hcju/isabelleps huggingface dataset:
intro: fixes type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool" assumes "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"   and "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf" shows "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
finite_ex_and1: fixes P :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> 'd \<Rightarrow> bool" assumes "finite {b. \<exists>a. P a b}" shows "finite {b. \<exists>a. P a b \<and> Q a b}"
lb_Suc: fixes n :: "nat"   and i :: "nat"   and k :: "nat"   and x :: "float" shows "lb (Suc n) i k x = float_plus_down prec (lapprox_rat prec 1 (int k)) (- float_round_up prec (x * ub n (F i) (G i k) x))"
bisim_Union: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set"   and Theta :: "('b \<times> 'b) set set" assumes "Bisim.mono Retr"   and "\<And>theta. theta \<in> Theta \<Longrightarrow> bisim Retr theta" shows "bisim Retr (\<Union> Theta)"
FAss_\<tau>red1r_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and v :: "'addr val"   and F :: "String.literal"   and D :: "String.literal" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (Val v\<bullet>F{D} := e, xs) (Val v\<bullet>F{D} := e', xs')"
assms_2: shows "a \<notin> span (S - {b})"
gbinomial_1: fixes a :: "'c" shows "a gchoose 1 = a"
fls_shift_const: fixes m :: "int"   and c :: "'a" shows "fls_shift m (fls_const c) = Abs_fls (\<lambda>n. if n = - m then c else (0::'a))"
tllist_all2_tnthD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist"   and n :: "nat" assumes "tllist_all2 P Q xs ys"   and "enat n < tlength xs" shows "P (tnth xs n) (tnth ys n)"
word_bw_assocs_2: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "(x OR y) OR z = x OR y OR z"
wbisim12: shows "delay_bisimulation_diverge trsys1 trsys2 bisim12 tlsim12 \<tau>move1 \<tau>move2"
cases: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and a :: "'a" assumes "SINKp f a" obtains x :: "'a" where "a = x" and "d_OUT f x = 0"
HPhase1or2Write_HInv4b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2Write s s' p d"   and "HInv4b s q" shows "HInv4b s' q"
pcr_cr_eq: shows "pcr_view\<^sub>h = cr_view\<^sub>h"
cases: fixes a :: "'d \<Rightarrow> real"   and X :: "'d set"   and I :: "'d \<Rightarrow> intv"   and r :: "('d \<times> 'd) set" assumes "a \<in> region X I r" obtains u :: "'d \<Rightarrow> real" and X\<^sub>0 :: "'d set" where "a = u" and "\<forall>x\<in>X. 0 \<le> u x" and "\<forall>x\<in>X. intv_elem x u (I x)" and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}" and "\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))"
snd_special_simp: fixes a :: "'a"   and x :: "'c \<times> 'b" shows "snd (case x of (s, m) \<Rightarrow> (s, a, m)) = (a, snd x)"
in_streams: fixes s :: "'d stream"   and S :: "'d set" assumes "stl s \<in> streams S"   and "shd s \<in> S" shows "s \<in> streams S"
orthogonal_complementI: fixes M :: "'a set"   and y :: "'a" assumes "\<And>x. x \<in> M \<Longrightarrow> is_orthogonal y x" shows "y \<in> orthogonal_complement M"
ij_le5_2: shows "j < dim_col (matricize I B)"
length_delete_le: fixes k :: "'d"   and al :: "('d \<times> 'e) list" shows "length (delete k al) \<le> length al"
card_insert_le_m1: fixes n :: "nat"   and y :: "'b set"   and x :: "'b" assumes "0 < n"   and "card y \<le> n - 1" shows "card (insert x y) \<le> n"
ldropn_Suc_LCons: fixes n :: "nat"   and x :: "'a"   and xs :: "'a llist" shows "ldropn (Suc n) (LCons x xs) = ldropn n xs"
execute_instr_sub1_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "execute_instr_sub1 instr \<equiv> do instr_name \<leftarrow> gets (\<lambda>s. fst instr); traps2 \<leftarrow> gets get_trap_set; if traps2 = {} \<and> instr_name \<notin> {call_type CALL, ctrl_type RETT, ctrl_type JMPL, bicc_type BE, bicc_type BNE, bicc_type BGU, bicc_type BLE, bicc_type BL, bicc_type BGE, bicc_type BNEG, bicc_type BG, bicc_type BCS, bicc_type BLEU, bicc_type BCC, bicc_type BA, bicc_type BN} then do npc_val \<leftarrow> gets (cpu_reg_val nPC); write_cpu npc_val PC; write_cpu (npc_val + 4) nPC; return () od else return () od"
nres_relI: fixes c :: "'c nres"   and R :: "('c \<times> 'd) set"   and a :: "'d nres" assumes "c \<le> \<Down> R a" shows "(c, a) \<in> \<langle>R\<rangle>nres_rel"
generate_valid_stateful_policy_IFSACS_2_def: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and edgesList :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_Algorithm.generate_valid_stateful_policy_IFSACS_2 G M edgesList \<equiv> \<lparr>hosts = nodes G, flows_fix = edges G, flows_state = set (TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations G M edgesList) \<inter> set (TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS G M edgesList)\<rparr>"
restrict_relE: fixes x :: "'b"   and y :: "'b"   and A :: "'b set"   and R :: "('b \<times> 'b) set" assumes "(x, y) \<in> restrict_rel A R" obtains "(x, y) \<in> R" and "x \<in> A" and "y \<in> A"
before_in_irefl: fixes x :: "'b"   and xs :: "'b list" shows "x < x in xs = False"
Suc_1: fixes t :: "'a bro" assumes "t \<in> B h" shows "ins x t \<in> Bp h"
m_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> UNIV"   and "y \<in> UNIV"   and "z \<in> UNIV" shows "x * y * z = x * (y * z)"
that: fixes x :: "'a" assumes "x \<in> C"   and "enat n \<le> esize x" shows "thesis"
worklist_aux_Nil: fixes succs :: "'d \<Rightarrow> 'c \<Rightarrow> 'c list"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'d" shows "worklist_aux succs f ([], s) = Some ([], s)"
bounded_Un: fixes S :: "'a set"   and T :: "'a set" shows "bounded (S \<union> T) = (bounded S \<and> bounded T)"
stutter_invariant_def: fixes \<phi> :: "'b pltl" shows "stutter_invariant \<phi> = (\<forall>\<sigma> \<tau>. \<sigma> \<approx> \<tau> \<longrightarrow> \<sigma> \<Turnstile>\<^sub>p \<phi> = \<tau> \<Turnstile>\<^sub>p \<phi>)"
transI: fixes r :: "('k \<times> 'k) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (y, z) \<in> r\<rbrakk> \<Longrightarrow> (x, z) \<in> r" shows "trans r"
HuntSands33: fixes p :: "L"   and G :: "Var \<Rightarrow> L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L" assumes "(p, G, c, H) \<in> HS" shows "secure p G c H"
U_single_in_FIN: shows "U_single \<in> FIN"
Un_least: fixes A :: "'e set"   and C :: "'e set"   and B :: "'e set" assumes "A \<subseteq> C"   and "B \<subseteq> C" shows "A \<union> B \<subseteq> C"
are_endpoints_in_S_ordered_cross_ratio: fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and b :: "proj2" assumes "are_endpoints_in_S p q a b"   and "B\<^sub>\<real> cart2_pt a cart2_pt b cart2_pt p" shows "1 \<le> cross_ratio p q a b"
Rf_mono_on_iia_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "Basis.mono_on (Pow A) (Rf f) = iia_on A f"
word_le_minus_mono: fixes a :: "'b word"   and c :: "'b word"   and d :: "'b word"   and b :: "'b word" assumes "a \<le> c"   and "d \<le> b"   and "a - b \<le> a"   and "c - d \<le> c" shows "a - b \<le> c - d"
comm_inv_char: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier V"   and "y \<in> carrier V"   and "x \<oplus>\<^bsub>V\<^esub> y = \<zero>\<^bsub>V\<^esub>" shows "inv\<^bsub>add_monoid V\<^esub> x = y"
Span_append: fixes ms :: "'b list"   and ns :: "'b list" shows "RSpan (ms @ ns) = RSpan ms + RSpan ns"
left_unique_OO: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "left_unique R"   and "left_unique S" shows "left_unique (R OO S)"
bh_1: shows "berlekamp_hensel p n f = hensel_lifting p n f gs"
HInit_HInv4: fixes s :: "state" assumes "HInit s" shows "HInv4 s"
jkbpCn_tFirst_init_inv: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and t :: "'e Trace"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview"   and "t \<in> PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n" shows "tFirst t \<in> set envInit"
subset_emptyI: fixes A :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> False" shows "A \<subseteq> {}"
dom_char: fixes f :: "'d" shows "Dop.dom f = D.cod f"
assms_12: shows "C = \<Union> (roots ` X (\<le>))"
e: fixes x :: "real" assumes "\<bar>x\<bar> < e" shows "\<bar>(f (x + t) - f t) / sin (x / 2)\<bar> \<le> 4 * (\<bar>M\<bar> * \<bar>x\<bar> powr (a - 1))"
equiintegrable_on_Un: fixes F :: "('a \<Rightarrow> 'b) set"   and I :: "'a set"   and G :: "('a \<Rightarrow> 'b) set" assumes "F equiintegrable_on I"   and "G equiintegrable_on I" shows "F \<union> G equiintegrable_on I"
mult_imp_le_div_pos: fixes y :: "'d"   and z :: "'d"   and x :: "'d" assumes "(0::'d) < y"   and "z * y \<le> x" shows "z \<le> x / y"
openin_Union: fixes K :: "'d set set"   and U :: "'d topology" assumes "\<And>S. S \<in> K \<Longrightarrow> openin U S" shows "openin U (\<Union> K)"
edges_nodes: shows "local.edges \<subseteq> local.nodes \<times> local.nodes"
in_cfl_in_cfls: fixes \<alpha> :: "('t + 'n) list"   and w :: "'t list" assumes "in_cfl G \<alpha> w" shows "in_cfls G {|\<alpha>|} w"
atoms_equiv_bounds_simps: fixes as :: "'a atom set"   and lb :: "nat \<Rightarrow> 'a option"   and ub :: "nat \<Rightarrow> 'a option" shows "as \<doteq> (lb, ub) \<equiv> \<forall>v. v \<Turnstile>\<^sub>a\<^sub>s as = v \<Turnstile>\<^sub>b (lb, ub)"
in_set_zip_map: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and f :: "'c \<Rightarrow> 'b"   and ys :: "'c list" assumes "(x, y) \<in> set (zip xs (map f ys))" obtains y' :: "'c" where "(x, y') \<in> set (zip xs ys)" and "f y' = y"
mset_le_mono_add_single: fixes a :: "'f"   and ys :: "'f multiset"   and b :: "'f"   and ws :: "'f multiset" assumes "a \<in># ys"   and "b \<in># ws" shows "{#a, b#} \<subseteq># ys + ws"
one_rel_def: shows "cq.one_rel = cq.frac \<one> \<one>"
sortedP_Cons_iff: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "linorder_list0.sortedP le (x # xs) = (Ball (set xs) (le x) \<and> linorder_list0.sortedP le xs)"
cc_restr_eqvt: fixes \<pi> :: "perm"   and S :: "Vars.var set"   and G :: "CoCalls" shows "\<pi> \<bullet> G G|` S = (\<pi> \<bullet> G) G|` (\<pi> \<bullet> S)"
Dynkin_lemma: fixes E :: "'a set set" assumes "Int_stable E"   and "E \<subseteq> {A \<in> sigma_sets \<Omega> G. P A}"   and "{A \<in> sigma_sets \<Omega> G. P A} \<subseteq> sigma_sets \<Omega> E" shows "sigma_sets \<Omega> E = {A \<in> sigma_sets \<Omega> G. P A}"
limproc_Sync_D: fixes Y :: "nat \<Rightarrow> 'c process"   and A :: "'c set"   and S :: "'c process" assumes "chain Y" shows "D (lim_proc (range Y) \<lbrakk>A\<rbrakk> S) = D (lim_proc (range (\<lambda>i. Y i \<lbrakk>A\<rbrakk> S)))"
mask_eq_0_eq_x: fixes x :: "'a word"   and w :: "'a word" shows "(x AND w = 0) = (x AND NOT w = x)"
not_gen_mono_on_subset: fixes T :: "'b set"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "T \<subseteq> S"   and "\<not> gen_mono_on f R T" shows "\<not> gen_mono_on f R S"
fps_conv_radius_inverse_pos: fixes f :: "'a fps" assumes "fps_nth f 0 \<noteq> (0::'a)"   and "0 < fps_conv_radius f" shows "0 < fps_conv_radius (inverse f)"
cod_simps_3: shows "gf.cod.leg1 = gf.leg1"
swap_id_common: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<noteq> c"   and "b \<noteq> c" shows "Fun.swap a b id \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id"
f: shows "index xs x = 0 \<and> index xs y = 1"
D0'_subset_DA0': shows "D0' \<subseteq># DA0'"
field_differentiable_at_within: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set" assumes "f field_differentiable at x" shows "f field_differentiable at x within S"
comp_arr_dom: fixes f :: "('a, 'a set) Maps.arr"   and a :: "('a, 'a set) Maps.arr" assumes "Maps.arr f"   and "Maps.dom f = a" shows "f \<odot> a = f"
P: shows "\<Gamma> \<Longrightarrow>\<^sub>t \<Gamma>1 * hn_val bool_rel a a'"
s_2: shows "t \<sqsubseteq> pair (s, s')"
matrix_vector_mult_add_rdistrib: fixes A :: "(('b, 'd) vec, 'c) vec"   and B :: "(('b, 'd) vec, 'c) vec"   and x :: "('b, 'd) vec" shows "(A + B) *v x = A *v x + B *v x"
are_isomorphic: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>" shows "\<exists>\<Phi>. invertible_functor S S' \<Phi> \<and> (\<forall>m. set_category.incl S \<AA> m \<longrightarrow> set_category.incl S' \<AA> (\<Phi> m))"
C: fixes z :: "'b" assumes "z \<in> Y" shows "dist y z \<le> C"
2: shows "finite (\<Union>y\<in>f ` A. A \<inter> f -` {y})"
less_irrefl: fixes x :: "'b word" shows "\<not> x <s x"
intros_1: fixes x :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and y :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "x = y" shows "abs.neval x 0 y"
pair: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1'"   and "g \<in> PrimRec1'" shows "c_f_pair f g \<in> PrimRec1'"
rbtreeify_g_code: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" shows "rbtreeify_g n kvs = (if n = 0 \<or> n = 1 then (Empty, kvs) else let (n', r) = divmod_nat n 2 in if r = 0 then case rbtreeify_g n' kvs of (t1, (k, v) # kvs') \<Rightarrow> apfst (Branch B t1 k v) (rbtreeify_g n' kvs') else case rbtreeify_f n' kvs of (t1, (k, v) # kvs') \<Rightarrow> apfst (Branch B t1 k v) (rbtreeify_g n' kvs'))"
disc_fct_def: fixes f :: "'b \<Rightarrow> 'c" shows "disc_fct f \<equiv> countable (range f)"
coprime_id_is_unit: fixes a :: "'b" shows "comm_monoid_mult_class.coprime a a = is_unit a"
pr_dup_1: fixes a :: "'a" assumes "ide a" shows "prj0' a a \<cdot> ECC.dup a = a"
ccw: fixes c :: "'a" assumes "c \<in> C" shows "ccw a b c"
subgroup_is_submonoid: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme" assumes "subgroup H G" shows "submonoid H G"
ExcessTable_cont_eq: fixes E :: "nat \<Rightarrow> nat"   and vs :: "nat list" shows "ExcessTable_cont E vs = map (\<lambda>v. (v, E v)) (filter (\<lambda>v. 0 < E v) vs)"
simps_30: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x51 :: "nat"   and x52 :: "nat"   and x53 :: "nat" shows "(case Pkt x51 x52 x53 of Rreq x xa xb xc xd xe xf xg xh \<Rightarrow> f1 x xa xb xc xd xe xf xg xh | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f5 x51 x52 x53"
extend_2: shows "v\<^sub>1 @ l =\<^sub>F w\<^sub>1"
strong_induct: fixes x1 :: "trm"   and x2 :: "trm"   and P :: "'a \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<longrightarrow>\<^sub>c x2"   and "\<And>M a N x y. \<lbrakk>x \<sharp> y; a \<sharp> y; \<not> fic M a; a \<sharp> N; x \<sharp> M\<rbrakk> \<Longrightarrow> P y (Cut <a>.M x.N) (M{a:=x.N})"   and "\<And>N x a M y. \<lbrakk>x \<sharp> y; a \<sharp> y; \<not> fin N x; a \<sharp> N; x \<sharp> M\<rbrakk> \<Longrightarrow> P y (Cut <a>.M x.N) (N{x:=<a>.M})" shows "P z x1 x2"
homologous_rel_set_eq_relboundary: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "(homologous_rel_set p X S c = singular_relboundary_set p X S) = singular_relboundary p X S c"
iterate_add_def: fixes n :: "nat"   and a :: "'a" shows "iterate_add n a = ((+) a ^^ n) (0::'a)"
Collect_cong: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
b_less_quarter: shows "\<bar>real_of_float (Float m2 e2)\<bar> < 1 / 4 * 2 powr real_of_int e1"
bigo_abs5: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(g)" shows "(\<lambda>x. \<bar>f x\<bar>) \<in> O(g)"
\<Delta>_simps_1: shows "seq (\<a>[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) TTfgh.tab"
well_ordered_strategy_def: shows "well_ordered_strategy \<equiv> override_on \<sigma>_arbitrary (\<lambda>v. choose v v) (winning_region p)"
rel_def: fixes Rv :: "('V \<times> 'a) set"   and Rw :: "('W \<times> 'b) set" shows "\<langle>Rv, Rw\<rangle>rel \<equiv> br \<alpha> invar O \<langle>Rv, Rw\<rangle>graph_rel"
FV_option_lem: fixes f :: "'a \<Rightarrow> sterm option" shows "\<forall>l\<in>dom f. FV (the (f l)) = FVoption (f l)"
deg_le_sect_def: fixes X :: "'b set"   and d :: "nat" shows "deg_le_sect X d = \<Union> (deg_sect X ` {..d})"
vwalk_verts_in_verts: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph"   and u :: "'a" assumes "vwalk p G"   and "u \<in> set p" shows "u \<in> verts G"
assms_2: shows "\<exists>c. hamlet (Rep_run r n c)"
continuous_map_into_subtopology: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "continuous_map X X' f"   and "f ` topspace X \<subseteq> T" shows "continuous_map X (subtopology X' T) f"
prime_nat_iff_prime: fixes k :: "int" shows "prime (nat k) = prime k"
thetaSeqZOD_sym: shows "sym thetaSeqZOD"
semiring_norm_78: fixes m :: "num"   and n :: "num" shows "(num.Bit0 m < num.Bit0 n) = (m < n)"
a: fixes a :: "'a" assumes "a \<in> A - {x}" shows "a \<in> A"
sum_list_triv: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and r :: "'c"   and xs :: "'d list" assumes "class.monoid_add plus zero" shows "(\<Sum>x\<leftarrow>xs. r) = of_nat (length xs) * r"
char_poly_matrix_hom: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "char_poly_matrix (mat\<^sub>h A) = map_poly_hom.mat_hom (char_poly_matrix A)"
zcount_empty: fixes a :: "'a" shows "zcount {#}\<^sub>z a = 0"
map_option_case: fixes f :: "'b \<Rightarrow> 'a"   and y :: "'b option" shows "map_option f y = (case y of None \<Rightarrow> None | Some x \<Rightarrow> Some (f x))"
simps_1: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> ('a, 'b, 'c) prog"   and d :: "'c" shows "fold_M f d [] = return d"
1: fixes t' :: "('fun, 'var) Term.term" assumes "t' \<in> \<Union> (ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>') \<union> ik\<^sub>e\<^sub>s\<^sub>t \<A>'" shows "t' \<in> \<Union> (ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>) \<union> ik\<^sub>e\<^sub>s\<^sub>t \<A>"
dterms_psimps_7: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and s\<^sub>m\<^sub>s\<^sub>g :: "'b \<Rightarrow> 'c"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "dterms \<Gamma> ({l}send(s\<^sub>m\<^sub>s\<^sub>g) . p) = sterms \<Gamma> p"
entt_disjI2_direct: fixes B :: "assn"   and A :: "assn" shows "B \<Longrightarrow>\<^sub>t A \<or>\<^sub>A B"
les_def: fixes R :: "'d \<Rightarrow> ('e \<times> 'f ifex) set"   and s :: "'d"   and s' :: "'d" shows "bdd_impl_pre.les R s s' \<equiv> \<forall>ni n. (ni, n) \<in> R s \<longrightarrow> (ni, n) \<in> R s'"
measure_increasing: shows "increasing (sets M) (Sigma_Algebra.measure M)"
a_rcos_module_rev: fixes x :: "'a"   and x' :: "'a" assumes "x \<in> carrier R"   and "x' \<in> carrier R"   and "x' \<oplus> \<ominus> x \<in> I" shows "x' \<in> I +> x"
secrets_are_homogeneous: fixes Sec :: "('fun, 'var) Term.term set"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool"   and s :: "('fun, 'var) Term.term"   and \<A> :: "('a strand_label \<times> ('fun, 'var) strand_step) list" assumes "\<forall>s\<in>Sec. P s \<longrightarrow> (\<forall>s'\<in>subterms s. {} \<turnstile>\<^sub>c s' \<or> s' \<in> Sec)"   and "s \<in> Sec"   and "P s" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t s \<A> Sec"
null_sets_UN': fixes I :: "'b set"   and N :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> N i \<in> null_sets M" shows "\<Union> (N ` I) \<in> null_sets M"
cases: fixes a1 :: "state"   and a2 :: "com"   and a3 :: "nat"   and a4 :: "state" assumes "a1 -a2-a3\<rightarrow> a4" obtains t :: "state" and f :: "state \<Rightarrow> state set" and s :: "state" and n :: "nat" where "a1 = sa__" and "a2 = Do f__" and "a3 = na__" and "a4 = ta__" and "ta__ \<in> f__ sa__"   | s0 :: "state" and c1 :: "com" and n :: "nat" and s1 :: "state" and c2 :: "com" and s2 :: "state" where "a1 = s0" and "a2 = (c1__; c2__)" and "a3 = nb__" and "a4 = s2" and "s0 -c1__-nb__\<rightarrow> s1" and "s1 -c2__-nb__\<rightarrow> s2"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and n :: "nat" and t :: "state" and c2 :: "com" where "a1 = sb__" and "a2 = IF ba__ THEN c1a__ ELSE c2a__" and "a3 = nc__" and "a4 = tb__" and "ba__ sb__" and "sb__ -c1a__-nc__\<rightarrow> tb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c2 :: "com" and n :: "nat" and t :: "state" and c1 :: "com" where "a1 = sc__" and "a2 = IF bb__ THEN c1 ELSE c2" and "a3 = nd__" and "a4 = tc__" and "\<not> bb__ sc__" and "sc__ -c2-nd__\<rightarrow> tc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and n :: "nat" where "a1 = sd__" and "a2 = WHILE bc__ DO ca__" and "a3 = ne__" and "a4 = sd__" and "\<not> bc__ sd__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and n :: "nat" and t :: "state" and u :: "state" where "a1 = se__" and "a2 = WHILE b DO cb__" and "a3 = nf__" and "a4 = u" and "b se__" and "se__ -cb__-nf__\<rightarrow> td__" and "td__ -WHILE b DO cb__-nf__\<rightarrow> u"   | s :: "state" and n :: "nat" and t :: "state" where "a1 = sf__" and "a2 = CALL" and "a3 = Suc ng__" and "a4 = te__" and "sf__ -body-ng__\<rightarrow> te__"   | f :: "state \<Rightarrow> state" and s :: "state" and c :: "com" and n :: "nat" and t :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = s" and "a2 = LOCAL f; c; g" and "a3 = n" and "a4 = g s t" and "f s -c-n\<rightarrow> t"
hnorm_less_Infinitesimal: fixes e :: "real star"   and x :: "'a star" assumes "e \<in> Infinitesimal"   and "hnorm x < e" shows "x \<in> Infinitesimal"
integral_spike: fixes S :: "'a set"   and T :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "integral T f = integral T g"
ttt: shows "\<forall>n\<ge>t. ff (n + 1) < d (n + 1) * Max (ff ` {s..n})"
not_open_singleton: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.perfect_space open" shows "\<not> open {x}"
ZFact_def: fixes k :: "int" shows "ZFact k = \<Z> Quot Idl\<^bsub>\<Z>\<^esub> {k}"