hcju/isabelleps huggingface dataset:
simps_1: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and accu :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations_accu G M accu [] = accu"
zmult_div_leq_mono_neg: fixes x :: "int"   and a :: "int"   and b :: "int"   and d :: "int" assumes "x \<le> 0"   and "a \<le> b"   and "0 < d" shows "x * b div d \<le> x * a div d"
I_r_closed: fixes a :: "'a"   and x :: "'a" assumes "a \<in> I"   and "x \<in> carrier R" shows "a \<otimes> x \<in> I"
eucl_rel_poly_unique_div: fixes x :: "'b poly"   and y :: "'b poly"   and q1 :: "'b poly"   and r1 :: "'b poly"   and q2 :: "'b poly"   and r2 :: "'b poly" assumes "eucl_rel_poly x y (q1, r1)"   and "eucl_rel_poly x y (q2, r2)" shows "q1 = q2"
hyps_4: shows "AbstForm w 0 (q_Imp x (q_Imp xw xevw)) allw"
rref_upt_condition4: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> (\<forall>j. i \<noteq> j \<longrightarrow> A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) = (0::'b))"
simps_2: fixes y :: "name"   and ys :: "name list"   and n :: "nat"   and x :: "name" shows "lookup (y # ys) n x = (if x = y then DBInd n else lookup ys (Suc n) x)"
Lim_cong_within: fixes a :: "'a"   and b :: "'a"   and x :: "'b"   and y :: "'b"   and S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "a = b"   and "x = y"   and "S = T"   and "\<And>x. \<lbrakk>x \<noteq> b; x \<in> T\<rbrakk> \<Longrightarrow> f x = g x" shows "(f \<longlongrightarrow> x) (at a within S) = (g \<longlongrightarrow> y) (at b within T)"
prob_proj_def: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs :: "'d set" shows "prob_proj PROB vs \<equiv> (\<lambda>a. action_proj a vs) ` PROB"
connected_clopen: fixes S :: "'b set" shows "connected S = (\<forall>T. openin (top_of_set S) T \<and> closedin (top_of_set S) T \<longrightarrow> T = {} \<or> T = S)"
cont_eff_ParT_2: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and i :: "nat"   and s :: "'state" assumes "properL cl"   and "n < length cl"   and "i < brn (cl ! n)" shows "eff (ParT cl) s (brnL cl n + i) = eff (cl ! n) s i"
plus_interval_tendsto: fixes X :: "nat \<Rightarrow> 'a interval"   and x :: "'a"   and Y :: "nat \<Rightarrow> 'a interval"   and y :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>i x"   and "Y \<longlonglongrightarrow>\<^sub>i y" shows "(\<lambda>i. X i + Y i) \<longlonglongrightarrow>\<^sub>i x + y"
repeat_mset_distrib2: fixes n :: "nat"   and A :: "'k multiset"   and B :: "'k multiset" shows "repeat_mset n (A + B) = repeat_mset n A + repeat_mset n B"
group_l_invI: assumes "\<And>x. x \<in> carrier \<G> \<Longrightarrow> \<exists>y\<in>carrier \<G>. y \<otimes> x = \<one>" shows "Group.group \<G>"
HFinite_minus_iff: fixes x :: "'a star" shows "(- x \<in> HFinite) = (x \<in> HFinite)"
ovr_overshadow_right: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "overrider F" shows "F x (F y z) = F x z"
funion_fempty_left: fixes B :: "'c fset" shows "{||} |\<union>| B = B"
IH_2: assumes "matches \<gamma> m p"   and "all_chains (no_call_to c) \<Gamma> rrs" shows "\<Gamma>(c \<mapsto> rrs),\<gamma>,p\<turnstile> [Rule m (Call c)] \<Rightarrow>\<^sub>z t"
some_in_eq: fixes A :: "'b set" shows "((SOME x. x \<in> A) \<in> A) = (A \<noteq> {})"
inject_5: fixes x51 :: "Core.term"   and x52 :: "Core.term"   and y51 :: "Core.term"   and y52 :: "Core.term" shows "(x51 $ x52 = y51 $ y52) = (x51 = y51 \<and> x52 = y52)"
tOp_assocTr41: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G"   and "d \<in> carrier G" shows "a \<cdot> b \<cdot> c \<cdot> d = a \<cdot> b \<cdot> (c \<cdot> d)"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "ty"   and a2 :: "val"   and a3 :: "val" assumes "P \<turnstile> a1 casts a2 to a3 " obtains T :: "ty" and v :: "val" where "a1 = T" and "a2 = v" and "a3 = v" and "\<forall>C. T \<noteq> Class C"   | C :: "char list" where "a1 = Class C__" and "a2 = Null" and "a3 = Null"   | Cs :: "char list list" and C :: "char list" and Cs' :: "char list list" and Ds :: "char list list" and a :: "nat" where "a1 = Class C" and "a2 = Ref (a, Cs)" and "a3 = Ref (a, Ds)" and "P \<turnstile> Path last Cs to C via Cs' " and "Ds = Cs @\<^sub>p Cs'"
irreducible\<^sub>d_factorization_exists_2: fixes p :: "'b poly" assumes "0 < degree p"   and "\<not> irreducible\<^sub>d p" shows "\<exists>fs. 1 < length fs \<and> (\<forall>f\<in>set fs. irreducible\<^sub>d f \<and> degree f < degree p) \<and> p = prod_list fs"
rtrancl_idemp_self_comp: fixes R :: "('b \<times> 'b) set" shows "R\<^sup>* O R\<^sup>* = R\<^sup>*"
strict_implies_order: fixes b :: "'a Word.word"   and a :: "'a Word.word" assumes "b <s a" shows "b \<le>s a"
card_UNIV_nat_def: shows "card_UNIV_class.card_UNIV = Phantom(nat) 0"
nodeToList_empty: fixes nd :: "('a, 'b) Node" shows "FingerTreeStruc.nodeToList nd \<noteq> []"
complex_eq_cvec_transp: shows "transp (\<approx>\<^sub>v)"
power_0: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" shows "power.power one times a 0 = one"
bij_betw_inv_into: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "bij_betw (inv_into A f) B A"
Auth_Fst: fixes x :: "rat"   and y :: "rat"   and H :: "msg set" assumes "Auth_Data x y \<in> parts H" shows "Pri_AgrK x \<in> parts H"
inject_3: fixes x41 :: "'a pfm"   and x42 :: "'a pfm"   and y41 :: "'a pfm"   and y42 :: "'a pfm" shows "((x41 \<^bold>\<and>\<^sub>! x42) = (y41 \<^bold>\<and>\<^sub>! y42)) = (x41 = y41 \<and> x42 = y42)"
transitions_cons_eq: fixes root :: "(att \<times> char list, att, nat) env"   and x :: "operation"   and xs :: "operation list"   and root'' :: "(att \<times> char list, att, nat) env" shows "root \<Midarrow>(x # xs)\<Rightarrow> root'' = (\<exists>root'. root \<midarrow>x\<rightarrow> root' \<and> root' \<Midarrow>xs\<Rightarrow> root'')"
orderI: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a = linordered_monoid.min a b" shows "a \<preceq> b"
subcocycle_lim_meas_Inv_2: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "subcocycle_lim u \<in> borel_measurable Invariants"
Says_K5: fixes servK :: "nat"   and A :: "agent"   and T3 :: "nat"   and evs :: "event list"   and authK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg" assumes "Crypt servK \<lbrace>Agent A, Number T3\<rbrace> \<in> parts (knows Spy evs)"   and "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "Says A B \<lbrace>servTicket, Crypt servK \<lbrace>Agent A, Number T3\<rbrace>\<rbrace> \<in> set evs"
v2: fixes u :: "'t" assumes "v2 \<prec>\<^sub>t u" shows "lookup b u = lookup c u"
padic_set_zero_res: fixes p :: "int"   and f :: "nat \<Rightarrow> int" assumes "prime p"   and "f \<in> padic_set p" shows "f 0 = 0"
trans: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "(a::'b) \<le> (b::'b)"   and "(b::'b) \<le> (c::'b)" shows "(a::'b) \<le> (c::'b)"
While_bury: fixes b :: "bexp"   and bc' :: "com"   and c :: "com"   and X :: "char list set" shows "(WHILE b DO bc' = bury c X) = (\<exists>c'. c = WHILE b DO c' \<and> bc' = bury c' (L (WHILE b DO c') X))"
neq_le_trans: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a \<noteq> b"   and "a \<preceq> b" shows "a \<prec> b"
nofail_antimono_fun: fixes f :: "'b \<Rightarrow> 'c nres"   and g :: "'b \<Rightarrow> 'c nres"   and x :: "'b" assumes "f \<le> g" shows "nofail (g x) \<longrightarrow> nofail (f x)"
binomial_eq_0_iff: fixes n :: "nat"   and k :: "nat" shows "(n choose k = 0) = (n < k)"
assms: shows "\<AA> \<in>\<^sub>\<circ> dg_GRPH \<alpha>\<lparr>Obj\<rparr>"
M_def: fixes x :: "int" shows "pll.M x = x mod p ^ local.ll"
safe_step_preserves_simple_ownership_distinct: fixes ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and ts' :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "(ts, m, \<S>) \<Rightarrow>\<^sub>d (ts', m', \<S>')"   and "safe_delayed (ts, m, \<S>)"   and "simple_ownership_distinct ts" shows "simple_ownership_distinct ts'"
CD_on_path_independent: fixes ds :: "'b set" shows "path_independent (CD_on ds)"
hta_prod'_def: fixes H1 :: "('a, 'c) hashedTa"   and H2 :: "('b, 'c) hashedTa" shows "hta_prod' H1 H2 \<equiv> let (Q, W, \<delta>d) = while pa_cond (pa_step H1 H2) (pa_initial H1 H2) in init_hta (hhh_cart.cart (hta_Qi H1) (hta_Qi H2)) \<delta>d"
A: fixes V :: "'a set" shows "U \<subseteq> V \<longrightarrow> sinks_aux I D U (ipurge_tr_aux_foldr I D P V xs) = U"
wf_trm_subtermeq: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and t :: "('a, 'c) Term.term"   and s :: "('a, 'c) Term.term" assumes "intruder_model arity Ana"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t"   and "s \<sqsubseteq> t" shows "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity s"
matrix_vector_mul_assoc: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and x :: "('a, 'd) Finite_Cartesian_Product.vec" shows "A *v (B *v x) = A ** B *v x"
coprime_quot_of_fract: fixes x :: "'b fract" shows "coprime (numerator x) (denominator x)"
ucast_1: shows "UCAST('c \<rightarrow> 'b) 1 = 1"
diagonal_to_Smith_i_PQ_eq: fixes P' :: "(('a, 'rows) vec, 'rows) vec"   and B' :: "(('a, 'cols) vec, 'rows) vec"   and Q' :: "(('a, 'cols) vec, 'cols) vec"   and xs :: "nat list"   and i :: "nat" assumes "(P', B', Q') = diagonal_to_Smith_i_PQ xs i bezout (P, B, Q)"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"   and "isDiagonal B"   and "i \<notin> set xs"   and "i < min (nrows A) (ncols A)" shows "B' = diagonal_to_Smith_i xs B i bezout"
intensional_simps_31: fixes y :: "'b \<Rightarrow> bool" shows "\<exists>x. y \<equiv> y"
2: fixes x :: "'a"   and y :: "'a" assumes "fst (rep_nat_term x) = fst (rep_nat_term y)"   and "snd (rep_nat_term x) < snd (rep_nat_term y)" shows "x < y"
s_Cons: fixes c' :: "('p, 's, 'm) configuration"   and xs :: "('p, 's, 'm) event list"   and c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and i :: "nat" assumes "\<exists>c''. trace c' xs c''"   and "c \<turnstile> ev \<mapsto> c'" shows "s c (ev # xs) (Suc i) = s c' xs i"
isPartition_def: fixes xs :: "'c list"   and lo :: "nat"   and hi :: "nat"   and p :: "nat" shows "isPartition xs lo hi p \<equiv> isPartition_wrt (\<le>) xs lo hi p"
\<sigma>_eq: shows "lst \<sigma>3' = fst \<sigma>3''"
both_above_arg2: fixes t' :: "'a ttree"   and t :: "'a ttree" shows "t' \<sqsubseteq> t \<otimes>\<otimes> t'"
distinct_rotate: fixes n :: "nat"   and xs :: "'b list" shows "distinct (rotate n xs) = distinct xs"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "Ik.wtE \<xi> \<equiv> \<forall>x. Ik_intT (tpOfV x) (\<xi> x)"
moebius_mu_smallo: shows "sum_upto moebius_mu \<in> o(\<lambda>x. x)"
Valid_def: fixes A :: "'a \<Rightarrow> bool" shows "\<turnstile> A \<equiv> \<forall>w. A w"
leg_simps_3: shows "C.arr \<pi>.dom.leg1"
subgroupE_1: fixes H :: "'a set" assumes "subgroup H (add_monoid R)" shows "H \<subseteq> carrier R"
distinct_odd: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "\<And>i. p i \<noteq> 0 \<Longrightarrow> 1 \<le> i \<and> i \<le> n"   and "\<And>i. p i \<le> 1" shows "distinct_of_odd (odd_of_distinct p) = p"
conf_def: fixes P :: "'b prog"   and h :: "'heap"   and v :: "'addr val"   and T :: "ty" shows "h.conf P h v T \<equiv> \<exists>T'. h.typeof_h h v = \<lfloor>T'\<rfloor> \<and> P \<turnstile> T' \<le> T"
exhaust: fixes y :: "comp" obtains "y = Less"   | "y = Equiv"   | "y = Greater"
higher_eq_iff: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "(punit.higher p v = punit.higher q v) = (\<forall>u. v \<prec> u \<longrightarrow> lookup p u = lookup q u)"
map_snd_zip_take: fixes xs :: "'c list"   and ys :: "'b list" shows "map snd (zip xs ys) = take (min (length xs) (length ys)) ys"
one_le_mult_iff: fixes m :: "nat"   and n :: "nat" shows "(Suc 0 \<le> m * n) = (Suc 0 \<le> m \<and> Suc 0 \<le> n)"
fds_logderiv_completely_multiplicative: fixes f :: "'b fds" assumes "completely_multiplicative_function (fds_nth f)"   and "fds_nth f 1 \<noteq> (0::'b)" shows "fds_deriv f / f = - fds (\<lambda>n. fds_nth f n * mangoldt n)"
language_def: shows "NR.language \<equiv> {xs. NR.nextl (nfa.init (Reverse_nfa M)) xs \<inter> nfa.final (Reverse_nfa M) \<noteq> {}}"
product_over_sum_left: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "CEM"   and "O x y \<and> O x z" shows "P (x \<otimes> (y \<oplus> z)) (x \<otimes> y \<oplus> x \<otimes> z)"
less_vec_def: fixes v :: "'b vec"   and w :: "'b vec" shows "(v <\<^sub>v w) = (dim_vec v = dim_vec w \<and> (\<forall>i<dim_vec w. v $ i < w $ i))"
congruent_triangleD_9: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "dist b1 c1 = dist c2 b2"
th: fixes w :: "'a" shows "poly q (w - z) = poly p w"
vote_set_upd: fixes v_hist :: "'c \<Rightarrow> 'e \<Rightarrow> 'd option"   and r :: "'c"   and v_f :: "'e \<Rightarrow> 'd option"   and p :: "'e" shows "vote_set (v_hist(r := v_f)) {p} = (if p \<in> dom v_f then insert (r, the (v_f p)) else id) (if v_hist r p = None then vote_set v_hist {p} else vote_set v_hist {p} - {(r, the (v_hist r p))})"
assms: fixes x :: "Preliminaries.var" assumes "x \<in> varsL l" shows "\<pi>1 x = \<pi>2 x"
sconst_cycle: fixes x :: "'d" shows "sconst x = cycle [x]"
irsSp_Cons_cdep: fixes n'' :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' irs-ns\<rightarrow>\<^sub>d* n'"   and "n s\<longrightarrow>\<^bsub>cd\<^esub> n''" shows "n irs-n # ns\<rightarrow>\<^sub>d* n'"
nrename_swap: fixes x :: "name"   and M :: "trm"   and y :: "name" assumes "x \<sharp> M" shows "[(x, y)] \<bullet> M = M[y\<turnstile>n>x]"
\<phi>_def: fixes y :: "'d"   and h :: "'c" shows "\<phi> y h = G h \<cdot>\<^sub>D \<eta>.map y"
typeof_conf: fixes h :: "heap"   and v :: "val"   and T :: "ty"   and P :: "'b prog" assumes "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> v :\<le> T"
compact_imp_bounded: fixes U :: "'c set" assumes "compact U" shows "bounded U"
aneq: assumes "a \<noteq> 0" shows "\<exists>y'>- r. \<forall>x\<in>{- r<..y'}. a * x\<^sup>2 + b * x + c < 0"
sorted_is_sorted_list_of_set_1: fixes A :: "'b set"   and xs :: "'b list" assumes "is_sorted_list_of_set A xs" shows "sorted xs"
NS_O_S_1: fixes S :: "('b \<times> 'b) set"   and NS :: "('b \<times> 'b) set" assumes "order_pair S NS" shows "NS O S = S"
D: shows "(\<tau> \<Turnstile> PRE (\<lambda>_. self \<tau>) (f_lam a1 \<tau>)) = (\<tau> \<Turnstile> PRE self a1)"
learn_bc_V_BCLIM: shows "learn_bc \<psi>\<^sup>\<tau> V\<^bsub>BC-LIM\<^esub> Some"
ww'_2: shows "- 1 \<le> w' \<and> w' * exp w' = x"
g: fixes y :: "'a" assumes "norm y < \<delta> / 2" shows "norm (g y) < \<delta> \<and> f (g y) = y"
j_le3: shows "pick UNIV j < card {i. i < dim_col A \<and> i \<in> J}"
not_fin_sat_extended_UNIV: fixes S :: "'b formula set" assumes "\<not> fin_sat S" shows "extended S = UNIV"
aform_err_def: fixes e :: "nat \<Rightarrow> real"   and Xe :: "(real \<times> real pdevs) \<times> real" shows "aform_err e Xe = {aform_val e (fst Xe) - snd Xe..aform_val e (fst Xe) + snd Xe}"
backward_subsumption_Q: fixes D :: "'a literal multiset"   and N :: "'a literal multiset set"   and C :: "'a literal multiset"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" assumes "D \<in> N"   and "strictly_subsumes D C" shows "(N, P, Q \<union> {C}) \<leadsto> (N, P, Q)"
colossless_gpv_2: shows "\<I> \<turnstile>g gpv \<surd>"
diff_eq_0_iff_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "(a - b = 0) = (a < top \<and> a \<le> b)"
\<K>_Suc: fixes j0 :: "nat"   and j :: "nat" shows "\<K> j0 (Suc j) = USigma (\<K> j0 j) (\<lambda>K. {Max (insert j0 K)<..})"
repeatable_both_both_nxt: fixes t' :: "'b ttree"   and t :: "'b ttree"   and t'' :: "'b ttree" assumes "t' \<otimes>\<otimes> t = t'" shows "t' \<otimes>\<otimes> t'' \<otimes>\<otimes> t = t' \<otimes>\<otimes> t''"
inv_op_closed2: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and h :: "'c" assumes "H \<lhd> G"   and "x \<in> carrier G"   and "h \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x \<in> H"
assms_3: shows "local.span A \<inter> local.span B = {0::'b}"
Cons_eq_appendI: fixes x :: "'c"   and xs1 :: "'c list"   and ys :: "'c list"   and xs :: "'c list"   and zs :: "'c list" assumes "x # xs1 = ys"   and "xs = xs1 @ zs" shows "x # xs = ys @ zs"
the_inv_into_f_f: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b" assumes "inj_on f A"   and "x \<in> A" shows "the_inv_into A f (f x) = x"
Diagonalize_preserves_Ide: fixes a :: "'a local.term" assumes "Ide a" shows "Ide \<^bold>\<lfloor>a\<^bold>\<rfloor>"
distinct_40: fixes x71 :: "key"   and x72 :: "msg"   and x5 :: "msg" shows "Crypt x71 x72 \<noteq> Hash x5"
diagonal_step_PQ_def: fixes A :: "(('a, 'c) vec, 'b) vec"   and i :: "nat"   and k :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_step_PQ A i k bezout = (let i_row = from_nat i; k_row = from_nat k; i_col = from_nat i; k_col = from_nat k; (p, q, u, v, d) = bezout (A $ i_row $ from_nat i) (A $ k_row $ from_nat k); P = row_add (interchange_rows (row_add (mat (1::'a)) k_row i_row p) i_row k_row) k_row i_row (- v) in Let (mult_column (column_add (column_add (mat (1::'a)) i_col k_col q) k_col i_col u) k_col (- (1::'a))) (Pair P))"
B: fixes X :: "'b set" shows "ipurge_ref_aux_less I D U (x # xs) X = ipurge_ref_aux_less I D U (ipurge_tr I D (D x) xs) (ipurge_ref I D (D x) xs X)"
axioms_2: fixes G :: "('c, 'd) monoid_scheme" assumes "comm_group G" shows "Group.group G"
conv_add: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<smile> = x\<^sup>\<smile> + y\<^sup>\<smile>"
flow0_def: fixes x0 :: "'a"   and t :: "real" shows "Y.flow0 x0 t = ll_on_open.flow UNIV (\<lambda>_. g) Y 0 x0 t"
assms: shows "\<forall>u\<in>V - {s, t}. excess f u = (0::'a)"
vv: shows "v \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom x])"
Union_Un_distrib: fixes A :: "'a set set"   and B :: "'a set set" shows "\<mu> (A \<union> B) = \<mu> A \<union> \<mu> B"
has_bochner_integral_discrete_difference: fixes X :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "has_bochner_integral M f x = has_bochner_integral M g x"
list_all2_check_before: fixes f :: "'c \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "(nat \<times> 'a \<times> 'b) list"   and ys :: "nat list"   and zs :: "'c list"   and I :: "\<I>"   and nt :: "nat"   and x :: "nat \<times> 'a \<times> 'b" assumes "list_all2 (\<lambda>x y. triple_eq_pair x y f g) xs (zip ys zs)"   and "\<And>y. y \<in> set ys \<Longrightarrow> \<not> enat y + right I < enat nt"   and "x \<in> set xs" shows "\<not> check_before I nt x"
lists_mono: fixes A :: "'a set"   and B :: "'a set" assumes "order.greater_eq B A" shows "order.greater_eq (lists B) (lists A)"
terminals_append: fixes a :: "(symbol \<times> character list) list"   and b :: "(symbol \<times> character list) list" shows "terminals (a @ b) = terminals a @ terminals b"
wf_trm_arity: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and f :: "'a"   and T :: "('a, 'c) Term.term list" assumes "intruder_model arity Ana"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity (Fun f T)" shows "length T = arity f"
univ_commute': fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "equiv A r"   and "f respects r"   and "x \<in> A" shows "univ f (r `` {x}) = f x"
src_src: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "src\<^sub>D (src\<^sub>D \<mu>) = src\<^sub>D \<mu>"
fset_to_im: fixes f :: "'d \<Rightarrow> 'c"   and X :: "'d set" shows "{f x |x. x \<in> X} = f ` X"
VIS_eq: shows "dom PRED = N \<union> {u. \<exists>i\<le>d. u \<in> Vd i}"
scale_sum_right: fixes a :: "complex"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "a *\<^sub>C sum f A = (\<Sum>x\<in>A. a *\<^sub>C f x)"
fps_lr_inverse_gp_ring1'_1: shows "fps_left_inverse (Abs_fps (\<lambda>n. 1::'b)) (1::'b) = 1 - fps_X"
Subst_all_preserves_finite: fixes ES :: "('b list set \<times> 'b trm set) set"   and Y :: "'b list set"   and yrhs :: "'b trm set" assumes "finite ES" shows "finite (Subst_all ES Y yrhs)"
a: shows "<e1,Es1> \<mapsto> <e2,Es2>"
residue_1_zero: fixes n :: "int" shows "residue 1 n = 0"
spoly_repE: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and g1 :: "'t \<Rightarrow>\<^sub>0 'c"   and g2 :: "'t \<Rightarrow>\<^sub>0 'c" assumes "spoly_rep d m G g1 g2" obtains q :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c" where "spoly g1 g2 = (\<Sum>g\<in>G. q g \<odot> g)" and "\<And>g. q g \<in> punit.dgrad_p_set d m" and "\<And>g. q g \<odot> g \<noteq> 0 \<Longrightarrow> lt (q g \<odot> g) \<prec>\<^sub>t term_of_pair (lcs (lp g1) (lp g2), component_of_term (lt g2))"
horner_bounds': fixes x :: "float"   and f :: "nat \<Rightarrow> nat"   and G :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and F :: "nat \<Rightarrow> nat"   and s :: "nat"   and lb :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"   and prec :: "nat"   and ub :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float \<Rightarrow> float"   and n :: "nat"   and j' :: "nat" assumes "0 \<le> real_of_float x"   and "\<And>n. f (Suc n) = G ((F ^^ n) s) (f n)"   and "\<And>i k x. lb 0 i k x = 0"   and "\<And>n i k x. lb (Suc n) i k x = float_plus_down prec (lapprox_rat prec 1 (int k)) (- float_round_up prec (x * ub n (F i) (G i k) x))"   and "\<And>i k x. ub 0 i k x = 0"   and "\<And>n i k x. ub (Suc n) i k x = float_plus_up prec (rapprox_rat prec 1 (int k)) (- float_round_down prec (x * lb n (F i) (G i k) x))" shows "real_of_float (lb n ((F ^^ j') s) (f j') x) \<le> horner F G n ((F ^^ j') s) (f j') (real_of_float x) \<and> horner F G n ((F ^^ j') s) (f j') (real_of_float x) \<le> real_of_float (ub n ((F ^^ j') s) (f j') x)"
bounded_linear: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_clinear f" shows "bounded_linear f"
Deduction2: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" shows "(F \<^bold>\<rightarrow> \<bottom> \<triangleright> AX1 \<union> \<Gamma> \<turnstile>\<^sub>H \<bottom>) = (AX1 \<union> \<Gamma> \<turnstile>\<^sub>H F)"
permute_atom_def: fixes p :: "'b perm"   and a :: "'b" shows "permute_atom p a = Rep_perm p a"
le_less: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and y :: "'t \<Rightarrow>\<^sub>0 'c" shows "(x \<preceq>\<^sub>p y) = (x \<prec>\<^sub>p y \<or> x = y)"