hcju/isabelleps huggingface dataset:
assoc_match_Type_conds_2: fixes a :: "String.literal"   and Ts :: "typ list"   and b :: "String.literal"   and Us :: "typ list"   and subs :: "((variable \<times> String.literal set) \<times> typ) list"   and subs' :: "((variable \<times> String.literal set) \<times> typ) list" assumes "assoc_match (Ty a Ts) (Ty b Us) subs = Some subs'" shows "length Ts = length Us"
leq_path1_def: shows "leq_path1 \<equiv> \<lambda>P C. {(xa, x). leq_path1p P C xa x}"
dce_sum: fixes A :: "'a"   and B :: "'a"   and C :: "'a" shows "(A \<rightarrow>\<^sub>\<E> B + C) = (A \<rightarrow>\<^sub>\<E> B \<or> A \<rightarrow>\<^sub>\<E> C)"
set_iterator_filter_alt_def: fixes P :: "'e \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "set_iterator_filter P it = (\<lambda>c f. it c (\<lambda>x \<sigma>. if P x then f x \<sigma> else \<sigma>))"
sig_red_regular_lc: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red (\<prec>\<^sub>t) top_tail F p q" shows "lc q = lc p"
Mp_smult_2: fixes a :: "int"   and f :: "int poly" shows "q.eq_m (Polynomial.smult a (q.Mp f)) (Polynomial.smult a f)"
delete_edges_nodes: fixes G :: "'b list_graph"   and es :: "('b \<times> 'b) list" shows "nodesL (FiniteListGraph.delete_edges G es) = nodesL G"
shadow_root_ptr_kinds_eq_h3: shows "shadow_root_ptr_kinds h3 = shadow_root_ptr_kinds h'"
TensorDiag_Diag: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u = t \<^bold>\<otimes> u"
card_proots_within_leq: fixes p :: "'a poly"   and s :: "'a set" assumes "p \<noteq> 0" shows "card (proots_within p s) \<le> proots_count p s"
of_nat_power_eq_of_nat_cancel_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and x :: "nat"   and b :: "nat"   and w :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero x = power.power one times (semiring_1.of_nat one plus zero b) w) = (x = b ^ w)"
dsr4: shows "ad (ad (zero_class.zero::'b)) = (zero_class.zero::'b)"
syzygy_module_list_subset: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "distinct bs" shows "syzygy_module_list bs \<subseteq> pmdl (set (init_syzygy_list bs))"
times_gt_mono: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "delta_gt \<delta> y z"   and "(1::'a) \<le> x" shows "delta_gt \<delta> (y * x) (z * x)"
conga_refl: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "C \<noteq> B" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A B C"
iT_Plus_Plus_neg_inverse: fixes I :: "nat set"   and k :: "nat" shows "I \<oplus> k \<oplus>- k = I"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Dom g)"
isin_set_inorder: fixes t :: "('b \<times> 'c) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "isin t x = (x \<in> set (Tree2.inorder t))"
join_ti_hint1: fixes s :: "nat \<Rightarrow> 'a list"   and x :: "nat"   and i :: "nat" assumes "join_ti s x (Suc i) = []" shows "join_ti s x i = []"
finite_extension_consistent: fixes K :: "'a set" assumes "subring K R" shows "ring.finite_extension (R\<lparr>carrier := K\<rparr>) = finite_extension"
SAStates_ChiRel: fixes S :: "'a"   and SA :: "('a, 'b, 'c) seqauto"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<in> States SA"   and "T \<in> States SA"   and "SA \<in> SAs A" shows "(S, T) \<notin> ChiRel A"
measurable_cong: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and M' :: "'b measure" assumes "\<And>w. w \<in> space M \<Longrightarrow> f w = g w" shows "(f \<in> M \<rightarrow>\<^sub>M M') = (g \<in> M \<rightarrow>\<^sub>M M')"
ClassicFinal_def: shows "ClassicFinal = {(p, t, left, right, atom, mrk). atom = atom0 \<and> left = left0 \<and> right = right0 \<and> mrk = reach root}"
row_mat_of_row_fun: fixes i :: "nat"   and nr :: "nat"   and f :: "nat \<Rightarrow> 'b vec"   and nc :: "nat" assumes "i < nr"   and "dim_vec (f i) = nc" shows "row (mat\<^sub>r nr nc f) i = f i"
\<R>\<^sub>1p_\<R>\<^sub>1_eq: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "\<R>\<^sub>1p \<Gamma>' = (\<lambda>x xa xb xc xd xe. \<langle>x, xa, xb\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>'\<^esub> \<langle>xc, xd, xe\<rangle>)"
digraphD_8: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
unitWT_def: shows "unitWT = (\<Lambda> x. WriterT\<cdot>(return\<cdot>(Writer\<cdot>mempty\<cdot>x)))"
Aeq: shows "A\<^sub>P\<^sub>Q = p \<bullet> A\<^sub>P @ p \<bullet> A\<^sub>Q"
signed_take_bit_rec: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = (if n = 0 then uminus (modulo a (numeral.numeral one plus (num.Bit0 num.One))) else plus (modulo a (numeral.numeral one plus (num.Bit0 num.One))) (times (numeral.numeral one plus (num.Bit0 num.One)) (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not (n - 1) (divide a (numeral.numeral one plus (num.Bit0 num.One))))))"
select_convs_2: fixes types_t :: "tf list"   and func_t :: "tf list"   and global :: "tg list"   and table :: "nat option"   and memory :: "nat option"   and local :: "t list"   and label :: "t list list"   and return :: "t list option"   and more :: "'a" shows "func_t \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr> = func_t"
list1: shows "[0..<length fs] = [0..<Suc i] @ [Suc i..<length fs]"
Re_def: fixes quat :: "quat" shows "quat.Re quat = (case quat of Quat x1 x2 x3 x4 \<Rightarrow> x1)"
mirror_ex: fixes p :: "fm"   and i :: "int"   and bs :: "real list" assumes "iszlfm p (real_of_int i # bs)" shows "(\<exists>x. Ifm (real_of_int x # bs) (mirror p)) = (\<exists>x. Ifm (real_of_int x # bs) p)"
valid_shallow_model: fixes Y :: "nat"   and Z :: "nat"   and M :: "nat"   and N :: "nat" shows "valid_net (shallow_model Y Z M N)"
assms: fixes e :: "real" assumes "0 < e" shows "\<exists>g h i j. (g has_integral i) (cbox a b) \<and> (h has_integral j) (cbox a b) \<and> \<bar>i - j\<bar> < e \<and> (\<forall>x\<in>cbox a b. g x \<le> f x \<and> f x \<le> h x)"
morph: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'c" assumes "SimplicialComplexIsomorphism X f" shows "SimplicialComplex (f \<turnstile> X)"
plus_int_eqvt: fixes pi :: "('a \<times> 'a) list"   and x :: "int"   and y :: "int" shows "pi \<bullet> (x + y) = pi \<bullet> x + pi \<bullet> y"
gamma2_def: shows "gamma2 = (\<lambda>x. (b, g2 b + x *\<^sub>R (g1 b - g2 b)))"
def_hashmap_size_nat_def: shows "def_hashmap_size = (\<lambda>_. 16)"
multiplicity_unit_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times p one" shows "factorial_semiring.multiplicity times one p x = 0"
Pair_inject: fixes a :: "'a"   and b :: "'b"   and a' :: "'a"   and b' :: "'b" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
offending_flows_union_mono: fixes m :: "'b SecurityInvariant_configured"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set" assumes "configured_SecurityInvariant m"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E" shows "\<Union> (c_offending_flows m \<lparr>nodes = V, edges = E'\<rparr>) \<subseteq> \<Union> (c_offending_flows m \<lparr>nodes = V, edges = E\<rparr>)"
comp_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and g :: "'b"   and f :: "'b" assumes "subcategory C Arr" shows "subcategory.comp C Arr g f = (if Arr f \<and> Arr g \<and> partial_magma.cod C f = partial_magma.dom C g then C g f else partial_magma.null C)"
push_forall_eval_AllQ: fixes \<phi> :: "atom fm" shows "\<forall>xs. eval (AllQ \<phi>) xs = eval (push_forall (AllQ \<phi>)) xs"
JFset_le: fixes a :: "'d JF1"   and b :: "'d JF2" shows "JF1set a \<subseteq> F1set1 (dtor1 a) \<union> (\<Union> (JF1set ` F1set2 (dtor1 a)) \<union> \<Union> (JF2set ` F1set3 (dtor1 a))) \<and> JF2set b \<subseteq> F2set1 (dtor2 b) \<union> (\<Union> (JF1set ` F2set2 (dtor2 b)) \<union> \<Union> (JF2set ` F2set3 (dtor2 b)))"
primorial_neq_zero: fixes x :: "real" shows "primorial x \<noteq> 0"
valuate_nonpos: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" assumes "\<forall>x\<in>vars p. (0 < coeff p x \<longrightarrow> v x \<le> (0::'a)) \<and> (coeff p x < 0 \<longrightarrow> (0::'a) \<le> v x)" shows "p \<lbrace> v \<rbrace> \<le> (0::'a)"
mult_closed: fixes U :: "'b set"   and V :: "'b set"   and x :: "'b"   and a :: "real" assumes "Subspace.subspace U V"   and "x \<in> U" shows "a \<cdot> x \<in> U"
g: fixes w :: "'a" shows "g w \<ge>p zero_poly"
right_neutral: fixes a :: "'a upred" shows "(a \<and> true) = a"
assms_4: shows "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> M"
wf_send_compose: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and X :: "('d, 'c) Term.term list"   and S' :: "('d, 'c) strand_step list"   and f :: "'d" shows "wf\<^sub>s\<^sub>t V (S @ map Send X @ S') = wf\<^sub>s\<^sub>t V (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S')"
restr_delete_twist: fixes A :: "'c set"   and a :: "'c"   and ps :: "('c \<times> 'd) list" shows "restrict A (delete a ps) = delete a (restrict A ps)"
normalY: shows "distributed M lborel Y (\<lambda>x. ennreal (normal_density \<nu> \<tau> x))"
ipd_impl_not_cd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and j :: "nat"   and k :: "nat"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a" assumes "IFC prob"   and "j \<in> {k..i}"   and "\<pi> j = IFC_def.ipd prob (\<pi> k)" shows "\<not> IFC_def.is_cdi prob i \<pi> k"
ring_iso_memI: fixes R :: "('c, 'd) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and S :: "('e, 'f) ring_scheme" assumes "\<And>x. x \<in> carrier R \<Longrightarrow> h x \<in> carrier S"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"   and "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"   and "bij_betw h (carrier R) (carrier S)" shows "h \<in> ring_iso R S"
cobounded2: fixes a :: "'a"   and b :: "'a" shows "min a b \<le> b"
PolysI_alt: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set" assumes "indets p \<subseteq> X" shows "p \<in> P[X]"
Dube_aux_ge_d: fixes d :: "nat"   and n :: "nat"   and j :: "nat" shows "d \<le> Dube_aux n d j"
crel_vs1_state_of: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g"   and b :: "'h Heap" assumes "crel_vs1 R a b" shows "state_dp_consistency.crel_vs R a (state_of b)"
bchoice_iff: fixes S :: "'a set"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<forall>x\<in>S. \<exists>y. Q x y) = (\<exists>f. \<forall>x\<in>S. Q x (f x))"
ch_by_ord_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and X :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.ch_by_ord \<P> betw f X \<equiv> MinkowskiBetweenness.short_ch \<P> X \<or> MinkowskiBetweenness.long_ch_by_ord betw f X"
mult_isol_var: fixes u :: "'a"   and x :: "'a"   and v :: "'a"   and y :: "'a" assumes "u \<le> x"   and "v \<le> y" shows "u \<parallel> v \<le> x \<parallel> y"
h3: shows "distinct (get_signs (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
sin_x_sin_y: fixes x :: "'a"   and y :: "'a" shows "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> odd n then - (real_of_int ((- 1) ^ (p div 2) * int (p choose n)) / fact p) *\<^sub>R x ^ n * y ^ (p - n) else (0::'a)) sums (sin x * sin y)"
Ssegment_inc1: fixes D :: "'c Order"   and b :: "'c"   and a :: "'c" assumes "Order D"   and "b \<in> carrier D" shows "(a \<prec>\<^bsub>D\<^esub> b \<and> a \<in> carrier D) = (a \<in> Ssegment D b)"
suminf_add: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "summable f"   and "summable g" shows "suminf f + suminf g = (\<Sum>n. f n + g n)"
build_rel_SPEC: fixes M :: "'a nres"   and \<Phi> :: "'b \<Rightarrow> bool"   and \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" assumes "M \<le> SPEC (\<lambda>x. \<Phi> (\<alpha> x) \<and> I x)" shows "M \<le> \<Down> (br \<alpha> I) (SPEC \<Phi>)"
\<gamma>\<alpha>_infl: fixes x :: "'a" shows "x \<le> \<gamma> (\<alpha> x)"
finite_intersection_of_empty: fixes P :: "'c set \<Rightarrow> bool" shows "(finite intersection_of P) UNIV"
R2_a_def: shows "R2_a \<equiv> {: stk, mrk \<leadsto> X, mrk' . X = set stk \<and> (stk, mrk) \<in> Loop' \<and> mrk' = mrk :}"
valid_9: shows "enough_flushs ts\<^sub>s\<^sub>b'"
Gauss_Jordan_column_k_det_P_def: fixes A' :: "'b \<times> nat \<times> (('b, 'c) vec, 'd) vec"   and k :: "nat" shows "Gauss_Jordan_column_k_det_P A' k = (let det_P = fst A'; i = fst (snd A'); A = snd (snd A'); from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if (\<forall>m\<ge>from_nat_i. A $ m $ from_nat_k = (0::'b)) \<or> i = nrows A then (det_P, i, A) else let gauss = Gauss_Jordan_in_ij_det_P A from_nat_i from_nat_k in (fst gauss * det_P, i + 1, snd gauss))"
finite_ordLess_infinite2: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "infinite B" shows "|A| <o |B|"
gcsrp_l_one: fixes N :: "'a set"   and a :: "'a" assumes "Group G"   and "G \<triangleright> N"   and "a \<in> gcsrp_map G N ` set_rcs G N" shows "gcsrp_top G N (gcsrp_one G N) a = a"
between_simp1: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "before vs ram1 ram2"   and "pre_between vs ram1 ram2" shows "between vs ram1 ram2 = fst (splitAt ram2 (snd (splitAt ram1 vs)))"
gcd_lcm_factorial_5: fixes a :: "'a"   and b :: "'a" shows "lcm_factorial a b = normalize (a * b div gcd_factorial a b)"
lift_redundant_cl: fixes C' :: "'a literal set"   and D :: "'a literal set"   and \<theta> :: "('a \<times> 'a trm) list"   and C :: "'a eclause"   and S :: "'a eclause set"   and \<eta> :: "('a \<times> 'a trm) list"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "C' = subst_cl D \<theta>"   and "redundant_clause C S \<eta> C'"   and "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>"   and "finite D" shows "redundant_clause C S \<sigma> D"
Rep_name_cases: fixes y :: "atom" assumes "y \<in> {a. sort_of a = Sort ''name'' []}" obtains x :: "name" where "y = Rep_name x"
bchoice_iff: fixes S :: "'d set"   and Q :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(\<forall>x\<in>S. \<exists>y. Q x y) = (\<exists>f. \<forall>x\<in>S. Q x (f x))"
hyps: fixes i :: "nat"   and S :: "nat \<Rightarrow> bool option" assumes "i < length ls"   and "ls ! i = (p, Fence # is, \<theta>, [], \<D>, \<O>, \<R>)" shows "share_all_until_volatile_write (ls[i := (p, is, \<theta>, [], \<D>', \<O>, \<R>')]) S = share_all_until_volatile_write ls S"
locally_path_connected_space_discrete_topology: fixes U :: "'c set" shows "locally_path_connected_space (discrete_topology U)"
axioms_1: fixes G :: "('f, 'g) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
c_in_is_pr: shows "c_in \<in> PrimRec2"
SubstMakeForm_def: fixes y :: "hf"   and y' :: "hf"   and u :: "hf"   and u' :: "hf"   and w :: "hf"   and w' :: "hf" shows "SubstMakeForm y y' u u' w w' \<equiv> y = q_Disj u w \<and> y' = q_Disj u' w' \<or> y = q_Neg u \<and> y' = q_Neg u' \<or> y = q_Ex u \<and> y' = q_Ex u'"
cases: fixes x :: "('c \<times> 'd) list" obtains "x = []"   | a :: "'c" and b :: "'d" and ps :: "('c \<times> 'd) list" where "x = (a, b) # ps"
perpendicular_to_x_axis_y_axis: fixes z :: "complex" assumes "perpendicular_to_x_axis (poincare_line 0\<^sub>h (of_complex z))"   and "z \<noteq> 0" shows "is_imag z"
insert_msg_id: fixes i :: "'a"   and e :: "'a \<times> 'b \<times> bool"   and n :: "'a option"   and j :: "nat" assumes "Broadcast (i, Insert e n) \<in> set (history j)" shows "fst e = i"
encode12_inv: fixes k :: "nat" assumes "k < d" shows "encode12 (encode1 k, encode2 k) = k"
Cantor_nf_unique: fixes \<alpha>s :: "V list"   and ms :: "nat list"   and \<beta>s :: "V list"   and ns :: "nat list" assumes "Cantor_sum \<alpha>s ms = Cantor_sum \<beta>s ns"   and "list.set \<alpha>s \<subseteq> ON"   and "list.set \<beta>s \<subseteq> ON"   and "list.set ms \<subseteq> {0<..}"   and "list.set ns \<subseteq> {0<..}"   and "length \<alpha>s = length ms"   and "length \<beta>s = length ns"   and "sorted_wrt (\<lambda>x y. y < x) \<alpha>s"   and "sorted_wrt (\<lambda>x y. y < x) \<beta>s" shows "\<alpha>s = \<beta>s \<and> ms = ns"
allUses_def: fixes \<alpha>n :: "'c \<Rightarrow> 'd list"   and inEdges' :: "'c \<Rightarrow> 'd \<Rightarrow> ('d \<times> 'e) list"   and "uses" :: "'c \<Rightarrow> 'd \<Rightarrow> 'b set"   and phis :: "'c \<Rightarrow> 'd \<times> 'b \<Rightarrow> 'b list option"   and g :: "'c"   and n :: "'d" shows "CFG_SSA_base.allUses \<alpha>n inEdges' uses phis g n \<equiv> uses g n \<union> CFG_SSA_base.phiUses \<alpha>n inEdges' phis g n"
KL_eq_0_iff_eq: fixes D :: "'a \<Rightarrow> real" assumes "prob_space (density M (\<lambda>x. ennreal (D x)))"   and "random_variable borel D"   and "AE x in M. 0 \<le> D x"   and "integrable M (\<lambda>x. D x * log b (D x))" shows "(KL_divergence b M (density M (\<lambda>x. ennreal (D x))) = 0) = (density M (\<lambda>x. ennreal (D x)) = M)"
ImG: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleIso G smult V smult' T W" shows "T ` V = W"
unit_factor_1_imp_normalized: fixes x :: "'b" assumes "unit_factor x = (1::'b)" shows "normalize x = x"
simps_3: fixes a :: "'b"   and b :: "'b" shows "Some a + Some b = Some (a + b)"
all_unshared_append: fixes xs :: "'b memref list"   and ys :: "'b memref list" shows "all_unshared (xs @ ys) = all_unshared xs \<union> all_unshared ys"
normal_series_prefix_closed: fixes i :: "nat" assumes "i \<le> length (take n \<GG>)"   and "0 < i" shows "normal_series (G\<lparr>carrier := \<GG> ! (n - 1), carrier := take n \<GG> ! (i - 1)\<rparr>) (take i (take n \<GG>))"
trivial_hom_iff: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "group_hom G H h" shows "(h ` carrier G = {\<one>\<^bsub>H\<^esub>}) = (kernel G H h = carrier G)"
lmeasurable_iff_has_integral: fixes S :: "'a set" shows "(S \<in> lmeasurable) = (indicat_real S has_integral Sigma_Algebra.measure lebesgue S) UNIV"
eq_fold: fixes M :: "'b multiset" shows "\<Sum>\<^sub># M = fold_mset (+) (0::'b) M"
mod_eqE: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo a c = modulo b c" obtains d :: "'b" where "b = plus a (times c d)"
is_option_of_def: shows "is_option_of \<equiv> {(opt, g) |opt g. g \<in> games_gfp \<and> (left_option g opt \<or> right_option g opt)}"
is_measure_trivial: fixes f :: "'a \<Rightarrow> nat" shows "is_measure f"
distance_det: fixes n :: "'node"   and n' :: "'node"   and x :: "nat"   and x' :: "nat" assumes "distance n n' x"   and "distance n n' x'" shows "x = x'"
interior_of_subset_topspace: fixes X :: "'c topology"   and S :: "'c set" shows "X interior_of S \<subseteq> topspace X"
invar_app_single: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.invar bq"   and "\<forall>t\<in>set bq. SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank t'"   and "SkewBinomialHeapStruc.tree_invar t'" shows "SkewBinomialHeapStruc.invar (bq @ [t'])"
word_le_def: fixes a :: "'b word"   and b :: "'b word" shows "(a \<le> b) = (uint a \<le> uint b)"
has_codomain_iff_arr: fixes f :: "'a" shows "(L.codomains f \<noteq> {}) = L.arr f"
promela_props_ltl_def: fixes APs :: "expr iarray"   and g :: "gState" shows "promela_props_ltl APs g \<equiv> Collect (propValid APs g)"
maxi_imp_non_empty: fixes xs :: "'a list list" assumes "maxi xs" shows "[] \<notin> set xs"
\<epsilon>_in_terms_of_\<psi>: fixes x :: "'c" assumes "C.ide x" shows "\<epsilon> x = \<psi> x (G x)"
Union_supports_set: fixes S :: "'a set" shows "\<Union> (supp ` S) supports S"
less_one_multI: fixes e :: "real"   and x :: "real" assumes "e \<le> 1"   and "0 < x"   and "x < 1" shows "e * x < 1"
sigma_sets_superset_generator: fixes A :: "'c set set"   and X :: "'c set" shows "A \<subseteq> sigma_sets X A"
elims_1: fixes x :: "nat"   and xa :: "nat"   and y :: "bool" assumes "inside x xa = y" obtains z :: "nat" where "x = za__" and "xa = 0" and "y = (0 < za__)"   | z :: "nat" and n :: "nat" where "x = z" and "xa = Suc n" and "y = inside (pdec2 z) n"
fract_expand_1: shows "0 = Fract (0::'b) (1::'b)"
MGT_def: fixes c :: "stmt"   and Z :: "state" shows "MGT c Z \<equiv> ((=) Z, c, \<lambda>t. \<exists>n. Z -c-n\<rightarrow> t)"
s: shows "s \<in> real_of_ereal ` {a<..<b}"
sup0: fixes a :: "'b" shows "a \<equiv> a \<squnion> \<bottom>"
elims_3: fixes x :: "('c \<times> nat) option"   and xa :: "'c"   and xb :: "lock_action list" assumes "\<not> lock_actions_ok' x xa xb" obtains l :: "('c \<times> nat) option" and t :: "'c" and L :: "lock_action" and Ls :: "lock_action list" where "x = l" and "xa = t" and "xb = L # Ls" and "\<not> (L = Lock \<and> \<not> may_lock l t \<or> lock_action_ok l t L \<and> lock_actions_ok' (upd_lock l t L) t Ls)"
unitary_adj_eq_inv: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary M = (mat_det M \<noteq> 0 \<and> mat_adj M = mat_inv M)"