hcju/isabelleps huggingface dataset:
CD: fixes k :: "nat" shows "\<exists>m n. from_nat_into (\<Union>m. range (D m)) k = D m n"
rell_F_mono_strong: fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L3 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and x :: "('n, 'p, 'r, 't, 'u, 'v, 'w, 'x, 'y, 'z) F"   and y :: "('o, 'q, 's, 't, 'u, 'v, 'w, 'x, 'y, 'z) F"   and L1' :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L3' :: "'r \<Rightarrow> 's \<Rightarrow> bool" assumes "rell_F L1 L2 L3 x y"   and "\<And>a b. \<lbrakk>a \<in> set1_F x; b \<in> set1_F y; L1 a b\<rbrakk> \<Longrightarrow> L1' a b"   and "\<And>a b. \<lbrakk>a \<in> set2_F x; b \<in> set2_F y; L2 a b\<rbrakk> \<Longrightarrow> L2' a b"   and "\<And>a b. \<lbrakk>a \<in> set3_F x; b \<in> set3_F y; L3 a b\<rbrakk> \<Longrightarrow> L3' a b" shows "rell_F L1' L2' L3' x y"
C_merge_adj: fixes xss :: "'c list list"   and m :: "nat" assumes "\<forall>xs\<in>set xss. length xs = m" shows "C_merge_adj xss \<le> m * length xss"
fmempty_def: shows "fmempty \<equiv> Abs_fmap Map.empty"
alpha_abs_res_minimal: fixes as :: "atom set"   and x :: "'a"   and p :: "perm"   and as' :: "atom set"   and x' :: "'a" assumes "(as, x) \<approx>res (=) supp p (as', x')" shows "(as \<inter> supp x, x) \<approx>res (=) supp p (as' \<inter> supp x', x')"
dom_tensor: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g" shows "local.dom (f \<otimes> g) = local.dom f \<otimes> local.dom g"
smooth_on_open_Un: fixes k :: "enat"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'a set" assumes "k-smooth_on S f"   and "k-smooth_on T f"   and "open S"   and "open T" shows "k-smooth_on (S \<union> T) f"
SN_on_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "SN_on r A = (\<nexists>f. f 0 \<in> A \<and> (\<forall>i. (f i, f (Suc i)) \<in> r))"
that: fixes s' :: "predicate \<times> object list \<Rightarrow> bool" assumes "compose_actions (map f \<alpha>s) s\<^sub>0 = Some s'"   and "\<forall>fmla\<in>close_world M'. close_eq s' \<Turnstile> fmla" shows "thesis"
f_def: shows "f = (\<lambda>x\<in>\<^sub>\<circ>A. SOME a. a \<in>\<^sub>\<circ> x \<or> x = []\<^sub>\<circ> \<and> a = []\<^sub>\<circ>)"
cUNION_csingle: fixes f :: "'e \<Rightarrow> 'd cset"   and x :: "'e" shows "cUnion (cimage f (csingle x)) = f x"
list: shows "filter (\<lambda>(n, e). e = l) n_as = list"
cvalidt_termD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and s :: "'e" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P Call p Q,A"   and "s \<in> P" shows "\<Gamma>\<turnstile>c \<down> Normal s"
INF_UNIV_bool_expand: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "bool \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (range A) = inf (A True) (A False)"
sublist_valid_plan__alt: fixes as1 :: "'a list"   and PROB :: "'a set"   and as2 :: "'a list" assumes "as1 \<in> valid_plans PROB"   and "subseq as2 as1" shows "as2 \<in> valid_plans PROB"
symm_closure_of_indRelRT_1: fixes TRel :: "('procT \<times> 'procT) set" assumes "refl TRel"   and "sym TRel" shows "indRelT TRel = symcl (indRelRT TRel)"
cnf_BigAnd: fixes ls :: "'a formula list" shows "cnf \<^bold>\<And>ls = \<Union> (cnf ` set ls)"
mkCone_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f0 :: "'a"   and f1 :: "'a"   and p0 :: "'a"   and p1 :: "'a" assumes "cospan_diagram C f0 f1" shows "cospan_diagram.mkCone C f0 f1 p0 p1 \<equiv> \<lambda>j. if j = cospan_shape.AA then p0 else if j = cospan_shape.BB then p1 else if j = cospan_shape.AT then C f0 p0 else if j = cospan_shape.BT then C f1 p1 else if j = cospan_shape.TT then C f0 p0 else partial_magma.null C"
index_mat_2: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'a vec" assumes "i < nr"   and "j < nc" shows "mat\<^sub>r nr nc g $$ (i, j) = g i $ j"
U0\<alpha>_def: shows "U0\<alpha> = \<Sigma> \<times> mrg_left"
ide_dom: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.ide (B.VV.dom f)"
hom_zero: fixes x :: "'c"   and y :: "'c" shows "poly2 0 x y = (0::'c)"
aubs_ex_AMax: fixes A :: "ant set"   and z :: "int" assumes "A \<subseteq> UBset (ant z)"   and "A \<noteq> {}" shows "\<exists>!m. m \<in> A \<and> (\<forall>x\<in>A. x \<le> m)"
finite_nominals: fixes p :: "('c, 'b) fm" shows "finite (nominals p)"
spec1_assoc: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point"   and p\<^sub>3 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "is_generic p\<^sub>1 p\<^sub>2"   and "is_generic p\<^sub>2 p\<^sub>3"   and "is_generic (local.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"   and "is_generic p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3)" shows "local.add a p\<^sub>1 (local.add a p\<^sub>2 p\<^sub>3) = local.add a (local.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
notin_outstanding_non_volatile_takeWhile_lem: fixes a :: "nat"   and sb :: "'b memref list" assumes "a \<notin> outstanding_refs (Not \<circ> is_volatile) sb" shows "a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
simps_67: fixes f1 :: "'f \<Rightarrow> 'e"   and f2 :: "nat set \<Rightarrow> 'f \<Rightarrow> 'e"   and f3 :: "nat \<Rightarrow> 'f \<Rightarrow> 'e"   and f4 :: "nat \<Rightarrow> 'e"   and f5 :: "'f \<Rightarrow> 'e"   and f6 :: "nat \<Rightarrow> 'e"   and f7 :: "'f \<Rightarrow> 'e"   and f8 :: "'e"   and x4 :: "nat" shows "(case \<not>unicast x4 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f4 x4"
gama2_cont_comp: shows "continuous_on {0..1} (\<lambda>x. \<gamma> x \<bullet> b)"
arr_Rel_def: fixes \<alpha> :: "V"   and T :: "V" shows "arr_Rel \<alpha> T \<equiv> (\<Z> \<alpha> \<and> vfsequence T) \<and> vbrelation (T\<lparr>ArrVal\<rparr>) \<and> arr_Rel_axioms \<alpha> T"
pos_def: fixes X :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "pos X = (\<forall>\<psi>. 0 \<le> \<psi> \<bullet>\<^sub>C (X *\<^sub>V \<psi>))"
inject_2: fixes x21 :: "net_tree"   and x22 :: "net_tree"   and y21 :: "net_tree"   and y22 :: "net_tree" shows "(x21 \<parallel> x22 = y21 \<parallel> y22) = (x21 = y21 \<and> x22 = y22)"
sim_cpoke: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and e :: "'event" assumes "S p q"   and "e \<in> E" shows "S (mk_lossless (p \<bind> (\<lambda>s. cpoke core1 s e))) (mk_lossless (q \<bind> (\<lambda>s. cpoke core2 s e)))"
VSigmaD1: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> VSigma A B" shows "a \<in>\<^sub>\<circ> A"
jmk: shows "jordan_matrix (map (\<lambda>(n, a). (n, k * a)) (na # n_as)) = four_block_mat (jordan_block n (k * a)) (0\<^sub>m n (sum_list (map fst n_as))) (0\<^sub>m (sum_list (map fst n_as)) n) (jordan_matrix (map (\<lambda>(n, a). (n, k * a)) n_as))"
simps_1: fixes A :: "'a \<Rightarrow> 'b"   and uu :: "'a \<Rightarrow> 'b"   and uv :: "'a \<Rightarrow> 'b" shows "Dom (MkArr A uu uv) = A"
reach_a: shows "tail G a \<rightarrow>\<^sup>*\<^bsub>H\<^esub> head G a"
theorem3_part2: assumes "\<forall>A. admits_triangular_reduction A" shows "\<forall>a b. \<exists>a1 b1 d. a = a1 * d \<and> b = b1 * d \<and> ideal_generated {a1, b1} = ideal_generated {1::'a}"
app_vrestrictionE: fixes x :: "V"   and r :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<circ> A" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> A" and "b \<in>\<^sub>\<circ> A" and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
BEGIN_END_before: fixes i :: "'b"   and j :: "'b" assumes "\<I> i"   and "\<I> j" shows "BEGIN i \<lless> END j = ((i, j) \<in> e \<union> b \<union> m \<union> ov \<union> ov\<inverse> \<union> s \<union> s\<inverse> \<union> f \<union> f\<inverse> \<union> d \<union> d\<inverse>)"
dagger_unfoldl: fixes x :: "'a" shows "1' + x\<^sup>+ \<le> x\<^sup>\<star>"
right_idem: fixes a :: "'c"   and b :: "'c" shows "a \<squnion> b \<squnion> b = a \<squnion> b"
PiM_component: fixes i :: "'b" assumes "i \<in> UNIV" shows "distr (Pi\<^sub>M UNIV (\<lambda>_. M)) M (\<lambda>\<omega>. \<omega> i) = M"
simps_5: fixes ODE :: "('a, 'c) ODE"   and p :: "('a, 'b, 'c) formula" shows "SIGP (EvolveODE ODE p) = SIGF p \<union> {Inl x |x. Inl x \<in> SIGO ODE} \<union> {Inr (Inr x) |x. Inr x \<in> SIGO ODE}"
tensor_mat_def: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "tensor_mat m1 m2 = mat d d (\<lambda>(i, j). m1 $$ (encode1 i, encode1 j) * m2 $$ (encode2 i, encode2 j))"
im_d_gchains1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and i :: "nat"   and j :: "nat" assumes "0 < n"   and "d_gchain G n f"   and "i \<le> n"   and "f i \<in> f ` {i. i \<le> n} - {f 0}"   and "(LEAST j. f j \<in> f ` {i. i \<le> n} - {f 0}) = j" shows "f ` {i. i \<le> n} = {f 0} \<union> {f i |i. j \<le> i \<and> i \<le> n}"
assms: fixes s :: "char list"   and x :: "'a"   and r :: "char list" assumes "p s = Inr (x, r)" shows "length r < length s"
intros_2: fixes \<Gamma> :: "'b formula multiset" shows "\<bottom>, \<Gamma> \<Rightarrow>\<^sub>n"
card_states: shows "card (states MN.DFA) = card (UNIV // R)"
Reals_of_nat: fixes n :: "nat" shows "of_nat n \<in> \<real>"
allUses_def: fixes g :: "'g"   and n :: "'node" shows "braun_ssa.allUses g n \<equiv> uses' g n \<union> braun_ssa.phiUses g n"
update_map_def: fixes M :: "('b, 'c) Mapping_Str.map"   and k :: "'b"   and v :: "'c" shows " M { k \<rightarrow> v } = Map (\<lambda>x. if x = k then Some v else M\<langle>x\<rangle>)"
restrict_upd_same: fixes m :: "'c \<Rightarrow> 'd option"   and x :: "'c"   and y :: "'d" shows "m(x \<mapsto> y) |` (- {x}) = m |` (- {x})"
residue_one_dvd: fixes a :: "nat"   and n :: "nat" assumes "a mod n = 1" shows "n dvd a - 1"
REP_def: fixes F :: "('a, 'a set) arr" shows "REP F \<equiv> if arr F then SOME f. f \<in> Map F else B.null"
LDF: shows "LeftDerivationFix (b1 @ [X] @ b2) (length b1) D j c"
power2_norm_eq_cinner': fixes x :: "'a" shows "(norm x)\<^sup>2 = Re (cinner x x)"
q_def: fixes x :: "nat"   and y :: "nat" shows "q x y = p x"
Mapping_lookup_map_values: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c option"   and m :: "('d, 'e) mapping"   and k :: "'d" shows "Mapping.lookup (map_values f m) k = Option.bind (Mapping.lookup m k) (f k)"
MapsToObj_2: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and X :: "'a"   and Y :: "'a" assumes "Category C"   and "f maps\<^bsub>C\<^esub> X to Y" shows "Y \<in> obj\<^bsub>C\<^esub>"
StaticCastReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow>* \<langle>\<lparr>C\<rparr>e',s'\<rangle>"
sum_telescope: fixes f :: "nat \<Rightarrow> 'd"   and i :: "nat" shows "(\<Sum>i\<le>i. f i - f (Suc i)) = f 0 - f (Suc i)"
simps_4: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and va :: "'v"   and vb :: "'v" shows "split f (Vertex va, Vertex vb) = 0"
nofail_simps_2: fixes X :: "'a set" shows "nofail (RES X) = True"
cases: fixes a :: "'var"   and S :: "'node set"   and n :: "'node" assumes "a \<in> rv S n" obtains as :: "'edge list" and n' :: "'node" and V :: "'var" where "a = V" and "n -as\<rightarrow>* n'" and "n' \<in> backward_slice S" and "V \<in> Use n'" and "\<forall>nx\<in>set (sourcenodes as). V \<notin> Def nx"
abs_numeral: 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 n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (numeral.numeral one plus n) = numeral.numeral one plus n"
fdomain_def: fixes r :: "V" shows "\<D>\<^sub>\<bullet> r = set {a. \<exists>b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
a_get_shadow_root_locs_def: fixes element_ptr :: "(_) element_ptr" shows "get_shadow_root_locs element_ptr \<equiv> {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr shadow_root_opt)}"
bl_bin_bl_rep_drop: fixes n :: "nat"   and bl :: "bool list" shows "bin_to_bl n (bl_to_bin bl) = replicate (n - length bl) False @ drop (length bl - n) bl"
NES: fixes \<AA> :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<AA> # \<Gamma> \<Rightarrow> \<Theta>" shows "\<Gamma> \<Rightarrow> \<^bold>\<not> \<AA> # \<Theta>"
sub_incidence_system_def: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" shows "sub_incidence_system \<U> \<A> \<V> \<B> \<equiv> sub_set_system \<U> \<A> \<V> \<B> \<and> incidence_system \<U> \<A>"
markedElementsTrailMemPrefixAreMarkedElementsPrefix: fixes M :: "('b \<times> bool) list"   and p :: "('b \<times> bool) list"   and m :: "'b" assumes "uniq (elements M)"   and "isPrefix p M"   and "m \<in> set (elements p)"   and "m \<in> set (markedElements M)" shows "m \<in> set (markedElements p)"
intS: shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) integrable_on S"
simps_2: fixes e :: "char list exp"   and es :: "char list exp list" shows "fvs (e # es) = fv e \<union> fvs es"
collapse: fixes enat :: "enat" assumes "enat \<noteq> 0" shows "eSuc (epred enat) = enat"
producesD: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C = {A}" shows "C \<in> N \<and> C \<noteq> {#} \<and> Pos A = Max_mset C \<and> \<not> ground_resolution_with_selection.interp S N C \<Turnstile> C \<and> S C = {#}"
hom_group_mult: fixes f :: "'b \<Rightarrow> 'a"   and H :: "('b, 'c) monoid_scheme"   and g :: "'b \<Rightarrow> 'a" assumes "f \<in> hom H G"   and "g \<in> hom H G" shows "(\<lambda>x. f x \<otimes>\<^bsub>G\<^esub> g x) \<in> hom H G"
ccLinear_def: fixes S :: "Vars.var set"   and G :: "CoCalls" shows "ccLinear S G = (\<forall>x\<in>S. \<forall>y\<in>S. x--y\<notin>G)"
remdups_concat_remdups: fixes xss :: "'a list list" shows "remdups (concat (remdups xss)) = remdups (concat xss)"
amor_le: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "int (T_merge t1 t2) + \<Phi> (merge t1 t2) - \<Phi> t1 - \<Phi> t2 \<le> int (lrh (merge t1 t2) + rlh t1 + rlh t2 + 1)"
in_set_conv_decomp_first: fixes x :: "'g"   and xs :: "'g list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
dist_0_norm: fixes x :: "'c" shows "dist (0::'c) x = norm x"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>) (\<cdot>) (\<lambda>f. fst (w, f) \<star> snd (w, f))"
drinks_no_possible_steps_1: fixes a :: "String.literal"   and b :: "value list"   and r :: "nat \<Rightarrow>f value option" assumes "\<not> (a = STR ''coin'' \<and> length b = 1)"   and "\<not> (a = STR ''vend'' \<and> b = [])" shows "possible_steps drinks 1 r a b = {||}"
Nadd_commute: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "x +\<^sub>N y = y +\<^sub>N x"
ideD_1: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "arr a"
arith_prog_dvd_mono: fixes b :: "nat"   and b' :: "nat"   and a :: "int" assumes "b dvd b'" shows "arith_prog a b' \<subseteq> arith_prog a b"
member_is_static_simp: fixes m :: "'b member_scheme" shows "is_static m = static m"
redundant_def: fixes C :: "'b Literal set"   and S :: "'b Literal set set" shows "redundant C S = (tautology C \<or> (\<exists>D. D \<in> S \<and> subsumes D C))"
surj_to_test: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "\<forall>b\<in>B. \<exists>a\<in>A. f a = b" shows "surj_to f A B"
disj: shows "\<forall>i\<in>{0..<Suc l}. \<forall>i'\<in>{0..<Suc l}. i \<noteq> i' \<longrightarrow> {p. p permutes {0..<Suc l} \<and> p i = j} \<inter> {p. p permutes {0..<Suc l} \<and> p i' = j} = {}"
analz_crypt_in: fixes K :: "nat"   and H :: "msg set"   and X :: "msg" assumes "Key (invK K) \<in> analz H" shows "analz (insert (Crypt K X) H) = insert (Crypt K X) (analz (insert X H))"
C1_differentiable_on_eq: fixes f :: "real \<Rightarrow> 'b"   and S :: "real set" shows "(f C1_differentiable_on S) = ((\<forall>x\<in>S. f differentiable at x) \<and> continuous_on S (\<lambda>x. vector_derivative f (at x)))"
t_def: shows "t = (\<lambda>x. if s x \<down> \<and> c (the (s x)) \<down> then Some (the (c (the (s x)))) else None)"
substT: fixes t1 :: "'trm"   and t :: "'trm"   and x :: "'var" assumes "t1 \<in> trm"   and "t \<in> trm"   and "x \<in> var" shows "substT t1 t x \<in> trm"
negligible_outer: fixes S :: "'c set" shows "negligible S = (\<forall>e>0. \<exists>T. S \<subseteq> T \<and> T \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue T < e)"
HFinite_st_Infinitesimal_add: fixes x :: "real star" assumes "x \<in> HFinite" shows "\<exists>e\<in>Infinitesimal. x = st x + e"
total_on_lexord_less_than_char_linear: fixes xs :: "char list list"   and ys :: "char list list" assumes "xs \<noteq> ys" shows "((xs, ys) \<notin> lexord (lexord less_than_char)) = ((ys, xs) \<in> lexord (lexord less_than_char))"
counit_in_hom_2: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>counit a : \<tau>\<^sub>0 a \<star>\<^sub>D map\<^sub>0 a \<Rightarrow>\<^sub>D G.map\<^sub>0 a\<guillemotright>"
dist_le_\<mu>: fixes v :: "'a" assumes "v \<in> verts G" shows "dist v \<le> \<mu> c s v"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []"   and "\<And>x xs y ys. (\<not> x \<noteq> y \<Longrightarrow> P xs ys) \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
1: shows "|\<Union> (range V)| <o |UNIV|"
intro: fixes U :: "'a set"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bot :: "'a" assumes "semilattice_sup_ow U sup le ls"   and "order_bot_ow U bot le ls" shows "bounded_semilattice_sup_bot_ow U sup le ls bot"
NullPointer_def: shows "NullPointer = STR ''java/lang/NullPointerException''"
ereal_between_1: fixes x :: "ereal"   and e :: "ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>"   and "0 < e" shows "x - e < x"
succ_ss_V: fixes u :: "nat" shows "E `` {u} \<subseteq> V"
unitary11_gen_cis_inversion: fixes k :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex"   and b :: "complex" assumes "k \<noteq> 0"   and "M = k *\<^sub>s\<^sub>m (0, b, cnj b, 0)"   and "b \<noteq> 0" shows "\<exists>k' \<phi>. k' \<noteq> 0 \<and> M = k' *\<^sub>s\<^sub>m (cis \<phi>, 0, 0, 1) *\<^sub>m\<^sub>m (0, 1, 1, 0)"
gcd_smult: fixes a :: "'b"   and f :: "'b poly"   and g :: "'b poly" shows "gcd (smult a f) g = (if a = (0::'b) \<or> f = 0 then normalize g else smult (gcd a (content g div gcd (content f) (content g))) (gcd f g))"
case_prod_unfold: shows "case_prod = (\<lambda>c p. c (cc p) (snd p))"
callee_in_procs: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "\<exists>ins outs. (p, ins, outs) \<in> set procs"
inint: fixes e :: "real" assumes "0 \<le> e"   and "e < d" shows "a + e *\<^sub>R l \<in> rel_interior S"
ARR_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.ARR C f \<equiv> f \<noteq> {} \<and> (\<forall>t. t \<in> f \<longrightarrow> f = {u. (monoidal_language.Arr C t \<and> monoidal_language.Arr C u \<and> monoidal_language.Dom C t = monoidal_language.Dom C u \<and> monoidal_language.Cod C t = monoidal_language.Cod C u) \<and> monoidal_language.Diagonalize C t = monoidal_language.Diagonalize C u})"
dom_lhs: shows "\<D>\<^sub>\<circ> T = 3\<^sub>\<nat>"
eval_fps_deriv: fixes z :: "'a"   and f :: "'a fps" assumes "ereal (norm z) < fps_conv_radius f" shows "eval_fps (fps_deriv f) z = deriv (eval_fps f) z"
sum_abs_ge_zero: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(0::'b) \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
is_glb_def: fixes s :: "'a"   and s1 :: "'a"   and s2 :: "'a" shows "is_glb s s1 s2 \<equiv> is_lb s s1 s2 \<and> (\<forall>s'. is_lb s' s1 s2 \<longrightarrow> s' \<preceq> s)"
zero_one_inf_x_axis: fixes H :: "circline" assumes "0\<^sub>h \<in> circline_set H"   and "1\<^sub>h \<in> circline_set H"   and "\<infinity>\<^sub>h \<in> circline_set H" shows "H = x_axis"
distinct_1: fixes x1 :: "'b"   and x21 :: "'a"   and x22 :: "('a, 'b) tllist" shows "TNil x1 \<noteq> TCons x21 x22"
R_join: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "x is R healthy"   and "y is R healthy" shows "x \<or> y is R healthy"
countable_separating_set_linorder2: shows "\<exists>B. countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b\<in>B. x \<le> b \<and> b < y))"
valid_deep_model: fixes Y :: "nat"   and r :: "nat"   and rs :: "nat list" shows "valid_net (deep_model Y r rs)"
iMOD_subset_conv: fixes m :: "nat"   and r' :: "nat"   and m' :: "nat"   and r :: "nat" assumes "0 < m" shows "([ r', mod m' ] \<subseteq> [ r, mod m ]) = (r' mod m = r mod m \<and> r \<le> r' \<and> m' mod m = 0)"
lift_prefs_wf: fixes alts :: "'a set"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and alts' :: "'a set" assumes "total_preorder_on alts R"   and "alts \<subseteq> alts'" shows "total_preorder_on alts' (lift_prefs alts alts' R)"
common_range_limit: fixes x :: "nat \<Rightarrow> 'd"   and y :: "nat \<Rightarrow> 'e" assumes "finite (range x)"   and "finite (range y)" obtains i :: "nat" where "limit x = range (suffix i x)" and "limit y = range (suffix i y)"