hcju/isabelleps huggingface dataset:
bind_eq_return_spmf: fixes p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf"   and x :: "'c" shows "(p \<bind> f = return_spmf x) = ((\<forall>y\<in>set_spmf p. f y = return_spmf x) \<and> lossless_spmf p)"
simps_2: fixes left :: "'c \<Rightarrow> 'd \<Rightarrow> ('a option \<times> 'c) spmf"   and right :: "'c \<Rightarrow> 'e \<Rightarrow> ('b option \<times> 'c) spmf"   and s :: "'c"   and b :: "'e" shows "(left \<oplus>\<^sub>O\<^sup>S right) s (Inr b) = map_spmf (apfst (map_option Inr)) (right s b)"
simps: fixes P :: "nat list \<Rightarrow> bool"   and s :: "nat"   and lm :: "nat list" shows "P abc_holds_for (s, lm) = P lm"
big_clocked_unclocked_equiv: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r1 :: "(v, v) result" shows "evaluate False env s e (s', r1) = (\<exists>c. evaluate True env (update_clock (\<lambda>_. c) s) e (update_clock (\<lambda>_. 0) s', r1) \<and> r1 \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock s')"
gcsrp_iop_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "gcsrp_iop G H = (\<lambda>x\<in>gcsrp_map G H ` set_rcs G H. gcsrp_map G H (c_iop G H ((gcsrp_map G H\<inverse>\<^bsub>gcsrp_map G H ` set_rcs G H,set_rcs G H\<^esub>) x)))"
extend_lub: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and u :: "'b"   and x' :: "'b" assumes "single_valued r"   and "is_lub (r\<^sup>*) x y u"   and "(x', x) \<in> r" shows "\<exists>v. is_lub (r\<^sup>*) x' y v"
UnionE: fixes A :: "'a"   and C :: "'a set set" assumes "s2p (\<mu> C) A" obtains X :: "'a set" where "s2p X A" and "s2p C X"
cases: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'b + 'd"   and a2 :: "'c + 'e" assumes "rel_sum R1 R2 a1 a2" obtains a :: "'b" and c :: "'c" where "a1 = Inl a" and "a2 = Inl c" and "R1 a c"   | b :: "'d" and d :: "'e" where "a1 = Inr b" and "a2 = Inr d" and "R2 b d"
subst_bvs1'_empty: fixes t :: "Core.term"   and lev :: "nat" shows "subst_bvs1' t lev [] = t"
mset_subset_eqD: fixes A :: "'f multiset"   and B :: "'f multiset"   and x :: "'f" assumes "A \<subseteq># B"   and "x \<in># A" shows "x \<in># B"
assms_1: shows "X \<in> obj\<^bsub>C\<^esub>"
that: assumes "f u \<circ> f v = f v \<circ> f u" shows "thesis"
c_snd_snd_lt: fixes x :: "nat" shows "c_snd (c_snd (Suc (Suc x))) < Suc (Suc x)"
ab_1: shows "ball (0::'n) B1 \<subseteq> cbox a b"
nth_bin_to_bl: fixes n :: "nat"   and m :: "nat"   and w :: "int" assumes "n < m" shows "bin_to_bl m w ! n = bit w (m - Suc n)"
trans_node_comp: fixes i :: "nat"   and np :: "('c, 'd seq_action) automaton"   and R\<^sub>i :: "nat set" shows "automaton.trans (\<langle>i : np : R\<^sub>i\<rangle>) = node_sos (automaton.trans np)"
oalist_inv_raw_induct: fixes xs :: "('a \<times> 'c) list"   and P :: "('a \<times> 'c) list \<Rightarrow> bool" assumes "oalist_inv_raw xs"   and "P []"   and "\<And>k v xs. \<lbrakk>oalist_inv_raw ((k, v) # xs); oalist_inv_raw xs; v \<noteq> (0::'c); \<And>k'. k' \<in> fst ` set xs \<Longrightarrow> lt k k'; P xs\<rbrakk> \<Longrightarrow> P ((k, v) # xs)" shows "P xs"
add_comm: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.plus_ord leq add" shows "add x y = add y x"
wf_action_translation_on_def: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and A :: "nat set"   and f :: "nat \<Rightarrow> nat" shows "wf_action_translation_on E E' A f = (inj_on f (actions E) \<and> (\<forall>a\<in>A. action_tid E a = action_tid E' (f a) \<and> action_obs E a \<approx> action_obs E' (f a)))"
get_child_nodes_new_element: fixes ptr' :: "(_) object_ptr"   and new_element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "r \<in> Shadow_DOM.get_child_nodes_locs ptr'" shows "r h h'"
interpret_op_def: fixes oper :: "('d, 'c) operation"   and state :: "'c \<Rightarrow> 'd set \<times> 'd set" shows "\<langle>oper\<rangle> state \<equiv> let metadata = fst (state (op_elem oper)); files = snd (state (op_elem oper)); after = case oper of Create i e \<Rightarrow> (metadata \<union> {i}, files) | Delete is e \<Rightarrow> (metadata - is, files - is) | Append i e \<Rightarrow> (metadata, files \<union> {i}) | Expunge e mo i \<Rightarrow> (metadata \<union> {i}, files - {mo}) | Store e mo i \<Rightarrow> (metadata, insert i (files - {mo})) in Some (state(op_elem oper := after))"
induct: fixes x1 :: "Var \<Rightarrow> Val"   and x2 :: "IMP"   and x3 :: "nat"   and x4 :: "Var \<Rightarrow> Val"   and P :: "(Var \<Rightarrow> Val) \<Rightarrow> IMP \<Rightarrow> nat \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes " x1 , x2 \<rightarrow>\<^sub>x3  x4 "   and "\<And>s. P s Skip 1 s"   and "\<And>t s x e. t = update s x (evalE e s) \<Longrightarrow> P s (Assign x e) 1 t"   and "\<And>s c1 n r c2 m t k. \<lbrakk> s , c1 \<rightarrow>\<^sub>n  r ; P s c1 n r;  r , c2 \<rightarrow>\<^sub>m  t ; P r c2 m t; k = max n m + 1\<rbrakk> \<Longrightarrow> P s (Comp c1 c2) k t"   and "\<And>b s c n r m t k. \<lbrakk>evalB b s;  s , c \<rightarrow>\<^sub>n  r ; P s c n r;  r , While b c \<rightarrow>\<^sub>m  t ; P r (While b c) m t; k = max n m + 1\<rbrakk> \<Longrightarrow> P s (While b c) k t"   and "\<And>b s t c. \<lbrakk>\<not> evalB b s; t = s\<rbrakk> \<Longrightarrow> P s (While b c) 1 t"   and "\<And>b s c1 n t c2. \<lbrakk>evalB b s;  s , c1 \<rightarrow>\<^sub>n  t ; P s c1 n t\<rbrakk> \<Longrightarrow> P s (Iff b c1 c2) (n + 1) t"   and "\<And>b s c2 n t c1. \<lbrakk>\<not> evalB b s;  s , c2 \<rightarrow>\<^sub>n  t ; P s c2 n t\<rbrakk> \<Longrightarrow> P s (Iff b c1 c2) (n + 1) t"   and "\<And>s n t. \<lbrakk> s , body \<rightarrow>\<^sub>n  t ; P s body n t\<rbrakk> \<Longrightarrow> P s Call (n + 1) t" shows "P x1 x2 x3 x4"
wf_iff_no_infinite_down_chain: fixes r :: "('c \<times> 'c) set" shows "wf r = (\<nexists>f. \<forall>i. (f (Suc i), f i) \<in> r)"
prime_card_elements_with_ord_eq_totient: fixes d :: "nat" assumes "prime n" shows "\<psi> d = (if d dvd n - 1 then totient d else 0)"
bot_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b" assumes "class.linordered_comm_monoid_add_bot plus less_eq less bot" shows "plus bot x = x"
set_band_filter: fixes p\<^sub>0 :: "int \<times> int"   and ps :: "(int \<times> int) list"   and p\<^sub>1 :: "int \<times> int"   and \<delta> :: "real"   and ps\<^sub>L :: "(int \<times> int) set"   and ps\<^sub>R :: "(int \<times> int) set"   and l :: "int"   and ps' :: "(int \<times> int) list" assumes "p\<^sub>0 \<in> set ps"   and "p\<^sub>1 \<in> set ps"   and "p\<^sub>0 \<noteq> p\<^sub>1"   and "dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < \<delta>"   and "set ps = ps\<^sub>L \<union> ps\<^sub>R"   and "sparse \<delta> ps\<^sub>L"   and "sparse \<delta> ps\<^sub>R"   and "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"   and "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"   and "ps' = filter (\<lambda>p. real_of_int l - \<delta> < real_of_int (fst p) \<and> real_of_int (fst p) < real_of_int l + \<delta>) ps" shows "p\<^sub>0 \<in> set ps' \<and> p\<^sub>1 \<in> set ps'"
one_fun_var1: fixes x :: "'c" assumes "x \<in> E" shows "1 x = (1::'d)"
SUP_const_minus_ennreal: fixes I :: "'b set"   and c :: "ennreal"   and f :: "'b \<Rightarrow> ennreal" assumes "I \<noteq> {}" shows "(SUP x\<in>I. c - f x) = c - Inf (f ` I)"
merge_not_top: fixes ss :: "(nat \<times> 'a) list"   and c :: "'a list"   and pc :: "nat"   and x :: "'a" assumes "snd ` set ss \<subseteq> A"   and "merge c pc ss x \<noteq> \<top>" shows "\<forall>(pc', s')\<in>set ss. pc' \<noteq> pc + 1 \<longrightarrow> s' \<sqsubseteq>\<^bsub>r\<^esub> c ! pc'"
assms_2: shows "\<forall>c. c \<in> Com X \<longrightarrow> vars (strip c) \<subseteq> X \<longrightarrow> f c \<in> Com X"
bounded_sesquilinear_def: fixes prod :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "bounded_sesquilinear prod \<equiv> ((\<forall>a a' b. prod (a + a') b = prod a b + prod a' b) \<and> (\<forall>a b b'. prod a (b + b') = prod a b + prod a b')) \<and> (\<forall>r a b. prod (r *\<^sub>C a) b = cnj r *\<^sub>C prod a b) \<and> (\<forall>a r b. prod a (r *\<^sub>C b) = r *\<^sub>C prod a b) \<and> (\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K)"
f: shows "recfn 1 f \<and> (\<forall>x. (eval f [x] \<down>) = (x \<in> X))"
ag_eq_sol1: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and x :: "'c"   and b :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "x \<in> carrier A"   and "b \<in> carrier A"   and "a \<plusminus>\<^bsub>A\<^esub> x = b" shows "x = -\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> b"
wf_hull_rel_relcomp: fixes R :: "('b \<times> 'b) set" assumes "wf R"   and "eqvt R" shows "wf (hull_rel O R)"
no_\<tau>moves2: fixes s2'' :: "'s2" shows "\<not> s2' -\<tau>2\<rightarrow> s2''"
ij_le2_1: shows "i < dim_row (matricize I (tensor0 ds))"
inner_mult_right': fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'b"   and b :: "'b"   and m :: "real" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "a \<bullet> (b * of_real m) = a \<bullet> b * m"
unitaryD2: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "unitary A" shows "inverts_mat (adjoint A) A"
less_add_order_eq_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and k :: "nat" assumes "class.monoid_add plus zero"   and "power.power zero plus a k = zero"   and "order.greater (monoid_add.add_order plus zero a) k" shows "k = 0"
greaterThanAtMost_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and u :: "'b" shows "ord.greaterThanAtMost less_eq less l u \<equiv> ord.greaterThan less l \<inter> ord.atMost less_eq u"
find_handler_prealloc_pres: fixes h :: "heap"   and P :: "jvm_prog"   and a :: "nat"   and frs :: "frame list"   and sh :: "sheap"   and xp' :: "nat option"   and h' :: "heap"   and frs' :: "frame list"   and sh' :: "sheap" assumes "preallocated h"   and "find_handler P a h frs sh = (xp', h', frs', sh')" shows "preallocated h'"
simps_2: fixes f :: "'c \<Rightarrow> ('c, 'd) Nested_Datatype.term"   and t :: "('c, 'd) Nested_Datatype.term"   and ts :: "('c, 'd) Nested_Datatype.term list" shows "subst_term_list f (t # ts) = subst_term f t # subst_term_list f ts"
infinite_params_available: fixes S :: "(nat, 'a) form set"   and C :: "(nat, 'a) form set set"   and f :: "nat \<Rightarrow> (nat, 'a) form"   and n :: "nat" assumes "infinite (- \<Union> (params ` S))" shows "\<exists>x. x \<notin> \<Union> (params ` (extend S C f n \<union> {f n}))"
Au: shows "A \<in> Units (ring_mat TYPE('a) n b)"
steps_reaches': fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "steps xs"   and "hd xs = x"   and "last xs = y" shows "x \<rightarrow>* y"
fmfilter_true: fixes m :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> bool" assumes "\<And>x y. m $$ x = Some y \<Longrightarrow> P x" shows "fmfilter P m = m"
IPImp_def: fixes p :: "PrIds" shows "IPImp p = (Init \<not> Calling memCh p \<and> \<box>[ENext p]_(e p) \<and> MClkIPSpec memCh crCh cst p \<and> RPCIPSpec crCh rmCh rst p \<and> RPSpec rmCh mm ires p \<and> (\<forall>l. #l \<in> #MemLoc \<longrightarrow> MSpec rmCh mm ires l))"
interaction_bounded_by_lift_spmf: fixes "consider" :: "'d \<Rightarrow> bool"   and p :: "'e spmf"   and n :: "enat" shows "interaction_bounded_by consider (lift_spmf p) n"
Col_dep2: fixes a :: "(real, 'b) vec"   and b :: "(real, 'b) vec"   and c :: "(real, 'b) vec" shows "real_euclid.Col a b c = dep2 (b - a) (c - a)"
simps_2: fixes var :: "'f"   and args :: "'f \<Rightarrow> ('d, 'f) trm" shows "SIGF ($\<phi> var args) = {Inr (Inr var)} \<union> {Inl x |x. x \<in> (\<Union>i. SIGT (args i))}"
assms_4: fixes x :: "'a" assumes "x \<in> S" shows "g (f x) = x"
DA0'_def: shows "DA0' = DA0 \<cdot> \<rho>"
rawpsubst2_fresh_switch: fixes r :: "'trm"   and t :: "'trm"   and s :: "'trm"   and x :: "'var"   and y :: "'var" assumes "r \<in> atrm"   and "t \<in> trm"   and "s \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "x \<notin> FvarsT s"   and "y \<notin> FvarsT t" shows "rawpsubstT r [(s, y), (t, x)] = rawpsubstT r [(t, x), (s, y)]"
filterlim_int_sequentially: shows "filterlim int at_top sequentially"
diff_add_eq_diff_diff_swap: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "a -\<^sub>o\<^sub>w (b +\<^sub>o\<^sub>w c) = a -\<^sub>o\<^sub>w c -\<^sub>o\<^sub>w b"
inv_all_def: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.inv_all n p A \<equiv> \<forall>i j. i < n \<longrightarrow> j < n \<longrightarrow> p A i j"
iprev_min_step: fixes n :: "nat"   and I :: "nat set"   and k :: "nat" assumes "iprev n I < k"   and "k < n" shows "k \<notin> I"
closed_empty: fixes "open" :: "'a set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.closed open {}"
maximal_setTr: fixes S :: "'a set"   and mx :: "'a set"   and J :: "'a set" assumes "maximal_set {I. ideal R I \<and> S \<inter> I = {}} mx"   and "ideal R J"   and "mx \<subset> J" shows "S \<inter> J \<noteq> {}"
distinct_sort: fixes f :: "'d \<Rightarrow> 'e Word.word"   and xs :: "'d list" shows "distinct (signed.sort_key f xs) = distinct xs"
assms_5: shows "are_all_operators_applicable_in s (map (strips_op_to_sasp \<Psi>) ops') \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent (map (strips_op_to_sasp \<Psi>) ops')"
assms_7: fixes A :: "'a trm" assumes "P A" shows "P (Snd A)"
eval_Id: fixes n :: "nat"   and m :: "nat"   and xs :: "nat list" assumes "n < m"   and "m = length xs" shows "eval (recf.Id m n) xs \<down>= xs ! n"
arb_actI: fixes s :: "'s" shows "arb_act s \<in> K s"
expand_apply_infmatrix: fixes f :: "'c \<Rightarrow> 'b"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and j :: "nat"   and i :: "nat" shows "apply_infmatrix f A j i = f (A j i)"
i_shrink_i_drop_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" shows "f \<Up> (n * k) \<div> k = f \<div> k \<Up> n"
arr_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f"   and "\<F>\<^sub>SC.arr g" shows "\<F>\<^sub>SC.arr (f \<otimes>\<^sub>S g)"
quasi_transI: fixes r :: "('a \<times> 'a) set" assumes "\<And>x y z. \<lbrakk>x \<^bsub>r\<^esub>\<prec> y; y \<^bsub>r\<^esub>\<prec> z\<rbrakk> \<Longrightarrow> x \<^bsub>r\<^esub>\<prec> z" shows "quasi_trans r"
F3: shows "\<forall>x v. {R. v R \<in> x} = v -` x"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "ElementClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
join_table: fixes n :: "nat"   and A :: "nat set"   and X :: "'a option list set"   and B :: "nat set"   and Y :: "'a option list set"   and b :: "bool"   and C :: "nat set" assumes "table n A X"   and "table n B Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A"   and "A \<union> B = C" shows "table n C (join X b Y)"
left_cancel: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b = a + c) = (b = c)"
set_integrable_cong: fixes M :: "'d measure"   and M' :: "'d measure"   and A :: "'d set"   and A' :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e" assumes "M = M'"   and "A = A'"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' x" shows "set_integrable M A f = set_integrable M' A' f'"
icsSp_Append_param_in: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and V :: "'var"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "n ics-ns\<rightarrow>\<^sub>d* n''"   and "n'' s-p:V\<rightarrow>\<^bsub>in\<^esub> n'" shows "n ics-ns @ [n'']\<rightarrow>\<^sub>d* n'"
keys_minus: fixes a :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (- a) = keys a"
assms_3: assumes "(\<box>(F \<and> G)) s" shows "P"
threeway_partition_def: fixes x :: "'b"   and xs :: "'b list" shows "threeway_partition x xs = (filter (\<lambda>y. y < x) xs, filter (\<lambda>y. y = x) xs, filter ((<) x) xs)"
dim_singleton: fixes x :: "'a" shows "vector_space_over_itself.dim {x} = (if x = (0::'a) then 0 else 1)"
elt_set_plus_def: fixes a :: "'a"   and B :: "'a set" shows "a +o B = {c. \<exists>b\<in>B. c = a + b}"
dist_eq_dist_code_abs_le: fixes p\<^sub>0 :: "int \<times> int"   and p\<^sub>1 :: "int \<times> int"   and c :: "int" shows "(dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> \<bar>real_of_int c\<bar>) = (dist_code p\<^sub>0 p\<^sub>1 \<le> c\<^sup>2)"
f'_descending: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "0 \<le> x" shows "f' y \<le> f' x"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "C.ide (C.map a)"
irreflexive_conv_closed: fixes x :: "'a" assumes "irreflexive x" shows "irreflexive (x\<^sup>T)"
norm_conv_dist: fixes x :: "'b" shows "\<parallel>x\<parallel> = dist x (0::'b)"
snull_strict: shows "snull\<cdot>\<bottom> = \<bottom>"
zero_min: fixes t :: "'a" shows "(0::'a) \<preceq> t"
emeasure_restr_to_subalg: fixes M :: "'b measure"   and F :: "'b measure"   and A :: "'b set" assumes "subalgebra M F"   and "A \<in> sets F" shows "emeasure (restr_to_subalg M F) A = emeasure M A"
Pi_pmf_def: fixes A :: "'c set"   and dflt :: "'d"   and p :: "'c \<Rightarrow> 'd pmf" shows "Pi_pmf A dflt p = embed_pmf (\<lambda>f. if \<forall>x. x \<notin> A \<longrightarrow> f x = dflt then \<Prod>x\<in>A. pmf (p x) (f x) else 0)"
times_bit0_def: fixes x :: "'b bit0"   and y :: "'b bit0" shows "x * y = Abs_bit0' (Rep_bit0 x * Rep_bit0 y)"
lifelock_free_is_lifelock_free_v2: fixes P :: "'d process" assumes "lifelock_free P" shows "lifelock_free_v2 P"
simps: fixes a :: "Regions.intv" shows "Regions.isGreater a = (\<exists>uu_. a = Regions.intv.Greater uu_)"
bezout_ring_imp_diagonal_admits_SNF_mod_ring_admits_aux2: fixes n :: "nat" assumes "class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus" shows "\<forall>A\<in>carrier_mat n n. admits_SNF_JNF A"
quad_nextVertex_prevVertex: fixes f :: "face"   and a :: "nat" assumes "|vertices f| = 4"   and "a \<in> \<V> f"   and "distinct (vertices f)" shows "f \<bullet> (f \<bullet> (f \<bullet> a)) = f\<^bsup>-1\<^esup> \<bullet> a"
bound: shows "\<forall>y. exist_path_length n y \<longrightarrow> y \<le> card E"
abs_eq: fixes xa :: "'e \<Rightarrow> 'c"   and x :: "'d \<Rightarrow> 'd \<Rightarrow> 'e" shows "map_sq_matrix xa (of_fun x) = of_fun (\<lambda>i j. xa (x i j))"
iSince_iUntil_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "finite I"   and "Max I \<le> k" shows "(P t1. t1 \<S> t2 I. Q t2) = (P (k - t1). t1 \<U> t2 k \<ominus> I. Q (k - t2))"
Sup_bot_pres: fixes f :: "'c \<Rightarrow> 'd" assumes "Sup_pres f" shows "bot_pres f"
rel_fun_def_butlast: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and T :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e \<Rightarrow> 'f" shows "(R ===> S ===> T) f g = (\<forall>x y. R x y \<longrightarrow> (S ===> T) (f x) (g y))"
competE: fixes A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and c :: "real"   and S0 :: "'b list set"   and aoff :: "'b list \<Rightarrow> 'b list \<Rightarrow> (nat \<times> nat list) list"   and s0 :: "'b list" assumes "compet A c S0"   and "0 \<le> c"   and "\<forall>s0 rs. length (aoff s0 rs) = length rs"   and "s0 \<in> S0" shows "\<exists>b\<ge>0. \<forall>rs. static s0 rs \<longrightarrow> real (T_on A s0 rs) \<le> c * real (T_off aoff s0 rs) + b"
card_eq_bij: fixes A :: "'c set"   and B :: "'d set" assumes "card A = card B"   and "finite A"   and "finite B" obtains f :: "'c \<Rightarrow> 'd" where "bij_betw f A B"
subseteq_mset_def: fixes A :: "'b multiset"   and B :: "'b multiset" shows "(A \<subseteq># B) = (\<forall>a. count A a \<le> count B a)"
Union_atLeastAtMost: fixes N :: "nat" assumes "0 < N" shows "(\<Union>n\<in>{0..<N}. {real n..real (n + 1)}) = {0..real N}"
monom_one_Suc: fixes n :: "nat" shows "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
power_0: fixes a :: "'b list" shows "a \<^sup>@ 0 = \<epsilon>"
Rep_formula_compl: fixes x :: "'a formula" shows "Rep_formula (- x) = - Rep_formula x"
scale_cancel_right: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and x :: "'c"   and b :: "'b" assumes "vector_space scale" shows "(scale a x = scale b x) = (a = b \<or> x = (0::'c))"
TrustClientMsg: fixes M :: "nat"   and PMS :: "nat"   and NA :: "nat"   and NB :: "nat"   and evs :: "event list"   and A :: "agent"   and B :: "agent"   and Na :: "nat"   and Nb :: "nat"   and Y :: "msg" assumes "M = PRF (PMS, NA, NB)"   and "evs \<in> tls"   and "A \<notin> bad"   and "B \<notin> bad"   and "Says A Spy (Key (clientK (Na, Nb, M))) \<notin> set evs"   and "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"   and "Crypt (clientK (Na, Nb, M)) Y \<in> parts (knows Spy evs)" shows "Says A B (Crypt (clientK (Na, Nb, M)) Y) \<in> set evs"
assms_2: fixes n :: "nat" shows "C n \<in> sets M"
closed_segment_def: fixes a :: "'b"   and b :: "'b" shows "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b |u. 0 \<le> u \<and> u \<le> 1}"
lemmaPow3: fixes E :: "'a set" assumes "E \<noteq> {}" shows "\<exists>x. \<exists>X\<in>Pow E - {{}}. x \<in> X"
simps: fixes xs :: "'a list"   and i :: "nat" shows "nth_of xs i = (if length xs \<le> i then None else Some (xs ! i))"
weakTransitionI_1: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
intros_16: fixes x :: "name"   and a :: "name"   and b :: "name"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> b" shows "<\<nu>x>a{b}.P \<equiv>\<^sub>e a{b}.(<\<nu>x>P)"
prems_2: shows "P,E \<turnstile> s \<surd>"
presFreshAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'd, 'opSym, 'var, 'e, 'c, 'f) Recursion.model_scheme" shows "presFreshAbs hA MOD \<equiv> \<forall>ys y us s A. wlsAbs (us, s) A \<longrightarrow> freshAbs ys y A \<longrightarrow> gFreshAbs MOD ys y A (hA A)"
valid_W_inv_sys_ghg_empty_iff: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state" assumes "valid_W_inv s" shows "sys_ghost_honorary_grey s = {}"
b: shows "limsup (v \<circ> r \<circ> s) \<noteq> \<infinity>"