hcju/isabelleps huggingface dataset:
unfoldr_rbt_entries_generator: fixes t :: "('c, 'd) rbt" shows "list.unfoldr rbt_entries_generator (rbt_init t) = RBT_Impl.entries t"
card_transfer: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_set A) (=) card card"
test_mult: fixes x :: "'a"   and y :: "'a" shows "t (!x \<cdot> !y) = !x \<cdot> !y"
remove_nth_id: fixes xs :: "'c list"   and n :: "nat" assumes "length xs \<le> n" shows "remove_nth n xs = xs"
G_def: fixes p :: "'d \<Rightarrow> 'c"   and I :: "'d set" shows "prod' p I \<equiv> if finite {x \<in> I. p x \<noteq> (1::'c)} then prod p {x \<in> I. p x \<noteq> (1::'c)} else (1::'c)"
refine: fixes Xi :: "'b list"   and X :: "'c set"   and bi :: "real list"   and b :: "'c"   and yi :: "real"   and y :: "real" assumes "(Xi, X) \<in> appr_rel"   and "(bi, b) \<in> lv_rel"   and "(yi, y) \<in> rnv_rel" shows "(nres_of (project_set_appr Xi bi yi), op_project_set X b y) \<in> \<langle>appr_rel\<rangle>nres_rel"
least_order: fixes le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "\<forall>x y z. le x y \<and> le y z \<longrightarrow> le x z"   and "\<forall>x y. le x y \<or> le y x"   and "finite A"   and "A \<noteq> {}" shows "\<exists>x. x \<in> A \<and> (\<forall>y. y \<in> A \<longrightarrow> le x y)"
l_get_parent_axioms: shows "l_get_parent DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs"
preserves_ide: fixes a :: "'a \<times> 'a" assumes "B.VV.ide a" shows "B.ide (fst a \<star>\<^sub>B snd a)"
take_butlast: fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "take n (butlast xs) = take n xs"
tranclpD: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "R\<^sup>+\<^sup>+ x y" shows "\<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
block_complement_size: fixes b :: "'a set" assumes "b \<subseteq> \<V>" shows "card (b\<^sup>c) = card \<V> - card b"
col_ket_vec: fixes M :: "complex mat" assumes "dim_col M = 1" shows "|col M 0\<rangle> = M"
is_red_monomialI: fixes c :: "'b"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and v :: "'x \<Rightarrow>\<^sub>0 nat" assumes "c \<noteq> (0::'b)"   and "f \<in> F"   and "f \<noteq> 0"   and "local.punit.adds_term (lpp f) v" shows "punit.is_red F (monomial c v)"
empty_not_add_mset: fixes a :: "'a"   and A :: "'a multiset" shows "{#} \<noteq> a, A"
rel_intros_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x21 :: "'c"   and y21 :: "'d"   and x22 :: "'c list"   and y22 :: "'d list" assumes "R x21 y21"   and "list_all2 R x22 y22" shows "list_all2 R (x21 # x22) (y21 # y22)"
r_i_unique: fixes a :: "'a"   and x :: "'a" assumes "a \<in> carrier G"   and "x \<in> carrier G"   and "a \<cdot> x = \<one>" shows "x = \<rho> a"
qGoodBinp_iff_goodBinp_asBinp: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" shows "goodBinp (asBinp qbinp) = qGoodBinp qbinp"
Clean_Pf_encPf: fixes p :: "'trm"   and \<phi> :: "'fmla" shows "let PPf = \<lambda>t1 t2. psubst Pf [(t1, yy), (t2, xx)] in p \<in> num \<and> \<phi> \<in> fmla \<and> Fvars \<phi> = {} \<longrightarrow> p \<notin> encPf ` proof \<longrightarrow> bprv (neg (PPf p \<langle>\<phi>\<rangle>))"
subseq_append': fixes zs :: "'b list"   and xs :: "'b list"   and ys :: "'b list" shows "subseq (zs @ xs) (zs @ ys) = subseq xs ys"
adjoint_pair_antipar_4: fixes f :: "'b"   and g :: "'b" assumes "D.adjoint_pair f g" shows "src\<^sub>D g = trg\<^sub>D f"
eq_idx_for_lvar_def: fixes t :: "(nat \<times> linear_poly) list"   and x :: "nat" shows "eq_idx_for_lvar t x \<equiv> eq_idx_for_lvar_aux t x 0"
paths_nxt_eq: fixes xs :: "'a list"   and t :: "'a ttree"   and x :: "'a" shows "(xs \<in> paths (nxt t x)) = (xs = [] \<or> x # xs \<in> paths t)"
sol_strict: shows "A\<^sub>2 *\<^sub>v x <\<^sub>v b\<^sub>2"
Call_SoundAux: fixes \<Sigma> :: "((Class \<times> Method) \<times> nat) list"   and f :: "Class \<times> Method"   and n :: "nat"   and Anno2 :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and l :: "int"   and code :: "(int \<times> Instr) list"   and args :: "Var list"   and code1 :: "(int \<times> Instr) list"   and l1 :: "int"   and C :: "Class"   and m :: "Method"   and T :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and MI :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and k :: "nat" assumes "\<Sigma>\<down>f = Some n"   and "MST\<down>(fst f, snd f) = Some (mkSPEC (Cachera n) Anno2)"   and "\<exists>par body l0. mbody_is (fst f) (snd f) (par, body, l0)"   and "(l, code, CallPrim f args, code1, l1) \<in> compilePrim"   and "MST\<down>(C, m) = Some (T, MI, Anno)"   and "Segment C m l l1 code1"   and "derivAssum G C m l1 (Cachera k)" shows "deriv G C m l (Cachera (n + k))"
antisym_conv3: fixes y :: "'a"   and x :: "'a" assumes "\<not> y \<prec> x" shows "(\<not> x \<prec> y) = (x = y)"
dom_cod: fixes f :: "'a \<times> 'a" assumes "VoV.arr f" shows "VoV.dom (VoV.cod f) = VoV.cod f"
zoom_def: fixes l :: "'g \<Longrightarrow> 'e"   and m :: "('g, 'f) state" shows "zoom l m = State (\<lambda>b. case run_state m (get\<^bsub>l\<^esub> b) of (c, a) \<Rightarrow> (c, put\<^bsub>l\<^esub> b a))"
N_2: shows "{x \<in> space M. \<not> P x} \<subseteq> N"
sum_tail_nat: fixes b :: "nat"   and a :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "0 < b"   and "a \<le> b" shows "sum f {a..b} = f b + sum f {a..b - 1}"
fMin_Min: fixes l :: "'a list" shows "fMin (fset_of_list l) = Min (set l)"
inc_le: fixes i :: "'b word"   and m :: "'b word" assumes "i < m" shows "i + 1 \<le> m"
SA_Top: fixes \<Gamma> :: "binding list"   and S :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f S" shows "\<Gamma> \<turnstile> S <: Top"
ub_ln_lb_ln_bounds: fixes x :: "float"   and prec :: "nat" assumes "0 < x" shows "real_of_float (the (lb_ln prec x)) \<le> ln (real_of_float x) \<and> ln (real_of_float x) \<le> real_of_float (the (ub_ln prec x))"
ltake_is_lprefix: fixes n :: "enat"   and xs :: "'b llist" shows "lprefix (ltake n xs) xs"
mk_iso_def: fixes fv :: "'a \<Rightarrow> 'e"   and fa :: "'b \<Rightarrow> 'f" shows "mk_iso fv fa \<equiv> \<lparr>iso_verts = fv, iso_arcs = fa, iso_head = fv \<circ> head G \<circ> the_inv_into (arcs G) fa, iso_tail = fv \<circ> tail G \<circ> the_inv_into (arcs G) fa\<rparr>"
corec_gpv_parametric': fixes S :: "'g \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'h \<Rightarrow> 'l \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'm \<Rightarrow> bool"   and R :: "'j \<Rightarrow> 'n \<Rightarrow> bool" shows "((S ===> rel_spmf (rel_generat A C (R ===> rel_sum (rel_gpv'' A C R) S))) ===> S ===> rel_gpv'' A C R) corec_gpv corec_gpv"
sel_2: fixes x1 :: "'b"   and x2 :: "'b hp list" shows "hps (Hp x1 x2) = x2"
insertE: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" assumes "a \<in> Set.insert b A" obtains "a = b"   | "a \<in> A"
Max_wrt_among_total_preorder: fixes A :: "'alt set" shows "Max_wrt_among (R j) A = {x \<in> alts \<inter> A. \<forall>y\<in>alts \<inter> A. R j y x}"
a: shows "a \<sim> somegcd G a (c \<otimes> a)"
PowD: fixes A :: "'b set"   and B :: "'b set" assumes "A \<in> Pow B" shows "A \<subseteq> B"
assms_2: fixes i :: "nat"   and j :: "nat" assumes "i \<le> Suc n"   and "j \<le> Suc n"   and "i \<noteq> j" shows "I i <+> I j = carrier R"
iterates_raw_def: fixes f :: "'c \<Rightarrow> 'c"   and s :: "'c" shows "iterates_raw f s = Yield s (f s)"
abs: fixes x :: "'a word" shows "\<bar>sint x\<bar> < 2 ^ (size a - 1) + 1"
ex_\<kappa>: fixes i :: "nat"   and l :: "nat" assumes "i < length fs"   and "l \<le> i" shows "\<exists>\<kappa>. M.sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<l]) = M.sumlist (map (\<lambda>j. \<kappa> j \<cdot>\<^sub>v fs ! j) [0..<l])"
rel_symp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "symp R" shows "symp (rel_bset R)"
opnet_deliver2: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and t :: "'d net_state"   and i :: "nat"   and d :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and t' :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and s :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, t), i:deliver(d), \<sigma>', t') \<in> T" shows "((\<sigma>, SubnetS s t), i:deliver(d), \<sigma>', SubnetS s t') \<in> opnet_sos S T"
inf_commute: fixes inf :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and x :: "'x"   and y :: "'x" assumes "class.semilattice_inf inf less_eq less" shows "inf x y = inf y x"
field_hom_0'_def: fixes hom :: "'a \<Rightarrow> 'b" shows "field_hom_0' hom \<equiv> field_hom hom"
polyneg0: fixes p :: "poly"   and n :: "nat" assumes "isnpolyh p n" shows "(~\<^sub>p p = 0\<^sub>p) = (p = 0\<^sub>p)"
cut_bound'_cbnd: fixes n :: "nat"   and F :: "'b formula" shows "cut_bound' n F \<le> cbnd n (depth F)"
counit_in_vhom: shows "\<guillemotleft>\<tau> : h \<star> k \<Rightarrow> src k\<guillemotright>"
proj_active: fixes i :: "nat"   and t :: "cnf llist"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "enat i < llength t"   and "active c (lnth t i)" shows "dynamic_component.proj tCMP active c (ltake (enat (Suc i)) t) = dynamic_component.proj tCMP active c (ltake (enat i) t) @\<^sub>l (tCMP c (lnth t i) #\<^sub>l []\<^sub>l)"
eval_runiq_rel: fixes x :: "'c"   and R :: "('c \<times> 'd) set" assumes "x \<in> Domain R"   and "runiq R" shows "(x, R ,, x) \<in> R"
that_7: fixes x :: "'a" assumes "x \<in> U" shows "f x = g x"
nb: shows "\<forall>v. (\<Pi>\<^sub>G) v \<noteq> None \<longrightarrow> ListMem v (\<Pi>\<^sub>\<V>)"
distinct_25: fixes x41 :: "'a rexp"   and x42 :: "'a rexp"   and x51 :: "'a rexp"   and x52 :: "'a rexp" shows "Plus x41 x42 \<noteq> Times x51 x52"
pred_of_rel_def: fixes s :: "('g \<times> 'h) set" shows "pred_of_rel s \<equiv> \<lambda>a b. (a, b) \<in> s"
c1: fixes x :: "'a set"   and y :: "'a set" assumes "closed_csubspace x"   and "closed_csubspace y"   and "x \<subseteq> y" shows "x +\<^sub>M orthogonal_complement x \<inter> y \<subseteq> y"
indic_ident: fixes f :: "'a \<Rightarrow> real"   and s :: "'a set" shows "(\<lambda>x. f x * indicat_real s x) = (\<lambda>x. if x \<in> s then f x else 0)"
special_coset_singleton: fixes w :: "'w" assumes "w \<in> W" shows "{w} \<in> \<P>"
absolutely_integrable_on_combine: fixes a :: "real"   and c :: "real"   and f :: "real \<Rightarrow> 'b"   and b :: "real" assumes "f absolutely_integrable_on {a..c}"   and "f absolutely_integrable_on {c..b}"   and "a \<le> c"   and "c \<le> b" shows "f absolutely_integrable_on {a..b}"
sq_mtx_plus_eq: fixes A :: "'b sq_mtx"   and B :: "'b sq_mtx" shows "A + B = to_mtx (\<chi>i j. A $$ i $ j + B $$ i $ j)"
sup1: fixes A :: "'e"   and k :: "'e"   and a :: "'e"   and b :: "'e" assumes "A \<equiv> sup k a" shows "sup A b \<equiv> sup k (sup a b)"
less_enat_def: fixes m :: "enat"   and n :: "enat" shows "(m < n) = (case m of enat m1 \<Rightarrow> case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True | \<infinity> \<Rightarrow> False)"
3: shows "(w \<sqinter> - prim_EP w v e) * prim_P w v e\<^sup>T = bot"
cbox_Pair_iff: fixes x :: "'a"   and y :: "'b"   and a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b" shows "((x, y) \<in> cbox (a, c) (b, d)) = (x \<in> cbox a b \<and> y \<in> cbox c d)"
semantics_tm_id_map: fixes ts :: "'a tm list" shows "map \<lparr>\<^bold>#, \<^bold>\<dagger>\<rparr> ts = ts"
dim_span_eq_card_independent: fixes B :: "'d set" assumes "cindependent B" shows "cdim (cspan B) = card B"
elims: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "('a, 'c \<Rightarrow> 'b) monoid_ap"   and xa :: "('a, 'c) monoid_ap"   and y :: "('a, 'b) monoid_ap" assumes "plus.ap_monoid_add plus x xa = y" obtains a1 :: "'a" and f :: "'c \<Rightarrow> 'b" and a2 :: "'a" and xb :: "'c" where "x = Monoid_ap a1 f" and "xa = Monoid_ap a2 xb" and "y = Monoid_ap (plus a1 a2) (f xb)"
accept_def: fixes w :: "nat \<Rightarrow> 'L" shows "G'.accept w \<equiv> \<exists>r. G'.is_acc_run r \<and> (\<forall>i. G'.L (r i) (w i))"
in_homE: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "D.C.arr f" and "D.C.dom f = a" and "D.C.cod f = b"
1: shows "\<forall>y\<in>set (Abs_State.dom S2) \<inter> set (Abs_State.dom S1). m_ivl (fun S2 y) \<le> m_ivl (fun S1 y)"
IH: fixes \<alpha>s :: "V list" shows "\<omega>_sum \<alpha>s = \<omega>_sum (take k \<alpha>s) + \<omega>_sum (drop k \<alpha>s)"
mojmir_accept_to_Acc: fixes \<delta> :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and \<delta>\<^sub>M :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0 :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and M_fin :: "('b ltl \<Rightarrow> nat option) \<Rightarrow> (('b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) \<times> 'b set \<times> 'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set"   and \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<psi> :: "'b ltl"   and \<pi> :: "'b ltl \<Rightarrow> nat option"   and \<phi> :: "'b ltl" assumes "ltl_to_rabin_base \<delta> \<delta>\<^sub>M q\<^sub>0 q\<^sub>0\<^sub>M M_fin \<Sigma> w"   and "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "mojmir_def.accept \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"   and "\<pi> (G \<psi>) = mojmir_def.smallest_accepting_rank \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}" shows "accepting_pair\<^sub>R (ltl_to_rabin_base_def.delta \<delta> \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma>) (ltl_to_rabin_base_def.initial q\<^sub>0 q\<^sub>0\<^sub>M \<phi>) (ltl_to_rabin_base_def.Acc_fin \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma> \<pi> (G \<psi>), ltl_to_rabin_base_def.Acc_inf \<delta>\<^sub>M q\<^sub>0\<^sub>M \<pi> (G \<psi>)) w"
mset_pos_neg_partition: fixes M :: "'b zmultiset" shows "M = zmset_of (mset_pos M) - zmset_of (mset_neg M)"
card_of_ordIso_subst: fixes A :: "'a set"   and B :: "'a set" assumes "A = B" shows "|A| =o |B|"
sum_simp_setup_7: shows "\<forall>v\<in>V. sum (augment f') (incoming v) = (\<Sum>u\<in>V. augment f' (u, v))"
emptyLeftSelf: fixes t :: "'a event list"   and A :: "'a event set" shows "(\<forall>t1. t1 \<in> set t \<longrightarrow> t1 \<notin> A) \<longrightarrow> t setinterleaves (([], t), A)"
simps_2: fixes s :: "nat"   and lm :: "nat list"   and p :: "abc_inst list"   and n :: "nat" shows "abc_steps_l (s, lm) p (Suc n) = abc_steps_l (abc_step_l (s, lm) (abc_fetch s p)) p n"
no_nonpos_Int_in_ball_complex: fixes z :: "complex"   and t :: "complex" assumes "0 < Re z"   and "t \<in> ball z (Re z / 2)" shows "t \<notin> \<int>\<^sub>\<le>\<^sub>0"
IH: shows "\<forall>ys. length ys < length p \<longrightarrow> set ys \<subseteq> I \<longrightarrow> epath E u ys v \<longrightarrow> (\<exists>dp. depath E u dp v \<and> set dp \<subseteq> I)"
simps_2: fixes a :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Minus a) vs = - interpret_floatarith a vs"
trivial_def: fixes g :: "'g"   and v :: "'val" shows "trivial g v \<equiv> \<exists>v'\<in>allVars g. isTrivialPhi g v v'"
l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf set_tag_name set_tag_name_locs"
remove_nth_length: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "length (remove_nth n xs) = length xs - 1"
1: fixes i :: "nat" assumes "i \<le> n" shows "g i \<in> carrier P"
dbl_simps_4: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus (uminus one) = uminus (numeral.numeral one plus (num.Bit0 num.One))"
intro: fixes \<BB> :: "V"   and \<CC> :: "V" assumes "\<And>a. a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr> \<Longrightarrow> a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<And>a b f. f : a \<mapsto>\<^bsub>\<BB>\<^esub> b \<Longrightarrow> f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "subdigraph_axioms \<BB> \<CC>"
T'_def: fixes I :: "'s pmf" shows "T' I = measure_pmf I \<bind> (\<lambda>s. distr (T s) (stream_space (count_space UNIV)) ((##) s))"
amin_aminTr: fixes z :: "ant"   and z' :: "ant"   and w :: "ant" assumes "z \<le> z'" shows "amin z w \<le> amin z' w"
subst_cong: fixes t :: "'a"   and \<Gamma>\<^sub>1 :: "(name, 'a) fmap"   and \<Gamma>\<^sub>2 :: "(name, 'a) fmap" assumes "\<And>x. x |\<in>| frees t \<Longrightarrow> fmlookup \<Gamma>\<^sub>1 x = fmlookup \<Gamma>\<^sub>2 x" shows "subst t \<Gamma>\<^sub>1 = subst t \<Gamma>\<^sub>2"
notin: shows "fst x \<notin> set (map fst op_list)"
hoare_Sup: fixes P :: "'b Assertion set"   and x :: "'b"   and q :: "'b Assertion" shows "hoare (\<Squnion>P) x q = (\<forall>p\<in>P. hoare p x q)"
blinfun_apply_blinfun_compose: fixes a :: "'f \<Rightarrow>\<^sub>L 'd"   and b :: "'e \<Rightarrow>\<^sub>L 'f"   and c :: "'e" shows "blinfun_apply (a o\<^sub>L b) c = blinfun_apply a (blinfun_apply b c)"
tail_del_vert: fixes u :: "'a" shows "tail (del_vert u) = tail G"
R: shows "U - (V - {u}) = U - V \<union> {u}"
Typed_in_\<R>: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma> {c} \<Gamma>'"   and "mds_consistent mds \<Gamma>"   and "\<forall>x. to_total \<Gamma> x = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x" shows "\<langle>c, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>'\<^esub> \<langle>c, mds, mem\<^sub>2\<rangle>"
in_arcsD2: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs (list_digraph G)" shows "v \<in> pverts (list_digraph G)"
simps_4: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm" shows "push_forall (And \<phi> \<psi>) = and (push_forall \<phi>) (push_forall \<psi>)"
set_intros_2: fixes x21 :: "'a"   and x22 :: "nat" shows "x21 \<in> set_pvar (PCoef x21 x22)"
mat_extension_unitary: fixes m :: "'a mat"   and dims :: "nat list"   and vars :: "nat set" assumes "m \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "unitary m" shows "unitary (mat_extension dims vars m)"
comp_tuple_arr: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "span f g"   and "arr h"   and "local.dom f = cod h" shows "ECC.tuple f g \<cdot> h = ECC.tuple (f \<cdot> h) (g \<cdot> h)"
integrable_abs: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f" shows "integrable M (\<lambda>x. \<bar>f x\<bar>)"
h30: shows "w_vec_R p qs signs = solve_for_lhs_R p qs subsets (M_mat_R signs subsets)"
orbit_def: fixes G :: "('d, 'e) monoid_scheme"   and action :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f" assumes "orbit_stabiliser G action" shows "orbit_stabiliser.orbit G action x = {y. \<exists>g\<in>carrier G. y = action g x}"
respects_unit: fixes a :: "'c" assumes "C.obj a" shows "(\<tau>\<^sub>0 a \<star>\<^sub>D \<tau>.F.unit a) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[\<tau>\<^sub>0 a] \<cdot>\<^sub>D \<l>\<^sub>D[\<tau>\<^sub>0 a] = \<tau>\<^sub>1 a \<cdot>\<^sub>D (\<tau>.G.unit a \<star>\<^sub>D \<tau>\<^sub>0 a)"
A: shows "\<forall>y\<in>Q. r x \<le> y"
pequivD: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and prog' :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "prog \<simeq> prog'"   and "sound P" shows "wp prog P = wp prog' P"
givenBy_eq_extend_set: shows "givenBy f = range (extend_set h)"
iMinI: fixes k :: "'a"   and I :: "'a set" assumes "k \<in> I" shows "iMin I \<in> I"
"case": fixes f :: "'b \<Rightarrow> nat \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "nat" shows "(case ??.Time_Monad.tm.TM x1 x2 of ??.Time_Monad.tm.TM x xa \<Rightarrow> f x xa) = f x1 x2"
undual_cases: fixes y :: "'a" assumes "y \<in> UNIV" obtains x :: "'a dual" where "y = undual x"
fun_belowD: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "f \<sqsubseteq> g" shows "f x \<sqsubseteq> g x"
order_iff_strict: fixes a :: "'a list"   and b :: "'a list" shows "subseq a b = (strict_subseq a b \<or> a = b)"
simps_2: fixes x :: "'b"   and xs :: "['b]" shows "isPrefixOf\<cdot>(x : xs)\<cdot>[] = FF"
ex_fin_conv: fixes A :: "'c fset" shows "(\<exists>x. x |\<in>| A) = (A \<noteq> {||})"
y: fixes x :: "'a" assumes "x \<in> S" shows "dist z y \<le> dist z x"
parcontract_imp_reduce: fixes x :: "comb"   and y :: "comb" assumes "x \<Rrightarrow>\<^sup>1 y" shows "x \<rightarrow> y"
mUnionM_empty: shows "mUnionM {#} = return {#}"
inf_Assertion_def: fixes p :: "'a Assertion"   and q :: "'a Assertion" shows "p \<sqinter> q = abs_wpt ({\<cdot> p } \<sqinter> {\<cdot> q })"
wivs_shift_def: shows "wivs_shift = 2 ^ dflt_size"
A_unicity: fixes NRO :: "msg"   and A :: "agent"   and B :: "agent"   and L :: "nat"   and K :: "nat"   and M :: "msg"   and evs :: "event list"   and M' :: "msg"   and NRO' :: "msg" assumes "NRO = Crypt (priEK A) \<lbrace>Number f_nro, Agent B, Nonce L, Crypt K M\<rbrace>"   and "NRO \<in> parts (knows Spy evs)"   and "Says A B \<lbrace>Number f_nro, Agent B, Nonce L, Crypt K M', NRO'\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> zg" shows "M' = M"