hcju/isabelleps huggingface dataset:
source: fixes S1 :: "'a"   and S2 :: "'a"   and SRel :: "('a \<times> 'a) set"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" assumes "(S1, S2) \<in> SRel" shows "(SourceTerm S1, SourceTerm S2) \<in> encoding.indRelST Enc SRel TRel"
assms_3: shows "pdevs_domain x \<subseteq> g ` K"
mult_right_zero: fixes x :: "'a" shows "x \<sqinter> bot = bot"
a_distinct_lists_def: fixes h :: "(_) heap" shows "local.CD.a_distinct_lists h = distinct (concat (map (\<lambda>ptr. |h \<turnstile> get_child_nodes ptr|\<^sub>r) |h \<turnstile> object_ptr_kinds_M|\<^sub>r @ map (\<lambda>document_ptr. |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r) |h \<turnstile> document_ptr_kinds_M|\<^sub>r))"
rel_fundefs_None1: fixes f :: "'e \<Rightarrow> ('dyn, 'f, 'g, 'h, 'opinl) Inca.instr fundef option"   and g :: "'e \<Rightarrow> ('dyn, 'f, 'g, 'h, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef option"   and x :: "'e" assumes "rel_fundefs f g"   and "f x = None" shows "g x = None"
that_2: assumes "\<not> c"   and "effect e h h' r" shows "thesis"
sumE: fixes u :: "V"   and A :: "V"   and B :: "V" assumes "u \<in> elts (A \<Uplus> B)" obtains x :: "V" where "x \<in> elts A" and "u = ZFC_Cardinals.Inl x"   | y :: "V" where "y \<in> elts B" and "u = ZFC_Cardinals.Inr y"
empty: assumes "S = {}" shows "RETURN d \<le> SPEC P"
lbound_ofD1: fixes x :: "'b"   and y :: "'b"   and b :: "'b" assumes "dual_order.lbound_of x y b" shows "order.greater_eq b x"
list_assn_aux_append_conv1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and l1 :: "'a list"   and l2 :: "'a list"   and m :: "'b list" shows "list_assn R (l1 @ l2) m = (\<exists>\<^sub>Am1 m2. list_assn R l1 m1 * list_assn R l2 m2 * \<up> (m = m1 @ m2))"
Osum_Partial_order: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "Partial_order r"   and "Partial_order r'" shows "Partial_order (r \<union>o r')"
central_limit_theorem_zero_mean: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and \<sigma> :: "real"   and \<mu> :: "real measure" assumes "indep_vars (\<lambda>i. borel) X UNIV"   and "\<And>n. integrable M (X n)"   and "\<And>n. expectation (X n) = 0"   and "0 < \<sigma>"   and "\<And>n. integrable M (\<lambda>x. (X n x)\<^sup>2)"   and "\<And>n. expectation (\<lambda>x. (X n x - expectation (X n))\<^sup>2) = \<sigma>\<^sup>2"   and "\<And>n. distr M borel (X n) = \<mu>" shows "weak_conv_m (\<lambda>n. distr M borel (\<lambda>x. (\<Sum>i<n. X i x) / sqrt (real n * \<sigma>\<^sup>2))) std_normal_distribution"
ps_first: shows "sturm_squarefree p ! 0 = p'"
select_convs_1: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
typing_induct: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e :: "exp"   and C :: "nat"   and P :: "(nat \<Rightarrow> classDef option) \<Rightarrow> (nat \<Rightarrow> nat option) \<Rightarrow> exp \<Rightarrow> nat \<Rightarrow> bool" assumes "CT;\<Gamma> \<turnstile> e : C"   and "\<And>C CT \<Gamma> x. \<Gamma> x = Some C \<Longrightarrow> P CT \<Gamma> (Var x) C"   and "\<And>C0 CT Cf Ci \<Gamma> e0 fDef fi. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : C0; P CT \<Gamma> e0 C0; fields(CT,C0) = Cf; lookup Cf (\<lambda>fd. vdName fd = fi) = Some fDef; vdType fDef = Ci\<rbrakk> \<Longrightarrow> P CT \<Gamma> (FieldProj e0 fi) Ci"   and "\<And>C C0 CT Cs Ds \<Gamma> e0 es m. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : C0; P CT \<Gamma> e0 C0; mtype(CT,m,C0) = Ds \<rightarrow> C; CT;\<Gamma> \<turnstile>+ es : Cs; \<And>i. i < length es \<Longrightarrow> P CT \<Gamma> (es ! i) (Cs ! i); CT \<turnstile>+ Cs <: Ds; length es = length Ds\<rbrakk> \<Longrightarrow> P CT \<Gamma> (MethodInvk e0 m es) C"   and "\<And>C CT Cs Df Ds \<Gamma> es. \<lbrakk>fields(CT,C) = Df; length es = length Df; varDefs_types Df = Ds; CT;\<Gamma> \<turnstile>+ es : Cs; \<And>i. i < length es \<Longrightarrow> P CT \<Gamma> (es ! i) (Cs ! i); CT \<turnstile>+ Cs <: Ds\<rbrakk> \<Longrightarrow> P CT \<Gamma> (New C es) C"   and "\<And>C CT D \<Gamma> e0. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : D; P CT \<Gamma> e0 D; CT \<turnstile> D <: C\<rbrakk> \<Longrightarrow> P CT \<Gamma> (Cast C e0) C"   and "\<And>C CT D \<Gamma> e0. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : D; P CT \<Gamma> e0 D; CT \<turnstile> C <: D; C \<noteq> D\<rbrakk> \<Longrightarrow> P CT \<Gamma> (Cast C e0) C"   and "\<And>C CT D \<Gamma> e0. \<lbrakk>CT;\<Gamma> \<turnstile> e0 : D; P CT \<Gamma> e0 D; CT \<turnstile> C \<not><: D; CT \<turnstile> D \<not><: C\<rbrakk> \<Longrightarrow> P CT \<Gamma> (Cast C e0) C" shows "P CT \<Gamma> e C"
allocationMonotonicity: fixes N1 :: "'c set"   and N2 :: "'c set"   and \<Omega> :: "'d set" assumes "N1 \<subseteq> N2" shows "allAllocations'' N1 \<Omega> \<subseteq> allAllocations'' N2 \<Omega>"
fmlookup_dom'_iff: fixes x :: "'a"   and m :: "('a, 'b) fmap" shows "(x \<in> fmdom' m) = (\<exists>a. fmlookup m x = Some a)"
max_comm: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.non_strict_order less_eq" shows "ord.max less_eq x y = ord.max less_eq y x"
hyps: assumes "n < length (filter P xs)" shows "\<exists>m\<ge>n. m < length xs \<and> filter P xs ! n = xs ! m \<and> filter P (take m xs) = take n (filter P xs)"
chain_lmap: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and xs :: "'b llist" assumes "\<forall>x y. R x y \<longrightarrow> R' (f x) (f y)"   and "chain R xs" shows "chain R' (lmap f xs)"
of_int_1_less_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less one (ring_1.of_int one plus zero minus z) = (1 < z)"
fmdom'_map: fixes f :: "'f \<Rightarrow> 'e"   and m :: "('d, 'f) fmap" shows "fmdom' (fmmap f m) = fmdom' m"
t2_def: shows "t2 = punit.lt (rep_list q)"
mult_inf_right: fixes z :: "complex_homo" assumes "z \<noteq> 0\<^sub>h" shows "z *\<^sub>h \<infinity>\<^sub>h = \<infinity>\<^sub>h"
minim_Under_under: fixes A :: "'b set" assumes "A \<noteq> {}"   and "A \<subseteq> Field s" shows "s.Under A = s.under (s.minim A)"
notKSQ: shows "sKS m \<notin> specKeysSecrets Q"
is_path_simps_2: fixes v :: "'V"   and w :: "'W"   and v' :: "'V" shows "is_path v [(v, w, v')] v' = ((v, w, v') \<in> E)"
1: fixes b :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "b \<in> set bs" shows "spp_inv b"
trm_subst_ident'': fixes t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "\<And>x. x \<in> fv t \<Longrightarrow> \<theta> x = Var x" shows "t \<cdot> \<theta> = t"
Inf_option_def: fixes A :: "'a option set" shows "\<Sqinter> A = (if None \<in> A then None else Some (\<Sqinter> Option.these A))"
hom_decomp_exact: fixes X :: "'x set"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "finite X"   and "valid_decomp X qs"   and "standard_decomp k qs"   and "hom_decomp qs" shows "hom_decomp (exact X k qs)"
hadd_assoc: fixes x :: "hf"   and y :: "hf"   and z :: "hf" shows "x @+ y @+ z = x @+ (y @+ z)"
\<mu>'_def: fixes i :: "nat"   and j :: "nat" shows "fs.gs.\<mu>' i j \<equiv> fs.gs.d (Suc j) * fs.gs.\<mu> i j"
rec2_3: shows "aux = compl gs bs (ps -- sps) sps (snd data)"
keys_impl_of: fixes t :: "('b, 'c) mapping_rbt" shows "set (RBT_Impl.keys (mapping_rbt.impl_of t)) = RBT_Mapping2.keys t"
distinct_1: fixes x2 :: "'b" shows "Infty \<noteq> Num x2"
assms_2: shows "set_pmf p - {y} \<noteq> {}"
rep_eq: fixes x :: "float" shows "is_float_nonneg x = (0 \<le> real_of_float x)"
simply_connected_eq_continuous_log: fixes S :: "complex set" assumes "open S" shows "simply_connected S = (connected S \<and> (\<forall>f. continuous_on S f \<and> (\<forall>z\<in>S. f z \<noteq> 0) \<longrightarrow> (\<exists>g. continuous_on S g \<and> (\<forall>z\<in>S. f z = exp (g z)))))"
diff_Suc_less: fixes n :: "nat"   and i :: "nat" assumes "0 < n" shows "n - Suc i < n"
simps_7: fixes f1 :: "T1 \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and x1 :: "T1" shows "rec_T2 f1 f2 (C x1) = f1 x1"
sets_completion: fixes M :: "'b measure" shows "sets (completion M) = {uu_. \<exists>S N N'. uu_ = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'}"
s_subid_iff2: fixes R :: "('a \<times> 'a set) set" shows "(R \<subseteq> 1\<^sub>\<sigma>) = (Multirelations.d R = R)"
integrable_minus: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" shows "integrable M (\<lambda>x. - f x)"
funStart_prop2: fixes f :: "'a \<Rightarrow> 'a set"   and x :: "'a" shows "f x \<subseteq> funStar f x"
const_def: shows "const = (\<lambda>x. if x \<in> Field s then r.zero else undefined)"
nonneg_dirichlet_series_axioms: shows "nonneg_dirichlet_series g"
elims_2: fixes x :: "exp" assumes "is_value x" obtains n :: "nat" where "x = ENat n"   | xa :: "nat" and e :: "exp" where "x = ELam xa e" and "FV e = {}"
moebius_decomposition: fixes c :: "complex"   and a :: "complex"   and d :: "complex"   and b :: "complex" assumes "c \<noteq> 0"   and "a * d - b * c \<noteq> 0" shows "mk_moebius a b c d = moebius_translation (a / c) + moebius_rotation_dilatation ((b * c - a * d) / (c * c)) + moebius_reciprocal + moebius_translation (d / c)"
\<psi>\<^sub>2_values_even: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"   and "j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"   and "even i" shows "Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))) $$ (i, j) = complex_of_real ((- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1))"
Vagree_ror: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set"   and W :: "variable set" assumes "Vagree \<nu> \<nu>' (V \<inter> W)" shows "\<exists>\<omega>. Vagree \<nu> \<omega> V \<and> Vagree \<omega> \<nu>' W"
a: fixes x :: "'a \<times> nat"   and y :: "'a \<times> nat" assumes "happened_immediately_before\<^sup>+\<^sup>+ x y" shows "\<exists>i j. i < j \<and> j < length ids \<and> ids ! i = x \<and> ids ! j = y"
fstar_pred_char: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "mono f" shows "fiter_id f x = gfp (\<lambda>y. x \<sqinter> f y)"
field_differentiable_Polygamma: fixes z :: "complex"   and n :: "nat"   and A :: "complex set" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "Polygamma n field_differentiable at z within A"
rel_interior_sing: fixes a :: "'a" shows "rel_interior {a} = {a}"
SUP_mono: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'e set"   and B :: "'f set"   and f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. less_eq (f n) (g m)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
insert_def: fixes x :: "'b"   and t :: "'b tree234" shows "insert x t = tree\<^sub>i (ins x t)"
diffeomorphism_image_eq_1: fixes k :: "enat"   and S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "diffeomorphism k S T f g" shows "f ` S = T"
llist_all2_ldropI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and n :: "enat" assumes "llist_all2 P xs ys" shows "llist_all2 P (ldrop n xs) (ldrop n ys)"
iTILL_inext_if: fixes t :: "nat"   and n :: "nat" shows "inext t [\<dots>n] = (if t < n then Suc t else t)"
test_bit_wi: fixes x :: "int"   and n :: "nat" shows "word_of_int x !! n = (n < LENGTH('b) \<and> bit x n)"
hcmod_mult_i2: fixes y :: "real star" shows "hcmod (hcomplex_of_hypreal y * iii) = \<bar>y\<bar>"
inverse_dimvec_def: fixes x :: "('c, 'd) dimvec" shows "inverse x = dim_lambda (\<lambda>i. - dim_nth x i)"
rank_Gauss_Jordan_code: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A = (if A = 0 then 0 else let A' = Gauss_Jordan A in mod_type_class.to_nat (GREATEST a. row a A' \<noteq> 0) + 1)"
small_seq: fixes c :: "com"   and \<pi> :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and c' :: "com"   and s' :: "char list \<Rightarrow> int \<Rightarrow> int"   and cx :: "com" assumes "c \<noteq> SKIP"   and "small_step \<pi> (c, s) = Some (c', s')" shows "small_step \<pi> (c;; cx, s) = Some (c';; cx, s')"
f''_eq_f: fixes x :: "val"   and \<sigma> :: "nat \<Rightarrow> val" assumes "x \<in> space (stock_measure (\<Gamma> v))"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "extract_real (cexpr_sem (merge (insert v V) V' (\<sigma>(v := x), \<rho>)) e) = extract_real (cexpr_sem (merge V (insert v V') (\<sigma>, \<rho>(v := x))) e)"
assms_2: shows "\<forall>i\<in>I. AEeq M (f i) (g i)"
openin_Times: fixes S :: "'a set"   and S' :: "'a set"   and T :: "'b set"   and T' :: "'b set" assumes "openin (top_of_set S) S'"   and "openin (top_of_set T) T'" shows "openin (top_of_set (S \<times> T)) (S' \<times> T')"
nonces_Nonce: fixes N :: "nat" shows "nonces (Nonce N) = {N}"
g_bigtheta: shows "g \<in> \<Theta>(\<lambda>x. real x powr p')"
thetaSSeqI_Sretr: shows "thetaSSeqI \<subseteq> Sretr (thetaSSeqI \<union> Sbis)"
degen_infs: fixes cs :: "('a \<Rightarrow> bool) list"   and w :: "'a stream"   and k :: "nat" shows "infs (degen cs) (w ||| k ## sscan (count cs) w k) = gen infs cs w"
gap_\<alpha>_def: shows "gap_\<alpha> \<equiv> \<lambda>(l, r, buf). BUF l (take l buf @ drop r buf)"
FF_def: shows "F'.FF \<equiv> \<lambda>\<mu>\<nu>. if C'.VV.arr \<mu>\<nu> then (F (fst \<mu>\<nu>), F (snd \<mu>\<nu>)) else D'.VV.null"
pr_simps_5: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.ide C a"   and "partial_magma.ide C b" shows "partial_magma.dom C (pr1 a b) = elementary_category_with_binary_products.prod C pr0 pr1 a b"
emeasure_lborel_box_eq: fixes l :: "'b"   and u :: "'b" shows "emeasure lborel (box l u) = ennreal (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then prod ((\<bullet>) (u - l)) Basis else 0)"
real_power_down_fl: fixes p :: "nat"   and x :: "float"   and n :: "nat" shows "real_of_float (power_down_fl p x n) = power_down p (real_of_float x) n"
valid_tuple_nt_X: fixes tas :: "nat \<times> 'a option list" assumes "tas \<in> ts_tuple_rel {(nt, X)}" shows "valid_tuple tuple_since' tas"
div_mult_mult1_if: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "divide (times c a) (times c b) = (if c = zero then zero else divide a b)"
ide_compE: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" assumes "ide (COMP g f)" obtains "seq g f" and "seq f g" and "COMP g f = local.dom f" and "COMP g f = cod g"
newton_coefficients_main: fixes k :: "nat" assumes "k < n" shows "local.newton_coefficients_main (rev (map local.f [0..<Suc k])) (rev (map local.x [0..<Suc k])) = rev (map (\<lambda>i. map (\<lambda>j. local.xij_f j i) [0..<Suc i]) [0..<Suc k])"
Quotient3_rel_rep: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs :: "'g \<Rightarrow> 'h"   and Rep :: "'h \<Rightarrow> 'g"   and a :: "'h"   and b :: "'h" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep b) = (a = b)"
null_anga_null_angaP: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "(QCongANullAcute a) = (IsNullAngaP a)"
same_suffix_suffix: fixes ys :: "'a list"   and xs :: "'a list"   and zs :: "'a list" shows "suffix (ys @ xs) (zs @ xs) = suffix ys zs"
pderivs_Star: fixes s :: "'a list"   and r :: "'a rexp" assumes "s \<noteq> []" shows "pderivs s (Star r) \<subseteq> Timess (pderivs_lang (PSuf s) r) (Star r)"
classes_changed_class_changed: fixes cn :: "char list"   and P1 :: "'b prog"   and P2 :: "'b prog" shows "(cn \<in> classes_changed P1 P2) = class_changed P1 P2 cn"
fork: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and e :: "('r, 'l, 'v) expr"   and r' :: "'r" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Rfork e])"   and "r' \<notin> RID\<^sub>G s" shows "revision_step_relaxed r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r')]), r' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e)))"
Range_snd: fixes r :: "('e \<times> 'd) set" shows "Range r = snd ` r"
noVal2_def: fixes K :: "'state \<Rightarrow> 'value \<Rightarrow> bool"   and v :: "'value" shows "noVal2 K v \<equiv> \<forall>s a ou s'. reachNT s \<and> K s v \<and> step s a = (ou, s') \<and> \<phi> (Trans s a ou s') \<longrightarrow> f (Trans s a ou s') \<noteq> v"
Legendre_1mod6: fixes m :: "int" assumes "prime (6 * m + 1)" shows "Legendre (- 3) (6 * m + 1) = 1"
ArrayT: fixes ty :: "ty" assumes "statT = ArrayT ty"   and "dynmethd G Object dynC sig = x" shows "thesis"
Confidentiality_Auth_A: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> kerbIV"   and "\<not> expiredAK Ta evs"   and "A \<notin> bad"   and "evs \<in> kerbIV" shows "Key authK \<notin> analz (knows Spy evs)"
exhaust: fixes y :: "ty" obtains "y = Void"   | "y = Boolean"   | "y = Integer"   | "y = NT"   | x5 :: "char list" where "y = Class x5"
hom_relboundary: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set"   and T :: "'b set"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set" shows "hom_relboundary p X S T c \<in> carrier (relative_homology_group (p - 1) (subtopology X S) T)"
intros_35: fixes \<C> :: "t_context"   and es :: "b_e list"   and t1s :: "t list"   and t2s :: "t list"   and ts :: "t list" assumes "\<C> \<turnstile> es : t1s _> t2s" shows "\<C> \<turnstile> es : ts @ t1s _> ts @ t2s"
freshAbs_substAbs_E2: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y"   and "freshAbs zs z (A $[Y / y]_ys)" shows "freshAbs ys y A \<or> fresh zs z Y"
MOST_SucD: fixes P :: "nat \<Rightarrow> bool" assumes "\<forall>\<^sub>\<infinity>n. P (Suc n)" shows "\<forall>\<^sub>\<infinity>n. P n"
cd: shows "(c, d) \<in> set (half_segments_of_aform X)"
continuous_within_topological: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous (at x within s) f = (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
smc_eqI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "semicategory \<alpha> \<AA>"   and "semicategory \<alpha> \<BB>"   and "\<AA>\<lparr>Obj\<rparr> = \<BB>\<lparr>Obj\<rparr>"   and "\<AA>\<lparr>Arr\<rparr> = \<BB>\<lparr>Arr\<rparr>"   and "\<AA>\<lparr>Dom\<rparr> = \<BB>\<lparr>Dom\<rparr>"   and "\<AA>\<lparr>Cod\<rparr> = \<BB>\<lparr>Cod\<rparr>"   and "\<AA>\<lparr>Comp\<rparr> = \<BB>\<lparr>Comp\<rparr>" shows "\<AA> = \<BB>"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b pmf" shows "set_pmf (map_pmf f v) = f ` set_pmf v"
coprime_of_int_gauss_int: fixes a :: "int"   and b :: "int" shows "coprime (of_int a) (of_int b) = coprime a b"
RG_proj_sum_list_prod: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> ('a, 'g) aezfun"   and xys :: "('b \<times> 'c) list" shows "RG_proj (\<Sum>(x, y)\<leftarrow>xys. f x y) = (\<Sum>(x, y)\<leftarrow>xys. RG_proj (f x y))"
language_state_for_inputs_map_fst: fixes io :: "('d \<times> 'e) list"   and M :: "('d, 'e, 'f) FSM"   and q :: "'f"   and T :: "'d list set" assumes "io \<in> LS M q"   and "map fst io \<in> T" shows "io \<in> LS\<^sub>i\<^sub>n M q T"
essentially_surjective: fixes b :: "'c" assumes "G.B.ide b" shows "\<exists>a. F.B.ide a \<and> G.B.isomorphic (G a) b"
state_eqI: fixes s :: "('b, 'c) state"   and s' :: "('b, 'c) state" assumes "\<B>\<^sub>i\<^sub>l s = \<B>\<^sub>i\<^sub>l s'"   and "\<B>\<^sub>i\<^sub>u s = \<B>\<^sub>i\<^sub>u s'"   and "\<T> s = \<T> s'"   and "\<V> s = \<V> s'"   and "\<U> s = \<U> s'"   and "\<U>\<^sub>c s = \<U>\<^sub>c s'" shows "s = s'"
elims: fixes x :: "'f \<Rightarrow> 'g option"   and xa :: "'g \<Rightarrow> 'f \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and xb :: "('f, 'h, 'i) new_thread_action"   and y :: "'f \<Rightarrow> 'g option" assumes "upd_inv x xa xb = y" obtains I :: "'f \<Rightarrow> 'g option" and P :: "'g \<Rightarrow> 'f \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool" and t :: "'f" and xc :: "'h" and m :: "'i" where "x = I__" and "xa = P__" and "xb = NewThread t xc m" and "y = I__(t \<mapsto> SOME i. P__ i t xc m)"   | I :: "'f \<Rightarrow> 'g option" and P :: "'g \<Rightarrow> 'f \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool" and v :: "'f" and va :: "bool" where "x = I" and "xa = P" and "xb = ThreadExists v va" and "y = I"
AnalBinds_not_there: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "x \<notin> domA \<Gamma>" shows "(FBinds \<Gamma>\<cdot>ae) x = \<bottom>"
finite_dimensional_vector_space_pair_1_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B1 :: "'e set"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f" shows "finite_dimensional_vector_space_pair_1 s1 B1 s2 \<equiv> finite_dimensional_vector_space s1 B1 \<and> vector_space s2"
lexord_take_index_conv: fixes x :: "'b list"   and y :: "'b list"   and r :: "('b \<times> 'b) set" shows "((x, y) \<in> lexord r) = (length x < length y \<and> take (length x) y = x \<or> (\<exists>i<min (length x) (length y). take i x = take i y \<and> (x ! i, y ! i) \<in> r))"
reconstruct_reach_def: fixes Vr :: "'q set"   and Vl :: "'q set" shows "reconstruct_reach Vr Vl \<equiv> find_path (E \<inter> Vr \<times> UNIV) V0 (\<lambda>v. v \<in> Vl) \<bind> (\<lambda>res. ASSERT (res \<noteq> None) \<bind> (\<lambda>_. RETURN (the res)))"
sylvester_mat_sub_carrier: fixes m :: "nat"   and n :: "nat"   and p :: "'a poly"   and q :: "'a poly" shows "sylvester_mat_sub m n p q \<in> carrier_mat (m + n) (m + n)"