hcju/isabelleps huggingface dataset:
nhop_update_changed: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flg :: "f"   and hops :: "nat"   and sip :: "nat" assumes "update rt dip (dsn, dsk, flg, hops, sip) \<noteq> rt" shows "the (nhop (update rt dip (dsn, dsk, flg, hops, sip)) dip) = sip"
diagram_axioms: shows "diagram (\<cdot>\<^sub>J) (\<cdot>) D"
jj: shows "jj \<in> UNlift cl dl P F II"
lens_plus_left_unit: fixes X :: "'c \<Longrightarrow> 'd" shows "0\<^sub>L +\<^sub>L X \<approx>\<^sub>L X"
float_defs_4: fixes a :: "('a, 'b) IEEE.float" shows "is_nan a = (IEEE.exponent a = emax TYPE(('a, 'b) IEEE.float) \<and> fraction a \<noteq> 0)"
swap_commute: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "Fun.swap a b f = Fun.swap b a f"
vempty_nin: fixes a :: "V" shows "a \<notin>\<^sub>\<circ> 0"
transpI: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r y z\<rbrakk> \<Longrightarrow> r x z" shows "transp r"
wf_imp_SN: fixes A :: "('b \<times> 'b) set" assumes "wf (A\<inverse>)" shows "SN A"
post_split_max: fixes t :: "('c \<times> nat) tree"   and t' :: "('c \<times> nat) tree"   and x :: "'c" assumes "invar t"   and "(t', x) = split_max t"   and "t \<noteq> \<langle>\<rangle>" shows "post_del t t'"
scopeExtSum: fixes x :: "name"   and P :: "ccs"   and Q :: "ccs" assumes "x \<sharp> P" shows "\<lparr>\<nu>x\<rparr>(P \<oplus> Q) \<sim> P \<oplus> \<lparr>\<nu>x\<rparr>Q"
big_equal_iff_bigtheta: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(O[F](f) = O[F](g)) = (f \<in> \<Theta>[F](g))"
inject_2: fixes x2 :: "'d"   and y2 :: "'d" shows "(Inr x2 = Inr y2) = (x2 = y2)"
S_alt_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and A :: "('a \<times> 'a) set" assumes "Prim_Invar1_loc w g r A" shows "Prim.S r A = {r} \<union> fst ` A"
minus_right_distributivity: fixes p :: "int poly fract"   and q :: "int poly fract"   and r :: "int poly fract" shows "rat_poly_times (p - q) r = rat_poly_times p r - rat_poly_times q r"
prefs_wf'_1: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a" assumes "pref_profile_wf agents alts R"   and "i \<in> agents" shows "total_preorder_on alts (R i)"
atMost_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and u :: "'b" shows "ord.atMost less_eq u \<equiv> {x. less_eq x u}"
fst_eqD: fixes x :: "'b"   and y :: "'c"   and a :: "'b" assumes "fst (x, y) = a" shows "x = a"
mojmir_accept_iff_token_set_accept: shows "\<MM>.accept = (\<exists>i<\<MM>.max_rank. finite \<MM>.fail \<and> finite (\<MM>.merge i) \<and> infinite (\<MM>.succeed i))"
e_type_callcl_host: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and cl :: "cl"   and t1s' :: "t list"   and t2s' :: "t list"   and tf :: "tf"   and f :: "host" assumes "\<S>\<bullet>\<C> \<turnstile> [Callcl cl] : t1s' _> t2s'"   and "cl = Func_host tf f" shows "\<exists>t1s t2s ts_c. t1s' = ts_c @ t1s \<and> t2s' = ts_c @ t2s \<and> tf = t1s _> t2s"
intro: fixes G :: "('c, 'b) pre_digraph" assumes "\<And>e. e \<in> arcs G \<Longrightarrow> tail G e \<in> verts G"   and "\<And>e. e \<in> arcs G \<Longrightarrow> head G e \<in> verts G" shows "wf_digraph G"
assms_1: shows "safe_distance_3r < s\<^sub>o - s\<^sub>e"
image_r_vimage_l: fixes Q :: "'e set"   and ob :: "'f \<Rightarrow> 'e"   and P :: "'f set" assumes "Q \<subseteq> ob ` P"   and "observable ob P" shows "ob -` Q \<subseteq> P"
subst_free: fixes name :: "name"   and env :: "(name, 'a) fmap" shows "subst (free name) env = (case fmlookup env name of None \<Rightarrow> free name | Some t \<Rightarrow> t)"
linordered_field_no_ub: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<forall>x. \<exists>y. less x y"
is_rbt_rbt_sorted: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" assumes "ord.is_rbt less t" shows "ord.rbt_sorted less t"
prime_power_inj: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize a"   and "power.power one times a m = power.power one times a n" shows "m = n"
simps_5: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm"   and \<Gamma> :: "real list" shows "eval (Or \<phi> \<psi>) \<Gamma> = (eval \<phi> \<Gamma> \<or> eval \<psi> \<Gamma>)"
\<pi>_\<theta>_bigo: shows "(\<lambda>x. \<pi> x - \<theta> x / ln x) \<in> O(\<lambda>x. x / (ln x)\<^sup>2)"
alphaAbs_qAbs_iff_alphaAbs_ex_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_ex_qAFresh xs x X xs' x' X'"
unsigned_take_bit_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and w :: "'c word" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.unsigned one plus zero (take_bit n w) = take_bit n (semiring_1.unsigned one plus zero w)"
sup2I2: fixes B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c"   and A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "B x y" shows "sup A B x y"
congruent_triangle_aas_17: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "angle a1 b1 c1 = angle a2 b2 c2"   and "angle b1 c1 a1 = angle b2 c2 a2"   and "dist a1 b1 = dist a2 b2"   and "\<not> collinear {a1, b1, c1}" shows "angle c1 b1 a1 = angle a2 b2 c2"
disc_1: fixes A :: "'c llist set" assumes "\<forall>x\<in>A. lnull x" shows "lnull (lSup A)"
rv_Y: fixes i :: "'i" shows "random_variable borel (Y i)"
ennreal_of_enat_le_iff: fixes m :: "enat"   and n :: "enat" shows "(ennreal_of_enat m \<le> ennreal_of_enat n) = (m \<le> n)"
inj_on_one_iff': fixes h :: "'f \<Rightarrow> 'g"   and G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme" assumes "h \<in> hom G H"   and "Group.group G"   and "Group.group H" shows "inj_on h (carrier G) = (\<forall>x. x \<in> carrier G \<longrightarrow> h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>)"
maintained_holds_subset_iff: fixes G :: "('b, 'c) labeled_graph"   and e\<^sub>L :: "'b allegorical_term"   and e\<^sub>R :: "'b allegorical_term" assumes "graph G" shows "maintained (transl_rule (e\<^sub>L \<sqsubseteq> e\<^sub>R)) G = (:G:\<lbrakk>e\<^sub>L\<rbrakk> \<subseteq> :G:\<lbrakk>e\<^sub>R\<rbrakk>)"
app_vdomain_def: fixes r :: "V" shows "\<D>\<^sub>\<circ> r = set {a. \<exists>b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
Min_Di_le: fixes i :: "nat"   and e :: "nat"   and y :: "nat list" assumes "i < m"   and "e \<in> Di i y"   and "length y \<le> n" shows "Min (Di i y) \<le> Max (set b)"
user_accessible_mod_cpu_reg: fixes new_cpu_reg :: "CPU_register \<Rightarrow> word32"   and s :: "'b sparc_state"   and pa :: "word36" shows "user_accessible (s\<lparr>cpu_reg := new_cpu_reg\<rparr>) pa = user_accessible s pa"
linear_continuous_at: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "bounded_linear f" shows "isCont f a"
arcs_blocks_def: shows "arcs_blocks \<equiv> mset_set (arc_to_block ` arcs G)"
Un_C_fgs_sub: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "C \<subseteq> fsps R M N f H"   and "C \<noteq> {}"   and "\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a" shows "\<Union> {a. \<exists>b. (a, b) \<in> C} \<subseteq> fgs R M (\<Union> {a. \<exists>b. (a, b) \<in> C})"
axiom_addimp: fixes uv :: "fol fm"   and uw :: "fol fm" shows "\<turnstile> concl (axiom_addimp uv uw)"
intro: fixes t :: "nat \<Rightarrow> real"   and t1 :: "real"   and p :: "nat"   and L :: "real"   and B :: "real"   and r :: "real" assumes "grid t"   and "max_step_axioms t t1 p L B r" shows "max_step t t1 p L B r"
IH1_2: shows "factors_of_factor_tree (product_factor_tree p l) = factors_of_factor_tree l"
simps_11: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x11 :: "'a"   and x12 :: "'b" shows "x41 ;; x42 \<noteq> x11 \<leftarrow> x12"
l1_inv6I: fixes x :: "l1_state" assumes "\<And>A B Rb Ra gnx. \<lbrakk>guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; in_progressS (progress x Ra) {xnx, xgnx, xgny, xsk, xEnd}; guessed_frame Ra xgnx = Some gnx; guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny))); guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; progress x Rb = Some {xny, xgnx, xgny, xsk}; can_signal x A B\<rbrakk> \<Longrightarrow> signalsResp x (Commit A B (Exp gnx (NonceF (Rb $ ny)))) < signalsResp x (Running A B (Exp gnx (NonceF (Rb $ ny))))" shows "x \<in> l1_inv6"
mult2_ns_def: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "mult2_ns ns s \<equiv> multpw ns O (mult s)\<^sup>="
abs_eq: fixes x :: "'b \<Rightarrow> 'c option" shows "Mapping.keys (Mapping x) = dom x"
game3_bad: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv"   and q :: "nat" assumes "interaction_bounded_by' isl \<A> q" shows "Sigma_Algebra.measure (measure_spmf (game3 \<A>)) {(b, bad). bad} \<le> real q / real (card prf_domain) * real q"
big_small_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "g \<in> o[F](h)" shows "f \<in> o[F](h)"
sum_distrib_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and r :: "'b" assumes "class.ring plus zero minus uminus times" shows "times (comm_monoid_add.sum plus zero f A) r = comm_monoid_add.sum plus zero (\<lambda>n. times (f n) r) A"
hyps: shows "\<exists>M. \<forall>z. M \<le> norm z \<longrightarrow> norm (\<Sum>i\<le>n. c i * z ^ i) \<le> e * norm z ^ Suc n"
bmod_Sup_fun2_weak: fixes \<G> :: "('c \<Rightarrow> 'd) set"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and x :: "'e" assumes "\<G> \<noteq> {}" shows "\<otimes> R f (\<Squnion>\<G>) x = \<Squnion>{\<otimes> R f g x |g. g \<in> \<G>}"
is_Var_succ_iff: fixes x :: "hf" shows "is_Var (succ x) = Ord x"
new_shadow_root_disconnected_nodes: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "h' \<turnstile> get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr) disconnected_nodes \<rightarrow>\<^sub>r []"
disc_iff_2: fixes sys :: "'b \<Rightarrow> ('a, 'b) process_term"   and T :: "('a, 'b) process_term" shows "isChoice (solution sys T) = (\<not> isACT sys T)"
to_tree_order_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ptr |\<in>| object_ptr_kinds h"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "h \<turnstile> ok Shadow_DOM.to_tree_order ptr"
preserves_cod: fixes f :: "'c" assumes "Cop.arr f" shows "S.cod (F f) = F (Cop.cod f)"
set_eq_iff_mset_remdups_eq: fixes x :: "'b list"   and y :: "'b list" shows "(set x = set y) = (mset (remdups x) = mset (remdups y))"
sup_state_opt_Some_Some: fixes P :: "'a prog"   and a :: "ty list \<times> ty err list"   and b :: "ty list \<times> ty err list" shows "P \<turnstile> \<lfloor>a\<rfloor> \<le>' \<lfloor>b\<rfloor> = P \<turnstile> a \<le>\<^sub>i b"
list_rel_append1: fixes as :: "'d list"   and bs :: "'d list"   and l :: "'e list"   and R :: "('d \<times> 'e) set" shows "((as @ bs, l) \<in> \<langle>R\<rangle>list_rel) = (\<exists>cs ds. l = cs @ ds \<and> (as, cs) \<in> \<langle>R\<rangle>list_rel \<and> (bs, ds) \<in> \<langle>R\<rangle>list_rel)"
simps_1: fixes x :: "'b list" shows "list_diff x [] = x"
assms_4: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> F" shows "prv (imp \<chi> \<phi>)"
simps_6: fixes ux :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and uy :: "((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set \<times> (char list \<Rightarrow> char list set)"   and b :: "bexp"   and C :: "acom"   and s :: "char list \<Rightarrow> int" shows "postQ ({(ux, S, uy)} WHILE b DO C) s = S s"
prems_1: shows "list.set \<alpha>s \<subseteq> ON"
vertices_def: shows "vertices = Abs_fset (\<Union> (set (map (\<lambda>c. insert (c, []) ((\<lambda>p. (c, 0 # p)) ` it_paths (it' c))) conclusions)))"
induct: fixes X :: "'b llist \<Rightarrow> bool"   and x :: "'b llist"   and P :: "'b llist \<Rightarrow> bool" assumes "prepend_cong1 X x"   and "\<And>xs. X xs \<Longrightarrow> P xs"   and "\<And>ys xs. \<lbrakk>prepend_cong1 X ys; P ys\<rbrakk> \<Longrightarrow> P (prepend xs ys)" shows "P x"
fix_def2: fixes F :: "'d \<rightarrow> 'd" shows "fix\<cdot>F = (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
Eq_RESUMPTION_into_resumption: fixes r :: "('f + 'g + 'h, 'i) Old_Datatype.node set" assumes "Eq_RESUMPTION r r" shows "r \<in> resumption"
wf_arrow: fixes \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f U" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T \<rightarrow> U"
mono_Sync: assumes "P \<sqsubseteq> P'" shows "(P \<lbrakk>A\<rbrakk> Q') \<sqsubseteq> (P' \<lbrakk>A\<rbrakk> Q')"
representation_def: fixes basis :: "'b set"   and v :: "'b" shows "representation basis v = (if independent basis \<and> v \<in> span basis then SOME f. (\<forall>v. f v \<noteq> (0::'a) \<longrightarrow> v \<in> basis) \<and> finite {v. f v \<noteq> (0::'a)} \<and> (\<Sum>v | f v \<noteq> (0::'a). f v *s v) = v else (\<lambda>b. 0::'a))"
rtrancl_eta_AppR: fixes t :: "dB"   and t' :: "dB"   and s :: "dB" assumes "t \<rightarrow>\<^sub>\<eta>\<^sup>* t'" shows "s \<degree> t \<rightarrow>\<^sub>\<eta>\<^sup>* s \<degree> t'"
polyn_expr_tOp_c: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c"   and "pol_coeff S d" shows "\<exists>e. pol_coeff S e \<and> fst e = fst c + fst d \<and> snd e (fst e) = snd c (fst c) \<cdot>\<^sub>r\<^bsub>S\<^esub> snd d (fst d) \<and> polyn_expr R X (fst e) e = polyn_expr R X (fst c) c \<cdot>\<^sub>r polyn_expr R X (fst d) d"
distinct_fst_append: fixes x :: "'c"   and xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" assumes "x \<in> set (map fst xs)"   and "distinct (map fst (xs @ ys))" shows "x \<notin> set (map fst ys)"
NCSaux2: fixes p :: "('b, 'c) Combinators list"   and a :: "'b"   and b :: "'b"   and l :: "'b set list"   and u :: "'b"   and v :: "'b"   and w :: "'c" assumes "noDenyAll p"   and "{a, b} \<in> set l"   and "all_in_list p l"   and "singleCombinators p"   and "FWNormalisationCore.sorted (DenyAllFromTo a b # p) l"   and "{a, b} \<noteq> firstList p"   and "AllowPortFromTo u v w \<in> set p" shows "{a, b} \<noteq> {u, v}"
map_nondetT_def: shows "map_nondetT = map_nondetT' undefined"
quasinorm_subset_space: fixes N1 :: "'b quasinorm"   and N2 :: "'b quasinorm" assumes "N1 \<subseteq>\<^sub>N N2" shows "space\<^sub>N N1 \<subseteq> space\<^sub>N N2"
Rep_regular_inverse: fixes x :: "'c regular" shows "Abs_regular (Rep_regular x) = x"
liftT_liftT'_1: fixes i :: "nat"   and m :: "nat"   and j :: "nat"   and n :: "nat"   and T :: "POPLmarkRecord.type" assumes "i + m \<le> j" shows "\<up>\<^sub>\<tau> n j (\<up>\<^sub>\<tau> m i T) = \<up>\<^sub>\<tau> m i (\<up>\<^sub>\<tau> n (j - m) T)"
abs_eq: fixes x :: "(common_metadata \<times> participant_metadata) \<times> view list" shows "embed_transaction (Transaction' x) = Transaction\<^sub>m (Unblinded (map_prod (map_prod Unblinded Unblinded) (map embed_view) x))"
Nd: shows "(1::'a) / d \<le> of_int N"
below_C: fixes x :: "C" shows "x \<sqsubseteq> C\<cdot>x"
not_fresh_FDERIV_floatarith: fixes ode_e :: "floatarith"   and ds :: "floatarith list"   and i :: "nat" assumes "\<not> fresh_floatarith (FDERIV_floatarith ode_e [0..<DIM('a)] ds) i"   and "length ds = DIM('a)" shows "\<not> fresh_floatarith ode_e i \<or> \<not> fresh_floatariths ds i"
sub_mult_one_circ: fixes x :: "'a" shows "x \<sqinter> top \<le> top \<sqinter> x"
compatAbsSwap_def: fixes xs :: "'varSort"   and s :: "'b"   and f :: "'var \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm set \<Rightarrow> 'f" shows "compatAbsSwap xs s f \<equiv> \<forall>x X x' X'. (\<forall>y. (y = x \<or> fresh xs y X) \<and> (y = x' \<or> fresh xs y X') \<longrightarrow> X #[y \<and> x]_xs = X' #[y \<and> x']_xs) \<longrightarrow> f x X = f x' X'"
zero_prod_def: shows "0 \<equiv> (0::'a, 0::'b)"
ex_un_null: shows "\<exists>!n. \<forall>f. J n f = n \<and> J f n = n"
fresh_star_def: fixes as :: "atom set"   and x :: "'c" shows "as \<sharp>* x \<equiv> \<forall>a\<in>as. a \<sharp> x"
spectrum_imp_pr: fixes \<pp> :: "('a \<times> 'a) set set" assumes "\<pp> \<in> qr.spectrum" shows "pr_ideal qr.carrier_quotient_ring \<pp> qr.add_rel qr.mult_rel qr.zero_rel qr.one_rel"
eqd_pref: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v"   and "\<^bold>|x\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "x \<cdot> x\<inverse>\<^sup>>u = u \<and> x\<inverse>\<^sup>>u \<cdot> v = y"
hyps: fixes u :: "'a" assumes "(x, u) \<in> r" shows "\<exists>y. (u, y) \<in> r\<^sup>* \<and> y \<in> NF r"
less_finite_1_def: fixes x :: "finite_1"   and y :: "finite_1" shows "(x < y) = False"
separating: fixes A :: "'k set"   and B :: "'k set"   and G :: "('k, 'l) graph_scheme"   and S :: "'k set" assumes "\<And>x y p. \<lbrakk>x \<in> A; y \<in> B; path G x p y\<rbrakk> \<Longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S" shows "separating_gen G A B S"
append_eq_Cons_conv: fixes ys :: "'a list"   and zs :: "'a list"   and x :: "'a"   and xs :: "'a list" shows "(ys @ zs = x \<Colon> xs) = (ys = [] \<and> zs = x \<Colon> xs \<or> (\<exists>ys'. ys = x \<Colon> ys' \<and> ys' @ zs = xs))"
cases: fixes a1 :: "state"   and a2 :: "com"   and a3 :: "state" assumes "a1 -a2\<rightarrow> a3" obtains t :: "state" and f :: "state \<Rightarrow> state set" and s :: "state" where "a1 = sa__" and "a2 = Do f__" and "a3 = ta__" and "ta__ \<in> f__ sa__"   | s0 :: "state" and c1 :: "com" and s1 :: "state" and c2 :: "com" and s2 :: "state" where "a1 = s0" and "a2 = (c1__; c2__)" and "a3 = s2" and "s0 -c1__\<rightarrow> s1" and "s1 -c2__\<rightarrow> s2"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and t :: "state" and c2 :: "com" where "a1 = sb__" and "a2 = IF ba__ THEN c1a__ ELSE c2a__" and "a3 = tb__" and "ba__ sb__" and "sb__ -c1a__\<rightarrow> tb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c2 :: "com" and t :: "state" and c1 :: "com" where "a1 = sc__" and "a2 = IF bb__ THEN c1 ELSE c2" and "a3 = tc__" and "\<not> bb__ sc__" and "sc__ -c2\<rightarrow> tc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" where "a1 = sd__" and "a2 = WHILE bc__ DO ca__" and "a3 = sd__" and "\<not> bc__ sd__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and t :: "state" and u :: "state" where "a1 = se__" and "a2 = WHILE b DO cb__" and "a3 = u" and "b se__" and "se__ -cb__\<rightarrow> td__" and "td__ -WHILE b DO cb__\<rightarrow> u"   | s :: "state" and t :: "state" where "a1 = sf__" and "a2 = CALL" and "a3 = te__" and "sf__ -body\<rightarrow> te__"   | f :: "state \<Rightarrow> state" and s :: "state" and c :: "com" and t :: "state" and g :: "state \<Rightarrow> state \<Rightarrow> state" where "a1 = s" and "a2 = LOCAL f; c; g" and "a3 = g s t" and "f s -c\<rightarrow> t"
rewrite_X_sound: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n rewrite_X \<phi> = w \<Turnstile>\<^sub>n \<phi>"
if_False: fixes x :: "'c"   and y :: "'c" shows "(if False then x else y) = y"
disj2: shows "set (ap gs bs (ps -- sps) hs data') \<inter>\<^sub>p D' = {}"
simps: fixes a1 :: "('a \<times> 'b) list"   and a2 :: "'b \<Rightarrow> 'a set" shows "crdt_ops a1 a2 = ((\<exists>deps. a1 = [] \<and> a2 = deps) \<or> (\<exists>xs deps oid oper. a1 = xs @ [(oid, oper)] \<and> a2 = deps \<and> crdt_ops xs deps \<and> oid \<notin> set (map fst xs) \<and> (\<forall>ref\<in>deps oper. ref \<in> set (map fst xs) \<and> ref < oid)))"
remove_max_mset_lemma: fixes m :: "'a"   and l' :: "'a list"   and m' :: "'a"   and t' :: "'a list"   and l :: "'a list" assumes "(m, l') = foldl f (m', t') l" shows "mset (m # l') = mset (m' # t' @ l)"
path_image_def: fixes g :: "real \<Rightarrow> 'b" shows "path_image g = g ` {0..1}"
right_cancel: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> carrier S"   and "y \<in> carrier S"   and "z \<in> carrier S" shows "(y \<oplus>\<^bsub>S\<^esub> x = z \<oplus>\<^bsub>S\<^esub> x) = (y = z)"
inner_dirsum_el_decomp_def: fixes Gs :: "'a set list" shows "\<Oplus>Gs\<leftarrow> = (\<lambda>x. if x \<in> (\<Oplus>G\<leftarrow>Gs. G) then THE gs. gs \<in> listset Gs \<and> x = sum_list gs else [])"
set_mset_mono: fixes A :: "'k multiset"   and B :: "'k multiset" assumes "A \<subseteq># B" shows "set_mset A \<subseteq> set_mset B"
ind: shows "Missing_List.min_list xs \<le> Missing_List.min_list ys"