hcju/isabelleps huggingface dataset:
cont_def: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" shows "cont luba orda lubb ordb f = (\<forall>Y. Complete_Partial_Order.chain orda Y \<longrightarrow> Y \<noteq> {} \<longrightarrow> f (luba Y) = lubb (f ` Y))"
Testo_undef: fixes \<phi> :: "fml option" shows "(Testo \<phi> = undefg) = (\<phi> = undeff)"
assms_2: fixes x :: "'a" assumes "x \<in> m" shows "finite (f x)"
elims: fixes x :: "nat \<Rightarrow> 'b"   and xa :: "nat"   and y :: "nat \<Rightarrow> 'b" assumes "shift x xa = y" obtains f :: "nat \<Rightarrow> 'b" and j :: "nat" where "x = f" and "xa = j" and "y = (\<lambda>i. f (i + j))"
typeII_two_cubes_trace_vertical_boundaries: shows "two_chain_vertical_boundary two_chain_typeII \<subseteq> one_chain_typeII"
simps_1: fixes xs :: "'b list" shows "List.n_lists 0 xs = [[]]"
assms: shows "\<forall>i. (f i, f (Suc i)) \<in> r"
shadow_root_eq_h3: fixes ptr' :: "(_) element_ptr"   and shadow_root_ptr_opt :: "(_) shadow_root_ptr option" shows "h3 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt = h' \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt"
simps_2: shows "FinalDeny \<noteq> FinalAllow"
S_def: shows "S = {w \<in> space M. stake (Suc n) w = stake (Suc n) x}"
serial_on_finite_cycle: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "serial_on A r"   and "A \<noteq> {}"   and "finite A" obtains a :: "'b" where "a \<in> A" and "(a, a) \<in> r\<^sup>+"
simps_1: fixes \<rho> :: "'b run"   and K :: "clock" shows "#\<^sub>\<le> \<rho> K 0 = (if hamlet (Rep_run \<rho> 0 K) then 1 else 0)"
img_mkPoint_2: fixes a :: "'s"   and t :: "'s" assumes "S.ide a"   and "t \<in> S.set a" shows "S.img (S.mkPoint a t) = t"
add_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<sqinter> b \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
F_sup_cong: fixes f :: "'s \<Rightarrow> ennreal"   and g :: "'s \<Rightarrow> ennreal"   and s :: "'s" assumes "\<And>s. s \<in> S \<Longrightarrow> f s = g s" shows "F_sup f s = F_sup g s"
minus_vec_def: fixes v\<^sub>1 :: "'b vec"   and v\<^sub>2 :: "'b vec" shows "v\<^sub>1 - v\<^sub>2 \<equiv> vec (dim_vec v\<^sub>2) (\<lambda>i. v\<^sub>1 $ i - v\<^sub>2 $ i)"
algo_correct: fixes xs :: "int list" shows "all_left xs \<le> SPEC (all_left_spec xs)"
simps_1: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat" shows "reads_consistent pending_write \<O> m [] = True"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b + (a + c) = a + (b + c)"
row_add_code_nth: fixes A :: "(('b, 'c) vec, 'a) vec"   and a :: "'a"   and b :: "'a"   and q :: "'b" shows "($) (row_add A a b q) = row_add_row A a b q"
RETURN_leof_RES_iff: fixes x :: "'b"   and Y :: "'b set" shows "(RETURN x \<le>\<^sub>n RES Y) = (x \<in> Y)"
reachable_app_iso_eq: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'e"   and v :: "'e" assumes "digraph_isomorphism hom"   and "u \<in> iso_verts hom ` verts G"   and "v \<in> iso_verts hom ` verts G" shows "(u \<rightarrow>\<^sup>*\<^bsub>app_iso hom G\<^esub> v) = (iso_verts (inv_iso hom) u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> iso_verts (inv_iso hom) v)"
of_int_abs: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "ring_1.of_int one plus zero minus \<bar>x\<bar> = abs (ring_1.of_int one plus zero minus x)"
sublist_el: fixes i :: "nat"   and j :: "nat"   and xs :: "'c list"   and x :: "'c" assumes "i \<le> j"   and "j < length xs" shows "x \<in> set (WB_Sort.sublist xs i j) \<equiv> \<exists>k<Suc j - i. xs ! (i + k) = x"
add_mono: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry"   and c :: "'d DBMEntry"   and d :: "'d DBMEntry" assumes "a \<preceq> b"   and "c \<preceq> d" shows "dbm_add a c \<preceq> dbm_add b d"
integral_mult_left_zero: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'a"   and c :: "'a" shows "LINT x|M. f x * c = integral\<^sup>L M f * c"
Gcd_Lcm: fixes A :: "'a set" shows "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
IH: assumes "\<forall>a\<in>set L. freeIn var a" shows "freeIn var (list_disj L)"
A: shows "\<exists>xi. xi \<in> Gromov_boundary \<and> Gromov_extension f xi = xi \<and> 0 < additive_strength f xi"
intros_2: fixes c :: "('a, 'b) lit list"   and \<sigma> :: "'tp" assumes "\<forall>x\<in>nvC c. TE.tpOfV x \<noteq> \<sigma>" shows "\<sigma> \<turnstile> c"
pdevs_val_cmul: fixes u :: "real"   and e :: "nat \<Rightarrow> real"   and xs :: "'a pdevs" shows "pdevs_val (\<lambda>i. u * e i) xs = u *\<^sub>R pdevs_val e xs"
homotopy_eqv_separation: fixes S :: "'a set"   and T :: "'a set" assumes "S homotopy_eqv T"   and "compact S"   and "compact T" shows "connected (- S) = connected (- T)"
beukers_nn_integral1_finite: fixes r :: "nat"   and s :: "nat" shows "beukers_nn_integral1 r s < \<infinity>"
simps_1: fixes s :: "'a \<Rightarrow> bool option" shows "execute_serial_plan s [] = s"
DProp_def: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "DProp A = (\<exists>\<phi>. A = Sec (\<phi> (FIX \<phi>)) \<and> Monotone \<phi>)"
card_of_unique: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "card_order_on A r" shows "r =o |A|"
null_def: shows "V.null = (THE n. \<forall>f. n \<bullet> f = n \<and> f \<bullet> n = n)"
get_document_ptr_simp2: fixes document_ptr :: "(_) document_ptr"   and document_ptr' :: "(_) document_ptr"   and document :: "('o, 'p, 'q, 'r option) RDocument_scheme"   and h :: "(_) heap" assumes "document_ptr \<noteq> document_ptr'" shows "get document_ptr (put document_ptr' document h) = get document_ptr h"
set_S_char: shows "set S = {x. poly (poly_f qs) x = 0}"
app_term_def: fixes t :: "term"   and u :: "term" shows "app t u = t $ u"
unify_None: fixes E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and ss :: "('c \<times> ('d, 'c) Term.term) list" assumes "unify E ss = None" shows "\<exists>E'. E' \<noteq> {#} \<and> (mset E, E') \<in> unif\<^sup>!"
essential_def: fixes G :: "('k, 'l) graph_scheme"   and B :: "'k set"   and S :: "'k set"   and x :: "'k" shows "essential G B S x = (\<exists>p. \<exists>y\<in>B. path G x p y \<and> (x \<noteq> y \<longrightarrow> (\<forall>z\<in>set p. z = x \<or> z \<notin> S)))"
r_empty: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt"   and p\<^sub>1 :: "(real, 'b) vec"   and p\<^sub>0 :: "(real, 'b) vec" assumes "invar (Node k v l r)"   and "p\<^sub>1 $ k < v" shows "set_kdt r \<inter> cbox p\<^sub>0 p\<^sub>1 = {}"
cSup_upper2: 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 x :: "'c"   and X :: "'c set"   and y :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "less_eq y x"   and "preorder.bdd_above less_eq X" shows "less_eq y (Sup X)"
cap_dim: shows "rat_poly.row_length (brickmat cap) = 4"
prems_4: shows "((\<forall>nm\<in>incoming (fst x). nm < name (fst x)) \<and> 0 < name (fst x) \<and> (\<forall>q\<in>snd x. name q < name (fst x) \<and> (\<forall>nm\<in>incoming q. nm < name (fst x)))) \<and> expand_name_ident (snd x)"
P3_1: shows "action_needed_asses a s2 \<subseteq>\<^sub>f action_needed_asses a s1"
contk: fixes s :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and t :: "'b set"   and k :: "'b \<Rightarrow> 'a" assumes "Retracts s h t k" shows "continuous_on t k"
det_row_mul: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and c :: "'c"   and a :: "('c, 'd) vec" shows "Square_Matrix.det (upd_row M i (c *s a)) = c * Square_Matrix.det (upd_row M i a)"
inj_on_fst_set_slist: fixes xs :: "(nat, 'd) slist" shows "inj_on fst (set_slist xs)"
S2MClkFwd_enabled: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S2 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<longrightarrow> $Enabled (<MClkFwd memCh crCh cst p>_(c p))"
semigroup_def: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "semigroup f \<equiv> \<forall>a b c. f (f a b) c = f a (f b c)"
Int_UNIV_left: fixes B :: "'d set" shows "UNIV \<inter> B = B"
rotate_trailE': fixes u :: "'a"   and p :: "'b list"   and w :: "'a" assumes "trail u p u"   and "w \<in> set (awalk_verts u p)" obtains q :: "'b list" where "trail w q w" and "set q = set p" and "set (awalk_verts w q) = set (awalk_verts u p)"
promela_is_run_ltl_def: fixes promg :: "program \<times> expr iarray \<times> gState"   and r :: "nat \<Rightarrow> gState" shows "promela_is_run_ltl promg r \<equiv> let (prog, APs, g) = promg in promela_is_run' (prog, g) r"
UN_simps_5: fixes A :: "'b set"   and B :: "'c \<Rightarrow> 'b set"   and C :: "'c set" shows "(\<Union>x\<in>C. A \<inter> B x) = A \<inter> \<Union> (B ` C)"
cfuncset_iff: fixes f :: "'e \<Rightarrow> 'f"   and l :: "'e"   and u :: "'e"   and X :: "'f set" shows "(f \<in> Abstract_Rigorous_Numerics.cfuncset l u X) = ((\<forall>i\<in>{l..u}. f i \<in> X) \<and> continuous_on {l..u} f)"
tranclp_rtranclp_tranclp: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "r\<^sup>+\<^sup>+ a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
infinite: fixes A :: "'q set" assumes "infinite A" shows "card A = 0"
must_wait_elims: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and lt :: "'a + 'b + 'b"   and Ts :: "'b set" assumes "must_wait s t lt Ts" obtains l :: "'a" and t' :: "'b" where "lt = Inl l" and "has_lock (locks s $ l) t'__" and "t'__ \<noteq> t" and "t'__ \<in> Ts"   | t' :: "'b" where "lt = Inr (Inl t'a__)" and "not_final_thread s t'a__" and "t'a__ \<in> Ts"   | t' :: "'b" where "lt = Inr (Inr t')" and "all_final_except s Ts" and "t' \<notin> interrupts s"
imset: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set" assumes "\<forall>k\<le>Suc n. ideal R (J k)" shows "{I. \<exists>k\<le>Suc n. I = J k} = {J k |k. k \<in> {j. j \<le> Suc n}}"
step: shows "(s, fst t', snd t') \<in> ars"
ante: shows "\<exists>w. P w x \<and> P w y \<and> P w z"
IH1_2: assumes "M \<Turnstile> F" shows "M(n := False) \<Turnstile> F"
bigtheta_trans1': fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "h \<in> \<Theta>[F](g)" shows "f \<in> O[F](h)"
map_apply_def: shows "map_apply = (\<lambda>m. m)"
simps: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set"   and st :: "nat set"   and t :: "'b option list" assumes "getIJ getIJ" shows "New_max.newQuery V Q (st, t) = (\<lambda>tab. New_max.projectTable V (New_max.semiJoin tab (st, t))) ` Q"
continuous_map_in_subtopology: fixes X :: "'b topology"   and X' :: "'c topology"   and S :: "'c set"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map X (subtopology X' S) f = (continuous_map X X' f \<and> f ` topspace X \<subseteq> S)"
gram_schmidt_hd: fixes w :: "'a vec"   and ws :: "'a vec list" assumes "w \<in> carrier_vec n" shows "hd (gram_schmidt n (w # ws)) = w"
pred_case_conv: fixes x :: "nat" shows "x - Suc 0 = (case x of 0 \<Rightarrow> 0 | Suc m \<Rightarrow> m)"
3134: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (- y \<sqinter> - x) = - y \<sqinter> - x"
"case": fixes f :: "'b \<Rightarrow> 'd \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'d" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
butlast_conv_take: fixes xs :: "'g list" shows "butlast xs = take (length xs - 1) xs"
set_add_pairs_single_naive: fixes data :: "'g"   and flag :: "bool"   and gs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and h :: "('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and ps :: "(bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "set (punit.add_pairs_single_naive data flag gs bs h ps) = set ps \<union> Pair flag ` ({h} \<times> (set gs \<union> set bs))"
"1_2": shows "fmlookup (action_needed_asses a s1) v = fmlookup (fst a) v"
vec_tensor_prod_2_bis: fixes v :: "complex Matrix.vec"   and w :: "complex Matrix.vec" assumes "v \<in> state_qbit 1"   and "w \<in> state_qbit 1" shows "v \<otimes> w = Matrix.vec 4 (\<lambda>i. if i = 0 then v $ 0 * w $ 0 else if i = 3 then v $ 1 * w $ 1 else if i = 1 then v $ 0 * w $ 1 else v $ 1 * w $ 0)"
substexp_substmem: fixes e' :: "('d, 'c) Expr"   and x :: "'d"   and e :: "('d, 'c) Expr"   and m :: "'d \<Rightarrow> 'c"   and elist :: "('d, 'c) Expr list" shows "ExprEval e'<Var x\e> m = ExprEval e' (m(x := ExprEval e m)) \<and> ExprEvalL (SubstL elist (Var x) e) m = ExprEvalL elist (m(x := ExprEval e m))"
assms: shows "wf_formula n (FOr \<phi>\<^sub>1 \<phi>\<^sub>2)"
intro: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set" assumes "Gamma num' plus' \<gamma>" shows "Abs_Int num' plus' \<gamma>"
iterates_mcont: fixes f :: "'b \<Rightarrow> 'a"   and F :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'a" assumes "f \<in> ccpo.iterates (fun_lub Sup) (fun_ord (\<le>)) F"   and "\<And>f. mcont \<Or> (\<sqsubseteq>) Sup (\<le>) f \<Longrightarrow> mcont \<Or> (\<sqsubseteq>) Sup (\<le>) (F f)" shows "mcont \<Or> (\<sqsubseteq>) Sup (\<le>) f"
wls_Abs_subst_all: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and x' :: "'var" assumes "wls s X"   and "wls s X'" shows "(Abs xs x X = Abs xs x' X') = (\<forall>Y. wls (asSort xs) Y \<longrightarrow> X #[Y / x]_xs = X' #[Y / x']_xs)"
constant_on_subset: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "f constant_on A"   and "B \<subseteq> A" shows "f constant_on B"
l_redu_subst2: fixes M :: "trm"   and M' :: "trm"   and c :: "coname"   and y :: "name"   and P :: "trm" assumes "M \<longrightarrow>\<^sub>l M'" shows "M{c:=y.P} \<longrightarrow>\<^sub>a* M'{c:=y.P}"
lF_def: fixes f :: "'c" shows "lF f \<equiv> THE \<mu>. \<guillemotleft>\<mu> : F (trg\<^sub>C f) \<star>\<^sub>D F f \<Rightarrow>\<^sub>D F f\<guillemotright> \<and> F (trg\<^sub>C f) \<star>\<^sub>D \<mu> = (F \<i>\<^sub>C[trg\<^sub>C f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
assms: fixes "is" :: "nat list" assumes "is \<lhd> ds" shows "e\<^sub>1 is = e\<^sub>2 is"
ref: fixes v :: "('a, 'b, 'c) val" shows "redex (Ref (VE v))"
mB: fixes I :: "nat set" assumes "finite I" shows "Sigma_Algebra.measure M (\<Union> (S ` I)) \<le> B"
m_closed: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "x \<in> carrier (UP Zp)"   and "y \<in> carrier (UP Zp)" shows "x \<oplus>\<^bsub>UP Zp\<^esub> y \<in> carrier (UP Zp)"
simps_2: fixes L :: "('a, 'b) Term.term literal"   and Ls :: "('a, 'b) Term.term literal list"   and Ks :: "('a, 'c) Term.term literal list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term option" shows "subsumes_list (L # Ls) Ks \<sigma> = (\<exists>K\<in>set Ks. is_pos K = is_pos L \<and> (case match_term_list [(atm_of L, atm_of K)] \<sigma> of None \<Rightarrow> False | Some x \<Rightarrow> subsumes_list Ls (remove1 K Ks) x))"
a3: shows "\<exists>y'>r. \<forall>(d, e, f)\<in>set d. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0"
map_id: fixes t :: "('a, 'b) blindable\<^sub>m" shows "map_blindable\<^sub>m id id t = t"
Fax_satPB: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "GE.satPB \<xi> Fax"
even_or_odd: fixes n :: "nat" shows "\<exists>i. n = 2 * i \<or> n = 2 * i + 1"
is_nnf_BigOr: fixes ls :: "'b formula list" shows "is_nnf \<^bold>\<Or>ls = (\<forall>l\<in>set ls. is_nnf l)"
tauChainAddTau_1: fixes P :: "pi"   and P' :: "pi"   and P'' :: "pi" assumes "(P, P') \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*"   and "P' \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P''" shows "(P, P'') \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*"
double_compl: fixes x :: "'d upred" shows "(\<not> \<not> x) = x"
scale_left_commute: fixes scale :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and a :: "'e"   and b :: "'e"   and x :: "'f" assumes "module scale" shows "scale a (scale b x) = scale b (scale a x)"
laplace_integrand_absolutely_integrable_on_Icc: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> complex"   and B :: "real"   and s :: "complex" assumes "AE x\<in>{a..b} in lebesgue. cmod (f x) \<le> B"   and "f integrable_on {a..b}" shows "complex_set_integrable lebesgue {a..b} (laplace_integrand f s)"
simps_2: fixes x :: "'a"   and n :: "nat" shows "p_power x (Suc n) = x \<cdot> p_power x n"
WHILE_bounded_aux2: fixes m :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres" shows "m \<bind> WHILE cond step \<le> filter_nb cond (Sup {bounded_WHILE n cond step m |n. True})"
simps_3: fixes f :: "Core.term"   and t :: "Core.term"   and lev :: "nat"   and u :: "Core.term list" shows "subst_bvs1' (f $ t) lev u = subst_bvs1' f lev u $ subst_bvs1' t lev u"
llist_all2_lsetD1: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e llist"   and ys :: "'f llist"   and x :: "'e" assumes "llist_all2 P xs ys"   and "x \<in> lset xs" shows "\<exists>y\<in>lset ys. P x y"
integrable_on_meissel_mertens: fixes A :: "real set" assumes "A \<subseteq> {1..}"   and "1 < Inf A"   and "A \<in> sets borel" shows "(\<lambda>t. (\<MM> t - ln t) / (t * (ln t)\<^sup>2)) integrable_on A"
intro: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "comm_semigroup plus" shows "comm_semigroup_add plus"
simps_1: fixes x21 :: "'b"   and x22 :: "'b"   and y21 :: "'b"   and y22 :: "'b" shows "(Point x21 x22 = Point y21 y22) = (x21 = y21 \<and> x22 = y22)"
inputChainEqvt: fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "p \<bullet> inputChain xvec N P = inputChain (p \<bullet> xvec) (p \<bullet> N) (p \<bullet> P)"
power_neg_one_If: fixes n :: "nat" shows "(- (1::'a)) ^ n = (if even n then 1::'a else - (1::'a))"
Bernstein_changes_01_eq_changes: fixes P :: "real poly"   and p :: "nat" assumes "degree P \<le> p" shows "Bernstein_changes_01 p P = changes (coeffs (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:]))"
assms_2: shows "\<exists>n''<n'. \<exists>nid'. (n'', nid') \<in> his t n nid"
getter_functions_R: fixes p :: "real poly"   and qs :: "real poly list" shows "calculate_data_R p qs = (get_matrix_R (calculate_data_R p qs), get_subsets_R (calculate_data_R p qs), get_signs_R (calculate_data_R p qs))"
abs_summable_infsetsum'_converges: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "f abs_summable_on A" shows "infsetsum'_converges f A"
length_greater_0_conv: fixes xs :: "'a list" shows "order.greater (length xs) 0 = (xs \<noteq> [])"
elims: fixes x :: "(dynamic, nat, bool) unboxed list"   and y :: "(dynamic, nat, bool) unboxed option" assumes "eval_AddNumUbx x = y" obtains xa :: "nat" and ya :: "nat" where "x = [OpUbx1 xa, OpUbx1 ya]" and "y = Some (OpUbx1 (xa + ya))"   | "x = []" and "y = None"   | vb :: "dynamic" and va :: "(dynamic, nat, bool) unboxed list" where "x = OpDyn vb__ # va__" and "y = None"   | vb :: "bool" and va :: "(dynamic, nat, bool) unboxed list" where "x = OpUbx2 vba__ # va" and "y = None"   | v :: "(dynamic, nat, bool) unboxed" where "x = [v__]" and "y = None"   | v :: "(dynamic, nat, bool) unboxed" and vd :: "dynamic" and vc :: "(dynamic, nat, bool) unboxed list" where "x = vc__ # OpDyn vd__ # vca__" and "y = None"   | v :: "(dynamic, nat, bool) unboxed" and vd :: "bool" and vc :: "(dynamic, nat, bool) unboxed list" where "x = ve__ # OpUbx2 vda__ # vc" and "y = None"   | v :: "(dynamic, nat, bool) unboxed" and vb :: "(dynamic, nat, bool) unboxed" and vd :: "(dynamic, nat, bool) unboxed" and ve :: "(dynamic, nat, bool) unboxed list" where "x = v # vb # vd # ve" and "y = None"
ordLeq_transitive: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and r'' :: "('f \<times> 'f) set" assumes "r \<le>o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
close_Var_open_Var: fixes xT :: "char list \<times> Unary_PCF.type"   and t :: "expr"   and i :: "nat" assumes "fresh xT t" shows "close_Var i xT (open_Var i xT t) = t"
unat_arith_simps_1: fixes a :: "'b word"   and b :: "'b word" shows "(a \<le> b) = (unat a \<le> unat b)"
scaleR_collapse: fixes u :: "real"   and a :: "'b" shows "(1 - u) *\<^sub>R a + u *\<^sub>R a = a"
1: shows "2 * pi * real_of_int (a * int d) / real k = 2 * pi * real_of_int a / real (k div d)"
distinct_1: fixes x2 :: "num" shows "num.One \<noteq> num.Bit0 x2"
gen_eqvt: fixes pi :: "(W.tvar \<times> W.tvar) list"   and T :: "ty"   and Xs :: "W.tvar list" shows "pi \<bullet> gen T Xs = gen (pi \<bullet> T) (pi \<bullet> Xs)"
Rep_Abs_prop_entailment: fixes \<A> :: "'a ltln set"   and \<phi> :: "'a ltln" shows "\<A> \<Turnstile>\<^sub>P rep_ltln\<^sub>P (abs_ltln\<^sub>P \<phi>) = \<A> \<Turnstile>\<^sub>P \<phi>"
in_pullback_algebra: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'c \<Rightarrow> 'b"   and \<Omega>' :: "'c set" assumes "A \<in> sets M" shows "f -` A \<inter> \<Omega>' \<in> sets.pullback_algebra M f \<Omega>'"
FAcc_\<tau>execI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and F :: "String.literal"   and D :: "String.literal" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e\<bullet>F{D}) h s s'"
ideD_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.dom C a = a"
split_cong: fixes q :: "'x \<times> 'y"   and f :: "'x \<Rightarrow> 'y \<Rightarrow> 'z"   and g :: "'x \<Rightarrow> 'y \<Rightarrow> 'z"   and p :: "'x \<times> 'y" assumes "\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y"   and "p = q" shows "(case p of (x, xa) \<Rightarrow> f x xa) = (case q of (x, xa) \<Rightarrow> g x xa)"
simps_2: fixes e :: "aexp"   and S :: "'av st" shows "aval'' e (Some S) = aval' e S"
finite_g: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "f_i i e \<noteq> \<top>"