hcju/isabelleps huggingface dataset:
case_1: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f5 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f6 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f7 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x1 :: "('d, 'e) Term.term" shows "(case send\<langle>x1\<rangle> of send\<langle>x\<rangle> \<Rightarrow> f1 x | receive\<langle>x\<rangle> \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle> \<Rightarrow> f3 x xa xb | insert\<langle>x,xa\<rangle> \<Rightarrow> f4 x xa | delete\<langle>x,xa\<rangle> \<Rightarrow> f5 x xa | \<langle>x: xa \<in> xb\<rangle> \<Rightarrow> f6 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa \<or>\<notin>: xb\<rangle> \<Rightarrow> f7 x xa xb) = f1 x1"
norm_sum: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (comm_monoid_add.sum plus zero f A) \<le> (\<Sum>i\<in>A. norm (f i))"
pred_stateI: fixes m :: "('b, 'a) state"   and P :: "'a \<Rightarrow> bool" assumes "\<And>a s s'. run_state m s = (a, s') \<Longrightarrow> P a" shows "pred_state P m"
compute_sqrt_iteration_base: fixes prec :: "nat"   and n :: "nat"   and m :: "int"   and e :: "int" shows "sqrt_iteration prec n (Float m e) = (if n = 0 then Float 1 ((if m = 0 then 0 else bitlen \<bar>m\<bar> + e) div 2 + 1) else let y = sqrt_iteration prec (n - 1) (Float m e) in Float 1 (- 1) * float_plus_up prec y (float_divr prec (Float m e) y))"
normalized_vec_dim: fixes v :: "complex vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "vec_normalize v \<in> carrier_vec n"
leftderives_implies_derives: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.leftderives \<NN> \<TT> \<RR> u v" shows "CFG.derives \<NN> \<TT> \<RR> u v"
partition_on_empty: fixes P :: "'a set set" shows "partition_on {} P = (P = {})"
COD_in_cod: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.COD f \<in> \<F>C.cod f"
compass_defs_3: shows "north = proj2_abs (vector [0, 1, 1])"
adj_def: fixes g :: "'m"   and v :: "'v" shows "adj g v = the_default [] (M_lookup g v)"
Rep_N_inverse: fixes x :: "'b N" shows "Abs_N (Rep_N x) = x"
possible_steps_singleton: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and d :: "nat"   and t :: "transition" shows "(possible_steps e s r l i = {|(d, t)|}) = ({((origin, dest), t). ((origin, dest), t) \<in> fset e \<and> origin = s \<and> Label t = l \<and> length i = Arity t \<and> apply_guards (Guards t) (join_ir i r)} = {((s, d), t)})"
nth_enumerate_eq: fixes m :: "nat"   and xs :: "'a list"   and n :: "nat" assumes "m < length xs" shows "enumerate n xs ! m = (n + m, xs ! m)"
Refl_Field_Restr: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" assumes "Refl r" shows "Field (Restr r A) = Field r \<inter> A"
openin_clauses_3: fixes K :: "'c set set"   and U :: "'c topology" assumes "\<forall>S\<in>K. openin U S" shows "openin U (\<Union> K)"
tab\<^sub>1_simps_2: fixes r :: "'a" assumes "ide r" shows "is_left_adjoint (tab\<^sub>1 r)"
proj_not_same_active: fixes n :: "nat"   and n' :: "enat"   and t :: "cnf llist"   and tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b" assumes "enat n \<le> n'"   and "\<not> lfinite t \<or> n' - 1 < llength t"   and "dynamic_component.proj tCMP active c (ltake n' t) \<noteq> dynamic_component.proj tCMP active c (ltake (enat n) t)" shows "\<exists>k\<ge>n. enat k < n' \<and> enat k < llength t \<and> active c (lnth t k)"
remove: fixes x :: "'k"   and A :: "'k multiset" assumes "x \<in># A" shows "\<Sum>\<^sub># A = x + \<Sum>\<^sub># (A - {#x#})"
power_int_minus: fixes x :: "'a"   and n :: "int" shows "x powi - n = inverse (x powi n)"
elims: fixes x :: "'b tree"   and y :: "nat" assumes "min_height x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "'b tree" and uu_ :: "'b" and r :: "'b tree" where "x = \<langle>l, uu_, r\<rangle>" and "y = min (min_height l) (min_height r) + 1"
reductio_aa_4: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi> in v]"   and "[\<phi> \<^bold>\<rightarrow> \<psi> in v]" shows "[\<^bold>\<not>\<phi> in v]"
segm_def: fixes x :: "'b segment" shows "segm x = {y. fst (Rep_segment x) \<le> y \<and> y \<le> snd (Rep_segment x)}"
WTrt_hext_mono: fixes h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and T :: "ty"   and h' :: "'heap" assumes "P,E,h \<turnstile> e : T"   and "h \<unlhd> h'" shows "P,E,h' \<turnstile> e : T"
TS: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs = 2"
exsplit_qf: fixes p :: "fm" assumes "qfree p" shows "qfree (exsplit p)"
commute: fixes a :: "'b"   and b :: "'b" shows "a + b = b + a"
empty_continuous_imp_continuous_from_below: fixes f :: "'a set \<Rightarrow> ennreal"   and A :: "nat \<Rightarrow> 'a set" assumes "positive M f"   and "Measure_Space.additive M f"   and "\<forall>A\<in>M. f A \<noteq> \<infinity>"   and "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> \<Inter> (range A) = {} \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"   and "range A \<subseteq> M"   and "incseq A"   and "\<Union> (range A) \<in> M" shows "(\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union> (range A))"
algebraic_int_plus: fixes x :: "'b"   and y :: "'b" assumes "algebraic_int x"   and "algebraic_int y" shows "algebraic_int (x + y)"
parts_insert_eq_I: fixes X :: "msg"   and H :: "msg set" assumes "\<And>x. x \<in> parts (insert X H) \<Longrightarrow> x \<in> insert X (parts H)" shows "parts (insert X H) = insert X (parts H)"
elims_2: fixes x :: "'form \<times> nat list" assumes "terminal_vertex x" obtains v :: "'form \<times> nat list" where "x = v" and "v |\<in>| vertices \<and> terminal_node (nodeOf v)"
IH: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and LVQ :: "char list set"   and LVE :: "char list set"   and upds :: "(char list \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set) list" assumes "vc C Q LVQ LVE"   and "finite (support Q)"   and "lesvars upds \<inter> varacom C = {}"   and "distinct (map fst upds)"   and "finite (varacom C)"   and "\<forall>l s1 s2. s1 = s2 on LVQ \<longrightarrow> Q l s1 = Q l s2"   and "\<forall>l s1 s2. s1 = s2 on LVE \<longrightarrow> postList upds l s1 = postList upds l s2"   and "\<forall>a\<in>set upds. case a of (a, ba, c) \<Rightarrow> \<forall>s1 s2. s1 = s2 on c \<longrightarrow> ba s1 = ba s2"   and "(\<Union>a\<in>set upds. case a of (a, ba, c) \<Rightarrow> c) \<subseteq> LVE" shows "\<turnstile>\<^sub>1 {\<lambda>l s. pre C Q l s \<and> preList upds C l s} strip C { time C \<Down> \<lambda>l s. Q l s \<and> postList upds l s} \<and> (\<forall>l s. pre C Q l s \<longrightarrow> Q l (postQ C s)) \<and> K4 upds LVQ C Q"
map_iterator_dom_def: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "map_iterator_dom it = set_iterator_image fst it"
ValidTC_def: fixes p :: "'a set"   and c :: "'a com"   and q :: "'a set" shows "[p]   c   [q] \<equiv> \<forall>s. s \<in> p \<longrightarrow> (\<exists>t. Sem c (Some s) (Some t) \<and> t \<in> q)"
lsu_lsl_prop_eq: shows "\<nu>\<^sup>\<natural> \<circ> \<nu> = \<nu>"
Enabled_disjD: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> Enabled (F \<or> G) \<longrightarrow> Enabled F \<or> Enabled G"
vec_dim_vec_of_poly_n_1: fixes p :: "'a poly"   and n :: "nat" shows "dim_vec (vec_of_poly_n p n) = n"
lemma_R_for_BC_simple: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_bc \<psi> U s" shows "\<exists>s'\<in>\<R>. learn_bc \<psi> U s'"
le_imp_power_dvd: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "m \<le> n" shows "dvd.dvd times (power.power one times a m) (power.power one times a n)"
atLeast0_atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "prod g {0..Suc n} = g 0 * prod (g \<circ> Suc) {0..n}"
Conj\<^sub>\<alpha>_def: shows "Conj\<^sub>\<alpha> \<equiv> (map_bset rep_Tree\<^sub>\<alpha> ---> abs_Tree\<^sub>\<alpha>) tConj"
insort_middle: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list" shows "sort (xs @ x # ys) = insort x (sort (xs @ ys))"
CN_1: fixes l :: "float poly"   and r :: "float poly"   and ord :: "nat" assumes "(l, r) = split_by_degree ord c" shows "(maxdegree l \<le> ord \<and> Ipoly x (map_poly real_of_float c) = Ipoly x (map_poly real_of_float l) + Ipoly x (map_poly real_of_float r)) \<and> num_params l \<le> num_params c \<and> num_params r \<le> num_params c"
pair_of_map_component: fixes f :: "nat \<Rightarrow> nat"   and v :: "'t" shows "pair_of_term (map_component f v) = (pp_of_term v, f (component_of_term v))"
in_idem: fixes x :: "'a"   and A :: "'a fset" assumes "x |\<in>| A" shows "min x (fMin A) = fMin A"
assms_2: shows "y \<cdot> p \<cdot> l \<le> l \<cdot> y"
P: shows "P = (`) f ` (`) (the_inv_into A f) ` P"
statelist'_TAKE_i: fixes m :: "nat"   and a :: "'c"   and as :: "'c list" assumes "Suc m \<le> length (a # as)" shows "m \<le> length as"
heap_is_wellformed_children_disc_nodes_different: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "set children \<inter> set disc_nodes = {}"
elims: fixes x :: "unit"   and y :: "('a list, 'a, unit, unit label) seqp" assumes "\<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G x = y" obtains "x = ()" and "y = labelled () (receive(\<lambda>msg msgs. msgs @ [msg]) . call(()) \<oplus> \<langle>\<lambda>msgs. if msgs \<noteq> [] then {msgs} else {}\<rangle> (send(hd) . (\<lbrakk>tl\<rbrakk> call(()) \<oplus> receive(\<lambda>msg msgs. tl msgs @ [msg]) . call(())) \<oplus> receive(\<lambda>msg msgs. msgs @ [msg]) . call(())))"
in_homE: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VV.in_hom f a b" obtains "VV.arr f" and "VV.dom f = a" and "VV.cod f = b"
Ord_iff_rank: fixes x :: "V" shows "Ord x = (rank x = x)"
nofail_inf_serialize: fixes a :: "'a nres"   and b :: "'a nres" assumes "nofail a"   and "nofail b" shows "inf a b = a \<bind> (\<lambda>x. ASSUME (inres b x) \<bind> (\<lambda>_. RETURN x))"
sub_4: shows "\<Gamma>\<^sub>1' \<sqsubseteq>\<^sub>c \<Gamma>\<^sub>2'"
start_proof_sound: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and S :: "('b, 'c, 'a) formula list \<times> ('b, 'c, 'a) formula list" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.sound (ids.start_proof S)"
monad_fail_nondetT': assumes "monad_commute Monad_Overloading.return Monad_Overloading.bind" shows "monad_fail Monad_Overloading.return Monad_Overloading.bind fail"
poly_gt_def: fixes default :: "'c"   and gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and power_mono :: "bool"   and discrete :: "bool"   and p :: "('d monom \<times> 'c) list"   and q :: "('d monom \<times> 'c) list" assumes "poly_order_carrier default gt power_mono discrete" shows "poly_order_carrier.poly_gt gt p q = (\<forall>\<alpha>. pos_assign \<alpha> \<longrightarrow> gt (eval_poly \<alpha> p) (eval_poly \<alpha> q))"
wls_disjoint: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and s' :: "'sort" assumes "wls s X"   and "wls s' X" shows "s = s'"
iso2Inv: fixes f :: "'b" assumes "ciso f" shows "cinv f Cinv f"
functions_of_number_partition_of: fixes A :: "'c set"   and B :: "'d set"   and F :: "('c \<Rightarrow> 'd) set" assumes "finite A"   and "finite B"   and "F \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B" shows "functions_of A B (number_partition_of A B F) = F"
intros_16: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T\<^sub>1 :: "ty"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and T :: "ty"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<^sub>1\<rfloor>"   and "P \<turnstile> T\<^sub>1 \<le> T"   and "P,E(V \<mapsto> T),h \<turnstile> e\<^sub>2 :' T\<^sub>2" shows "P,E,h \<turnstile> {V:T; V:=Val v;; e\<^sub>2} :' T\<^sub>2"
add_inf_distrib_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "inf a b + c = inf (a + c) (b + c)"
filterlim_at: fixes "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and b :: "'c"   and s :: "'c set"   and F :: "'d filter" assumes "class.topological_space open" shows "filterlim f (topological_space.at_within open b s) F = ((\<forall>\<^sub>F x in F. f x \<in> s \<and> f x \<noteq> b) \<and> filterlim f (topological_space.nhds open b) F)"
dup_simps_3: fixes f :: "'a" assumes "arr f" shows "cod (dup f) = local.prod (cod f) (cod f)"
relImage_def: fixes R :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'c" shows "relImage R f \<equiv> {(f a1, f a2) |a1 a2. (a1, a2) \<in> R}"
H_def: shows "H = P2 * D * Q2 @\<^sub>c P2 * E"
rep_eq: fixes x :: "mini_alg" shows "real_of x = (case Rep_mini_alg x of (p, q, b) \<Rightarrow> real_of_rat p + real_of_rat q * sqrt (real b))"
mem_diff: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and c' :: "'c"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and lc :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and mem'' :: "'a \<Rightarrow> 'b" assumes "sifum_security_init dma \<C>_vars eval"   and "((c', mds'), mem') \<in> sifum_security_init.loc_reach dma eval lc"   and "\<forall>x. var_asm_not_written mds' x \<longrightarrow> mem' x = mem'' x \<and> dma mem' x = dma mem'' x" shows "((c', mds'), mem'') \<in> sifum_security_init.loc_reach dma eval lc"
RHS_in_hom: shows "\<guillemotleft>\<lparr>Chn = \<mu>_\<nu>_\<pi>\<rho>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi> \<star> \<rho>)\<rparr> \<bullet> \<lparr>Chn = \<mu>\<nu>_\<pi>_\<rho>.chine_assoc, Dom = Dom (((\<mu> \<star> \<nu>) \<star> \<pi>) \<star> \<rho>), Cod = Cod ((\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho>)\<rparr> : ((\<mu> \<star> \<nu>) \<star> \<pi>) \<star> \<rho> \<Rightarrow> \<mu> \<star> \<nu> \<star> \<pi> \<star> \<rho>\<guillemotright>"
var_fresh_for_ty: fixes x :: "W.var"   and T :: "ty" shows "x \<sharp> T"
uplimit_loR: fixes x :: "nat"   and y :: "nat"   and xa :: "nat" assumes "Suc 0 < x"   and "Suc 0 < y"   and "loR [x, y, xa]" shows "xa \<le> x"
IH: fixes t3s :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> xs : t2s _> t3s" shows "\<S>\<bullet>\<C> \<turnstile> es @ xs : t1s _> t3s"
alw_InvInfoAtBetaUpright: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw (holds InvInfoAtBetaUpright) s"
ffact_def: fixes n :: "nat"   and a :: "'b" shows "ffact n a = pochhammer (a + (1::'b) - of_nat n) n"
transpose_mat_def: fixes A :: "'a mat" shows "A\<^sup>T \<equiv> Matrix.mat (dim_col A) (dim_row A) (\<lambda>(i, j). A $$ (j, i))"
subdegree_eq_0_iff: fixes f :: "'a fps" shows "(subdegree f = 0) = (f = 0 \<or> fps_nth f 0 \<noteq> (0::'a))"
cod_simp: fixes f :: "'a" assumes "Right_a.arr f" shows "Right_a.cod f = cod f"
mod2_eq_if: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "modulo a (numeral.numeral one plus (num.Bit0 num.One)) = (if dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a then zero else one)"
P: shows "P \<subseteq> {q. irreducible q \<and> monic q}"
subset_diff: fixes B :: "'c set"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "B \<subseteq> A"   and "finite A" shows "sum_0 g A = sum_0 g (A - B) + sum_0 g B"
pow_ge_zero: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and n :: "nat" assumes "class.ordered_semiring_1 one times plus zero less_eq"   and "less_eq zero a" shows "less_eq zero (power.power one times a n)"
Int_UNIV_right: fixes A :: "'a set" shows "A \<inter> UNIV = A"
collect_clkvt_def: fixes S :: "('f \<times> ('e, 'g) cconstraint \<times> 'h \<times> 'e list \<times> 'f) set" shows "collect_clkvt S = \<Union> {set ((fst \<circ> snd \<circ> snd \<circ> snd) t) |t. t \<in> S}"
dvd_0_right: fixes times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times a zero"
space_null_measure: fixes M :: "'c measure" shows "space (null_measure M) = space M"
find_simps_3: fixes a :: "'c"   and t :: "('c, 'd) bintree"   and xs :: "('c, 'd) bintree option list" shows "Binomial_Queue.find a (Some t # xs) = (if priority t = a then Some t else Binomial_Queue.find a xs)"
orbit_eqI_1: fixes y :: "'a"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "y = f x" shows "y \<in> orbit f x"
exhaust: fixes y :: "ref_ty" obtains "y = NullT"   | x2 :: "qtname" where "y = IfaceT x2"   | x3 :: "qtname" where "y = ClassT x3"   | x4 :: "ty" where "y = ArrayT x4"
vec_index_is_linear: fixes u :: "'a vec"   and v :: "'a vec"   and j :: "nat"   and k :: "'a"   and l :: "'a" assumes "dim_vec u = dim_vec v"   and "j < dim_vec u" shows "(k \<cdot>\<^sub>v u + l \<cdot>\<^sub>v v) $ j = k * u $ j + l * v $ j"
seq_nil_left: fixes a :: "'b::refinement_lattice" shows "nil ; (a::'b::refinement_lattice) = a"
product_sigma_finite_def: fixes M :: "'a \<Rightarrow> 'b measure" shows "product_sigma_finite M \<equiv> \<forall>i. sigma_finite_measure (M i)"
wf23: shows "mat n1 nc (mat_mult n1 m2 m3)"
measure_space_inter: fixes s :: "'b set"   and M :: "'b pmf"   and t :: "'b set" assumes "s \<in> measure_pmf.events M"   and "t \<in> measure_pmf.events M"   and "measure_pmf.prob M t = measure_pmf.prob M (space (measure_pmf M))" shows "measure_pmf.prob M (s \<inter> t) = measure_pmf.prob M s"
tr_def: shows "tr \<equiv> trace (ioa.asig A) e"
lexLessBacktrack: fixes p :: "(Literal \<times> bool) list"   and a :: "(Literal \<times> bool) list"   and x :: "Literal" assumes "p = prefixBeforeLastDecision a"   and "decisions a \<noteq> []" shows "(p @ [(x, False)], a) \<in> lexLess"
18: fixes x :: "'a" shows "- x \<squnion> - - x = top"
rep_eq: fixes x :: "'e"   and xa :: "'c" shows "Rep_formula (Pred x xa) = Pred\<^sub>\<alpha> x xa"
intros: fixes q :: "'d"   and f :: "'e"   and qs :: "'d list"   and \<delta> :: "('d, 'e) ta_rule set"   and ts :: "'e tree list" assumes "q \<rightarrow> f qs \<in> \<delta>"   and "length ts = length qs"   and "\<And>i. i < length qs \<Longrightarrow> accs \<delta> (ts ! i) (qs ! i)" shows "accs \<delta> (NODE f ts) q"
Map_in_Hom: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "Map f \<in> HOM (Dom f) (Cod f)"
measure_measure_6: shows "measure_space \<Omega>' A' \<mu>'"
Min_ge_iff: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
distinct_map_fst_filter: fixes xs :: "('a \<times> 'b) list"   and P :: "'a \<times> 'b \<Rightarrow> bool" assumes "distinct (list.map fst xs)" shows "distinct (list.map fst (List.filter P xs))"
poly_root_factor_2: fixes q :: "'a poly"   and r :: "'a"   and k :: "'a" shows "(poly (q * [:r, 1::'a:]) k = (0::'a)) = (k = - r \<or> poly q k = (0::'a))"
uconnected_symI: fixes u :: "'b"   and v :: "'b"   and E :: "'b uprod set" assumes "(u, v) \<in> uconnected E" shows "(v, u) \<in> uconnected E"
bij_def: fixes f :: "'b \<Rightarrow> 'c" shows "bij f = (inj f \<and> top_pres ((`) f))"
in_square2: fixes a :: "nat"   and b :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" shows "((a, b) \<in> square2 n i j) = (i \<le> a \<and> a < 2 ^ n + i \<and> j \<le> b \<and> b < 2 ^ n + j)"
fw_upd_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and k :: "nat"   and i :: "nat"   and j :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "linordered_ab_monoid_add.fw_upd plus less_eq m k i j \<equiv> upd m i j (ord.min less_eq (m i j) (plus (m i k) (m k j)))"
left_unique_vec_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "left_unique R" shows "left_unique (vec_rel R)"
unifiers_def: fixes E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "unifiers E = {\<sigma>. \<forall>p\<in>E. subst_atm_abbrev (fst p) \<sigma> = subst_atm_abbrev (snd p) \<sigma>}"
af\<^sub>G_sat_core: fixes \<G> :: "'a ltl set"   and \<phi> :: "'a ltl"   and w :: "'a set list" assumes "Only_G \<G>"   and "\<G> \<Turnstile>\<^sub>P \<phi>" shows "\<G> \<Turnstile>\<^sub>P af\<^sub>G \<phi> w"
L2_set_le_sum_abs: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" shows "L2_set f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
theNFTBoth_def: fixes cl :: "('a, 'b, 'c) cmd list"   and dl :: "('d, 'e, 'f) cmd list" shows "theNFTBoth cl dl \<equiv> theNFT cl \<inter> theNFT dl"
filterlim_within_subset: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and l :: "'b filter"   and x :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "filterlim f l (topological_space.at_within open x S)"   and "T \<subseteq> S" shows "filterlim f l (topological_space.at_within open x T)"