hcju/isabelleps huggingface dataset:
fp_cop_F_range_inv': fixes x :: "'a"   and ds :: "'b set" assumes "x \<in> ManiPref.fp_cop_F ds" shows "x \<in> Field (override_on Pd Pds' ds' (Xd x)) \<and> Xd x \<in> ds"
rel_is_equivalence: shows "equivalence (Sigma (top.neighborhoods \<pp>) sheaf_spec) {(x, y). dl.rel x y}"
list_comb_app: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app f (xs @ ys) = term.list_comb app (term.list_comb app f xs) ys"
UN: fixes K :: "'a set set"   and S :: "'a set"   and B :: "'a set set" assumes "\<And>U. U \<in> K \<Longrightarrow> generated_topology S B U" shows "generated_topology S B (\<Union> K)"
x1: shows "x [^]\<^bsub>R\<^esub> CARD('a) ^ m' = x"
sums_le: fixes f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c" assumes "\<And>n. f n \<le> g n"   and "f sums s"   and "g sums t" shows "s \<le> t"
simps_5: fixes n :: "nat"   and f :: "recf"   and g :: "recf" shows "arity (Pr n f g) = Suc n"
f1: shows "low_equal s1 s2 \<and> t1 = snd (fst (sub_instr_sub1 (fst instr) (if fst instr = arith_type SUB \<or> fst instr = arith_type SUBcc then user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1))) - get_operand2 (snd instr) s1 else user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1))) - get_operand2 (snd instr) s1 - ucast (get_icc_C (cpu_reg_val PSR (snd (fst (get_curr_win () s1)))))) (user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1)))) (get_operand2 (snd instr) s1) (snd (fst (write_reg (if get_operand_w5 (snd instr ! 3) \<noteq> 0 then if fst instr = arith_type SUB \<or> fst instr = arith_type SUBcc then user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1))) - get_operand2 (snd instr) s1 else user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s1))) - get_operand2 (snd instr) s1 - ucast (get_icc_C (cpu_reg_val PSR (snd (fst (get_curr_win () s1))))) else user_reg_val (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s1))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s1)))))))) \<and> t2 = snd (fst (sub_instr_sub1 (fst instr) (if fst instr = arith_type SUB \<or> fst instr = arith_type SUBcc then user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2))) - get_operand2 (snd instr) s2 else user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2))) - get_operand2 (snd instr) s2 - ucast (get_icc_C (cpu_reg_val PSR (snd (fst (get_curr_win () s2)))))) (user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2)))) (get_operand2 (snd instr) s2) (snd (fst (write_reg (if get_operand_w5 (snd instr ! 3) \<noteq> 0 then if fst instr = arith_type SUB \<or> fst instr = arith_type SUBcc then user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2))) - get_operand2 (snd instr) s2 else user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! Suc 0)) (snd (fst (get_curr_win () s2))) - get_operand2 (snd instr) s2 - ucast (get_icc_C (cpu_reg_val PSR (snd (fst (get_curr_win () s2))))) else user_reg_val (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s2))))))))"
partition_on_eq_implies_eq_carrier: fixes A :: "'b set"   and Q :: "'b set set"   and B :: "'b set" assumes "partition_on A Q"   and "partition_on B Q" shows "A = B"
excess_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and v :: "nat" assumes "Network c s t" shows "Network.excess c f v \<equiv> sum f (Graph.incoming c v) - sum f (Graph.outgoing c v)"
mcalc_iff: fixes \<sigma> :: "'tp"   and c :: "('b, 'c) lit list" shows "(\<sigma> \<turnstile> c) = (infTp \<sigma> \<or> (\<forall>x\<in>nvC c. TE.tpOfV x \<noteq> \<sigma>))"
map_sings: fixes xs :: "'b list"   and B :: "'b set" assumes "xs \<in> lists B" shows "map (\<lambda>x. [x]) xs \<in> lists (sings B)"
A_RCOSETS_def': fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "a_rcosets\<^bsub>G\<^esub> H \<equiv> \<Union>a\<in>carrier G. {H +>\<^bsub>G\<^esub> a}"
zero_cblinfun_image: fixes S :: "'b ccsubspace" shows "0 *\<^sub>S S = 0"
emeasure_return_pmf: fixes x :: "'c"   and X :: "'c set" shows "emeasure (measure_pmf (return_pmf x)) X = indicator X x"
sup_aci_2: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup (sup x y) z = sup x (sup y z)"
object_ptr_kinds_eq: shows "object_ptr_kinds h = object_ptr_kinds h'"
cong: fixes charts :: "('e, 'f) chart set"   and k :: "enat" shows "c_manifold.atlas charts k \<equiv> c_manifold.atlas charts k"
asIMOD_igSubstInp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and val :: "'varSort \<Rightarrow> 'var \<Rightarrow> 'b"   and ys :: "'varSort"   and Y :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and y :: "'var"   and inp :: "'c \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b) option" assumes "sWlsVal SEM val" shows "lift (\<lambda>X. X val) (igSubstInp (asIMOD SEM) ys Y y inp) = lift (\<lambda>X. X (val (y := Y val)_ys)) inp"
cons_post_rule: fixes P :: "assn"   and c :: "'b Heap"   and Q :: "'b \<Rightarrow> assn"   and Q' :: "'b \<Rightarrow> assn" assumes "<P> c <Q>"   and "\<And>x. Q x \<Longrightarrow>\<^sub>A Q' x" shows "<P> c <Q'>"
sg3: shows "\<not> t mod cycleLength (nC i) mem schedule (nC j)"
set_restrict_spmf: fixes p :: "'c spmf"   and A :: "'c set" shows "set_spmf (p \<upharpoonleft> A) = set_spmf p \<inter> A"
OstrowskiTr2: fixes v :: "'a \<Rightarrow> ant"   and s :: "'a"   and t :: "'a" assumes "valuation K v"   and "s \<in> carrier K"   and "t \<in> carrier K"   and "0 \<le> v s"   and "v t < 0" shows "0 < v (1\<^sub>r \<plusminus> -\<^sub>a t \<cdot>\<^sub>r (s \<plusminus> t)\<^bsup>\<hyphen> K\<^esup>)"
lmap_eq_LCons_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist"   and y :: "'a"   and ys :: "'a llist" shows "(lmap f xs = LCons y ys) = (\<exists>x xs'. xs = LCons x xs' \<and> y = f x \<and> ys = lmap f xs')"
disj: shows "\<forall>c. re c \<sqinter> cl c = bot"
prj_in_hom_6: shows "\<guillemotleft>doms.Prj\<^sub>0\<^sub>0 : Chn (local.dom \<mu> \<star> local.dom \<nu> \<star> local.dom \<pi>) \<rightarrow>\<^sub>C doms.\<pi>.apex\<guillemotright>"
is_reduced_GB_singleton: fixes f :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_reduced_GB {f} = (lc f = (1::'c))"
P4: shows "\<lfloor>\<lambda>w. \<forall>x. Dependent x w \<longrightarrow> (\<exists>xa. (Independent xa \<^bold>\<and> xa explains x) w)\<rfloor>"
HSem_def': fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrace>\<Gamma>\<rbrace>\<rho> = (\<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<N>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>x\<^esub>)"
Iff_MP_left: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and C :: "fmla" assumes "H \<turnstile> A IFF B"   and "insert A H \<turnstile> C" shows "insert B H \<turnstile> C"
Un_Int_eq_4: fixes T :: "'b set"   and S :: "'b set" shows "T \<inter> (S \<union> T) = T"
integrable_combine_division: fixes \<D> :: "'a set set"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<D> division_of S"   and "\<And>i. i \<in> \<D> \<Longrightarrow> f integrable_on i" shows "f integrable_on S"
eball_empty: fixes r :: "ereal"   and z :: "'a" assumes "r \<le> 0" shows "eball z r = {}"
a_smult: shows "a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"
if_sets: fixes a :: "'a" shows "(\<lambda>x. if x \<in> S then f x else 0) -` {f a} \<in> sets lebesgue"
PSimQ: shows "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
refl: fixes a :: "'b zmultiset" shows "a \<subseteq>#\<^sub>z a"
basis_of_hull: fixes G :: "'b list set" shows "\<BB> G = \<BB> \<langle>G\<rangle>"
majsI: fixes S :: "process set" assumes "N div 2 < card S" shows "S \<in> majs"
map_prod_surj_on: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and A' :: "'c set"   and g :: "'f \<Rightarrow> 'e"   and B :: "'f set"   and B' :: "'e set" assumes "f ` A = A'"   and "g ` B = B'" shows "map_prod f g ` (A \<times> B) = A' \<times> B'"
in_span_delete: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and S :: "'e set"   and b :: "'e" assumes "vector_space scale"   and "a \<in> module.span scale S"   and "a \<notin> module.span scale (S - {b})" shows "b \<in> module.span scale (insert a (S - {b}))"
pseudo_proj_True_stake_image: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat"   and w :: "bool stream"   and x :: "bool stream" assumes "infinite_coin_toss_space p M"   and "stake n w = stake n x" shows "pseudo_proj_True n w = pseudo_proj_True n x"
le_ordertype_obtains_subset: fixes \<beta> :: "V"   and \<alpha> :: "V"   and H :: "V set" assumes "\<beta> \<le> \<alpha>"   and "ordertype H VWF = \<alpha>"   and "small H"   and "Ord \<beta>" obtains G :: "V set" where "G \<subseteq> H" and "ordertype G VWF = \<beta>"
action_simple: fixes s :: "'s"   and ct :: "'s \<Rightarrow> 's pmf" assumes "s \<in> S" shows "action (simple ct s) = ct s"
thd0: shows "det (\<chi>i. if i = k then row k A + (\<Sum>i\<in>UNIV - {k}. x $ i *s row i A) else row i A) = det A"
inv_mult_units: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Units G"   and "y \<in> Units G" shows "inv (x \<otimes> y) = inv y \<otimes> inv x"
m_def: fixes x :: "'a" shows "m x = nat \<lfloor>c * er x\<rfloor>"
sorted_wrt_dist_nbors_diff: fixes p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list"   and q :: "(real, 'b) vec"   and n :: "nat" assumes "sorted_wrt_dist p ps" shows "\<forall>r\<in>set ps \<union> {q} - set (upd_nbors n p q ps). \<forall>s\<in>set (upd_nbors n p q ps). dist s p \<le> dist r p"
fitsD: fixes G :: "prog"   and s :: "st"   and a' :: "val"   and T :: "ty" assumes "G,s\<turnstile>a' fits T" shows "(\<exists>pt. T = PrimT pt) \<or> (\<exists>t. T = RefT t) \<and> a' = Null \<or> (\<exists>t. T = RefT t) \<and> a' \<noteq> Null \<and> G\<turnstile>obj_ty (lookup_obj s a')\<preceq>T"
sq_mtx_diag_def: shows "sq_mtx_diag \<equiv> map_fun id to_mtx diag_mat"
l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas_axioms: shows "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas ShadowRootClass.type_wf"
a2: shows "j < dim_col (M\<^sup>\<star> * N\<^sup>\<star>)"
is_extensional: fixes f :: "'c" assumes "\<not> \<tau>.A.arr f" shows "(K \<circ> \<tau>'\<tau>.map) f = K.B.null"
add_incr: fixes b :: "'a"   and a :: "'a" assumes "(0::'a) \<le> b" shows "a \<le> a + b"
a: shows "[a ^ (p - 1) = 1] (mod p) \<and> (\<forall>x>0. x \<le> p - 2 \<longrightarrow> [a ^ x \<noteq> 1] (mod p)) \<and> 0 < a \<and> a < p"
Nonce_secrecy: fixes evs :: "event list"   and NB :: "nat" assumes "evs \<in> yahalom" shows "\<forall>KK\<subseteq>- range shrK. (\<forall>K\<in>KK. K \<in> symKeys \<longrightarrow> \<not> KeyWithNonce K NB evs) \<longrightarrow> (Nonce NB \<in> analz (Key ` KK \<union> knows Spy evs)) = (Nonce NB \<in> analz (knows Spy evs))"
gcd_eq_0_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(gcd a b = zero) = (a = zero \<and> b = zero)"
inv_lc_q_2: shows "inv lead_coeff q \<otimes> lead_coeff q = \<one>"
max_fun_diff_alt: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" shows "s.max_fun_diff f g = s.maxim ((SUPP f \<union> SUPP g) \<inter> {a. f a \<noteq> g a})"
conjugate_conjugate_comp: shows "conjugate \<circ> conjugate = id"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Dom (t \<^bold>\<star> u) = Dom t \<^bold>\<star> Dom u"
chain_lemma: fixes S :: "nat \<Rightarrow> 'a"   and i :: "nat"   and k :: "nat" assumes "chain S" shows "S i \<sqsubseteq> S k \<or> S k \<sqsubseteq> S i"
stable_decision: fixes Quorum :: "process set set"   and tr :: "v_state list"   and i :: "nat"   and s :: "v_state"   and t :: "v_state"   and j :: "nat"   and p :: "process"   and v :: "val" assumes "quorum_process Quorum"   and "tr \<in> beh (quorum_process.v_TS Quorum)"   and "i < length tr"   and "s = tr ! i"   and "t = tr ! (i - j)"   and "decisions s p = Some v" shows "decisions t p = Some v"
pCons_eq_0_iff: fixes a :: "'c"   and p :: "'c poly" shows "(pCons a p = 0) = (a = (0::'c) \<and> p = 0)"
inject_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(LCall x2 = LCall y2) = (x2 = y2)"
le_iff_sup: fixes x :: "'g multiset"   and y :: "'g multiset" shows "(x \<subseteq># y) = (x \<union># y = y)"
memo_divisors_nat_def: shows "memo_divisors_nat \<equiv> memo_nat 0 100 divisors_nat_copy"
listset_length: fixes xs :: "'a list"   and Xs :: "'a set list" assumes "xs \<in> listset Xs" shows "length xs = length Xs"
card_lucas_lehmer_Units: fixes m :: "nat" assumes "1 < m" shows "card (Units (lucas_lehmer_ring_mod m)) < m\<^sup>2"
noFaultnI: fixes \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and s :: "('d, 'e) xstate"   and n :: "nat"   and f :: "'e" assumes "\<And>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t \<Longrightarrow> t \<noteq> Fault f" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>{Fault f}"
green_typeI_chain_axioms: shows "green_typeI_chain i j F two_chain_typeI s"
ConstrainsI: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set" assumes "\<And>act s s'. \<lbrakk>act \<in> Acts F; (s, s') \<in> act; s \<in> A\<rbrakk> \<Longrightarrow> s' \<in> A'" shows "F \<in> A Co A'"
exec_While\<^sub>C': fixes \<sigma> :: "'a control_state_scheme"   and P :: "'a control_state_scheme \<Rightarrow> bool"   and B\<^sub>1 :: "'a control_state_scheme \<Rightarrow> (unit \<times> 'a control_state_scheme) option"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> _while_C P B\<^sub>1 ;- M) = (\<sigma> \<Turnstile> M)"
dim_eq_0: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and Basis :: "'c set"   and S :: "'c set" assumes "finite_dimensional_vector_space scale Basis" shows "(vector_space.dim scale S = 0) = (S \<subseteq> {0::'c})"
simps: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c stream" shows "holds P xs = P (shd xs)"
distinct_11: shows "cospan_shape.AA \<noteq> cospan_shape.BB"
a_get_root_node_si_locs_def: shows "get_root_node_si_locs = get_ancestors_si_locs"
pickEff: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and r :: "'a"   and s :: "'b" assumes "RuleSystem_Defs.enabled eff r s" shows "eff r s (RuleSystem_Defs.pickEff eff r s)"
dc: shows "d 1\<^sub>\<pi> = 1\<^sub>\<sigma>"
Diagonalize_Diag: fixes t :: "'a local.term" assumes "Diag t" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> = t"
bl_to_bin_aux_app_cat: fixes bsa :: "bool list"   and bs :: "bool list"   and w :: "int" shows "bl_to_bin_aux (bsa @ bs) w = concat_bit (length bs) (bl_to_bin bs) (bl_to_bin_aux bsa w)"
INF_principal_finite: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c set" assumes "finite X" shows "(INF x\<in>X. principal (f x)) = principal (\<Inter> (f ` X))"
fpairs_vempty: shows "fpairs []\<^sub>\<circ> = []\<^sub>\<circ>"
update_assignment: fixes v :: "'a"   and u :: "'b"   and us :: "('a \<times> 'b) list"   and as :: "'a \<Rightarrow> 'b" assumes "distinct (map fst ((v, u) # us))" shows "update_assignment ((v, u) # us) as = update_assignment us (as(v := u))"
converse_inject: fixes r :: "('b \<times> 'a) set"   and s :: "('b \<times> 'a) set" shows "(\<smile> r = \<smile> s) = (r = s)"
prefix_length_le_eq_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(prefix xs ys \<and> length ys \<le> length xs) = (xs = ys)"
is_sig_redD1: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg top_tail F p" shows "ord_term_lin.is_le_rel sing_reg"
comp_simp: fixes g :: "'c local.term set \<times> 'c local.term set"   and f :: "'c local.term set \<times> 'c local.term set" assumes "\<F>C.seq (fst g) (fst f)"   and "\<F>C.seq (snd g) (snd f)" shows "\<F>C.CC.comp g f = (fst g \<cdot> fst f, snd g \<cdot> snd f)"
cnj_units_1: shows "Quaternions.cnj \<i> = - \<i>"
complex_of_real_integrable_eq: fixes M :: "'e measure"   and f :: "'e \<Rightarrow> real" shows "complex_integrable M (\<lambda>x. complex_of_real (f x)) = integrable M f"
qe: fixes as :: "'a list" assumes "\<forall>a\<in>set as. depends\<^sub>0 a \<and> \<not> solvable\<^sub>0 a" shows "I (qe as) xs = (\<exists>x. \<forall>a\<in>set as. I\<^sub>a a (x # xs))"
binary_functor_def: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'c \<Rightarrow> 'd" shows "binary_functor A1 A2 B F \<equiv> (category A1 \<and> category A2) \<and> category B \<and> product_category A1 A2 \<and> functor (product_category.comp A1 A2) B F"
ivl_rep_of_sets_def: fixes XS :: "'c set set" shows "ivl_rep_of_sets XS = SPEC (\<lambda>(i, s). i \<le> s \<and> (\<forall>X\<in>XS. X \<subseteq> {i..s}))"
step_desc: fixes r :: "('Q, 'L) ta_rule"   and it :: "('Q, 'L) ta_rule set"   and \<Sigma> :: "'\<Sigma>" assumes "r \<in> it"   and "it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}"   and "cinvar it \<Sigma>" shows "(\<alpha>s \<Sigma>, \<alpha>s (cstep r \<Sigma>)) \<in> brw_inner_step r"
inf_commute: fixes x :: "'c multiset"   and y :: "'c multiset" shows "x \<inter># y = y \<inter># x"
split_mult_neg_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a \<and> less_eq b zero \<or> less_eq a zero \<and> less_eq zero b" shows "less_eq (times a b) zero"
that: shows "linear_on src.diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X"
adaptive_rtol_spec_def: shows "adaptive_rtol_spec = SPEC (\<lambda>x. True)"
intros_3: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and u :: "('c, 'd) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "App t u \<rightarrow>\<^sub>e\<^sub>m\<^sub>b App s u"
distinct_19: fixes x41 :: "bexp"   and x42 :: "com"   and x43 :: "com"   and x51 :: "bexp"   and x52 :: "com" shows "IF x41 THEN x42 ELSE x43 \<noteq> WHILE x51 DO x52"
ide_char': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C" shows "partial_magma.ide (span_vertical_category.vcomp C) \<mu> = identity_arrow_of_spans C \<mu>"
index_mat_multrow_mat_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and k :: "nat"   and a :: "'b" assumes "i < n"   and "j < n" shows "multrow_mat n k a $$ (i, j) = (if k = i \<and> k = j then a else if i = j then 1::'b else (0::'b))"
lb1'_def: shows "lb1' = (\<lambda>x. C / ln (ln (real x)) - 10 * C / (ln (ln (real x)))\<^sup>2)"
li_smaller_than_gen: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "A \<subseteq> carrier V"   and "B \<subseteq> carrier V"   and "lin_indpt A"   and "gen_set B" shows "card A \<le> card B"
PiE_def: fixes S :: "'g set"   and T :: "'g \<Rightarrow> 'h set" shows "Pi\<^sub>E S T = Pi S T \<inter> extensional S"
is_nextElem_edges2: fixes f :: "face"   and a :: "nat"   and b :: "nat" assumes "distinct (vertices f)"   and "is_nextElem (vertices f) a b" shows "(a, b) \<in> \<E> f"
lem2: fixes x :: "'b"   and y :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "x \<noteq> y" shows "(x, M = y, N) = (y \<in># M \<and> N = x, M - (y, {#}))"
g: shows "pi \<bullet> (\<parallel>(A IMP B)\<parallel>) = \<parallel>(A IMP B)\<parallel>"
exec_1_eq: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list" shows "exec_1 P = {(\<sigma>, \<sigma>'). exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>}"
more_temp_simps3_5: fixes F :: "'a \<Rightarrow> bool" shows "\<not> \<diamond>F \<equiv> \<box>\<not> F"
count_roots_interval_rat_code: fixes p :: "int poly" shows "count_roots_interval_rat p = (let rp = of_int_poly p; ps = sturm_rat rp in Root_Info (\<lambda>a b. sign_changes_rat ps a - sign_changes_rat ps b + (if poly rp a = 0 then 1 else 0)) (\<lambda>a. sign_changes_neg_number_rootat ps - sign_changes_rat ps a))"
midpoint_distinct_3: fixes I :: "'p"   and B :: "'p"   and A :: "'p" assumes "I \<noteq> B"   and "I Midpoint A B" shows "A \<noteq> B \<and> I \<noteq> A"
pres_typeD: fixes step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and n :: "nat"   and A :: "'c set"   and s :: "'c"   and p :: "nat"   and q :: "nat"   and s' :: "'c" assumes "pres_type step n A"   and "s \<in> A"   and "p < n"   and "(q, s') \<in> set (step p s)" shows "s' \<in> A"
irreflexive_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "irreflexive x"   and "irreflexive y" shows "irreflexive (x \<squnion> y)"
distinct_1: fixes x1 :: "'e"   and x21 :: "nat set"   and x22 :: "'e" shows "broadcast x1 \<noteq> groupcast x21 x22"
mset_subsetD: fixes A :: "'a multiset"   and B :: "'a multiset"   and x :: "'a" assumes "A \<subset># B"   and "x \<in># A" shows "x \<in># B"
inv_nsqn_sqn: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> kD rt"   and "the (flag rt ip) = Aodv_Basic.inv" shows "nsqn rt ip = sqn rt ip - 1"
edges_disj_def: fixes g :: "graph" shows "edges_disj g \<equiv> \<forall>f\<in>\<F> g. \<forall>f'\<in>\<F> g. f \<noteq> f' \<longrightarrow> \<E> f \<inter> \<E> f' = {}"
bind_in_space: fixes M :: "'a measure"   and A :: "'a \<Rightarrow> 'b measure"   and N :: "'b measure" assumes "subprob_space M"   and "A \<in> M \<rightarrow>\<^sub>M subprob_algebra N" shows "M \<bind> A \<in> space (subprob_algebra N)"
run_bind_env: fixes bind :: "'a \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and x :: "('b, 'a) envT"   and f :: "'c \<Rightarrow> ('b, 'a) envT"   and r :: "'b" shows "run_env (bind_env bind x f) r = bind (run_env x r) (\<lambda>y. run_env (f y) r)"