hcju/isabelleps huggingface dataset:
comp: fixes S :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "swapidseq_ext S n p"   and "a \<noteq> b" shows "swapidseq_ext (insert a (insert b S)) (Suc n) (Fun.swap a b id \<circ> p)"
elims_3: fixes x :: "'d \<Rightarrow> bool"   and xa :: "'d list" assumes "\<not> successive x xa" obtains P :: "'d \<Rightarrow> bool" and xb :: "'d" and y :: "'d" and xs :: "'d list" where "x = P" and "xa = xb # y # xs" and "\<not> (\<not> P y \<and> successive P xs \<or> \<not> P xb \<and> successive P (y # xs))"
csum_cong2: fixes p2 :: "('c \<times> 'c) set"   and r2 :: "('d \<times> 'd) set"   and q :: "('e \<times> 'e) set" assumes "p2 =o r2" shows "q +c p2 =o q +c r2"
simps_6: fixes x :: "nat"   and v :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and e3 :: "exp" shows "subst x v (EIf e1 e2 e3) = EIf (subst x v e1) (subst x v e2) (subst x v e3)"
assoc: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
hom_add: fixes x :: "'pq mod_ring poly"   and y :: "'pq mod_ring poly" shows "#(x + y) = #x + #y"
bind_mplus: fixes a :: "'c\<cdot>'b"   and b :: "'c\<cdot>'b"   and k :: "'c \<rightarrow> 'a\<cdot>'b" shows "mplus\<cdot>a\<cdot>b \<bind> k = mplus\<cdot>(a \<bind> k)\<cdot>(b \<bind> k)"
enab_gamma1: shows "\<turnstile> $pc1 = #g \<longrightarrow> Enabled gamma1"
step2: fixes B :: "'a set"   and cs :: "'a list"   and C :: "'a set"   and b :: "'a" assumes "B \<subseteq> A"   and "cs \<in> local.mk_linear_orders C (B - {SOME x. x \<in> f B - C})"   and "b \<in> f B"   and "b \<notin> C"   and "P C (B - {SOME x. x \<in> f B - C}) cs" shows "P C B ((SOME x. x \<in> f B - C) # cs)"
r_cancel_one': fixes x :: "nat"   and a :: "nat" assumes "x \<in> carrier (units_of R)"   and "a \<in> carrier (units_of R)" shows "(x = a \<otimes>\<^bsub>units_of R\<^esub> x) = (a = \<one>\<^bsub>units_of R\<^esub>)"
wpQ_WhileFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com" assumes "\<not> bval b s" shows "Q s + 1 \<le> wp\<^sub>Q (WHILE b DO c) Q s"
SN_on_all_reducts_SN_on_conv: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "SN_on r {a} = (\<forall>b. (a, b) \<in> r \<longrightarrow> SN_on r {b})"
BinOpThrow2: fixes 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 e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "expr"   and e :: "expr"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and bop :: "bop" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>2\<rangle>"
indifferent_prefI: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set" assumes "x \<^bsub>r\<^esub>\<preceq> y"   and "y \<^bsub>r\<^esub>\<preceq> x" shows "x \<^bsub>r\<^esub>\<approx> y"
len: shows "\<^bold>|r\<^bold>| + \<^bold>|s\<^bold>| - gcd \<^bold>|r\<^bold>| \<^bold>|s\<^bold>| \<le> \<^bold>|u\<^bold>|"
impl_of_update_with: fixes v :: "'c"   and k :: "'b"   and f :: "'c \<Rightarrow> 'c"   and al :: "('b, 'c) assoc_list" shows "assoc_list.impl_of (Assoc_List.update_with v k f al) = AList.update_with_aux v k f (assoc_list.impl_of al)"
indmhom_surjec1: fixes N :: "('e, 'b, 'f) Module_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "R module N"   and "surjec\<^bsub>M,N\<^esub> f"   and "f \<in> mHom R M N" shows "surjec\<^bsub>M /\<^sub>m (ker\<^bsub>M,N\<^esub> f),N\<^esub> f\<^sup>\<flat>\<^bsub>R M, N\<^esub>"
Zp_defs_1: shows "\<one> = padic_one p"
it_paths_cases: fixes a2 :: "nat list"   and a1 :: "'a rose_tree" assumes "a2 \<in> it_paths a1" obtains t :: "'a rose_tree" where "a1 = t__" and "a2 = []"   | i :: "nat" and t :: "'a rose_tree" and t' :: "'a rose_tree" and "is" :: "nat list" where "a1 = t" and "a2 = i # is" and "i < length (children t)" and "children t ! i = t'" and "is \<in> it_paths t'"
r_3: shows "(\<lambda>i j x. eval r [i, j, x]) = local.adverse z"
bi_unique_fun: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool" assumes "bi_total A"   and "bi_unique B" shows "bi_unique (A ===> B)"
simps_1: shows "pos_arctic_delta MinInfty_delta = False"
xsup: fixes y :: "'a" assumes "y \<in> S" shows "norm y \<le> norm x"
connected_iff_const: fixes S :: "'b set" shows "connected S = (\<forall>P. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c))"
sum_in_bigo_1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](h)"   and "g \<in> O[F](h)" shows "(\<lambda>x. f x + g x) \<in> O[F](h)"
domsdisj: fixes p :: "((int \<times> int) set set, int) Combinators list"   and A :: "((int \<times> int) set set, int) Combinators"   and y :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "p \<noteq> []"   and "\<And>x s. s \<in> set p \<and> x \<in> dom (C A) \<Longrightarrow> x \<notin> dom (C s)"   and "y \<in> dom (C A)" shows "y \<notin> dom (C (list2FWpolicy p))"
a2: shows "\<forall>p\<in>set Ps. \<exists>\<Phi>' \<Psi>' m. m \<le> n' \<and> ( \<Phi>' + \<Gamma>' \<Rightarrow>* \<Psi>' + \<Delta>', m) \<in> derivable (ext R R2 M1 M2) \<and> p = ( \<Phi>' \<oplus> Modal M Ms \<Rightarrow>* \<Psi>')"
butterfly_comp_cblinfun: fixes \<psi> :: "'b"   and \<phi> :: "'c"   and a :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "butterfly \<psi> \<phi> o\<^sub>C\<^sub>L a = butterfly \<psi> (a* *\<^sub>V \<phi>)"
next_action_consistent: shows "\<forall>s t execs. vpeq (current s) s t \<and> (\<forall>d\<in>int.involved (int.next_action s execs). vpeq d s t) \<and> current s = current t \<longrightarrow> int.next_action s execs = int.next_action t execs"
inorder_delete: fixes t :: "'b tree"   and x :: "'b" assumes "True \<and> Sorted_Less.sorted (inorder t)" shows "inorder (delete x t) = del_list x (inorder t)"
pred_inject_1: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" shows "pred_Bonk_Schramm_extension_unfolded P (basepoint a) = P a"
equiv_def: fixes A :: "'e set"   and r :: "('e \<times> 'e) set" shows "equiv A r = (refl_on A r \<and> sym r \<and> trans r)"
list_emb_append2: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "list_emb P xs ys" shows "list_emb P xs (zs @ ys)"
commute: fixes a :: "'b"   and b :: "'b" shows "a OR b = b OR a"
clof: fixes T :: "'a set" assumes "T \<in> \<F>" shows "closed T"
Sup_to_Inf: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and dual :: "'b \<Rightarrow> 'b" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Sup = dual \<circ> Inf \<circ> (`) dual"
ETP_base: fixes n :: "nat" shows "ETP_base (I n) domain range F F\<^sub>i\<^sub>n\<^sub>v"
nn_integral_monotone_convergence_SUP_AE: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "\<And>i. AE x in M. f i x \<le> f (Suc i) x"   and "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M = (SUP i. integral\<^sup>N M (f i))"
decisive1: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} x y" shows "decisive swf A Is {j} x z"
publicKey_neq_privateKey: fixes A :: "agent"   and A' :: "agent" shows "pubK A \<noteq> priK A'"
reduc_fair_imp_Liminf_reduc_sat: fixes Ns :: "'a set llist" assumes "chain (\<rhd>) Ns"   and "reduc_fair Ns" shows "reduc_saturated (Liminf_llist Ns)"
f1: shows "\<pi>\<^sub>* y \<in> carrier \<X>"
hit_bound_def: shows "hit_bound = Lazy_Sequence (\<lambda>_. Some (None, Lazy_Sequence.empty))"
sorted_wrt_upt: fixes m :: "nat"   and n :: "nat" shows "sorted_wrt (<) [m..<n]"
\<Delta>_naturality: shows "(f \<star> tab\<^sub>0 f) \<cdot> f.tab = (f \<star> f.the_\<theta>) \<cdot> \<a>[f, tab\<^sub>0 f, f.chine] \<cdot> (f.tab \<star> f.chine) \<cdot> f.the_\<nu>"
mismatchChain: fixes P :: "pi"   and P' :: "pi"   and a :: "name"   and b :: "name" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "a \<noteq> b"   and "P \<noteq> P'" shows "[a\<noteq>b]P \<Longrightarrow>\<^sub>\<tau> P'"
coclop_def: fixes f :: "'b \<Rightarrow> 'b" shows "coclop f = (f \<le> id \<and> mono f \<and> f \<le> f \<circ> f)"
wo_aux_exists: shows "Card_order wo_aux \<and> infinite (Field wo_aux) \<and> regularCard wo_aux \<and> |UNIV| <o wo_aux"
trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "b \<succeq> a"   and "c \<succeq> b" shows "c \<succeq> a"
struct_map_comp: fixes g :: "'a \<Rightarrow> 'b"   and T :: "('a, 'c) partial_object_scheme"   and S :: "('b, 'd) partial_object_scheme"   and f :: "'b \<Rightarrow> 'e"   and U :: "('e, 'f) partial_object_scheme" assumes "g \<in> struct_maps T S"   and "f \<in> struct_maps S U" shows "compose (carrier T) f g \<in> struct_maps T U"
FF_def: shows "SPN.FF \<equiv> \<lambda>\<mu>\<nu>. if VV.arr \<mu>\<nu> then (SPN (fst \<mu>\<nu>), SPN (snd \<mu>\<nu>)) else Span.VV.null"
B_2: shows "b \<in># A \<or> b \<in># B"
ortho_gs: shows "is_ortho_set (set (map basis_enum_of_vec gs))"
is_extensional: fixes f :: "'a \<times> 'a" assumes "\<not> VV.arr f" shows "fst f \<otimes> snd f = null"
finite_Diff: fixes A :: "'e set"   and B :: "'e set" assumes "finite A" shows "finite (A - B)"
frameResFresh: fixes x :: "name"   and F :: "'b frame" assumes "x \<sharp> F" shows "\<lparr>\<nu>x\<rparr>F \<simeq>\<^sub>F F"
zero_cong: shows "\<zero>\<^bsub>residue_ring (int p)\<^esub> = 0"
B'_subset: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "a \<le> b" shows "B' a \<subseteq> B' b"
refl_onD: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and a :: "'b" assumes "refl_on A r"   and "a \<in> A" shows "(a, a) \<in> r"
VInterD: fixes a :: "V"   and A :: "V"   and x :: "V" assumes "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> A"   and "x \<in>\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> x"
b: shows "\<gamma>\<^sub>T\<^sub>R K (\<phi> .\<and>. \<psi>) \<inter> (K .\<bottom>. \<phi>) \<noteq> {}"
S_o_alpha_type_scheme_list: fixes S :: "nat \<Rightarrow> typ"   and alpha :: "nat \<Rightarrow> nat"   and A :: "type_scheme list" shows "$ (S \<circ> alpha) A = $ S ($ (\<lambda>x. TVar (alpha x)) A)"
assms_2: shows "x \<in> normalize_set_of_vec X"
R'_no_Q: fixes v :: "'a" assumes "v \<in> set R'"   and "v \<noteq> z" shows "\<not> Q.hitting_paths v"
lm076: fixes g :: "('c \<times> 'd) set"   and A :: "'c set"   and B :: "'d set"   and C :: "'c set" assumes "g \<in> injections A B" shows "Outside' C g \<in> injections (A - C) B"
\<xi>_Inf_many: shows "\<exists>\<^sub>\<infinity>k. \<bar>\<xi> - real_of_int (p k) / real_of_int (q k)\<bar> < 1 / real_of_int (q k) powr (2 + \<delta> * \<epsilon> / (1 + \<epsilon>))"
of_char_def: fixes c :: "char" shows "of_char c = horner_sum of_bool (2::'a) [digit0 c, digit1 c, digit2 c, digit3 c, digit4 c, digit5 c, digit6 c, digit7 c]"
id: shows "jnf_vector A = jnf_vector_main n A"
Ext_foo_permits_acc: fixes S :: "qtname" shows "tprg \<turnstile> (Ext, mdecl Ext_foo) in Ext permits_acc_from S"
valid_write_sops_axioms: shows "valid_write_sops ts\<^sub>s\<^sub>b"
simps_1: fixes x11 :: "nat"   and x12 :: "nat"   and y11 :: "nat"   and y12 :: "nat" shows "(picks x11 x12 = picks y11 y12) = (x11 = y11 \<and> x12 = y12)"
3: assumes "\<lbrakk>M; \<A>\<rbrakk>\<^sub>d' \<I>" shows "\<lbrakk>M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>; \<A>\<rbrakk>\<^sub>d \<I>"
row_echelon_def: fixes A :: "'c mat" shows "row_echelon A = fst (gauss_jordan A (1\<^sub>m (dim_row A)))"
right_neutral: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and a :: "'a" assumes "monoid f z" shows "f a z = a"
bddh_bdd_map: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'a"   and bdd :: "'b bdd" shows "bddh n (bdd_map f bdd) = bddh n bdd"
rref_upt_condition4: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'d)) = (0::'d))"
Rep_S: fixes x :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) S" shows "Rep_S x \<in> {x. P x}"
Gauss_Jordan_column_k_def: fixes A' :: "nat \<times> (('d, 'e) vec, 'f) vec"   and k :: "nat" shows "Gauss_Jordan_column_k A' k = (let i = fst A'; A = 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::'d)) \<or> i = nrows A then (i, A) else (i + 1, Gauss_Jordan_in_ij A from_nat_i from_nat_k))"
continuous_map_atin: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map X Y f = (\<forall>x\<in>topspace X. limitin Y f (f x) (atin X x))"
prems_1: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
iT_Div_mem_iff2: fixes k :: "nat"   and x :: "nat"   and I :: "nat set" assumes "0 < k" shows "(x div k \<in> I \<oslash> k) = (\<exists>y\<in>I. y div k = x div k)"
matrix_mul_mat: fixes A :: "(('b, 'c) vec, 'd) vec"   and x :: "'b" shows "A ** mat x = x *k A"
mu1_def: shows "mu1 = monomial ((1::'b) / lc g1) (lcs (lp g1) (lp g2) - lp g1)"
exists_vector_in_Berlekamp_subspace_dvd: fixes P :: "'a mod_ring poly set"   and p_i :: "'a mod_ring poly"   and p_j :: "'a mod_ring poly"   and w :: "'a mod_ring poly" assumes "finite P"   and "u = \<Prod>P"   and "P \<subseteq> {q. irreducible q \<and> monic q}"   and "p_i \<in> P"   and "p_j \<in> P"   and "p_i \<noteq> p_j"   and "monic u"   and "square_free u"   and "\<not> irreducible w"   and "w dvd u"   and "monic w"   and "p_i dvd w"   and "p_j dvd w" shows "\<exists>v. v \<in> {h. [h ^ CARD('a) = h] (mod u) \<and> degree h < degree u} \<and> v mod p_i \<noteq> v mod p_j \<and> degree (v mod p_i) = 0 \<and> degree (v mod p_j) = 0 \<and> (\<exists>s. gcd w (v - [:s:]) \<noteq> w \<and> gcd w (v - [:s:]) \<noteq> 1)"
finite_incidence_system_axioms: shows "finite_incidence_system (\<V> - b) (str_del_block b)"
simps_3: fixes m :: "'a"   and x :: "nat" shows "parent_node (Formal_out (m, x)) = m"
comp: shows "(\<circ>) = (\<lambda>f g x. f (g x))"
assms_2: shows "\<lbrace>\<alpha> u \<cdot> \<tau> w\<rbrace>y\<lbrace>z\<rbrace>"
irfb_inv1: shows "fb\<^sup>-\<^sub>\<R> \<circ> fb\<^sub>\<R> = id"
inclosed_simps_2: fixes \<sigma> :: "'a"   and i :: "nat"   and d :: "nat"   and di :: "nat" shows "inoclosed \<sigma> (i:newpkt(d, di)) = False"
adds_poly_mappingI: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" assumes "lookup s \<le> lookup t" shows "s adds t"
mat_equal_def: fixes A :: "'b mat"   and B :: "'b mat" shows "mat_equal A B = (dim_row A = dim_row B \<and> dim_col A = dim_col B \<and> mat_to_list A = mat_to_list B)"
hyps: fixes es' :: "char list exp list" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"   and "finals es" shows "es = es' \<and> s = s'"
orthogonal_comp_def: fixes W :: "'a set" shows "W\<^sup>\<bottom> \<equiv> {x. \<forall>y\<in>W. orthogonal y x}"
card_less_Suc2: fixes M :: "nat set"   and i :: "nat" assumes "0 \<notin> M" shows "card {k. Suc k \<in> M \<and> k < i} = card {k \<in> M. k < Suc i}"
assms: shows "((f, l), p, f', l') \<in> trcl pr_algo_lts"
ins_order: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "order k t" shows "order_up\<^sub>i k (ins k x t)"
rell_G_def: fixes L1 :: "'j \<Rightarrow> 'q \<Rightarrow> bool"   and L2 :: "'k \<Rightarrow> 'r \<Rightarrow> bool" shows "rell_G L1 L2 = rel_G L1 L2 (=) (=) (=) (=)"
column_gr_k_Gram_Schmidt_upt: fixes k :: "nat"   and i :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "k < i"   and "i < ncols A" shows "column (mod_type_class.from_nat i) (Gram_Schmidt_upt_k A k) = column (mod_type_class.from_nat i) A"
fps_compose_prod_distrib: fixes c :: "'a fps"   and a :: "'b \<Rightarrow> 'a fps"   and S :: "'b set" assumes "fps_nth c 0 = (0::'a)" shows "prod a S oo c = (\<Prod>k\<in>S. a k oo c)"
BVG_nonelem: fixes x :: "variable"   and \<alpha> :: "game" shows "(x \<notin> BVG \<alpha>) = (\<forall>I \<omega> X. (\<omega> \<in> game_sem I \<alpha> X) = (\<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> {x})))"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and el_step :: "'b el_step" shows "P (case el_step of LOC x \<Rightarrow> f1 x | ENV x \<Rightarrow> f2 x) = ((\<forall>x1. el_step = LOC x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. el_step = ENV x2 \<longrightarrow> P (f2 x2)))"
meet_comm: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" shows "x \<sqinter>\<^bsub>L\<^esub> y = y \<sqinter>\<^bsub>L\<^esub> x"
subprob_measurableD_3: fixes N :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and S :: "'d measure"   and x :: "'c"   and K :: "'e measure" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S"   and "x \<in> space M" shows "N x \<rightarrow>\<^sub>M K = S \<rightarrow>\<^sub>M K"
finite_UnionD: fixes A :: "'c set set" assumes "finite (\<Union> A)" shows "finite A"
comp_char: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a" assumes "dual_category C" shows "dual_category.comp C g f = C f g"
generate_valid_topology_def_alt: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "generate_valid_topology M G = delete_edges G (\<Union> (get_offending_flows M G))"
assms_3: shows "\<forall>b\<in>basis. continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"
comp_fun_idem_axioms: shows "comp_fun_idem (\<squnion>)"
mapi_block_id: fixes v :: "nat"   and f :: "('c, 'd) fm \<Rightarrow> ('c, 'd) fm"   and ps :: "('c, 'd) fm list"   and i :: "'d" shows "mapi_block (mapper f {} v) (ps, i) = (ps, i)"
quad_form_scale_v: fixes k :: "complex"   and v :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "quad_form (k *\<^sub>s\<^sub>v v) H = cor ((cmod k)\<^sup>2) * quad_form v H"
degree_smult_not_zero_divisor_left: fixes c :: "'a"   and p :: "'a poly" assumes "\<not> zero_divisor c" shows "degree (Polynomial.smult c p) = degree p"
doms_append_2: fixes \<Gamma> :: "binding list"   and \<Delta> :: "binding list" shows "trm_dom (\<Gamma> @ \<Delta>) = trm_dom \<Gamma> \<union> trm_dom \<Delta>"
inv_in_hom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VV.iso f"   and "VV.in_hom f a b" shows "VV.in_hom (VV.inv f) b a"
strip_Step: fixes f :: "char list \<Rightarrow> aexp \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "bexp \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'a"   and C :: "'a acom" shows "strip (Step f g S C) = strip C"
sigma_sets_pair_measure_generator_finite: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "sigma_sets (A \<times> B) {a \<times> b |a b. a \<subseteq> A \<and> b \<subseteq> B} = Pow (A \<times> B)"
C_eq_Until_InsertDenies: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "Cp (list2FWpolicy (insertDenies (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules Cp (insertDeny (removeShadowRules1 (policy2list p)))))) l)))) = Cp p"
A_authenticates_B: fixes servK :: "nat"   and T3 :: "nat"   and evs :: "event list"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and A :: "agent"   and Ta :: "nat"   and authTicket :: "msg" assumes "Crypt servK (Number T3) \<in> parts (knows Spy evs)"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace> \<in> parts (knows Spy evs)"   and "Key authK \<notin> analz (knows Spy evs)"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "Says B A (Crypt servK (Number T3)) \<in> set evs"
induct: fixes x1 :: "com"   and x2 :: "char list \<Rightarrow> val option"   and x3 :: "nat"   and x4 :: "com"   and x5 :: "char list \<Rightarrow> val option"   and P :: "com \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> nat \<Rightarrow> com \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> bool" assumes "\<langle>x1,x2\<rangle> \<rightarrow>\<^bsup>x3\<^esup> \<langle>x4,x5\<rangle>"   and "\<And>c s. P c s 0 c s"   and "\<And>c s c'' s'' n c' s'. \<lbrakk>\<langle>c,s\<rangle> \<rightarrow> \<langle>c'',s''\<rangle>; \<langle>c'',s''\<rangle> \<rightarrow>\<^bsup>n\<^esup> \<langle>c',s'\<rangle>; P c'' s'' n c' s'\<rbrakk> \<Longrightarrow> P c s (Suc n) c' s'" shows "P x1 x2 x3 x4 x5"
deg_pp_of_exps: fixes a :: "nat"   and b :: "nat"   and i :: "nat" shows "deg_pp (fst (rep_nat_term (Term_Order.of_exps a b i))) = a + b"
set_of_sub_inc_right: fixes B :: "'b interval"   and B' :: "'b interval"   and A :: "'b interval" assumes "set_of B \<subseteq> set_of B'" shows "set_of (A - B) \<subseteq> set_of (A - B')"
frechet_derivative_unique_within_closed_interval: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and f'' :: "'a \<Rightarrow> 'b" assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i < b \<bullet> i"   and "x \<in> cbox a b"   and "(f has_derivative f') (at x within cbox a b)"   and "(f has_derivative f'') (at x within cbox a b)" shows "f' = f''"
ground_SMP_disjointI_aux2: fixes f :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "f \<equiv> \<lambda>M. {t \<cdot> \<delta> |t \<delta>. t \<in> M \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> fv (t \<cdot> \<delta>) = {}}"   and "finite_SMP_representation arity Ana \<Gamma> M" shows "f (set M) = f (SMP (set M))"
un: fixes x :: "'a" assumes "x \<in> carrier R - {\<zero>}" shows "x \<in> Units R"
fds_abs_convergent_zero_iff: fixes f :: "'a fds"   and s :: "'a" assumes "completely_multiplicative_function (fds_nth f)"   and "fds_abs_converges f s" shows "(eval_fds f s = (0::'a)) = (\<exists>p. prime p \<and> fds_nth f p = nat_power p s)"
Class_closed: fixes a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "a \<in> rel.Class b"   and "b \<in> R \<times> S" shows "a \<in> R \<times> S"
dfa_of_pf_well_formed: fixes n :: "nat"   and p :: "pf" shows "wf_dfa (dfa_of_pf n p) n"