hcju/isabelleps huggingface dataset:
parPres: fixes P :: "ccs"   and Q :: "ccs"   and R :: "ccs" assumes "P \<sim> Q" shows "P \<parallel> R \<sim> Q \<parallel> R"
bdt_Some_var1_One: fixes t :: "dag"   and var :: "ref \<Rightarrow> nat"   and x :: "bdt" assumes "bdt t var = Some x"   and "var (root t) = 1" shows "x = One \<and> t = Node Tip (root t) Tip"
AT_def: fixes s :: "('g, 'f, 'h, 'i, 'j) system_state"   and p :: "'h" shows "AT s p = atCs (cPGM (GST s p))"
geodesic_segment_endpoints_1: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "x \<in> G"
matrix_vector_mul_diag_axis: fixes f :: "'b \<Rightarrow> 'a"   and i :: "'b"   and k :: "'a" shows "diag_mat f *v axis i k = axis i (f i * k)"
bintrunc_bintrunc: fixes m :: "nat"   and w :: "int" shows "take_bit m (take_bit m w) = take_bit m w"
sets_Collect_const: fixes M :: "'c measure"   and P :: "bool" shows "{x \<in> space M. P} \<in> sets M"
vinsert_commutativity: fixes a :: "V"   and b :: "V"   and C :: "V" shows "vinsert a (vinsert b C) = vinsert b (vinsert a C)"
sub: fixes a :: "'a" assumes "a \<in> S" shows "a \<in> R"
simps_1: fixes P :: "('a \<times> 'a) set"   and Pw :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and Rw :: "('a \<times> 'a) set" shows "SN_rel_ext_step P Pw R Rw top_s = P"
b1: shows "real_of_int (sint ra) < real_of_int (sint POS_INF)"
rep_eq: fixes x :: "'a poly" shows "coeff (- x) = (\<lambda>n. - coeff x n)"
nn_integral_mono: fixes M :: "'b measure"   and u :: "'b \<Rightarrow> ennreal"   and v :: "'b \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
\<Phi>_hp_def: fixes h :: "'b hp" shows "\<Phi>_hp h = \<Phi>_hps (hps h) + log 2 (real (size_hps (hps h) + 1))"
abs_homogenous: fixes x :: "'a"   and a :: "real" assumes "x \<in> E" shows "norm (a \<cdot> x) = \<bar>a\<bar> * norm x"
les_def: fixes s :: "'a"   and s' :: "'a" shows "les s s' \<equiv> \<forall>ni n. (ni, n) \<in> R s \<longrightarrow> (ni, n) \<in> R s'"
fiter_induct_nil: fixes x :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice" assumes "(x::'a::refinement_lattice) \<sqsubseteq> nil \<sqinter> (c::'a::refinement_lattice) ; x" shows "(x::'a::refinement_lattice) \<sqsubseteq> (c::'a::refinement_lattice)\<^sup>\<star>"
covd: shows "ov O d \<subseteq> s \<union> ov \<union> d"
hyps_3: shows "M (\<Sum>x\<in>A. M (f x)) = M (sum f A)"
preserves_ide: fixes a :: "'d" assumes "D.ide a" shows "D.ide (src\<^sub>D a)"
superposition_preserves_finiteness: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and P1 :: "'b eclause"   and P2 :: "'b eclause"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "finite (cl_ecl P1)"   and "finite (cl_ecl P2)"   and "basic_superposition.superposition trm_ord sel pos_ord filter_trms P1 P2 C \<sigma> k C'" shows "finite (cl_ecl C) \<and> finite C'"
Rep_myopt: fixes x :: "'d myopt" shows "Rep_myopt x \<in> {X. finite X \<and> card X \<le> 1}"
insert: fixes g :: "'b \<Rightarrow> 'a"   and x :: "'b"   and A :: "'b multiset" shows "\<Prod>\<^sub># (image_mset g (add_mset x A)) = g x * \<Prod>\<^sub># (image_mset g A)"
finite_enumerate_mono_iff: fixes S :: "'a set"   and m :: "nat"   and n :: "nat" assumes "finite S"   and "m < card S"   and "n < card S" shows "(enum S m < enum S n) = (m < n)"
vars_mset_subst_db: fixes i :: "nat"   and x :: "'a"   and s :: "('s, 'a) tm" shows "vars_mset (subst_db i x s) = {#y \<in># vars_mset s. y \<noteq> x#}"
degree_map_poly: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d poly" assumes "(f (lead_coeff p) = (0::'c)) = (p = 0)" shows "degree (map_poly f p) = degree p"
elims: fixes size0 :: "'a bound \<Rightarrow> nat"   and size1 :: "'b bound \<Rightarrow> nat"   and x :: "('a \<times> 'b) bound"   and y :: "nat" assumes "prod_index_def.size size0 size1 x = y" obtains l0 :: "'a" and r0 :: "'b" and l1 :: "'a" and r1 :: "'b" where "x = Bound (l0, r0) (l1, r1)" and "y = size0 (Bound l0 l1) * size1 (Bound r0 r1)"
uk_norm_uk_qk: shows "column k (Gram_Schmidt_matrix A) = norm (column k (Gram_Schmidt_matrix A)) *\<^sub>R column k (fst (QR_decomposition A))"
rel_set_strong_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'a set"   and B :: "'b set" shows "PTA.rel_set_strong R A B = (\<forall>x y. R x y \<longrightarrow> (x \<in> A) = (y \<in> B))"
card_not_ncols: shows "card (columns A) \<noteq> ncols A"
\<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>_correct: fixes opubx :: "'opubx"   and xs :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and \<tau> :: "Unboxed.type option" assumes "\<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> opubx = (map typeof xs, \<tau>)" shows "\<exists>y. \<UU>\<bb>\<xx>\<OO>\<pp> opubx xs = Some y \<and> typeof y = \<tau>"
pred_mono_strong: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b list"   and Pa :: "'b \<Rightarrow> bool" assumes "list_all P x"   and "\<And>z. \<lbrakk>z \<in> set x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "list_all Pa x"
sums_Suc_iff: fixes f :: "nat \<Rightarrow> 'b"   and s :: "'b" shows "(\<lambda>n. f (Suc n)) sums s = f sums (s + f 0)"
A2': fixes a :: "'p"   and b :: "'p"   and p :: "'p"   and q :: "'p"   and r :: "'p"   and s :: "'p" assumes "a b \<equiv> p q"   and "a b \<equiv> r s" shows "p q \<equiv> r s"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state_altc return bind get put altc" shows "monad_state return bind get put"
altc_get: fixes C :: "'c cset"   and f :: "'c \<Rightarrow> 's \<Rightarrow> 'm" shows "altc C (\<lambda>c. get (f c)) = get (\<lambda>s. altc C (\<lambda>c. f c s))"
close_opnet: fixes n :: "net_tree"   and U :: "(nat \<Rightarrow> state) \<Rightarrow> (nat \<Rightarrow> state) \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> state) \<Rightarrow> bool" assumes "wf_net_tree n"   and "oclosed (opnet (\<lambda>i. opaodv i \<langle>\<langle>\<^bsub>i\<^esub> qmsg) n) \<Turnstile> (\<lambda>_ _ _. True, U \<rightarrow>) global P" shows "closed (pnet (\<lambda>i. paodv i \<langle>\<langle> qmsg) n) \<TTurnstile> netglobal P"
F_apply: fixes a :: "'a"   and b :: "'b" shows "(*\<^sub>V) (F a b) = f a b"
adm_conj: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "adm P"   and "adm Q" shows "adm (\<lambda>x. P x \<and> Q x)"
set_offending_flows_def: fixes G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" shows "set_offending_flows G nP = {F. F \<subseteq> edges G \<and> is_offending_flows_min_set F G nP}"
mono_seq_DT_left: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq>\<^sub>D\<^sub>T P'" shows "(P `;` S) \<sqsubseteq>\<^sub>D\<^sub>T (P' `;` S)"
vec_last_carrier: fixes v :: "'a vec"   and n :: "nat" shows "vec_last v n \<in> carrier_vec n"
case_eq_if: fixes f1 :: "'l"   and f2 :: "'m \<Rightarrow> 'l"   and option :: "'m option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
list_slice2_nth_nth_eq1: fixes m :: "nat"   and xs :: "'b list"   and k :: "nat"   and n :: "nat" assumes "m < length xs div k"   and "n < k" shows "list_slice2 xs k ! m ! n = xs ! (m * k + n)"
abs_eq: fixes xa :: "int"   and x :: "nat" shows "setBit (Word.Word xa) x = Word.Word (set_bit x xa)"
os_reverse_aux_rule: fixes xs :: "'c list"   and p :: "'c node ref option"   and ys :: "'c list"   and q :: "'c node ref option" shows "<os_list xs p * os_list ys q> os_reverse_aux q p <os_list (rev xs @ ys)>"
mult_less_cancel_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times c a) (times c b) = ((less_eq zero c \<longrightarrow> less a b) \<and> (less_eq c zero \<longrightarrow> less b a))"
prems_2: shows "char_poly A = (\<Prod>e\<leftarrow>e # es. [:- e, 1::'a:])"
diff_diff_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c - (b - a) = c + a - b"
f_aggregate_def: fixes s :: "'b list"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" shows "f_aggregate s k ag \<equiv> map ag (list_slice s k)"
GSgame_def: fixes A :: "'b llist set"   and N :: "enat" shows "GSgame A N \<equiv> \<forall>e\<in>A. llength e = 2 * N"
asm: shows "(\<Sum>n. real (nth_prime n) / real_of_int (prod a {..n})) \<in> \<rat>"
induct: fixes P :: "C \<Rightarrow> bool"   and a0 :: "C" assumes "P (C [])"   and "\<And>n ns. \<lbrakk>P (C ns); P n\<rbrakk> \<Longrightarrow> P (C (n # ns))" shows "P a0"
intro: fixes A :: "'b llist set"   and N :: "enat" assumes "GSgame A N"   and "GaleStewartDefensiveStrategies.closed_GSgame_axioms A N" shows "GaleStewartDefensiveStrategies.closed_GSgame A N"
power_diff_rev_if: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "a \<noteq> (0::'a)" shows "a ^ m / a ^ n = (if n \<le> m then a ^ (m - n) else ((1::'a) / a) ^ (n - m))"
fls_delta0_subdegree: fixes a :: "'a" shows "fls_subdegree (Abs_fls (\<lambda>n. if n = 0 then a else (0::'a))) = 0"
rtrancl_trancl_trancl: fixes x :: "'c"   and y :: "'c"   and r :: "('c \<times> 'c) set"   and z :: "'c" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
mem_CD_on_Cd: fixes x :: "'a"   and ds :: "'b set"   and X :: "'a set" shows "(x \<in> CD_on ds X) = (x \<in> Cd (Xd x) X \<and> Xd x \<in> ds)"
length_def: fixes h :: "heap"   and a :: "'d array" shows "Array.length h a = length (Array.get h a)"
LI_acyclic_trans: shows "acyclic (LI_rel\<^sup>+)"
int_simps_4: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<not> \<not> P) = P"
prems_8: shows "InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state')"
silent_move_fromI: fixes s0 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "mthr.silent_moves s0 s1"   and "\<tau>mredT s1 s2" shows "mthr.silent_move_from s0 s1 s2"
ttm_children: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.tree_to_multiset t = {#(SkewBinomialHeapStruc.val t, SkewBinomialHeapStruc.prio t)#} + SkewBinomialHeapStruc.queue_to_multiset (SkewBinomialHeapStruc.children t)"
Par1Chain: fixes P :: "pi"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P' \<parallel> Q"
RedStaticUpDynCast: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Cs :: "char list list"   and C :: "char list"   and Cs' :: "char list list"   and Ds :: "char list list"   and E :: "char list \<Rightarrow> ty option"   and a :: "nat"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> Path last Cs to C unique"   and "P \<turnstile> Path last Cs to C via Cs' "   and "Ds = Cs @\<^sub>p Cs'" shows "redp P E (Cast C (ref (a, Cs))) s (ref (a, Ds)) s"
PBQ: shows "(P, S, Q) = Smith_2x2_JNF_eucl A"
refl_ge: shows "refl {(x, y). y \<le> x}"
base: fixes r :: "'a sequent list \<times> 'a sequent"   and R :: "('a sequent list \<times> 'a sequent) set"   and S :: "'a sequent"   and \<Gamma> :: "'a form multiset"   and \<Delta> :: "'a form multiset" assumes "r \<in> Ax"   and "Ax \<subseteq> R"   and "snd (extendRule S r) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "lastRule ( \<Gamma> \<Rightarrow>* \<Delta>, 0) r R"
assert_Sup_range: fixes p :: "'b \<Rightarrow> 'a" shows "{. \<Squnion>range p .} = \<Squnion>range (assert \<circ> p)"
simDef_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "P \<leadsto>\<^sup>^<Rel> Q \<equiv> \<forall>Rs. Q \<longmapsto> Rs \<longrightarrow> weakSimAct P Rs P Rel"
backflows_subseteq: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" shows "(X \<subseteq> Y) = (backflows X \<subseteq> backflows Y)"
simps_2: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "restricted_subformulas (\<phi> or\<^sub>n \<psi>) = restricted_subformulas \<phi> \<union> restricted_subformulas \<psi>"
simps_1: fixes x :: "Eg1.var"   and m :: "Mode"   and mds :: "Mode \<Rightarrow> Eg1.var set" shows "A.update_modes (x +=\<^sub>m m) mds = mds(m := insert x (mds m))"
identity_pseudofunctor_axioms: shows "identity_pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
case_1: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'd"   and x1 :: "'e" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
intro: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>M\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" assumes "module_ow U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1"   and "module_ow U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2" shows "module_pair_ow U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
dominant_term_ms_aux_2: fixes basis :: "(real \<Rightarrow> real) list"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "trimmed_ms_aux xs"   and "is_expansion_aux xs f basis" shows "\<forall>\<^sub>F x in at_top. sgn (f x) = sgn (fst (dominant_term_ms_aux xs))"
2: shows "gtarget (stake (Suc i) r) v = u"
morgan: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> .\<not> (p .\<and>. q)) = (A \<turnstile> .\<not> p .\<or>. .\<not> q)"
binder_def: fixes v :: "nat \<times> char list" shows "binder v = fst v"
delta_remove: fixes S :: "'b set"   and a :: "'b"   and b :: "'b \<Rightarrow> 'c"   and c :: "'b \<Rightarrow> 'c" assumes "finite S" shows "(\<Sum>k\<in>S. if k = a then b k else c k) = (if a \<in> S then b a + sum c (S - {a}) else sum c (S - {a}))"
terminates_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>Seq c1 c2 \<down> s" obtains f :: "'f" where "sb__ = Fault f"   | s :: "'e" where "sb__ = Normal sc__" and "\<Gamma>\<turnstile>c1 \<down> Normal sc__" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal sc__\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> s'"   | "sb__ = Stuck"   | s :: "'e" where "sb__ = Abrupt s"
class_Object: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "wf_prog wf_md P" shows "\<exists>C fs ms. class P Object = \<lfloor>(C, fs, ms)\<rfloor>"
steps_epsilon: fixes p :: "bool list"   and q :: "bool list"   and w :: "'b list" shows "((p, q) \<in> steps epsilon w) = (w = [] \<and> p = q)"
simps: fixes ip_start :: "'b word"   and ip_end :: "'b word" shows "iprange_interval (ip_start, ip_end) = WordInterval ip_start ip_end"
m_def: shows "m = order f a - n mod order f a - 1"
mat_of_row_dim_2: fixes y :: "'b Matrix.vec" shows "dim_col (mat_of_row y) = dim_vec y"
Start_nsys_xcpts1_1: shows "Start \<noteq> NullPointer"
Schottky: fixes f :: "complex \<Rightarrow> complex"   and r :: "real"   and t :: "real"   and z :: "complex" assumes "f holomorphic_on cball 0 1"   and "cmod (f 0) \<le> r"   and "\<And>z. z \<in> cball 0 1 \<Longrightarrow> \<not> (f z = 0 \<or> f z = 1)"   and "0 < t"   and "t < 1"   and "cmod z \<le> t" shows "cmod (f z) \<le> exp (pi * exp (pi * (2 + 2 * r + 12 * t / (1 - t))))"
bit_word_ucast_iff: fixes w :: "'c word"   and n :: "nat" shows "bit (ucast w) n = (n < LENGTH('c) \<and> n < LENGTH('b) \<and> bit w n)"
finite_Plus_iff: fixes A :: "'a set"   and B :: "'b set" shows "finite (A <+> B) = (finite A \<and> finite B)"
substitution_code_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and "next" :: "'node \<times> 'val" shows "uninst_code.substitution_code p g next = the (the_trivial (snd next) (the (Mapping.lookup (p g) next)))"
ys_5: shows "ys \<in> aforms_err e XS"
order_apply_inverse: fixes f :: "'a perm"   and a :: "'a" shows "order f (inverse f \<langle>$\<rangle> a) = order f a"
Collect_inj: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "Collect P = Collect Q" shows "P = Q"
P_def: shows "P \<equiv> \<lambda>x. if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then Px x else 0"
snoc_suffix_snoc: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list"   and y :: "'c" shows "suffix (xs @ [x]) (ys @ [y]) = (x = y \<and> suffix xs ys)"
mult_monom: fixes a :: "'b"   and m :: "nat"   and b :: "'b"   and n :: "nat" shows "Polynomial.monom a m * Polynomial.monom b n = Polynomial.monom (a * b) (m + n)"
prefix_prepend_less: fixes n :: "nat"   and vs :: "nat list"   and f :: "nat \<Rightarrow> nat option" assumes "n < length vs" shows "prefix (vs \<odot> f) n = take (Suc n) vs"
finish_simps_9: fixes u :: "'v"   and s :: "'v tarjan_state" shows "state.more (finish u s) = state.more s"
C_def: shows "(C\<^sub>0, C\<^sub>1) = find_closest_pair (C\<^sub>0', C\<^sub>1') ps'"
map_ide_simp: fixes a :: "'a" assumes "ide a" shows "\<rho>' a = local.inv (\<rho> a)"
ALPHA: shows "A $[[y \<and> x]]_zs $= A' $[[y' \<and> x']]_zs'"
lfp_unfold: fixes f :: "'a acom \<Rightarrow> 'a acom"   and c :: "com" assumes "\<And>x. (f x \<in> {C. strip C = c}) = (x \<in> {C. strip C = c})"   and "mono f" shows "Collecting.lfp c f = f (Collecting.lfp c f)"
diff_0: fixes a :: "'a" shows "(0::'a) - a = - a"
independent_vertices_def: fixes E :: "'b set set"   and S :: "'b set" shows "independent_vertices E S = (S \<subseteq> \<Union> E \<and> (\<forall>v1 v2. v1 \<in> S \<and> v2 \<in> S \<longrightarrow> {v1, v2} \<notin> E))"
set_integral_Un_AE: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "AE x in M. \<not> (x \<in> A \<and> x \<in> B)"   and "A \<in> sets M"   and "B \<in> sets M"   and "set_integrable M A f"   and "set_integrable M B f" shows "set_lebesgue_integral M (A \<union> B) f = set_lebesgue_integral M A f + set_lebesgue_integral M B f"
chebyshev_inequality: fixes Y :: "'a \<Rightarrow> real"   and s :: "real" assumes "integrable M (\<lambda>y. (Y y)\<^sup>2)"   and "random_variable borel Y"   and "0 < s" shows "prob {a \<in> space M. s \<le> \<bar>Y a - expectation Y\<bar>} \<le> expectation (\<lambda>x. (Y x - expectation Y)\<^sup>2) / s\<^sup>2"
vimage2pI: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'e"   and x :: "'g"   and g :: "'h \<Rightarrow> 'f"   and y :: "'h" assumes "R (f x) (g y)" shows "vimage2p f g R x y"
bit_setBit_iff: fixes w :: "'b word"   and m :: "nat"   and n :: "nat" shows "bit (setBit w m) n = (m = n \<and> n < LENGTH('b) \<or> bit w n)"
simps_1: fixes k :: "'c"   and v :: "'d" shows "update k v [] = [(k, v)]"
disj_qf: fixes p :: "fm"   and q :: "fm" assumes "qfree p"   and "qfree q" shows "qfree (Cooper.disj p q)"
type_less_eq_x_Optional_intro_1: fixes \<tau> :: "'a OCL_Types.type"   and \<rho> :: "'a basic_type"   and \<sigma> :: "'a basic_type" assumes "\<tau> = \<rho>[1]"   and "\<rho> \<le> \<sigma>" shows "\<tau> \<le> \<sigma>[?]"
smult_diff_right: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "Polynomial.smult a (p - q) = Polynomial.smult a p - Polynomial.smult a q"
Act_weak_tau_modality_eq_iff: fixes \<alpha>1 :: "'act"   and x1 :: "('idx, 'pred, 'act) formula"   and \<alpha>2 :: "'act"   and x2 :: "('idx, 'pred, 'act) formula" shows "(Act \<alpha>1 (weak_tau_modality x1) = Act \<alpha>2 (weak_tau_modality x2)) = (Act \<alpha>1 x1 = Act \<alpha>2 x2)"
id_hom: fixes A :: "'b" assumes "A \<in> Ob" shows "Id A \<in> Hom A A"
vconverse_vtimes: fixes A :: "V"   and B :: "V" shows "(A \<times>\<^sub>\<circ> B)\<inverse>\<^sub>\<circ> = B \<times>\<^sub>\<circ> A"
iT_Div_singleton: fixes a :: "nat"   and k :: "nat" shows "{a} \<oslash> k = {a div k}"
zero_aux: fixes f :: "nat \<Rightarrow> 'a" assumes "completely_multiplicative_function f" shows "f 0 = (0::'a)"
dvd_imp_multiplicity_le: fixes a :: "'a"   and b :: "'a"   and p :: "'a" assumes "a dvd b"   and "b \<noteq> (0::'a)" shows "multiplicity p a \<le> multiplicity p b"
ncoll: fixes x :: "'b" assumes "x \<in> X" shows "ccw (0::'a) a (f x)"
subgroup_generated_group_carrier: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "subgroup_generated G (carrier G) = G"
pnat_min_simp_3: fixes y :: "pnat" shows "pnat_min pnat.PInfty y = y"