hcju/isabelleps huggingface dataset:
assms_2: shows "xs \<in> valid_lists S (\<Squnion>i. Y i)"
ask_nondet_def: fixes ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> ('a, 'b) nondetT" shows "nondetM_base.ask_nondet ask f = NondetT (ask (\<lambda>r. run_nondet (f r)))"
reflp_def: fixes r :: "'m \<Rightarrow> 'm \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
init_eqI: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat option"   and g :: "nat \<Rightarrow> nat option" assumes "\<And>x. x \<le> n \<Longrightarrow> f x = g x" shows "f \<triangleright> n = g \<triangleright> n"
SetMarkInvFinal_def: fixes i :: "I" shows "SetMarkInvFinal i = (case i of final \<Rightarrow> Final | _ \<Rightarrow> {})"
mset_ran_cong: fixes a :: "int \<Rightarrow> 'b"   and b :: "int \<Rightarrow> 'b"   and r :: "int set" assumes "a = b on r" shows "mset_ran a r = mset_ran b r"
na_def: shows "next_execs s execs d = aseq # aseqs"
V_def: fixes \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd" shows "BD_Security.V \<phi> f \<equiv> BD_Security.filtermap \<phi> f"
summands_equal: fixes a :: "'a vec"   and b :: "'a vec"   and c :: "'a vec"   and d :: "'a vec" assumes "a = b"   and "c = d" shows "a + c = b + d"
select_convs_3: fixes access :: "acc_modi"   and cfields :: "fdecl list"   and methods :: "mdecl list"   and init :: "stmt"   and more :: "'b" shows "init \<lparr>access = access, cfields = cfields, methods = methods, init = init, \<dots> = more\<rparr> = init"
extremum_uniqueI: fixes a :: "'d" assumes "a \<le> bot" shows "a = bot"
simps: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" shows "rb_aux_inv (bs, ss, ps) = (rb_aux_inv1 bs \<and> Ball (set ss) (is_syz_sig dgrad) \<and> (\<forall>p q. Inl (p, q) \<in> set ps \<longrightarrow> is_regular_spair p q \<and> p \<in> set bs \<and> q \<in> set bs) \<and> (\<forall>j. Inr j \<in> set ps \<longrightarrow> (j < length fs \<and> (\<forall>b\<in>set bs. lt b \<prec>\<^sub>t term_of_pair (0::'a, j))) \<and> length (filter (\<lambda>q. sig_of_pair q = term_of_pair (0::'a, j)) ps) \<le> 1) \<and> sorted_wrt pair_ord ps \<and> (\<forall>p\<in>set ps. (\<forall>b1\<in>set bs. \<forall>b2\<in>set bs. is_regular_spair b1 b2 \<longrightarrow> sig_of_pair p \<prec>\<^sub>t lt (spair b1 b2) \<longrightarrow> Inl (b1, b2) \<in> set ps \<or> Inl (b2, b1) \<in> set ps) \<and> (\<forall>j<length fs. sig_of_pair p \<prec>\<^sub>t term_of_pair (0::'a, j) \<longrightarrow> Inr j \<in> set ps)) \<and> (\<forall>b\<in>set bs. \<forall>p\<in>set ps. lt b \<preceq>\<^sub>t sig_of_pair p) \<and> (\<forall>a\<in>set bs. \<forall>b\<in>set bs. is_regular_spair a b \<longrightarrow> Inl (a, b) \<notin> set ps \<longrightarrow> Inl (b, a) \<notin> set ps \<longrightarrow> \<not> is_RB_in dgrad rword (set bs) (lt (spair a b)) \<longrightarrow> (\<exists>p\<in>set ps. sig_of_pair p = lt (spair a b) \<and> \<not> sig_crit' bs p)) \<and> (\<forall>j<length fs. Inr j \<notin> set ps \<longrightarrow> 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)))"
basic: fixes a :: "'b set"   and A :: "'b set set" assumes "a \<in> A" shows "a \<in> sigma A"
cptn_if_cptn_mod: fixes c :: "('b com option \<times> 'b) list" assumes "c \<in> cptn_mod" shows "c \<in> cptn"
assms_1: shows "(u, j, i) \<in> pair_less"
surjec_ind_bijec: fixes A :: "('d, 'e) Ring_scheme"   and R :: "('f, 'g) Ring_scheme"   and f :: "'d \<Rightarrow> 'f" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R"   and "surjec\<^bsub>A,R\<^esub> f" shows "bijec\<^bsub>A /\<^sub>r (ker\<^bsub>A,R\<^esub> f),R\<^esub> (f\<degree>\<^bsub>A,R\<^esub>)"
sameDom_lift2: fixes f :: "'g \<Rightarrow> 'f"   and inp :: "'e \<Rightarrow> 'g option" shows "sameDom (lift f inp) inp"
fml_sem_equals: fixes \<omega> :: "variable \<Rightarrow> real"   and I :: "interp"   and \<theta> :: "trm"   and \<eta> :: "trm" shows "(\<omega> \<in> fml_sem I (Equals \<theta> \<eta>)) = (term_sem I \<theta> \<omega> = term_sem I \<eta> \<omega>)"
eval_norm: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>D :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>D :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>D :: "'d"   and V :: "'b \<Rightarrow> 'd"   and t :: "'b local.term" assumes "FreeMonoidalCategory.evaluation_functor C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V"   and "monoidal_language.Arr C t" shows "evaluation_map.eval D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V (free_monoidal_category.norm C t) = evaluation_map.eval D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V t"
fmfilter_empty: fixes P :: "'c \<Rightarrow> bool" shows "fmfilter P {$$} = {$$}"
ring_simprules_3: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> x \<in> carrier P"
contractible_imp_simply_connected: fixes S :: "'b set" assumes "contractible S" shows "simply_connected S"
epigraph_invertible: fixes f :: "'b \<Rightarrow> ereal"   and g :: "'b \<Rightarrow> ereal" assumes "Epigraph UNIV f = Epigraph UNIV g" shows "f = g"
intros: fixes a :: "'c"   and b :: "'d"   and r :: "('c \<times> 'd) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
analz_insert_freshK2: fixes evs :: "event list"   and servK :: "nat"   and K :: "nat" assumes "evs \<in> kerbIV"   and "servK \<notin> authKeys evs"   and "servK \<notin> range shrK"   and "K \<in> symKeys" shows "(Key K \<in> analz (insert (Key servK) (knows Spy evs))) = (K = servK \<or> Key K \<in> analz (knows Spy evs))"
closed_segment_eq_sing: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(closed_segment a b = {c}) = (a = c \<and> b = c)"
gt_half_sum: 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 a b" shows "less (divide (plus a b) (plus one one)) b"
distinguish_trace_eq: fixes \<I> :: "('h, 'i) \<I>"   and res :: "('h, 'i) resource"   and res' :: "('h, 'i) resource" assumes "\<And>distinguisher. \<I> \<turnstile>g distinguisher \<surd> \<Longrightarrow> connect distinguisher res = connect distinguisher res'" shows "outs_\<I> \<I> \<turnstile>\<^sub>R res \<approx> res'"
rep_in_span: fixes r :: "'b \<Rightarrow>\<^sub>0 'a" shows "rep r \<in> local.span (keys r)"
validTauTransition: fixes P :: "pi"   and P' :: "pi" assumes "P \<longmapsto> \<tau> \<prec> P'"   and "valid P" shows "valid P'"
new_character_data_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'v" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
cases: fixes x :: "opubx" obtains "x = OpAddNumUbx"
"1_2": shows "i < nth_least_ext s (Suc k)"
to_nat_def: fixes a :: "'a" shows "to_nat a = (SOME i. univ_list ! i = a \<and> i < length univ_list)"
nat_discrete_filtration: shows "Filtration.filtration M nat_filtration"
atLeastLessThan_empty_iff: fixes a :: "'a word"   and b :: "'a word" shows "(signed.atLeastLessThan a b = {}) = (\<not> a <s b)"
that: fixes a1 :: "floatarith"   and a2 :: "floatarith" assumes "fa = Add a1 a2"   and "fa1 = subst_floatarith (\<lambda>x. Var (s x)) a1"   and "fa2 = subst_floatarith (\<lambda>x. Var (s x)) a2" shows "thesis"
dist_fun_le_dist_first_terms: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b"   and N :: "nat" shows "dist x y \<le> 2 * Max {dist (x (from_nat n)) (y (from_nat n)) |n. n \<le> N} + (1 / 2) ^ N"
length_power_polys_i: fixes ff_ops :: "'a arith_ops_record"   and x :: "'a list"   and y :: "'a list"   and z :: "'a list"   and n :: "nat" shows "length (power_polys_i ff_ops x y z n) = n"
rw4: shows "IArray (map (\<lambda>x. IArray.of_fun (\<lambda>i. A $ mod_type_class.from_nat x $ mod_type_class.from_nat i) CARD('c)) [0..<CARD('b)]) !! mod_type_class.to_nat b !! mod_type_class.to_nat j = A $ b $ j"
smc_Comp_vdomainI: fixes b :: "V"   and c :: "V"   and g :: "V"   and a :: "V"   and f :: "V"   and gf :: "V" assumes "g : b \<mapsto>\<^bsub>\<BB>\<^esub> c"   and "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b"   and "gf = [g, f]\<^sub>\<circ>" shows "gf \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<BB>\<lparr>Comp\<rparr>)"
rank_drop_cols: shows "vec_space.rank (length signs) (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) = dim_col (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
eval_hom': fixes a :: "'a"   and p :: "'a list" assumes "a \<in> carrier R"   and "set p \<subseteq> carrier R" shows "h (R.eval p a) = S.eval (map h p) (h a)"
arc_join_eq: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "pathfinish g1 = pathstart g2" shows "arc (g1 +++ g2) = (arc g1 \<and> arc g2 \<and> path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2})"
vars_monom_single_cases: fixes v :: "nat"   and k :: "nat"   and a :: "'a" shows "vars (MPoly_Type.monom (Poly_Mapping.single v k) a) = (if k = 0 \<or> a = (0::'a) then {} else {v})"
assms: shows "z' \<in> set (zipper_children\<^sub>m z)"
R1_def: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R1 P \<equiv> \<lambda>(A, A'). P (A, A') \<and> tr A \<le> tr A'"
Cd_single: fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> X"   and "inj_on Xd X"   and "x \<in> Field (Pd_shuffle_to_top ds' Y d)" shows "x \<in> Pds'.Cd d X"
set_incl_pred2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(A \<le> B) = (\<forall>a1 a2. A a1 a2 \<longrightarrow> B a1 a2)"
Collect_subset: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "{x \<in> A. P x} \<subseteq> A"
distr_PiM_reindex: fixes K :: "'b set"   and M :: "'b \<Rightarrow> 'c measure"   and f :: "'d \<Rightarrow> 'b"   and I :: "'d set" assumes "\<And>i. i \<in> K \<Longrightarrow> prob_space (M i)"   and "inj_on f I"   and "f \<in> I \<rightarrow> K" shows "distr (Pi\<^sub>M K M) (Pi\<^sub>M I (\<lambda>i. M (f i))) (\<lambda>\<omega>. \<lambda>n\<in>I. \<omega> (f n)) = Pi\<^sub>M I (\<lambda>i. M (f i))"
r_rp: shows "bs_env \<rho> \<rho>'"
repr_independenceD: fixes H :: "nat set"   and y :: "nat"   and x :: "nat" assumes "subgroup H G"   and "y \<in> carrier G"   and "H #>\<^bsub>G\<^esub> x = H #>\<^bsub>G\<^esub> y" shows "y \<in> H #>\<^bsub>G\<^esub> x"
sg4: shows "lose t = [True] \<or> lose t = [False]"
norm_round_complex_le: fixes z :: "complex" shows "(cmod (z - gauss2complex (round_complex z)))\<^sup>2 \<le> 1 / 2"
ratrel_iff: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "ratrel x y = (snd x \<noteq> 0 \<and> snd y \<noteq> 0 \<and> fst x * snd y = fst y * snd x)"
effect_heapI: fixes h' :: "heap"   and f :: "heap \<Rightarrow> 'a \<times> heap"   and h :: "heap"   and r :: "'a" assumes "h' = snd (f h)"   and "r = fst (f h)" shows "effect (heap f) h h' r"
invc_ins: fixes t :: "('b \<times> color) tree"   and x :: "'b" shows "invc t \<longrightarrow> invc2 (ins x t) \<and> (color t = Black \<longrightarrow> invc (ins x t))"
word_mod_less_divisor: fixes n :: "'a word"   and m :: "'a word" assumes "0 < n" shows "m mod n < n"
fi_is_sum_of_mu_gso: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "map of_int_hom.vec_hom fs ! i = gs.M.sumlist (map (\<lambda>j. gs.\<mu> i j \<cdot>\<^sub>v gs.gso j) [0..<Suc i])"
kex: shows "\<exists>k\<le>n. \<pi> k = y"
proper_map_polyfun_univ: fixes K :: "'a set"   and c :: "nat \<Rightarrow> 'a"   and i :: "nat"   and n :: "nat" assumes "compact K"   and "c i \<noteq> (0::'a)"   and "1 \<le> i"   and "i \<le> n" shows "compact {z. (\<Sum>i\<le>n. c i * z ^ i) \<in> K}"
multi_union_self_other_eq: fixes A :: "'d multiset"   and X :: "'d multiset"   and Y :: "'d multiset" assumes "A + X = A + Y" shows "X = Y"
fn_comm_power': fixes fa :: "'c \<Rightarrow> 'c"   and tr :: "'b \<Rightarrow> 'c"   and fr :: "'b \<Rightarrow> 'b"   and n :: "nat"   and x :: "'b" assumes "\<And>x. fa (tr x) = tr (fr x)" shows "(fa ^^ n) (tr x) = tr ((fr ^^ n) x)"
vwalk_join_def: fixes p :: "'b list"   and q :: "'b list" shows "p \<oplus> q \<equiv> p @ tl q"
is_RELABEL_project_conv: shows "is_RELABEL \<circ> project_operation = is_RELABEL'"
Crypt_def: fixes K :: "nat"   and X :: "msg" shows "Crypt K X = Abs_Msg (\<Union>U\<in>Rep_Msg X. msgrel `` {CRYPT K U})"
simps_9: fixes p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp" shows "ctermsl (p1 \<oplus> p2) = ctermsl p1 \<union> ctermsl p2"
Un_def: fixes A :: "'d set"   and B :: "'d set" shows "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
decisionD: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b"   and r :: "'a rule list"   and s :: "state"   and t :: "state"   and X :: "final_decision" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "s = Decision X" shows "t = Decision X"
nataction_Suc2: fixes a :: "'a"   and n :: "nat" shows "a +^ Suc n = a +^ n + a"
h_diff_def: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_diff i e = h_plus i e - (if cycles \<Delta> \<noteq> {} then h_minus i e else 0)"
\<Z>_srng_axioms_def: fixes \<alpha> :: "V"   and \<SS> :: "V" shows "\<Z>_srng_axioms \<alpha> \<SS> \<equiv> (vcard \<SS> = 3\<^sub>\<nat> \<and> \<Z>_csgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vplus\<rparr>]\<^sub>\<circ>) \<and> \<Z>_sgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vmult\<rparr>]\<^sub>\<circ> \<and> (\<forall>a b c. a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> (a +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b) *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c) \<and> (\<forall>a b c. a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr> \<longrightarrow> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c) = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c)"
fmpt_I: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "finite_measure M"   and "T \<in> M \<rightarrow>\<^sub>M M"   and "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M (T -` A \<inter> space M) = emeasure M A" shows "fmpt M T"
distinct_remove1: fixes xs :: "'c list"   and x :: "'c" assumes "cycle xs" shows "cycle (remove1 x xs)"
scaleC_right: fixes a :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and r :: "complex"   and b :: "'c" shows "a *\<^sub>V r *\<^sub>C b = r *\<^sub>C (a *\<^sub>V b)"
X_distrib: fixes n :: "nat" shows "distr M borel (X n) = \<mu>"
SC_refl: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "tarski_first3 C" shows "C a b a b"
ACI_ACI_norm: fixes r :: "'b rexp" shows "ACI \<guillemotleft>r\<guillemotright> r"
inf_filter_def: fixes F :: "'a filter"   and F' :: "'a filter" shows "inf F F' = Abs_filter (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
strict_implies_not_eq: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<prec> b" shows "a \<noteq> b"
finite_on_def: fixes S :: "'b set"   and f :: "'b \<Rightarrow> ereal" shows "finite_on S f = (\<forall>x\<in>S. f x \<noteq> \<infinity> \<and> f x \<noteq> - \<infinity>)"
fps_inv_fps_exp_compose_2: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "fps_exp a - 1 oo fps_inv (fps_exp a - 1) = fps_X"
vars_snd_rcv_strand_subset2_4: fixes S :: "('d, 'c) strand_step list" shows "wfvarsoccs\<^sub>s\<^sub>t S \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
interior_of_complement: fixes X :: "'a topology"   and S :: "'a set" shows "X interior_of (topspace X - S) = topspace X - X closure_of S"
exec_step_ind_New_Init: fixes sh :: "sheap"   and C :: "char list"   and ics :: "init_call_status"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list" assumes "\<forall>obj. sh C \<noteq> \<lfloor>(obj, Done)\<rfloor>"   and "\<forall>Cs. ics \<noteq> Called Cs" shows "exec_step_ind (StepI (New C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Calling C []) # frs, sh)"
rep_eq: fixes x :: "'b poly" shows "count (poly_roots x) = (\<lambda>x. if x = 0 then 0 else order x x)"
intro_prgR: fixes c :: "'c nres"   and R :: "('c \<times> 'd) set"   and a :: "'d nres" assumes "c \<le> \<Down> R a" shows "c \<le> \<Down> R a"
bij_betw_restrict_eq: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" shows "bij_betw (restrict f A) A B = bij_betw f A B"
hypreal_three_squares_add_zero_iff: fixes x :: "real star"   and y :: "real star"   and z :: "real star" shows "(x * x + y * y + z * z = 0) = (x = 0 \<and> y = 0 \<and> z = 0)"
lc: fixes s :: "'a"   and t :: "'a"   and a :: "'b"   and u :: "'a"   and b :: "'b" assumes "(s, t) \<in> L a"   and "(s, u) \<in> E b" shows "(t, u) \<in> local.conversion' (under r a) O E b O local.conversion' (under r a) O ((L a)\<inverse>)\<^sup>= O local.conversion' (under r a \<inter> under r b) \<or> (t, u) \<in> local.conversion' (under r a \<union> under r b) O ((L a)\<inverse>)\<^sup>= O local.conversion' (under r b)"
simps_4: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and tst :: "'b"   and c :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_ZObis tval aval indis (while tst do c) = (PL_Indis.SC_discr aval indis (while tst do c) \<or> PL_Indis.SC_siso tval aval indis (while tst do c))"
simps: fixes xs :: "'f"   and x :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "qSkelAbs (qAbs xs x X) = Branch (\<lambda>i. Some (qSkel X)) Map.empty"
assms_6: shows "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"
prod_unit_l: fixes a :: "'a"   and b :: "'a" assumes "a \<otimes> b \<in> Units G"   and "a \<in> Units G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "b \<in> Units G"
nummul_nb: fixes t :: "Cooper.num"   and i :: "int" assumes "numbound0 t" shows "numbound0 (nummul i t)"
cospan_\<mu>\<nu>: shows "C.cospan trg.leg0 trg.leg1"
inner_sum_right: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (comm_monoid_add.sum plus zero f A) = (\<Sum>y\<in>A. inner x (f y))"
listrel_def: shows "listrel \<equiv> \<lambda>r. {(xa, x). listrelp (\<lambda>x xa. (x, xa) \<in> r) xa x}"
simps_1: fixes f :: "'b \<Rightarrow> 'a pred" shows "apply f Empty = Empty"
inv_hcomp: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "horizontal_composition V H src trg"   and "category.iso V \<nu>"   and "category.iso V \<mu>"   and "src \<nu> = trg \<mu>" shows "category.inv V (H \<nu> \<mu>) = H (category.inv V \<nu>) (category.inv V \<mu>)"
Int_Diff_both: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<inter> B - C = (A - C) \<inter> (B - C)"
SeqThrow: fixes e :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>throw e;; e\<^sub>2,s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
upper_triangular_four_block: fixes A :: "'c mat"   and n :: "nat"   and D :: "'c mat"   and m :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat n n"   and "D \<in> carrier_mat m m"   and "upper_triangular A"   and "upper_triangular D" shows "upper_triangular (four_block_mat A B (0\<^sub>m m n) D)"
assms_4: shows "\<bar>r\<bar> \<le> b"
bind_return_pmf: fixes x :: "'f"   and f :: "'f \<Rightarrow> 'e pmf" shows "return_pmf x \<bind> f = f x"
intro: fixes np :: "nat \<Rightarrow> ('a, 'b seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'c) \<times> 'd, 'b seq_action) automaton"   and sr :: "'a \<Rightarrow> 'c \<times> 'd"   and qp :: "('e, 'b seq_action) automaton" assumes "openproc np onp sr"   and "openproc_parq_axioms qp" shows "openproc_parq np onp sr qp"
len_leaves: fixes as :: "'a list" shows "len (leaves as) = length as + 1"
DomainPI: fixes r :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'b"   and b :: "'d" assumes "r a b" shows "Domainp r a"
keys_sum: fixes b :: "'e \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'd"   and I :: "'e set" shows "Poly_Mapping.keys (sum b I) \<subseteq> (\<Union>i\<in>I. Poly_Mapping.keys (b i))"
foldl_node_correct: fixes f :: "'d \<Rightarrow> 'e \<times> 'f \<Rightarrow> 'd"   and \<sigma> :: "'d"   and nd :: "('e, 'f) Node" shows "foldl_node f \<sigma> nd = List.foldl f \<sigma> (nodeToList nd)"
Sup_insert: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and a :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (insert a A) = sup a (Sup A)"
sinr_strict: shows "sinr\<cdot>\<bottom> = \<bottom>"
wellorder_Least_lemma_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and k :: "'b" assumes "class.wellorder less_eq less"   and "P k" shows "P (ord.Least less_eq P)"
inject_3: fixes x31 :: "exp"   and x32 :: "nat"   and x33 :: "exp list"   and y31 :: "exp"   and y32 :: "nat"   and y33 :: "exp list" shows "(MethodInvk x31 x32 x33 = MethodInvk y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
numsub: fixes bs :: "int list"   and a :: "Cooper.num"   and b :: "Cooper.num" shows "Inum bs (numsub a b) = Inum bs (Sub a b)"