hcju/isabelleps huggingface dataset:
assms_3: shows "\<one>\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>"
fun_add_closed: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "f \<in> carrier F"   and "g \<in> carrier F" shows "f \<oplus>\<^bsub>F\<^esub> g \<in> carrier F"
match_final_backward: fixes s1 :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state"   and s2 :: "('fenv_ubx, 'henv, ('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame) state" assumes "s1 \<sim> s2"   and "Subx.final s2" shows "Sinca.final s1"
lang_dual_lderiv_dual: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp_dual"   and w :: "'a \<times> bool list" assumes "wf_dual \<Sigma> n r"   and "w \<in> (set \<circ> \<sigma> \<Sigma>) n" shows "lang_dual \<Sigma> n (Co\<DD> \<Sigma> w r) = lQuot w (lang_dual \<Sigma> n r)"
headconst_isnormNum: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "isnormNum (headconst p)"
simps_2: fixes v :: "'c option"   and m :: "('d \<times> ('d, 'c) trie) list"   and k :: "'d"   and ks :: "'d list" shows "lookup_trie (Trie v m) (k # ks) = (case map_of m k of None \<Rightarrow> None | Some st \<Rightarrow> lookup_trie st ks)"
sum_list_const_mult_prod: fixes r :: "'a"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xys :: "('b \<times> 'c) list" shows "r * (\<Sum>(x, y)\<leftarrow>xys. f x y) = (\<Sum>(x, y)\<leftarrow>xys. r * f x y)"
cases: fixes a1 :: "'a sequent list \<times> 'a sequent"   and a2 :: "'a form" assumes "leftPrincipal a1 a2" obtains C :: "'a sequent" and F :: "'a" and Fs :: "'a form list" and Ps :: "'a sequent list" where "a1 = (Ps, C)" and "a2 = Compound F Fs" and "C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* Em)"
trss_c_cases: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" obtains csp :: "'a list multiset" where "c' = csp + c" and "\<And>s. s \<in># csp \<Longrightarrow> \<exists>p u v. s = [entry fg p] \<and> (u, Spawn p, v) \<in> edges fg \<and> initialproc fg p"
iso_tuple_snd_def: fixes isom :: "('a, 'c, 'b) tuple_isomorphism" shows "Record.iso_tuple_snd isom = snd \<circ> Record.repr isom"
fn_norm_le_cong: fixes V :: "'c set"   and norm :: "'c \<Rightarrow> real"   and f :: "'c \<Rightarrow> real"   and x :: "'c" assumes "normed_vectorspace_with_fn_norm V norm"   and "Function_Norm.continuous V f norm"   and "linearform V f"   and "x \<in> V" shows "\<bar>f x\<bar> \<le> \<Squnion>({0} \<union> {\<bar>f x\<bar> / norm x |x. x \<noteq> (0::'c) \<and> x \<in> V}) * norm x"
transpose_def: shows "Square_Matrix.transpose \<equiv> (to_fun ---> of_fun) (\<lambda>M i j. M j i)"
distinct_def: shows "distinct \<equiv> rec_list True (\<lambda>x xs. (\<and>) (x \<notin> set xs))"
fls_inverse_X_power': fixes n :: "nat" assumes "inverse (1::'a) = (1::'a)" shows "inverse (fls_X ^ n) = fls_X_inv ^ n"
cond_1: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"   and "movement.s a\<^sub>e v\<^sub>e s\<^sub>e (movement.t_stop a\<^sub>e v\<^sub>e) < s\<^sub>o" shows "\<not> safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o {0..}"
theta_def: fixes n :: "nat" shows "theta n = (\<Sum>p = 1..n. if prime p then ln (real p) else 0)"
atLeastatMost_subset_iff: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word"   and d :: "'a word" shows "(signed.atLeastAtMost a b \<subseteq> signed.atLeastAtMost c d) = (\<not> a \<le>s b \<or> c \<le>s a \<and> b \<le>s d)"
sym_digraphI_app_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "iG.digraph_isomorphism hom" shows "sym_digraph (app_iso hom (app_iso hom G))"
monad_prob_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and sample :: "'c pmf \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" shows "monad_prob return bind sample \<equiv> monad return bind \<and> monad_prob_axioms bind sample"
kSup_distl: fixes f :: "'b \<Rightarrow> 'd set"   and G :: "('d \<Rightarrow> 'c set) set" shows "f \<circ>\<^sub>K \<Squnion> G = \<Squnion> \<P> ((\<circ>\<^sub>K) f) G"
StrictRefEq\<^sub>I\<^sub>n\<^sub>t\<^sub>e\<^sub>g\<^sub>e\<^sub>r: fixes x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "x \<doteq> y \<equiv> \<lambda>\<tau>. if (\<upsilon> x) \<tau> = true \<tau> \<and> (\<upsilon> y) \<tau> = true \<tau> then (x \<triangleq> y) \<tau> else invalid \<tau>"
AE_prob_1: fixes s :: "'s"   and A :: "'s stream set" assumes "T.prob s A = 1" shows "AE x in T s. x \<in> A"
Suc: fixes acc :: "nat" shows "totient_naive k acc n = card {x \<in> {0<..k}. coprime x n} + acc"
loopfree_digraphI_app_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "iG.digraph_isomorphism hom" shows "loopfree_digraph (app_iso hom (app_iso hom G))"
gen_akiyama_tanigawa_row_def: fixes f :: "nat \<Rightarrow> real"   and n :: "nat"   and l :: "nat"   and u :: "nat" shows "gen_akiyama_tanigawa_row f n l u = map (gen_akiyama_tanigawa f n) [l..<u]"
hyps: fixes u :: "'a"   and r :: "'a list"   and ce :: "'a list multiset"   and w :: "('b, 'c) label list"   and r' :: "'a list"   and ce' :: "'a list multiset" assumes "({#[entry fg p]#}, ww, {#u # r#} + ce) \<in> trcl (ntr fg)"   and "(([u], ce), w, r', ce') \<in> trcl (trss fg)"   and "initialproc fg p"   and "length r' \<le> 1"   and "w \<noteq> []" shows "\<exists>ww'. ({#[entry fg p]#}, ww', {#r' @ r#} + ce') \<in> trcl (ntr fg)"
norm_exp_eq_Re: fixes z :: "complex" shows "cmod (exp z) = exp (Re z)"
Zn_iso: fixes n :: "nat" assumes "order (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)) = n" shows "DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n) \<cong> Z n"
arb_pf_def: shows "arb_pf = (\<lambda>x n w. 0)(stk := \<lambda>n w. 1, risk_free_asset := \<lambda>n w. - geom_proc 0 w)"
select_convs_2: fixes name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "incoming \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = incoming"
plus_1_not_0: fixes n :: "nat" shows "n + 1 \<noteq> 0"
conditional_entropy_nonneg: fixes X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e" assumes "simple_function M X"   and "simple_function M Y" shows "0 \<le> \<H>(X | Y)"
I_subst: fixes \<phi> :: "atom fm"   and i :: "int"   and ks :: "int list"   and xs :: "int list" assumes "qfree \<phi>" shows "Z.I \<phi> ((i + \<langle>ks,xs\<rangle>) # xs) = Z.I (subst i ks \<phi>) xs"
wide_subdigraph_axioms: shows "\<AA> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<BB>"
chine_assoc_def: shows "f_src_src.chine_assoc \<equiv> \<langle>f_src_src.Prj\<^sub>1\<^sub>1 \<lbrakk>f.leg0, src.leg1 \<cdot> src_src.prj\<^sub>1\<rbrakk> \<langle>f_src_src.Prj\<^sub>0\<^sub>1 \<lbrakk>src.leg0, src.leg1\<rbrakk> f_src_src.Prj\<^sub>0\<rangle>\<rangle>"
Poly_replicate_0: fixes n :: "nat" shows "Poly (replicate n (0::'b)) = 0"
mkM_def: fixes T :: "'s Trace set" shows "Clock.mkM T \<equiv> \<lparr>worlds = T, relations = \<lambda>a. rel_ext (\<lambda>(t, t'). {t, t'} \<subseteq> T \<and> clock_jview a t = clock_jview a t'), valuation = envVal \<circ> tLast\<rparr>"
image_lt_monomial_lt: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set" shows "lt ` monomial (1::'c) ` lt ` F = lt ` F"
apx_empty_iff': fixes Z :: "('c \<Rightarrow> real) set" assumes "Z \<in> V'" shows "(Z = {}) = (Approx\<^sub>\<beta> Z = {})"
distinct_5: fixes x1 :: "'a"   and x4 :: "nat" shows "broadcast x1 \<noteq> \<not>unicast x4"
sc_completionI: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option" assumes "\<And>ttas s' t x ta x' m'. \<lbrakk>s -\<triangleright>ttas\<rightarrow>* s'; ta_seq_consist P vs (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))); thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor>; t \<turnstile> \<langle>x, shr s'\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>; actions_ok s' t ta\<rbrakk> \<Longrightarrow> \<exists>ta' x'' m''. t \<turnstile> \<langle>x, shr s'\<rangle> -ta'\<rightarrow> \<langle>x'', m''\<rangle> \<and> actions_ok s' t ta' \<and> ta_seq_consist P (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>)" shows "sc_completion s vs"
mr_cosmOp_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set" shows "mr_cosmOp M H = c_iop (b_ag M) H"
eucl_rel_poly: fixes x :: "'b poly"   and y :: "'b poly" shows "eucl_rel_poly x y (x div y, x mod y)"
nths_intersection_eq: fixes xs :: "'a list"   and A :: "nat set"   and B :: "nat set" assumes "{0..<length xs} \<subseteq> A" shows "nths xs B = nths xs (A \<inter> B)"
power_Suc: fixes a :: "'a"   and n :: "nat" shows "power.power 1' (;) a (Suc n) = a ; power.power 1' (;) a n"
rel_sel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d Monomorphic_Monad.id"   and b :: "'e Monomorphic_Monad.id" shows "rel_id R a b = R (extract a) (extract b)"
continuous_at_Arg2pi: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<ge>\<^sub>0" shows "isCont Arg2pi z"
nths_singleton: fixes x :: "'b"   and A :: "nat set" shows "nths [x] A = (if 0 \<in> A then [x] else [])"
sorted_less_rev_set_unique: fixes xs :: "'a list"   and ys :: "'a list" assumes "Sorted_Less.sorted (rev xs)"   and "Sorted_Less.sorted (rev ys)"   and "set xs = set ys" shows "xs = ys"
Eq_Zero_thm: fixes i :: "name"   and t :: "tm" assumes "atom i \<sharp> t" shows "{SyntaxN.All i (Neg (Var i IN t))} \<turnstile> t EQ Zero"
add_mono_thms_linordered_semiring_4: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i = j \<and> k = l" shows "i + k = j + l"
o_objects_exist_1: fixes v :: "i" shows "[\<^bold>\<diamond>(\<^bold>\<exists>x. \<lparr>E!,x\<^sup>P\<rparr>) in v]"
open_minus_countable: fixes A :: "real set"   and S :: "real set" assumes "countable A"   and "S \<noteq> {}"   and "open S" shows "\<exists>x\<in>S. x \<notin> A"
E5: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> \<box>\<diamond>F"
VInter_vdoubleton: fixes a :: "V"   and b :: "V" shows "\<Inter>\<^sub>\<circ> (set {a, b}) = a \<inter>\<^sub>\<circ> b"
iUntil_singleton: fixes t :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t0. t0 \<U> t1 {t}. Q t1) = Q t"
cases: fixes a1 :: "'a \<Rightarrow> 'a list"   and a2 :: "('a, 'b) Term.term"   and a3 :: "('a, 'b) Term.term" assumes "term_variants_pred a1 a2 a3" obtains P :: "'a \<Rightarrow> 'a list" and x :: "'b" where "a1 = P__" and "a2 = Var x" and "a3 = Var x"   | T :: "('a, 'b) Term.term list" and S :: "('a, 'b) Term.term list" and P :: "'a \<Rightarrow> 'a list" and g :: "'a" and f :: "'a" where "a1 = Pa__" and "a2 = Fun f__ T__" and "a3 = Fun g S__" and "length T__ = length S__" and "\<And>i. i < length T__ \<Longrightarrow> term_variants_pred Pa__ (T__ ! i) (S__ ! i)" and "g \<in> set (Pa__ f__)"   | T :: "('a, 'b) Term.term list" and S :: "('a, 'b) Term.term list" and P :: "'a \<Rightarrow> 'a list" and f :: "'a" where "a1 = P" and "a2 = Fun f T" and "a3 = Fun f S" and "length T = length S" and "\<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i)"
int_pow_neg: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and i :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> - i = inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> i)"
strand_vars_unfold: fixes v :: "'c"   and S :: "('d, 'c) strand_step list" assumes "v \<in> vars\<^sub>s\<^sub>t S" shows "\<exists>S' x S''. S = S' @ x # S'' \<and> v \<in> vars\<^sub>s\<^sub>t\<^sub>p x"
ascP_Cons: fixes x :: "'b" shows "ascP ((#) x)"
Euler: fixes z :: "complex" shows "exp z = complex_of_real (exp (Re z)) * (complex_of_real (cos (Im z)) + \<i> * complex_of_real (sin (Im z)))"
bind_def: shows "bind = COERCE(udom\<cdot>'e \<rightarrow> (udom \<rightarrow> udom\<cdot>'e) \<rightarrow> udom\<cdot>'e, 'd\<cdot>'e \<rightarrow> ('d \<rightarrow> 'f\<cdot>'e) \<rightarrow> 'f\<cdot>'e)\<cdot>bindU"
hash: shows "boa \<le> vimage2p rha rha (=)"
no_meet1: fixes A :: "'c set"   and B :: "'c set" assumes "A \<inter> B = {}" shows "\<forall>a\<in>A. a \<notin> B"
le_imp_add: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.semi_canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" shows "\<exists>c. b = plus a c"
empty: shows "{b. representation B v b \<noteq> 0} = {}"
low_mds_eq_def: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2 \<equiv> \<forall>x. dma mem\<^sub>1 x = Low \<and> (x \<in> \<C> \<or> x \<notin> mds AsmNoReadOrWrite) \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
assms_3: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
Greatest_ge: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "P k" shows "k \<le> (GREATEST x. P x)"
i_drop_Suc_Cons: fixes n :: "nat"   and x :: "'d"   and f :: "nat \<Rightarrow> 'd" shows "([x] \<frown> f) \<Up> Suc n = f \<Up> n"
countable_iff_bij: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "bij_betw f A B" shows "countable A = countable B"
sbh_step_preserves_tmps_bound': fixes "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "'a memref list"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')" shows "dom \<theta> \<union> load_tmps is = dom \<theta>' \<union> load_tmps is'"
delete_order: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "0 < k"   and "bal t"   and "root_order k t" shows "root_order k (delete k x t)"
u2: shows "is_projection_on ((*\<^sub>V) (Proj M)) (space_as_set M)"
add_rel_well_defined_llbound: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b"   and z :: "'a set" assumes "x \<in> Sigma I \<FF>"   and "y \<in> Sigma I \<FF>"   and "z \<in> I"   and "z \<subseteq> fst x"   and "z \<subseteq> fst y" shows "\<forall>\<^sub>F w in llbound. \<lfloor>z, +\<^bsub>z\<^esub> (\<rho> (fst x) z (snd x)) (\<rho> (fst y) z (snd y))\<rfloor> = \<lfloor>w, +\<^bsub>w\<^esub> (\<rho> (fst x) w (snd x)) (\<rho> (fst y) w (snd y))\<rfloor>"
connected_closed_monotone_preimage: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and C :: "'b set" assumes "continuous_on S f"   and "f ` S = T"   and "\<And>C. closedin (top_of_set S) C \<Longrightarrow> closedin (top_of_set T) (f ` C)"   and "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f -` {y})"   and "connected C"   and "C \<subseteq> T" shows "connected (S \<inter> f -` C)"
2: shows "\<exists>!m. m \<in> mor \<and> m2zm = f"
Units_r_inv_ex: fixes x :: "'a" assumes "x \<in> Units R" shows "\<exists>y\<in>carrier R. x \<otimes> y = \<one>"
not_finite_existsD: fixes P :: "'b \<Rightarrow> bool" assumes "infinite {a. P a}" shows "\<exists>a. P a"
m'_props_2: shows "m = map_option \<alpha> \<circ> m'"
row_rank_eq_col_rank_rref: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "reduced_row_echelon_form A" shows "row_rank A = col_rank A"
linear_inj_bounded_below_pos: fixes f :: "'c \<Rightarrow> 'd" assumes "linear f"   and "inj f" obtains B :: "real" where "0 < B" and "\<And>x. B * norm x \<le> norm (f x)"
nonneg_cexpr_sem_marg_dens: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and x :: "nat"   and \<rho> :: "nat \<Rightarrow> val"   and v :: "val" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"   and "x \<in> set vs"   and "\<rho> \<in> space (state_measure (set vs') \<Gamma>)"   and "v \<in> type_universe (\<Gamma> x)" shows "0 \<le> extract_real (cexpr_sem (case_nat v \<rho>) (marg_dens_cexpr \<Gamma> vs x \<delta>))"
simps_2: fixes i :: "nat"   and P :: "pol" shows "neg (Pinj i P) = Pinj i (neg P)"
f_approx_nonneg: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and integrable :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> bool"   and integral :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and C :: "real"   and x :: "real" assumes "akra_bazzi_real as bs hs k x\<^sub>0 x\<^sub>1 hb e p integrable integral f g C"   and "x\<^sub>0 \<le> x" shows "0 \<le> akra_bazzi_real.f_approx x\<^sub>0 p integral g x"
surjective: fixes r :: "'a trace_scheme" shows "r = \<lparr>initial = initial r, final = final r, \<dots> = trace.more r\<rparr>"
subspace_def: fixes S :: "'b set" shows "subspace S = ((0::'b) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *s x \<in> S))"
b: shows "(c::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice) \<parallel> (i::'a::refinement_lattice)"
f: shows "trace (s init t n) (take (Suc n - n) (drop n t)) (s init t (Suc n))"
inv: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "j < i \<longrightarrow> A $$ (i, j) = (0::'a)"
IH2_4: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M2" shows "the (Mapping.lookup M2 f) < length slp2"
RedVar: fixes l :: "sfields"   and V :: "char list"   and v :: "val"   and h :: "heap"   and sh :: "sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "l V = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>Var V,(h, l, sh),b\<rangle> \<rightarrow> \<langle>Val v,(h, l, sh),b\<rangle>"
delete_edge_wf: fixes G :: "'b graph"   and v :: "'b"   and v' :: "'b" assumes "wf_graph G" shows "wf_graph (FiniteGraph.delete_edge v v' G)"
lem: fixes x :: "'b"   and xs :: "'b list" shows "x \<squnion>\<^bsub>(#)\<^esub> xs = x # xs"
vseq_implies_hpar_2: fixes \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.seq \<nu> \<mu>" shows "S.trg \<nu> = S.trg \<mu>"
wfE_min: fixes R :: "('d \<times> 'd) set"   and x :: "'d"   and Q :: "'d set" assumes "wf R"   and "x \<in> Q" obtains z :: "'d" where "z \<in> Q" and "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
single_valued_def: fixes r :: "('b \<times> 'c) set" shows "single_valued r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
ord_of_nat_equality: fixes n :: "nat" shows "ord_of_nat n = \<Squnion> ((ZFC_in_HOL.succ \<circ> ord_of_nat) ` {..<n})"
l8: shows "\<not> (- (ordinate A - ordinate B) = 0 \<and> abscissa A - abscissa B = 0 \<and> - abscissa A * (ordinate A - ordinate B) + ordinate A * (abscissa A - abscissa B) = 0)"
f_preserves': fixes as_pos :: "'b list"   and as_neg :: "'b list" assumes "f as = Some (as_pos, as_neg)" shows "as_neg = []"
assms_4: shows "f ` C \<noteq> f ` D"
acc_lengths_merge2: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list"   and k :: "nat" assumes "merge as bs us vs" shows "list.set (acc_lengths k vs) \<subseteq> list.set (acc_lengths k bs)"
lexext_rev_compat_list: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and xs :: "'b list"   and xs' :: "'b list" assumes "gt y x" shows "lexext_rev gt (xs @ y # xs') (xs @ x # xs')"
trm_tensor: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<t>[local.prod a b] = \<iota> \<cdot> local.prod \<t>[a] \<t>[b]"
cone_decomp_concat: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ss :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set list"   and pss :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list list" assumes "direct_decomp T ss"   and "length pss = length ss"   and "\<And>i. i < length ss \<Longrightarrow> cone_decomp (ss ! i) (pss ! i)" shows "cone_decomp T (concat pss)"
semilattice_order_set_axioms: shows "semilattice_order_set (\<sqinter>) (\<preceq>) (\<prec>)"
hassoc_map_rel_raw_def: shows "hassoc_map_rel_raw = br map_of (\<lambda>_. True)"
Mp_smult_1: fixes a :: "int"   and f :: "int poly" shows "smult (M a) f =m smult a f"
sample_bits_fusion: fixes v :: "nat"   and f :: "nat \<Rightarrow> 'a pmf" assumes "0 < v" shows "pmf_of_set {..<v} \<bind> (\<lambda>c. pmf_of_set UNIV \<bind> (\<lambda>b. f (2 * c + (if b then 1 else 0)))) = pmf_of_set {..<2 * v} \<bind> f"
heap_upds_ok_append: fixes \<Delta> :: "(Vars.var \<times> 'b) list"   and S :: "stack_elem list"   and \<Gamma> :: "(Vars.var \<times> 'b) list" assumes "domA \<Delta> \<inter> upds S = {}"   and "heap_upds_ok (\<Gamma>, S)" shows "heap_upds_ok (\<Delta> @ \<Gamma>, S)"
clock_monotone_1: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and r1 :: "'b state \<times> (v, v) result" assumes "evaluate_match ck env s v pes err_v r1" shows "\<forall>s' r'. r1 = (s', r') \<and> ck = True \<longrightarrow> clock s' \<le> clock s"
distinct_mapI: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'b list" assumes "distinct (map f l)" shows "distinct l"
product_run_embed_snd: fixes \<delta>\<^sub>1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and \<delta>\<^sub>2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>1 :: "'a"   and q\<^sub>2 :: "'b"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" shows "run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w i \<in> embed_transition_snd (run\<^sub>t \<delta>\<^sub>2 q\<^sub>2 w i)"
tensor_compose_distribution1: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "mat (row_length A1) (length A1) A1"   and "mat (row_length A2) (length A2) A2"   and "mat (row_length B1) (length B1) B1"   and "mat (row_length B2) (length B2) B2"   and "length A1 = row_length A2"   and "length B1 = row_length B2"   and "A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> []" shows "mat (row_length A1 * row_length B1) (length A2 * length B2) (A1 \<circ> A2 \<otimes> B1 \<circ> B2)"
assms: shows "snd ` set (pivot_positions A) \<noteq> {0..<nc}"
nocall_stermsl_max: fixes r :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and q :: "('a, 'b, 'c, 'd) seqp" assumes "r \<in> stermsl p"   and "not_call r" shows "\<not> r \<leadsto>\<^bsub>\<Gamma>\<^esub> q"
isomorphic_reflexive: fixes f :: "'a" assumes "B.ide f" shows "f \<cong>\<^sub>B f"