hcju/isabelleps huggingface dataset:
step: fixes m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and x :: "'c"   and Y :: "'c set"   and y :: "'c" assumes "rwof m0 cond step x"   and "cond x"   and "step x = RES Y"   and "y \<in> Y" shows "rwof m0 cond step y"
rel_interior_cball: fixes S :: "'a set" shows "rel_interior S = {x \<in> S. \<exists>e>0. cball x e \<inter> affine hull S \<subseteq> S}"
simps_8: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "\<xi> \<Turnstile>\<^sub>r \<phi> U\<^sub>r \<psi> = (\<exists>i. suffix i \<xi> \<Turnstile>\<^sub>r \<psi> \<and> (\<forall>j<i. suffix j \<xi> \<Turnstile>\<^sub>r \<phi>))"
sum3: shows "(\<lambda>x. \<Sum>\<^sub>ay. - ((fds_nth f x / nat_power x s) ^ Suc y * of_real (ln (real x)))) abs_summable_on {p. prime p}"
simps_10: fixes r :: "'a rexp"   and s :: "'a rexp"   and v :: "'a rexp"   and va :: "'a rexp" shows "pnPlus (Plus r s) (Times v va) = pnPlus r (pnPlus s (Times v va))"
det_state_cover_ass_diff: fixes M :: "('d, 'e, 'f) FSM"   and f :: "'f \<Rightarrow> 'd list"   and s1 :: "'f"   and s2 :: "'f" assumes "is_det_state_cover_ass M f"   and "s1 \<in> d_reachable M (initial M)"   and "s2 \<in> d_reachable M (initial M)"   and "s1 \<noteq> s2" shows "f s1 \<noteq> f s2"
homogeneous_monomial: fixes t :: "'a \<Rightarrow>\<^sub>0 nat"   and c :: "'b" shows "homogeneous (monomial c t)"
apply_cltn2_right_abs_in_S: fixes J :: "((real, 3) vec, 3) vec"   and p :: "proj2" assumes "invertible J" shows "(apply_cltn2 p (cltn2_abs J) \<in> S) = (proj2_rep p \<bullet> (J ** M ** Finite_Cartesian_Product.transpose J *v proj2_rep p) = 0)"
ask_writer_def: fixes ask :: "('l \<Rightarrow> 'k) \<Rightarrow> 'k"   and f :: "'l \<Rightarrow> ('i, 'j, 'k) writerT" shows "ask_writer ask f = WriterT (ask (\<lambda>r. run_writer (f r)))"
power_mult_distrib: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.comm_monoid_mult times one" shows "power.power one times (times a b) n = times (power.power one times a n) (power.power one times b n)"
modif_clause: shows "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call q (Modif \<sigma>),(ModifAbr \<sigma>)"
mem_next: fixes x :: "ordinal" shows "(LEAST z. z \<in> A \<and> x < z) \<in> A"
order_iff_strict: fixes a :: "'loc \<times> 't"   and b :: "'loc \<times> 't" shows "(a\<le>\<^sub>pb) = (a<\<^sub>pb \<or> a = b)"
lems_4: fixes t :: "'b tree"   and a :: "'b" shows "depth t a \<le> height t"
set_intros_2: fixes x23 :: "'a"   and x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "x23 \<in> set1_rbt (Node x21 x22 x23 x24 x25)"
eventually_at_split: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "class.linorder_topology less_eq less open" shows "eventually P (topological_space.at_within open x UNIV) = (eventually P (topological_space.at_within open x (ord.lessThan less x)) \<and> eventually P (topological_space.at_within open x (ord.greaterThan less x)))"
monocontlub2cont: fixes f :: "'b \<Rightarrow> 'c" assumes "monofun f"   and "contlub f" shows "cont f"
th: shows "Abs_fps (\<lambda>n. 1::'a) = 1 / (1 - fps_X)"
oinvariantD: fixes A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<times> 'e \<Rightarrow> bool"   and s :: "'d \<times> 'e" assumes "A \<Turnstile> (S, U \<rightarrow>) P"   and "s \<in> oreachable A S U" shows "P s"
LCons_3: shows "\<not> lnull (lSup (fst ` Y))"
strict_implies_order: fixes a :: "'a option \<Rightarrow>\<^sub>0 nat"   and b :: "'a option \<Rightarrow>\<^sub>0 nat" assumes "extended_ord_strict a b" shows "extended_ord a b"
moebius_circline_neq_I: fixes H1 :: "circline"   and H2 :: "circline"   and M :: "moebius" assumes "H1 \<noteq> H2" shows "moebius_circline M H1 \<noteq> moebius_circline M H2"
L7_Controller: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "y 0 = [500]" shows "last (fin_make_untimed (inf_truncate z t)) = l t"
function_zero_closed: shows "\<zero>\<^bsub>function_ring (carrier R) R\<^esub> \<in> carrier (function_ring (carrier R) R)"
prefix_prefix: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<le> ys" shows "xs \<le> ys @ zs"
is_iso_def: fixes G :: "('g, 'h) monoid_scheme"   and H :: "('i, 'j) monoid_scheme" shows "G \<cong> H = (Group.iso G H \<noteq> {})"
difference_closure: fixes x :: "i"   and y :: "i" assumes "\<exists>w. P w x \<and> \<not> O w y" shows "\<exists>z. \<forall>w. P w z = (P w x \<and> \<not> O w y)"
invariant_reaches: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow>* b"   and "P a" shows "P b"
wf_ipassmt1: shows "ipassmt_sanity_nowildcards (map_of ipassmt)"
not_in_interior_convex_hull: fixes S :: "'a set"   and x :: "'a" assumes "finite S"   and "card S \<le> Suc DIM('a)"   and "x \<in> S" shows "x \<notin> interior (convex hull S)"
distinct_19: shows "ToUpperCaseOp \<noteq> string_unop.ToIntegerOp"
Digamma_approx: fixes x :: "real" assumes "0 < x" shows "Digamma x = ln x - 1 / (2 * x) - P x"
assms_6: shows "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"
is_pred_syz_def: fixes ss :: "'t list"   and u :: "'t" shows "is_pred_syz ss u = (\<exists>v\<in>set ss. v adds\<^sub>t u)"
ale_natle: fixes n :: "nat"   and m :: "nat" shows "(an n \<le> an m) = (n \<le> m)"
trace_eq_callee_of_coreI: fixes core1 :: "('a, 'b, 'c, 'd, 'e, 'f) core"   and core2 :: "('g, 'b, 'c, 'd, 'e, 'f) core"   and E :: "'b set"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'g spmf" assumes "trace_core_eq core1 core2 E IA IU p q" shows "trace_callee_eq (callee_of_core core1) (callee_of_core core2) (E <+> IA <+> IU) p q"
dvd_productE: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times p (times a b)" obtains x :: "'a" and y :: "'a" where "p = times x y" and "dvd.dvd times x a" and "dvd.dvd times y b"
All_transfer: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "bi_total A" shows "((A ===> (=)) ===> (=)) All All"
f_def: shows "f = (\<lambda>k x. if x \<in> {inverse (real (Suc k))..c} then x powr a else 0)"
emeasure_restrict_space: fixes \<Omega> :: "'a set"   and M :: "'a measure"   and A :: "'a set" assumes "\<Omega> \<inter> space M \<in> sets M"   and "A \<subseteq> \<Omega>" shows "emeasure (restrict_space M \<Omega>) A = emeasure M A"
any_Err: fixes P :: "'b prog"   and T :: "ty err" shows "P \<turnstile> T \<le>\<^sub>\<top> Err"
replicate_eq_replicate: fixes m :: "nat"   and x :: "'a"   and n :: "nat"   and y :: "'a" shows "(replicate m x = replicate n y) = (m = n \<and> (m \<noteq> 0 \<longrightarrow> x = y))"
rel_with_target_impl_transC_TRel_is_strong_reduction_simulation_rev: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "strong_reduction_simulation (Rel\<inverse>) (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+" shows "strong_reduction_simulation ((TRel\<^sup>+)\<inverse>) Target"
preserves_arr: fixes f :: "'a" assumes "V.arr f" shows "V.arr (H.R f)"
finfun_update_twist: fixes a :: "'a"   and a' :: "'a"   and f :: "'a \<Rightarrow>f 'b"   and b :: "'b"   and b' :: "'b" assumes "a \<noteq> a'" shows "f(a $:= b, a' $:= b') = f(a' $:= b', a $:= b)"
add_diff_inverse: fixes a :: "'k multiset"   and b :: "'k multiset" assumes "a \<subseteq># b" shows "a + (b - a) = b"
map_eq_on_tvsT_imp_map_eq_on_typ: fixes T :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "\<And>p. p \<in> tvsT T \<Longrightarrow> subs p = subs' p" shows "tsubstT T (convert_subs subs) = tsubstT T (convert_subs subs')"
add_le_same_cancel1: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and b :: "'g"   and a :: "'g" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus b a) b = less_eq a zero"
connected_in_V_2: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Graph G"   and "Graph.connected G v w" shows "w \<in> V\<^bsub>G\<^esub>"
summable': fixes z :: "complex" assumes "cmod z < K" shows "summable (\<lambda>n. diffs ((gchoose) a) n * z ^ n)"
index_mat_addcol_5: fixes a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "dim_col (addcol a k l A) = dim_col A"
that: fixes y :: "real" assumes "y \<in> open_segment a b"   and "f b - f a = (b - a) * f' y" shows "thesis"
Infi_lower: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> X" shows "\<Sqinter> X \<le> x"
pt_set_append: fixes pi1 :: "('b \<times> 'b) list"   and pi2 :: "('b \<times> 'b) list"   and Xs :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "(pi1 @ pi2) \<bullet> Xs = pi1 \<bullet> pi2 \<bullet> Xs"
is_path_undir_split': fixes G :: "('c, 'd) graph"   and v :: "'c"   and p1 :: "('c \<times> 'd \<times> 'c) list"   and u :: "'c"   and w :: "'d"   and u' :: "'c"   and p2 :: "('c \<times> 'd \<times> 'c) list"   and v' :: "'c" assumes "valid_graph G" shows "is_path_undir G v (p1 @ (u, w, u') # p2) v' = (is_path_undir G v p1 u \<and> ((u, w, u') \<in> edges G \<or> (u', w, u) \<in> edges G) \<and> is_path_undir G u' p2 v')"
limitin_subtopology: fixes X :: "'b topology"   and S :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" shows "limitin (subtopology X S) f l F = (l \<in> S \<and> (\<forall>\<^sub>F a in F. f a \<in> S) \<and> limitin X f l F)"
L_transform_Tree_eqvt: fixes p :: "perm"   and t :: "('a, 'b, 'c, 'd) FL_Formula.Tree" shows "p \<bullet> L_transform_Tree t = L_transform_Tree (p \<bullet> t)"
HFail_HInv4c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "HInv4c s q" shows "HInv4c s' q"
bind_commute: fixes m :: "'m"   and m' :: "'m"   and f :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> 'm" shows "bind m (\<lambda>x. bind m' (f x)) = bind m' (\<lambda>y. bind m (\<lambda>x. f x y))"
strict_trans2: fixes a :: "'t"   and b :: "'t"   and c :: "'t" assumes "a \<prec>\<^sub>t b"   and "b \<preceq>\<^sub>t c" shows "a \<prec>\<^sub>t c"
d'_4: shows "\<forall>t\<in>S_Max_Lt. d' < t"
ANDLEFT2_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and A :: "ty"   and B :: "ty"   and X :: "ntrm set" shows "pi \<bullet> ANDLEFT2 (A AND B) X = ANDLEFT2 (A AND B) (pi \<bullet> X)"
distinct_206: fixes x13 :: "Syntax.term"   and x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term" shows "Roll x13 \<noteq> Syntax.Let x81 x82 x83"
is_irreducible_iff_nexists_is_reducible_lit: fixes Ds :: "'a literal list list"   and C :: "'a literal list"   and C' :: "'a literal list" shows "(reduce Ds C C' = C') = (\<not> (\<exists>L\<in>set C'. is_reducible_lit Ds (C @ remove1 L C') L))"
assms_2: fixes y :: "'a vec" assumes "[A*\<^sub>vy]\<le>b" shows "y \<bullet> c \<le> x \<bullet> c"
rev_induct: fixes P :: "'e list \<Rightarrow> bool"   and xs :: "'e list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (xs @ [x])" shows "P xs"
idem: fixes a :: "'a upred" shows "(a \<or> a) = a"
monad_prob_prob: shows "monad_prob return_pmf bind_pmf bind_pmf"
bdd_above_finite: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set" assumes "class.lattice inf less_eq less sup"   and "finite A" shows "preorder.bdd_above less_eq A"
weight_less_imp_le: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term" assumes "weight_less s t" shows "weight_le s t"
v_LAM: fixes x :: "name"   and e :: "lam" shows "val LAM [x].e"
T_vrestr_composed_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and n :: "nat"   and m :: "nat" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.vimage_restr M (T ^^ n) (qmpt.vimage_restr M (T ^^ m) A) = qmpt.vimage_restr M (T ^^ (n + m)) A"
iso_in_homE: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<cong>\<^sub>D b\<guillemotright>" obtains "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" and "D.iso f"
psubstt_id_1: fixes t :: "'a FOL_Fitting.term" shows "psubstt id t = t"
set_fold_remove1': fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<in> set xs - set ys" shows "x \<in> set (fold remove1 ys xs)"
empty_foldSetDE: fixes x :: "'f"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and e :: "'f" assumes "({}, x) \<in> foldSetD D f e" obtains "x = e" and "e \<in> D"
int_eq: fixes i :: "nat" shows "integral UNIV (U i) = p i"
bogus_path_def: shows "local.bogus_path = [user\<^sub>1, name\<^sub>1, name\<^sub>2]"
assms_2: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool" shows "R e h l sh b e h l sh b"
path_B: shows "path B (w || map snd tr) qB \<and> length w = length (map snd tr)"
equiv_test: fixes b1 :: "'d bool_expr"   and b2 :: "'d bool_expr" shows "equiv_test ifex_of b1 b2 = (\<forall>s. val_bool_expr b1 s = val_bool_expr b2 s)"
call_source_Def_empty: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "Def (sourcenode a) = {}"
Cons_ct_prefix_Cons: fixes x :: "ct"   and xs :: "ct list"   and y :: "ct"   and ys :: "ct list" shows "ct_prefix (x # xs) (y # ys) = (ct_eq x y \<and> ct_prefix xs ys)"
sel: fixes x2 :: "'a" shows "the \<lfloor>x2\<rfloor> = x2"
iarray_to_vec_nth: fixes A :: "'d iarray"   and i :: "'e" shows "iarray_to_vec A $ i = A !! mod_type_class.to_nat i"
r2f_def: fixes R :: "('d \<times> 'e) set" shows "\<F> R = (``) R \<circ> \<eta>"
tr_s_field_readI: fixes var :: "char list"   and T :: "x \<Rightarrow> x option"   and var' :: "char list"   and x :: "x"   and x' :: "x"   and f :: "char list" assumes "(case T (x_var var) of None \<Rightarrow> var | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var) = var'"   and "(case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'" shows "tr_s T (s_read var x f) (s_read var' x' f)"
subgame_V': fixes V' :: "'a set" assumes "V' \<subseteq> V\<^bsub>G'\<^esub>" shows "V\<^bsub>G'.subgame V'\<^esub> = V'"
divide_eq_0_iff: fixes a :: "'a"   and b :: "'a" shows "(a / b = (0::'a)) = (a = (0::'a) \<or> b = (0::'a))"
CP_UNCURRY_def: fixes uu :: "'h"   and uv :: "'i" shows "CP_UNCURRY uu uv \<equiv> True"
nat_pow_eq: fixes x :: "nat"   and k :: "nat" shows "x [^]\<^bsub>G\<^esub> k = x ^ k mod n"
ext_inject: fixes role :: "role_t"   and owner :: "agent"   and partner :: "agent"   and more :: "'a"   and role' :: "role_t"   and owner' :: "agent"   and partner' :: "agent"   and more' :: "'a" shows "(\<lparr>role = role, owner = owner, partner = partner, \<dots> = more\<rparr> = \<lparr>role = role', owner = owner', partner = partner', \<dots> = more'\<rparr>) = (role = role' \<and> owner = owner' \<and> partner = partner' \<and> more = more')"
inf0: fixes a :: "'a" shows "a \<equiv> inf a top_class.top"
exhaust: fixes y :: "proc" obtains "y = p0"   | "y = p1"
section_retraction_of_iso_2: fixes f :: "'a1"   and g :: "'a1" assumes "C1.iso f"   and "C1.ide (f \<cdot>\<^sub>1 g)" shows "C1.inverse_arrows f g"
simps_4: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and d :: "real mpoly"   and p :: "real mpoly" shows "quadratic_part_1 var a b d (Neq p) = (let deg = MPoly_Type.degree p var in \<Sum>i = 0..<deg + 1. isolate_variable_sparse p var i * (a + b * Var var) ^ i * d ^ (deg - i))"
proj_id: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "intT \<sigma> a" shows "proj \<sigma> a = a"
perp_col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and E :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> E"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D"   and "Tarski_neutral_dimensionless.Col Bet A B E" shows "Tarski_neutral_dimensionless.Perp Bet Cong A E C D"
tp_sincos: fixes i :: "nat"   and j :: "nat" shows "trigpoly (\<lambda>x. sin (real i * x) * sin (real j * x)) \<and> trigpoly (\<lambda>x. sin (real i * x) * cos (real j * x)) \<and> trigpoly (\<lambda>x. cos (real i * x) * sin (real j * x)) \<and> trigpoly (\<lambda>x. cos (real i * x) * cos (real j * x))"
mult_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "(0::'a) \<le> c" shows "a * c \<le> b * c"
induct: fixes P :: "Arity list \<Rightarrow> stack_elem list \<Rightarrow> bool"   and a0 :: "Arity list"   and a1 :: "stack_elem list" assumes "\<And>uu_. P uu_ []"   and "\<And>a as e1 e2 S. P as S \<Longrightarrow> P (a # as) (Alts e1 e2 # S)"   and "\<And>x S. P [] S \<Longrightarrow> P [] (x # S)"   and "\<And>as v S. P as S \<Longrightarrow> P as (Arg v # S)"   and "\<And>as v S. P as S \<Longrightarrow> P as (Upd v # S)"   and "\<And>as v S. P as S \<Longrightarrow> P as (Dummy v # S)" shows "P a0 a1"
index_uminus_mat_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "(- A) $$ (i, j) = - A $$ (i, j)"
NSLIMSEQ_inverse: fixes X :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a"   and "a \<noteq> (0::'a)" shows "(\<lambda>n. inverse (X n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S inverse a"
induct: fixes P :: "((('a, 'b, 'c) prot_fun, 'd) Term.term \<times> (('e, 'f, 'g) prot_fun, 'h) Term.term) list \<Rightarrow> nat \<times> bool \<Rightarrow> bool"   and a0 :: "((('a, 'b, 'c) prot_fun, 'd) Term.term \<times> (('e, 'f, 'g) prot_fun, 'h) Term.term) list"   and a1 :: "nat \<times> bool" assumes "\<And>uu_. P [] uu_"   and "\<And>m s S D n. \<lbrakk>m = n \<Longrightarrow> P D n; m \<noteq> n \<Longrightarrow> P D n\<rbrakk> \<Longrightarrow> P ((Fun (Val m) [], Fun (Set s) S) # D) n"   and "\<And>vb va D n. P D n \<Longrightarrow> P ((Var vb, va) # D) n"   and "\<And>vd vc va D n. P D n \<Longrightarrow> P ((Fun (Fu vd) vc, va) # D) n"   and "\<And>vd vc va D n. P D n \<Longrightarrow> P ((Fun (Set vd) vc, va) # D) n"   and "\<And>vd vc va D n. P D n \<Longrightarrow> P ((Fun (Abs vd) vc, va) # D) n"   and "\<And>vc va D n. P D n \<Longrightarrow> P ((Fun prot_fun.Pair vc, va) # D) n"   and "\<And>vd vc va D n. P D n \<Longrightarrow> P ((Fun (Attack vd) vc, va) # D) n"   and "\<And>vd ve vc va D n. P D n \<Longrightarrow> P ((Fun (PubConstAtom vd ve) vc, va) # D) n"   and "\<And>vd vc va D n. P D n \<Longrightarrow> P ((Fun (PubConstSetType vd) vc, va) # D) n"   and "\<And>vd vc va D n. P D n \<Longrightarrow> P ((Fun (PubConstAttackType vd) vc, va) # D) n"   and "\<And>vd vc va D n. P D n \<Longrightarrow> P ((Fun (PubConstBottom vd) vc, va) # D) n"   and "\<And>vd vc va D n. P D n \<Longrightarrow> P ((Fun (PubConstOccursSecType vd) vc, va) # D) n"   and "\<And>vc va D n. P D n \<Longrightarrow> P ((Fun OccursFact vc, va) # D) n"   and "\<And>vc va D n. P D n \<Longrightarrow> P ((Fun OccursSec vc, va) # D) n"   and "\<And>vb vd ve va D n. P D n \<Longrightarrow> P ((Fun vb (vd # ve), va) # D) n"   and "\<And>v vb D n. P D n \<Longrightarrow> P ((v, Var vb) # D) n"   and "\<And>v vd vc D n. P D n \<Longrightarrow> P ((v, Fun (Fu vd) vc) # D) n"   and "\<And>v vd vc D n. P D n \<Longrightarrow> P ((v, Fun (Val vd) vc) # D) n"   and "\<And>v vd vc D n. P D n \<Longrightarrow> P ((v, Fun (Abs vd) vc) # D) n"   and "\<And>v vc D n. P D n \<Longrightarrow> P ((v, Fun prot_fun.Pair vc) # D) n"   and "\<And>v vd vc D n. P D n \<Longrightarrow> P ((v, Fun (Attack vd) vc) # D) n"   and "\<And>v vd ve vc D n. P D n \<Longrightarrow> P ((v, Fun (PubConstAtom vd ve) vc) # D) n"   and "\<And>v vd vc D n. P D n \<Longrightarrow> P ((v, Fun (PubConstSetType vd) vc) # D) n"   and "\<And>v vd vc D n. P D n \<Longrightarrow> P ((v, Fun (PubConstAttackType vd) vc) # D) n"   and "\<And>v vd vc D n. P D n \<Longrightarrow> P ((v, Fun (PubConstBottom vd) vc) # D) n"   and "\<And>v vd vc D n. P D n \<Longrightarrow> P ((v, Fun (PubConstOccursSecType vd) vc) # D) n"   and "\<And>v vc D n. P D n \<Longrightarrow> P ((v, Fun OccursFact vc) # D) n"   and "\<And>v vc D n. P D n \<Longrightarrow> P ((v, Fun OccursSec vc) # D) n" shows "P a0 a1"
simps: fixes a1 :: "'a"   and a2 :: "'a"   and r :: "('a \<times> 'a) set" shows "((a1, a2) \<in> r\<^sup>*) = ((\<exists>a. a1 = a \<and> a2 = a) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>* \<and> (b, c) \<in> r))"
subst_elim_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and v :: "'c" shows "subst_elim \<sigma> v \<equiv> \<forall>t. v \<notin> fv (t \<cdot> \<sigma>)"
ideal_mult_in_subgroup_generated: fixes \<aa> :: "'a set"   and \<bb> :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal \<aa> R (+) (\<cdot>) \<zero> \<one>"   and "ideal \<bb> R (+) (\<cdot>) \<zero> \<one>"   and "a \<in> \<aa>"   and "b \<in> \<bb>" shows "a \<cdot> b \<in> ideal_gen_by_prod \<aa> \<bb>"
s: shows "\<Turnstile>\<^sub>3 {em P'} c { em Q'}"
mset_le_incr_right1: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b" shows "a \<subseteq># b + c"
assms_7: shows "\<b> ps (Suc 0) \<le> d"
spec_ops_empty: fixes deps :: "'d \<Rightarrow> 'c set" shows "spec_ops [] deps"
bet2_le2__le2356: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Bet A B C"   and "Bet A' B' C'"   and "A B Le A' B'"   and "A' C' Le A C" shows "B' C' Le B C"
edges_it_def: fixes g :: "'G" shows "edges_it g \<equiv> it_to_it (edges_list_it g)"
that_1: assumes "phiDefNodes_aux g v unvisited n = {}"   and "n \<notin> set unvisited \<or> v \<in> defs g n" shows "thesis"
long_divides_def: fixes p :: "'a list"   and q :: "'a list"   and t :: "'a list \<times> 'a list" shows "long_divides p q t = (t \<in> carrier (poly_ring R) \<times> carrier (poly_ring R) \<and> p = q \<otimes>\<^bsub>poly_ring R\<^esub> fst t \<oplus>\<^bsub>poly_ring R\<^esub> snd t \<and> (snd t = [] \<or> degree (snd t) < degree q))"
Aizerman_on_def: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" shows "Aizerman_on A f = (\<forall>B\<subseteq>A. \<forall>C\<subseteq>B. f B \<subseteq> C \<longrightarrow> f C \<subseteq> f B)"
le_mid2__le13: fixes M :: "'p"   and A :: "'p"   and B :: "'p"   and M' :: "'p"   and A' :: "'p"   and B' :: "'p" assumes "M Midpoint A B"   and "M' Midpoint A' B'"   and "A M Le A' M'" shows "A B Le A' B'"
hom_dvd: fixes p :: "'a poly"   and q :: "'a poly"   and a :: "'a" assumes "p dvd q" shows "poly p a dvd poly q a"