hcju/isabelleps huggingface dataset:
simps_2: fixes prec :: "nat"   and n :: "nat"   and s :: "nat"   and c :: "float"   and k :: "nat"   and f :: "floatarith"   and bs :: "float interval option list" shows "approx_tse prec n (Suc s) c k f bs = (if isDERIV_approx prec n f bs then lift_bin (approx prec f (bs[n := Some (interval_of c)])) (approx_tse prec n s c (Suc k) (DERIV_floatarith n f) bs) (\<lambda>ivl1 ivl2. approx prec (Add (Var 0) (Mult (Inverse (Num (Float (int k) 0))) (Mult (Add (Var (Suc (Suc 0))) (Minus (Num c))) (Var (Suc 0))))) [Some ivl1, Some ivl2, bs ! n]) else approx prec f bs)"
non_neg_vec_st_vec: fixes v :: "'a st_vec" shows "non_neg_vec (st_vec v)"
has_white_path_to_eq_imp: fixes x :: "'a"   and y :: "'a" shows "eq_imp (\<lambda>r' s. (sys_fM s, \<Union> (ran ` obj_fields ` set_option (sys_heap s r')), map_option obj_mark (sys_heap s r'))) (x has_white_path_to y)"
simps_2: fixes \<C> :: "t_context"   and t :: "t"   and uu :: "unop_i"   and ts :: "checker_type" shows "check_single \<C> (Unop_i t uu) ts = (if is_int_t t then type_update ts [TSome t] (Type [t]) else Bot)"
assms_2: shows "snd (rep_nat_term u) = snd (rep_nat_term v)"
noFault_startD': fixes t :: "('d, 'e) xstate"   and f :: "'e"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and s :: "('d, 'e) xstate" assumes "t \<noteq> Fault f"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "s \<noteq> Fault f"
b_empty: fixes T :: "('a, 'b) TS_scheme" shows "[] \<in> beh T"
well_related_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "well_related_set B (\<sqsubseteq>)"
lookup_single: fixes k :: "'e"   and v :: "'d"   and k' :: "'e" shows "lookup (monomial v k) k' = (v when k = k')"
word_not_dist_2: fixes x :: "'b word"   and y :: "'b word" shows "NOT (x AND y) = NOT x OR NOT y"
integrable_bound: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "integrable M f"   and "g \<in> borel_measurable M"   and "AE x in M. norm (g x) \<le> norm (f x)" shows "integrable M g"
mod_poly_less: fixes x :: "'a poly"   and y :: "'a poly" assumes "degree x < degree y" shows "x mod y = x"
eucl_truncate_down_vec_def: fixes p :: "nat"   and x :: "('a, 'b) vec" shows "eucl_truncate_down p x = (\<chi>i. eucl_truncate_down p (x $ i))"
preserves_cod: fixes f :: "'c" assumes "Cop.arr f" shows "S.cod (\<tau> f) = F (Cop.cod f)"
serial_strips_equivalent_to_serial_sas_plus: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<pi> :: "('c \<times> 'd) strips_operator list" assumes "is_valid_problem_sas_plus \<Psi>"   and "STRIPS_Semantics.is_serial_solution_for_problem (\<phi> \<Psi> ) \<pi>" shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> (map (strips_op_to_sasp \<Psi>) \<pi>)"
integral_empty: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "space M = {}" shows "integral\<^sup>L M f = (0::'b)"
IDirProds_empty: fixes S :: "'d \<Rightarrow> 'a set" shows "IDirProds G S {} = {\<one>}"
map_S_def: shows "map_S = (id ---> id ---> id ---> id ---> id ---> id ---> Rep_S ---> Abs_S) map_G"
genideal_self: fixes R :: "('c, 'd) ring_scheme"   and S :: "'c set" assumes "ring R"   and "S \<subseteq> carrier R" shows "S \<subseteq> Idl\<^bsub>R\<^esub> S"
assms: shows "(Atm atm, s) \<rightarrow>c cf"
dags_in_mixed_levels: fixes t1 :: "dag"   and t2 :: "dag" assumes "t1 \<in> Dags (fnct ` Nodes n levellista) low high"   and "t2 \<in> Dags (fnct ` Nodes (Suc n) levellista) low high"   and "t2 \<notin> Dags (fnct ` Nodes n levellista) low high" shows "P t1 t2 var"
not_coprimeE: fixes a :: "'a"   and b :: "'a" assumes "\<not> coprime a b" obtains c :: "'a" where "c dvd a" and "c dvd b" and "\<not> is_unit c"
finite_measure_subadditive_finite: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 's stream set"   and s :: "'s" assumes "finite I"   and "A ` I \<subseteq> T.events s" shows "T.prob s (\<Union> (A ` I)) \<le> (\<Sum>i\<in>I. T.prob s (A i))"
residuated_conjl2: fixes y :: "'a" shows "residuated ((\<lhd>) y)"
igSubstInp_def: fixes MOD :: "('m, 'n, 'o, 'p, 'q, 'r, 'l, 's, 't) model_scheme"   and ys :: "'o"   and Y :: "'l"   and y :: "'r"   and inp :: "'k \<Rightarrow> 'l option" shows "igSubstInp MOD ys Y y inp \<equiv> lift (igSubst MOD ys Y y) inp"
h1: fixes F :: "real list \<Rightarrow> bool" shows "(\<exists>x xs. length xs = amount \<and> F (xs @ x # \<Gamma>)) = (\<exists>xs. length xs = Suc amount \<and> F (xs @ \<Gamma>))"
sees_field_i_i_i_o_o_i_def: shows "sees_field_i_i_i_o_o_i \<equiv> \<lambda>x xa xb xe. pred.Pred (\<lambda>(xc, xd). x \<turnstile> xa sees xb:xc (xd) in xe)"
dim_zero_eq': fixes A :: "'b set" assumes "A = {} \<or> A = {0::'b}" shows "local.dim A = 0"
cut_less_all_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>< t = I) = (\<forall>x\<in>I. x < t)"
meet_aboveI: fixes z :: "'c"   and x :: "'c"   and y :: "'c" assumes "z \<sqsubseteq> x"   and "z \<sqsubseteq> y" shows "z \<sqsubseteq> x \<sqinter> y"
nz: shows "\<forall>\<^sub>F x in at_top. eval C x \<noteq> 0"
E: shows "s \<circ> a = a \<and> \<lambda> (s, a) = \<dd>"
leg1_iso: shows "\<exists>\<nu>. \<guillemotleft>\<nu> : local.dom \<sigma> \<Rightarrow> s\<^sub>1 \<star> src s\<^sub>0\<guillemotright> \<and> local.iso \<nu>"
sup2I1: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "A x y" shows "sup A B x y"
ResF: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and x :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'"   and "x \<sharp> \<alpha>" shows "<\<nu>x>P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> <\<nu>x>P'"
ide_leg1: shows "ide t\<^sub>0u\<^sub>1.p\<^sub>1"
pos_decomp_def: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "ps\<^sub>+ = filter (\<lambda>p. snd p \<noteq> {}) ps"
in_minimize_wrt_iff: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "\<And>x y. Q x y \<Longrightarrow> \<not> Q y x"   and "sorted_wrt Q xs"   and "\<And>x y. \<not> P x y \<Longrightarrow> Q x y"   and "\<And>x. P x x" shows "(x \<in> set (minimize_wrt P xs)) = (x \<in> set xs \<and> (\<forall>y\<in>set xs. P y x))"
u: fixes x :: "'a" shows "(\<lambda>i. f i x) \<longlonglongrightarrow> u x"
is_red_map_scale: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red F (c \<cdot> p)" shows "is_red F p"
H_into_carrier_G: fixes x :: "'a" assumes "x \<in> H" shows "x \<in> carrier G"
valid_path_def: fixes f :: "real \<Rightarrow> 'b" shows "valid_path f \<equiv> f piecewise_C1_differentiable_on {0..1}"
abelian_subgroupI: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "H \<lhd> add_monoid G"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x" shows "abelian_subgroup H G"
apfst_apsnd: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and x :: "'d \<times> 'e" shows "apfst f (apsnd g x) = (f (fst x), g (snd x))"
S5Fail: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> RPCFail crCh rmCh rst p \<and> $S5 rmhist p \<and> unchanged (e p, c p, m p, rmhist!p) \<longrightarrow> S6 rmhist p$"
mcont_def: fixes luba :: "'h set \<Rightarrow> 'h"   and orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and lubb :: "'i set \<Rightarrow> 'i"   and ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i" shows "mcont luba orda lubb ordb f = (monotone orda ordb f \<and> cont luba orda lubb ordb f)"
less_infI2: fixes b :: "'b multiset"   and x :: "'b multiset"   and a :: "'b multiset" assumes "b \<subset># x" shows "a \<inter># b \<subset># x"
strong_cases: fixes a1 :: "lam"   and a2 :: "lam"   and a :: "name"   and aa :: "name" assumes " a1 \<longrightarrow>\<^sub>1 a2" obtains M :: "lam" where "a1 = M" and "a2 = M"   | t1 :: "lam" and t2 :: "lam" and s1 :: "lam" and s2 :: "lam" where "a1 = App t1a__ s1__" and "a2 = App t2a__ s2__" and " t1a__ \<longrightarrow>\<^sub>1 t2a__" and " s1__ \<longrightarrow>\<^sub>1 s2__"   | s1 :: "lam" and s2 :: "lam" where "\<lbrakk>a \<sharp> a1; a \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = Lam [a].s1a__ \<and> a2 = Lam [a].s2a__ \<and>  s1a__ \<longrightarrow>\<^sub>1 s2a__"   | s1 :: "lam" and s2 :: "lam" and t1 :: "lam" and t2 :: "lam" where "\<lbrakk>aa \<sharp> a1; aa \<sharp> a2\<rbrakk> \<Longrightarrow> a1 = App (Lam [aa].t1) s1 \<and> a2 = t2[aa::=s2] \<and> aa \<sharp> (s1, s2) \<and>  s1 \<longrightarrow>\<^sub>1 s2 \<and>  t1 \<longrightarrow>\<^sub>1 t2"
tNDComm: fixes a :: "'d set set"   and b :: "'e set set"   and c :: "'d set set"   and d :: "'e set set" assumes "twoNetsDistinct a b c d" shows "twoNetsDistinct c d a b"
reflects_iso: fixes \<mu> :: "'b" assumes "D.arr \<mu>"   and "C.iso (F.right_map \<mu>)" shows "D.iso \<mu>"
impl_of_inverse: fixes x :: "('c, 'd) trie" shows "Trie2.trie.Trie (impl_of x) = x"
ccpo_Sup_singleton: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.ccpo Sup less_eq less" shows "Sup {x} = x"
bvars_subst_ident: fixes S :: "('b, 'a) strand_step list"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "bvars\<^sub>s\<^sub>t S = bvars\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>)"
uv_1: shows "c *\<^sub>R r + b *\<^sub>R q + a *\<^sub>R p = r + u *\<^sub>R (q - r) + v *\<^sub>R (p - r)"
all_simps_5: fixes P :: "'d \<Rightarrow> bool"   and Q :: "bool" shows "(\<forall>x. P x \<longrightarrow> Q) = ((\<exists>x. P x) \<longrightarrow> Q)"
decr_grading_term_def: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and v :: "'t" shows "decr_grading_term d n v = term_of_pair (decr_grading d n (pp_of_term v), component_of_term v)"
integral_combine_division_topdown: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and \<D> :: "'c set set" assumes "f integrable_on S"   and "\<D> division_of S" shows "integral S f = (\<Sum>i\<in>\<D>. integral i f)"
valid: fixes g :: "'G" assumes "invar g" shows "valid_graph (\<alpha> g)"
restrictedDomain: fixes R :: "('a \<times> 'b) set"   and X :: "'a set" shows "Domain (R || X) = Domain R \<inter> X"
mgf: shows "MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))) \<noteq> []"
cycle_is_surj: fixes cs :: "'b list" assumes "cycle cs" shows "cycle_of_list cs ` set cs = set cs"
a_gla2: fixes x :: "'a"   and y :: "'a" assumes "ad x \<le> ad y" shows "ad x \<cdot> y = (zero_class.zero::'a)"
mat_swapcols_def: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "swapcols k l A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = j then A $$ (i, l) else if l = j then A $$ (i, k) else A $$ (i, j))"
subtopology_restrict: fixes X :: "'b topology"   and S :: "'b set" shows "subtopology X (topspace X \<inter> S) = subtopology X S"
pair1: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm" assumes "A \<rightarrow>\<beta> A'" shows "SimplyTyped.Pair A B \<rightarrow>\<beta> SimplyTyped.Pair A' B"
Interleaves_prefix_fst: fixes ws :: "'b list"   and P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "\<And>n. n < length ws \<Longrightarrow> P (ws ! n) (drop (Suc n) ws @ xs)" shows "xs \<cong> {ys, zs, P} = ws @ xs \<cong> {ws @ ys, zs, P}"
K_def: fixes off :: "nat"   and k :: "nat" shows "K off k = D * k + (m - 1) + off"
comp_null_1: fixes f :: "'a" shows "S.comp S.null f = S.null"
PO_a0i_refines_trans_a0n: shows "{R0n0i} TS.trans a0n, TS.trans a0i {> R0n0i}"
to_nat_on_inj: fixes A :: "'c set"   and a :: "'c"   and b :: "'c" assumes "countable A"   and "a \<in> A"   and "b \<in> A" shows "(to_nat_on A a = to_nat_on A b) = (a = b)"
l7_15: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and P' :: "'a"   and Q :: "'a"   and Q' :: "'a"   and R :: "'a"   and R' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A P P'"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A Q Q'"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A R R'"   and "Bet P Q R" shows "Bet P' Q' R'"
dg_is_arrD_3: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"
Sup_least: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and A :: "'e set"   and z :: "'e" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
simps: fixes rep_key_order :: "'c \<Rightarrow> 'a key_order"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and xs :: "('a \<times> 'b) list"   and ko :: "'c" shows "oalist_raw.update_by_fun_raw rep_key_order k f (xs, ko) = (comparator.update_by_fun_pair (key_compare (rep_key_order ko)) k f xs, ko)"
weak_get_put: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e" assumes "mwb_lens x"   and "\<sigma> \<in> \<S>\<^bsub>x\<^esub>" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
Conj_EH_1: fixes A :: "fm"   and B :: "fm"   and H :: "fm set"   and C :: "fm" assumes "insert A (insert B H) \<turnstile> C" shows "insert (A AND B) H \<turnstile> C"
lcp_distinct_hd: fixes u :: "'b list"   and v :: "'b list" assumes "hd u \<noteq> hd v" shows "u \<and>\<^sub>p v = \<epsilon>"
maddux_141: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x ; y \<cdot> z = 0) = (x\<^sup>T ; z \<cdot> y = 0)"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and pause :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind"   and "monad_resumption_axioms bind pause" shows "monad_resumption return bind pause"
v11_vconverse_app: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "v11 r"   and "r\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r\<inverse>\<^sub>\<circ>\<lparr>b\<rparr> = a"
distinct_disc_4: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and pr_operation' :: "pr_operation'" assumes "Network c s t"   and "is_SAT_PUSH' pr_operation'" shows "\<not> is_RELABEL' pr_operation'"
run_owner_iii_ex: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "pred_owner_iii n (run_asset_ii n s) (run_owner_iii n s)"
swap\<^sub>f_def: shows "swap\<^sub>f \<equiv> map_fun id (map_fun id (map_fun fmlookup Abs_fmap)) (\<lambda>b i p x. p (swap b i x))"
strategy_conforming_path_exists: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "v0 \<in> V\<^bsub>G'\<^esub>"   and "G'.strategy p \<sigma>"   and "G'.strategy p** \<sigma>'" obtains P :: "'a llist" where "vmc2_path G' P v0 p \<sigma> \<sigma>'"
Int_lower1: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B \<subseteq> A"
idom_potent_unit: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "0 < n" shows "Unit R a = Unit R (a^\<^bsup>R n\<^esup>)"
lhd_ldrop: fixes n :: "enat"   and xs :: "'b llist" assumes "n < llength xs" shows "lhd (ldrop n xs) = xs $ the_enat n"
inter_scheme_def: fixes l :: "nat"   and U :: "nat list set" shows "inter_scheme l U \<equiv> SOME zs. \<exists>k xs ys. 0 < l \<and> (l = 2 * k - 1 \<and> U = {xs, ys} \<and> Form_Body k k xs ys zs \<or> l = 2 * k \<and> U = {xs, ys} \<and> Form_Body (Suc k) k xs ys zs)"
starlike_compact_projective2: fixes S :: "'c set"   and a :: "'c" assumes "compact S"   and "a \<in> rel_interior S"   and "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S" shows "S homeomorphic cball a 1 \<inter> affine hull S"
CQ_holds_def: fixes p :: "'sz"   and \<theta> :: "('sf, 'sz) trm"   and \<theta>' :: "('sf, 'sz) trm" shows "CQ_holds p \<theta> \<theta>' \<equiv> valid (Equals \<theta> \<theta>') \<longrightarrow> valid ($\<phi> p (singleton \<theta>) \<leftrightarrow> $\<phi> p (singleton \<theta>'))"
monom_def: fixes a :: "'a list"   and n :: "nat" shows "UP.monom a n = a # replicate n \<zero>\<^bsub>K [X]\<^esub>"
simps: fixes a1 :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and a2 :: "'d"   and a3 :: "('e \<Rightarrow> 'f) set"   and a4 :: "'d"   and a5 :: "('e \<Rightarrow> 'f) set" shows "a1 \<turnstile> \<langle>a2, a3\<rangle> \<leadsto> \<langle>a4, a5\<rangle> = ((\<exists>A l Z. a1 = A \<and> a2 = l \<and> a3 = Z \<and> a4 = l \<and> a5 = (Z \<inter> {u. u \<turnstile> inv_of A l})\<^sup>\<up> \<inter> {u. u \<turnstile> inv_of A l}) \<or> (\<exists>A l g a r l' Z. a1 = A \<and> a2 = l \<and> a3 = Z \<and> a4 = l' \<and> a5 = (Z \<inter> {u. u \<turnstile> g})\<^bsub>r \<rightarrow> 0\<^esub> \<inter> {u. u \<turnstile> inv_of A l'} \<and> A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'))"
adds_pp_def: fixes t :: "'a"   and v :: "'t" shows "(t adds\<^sub>p v) = (t adds pp_of_term v)"
simps_5: fixes x :: "name"   and \<theta> :: "(name \<times> trm) list"   and t :: "trm"   and s :: "trm" assumes "x \<sharp> \<theta>"   and "x \<sharp> t" shows "\<theta><t to x in s> = \<theta><t> to x in \<theta>
"
a: shows "<a>:M \<in> ORRIGHT2 (B OR C) (\<parallel><C>\<parallel>)"
is_weakly_dom_component_safe_def: shows "Core_DOM_DOM_Components.is_weakly_dom_component_safe \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe Core_DOM_Functions.get_root_node Core_DOM_Functions.to_tree_order"
InvariantUniqQAfterNotifyWatchesLoop: fixes state :: "State"   and Wl :: "nat list"   and literal :: "Literal"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "InvariantUniqQ (getQ state)" shows "let state' = notifyWatches_loop literal Wl newWl state in InvariantUniqQ (getQ state')"
assms_2: shows "B \<subseteq> {i. i < 2 ^ n}"
hom_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "h x \<in> carrier H"
connected_nest: fixes S :: "'a \<Rightarrow> 'b set" assumes "\<And>n. compact (S n)"   and "\<And>n. connected (S n)"   and "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m" shows "connected (\<Inter> (range S))"
map_of_zip_dom_to_range: fixes a :: "'b"   and A :: "'b list"   and B :: "'c list" assumes "a \<in> set A"   and "length B = length A" shows "the (map_of (zip A B) a) \<in> set B"
holomorphic_on_diff: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "f holomorphic_on s"   and "g holomorphic_on s" shows "(\<lambda>z. f z - g z) holomorphic_on s"
mono2: fixes f :: "'c" assumes "monotone ord (\<le>) (U f)" shows "monotone ord (\<le>) (U (F f))"
m_ac_3: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z) = y \<otimes>\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> z)"
is_ord_l_bal: fixes x :: "'a"   and l :: "'a tree\<^sub>0"   and r :: "'a tree\<^sub>0" assumes "is_ord (MKT\<^sub>0 x l r)"   and "height l = Suc (Suc (height r))" shows "is_ord (l_bal\<^sub>0 x l r)"
mtf2_effect: fixes q :: "'b"   and xs :: "'b list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs" shows "(index xs q < i \<and> i < length xs \<longrightarrow> index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index xs q < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < length xs) \<and> (index xs q = i \<longrightarrow> index (mtf2 n q xs) (xs ! i) = index xs q - n \<and> index (mtf2 n q xs) (xs ! i) = index xs q - n) \<and> (index xs q - n \<le> i \<and> i < index xs q \<longrightarrow> index (mtf2 n q xs) (xs ! i) = Suc (index xs (xs ! i)) \<and> index xs q - n < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) \<le> index xs q) \<and> (i < index xs q - n \<longrightarrow> index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < index xs q - n)"
inf_compl_bot_left2: fixes x :: "'a upred"   and y :: "'a upred" shows "(x \<and> \<not> x \<and> y) = false"
wtk: shows "wt_kildall P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt"
supp_fun_app_eqvt2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and x :: "'b"   and y :: "'c" assumes "eqvt f" shows "supp (f x y) \<subseteq> supp x \<union> supp y"
correct1: fixes \<D> :: "nat \<Rightarrow> ('e + 'b, 'f + 'd) distinguisher" assumes "\<And>\<eta>. \<I>_inner1 \<eta> \<oplus>\<^sub>\<I> \<I>_common1 \<eta> \<turnstile>g \<D> \<eta> \<surd>"   and "\<And>\<eta>. interaction_any_bounded_by (\<D> \<eta>) (bound1 \<eta>)"   and "\<And>\<eta>. lossless1 \<Longrightarrow> plossless_gpv (\<I>_inner1 \<eta> \<oplus>\<^sub>\<I> \<I>_common1 \<eta>) (\<D> \<eta>)" shows "(\<forall>\<eta>. wiring (\<I>_inner1 \<eta>) (\<I>_real1 \<eta>) (cnv1 \<eta>) (w1 \<eta>)) \<and> Negligible.negligible (\<lambda>\<eta>. advantage (\<D> \<eta>) (ideal1 \<eta>) (cnv1 \<eta> |\<^sub>= 1\<^sub>C \<rhd> real1 \<eta>))"
insert_subsetI: fixes x :: "'d"   and A :: "'d set"   and X :: "'d set" assumes "x \<in> A"   and "X \<subseteq> A" shows "insert x X \<subseteq> A"
strand_subst_set: fixes S :: "('c, 'd) strand_step list"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "set (S \<cdot>\<^sub>s\<^sub>t \<theta>) = (\<lambda>x. x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta>) ` set S"
vcomp_obtain_middle: fixes a :: "V"   and c :: "V"   and r :: "V"   and s :: "V" assumes "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r \<circ>\<^sub>\<circ> s" obtains b :: "V" where "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> s" and "\<langle>b, c\<rangle> \<in>\<^sub>\<circ> r"
union_filter_mset_complement: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and M :: "'c multiset" assumes "\<forall>x. P x = (\<not> Q x)" shows "filter_mset P M + filter_mset Q M = M"
indRelTEQ_refl: fixes TRel :: "('procT \<times> 'procT) set" assumes "refl TRel" shows "refl (indRelTEQ TRel)"
semigroup_axioms: shows "Type_Semigroups.semigroup plus"
poly_of_focus_eq_zero_iff: fixes x :: "'a"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "(poly_of_focus x p = 0) = (p = 0)"
1: shows "L f = trg\<^sub>B f \<star>\<^sub>B f \<and> L f' = trg\<^sub>B f' \<star>\<^sub>B f'"
M: fixes m :: "nat"   and n :: "nat" assumes "M \<le> m"   and "M \<le> n" shows "dist (G (real m)) (G (real n)) < \<epsilon>"
obtain_sorted_carrier'''_def: shows "obtain_sorted_carrier''' = getEdges \<bind> (\<lambda>l. RETURN (quicksort_by_rel edges_less_eq [] l, max_node l))"
assms_2: shows "ereal (1 - s \<bullet> (1::'a)) \<le> c"
span_base: fixes a :: "'b"   and S :: "'b set" assumes "a \<in> S" shows "a \<in> m1.span S"
permutes_orbit_subset: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" assumes "f permutes S"   and "x \<in> S" shows "orbit f x \<subseteq> S"
Lunit_in_lunit: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "t \<in> f" shows "\<^bold>\<l>\<^bold>[t\<^bold>] \<in> free_monoidal_category.lunit\<^sub>F\<^sub>M\<^sub>C C f"