hcju/isabelleps huggingface dataset:
eq_integralD: fixes k :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "integral k f = y" shows "(f has_integral y) k \<or> \<not> f integrable_on k \<and> y = (0::'a)"
has_vderiv_eq: fixes f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'a"   and T :: "real set" assumes "(f has_vderiv_on f') S"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x"   and "\<And>x. x \<in> S \<Longrightarrow> f' x = g' x"   and "S = T" shows "(g has_vderiv_on g') T"
extp_ext_eq: fixes R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M :: "'b"   and N :: "'b" shows "extp (\<lambda>x. x \<in> R) (\<lambda>x xa. (x, xa) \<in> R') M N = (\<lambda>x. x \<in> ext R R' M N)"
set_map0_1: fixes f1 :: "'l \<Rightarrow> 'o"   and f2 :: "'m \<Rightarrow> 'p"   and f3 :: "'n \<Rightarrow> 'q" shows "F'set3 \<circ> F'map f2 f3 f1 = (`) f1 \<circ> F'set3"
that: fixes b :: "'a" assumes "b \<in> set bs" shows "b dvd a"
lm_to_map_vardict_inv: fixes xs :: "(String.literal \<times> variable) list"   and ss :: "(integer \<times> edge list) iarray"   and proc :: "nat \<times> edgeIndex \<times> procArg list \<times> varDecl list" assumes "\<forall>(k, v)\<in>set xs. k \<in> process_names ss proc \<and> variable_inv v" shows "vardict_inv ss proc (lm.to_map xs)"
sum_negf: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(\<Sum>x\<in>A. - f x) = - sum f A"
2: shows "(g', p') = upd (lm.update k v') (g, p)"
prime_factorization_times_prime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and p :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "normalization_semidom.prime zero times one normalize p" shows "factorial_semiring.prime_factorization zero times one normalize (times p x) = {#p#} + factorial_semiring.prime_factorization zero times one normalize x"
head_\<omega>_1: shows "head_\<omega> 1 = 1"
C_def: shows "C = {c. Marker \<in> set (msgs (coS t j) c)}"
roundup_le_ub: fixes emBits :: "nat" assumes "176 + sLen \<le> emBits"   and "emBits \<le> 2 ^ 32 * 160" shows "roundup emBits 8 * 8 - 168 \<le> 2 ^ 32 * 160"
alphaAbsGood_equivP: shows "equivP qGoodAbs alphaAbsGood"
valid_before: shows "valid_mmsaux args cur (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) ys"
pmf_bind: fixes N :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf"   and i :: "'c" shows "pmf (N \<bind> f) i = measure_pmf.expectation N (\<lambda>x. pmf (f x) i)"
of_complex_eq_0_iff: fixes x :: "complex" shows "(of_complex x = (0::'b)) = (x = 0)"
trie_update_def: fixes k :: "'b list"   and v :: "'c"   and t :: "('b, 'c) trie" shows "trie_update k v t = trie_update_with' k t v (\<lambda>v'. v)"
exp_minus_inverse: fixes x :: "'b" shows "exp x * exp (- x) = (1::'b)"
A1_def: shows "A1 = {x. jumpF f (at_right x) \<noteq> 0 \<and> a \<le> x \<and> x < b}"
s3: shows "ccw'.sortedP 0 (u # v # (w + y) # ys)"
eval_Callee: fixes G :: "prog"   and s :: "st"   and l :: "lname \<Rightarrow> val option"   and e :: "expr"   and v :: "val"   and s' :: "state" shows "G\<turnstile>Norm s \<midarrow>Callee l e-\<succ>v\<rightarrow> s' = False"
assms_2: shows "ide w \<and> \<guillemotleft>\<theta> : f' \<star> w \<Rightarrow> f\<guillemotright> \<and> \<guillemotleft>\<nu> : g \<Rightarrow> g' \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (r \<star> \<theta>) \<cdot> \<a>[r, f', w] \<cdot> (\<rho>' \<star> w) \<cdot> \<nu> = \<rho>"
order_tendsto_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c"   and F :: "'d filter" assumes "class.order_topology less_eq less open" shows "filterlim f (topological_space.nhds open x) F = ((\<forall>l. less l x \<longrightarrow> (\<forall>\<^sub>F x in F. less l (f x))) \<and> (\<forall>u. less x u \<longrightarrow> (\<forall>\<^sub>F x in F. less (f x) u)))"
check_code_def: shows "check_code = PivotUpdateMinVars.check eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update_code"
zero_le_pprt: fixes a :: "'a" shows "(0::'a) \<le> pprt a"
Eps_cong: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "Eps P = Eps Q"
Gamma_series_def: fixes z :: "'b"   and n :: "nat" shows "Gamma_series z n = fact n * exp (z * of_real (ln (real n))) / pochhammer z (n + 1)"
2: shows "Sigma_Algebra.measure (measure_spmf p) {x. bad1 x} = Sigma_Algebra.measure (measure_spmf q) {y. bad2 y}"
simps_2: fixes c :: "'b \<Rightarrow> 'b \<Rightarrow> 'a + unit"   and x :: "'b"   and xs :: "'b list" shows "check_pairwise c (x # xs) = (try forallM (c x) xs catch (\<lambda>x. Inl (snd x))) \<bind> (\<lambda>_. check_pairwise c xs)"
preserves_equivalence_maps: fixes f :: "'c" assumes "C.equivalence_map f" shows "D.equivalence_map (F f)"
path_eventually_conforms_to_\<sigma>_map_n: fixes P :: "'a llist" assumes "lset P \<subseteq> winning_region p"   and "valid_path P"   and "path_conforms_with_strategy p P well_ordered_strategy" shows "\<exists>n. path_conforms_with_strategy p (ldropn n P) (path_strategies P $ n)"
simple_bochner_integral_def: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" shows "simple_bochner_integral M f = (\<Sum>y\<in>f ` space M. Sigma_Algebra.measure M {x \<in> space M. f x = y} *\<^sub>R y)"
region_set'_eq_1: fixes R :: "('c \<Rightarrow> real) set"   and u :: "'c \<Rightarrow> real"   and l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "R \<in> \<R>"   and "u \<in> R"   and "A \<turnstile> l \<longrightarrow>\<^bsup>g,\<mu>,X\<^esup> l'" shows "[[(SOME r. set r = X)\<rightarrow>0]u]\<^sub>\<R> = [X::=0]R"
2: shows "C.commutative_square \<nu>.leg0 \<pi>\<rho>.leg1 \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0\<^sub>1 \<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0"
assms: shows "wffp (sl1 @ [Idle st1] @ [State st2 u2 c2] @ sl2)"
LeastI2_ex: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>a. P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (ord.Least less_eq P)"
conway_assm1_def: fixes x :: "('c, 'd) boffa_mon" shows "conway_assm1 x = (\<forall>i j. x\<^bsub>i\<^esub> \<cdot> x\<^bsub>j\<^esub> \<le> x\<^bsub>i \<cdot> j\<^esub>)"
cases_Neg_iff: fixes P :: "Formula.formula \<Rightarrow> bool"   and \<phi> :: "Formula.formula" shows "(case \<phi> of formula.Neg \<psi> \<Rightarrow> P \<psi> | _ \<Rightarrow> False) = (\<exists>\<psi>. \<phi> = formula.Neg \<psi> \<and> P \<psi>)"
automgroupTr1: fixes G :: "('e, 'f) Group_scheme"   and f :: "'e \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'e"   and h :: "'e \<Rightarrow> 'e" assumes "Group G"   and "gbij\<^bsub>G,G\<^esub> f"   and "gbij\<^bsub>G,G\<^esub> g"   and "gbij\<^bsub>G,G\<^esub> h" shows "h \<circ>\<^bsub>G\<^esub> g \<circ>\<^bsub>G\<^esub> f = h \<circ>\<^bsub>G\<^esub> (g \<circ>\<^bsub>G\<^esub> f)"
DirProd_iso_trans: fixes G :: "('b, 'c) monoid_scheme"   and G2 :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and I :: "('h, 'i) monoid_scheme" assumes "Group.group G"   and "G \<cong> G2"   and "H \<cong> I" shows "G \<times>\<times> H \<cong> G2 \<times>\<times> I"
cases: fixes a1 :: "val"   and a2 :: "val" assumes "a1 \<sqsubseteq> a2" obtains n :: "nat" where "a1 = VNat n" and "a2 = VNat n"   | t1 :: "(val \<times> val) fset" and t2 :: "(val \<times> val) fset" where "a1 = VFun t1" and "a2 = VFun t2" and "fset t1 \<subseteq> fset t2"
simps: fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and f :: "'c \<Rightarrow> ennreal"   and gpv :: "('c, 'a, 'b) gpv" shows "expectation_gpv fail \<I> f gpv = \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. expectation_gpv fail \<I> f (c r)) \<partial>measure_spmf (the_gpv gpv) + fail * ennreal (pmf (the_gpv gpv) None)"
orderI: fixes a :: "'b"   and b :: "'b" assumes "a = min a b" shows "a \<le> b"
Smith_1xn_def: fixes A :: "'a mat" shows "Smith_1xn A = (if dim_col A = 0 then (A, 1\<^sub>m (dim_col A)) else Smith_1xn_aux (dim_col A - 1) A (A, 1\<^sub>m (dim_col A)))"
filter_compliant_stateful_ACS_def: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and Es :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS G M Es = TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS_accu G M [] Es"
surjI: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. g (f x) = x" shows "top_pres (\<P> g)"
butlast_conv_take: fixes xs :: "'f list" shows "butlast xs = take (length xs - 1) xs"
insert_ops_exist: fixes xs :: "('b \<times> 'b option) list" assumes "rga_ops xs" shows "\<exists>ys. set xs = set ys \<and> insert_ops ys"
minus_eq: fixes x :: "'c"   and y :: "'c" shows "x \<ominus>\<^bsub>V\<^esub> y = x \<oplus>\<^bsub>V\<^esub> \<ominus>\<^bsub>V\<^esub> y"
is_scalar_pp_def: shows "is_scalar = (\<lambda>_. True)"
total_def: fixes ES :: "'a ES_rec"   and E :: "'a set" shows "EventSystems.total ES E \<equiv> E \<subseteq> E\<^bsub>ES\<^esub> \<and> (\<forall>\<tau>\<in>Tr\<^bsub>ES\<^esub>. \<forall>e\<in>E. \<tau> @ [e] \<in> Tr\<^bsub>ES\<^esub>)"
simulation_jAction_eq: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and M :: "('b, 'c, 'e) KripkeStructure"   and M' :: "('b, 'c, 'f) KripkeStructure"   and f :: "'e \<Rightarrow> 'f"   and w :: "'e" assumes "JKBP jkbp"   and "kripke M"   and "sim M M' f"   and "w \<in> worlds M" shows "JKBP.jAction jkbp M w = JKBP.jAction jkbp M' (f w)"
map: shows "G\<^sub>2 = map_ugraph f G\<^sub>1"
xeq1_sym': fixes x :: "Value"   and s :: "Store"   and t :: "Store" assumes "s \<equiv>[x] t" shows "t \<equiv>[x] s"
hom_dvd_iff: fixes p :: "'a poly"   and q :: "'a poly" shows "(map_poly hom p dvd map_poly hom q) = (p dvd q)"
intro: fixes G :: "('c, 'd) pre_digraph"   and S :: "'c set" assumes "wf_digraph G"   and "max_reachable_set_axioms G S" shows "max_reachable_set G S"
quantale_homset_iff: fixes f :: "'c \<Rightarrow> 'd" shows "(f \<in> quantale_homset) = (comp_pres f \<and> Sup_pres f)"
no_overlaps_lroft_s3_hlp_hlp: fixes amr :: "('b \<times> 32 simple_match \<times> char list \<times> simple_action) list"   and ab :: "of_match_field set"   and p :: "(32, 'c) simple_packet_ext_scheme"   and ad :: "of_match_field set"   and ba :: "of_action list"   and bb :: "of_action list"   and ac :: "'b"   and ifs :: "char list list" assumes "distinct (map fst amr)"   and "OF_match_fields_unsafe ab p"   and "ab \<noteq> ad \<or> ba \<noteq> bb"   and "OF_match_fields_unsafe ad p"   and "(ac, ab, ba) \<in> set (lr_of_tran_s3 ifs amr)"   and "(ac, ad, bb) \<in> set (lr_of_tran_s3 ifs amr)" shows "False"
assms: shows "\<nu> \<star> \<mu> \<noteq> null"
refine_iter: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<omega> \<sqsubseteq> c\<^sup>\<star>"
simps: fixes w1 :: "32 Word.word"   and w2 :: "32 Word.word" shows "wtimes w1 w2 = (if w1 = POS_INF \<and> w2 = POS_INF then POS_INF else if w1 = NEG_INF \<and> w2 = POS_INF then NEG_INF else if w1 = POS_INF \<and> w2 = NEG_INF then NEG_INF else if w1 = NEG_INF \<and> w2 = NEG_INF then POS_INF else if w1 = POS_INF \<and> w2 <s 0 then NEG_INF else if w1 = POS_INF \<and> 0 <s w2 then POS_INF else if w1 = POS_INF \<and> 0 = w2 then 0 else if w1 = NEG_INF \<and> w2 <s 0 then POS_INF else if w1 = NEG_INF \<and> 0 <s w2 then NEG_INF else if w1 = NEG_INF \<and> 0 = w2 then 0 else if w1 <s 0 \<and> w2 = POS_INF then NEG_INF else if 0 <s w1 \<and> w2 = POS_INF then POS_INF else if 0 = w1 \<and> w2 = POS_INF then 0 else if w1 <s 0 \<and> w2 = NEG_INF then POS_INF else if 0 <s w1 \<and> w2 = NEG_INF then NEG_INF else if 0 = w1 \<and> w2 = NEG_INF then 0 else let prod = SCAST(32 \<rightarrow> 64) w1 * SCAST(32 \<rightarrow> 64) w2 in if prod \<le>s SCAST(32 \<rightarrow> 64) NEG_INF then NEG_INF else if SCAST(32 \<rightarrow> 64) POS_INF \<le>s prod then POS_INF else SCAST(64 \<rightarrow> 32) prod)"
IH_1: fixes u :: "('a \<times> bool) list"   and ass :: "'a \<Rightarrow> bool"   and ass' :: "'a \<Rightarrow> bool" assumes "ifex_no_twice t"   and "ifex_sat_list t = Some u"   and "ass = update_assignment u ass'" shows "val_ifex t ass = True"
starfun_exp_ge_add_one_self: fixes x :: "real star" assumes "0 \<le> x" shows "1 + x \<le> (*f* exp) x"
commute: fixes a :: "'k multiset"   and b :: "'k multiset" shows "a \<union># b = b \<union># a"
inject_1: fixes x2 :: "'g"   and y2 :: "'g" shows "(sum_bot.Left x2 = sum_bot.Left y2) = (x2 = y2)"
at_within_union: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open" shows "topological_space.at_within open x (S \<union> T) = sup (topological_space.at_within open x S) (topological_space.at_within open x T)"
mset_lt_single_right_iff: fixes M :: "'b multiset"   and y :: "'b" shows "(M < {#y#}) = (\<forall>x\<in>#M. x < y)"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'j" assumes "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| element_ptr_kinds h"
ex_img: shows "\<exists>img. set_category_given_img (\<cdot>) img \<AA>"
PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1_dir2: fixes N :: "nat"   and ps :: "nat list" assumes "DiningPhilosophers N"   and "0 < length ps" shows "DiningPhilosophers.LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m N (ps ! 0) \<sqsubseteq> DiningPhilosophers.PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m N 1 ps"
stake_append: fixes n :: "nat"   and u :: "'d list"   and s :: "'d stream" shows "stake n (u @- s) = take (min (length u) n) u @ stake (n - length u) s"
prems: shows "\<not> gexp_constrains (In x1a x2) (V (I v))"
states_Power_dfa: shows "dfa.states Power_dfa = HF ` epsclo ` Pow (nfa.states M)"
takefill_same: fixes fill :: "'b"   and xs :: "'b list" shows "takefill fill (length xs) xs = xs"
dominates_antitrans: fixes g :: "'g"   and n\<^sub>1 :: "'node"   and m :: "'node"   and n\<^sub>2 :: "'node" assumes "invar g"   and "old.dominates g n\<^sub>1 m"   and "old.dominates g n\<^sub>2 m" obtains "old.dominates g n\<^sub>1 n\<^sub>2"   | "old.dominates g n\<^sub>2 n\<^sub>1"
NS_all_least: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'a"   and t :: "('a, 'b) Term.term" assumes "admissible_kbo w w0 pr_strict pr_weak least scf"   and "least f" shows "snd (kbo.kbo w w0 scf least pr_strict pr_weak t (Fun f []))"
select_convs_3: fixes ext :: "real_int"   and lan :: "nat_int"   and own :: "cars"   and more :: "'a" shows "own \<lparr>ext = ext, lan = lan, own = own, \<dots> = more\<rparr> = own"
UP_mapsto: shows "S.UP \<in> UNIV \<rightarrow> S.Univ"
is_run_conv: fixes r :: "nat \<Rightarrow> (cmd, local_state, global_state) Gen_Scheduler.global_config option" shows "ref_is_run prog r = (\<exists>r'. r = Some \<circ> r' \<and> li'.sa.is_run prog r')"
presSwap_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'd, 'opSym, 'var, 'c, 'e, 'f) Recursion.model_scheme" shows "presSwap h MOD \<equiv> \<forall>zs z1 z2 s X. wls s X \<longrightarrow> h (X #[z1 \<and> z2]_zs) = gSwap MOD zs z1 z2 X (h X)"
degree_1_dvd_root: fixes q :: "'a poly"   and p :: "'a poly" assumes "degree q = 1"   and "\<And>x. poly p x \<noteq> (0::'a)" shows "\<not> q dvd p"
assms_3: shows "linear_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 (*s\<^sub>1) (*s\<^sub>2) f"
hull_redundant: fixes a :: "'a"   and S :: "'a set \<Rightarrow> bool"   and s :: "'a set" assumes "a \<in> S hull s" shows "S hull insert a s = S hull s"
Dom_dom: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" shows "Dom (local.dom f) = Dom f"
ts_lose_ack_st1ts: fixes lose :: "nat \<Rightarrow> bool list"   and t :: "nat"   and ack :: "nat \<Rightarrow> 'a list"   and x :: "'a"   and st_out :: "nat \<Rightarrow> 'a"   and y :: "'a" assumes "ts lose"   and "lose t = [True] \<longrightarrow> ack t = [x] \<and> st_out t = x"   and "lose t = [False] \<longrightarrow> ack t = [y] \<and> st_out t = y" shows "ack t = [st_out t]"
risk_neutral_prob_def: fixes N :: "'a measure" shows "risk_neutral_prob N = (prob_space N \<and> (\<forall>asset\<in>stocks Mkt. martingale N F (discounted_value r (prices Mkt asset))))"
has_integral_Union: fixes \<T> :: "'a set set"   and f :: "'a \<Rightarrow> 'b"   and i :: "'a set \<Rightarrow> 'b" assumes "finite \<T>"   and "\<And>S. S \<in> \<T> \<Longrightarrow> (f has_integral i S) S"   and "pairwise (\<lambda>S S'. negligible (S \<inter> S')) \<T>" shows "(f has_integral sum i \<T>) (\<Union> \<T>)"
poly_gcd_0_iff: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" shows "(poly (gcd p q) x = (0::'a)) = (poly p x = (0::'a) \<and> poly q x = (0::'a))"
loc_lm_1: fixes n :: "nat" assumes "c_fst n mod 7 \<noteq> 0" shows "0 < n"
intros_3: fixes G :: "'b formula"   and F :: "'b formula" shows "(G::'b formula) \<^bold>\<rightarrow> (F::'b formula) \<^bold>\<or> G \<in> AX10"
uconnected_transcl: fixes E :: "'a uprod set" shows "(uconnected E)\<^sup>* = uconnected E"
injectivity: fixes x :: "'a poly"   and y :: "'a poly" assumes "pCons (0::'a) x = pCons (0::'a) y" shows "x = y"
exhaust: fixes y :: "('c, 'd) chain" obtains x1 :: "'c" where "y = \<lbrace> x1 \<rbrace>"   | x21 :: "'c" and x22 :: "'d" and x23 :: "('c, 'd) chain" where "y = \<lbrace> x21 \<rbrace> \<cdot> x22 \<cdot> x23"
skl1_inv2_init: shows "init skl1 \<subseteq> skl1_inv2"
y: shows "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < r"
restrict_apply': fixes x :: "'h"   and A :: "'h set"   and f :: "'h \<Rightarrow> 'i" assumes "x \<in> A" shows "restrict f A x = f x"
inf_l_bl_def: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
c_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "c_aux a b c = a_aux c a b"
prod_2_id_def: fixes p :: "'e \<Rightarrow> 'f decision option"   and q :: "'e \<Rightarrow> 'g decision option" shows "p \<Otimes>\<^sub>2\<^sub>I q = p \<Otimes>\<^sub>2 q \<circ> (\<lambda>x. (x, x))"
get_child_nodes_ok: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "DocumentClass.known_ptr ptr"   and "DocumentClass.type_wf h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_child_nodes ptr"
Pdevs_raw_def: fixes xs :: "('e \<times> 'd) list"   and i :: "'e" shows "Pdevs_raw xs i = the_default (0::'d) (map_of xs i)"
verts_slim_in_G: shows "pverts H.slim \<subseteq> pverts slim"
tensor_from_lookup_eqI: fixes ds :: "nat list"   and e\<^sub>1 :: "nat list \<Rightarrow> 'a"   and e\<^sub>2 :: "nat list \<Rightarrow> 'a" assumes "\<And>is. is \<lhd> ds \<Longrightarrow> e\<^sub>1 is = e\<^sub>2 is" shows "tensor_from_lookup ds e\<^sub>1 = tensor_from_lookup ds e\<^sub>2"
refl_on_singleton: fixes x :: "'d" shows "refl_on {x} {(x, x)}"
iso_1: shows "hom_boundary (p + 1) X S \<in> Group.iso (relative_homology_group (p + 1) X S) (subgroup_generated (homology_group p (subtopology X S)) H)"
emeasure_space_1: fixes s :: "'s" shows "emeasure (T s) (space (T s)) = 1"
undefined: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and a :: "'a"   and b :: "'a" assumes "merge_on A h bo m"   and "h a \<noteq> h b"   and "a \<in> A" shows "m a b = None"
CollectD: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "a \<in> \<lbrace>\<acute>P\<rbrace>" shows "P a"
dominate_sdominate: fixes G :: "nat graph_rec"   and n1 :: "nat"   and n2 :: "nat" assumes "cfg_doms G"   and "cfg_doms.dominate G n1 n2"   and "n1 \<noteq> n2" shows "cfg_doms.strict_dominate G n1 n2"
lift_imp_neg: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" assumes "\<turnstile> A \<longrightarrow> B" shows "\<turnstile> \<not> B \<longrightarrow> \<not> A"
ccw_antisym: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and p :: "'b"   and q :: "'b"   and r :: "'b" assumes "ccw_system12 ccw S"   and "distinct3 p q r"   and "in3 S p q r"   and "ccw p q r" shows "\<not> ccw p r q"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_loop_on_left_moving m rs (l, r) = (wadjust_loop_on_left_moving_O m rs (l, r) \<or> wadjust_loop_on_left_moving_B m rs (l, r))"
ghd_UNIV: fixes x :: "'v" shows "ground_heads_var x = UNIV"
assms_4: shows "h' \<turnstile> getter \<rightarrow>\<^sub>r x"
t: shows "t = Node w t\<^sub>1 t\<^sub>2"
termination_0_1_immediate: fixes p :: "real"   and s :: "'a" assumes "\<And>s. s \<notin> B \<Longrightarrow> p \<le> spmf (map_spmf (\<lambda>x. x \<notin> B) (spmf_of_set A)) False"   and "0 < p"   and "\<And>s. s \<notin> B \<Longrightarrow> lossless_spmf (spmf_of_set A)" shows "lossless_spmf (local.while s)"
inject_1: fixes x11 :: "'a"   and x12 :: "nat"   and x13 :: "'a interval"   and y11 :: "'a"   and y12 :: "nat"   and y13 :: "'a interval" shows "(INS x11 x12 x13 = INS y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
match_fv_cong: fixes r :: "formula Regex.regex"   and v :: "event_data list"   and v' :: "event_data list"   and \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option" assumes "\<forall>x\<in>fv_regex r. v ! x = v' ! x" shows "Regex.match (sat \<sigma> V v) r = Regex.match (sat \<sigma> V v') r"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "G (C g f) = D (G g) (G f)"
dd_commute_modulo: fixes r :: "('a \<times> 'a) set"   and L :: "'a \<Rightarrow> ('c \<times> 'c) set"   and R :: "'a \<Rightarrow> ('c \<times> 'c) set"   and E :: "'a \<Rightarrow> ('c \<times> 'c) set" assumes "wf r"   and "trans r"   and "\<And>a b s t u. \<lbrakk>(s, t) \<in> L a; (s, u) \<in> R b\<rbrakk> \<Longrightarrow> (t, u) \<in> conversion' L R E (under r a) O (R b)\<^sup>= O conversion' L R E (under r a \<union> under r b) O ((L a)\<inverse>)\<^sup>= O conversion' L R E (under r b)"   and "\<And>a b s t u. \<lbrakk>(s, t) \<in> L a; (s, u) \<in> E b\<rbrakk> \<Longrightarrow> (t, u) \<in> conversion' L R E (under r a) O E b O conversion' L R E (under r a) O ((L a)\<inverse>)\<^sup>= O conversion' L R E (under r a \<inter> under r b) \<or> (t, u) \<in> conversion' L R E (under r a \<union> under r b) O ((L a)\<inverse>)\<^sup>= O conversion' L R E (under r b)"   and "\<And>a b s t u. \<lbrakk>(s, t) \<in> (E a)\<inverse>; (s, u) \<in> R b\<rbrakk> \<Longrightarrow> (t, u) \<in> conversion' L R E (under r a \<inter> under r b) O (R b)\<^sup>= O conversion' L R E (under r b) O E a O conversion' L R E (under r b) \<or> (t, u) \<in> conversion' L R E (under r a) O (R b)\<^sup>= O conversion' L R E (under r a \<union> under r b)" shows "(\<Union> (range R) \<union> \<Union> (range E) \<union> (\<Union> (range L))\<inverse>)\<^sup>* \<subseteq> (\<Union> (range R))\<^sup>* O (\<Union> (range E))\<^sup>* O ((\<Union> (range L))\<inverse>)\<^sup>*"