hcju/isabelleps huggingface dataset:
conc_trans_additional_5: fixes A :: "'g nres"   and B :: "'g nres"   and C :: "'g nres" assumes "A \<le> \<Down> Id B"   and "B \<le> C" shows "A \<le> C"
length_removeKeyList: fixes ws :: "'a list"   and ps :: "('a \<times> 'b) list" shows "|removeKeyList ws ps| \<le> |ps|"
assms: shows "x \<in> at_least_as_good y B Pr"
CD_on_def2: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and A :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.CD_on Pd ds A = (\<Union>d\<in>ds. Contracts.Cd Pd d (A \<inter> Field (Pd d)))"
out_arcs_add_arc_iff: fixes a :: "'b"   and u :: "'a" shows "out_arcs (G.add_arc a) u = (if tail G a = u then insert a (out_arcs G u) else out_arcs G u)"
fin: shows "finite ({q. \<exists>t. (q, t) \<in> set (step (hd b) (a ! hd b)) \<and> t \<squnion>\<^bsub>f\<^esub> a ! q \<noteq> a ! q} \<union> set (tl b))"
brnL_nchotomy: fixes cl :: "('test, 'atom, 'choice) cmd list"   and ii :: "nat" assumes "properL cl"   and "ii < brnL cl (length cl)" shows "\<exists>n i. n < length cl \<and> i < brn (cl ! n) \<and> ii = brnL cl n + i"
functor_axioms: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map"
F1'_antimono: shows "antimono (\<lambda>XH. - Rf CH XH)"
line_S_two_intersections_only: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and l :: "proj2_line" assumes "p \<noteq> q"   and "p \<in> S"   and "q \<in> S"   and "r \<in> S"   and "proj2_incident p l"   and "proj2_incident q l"   and "proj2_incident r l" shows "r = p \<or> r = q"
map_add_dom_app_simps_3: fixes m :: "'b"   and l2 :: "'b \<Rightarrow> 'c option"   and l1 :: "'b \<Rightarrow> 'c option" assumes "m \<notin> dom l2" shows "(l2 \<Oplus> l1) m = l1 m"
interchange_same_rows: fixes A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c" shows "interchange_rows A a a = A"
is_root_pass2: fixes hs :: "'b heap list" assumes "no_Emptys hs" shows "is_root (pass\<^sub>2 hs)"
arg_min_equality: fixes P :: "'b \<Rightarrow> bool"   and k :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "P k"   and "\<And>x. P x \<Longrightarrow> order.greater_eq (f x) (f k)" shows "f (arg_min f P) = f k"
self_append_conv2: fixes ys :: "'b list"   and xs :: "'b list" shows "(ys = xs \<cdot> ys) = (xs = \<epsilon>)"
subst_atmss_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and AAA :: "'b set set"   and \<sigma> :: "'c" shows "substitution_ops.subst_atmss subst_atm AAA \<sigma> = (\<lambda>AA. substitution_ops.subst_atms subst_atm AA \<sigma>) ` AAA"
intros: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "'e"   and d :: "'f" assumes "R1 a b"   and "R2 c d" shows "rel_prod R1 R2 (a, c) (b, d)"
intro: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>v. v \<notin> \<Union> X \<Longrightarrow> f v = v" shows "ChamberComplexEndomorphism_axioms X f"
simps_3: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and L :: "lrule"   and j :: "nat" shows "step_result (SG, C) (i, Lrule L j) = (close (SG @ Lrule_result L j (SG ! i)) (SG ! i), C)"
mcont2mcont': fixes lub' :: "'d set \<Rightarrow> 'd"   and ord' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e spmf"   and lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd" assumes "\<And>x. mcont lub' ord' lub_spmf (ord_spmf (=)) (f x)"   and "\<And>y. mcont lub ord lub_spmf (ord_spmf (=)) (\<lambda>x. f x y)"   and "mcont lub ord lub' ord' t" shows "mcont lub ord lub_spmf (ord_spmf (=)) (\<lambda>x. f x (t x))"
grid_length: fixes p' :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list"   and ds :: "nat set" assumes "p' \<in> grid p ds" shows "length p' = length p"
bset_map_closed: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'e" assumes "finite A \<and> card A \<le> CARD('d)" shows "finite (f ` A) \<and> card (f ` A) \<le> CARD('d)"
star_of_minus: fixes x :: "'b" shows "star_of (- x) = - star_of x"
pmc_completion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div>\<^bsub>* \<gamma>\<^sub>A\<^sub>G\<^sub>M\<^esub> \<phi> = K \<div>\<^bsub>\<gamma>\<^sub>A\<^sub>G\<^sub>M\<^esub> \<phi>"
u_def: fixes n :: "nat"   and t :: "'a" shows "u n t = (\<Sum>i\<in>{..<n * 2 ^ n} - {0}. real i / 2 ^ n * \<chi> (A n i) t)"
canonical_proj_vimage_mem_iff: fixes I :: "'a set"   and J :: "'a set set"   and a :: "'a" assumes "ideal I R"   and "J \<subseteq> carrier (R Quot I)"   and "a \<in> carrier R" shows "(a \<in> \<Union> J) = (I +> a \<in> J)"
atLeastLessThan_injective2: fixes f :: "nat \<Rightarrow> nat"   and m :: "nat"   and n :: "nat" assumes "bij_betw f {0..<m} {0..<n}" shows "m = n"
distinct_1: fixes x1 :: "'e" shows "prot_atom.Atom x1 \<noteq> Value"
lm151: fixes f :: "('c \<times> 'd) set"   and g :: "('c \<times> 'd) set" assumes "f \<subseteq> g"   and "runiq g"   and "Domain f = Domain g" shows "g \<subseteq> f"
cod_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.cod (g \<cdot>\<^sub>C f) = C.cod g"
scale_sum_right: fixes a :: "rat"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "a *R sum f A = (\<Sum>x\<in>A. a *R f x)"
case_prod_beta: fixes f :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and p :: "'l \<times> 'm" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
all_zero: shows "\<forall>m\<ge>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'b)"
bij_betw_quotient_range_permutation_eq: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A = card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // range_permutation A B = {{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B}}"
med'_def: shows "med' = (\<lambda>xs. to_dual_ord (med (map of_dual_ord xs)))"
subst_apply_extstrand_def: fixes S :: "('c, 'd) extstrand_step list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "S \<cdot>\<^sub>e\<^sub>s\<^sub>t \<theta> \<equiv> map (\<lambda>x. x \<cdot>\<^sub>e\<^sub>s\<^sub>t\<^sub>p \<theta>) S"
slpa_get_return_edges: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list"   and as :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.same_level_path_aux kind get_return_edges cs as"   and "cs \<noteq> []"   and "CFG.upd_cs kind cs as = []"   and "\<forall>xs ys. as = xs @ ys \<and> ys \<noteq> [] \<longrightarrow> CFG.upd_cs kind cs xs \<noteq> []" shows "last as \<in> get_return_edges (last cs)"
Elem_Empty_Nat: shows "Elem Empty HOLZF.Nat"
Gets_certificate_valid: fixes A :: "agent"   and X :: "msg"   and C :: "agent"   and EKi :: "nat"   and SKi :: "nat"   and evs :: "event list" assumes "Gets A \<lbrace>X, cert C EKi onlyEnc (priSK RCA), cert C SKi onlySig (priSK RCA)\<rbrace> \<in> set evs"   and "evs \<in> set_cr" shows "EKi = pubEK C \<and> SKi = pubSK C"
r7: fixes x :: "name"   and s :: "trm"   and t :: "trm"   and t' :: "trm" assumes "x \<sharp> s"   and "t \<mapsto> t'" shows "s to x in t \<mapsto> s to x in t'"
Dom_Trg: fixes t :: "'c local.term" assumes "Arr t" shows "Dom (Trg t) = Trg t"
intro: fixes G :: "'b set"   and V :: "'c set"   and smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c" assumes "of_nat (card G) \<noteq> (0::'a)"   and "\<exists>ms. set ms \<subseteq> V \<and> R_scalar_mult.RSpan UNIV (aezfun_scalar_mult.fsmult smult) ms = V" shows "FinGroupRepresentation_axioms G smult V"
aform_rel_def: shows "\<langle>rnv_rel\<rangle>aform_rel = br Affine top"
scaleR_right_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "real"   and b :: "real"   and x :: "'b" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "a \<le> b"   and "less_eq zero x" shows "less_eq (scaleR a x) (scaleR b x)"
list_all2_rev1: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'c list" shows "list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
List_distinct: fixes h :: "'c \<Rightarrow> 'c"   and x :: "'c"   and as :: "'c list" assumes "List h x as" shows "distinct as"
atLeastatMost_psubset_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b \<subset> ord.atLeastAtMost less_eq c d) = ((\<not> less_eq a b \<or> less_eq c a \<and> less_eq b d \<and> (less c a \<or> less b d)) \<and> less_eq c d)"
igAbsRen_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igAbsRen MOD \<equiv> \<forall>xs y x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD xs y X \<longrightarrow> igAbs MOD xs y (igSubst MOD xs (igVar MOD xs y) x X) = igAbs MOD xs x X"
pdevs_val_add: fixes e :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real"   and xs :: "'a pdevs" shows "pdevs_val (\<lambda>i. e i + f i) xs = pdevs_val e xs + pdevs_val f xs"
lm016: fixes G :: "'c set"   and N :: "'d set" shows "possible_allocations_rel G N \<subseteq> injectionsUniverse \<inter> {a. Range a \<subseteq> N \<and> Domain a \<in> all_partitions G}"
star_unfoldl: fixes x :: "'a" shows "(1::'a) + x * x\<^sup>* \<le> x\<^sup>*"
mulfact: fixes c :: "real"   and a :: "complex"   and b :: "complex" assumes "0 \<le> c"   and "a \<le> b" shows "complex_of_real c * a \<le> complex_of_real c * b"
initial\<^sub>v_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list"   and \<S> :: "nat \<Rightarrow> bool option"   and valid :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'f list \<times> 'd \<times> nat set \<times> ('g \<Rightarrow> 'h option)) list \<Rightarrow> bool" shows "initial\<^sub>v ts \<S> valid \<equiv> (simple_ownership_distinct ts \<and> read_only_unowned \<S> ts) \<and> unowned_shared \<S> ts \<and> initial\<^sub>v_axioms ts valid"
cases: fixes a1 :: "'c net_state"   and a2 :: "'d node_action"   and a3 :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(a1, a2, a3) \<in> node_sos S" obtains s :: "'c" and m :: "'d" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS ia__ sb__ R__" and "a2 = R__:*cast(m__)" and "a3 = NodeS ia__ s'b__ R__" and "(sb__, broadcast m__, s'b__) \<in> S"   | s :: "'c" and D :: "nat set" and m :: "'d" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS ib__ sc__ Ra__" and "a2 = (Ra__ \<inter> D):*cast(ma__)" and "a3 = NodeS ib__ s'c__ Ra__" and "(sc__, groupcast D ma__, s'c__) \<in> S"   | s :: "'c" and d :: "nat" and m :: "'d" and s' :: "'c" and R :: "nat set" and i :: "nat" where "a1 = NodeS ic__ sd__ Rb__" and "a2 = {da__}:*cast(mb__)" and "a3 = NodeS ic__ s'd__ Rb__" and "(sd__, unicast da__ mb__, s'd__) \<in> S" and "da__ \<in> Rb__"   | s :: "'c" and d :: "nat" and s' :: "'c" and R :: "nat set" and i :: "nat" where "a1 = NodeS id__ se__ Rc__" and "a2 = \<tau>" and "a3 = NodeS id__ s'e__ Rc__" and "(se__, \<not>unicast db__, s'e__) \<in> S" and "db__ \<notin> Rc__"   | s :: "'c" and d :: "nat" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS ie__ sf__ Rd__" and "a2 = ie__:deliver(d)" and "a3 = NodeS ie__ s'f__ Rd__" and "(sf__, deliver d, s'f__) \<in> S"   | s :: "'c" and m :: "'d" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS if__ sg__ Re__" and "a2 = {if__}\<not>{}:arrive(mc__)" and "a3 = NodeS if__ s'g__ Re__" and "(sg__, receive mc__, s'g__) \<in> S"   | s :: "'c" and s' :: "'c" and i :: "nat" and R :: "nat set" where "a1 = NodeS ig__ sh__ Rf__" and "a2 = \<tau>" and "a3 = NodeS ig__ s' Rf__" and "(sh__, \<tau>, s') \<in> S"   | i :: "nat" and s :: "'c" and R :: "nat set" and m :: "'d" where "a1 = NodeS ih__ si__ Rg__" and "a2 = {}\<not>{ih__}:arrive(m)" and "a3 = NodeS ih__ si__ Rg__"   | i :: "nat" and s :: "'c" and R :: "nat set" and i' :: "nat" where "a1 = NodeS ii__ sj__ Rh__" and "a2 = connect(ii__, i'__)" and "a3 = NodeS ii__ sj__ (Rh__ \<union> {i'__})"   | i :: "nat" and s :: "'c" and R :: "nat set" and i' :: "nat" where "a1 = NodeS ij__ sk__ Ri__" and "a2 = connect(i'a__, ij__)" and "a3 = NodeS ij__ sk__ (Ri__ \<union> {i'a__})"   | i :: "nat" and s :: "'c" and R :: "nat set" and i' :: "nat" where "a1 = NodeS ik__ sl__ Rj__" and "a2 = disconnect(ik__, i'b__)" and "a3 = NodeS ik__ sl__ (Rj__ - {i'b__})"   | i :: "nat" and s :: "'c" and R :: "nat set" and i' :: "nat" where "a1 = NodeS il__ sm__ Rk__" and "a2 = disconnect(i'c__, il__)" and "a3 = NodeS il__ sm__ (Rk__ - {i'c__})"   | i :: "nat" and i' :: "nat" and i'' :: "nat" and s :: "'c" and R :: "nat set" where "a1 = NodeS im__ sn__ Rl__" and "a2 = connect(i'd__, i''__)" and "a3 = NodeS im__ sn__ Rl__" and "im__ \<noteq> i'd__" and "im__ \<noteq> i''__"   | i :: "nat" and i' :: "nat" and i'' :: "nat" and s :: "'c" and R :: "nat set" where "a1 = NodeS i s R" and "a2 = disconnect(i', i'')" and "a3 = NodeS i s R" and "i \<noteq> i'" and "i \<noteq> i''"
CommuteWithIntervalCut2_def: fixes "fun" :: "'b set \<Rightarrow> 'b set" shows "CommuteWithIntervalCut2 fun \<equiv> \<forall>t fun2 I. fun2 = (\<lambda>I. I \<down>< t) \<or> fun2 = (\<lambda>I. I \<down>> t) \<longrightarrow> fun (fun2 I) = fun2 (fun I)"
Q_def: fixes i :: "nat" shows "Q i \<equiv> Polynomial.smult (\<alpha> i) (fst (pdivmod (F (i - 2)) (F (i - 1))))"
rel_eq_onp: fixes P1 :: "'b \<Rightarrow> bool"   and P2 :: "'d \<Rightarrow> bool" shows "rel_gpv (eq_onp P1) (eq_onp P2) = eq_onp (pred_gpv P1 P2)"
\<sigma>_\<rho>: fixes p :: "fm"   and x :: "int"   and bs :: "real list"   and k :: "int"   and t :: "MIR.num"   and b' :: "real" assumes "iszlfm p (real_of_int x # bs)"   and "0 < real_of_int k"   and "numbound0 t"   and "isint t (real_of_int x # bs)"   and "k dvd \<lfloor>Inum (b' # bs) t\<rfloor>" shows "Ifm (real_of_int x # bs) (\<sigma>_\<rho> p (t, k)) = Ifm (real_of_int (\<lfloor>Inum (b' # bs) t\<rfloor> div k) # bs) p"
mem_cone: fixes S :: "'b set"   and x :: "'b"   and c :: "real" assumes "cone S"   and "x \<in> S"   and "0 \<le> c" shows "c *\<^sub>R x \<in> S"
favorites_def: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'d" shows "favorites R i = Max_wrt (R i)"
Zm_carrier: fixes e :: "'f" shows "carrier (Zm R e) = {e}"
fst_below_iff: fixes x :: "'b \<times> 'c"   and y :: "'b" shows "(fst x \<sqsubseteq> y) = (x \<sqsubseteq> (y, snd x))"
elims_3: fixes x :: "'b"   and xa :: "'b list" assumes "\<not> one_call_in_path x xa" obtains x :: "'b" and y :: "'b" and xs :: "'b list" where "xa__ = x" and "xa = y # xs" and "\<not> (if x = y then no_call_in_path x xs else one_call_in_path x xs)"
ev_unfold: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "ev \<phi> \<pi>l = (\<phi> \<pi>l \<or> ev \<phi> (map stl \<pi>l))"
get_parent_pure: fixes node_ptr :: "(_) node_ptr"   and h :: "(_) heap" shows "pure (get_parent node_ptr) h"
is_SDG_path_ics_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'" shows "n ics-ns\<rightarrow>\<^sub>d* n'"
coinduct: fixes R :: "'a Stream \<Rightarrow> 'a Stream \<Rightarrow> bool"   and x :: "'a Stream"   and y :: "'a Stream" assumes "Stream_bisim R"   and "R x y" shows "x = y"
FG_constituents_constituents: assumes "of_nat (card G) \<noteq> (0::'d)" shows "ActingGroup.group_ring = (\<Oplus>U\<leftarrow>ActingGroup.FG_constituents. U)"
assms_1: shows "fst h \<subseteq>\<^sub>f s"
quietInput: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "quiet P" shows "False"
entails_concl_compact: fixes EE :: "'a set"   and CC :: "'a set" assumes "finite EE"   and "CC \<Turnstile> EE" shows "\<exists>CC'\<subseteq>CC. finite CC' \<and> CC' \<Turnstile> EE"
k_def: fixes G1 :: "'b poly"   and G2 :: "'b poly" shows "k G1 G2 = (LEAST i. F G1 G2 (Suc i) = 0)"
open_discrete: fixes "open" :: "'b set \<Rightarrow> bool"   and A :: "'b set" assumes "class.discrete_topology open" shows "open A"
prod_list_coprime_left: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and xs :: "'b list"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>x. x \<in> set xs \<Longrightarrow> algebraic_semidom.coprime times one x a" shows "algebraic_semidom.coprime times one (monoid_mult.prod_list one times xs) a"
log2_lm3: fixes u :: "nat"   and x :: "nat" assumes "log2 u < x" shows "c_in x u = 0"
distributed_simple_function_superset: fixes X :: "'a \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> real"   and A :: "'d set" assumes "simple_function M X"   and "\<And>x. x \<in> X ` space M \<Longrightarrow> P x = prob (X -` {x} \<inter> space M)"   and "X ` space M \<subseteq> A"   and "finite A" shows "distributed M (count_space A) X (\<lambda>x. ennreal (if x \<in> X ` space M then P x else 0))"
iLast_iff: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLast t0 I P = (\<box> t [\<dots>0] \<oplus> iprev t0 I. P t)"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (B.L f) = B.arr f"
simps_4: shows "ccode.Eq \<noteq> ccode.Gt"
min_worst_chain_Suc: fixes x :: "'a"   and i :: "nat" assumes "local.bad x"   and "p (local.worst_chain i) x" shows "\<not> gt (local.worst_chain (Suc i)) x"
comp_arr_ide: fixes a :: "'b"   and f :: "'b" assumes "B.ide a"   and "B.seq f a" shows "f \<cdot>\<^sub>B a = f"
invar_rderiv_and_add_step: fixes as :: "'a"   and r :: "'b rexp"   and brs :: "bool \<times> 'b rexp list" assumes "invar_rderiv_and_add as r brs"   and "fst brs" shows "invar_rderiv_and_add as r (rderiv_and_add as brs)"
Init_ES_satisfies_invariant: fixes A :: "'b list set" assumes "finite (UNIV // \<approx>A)" shows "invariant (Init (UNIV // \<approx>A))"
A: fixes i :: "'i" assumes "i \<in> J" shows "A i \<in> sets (M i)"
inverse_matrix_ring_iarray_def: fixes A :: "'b iarray iarray" shows "inverse_matrix_ring_iarray A = (let d = det_iarrays_rings A in if is_unit d then Some ((1::'b) div d *ssi adjugate_iarrays A) else None)"
Rep_Abs_Seq: fixes P :: "'c process"   and Q :: "'c process" shows "Rep_process (Abs_process ({(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). t \<in> div_seq P Q}, div_seq P Q)) = ({(t, X). (t, X \<union> {tick}) \<in> F P \<and> tickFree t} \<union> {(t, X). \<exists>t1 t2. t = t1 @ t2 \<and> t1 @ [tick] \<in> T P \<and> (t2, X) \<in> F Q} \<union> {(t, X). t \<in> div_seq P Q}, div_seq P Q)"
geseq_def: shows "geseq = {(f, g). f \<in> SEQ A \<and> g \<in> SEQ A \<and> (f = g \<or> (\<exists>i. size (g i) < size (f i) \<and> (\<forall>j<i. f j = g j)))}"
Right_Coset_cardinality: fixes x :: "'a" assumes "x \<in> G" shows "card (H |\<cdot> x) = card H"
sset_simp_3: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "sset (x :# xs) = insert x (sset xs)"
OpposedThinChamberComplexFoldings_axioms: shows "OpposedThinChamberComplexFoldings X f g C0"
lset_ltl: fixes xs :: "'a llist" shows "lset (ltl xs) \<subseteq> lset xs"
mem_Times_iff: fixes x :: "'b \<times> 'c"   and A :: "'b set"   and B :: "'c set" shows "(x \<in> A \<times> B) = (fst x \<in> A \<and> snd x \<in> B)"
insert_disjoint_1: fixes a :: "'c"   and A :: "'c set"   and B :: "'c set" shows "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
comm_rev_iff: fixes u :: "'b list"   and v :: "'b list" shows "(rev u \<cdot> rev v = rev v \<cdot> rev u) = (u \<cdot> v = v \<cdot> u)"
add_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b"   and "less_eq c d" shows "less_eq (plus a c) (plus b d)"
as0_5: shows "S DA0 \<cdot> \<eta>0 = S_M S M DA"
intro: fixes G :: "('a, 'b) Graph_scheme" assumes "Graph G"   and "Tree_axioms G" shows "Tree G"
outSetCorelated_def: fixes x :: "chanID" shows "outSetCorelated x \<equiv> {y. \<exists>v. v \<in> OUTfromV x \<and> y \<in> VARto v}"
assertion_def: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "mbt_algebra.assertion one times inf less_eq dual top = {x. less_eq x one \<and> inf (times x top) (dual x) = x}"
fflip_def: fixes f :: "V" shows "fflip f = (\<lambda>ab\<in>\<^sub>\<circ>(\<D>\<^sub>\<circ> f)\<inverse>\<^sub>\<bullet>. f \<lparr>ab\<lparr>1\<^sub>\<nat>\<rparr>, ab\<lparr>[]\<^sub>\<circ>\<rparr>\<rparr>\<^sub>\<bullet>)"
T3_def: shows "T3 \<equiv> unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_checks T))"
trm_simps_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category_with_terminal_object C"   and "partial_magma.arr C f" shows "partial_magma.dom C (category_with_terminal_object.trm C f) = partial_magma.dom C f"
hyps: assumes "m < length paid_A'" shows "s_A' m = s_A m"
Quotient3_Tree\<^sub>\<alpha>: shows "Quotient3 (=\<^sub>\<alpha>) abs_Tree\<^sub>\<alpha> rep_Tree\<^sub>\<alpha>"
cases: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a1 :: "'b"   and a2 :: "'b"   and a3 :: "nat" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.distance sourcenode targetnode kind valid_edge a1 a2 a3" obtains n :: "'b" and as :: "'a list" and n' :: "'b" and x :: "nat" where "a1 = n" and "a2 = n'" and "a3 = x" and "CFG.intra_path sourcenode targetnode kind valid_edge n as n'" and "length as = x" and "\<forall>as'. CFG.intra_path sourcenode targetnode kind valid_edge n as' n' \<longrightarrow> x \<le> length as'"