hcju/isabelleps huggingface dataset:
l1_inv2I: fixes x :: "l1_state" assumes "\<And>gnx A B Rb. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; in_progressS (progress x Rb) {xny, xgnx, xgny, xsk}; guessed_frame Rb xgnx = Some gnx; can_signal x A B\<rbrakk> \<Longrightarrow> 0 < signalsInit x (Running A B (Exp gnx (NonceF (Rb $ ny))))" shows "x \<in> l1_inv2"
beta_cases_1: fixes i :: "nat"   and t :: "dB"   and P :: "bool" assumes "Var i \<rightarrow>\<^sub>\<beta> t" shows "P"
elims_1: fixes x :: "'b \<Rightarrow> bool"   and xa :: "'b list"   and y :: "bool" assumes "\<guillemotleft>x\<guillemotright> xa = y" obtains p :: "'b \<Rightarrow> bool" where "x = pa__" and "xa = []" and "\<not> y"   | p :: "'b \<Rightarrow> bool" and a :: "'b" and S :: "'b list" where "x = p" and "xa = a # S" and "y = p a"
proper_sub_set_system_axioms_def: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" shows "proper_sub_set_system_axioms \<U> \<A> \<V> \<B> \<equiv> \<U> \<subset> \<V> \<and> \<A> \<subseteq># \<B>"
WbisT_matchT_MT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>wT d" shows "matchT_MT c d"
state_diverges: fixes s1 :: "'state1" assumes "L1.inf_step s1" shows "L1.sem_behaves s1 Diverges"
of_bool_odd_eq_mod_2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "zero_neq_one.of_bool one zero (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a) = modulo a (numeral.numeral one plus (num.Bit0 num.One))"
hom_0: fixes x :: "int" assumes "of_int x = (0::'b)" shows "x = 0"
sum_eq_Suc0_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "finite A" shows "(sum f A = Suc 0) = (\<exists>a\<in>A. f a = Suc 0 \<and> (\<forall>b\<in>A. a \<noteq> b \<longrightarrow> f b = 0))"
wf_graph_union_edges: fixes V :: "'c set"   and E :: "('c \<times> 'c) set"   and E' :: "('c \<times> 'c) set" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "wf_graph \<lparr>nodes = V, edges = E'\<rparr>" shows "wf_graph \<lparr>nodes = V, edges = E \<union> E'\<rparr>"
simps_2: fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" shows "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
not_ev_iff: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "(\<not> ev P \<omega>) = alw (\<lambda>xs. \<not> P xs) \<omega>"
sum_fun_distl: fixes Y :: "'a set"   and x :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "finite Y" shows "x \<cdot> \<Sum> (f ` Y) = \<Sum> {x \<cdot> f y |y. y \<in> Y}"
transfer: shows "state_val (transfer (kind a) s) = fst cfx(ParamDefs (targetnode a) [:=] map (fst cf) outs)"
Units_r_inv: fixes x :: "int" assumes "x \<in> Units R" shows "x \<otimes> inv x = \<one>"
polychain_appendI: fixes xs :: "('a \<times> 'a) list"   and ys :: "('a \<times> 'a) list" assumes "polychain xs"   and "polychain ys"   and "\<lbrakk>xs \<noteq> []; ys \<noteq> []\<rbrakk> \<Longrightarrow> snd (last xs) = fst (hd ys)" shows "polychain (xs @ ys)"
comp_eq_elim: fixes a :: "'c \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'c"   and c :: "'d \<Rightarrow> 'b"   and d :: "'a \<Rightarrow> 'd" assumes "a \<circ> b = c \<circ> d" obtains "\<And>v. a (b v) = c (d v)"
nth_Abs_fls_nat_lower_bound: fixes m :: "nat"   and f :: "int \<Rightarrow> 'a"   and n :: "int" assumes "\<forall>n>m. f (- int n) = (0::'a)" shows "fls_nth (Abs_fls f) n = f n"
seq_assoc: fixes a :: "'b::refinement_lattice"   and b :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" shows "(a::'b::refinement_lattice) ; (b::'b::refinement_lattice) ; (c::'b::refinement_lattice) = a ; (b ; c)"
absorb2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "semilattice_order f less_eq less"   and "less_eq b a" shows "f a b = b"
hid_def: fixes P :: "('c, 'd) action"   and cs :: "'c set" shows "hid P cs \<equiv> R (\<lambda>(S, S'). \<exists>s. diff_tr S' S = tr_filter (s - tr S) cs \<and> action.relation_of P (S, S'\<lparr>tr := s, ref := ref S' \<union> cs\<rparr>)) ;; action.relation_of Skip"
gState_progress_rel_exclusive: fixes g :: "'a gState\<^sub>I_scheme"   and g' :: "'a gState\<^sub>I_scheme"   and prog :: "program"   and p :: "nat" assumes "(g, g') \<in> gState_progress_rel prog" shows "(g, g'\<lparr>exclusive := p\<rparr>) \<in> gState_progress_rel prog"
RN_deriv_finite: fixes N :: "'a measure" assumes "sigma_finite_measure N"   and "absolutely_continuous N N"   and "sets N = sets N" shows "AE x in N. RN_deriv N N x \<noteq> \<infinity>"
trms\<^sub>l\<^sub>s\<^sub>t_append: fixes A :: "('f \<times> ('d, 'e) strand_step) list"   and B :: "('f \<times> ('d, 'e) strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>t (A @ B) = trms\<^sub>l\<^sub>s\<^sub>t A \<union> trms\<^sub>l\<^sub>s\<^sub>t B"
span_neg: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and S :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "x \<in> pmdl S" shows "- x \<in> pmdl S"
pat_fmap_empty: shows "fmempty \<equiv> op_fmap_empty"
measure_Union': fixes F :: "'a set set"   and M :: "'a measure" assumes "finite F"   and "\<And>S. S \<in> F \<Longrightarrow> S \<in> fmeasurable M"   and "disjoint F" shows "Sigma_Algebra.measure M (\<Union> F) = sum (Sigma_Algebra.measure M) F"
assms_3: shows "\<forall>i. p = Nom i \<or> p = (\<^bold>\<diamond> Nom i) \<longrightarrow> i \<in> A"
\<Y>_def: fixes T :: "(symbol \<times> character list) set"   and P :: "(symbol \<times> character list) list set"   and k :: "nat" shows "\<Y> T P k = Sel T (\<W> P k)"
unit_factor_1: 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" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "unit_factor one = one"
deterministic: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc'' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "lc \<leadsto> lc'"   and "lc \<leadsto> lc''" shows "lc' = lc''"
scaleR_nonneg_nonpos: fixes a :: "real"   and x :: "'a" assumes "0 \<le> a"   and "x \<le> (0::'a)" shows "a *\<^sub>R x \<le> (0::'a)"
simps_5: fixes t1 :: "pdf_type"   and t2 :: "pdf_type" shows "stock_measure (PRODUCT t1 t2) = embed_measure (stock_measure t1 \<Otimes>\<^sub>M stock_measure t2) (\<lambda>(a, b). <|a, b|>)"
map_eq: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'d set"   and T :: "'e set"   and g :: "'d \<Rightarrow> 'e" assumes "Set_Theory.map f S T"   and "Set_Theory.map g S T"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "f = g"
isCont_swap: fixes a :: "'a \<times> 'b" shows "isCont prod.swap a"
t_A_A'_leq: fixes n :: "nat" assumes "n < length paid_A'" shows "t_A n \<le> t_A' n"
scast_of_nat_to_signed: fixes x :: "nat" shows "SCAST('b \<rightarrow> 'b signed) (word_of_nat x) = word_of_nat x"
assms_4: shows "A - (\<Union>i. (T ^^ i) --` B) \<in> null_sets M"
map_add_assoc: fixes m1 :: "'b \<Rightarrow> 'c option"   and m2 :: "'b \<Rightarrow> 'c option"   and m3 :: "'b \<Rightarrow> 'c option" shows "m1 ++ (m2 ++ m3) = m1 ++ m2 ++ m3"
G'_2: shows "V \<subseteq> V\<^bsub>G'\<^esub>"
hs_is_empty_def: fixes t :: "'c Tree" shows "hs_is_empty t = (t = E)"
mkKripke_kripke: fixes ws :: "'c set"   and rels :: "'a \<Rightarrow> ('c \<times> 'c) set"   and val :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "kripke (mkKripke ws rels val)"
lossless_bernoulli: fixes p :: "real" shows "lossless_spmf (bernoulli p)"
that: fixes q :: "real \<Rightarrow> complex" assumes "path q"   and "pathfinish q - pathstart q = 2 * complex_of_real pi * \<i> * winding_number p \<zeta>"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = \<zeta> + exp (q t)" shows "thesis"
pref_iffD1: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" assumes "|~ F = G" shows "|~ F \<longrightarrow> G"
simps_2: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> (\<phi> or\<^sub>n \<psi>) w = \<G>\<F> \<phi> w \<union> \<G>\<F> \<psi> w"
max_idx_vars: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "pre_wf_formula n \<phi>" shows "\<forall>p\<in>FOV \<phi> \<union> SOV \<phi>. p < n"
sorted_wrt_mono_rel: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "sorted_wrt P xs" shows "sorted_wrt Q xs"
add_0_right: fixes a :: "'ag" assumes "a \<in> U" shows "a +\<^sub>o\<^sub>w 0\<^sub>o\<^sub>w = a"
unrestrict_path_edges: fixes g :: "'b ugraph"   and E :: "('b \<times> 'b) set"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "path (restrict_edges g E) u p v" shows "path g u p v"
measurable_def: fixes A :: "'c measure"   and B :: "'d measure" shows "A \<rightarrow>\<^sub>M B = {f \<in> space A \<rightarrow> space B. \<forall>y\<in>sets B. f -` y \<inter> space A \<in> sets A}"
numeral_Bit1_div_2: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and n :: "num" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 num.One)) = numeral.numeral one plus n"
eq_up_to_def: fixes \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'a" shows "IFC_def.eq_up_to \<pi> k \<pi>' = (\<forall>i\<le>k. \<pi> i = \<pi>' i)"
plus_eq_lift: fixes x :: "V"   and y :: "V" shows "x + y = x \<squnion> lift x y"
eq: shows "set xs = {x. finfun_dom f(a $:= b) $ x}"
predcong: fixes l :: "tm list"   and l' :: "tm list"   and e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and i :: "String.literal" assumes "equal_length l l'" shows "semantics e f g (foldr Imp (zip_eq l l') (Imp (Atom (Rl i l)) (Atom (Rl i l'))))"
assms_2: shows "\<forall>\<^sub>F x in F. 0 < f x"
drop_nonempty_def: fixes v :: "'a"   and n :: "nat"   and xs :: "'a list" shows "drop_nonempty v n xs = (if n < length xs then drop n xs else [last (v # xs)])"
fdemodalisation2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(antirange_semiring.bdia (\<cdot>) ar x y \<le> ar (ar z)) = (ar (ar y) \<cdot> x \<le> x \<cdot> ar (ar z))"
ji: shows "(j, i) \<in> ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>"
poly_eqI: fixes p :: "'b poly"   and q :: "'b poly" assumes "\<And>n. coeff p n = coeff q n" shows "p = q"
prod_casesK: fixes k :: "'c" shows "(\<lambda>(uu_, uu_). k) = (\<lambda>_. k)"
is_orthogonal_sym: fixes \<psi> :: "'a"   and \<phi> :: "'a" shows "is_orthogonal \<psi> \<phi> = is_orthogonal \<phi> \<psi>"
dom_inv: fixes f :: "'a \<times> 'a" assumes "VoV.iso f" shows "VoV.dom (VoV.inv f) = VoV.cod f"
last: shows "envObs a (tLast t) = envObs a (tLast t')"
assms_2: fixes c :: "'b"   and t :: "'a" shows "f (monom_mult c t p) = monom_mult c t (f p)"
valid_action_valid_succ: fixes h :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" assumes "h \<in> PROB"   and "s \<in> valid_states PROB" shows "state_succ s h \<in> valid_states PROB"
CESem_simps_no_tick_5: fixes as :: "(Vars.var \<times> exp) list"   and body :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrakk> Terms.Let as body \<rbrakk>\<^bsub>\<rho>\<^esub> \<sqsubseteq> \<N>\<lbrakk> body \<rbrakk>\<^bsub>\<N>\<lbrace>as\<rbrace>\<rho>\<^esub>"
B: shows "\<forall>i\<in>domain x. (x)\<^sub>F i \<in> B i \<and> B i \<subseteq> a i \<and> B i \<in> local.basis_proj"
resolution_example1: shows "resolution_deriv {{NP, PQ}, {NQ}, {PP, PQ}} {{NP, PQ}, {NQ}, {PP, PQ}, {NP}, {PP}, {}}"
sup_state_conv2: fixes P :: "'a prog"   and s1 :: "ty list \<times> ty err list"   and s2 :: "ty list \<times> ty err list" shows "P \<turnstile> s1 \<le>\<^sub>i s2 = (P \<turnstile> fst s1 [\<le>] fst s2 \<and> P \<turnstile> snd s1 [\<le>\<^sub>\<top>] snd s2)"
Inv_empty3: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and c :: "'c list"   and n :: "nat" assumes "BIT_Off acts qs init"   and "length init = 0" shows "Inv c (BIT_Off.s_A acts qs init n) = {}"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext_irrefl ext"   and "ext_irrefl_trans_strong_axioms ext" shows "ext_irrefl_trans_strong ext"
chamber_vertices: fixes X :: "'b set set"   and x :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X x" shows "order.greater_eq (\<Union> X) x"
int_nat_card: shows "int (nat (int CARD('a))) = int CARD('a)"
at_to_0: fixes a :: "'a" shows "at a = filtermap (\<lambda>x. x + a) (at (0::'a))"
distinct_map_ran: fixes xs :: "'c list"   and y :: "'d"   and ys :: "'d list" assumes "distinct xs"   and "y \<notin> set ys"   and "length xs = length ys" shows "y \<notin> ran [xs [\<mapsto>] ys]"
lossless_oracle: fixes len :: "nat"   and \<sigma> :: "'b \<Rightarrow> bool list option"   and x :: "'b" shows "lossless_spmf (oracle len \<sigma> x)"
assms_1: fixes n :: "nat" shows "f n \<le> x"
\<J>_induct: fixes x :: "item"   and k :: "nat"   and u :: "nat"   and P :: "item \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" assumes "x \<in> \<J> k u"   and "\<And>x k u. \<lbrakk>\<And>x' k' u'. \<lbrakk>x' \<in> \<J> k' u'; indexlt k' u' k u\<rbrakk> \<Longrightarrow> P x' k' u'; x \<in> \<J> k u\<rbrakk> \<Longrightarrow> P x k u" shows "P x k u"
externals_def: fixes asig :: "'a set \<times> 'a set \<times> 'a set" shows "externals asig \<equiv> inputs asig \<union> outputs asig"
refl_ax_holds: fixes e :: "(name, nat) finfun" shows "eval_fmla e refl_ax"
boundary_is_vert_hor: shows "two_chain_boundary two_chain = two_chain_vertical_boundary two_chain \<union> two_chain_horizontal_boundary two_chain"
uint_bintrunc: fixes bin :: "num" shows "uint (numeral bin) = take_bit LENGTH('c) (numeral bin)"
dominates_def: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" shows "dominates F f g = (\<forall>p. (\<lambda>x. g x powr p) \<in> o[F](f))"
bisim1_inline_call_Val: fixes P :: "'addr J1_prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and n0 :: "nat"   and v :: "'addr val" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, None)"   and "call1 e' = \<lfloor>(a, M, vs)\<rfloor>"   and "compE2 e ! pc = Invoke M n0" shows "Suc (length vs) \<le> length stk \<and> n0 = length vs \<and> P,e,h \<turnstile> (inline_call (Val v) e', xs) \<leftrightarrow> (v # drop (Suc (length vs)) stk, loc, Suc pc, None)"
to_cnf_eq_Cons: fixes x :: "ordinal"   and a :: "ordinal"   and list :: "ordinal list" assumes "to_cnf x = a # list" shows "a = oLog \<omega> x"
disjoint_edges_def: fixes e1 :: "'b \<times> 'b"   and e2 :: "'b \<times> 'b" shows "disjoint_edges e1 e2 = (fst e1 \<noteq> fst e2 \<and> fst e1 \<noteq> snd e2 \<and> snd e1 \<noteq> fst e2 \<and> snd e1 \<noteq> snd e2)"
elims_3: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal set" assumes "\<not> validate_clause x xa" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and C :: "'b literal set" where "x = I" and "xa = C" and "\<not> (\<forall>s. ground_clause (subst_cl C s) \<longrightarrow> validate_ground_clause I (subst_cl C s))"
vec_uniqueness: fixes m :: "nat"   and v :: "'c list"   and n :: "nat" assumes "vec m v"   and "vec n v" shows "m = n"
section_retraction_of_iso_2: fixes f :: "'s"   and g :: "'s" assumes "local.iso f"   and "ide (f \<cdot> g)" shows "inverse_arrows f g"
Res: fixes y :: "name"   and R :: "pi"   and S :: "pi" assumes "y \<sharp> R" shows "(<\<nu>y>(R \<parallel> S), R \<parallel> <\<nu>y>S) \<in> Rel"
assms_1: shows "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>"
prems_2: assumes "length q \<le> length p" shows "take (length q) p \<noteq> q"
Vset: fixes i :: "V" shows "Vset i = (SUP j\<in>elts i. VPow (Vset j))"
ftf_syn: fixes u :: "'b event list"   and t :: "'b event list"   and s :: "'b event list"   and A :: "'b set" assumes "front_tickFree u"   and "front_tickFree t"   and "s setinterleaves ((t, u), insert tick (ev ` A))" shows "front_tickFree s"
o_case_sum: fixes h :: "'d \<Rightarrow> 'c"   and f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" shows "h \<circ> f (+) g = (h \<circ> f) (+) (h \<circ> g)"
leadsTo_refl: fixes F :: "'c program"   and A :: "'c set" shows "F \<in> A \<longmapsto> A"
l1_inv2_trans: shows "{l1_inv2} TS.trans l1 {> l1_inv2}"
simps_21: fixes g1 :: "'b gexp"   and g2 :: "'b gexp"   and g1' :: "'b gexp"   and g2' :: "'b gexp" shows "GExp_Lexorder.less_gexp_aux (Nor g1 g2) (Nor g1' g2') = (GExp_Lexorder.less_gexp_aux g1 g1' \<or> g1 = g1' \<and> GExp_Lexorder.less_gexp_aux g2 g2')"
single_Nonce_secrecy: fixes A :: "agent"   and B :: "agent"   and KAB :: "nat"   and na :: "msg"   and NB' :: "nat"   and X :: "msg"   and evs :: "event list"   and NB :: "nat" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key KAB, na, Nonce NB'\<rbrace>, X\<rbrace> \<in> set evs"   and "NB \<noteq> NB'"   and "KAB \<notin> range shrK"   and "evs \<in> yahalom" shows "(Nonce NB \<in> analz (insert (Key KAB) (knows Spy evs))) = (Nonce NB \<in> analz (knows Spy evs))"
l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" shows "l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs known_ptrs get_parent \<equiv> (l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs \<and> l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent) \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_get_host get_host get_host_locs \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
3: shows "card X - (card X - Suc 0) = Suc 0"
SUP_mono: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. less_eq (f n) (g m)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
sgn_power: fixes x :: "'a"   and n :: "nat" shows "sgn (x ^ n) = (if n = 0 then 1::'a else if even n then \<bar>sgn x\<bar> else sgn x)"
set_cases_1: fixes e :: "'c"   and a :: "('c, 'd) step" assumes "e \<in> set1_step a" obtains z2 :: "'d" where "a = Yield e z2"
vangle_nonneg: fixes u :: "'b"   and v :: "'b" shows "0 \<le> vangle u v"
mkArr_restrict_eq: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and B :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and F :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr (mkArr A B F)" shows "mkArr A B (restrict F A) = mkArr A B F"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and compare :: "RBT_Impl.compare" shows "P (case compare of compare.LT \<Rightarrow> f1 | compare.GT \<Rightarrow> f2 | compare.EQ \<Rightarrow> f3) = ((compare = compare.LT \<longrightarrow> P f1) \<and> (compare = compare.GT \<longrightarrow> P f2) \<and> (compare = compare.EQ \<longrightarrow> P f3))"
matrix_sum_smult: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a mat"   and d :: "nat"   and c :: "'a" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d" shows "matrix_sum d (\<lambda>k. c \<cdot>\<^sub>m f k) n = c \<cdot>\<^sub>m matrix_sum d f n"
Lr_def: fixes F'' :: "'b filter"   and g'' :: "'b \<Rightarrow> real" shows "O[F''](g'') = {f. \<exists>c>0. \<forall>\<^sub>F x in F''. norm (f x) \<le> c * norm (g'' x)}"