hcju/isabelleps huggingface dataset:
length_list_of_eucl: fixes x :: "'e" shows "length (list_of_eucl x) = DIM('e)"
BSI_def: fixes \<V> :: "'a V_rec"   and Tr :: "'a list set" shows "BSI \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. \<beta> @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> (\<exists>\<alpha>'. \<beta> @ [c] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
phi': fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat"   and z :: "'c list \<times> bool list \<times> 'c list" assumes "BIT_Off acts qs init" shows "BIT_Off.phi acts qs init n z = (\<Sum>(x, y)\<in>Inv (fst z) (BIT_Off.s_A acts qs init n). if fst (snd z) ! index init y then 2 else 1)"
sminus_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "'a"   and t :: "'b" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.sminus pair_of_term term_of_pair v t = term_of_pair (term_powerprod.pp_of_term pair_of_term v - t, term_powerprod.component_of_term pair_of_term v)"
mkCone_def: fixes p0 :: "'c"   and p1 :: "'c" shows "mkCone p0 p1 \<equiv> \<lambda>j. if j = J.AA then p0 else if j = J.BB then p1 else if j = J.AT then f0 \<cdot> p0 else if j = J.BT then f1 \<cdot> p1 else if j = J.TT then f0 \<cdot> p0 else C.null"
x: shows "x = (1 - u) *\<^sub>R a + u *\<^sub>R b"
step_cjump_true: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and n :: "nat"   and d :: "'dyn"   and H :: "'henv"   and \<Sigma> :: "'dyn list"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.ICJump n"   and "is_true d" shows "Sinca.step (State F H (Frame f pc (d # \<Sigma>) # st)) (State F H (Frame f n \<Sigma> # st))"
distinct_1: fixes x1 :: "'e"   and x2 :: "'f" shows "Inl x1 \<noteq> Inr x2"
eq_as_subsets: fixes G :: "('c, 'b) labeled_graph"   and e\<^sub>1 :: "'c allegorical_term"   and e\<^sub>2 :: "'c allegorical_term" shows "G \<Turnstile> (e\<^sub>1, e\<^sub>2) = (G \<Turnstile> e\<^sub>1 \<sqsubseteq> e\<^sub>2 \<and> G \<Turnstile> e\<^sub>2 \<sqsubseteq> e\<^sub>1)"
acc_modi_top: fixes a :: "acc_modi" assumes "Public \<le> a" shows "a = Public"
tcontra_def: shows "tcontra \<equiv> tcopy |+| H |+| dither"
wset_thread_ok_conv_dom: fixes ws :: "'a \<Rightarrow> 'b wait_set_status option"   and ts :: "'a \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option" shows "wset_thread_ok ws ts = (dom ws \<subseteq> dom ts)"
nat_less_induct: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "\<And>n. \<forall>m<n. P m \<Longrightarrow> P n" shows "P n"
distinct_is_map: fixes m :: "('a \<times> 'b) list" assumes "distinct (map fst m)" shows "is_map (set m)"
infinite_chain_intro: fixes r :: "('a \<times> 'a) set" assumes "\<forall>(a, b)\<in>r. \<exists>c. (b, c) \<in> r"   and "r \<noteq> {}" shows "\<exists>f. \<forall>i. (f i, f (Suc i)) \<in> r"
simps_11: fixes x41 :: "typ"   and x42 :: "Core.term"   and x11 :: "String.literal"   and x12 :: "typ" shows "Abs x41 x42 \<noteq> Ct x11 x12"
Ln_eq_iff: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0" shows "(Ln w = Ln z) = (w = z)"
assms_2: shows "q \<in> dgrad_p_set d m"
convex_in_outside: fixes S :: "'a set"   and z :: "'a" assumes "convex S"   and "z \<notin> S" shows "z \<in> outside S"
Rep_preal: fixes x :: "preal" shows "Rep_preal x \<in> Collect Dedekind_Real.cut"
st2dims0: shows "st2.dims0 = replicate (Suc (Suc k)) 2"
phi_5: shows "\<phi> ` {0..1} = {0..1}"
bal_i1: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool" assumes "RBTi1 TYPE('b) bal_i" shows "bal_i 1 1"
a': shows "( \<Gamma> \<Rightarrow>* \<Delta>, n) \<in> derivable (ext R R2 M N)"
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 :: "('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> 'ag" 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"
fb_impossibility_3_3_axioms: shows "fb_impossibility_3_3 agents alts scf A1 A2 A3 a b c"
indicator_times: fixes A :: "'c set"   and B :: "'d set"   and x :: "'c \<times> 'd" shows "indicator (A \<times> B) x = indicator A (fst x) * indicator B (snd x)"
map_disj_eq_dom_left: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h\<^sub>0' :: "'c \<Rightarrow> 'd option" assumes "h\<^sub>0 \<bottom> h\<^sub>1"   and "dom h\<^sub>0' = dom h\<^sub>0" shows "h\<^sub>0' \<bottom> h\<^sub>1"
simps_1: fixes t1 :: "'b trm"   and t2 :: "'b trm" shows "safe_formula (Eq t1 t2) = (is_Const t1 \<or> is_Const t2)"
map_of_eq_empty_iff: fixes xys :: "('a \<times> 'b) list" shows "(table_of xys = Map.empty) = (xys = [])"
rcis_plus_2pi: fixes y :: "real"   and x :: "real" shows "rcis y (x + 2 * pi) = rcis y x"
app\<^sub>i_Pop: fixes P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and T :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Pop, P, pc, mxs, T\<^sub>r, T # ST, LT) = True"
set_mode_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_mode_locs shadow_root_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
fun_def: fixes X :: "exp" shows "fun X = the_elem (\<Union>U\<in>Rep_Exp X. {freefun U})"
recurse_Leaf: fixes f :: "'e \<Rightarrow> ('c, 'd) state"   and x :: "'e" shows "recurse f (Tree_Relabelling.tree.Leaf x) = pure Tree_Relabelling.tree.Leaf \<diamondop> f x"
wf_ast_problem_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "wf_ast_problem problem \<equiv> ast_problem.well_formed problem"
fst_cs_changed_by_recv_recording: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "fst (cs c cid) \<noteq> fst (cs c' cid)"   and "channel cid = Some (p, q)" shows "snd (cs c cid) = Recording \<and> (\<exists>p q u u' m. ev = Recv cid q p u u' m)"
matrix_vector_mult_in_columnspace: fixes A :: "((real, 'c) vec, 'b) vec"   and x :: "(real, 'c) vec" shows "A *v x \<in> span (columns A)"
o_substC: fixes \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm" shows "substC \<pi>1 \<circ> substC \<pi>2 = substC (subst \<pi>1 \<circ> \<pi>2)"
elims_3: fixes x :: "nat \<Rightarrow> bool"   and xa :: "lit uprod" assumes "\<not> sat_clause x xa" obtains \<sigma> :: "nat \<Rightarrow> bool" and l1 :: "lit" and l2 :: "lit" where "x = \<sigma>" and "xa = Upair l1 l2" and "\<not> (sat_lit \<sigma> l1 \<or> sat_lit \<sigma> l2)"
ltake_prepend: fixes i :: "nat"   and xs :: "'d list"   and t :: "'d llist" shows "Co_Snapshot.ltake i (prepend xs t) = (if i \<le> length xs then take i xs else xs @ Co_Snapshot.ltake (i - length xs) t)"
outs_\<I>_mono: fixes \<I> :: "('d, 'e) \<I>"   and \<I>' :: "('d, 'e) \<I>" assumes "\<I> \<le> \<I>'" shows "outs_\<I> \<I> \<subseteq> outs_\<I> \<I>'"
obs_singleton: fixes n :: "'node"   and S :: "'node set" shows "(\<exists>m. obs n (wod_backward_slice S) = {m}) \<or> obs n (wod_backward_slice S) = {}"
fminusE: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset" assumes "c |\<in>| A |-| B" obtains "c |\<in>| A" and "c |\<notin>| B"
le_bet: fixes C :: "'p"   and D :: "'p"   and A :: "'p"   and B :: "'p" assumes "C D Le A B" shows "\<exists>X. Bet A X B \<and> Cong A X C D"
inter_guards_exec_noFault: fixes c1 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "(c1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "\<not> isFault t" shows "\<Gamma>\<turnstile> \<langle>c1,s\<rangle> \<Rightarrow> t \<and> \<Gamma>\<turnstile> \<langle>c2,s\<rangle> \<Rightarrow> t"
I_SucSuc_I0: shows "ideal (I (Suc (Suc n))) R \<and> ideal (I 0) R"
bound0at_l: fixes p :: "fm" assumes "isatom p"   and "bound0 p" shows "isrlfm p"
necessity_2: assumes "\<forall>A. admits_diagonal_reduction A" shows "\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})"
filtermap_nhds_shift: fixes d :: "'a"   and a :: "'a" shows "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d)"
dom_simp: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VxVxV.arr f" shows "B.VxVxV.dom f = (B.dom (fst f), B.VxV.dom (snd f))"
nu_fusion_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq h"   and "order.has_greatest_postfixpoint less_eq g"   and "order.has_greatest_fixpoint less_eq h"   and "ord.lifted_less_eq less_eq (h \<circ> u) (u \<circ> g)" shows "less_eq (order.the_greatest_fixpoint less_eq h) (u (order.the_greatest_postfixpoint less_eq g))"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and option :: "'d option" shows "h (case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (case option of None \<Rightarrow> h f1 | \<lfloor>x\<rfloor> \<Rightarrow> h (f2 x))"
Enc_synth: fixes X :: "msg"   and K :: "msg"   and H :: "msg set" assumes "Enc X K \<in> synth H" obtains "Enc X K \<in> H"   | "X \<in> synth H" and "K \<in> synth H"
ca_cdux: shows "c \<otimes> a = c \<otimes> somegcd G a b \<otimes> u \<otimes> x__"
f_def: shows "f = (\<lambda>n. of_nat n *s x)"
map_le_imp_upd_le: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b" assumes "m1 \<subseteq>\<^sub>m m2" shows "m1(x := None) \<subseteq>\<^sub>m m2(x \<mapsto> y)"
P5: shows "M' Midpoint S R \<and> M' Midpoint C D"
res1: shows "min r1 r2 = real_of_int (sint (wmin w1 w2))"
irrefl_prec_F: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Q :: "'c set"   and entails_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G_q :: "'c \<Rightarrow> 'b inference set"   and Red_I_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b inference set"   and Red_F_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set"   and \<G>_F_q :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and \<G>_I_q :: "'c \<Rightarrow> 'a inference \<Rightarrow> 'b inference set option"   and Inf_FL :: "('a \<times> 'd) inference set"   and Equiv_F :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Prec_F :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Prec_L :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and active :: "'d"   and C :: "'a" assumes "given_clause_basis Bot_F Inf_F Bot_G Q entails_q Inf_G_q Red_I_q Red_F_q \<G>_F_q \<G>_I_q Inf_FL Equiv_F Prec_F Prec_L active" shows "\<not> Prec_F C C"
le_INF_iff: fixes u :: "'a::complete_lattice"   and f :: "'b::type \<Rightarrow> 'a::complete_lattice"   and A :: "'b::type set" shows "((u::'a::complete_lattice) \<le> \<Sqinter> ((f::'b::type \<Rightarrow> 'a::complete_lattice) ` (A::'b::type set))) = (\<forall>i::'b::type\<in>A. u \<le> f i)"
neg_eq_iff_add_eq_0: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "(uminus a = b) = (plus a b = zero)"
underS_suc: fixes b :: "'b"   and A :: "'b set" assumes "b \<in> s.underS (s.suc A)"   and "A \<subseteq> Field s" shows "\<exists>a\<in>A. b \<in> s.under a"
N1: shows "nonneg_cexpr V \<Gamma> e1"
nprv_dsjIL: fixes F :: "'fmla set"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F \<phi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F (dsj \<phi> \<psi>)"
less_asym: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c"   and C :: "bool" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b"   and "b \<prec>\<^bsub>D\<^esub> a" shows "C"
prems_2: shows "Broadcast m3 \<in> set (history i)"
C: shows "\<forall>x\<in>cball 0 pi. cmod (g x) \<le> C"
simps_6: fixes uw :: "pdf_dist"   and e :: "expr" shows "free_vars_expr_code (Random uw e) = free_vars_expr_code e"
find_min_single: fixes x :: "('c, 'd) bintree option" shows "find_min [x] = x"
dom_eqI': fixes a :: "'a"   and f :: "'a" assumes "a \<in> C.domains f" shows "a = C.dom f"
simps_5: fixes v :: "'d list"   and va :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and S :: "('c, 'd) strand_step list" shows "ik\<^sub>s\<^sub>t (\<forall>v\<langle>\<or>\<noteq>: va\<rangle>\<^sub>s\<^sub>t # S) = ik\<^sub>s\<^sub>t S"
simps_2: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "pre (x ::= a) Q = ((\<lambda>(ps, t). x \<in> dom ps \<and> vars a \<subseteq> dom ps \<and> Q (ps(x \<mapsto> paval a ps), t)) \<and>* $ 1)"
multiset_concat_gen: fixes M :: "'c multiset"   and xs :: "'c list list" shows "M + mset (concat xs) = fold (\<lambda>x M. M + mset x) xs M"
add_carrier_vec: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 + v\<^sub>2 \<in> carrier_vec n"
rep_eq: fixes x :: "nat"   and xa :: "uint32" shows "Rep_uint32 (push_bit x xa) = push_bit x (Rep_uint32 xa)"
Liminf_compose_continuous_mono: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'a" assumes "continuous_on UNIV f"   and "mono f"   and "F \<noteq> \<bottom>" shows "Liminf F (\<lambda>n. f (g n)) = f (Liminf F g)"
bound_intersect_2d: fixes prec :: "nat"   and Xs :: "((real \<times> real) \<times> real \<times> real) list"   and x :: "real"   and m :: "real"   and M :: "real" assumes "bound_intersect_2d prec Xs x = Some (m, M)" shows "(\<Union>(p1, p2)\<in>set Xs. closed_segment p1 p2) \<inter> {p. fst p = x} \<subseteq> {(x, m)..(x, M)}"
order_trans: fixes y :: "('d, 'e) Term.term"   and x :: "('d, 'e) Term.term"   and z :: "('d, 'e) Term.term" assumes "y \<unrhd> x"   and "z \<unrhd> y" shows "z \<unrhd> x"
lincomb_sum: fixes B :: "'a set"   and A :: "complex vec set"   and c' :: "'a \<Rightarrow> complex"   and c :: "complex vec \<Rightarrow> complex" assumes "B = basis_enum_of_vec ` A"   and "finite A"   and "A \<subseteq> vec_of_basis_enum ` X"   and "\<And>b. c' b = c (vec_of_basis_enum b)" shows "lincomb c A = vec_of_basis_enum (\<Sum>b\<in>B. c' b *\<^sub>C b)"
IT: shows "RETURN tsl' \<le> it_to_sorted_list ordR s'"
cases: fixes a1 :: "'b nmultiset"   and a2 :: "'b nmultiset" assumes "sub_nmsetp a1 a2" obtains X :: "'b nmultiset" and M :: "'b nmultiset multiset" where "a1 = X" and "a2 = MSet M" and "X \<in># M"
exhaust: fixes y :: "poscheckvariant" obtains "y = assign"   | "y = check"
ccw'_def: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" shows "ccw' p q r = (0 < det3 p q r)"
sign_extend_bitwise_if': fixes i :: "nat"   and e :: "nat"   and w :: "'b word" assumes "i < LENGTH('b)" shows "sign_extend e w !! i = (if i < e then w !! i else w !! e)"
condition2_part1: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f"   and b :: "'f"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" assumes "echelon_form_upt_k A k"   and "a < b"   and "\<not> is_zero_row_upt_k a (Suc k) (fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k))"   and "\<not> is_zero_row_upt_k b (Suc k) (fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k))"   and "\<forall>m\<ge>mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1). A $ m $ mod_type_class.from_nat k = (0::'d)" shows "(LEAST n. A $ a $ n \<noteq> (0::'d)) < (LEAST n. A $ b $ n \<noteq> (0::'d))"
test_import_3: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.il_semiring inf sup bot less_eq less top"   and "\<exists>y. sup x y = top \<and> inf x y = bot \<and> inf y x = bot" shows "inf (sup (il_semiring.neg inf sup bot top x) y) x = inf y x"
scount_eq_card: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" assumes "ev (alw (\<lambda>xs. \<not> P xs)) \<omega>" shows "scount P \<omega> = enat (card {i. P (sdrop i \<omega>)})"
mset_rev: fixes xs :: "'e list" shows "mset (rev xs) = mset xs"
invar: fixes s :: "('b, 'c) RBT.rbt" shows "rm.invar s"
in_listsI: fixes xs :: "'d list"   and A :: "'d set" assumes "\<forall>x\<in>set xs. x \<in> A" shows "xs \<in> lists A"
Table_def: fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real"   and l0 :: "real" shows "Table f1 f2 f1' f2' e c l0 \<equiv> Table0 f1 f2 f1' f2' e c \<and> Table_axioms f1 f2 f1' f2' e c l0"
ordstNS_trans: fixes X :: "char list"   and Y :: "char list"   and Z :: "char list" assumes "ordstNS X Y"   and "ordstNS Y Z" shows "ordstNS X Z"
sig_red_singleD3: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "q = p - monom_mult (lookup (rep_list p) (t + punit.lt (rep_list f)) / punit.lc (rep_list f)) t f"
cases: fixes a1 :: "'b poly"   and a2 :: "'b poly"   and a3 :: "'b poly \<times> 'b poly" assumes "eucl_rel_poly a1 a2 a3" obtains x :: "'b poly" where "a1 = x__" and "a2 = 0" and "a3 = (0, x__)"   | y :: "'b poly" and x :: "'b poly" and q :: "'b poly" where "a1 = xa__" and "a2 = y__" and "a3 = (qa__, 0)" and "y__ \<noteq> 0" and "xa__ = qa__ * y__"   | y :: "'b poly" and r :: "'b poly" and x :: "'b poly" and q :: "'b poly" where "a1 = x" and "a2 = y" and "a3 = (q, r)" and "y \<noteq> 0" and "degree r < degree y" and "x = q * y + r"
positive_def: shows "Rat.positive \<equiv> map_fun Rep_Rat id (\<lambda>x. 0 < fst x * snd x)"
Ex_omega_sum: fixes \<gamma> :: "V"   and n :: "nat" assumes "\<gamma> \<in> elts (\<omega> \<up> ord_of_nat n)" shows "\<exists>ns. \<gamma> = omega_sum ns \<and> length ns = n"
subtuple_mono: fixes xm :: "'c \<rightharpoonup>\<^sub>f 'a"   and ym :: "'c \<rightharpoonup>\<^sub>f 'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> fmran' xm; y \<in> fmran' ym\<rbrakk> \<Longrightarrow> f x y \<longrightarrow> g x y" shows "subtuple f xm ym \<longrightarrow> subtuple g xm ym"
Gcd_minors_dvd: fixes P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat"   and B :: "'b mat"   and m :: "nat"   and n :: "nat"   and I :: "nat set"   and J :: "nat set"   and k :: "nat" assumes "P * A * Q = B"   and "P \<in> carrier_mat m m"   and "A \<in> carrier_mat m n"   and "Q \<in> carrier_mat n n"   and "I \<subseteq> {0..<dim_row A}"   and "J \<subseteq> {0..<dim_col A}"   and "card I = k"   and "card J = k" shows "Gcd (minors A k) dvd Determinant.det (submatrix B I J)"
length_augment_list: fixes n :: "nat"   and f :: "'c list \<Rightarrow> 'c"   and p :: "'c list" shows "length ((augment_list f ^^ n) p) = n + length p"
mem_compact_implies_subset_existence_interval: fixes t0 :: "real"   and x0 :: "'a"   and K :: "'a set" assumes "t0 \<in> UNIV"   and "x0 \<in> X"   and "\<And>t. \<lbrakk>t \<in> UNIV; t - t0 \<in> existence_ivl0 x0\<rbrakk> \<Longrightarrow> flow0 x0 (t - t0) \<in> K"   and "compact K"   and "K \<subseteq> X" shows "UNIV \<subseteq> (+) t0 ` existence_ivl0 x0"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g f (local.inv \<epsilon>) (local.inv \<eta>)"
trg_trg: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "trg\<^sub>C (trg\<^sub>C \<mu>) = trg\<^sub>C \<mu>"
contains_def: fixes s :: "val option"   and r :: "proc \<times> val" shows "contains s r \<equiv> \<exists>rs. r \<in> set rs \<and> s = \<bottom> \<star> rs"
list_all2_cons_def: fixes g :: "('e, 'f) generator"   and h :: "('g, 'h) generator"   and P :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and sg :: "'f"   and sh :: "'h" shows "list_all2_cons g h P sg sh = list_all2 P (unstream g sg) (unstream h sh)"
basis_reduction_add_row_main_3: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and fs :: "int vec list"   and i :: "nat"   and j :: "nat"   and fs' :: "int vec list"   and c :: "int" assumes "LLL.LLL_invariant_weak n m fs_init fs"   and "i < m"   and "j < i"   and "fs' = fs[i := fs ! i - c \<cdot>\<^sub>v fs ! j]"   and "c = round (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)"   and "LLL.\<mu>_small_row n i fs (Suc j)" shows "LLL.\<mu>_small_row n i fs' j"
natural_transformation_axioms: shows "natural_transformation A B F G \<rho>"
Cn_total: fixes gs :: "recf list"   and f :: "recf"   and n :: "nat" assumes "\<forall>g\<in>set gs. Partial_Recursive.total g"   and "Partial_Recursive.total f"   and "recfn n (Cn n f gs)" shows "Partial_Recursive.total (Cn n f gs)"
surjective: fixes r :: "('b, 'c, 'd) category_scheme" shows "r = \<lparr>ob = Ob\<^bsub>r\<^esub>, ar = Ar\<^bsub>r\<^esub>, dom = category.dom r, cod = cod r, id = category.id r, comp = (\<bullet>\<^bsub>r\<^esub>), \<dots> = category.more r\<rparr>"
map_compatI: fixes f :: "'a \<Rightarrow> 'b option"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b option" assumes "f \<subseteq>\<^sub>m Some \<circ> h"   and "g \<subseteq>\<^sub>m Some \<circ> h" shows "f \<subseteq>\<^sub>m f ++ g"
Arity_above_top: fixes a :: "Arity" shows "(0 \<sqsubseteq> a) = (a = 0)"
map_definedness: fixes f :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and y :: "'a" assumes "f x = Some y" shows "x \<in> dom f"
le_antisym: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<preceq>\<^bsub>D\<^esub> b"   and "b \<preceq>\<^bsub>D\<^esub> a" shows "a = b"
jumpF_pathfinish_eq_0: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path g"   and "Re (pathfinish g) \<noteq> Re z" shows "jumpF_pathfinish g z = 0"