hcju/isabelleps huggingface dataset:
cross_ratio_inversion: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo"   and z :: "complex_homo" assumes "u \<noteq> v"   and "v \<noteq> w"   and "u \<noteq> w" shows "cross_ratio (inversion z) (inversion u) (inversion v) (inversion w) = conjugate (cross_ratio z u v w)"
predicate1I: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x \<Longrightarrow> Q x" shows "P \<le> Q"
poly_IVT_pos: fixes a :: "real"   and b :: "real"   and p :: "real Polynomial.poly" assumes "a < b"   and "poly p a < 0"   and "0 < poly p b" shows "\<exists>x>a. x < b \<and> poly p x = 0"
Ndet_def: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "P \<sqinter> Q \<equiv> action_of (action.relation_of P \<or> action.relation_of Q)"
frontier_empty: shows "frontier {#}\<^sub>z = {}\<^sub>A"
flip_altsublist_adjacent_sym: fixes ss :: "'w list"   and ts :: "'w list" assumes "flip_altsublist_adjacent ss ts" shows "flip_altsublist_adjacent ts ss"
closed_simplex: fixes n :: "int"   and S :: "'b set" assumes "n simplex S" shows "closed S"
lemma_B: fixes B :: "agent"   and evs :: "event list"   and K :: "nat"   and A :: "agent"   and Tk :: "nat"   and X :: "msg"   and Ta :: "nat" assumes "B \<notin> bad"   and "evs \<in> bankerberos"   and "Key K \<notin> analz (knows Spy evs)"   and "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "Crypt K (Number Ta) \<in> parts (knows Spy evs)" shows "Says B A (Crypt K (Number Ta)) \<in> set evs"
prime_ge_1_int: fixes p :: "int" assumes "prime p" shows "1 \<le> p"
rel_map_1: fixes Sb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e \<Rightarrow> 'c"   and x :: "'e list"   and y :: "'d list" shows "list_all2 Sb (map i x) y = list_all2 (\<lambda>x. Sb (i x)) x y"
Res: fixes S :: "pi"   and T :: "pi"   and x :: "name" assumes "(S, T) \<in> Rel''" shows "(<\<nu>x>S, <\<nu>x>T) \<in> Rel''"
IH_1: fixes env :: "('a, bool) mapping"   and t1 :: "'a ifex"   and t2 :: "'a ifex" shows "reduced (normif env s1 t1 t2) (Mapping.keys env)"
set_mkIde: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set" assumes "set_category_given_img S img \<AA>"   and "A \<subseteq> Collect (category.terminal S)"   and "|A| <o \<AA>" shows "set_category_data.set S img (set_category_given_img.mkIde S img \<AA> A) = A"
sums_Cons_conv_append_tl: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'b"   and xs :: "'b list" assumes "class.monoid_add plus zero" shows "monoid_add.sums plus zero (x # xs) = zero # x # map (plus x) (tl (monoid_add.sums plus zero xs))"
inject: fixes x1 :: "nat list"   and x2 :: "facetype"   and y1 :: "nat list"   and y2 :: "facetype" shows "(Face x1 x2 = Face y1 y2) = (x1 = y1 \<and> x2 = y2)"
not_exp_less_zero: fixes x :: "real" shows "\<not> exp x < 0"
Domainp_set: fixes T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
assert_bind_spec_conv: fixes \<Phi> :: "bool"   and m :: "'a nres"   and \<Psi> :: "'a \<Rightarrow> bool" shows "(assert \<Phi> \<bind> (\<lambda>_. m) \<le> SPEC \<Psi>) = (\<Phi> \<and> m \<le> SPEC \<Psi>)"
RE: assumes "\<not> a" shows "hn_refine (\<Gamma>1 * hn_val bool_rel a a') c' \<Gamma>2c R c"
DataSpace_DataSpace_Update: fixes U :: "'b update"   and DP :: "'b data" shows "Data.DataSpace (Rep_update U DP) = Data.DataSpace DP"
setsetmapim_comp: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set set" shows "(f \<circ> g) \<turnstile> A = f \<turnstile> (g \<turnstile> A)"
simps_15: fixes x21 :: "'d"   and x22 :: "'d list" shows "set (x21 # x22) = insert x21 (set x22)"
int_of_uint64_code: fixes x :: "uint64" shows "int_of_uint64 x = int_of_integer (integer_of_uint64 x)"
lookup_update: fixes k :: "'b"   and v :: "'a"   and f :: "'b \<Rightarrow>\<^sub>0 'a"   and k' :: "'b" shows "lookup (Poly_Mapping.update k v f) k' = (if k = k' then v else lookup f k')"
seq_nil_right: fixes a :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) ; nil = a"
less_le_not_le_type: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "(\<tau> < \<sigma>) = (\<tau> \<le> \<sigma> \<and> \<not> \<sigma> \<le> \<tau>)"
wfP_iff_no_infinite_down_chain_llist: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "wfP r = (\<nexists>c. \<not> lfinite c \<and> chain r\<inverse>\<inverse> c)"
assms_2: shows "summable (\<lambda>k. norm (b k))"
update_convs_1: fixes heap' :: "heap \<Rightarrow> heap"   and heap :: "heap"   and locals :: "locals"   and more :: "'a" shows "heap_update heap' \<lparr>heap = heap, locals = locals, \<dots> = more\<rparr> = \<lparr>heap = heap' heap, locals = locals, \<dots> = more\<rparr>"
map_ran_restrictA: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'b set"   and \<Gamma> :: "('b \<times> 'c) list" shows "map_ran f (restrictA V \<Gamma>) = restrictA V (map_ran f \<Gamma>)"
more_arith_simps_11: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a * b * c = a * (b * c)"
filter_obtain_two: fixes i :: "nat"   and j :: "nat"   and P :: "'b \<Rightarrow> bool"   and lst :: "'b list" assumes "i < j"   and "j < length (filter P lst)" shows "\<exists>i2 j2. i2 < j2 \<and> j2 < length lst \<and> lst ! i2 = filter P lst ! i \<and> lst ! j2 = filter P lst ! j"
dg_prod_2_Cod_vsv: shows "vsv ((\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Cod\<rparr>)"
hyps_1: assumes "max_reg a1 = None" shows "aval a1 (join_ir i r) = aval a1 (join_ir i r')"
unimodular_reduce_norm: fixes z :: "complex" assumes "cmod z = 1" shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + \<i>) < 1 \<or> cmod (z - \<i>) < 1"
weak_map_of_SomeI: fixes k :: "'b"   and x :: "'c"   and l :: "('b \<times> 'c) list" assumes "(k, x) \<in> set l" shows "\<exists>x. map_of l k = Some x"
q: shows "q = smult a [:- (- b / a), 1::'a:]"
directed: shows "directed (Collect derivable) (\<sqsubseteq>)"
leg1_simps_3: shows "src\<^sub>D (F g) = src\<^sub>D (F f)"
is_app_def: fixes t :: "'a" shows "is_app t = (unapp t \<noteq> None)"
vector_derivative_def: fixes f :: "real \<Rightarrow> 'c"   and net :: "real filter" shows "vector_derivative f net = (SOME f'. (f has_vector_derivative f') net)"
dma_\<C>: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<forall>x\<in>\<C>. mem\<^sub>1 x = mem\<^sub>2 x" shows "dma mem\<^sub>1 = dma mem\<^sub>2"
\<phi>_def: shows "\<phi> \<equiv> \<lambda>S. S \<subseteq> {0..1} \<and> 0 \<in> S \<and> 1 \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. open_segment x y \<inter> S = {} \<longrightarrow> p x = p y)"
assms_2: shows "demand f \<noteq> C\<^sup>\<infinity>"
hs_memb_def: fixes x :: "'b"   and s :: "('b, unit) hashtable" shows "hs_memb x s \<equiv> hm_lookup x s \<bind> (\<lambda>r. return (case r of None \<Rightarrow> False | Some x \<Rightarrow> True))"
elims_2: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "'f"   and xb :: "'d list"   and xc :: "'f" assumes "d_reaches x xa xb xc" obtains M :: "('d, 'e, 'f) FSM" and p :: "'f" and xs :: "'d list" and q :: "'f" where "x = M" and "xa = p" and "xb = xs" and "xc = q" and "\<exists>tr ys. d_reached_by M p xs q tr ys"
hyper_subset_def: fixes A :: "'f set option"   and B :: "'f set option" shows "A \<sqsubseteq> B \<equiv> case B of None \<Rightarrow> True | \<lfloor>B\<rfloor> \<Rightarrow> case A of None \<Rightarrow> False | \<lfloor>A\<rfloor> \<Rightarrow> A \<subseteq> B"
Assign': fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and a :: "aexp" assumes "\<forall>l s. P l s \<longrightarrow> Q l (s[a/x])" shows "\<turnstile>\<^sub>t {P} x ::= a {Q}"
assms: shows "[\<^bold>\<exists>x. \<phi> x in v]"
holomorphic_on_Un: fixes f :: "complex \<Rightarrow> complex"   and A :: "complex set"   and B :: "complex set" assumes "f holomorphic_on A"   and "f holomorphic_on B"   and "open A"   and "open B" shows "f holomorphic_on A \<union> B"
i3: shows "\<forall>d\<in>D. \<not> hasRead s p d q"
list_all_filter_nil: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list"   and Q :: "'e \<Rightarrow> bool" assumes "list_all P xs"   and "\<And>x. P x \<Longrightarrow> \<not> Q x" shows "filter Q xs = []"
rel_fun_eq: shows "rel_fun (=) (=) = (=)"
that_2: fixes r :: "'a"   and s :: "'a" assumes "coprime r s"   and "r dvd a"   and "s dvd b" shows "f r s = g r s"
lconjby_closed: fixes G :: "'a set"   and g :: "'a"   and x :: "'a" assumes "Group G"   and "g \<in> G"   and "x \<in> G" shows "lconjby g x \<in> G"
1: fixes y :: "'a" shows "y \<in> P \<longrightarrow> {x. (x = f p \<or> x \<in> P) \<and> x \<sqsubset> y} = {x \<in> P. x \<sqsubset> y}"
run_owner_iii_def: fixes n :: "nat"   and s :: "(agent \<times> msg) set" shows "run_owner_iii n s \<equiv> SOME x. pred_owner_iii n (run_asset_ii n s) x"
wf_prog_def: shows "wf_prog = {(prog, procs). WellFormProgs.wf prog procs}"
length_block: fixes i :: "nat"   and t :: "nat" shows "e_length (block i t) = Suc (Suc t)"
c1: fixes nhip :: "nat" shows "P (0, unk, val, Suc 0, nhip)"
Gram_Schmidt_step_def: fixes a :: "('a, 'b) vec"   and ys :: "('a, 'b) vec list" shows "Gram_Schmidt_step a ys = ys @ [a - proj_onto a (set ys)]"
f1_def: fixes u :: "nat"   and v :: "nat" shows "f1 u v = (\<Sum>x\<in>{i. i < u}. g1 x v)"
\<tau>multithreaded_wf_axioms: shows "\<tau>multithreaded_wf final2 r2 \<tau>move2"
epath_diff_edge: fixes A :: "'b uprod set"   and A' :: "'b uprod set"   and u :: "'b"   and p :: "'b uprod list"   and v :: "'b" assumes "A \<subseteq> A'"   and "\<not> epath A u p v"   and "epath A' u p v" shows "\<exists>e. e \<in> set p - A"
elims_2: fixes x :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "'d list" assumes "top_sorted_abs x xa" obtains R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" where "x = Ra__" and "xa = []"   | R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" and h :: "'d" and l :: "'d list" where "x = R" and "xa = h # l" and "list_all (\<lambda>x. \<not> R x h) l \<and> top_sorted_abs R l"
stkOk_cons: fixes c :: "'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b ref"   and r :: "'b \<Rightarrow> 'b ref"   and iL :: "'b \<Rightarrow> 'b ref"   and iR :: "'b \<Rightarrow> 'b ref"   and t :: "'b ref"   and p :: "'b"   and stk :: "'b list" shows "stkOk c l r iL iR t (p # stk) = (stkOk c l r iL iR (Ref p) stk \<and> iL p = (if c p then l p else t) \<and> iR p = (if c p then t else r p))"
cexpr_sem_integrate_vars: fixes \<rho> :: "nat \<Rightarrow> val"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and vs :: "nat list"   and e :: "cexpr" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "distinct vs"   and "set vs \<inter> V' = {}"   and "integrable (state_measure (set vs) \<Gamma>) (\<lambda>\<sigma>. extract_real (cexpr_sem (merge (set vs) V' (\<sigma>, \<rho>)) e))"   and "\<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> set vs \<union> V'" shows "extract_real (cexpr_sem \<rho> (integrate_vars \<Gamma> vs e)) = LINT \<sigma>|state_measure (set vs) \<Gamma>. extract_real (cexpr_sem (merge (set vs) V' (\<sigma>, \<rho>)) e)"
closure_of: fixes X :: "'a topology"   and S :: "'a set" shows "X closure_of S = topspace X \<inter> (S \<union> X derived_set_of S)"
prj_simps_4: shows "C.arr \<mu>_\<nu>\<pi>.prj\<^sub>0"
rt_fresher_trans_rt_strictly_fresher': fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt3 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubset>\<^bsub>dip\<^esub> rt3" shows "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt3"
Tensor_Invariance: fixes ws1 :: "wall"   and ws2 :: "wall" assumes "is_tangle_diagram ws1 \<and> is_tangle_diagram ws2" shows "kauff_mat (ws1 \<otimes> ws2) = kauff_mat ws1 \<otimes> kauff_mat ws2"
order_iff_strict: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "ordering less_eq less" shows "less_eq a b = (less a b \<or> a = b)"
ereal_closed_levels: fixes f :: "'b \<Rightarrow> ereal" shows "(\<forall>y. closed {x. f x \<le> y}) = (\<forall>r. closed {x. f x \<le> ereal r})"
small_all_tdghms: fixes \<alpha> :: "V" shows "small {\<NN>. \<exists>\<FF> \<GG> \<AA> \<BB>. \<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>}"
btree_map_def: shows "btree_map \<equiv> \<Lambda> a. \<mu> uu. btree_abs oo ssum_map\<cdot>(u_map\<cdot>a)\<cdot>(sprod_map\<cdot>(u_map\<cdot>uu)\<cdot>(u_map\<cdot>uu)) oo btree_rep"
map_idI: fixes xs :: "'k list"   and f :: "'k \<Rightarrow> 'k" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = x" shows "map f xs = xs"
space_tail_algebra: fixes n :: "nat" shows "space (tail_algebra n) = UNIV \<rightarrow>\<^sub>E space M"
tauChainSum2: fixes Q :: "ccs"   and Q' :: "ccs"   and P :: "ccs" assumes "Q \<Longrightarrow>\<^sub>\<tau> Q'"   and "Q \<noteq> Q'" shows "P \<oplus> Q \<Longrightarrow>\<^sub>\<tau> Q'"
and_def: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" shows "and \<phi>\<^sub>1 \<phi>\<^sub>2 = (if \<phi>\<^sub>1 = TrueF then \<phi>\<^sub>2 else if \<phi>\<^sub>2 = TrueF then \<phi>\<^sub>1 else if \<phi>\<^sub>1 = FalseF \<or> \<phi>\<^sub>2 = FalseF then FalseF else And \<phi>\<^sub>1 \<phi>\<^sub>2)"
et_1: fixes v :: "'a"   and e :: "'a"   and t :: "'a" assumes "vector v"   and "e \<le> v * - v\<^sup>T"   and "t \<le> v * v\<^sup>T" shows "e * t = bot"
quietSim: fixes P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b" assumes "quiet P"   and "eqvt Rel"   and "\<And>P. quiet P \<Longrightarrow> (\<Psi>, \<zero>, P) \<in> Rel" shows "\<Psi> \<rhd> \<zero> \<leadsto><Rel> P"
3: shows "\<Union> (Range a) \<inter> \<Union> (Range ({seller} \<times> ({\<Omega> - \<Union> (Range a)} - {{}}))) = {}"
change_ord_def: fixes to :: "'a nat_term_order" shows "change_ord to = (\<lambda>x. x)"
prod_impl_aux_alt: shows "local.prod = \<lparr>g_V = {(q, s). q \<in> igba.V \<and> s \<in> sa.V}, g_E = E_of_succ (\<lambda>(q, s). if local.igba.L q (local.sa.L s) then succ_of_E igba.E q \<times> succ_of_E sa.E s else {}), g_V0 = igba.V0 \<times> sa.V0, igbg_num_acc = igba.num_acc, igbg_acc = \<lambda>(q, s). if s \<in> sa.V then igba.acc q else {}\<rparr>"
map2_val_compat'_def: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and f :: "('b \<times> 'c) list \<times> 'a \<Rightarrow> ('b \<times> 'd) list \<times> 'a" shows "oalist_raw.map2_val_compat' rep_key_order f = (\<forall>zs. oalist_raw.oalist_inv rep_key_order zs \<longrightarrow> oalist_raw.oalist_inv rep_key_order (f zs) \<and> snd (f zs) = snd zs \<and> fst ` set (fst (f zs)) \<subseteq> fst ` set (fst zs))"
igAbsIPresIGWlsSTR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igAbsIPresIGWlsSTR MOD \<equiv> \<forall>xs s x X. isInBar (xs, s) \<longrightarrow> igWlsAbs MOD (xs, s) (igAbs MOD xs x X) = igWls MOD s X"
inv_resr_neg: fixes y :: "'a"   and x :: "'a" shows "inv_resr (- x) (- y) = inv_resl x y"
mset_set_subseteq_mset_set: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "A \<subseteq> B" shows "mset_set A \<subseteq># mset_set B"
P'_add_Q_le_one: shows "P' + Q \<le>\<^sub>L 1\<^sub>m d"
mdelay_bisimulation_final_base: shows "delay_bisimulation_final_base r1.redT r2.redT mbisim m\<tau>move1 m\<tau>move2 r1.mfinal r2.mfinal"
cut_def: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a \<times> 'a) set"   and x :: "'a" shows "Wfrec.cut f R x = (\<lambda>y. if (y, x) \<in> R then f y else undefined)"
dghm_id_components_2: fixes \<CC> :: "V" shows "dghm_id \<CC>\<lparr>ArrMap\<rparr> = vid_on (\<CC>\<lparr>Arr\<rparr>)"
assms_4: shows "prv (imp \<phi> \<chi>1)"
Mapping_eq_lookup: fixes m :: "('c, 'd) mapping"   and m' :: "('c, 'd) mapping" shows "(m = m') = (Mapping.lookup m = Mapping.lookup m')"
eq_elems_eq_val: fixes x :: "'c"   and y :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "x = y" shows "f x = f y"
Subject_reductions: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"   and "P,E,s \<turnstile> e : T \<surd>" shows "\<exists>T'. P,E,s' \<turnstile> e' : T' \<surd> \<and> P \<turnstile> T' \<le> T"
akra_bazzi_term_floor_add: fixes b :: "real"   and x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and c :: "real" assumes "0 < b"   and "b < 1"   and "real x\<^sub>0 \<le> b * real x\<^sub>1 + c"   and "c < (1 - b) * real x\<^sub>1"   and "0 < x\<^sub>1" shows "akra_bazzi_term x\<^sub>0 x\<^sub>1 b (\<lambda>x. nat \<lfloor>b * real x + c\<rfloor>)"
rep_eq: fixes x :: "'a program"   and xa :: "'a program" shows "(x < xa) = (rep_program x \<sqsubseteq> rep_program xa \<and> \<not> rep_program xa \<sqsubseteq> rep_program x)"
invocation_class_IntVir: fixes s :: "st"   and a' :: "val"   and statT :: "ref_ty" shows "invocation_class IntVir s a' statT = obj_class (lookup_obj s a')"
functor_axioms: shows "functor B D (M \<circ> H)"
take_bit_tightened: fixes n :: "nat"   and a :: "'a"   and b :: "'a"   and m :: "nat" assumes "take_bit n a = take_bit n b"   and "m \<le> n" shows "take_bit m a = take_bit m b"
hom_one: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h" shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>H\<^esub>"
fmap_eqdom_Cons1: fixes xm :: "'f \<rightharpoonup>\<^sub>f 'e"   and i :: "'f"   and x :: "'e"   and ym :: "'f \<rightharpoonup>\<^sub>f 'g"   and R :: "'e \<Rightarrow> 'g \<Rightarrow> bool" assumes "fmlookup xm i = None"   and "fmdom (xm(i \<mapsto>\<^sub>f x)) = fmdom ym"   and "fmrel R (xm(i \<mapsto>\<^sub>f x)) ym" shows "\<exists>z zm. fmlookup zm i = None \<and> ym = zm(i \<mapsto>\<^sub>f z) \<and> R x z \<and> fmrel R xm zm"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and option :: "'d option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = Some x2 \<and> \<not> P (f2 x2))))"
ZFfunDom_def: fixes f :: "ZF" shows "|dom|f \<equiv> Fst (Fst f)"
impossibility_of_doubling_the_cube_lemma: fixes x :: "real" assumes "x \<in> radical_sqrt"   and "x ^ 3 = 2" shows "False"
cod_eqI': fixes a :: "'a"   and f :: "'a" assumes "a \<in> C.codomains f" shows "a = C.cod f"
add_set_def: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "class.group_add minus plus zero uminus" shows "group_add.add_set plus A B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = plus a b}"
simps_15: fixes p :: "nat"   and a :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Arctan a) vs = approx_floatarith p a vs \<bind> arctan_aform_err p"
por_def: fixes x :: "ValD"   and y :: "ValD" shows "x por y \<equiv> if x = ValTT then ValTT else if y = ValTT then ValTT else if x = ValFF \<and> y = ValFF then ValFF else \<bottom>"
differing_vars_def: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.differing_vars mem\<^sub>1 mem\<^sub>2 \<equiv> {x. mem\<^sub>1 x \<noteq> mem\<^sub>2 x}"
sublist_rev: fixes xs :: "'c list"   and ys :: "'c list" shows "sublist (rev xs) (rev ys) = sublist xs ys"
take_bit_unset_bit_eq: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat"   and a :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "take_bit n (ring_bit_operations.unset_bit one push_bit and not m a) = (if n \<le> m then take_bit n a else ring_bit_operations.unset_bit one push_bit and not m (take_bit n a))"
drest_exec_as_proj_eq_drest_exec: fixes s :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and vs :: "'a set" assumes "sat_precond_as s as" shows "fmrestrict_set vs (exec_plan s (as_proj as vs)) = fmrestrict_set vs (exec_plan s as)"
inf_exists: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L" shows "\<exists>i. is_glb L i A"
less_prod_Pair_conv: fixes a\<^sub>1 :: "'c"   and b\<^sub>1 :: "'d"   and r\<^sub>A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r\<^sub>B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a\<^sub>2 :: "'c"   and b\<^sub>2 :: "'d" shows "((a\<^sub>1, b\<^sub>1) \<sqsubset>\<^bsub>Product.le r\<^sub>A r\<^sub>B\<^esub> (a\<^sub>2, b\<^sub>2)) = (a\<^sub>1 \<sqsubset>\<^bsub>r\<^sub>A\<^esub> a\<^sub>2 \<and> b\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>B\<^esub> b\<^sub>2 \<or> a\<^sub>1 \<sqsubseteq>\<^bsub>r\<^sub>A\<^esub> a\<^sub>2 \<and> b\<^sub>1 \<sqsubset>\<^bsub>r\<^sub>B\<^esub> b\<^sub>2)"
induct: fixes x :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and P :: "(nat \<times> int) list \<Rightarrow> bool" assumes "x \<in> grid b ds"   and "P b"   and "\<And>p d dir. \<lbrakk>p \<in> grid b ds; P p; d \<in> ds\<rbrakk> \<Longrightarrow> P (child p dir d)" shows "P x"
iiface_rewrite_matches_Primitive: fixes \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and x :: "'c common_primitive"   and a :: "action"   and p :: "('c, 'd) tagged_packet_scheme" shows "(matches (common_matcher, \<alpha>) (MatchNot (iiface_rewrite ipassmt (Match x))) a p = matches (common_matcher, \<alpha>) (MatchNot (Match x)) a p) = (matches (common_matcher, \<alpha>) (iiface_rewrite ipassmt (Match x)) a p = matches (common_matcher, \<alpha>) (Match x) a p)"
cyclic_group_axioms: shows "Cyclic_Group.cyclic_group \<G>"
augmenting_path_imp_shortest: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and p :: "(nat \<times> nat) list" assumes "NFlow c s t f"   and "NPreflow.isAugmentingPath c s t f p" shows "\<exists>p. Graph.isShortestPath (residualGraph c f) s p t"