hcju/isabelleps huggingface dataset:
monic_factor: fixes p :: "'c poly"   and q :: "'c poly" assumes "monic (p * q)"   and "monic p" shows "monic q"
prems: shows "bvars\<^sub>s\<^sub>s\<^sub>t (a # S) \<inter> subst_domain \<theta> = {}"
intro: fixes r :: "'b \<Rightarrow> 'c"   and subsumes :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and set_of :: "'c \<Rightarrow> 'b set"   and sel :: "'c \<Rightarrow> 'b \<times> 'c"   and un :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and size :: "'c \<Rightarrow> nat" assumes "subsumption_impl r subsumes set_of sel un size"   and "relation_subsumption_impl_axioms r set_of" shows "relation_subsumption_impl r subsumes set_of sel un size"
zero_flow: shows "flow (residual_network f) (\<lambda>_. 0)"
right_idem: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b \<sqinter> b = a \<sqinter> b"
AE_measure_pmf: fixes M :: "'c pmf" shows "AE x in measure_pmf M. x \<in> set_pmf M"
varMonotoneAux: fixes b :: "BExpr"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val"   and \<Psi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "(b, A, \<Phi>, s, t) \<in> var"   and "\<And>s t. \<Phi> (s, t) \<Longrightarrow> \<Psi> (s, t)" shows "(b, A, \<Psi>, s, t) \<in> var"
mat2matofpoly_row_def: fixes A :: "(('c, 'd) vec, 'e) vec"   and i :: "'e" shows "mat2matofpoly_row A i = (\<chi>j. [:A $ i $ j:])"
lnth_lalternate: fixes n :: "nat"   and xs :: "'b llist" assumes "2 * enat n < llength xs" shows "lalternate xs $ n = xs $ 2 * n"
SuppTransLet: fixes b :: "'b"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "supp (TransLet b \<Gamma> e) \<subseteq> supp (\<Gamma>, e) - atom ` domA \<Gamma>"
insert_rga_nonexistent: fixes i :: "'a"   and xs :: "'a list"   and e :: "'a" assumes "i \<notin> set xs" shows "insert_rga xs (e, Some i) = xs"
cp0: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and \<tau> :: "'a state \<times> 'a state" assumes "profile_mono_scheme_defined f g" shows "f X \<tau> = f (\<lambda>_. X \<tau>) \<tau>"
ptr_kinds_ptr_kinds_M: fixes ptr :: "(_) node_ptr"   and h :: "(_) heap" shows "(ptr \<in> set |h \<turnstile> node_ptr_kinds_M|\<^sub>r) = (ptr |\<in>| node_ptr_kinds h)"
Icc_eq_insert_lb_nat: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "{m..n} = insert m {Suc m..n}"
measurable_compose_rev: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'a measure"   and N :: "'b measure"   and g :: "'c \<Rightarrow> 'a"   and M :: "'c measure" assumes "f \<in> L \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M L" shows "(\<lambda>x. f (g x)) \<in> M \<rightarrow>\<^sub>M N"
OT_12_P2_assm: fixes m0 :: "bool"   and m1 :: "bool"   and \<sigma> :: "bool" shows "R2_OT12 (m0, m1) \<sigma> = funct_OT_12 (m0, m1) \<sigma> \<bind> (\<lambda>(out1, out2). S2_OT12 \<sigma> out2)"
t_def: shows "t \<equiv> \<langle>B, a, ra\<rangle>"
node_sos_states: fixes ns :: "'c net_state"   and a :: "'d node_action"   and ns' :: "'c net_state"   and S :: "('c \<times> 'd seq_action \<times> 'c) set" assumes "(ns, a, ns') \<in> node_sos S" obtains i :: "nat" and s :: "'c" and R :: "nat set" and s' :: "'c" and R' :: "nat set" where "ns = NodeS i s R" and "ns' = NodeS i s' R'"
simps_13: fixes Vs :: "char list list"   and V :: "char list"   and T :: "ty"   and e :: "char list exp" shows "compE\<^sub>1 Vs {V:T; e} = {length Vs:T; compE\<^sub>1 (Vs @ [V]) e}"
atUV_empty: fixes U :: "'f set"   and V :: "'f set" shows "\<not> atUV U V {#}"
cong_le_per2__le: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Per Bet Cong A' B' C'"   and "Cong A B A' B'"   and "Tarski_neutral_dimensionless.Le Bet Cong B C B' C'" shows "Tarski_neutral_dimensionless.Le Bet Cong A C A' C'"
assms_3: shows "eint 2 * val_Zp (pderiv f \<bullet> a) < val_Zp (f \<bullet> a)"
is_free_simps_1: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.is_free unfree (free name)"
pred_envirI: fixes p :: "'f \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" assumes "\<And>x. p (f x)" shows "pred_envir p f"
weak_transition_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and \<alpha> :: "'c"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>" shows "weak_nominal_ts.weak_transition transition \<tau> P \<alpha> P' \<equiv> if \<alpha> = \<tau> then weak_nominal_ts.tau_transition transition \<tau> P P' else weak_nominal_ts.observable_transition transition \<tau> P \<alpha> P'"
domain_K: fixes k :: "nat \<Rightarrow>\<^sub>F 'a"   and n :: "nat" assumes "k \<in> K n" shows "domain k = domain (f n)"
add_le_same_cancel2: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a + b \<subseteq># b) = (a \<subseteq># {#})"
dt: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and G :: "'b formula" assumes "(F::'b formula) \<triangleright> (\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H (G::'b formula)" shows "(\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H (F::'b formula) \<^bold>\<rightarrow> (G::'b formula)"
map_of_remdups_wrt_rev: fixes xs :: "('a \<times> 'b) list" shows "map_of (remdups_wrt_rev fst xs []) = map_of xs"
Torder_Chain: fixes X :: "'a set" assumes "X \<subseteq> carrier D"   and "Torder (Iod D X)" shows "Chain D X"
Seq_decomp_star_Stuck: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p\<^sub>1 :: "('d, 'e, 'f) com"   and p\<^sub>2 :: "('d, 'e, 'f) com"   and s :: "'d" assumes "\<Gamma> \<turnstile> (Seq p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>* (Skip, Stuck)" shows "\<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Skip, Stuck) \<or> (\<exists>s'. \<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Skip, Normal s') \<and> \<Gamma> \<turnstile> (p\<^sub>2, Normal s') \<rightarrow>\<^sup>* (Skip, Stuck))"
cont2cont_disj: fixes f :: "'d \<Rightarrow> bool"   and g :: "'d \<Rightarrow> bool" assumes "cont f"   and "cont g" shows "cont (\<lambda>x. f x \<or> g x)"
image_eq: fixes g :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on g A" shows "\<Prod>(g ` A) = prod g A"
spr_jview_tObsC: fixes a :: "'a"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" assumes "spr_jview a t = spr_jview a t'" shows "tObsC t = tObsC t'"
Bdec: shows "set_nn_integral M (\<Union>n. (T ^^ n) --` A) f = set_nn_integral M (B 0) f + (\<Sum>n. set_nn_integral M (B (n + 1)) f)"
power_dvd_imp_le: fixes i :: "nat"   and m :: "nat"   and n :: "nat" assumes "i ^ m dvd i ^ n"   and "1 < i" shows "m \<le> n"
simps_18: fixes v :: "nat"   and va :: "atom fm" shows "push_forall (AllQ (AllN v va)) = (if freeIn 0 (AllN v va) then lowerFm 0 1 (AllN v va) else AllQ (AllN v va))"
dgrad_p_set_def: fixes d :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> nat"   and m :: "nat" shows "punit.dgrad_p_set d m = {p. local.punit.pp_of_term ` keys p \<subseteq> dgrad_set d m}"
a2: shows "(\<lambda>x. norm (f x)) abs_summable_on S"
strassen_measure_def: shows "strassen_measure \<equiv> \<lambda>(A, B). dim_row A + dim_col A + dim_col B + (dim_row A + dim_col A + dim_col B) + (if strassen_even A B then 0 else 1)"
Limsup_le_iff: fixes F :: "'d filter"   and X :: "'d \<Rightarrow> 'c"   and C :: "'c" shows "(Limsup F X \<le> C) = (\<forall>y>C. \<forall>\<^sub>F x in F. X x < y)"
moebius_similarity_inf: fixes a :: "complex"   and b :: "complex" assumes "a \<noteq> 0" shows "moebius_pt (moebius_similarity a b) \<infinity>\<^sub>h = \<infinity>\<^sub>h"
primitive_part_alt_def: fixes p :: "'a poly" shows "primitive_part p = sdiv_poly p (content p)"
s_1: shows "Suc (i - 2) = i - 1"
dexn2: fixes mn :: "char list list"   and cn :: "char list"   and s :: "'b state"   and ck :: "bool"   and env :: "v sem_env"   and locs :: "locn \<times> locn"   and ts :: "t list" assumes "TypeExn (mk_id mn cn) \<in> defined_types s" shows "evaluate_dec ck mn env s (Dexn locs cn ts) (s, Rerr (Rabort Rtype_error))"
lastOvl: shows "last (natToVertexList v f es) = Some (last (verticesFrom f v))"
set_lub_spmf: fixes Y :: "'c spmf set" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y" shows "set_spmf (lub_spmf Y) = \<Union> (set_spmf ` Y)"
retract_inverse: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and x :: "'d" assumes "retract G"   and "x \<in> carrier \<X>\<^bsub>G\<^esub>" shows "\<pi>\<^sub>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>G\<^esub> x) = x"
map_closed: fixes a :: "'a" assumes "a \<in> M" shows "\<eta> a \<in> M'"
containsE: fixes A :: "'a set"   and x :: "'a" assumes "Predicate_Compile.contains A x" obtains A' :: "'a set" and x' :: "'a" where "A = A'" and "x = x'" and "x \<in> A"
assms: shows "e \<in> UNIV \<rightarrow> {- 1..1}"
star_rtc1: fixes x :: "'a" shows "(1::'a) \<squnion> x \<squnion> qstar x \<cdot> qstar x \<le> qstar x"
invs: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "prop_invs c0"   and "next_change_multiplicity' c0 c1 loc t n" shows "prop_invs c1"
rel_funI: fixes A :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and B :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and f :: "'q \<Rightarrow> 's"   and g :: "'r \<Rightarrow> 't" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
assms_2: shows "g \<midarrow>b\<rightarrow> c"
assms_2: shows "finite {t. l2 t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}"
abs_eq: fixes x :: "nat \<Rightarrow> bool" shows "set_bits x = Abs_uint (set_bits x)"
geodesic_segment_union_2: fixes x :: "'b"   and z :: "'b"   and y :: "'b"   and G :: "'b set"   and H :: "'b set" assumes "dist x z = dist x y + dist y z"   and "geodesic_segment_between G x y"   and "geodesic_segment_between H y z" shows "G \<inter> H = {y}"
lth: fixes xsa :: "'b list"   and xs :: "'c list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and na :: "nat"   and n :: "nat" assumes "length xsa = length xs" shows "rotate1 (map2 f (rotate na xsa) (rotate n xs)) = map2 f (rotate1 (rotate na xsa)) (rotate1 (rotate n xs))"
division_of_finite: fixes s :: "'b set set"   and i :: "'b set" assumes "s division_of i" shows "finite s"
assms: shows "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)"
distrib_lattice_axioms_def: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.distrib_lattice_axioms inf sup \<equiv> \<forall>x y z. sup x (inf y z) = inf (sup x y) (sup x z)"
is_subprob_densityI: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "f \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x"   and "space M \<noteq> {}"   and "integral\<^sup>N M f \<le> 1" shows "is_subprob_density M f"
ceiling_altdef: fixes x :: "'b" shows "\<lceil>x\<rceil> = (if x = of_int \<lfloor>x\<rfloor> then \<lfloor>x\<rfloor> else \<lfloor>x\<rfloor> + 1)"
par_by_merge_commute_swap: fixes P :: "('d, 'f) urel"   and M :: "(('d, 'f, 'f) mrg, 'e) urel"   and Q :: "('d, 'f) urel" shows "P \<parallel>\<^bsub>M\<^esub> Q = Q \<parallel>\<^bsub>swap\<^sub>m ;; M\<^esub> P"
coplanar_perm_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D" shows "Coplanar A C B D"
distinct_33: fixes x4 :: "key"   and x61 :: "msg"   and x62 :: "msg" shows "Key x4 \<noteq> \<lbrace>x61, x62\<rbrace>"
quotient_imp_surjective_map: fixes X :: "'a topology"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "quotient_map X X' f" shows "f ` topspace X = topspace X'"
n2: shows "nprv (insert (subst \<phi> (Var y) x) F) (eql (Var y) t)"
lstrict_prefix_code_2: fixes y :: "'c"   and ys :: "'c llist" shows "lstrict_prefix LNil (LCons y ys) = True"
equivalence_functor_L: shows "equivalence_functor (\<bullet>) (\<bullet>) L"
IH1: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "s \<in> reachable (pnet np p1) TT"   and "(s, disconnect(i, i'), s') \<in> automaton.trans (pnet np p1)" shows "netmap s' = netmap s"
simps_18: fixes sh :: "sheap"   and e\<^sub>1 :: "'a exp"   and C :: "char list"   and V :: "'a"   and e\<^sub>2 :: "'a exp" shows "iconf sh (try e\<^sub>1 catch(C V) e\<^sub>2) = (iconf sh e\<^sub>1 \<and> \<not> sub_RI e\<^sub>2)"
continuous_at_within_inverse: fixes a :: "'c"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous (at a within s) f"   and "f a \<noteq> (0::'d)" shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
degree_power_le: fixes p :: "'b poly"   and n :: "nat" shows "degree (p ^ n) \<le> degree p * n"
dirichlet_prod_altdef2: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "f \<star> g = (\<lambda>n. \<Sum>(r, d)\<in>{(r, d). r * d = n}. f r * g d)"
ediv_mono: fixes a :: "enat"   and b :: "enat"   and k :: "nat" assumes "a \<le> b" shows "ediv a k \<le> ediv b k"
wf_mdecl_def: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "String.literal" shows "wf_mdecl wf_md P C \<equiv> \<lambda>(M, Ts, T, m). (\<forall>T\<in>set Ts. is_type P T) \<and> is_type P T \<and> (case m of None \<Rightarrow> C\<bullet>M(Ts) :: T | \<lfloor>mb\<rfloor> \<Rightarrow> wf_md P C (M, Ts, T, mb))"
insert_noteq_member: fixes b :: "'b"   and B :: "'b multiset"   and c :: "'b"   and C :: "'b multiset" assumes "add_mset b B = add_mset c C"   and "b \<noteq> c" shows "c \<in># B"
ba_bot: shows "bot = (THE x. \<forall>z. x = - (z \<squnion> - z))"
skl1_step2_refines_a0i_running_skip_i: fixes A :: "agent"   and B :: "agent"   and Ni :: "msg"   and Rb :: "fid_t"   and gnx :: "msg" shows "{R0sk1iai} (a0i_running A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>)\<^sup>=, skl1_step2 Rb A B Ni gnx {> R0sk1iai}"
Rep_Arity_cases: fixes y :: "nat" assumes "y \<in> UNIV" obtains x :: "Arity" where "y = Rep_Arity x"
edge_succ_in_arcs: fixes a :: "'b" shows "(edge_succ M a \<in> arcs G) = (a \<in> arcs G)"
finprod_closed: fixes G :: "('e, 'f) monoid_scheme"   and f :: "'g \<Rightarrow> 'e"   and A :: "'g set" assumes "Group.comm_monoid G"   and "f \<in> A \<rightarrow> carrier G" shows "finprod G f A \<in> carrier G"
prems_1: shows "outs_\<I> \<I> \<turnstile>\<^sub>R res \<sim> res'"
measure_uniform_measure_eq_cond_prob: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "random_variable (count_space UNIV) P"   and "random_variable (count_space UNIV) Q" shows "\<P>(x in uniform_measure M {x \<in> space M. Q x}. P x) = cond_prob M P Q"
slice_intra_kind_in_slice: fixes a :: "'edge"   and S :: "'node SDG_node set" assumes "sourcenode a \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "intra_kind (kind a)" shows "slice_kind S a = kind a"
red2_rtrancl_\<tau>_into_RedT_\<tau>: fixes t :: "'t"   and x2 :: "'x2"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x2' :: "'x2"   and m2' :: "'m2"   and x1 :: "'x1"   and m1 :: "'m1" assumes "r2.silent_moves t (x2, shr s2) (x2', m2')"   and "t \<turnstile> (x1, m1) \<approx> (x2, shr s2)"   and "thr s2 t = \<lfloor>(x2, no_wait_locks)\<rfloor>"   and "wset s2 t = None" shows "\<tau>mRed2 s2 (redT_upd_\<epsilon> s2 t x2' m2')"
simps_1: fixes \<rho> :: "'b run"   and K :: "clock" shows "#\<^sub>< \<rho> K 0 = 0"
ipurge_tr_rev_aux_all: fixes D :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and U :: "'a set"   and I :: "('a \<times> 'a) set" assumes "\<And>v. v \<in> D ` set xs \<Longrightarrow> \<exists>u\<in>U. (v, u) \<in> I" shows "ipurge_tr_rev_aux I D U xs = xs"
down_eq_id_list_rel: fixes x :: "'a list nres" shows "\<Down> (\<langle>Id\<rangle>list_rel) x = x"
in_wN_of_wstate_in_N_of_wstate: fixes C :: "'a literal multiset"   and i :: "nat"   and St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat" assumes "(C, i) \<in># wN_of_wstate St" shows "C \<in> N_of_wstate St"
nat_le_real_less: fixes n :: "nat"   and m :: "nat" shows "(n \<le> m) = (real n < real m + 1)"
isLub_def: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" shows "isLub R S x = leastP (isUb R S) x"
assms_1: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"
floorlog_geI: fixes b :: "nat"   and w :: "nat"   and x :: "nat" assumes "b ^ (w - 1) \<le> x"   and "1 < b" shows "w \<le> floorlog b x"
bisim1FAss3: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and e2 :: "'addr expr1"   and xs :: "'addr val list" shows "P,e\<bullet>F{D} := e2,h \<turnstile> (unit, xs) \<leftrightarrow> ([], xs, Suc (length (compE2 e) + length (compE2 e2)), None)"
composite_simps_4: shows "cod (g \<star> h) = cod g \<star> cod h"
surj_fun_eq: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set"   and g1 :: "'a \<Rightarrow> 'c"   and g2 :: "'a \<Rightarrow> 'c" assumes "\<P> f X = UNIV"   and "\<forall>x\<in>X. (g1 \<circ> f) x = (g2 \<circ> f) x" shows "g1 = g2"
dual_Inf: fixes X :: "'a set" shows "\<Sqinter>X ^ o = \<Squnion>(dual ` X)"
eq: fixes q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "monomial (lookup (poly_subst (\<lambda>y. monomial (a y) 0) q) 0) 0 = poly_subst (\<lambda>y. monomial (a y) 0) q"
step_desc: fixes r :: "('Q, 'L) ta_rule"   and it :: "('Q, 'L) ta_rule set"   and \<Sigma> :: "'\<Sigma>" assumes "r \<in> it"   and "it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}"   and "cinvar it \<Sigma>" shows "\<alpha>s (cstep r \<Sigma>) = br'_inner_step r (\<alpha>s \<Sigma>)"
af_nested_propos: fixes \<phi> :: "'b ltl"   and \<nu> :: "'b set" shows "nested_propos (af_letter \<phi> \<nu>) \<subseteq> nested_propos \<phi>"
ucast_mask_eq: fixes n :: "nat" shows "ucast (mask n) = mask (min LENGTH('c) n)"
heap_is_wellformed_children_disc_nodes_different: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "set children \<inter> set disc_nodes = {}"
homeomorphic_map_involution: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'a" assumes "continuous_map X X f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f (f x) = x" shows "homeomorphic_map X X f"
Pair_synth_analz: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" shows "(\<langle>X, Y\<rangle> \<in> synth (analz H)) = (X \<in> synth (analz H) \<and> Y \<in> synth (analz H))"
diverge1: shows "s1 -\<tau>1\<rightarrow> \<infinity>"
convex_halfspace_Im_gt: fixes b :: "real" shows "convex {x. b < Im x}"
of_match_fields_safe_eq2: fixes m :: "of_match_field set"   and p :: "(32, 'a) simple_packet_ext_scheme" assumes "all_prerequisites m" shows "OF_match_fields_safe m p = (OF_match_fields m p = Some True)"
row_spaceE: fixes x :: "'c vec"   and A :: "'c mat" assumes "x \<in> row_space A" obtains v :: "'c vec" where "v \<in> carrier_vec (dim_row A)" and "x = v \<^sub>v* A"
lift_CFGExit: 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 Exit :: "'h"   and Def :: "'h \<Rightarrow> 'i set"   and Use :: "'h \<Rightarrow> 'i set"   and ParamDefs :: "'h \<Rightarrow> 'i list"   and ParamUses :: "'h \<Rightarrow> 'i set list" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit" shows "CFGExit src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main NewExit"
nhds_def: fixes "open" :: "'d set \<Rightarrow> bool"   and a :: "'d" assumes "class.topological_space open" shows "topological_space.nhds open a = Inf (principal ` {S. open S \<and> a \<in> S})"
prems_2: shows "c \<subseteq>\<^sub>g Catch c1' c2'"
closure_of_interior_of: fixes X :: "'b topology"   and S :: "'b set" shows "X closure_of S = topspace X - X interior_of (topspace X - S)"
irreducible\<^sub>d_as_irreducible: fixes p :: "int poly" shows "irreducible\<^sub>d p = irreducible (primitive_part p)"
strict_trans1: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "b \<preceq> a"   and "c \<prec> b" shows "c \<prec> a"
cases: fixes a1 :: "nat"   and a2 :: "(nat \<times> form) list"   and s :: "(nat \<times> form) list" assumes "(a1, a2) \<in> deriv s" obtains "a1 = 0" and "a2 = s"   | n :: "nat" and x :: "(nat \<times> form) list" and y :: "(nat \<times> form) list" where "a1 = Suc n" and "a2 = y" and "(n, x) \<in> deriv s" and "y \<in> set (subs x)"
icheck_init_class: fixes P :: "'c prog"   and C :: "char list"   and e :: "'d exp" assumes "icheck P C e" shows "init_class P e = \<lfloor>C\<rfloor>"
invP: shows "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"
hyps_2: shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q"
lim: shows "(sum (\<lambda>x. f x + g x) \<longlongrightarrow> infsetsum' f A + infsetsum' g A) (finite_subsets_at_top A)"
refine: fixes ops :: "'c approximate_set_ops"   and set_of_appr :: "'c list \<Rightarrow> real list set"   and appr_rell :: "('c list \<times> real list set) set"   and optns :: "'c numeric_options"   and ai :: "'c list"   and X :: "'d set" assumes "approximate_sets ops set_of_appr appr_rell optns"   and "(ai, X) \<in> approximate_sets.appr_rel appr_rell" shows "(RETURN (approximate_sets.ivl_rep_of_set_impl ops optns ai), op_ivl_rep_of_set X) \<in> \<langle>lv_rel \<times>\<^sub>r lv_rel\<rangle>nres_rel"
Un_Diff_Diff: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> B - (A - B) = B"
walk_in_V: fixes xs :: "'c list" assumes "T.walk xs" shows "set xs \<subseteq> V\<^bsub>T\<^esub>"
pdigraph_base_axioms: shows "pdigraph_base \<alpha> I \<AA>"
C: shows "0\<^sub>m (dim_row (diag_block_mat As)) (dim_col A) \<in> carrier_mat (dim_row (diag_block_mat As)) (dim_col A)"
assms_1: shows "InvariantEquivalentZL (getF state) (getM state) Phi"
vector_N_pp: fixes x :: "'a" assumes "x \<sqinter> top = x" shows "- (- x \<sqinter> top) \<sqinter> top = - - x \<sqinter> top"