hcju/isabelleps huggingface dataset:
euclidean_size_numeral: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and k :: "num" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size (numeral.numeral one plus k) = numeral k"
signed_imp_used: fixes M :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Crypt (priSK M) (Hash X) \<in> parts (knows Spy evs)"   and "M \<notin> bad"   and "evs \<in> set_pur" shows "parts {X} \<subseteq> used evs"
I: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option" assumes "Prim_Invar2_ctd_Presentation_Loc w g r Q \<pi>" shows "Prim2.prim_invar2_ctd w g r Q \<pi>"
cSup_eq: fixes X :: "'d set"   and c :: "'d" assumes "\<forall>x\<in>X. x \<le> c"   and "\<exists>x\<in>X. c \<le> x" shows "Sup X = c"
Awhile_6: shows "finite (varacom ({A} WHILE b DO C))"
intro: fixes x :: "'b \<Longrightarrow> 'c" assumes "weak_lens x"   and "bij_lens_axioms x" shows "bij_lens x"
galois: fixes c :: "'a"   and a :: "'b" shows "(c \<le> \<gamma> a) = (\<alpha> c \<le> a)"
R_is_R2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "P is R healthy" shows "P is R2 healthy"
l3_inv1D: fixes x :: "l3_state" assumes "x \<in> l3_inv1" shows "parts (ik x) \<inter> range LtK \<subseteq> ik x"
between_next_empty: fixes f :: "face"   and v :: "nat" assumes "distinct (vertices f)" shows "between (vertices f) v (f \<bullet> v) = []"
ord_strict_pE: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<prec>\<^sub>p q" obtains v :: "'t" where "lookup p v = (0::'c)" and "lookup q v \<noteq> (0::'c)" and "\<And>u. v \<prec>\<^sub>t u \<Longrightarrow> lookup p u = lookup q u"
prime_factorization_unique_nat: fixes S :: "nat set"   and f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "S = {p. 0 < f p}"   and "finite S"   and "\<forall>p\<in>S. prime p"   and "n = (\<Prod>p\<in>S. p ^ f p)" shows "S = prime_factors n \<and> (\<forall>p. prime p \<longrightarrow> f p = multiplicity p n)"
homeomorphic_imp_retraction_maps: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "homeomorphic_maps X Y f g" shows "retraction_maps X Y f g"
interior_def: fixes S :: "'c set" shows "interior S = \<Union> {T. open T \<and> T \<subseteq> S}"
iso_homology_contractible_space_subtopology1: fixes X :: "'a topology"   and S :: "'a set" assumes "contractible_space X"   and "S \<subseteq> topspace X"   and "S \<noteq> {}" shows "homology_group 0 (subtopology X S) \<cong> integer_group \<times>\<times> relative_homology_group 1 X S"
full_elims_3: fixes n :: "nat"   and l :: "'b tree234"   and p :: "'b"   and m :: "'b tree234"   and q :: "'b"   and r :: "'b tree234" assumes "full n \<langle>l, p, m, q, r\<rangle>" obtains na :: "nat" where "n = Suc na" and "full na l" and "full na m" and "full na r"
bex_triv_one_point2: fixes A :: "'g set"   and a :: "'g" shows "(\<exists>x\<in>A. a = x) = (a \<in> A)"
finite_cosets_H: shows "finite (rcosets\<^bsub>G\<^esub> H)"
simps: fixes a1 :: "real"   and a2 :: "(real \<times> 'a) stream" shows "wf_times a1 a2 = (\<exists>t t' \<omega> s'. a1 = t \<and> a2 = (t', s') ## \<omega> \<and> t < t' \<and> wf_times t' \<omega>)"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.arr f"   and "C.dom f = a"   and "C.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"
vimage2pI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and x :: "'e"   and g :: "'f \<Rightarrow> 'd"   and y :: "'f" assumes "R (f x) (g y)" shows "vimage2p f g R x y"
subspace_dim_equal: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" assumes "vec.subspace S"   and "vec.subspace T"   and "S \<subseteq> T"   and "vec.dim T \<le> vec.dim S" shows "S = T"
f_id: fixes s :: "('a, 'U, 'b) state" shows "f s s"
nat_mult_min_left: fixes m :: "nat"   and n :: "nat"   and q :: "nat" shows "min m n * q = min (m * q) (n * q)"
leg1_commutes: shows "\<nu>\<mu>.cod.leg1 \<cdot> \<nu>\<mu>.chine = \<nu>\<mu>.dom.leg1"
lincomb_list_indpt_distinct: fixes ls :: "'a vec list" assumes "\<And>v. v \<in> set ls \<Longrightarrow> dim_vec v = n"   and "\<And>c. lincomb_list c ls = 0\<^sub>v n \<Longrightarrow> \<forall>i<length ls. c i = (0::'a)" shows "distinct ls"
sin_Arg2pi: fixes z :: "complex" shows "cmod z * sin (Arg2pi z) = Im z"
Diag_implies_Arr: fixes t :: "'c local.term" assumes "Diag t" shows "Arr t"
pred_alt_def_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "vwellformed (Vabs cs \<Gamma>) = (wellformed_venv \<Gamma> \<and> wellformed_clauses cs)"
emeasure_pmf_of_set: fixes A :: "'a set" shows "emeasure (measure_pmf local.pmf_of_set) A = ennreal (real (card (S \<inter> A)) / real (card S))"
facI: fixes u :: "'b list"   and p :: "'b list"   and s :: "'b list" shows "u \<le>f p \<cdot> u \<cdot> s"
tau_seq_prop: fixes x :: "'a"   and y :: "'a" assumes "\<tau> x = x" shows "x \<cdot> y = x"
hd_stake: fixes i :: "nat"   and \<pi> :: "'a stream" assumes "0 < i" shows "hd (stake i \<pi>) = shd \<pi>"
ex_in_SPRAY_path: shows "\<exists>y. \<exists>R\<in>SPRAY x. y \<in> R"
A1: shows "\<forall>ls. drop n ls = nat_to_list (c_drop n (list_to_nat ls))"
det2_def: fixes a11 :: "'a"   and a12 :: "'a"   and a21 :: "'a"   and a22 :: "'a" shows "det2 a11 a12 a21 a22 \<equiv> a11 * a22 - a12 * a21"
fold_congs_7: fixes r :: "('b, 'c) sparc_state_scheme"   and r' :: "('b, 'c) sparc_state_scheme"   and v' :: "(int \<times> word32 \<times> CPU_register) list"   and f :: "(int \<times> word32 \<times> CPU_register) list \<Rightarrow> (int \<times> word32 \<times> CPU_register) list"   and f' :: "(int \<times> word32 \<times> CPU_register) list \<Rightarrow> (int \<times> word32 \<times> CPU_register) list" assumes "r = r'"   and "dwrite r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "dwrite_update f r = dwrite_update f' r'"
antidom_dom: fixes x :: "'a" shows "a (d x) = a x"
node_ptr_kinds_eq3: shows "node_ptr_kinds h = node_ptr_kinds h'"
axioms_2: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "ring_of_sets \<Omega> M" shows "ring_of_sets_axioms M"
C: shows "C = ''y'' ::= N 0;; ''z'' ::= N 0;; {(I1, S1, E1)} WHILE Less (V ''z'') (V ''a'') DO (''g'' ::= N 0;; {(I2, S2, E2)} WHILE Less (V ''g'') (V ''b'') DO (''y'' ::= Plus (V ''y'') (N 1);; ''g'' ::= Plus (V ''g'') (N 1));; ''z'' ::= Plus (V ''z'') (N 1))"
evaluate_list_clock_monotone: fixes eval :: "'b state \<Rightarrow> exp0 \<Rightarrow> 'b state \<times> (v, v) result"   and s :: "'b state"   and es :: "exp0 list" shows "clock (fst (Evaluate_Single.evaluate_list eval s es)) \<le> clock s"
oalist_inv_raw_map_pair: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and f :: "'a \<times> 'd \<Rightarrow> 'a \<times> 'e" assumes "oalist_inv_raw x xs"   and "\<And>a b. key_compare (rep_key_order x) (fst (f a)) (fst (f b)) = key_compare (rep_key_order x) (fst a) (fst b)" shows "oalist_inv_raw x (map_pair f xs)"
simps: fixes xs :: "bool list" shows "harry_meas xs = (if xs = [] then 0 else if hd xs then 1 + harry_meas (tl xs) else if \<not> last xs then harry_meas (butlast xs) else let n = length xs in harry_meas (take (n - 2) (tl xs)) + 2 * n - 1)"
linear_map_def: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and W :: "('b, 'f, 'g) module_scheme"   and T :: "'d \<Rightarrow> 'f" shows "linear_map K V W T \<equiv> vectorspace K V \<and> vectorspace K W \<and> mod_hom K V W T"
X_poly_minus_nat_pow_closed: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "X_poly_minus R a [^]\<^bsub>UP R\<^esub> n \<in> carrier (UP R)"
cod_simp: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.arr f" shows "VVV.cod f = VxVxV.cod f"
these_def: fixes A :: "'a option set" shows "Option.these A = the ` {x \<in> A. x \<noteq> None}"
intro_part: fixes \<Gamma>\<^sub>1 :: "'c constr list"   and n\<^sub>1 :: "nat"   and \<Psi>\<^sub>1 :: "'c TESL_atomic list"   and \<Phi>\<^sub>1 :: "'c TESL_atomic list"   and \<Gamma>\<^sub>2 :: "'c constr list"   and n\<^sub>2 :: "nat"   and \<Psi>\<^sub>2 :: "'c TESL_atomic list"   and \<Phi>\<^sub>2 :: "'c TESL_atomic list" assumes "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>i \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2" shows "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow> \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2"
forward_terminating_iff4: fixes x :: "'a" assumes "path x" shows "forward_terminating x = (x \<le> - (1 ; x\<^sup>T) ; x\<^sup>T\<^sup>\<star>)"
gf_def: shows "gf \<equiv> restrict1 (g \<circ> f) (\<Union> A)"
getFresh: fixes V :: "'var set" assumes "finite V" shows "getFresh V \<in> var \<and> getFresh V \<notin> V"
B_norm: fixes i :: "'n"   and j :: "'n" shows "local.B $h i $h j = cmod (A $h i $h j)"
bind_pair_pmf: fixes M :: "'c \<times> 'd \<Rightarrow> 'e measure"   and N :: "'e measure"   and A :: "'c pmf"   and B :: "'d pmf" assumes "M \<in> count_space UNIV \<Otimes>\<^sub>M count_space UNIV \<rightarrow>\<^sub>M subprob_algebra N" shows "measure_pmf (pair_pmf A B) \<bind> M = measure_pmf A \<bind> (\<lambda>x. measure_pmf B \<bind> (\<lambda>y. M (x, y)))"
swap_preserves_good: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "good X" shows "good (X #[x \<and> y]_xs)"
isoI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "OrderingSetMap less_eq less less_eq' less' P f"   and "inj_on f P"   and "\<And>a b. \<lbrakk>a \<in> P; b \<in> P; less_eq' (f a) (f b)\<rbrakk> \<Longrightarrow> less_eq a b" shows "OrderingSetIso less_eq less less_eq' less' P f"
frag_cmul_sum: fixes a :: "int"   and b :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and I :: "'d set" shows "frag_cmul a (sum b I) = (\<Sum>i\<in>I. frag_cmul a (b i))"
MakeNT_comp: fixes X :: "'d"   and f :: "('d, 'g, 'e, 'h, 'f, 'i) NatTrans" assumes "X \<in> obj\<^bsub>NTCatDom f\<^esub>" shows "MakeNT f $$ X = f $$ X"
inf2: fixes B :: "'b::semilattice_inf"   and k :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf"   and a :: "'b::semilattice_inf" assumes "B::'b::semilattice_inf \<equiv> (k::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf)" shows "(a::'b::semilattice_inf) \<sqinter> (B::'b::semilattice_inf) \<equiv> (k::'b::semilattice_inf) \<sqinter> (a \<sqinter> (b::'b::semilattice_inf))"
nakk': shows "isnpolyh (a ^\<^sub>p k' - k) 0"
pcorresI: fixes G :: "'d \<Rightarrow> bool"   and A :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and \<phi> :: "'d \<Rightarrow> 'c"   and B :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real" assumes "\<And>Q. unitary Q \<Longrightarrow> \<guillemotleft> G \<guillemotright> && (wp A Q \<circ> \<phi>) = \<guillemotleft> G \<guillemotright> && wp B (Q \<circ> \<phi>)" shows "pcorres \<phi> G A B"
fold_insert_idem2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b set"   and z :: "'c"   and x :: "'b" assumes "comp_fun_idem f"   and "finite A" shows "Finite_Set.fold f z (insert x A) = Finite_Set.fold f (f x z) A"
finprod_Suc: fixes f :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "f \<in> {..Suc n} \<rightarrow> carrier (ideals_set R)" shows "finprod (ideals_set R) f {..Suc n} = f (Suc n) \<otimes>\<^bsub>ideals_set R\<^esub> finprod (ideals_set R) f {..n}"
simps_4: fixes a1 :: "'a aexp"   and a2 :: "'a aexp"   and s :: "'a \<Rightarrow> value option" shows "aval (Minus a1 a2) s = value_minus (aval a1 s) (aval a2 s)"
timestamp_le_weight: fixes n :: "nat"   and C :: "'a literal multiset" shows "n \<le> weight (C, n)"
assms_1: shows "nprv F (imp \<phi> \<psi>)"
not_less: fixes x :: "'a list"   and y :: "'a list" shows "(\<not> x <lex y) = (y \<le>lex x)"
prod_vwb_lens: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'g" assumes "vwb_lens X"   and "vwb_lens Y" shows "vwb_lens (X \<times>\<^sub>L Y)"
0: shows "(length ps - 1) div 2 < length (map (\<lambda>p. p $ k) ps)"
span_UNIV: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "module scale" shows "module.span scale UNIV = UNIV"
semiring_norm_160: shows "(1::'c) \<noteq> (0::'c)"
simps: fixes l :: "rat"   and r :: "rat"   and sr :: "rat" shows "local.tighten_poly_bounds_for_x l r sr = (if x < l \<or> r < x then (l, r, sr) else case local.tighten_poly_bounds l r sr of (l', r', sr') \<Rightarrow> local.tighten_poly_bounds_for_x l' r' sr')"
eSuc_numeral: fixes k :: "num" shows "eSuc (numeral k) = numeral (k + num.One)"
split_sel_asm: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and option :: "'e option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> option = Some (the option) \<and> \<not> P (f2 (the option))))"
Uf: fixes t :: "'a" assumes "t \<in> S - U" shows "open (Uf t) \<and> Uf t \<inter> S = {x \<in> S. 0 < pf t x}"
l_new_element_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs \<equiv> l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
tm_wf_t_wcode_main: shows "tm_wf0 t_wcode_main"
rank_invar_cons_down: fixes t :: "('c, 'd) SkewBinomialTree"   and bq :: "('c, 'd) SkewBinomialTree list" assumes "rank_invar (t # bq)" shows "rank_invar bq"
safe_distance_no_collsion_delta_axioms: shows "safe_distance_no_collsion_delta a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"
is_sorting_def: fixes R :: "('a \<times> 'a) set"   and xs :: "'a list"   and ys :: "'a list" shows "is_sorting R xs ys = (mset xs = mset ys \<and> Linorder_Relations.sorted_wrt R ys)"
eventually_going_toI: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and f :: "'b \<Rightarrow> 'a" assumes "eventually P F" shows "\<forall>\<^sub>F x in f going_to F. P (f x)"
Joints_mapE: fixes ys :: "'b list"   and YS :: "('b \<times> 'b pdevs) list" assumes "ys \<in> Joints YS" obtains e :: "nat \<Rightarrow> real" where "ys = map (aform_val e) YS" and "\<And>i. e i \<in> {- 1..1}"
anonymous_profile_def: shows "anonymous_profile (prefs_from_table xs) = image_mset (weak_ranking \<circ> prefs_from_table xs) (mset_set agents)"
lcoeff_nonzero_deg: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a" assumes "UP_ring R"   and "deg R p \<noteq> 0"   and "p \<in> carrier (UP R)" shows "coeff (UP R) p (deg R p) \<noteq> \<zero>\<^bsub>R\<^esub>"
sym: fixes x :: "'b"   and y :: "'b" shows "invert_order (c1 x y) = c1 y x"
drop_Suc_Cons: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "drop (Suc n) (x # xs) = drop n xs"
Seq_ZObisT_ZObis: fixes indis :: "('a \<times> 'a) set"   and c1 :: "('b, 'c) com"   and d1 :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c2 :: "('b, 'c) com"   and d2 :: "('b, 'c) com" assumes "PL_Indis indis"   and "(c1, d1) \<in> PL_Indis.ZObisT tval aval indis"   and "(c2, d2) \<in> PL_Indis.ZObis tval aval indis" shows "(c1 ;; c2, d1 ;; d2) \<in> PL_Indis.ZObis tval aval indis"
funionM_fUnionM2: fixes x :: "'m"   and A :: "'m fset" shows "funionM x (fUnionM A) = ffold funionM x A"
echelon_form_upt_k_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "echelon_form_upt_k A k = ((\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)) \<and> (\<forall>i j. i < j \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k j k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'a)) < (LEAST n. A $ j $ n \<noteq> (0::'a))))"
sym_shrK: fixes X :: "agent" shows "shrK X \<in> symKeys"
bin_sc_eq_2: fixes n :: "nat" shows "bin_sc n True = ring_bit_operations_class.set_bit n"
is_inf_def: fixes x :: "nat"   and y :: "nat"   and i :: "nat" shows "nat.is_inf x y i = (i \<le> x \<and> i \<le> y \<and> (\<forall>z. z \<le> x \<and> z \<le> y \<longrightarrow> z \<le> i))"
inf_K_S_next_stimulus: fixes a :: "'a"   and b :: "'a"   and s :: "'b"   and t :: "'b" assumes "a \<le>\<^sub>\<K> b \<and> s \<le>\<^sub>\<S> t" shows "\<lambda> (s, a) \<le>\<^sub>\<S> \<lambda> (t, b)"
oalist_inv_sort_oalist_aux: fixes xs :: "('a \<times> 'd) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "oalist_inv (sort_oalist_aux ko xs, ko)"
Label_def: fixes T :: "('a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) \<times> 'a) set" shows "Label T = label ` T"
mult_isol: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<cdot> z \<le> y \<cdot> z"
that: fixes d :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int"   and e :: "((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int"   and n :: "nat" assumes "singular_chain p (subtopology X (S - U)) d"   and "singular_chain p (subtopology X T) e"   and "(singular_subdivision p ^^ n) c = d + e" shows "thesis"
add_diff_cancel_right': fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
simps: fixes y :: "name"   and x :: "tm"   and z :: "name" assumes "atom y \<sharp> (x, z)"   and "atom z \<sharp> x" shows "OrdP x = All2 y x (Var y SUBS x AND All2 z (Var y) (Var z SUBS Var y))"
simps_3: fixes x21 :: "'b"   and x22 :: "'b list" shows "x21 # x22 \<noteq> []"
cont_plus1: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a" assumes "class.plus_cpo below" shows "cont (\<lambda>x. x + y)"
span_explicit: fixes b :: "'b set" shows "vs1.span b = {\<Sum>a\<in>t. r a *a a |t r. finite t \<and> t \<subseteq> b}"
fn_norm_least: fixes f :: "'a \<Rightarrow> real"   and c :: "real" assumes "Function_Norm.continuous E f norm"   and "\<And>x. x \<in> E \<Longrightarrow> \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"   and "0 \<le> c" shows "\<parallel>f\<parallel>\<hyphen>E \<le> c"
gauss_int_code_post_16: fixes b :: "num" shows "Gauss_Int 0 (numeral b) = numeral b * \<i>\<^sub>\<int>"
cotrace_trace: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event llist"   and i :: "nat" assumes "cotrace c t" shows "\<exists>!c'. trace c (Co_Snapshot.ltake i t) c'"
FinGroupRepresentation_FG: assumes "of_nat (card G) \<noteq> (0::'a)" shows "FinGroupRepresentation G (*) group_ring"
wr: shows "well_ordered_set (P \<union> {f p}) (\<sqsubseteq>)"
pair_map_pmf1: fixes f :: "'g \<Rightarrow> 'e"   and A :: "'g pmf"   and B :: "'f pmf" shows "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
Zr_def: shows "Zr = \<lparr>carrier = Zset, pop = \<lambda>n\<in>Zset. \<lambda>m\<in>Zset. m + n, mop = restrict uminus Zset, zero = 0, tp = \<lambda>m\<in>Zset. restrict ((*) m) Zset, un = 1\<rparr>"
parZero: fixes P :: "pi" shows "P \<parallel> \<zero> \<approx> P"
map_sel: fixes f1 :: "'f \<Rightarrow> 'e"   and f2 :: "'g \<Rightarrow> 'd"   and a :: "('f, 'g) optionT" shows "run_option (map_optionT' f1 f2 a) = f2 (run_option a)"
exhaust: fixes y :: "('d, 'e, 'f) generat" obtains x1 :: "'d" where "y = Pure x1"   | x21 :: "'e" and x22 :: "'f" where "y = IO x21 x22"
inj_image_subset_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj f" shows "(f ` A \<subseteq> f ` B) = (A \<subseteq> B)"
bconf_def: fixes P :: "'c prog"   and sh :: "sheap"   and e :: "'d exp"   and b :: "bool" shows "P,sh \<turnstile>\<^sub>b (e,b) \<surd> \<equiv> b \<longrightarrow> (\<exists>C. icheck P C (ss_exp e) \<and> initPD sh C)"
gt'_trans: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "gt' z y"   and "gt' y x" shows "gt' z x"
feq: fixes x :: "'a" assumes "x \<in> S" shows "f x = exp (g x)"
add_eq_0_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "(plus a b = zero) = (b = uminus a)"
dom_def: fixes m :: "'a \<Rightarrow> 'b option" shows "dom m = {a. m a \<noteq> \<bottom>}"
take_eq_Nil: fixes n :: "nat"   and xs :: "'g list" shows "(take n xs = []) = (n = 0 \<or> xs = [])"
T_APP: fixes \<Gamma> :: "(W.var \<times> tyS) list"   and t\<^sub>1 :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty"   and t\<^sub>2 :: "trm" assumes " \<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1\<rightarrow>T\<^sub>2 "   and " \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>1 " shows " \<Gamma> \<turnstile> App t\<^sub>1 t\<^sub>2 : T\<^sub>2 "
cqt_further_5: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>"   and \<psi> :: "'b \<Rightarrow> \<o>" shows "[(\<^bold>\<exists>\<alpha>. \<phi> \<alpha> \<^bold>& \<psi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<exists>\<alpha>. \<phi> \<alpha>) \<^bold>& (\<^bold>\<exists>\<alpha>. \<psi> \<alpha>) in v]"
l: shows "limsup (\<lambda>n. ereal (norm (f n * of_real r ^ n) / norm (f (Suc n) * of_real r ^ Suc n))) = c * ereal (inverse r)"
FIXvarFIX': fixes b :: "BExpr"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" shows "PhiWhile b \<Phi> = (\<lambda>(s, t, \<beta>). (b, \<Phi>, \<beta>, s, t) \<in> var)"
simps: fixes l :: "name"   and s :: "tm"   and k :: "tm"   and sl :: "name"   and m :: "name"   and n :: "name"   and sm :: "name"   and sn :: "name"   and vf :: "bool"   and t :: "tm" assumes "atom l \<sharp> (s, k, sl, m, n, sm, sn)"   and "atom sl \<sharp> (s, m, n, sm, sn)"   and "atom m \<sharp> (s, n, sm, sn)"   and "atom n \<sharp> (s, sm, sn)"   and "atom sm \<sharp> (s, sn)"   and "atom sn \<sharp> s" shows "SeqCTermP vf s k t = LstSeqP s k t AND All2 l (SUCC k) (SyntaxN.Ex sl (HPair (Var l) (Var sl) IN s AND (Var sl EQ Zero OR (if vf then VarP (Var sl) else Fls) OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sn (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (Var sm) IN s AND HPair (Var n) (Var sn) IN s AND Var sl EQ Q_Eats (Var sm) (Var sn))))))))"