hcju/isabelleps huggingface dataset:
simple_distributed_def: fixes M :: "'c measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> real" shows "simple_distributed M X f = ((\<forall>x. 0 \<le> f x) \<and> distributed M (count_space (X ` space M)) X (\<lambda>x. ennreal (f x)) \<and> finite (X ` space M))"
normal_def: fixes \<phi> :: "atom fm" shows "Z.normal \<phi> = (\<forall>a\<in>atoms \<phi>. divisor a \<noteq> 0)"
drop_shiftr: fixes n :: "nat"   and w :: "'b word" shows "drop n (to_bl (w >> n)) = take (size w - n) (to_bl w)"
zero_in_group_closure: fixes S :: "'a set" shows "(0::'a) \<in> group_closure S"
distinguishing_formula_eqvt: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and Q :: "'b"   and p :: "perm" assumes "indexed_nominal_ts TYPE('a) satisfies transition"   and "\<not> indexed_nominal_ts.logically_equivalent TYPE('a) satisfies transition P Q" shows "p \<bullet> indexed_nominal_ts.distinguishing_formula satisfies transition P Q = indexed_nominal_ts.distinguishing_formula satisfies transition (p \<bullet> P) (p \<bullet> Q)"
pred_def: fixes P :: "bool" shows "pred P \<equiv> \<lambda>t n. P"
leg0_in_hom_1: shows "\<guillemotleft>f : src f \<rightarrow> src r\<guillemotright>"
homeomorphic_imp_closed_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphic_map X Y f" shows "closed_map X Y f"
incid_inter_right: fixes l :: "Lines"   and m :: "Lines" shows "incid (Pascal_Property.inter l m) m"
map_subseqs: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "map (map f) (subseqs xs) = subseqs (map f xs)"
cong_trivial_identity: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Cong A A B B"
mult_nonpos_nonneg: 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 a zero"   and "less_eq zero b" shows "less_eq (times a b) zero"
dist_triangle_add_half: fixes x :: "'b"   and x' :: "'b"   and e :: "real"   and y :: "'b"   and y' :: "'b" assumes "dist x x' < e / 2"   and "dist y y' < e / 2" shows "dist (x + y) (x' + y') < e"
prim_rotate_conv: fixes w :: "'b list"   and n :: "nat" shows "primitive w = primitive (rotate n w)"
bot_flat_complete_lattice_def: shows "bot = flat_complete_lattice.Bot"
simps_1: fixes p :: "'b"   and b :: "bool" shows "unpackMessage <p, inM b> = Bool b"
simps_4: fixes a :: "tm" shows "tmbound0 (tm.Neg a) = tmbound0 a"
axioms_1: fixes R :: "'a set" assumes "Ring1 R" shows "Group R"
intro: fixes Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set" assumes "\<And>N. Ri N \<subseteq> \<Gamma>"   and "\<And>N N'. N \<subseteq> N' \<Longrightarrow> Rf N \<subseteq> Rf N'"   and "\<And>N N'. N \<subseteq> N' \<Longrightarrow> Ri N \<subseteq> Ri N'"   and "\<And>N' N. N' \<subseteq> Rf N \<Longrightarrow> Rf N \<subseteq> Rf (N - N')"   and "\<And>N' N. N' \<subseteq> Rf N \<Longrightarrow> Ri N \<subseteq> Ri (N - N')"   and "\<And>N. satisfiable (N - Rf N) \<Longrightarrow> satisfiable N" shows "redundancy_criterion \<Gamma> Rf Ri"
var_rename_wt_1: fixes n :: "nat" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (var_rename n)"
elts_E: fixes k :: "nat" assumes "k < length \<alpha>s" shows "elts (E k) \<subseteq> elts (\<omega>_sum \<alpha>s)"
fst_snd_arg_min_on: fixes B :: "'c set"   and p :: "'c \<Rightarrow> 'd" assumes "finite B"   and "inj_on p B"   and "B \<noteq> {}" shows "fst (arg_min_on snd ((\<lambda>x. (x, p x)) ` B)) = arg_min_on p B"
sp_it_is_post_condition: fixes p :: "'b upred"   and C :: "'b hrel" shows "\<lbrace>p\<rbrace> C \<lbrace>p sp C\<rbrace>\<^sub>u"
enum_sdim_def: shows "enum_class.enum = [Length, Mass, Time, Current, Temperature, Amount, Intensity]"
cone_determines_arrow_2: fixes d :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and a :: "'c"   and \<chi> :: "'j \<Rightarrow> 'c" assumes "J_C.ide d"   and "Limit.cone (\<cdot>\<^sub>J) (\<cdot>) (\<chi>.\<Phi>.Ya.Cop_S.Map d) a \<chi>" shows "\<chi>.\<Phi>.Ya.Cop_S.Map (\<chi>.\<Phi>.Ya.Cop_S.MkArr (constant_functor.map (\<cdot>\<^sub>J) (\<cdot>) a) (\<chi>.\<Phi>.Ya.Cop_S.Map d) \<chi>) = \<chi>"
f_def: shows "f = (\<lambda>x S y. if (x, y) \<in> r then g x S y else g y S x)"
connected_def: fixes u :: "nat"   and v :: "nat" shows "cf.connected u v \<equiv> \<exists>p. cf.isPath u p v"
UN_ball_bex_simps_4: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>\<Union> (B ` A). P x) = (\<exists>a\<in>A. \<exists>x\<in>B a. P x)"
homeomorphic_fixpoint_property: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T" shows "(\<forall>f. continuous_on S f \<and> f ` S \<subseteq> S \<longrightarrow> (\<exists>x\<in>S. f x = x)) = (\<forall>g. continuous_on T g \<and> g ` T \<subseteq> T \<longrightarrow> (\<exists>y\<in>T. g y = y))"
contour_integral_part_circlepath_reverse: fixes c :: "complex"   and r :: "real"   and a :: "real"   and b :: "real"   and f :: "complex \<Rightarrow> complex" shows "contour_integral (part_circlepath c r a b) f = - contour_integral (part_circlepath c r b a) f"
dom_dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.dom C (partial_magma.dom C f) = partial_magma.dom C f"
duh: fixes A :: "'b set"   and B :: "'b set" assumes "A \<inter> B = {}" shows "A \<subseteq> - B"
unfoldr_eq_Nil_iff: fixes g :: "('a, 'b) generator"   and s :: "'b" shows "(list.unfoldr g s = []) = (\<not> list.has_next g s)"
theorem10ps: fixes g :: "'c \<rightarrow> 'd set"   and r :: "'c \<rightarrow> 'c set" shows "(\<mu> f. \<Lambda> x. \<^ps> g\<cdot>x \<union> f\<cdot>(\<^ps> r\<cdot>x)) = (\<Lambda> x. \<Union>i. \<^ps> g\<cdot>((\<^ps> r)\<^bsup>i\<^esup>\<cdot>x))"
split_pairs: fixes A :: "'d"   and B :: "'e"   and X :: "'d \<times> 'e" shows "((A, B) = X) = (fst X = A \<and> snd X = B)"
lhd_subset_Sup_llist: fixes Xs :: "'b set llist" assumes "\<not> lnull Xs" shows "lhd Xs \<subseteq> Sup_llist Xs"
that: fixes M :: "'a dfa" assumes "dfa M"   and "dfa.language M = L"   and "card (states M) = card (UNIV // R)" shows "thesis"
Can_and_Diag_implies_Ide: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t"   and "monoidal_language.Diag C t" shows "monoidal_language.Ide C t"
ifbd_inf_pres: shows "inf_pres bd\<^sup>-\<^sub>\<F>"
gauss_jordan_3: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat"   and nc2 :: "nat"   and C :: "'c mat"   and D :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc2"   and "gauss_jordan A B = (C, D)" shows "C \<in> carrier_mat nr nc"
simps: fixes a1 :: "('a, 'b) ta_rule set"   and a2 :: "'b tree"   and a3 :: "'a" shows "accs_laz a1 a2 a3 = (\<exists>q f qs \<delta> ts. a1 = \<delta> \<and> a2 = NODE f ts \<and> a3 = q \<and> q \<rightarrow> f qs \<in> \<delta> \<and> list_all_zip (accs_laz \<delta>) ts qs)"
4: fixes x :: "'a"   and v :: "'a" assumes "x \<in> S" shows "frechet_derivative f (at x) v = frechet_derivative g (at x) v"
shiftr_less_t2n': fixes x :: "'b word"   and n :: "nat"   and m :: "nat" assumes "x AND mask (n + m) = x"   and "m < LENGTH('b)" shows "x >> n < 2 ^ m"
fold_insert: fixes dxs :: "'c set_dlist"   and x :: "'c"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d" assumes "\<not> member dxs x" shows "DList_Set.fold f (DList_Set.insert x dxs) b = DList_Set.fold f dxs (f x b)"
Nml_Nmlize_2: fixes t :: "'c local.term" assumes "Arr t" shows "Src \<^bold>\<lfloor>t\<^bold>\<rfloor> = Src t"
WT_callee: fixes \<I> :: "('a, 'b) \<I>"   and s :: "('a, 'b) resource" assumes "\<I> \<turnstile>res s \<surd>" shows "\<I> \<turnstile>c run_resource s \<surd>"
f_int: fixes a :: "'n"   and b :: "'n" shows "f absolutely_integrable_on cbox a b"
weight_spmf_eq_nn_integral_support: fixes p :: "'c spmf" shows "ennreal (weight_spmf p) = \<integral>\<^sup>+ x. ennreal (spmf p x) \<partial>count_space (set_spmf p)"
assms: shows "comp_tfr\<^sub>s\<^sub>s\<^sub>t arity Ana \<Gamma> Pair M S"
orderE: fixes a :: "'g multiset"   and b :: "'g multiset" assumes "a \<subseteq># b" obtains "a = a \<inter># b"
satFlagLessState_def: shows "satFlagLessState = {(state1, state2). getSATFlag state1 \<noteq> UNDEF \<and> getSATFlag state2 = UNDEF}"
fds_deriv_exp: fixes f :: "'a fds" shows "fds_deriv (fds_exp f) = fds_exp f * fds_deriv f"
H_loop: fixes p :: "'a"   and r :: "'a"   and x :: "'a" assumes "H (d p \<cdot> d r) x p" shows "H p (while r do x od) (d p \<cdot> ad r)"
sep_conj_assoc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj (sep_algebra.sep_conj plus sep_disj P Q) R = sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_conj plus sep_disj Q R)"
jth: shows "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)"
step_1_main_inv: fixes i :: "nat"   and j :: "nat"   and A :: "'a mat" assumes "i \<le> j"   and "A \<in> carrier_mat n n"   and "local.inv_all local.uppert A"   and "local.inv_part local.diff_ev A i j" shows "local.inv_all local.uppert (step_1_main n i j A) \<and> local.inv_all local.diff_ev (step_1_main n i j A)"
abs_eq: shows "y_axis = abs_circline y_axis_clmat"
simulation_reaches: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and a :: "'c"   and a' :: "'c"   and b :: "'d" assumes "Simulation_Invariant A B sim PA PB"   and "A\<^sup>*\<^sup>* a a'"   and "sim a b"   and "PA a"   and "PB b" shows "\<exists>b'. B\<^sup>*\<^sup>* b b' \<and> sim a' b' \<and> PA a' \<and> PB b'"
formulas_insert: fixes x :: "'a formula"   and S :: "'a set"   and a :: "'a" assumes "x \<in> formulas S" shows "x \<in> formulas (insert a S)"
resCap_cf_impl_refine: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'a"   and p :: "(nat \<times> nat) list" assumes "RGraph c s t cf"   and "Graph.isSimplePath cf s p t" shows "Network.resCap_cf_impl c cf p \<le> (spec r. r = resCap_cf cf p)"
"2_4": shows "map fst r = stake (Suc n) w"
inprod_cvec_ge_zero: fixes z :: "complex \<times> complex" shows "0 \<le> Re \<langle>z,z\<rangle>"
CSHORun_def: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f" shows "CSHORun A rho HOs SHOs coords \<equiv> CHOinitConfig A (rho 0) (coords 0) \<and> (\<forall>r. CSHOnextConfig A r (rho r) (HOs r) (SHOs r) (coords (Suc r)) (rho (Suc r)))"
abs_inverse: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Abs :: "'d set \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'd set"   and c :: "'d set" assumes "quot_type R Abs Rep"   and "\<exists>x. R x x \<and> c = Collect (R x)" shows "Rep (Abs c) = c"
eq_mod_iff: fixes n :: "int"   and b :: "int" assumes "0 < n" shows "(b = b mod n) = (0 \<le> b \<and> b < n)"
arrI: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "D.arr f"
not_deterministic_conv: fixes e :: "((nat \<times> nat) \<times> transition) fset" assumes "\<not> deterministic e" shows "\<exists>s l i r d1 d2 t1 t2. (d1 \<noteq> d2 \<or> t1 \<noteq> t2) \<and> ((s, d1), t1) |\<in>| e \<and> ((s, d2), t2) |\<in>| e \<and> Label t1 = Label t2 \<and> can_take_transition t1 i r \<and> can_take_transition t2 i r"
strip_anno: fixes a :: "'b"   and c :: "com" shows "strip (anno a c) = c"
lab_eq: shows "lrs = {(a, c, b). c = a \<and> (a, b) \<in> ars}"
get_parent_ok: fixes h :: "(_) heap"   and ptr :: "(_) node_ptr" assumes "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "ptr |\<in>| node_ptr_kinds h" shows "h \<turnstile> ok Shadow_DOM.get_parent ptr"
bounded_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a \<le> b \<sqinter> c) = (a \<le> b \<and> a \<le> c)"
Phase1or2Write_HInv4a2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "Phase1or2Write s s' p d"   and "HInv4a2 s q" shows "HInv4a2 s' q"
closed: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a set" assumes "semilattice_set f"   and "finite A"   and "A \<noteq> {}"   and "\<And>x y. f x y \<in> {x, y}" shows "semilattice_set.F f A \<in> A"
dual_unlabel_transaction_strand: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" assumes "wellformed_transaction T" shows "unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_strand T)) = unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T)) @ unlabel (transaction_selects T) @ unlabel (transaction_checks T) @ unlabel (transaction_updates T) @ unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T))"
get_M_Mshadow_root_preserved3: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and setter :: "('bc \<Rightarrow> 'bc) \<Rightarrow> ('as, 'aq, 'ar, 'ba option) RShadowRoot_scheme \<Rightarrow> ('as, 'aq, 'ar, 'ba option) RShadowRoot_scheme"   and v :: "'bc"   and h' :: "(_) heap"   and getter :: "(_) RObject_ext \<Rightarrow> 'bd" assumes "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr \<noteq> object_ptr"   and "h \<turnstile> put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
"6_4": shows "v \<in> E (EIf e1 e2 e3) \<rho>"
sees: shows "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"
many_strongly_connected_implies_no_start_end_points: fixes x :: "'a" assumes "many_strongly_connected x" shows "no_start_end_points x"
calpha1_alpha1: shows "(b \<union> m \<union> ov \<union> s \<union> d) O (b \<union> m \<union> ov \<union> s \<union> d) \<subseteq> b \<union> m \<union> ov \<union> s \<union> d"
cases: fixes x :: "'b interrupt_action \<times> 'b set" obtains t :: "'b" and Ts :: "'b set" where "x = (IsInterrupted ta__ True, Ts__)"   | t :: "'b" and Ts :: "'b set" where "x = (Interrupt t, Tsa__)"   | v :: "'b" and Ts :: "'b set" where "x = (IsInterrupted v__ False, Tsb__)"   | v :: "'b" and Ts :: "'b set" where "x = (ClearInterrupt v, Ts)"
emb_step_at_remove_arg: fixes p :: "dir list"   and t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" assumes "emb_step_at p dir.Left t = s"   and "position_of t (p @ [dir.Left])"   and "list_all (\<lambda>x. x = dir.Left) p" shows "let i = num_args t - Suc (length p) in head t = head s \<and> i < num_args t \<and> args s = take i (args t) @ drop (Suc i) (args t)"
prod_mset_primes_dvd_imp_subset: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "\<Prod>\<^sub># A dvd \<Prod>\<^sub># B"   and "\<And>p. p \<in># A \<Longrightarrow> prime p"   and "\<And>p. p \<in># B \<Longrightarrow> prime p" shows "A \<subseteq># B"
load_tmps_append: fixes xs :: "instr list"   and ys :: "instr list" shows "load_tmps (xs @ ys) = load_tmps xs \<union> load_tmps ys"
map_surj: shows "set (map (L.rupture_surj (carrier L) q \<circ> L.poly_of_const) q) \<subseteq> carrier (Rupt\<^bsub>L\<^esub> (carrier L) q)"
hoare_triple_def: fixes P :: "assn"   and c :: "'b Heap"   and Q :: "'b \<Rightarrow> assn" shows "<P> c <Q> = (\<forall>h as \<sigma> r. pHeap h as \<Turnstile> P \<longrightarrow> run c (Some h) \<sigma> r \<longrightarrow> \<sigma> \<noteq> None \<and> pHeap (the \<sigma>) (new_addrs h as (the \<sigma>)) \<Turnstile> Q r \<and> relH {a. a < lim h \<and> a \<notin> as} h (the \<sigma>) \<and> lim h \<le> lim (the \<sigma>))"
div_rat_poly_step_sur: fixes a :: "int poly \<times> int poly"   and q :: "int poly"   and x :: "int"   and pair :: "rat poly \<times> rat poly" assumes "(case a of (a, b) \<Rightarrow> (of_int_poly a, of_int_poly b)) = (div_rat_poly_step (of_int_poly q) \<circ> rat_of_int) x pair" shows "\<exists>c'. pair = (case c' of (a, b) \<Rightarrow> (of_int_poly a, of_int_poly b))"
obj_simps_5: fixes a :: "'a" assumes "obj a" shows "cod a = a"
assms: shows "A *\<^sub>V x = (0::'a)"
index_vec: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "i < n" shows "vec n f $ i = f i"
rat_to_int_poly_def: fixes p :: "rat poly" shows "rat_to_int_poly p \<equiv> let ais = coeffs p; d = fst (common_denom ais) in (d, map_poly (\<lambda>x. case quotient_of x of (p, q) \<Rightarrow> p * d div q) p)"
sign_changes_poly_times_one_minus_x: fixes g :: "'a poly" assumes "g \<noteq> 0" shows "0 < sign_changes (coeffs ([:1::'a, - (1::'a):] * g)) - sign_changes (coeffs g) \<and> odd (sign_changes (coeffs ([:1::'a, - (1::'a):] * g)) - sign_changes (coeffs g))"
trans: fixes x :: "'o"   and b :: "'a"   and a :: "'a"   and c :: "'a" assumes "le x b a"   and "le x c b" shows "le x c a"
showsp_ratfps_def: fixes p :: "nat"   and x :: "'b ratfps"   and y :: "char list" shows "showsp_ratfps p x y = show_ratfps x @ y"
relcomp_mono: fixes r' :: "('a \<times> 'b) set"   and r :: "('a \<times> 'b) set"   and s' :: "('b \<times> 'c) set"   and s :: "('b \<times> 'c) set" assumes "r' \<subseteq> r"   and "s' \<subseteq> s" shows "r' ; s' \<subseteq> r ; s"
mat_inv_l: fixes A :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det A \<noteq> 0" shows "mat_inv A *\<^sub>m\<^sub>m A = eye"
assms_1: shows "sorted_snd (p\<^sub>2 # ps)"
whiletrue: fixes BMap :: "'a \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and b :: "'b"   and m :: "'c \<Rightarrow> 'a"   and \<iota> :: "nat"   and c :: "('b, 'c) MWLsCom" assumes "BMap (E b m) = True" shows "((while\<^bsub>\<iota>\<^esub> b do c od, m), [], Some (c;while\<^bsub>\<iota>\<^esub> b do c od), m) \<in> MWLs_semantics.MWLsSteps_det E BMap"
height_balance: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and t :: "'b tree"   and h :: "nat" assumes "RBTi1 TYPE('b) bal_i"   and "\<not> bal_i (size t) h" shows "height (balance_tree t) < h"
cyc_free_subs_diag: 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 n :: "nat"   and I :: "nat set"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> I \<longrightarrow> less_eq zero (linordered_ab_monoid_add.len plus m i i xs)"   and "i \<le> n" shows "less_eq zero (m i i)"
F1in_mono23: fixes A2 :: "'d set"   and A2a :: "'d set"   and A3 :: "'e set"   and A3a :: "'e set"   and A1 :: "'f set" assumes "A2 \<subseteq> A2a"   and "A3 \<subseteq> A3a" shows "F1in A1 A2 A3 \<subseteq> F1in A1 A2a A3a"
inner_add_2: 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 inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (plus y z) = inner x y + inner x z"
Par_ZObis_discrR1: fixes c :: "('test, 'atom) com"   and d1 :: "('test, 'atom) com"   and d2 :: "('test, 'atom) com" assumes "c \<approx>01 d1"   and "discr d2" shows "c \<approx>01 d1 | d2"
inside_Int_outside: fixes S :: "'b set" shows "inside S \<inter> outside S = {}"
uminus_in_iff: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. - f x) \<in> O[F](g)) = (f \<in> O[F](g))"
nonempty_alts: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "pref_profile_wf agents alts R" shows "alts \<noteq> {}"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> items H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> items H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> items H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P Y"   and "\<And>K X. \<lbrakk>Crypt K X \<in> items H; P (Crypt K X)\<rbrakk> \<Longrightarrow> P X"   and "\<And>x y. \<lbrakk>Auth_Data x y \<in> items H; P (Auth_Data x y); Pri_AgrK y \<in> items H; P (Pri_AgrK y)\<rbrakk> \<Longrightarrow> P (Pri_AgrK x)"   and "\<And>x y. \<lbrakk>Auth_Data x y \<in> items H; P (Auth_Data x y); Pri_AgrK x \<in> items H; P (Pri_AgrK x)\<rbrakk> \<Longrightarrow> P (Pri_AgrK y)" shows "P x"
ev: shows "(\<chi>i j. complex_of_real (st_mat A $h i $h j)) *v v = complex_of_real a *s v"
Catch: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "'b"   and c\<^sub>2 :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile>c\<^sub>1 \<down> Normal s"   and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c\<^sub>2 \<down> Normal s'" shows "\<Gamma>\<turnstile>Catch c\<^sub>1 c\<^sub>2 \<down> Normal s"
E22: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "\<turnstile> F = G" shows "\<turnstile> \<diamond>F = \<diamond>G"
index_vec: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "i < n" shows "Matrix.vec n f $v i = f i"
perp_down_def: fixes a :: "proj2"   and l :: "proj2_line" shows "perp_down a l \<equiv> endpoint_in_S (perp_up a l) a"
restrict_out: fixes x :: "'a"   and A :: "'a set"   and m :: "'a \<Rightarrow> 'b option" assumes "x \<notin> A" shows "(m |` A) x = None"
trivial_1: fixes a :: "'b" assumes "a \<in> C - A" shows "g a = \<^bold>1"
sq_norm_gso_le_f: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat" assumes "gram_schmidt_fs_lin_indpt n fs"   and "i < length fs" shows "\<parallel>gram_schmidt_fs.gso n fs i\<parallel>\<^sup>2 \<le> \<parallel>fs ! i\<parallel>\<^sup>2"
bind_def: fixes x :: "('b, 'd) state"   and f :: "'d \<Rightarrow> ('b, 'c) state" shows "x \<bind> f = State (\<lambda>s. case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
absolutely_integrable_on_def: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'c" shows "(f absolutely_integrable_on S) = (f integrable_on S \<and> (\<lambda>x. norm (f x)) integrable_on S)"
of_int_power: fixes z :: "int"   and n :: "nat" shows "of_int (z ^ n) = of_int z ^ n"
carrier_singleton_group: fixes a :: "'e" shows "carrier (singleton_group a) = {a}"