hcju/isabelleps huggingface dataset:
cod_src_eq_dom_src: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "partial_magma.cod C (Leg0 (Cod \<mu>)) = partial_magma.cod C (Leg0 (Dom \<mu>))"
count_roots_above_correct: fixes p :: "real poly"   and a :: "real" shows "count_roots_above p a = card {x. a < x \<and> poly p x = 0}"
reflp_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
not_bet_and_out: fixes A :: "'p"   and B :: "'p"   and C :: "'p" shows "\<not> (Bet A B C \<and> B Out A C)"
assms_4: fixes j :: "nat" assumes "j < length fs"   and "p = Inr j"   and "Inr j \<notin> set ps" shows "is_RB_in dgrad rword (set bs) (term_of_pair (0::'a, j)) \<and> rep_list (monomial (1::'b) (term_of_pair (0::'a, j))) \<in> ideal (rep_list ` set bs)"
Max_in: fixes A :: "'b set" assumes "finite A"   and "A \<noteq> {}" shows "Max A \<in> A"
u_def: shows "u = (\<lambda>x. u2 x - u3 x)"
lambda_one: shows "(\<lambda>x. x) = (\<cdot>) (1::'b)"
length_interpret_floatariths: fixes fas :: "floatarith list"   and xs :: "real list" shows "length (interpret_floatariths fas xs) = length fas"
continuous_map_coordinatewise_then_product: fixes I :: "'b set"   and T1 :: "'c topology"   and T :: "'b \<Rightarrow> 'd topology"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd" assumes "\<And>i. i \<in> I \<Longrightarrow> continuous_map T1 (T i) (\<lambda>x. f x i)"   and "\<And>i x. \<lbrakk>i \<notin> I; x \<in> topspace T1\<rbrakk> \<Longrightarrow> f x i = undefined" shows "continuous_map T1 (product_topology T I) f"
hereditarily_fs_eqvt: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and p :: "perm" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (p \<bullet> t\<^sub>\<alpha>)"
divide_less_eq_1_pos: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less (divide b a) one = less b a"
exprel_imp_eq_freediscrim: fixes U :: "freeExp"   and V :: "freeExp" assumes "U \<sim> V" shows "freediscrim U = freediscrim V"
is_subprob_densityD_1: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "is_subprob_density M f" shows "f \<in> borel_measurable M"
refl: fixes x :: "'a" assumes "x \<in> carrier S" shows "x .= x"
Sim: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel" shows "\<Psi>' \<rhd> S \<leadsto>[Rel] T"
mk_globally_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (mk_globally \<phi>) \<subseteq> atoms_ltln \<phi>"
is_elsewhere_def: fixes p :: "('c, 'd) fm"   and i :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and xs :: "(nat \<times> nat) set" shows "is_elsewhere p i branch xs \<equiv> \<exists>w w' ps. (w, w') \<notin> xs \<and> branch !. w = Some (ps, i) \<and> (p = Nom i \<or> ps !. w' = Some p)"
array_length'_def: shows "array_length' \<equiv> integer_of_nat \<circ> array_length"
Fake_analz_eq: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> synth (analz H)" shows "synth (analz (insert X H)) = synth (analz H)"
fps_middle_coeffs: fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "n \<noteq> 0 \<and> n \<noteq> p" shows "(1 + fps_X) ^ p $ n mod int p = int (0 mod p)"
rwof_leof_init: fixes m :: "'c nres"   and c :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres" shows "m \<le>\<^sub>n SPEC (rwof m c f)"
eq: shows "Collect ((\<lambda>v. v = signs ! i) \<circ> consistent_sign_vec_copr qs) \<inter> set (sorted_list_of_set {x. poly p x = 0}) = {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = signs ! i}"
iTILL_subset_conv: fixes n' :: "nat"   and n :: "nat" shows "([\<dots>n'] \<subseteq> [\<dots>n]) = (n' \<le> n)"
memberid_mdecl_simp1: fixes m :: "mdecl" shows "memberid (mdecl m) = mid (fst m)"
add: shows "degree_m (\<Prod>\<^sub># fs) \<le> \<Sum>\<^sub># (image_mset degree_m fs)"
assms_2: fixes t' :: "sterm"   and L :: "char list set"   and u' :: "sterm" assumes "t => t'"   and "finite L"   and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. u\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> u' = \<sigma>[s,p] t'')"   and "z = Upd t' l u'" shows "Q"
prems_1: shows "set (v # vs) \<subseteq> carrier M"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_loop_on_left_moving_O m rs (l, r) = (\<exists>ml mr ln rn. l = Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> r = Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc rs \<and> 0 < mr)"
C: fixes r :: "'d rexp"   and r' :: "'d rexp"   and s :: "'d rexp"   and s' :: "'d rexp" assumes "r ~ r'"   and "s ~ s'" shows "Conc r s ~ Conc r' s'"
assms_1: shows "B \<subseteq> U\<^sub>2"
simps_9: fixes f :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "map_option f (\<lfloor>x2\<rfloor>) = \<lfloor>f x2\<rfloor>"
parent_edge_root: fixes G :: "('a, 'b) Graph_scheme"   and root :: "'a"   and v :: "'a" assumes "RootedTree G root"   and "v \<rightarrow>\<^bsub>G\<^esub> root" shows "RootedTree.parent G root v = root"
assms_2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [e1]))"
iso_lunit: fixes a :: "'a" assumes "ide a" shows "local.iso \<l>\<^sub>1[a]"
distrib_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
set_mset_inter: fixes A :: "'a multiset"   and B :: "'a multiset" shows "set_mset (A \<inter># B) = set_mset A \<inter> set_mset B"
eventually_sequentially_seg: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat" shows "(\<forall>\<^sub>F n in sequentially. P (n + k)) = eventually P sequentially"
test_comp_add1: fixes p :: "'a" shows "! (\<iota> p) + \<iota> p = (1::'b)"
append_Nil: fixes ys :: "'b list" shows "[] @ ys = ys"
closest_pair_rec_distinct: fixes xs :: "(int \<times> int) list"   and ys :: "(int \<times> int) list"   and p :: "(int \<times> int) \<times> int \<times> int" assumes "distinct xs"   and "(ys, p) = closest_pair_rec xs" shows "distinct ys"
not_limit_ordinal_nat: fixes n :: "nat" shows "\<not> limit_ordinal (ordinal_of_nat n)"
tso_lock_invL_def: shows "gc.tso_lock_invL = (\<lambda>s. (atS gc gc.tso_lock_locs s \<longrightarrow> sys_mem_lock s\<down> = Some gc) \<and> (atS gc (- gc.tso_lock_locs) s \<longrightarrow> sys_mem_lock s\<down> \<noteq> Some gc))"
dvd_div_div_eq_mult: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<noteq> (0::'a)"   and "c \<noteq> (0::'a)"   and "a dvd b"   and "c dvd d" shows "(b div a = d div c) = (b * c = a * d)"
RHS: shows "integral UNIV (\<lambda>x. integral UNIV (\<lambda>y. f (x, y))) = integral (cbox a b) (\<lambda>x. integral (cbox (g1 x) (g2 x)) (\<lambda>y. g (x, y)))"
apply_ocrit_def: fixes crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps' :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "apply_ocrit crit data hs ps' ps = (let c = crit data hs ps' in filter (\<lambda>(p, q). \<not> c p q) ps)"
src_def: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and i :: "nat"   and p :: "'a" assumes "distributed_system channel" shows "distributed_system.src channel i p \<equiv> \<exists>q. channel i = Some (p, q)"
val_def: shows "val \<equiv> sum f (outgoing s) - sum f (incoming s)"
lta_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LtA D E F" shows "C B A LtA F E D"
showsp_int_def: fixes p :: "nat"   and i :: "int" shows "showsp_int p i = (if i < 0 then shows_string ''-'' \<circ> showsp_nat p (nat (- i)) else showsp_nat p (nat i))"
DynCastThrow: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and r :: "nat \<times> char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P,E \<turnstile> \<langle>Cast C (Throw r),s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
allAllocationsBridgingLemma: fixes N :: "'d set"   and G :: "'e list" assumes "0 < card N"   and "distinct G" shows "allAllocations N (set G) = set (allAllocationsAlg N G)"
dilated_ticks: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and m :: "nat"   and n :: "nat"   and c :: "clock" assumes "dilating f sub r" shows "{i. f m \<le> i \<and> i \<le> f n \<and> hamlet (Rep_run r i c)} = f ` {i. m \<le> i \<and> i \<le> n \<and> hamlet (Rep_run sub i c)}"
p0: shows "hom (lead_coeff p) \<noteq> (0::'b)"
remove_vertex_V': fixes x :: "'a" shows "V\<^bsub>remove_vertex x\<^esub> \<subseteq> V"
ra_ia_rb_ib_rc_ic_def_6: shows "ic = Im c"
simps_5: fixes types_t :: "tf list"   and func_t :: "tf list"   and global :: "tg list"   and table :: "nat option"   and memory :: "nat option"   and local :: "t list"   and label :: "t list list"   and return :: "t list option"   and more :: "'a" shows "memory \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr> = memory"
small_big_asymmetric: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> l F g"   and "g \<in> L F f" shows "\<forall>\<^sub>F x in F. f x = (0::'b)"
FF_def: shows "\<tau>.F.FF \<equiv> \<lambda>\<mu>\<nu>. if B.VV.arr \<mu>\<nu> then (F (fst \<mu>\<nu>), F (snd \<mu>\<nu>)) else C.VV.null"
hyps_2: shows "h' [succs]\<rightarrow>* h''"
Beta_def: fixes a :: "'b"   and b :: "'b" shows "Beta a b = Gamma a * Gamma b / Gamma (a + b)"
assms_6: shows "v' \<in> step.allDefs g x"
chooseNext_all_code: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g" assumes "CFG_SSA_Transformed_code \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs u p var"   and "uninst_code.ssa.redundant_code p g" shows "chooseNext_all (usesOf (u g)) (p g) g = Max (uninst_code.ssa.trivial_phis p g)"
int_less_eq: fixes a :: "'a interval"   and b :: "'a interval" shows "(a \<le> b) = (low a < low b \<or> low a = low b \<and> high a \<le> high b)"
select_rec_partition: fixes k :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "k < length xs" shows "select k xs = (let (ls, es, gs) = partition3 x xs in if k < length ls then select k ls else if k < length ls + length es then x else select (k - length ls - length es) gs)"
JVM_le_unfold: fixes P :: "'b prog"   and m :: "nat"   and n :: "nat" shows "JVM_SemiType.le P m n \<equiv> Err.le (Opt.le (Product.le (Listn.le (subtype P)) (Listn.le (Err.le (subtype P)))))"
secrecy_lemma: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and nb :: "msg"   and K :: "nat"   and na :: "msg" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Says Server A \<lbrace>nb, Crypt (shrK A) \<lbrace>Agent B, Key K, na\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key K, nb\<rbrace>\<rbrace> \<in> set evs \<longrightarrow> Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs \<longrightarrow> Key K \<notin> analz (knows Spy evs)"
simps_2: fixes Vs :: "char list list"   and C :: "char list"   and e :: "char list exp" shows "compE\<^sub>1 Vs (Cast C e) = Cast C (compE\<^sub>1 Vs e)"
IA1: shows "Iagree I J {Inl x |x. x \<in> SIGT \<theta>\<^sub>1}"
fun_meet_simp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(f \<squnion> g) x = f x \<squnion> g x"
that: fixes u :: "'t" assumes "v = t \<oplus> u" shows "thesis"
bind_gpv_assoc: fixes f :: "('e, 'b, 'c) gpv"   and g :: "'e \<Rightarrow> ('d, 'b, 'c) gpv"   and h :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "bind_gpv (bind_gpv f g) h = bind_gpv f (\<lambda>x. bind_gpv (g x) h)"
n_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set" assumes "Reachability_Problem K S S1 S2" shows "Reachability_Problem.n K S S1 S2 = (\<lambda>s\<in>S. Markov_Decision_Process.P_inf K s (\<lambda>\<omega>. (HLD S1 suntil HLD S2) (s ## \<omega>)))"
rep_eq: fixes x :: "circline_mat" shows "Rep_circline_mat (of_circline_clmat x) = of_circline_cmat (Rep_circline_mat x)"
eq: shows "h\<^sub>0 ++ h = h\<^sub>1 ++ h"
sba1_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
reads_consistent_drop: fixes \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list" assumes "reads_consistent False \<O> m sb" shows "reads_consistent True (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>) (flush (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) m) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
minus_log_convex: fixes b :: "real" assumes "1 < b" shows "convex_on {0<..} (\<lambda>x. - log b x)"
step_deterministic: fixes s1 :: "('fenv, 'henv, ('fun, ('dyn, 'num, 'bool) unboxed) frame) state"   and s2 :: "('fenv, 'henv, ('fun, ('dyn, 'num, 'bool) unboxed) frame) state"   and s3 :: "('fenv, 'henv, ('fun, ('dyn, 'num, 'bool) unboxed) frame) state" assumes "s1 \<rightarrow> s2"   and "s1 \<rightarrow> s3" shows "s2 = s3"
da_elim_cases_31: fixes Env :: "env"   and B :: "lname set"   and v :: "val"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Lit v\<rangle>\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
distinct_209: fixes x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term"   and x15 :: "Syntax.term" shows "Syntax.Let x81 x82 x83 \<noteq> Auth x15"
Broadcast_Store_Deliver_prefix_closed: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "'a"   and e :: "'b"   and mo :: "'a"   and j :: "nat" assumes "xs @ [Broadcast (i, Store e mo i)] prefix of j" shows "Deliver (mo, Append mo e) \<in> set xs \<or> (\<exists>mo2. Deliver (mo, Store e mo2 mo) \<in> set xs)"
convex_hull_translation: fixes a :: "'a"   and S :: "'a set" shows "convex hull (+) a ` S = (+) a ` (convex hull S)"
completely_specified_from_rel: fixes rel :: "('a \<times> ('b \<times> 'c) \<times> 'a) set"   and q0 :: "'a" assumes "completely_specified_rel_over rel (nodes (from_rel rel q0))" shows "completely_specified (from_rel rel q0)"
under_zero: assumes "Field r \<noteq> {}" shows "r.under r.zero = {r.zero}"
distrib_inf_sup_eq: shows "class.distrib_lattice (\<sqinter>) (\<le>) (<) (\<squnion>) = (\<forall>x y z. x \<sqinter> z = y \<sqinter> z \<and> x \<squnion> z = y \<squnion> z \<longrightarrow> x = y)"
every_position_is_determined: fixes p :: "'a list" assumes "position p" shows "winning_position_Even p \<or> winning_position_Odd p"
rtrancl_Int1: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set"   and B :: "('b \<times> 'b) set" assumes "(a, b) \<in> (A \<inter> B)\<^sup>*" shows "(a, b) \<in> A\<^sup>*"
parallel_fixp_induct_1_1: fixes orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and luba :: "'d set \<Rightarrow> 'd"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and lubb :: "'e set \<Rightarrow> 'e"   and F :: "('f \<Rightarrow> 'd) \<Rightarrow> 'f \<Rightarrow> 'd"   and G :: "('g \<Rightarrow> 'e) \<Rightarrow> 'g \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'e"   and P :: "('f \<Rightarrow> 'd) \<Rightarrow> ('g \<Rightarrow> 'e) \<Rightarrow> bool" assumes "partial_function_definitions orda luba"   and "partial_function_definitions ordb lubb"   and "\<And>x. monotone (fun_ord orda) orda (\<lambda>f. F f x)"   and "\<And>y. monotone (fun_ord ordb) ordb (\<lambda>g. G g y)"   and "f \<equiv> ccpo.fixp (fun_lub luba) (fun_ord orda) F"   and "g \<equiv> ccpo.fixp (fun_lub lubb) (fun_ord ordb) G"   and "ccpo.admissible (prod_lub (fun_lub luba) (fun_lub lubb)) (rel_prod (fun_ord orda) (fun_ord ordb)) (\<lambda>x. P (fst x) (snd x))"   and "P (\<lambda>_. luba {}) (\<lambda>_. lubb {})"   and "\<And>f g. P f g \<Longrightarrow> P (F f) (G g)" shows "P f g"
NEG_cast: shows "sint (SCAST(32 \<rightarrow> 64) NEG_INF) = sint NEG_INF"
worker_wrapper_body: fixes wrap :: "'b \<rightarrow> 'a"   and unwrap :: "'a \<rightarrow> 'b"   and body :: "'a \<rightarrow> 'a"   and computation :: "'a" assumes "wrap oo unwrap oo body = body"   and "computation = fix\<cdot>body" shows "computation = wrap\<cdot>(fix\<cdot>(unwrap oo body oo wrap))"
subsumes_transitive: fixes t1 :: "transition"   and c :: "nat \<Rightarrow>f value option"   and t2 :: "transition"   and t3 :: "transition" assumes "subsumes t1 c t2"   and "subsumes t2 c t3" shows "subsumes t1 c t3"
finite_succeed_t: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F" shows "finite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i) = finite (mojmir_def.succeed_t \<Sigma> \<delta> q\<^sub>0 w F i)"
lfilled_eq: fixes j :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and LI :: "e list"   and es' :: "e list" assumes "Lfilled j lholed es LI"   and "Lfilled j lholed es' LI" shows "es = es'"
val_return: fixes x :: "'b" shows "Time_Monad.val (return x) = x"
equivalence_functor_axioms: shows "equivalence_functor (\<bullet>) (\<bullet>) L"
approx_def: shows "approx = (\<Sum>k<n. x ^ k / fact k)"
lm054: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "argmax f A = A \<inter> {x. f x = Max (f ` A)}"
confluent_def: fixes R :: "('b \<times> 'b) set" shows "confluent R = (\<forall>x y y'. (x, y) \<in> R \<and> (x, y') \<in> R \<longrightarrow> (y, y') \<in> joinable R)"
invD_1: fixes w :: "nat \<Rightarrow> real"   and m :: "nat"   and S :: "nat \<Rightarrow> 'b set"   and C :: "nat set"   and R :: "'b set" assumes "Set_Cover w m S"   and "Set_Cover.inv w m S C R" shows "Set_Cover.sc m S C (Set_Cover.U m S - R)"
rel_option_mono: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g option"   and y :: "'h option"   and R' :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "rel_option R x y"   and "\<And>x y. R x y \<Longrightarrow> R' x y" shows "rel_option R' x y"
fg_id: fixes xa :: "'V + 'V'" assumes "xa \<in> vertices G" shows "(f \<circ> g) xa = xa"
run_tell_env: fixes tell :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and w :: "'c"   and m :: "('b, 'a) envT"   and r :: "'b" shows "run_env (tell_env tell w m) r = tell w (run_env m r)"
P'_def: shows "P' = {p. p \<in># prime_factorization n \<and> even p}"
irreducible_def: fixes G :: "('b, 'c) monoid_scheme"   and a :: "'b" shows "Divisibility.irreducible G a = (a \<notin> Units G \<and> (\<forall>b\<in>carrier G. Divisibility.properfactor G b a \<longrightarrow> b \<in> Units G))"
nrows_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "nrows A = CARD('d)"
transitionRelation_def: fixes decisionVars :: "nat set" shows "transitionRelation decisionVars \<equiv> {(stateA, stateB). transition stateA stateB decisionVars}\<^sup>*"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>)"
prems: fixes v :: "'a" assumes "v \<in> set (v__ # V)" shows "f v \<le> g v"
fun_ord_def: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd" shows "fun_ord ord f g = (\<forall>x. ord (f x) (g x))"
EQ_def: fixes f :: "atom fm"   and xs :: "'a list" shows "EQ f xs = {xs ! k |k. Eq (Suc k) 0 \<in> set (DLO.atoms\<^sub>0 f) \<or> Eq 0 (Suc k) \<in> set (DLO.atoms\<^sub>0 f)}"
associated_sym: fixes a :: "'a"   and b :: "'a" assumes "a \<sim>\<^bsub>add_monoid R\<^esub> b" shows "b \<sim>\<^bsub>add_monoid R\<^esub> a"
2: shows "t\<^sup>+ ; e + e \<le> - 1'"
assms: fixes x :: "'a"   and y :: "'a" assumes "le x y" shows "f x \<le> f y"
butlast_conv_take: fixes xs :: "'b list" shows "butlast xs = take (length xs - 1) xs"
elims: fixes x :: "D"   and y :: "int option" assumes "fromA x = y" obtains xa :: "int" where "x = A xa" and "y = Some xa"   | v :: "char list" where "x = B v__" and "y = None"   | v :: "bool" where "x = C v" and "y = None"
l_coset_swap: fixes y :: "'a"   and x :: "'a"   and H :: "'a set" assumes "y \<in> x <#\<^bsub>add_monoid G\<^esub> H"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "x \<in> y <#\<^bsub>add_monoid G\<^esub> H"
epsilon: fixes e :: "'a" assumes "(0::'a) < e" shows "\<bar>x\<bar> \<le> e"
outstanding_non_volatile_refs_owned_or_read_only: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "non_volatile_owned_or_read_only False \<S> \<O> sb"
subset_refl: fixes A :: "'e set" shows "A \<subseteq> A"
sum_mset_eq_sum_on_supersets: fixes A :: "nat set"   and M :: "nat multiset" assumes "finite A"   and "set_mset M \<subseteq> A" shows "(\<Sum>i\<in>set_mset M. count M i * i) = (\<Sum>i\<in>A. count M i * i)"
pause_env_def: fixes out :: "'o"   and c :: "'i \<Rightarrow> ('c, 'm) envT" shows "local.pause_env out c = EnvT (\<lambda>r. pause out (\<lambda>i. run_env (c i) r))"
AE_pair_iff: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "{x \<in> space (lborel \<Otimes>\<^sub>M lborel). P (fst x) (snd x)} \<in> sets (lborel \<Otimes>\<^sub>M lborel)" shows "(AE x in lborel. almost_everywhere lborel (P x)) = (AE x in lborel \<Otimes>\<^sub>M lborel. P (fst x) (snd x))"
less_prefix_match_def: fixes a :: "'a prefix_match"   and b :: "'a prefix_match" shows "(a < b) = (a \<noteq> b \<and> (if pfxm_length a = pfxm_length b then pfxm_prefix a \<le> pfxm_prefix b else pfxm_length b < pfxm_length a))"
sparse_identity: fixes \<delta> :: "real"   and ps :: "(int \<times> int) list"   and p\<^sub>0 :: "int \<times> int" assumes "sparse \<delta> (set ps)"   and "\<forall>p\<in>set ps. \<delta> \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))" shows "sparse \<delta> (set (p\<^sub>0 # ps))"
simps: fixes io :: "('e \<times> 'f) list"   and M :: "('e, 'f, 'g) FSM"   and V :: "'e list set" shows "N io M V = {V'' \<in> Perm V M. map fst (mcp' io V'') = mcp' (map fst io) V}"
assert_def: fixes return :: "'f \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and fail :: "'e"   and P :: "'f \<Rightarrow> bool"   and m :: "'e" shows "monad_fail_base.assert return bind fail P m = bind m (\<lambda>x. if P x then return x else fail)"