hcju/isabelleps huggingface dataset:
power_mono_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a"   and "less_eq zero b"   and "0 < n" shows "less_eq (power.power one times a n) (power.power one times b n) = less_eq a b"
mk_strong_release_def: fixes x :: "'a ltln"   and y :: "'a ltln" shows "mk_strong_release x y \<equiv> case y of true\<^sub>n \<Rightarrow> mk_finally x | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> case x of true\<^sub>n \<Rightarrow> y | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> x M\<^sub>n y"
UN_equiv_class2: fixes A1 :: "'a set"   and r1 :: "('a \<times> 'a) set"   and A2 :: "'b set"   and r2 :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c set"   and a1 :: "'a"   and a2 :: "'b" assumes "equiv A1 r1"   and "equiv A2 r2"   and "congruent2 r1 r2 f"   and "a1 \<in> A1"   and "a2 \<in> A2" shows "(\<Union>x1\<in>r1 `` {a1}. \<Union> (f x1 ` r2 `` {a2})) = f a1 a2"
mult_sign_intros_5: fixes a :: "'d"   and b :: "'d" assumes "(0::'d) < a"   and "(0::'d) < b" shows "(0::'d) < a * b"
induct: fixes f :: "nat \<Rightarrow> real" assumes "polynomial f" obtains n :: "real" where "f \<in> O(\<lambda>x. real x powr n)"
mem_imp_VWF: fixes x :: "V"   and y :: "V" assumes "x \<in> elts y" shows "(x, y) \<in> VWF"
pickQFreshEnv_card_of: fixes V :: "'var set"   and XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Rho :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option) set"   and xs :: "'varSort" assumes "|V| <o |UNIV|"   and "|XS| <o |UNIV|"   and "Ball XS qGood"   and "|Rho| <o |UNIV|"   and "Ball Rho qGoodEnv" shows "pickQFreshEnv xs V XS Rho \<notin> V \<and> (\<forall>X\<in>XS. qFresh xs (pickQFreshEnv xs V XS Rho) X) \<and> (\<forall>rho\<in>Rho. qFreshEnv xs (pickQFreshEnv xs V XS Rho) rho)"
reachable_neq_reachable1: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>with_proj (subdivide G e w)\<^esub> w"   and "v \<noteq> w" shows "(v, w) \<in> (parcs (subdivide G e w))\<^sup>+"
atomic_ev_wait_all_preserves_invariants: fixes s :: "state_t"   and tid :: "thread_id_t" assumes "atomic_step_invariant s" shows "atomic_step_invariant (atomic_step_ev_wait_all tid s)"
snth_of_seq: fixes f :: "nat \<Rightarrow> 'a" shows "(!!) (of_seq f) = f"
iT_Plus_singleton: fixes a :: "nat"   and k :: "nat" shows "{a} \<oplus> k = {a + k}"
frontier_subset_compact: fixes S :: "'a set" assumes "compact S" shows "frontier S \<subseteq> S"
intros_3: fixes d :: "int"   and u :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "real_of_int d < u x - u y"   and "u x - u y < real_of_int (d + 1)" shows "intv'_elem x y u (Intv' d)"
range: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "MaxR r" shows "MaxR.MaxR_opt r A \<in> Some ` (A \<inter> Field r) \<union> {None}"
mset_subset_eq_multiset_union_diff_commute: fixes B :: "'k multiset"   and A :: "'k multiset"   and C :: "'k multiset" assumes "B \<subseteq># A" shows "A - B + C = A + C - B"
ldrop_eSuc_conv_ltl: fixes n :: "enat"   and xs :: "'b llist" shows "ldrop (eSuc n) xs = ltl (ldrop n xs)"
sortedP_append_iff: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "linorder_list0.sortedP le (xs @ ys) = (linorder_list0.sortedP le xs \<and> linorder_list0.sortedP le ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. le x y))"
get_disconnected_nodes_new_element: fixes h :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) document_ptr" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "r \<in> get_disconnected_nodes_locs ptr'" shows "r h h'"
wf_constr_bvars_disj: fixes S :: "('a, 'b) strand_step list"   and \<theta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>" shows "(subst_domain \<theta> \<union> range_vars \<theta>) \<inter> bvars\<^sub>s\<^sub>t S = {}"
col_A_B_ABl: fixes A :: "Points"   and B :: "Points"   and l :: "Lines" shows "col A B (Pascal_Property.inter (line A B) l)"
B2_def: shows "B2 = (\<Sum>((i, j), k)\<in>J. real_of_int (2 * poly.coeff p i * poly.coeff p j) / real_of_int (int (k\<^sup>2 * (i - j))))"
halloc_conforms: fixes G :: "prog"   and s1 :: "state"   and oi :: "obj_tag"   and a :: "loc"   and s2 :: "state"   and L :: "lenv"   and fs :: "vn \<Rightarrow> val option" assumes "G\<turnstile>s1 \<midarrow>halloc oi\<succ>a\<rightarrow> s2"   and "wf_prog G"   and "s1\<Colon>\<preceq>(G, L)"   and "is_type G (obj_ty \<lparr>tag = oi, values = fs\<rparr>)" shows "s2\<Colon>\<preceq>(G, L)"
bf': shows "\<forall>x\<in>S. f' x = blinfun_apply (bf' x)"
sq_norm_vec_of_poly: fixes p :: "'a poly" shows "\<parallel>vec_of_poly p\<parallel>\<^sup>2 = \<parallel>p\<parallel>\<^sup>2"
frequently_filtercomapI: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and F :: "'b filter"   and G :: "'a filter"   and P :: "'c \<Rightarrow> bool" assumes "LIM n G. f (g n) :> F"   and "\<forall>\<^sub>F n in G. P (g n)"   and "G \<noteq> bot" shows "frequently P (filtercomap f F)"
duplicateEdge_def: fixes g :: "graph"   and f :: "face"   and a :: "nat"   and b :: "nat" shows "duplicateEdge g f a b \<equiv> 2 \<le> directedLength f a b \<and> 2 \<le> directedLength f b a \<and> b \<in> set (neighbors g a)"
wffp_hd_purgeIdle: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list"   and s :: "('a, 'b, 'c) state" assumes "GM_sec_model.wffp do sl"   and "GM_sec_model.purgeIdle sl \<noteq> []"   and "isState s"   and "hd sl \<in> GM_sec_model.\<delta> do s" shows "hd (GM_sec_model.purgeIdle sl) \<in> GM_sec_model.\<delta> do s"
proj2_another_point_on_line: fixes p :: "proj2"   and l :: "proj2_line" shows "\<exists>q. q \<noteq> p \<and> proj2_incident q l"
lcoset_0: fixes w :: "'b" shows "w +o 0 = {w}"
assms: shows "\<forall>i\<le>n. \<forall>j\<le>n. M i j \<noteq> \<infinity> \<longrightarrow> get_const (M i j) \<in> \<int>"
dghm_ArrMap_vsubset_Vset: shows "\<FF>\<lparr>ArrMap\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
i_Exec_Stream_Acc_LocalState_append: fixes k :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" shows "i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun (xs \<frown> input) c = f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c \<frown> i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input (f_Exec_Comp trans_fun (xs \<odot> k) c)"
shadow_root_ptr_kinds_def: fixes heap :: "(_) heap" shows "shadow_root_ptr_kinds heap = the |`| cast |`| ffilter is_shadow_root_ptr_kind (document_ptr_kinds heap)"
is_inf_def: fixes x :: "'a"   and y :: "'a"   and i :: "'a" shows "is_inf x y i = (i \<sqsubseteq> x \<and> i \<sqsubseteq> y \<and> (\<forall>z. z \<sqsubseteq> x \<and> z \<sqsubseteq> y \<longrightarrow> z \<sqsubseteq> i))"
le_numeral_extra_4: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one one"
edge'_simps_3: fixes y :: "'v node" shows "edge' y SOURCE = False"
measurable_inequality_set_3: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{x \<in> space M. g x \<le> f x} \<in> sets M"
simps_2: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'c list"   and x :: "'c" shows "sinks_aux I D U (xs @ [x]) = (if \<exists>v\<in>sinks_aux I D U xs. (v, D x) \<in> I then insert (D x) (sinks_aux I D U xs) else sinks_aux I D U xs)"
cimage_cong: fixes M :: "'d cset"   and N :: "'d cset"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "M = N"   and "\<And>x. cin x N \<Longrightarrow> f x = g x" shows "cimage f M = cimage g N"
list_def: fixes A :: "'b set" shows "list A \<equiv> SOME al. distinct al \<and> set al = A"
strict_prefix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs <p ys) = (xs \<le>p ys \<and> xs \<noteq> ys)"
infsetsum_def: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "infsetsum f A = integral\<^sup>L (count_space A) f"
sds_lowering_strategyproof_axioms: shows "sds_lowering_strategyproof agents alts sds agents' alts'"
P: shows "P = (\<lambda>x y. x \<in> B \<longrightarrow> y \<in> B \<longrightarrow> R\<^sup>+\<^sup>+ (gr x) (gr y))"
basic_just_sp_pred_def: fixes G :: "('a, 'b) pre_digraph"   and dist :: "'a \<Rightarrow> ereal"   and c :: "'b \<Rightarrow> real"   and s :: "'a"   and num :: "'a \<Rightarrow> enat"   and pred :: "'a \<Rightarrow> 'b option" shows "basic_just_sp_pred G dist c s num pred \<equiv> basic_sp G dist c s \<and> basic_just_sp_pred_axioms G dist c s num pred"
L': shows "(\<lambda>n. \<Prod>i\<le>Suc n. f (i + M)) \<longlonglongrightarrow> L"
that: fixes Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> !P \<Longrightarrow>\<^sub>\<tau> Q"   and "\<Psi> \<rhd> Q \<sim> P' \<parallel> !P" shows "thesis"
Gram_Schmidt_step: fixes S :: "'a set"   and x :: "'a"   and a :: "'a" assumes "pairwise orthogonal S"   and "x \<in> span S" shows "orthogonal x (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b))"
abs_eq: fixes x :: "'b list" shows "Poly_Mapping.nth x = Abs_poly_mapping (nth_default (0::'b) x)"
Nil: fixes r :: "('a \<times> 'a) set" shows "([], []) \<in> genPrefix r"
IH: fixes V :: "'a set" assumes "wf'\<^sub>s\<^sub>s\<^sub>t V S" shows "wf'\<^sub>s\<^sub>s\<^sub>t (V \<union> W) S"
pos_le_minus_divideC_eq: fixes c :: "complex"   and a :: "'a"   and b :: "'a" assumes "0 < c" shows "(a \<le> - (b /\<^sub>C c)) = (c *\<^sub>C a \<le> - b)"
atoms_list_correct: fixes \<phi> :: "'a ltln" shows "set (atoms_list \<phi>) = atoms_ltln \<phi>"
conc_fun_strict: fixes R :: "('b \<times> 'c) set" shows "\<Down> R SUCCEED = SUCCEED"
SucFbd_Cnotzero: fixes r2a :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "(cardSuc ((bd_F1 +c r2a) +c r2), czero) \<notin> ordIso \<and> Card_order (cardSuc ((bd_F1 +c r2a) +c r2))"
BRANCH_def: fixes P :: "'a" shows "BRANCH P \<equiv> P"
UN: fixes A :: "nat \<Rightarrow> 'a set" assumes "disjoint_family A"   and "range A \<subseteq> M" shows "\<Union> (range A) \<in> M"
map_id: fixes t :: "'c pmf" shows "map_pmf id t = t"
conjugation_subgroup: fixes H :: "'a set"   and g :: "'a" assumes "subgroup H G"   and "g \<in> carrier G" shows "subgroup (g <# (H #> inv g)) G"
Bk: shows "B ^\<^sub>m k \<in> carrier_mat m m"
set_deg_shifts_singleton: fixes d :: "nat"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "set (deg_shifts d [f]) = (\<lambda>t. local.punit.monom_mult (1::'b) t f) ` deg_le_sect X (d - poly_deg f)"
distinct_47: fixes pi :: "pi"   and pi' :: "pi" shows "\<tau>.pi \<noteq> !pi'"
uminus_image_real_of_ereal_image_greaterThanLessThan: fixes l :: "ereal"   and u :: "ereal" shows "uminus ` real_of_ereal ` {l<..<u} = real_of_ereal ` {- u<..<- l}"
Upper_dual: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "Upper (inv_gorder L) A = Lower L A"
imply_lift_Imp: fixes p :: "('a, 'b) fm"   and q :: "('a, 'b) fm"   and ps :: "('a, 'b) fm list" assumes "\<turnstile> p \<^bold>\<longrightarrow> q" shows "\<turnstile> p \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> q"
edom_iff2: fixes m :: "'d \<Rightarrow> 'c"   and a :: "'d" shows "(m a = \<bottom>) = (a \<notin> edom m)"
fset_inject: fixes x :: "'b fset"   and y :: "'b fset" shows "(\<langle>x\<rangle>\<^sub>f = \<langle>y\<rangle>\<^sub>f) = (x = y)"
elims: fixes x :: "'a \<Rightarrow> atom set"   and xa :: "'b \<Rightarrow> atom set"   and xb :: "'a \<times> 'b"   and y :: "atom set" assumes "prod_fv x xa xb = y" obtains fv1 :: "'a \<Rightarrow> atom set" and fv2 :: "'b \<Rightarrow> atom set" and xc :: "'a" and ya :: "'b" where "x = fv1" and "xa = fv2" and "xb = (xc, ya)" and "y = fv1 xc \<union> fv2 ya"
map_poly_inj_comm_ring_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "map_poly_inj_comm_ring_hom hom \<equiv> inj_comm_ring_hom hom"
Abs_lifted_distrib_lattice_top_inject: fixes x :: "('c, 'd) phi \<Rightarrow> 'd"   and y :: "('c, 'd) phi \<Rightarrow> 'd" assumes "x \<in> {xf. True}"   and "y \<in> {xf. True}" shows "(Abs_lifted_distrib_lattice_top x = Abs_lifted_distrib_lattice_top y) = (x = y)"
decl: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mu :: "'Var ModeUpd"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and E :: "('Var, 'AExp, 'BExp) Stmt list" assumes "\<langle>c, update_modes mu mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" shows "\<langle>cxt_to_stmt E (c@[mu]), mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>cxt_to_stmt E c', mds', mem'\<rangle>\<^sub>w"
seq_char: fixes g :: "'a"   and f :: "'a" shows "S.seq g f = (S.arr f \<and> S.arr g \<and> seq g f)"
func_le_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and f :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" assumes "sifum_security stop eval some_val some_val'" shows "f \<preceq> g = (\<forall>x\<in>dom f. f x = g x)"
EmptyLeftSync: fixes s :: "'a event list"   and A :: "'a event set"   and t :: "'a event list" assumes "s setinterleaves (([], t), A)" shows "s = t \<and> set t \<inter> A = {}"
oassert_iff_1: fixes \<Phi> :: "bool"   and x :: "unit" shows "(oassert \<Phi> = Some x) = \<Phi>"
case_eq_if: fixes f :: "('b \<Rightarrow> bool) \<Rightarrow> 'a"   and pred :: "'b pred" shows "(case pred of Pred x \<Rightarrow> f x) = f (eval pred)"
codomains_char: fixes f :: "'a arrow_of_spans_data" shows "V.codomains f = (if V.arr f then {V.cod f} else {})"
rank_matrix_dim_range: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "linear (*s) (*s) f" shows "rank (matrix f) = vec.dim (range f)"
evals: shows "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>map Val (v # vs),(h, l)\<rangle>"
asAbs_pick: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodAbs A" shows "asAbs (pick A) = A"
irreducible\<^sub>d_m_def: fixes m :: "int"   and f :: "int poly" shows "poly_mod.irreducible\<^sub>d_m m f \<equiv> 0 < degree (poly_mod.Mp m f) \<and> (\<forall>g h. degree (poly_mod.Mp m g) < degree (poly_mod.Mp m f) \<longrightarrow> degree (poly_mod.Mp m h) < degree (poly_mod.Mp m f) \<longrightarrow> poly_mod.Mp m f \<noteq> poly_mod.Mp m (g * h))"
mk_lossless_lossless: fixes p :: "'c spmf" assumes "lossless_spmf p" shows "mk_lossless p = p"
Rf_sat: fixes N :: "'b literal multiset set" assumes "satisfiable (N - standard_redundancy_criterion.Rf N)" shows "satisfiable N"
typedef_lens_def: shows "typedef\<^sub>L = \<lparr>lens_get = Abs, lens_put = \<lambda>s. Rep\<rparr>"
neg_numeral_less_signed_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num"   and w :: "'b Word.word" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus (numeral.numeral one plus n)) (ring_1.signed one plus zero minus w) = (- numeral n < sint w)"
assms_4: shows "n \<in>\<^sub>\<circ> \<omega>"
lnth_ldropn: fixes n :: "nat"   and m :: "nat"   and xs :: "'a llist" assumes "enat (n + m) < llength xs" shows "lnth (ldropn n xs) m = lnth xs (m + n)"
less_char: fixes x :: "char"   and y :: "char" shows "x < y \<equiv> int_of_char x < int_of_char y"
nat_pow_closed: fixes x :: "int \<times> int"   and n :: "nat" assumes "x \<in> carrier R" shows "x [^]\<^bsub>R\<^esub> n \<in> carrier R"
U0: shows "u \<in> carrier (residue_ring (p ^ k)) \<and> u * p ^ m = x (m + k)"
bounded_linear_linepath: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and x :: "real" assumes "bounded_linear f" shows "f (linepath a b x) = linepath (f a) (f b) x"
preserves_seq: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.seq (trg\<^sub>C g) (trg\<^sub>C f)"
Pre_element: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "ExPre D a" shows "Pre D a \<in> carrier D \<and> Pre D a \<prec>\<^bsub>D\<^esub> a \<and> \<not> (\<exists>y\<in>carrier D. Pre D a \<prec>\<^bsub>D\<^esub> y \<and> y \<prec>\<^bsub>D\<^esub> a)"
cycle_map: fixes r :: "Vertex \<Rightarrow> Vertex"   and a :: "Vertex"   and b :: "Vertex"   and c :: "Vertex"   and d :: "Vertex" assumes "r \<in> complex_rotations"   and "distinct [a, b, c, d]"   and "r a = b"   and "r b = a"   and "r c = d"   and "r d = c" shows "\<forall>v w. r v = w \<longrightarrow> r w = v"
GuaranteeStrip: fixes P :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f"   and g :: "'d set" assumes "P \<subseteq> R"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c Q,A"   and "f \<in> F" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P guaranteeStrip f g c Q,A"
ereal_minus_3: fixes r :: "real" shows "ereal r - \<infinity> = - \<infinity>"
relpow_Suc_I: fixes x :: "'b"   and y :: "'b"   and n :: "nat"   and R :: "('b \<times> 'b) set"   and z :: "'b" assumes "(x, y) \<in> R ^^ n"   and "(y, z) \<in> R" shows "(x, z) \<in> R ^^ Suc n"
distinct_adj_ConsD: fixes x :: "'g"   and xs :: "'g list" assumes "distinct_adj (x # xs)" shows "distinct_adj xs"
fls_lr_inverse_X_intpow_2: fixes i :: "int"   and y :: "'a" shows "fls_right_inverse (fls_X_intpow i) y = fls_shift i (fls_const y)"
counit_simps_1: shows "arr \<epsilon>\<^sub>0"
Mh': shows "h \<in> state_measure (V \<union> V') \<Gamma> \<rightarrow>\<^sub>M subprob_algebra R"
option_assn_simps_2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and v :: "'a option" shows "option_assn P v None = \<up> (v = None)"
snd_image_times: fixes A :: "'e set"   and B :: "'d set" shows "snd ` (A \<times> B) = (if A = {} then {} else B)"
ceuclidean_representation_sum: fixes f :: "'a \<Rightarrow> complex"   and b :: "'a" shows "((\<Sum>i\<in>CBasis. f i *\<^sub>C i) = b) = (\<forall>i\<in>CBasis. f i = cnj \<langle>b, i\<rangle>)"
in_var_def: fixes x :: "'c \<Longrightarrow> 'd" shows "$x = x ;\<^sub>L fst\<^sub>L"
d_indistinguishable_trans: fixes d :: "'d"   and e :: "'exp"   and e' :: "'exp"   and e'' :: "'exp" assumes "e \<equiv>\<^bsub>d\<^esub> e'"   and "e' \<equiv>\<^bsub>d\<^esub> e''" shows "e \<equiv>\<^bsub>d\<^esub> e''"
slpa_snoc_Call: fixes cs :: "'edge list"   and as :: "'edge list"   and 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 "same_level_path_aux cs as"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "same_level_path_aux cs (as @ [a])"
pt_noption_inst: assumes "pt TYPE('a) TYPE('b)" shows "pt TYPE('a noption) TYPE('b)"
index_of_aux_lookup_1: fixes p :: "'a"   and ls :: "'a list"   and n :: "nat" assumes "p \<in> set ls" shows "index_of_aux ls p n - n < length ls"
wtPB_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "wtPB \<Phi> \<equiv> Ball \<Phi> wtC"
bv_to_nat_extend: fixes n :: "nat"   and w :: "bit list" shows "bv_to_nat (bv_extend n \<zero> w) = bv_to_nat w"
nplarger: shows "3 \<le> 2 ^ (multiplicity 2 m + 1) - 1"
length_rs: shows "length rs = length ts - length ts div 2 - 1"
t'_1: shows "t' \<in>\<^sub>A frontier (c_imp c loc1)"
R_def: shows "R = (\<lambda>x. {y \<in> A. x < y})"
simps: fixes a1 :: "'d option"   and a2 :: "'d option"   and C :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd option" shows "((a1, a2) \<in> option_while' C B) = ((\<exists>r. a1 = Some r \<and> a2 = Some r \<and> \<not> C r) \<or> (\<exists>r. a1 = Some r \<and> a2 = None \<and> C r \<and> B r = None) \<or> (\<exists>r r' sr''. a1 = Some r \<and> a2 = sr'' \<and> C r \<and> B r = Some r' \<and> (Some r', sr'') \<in> option_while' C B))"
cases: fixes x :: "('a \<Rightarrow> 'b) \<times> 'a list" obtains f :: "'a \<Rightarrow> 'b" and x :: "'a" where "xa__ = (f__, [xb__])"   | f :: "'a \<Rightarrow> 'b" and x :: "'a" and y :: "'a" and zs :: "'a list" where "xa__ = (f, x # y # zs)"   | a :: "'a \<Rightarrow> 'b" where "xa__ = (a, [])"
one_eq_of_rat_iff: fixes a :: "rat" shows "((1::'a) = of_rat a) = (1 = a)"
lcm_D: fixes a :: "'b" shows "a r\<rightarrow> a = (1::'b)"
distinct_zip_inj: fixes ys :: "'b list"   and a :: "'c"   and b :: "'b"   and xs :: "'c list"   and c :: "'c" assumes "distinct ys"   and "(a, b) \<in> set (zip xs ys)"   and "(c, b) \<in> set (zip xs ys)" shows "a = c"
swap_typeI_is_typeII: fixes C :: "real \<times> real \<Rightarrow> real \<times> real" assumes "typeI_twoCube C" shows "typeII_twoCube (prod.swap \<circ> C \<circ> prod.swap)"
simps: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set" shows "orthogonal_current \<Gamma> f S = ((\<forall>x. x \<in> S \<longrightarrow> x \<notin> A \<Gamma> \<longrightarrow> weight \<Gamma> x \<le> d_IN f x) \<and> (\<forall>x. x \<in> S \<longrightarrow> x \<in> A \<Gamma> \<longrightarrow> x \<notin> B \<Gamma> \<longrightarrow> d_OUT f x = weight \<Gamma> x) \<and> (\<forall>x xa. xa \<in> RF\<^bsub>\<Gamma>\<^esub> S \<longrightarrow> x \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S \<longrightarrow> f (x, xa) = 0))"
assms: shows "n = n' \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. k < n \<and> \<parallel>c\<parallel>\<^bsub>t k\<^esub>)"
simps_5: fixes v :: "'c match_expr"   and rs :: "'c rule list" shows "process_ret (Rule v Log # rs) = Rule v Log # process_ret rs"
assms_3: shows "\<forall>\<^sub>F x in at a. f x \<noteq> b"
indexed_nominal_ts_axioms_def: shows "indexed_nominal_ts_axioms TYPE('a) TYPE('b) \<equiv> |UNIV| <o |UNIV| \<and> |UNIV| <o |UNIV|"
WTrtALength: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'addr expr"   and T :: "ty" assumes "P,E,h \<turnstile> a : T\<lfloor>\<rceil>" shows "P,E,h \<turnstile> a\<bullet>length : Integer"
drop_butlast: fixes n :: "nat"   and xs :: "'c list" shows "drop n (butlast xs) = butlast (drop n xs)"
real_root_pow_pos2: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 \<le> x" shows "root n x ^ n = x"
set_bd_1: fixes x :: "('l, 'k, 'm) F1" shows "|F1set1 x| \<le>o bd_F1"
assms_4: shows "x \<longlonglongrightarrow> x'"
sig_inv_setI: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "keys (vectorize_poly r) \<subseteq> {0..<j}" shows "r \<in> sig_inv_set' j"