hcju/isabelleps huggingface dataset:
simps_9: fixes v :: "Term.var" shows "constVal (Acc v) = None"
FIXvarFIX': fixes b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "PhiWhile b \<Phi> = (\<lambda>(s, t). (b, \<Phi>, s, t) \<in> var)"
con: shows "(if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'a) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'a)) + 1 else 0) \<le> rank A"
conjugate_cancel_iff: fixes conjugate :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.conjugate conjugate" shows "(conjugate a = conjugate b) = (a = b)"
mult_scaleQ_right: fixes x :: "'a['b, 'd]"   and a :: "'a"   and y :: "'a['c, 'd]" shows "x \<^bold>\<cdot> (a *\<^sub>Q y) = a *\<^sub>Q x \<^bold>\<cdot> y"
refine: fixes E :: "'a \<Rightarrow> 'a list"   and onstack :: "('a, unit) RBT_Impl.rbt"   and V :: "('a, unit) RBT_Impl.rbt"   and u :: "'a" shows "RETURN (code_red_dfs E onstack V u) \<le> impl_red_dfs E onstack V u"
with_proj_def: fixes G :: "'b pair_pre_digraph" shows "with_proj G = \<lparr>verts = pverts G, arcs = parcs G, tail = fst, head = snd\<rparr>"
StartBallot_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "StartBallot s s' p = (phase s p \<in> {1, 2} \<and> phase s' = (phase s)(p := 1) \<and> (\<exists>b\<in>Ballot p. mbal (dblock s p) < b \<and> dblock s' = (dblock s)(p := dblock s p\<lparr>mbal := b\<rparr>)) \<and> InitializePhase s s' p \<and> inpt s' = inpt s \<and> outpt s' = outpt s \<and> disk s' = disk s)"
vec_def: shows "vec \<equiv> map_fun id (map_fun id Abs_vec) (\<lambda>n f. (n, mk_vec n f))"
Quotient_symp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "symp R"
catch_state_def: fixes m1 :: "('s, 'm) stateT"   and m2 :: "('s, 'm) stateT" shows "catch_state m1 m2 = StateT (\<lambda>s. catch (run_state m1 s) (run_state m2 s))"
unifiers_Fun: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and f :: "'d"   and ss :: "('d, 'c) Term.term list"   and g :: "'d"   and ts :: "('d, 'c) Term.term list" shows "(\<sigma> \<in> unifiers {(Fun f ss, Fun g ts)}) = (length ss = length ts \<and> f = g \<and> \<sigma> \<in> unifiers (set (zip ss ts)))"
dvd_imp_multiplicity_le: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a b"   and "b \<noteq> zero" shows "factorial_semiring.multiplicity times one p a \<le> factorial_semiring.multiplicity times one p b"
collapse: fixes strand_label :: "'d strand_label" assumes "strand_label \<noteq> \<star>" shows "ln the_LabelN strand_label = strand_label"
length_rotate: fixes n :: "nat"   and xs :: "'b list" shows "length (rotate n xs) = length xs"
set_simps_2: fixes x21 :: "'b"   and x22 :: "'b list" shows "set (x21 # x22) = insert x21 (set x22)"
simps: fixes n :: "nat"   and p :: "int poly"   and l :: "rat"   and r :: "rat" shows "root_pos_1 n (p, l, r) = select_correct_factor_int_poly (tighten_bound_root n (compare_1_rat (p, l, r))) (\<lambda>x. x) (initial_lower_bound n l, initial_upper_bound n r) (poly_nth_root n p)"
mem_Ch_CH: fixes x :: "'a"   and h :: "'c"   and X :: "'a set" assumes "x \<in> BCh' h X" shows "x \<in> Bossy''.CH X"
prm_set_notmembership: fixes x :: "'b"   and S :: "'b set"   and \<pi> :: "'b prm" assumes "x \<notin> S" shows "\<pi> $ x \<notin> \<pi> {$} S"
B23: fixes x :: "'a"   and y :: "'a" assumes "(1::'a) + x \<le> y"   and "y \<odot> y = y" shows "x\<^sup>\<star> \<le> y"
on_curvep_infinity: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and y :: "'a" shows "on_curvep a b (x, y, \<zero>) = (x \<in> carrier R \<and> y \<in> carrier R)"
61231: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<sqinter> (- - y \<sqinter> - (x \<sqinter> z)) = - x \<sqinter> - - y"
simps: fixes p :: "'c \<times> 'd"   and a :: "'c" shows "fstsp p a = (a = fst p)"
stream_enc_unique: fixes i :: "nat"   and I :: "(nat + nat set) list"   and w :: "'a list" assumes "i < length I"   and "\<exists>p. I ! i = Inl p" shows "\<exists>!p. snd (stream_enc (w, I) !! p) ! i"
exec_gpv_oracle_bisim_bad_full: fixes X :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and s1 :: "'b"   and s2 :: "'c"   and bad1 :: "'b \<Rightarrow> bool"   and bad2 :: "'c \<Rightarrow> bool"   and oracle1 :: "'b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'b) spmf"   and oracle2 :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf"   and gpv :: "('f, 'd, 'e) gpv"   and R :: "'f \<times> 'b \<Rightarrow> 'f \<times> 'c \<Rightarrow> bool" assumes "X s1 s2"   and "bad1 s1 = bad2 s2"   and "\<And>s1 s2 x. X s1 s2 \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (\<not> bad2 s2' \<longrightarrow> a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"   and "callee_invariant oracle1 bad1"   and "callee_invariant oracle2 bad2"   and "\<And>s1 x. bad1 s1 \<Longrightarrow> lossless_spmf (oracle1 s1 x)"   and "\<And>s2 x. bad2 s2 \<Longrightarrow> lossless_spmf (oracle2 s2 x)"   and "lossless_gpv \<I>_full gpv"   and "\<And>a s1 b s2. \<lbrakk>bad1 s1 = bad2 s2; \<not> bad2 s2 \<Longrightarrow> a = b \<and> X s1 s2\<rbrakk> \<Longrightarrow> R (a, s1) (b, s2)" shows "rel_spmf R (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
time_filter_tm: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "time (filter_tm P xs) = length xs + 1"
hyps: shows "map_poly Re (p ^ n) = map_poly Re p ^ n"
attach_shadow_root_element_ptr_in_heap: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and shadow_root_mode :: "shadow_root_mode" assumes "h \<turnstile> ok attach_shadow_root element_ptr shadow_root_mode" shows "element_ptr |\<in>| element_ptr_kinds h"
hyps_3: assumes "f \<in> A \<rightarrow> carrier R"   and "\<exists>a\<in>A. f a = \<zero>" shows "finprod R f A = \<zero>"
simps_6: fixes f1 :: "\<omega> \<Rightarrow> 'a"   and f2 :: "\<Pi>\<^sub>1 set \<Rightarrow> 'a"   and x2 :: "\<Pi>\<^sub>1 set" shows "(case \<alpha>\<nu> x2 of \<omega>\<nu> x \<Rightarrow> f1 x | \<alpha>\<nu> x \<Rightarrow> f2 x) = f2 x2"
cCons_not_0_eq: fixes x :: "'b"   and xs :: "'b list" assumes "x \<noteq> (0::'b)" shows "x ## xs = x # xs"
post_before_pre_different_processes: fixes i :: "nat"   and j :: "nat"   and t :: "('a, 'b, 'c) event list" assumes "i < j"   and "j < length t"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "postrecording_event t i"   and "prerecording_event t j"   and "trace init t final" shows "occurs_on (t ! i) \<noteq> occurs_on (t ! j)"
equiv_partition_on: fixes A :: "'b set"   and P :: "'b set set" assumes "partition_on A P" shows "equiv A {(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p}"
DomUnion: fixes a :: "'a"   and b :: "'b"   and L :: "('a \<times> 'b) list" shows "Dom ((a, b) # L) = {a} \<union> Dom L"
isGlb_le_isLb: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "isGlb R S x"   and "isLb R S y" shows "y \<le> x"
gAbsRen_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gAbsRen MOD \<equiv> \<forall>xs y x s X' X. isInBar (xs, s) \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> fresh xs y X' \<and> gFresh MOD xs y X' X \<longrightarrow> gAbs MOD xs y (X' #[y // x]_xs) (gSubst MOD xs (Var xs y) (gVar MOD xs y) x X' X) = gAbs MOD xs x X' X"
drop_bv_mapzip: fixes n :: "nat"   and x :: "bit list"   and y :: "bit list"   and f :: "bit \<Rightarrow> bit \<Rightarrow> bit" assumes "n \<le> length x"   and "length x = length y" shows "drop n (bv_mapzip f x y) = bv_mapzip f (drop n x) (drop n y)"
CollectE: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "a \<in> {x. P x}" obtains "P a"
below_bottom_iff: fixes x :: "'a" shows "(x \<sqsubseteq> \<bottom>) = (x = \<bottom>)"
hausdorff_distance_closures: fixes A :: "'b set"   and B :: "'b set" shows "hausdorff_distance (closure A) (closure B) = hausdorff_distance A B"
sup_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd err" shows "Listn.sup f = (\<lambda>xs ys. if length xs = length ys then coalesce (xs [+f] ys) else Err)"
rref_upt_condition1: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. is_zero_row_upt_k i k A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row_upt_k j k A)"
bit_minus_1_iff: 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" 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 "bit (uminus one) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero)"
ivl_disj_un_one_4: fixes l :: "'a"   and u :: "'a" assumes "l \<le> u" shows "{..<l} \<union> {l..u} = {..u}"
bind_spmf_pmf_assoc: fixes p :: "'f pmf"   and f :: "'f \<Rightarrow> 'e spmf"   and g :: "'e \<Rightarrow> 'd spmf" shows "p \<bind> f \<bind> g = p \<bind> (\<lambda>x. f x \<bind> g)"
assms_5: shows "distinct [u, v, w] \<longrightarrow> perpendicular (poincare_line u v) (poincare_line u w)"
vals_def: shows "vals \<equiv> fin_string_upto MAX_STR"
l_distr_rng_of_frac: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a"   and r'' :: "'a"   and s'' :: "'a" assumes "(r, s) \<in> carrier rel"   and "(r', s') \<in> carrier rel"   and "(r'', s'') \<in> carrier rel" shows "(r'' |\<^bsub>rel\<^esub> s'') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> ((r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s')) = (r'' |\<^bsub>rel\<^esub> s'') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s) \<oplus>\<^bsub>rec_rng_of_frac\<^esub> (r'' |\<^bsub>rel\<^esub> s'') \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s')"
assms_2: shows "B \<subseteq>\<^sub>\<circ> A"
fresh_vsubst_E2: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "fresh zs z (X #[y1 // y]_ys)" shows "fresh ys y X \<or> zs \<noteq> ys \<or> z \<noteq> y1"
disjoint_def: fixes A :: "'a set set" shows "disjoint A = (\<forall>a\<in>A. \<forall>b\<in>A. a \<noteq> b \<longrightarrow> a \<inter> b = {})"
sums_mult: fixes a :: "'a"   and c :: "'a" assumes "f sums a" shows "(\<lambda>n. c * f n) sums (c * a)"
pos_mod_sign: fixes b :: "'c"   and a :: "'c" assumes "(0::'c) < b" shows "(0::'c) \<le> a mod b"
le_F_T: fixes P :: "'b process"   and Q :: "'b process" assumes "P \<sqsubseteq>\<^sub>F Q" shows "P \<sqsubseteq>\<^sub>T Q"
IH_1: fixes t' :: "'a tree" assumes "del x l = Some t'" shows "\<Phi> t' - \<Phi> l \<le> 6 * e * real (height l)"
vars_in_instance_in_range_lit: fixes L :: "('b, 'a) Term.term literal"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "vars_lit (subst_lit L \<sigma>) \<subseteq> \<Union> (vars_term ` range \<sigma>)"
uv_3: shows "\<not> (u1, u2) \<approx>\<^sub>v \<infinity>\<^sub>v"
wh: shows "(f \<circ> pseudo_proj_True n) w \<noteq> f w"
coeff_monom: fixes a :: "'c"   and m :: "nat"   and n :: "nat" shows "coeff (monom a m) n = (if m = n then a else (0::'c))"
zero_rat_def: shows "0 \<equiv> Abs_Rat (0, 1)"
bij_perm_swap_iff: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'a" shows "bij (perm_swap a b f) = bij f"
inject_1: fixes x5 :: "char list"   and y5 :: "char list" shows "(Call x5 = Call y5) = (x5 = y5)"
lookup_dflt_eq_lookup_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "lookup_dflt xs = comparator.lookup_pair comp xs"
inv_from_bot_def: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and mi :: "nat" shows "local.inv_from_bot p A mi \<equiv> \<forall>i\<ge>mi. i < n \<longrightarrow> p A i"
lm021: fixes bids :: "'d \<times> 'e set \<Rightarrow> 'f"   and X :: "('d \<times> 'e set) set" shows "inj_on (summedBid bids) X"
of_nat_le_hmset_of_enat_iff: fixes m :: "nat"   and n :: "enat" shows "(of_nat m \<le> hmset_of_enat n) = (enat m \<le> n)"
face_of_convex_hulls: fixes S :: "'b set"   and T :: "'b set" assumes "finite S"   and "T \<subseteq> S"   and "affine hull T \<inter> convex hull (S - T) = {}" shows "convex hull T face_of convex hull S"
assms_3: shows "z \<equiv> the_elem (Set.filter (\<lambda>(x, y). x\<^sup>2 + y\<^sup>2 = p) (SIGMA x:{0..p}. {x..p}))"
enat_sub_add: fixes y :: "enat"   and x :: "enat"   and z :: "enat" assumes "y \<le> x" shows "x - y + z = x + z - y"
globally_consistent_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and A :: "'b \<Rightarrow> ('c \<times> 'c) option"   and mds :: "Mode \<Rightarrow> 'b set" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.globally_consistent dma A mds \<equiv> finite (dom A) \<and> (\<forall>x\<in>dom A. sifum_security.globally_consistent_var dma A mds x)"
disc_8: fixes x31 :: "nat"   and x32 :: "'d"   and x33 :: "'d"   and x34 :: "'e"   and x35 :: "'e"   and x36 :: "'f" shows "\<not> isSend (Recv x31 x32 x33 x34 x35 x36)"
dis_rews_4: fixes x :: "[:'a:]" shows "(is_scons\<cdot>x = \<bottom>) = (x = \<bottom>)"
Its_fun: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int"   and n :: "nat"   and n' :: "nat" assumes "Its b c s n"   and "Its b c s n'" shows "n = n'"
H: shows "\<Gamma> \<turnstile> Rcd fs..l : T"
countable_UN'': fixes X :: "'a set"   and A :: "'a \<Rightarrow> 'c set"   and M :: "'c measure" assumes "countable X"   and "\<And>x y. x \<in> X \<Longrightarrow> A x \<in> sets M" shows "\<Union> (A ` X) \<in> sets M"
Ae_def: shows "Ae = A(decode_Var v::=Eats (Var (decode_Var v)) (Var (decode_Var w)))"
hb_consistent_append_porder: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and x :: "'c"   and y :: "'c" assumes "happens_before hb_weak hb"   and "happens_before.hb_consistent hb (xs @ ys)"   and "x \<in> set xs"   and "y \<in> set ys" shows "\<not> hb y x"
ParCompose: fixes P :: "pi"   and Q :: "pi"   and R :: "pi"   and T :: "pi" assumes "P \<approx> Q"   and "(R, T) \<in> bisim O bangRel weakBisim O bisim" shows "(P \<parallel> R, Q \<parallel> T) \<in> bisim O bangRel weakBisim O bisim"
OrdI: fixes i :: "V" assumes "Transset i"   and "\<And>x. x \<in> elts i \<Longrightarrow> Transset x" shows "Ord i"
rtos_def: fixes r :: "nat"   and s :: "nat"   and i :: "nat" shows "rtos r s i = (if i < r * s then i mod s * r + i div s else r * s)"
eq_iff_diff_eq_0: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "(a = b) = (minus a b = zero)"
las_trans_l: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset" assumes "locally_antisym R (A + B)" shows "locally_antisym R A"
parallel_oracle_def: fixes left :: "'h \<Rightarrow> 'j \<Rightarrow> ('l \<times> 'h) spmf"   and right :: "'i \<Rightarrow> 'k \<Rightarrow> ('m \<times> 'i) spmf" shows "left \<ddagger>\<^sub>O right = (\<lambda>(s1, s2). case_sum (\<lambda>a. map_spmf (map_prod Inl (\<lambda>s1'. (s1', s2))) (left s1 a)) (\<lambda>b. map_spmf (map_prod Inr (Pair s1)) (right s2 b)))"
concurrent_direct_execution_simulates_store_buffer_history_execution: fixes ts\<^sub>s\<^sub>b :: "('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 m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts\<^sub>s\<^sub>b' :: "('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 m\<^sub>s\<^sub>b' :: "nat \<Rightarrow> nat"   and \<S>\<^sub>s\<^sub>b' :: "nat \<Rightarrow> bool option"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "ini.sbh_concurrent_steps program_step (ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b')"   and "initial\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b"   and "valid ts\<^sub>s\<^sub>b"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"   and "ini.safe_reach_direct program_step safe_free_flowing (ts, m, \<S>)" shows "\<exists>ts' m' \<S>'. ini.direct_concurrent_steps program_step (ts, m, \<S>) (ts', m', \<S>') \<and> (ts\<^sub>s\<^sub>b', m\<^sub>s\<^sub>b', \<S>\<^sub>s\<^sub>b') \<sim> (ts', m', \<S>')"
main_1: shows "(\<lambda>i. cis (real i * 2 * pi / real n)) ` {0..<n} = {x. x ^ n = 1}"
hconf_upd_obj: fixes P :: "'a prog"   and h :: "heap"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields"   and fs' :: "Objects.fields" assumes "P \<turnstile> h \<surd>"   and "h a = \<lfloor>(C, fs)\<rfloor>"   and "P,h \<turnstile> (C, fs') \<surd>" shows "P \<turnstile> h(a \<mapsto> (C, fs')) \<surd>"
coprime_1_right: fixes a :: "'a" shows "coprime a (1::'a)"
munionM_mUnionM1: fixes A :: "'m multiset"   and x :: "'m" shows "munionM (mUnionM A) x = fold_mset munionM x A"
refine: fixes c :: "nat \<times> nat \<Rightarrow> int"   and s :: "nat"   and t :: "nat"   and N :: "nat"   and am :: "nat \<Rightarrow> nat list"   and psi :: "nat list Heap.array"   and cf :: "nat \<times> nat \<Rightarrow> int"   and cfi :: "int Heap.array"   and u :: "nat"   and ui :: "nat" assumes "Edka_Impl c s t N" shows "hn_refine (hn_ctxt (Edka_Impl.is_am N) am psi * hn_ctxt (asmtx_assn N int_assn) cf cfi * hn_val nat_rel u ui) (succ_imp N psi cfi ui) (hn_ctxt (asmtx_assn N int_assn) cf cfi * hn_val nat_rel u ui * hn_ctxt (Edka_Impl.is_am N) am psi) (list_assn nat_assn) (Network.rg_succ2 c am cf u)"
le_cases3: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a"   and z :: "'a" obtains "ko.le ko x y" and "ko.le ko y z"   | "ko.le ko y x" and "ko.le ko x z"   | "ko.le ko x z" and "ko.le ko z y"   | "ko.le ko z y" and "ko.le ko y x"   | "ko.le ko y z" and "ko.le ko z x"   | "ko.le ko z x" and "ko.le ko x y"
is_idirprod_def: fixes A :: "'a set"   and S :: "'c \<Rightarrow> 'a set"   and I :: "'c set" shows "is_idirprod A S I = ((\<forall>i\<in>I. S i \<lhd> G) \<and> A = IDirProds G S I \<and> compl_fam S I)"
nn_integral_monotone_convergence_INF_AE: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure"   and i :: "nat" assumes "\<And>i. AE x in M. f (Suc i) x \<le> f i x"   and "\<And>i. f i \<in> borel_measurable M"   and "integral\<^sup>N M (f i) < \<infinity>" shows "\<integral>\<^sup>+ x. (INF i. f i x) \<partial>M = (INF i. integral\<^sup>N M (f i))"
card_UNIV_tau_i_below_zero: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "finite UNIV"   and "mono \<tau>"   and "\<forall>i. (\<tau> ^^ (i + 1)) UNIV \<subset> (\<tau> ^^ i) UNIV" shows "card ((\<tau> ^^ card UNIV) UNIV) \<le> 0"
CTYPE_ANNOT_def: fixes c :: "'a"   and I :: "'b itself" shows "c :::\<^sub>i I \<equiv> c"
primepow_mult_aprimedivisorI: fixes n :: "'a" assumes "primepow n" shows "primepow (aprimedivisor n * n)"
aristo_conversion3: fixes A :: "'a set"   and B :: "'a set" assumes "A Q B"   and "B \<noteq> {}" shows "B I A"
frameChainAlpha': fixes p :: "(name \<times> name) list"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'d" assumes "(p \<bullet> A\<^sub>P) \<sharp>* \<Psi>\<^sub>P"   and "set p \<subseteq> set A\<^sub>P \<times> set (p \<bullet> A\<^sub>P)" shows "\<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle> = \<langle>p \<bullet> A\<^sub>P, p \<bullet> \<Psi>\<^sub>P\<rangle>"
generate_eq_finprod_Pi_image: fixes gs :: "'a set" assumes "finite gs"   and "gs \<subseteq> carrier G" shows "generate G gs = (\<lambda>x. finprod G x gs) ` (\<Pi> a\<in>gs. generate G {a})"
hyps: assumes "k < n" shows "H_k k *\<^sub>v ket_zero_k k = ket_plus_k k"
rbtreeify_f_conv_drop: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> length kvs" shows "snd (rbtreeify_f n kvs) = drop n kvs"
le_divide_eq_1_pos: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less_eq one (divide b a) = less_eq a b"
equiv_sym: fixes V :: "'b set"   and R :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "equiv V R"   and "(x, y) \<in> R" shows "(y, x) \<in> R"
limited_preimages: fixes f :: "'b \<Rightarrow> 'a"   and D :: "'b set"   and k :: "nat" assumes "\<forall>x\<in>f ` D. card (f -` {x} \<inter> D) \<le> k"   and "finite D" shows "card D \<le> card (f ` D) * k"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Unblinded x1 = Unblinded y1) = (x1 = y1)"
message_must_be_delivered_2_trace: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and m :: "'c message"   and i :: "nat"   and q :: "'a"   and p :: "'a" assumes "trace c t c'"   and "m \<in> set (msgs c i)"   and "m \<notin> set (msgs c' i)"   and "channel i = Some (q, p)" shows "\<exists>ev\<in>set t. (\<exists>p q. ev = RecvMarker i p q \<and> m = Marker) \<or> (\<exists>p q s s' m'. ev = Recv i q p s s' m' \<and> m = Msg m')"
simps_4: fixes v :: "trm"   and va :: "trm" shows "get_and_list (formula.Eq v va) = [formula.Eq v va]"
hyps: shows "block_nominals (mapi_block (bridge k j xs v) (ps, a)) \<union> {k, j} = block_nominals (ps, a) \<union> {k, j}"
assms_4: assumes "\<gamma> (Trans s a ou s')" shows "g (Trans s a ou s') = g (Trans s1 a1 ou1 s1')"