) ## bT) = enat n \<and> snd (((c, s2) ## bT) !! n) \<approx> t})"
indiscrete_top_def: fixes X :: "'b set" shows "indiscrete_top X = {{}, X}"
enabledE: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" assumes "(Enabled F) s" obtains u :: "nat \<Rightarrow> 'b" where "F (first s ## u)"
Cons_prefix_Cons: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "x # xs \<le> y # ys = (x = y \<and> xs \<le> ys)"
alg_dvd_of_nat_iff: fixes x :: "nat"   and y :: "nat" shows "(of_nat x alg_dvd of_nat y) = (x dvd y)"
orbit_partition: shows "Congruence.partition E (orbits G E \<phi>)"
nth_mem: fixes n :: "nat"   and xs :: "'k list" assumes "n < length xs" shows "xs ! n \<in> set xs"
find_first_none: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "find_first x xs = None"
wlp_def: fixes pr :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "wlp pr \<equiv> pr False"
inverse: fixes f :: "'b \<Rightarrow> 'a" shows "U (C f) = f"
finite_ex_and2: fixes Q :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite {b. \<exists>a. Q a b}" shows "finite {b. \<exists>a. P a b \<and> Q a b}"
rc: shows "root_cond (abs_int_poly (reflect_poly p), inverse r, inverse l) (inverse (real_of_1 (p, l, r)))"
\<pi>0: shows "product_cone J0 (\<cdot>) D0 a0 \<pi>0"
ucast_down_ucast_id: fixes d :: "'b word \<Rightarrow> 'c word"   and w :: "'c word" assumes "is_down d" shows "UCAST('b \<rightarrow> 'c) (UCAST('c \<rightarrow> 'b) w) = w"
lemma4: fixes r :: "lam"   and r' :: "lam"   and r'' :: "lam" assumes "r \<rightarrow>\<^sub>s r'"   and "r' \<rightarrow>\<^sub>\<beta> r''" shows "r \<rightarrow>\<^sub>s r''"
parents_def: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list" shows "parents d b p = {x \<in> grid b {d}. p \<in> grid x {d}}"
rbt_union_with_key_def: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt" shows "rbt_union_with_key f t1 t2 = (case RBT_Impl.compare_height t1 t1 t2 t2 of compare.LT \<Rightarrow> RBT_Impl.fold (rbt_insert_with_key (\<lambda>k v w. f k w v)) t1 t2 | compare.GT \<Rightarrow> RBT_Impl.fold (rbt_insert_with_key f) t2 t1 | compare.EQ \<Rightarrow> rbtreeify (sunion_with f (entries t1) (entries t2)))"
preserves_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "trg\<^sub>D (F \<mu>) = F.map\<^sub>0 (trg\<^sub>C \<mu>)"
2: assumes "\<lbrakk>M; D; A @ B\<rbrakk>\<^sub>s \<I>" shows "\<lbrakk>M \<union> (ik\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>); dbupd\<^sub>s\<^sub>s\<^sub>t A \<I> D; B\<rbrakk>\<^sub>s \<I>"
de_morgan1': fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "!p + !q = !(p \<cdot> q)"
end_point_char: fixes x :: "'a"   and p :: "'a" shows "(x\<^sup>T ; p = 0) = (p \<le> - (x ; 1))"
shrK_in_knows_Server: fixes A :: "agent"   and evs :: "event list" shows "Key (shrK A) \<in> knows Kas evs"
PARORD: fixes r :: "('a \<times> 'a) set" assumes "wo_rel r" shows "Partial_order r"
of_nat_0: fixes m :: "nat" shows "(word_of_nat m = 0) = (\<exists>q. m = q * 2 ^ LENGTH('b))"
trivial_same: fixes X :: "'b set" shows "trivial X = (\<forall>x1\<in>X. \<forall>x2\<in>X. x1 = x2)"
box_stp_actI: fixes P :: "state \<Rightarrow> bool"   and sigma :: "behavior" assumes "sigma \<Turnstile> \<box>P" shows "sigma \<Turnstile> \<box>$P"
cut_greater_ge_conv_if: fixes I :: "'a set"   and t :: "'a" shows "I \<down>> t = (if t \<in> I then I \<down>\<ge> t - {t} else I \<down>\<ge> t)"