hcju/isabelleps huggingface dataset:
atLeastatMost_subset_iff: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry"   and d :: "'c DBMEntry" shows "(linordered_monoid.atLeastAtMost a b \<subseteq> linordered_monoid.atLeastAtMost c d) = (\<not> a \<preceq> b \<or> c \<preceq> a \<and> b \<preceq> d)"
simps_3: fixes va :: "nat" shows "sys' (Suc (Suc va)) = ACT ''a'' (VAR (0::'a))"
pdevs_val_nlex_pdevs2: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and x :: "(real \<times> real) pdevs" assumes "e \<in> UNIV \<rightarrow> I"   and "uminus ` I = I" obtains e' :: "nat \<Rightarrow> real" where "e' \<in> UNIV \<rightarrow> I" and "pdevs_val e (nlex_pdevs x) = pdevs_val e' x"
borel_measurable_vimage_halfspace_component_lt: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S))"
assms: fixes z :: "int" assumes "x = of_int z"   and "0 \<le> z"   and "z < n" shows "P"
ln_loop: fixes \<sigma> :: "nat \<Rightarrow> bool"   and cnf :: "lit uprod set"   and l :: "lit" assumes "sat_cnf \<sigma> cnf"   and "(l, negate l) \<in> (imp_graph cnf)\<^sup>*" shows "\<not> sat_lit \<sigma> l"
eval_fps_eqD: fixes f :: "complex fps"   and g :: "complex fps" assumes "0 < fps_conv_radius f"   and "0 < fps_conv_radius g"   and "\<forall>\<^sub>F z in nhds 0. eval_fps f z = eval_fps g z" shows "f = g"
relation_subsumption_impl_axioms_def: fixes r :: "'b \<Rightarrow> 'c"   and set_of :: "'c \<Rightarrow> 'b set" shows "relation_subsumption_impl_axioms r set_of \<equiv> \<forall>a. finite {b. (a, b) \<in> {(a, b). b \<in> set_of (r a)}\<^sup>*}"
Label_Delta_subset: fixes SA :: "('d, 'e, 'f) seqauto" shows "Label (Delta SA) \<subseteq> Labels SA"
finsum_closed: fixes f :: "'c \<Rightarrow> nat \<Rightarrow> 'a"   and A :: "'c set" assumes "f \<in> A \<rightarrow> carrier P" shows "finsum P f A \<in> carrier P"
subterm_inj_on_def: fixes f :: "'d \<Rightarrow> ('e, 'f) Term.term"   and A :: "'d set" shows "subterm_inj_on f A \<equiv> \<forall>x\<in>A. \<forall>y\<in>A. (\<exists>v. v \<sqsubseteq> f x \<and> v \<sqsubseteq> f y) \<longrightarrow> x = y"
suffix_plus: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" shows "s |\<^sub>s n |\<^sub>s m = s |\<^sub>s m + n"
power_sub: fixes n :: "nat"   and m :: "nat"   and a :: "nat" assumes "n \<le> m"   and "0 < a" shows "a ^ (m - n) = a ^ m div a ^ n"
convex_segment_1: fixes a :: "'a"   and b :: "'a" shows "convex (closed_segment a b)"
less_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f < g) = strict (\<le>) f g"
decide_existential_def: fixes fml :: "rat poly fml" shows "decide_existential fml = (let (fml_struct, polys) = convert fml; conds = find_consistent_signs polys in find (lookup_sem fml_struct) conds \<noteq> None)"
dvd_def: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" shows "dvd.dvd times b a = (\<exists>k. a = times b k)"
stutter_extend_edgesI_stutter: fixes v :: "'a"   and V :: "'a set"   and E :: "('a \<times> 'a) set" assumes "v \<in> V"   and "v \<notin> Domain E" shows "(v, v) \<in> stutter_extend_edges V E"
mkArr'_def: fixes a :: "'s"   and b :: "'s"   and F :: "'s \<Rightarrow> 's" shows "mkArr' a b F = mkArr (local.set a) (local.set b) (img \<circ> F \<circ> mkPoint a)"
isomorphic_to_zero_left: fixes G :: "'b set"   and smult :: "('c, 'b) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and V :: "'d set"   and smult' :: "('c, 'b) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and T :: "'d \<Rightarrow> 'e"   and W :: "'e set" assumes "FGModuleIso G smult V smult' T W"   and "V = 0" shows "W = 0"
Union_fold: fixes xs :: "('b \<times> unit) list" shows "\<Union> (f ` {x. (x, ()) \<in> set xs}) = fold (\<lambda>(a, uu_). (\<union>) (f a)) xs {}"
simps_42: fixes x7 :: "'b ptrm"   and x41 :: "'b"   and x42 :: "PreSimplyTyped.type"   and x43 :: "'b ptrm" shows "PSnd x7 \<noteq> PFn x41 x42 x43"
liouville_def: fixes x :: "real"   and p :: "nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> int" shows "liouville x p q \<equiv> (\<forall>n. 0 < \<bar>x - real_of_int (p n) / real_of_int (q n)\<bar>) \<and> (\<forall>n. 1 < q n) \<and> (\<forall>n. \<bar>x - real_of_int (p n) / real_of_int (q n)\<bar> < 1 / real_of_int (q n) ^ n)"
le_neq_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.order less_eq less"   and "less_eq a b"   and "a \<noteq> b" shows "less a b"
lappend_ltake_enat_ldropn: fixes n :: "nat"   and xs :: "'a llist" shows "ltake (enat n) xs @\<^sub>l ldropn n xs = xs"
consume_type: fixes ts :: "t list"   and ts' :: "ct list"   and c_t :: "checker_type" assumes "consume (Type ts) ts' = c_t"   and "c_t \<noteq> Bot" shows "\<exists>ts''. ct_list_eq (to_ct_list ts) (to_ct_list ts'' @ ts') \<and> c_t = Type ts''"
same_size: fixes signs :: "rat list list"   and subsets :: "nat list list" assumes "invertible_mat (M_mat signs subsets)" shows "length subsets = length signs"
balL_bd: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "bd_inv l"   and "bd_inv r"   and "cl r = B"   and "black_depth l + 1 = black_depth r" shows "bd_inv (balL l k v r) \<and> black_depth (balL l k v r) = black_depth l + 1"
subst_type_aux: fixes \<Delta> :: "(name \<times> ty) list"   and x :: "name"   and U :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and u :: "trm" assumes "\<Delta> @ [(x, U)] @ \<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> u : U" shows "\<Delta> @ \<Gamma> \<turnstile> t[x\<mapsto>u] : T"
add_less_imp_less_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less (plus a c) (plus b c)" shows "less a b"
assms: fixes y :: "'a" shows "summable (\<lambda>n. c n * y ^ n)"
assms_3: shows "(f \<sqinter> q)\<^sup>T * f\<^sup>\<star> * e = bot"
hadamard_dim: shows "hadamard \<in> carrier_mat 2 2"
CallObjThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and M :: "char list"   and ps :: "char list exp list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
in_normalized_matches: fixes ls :: "'b match_expr"   and m :: "'b match_expr"   and \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a :: "action"   and p :: "'c" assumes "ls \<in> set (normalize_match m) \<and> matches \<gamma> ls a p" shows "matches \<gamma> m a p"
asq_def: shows "asq = rat_of_int (sgn a) * rat_of_nat sq / rat_of_int b"
tm_tdghm_NTMap_in_Vset: shows "\<NN>\<lparr>NTMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
p_fun_def_var: fixes x :: "'a" shows "is_p_fun x = (x ; - 1' \<le> - x)"
tan_def: shows "tan = (\<lambda>x. sin x / cos x)"
assms_3: shows "A \<inter> B \<in> null_sets M"
if_weak_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'a"   and y :: "'a" assumes "b = c" shows "(if b then x else y) = (if c then x else y)"
check: fixes a :: "nat list"   and b :: "nat list" assumes "hlde a b" shows "check' a b = filter (\<lambda>(x, y). static_bounds a b x y \<and> a \<bullet> x = b \<bullet> y \<and> hlde.boundr a b x y \<and> hlde.subdprodl a b x y \<and> hlde.subdprodr a b y)"
simps_2: fixes t1 :: "lam"   and t2 :: "lam" shows "fst_app_aux (App t1 t2) = Some t1"
dg_dghm_id_is_iso_dghm: shows "dghm_id \<CC> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<CC>"
cols_mat_of_cols: fixes vs :: "'b vec list"   and n :: "nat" assumes "set vs \<subseteq> carrier_vec n" shows "cols (mat_of_cols n vs) = vs"
path_valid_node_2: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and n :: "'h"   and as :: "'g list"   and n' :: "'h" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "CFG.valid_node sourcenode targetnode valid_edge n'"
l_heap_is_wellformed_axioms: shows "l_heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes get_disconnected_nodes"
E_ss_VxV: shows "E \<subseteq> V \<times> V"
theI_unique: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "\<exists>!x. P x" shows "P x = (x = The P)"
is_found_def: fixes status :: "status" shows "is_found status = (status = FOUND)"
exec_exec_max: fixes s1 :: "state"   and c1 :: "stmt"   and n1 :: "nat"   and t1 :: "state"   and s2 :: "state"   and c2 :: "stmt"   and n2 :: "nat"   and t2 :: "state" assumes "s1 -c1-n1\<rightarrow> t1"   and "s2 -c2-n2\<rightarrow> t2" shows "s1 -c1-max n1 n2\<rightarrow> t1 \<and> s2 -c2-max n1 n2\<rightarrow> t2"
subset_CollectI: fixes B :: "'a set"   and A :: "'a set"   and Q :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "B \<subseteq> A"   and "\<And>x. \<lbrakk>x \<in> B; Q x\<rbrakk> \<Longrightarrow> P x" shows "{x \<in> B. Q x} \<subseteq> {x \<in> A. P x}"
foldr_conv_fold: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'d list" shows "foldr f xs = fold f (rev xs)"
pred_eqI: fixes P :: "'b Predicate.pred"   and Q :: "'b Predicate.pred" assumes "\<And>w. pred.eval P w = pred.eval Q w" shows "P = Q"
nh: assumes "w1 = NEG_INF \<and> w2 \<noteq> POS_INF \<and> 0 <s w2" shows "P w1 w2"
IN: fixes x :: "'v vertex" shows "d_IN f x \<le> value_flow \<Delta>'' f"
prev_N_def: shows "prev_N = (\<lambda>n. Max {N. Nf N \<le> n})"
merge_interact: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list" assumes "merge as bs us vs"   and "strict_sorted (concat as)"   and "strict_sorted (concat bs)"   and "bs \<in> lists (- {[]})" shows "strict_sorted (interact us vs)"
sqrt_C1_differentiable: fixes f :: "real \<Rightarrow> real"   and S :: "real set" assumes "f C1_differentiable_on S"   and "f ` S \<subseteq> {0<..}" shows "(\<lambda>x. sqrt (f x)) C1_differentiable_on S"
in_span_Basis: fixes x :: "'a" shows "x \<in> span Basis"
P1_xor_inf_th: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "R1_xor A B = xor_funct A B \<bind> (\<lambda>(out1, out2). S1_xor A out1)"
simplical_face_def: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'b" shows "simplical_face k x \<equiv> \<lambda>i. if i < k then x i else if i = k then 0::'b else x (i - 1)"
card_pick: fixes n :: "nat"   and S :: "nat set" assumes "n < card S \<or> infinite S" shows "card {a \<in> S. a < pick S n} = n"
commute: fixes a :: "'c option"   and b :: "'c option" shows "least a b = least b a"
awalk_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "S.awalk u p v \<equiv> u \<in> pverts slim \<and> set p \<subseteq> parcs slim \<and> pcas u p v"
distinct_21: fixes x2 :: "('d, 'e) Term.term"   and x71 :: "'e list"   and x72 :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list"   and x73 :: "(('d, 'e) Term.term \<times> ('d, 'e) Term.term) list" shows "receive\<langle>x2\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
Ri_eq_Ri_diff_Rf: fixes N :: "'a literal multiset set" shows "Ri N = Ri (N - Rf N)"
gebub_def: fixes n :: "nat"   and m :: "nat" shows "gebub n m = index init (s'_A n m ! Suc (paid_A ! n ! (length (paid_A ! n) - Suc m)))"
nat_ivl_Suc': fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> Suc n" shows "sum g {m..Suc n} = g (Suc n) + sum g {m..n}"
transpose_columnvector: fixes v :: "('b, 'c) vec" shows "Finite_Cartesian_Product.transpose (columnvector v) = rowvector v"
flipped_reflections_reduced_eq_lconjseq: fixes ss :: "'w list" assumes "S_reduced ss" shows "flipped_reflections (sum_list ss) = set (lconjseq ss)"
interior_of_subset_closure_of: fixes X :: "'c topology"   and S :: "'c set" shows "X interior_of S \<subseteq> X closure_of S"
servK_not_AKcryptSK: fixes B :: "agent"   and A :: "agent"   and SK :: "nat"   and Ts :: "nat"   and evs :: "event list"   and K :: "nat" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key SK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Key SK \<notin> analz (knows Spy evs)"   and "SK \<in> symKeys"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV_gets" shows "\<not> AKcryptSK SK K evs"
fid_on_ffield: fixes A :: "V" shows "\<F>\<^sub>\<bullet> (fid_on A) = A"
exp_approx: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 \<le> x"   and "x < 2" shows "exp x - (\<Sum>k<n. x ^ k / fact k) \<in> {0..2 * x ^ n / (2 - x) / fact n}"
aact''_snd: shows "snd \<circ> aact'' = (snd \<circ> aact)(a := ps s' a)"
preserves_arr: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.arr (B.HoVH f)"
is_processT5_SYNC2: fixes t :: "'c event list"   and Xa :: "'c event set"   and P :: "'c process"   and u :: "'c event list"   and Ya :: "'c event set"   and Q :: "'c process"   and sa :: "'c event list"   and A :: "'c set"   and Y :: "'c event set" assumes "(t, Xa) \<in> F P \<and> (u, Ya) \<in> F Q \<and> sa setinterleaves ((t, u), insert tick (ev ` A))"   and "\<forall>c. c \<in> Y \<longrightarrow> (\<forall>t1 u1. sa @ [c] setinterleaves ((t1, u1), insert tick (ev ` A)) \<longrightarrow> ((t1, {}) \<in> F P \<longrightarrow> (u1, {}) \<notin> F Q) \<and> ((t1, {}) \<in> F Q \<longrightarrow> (u1, {}) \<notin> F P))" shows "\<exists>t2 u2 Xb. (t2, Xb) \<in> F P \<and> (\<exists>Yb. (u2, Yb) \<in> F Q \<and> sa setinterleaves ((t2, u2), insert tick (ev ` A)) \<and> (Xa \<union> Ya) \<inter> insert tick (ev ` A) \<union> Xa \<inter> Ya \<union> Y = (Xb \<union> Yb) \<inter> insert tick (ev ` A) \<union> Xb \<inter> Yb)"
per_union_def: fixes R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" shows "per_union R a b = R \<union> {(x, y). (x, a) \<in> R \<and> (b, y) \<in> R} \<union> {(x, y). (x, b) \<in> R \<and> (a, y) \<in> R}"
mult_strict_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less c d"   and "less zero b"   and "less_eq zero c" shows "less (times a c) (times b d)"
final_threadE: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "r1.final_thread s t" obtains x :: "'x1" where "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" and "final1 x" and "wset s t = None"
Seq': fixes l :: "nat"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" assumes "l \<turnstile>' c\<^sub>1"   and "l \<turnstile>' c\<^sub>2" shows "l \<turnstile>' c\<^sub>1;; c\<^sub>2"
LocalSecretsComposition_neg_loc_k: fixes key :: "Keys"   and P :: "specID"   and ch :: "chanID" assumes "kKS key \<notin> LocalSecrets P"   and "exprChannel ch (kE key)"   and "kKS key \<notin> specKeysSecrets P" shows "ch \<notin> loc P"
Always_def: fixes A :: "'d set" shows "Always A \<equiv> {F. Init F \<subseteq> A} \<inter> Stable A"
deriv: fixes za :: "real" assumes "norm za < K" shows "a * f za = (1 + za) * f' za"
linear_indep_image_lemma: fixes f :: "'b \<Rightarrow> 'b"   and B :: "'b set"   and x :: "'b" assumes "linear (*s) (*s) f"   and "finite B"   and "independent (f ` B)"   and "inj_on f B"   and "x \<in> span B"   and "f x = (0::'b)" shows "x = (0::'b)"
accmethd_SomeD: fixes G :: "prog"   and S :: "qtname"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd" assumes "accmethd G S C sig = Some m" shows "methd G C sig = Some m \<and> G \<turnstile>Methd sig m of C accessible_from S"
ord: shows "order (complex_of_real lam) (char_poly (of_real_hom.mat_hom A)) = 1"
set_mset_sup: fixes A :: "'b multiset"   and B :: "'b multiset" shows "set_mset (A \<union># B) = set_mset A \<union> set_mset B"
proots_count_reciprocal': fixes P :: "real poly"   and p :: "nat" assumes "degree P \<le> p"   and "P \<noteq> 0" shows "proots_count P {x. 0 < x \<and> x < 1} = proots_count (reciprocal_poly p P) {x. 1 < x}"
rep_eq: fixes x :: "'c cset"   and xa :: "'c \<Rightarrow> bool" shows "rcset (cset_Coll x xa) = {xb \<in> rcset x. xa xb}"
mult_right_le_one_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x"   and "less_eq zero y"   and "less_eq y one" shows "less_eq (times x y) x"
assms_2: assumes "x \<in>\<^sub>\<circ> A"   and "x \<in>\<^sub>\<circ> B" shows "P"
fld_simp: fixes C :: "'c"   and f :: "'b decl_scheme" shows "fld (C, f) = f"
simps: fixes b :: "'a" shows "the_Inr (Inr b) = b"
assms_9: shows "f \<bullet> \<beta> = \<zero>"
next_p_frontier_change: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t"   and loc' :: "'loc" assumes "next_propagate' c0 c1 loc t"   and "summary loc loc' \<noteq> {}\<^sub>A" shows "c_work c1 loc' = c_work c0 loc' + after_summary (frontier_changes (c_imp c1 loc) (c_imp c0 loc)) (summary loc loc')"
sdrop_strict_1: shows "sdrop\<cdot>\<bottom> = \<bottom>"
simps_5: fixes v :: "Cooper.num" shows "not (Le v) = NOT (Le v)"
prefix_length_le: fixes xs :: "'d list"   and ys :: "'d list" assumes "prefix xs ys" shows "length xs \<le> length ys"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p_in_subst: fixes p :: "'d strand_label \<times> ('e, 'f) Term.term \<times> ('e, 'f) Term.term"   and a :: "'d strand_label \<times> ('e, 'f) stateful_strand_step"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" assumes "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>)" shows "\<exists>q\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p a. fst p = fst q \<and> snd p = snd q \<cdot>\<^sub>p rm_vars (set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (snd a))) \<delta>"
valid_length_muaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "'muaux"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list" assumes "valid_muaux args cur aux auxlist" shows "length_muaux args aux = length auxlist"
mono_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> less_eq (f x) (f y))"
used_Gets_rev: fixes evs :: "event list"   and B :: "agent"   and X :: "msg" shows "used (evs @ [Gets B X]) = used evs"
simps_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "monoidal_language C" shows "monoidal_language.Diag C \<^bold>\<langle>f\<^bold>\<rangle> = partial_magma.arr C f"
CDMode_CTRL_Labels_def: shows "CDMode_CTRL_Labels = {(En ''Next'', defaultguard, defaultaction), (En ''Back'', defaultguard, defaultaction), (En ''Ready'', defaultguard, CDMode_CTRL_Action1), (En ''Ready'', defaultguard, CDMode_CTRL_Action2), (En ''EndOfTitle'', \<lambda>d. Select0 d < Select1 d, defaultaction)}"
continuous_on_swap: fixes A :: "('b \<times> 'c) set" shows "continuous_on A prod.swap"
ResOutput: fixes x :: "name"   and a :: "name"   and b :: "name"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> b" shows "<\<nu>x>a{b}.P \<equiv>\<^sub>e a{b}.(<\<nu>x>P)"
index_set_closed_not_in: fixes I :: "nat set"   and i :: "nat"   and j :: "nat" assumes "index_set I"   and "i \<notin> I"   and "\<forall>x. eval r_phi [i, x] = eval r_phi [j, x]" shows "j \<notin> I"
lUnionM_append: fixes A :: "'m list"   and B :: "'m list" shows "lUnionM (A @ B) = lunionM (lUnionM A) (lUnionM B)"
fa_intros_4: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "Transformers.additive (wp a)"   and "Transformers.additive (wp b)"   and "well_def b" shows "Transformers.additive (wp (a ;; b))"
measurable_embed_measure2': fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "inj_on f (space M)" shows "f \<in> M \<rightarrow>\<^sub>M embed_measure M f"
set_rev_mp: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
empty_path2: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)"   and "invar g" shows "g \<turnstile> n-[n]\<rightarrow>n"
runit'_simps_2: fixes f :: "'d" assumes "D.ide f" shows "src\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[f] = src\<^sub>D f"
compose_Bij: fixes x :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and y :: "'c \<Rightarrow> 'c" assumes "x \<in> Bij S"   and "y \<in> Bij S" shows "compose S x y \<in> Bij S"
that: fixes a :: "V"   and b :: "V" assumes "x = [a, b]\<^sub>\<circ>" shows "thesis"
Arccos_cos: fixes z :: "complex" assumes "0 < Re z \<and> Re z < pi \<or> Re z = 0 \<and> 0 \<le> Im z \<or> Re z = pi \<and> Im z \<le> 0" shows "Arccos (cos z) = z"
add_ac_3: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b + (a + c) = a + (b + c)"
dsn_rrep: fixes i :: "nat" shows "paodv i \<TTurnstile> (recvmsg rreq_rrep_sn \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:6, PAodv-:7} \<union> {PRrep-:n |n. True} \<longrightarrow> 1 \<le> dsn \<xi>)"
paint_bd_inv: fixes t :: "('a, 'b) rbt"   and c :: "color" assumes "bd_inv t" shows "bd_inv (paint c t)"
comb_planar_iso: fixes hom :: "('a, 'b, 'c, 'd) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "comb_planar (app_iso hom G) = comb_planar G"
li: shows "local.independent K (map (\<lambda>u'. u' \<otimes> u) Us)"
wp_loop_step_unitary: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "unitary P"   and "unitary Q" shows "unitary (\<lambda>s. \<guillemotleft> G \<guillemotright> s * wp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s)"
elims_2: fixes x :: "atom list \<times> 'a"   and xa :: "atom list \<times> 'a" assumes "x \<approx>abs_lst xa" obtains bs :: "atom list" and xb :: "'a" and cs :: "atom list" and y :: "'a" where "x = (bs, xb)" and "xa = (cs, y)" and "\<exists>p. (bs, xb) \<approx>lst (=) supp p (cs, y)"
simps: fixes x :: "'sz"   and y :: "'sz"   and z :: "'sz" shows "swap x y z = (if z = x then y else if z = y then x else z)"
Ints_det: fixes A :: "'a mat" assumes "\<And>i j. \<lbrakk>i < dim_row A; j < dim_col A\<rbrakk> \<Longrightarrow> A $$ (i, j) \<in> \<int>" shows "det A \<in> \<int>"