hcju/isabelleps huggingface dataset:
generate_valid_topology_SOME_def_alt: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "generate_valid_topology_SOME M G = delete_edges G (\<Union>m\<in>set M. if c_sinvar m G then {} else SOME F. F \<in> c_offending_flows m G)"
canon_term_pp_rel_id: fixes a :: "'t \<times> 'a"   and b :: "'t \<times> 'a" assumes "term_pp_rel (=) a b"   and "canon_term_pp_pair a"   and "canon_term_pp_pair b" shows "a = b"
frontier_subset_closed: fixes S :: "'c set" assumes "closed S" shows "frontier S \<subseteq> S"
rightUniqueSetCardinality: fixes x :: "'a"   and R :: "('a \<times> 'b) set" assumes "x \<in> Domain R"   and "runiq R" shows "card (R `` {x}) = 1"
fct_gen_subalgebra_is_subalgebra: fixes X :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "X \<in> M \<rightarrow>\<^sub>M N" shows "subalgebra M (fct_gen_subalgebra M N X)"
HInfinite_hypreal_sqrt: fixes x :: "real star" assumes "0 \<le> x"   and "x \<in> HInfinite" shows "(*f* sqrt) x \<in> HInfinite"
borel_singleton: fixes A :: "'d set"   and x :: "'d" assumes "A \<in> sets borel" shows "insert x A \<in> sets borel"
exact4_exact3: fixes M :: "('f, 'a) Module"   and N :: "('g, 'a) Module"   and z :: "'h \<Rightarrow> 'f"   and e :: "'h"   and f :: "'f \<Rightarrow> 'g"   and z1 :: "'g \<Rightarrow> 'h" assumes "R module M"   and "R module N"   and "z \<in> mHom R (Zm R e) M"   and "f \<in> mHom R M N"   and "z1 \<in> mHom R N (Zm R e)"   and "exact4 R (Zm R e) z M f N z1 (Zm R e)" shows "exact3 R (Zm R e) z M f N \<and> exact3 R M f N z1 (Zm R e)"
1: shows "goodEnv (rho [x \<leftarrow> X]_xs)"
ACI_norm_lang: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp" assumes "project \<Sigma> project" shows "project.lang \<Sigma> project lookup n \<guillemotleft>r\<guillemotright> = project.lang \<Sigma> project lookup n r"
igSwapIPresIGWls_def: fixes MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'p, 'q, 'r) model_scheme" shows "igSwapIPresIGWls MOD \<equiv> \<forall>zs z1 z2 s X. igWls MOD s X \<longrightarrow> igWls MOD s (igSwap MOD zs z1 z2 X)"
ide_dom: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "ide (local.dom f)"
pred_rel: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e list" shows "list_all P x = list_all2 (eq_onp P) x x"
nonneg: fixes x :: "'a" shows "0 \<le> supp_sum (\<lambda>V. setdist {x} (S - V)) \<C>"
get_correct: fixes s :: "'a"   and s' :: "'b list"   and n :: "nat" assumes "(s, s') \<in> arel" shows "get s n = s' ! n"
IH: fixes n :: "nat" shows "horner_sum of_bool (2::'a) (map (bit a) [0..<n]) = take_bit n a"
UNpart01_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list"   and P :: "nat \<Rightarrow> nat set set" shows "UNpart01 cl dl P \<equiv> {BrnFT cl dl} \<union> UNpart1 cl dl P"
index_I: fixes s :: "'b" assumes "s \<in> \<Union> (S ` I)" shows "index s \<in> I"
app_deterministic: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and x :: "'c"   and e :: "('a, 'b, 'c) expr"   and v :: "('a, 'b, 'c) val"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Apply (VE (Lambda x e)) (VE v)])" shows "substitution.revision_step subst r s s' = (s' = s(r \<mapsto> (\<sigma>, \<tau>, \<E> [subst (VE v) x e])))"
simps_7: fixes \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option"   and P :: "'b ptrm" shows "ptrm_infer_type \<Gamma> (PSnd P) = (case ptrm_infer_type \<Gamma> P of None \<Rightarrow> None | Some TUnit \<Rightarrow> None | Some (TVar xa) \<Rightarrow> Map.empty xa | Some (TArr type1 xa) \<Rightarrow> Map.empty xa | Some (TPair \<tau> xa) \<Rightarrow> Some xa)"
qgood_qXyx: shows "\<forall>y. qGood (pick X #[[y \<and> x]]_xs)"
domain_unique: fixes a :: "'a"   and f :: "'a"   and a' :: "'a" assumes "a \<in> domains f"   and "a' \<in> domains f" shows "a = a'"
ae_geq: shows "(AE w in M. 0 \<le> cls_val_process Mkt2 p m w) = (AE w in M. 0 \<le> cls_val_process Mkt p m w)"
CarAudioSystem_def: shows "CarAudioSystem = (PseudoHA Root_CTRL (LiftInitData [V0 0, V1 0]) [++] (''CarAudioSystem'', CDPlayer_CTRL) [++] (''CarAudioSystem'', AudioPlayer_CTRL) [++] (''On'', TunerMode_CTRL) [++] (''On'', CDMode_CTRL))"
rat_fun_lem: fixes F :: "'b filter"   and l :: "real"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real"   and c :: "real"   and d :: "real"   and u :: "real" assumes "F \<noteq> \<bottom>"   and "0 < l"   and "\<forall>\<^sub>F n in F. l \<le> f n / g n"   and "\<forall>\<^sub>F n in F. (f n + c) / (g n + d) \<le> u"   and "filterlim g at_top F" shows "l \<le> u"
discr_finished: fixes c :: "('test, 'atom, 'choice) cmd" assumes "proper c"   and "finished c" shows "discr c"
xsx1_last: shows "(xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1)\<lparr>vcard xs\<^sub>1\<rparr> = x\<^sub>1"
cases: fixes x :: "('a \<times> 'c) list \<times> 'o" obtains xs :: "('a \<times> 'c) list" and ko :: "'o" where "x = (xs, ko)"
distributed_borel_measurable': fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> ennreal"   and h :: "'e \<Rightarrow> 'd"   and L :: "'e measure" assumes "distributed M N X f"   and "h \<in> L \<rightarrow>\<^sub>M N" shows "(\<lambda>x. f (h x)) \<in> borel_measurable L"
Int_lower2: fixes A :: "'e set"   and B :: "'e set" shows "A \<inter> B \<subseteq> B"
acc_point_cres: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "padic_integers p"   and "s \<in> carrier padic_int p\<^bsup>\<omega>\<^esup>" shows "acc_point s (Suc k) = const_res (Suc k) (res_seq s k)"
simps: fixes cstate :: "(call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a) discr" shows "\<aC>\<cdot>cstate = (case undiscr cstate of (App lab f vs, \<beta>, ve, b) \<Rightarrow> let fs = \<aA> f \<beta> ve; as = map (\<lambda>v. \<aA> v \<beta> ve) vs; b' = \<anb> b lab in (\<Union>f'\<in>fs. \<aF>\<cdot>(Discr (f', as, ve, b'))) \<union> {((lab, \<beta>), f') |f'. f' \<in> fs} | (call.Let lab ls c', \<beta>, ve, b) \<Rightarrow> let b' = \<anb> b lab; \<beta>' = \<beta>(lab \<mapsto> b'); ve' = ve \<union>. \<Union>.map (\<lambda>(v, l). { (v, b') := \<aA> (L l) \<beta>' ve}.) ls in \<aC>\<cdot>(Discr (c', \<beta>', ve', b')))"
slg_rel_sv: fixes R :: "('a \<times> 'b) set" assumes "single_valued R"   and "Range R = UNIV" shows "single_valued (\<langle>R\<rangle>slg_rel)"
V: shows "(f \<inter> V \<times> UNIV) `` V \<subseteq> vertices (S (max i j))"
de_Morgan_conj: fixes x :: "'a"   and y :: "'a" shows "NOT (x AND y) = NOT x OR NOT y"
nat_eq_numeral_power_cancel_iff: fixes y :: "int"   and x :: "num"   and n :: "nat" shows "(nat y = numeral x ^ n) = (y = numeral x ^ n)"
lookup_lift_poly_fun_homogenous: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and f :: "('b \<Rightarrow>\<^sub>0 'd) \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'd"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair"   and "f 0 = 0" shows "lookup (term_powerprod.lift_poly_fun pair_of_term term_of_pair f p) v = lookup (f (term_powerprod.proj_poly term_of_pair (term_powerprod.component_of_term pair_of_term v) p)) (term_powerprod.pp_of_term pair_of_term v)"
case_sum_inject: fixes f1 :: "'c \<Rightarrow> 'e"   and f2 :: "'d \<Rightarrow> 'e"   and g1 :: "'c \<Rightarrow> 'e"   and g2 :: "'d \<Rightarrow> 'e" assumes "case_sum f1 f2 = case_sum g1 g2" obtains "f1 = g1" and "f2 = g2"
fun_dom_scene_empty: shows "fds {} = \<bottom>\<^sub>S"
th': shows "\<forall>x\<in>set (concat (map (\<lambda>b. map (Pair b) [1..snd (snd (unit p))]) (fst (snd (unit p))))). bound0 (case x of (b, j) \<Rightarrow> simpfm (subst0 (Add b (C j)) (fst (unit p))))"
assms_3: shows "C\<^sub>1 \<subseteq> C\<^sub>2"
srtc_symmetric: fixes a :: "'a"   and b :: "'a" assumes "a \<leftrightarrow>\<^sup>* b" shows "b \<leftrightarrow>\<^sup>* a"
Gk_def: shows "Gk = mat k n (\<lambda>(i, j). fs ! i $ j)"
cSup_insert_If: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and a :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_above less_eq X" shows "Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
locally_compact_compact_subopen: fixes S :: "'b set" shows "locally compact S = (\<forall>K T. K \<subseteq> S \<and> compact K \<and> open T \<and> K \<subseteq> T \<longrightarrow> (\<exists>U V. K \<subseteq> U \<and> U \<subseteq> V \<and> U \<subseteq> T \<and> V \<subseteq> S \<and> openin (top_of_set S) U \<and> compact V))"
preserves_arr: fixes f :: "(bool, bool list) concrete_category.arr" assumes "J.arr f" shows "C.arr (D.map f)"
append: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "monoid_list f z" shows "monoid_list.F f z (xs @ ys) = f (monoid_list.F f z xs) (monoid_list.F f z ys)"
emeasure_eq_0_AE: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a measure" assumes "AE x in M. \<not> P x" shows "emeasure M {x \<in> space M. P x} = 0"
subset_diff: fixes B :: "'b set"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "B \<subseteq> A"   and "finite A" shows "sum g A = sum g (A - B) + sum g B"
LLL_inv_wI: fixes fs :: "int vec list" assumes "set fs \<subseteq> carrier_vec n"   and "length fs = m"   and "lattice_of fs = L"   and "gs.lin_indpt_list (map of_int_hom.vec_hom fs)" shows "LLL_invariant_weak fs"
hom_functor_axioms: shows "hom_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>S) \<phi>D"
Red_I_\<G>_q_def: fixes q :: "'q"   and N :: "'f set" shows "Red_I_\<G>_q q N = {\<iota> \<in> Inf_F. \<G>_I_q q \<iota> \<noteq> None \<and> the (\<G>_I_q q \<iota>) \<subseteq> Red_I_q q (\<G>_Fset_q q N) \<or> \<G>_I_q q \<iota> = None \<and> \<G>_F_q q (concl_of \<iota>) \<subseteq> \<G>_Fset_q q N \<union> Red_F_q q (\<G>_Fset_q q N)}"
prem: shows "fv (as @ \<Gamma>, body, S) \<subseteq> set L \<union> domA (as @ \<Gamma>)"
cr_ennreal_def: shows "cr_ennreal \<equiv> \<lambda>x y. x = enn2ereal y"
fresh_star_Cons: fixes as :: "atom set"   and x :: "'a"   and xs :: "'a list" shows "as \<sharp>* (x # xs) = (as \<sharp>* x \<and> as \<sharp>* xs)"
trans: fixes b :: "'a word"   and a :: "'a word"   and c :: "'a word" assumes "b \<le>s a"   and "c \<le>s b" shows "c \<le>s a"
older_seniors_psubset_card_mono: fixes A :: "nat set"   and x :: "nat"   and n :: "nat" assumes "A \<subset> older_seniors x n" shows "card A < card (older_seniors x n)"
k_def: shows "k = length (filter (\<lambda>i. f i = 0) ys)"
poly_primes: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and p :: "'b list"   and q :: "'b list" assumes "class.idom times one plus zero minus uminus" shows "semiring_0.divides plus zero times [a, one] (semiring_0.pmult plus zero times p q) = (semiring_0.divides plus zero times [a, one] p \<or> semiring_0.divides plus zero times [a, one] q)"
0: shows "rowvector w *v (sr *s y - A *v y) = 0"
add_tvars'_def: shows "add_tvars' = fold_types add_tvarsT'"
sqrt_nat_iff_is_square: fixes n :: "nat" shows "(sqrt (real n) \<in> \<nat>) = is_square n"
frechet_derivative_id: fixes a :: "'a" shows "frechet_derivative id (at a) = id"
00: shows "\<forall>n\<in>num. prv (neg (subst \<phi> n x))"
disconnected_nodes_h': shows "h' \<turnstile> get_disconnected_nodes owner_document \<rightarrow>\<^sub>r child # disconnected_nodes_h"
diff_prime_power_imp_coprime: fixes p :: "nat"   and q :: "nat"   and n :: "nat"   and m :: "nat" assumes "p \<noteq> q"   and "normalization_semidom_class.prime p"   and "normalization_semidom_class.prime q" shows "coprime (p ^ n) (q ^ m)"
div_less_dividend: fixes n :: "nat"   and m :: "nat" assumes "1 < n"   and "0 < m" shows "m div n < m"
intro: fixes M1 :: "'b measure"   and M2 :: "'c measure" assumes "sigma_finite_measure M1"   and "sigma_finite_measure M2" shows "pair_sigma_finite M1 M2"
sel_3: fixes even :: "bool"   and t :: "'a tree" shows "right (even_odd_mirror even t) = even_odd_mirror (\<not> even) (if even then left t else right t)"
pick: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "R\<^sup>+\<^sup>+ x y" shows "chain R (llist_of (x # ??.Lazy_List_Chain.pick R x y @ [y]))"
cmp_rfn4: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set"   and l :: "nat" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g"   and "l \<le> s * r - Suc 0" shows "cmp_rfn G r f s g (Suc l) \<subseteq> cmp_rfn G r f s g l"
FINITE_ALL_DISTINCT_LISTS: fixes P :: "'c set" assumes "finite P" shows "finite {p. distinct p \<and> set p \<subseteq> P}"
one_element_vec_def: fixes n :: "nat"   and e :: "'a" shows "one_element_vec n e = vec n (\<lambda>i. e)"
measurable_measure_pmf: fixes M :: "'c \<Rightarrow> 'd pmf" shows "(\<lambda>x. measure_pmf (M x)) \<in> count_space UNIV \<rightarrow>\<^sub>M subprob_algebra (count_space UNIV)"
is_bintree_list_Cons: fixes l :: "nat"   and ts :: "('c, 'd) bintree list"   and t :: "('c, 'd) bintree" assumes "is_bintree_list l ts"   and "is_bintree_list l (children t)" shows "is_bintree_list (Suc l) (t # ts)"
assms_1: shows "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"
domNMT: fixes X :: "'b \<Rightarrow> 'c option" assumes "dom X \<noteq> {}" shows "X \<noteq> \<emptyset>"
int_eq_iff_numeral: fixes m :: "nat"   and v :: "num" shows "(int m = numeral v) = (m = numeral v)"
xyz_2: shows "(Y, Z) \<in> mult2_alt b2 ns s"
nonpos_Ints_altdef: shows "\<int>\<^sub>\<le>\<^sub>0 = {n \<in> \<int>. n \<le> (0::'a)}"
Fence: fixes \<O>s :: "nat set list"   and i :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" shows "\<O>s,i\<turnstile> (Fence # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
sphere_retract_of_punctured_universe_gen: fixes b :: "'a"   and a :: "'a"   and r :: "real" assumes "b \<in> ball a r" shows "sphere a r retract_of - {b}"
sorted_ins_list: fixes xs :: "'b list"   and x :: "'b" assumes "sorted_less xs" shows "sorted_less (ins_list x xs)"
emeasure_lborel_ball_finite: fixes c :: "'b"   and r :: "real" shows "emeasure lborel (ball c r) < \<infinity>"
rel_mono_strong: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g list"   and y :: "'h list"   and Ra :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "list_all2 R x y"   and "\<And>z yb. \<lbrakk>z \<in> set x; yb \<in> set y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "list_all2 Ra x y"
struct_spec_f4: shows "punit.struct_spec punit.f4_sel punit.add_pairs_canon punit.add_basis_canon punit.f4_red"
fun_below_iff: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f \<sqsubseteq> g) = (\<forall>x. f x \<sqsubseteq> g x)"
assms: shows "w\<^sub>1 =\<^sub>I w\<^sub>2"
vwalk_arcs_set_nil: fixes x :: "'a \<times> 'a"   and p :: "'a list" assumes "x \<in> set (vwalk_arcs p)" shows "p \<noteq> []"
assms_2: shows "(f, h) \<in> [P']\<^sub>a R \<rightarrow> S"
mult_subgroup_generated: fixes G :: "('e, 'f) monoid_scheme"   and S :: "'e set" shows "(\<otimes>\<^bsub>subgroup_generated G S\<^esub>) = (\<otimes>\<^bsub>G\<^esub>)"
intros_2: fixes c1 :: "'c \<Rightarrow> 'd"   and l :: "('c, 'd) label"   and c2 :: "'c \<Rightarrow> 'd"   and ls :: "('c, 'd) label list"   and c3 :: "'c \<Rightarrow> 'd" assumes "ce c1 l c2"   and "ce_star c2 ls c3" shows "ce_star c1 (l # ls) c3"
singular_chain_minus: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_chain p X (- c) = singular_chain p X c"
add_mono_thms_linordered_semiring_2: fixes i :: "'a"   and j :: "'a"   and k :: "'a"   and l :: "'a" assumes "i = j \<and> k \<le> l" shows "i + k \<le> j + l"
overlapping_var1I: fixes name :: "name"   and t :: "term" shows "overlapping (Free name) t"
prio_emb_step_def: fixes prio :: "dir list \<Rightarrow> nat"   and t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "prio_emb_step prio t s = emb_step_at' (prio_emb_pos prio t s) t"
invariantsHoldInInitialState: fixes state :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" assumes "isInitialState state F0" shows "invariantsHoldInState state F0 decisionVars"
linorder_from_keys_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "linorder_from_keys A f = {(x, y). (x, y) \<in> A \<times> A \<and> f x \<le> f y}"
qfree_lift_dnf_qe: fixes aneg :: "'c \<Rightarrow> 'c fm"   and anormal :: "'c \<Rightarrow> bool"   and I\<^sub>a :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and depends\<^sub>0 :: "'c \<Rightarrow> bool"   and decr :: "'c \<Rightarrow> 'c"   and qe :: "'c list \<Rightarrow> 'c fm"   and \<phi> :: "'c fm" assumes "ATOM aneg anormal I\<^sub>a depends\<^sub>0 decr"   and "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> qfree (qe as)" shows "qfree (ATOM.lift_dnf_qe aneg depends\<^sub>0 decr qe \<phi>)"
prj_chine_assoc_3: shows "\<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<^sub>0 \<cdot> \<mu>_\<nu>_\<pi>\<rho>.chine_assoc = \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0"
list_of_oalist_tc_inverse: fixes x :: "('a, 'b) oalist_tc" shows "Abs_oalist_tc (list_of_oalist_tc x) = x"
rsquarefree_def': fixes p :: "'c poly" shows "rsquarefree p = (p \<noteq> 0 \<and> (\<forall>a. order a p \<le> 1))"
approx_approx_zero_iff: fixes x :: "complex star" shows "(x \<approx> 0) = (hcmod x \<approx> 0)"
exhaust: fixes y :: "\<nu>" obtains x1 :: "\<omega>" where "y = \<omega>\<nu> x1"   | x2 :: "\<Pi>\<^sub>1 set" where "y = \<alpha>\<nu> x2"
IH: shows "card (pderivs_lang UNIV1 r) \<le> awidth r"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and v :: "'d llist" shows "lset (lmap f v) = f ` lset v"
max_decr: fixes X :: "nat set"   and k :: "nat" assumes "\<exists>x\<in>X. k \<le> x"   and "finite X" shows "(MAX x\<in>X. x - k) = Max X - k"
red_single_plus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "red_single p q f t" shows "red_single (p + r) (q + r) f t \<or> red_single (q + r) (p + r) f t \<or> (\<exists>s. red_single (p + r) s f t \<and> red_single (q + r) s f t)"
get_global: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (global \<C>)"   and "tg_t (global \<C> ! i) = t" shows "\<C> \<turnstile> [Get_global i] : [] _> [t]"
rows_mat_of_rows: fixes vs :: "'a vec list"   and n :: "nat" assumes "set vs \<subseteq> carrier_vec n" shows "rows (mat_of_rows n vs) = vs"
P1: shows "A0 \<noteq> B0 \<and> C0 \<noteq> B0 \<and> (\<forall>X Y Z. A0 B0 C0 CongA X Y Z \<longrightarrow> a X Y Z)"
rcResB: fixes P' :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "y \<noteq> a" shows "F (<\<nu>y>P')"
entt_star_true_simp_2: fixes A :: "assn"   and B :: "assn" shows "(A * true \<Longrightarrow>\<^sub>t B) = (A \<Longrightarrow>\<^sub>t B)"
max_Var_floatarith_le_max_Var_floatariths_nthI: fixes n :: "nat"   and fas :: "floatarith list"   and z :: "nat" assumes "n < length fas"   and "max_Var_floatariths fas \<le> z" shows "max_Var_floatarith (fas ! n) \<le> z"
fresh_star_list: fixes xs :: "'d list"   and c :: "'e" shows "xs \<sharp>* c \<equiv> \<forall>x\<in>set xs. x \<sharp> c"
not_less_iff_gr_or_eq: fixes x :: "'b Word.word"   and y :: "'b Word.word" shows "(\<not> x <s y) = (y <s x \<or> x = y)"
greek_less_rempty: fixes a :: "(accent \<times> 'a) list"   and r :: "('a \<times> 'a) set" shows "((a, []) \<in> greek_less r) = False"
d: shows "(x * e1 + (q - s1)) mod q = (x * y + x * q - x * b + (q * q + q * q + q - x * y - x * (q - b) - (q - r))) mod q"
compositionality: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and h :: "'e \<Rightarrow> 'c"   and i :: "'f \<Rightarrow> 'd"   and prod :: "'e \<times> 'f" shows "map_prod f g (map_prod h i prod) = map_prod (f \<circ> h) (g \<circ> i) prod"
assms_1: shows " t \<longrightarrow>\<^sub>1 t1"
join_cycle_free: fixes g\<^sub>1 :: "'b ugraph"   and g\<^sub>2 :: "'b ugraph"   and u :: "'b"   and v :: "'b" assumes "cycle_free g\<^sub>1"   and "cycle_free g\<^sub>2"   and "nodes g\<^sub>1 \<inter> nodes g\<^sub>2 = {}"   and "u \<in> nodes g\<^sub>1"   and "v \<in> nodes g\<^sub>2" shows "cycle_free (ins_edge (u, v) (graph_join g\<^sub>1 g\<^sub>2))"
islimpt_ball: fixes y :: "'a"   and x :: "'a"   and e :: "real" shows "y islimpt ball x e = (0 < e \<and> y \<in> cball x e)"
pmf_set_map: fixes f :: "'a \<Rightarrow> 'b" shows "set_pmf \<circ> map_pmf f = (`) f \<circ> set_pmf"
tha: shows "det (\<chi>i. if i \<in> insert z T then sum (a i) S else c i) = (\<Sum>(j, f)\<in>S \<times> {f. (\<forall>i\<in>T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}. det (\<chi>i. if i \<in> T then a i (f i) else if i = z then a i j else c i))"
Qg_tassoc: fixes N :: "'a set" assumes "G \<triangleright> N" shows "\<forall>X\<in>set_rcs G N. \<forall>Y\<in>set_rcs G N. \<forall>Z\<in>set_rcs G N. c_top G N X (c_top G N Y Z) = c_top G N (c_top G N X Y) Z"
mult_less_cancel2: fixes m :: "nat"   and k :: "nat"   and n :: "nat" shows "(m * k < n * k) = (0 < k \<and> m < n)"
induct: fixes P :: "'b upD \<Rightarrow> 'b \<Rightarrow> 'b tree23 \<Rightarrow> 'b \<Rightarrow> 'b tree23 \<Rightarrow> bool"   and a0 :: "'b upD"   and a1 :: "'b"   and a2 :: "'b tree23"   and a3 :: "'b"   and a4 :: "'b tree23" assumes "\<And>t1 a t2 b t3. P (TD t1) a t2 b t3"   and "\<And>t1 a t2 b t3 c t4. P (UF t1) a \<langle>t2, b, t3\<rangle> c t4"   and "\<And>t1 a t2 b t3 c t4 d t5. P (UF t1) a \<langle>t2, b, t3, c, t4\<rangle> d t5"   and "\<And>v b d e. P (UF v) b \<langle>\<rangle> d e" shows "P a0 a1 a2 a3 a4"
b_assn_subtyping_match_tR: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and y :: "'b"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "P x \<Longrightarrow> hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y" shows "hn_ctxt (b_assn A P) x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y"
setops\<^sub>s\<^sub>s\<^sub>t_mono: fixes M :: "('d, 'e) stateful_strand_step list"   and N :: "('d, 'e) stateful_strand_step list" assumes "set M \<subseteq> set N" shows "setops\<^sub>s\<^sub>s\<^sub>t M \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t N"
poly_rel_def: fixes x :: "'i list"   and x' :: "'a mod_ring poly" shows "poly_rel x x' = list_all2 R x (coeffs x')"
P_subset: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "P x \<subseteq> P y"
irreflI: fixes R :: "('b \<times> 'b) set" assumes "\<And>a. (a, a) \<notin> R" shows "irrefl R"
to_st_cons: fixes a :: "('fun, 'var) extstrand_step"   and B :: "('fun, 'var) extstrand_step list" shows "to_st (a # B) = to_st [a] @ to_st B"
times_tree_def: fixes x :: "'b tree"   and y :: "'b tree" shows "x * y = pure (*) \<diamondop> x \<diamondop> y"
max_job_lower_bound_makespan: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "lb T A j" shows "Max\<^sub>0 (t ` {1..j}) \<le> makespan T"