hcju/isabelleps huggingface dataset:
mfinal1_inv_simulation: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2" shows "\<exists>s2'. \<tau>mRed2 s2 s2' \<and> s1 \<approx>m s2' \<and> r1.final_threads s1 \<subseteq> r2.final_threads s2' \<and> shr s2' = shr s2"
left_neutral: fixes a :: "'b multiset" shows "{#} \<union># a = a"
MClkFwd_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkFwd send rcv cst p = ($Calling send p \<and> $(cst!p) = #clkA \<and> Call rcv p MClkRelayArg<arg<send!p>> \<and> (cst!p)$ = #clkB \<and> unchanged (rtrner send!p))"
in_idealE_rep_list_dgrad_sig_set: fixes d :: "'a \<Rightarrow> nat"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "hom_grading d"   and "p \<in> punit_dgrad_max_set d"   and "p \<in> ideal (set fs)" obtains r :: "'t \<Rightarrow>\<^sub>0 'b" where "r \<in> dgrad_sig_set' (length fs) d" and "p = rep_list r"
A: shows "\<forall>\<phi>\<in>C. \<A> \<Turnstile> \<phi>"
less_eqE: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<subseteq># b" obtains c :: "'k multiset" where "b = a + c"
crename_NotR: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and N :: "trm"   and c :: "coname" assumes "R[a\<turnstile>c>b] = NotR x.N c"   and "x \<sharp> R"   and "c \<sharp> (a, b)" shows "\<exists>N'. R = NotR x.N' c \<and> N'[a\<turnstile>c>b] = N"
exhaust_sel: fixes state :: "('a, 'b) state" obtains "state = State (run_state state)"
sum_mono_inv: fixes f :: "'d \<Rightarrow> 'c"   and I :: "'d set"   and g :: "'d \<Rightarrow> 'c"   and i :: "'d" assumes "sum f I = sum g I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<le> g i"   and "i \<in> I"   and "finite I" shows "f i = g i"
succs_mono: fixes p :: "'c prog \<Rightarrow> String.literal \<Rightarrow> 'c mdecl \<Rightarrow> bool"   and P :: "'c prog"   and i :: "'d instr"   and mxs :: "nat"   and mpc :: "nat"   and rT :: "ty"   and \<tau>' :: "ty list \<times> ty err list"   and \<tau> :: "ty list \<times> ty err list"   and pc :: "nat" assumes "wf_prog p P"   and "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>')"   and "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'" shows "set (succs i \<tau> pc) \<subseteq> set (succs i \<tau>' pc)"
gcd_list_greatest: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and bs :: "'a list"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>b. b \<in> set bs \<Longrightarrow> dvd.dvd times a b" shows "dvd.dvd times a (semiring_gcd.Gcd_fin one zero gcd (set bs))"
state_rank_eq_rank: fixes q :: "'b"   and n :: "nat" shows "state_rank q n = (case oldest_token q n of None \<Rightarrow> None | Some t \<Rightarrow> rank t n)"
length_row_mat_to_cols_list: fixes i :: "nat"   and A :: "complex Matrix.mat" assumes "i < dim_row A" shows "length (Matrix_Legacy.row (mat_to_cols_list A) i) = dim_col A"
is_trail_path: fixes G :: "('a, 'b) graph"   and v :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_unMultigraph G" shows "valid_unMultigraph.is_trail G v ps v' = (valid_graph.is_path G v ps v' \<and> distinct ps \<and> set ps \<inter> set (rev_path ps) = {})"
distinct_1: fixes x11 :: "name"   and x12 :: "ty"   and x21 :: "pat"   and x22 :: "pat" shows "PVar x11 x12 \<noteq> \<langle>\<langle>x21, x22\<rangle>\<rangle>"
altc_sample_option_def: fixes altc_sample :: "'b \<Rightarrow> ('d \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'d \<Rightarrow> ('a, 'm) optionT" shows "local.altc_sample_option altc_sample p f = OptionT (altc_sample p (\<lambda>x. run_option (f x)))"
is_process_def: fixes P :: "('c event list \<times> 'c event set) set \<times> 'c event list set" shows "is_process P = (([], {}) \<in> FAILURES P \<and> (\<forall>s X. (s, X) \<in> FAILURES P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<in> FAILURES P \<longrightarrow> (s, {}) \<in> FAILURES P) \<and> (\<forall>s X Y. (s, Y) \<in> FAILURES P \<and> X \<subseteq> Y \<longrightarrow> (s, X) \<in> FAILURES P) \<and> (\<forall>s X Y. (s, X) \<in> FAILURES P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> FAILURES P) \<longrightarrow> (s, X \<union> Y) \<in> FAILURES P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> FAILURES P \<longrightarrow> (s, X - {tick}) \<in> FAILURES P) \<and> (\<forall>s t. s \<in> DIVERGENCES P \<and> tickFree s \<and> front_tickFree t \<longrightarrow> s @ t \<in> DIVERGENCES P) \<and> (\<forall>s X. s \<in> DIVERGENCES P \<longrightarrow> (s, X) \<in> FAILURES P) \<and> (\<forall>s. s @ [tick] \<in> DIVERGENCES P \<longrightarrow> s \<in> DIVERGENCES P))"
fr_eq2: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list"   and x :: "'b" assumes "islin p" shows "(\<exists>x. Ifm vs (x # bs) p) = (Ifm vs (x # bs) (minusinf p) \<or> Ifm vs (x # bs) (plusinf p) \<or> Ifm vs ((0::'b) # bs) p \<or> (\<exists>(n, t)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> (0::'b) \<and> Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * (2::'b)) # bs) p) \<or> (\<exists>(n, t)\<in>set (uset p). \<exists>(m, s)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> (0::'b) \<and> \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> (0::'b) \<and> Ifm vs ((- Itm vs (x # bs) t / \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (2::'b) # bs) p))"
ineq_5: shows "l < (if l = mm then Suc j else Suc l)"
simps_1: fixes uv :: "String.literal \<Rightarrow> 'a"   and uw :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a" shows "semantics_list uv uw [] = []"
sym_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> X" shows "(x \<sim> y) = (x = y)"
map_pre_G_def: shows "map_pre_G \<equiv> \<lambda>f1 f2. BNF_Composition.id_bnf \<circ> map_G0 f1 (map_F f2) \<circ> BNF_Composition.id_bnf"
simps: fixes a :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "hereditarily_fs a = ((\<exists>tset\<^sub>\<alpha>. a = Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<and> finite (supp tset\<^sub>\<alpha>) \<and> (\<forall>x. x \<in> set_bset tset\<^sub>\<alpha> \<longrightarrow> hereditarily_fs x)) \<or> (\<exists>t\<^sub>\<alpha>. a = Not\<^sub>\<alpha> t\<^sub>\<alpha> \<and> hereditarily_fs t\<^sub>\<alpha>) \<or> (\<exists>\<phi>. a = Pred\<^sub>\<alpha> \<phi>) \<or> (\<exists>t\<^sub>\<alpha> \<alpha>. a = Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> \<and> hereditarily_fs t\<^sub>\<alpha>))"
empty_lepoll: fixes A :: "'b set" shows "{} \<lesssim> A"
keys_sum_list_subset: fixes ps :: "('b \<Rightarrow>\<^sub>0 'c) list" shows "keys (sum_list ps) \<subseteq> Keys (set ps)"
less_le_trans: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat"   and z :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x \<prec> y"   and "y \<preceq> z" shows "x \<prec> z"
induct: fixes P :: "('a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a \<times> 'c) list \<Rightarrow> ('a \<times> 'c) list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a1 :: "('a \<times> 'c) list"   and a2 :: "('a \<times> 'c) list" assumes "\<And>f k v as k' v' bs. \<lbrakk>c k' k = Eq \<Longrightarrow> P f as bs; c k' k = Lt \<Longrightarrow> P f ((k, v) # as) bs; c k' k = Gt \<Longrightarrow> P f as ((k', v') # bs)\<rbrakk> \<Longrightarrow> P f ((k, v) # as) ((k', v') # bs)"   and "\<And>f bs. P f [] bs"   and "\<And>f as. P f as []" shows "P a0 a1 a2"
sel: fixes res :: "('a', 'b) resource" shows "run_resource (local.map_resource res) = map_spmf (map_prod b local.map_resource) \<circ> run_resource res \<circ> a"
se_star_Cons: fixes c1 :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and ls :: "('c, 'd) label list"   and c2 :: "('c, 'd) conf" shows "se_star c1 (l # ls) c2 = (\<exists>c. se c1 l c \<and> se_star c ls c2)"
lnth_subset_Sup_llist: fixes i :: "nat"   and Xs :: "'b set llist" assumes "enat i < llength Xs" shows "lnth Xs i \<subseteq> Sup_llist Xs"
proj_preserves: fixes x :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "x \<in> A" shows "proj r x \<in> A // r"
simps: fixes fi :: "int vec"   and i :: "nat"   and dmus :: "int iarray iarray"   and j :: "nat" shows "dmu_array_row_main_cost fi i dmus j = (if i \<le> j then (dmus, 0) else let sj = Suc j; dmus_i = dmus !! i; djj = dmus !! j !! j; (sigma, cost_sigma) = sigma_array_cost dmus dmus_i (dmus !! sj) djj j; dmu_ij = djj * (fi \<bullet> fs !! sj) - sigma; dmus' = iarray_update dmus i (iarray_append dmus_i dmu_ij); (res, cost_rec) = dmu_array_row_main_cost fi i dmus' sj; local_cost = 2 * n + 2 in (res, cost_rec + cost_sigma + local_cost))"
rat_one_code: shows "quotient_of 1 = (1, 1)"
real_cond_exp_jensens_inequality_1: fixes X :: "'a \<Rightarrow> real"   and I :: "real set"   and a :: "real"   and b :: "real"   and q :: "real \<Rightarrow> real" assumes "integrable M X"   and "AE x in M. X x \<in> I"   and "I = {a<..<b} \<or> I = {a<..} \<or> I = {..<b} \<or> I = UNIV"   and "integrable M (\<lambda>x. q (X x))"   and "convex_on I q"   and "q \<in> borel_measurable borel" shows "AE x in M. real_cond_exp M F X x \<in> I"
insort_key_left_comm: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and y :: "'c"   and xs :: "'c list" assumes "f x \<noteq> f y" shows "insort_key f y (insort_key f x xs) = insort_key f x (insort_key f y xs)"
f_less: shows "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - I) < e"
refl_rewritesI: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s \<leadsto> s'" shows "(s, s') \<in> [\<leadsto>]\<^sup>="
simps_1: fixes ys :: "'a list" shows "[] @ ys = ys"
rtp1: shows "\<alpha> + \<beta> + \<gamma> = (1::'a)"
lex_ext_to_lex_ext_unbounded: fixes xs :: "'b list"   and n :: "nat"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" assumes "length xs \<le> n"   and "length ys \<le> n" shows "lex_ext f n xs ys = lex_ext_unbounded f xs ys"
fun_relD1: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'd"   and Ra :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set"   and x :: "'a"   and r :: "'b" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f x = r" shows "\<forall>x'. (x, x') \<in> Ra \<longrightarrow> (r, f' x') \<in> Rr"
neutral: shows "(\<Sum>a. (0::'a)) = (0::'a)"
SCallRedsNonStatic: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> char list exp"   and D :: "char list" assumes "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>map Val vs,s\<^sub>2,False\<rangle>"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s,b\<rangle> \<rightarrow>* \<langle>THROW IncompatibleClassChangeError,s\<^sub>2,False\<rangle>"
sample_env_def: fixes sample :: "'f pmf \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and p :: "'f pmf"   and f :: "'f \<Rightarrow> ('d, 'e) envT" shows "sample_env sample p f = EnvT (\<lambda>r. sample p (\<lambda>x. run_env (f x) r))"
cases: fixes a1 :: "('a, 'b) ta_rule set"   and a2 :: "'b tree"   and a3 :: "'a" assumes "accs_laz a1 a2 a3" obtains q :: "'a" and f :: "'b" and qs :: "'a list" and \<delta> :: "('a, 'b) ta_rule set" and ts :: "'b tree list" where "a1 = \<delta>" and "a2 = NODE f ts" and "a3 = q" and "q \<rightarrow> f qs \<in> \<delta>" and "list_all_zip (accs_laz \<delta>) ts qs"
cr_hmultiset_def: shows "cr_hmultiset \<equiv> \<lambda>x y. x = Hereditary_Multiset.Rep_hmultiset y"
incr_bv_combine: fixes m :: "nat"   and k :: "nat"   and n :: "nat"   and s :: "Core.term" shows "incr_bv m k (incr_bv n k s) = incr_bv (m + n) k s"
g1: shows "g 1 = f' (\<lambda>j. if j = 0 then 1 else 0)"
prod_coprime_left: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "\<And>i. i \<in> A \<Longrightarrow> coprime (f i) a" shows "coprime (prod f A) a"
subtensor0: fixes ds :: "nat list"   and i :: "nat" assumes "ds \<noteq> []"   and "i < hd ds" shows "subtensor (tensor0 ds) i = tensor0 (tl ds)"
assms_2: shows "\<forall>(uu_, x)\<in>set x. x \<noteq> []"
True_start_steps_starD: fixes A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and rr :: "bool list"   and w :: "'b list" assumes "(True # start A, rr) \<in> NAe.steps (RegExp2NAe.star A) w" shows "\<exists>us v. w = concat us @ v \<and> Ball (set us) (NAe.accepts A) \<and> (\<exists>r. (start A, r) \<in> NAe.steps A v \<and> rr = True # r)"
Ln_Arg: fixes z :: "complex" assumes "z \<noteq> 0" shows "Ln z = complex_of_real (ln (cmod z)) + \<i> * complex_of_real (Arg z)"
solve_system_imp_is_solution: fixes x :: "('b, 'c) vec"   and A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "is_solution x (fst (solve_system A b)) (snd (solve_system A b))" shows "is_solution x A b"
stake_finite_universe_finite: fixes A :: "'b set"   and n :: "nat" assumes "finite A"   and "A \<noteq> {}" shows "finite (stake n ` streams A)"
simps_6: fixes v :: "'b poly"   and va :: "'b poly" shows "degreen (Mul v va) = (\<lambda>m. 0)"
assms_5: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "g \<in> G" shows "homogeneous g"
Residues_nat_def: fixes m :: "nat" shows "Residues_nat m = \<lparr>carrier = {0..<m}, monoid.mult = \<lambda>x y. x * y mod m, one = 1, ring.zero = 0, add = \<lambda>x y. (x + y) mod m\<rparr>"
nhop_update: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "rt \<noteq> update rt dip (dsn, dsk, flg, hops, sip)" shows "the (nhop (update rt dip (dsn, dsk, flg, hops, sip)) dip) = sip"
ex_le_of_int: fixes x :: "'b" shows "\<exists>z. x \<le> of_int z"
some_quasi_inverse_def: fixes f :: "'a" shows "f\<^sup>~ = (SOME g. quasi_inverses f g)"
intros_1: fixes h :: "'var"   and \<sigma> :: "'var \<Rightarrow> 'val"   and n :: "nat" assumes "h \<in> local.hvars"   and "h \<in> local.reads (path \<sigma> n)"   and "\<forall>k<n. h \<notin> local.writes (path \<sigma> k)" shows "(path \<sigma>, n) \<in> scp"
Red1CondF: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>if (false) e1 else e2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>e2,s\<rangle>"
domain_def: fixes R :: "('c, 'd) ring_scheme" shows "domain R \<equiv> cring R \<and> domain_axioms R"
xvecFreshN: shows "(p \<bullet> xvec) \<sharp>* N"
prems_3: shows "n2 < length (al @ [a]) \<and> j2 < f ((al @ [a]) ! n2)"
image_def: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
nn_integral_pmf_of_set: fixes S :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "S \<noteq> {}"   and "finite S" shows "integral\<^sup>N (measure_pmf (local.pmf_of_set S)) f = sum f S / of_nat (card S)"
dis_is_hbl: shows "Height_Bounded_Labeling c s t f l"
sgnxr: shows "(f has_sgnx r) (at_right a)"
rep_eq: shows "pmf local.pmf_of_multiset = (\<lambda>x. real (count M x) / real (size M))"
cons2: shows "evaluate_dec ck_ mn_ env_ s1_ d_ (s2_, Rval new_env_) \<and> evaluate_decs ck_ mn_ (extend_dec_env new_env_ env_) s2_ ds_ (s3_, r_) \<and> (\<forall>s' ra. (s3_, r_) = (s', ra) \<and> ck_ = False \<longrightarrow> (\<exists>c. evaluate_decs True mn_ (extend_dec_env new_env_ env_) (update_clock (\<lambda>_. c) s2_) ds_ (update_clock (\<lambda>_. 0) s', ra)))"
iso_set_trans: fixes h :: "'g \<Rightarrow> 'h"   and G :: "('g, 'i) monoid_scheme"   and H :: "('h, 'j) monoid_scheme"   and i :: "'h \<Rightarrow> 'k"   and I :: "('k, 'l) monoid_scheme" assumes "h \<in> Group.iso G H"   and "i \<in> Group.iso H I" shows "i \<circ> h \<in> Group.iso G I"
hyps_5: assumes "lookup p ` keys p \<subseteq> P[- X]" shows "p \<in> range (focus X)"
pick_def: fixes A :: "'a set" shows "pick A \<equiv> SOME x. x \<in> A"
assms_1: shows "G \<in> set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)"
ahm_iteratei_correct: fixes hm :: "('a, 'b) ArrayHashMap_Impl.hashmap" assumes "ArrayHashMap_Impl.ahm_invar hm" shows "map_iterator (ArrayHashMap_Impl.ahm_iteratei hm) (ArrayHashMap_Impl.ahm_\<alpha> hm)"
singleton_set_some: fixes X :: "'a set" shows "(\<exists>Y. Y \<in> singleton_set X) = (\<exists>x. x \<in> X)"
NOUA2: shows "OUadmitFO \<sigma> ODE2 U"
Q_def: shows "Q = Pascal_Property.inter (line B C) (line E F)"
strict_trans2: fixes b :: "'d"   and a :: "'d"   and c :: "'d" assumes "b < a"   and "c \<le> b" shows "c < a"
assms: shows "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> A"
simps_2: fixes r :: "real"   and x :: "complex" shows "Im (r *\<^sub>R x) = r * Im x"
set_restriction_fun_singleton: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" shows "set_restriction_fun P {a} = (if P a then {a} else {})"
orbit_FOO: fixes a :: "'c"   and g :: "'c \<Rightarrow> 'c"   and g' :: "'d \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd" assumes "a \<in> orbit g a"   and "\<And>x. x \<in> orbit g a \<Longrightarrow> g' (f x) = f (g x)" shows "f ` orbit g a = orbit g' (f a)"
axioms: fixes G :: "('c \<Rightarrow> 'c) set"   and S :: "'c set" assumes "transformation_group G S" shows "subgroup G (monoid.Units (S \<rightarrow>\<^sub>E S) (compose S) (identity S)) (compose S) (identity S)"
llam_cases: fixes t :: "llam" shows "(\<exists>a. t = lPar a) \<or> (\<exists>n. t = lVar n) \<or> (\<exists>t1 t2. t = lApp t1 t2) \<or> (\<exists>t1. t = lLam t1)"
assms_2: shows "x \<in>\<^sub>\<circ> S"
lookup_edom: fixes m :: "'d \<Rightarrow> 'c"   and x :: "'d" assumes "m x \<noteq> \<bottom>" shows "x \<in> edom m"
simps_1: fixes P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "sorted_wrt P [] = True"
OrderingSetIsoI_orders_greater2less: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b set" assumes "inj_on f P"   and "\<And>a b. \<lbrakk>a \<in> P; b \<in> P\<rbrakk> \<Longrightarrow> order.greater_eq a b = order.greater_eq (f b) (f a)" shows "OrderingSetIso order.greater_eq order.greater dual_order.greater_eq dual_order.greater P f"
hom_closed: fixes x :: "'a \<Rightarrow>\<^sub>0 int" assumes "x \<in> carrier (free_Abelian_group S)" shows "h x \<in> carrier (free_Abelian_group T)"
minarc_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "minarc\<^sub>M f = (\<lambda>e. if f e \<noteq> bot \<and> (\<forall>d. f d \<noteq> bot \<longrightarrow> f e + bot \<le> f d + bot \<and> (enum_lex_less d e \<longrightarrow> f e + bot \<noteq> f d + bot)) then top else bot)"
mult_rng_of_frac_def: fixes X :: "('a \<times> 'a) set"   and Y :: "('a \<times> 'a) set" shows "mult_rng_of_frac X Y \<equiv> let x' = SOME x. x \<in> X; y' = SOME y. y \<in> Y in fst x' \<otimes> fst y' |\<^bsub>rel\<^esub> snd x' \<otimes> snd y'"
neg_0_less_iff_less_erea: fixes a :: "ereal" shows "(0 < - a) = (a < 0)"
icard_eq_enat_card_conv: fixes n :: "nat"   and A :: "'a set" assumes "0 < n" shows "(icard A = enat n) = (card A = n)"
analz_insert_Crypt_subset: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" shows "analz (insert (Crypt K X) H) \<subseteq> insert (Crypt K X) (analz (insert X H))"
PiE_fun_upd: fixes y :: "'b"   and T :: "'c \<Rightarrow> 'b set"   and x :: "'c"   and f :: "'c \<Rightarrow> 'b"   and S :: "'c set" assumes "y \<in> T x"   and "f \<in> Pi\<^sub>E S T" shows "f(x := y) \<in> Pi\<^sub>E (insert x S) T"
iv_in_preflect_image: fixes t0 :: "'b"   and T :: "'b set" assumes "t0 \<in> T" shows "t0 \<in> preflect t0 ` T"
intro\<^sub>3: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'd"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and mem\<^sub>2 :: "'b \<Rightarrow> 'd"   and c :: "('b, 'a, 'c) Stmt"   and \<Gamma>' :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "sifum_types.\<R>\<^sub>3_aux aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma> ((c\<^sub>1, mds), mem\<^sub>1) ((c\<^sub>2, mds), mem\<^sub>2)"   and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma> c \<Gamma>'" shows "sifum_types.\<R>\<^sub>3_aux aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>' ((c\<^sub>1 ;; c, mds), mem\<^sub>1) ((c\<^sub>2 ;; c, mds), mem\<^sub>2)"
scale_zero_right: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b" assumes "Modules.module scale" shows "scale a (0::'c) = (0::'c)"
invertible_Hermite_of_row_i: fixes ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "(('a, 'c) vec, 'b) vec"   and i :: "'b" assumes "ass_function ass" shows "\<exists>P. invertible P \<and> Hermite_of_row_i ass res A i = P ** A"
cInf_atLeastAtMost: fixes y :: "'b multiset"   and x :: "'b multiset" assumes "y \<subseteq># x" shows "Inf (subset_mset.atLeastAtMost y x) = y"
div_less_dividend_word: fixes x :: "'b word"   and n :: "'b word" assumes "x \<noteq> 0"   and "n \<noteq> 1" shows "x div n < x"
aux_lemma: fixes p1a :: "'a list"   and p2a :: "'a list" assumes "set p1a \<subseteq> carrier R"   and "set p2a \<subseteq> carrier R" shows "poly_mult p1a p2a = poly_mult (\<zero> # p1a) p2a"
strict_trans: fixes b :: "'c list"   and a :: "'c list"   and c :: "'c list" assumes "strict_prefix b a"   and "strict_prefix c b" shows "strict_prefix c a"
F_def: shows "F = (\<lambda>x. x powr (e + 1) * (1 - (e + 1) * ln x) / (e + 1)\<^sup>2)"
assms_4: shows "val_process Mkt pf n (spick w n False) \<le> 0"
map_add_SomeD: fixes m :: "'b \<Rightarrow> 'a option"   and n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a" assumes "Map.map_add m n k = Some x" shows "n k = Some x \<or> n k = None \<and> m k = Some x"
jump_poly0_2: fixes q :: "real poly"   and x :: "real" shows "jump_poly q 0 x = 0"
await_elim: fixes b :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Await b c\<^sub>1, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>2, mds', mem'\<rangle>" shows "ev\<^sub>B mem b \<and> no_await c\<^sub>1 \<and> is_final c\<^sub>2 \<and> \<langle>c\<^sub>1, mds, mem\<rangle> \<leadsto>\<^sup>+ \<langle>c\<^sub>2, mds', mem'\<rangle>"
fin_B: shows "finite {x. g x \<noteq> (0::'c)}"