hcju/isabelleps huggingface dataset:
intro: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat" assumes "\<forall>e. (0::'a) \<le> f e \<and> f e \<le> c e"   and "\<forall>v\<in>Graph.V c - {s, t}. sum f (Graph.outgoing c v) \<le> sum f (Graph.incoming c v)" shows "Preflow c s t f"
c_tl_c_cons: fixes x :: "nat"   and y :: "nat" shows "c_tl (c_cons x y) = y"
tms_cong1: fixes t :: "trm"   and t' :: "trm"   and u :: "trm"   and u' :: "trm" shows "{} \<turnstile> t EQ t' AND u EQ u' IMP tms t u EQ tms t' u'"
hyps: fixes i :: "nat"   and a :: "nat"   and j :: "nat"   and b :: "nat" assumes "i \<le> a"   and "a < 2 ^ n + i"   and "j \<le> b"   and "b < 2 ^ n + j" shows "square2 n i j - {(a, b)} \<in> tiling Ls"
distinct_7: fixes x1 :: "'a"   and x51 :: "'a"   and x52 :: "bexp"   and x53 :: "'a"   and x54 :: "'a acom"   and x55 :: "'a" shows "SKIP {x1} \<noteq> {x51} WHILE x52 DO {x53} x54 {x55}"
keys_monom_mult_subset: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and c :: "'f"   and t :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "term_powerprod pair_of_term term_of_pair" shows "keys (term_powerprod.monom_mult pair_of_term term_of_pair c t p) \<subseteq> term_powerprod.splus pair_of_term term_of_pair t ` keys p"
iso5: fixes x :: "('a \<times> 'a set) set" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> Multirelations.U \<cdot> 1\<^sub>\<pi> = x \<cdot> 1\<^sub>\<pi>"
b1i: shows "b1 \<inter> path_image (p +++ linepath a b) = {}"
span_explicit: fixes b :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "phull b = {\<Sum>a\<in>t. r a \<cdot> a |t r. finite t \<and> t \<subseteq> b}"
rOpen: fixes M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and y :: "name"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>([(x, y)] \<bullet> N)\<rangle> \<prec> [(x, y)] \<bullet> P'"   and "y \<in> supp N"   and "x \<sharp> N"   and "x \<sharp> P'"   and "x \<noteq> y"   and "y \<sharp> xvec"   and "y \<sharp> yvec"   and "y \<sharp> M"   and "distinct xvec"   and "distinct yvec"   and "xvec \<sharp>* \<Psi>"   and "y \<sharp> \<Psi>"   and "yvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "y \<sharp> P"   and "yvec \<sharp>* P"   and "xvec \<sharp>* M"   and "y \<sharp> M"   and "yvec \<sharp>* M"   and "xvec \<sharp>* yvec"   and "x\<alpha> = M\<lparr>\<nu>*(xvec @ y # yvec)\<rparr>\<langle>N\<rangle>"   and "xP' = P'" shows "Prop"
untame: shows "untame (\<lambda>g. P g \<and> U g)"
split_face_edges_f21_vs: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and f12 :: "face"   and f21 :: "face"   and vs2 :: "nat list" assumes "pre_split_face f ram1 ram2 []"   and "(f12, f21) = split_face f ram1 ram2 []"   and "vs2 = between (vertices f) ram2 ram1"   and "vs2 \<noteq> []" shows "\<E> f21 = {(last vs2, ram1), (ram1, ram2), (ram2, hd vs2)} \<union> Edges vs2"
Simulation_Invariant_axioms: shows "Simulation_Invariant A B (\<sim>) QA QB"
strict_implies_order: fixes b :: "'b word"   and a :: "'b word" assumes "b <s a" shows "b \<le>s a"
matrix_add_ldistrib: fixes A :: "(('a, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and B :: "(('a, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and C :: "(('a, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "A ** (B + C) = A ** B + A ** C"
abs_eq: fixes x :: "'c \<Rightarrow> 'd option"   and xa :: "'c \<Rightarrow> bool" assumes "eq_onp (\<lambda>m. finite (dom m)) x x" shows "fmfilter xa (Abs_fmap x) = Abs_fmap (map_filter xa x)"
less_eq_natpair_def: fixes A :: "natpair"   and B :: "natpair" shows "(A \<le> B) = (case A of natpair.Pair (a, b) \<Rightarrow> case B of natpair.Pair (c, d) \<Rightarrow> if a = c then b \<le> d else a < c)"
map_sum_sel_2: fixes s :: "'c + 'd"   and f :: "'c \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'e" assumes "\<not> isl s" shows "projr (map_sum f g s) = g (projr s)"
add_zero_left: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "(0::'b) + x = x"
isomorphic_REP_src: fixes A :: "('a, 'a set) arr" assumes "ide A" shows "REP A \<cong>\<^sub>B src (REP A)"
assms_1: shows "rga_ops (xs @ [(i1, r1), (i2, r2)])"
size_list_append: fixes f :: "'c \<Rightarrow> nat"   and xs :: "'c list"   and ys :: "'c list" shows "size_list f (xs @ ys) = size_list f xs + size_list f ys"
simps_1: fixes G :: "'b list_graph"   and M :: "'b SecurityInvariant list"   and accu :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_impl.filter_IFS_no_violations_accu G M accu [] = accu"
perm_simps_3: fixes p :: "perm"   and fm1 :: "fm"   and fm2 :: "fm" shows "p \<bullet> fm1 OR fm2 = (p \<bullet> fm1) OR (p \<bullet> fm2)"
gf: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g f \<eta> \<epsilon>"
interp_ops_list_equiv: fixes ops :: "('c \<times> ('c, 'd) list_op) list" shows "fst (interp_ops ops) = interp_ins (insertions ops)"
healthy_nnegD2: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and b :: "real"   and P :: "'b \<Rightarrow> real" assumes "healthy t"   and "bounded_by b P"   and "nneg P" shows "nneg (t P)"
distinct_27: fixes x41 :: "fm"   and x42 :: "fm"   and x61 :: "fm"   and x62 :: "fm" shows "Con' x41 x42 \<noteq> Eql' x61 x62"
epigraph_lsc_hull: fixes f :: "'b \<Rightarrow> ereal" shows "Epigraph UNIV (lsc_hull f) = closure (Epigraph UNIV f)"
Rf_mono_on_iia_on: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" shows "Basis.mono_on (Pow A) (Rf f) = iia_on A f"
Aan: shows "A $ a $ (LEAST n. A $ a $ n \<noteq> (0::'a)) \<noteq> (0::'a)"
lowest_tops_NoneD: fixes k :: "'a ifex list" assumes "lowest_tops k = None" shows "\<nexists>v t e. IF v t e \<in> set k"
mono_rtranclp: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "\<And>a b. x a b \<longrightarrow> y a b" shows "x\<^sup>*\<^sup>* a b \<longrightarrow> y\<^sup>*\<^sup>* a b"
foldl_node_correct: fixes f :: "'d \<Rightarrow> 'e \<times> 'f \<Rightarrow> 'd"   and \<sigma> :: "'d"   and nd :: "('e, 'f) Node" shows "foldl_node f \<sigma> nd = foldl f \<sigma> (nodeToList nd)"
fls_compose_power_0_left: fixes d :: "nat" shows "fls_compose_power 0 d = 0"
irred_inner_nz: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "(\<forall>b. b dvd x \<longrightarrow> \<not> x dvd b \<longrightarrow> b dvd (1::'a)) = (\<forall>a b. x = a * b \<longrightarrow> a dvd (1::'a) \<or> b dvd (1::'a))"
abstract_LP_def: fixes A :: "'b mat"   and b :: "'b vec"   and c :: "'b vec"   and m :: "nat"   and n :: "nat" shows "abstract_LP A b c m n \<equiv> b \<in> carrier_vec m \<and> c \<in> carrier_vec n \<and> A \<in> carrier_mat m n"
final_satisfies: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i" assumes "local.wf idx \<phi> \<and> lformula \<phi>"   and "Length \<AA> = 0"   and "#\<^sub>V \<AA> = idx" shows "final idx \<phi> = (\<AA> \<Turnstile> \<phi>)"
seq_split: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   and "rs = rs\<^sub>1 @ rs\<^sub>2" obtains t' :: "state" where "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t'" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t'\<rangle> \<Rightarrow> t"
star_left_induct: fixes z :: "'b"   and y :: "'b"   and x :: "'b" shows "z \<squnion> y * x \<le> x \<longrightarrow> y\<^sup>\<star> * z \<le> x"
Lm_empty: fixes r :: "'b rexp" shows "Lm (empty_mrexp r) = {}"
comm_ring_hom_axioms: fixes x :: "'c"   and y :: "'c" shows "comm_ring_hom (\<lambda>p. poly2 p x y)"
sisoL_def: fixes cl :: "('test, 'atom, 'choice) cmd list" shows "sisoL cl \<equiv> Ball (set cl) siso"
ring_hom_trans: fixes f :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme"   and g :: "'f \<Rightarrow> 'i"   and T :: "('i, 'j) ring_scheme" assumes "f \<in> ring_hom R S"   and "g \<in> ring_hom S T" shows "g \<circ> f \<in> ring_hom R T"
three_in_S_tangent_intersection_no_3_Col: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "p \<in> S"   and "q \<in> S"   and "r \<in> S"   and "p \<noteq> q"   and "r \<notin> {p, q}" shows "proj2_no_3_Col {proj2_intersection (polar p) (polar q), r, p, q}"
MutrecAux: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and n :: "nat"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "finite G \<and> card G = n \<and> verified G \<and> A \<in> G" shows "{} \<rhd> Call : A"
simps: fixes r :: "'b \<Rightarrow> nat option"   and \<nu> :: "'a"   and q :: "'b" shows "step r \<nu> q = (if \<not> sink q \<and> pre_ranks r \<nu> q \<noteq> {} then Some (card {q'. \<not> sink q' \<and> pre_ranks r \<nu> q' \<noteq> {} \<and> Min (pre_ranks r \<nu> q') < Min (pre_ranks r \<nu> q)}) else None)"
degree_poly_y_minus_x: fixes p :: "'a poly" shows "degree (poly_y_x (poly_x_minus_y p)) = degree p"
Int_Un_distrib2: fixes B :: "'e set"   and C :: "'e set"   and A :: "'e set" shows "(B \<union> C) \<inter> A = B \<inter> A \<union> C \<inter> A"
invertible_interchange_rows: fixes a :: "'b"   and b :: "'b" shows "invertible (interchange_rows (Finite_Cartesian_Product.mat (1::'a)) a b)"
sorted: shows "\<forall>x\<in>F. \<forall>y\<in>E \<union> {e}. weight x \<le> weight y"
derived_block_size: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>D" shows "int (card b) = \<Lambda>"
concat_set: fixes qs :: "nat set"   and cs :: "pratt list list" assumes "\<forall>q\<in>qs. \<exists>c\<in>set cs. Prime q \<in> set c" shows "\<forall>q\<in>qs. Prime q \<in> set (concat cs)"
ief_lens_def: fixes x :: "'b \<Longrightarrow> 'c" shows "ief_lens x \<equiv> weak_lens x \<and> ief_lens_axioms x"
simps_1: fixes f :: "'c" shows "Cod \<^bold>\<langle>f\<^bold>\<rangle> = \<^bold>\<langle>C.cod f\<^bold>\<rangle>"
iT_Div_iWeakSince_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "0 < k" shows "(P t1. t1 \<B> t2 I \<oslash> k. Q t2) = (P (t1 div k). t1 \<B> t2 I. Q (t2 div k))"
fls_shifted_times_simps_1: fixes f :: "'a fls"   and n :: "int"   and g :: "'a fls" shows "f * fls_shift n g = fls_shift n (f * g)"
in_outs'_restrict_gpvD: fixes x :: "'d"   and \<I> :: "('d, 'f) \<I>"   and gpv :: "('e, 'd, 'f) gpv" assumes "x \<in> outs'_gpv (restrict_gpv \<I> gpv)" shows "x \<in> outs_\<I> \<I>"
subst0: fixes p :: "fm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and t :: "tm" assumes "qfree p" shows "Ifm vs (x # bs) (subst0 t p) = Ifm vs (Itm vs (x # bs) t # bs) p"
lower_circle_rescale: fixes l :: "real"   and r :: "real" assumes "l < r" shows "lower_circle l r = (\<lambda>x. x * (complex_of_real r - complex_of_real l) + complex_of_real l) ` lower_circle_01"
cfun_eq_iff: fixes f :: "'r \<rightarrow> 's"   and g :: "'r \<rightarrow> 's" shows "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
t1_def: shows "Some t1 = approx_tm prec ord I a l env"
depath_def: fixes E :: "'b uprod set"   and u :: "'b"   and dp :: "'b uprod list"   and v :: "'b" shows "depath E u dp v \<equiv> epath E u dp v \<and> distinct dp"
delta_gt: fixes i :: "nat" assumes "2 \<le> i"   and "i < k" shows "0 < \<delta> i"
restriction_clm_v2_non_empty_v1_empty: fixes v :: "view"   and u :: "view"   and w :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=u--w \<and> restrict w (clm ts) c \<noteq> bot \<and> consec (lan u) (lan w) \<longrightarrow> restrict u (clm ts) c = bot"
less_asym: fixes x :: "'a Word.word"   and y :: "'a Word.word"   and P :: "bool" assumes "x <s y"   and "\<not> P \<Longrightarrow> y <s x" shows "P"
dbm_entry_dbm_min: fixes u :: "'c \<Rightarrow> 'd"   and c1 :: "'c"   and c2 :: "'c"   and a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "dbm_entry_val u (Some c1) (Some c2) (min a b)" shows "dbm_entry_val u (Some c1) (Some c2) b"
trans_NS: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "pre_order_pair S NS" shows "trans NS"
circ_mult_increasing: fixes x :: "'a" shows "x \<le> x\<^sup>+"
Shuffle_Plus_ShuffleLR_PlusR: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Times_Plus (ShuffleLR r (Plus s t)) = Plus (Times_Plus (ShuffleLR r s)) (Times_Plus (ShuffleLR r t))"
extension_2: shows "K \<lhd> G\<lparr>carrier := H\<rparr>"
usubst_game_loop: fixes \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<alpha> :: "game"   and I :: "interp"   and X :: "(variable \<Rightarrow> real) set" assumes "Uvariation \<nu> \<omega> U"   and "\<And>\<nu> \<omega> X. \<lbrakk>Uvariation \<nu> \<omega> (fst (usubstappp \<sigma> U \<alpha>)); snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>) \<noteq> undefg\<rbrakk> \<Longrightarrow> (\<nu> \<in> game_sem I (the (snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>))) X) = (\<nu> \<in> game_sem (USubst.adjoint \<sigma> I \<omega>) \<alpha> X)"   and "snd (usubstappp \<sigma> U \<alpha>**) \<noteq> undefg" shows "(\<nu> \<in> game_sem I (the (snd (usubstappp \<sigma> U \<alpha>**))) X) = (\<nu> \<in> game_sem (USubst.adjoint \<sigma> I \<omega>) \<alpha>** X)"
simps_35: fixes x31 :: "jump"   and x32 :: "stmt"   and x2 :: "expr" shows "x31\<bullet> x32 \<noteq> Expr x2"
bisim1While6: fixes P :: "'a prog"   and h :: "'heap"   and c :: "'addr expr1"   and e :: "'addr expr1"   and xs :: "'addr val list" shows "P,while (c) e,h \<turnstile> (while (c) e, xs) \<leftrightarrow> ([], xs, Suc (Suc (length (compE2 c) + length (compE2 e))), None)"
add_left_imp_eq: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g"   and b :: "'g"   and c :: "'g" assumes "class.cancel_semigroup_add plus"   and "plus a b = plus a c" shows "b = c"
T_def: fixes s :: "'a" shows "K1.T s = distr (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K1 x)))) (stream_space (count_space UNIV)) (K1.walk s)"
finite_convex_hull_iff_convex_hull_list: fixes Vs :: "'a vec set"   and Vsl' :: "'a vec list" assumes "Vs \<subseteq> carrier_vec n"   and "Vs = set Vsl'" shows "n.convex_hull Vs = n.convex_hull_list Vsl'"
coeff_degree: fixes p :: "'a list"   and i :: "nat" assumes "degree p < i" shows "local.coeff p i = \<zero>"
mQmpTr2: fixes H :: "'a set"   and N :: "'a set"   and X :: "'a set" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N"   and "X \<in> carrier (M /\<^sub>m H)" shows "(Mp\<^bsub>M  H,N\<^esub>) X \<in> carrier (M /\<^sub>m N)"
div_le_dividend: fixes m :: "nat"   and n :: "nat" shows "m div n \<le> m"
dom_runit: fixes a :: "'a" assumes "ide a" shows "local.dom (C.runit a) = C.tensor a C.unity"
one_beta_star: fixes t1 :: "lam"   and t2 :: "lam" assumes " t1 \<longrightarrow>\<^sub>1 t2" shows " t1 \<longrightarrow>\<^sub>\<beta>\<^sup>* t2"
space_vimage_algebra: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and M :: "'c measure" shows "space (vimage_algebra X f M) = X"
0: shows "Cring_Poly.compose R (T\<^bsub>a\<^esub> p) (X_poly_minus R a) = p"
axioms_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and e :: "'b" assumes "yoneda_lemma_fixed_e C S \<phi> F a e" shows "yoneda_lemma C S \<phi> F a"
suffix_def: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'a" shows "suffix k x \<equiv> \<lambda>n. x (k + n)"
lipschitz_on_def: fixes C :: "real"   and U :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "C-lipschitz_on U f = (0 \<le> C \<and> (\<forall>x\<in>U. \<forall>y\<in>U. dist (f x) (f y) \<le> C * dist x y))"
square_free_monic_poly_def: fixes Gcd :: "'c poly \<Rightarrow> 'c poly \<Rightarrow> 'c poly"   and p :: "'c poly" shows "yun_gcd.square_free_monic_poly Gcd p = p div Gcd p (pderiv p)"
F2set3_bd_incr: fixes x :: "('e, 'f, 'd) F2" shows "|F2set3 x| \<le>o F2bd'"
deg_out: shows "out_degree G v + 1 = in_degree G v"
constrains_weaken_R: fixes F :: "'d program"   and A :: "'d set"   and A' :: "'d set"   and B' :: "'d set" assumes "F \<in> A co A'"   and "A' \<subseteq> B'" shows "F \<in> A co B'"
simps_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and \<alpha> :: "'c \<Rightarrow> 'd"   and x :: "'c"   and p :: "nat"   and m :: "('c \<times> nat) list" assumes "class.linorder less_eq less" shows "linorder.eval_monom_list \<alpha> ((x, p) # m) = linorder.eval_monom_list \<alpha> m * \<alpha> x ^ p"
cases: fixes a1 :: "'tp"   and a2 :: "('fsym, 'psym) lit list" assumes "a1 \<turnstile>2 a2" obtains \<sigma> :: "'tp" and c :: "('fsym, 'psym) lit list" where "a1 = \<sigma>__" and "a2 = ca__" and "infTp \<sigma>__"   | c :: "('fsym, 'psym) lit list" and \<sigma> :: "'tp" where "a1 = \<sigma>" and "a2 = c" and "\<And>l x. \<lbrakk>l \<in> set c; x \<in> nv2L l; tpOfV x = \<sigma>\<rbrakk> \<Longrightarrow> isGuard x (grdOf c l x)"
msg_fresh_1: fixes \<sigma> :: "nat \<Rightarrow> state"   and hops :: "nat"   and rreqid :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat" shows "msg_fresh \<sigma> (Rreq hops rreqid dip dsn dsk oip osn sip) = (1 \<le> osn \<and> (sip \<noteq> oip \<longrightarrow> oip \<in> kD (rt (\<sigma> sip)) \<and> osn \<le> nsqn (rt (\<sigma> sip)) oip \<and> (nsqn (rt (\<sigma> sip)) oip = osn \<longrightarrow> the (dhops (rt (\<sigma> sip)) oip) \<le> hops \<or> the (flag (rt (\<sigma> sip)) oip) = Aodv_Basic.inv)))"
collinear_small: fixes s :: "'a set" assumes "finite s"   and "card s \<le> 2" shows "collinear s"
sub_inter_r: fixes T3 :: "funty"   and T1 :: "funty"   and T2 :: "funty" assumes "T3 <:: T1"   and "T3 <:: T2" shows "T3 <:: T1 \<sqinter> T2"
still_reachable: fixes B :: "'b set"   and Ra :: "('b \<times> 'b) set"   and A :: "'b set"   and Rb :: "('b \<times> 'b) set" assumes "B \<subseteq> Ra\<^sup>* `` A"   and "\<forall>(x, y)\<in>Rb - Ra. y \<in> Ra\<^sup>* `` A" shows "Rb\<^sup>* `` B \<subseteq> Ra\<^sup>* `` A"
arc_axiom: fixes x :: "'a" assumes "x \<noteq> bot" shows "\<exists>y. arc y \<and> y \<le> - - x"
F3: shows "F3 = filter (\<lambda>f. \<exists>v\<in>set V3. f \<in> set (facesAt g v)) (faces g)"
InitConf_status: fixes A :: "('a, 'b, 'c) hierauto" shows "(A, InitConf A, {}, HAInitValue A) \<in> status"
iso_is_arr: fixes f :: "'a \<times> 'a" assumes "B.VV.iso f" shows "B.VV.arr f"
times_vector_id: fixes v :: "'a" shows "local.times v [id] = [v]"
mult_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "x * y \<in> A"
bits_of_int_simps_1: shows "bits_of 0 = [False]"
maybe_arith_int_Not_Num: fixes f :: "int \<Rightarrow> int \<Rightarrow> int"   and a1 :: "value option"   and a2 :: "value option" shows "(\<forall>n. maybe_arith_int f a1 a2 \<noteq> Some (Num n)) = (maybe_arith_int f a1 a2 = None)"
Mason_Stothers: fixes A :: "'b poly"   and B :: "'b poly"   and C :: "'b poly" assumes "A \<noteq> 0"   and "B \<noteq> 0"   and "C \<noteq> 0"   and "\<exists>p\<in>{A, B, C}. pderiv p \<noteq> 0"   and "A + B + C = 0"   and "Gcd {A, B, C} = 1" shows "Max {degree A, degree B, degree C} < degree (radical (A * B * C))"
scale_right_distrib: fixes a :: "'a"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "a *s (x + y) = a *s x + a *s y"
wf_design: shows "design (del_point p) (del_point_blocks p)"
that: fixes h :: "'a set" assumes "h \<in> F"   and "C \<subseteq> S \<inter> {x. a h \<bullet> x = b h}" shows "thesis"
init_item_\<alpha>: fixes r :: "symbol \<times> symbol list"   and k :: "nat" shows "item_\<alpha> (init_item r k) = []"
invertible_left_inverse: fixes u :: "'a" assumes "additive.invertible u"   and "u \<in> R" shows "- u + u = \<zero>"
equalsffemptyD: fixes A :: "'a fset"   and a :: "'a" assumes "A = {||}" shows "a |\<notin>| A"
assms_1: shows "II \<in> UNpart01 cl dl P"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "S.UP (g \<cdot> f) = S.UP g \<cdot>\<^sub>S S.UP f"
wf_pair_negchecks_map': fixes X :: "'a set"   and A :: "('c \<times> ('b, 'a) strand_step) list"   and i :: "'c"   and Y :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and M :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list list" assumes "wf\<^sub>l\<^sub>s\<^sub>t X A" shows "wf\<^sub>l\<^sub>s\<^sub>t X (map (\<lambda>G. (i, \<forall>Y\<langle>\<or>\<noteq>: (F @ G)\<rangle>\<^sub>s\<^sub>t)) M @ A)"
leg_simps_4: shows "C.dom Cod\<mu>_Cod\<nu>.leg1 = Cod\<mu>_Cod\<nu>.apex"
carrier_quotient_ring_def: shows "carrier_quotient_ring \<equiv> rel.Partition"
wf23: shows "mat nr ncc (mat_mult nr m2 m3)"
commute: fixes a :: "'d"   and b :: "'d" shows "max a b = max b a"
h_q1_copr: shows "(\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> (\<forall>q. List.member (take (length qs div 2) qs) q \<longrightarrow> coprime p q)"
sum_eq_if_same_subterms: fixes i :: "nat"   and j :: "nat"   and f :: "nat \<Rightarrow> 'd"   and f' :: "nat \<Rightarrow> 'd" assumes "\<forall>k. i \<le> k \<and> k < j \<longrightarrow> f k = f' k" shows "sum f {i..<j} = sum f' {i..<j}"
rel_funE: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and x :: "'d"   and y :: "'e" assumes "rel_fun A B f g"   and "A x y" obtains "B (f x) (g y)"
assms_2: fixes A :: "'a set" assumes "A \<in> sets M"   and "0 < emeasure M A" shows "\<exists>n>0. 0 < emeasure M ((T ^^ n) -` A \<inter> A)"
Left_Coset_Class_unit: fixes g :: "'a" assumes "g \<in> G" shows "g \<cdot>| Normal = Class g"
renaming_distr_global_1: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and ls :: "('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr" assumes "bij \<alpha>" shows "\<R>\<^sub>G \<alpha> \<beta> (s(r \<mapsto> ls)) = \<R>\<^sub>G \<alpha> \<beta> s(\<alpha> r \<mapsto> \<R>\<^sub>L \<alpha> \<beta> ls)"
ordinal_not_less_0: fixes x :: "ordinal" shows "\<not> x < 0"
Assert'Precond_2: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<not> \<U> (assert_bound a s)" shows "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s (assert_bound a s) \<and> \<diamond> (assert_bound a s)"
r_inv_ex: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b" assumes "Group.group G"   and "x \<in> carrier G" shows "\<exists>y\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"
circ_sup_1: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<circ> = x\<^sup>\<circ> * (y * x\<^sup>\<circ>)\<^sup>\<circ>"