hcju/isabelleps huggingface dataset:
dual_order: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.order less_eq less" shows "class.order (\<lambda>x y. less_eq y x) (\<lambda>x y. less y x)"
FDERIV: fixes f :: "'e \<Rightarrow> 'a"   and f' :: "'e \<Rightarrow> 'a"   and x :: "'e"   and s :: "'e set"   and g :: "'e \<Rightarrow> 'b"   and g' :: "'e \<Rightarrow> 'b" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at x within s)" shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"
rename_Always: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set" assumes "bij h" shows "(rename h F \<in> Always (h ` A)) = (F \<in> Always A)"
lnull_llist_of: fixes xs :: "'c list" shows "lnull (llist_of xs) = (xs = [])"
is_ub_insert: fixes x :: "'a"   and A :: "'a set"   and y :: "'a" shows "insert x A <| y = (x \<sqsubseteq> y \<and> A <| y)"
v11_app_if_vconverse_app: fixes y :: "V"   and x :: "V" assumes "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"   and "r\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> = x" shows "r\<lparr>x\<rparr> = y"
diffeomorphism_imp_homeomorphism: fixes k :: "enat"   and s :: "'b set"   and t :: "'c set"   and p :: "'b \<Rightarrow> 'c"   and p' :: "'c \<Rightarrow> 'b" assumes "diffeomorphism k s t p p'" shows "homeomorphism s t p p'"
cases: fixes a :: "'c"   and g :: "'c \<Rightarrow> ('d, 'c) step" assumes "a \<in> productive_on g" obtains s :: "'c" where "a = sa__" and "g sa__ = Done"   | s :: "'c" and s' :: "'c" where "a = sb__" and "g sb__ = Skip s'__" and "s'__ \<in> productive_on g"   | s :: "'c" and x :: "'d" and s' :: "'c" where "a = s" and "g s = Yield x s'"
in_countE: fixes x :: "'d"   and M :: "'d multiset" assumes "x \<in># M" obtains n :: "nat" where "count M x = Suc n"
rep_eq: fixes x :: "'a coreflexive"   and xa :: "'a coreflexive" shows "Rep_coreflexive (x \<squnion> xa) = Rep_coreflexive x \<squnion> Rep_coreflexive xa"
s0_secrecy_trans: shows "{s0_secrecy} TS.trans s0 {> s0_secrecy}"
reconstruct_share: fixes x :: "bool" shows "share x \<bind> reconstruct = return_spmf x"
False: shows "f \<noteq> \<zero>\<^bsub>P\<^esub>"
SUP_insert: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and A :: "'b set" shows "\<Squnion> (f ` insert a A) = f a \<squnion> \<Squnion> (f ` A)"
neg_less_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) a = less zero a"
m_comm: fixes x :: "rat"   and y :: "rat" shows "x + y = y + x"
independent_pdevs_eq_Nil_iff: fixes xs :: "(real \<times> real) list" assumes "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs"   and "list_all (\<lambda>x. x \<noteq> 0) xs" shows "(independent_pdevs xs = []) = (xs = [])"
rep_eq: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'b \<Rightarrow>\<^sub>0 'c" assumes "inj f" shows "lookup (Poly_Mapping.map_key f x) = lookup x \<circ> f"
strongEqWeakCong: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sup>s Q" shows "P \<simeq>\<^sup>s Q"
cSym': fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel'" shows "(\<Psi>', T, S) \<in> Rel'"
subprob_not_empty: fixes M :: "'c measure" assumes "subprob_space M" shows "space M \<noteq> {}"
m2'eq: shows "m2' = m1'(x := ExprEval e m1')"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.R B.map \<phi>" shows "equivalence_functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.R"
B: assumes "P \<sigma>"   and "\<sigma> \<Turnstile> B\<^sub>1 ;- M" shows "Q"
berlekamp_resulting_mat_def: fixes u :: "'b mod_ring poly" shows "berlekamp_resulting_mat u = (let Q = berlekamp_mat u; n = dim_row Q; QI = mat n n (\<lambda>(i, j). if i = j then Q $$ (i, j) - 1 else Q $$ (i, j)) in gauss_jordan_single QI\<^sup>T)"
fds_abs_converges_shift: fixes c :: "'a"   and f :: "'a fds"   and z :: "'a" shows "fds_abs_converges (fds_shift c f) z = fds_abs_converges f (z - c)"
ex_card: fixes S :: "'a set" assumes "finite S" shows "\<exists>n. card S = n"
id_o: fixes g :: "'e \<Rightarrow> 'f" shows "id \<circ> g = g"
inext_nth_card_append_eq3: fixes A :: "nat set"   and B :: "nat set" assumes "finite A"   and "B \<noteq> {}"   and "Max A < iMin B" shows "(A \<union> B) \<rightarrow> card A = iMin B"
expand_big_forest: fixes H :: "'a"   and d :: "'a" assumes "big_forest H d" shows "(d[[H]])\<^sup>\<star> * (H * d)\<^sup>\<star> = (d[[H]])\<^sup>\<star> \<squnion> (H * d)\<^sup>\<star>"
simps_2: fixes V :: "char list"   and e :: "expr" shows "#:V:=e = 2"
poly_has_sgnx_values_3: fixes p :: "real poly" assumes "p \<noteq> 0" shows "(poly p has_sgnx 1) at_top \<or> (poly p has_sgnx - 1) at_top"
H_def: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "H p x q = (d p \<le> wp x q)"
CFG_edge_no_Def_equal: fixes a :: "'a"   and V :: "'d"   and s :: "'c" assumes "valid_edge a"   and "V \<notin> Def (sourcenode a)"   and "pred (kind a) s" shows "state_val (transfer (kind a) s) V = state_val s V"
v_cons: fixes \<Gamma> :: "(name \<times> ty) list"   and a :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "a \<sharp> \<Gamma>" shows "valid ((a, T) # \<Gamma>)"
conway_c2_var: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y)\<^sup>\<star> \<cdot> x \<cdot> y \<cdot> y\<^sup>\<star> \<le> (x \<cdot> y)\<^sup>\<star> \<cdot> y\<^sup>\<star>"
vors: shows "i < |rotate n ls| \<and> j < |rotate n ls| \<and> rotate n ls ! i = x \<and> rotate n ls ! j = y \<and> Suc i mod |rotate n ls| = j"
g: fixes x :: "'c" shows "norm (g x) \<le> norm x * Kg"
word_less_add_right: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" assumes "x < y - z"   and "z \<le> y" shows "x + z < y"
fv_supp_as: fixes as :: "assn" shows "supp as = atom ` fv as"
nonstop_lemma: fixes m :: "nat"   and r :: "nat"   and t :: "nat" shows "rec_exec rec_nonstop [m, r, t] = nonstop m r t"
lunit_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and b :: "'b"   and f :: "'b" assumes "left_hom_with_unit V H \<a> \<iota> b" shows "left_hom_with_unit.lunit V H \<a> \<iota> b f \<equiv> THE \<mu>. partial_magma.in_hom (subcategory.comp V (weak_composition.left V H b)) \<mu> (weak_composition.H\<^sub>L H b f) f \<and> weak_composition.H\<^sub>L H b \<mu> = subcategory.comp V (weak_composition.left V H b) (H \<iota> f) (category.inv V (\<a> b b f))"
letter_eq: fixes alphabet1 :: "'b list"   and init1 :: "'c \<Rightarrow> 'd"   and delta1 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and accept1 :: "'d \<Rightarrow> bool"   and wellformed1 :: "'d \<Rightarrow> bool"   and Language1 :: "'d \<Rightarrow> 'b language"   and wf1 :: "'c \<Rightarrow> bool"   and Lang1 :: "'c \<Rightarrow> 'b language"   and alphabet2 :: "'e list"   and init2 :: "'f \<Rightarrow> 'g"   and delta2 :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and accept2 :: "'g \<Rightarrow> bool"   and wellformed2 :: "'g \<Rightarrow> bool"   and Language2 :: "'g \<Rightarrow> 'e language"   and wf2 :: "'f \<Rightarrow> bool"   and Lang2 :: "'f \<Rightarrow> 'e language"   and letter_eq :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'b"   and b :: "'e" assumes "DFAs alphabet1 init1 delta1 accept1 wellformed1 Language1 wf1 Lang1 alphabet2 init2 delta2 accept2 wellformed2 Language2 wf2 Lang2 letter_eq"   and "letter_eq a b" shows "(a \<in> set alphabet1) = (b \<in> set alphabet2)"
ternary_lift1: fixes tv :: "ternaryvalue" shows "(eval_ternary_Not tv \<noteq> TernaryFalse) = (tv = TernaryFalse \<or> tv = TernaryUnknown)"
a_le: shows "\<forall>x y. \<alpha> y x \<le> y"
rep_eq: fixes x :: "'a mpoly"   and xa :: "'a mpoly" shows "mapping_of (x + xa) = mapping_of x + mapping_of xa"
yun_rel_gcd: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly"   and G :: "int poly"   and c' :: "rat"   and g :: "rat poly"   and d :: "rat" assumes "yun_rel F c f"   and "yun_wrel G c' g"   and "c' \<noteq> 0"   and "d = rat_of_int (lead_coeff (gcd F G))" shows "yun_rel (gcd F G) d (gcd f g)"
null_space_orthogonal_complement_row_space: fixes A :: "((real, 'a) vec, 'b) vec" shows "null_space A = real_matrix.v.orthogonal_complement (row_space (\<chi>i j. cnj_class.cnj (A $ i $ j)))"
hypnat_of_nat_mem_Nats: fixes n :: "nat" shows "hypnat_of_nat n \<in> \<nat>"
brc_inv_imp_WssQ: fixes Q :: "('a, unit) HashMap.hashmap"   and W :: "'a list"   and rcm :: "(('a, 'b) ta_rule, nat) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule set" assumes "brc_\<alpha> (Q, W, rcm) \<in> br'_invar \<delta>" shows "set W \<subseteq> hs.\<alpha> Q"
to_bl_revcast: fixes w :: "'b word" shows "to_bl (revcast w) = takefill False LENGTH('a) (to_bl w)"
power_Suc2: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (Suc n) = times (power.power one times a n) a"
incseq_SucI: fixes X :: "nat \<Rightarrow> 'c" assumes "\<And>n. X n \<le> X (Suc n)" shows "incseq X"
body_def: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig" shows "body G C sig = (let m = the (methd G C sig) in Body (declclass m) (stmt (mbody (mthd m))))"
enumerate_eq_zip: fixes n :: "nat"   and xs :: "'b list" shows "enumerate n xs = zip [n..<n + length xs] xs"
unat_bintrunc: fixes bin :: "num" shows "unat (numeral bin) = nat (take_bit LENGTH('a) (numeral bin))"
possible_steps_1_coin: fixes i :: "value list"   and r :: "nat \<Rightarrow>f value option" assumes "length i = 1" shows "possible_steps drinks 1 r STR ''coin'' i = {|(1, coin)|}"
mset_merge: fixes xs :: "'c list"   and ys :: "'c list" shows "mset (merge xs ys) = mset xs + mset ys"
asymp_equiv_refl_ev: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "f \<sim>[F] g"
assn_imp_def: fixes P :: "'b assn"   and Q :: "'b assn" shows "(P \<Rightarrow> Q) = (\<forall>Y s Z. P Y s Z \<longrightarrow> Q Y s Z)"
image_eqI: fixes b :: "'c"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and A :: "'d set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
assms: shows "pdec1 (the (eval r_leap [t, i, x])) = 0"
inject: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "'a list"   and y1 :: "'a"   and y2 :: "'b"   and y3 :: "'a list" shows "(x1 \<rightarrow> x2 x3 = y1 \<rightarrow> y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
pow_generator_mod_int: fixes k :: "int" shows "\<^bold>g [^] (k mod int (order G)) = \<^bold>g [^] k"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "\<^bold>\<lfloor>t \<^bold>\<star> u\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>"
convex_hull_convex_sum: fixes A :: "'a vec set"   and x :: "'a vec"   and y :: "'a vec"   and a :: "'a" assumes "A \<subseteq> carrier_vec n"   and "x \<in> convex_hull A"   and "y \<in> convex_hull A"   and "(0::'a) \<le> a"   and "a \<le> (1::'a)" shows "a \<cdot>\<^sub>v x + ((1::'a) - a) \<cdot>\<^sub>v y \<in> convex_hull A"
oseq_guardTE: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and g :: "'e \<Rightarrow> 'e set"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, {l}\<langle>g\<rangle> p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains "a = \<tau>" and "\<sigma>' i \<in> g (\<sigma> i)" and "q = p"
s2: shows "c - y * A \<le> c2 - y * A1"
card_bijections_domain_permutation_eq_0: fixes A :: "'c set"   and B :: "'d set" assumes "card A \<noteq> card B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_permutation A B) = 0"
IH_2: fixes \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list" assumes "lower_on \<sigma> \<eta> (vars_of t2)" shows "lower_or_eq (t2 \<lhd> \<sigma>) (t2 \<lhd> \<eta>)"
\<delta>_symbols_def: fixes \<delta> :: "('b, 'a) ta_rule set" shows "\<delta>_symbols \<delta> \<equiv> rhsl ` \<delta>"
H_llist_hom_append: fixes xs :: "'b llist"   and ys :: "'b llist" shows "list2H\<cdot>(xs :++ ys) = list2H\<cdot>xs oo list2H\<cdot>ys"
Seq_is_CSP: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "is_CSP_process (action.relation_of P ;; action.relation_of Q)"
mult_isor: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<parallel> z \<le> y \<parallel> z"
bisim1Sync9: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([Addr a], xs, 8 + length (compE2 e1) + length (compE2 e2), None)"
proper_design_axioms: shows "proper_design \<V> (multiple_blocks n)"
root_char_poly_smult: fixes A :: "complex mat"   and n :: "nat"   and k :: "complex"   and x :: "complex" assumes "A \<in> carrier_mat n n"   and "k \<noteq> 0" shows "(poly (char_poly (k \<cdot>\<^sub>m A)) x = 0) = (poly (char_poly A) (x / k) = 0)"
MinimalMethodDefs_def: 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 C :: "char list"   and M :: "char list" shows "MinimalMethodDefs P C M \<equiv> {(Cs, mthd). (Cs, mthd) \<in> MethodDefs P C M \<and> (\<forall>(Cs', mthd')\<in>MethodDefs P C M. P,C \<turnstile> Cs' \<sqsubseteq> Cs \<longrightarrow> Cs' = Cs)}"
CompleteLatticeI_simp: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<lparr>pset = A, order = r\<rparr> \<in> PartialOrder"   and "\<forall>S\<subseteq>A. \<exists>L. isLub S \<lparr>pset = A, order = r\<rparr> L" shows "\<lparr>pset = A, order = r\<rparr> \<in> CompleteLattice"
Iagree_Func: fixes I :: "('d, 'e, 'f) interp"   and J :: "('d, 'e, 'f) interp"   and V :: "('d + 'e + 'f) set"   and f :: "'d" assumes "Iagree I J V"   and "Inl f \<in> V" shows "Functions I f = Functions J f"
distinct_5: fixes name :: "name"   and trm :: "trm"   and trm1' :: "trm"   and trm2' :: "trm" shows "Lam [name].trm \<noteq> App trm1' trm2'"
subseqs_refl: fixes xs :: "'b list" shows "xs \<in> set (subseqs xs)"
simps: fixes \<phi> :: "'a ltlc" shows "ltlc_to_ltln \<phi> = ltlc_to_ltln' False \<phi>"
extended_Gromov_distance_inside: fixes x :: "'a"   and y :: "'a" shows "extended_Gromov_distance (to_Gromov_completion x) (to_Gromov_completion y) = ereal (dist x y)"
basic_monos_1: fixes A :: "'c set" shows "A \<subseteq> A"
tinres_step2: fixes r :: "cell list"   and r' :: "cell list"   and ss :: "nat"   and l :: "cell list"   and t :: "(action \<times> nat) list"   and sa :: "nat"   and la :: "cell list"   and ra :: "cell list"   and sb :: "nat"   and lb :: "cell list"   and rb :: "cell list" assumes "tinres r r'"   and "step0 (ss, l, r) t = (sa, la, ra)"   and "step0 (ss, l, r') t = (sb, lb, rb)" shows "la = lb \<and> tinres ra rb \<and> sa = sb"
in_cenum: fixes cEnum :: "('a list \<times> (('a \<Rightarrow> bool) \<Rightarrow> bool) \<times> (('a \<Rightarrow> bool) \<Rightarrow> bool)) option"   and enum :: "'a list"   and rest :: "(('a \<Rightarrow> bool) \<Rightarrow> bool) \<times> (('a \<Rightarrow> bool) \<Rightarrow> bool)"   and f :: "'a" assumes "class.cenum cEnum"   and "ID cEnum = Some (enum, rest)" shows "f \<in> set enum"
put_state_def: shows "put_state \<equiv> \<lambda>uu uua. rec_stateT (\<lambda>f s. StateT (\<lambda>_. f s)) uua uu"
mwb_lens_weak: fixes x :: "'d \<Longrightarrow> 'e" assumes "mwb_lens x" shows "weak_lens x"
equal_neg_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "(a = uminus a) = (a = zero)"
is_path_extend: fixes n :: "nat"   and S :: "(nat \<times> nat) set"   and p :: "nat list"   and T :: "(nat \<times> nat) set" assumes "is_path n S p"   and "S \<subseteq> T" shows "is_path n T p"
is_empty_lookup_empty: fixes t :: "('c, 'd) Trie.trie" assumes "invar_trie t" shows "is_empty_trie t = (lookup_trie t = Map.empty)"
IH: fixes as2 :: "('a \<times> ('b, 'c) fmap) list" assumes "subseq as1 as2"   and "no_effectless_act as2" shows "no_effectless_act as1"
mod_self: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a a = zero"
subst_rename: fixes c :: "name"   and t\<^sub>1 :: "trm"   and a :: "name"   and t\<^sub>2 :: "trm" assumes "c \<sharp> t\<^sub>1" shows "t\<^sub>1[a::=t\<^sub>2] = ([(c, a)] \<bullet> t\<^sub>1)[c::=t\<^sub>2]"
homeomorphism_apply1: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and x :: "'a" assumes "homeomorphism S T f g"   and "x \<in> S" shows "g (f x) = x"
assms_2: shows "f \<in> A \<rightarrow> B"
keys_for_monotone: fixes G :: "chmsg set"   and H :: "chmsg set" assumes "G \<subseteq> H" shows "keys_for G \<subseteq> keys_for H"
term_pp_rel_cancel: fixes r :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and a :: "'t \<times> 'a" assumes "reflp r" shows "term_pp_rel r a (cancel_term_pp_pair a)"
arctan_aform_err: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and p :: "nat"   and Y :: "(real \<times> real pdevs) \<times> real" assumes "x \<in> aform_err e X"   and "arctan_aform_err p X = Some Y"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "arctan x \<in> aform_err e Y"
distinct_set_conv_list: fixes xs :: "'c list"   and g :: "'c \<Rightarrow> 'a" assumes "distinct xs" shows "set.F g (set xs) = list.F (map g xs)"
index_non_zero_implies_no_group: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and ps :: "'b set" assumes "GDD point_set block_collection groups index"   and "ps \<subseteq> point_set"   and "card ps = 2"   and "int (block_collection index ps) = index" shows "size (filter_mset ((\<subseteq>) ps) (mset_set groups)) = 0"
poly_eval_update: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'a"   and x :: "nat"   and c :: "'a" shows "p \<lbrace> v(x := c) \<rbrace> = p \<lbrace> v \<rbrace> + coeff p x *R (c - v x)"
StoreBuffer: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and m' :: "nat \<Rightarrow> nat"   and sb' :: "'a memref list"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "(m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f (m', sb', \<O>', \<R>', \<S>')" shows "sbh_computation.concurrent_step (ts, m, \<S>) (ts[i := (p, is, \<theta>, sb', \<D>, \<O>', \<R>')], m', \<S>')"
assms_4: shows "u \<in> nodes g\<^sub>1"
row_carrier: fixes A :: "'a mat"   and i :: "nat" shows "row A i \<in> carrier_vec (dim_col A)"
simps_5: fixes Label :: "String.literal"   and Arity :: "nat"   and Guards :: "vname gexp list"   and Outputs :: "vname aexp list"   and Updates :: "(nat \<times> vname aexp) list"   and more :: "'a" shows "Updates \<lparr>Label = Label, Arity = Arity, Guards = Guards, Outputs = Outputs, Updates = Updates, \<dots> = more\<rparr> = Updates"
meet_greatest: fixes z :: "'b"   and x :: "'b"   and y :: "'b" assumes "z \<sqsubseteq> x"   and "z \<sqsubseteq> y" shows "z \<sqsubseteq> x \<sqinter> y"
vardict_inv_process_names: fixes ss :: "(integer \<times> edge list) iarray"   and proc :: "nat \<times> edgeIndex \<times> procArg list \<times> varDecl list"   and v :: "(String.literal, variable) assoc_list"   and k :: "String.literal"   and x :: "variable" assumes "vardict_inv ss proc v"   and "lm.lookup k v = Some x" shows "k \<in> process_names ss proc"
max_Sup_enat1: fixes Y :: "enat set"   and x :: "enat" assumes "Y \<noteq> {}" shows "max (\<Squnion> Y) x = (\<Squnion>y\<in>Y. max y x)"
box01_nonempty: shows "box (0::'b) One \<noteq> {}"
args_respects: shows "(\<lambda>U. {Abs_ExpList (freeargs U)}) respects exprel"
l_set_attribute_get_disconnected_nodes_axioms_def: fixes set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_attribute_get_disconnected_nodes_axioms set_attribute_locs get_disconnected_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_attribute_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
pdivides_iff_shell: fixes K :: "('c multiset \<Rightarrow> 'a) set"   and p :: "('c multiset \<Rightarrow> 'a) list"   and q :: "('c multiset \<Rightarrow> 'a) list" assumes "subfield K L"   and "p \<in> carrier (K [X]\<^bsub>L\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>L\<^esub>)" shows "p pdivides\<^bsub>L\<^esub> q = p divides\<^bsub>K [X]\<^bsub>L\<^esub>\<^esub> q"
1: fixes u1 :: "'a local.term"   and u2 :: "'a local.term" assumes "ide \<lbrace>u1\<rbrace>"   and "ide \<lbrace>u2\<rbrace>"   and "Arr u1"   and "Arr u2"   and "Dom u1 = Cod u2" shows "ide (\<lbrace>u1\<rbrace> \<cdot> \<lbrace>u2\<rbrace>)"
continuous: fixes F :: "'e filter"   and f :: "'e \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. f x ** g x)"
induct: fixes P :: "'b rule list \<Rightarrow> bool"   and a0 :: "'b rule list" assumes "P []"   and "\<And>m a rs. (\<not> (m = MatchAny \<and> (a = Accept \<or> a = Drop \<or> a = Reject)) \<Longrightarrow> P rs) \<Longrightarrow> P (Rule m a # rs)" shows "P a0"
top_left_normalize: fixes a :: "'a" shows "\<^bold>\<top> \<^bold>* a = normalize a"
poly_eval_def: fixes a :: "'c \<Rightarrow> 'b"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "poly_eval a p = lookup (poly_subst (\<lambda>y. monomial (a y) 0) p) 0"
Z_def: fixes n :: "nat" shows "Z n = emb I (J n) (B n)"
compact_Int_closed: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "topological_space.closed open T" shows "topological_space.compact open (S \<inter> T)"
maximal_set_def: fixes S :: "'c set set"   and mx :: "'c set" shows "maximal_set S mx = (mx \<in> S \<and> (\<forall>s\<in>S. mx \<subseteq> s \<longrightarrow> s = mx))"
mta_bisim_def: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and tta1 :: "'a \<times> ('f, 'a, 'b, 'c, 'g, 'h) thread_action"   and tta2 :: "'a \<times> ('f, 'a, 'd, 'e, 'g, 'h) thread_action" shows "FWbisimulation_base.mta_bisim bisim tta1 tta2 \<equiv> fst tta1 = fst tta2 \<and> ta_bisim bisim (snd tta1) (snd tta2)"