hcju/isabelleps huggingface dataset:
proves_eta_step_pre: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and vs :: "(variable \<times> typ) list"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "\<forall>(x, \<tau>)\<in>set vs. (x, \<tau>) \<notin> fv t \<union> FV \<Gamma>"   and "term_ok \<Theta> (subst_bvs (map (\<lambda>(x, y). Fv x y) vs) t)"   and "t \<rightarrow>\<^sub>\<eta> u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (subst_bvs (map (\<lambda>(x, y). Fv x y) vs) t) (subst_bvs (map (\<lambda>(x, y). Fv x y) vs) u)"
learn_bcE_1: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "learn_bc \<psi> U s" shows "environment \<psi> U s"
weak_sharing_consistent_preserves_distinct_share_all_until_volatile_write: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "ownership_distinct ts"   and "read_only_unowned \<S> ts"   and "weak_sharing_consis ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O> \<inter> read_only (share_all_until_volatile_write ts \<S>) = {}"
nz_empty_rs_stateD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> [] \<Rightarrow>\<^sub>z t" shows "t = Undecided"
assms_4: fixes f :: "'c"   and x :: "'b"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "\<And>x h h' r. effect (U f x) h h' r \<Longrightarrow> P x h h' r"   and "effect (U (F f) x) h h' r" shows "P x h h' r"
facts_3: fixes n :: "nat" assumes "2 * card (uverts H) \<le> n" shows "1 - probGn p n (\<lambda>es. \<not> H \<sqsubseteq> edge_space.edge_ugraph n es) = prob_in_class p {G. H \<sqsubseteq> G} n"
fqtran_def: fixes x :: "'b" shows "fqtran x \<equiv> (\<sqinter>) x"
lborel_prod: shows "lborel \<Otimes>\<^sub>M lborel = lborel"
connect_cong_trace: fixes A :: "'h set"   and res1 :: "('h, 'i) resource"   and res2 :: "('h, 'i) resource"   and \<I> :: "('h, 'i) \<I>"   and d :: "('h, 'i) distinguisher" assumes "A \<turnstile>\<^sub>R res1 \<approx> res2"   and "\<I> \<turnstile>g d \<surd>"   and "outs_gpv \<I> d \<subseteq> A"   and "\<I> \<turnstile>res res1 \<surd>"   and "\<I> \<turnstile>res res2 \<surd>" shows "connect d res1 = connect d res2"
lin_independentI_concat_all_scalars: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and mss :: "'b list list" assumes "VectorSpace smult V"   and "set (concat mss) \<subseteq> V"   and "\<And>rss. \<lbrakk>set (concat rss) \<subseteq> UNIV; list_all2 (\<lambda>xs ys. length xs = length ys) rss mss; scalar_mult.lincomb smult (concat rss) (concat mss) = (0::'b)\<rbrakk> \<Longrightarrow> \<forall>rs\<in>set rss. set rs \<subseteq> 0" shows "R_scalar_mult.R_lin_independent UNIV smult (concat mss)"
h2d: shows "set (characterize_consistent_signs_at_roots p (qs1 @ qs2)) \<subseteq> set (signs_smash signs1 signs2)"
numbers_of_marks_mark_out: fixes n :: "nat"   and m :: "nat"   and bs :: "bool list" shows "numbers_of_marks n (mark_out m bs) = {q \<in> numbers_of_marks n bs. \<not> Suc m dvd Suc q - n}"
HEndPhase1_blocksOf: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p" shows "blocksOf s' q \<subseteq> blocksOf s q \<union> {dblock s' q}"
map_getminE: fixes m :: "'m"   and k :: "'a"   and p :: "'b" assumes "getmin m = (k, p)"   and "invar m"   and "lookup m \<noteq> Map.empty" obtains "lookup m k = Some p" and "\<forall>k' p'. lookup m k' = Some p' \<longrightarrow> p \<le> p'"
f_2_def: fixes p :: "nat"   and q :: "nat"   and x :: "int" assumes "QR p q" shows "QR.f_2 p q x = int p * int q - x"
left_treeE: fixes G :: "('e, 'f) Graph_scheme"   and u :: "'e"   and s :: "'e"   and t :: "'e" assumes "Tree G"   and "u \<in> Tree.left_tree G s t" shows "u \<in> V\<^bsub>G\<^esub> \<and> s \<in> set (u \<leadsto>\<^bsub>G\<^esub> t)"
a2: shows "BusInterface activation1 (nReturn k) recv (nStore k) (nSend k) (nGet k)"
swap_only_swaps: fixes arr' :: "'a \<Rightarrow> 'b"   and arr :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "arr' = swap arr x y"   and "z \<noteq> x"   and "z \<noteq> y" shows "arr' z = arr z"
aforms_err_Cons: fixes e :: "nat \<Rightarrow> real"   and x :: "(real \<times> real pdevs) \<times> real"   and xs :: "((real \<times> real pdevs) \<times> real) list" shows "aforms_err e (x # xs) = set_Cons (aform_err e x) (aforms_err e xs)"
surjective: fixes r :: "'a prog_scheme" shows "r = \<lparr>ifaces = ifaces r, classes = classes r, \<dots> = prog.more r\<rparr>"
subu_fin: shows "- \<infinity> < I.subcocycle_avg_ereal vI"
quat_inverse_cnj: fixes q :: "quat" shows "inverse q = inverse (quat_of_real ((norm q)\<^sup>2)) * Quaternions.cnj q"
pd: shows "\<pi> n pd\<rightarrow> \<pi> 0"
normalized_n_primitive_alist_and: fixes disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and P :: "'c \<Rightarrow> bool"   and as :: "'b negation_type list" shows "normalized_n_primitive disc_sel P (alist_and as) = (\<forall>a\<in>set as. normalized_n_primitive disc_sel P (negation_type_to_match_expr a))"
l7_13_R1: fixes A :: "'p"   and P :: "'p"   and P' :: "'p"   and Q' :: "'p"   and Q :: "'p" assumes "A \<noteq> P"   and "A Midpoint P' P"   and "A Midpoint Q' Q" shows "Cong P Q P' Q'"
l_cancel_one': fixes x :: "'a list"   and a :: "'a list" assumes "x \<in> carrier (K [X])"   and "a \<in> carrier (K [X])" shows "(x = x \<oplus>\<^bsub>K [X]\<^esub> a) = (a = \<zero>\<^bsub>K [X]\<^esub>)"
prems_8: shows "LeftDerivationLadder \<alpha> D L (terminals (drop r p @ [X]))"
hd_butlast_last_id: fixes xs :: "'a list" assumes "1 < length xs" shows "hd xs # tl (butlast xs) @ [last xs] = xs"
upt_rev_upt: fixes j :: "nat"   and i :: "nat" shows "rev [j>..i] = [i..<j]"
pw_bind_leI: fixes S :: "'a nres"   and M :: "'b nres"   and f :: "'b \<Rightarrow> 'a nres" assumes "nofail S \<Longrightarrow> nofail M"   and "\<And>x. \<lbrakk>nofail M; inres M x\<rbrakk> \<Longrightarrow> f x \<le> S" shows "M \<bind> f \<le> S"
Dickson_term: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'d \<Rightarrow> nat"   and K :: "'e set"   and m :: "nat" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "dickson_grading d"   and "finite K" shows "almost_full_on (term_powerprod.adds_term pair_of_term) {t. term_powerprod.pp_of_term pair_of_term t \<in> dgrad_set d m \<and> term_powerprod.component_of_term pair_of_term t \<in> K}"
ln_primorial_asymp_equiv: shows "(\<lambda>x. ln (real (primorial x))) \<sim>[at_top] (\<lambda>x. x)"
set_iterate: shows "set_iteratei \<alpha> invar iteratei"
bitwise_inner_prod_with_zero: fixes m :: "nat"   and n :: "nat" assumes "m < 2 ^ n" shows "0 \<cdot>\<^bsub>n\<^esub>  m = 0"
card_UNIV_prod_def: shows "card_UNIV_class.card_UNIV = Phantom('a \<times> 'b) (of_phantom card_UNIV_class.card_UNIV * of_phantom card_UNIV_class.card_UNIV)"
coincidence_dterm: fixes \<theta> :: "('b, 'c) trm"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and \<nu>' :: "(real, 'c) vec \<times> (real, 'c) vec"   and I :: "('b, 'd, 'c) interp" assumes "dsafe \<theta>"   and "Vagree \<nu> \<nu>' (FVT \<theta>)" shows "dterm_sem I \<theta> \<nu> = dterm_sem I \<theta> \<nu>'"
simps: fixes a :: "('b, unit) Old_Datatype.node set"   and A :: "('b, unit) Old_Datatype.node set set" shows "(a \<in> list A) = (a = NIL \<or> (\<exists>aa M. a = CONS aa M \<and> aa \<in> A \<and> M \<in> list A))"
gaugeD_2: fixes \<gamma> :: "'a \<Rightarrow> 'a set"   and x :: "'a" assumes "gauge \<gamma>" shows "open (\<gamma> x)"
irreducible_imp_prime_elem: fixes x :: "'b poly" assumes "irreducible x" shows "prime_elem x"
assms: shows "\<forall>(t, t')\<in>set D. (fv t \<union> fv t') \<inter> set X = {}"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<cdot> z) = x \<cdot> y \<cdot> z"
intro: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "ring R"   and "ring S"   and "ring_hom_ring_axioms R S h" shows "ring_hom_ring R S h"
Rep_formula_eqvt: fixes x :: "perm"   and xa :: "('a, 'b, 'c) formula" shows "x \<bullet> Rep_formula xa = Rep_formula (x \<bullet> xa)"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "is_Proj x = (\<exists>M. closed_csubspace M \<and> is_projection_on ((*\<^sub>V) x) M)"
zero_in_finite_cone: fixes Vs :: "'a vec set" assumes "Vs \<subseteq> carrier_vec n" shows "0\<^sub>v n \<in> finite_cone Vs"
index_mat_addrow_mat_2: fixes n :: "nat"   and a :: "'c"   and k :: "nat"   and l :: "nat" shows "dim_row (addrow_mat n a k l) = n"
diagonal_step_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "nat"   and j :: "nat"   and d :: "'d"   and v :: "'d" shows "diagonal_step A i j d v = (\<chi>a b. if a = from_nat i \<and> b = from_nat i then d else if a = from_nat j \<and> b = from_nat j then v * A $ from_nat j $ from_nat j else A $ a $ b)"
eval_is_hom: fixes K :: "'a list set"   and a :: "'a list" assumes "subring K (K [X])"   and "a \<in> carrier (K [X])" shows "(\<lambda>p. UP.eval p a) \<in> ring_hom (K [X]\<^bsub>K [X]\<^esub>) (K [X])"
prob_Max_IID_geometric_greaterThan: fixes n :: "nat"   and i :: "nat" assumes "p \<in> {0<..1}" shows "measure_pmf.prob (H\<^sub>N n) {i<..} = 1 - (1 - q ^ (i + 1)) ^ n"
Zr_minus: fixes n :: "int" shows "-\<^sub>a\<^bsub>Zr\<^esub> n = - n"
drop_eq_Nil: fixes n :: "nat"   and xs :: "'b list" shows "(drop n xs = \<epsilon>) = (\<^bold>|xs\<^bold>| \<le> n)"
eq_\<I>_gpv_2: shows "pfinite_gpv \<I> gpv"
filterlim_power_at_infinity: fixes n :: "nat" assumes "n \<noteq> 0" shows "LIM x at_infinity. x ^ n :> at_infinity"
message_must_be_delivered: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat"   and m :: "'m message" assumes "c \<turnstile> ev \<mapsto> c'"   and "(msgs c i \<noteq> [] \<and> hd (msgs c i) = m) \<and> (msgs c' i = [] \<or> hd (msgs c' i) \<noteq> m)" shows "(\<exists>p q. ev = RecvMarker i p q \<and> m = Marker) \<or> (\<exists>p q s s' m'. ev = Recv i p q s s' m' \<and> m = Msg m')"
inext_min_step2: fixes I :: "nat set"   and n :: "nat" shows "\<not> (\<exists>k\<in>I. n < k \<and> k < inext n I)"
upper: fixes a :: "'a" assumes "a \<in> A" shows "a \<le> x"
eq_nb: fixes t :: "tm" assumes "tmbound0 t" shows "bound0 (eq t)"
wlog_unit_disc_fix: fixes M :: "moebius"   and P :: "moebius \<Rightarrow> bool" assumes "unit_disc_fix M"   and "\<And>k. cmod k < 1 \<Longrightarrow> P (blaschke k)"   and "\<And>\<phi>. P (moebius_rotation \<phi>)"   and "\<And>M1 M2. \<lbrakk>unit_disc_fix M1; P M1; unit_disc_fix M2; P M2\<rbrakk> \<Longrightarrow> P (M1 + M2)" shows "P M"
elims_3: fixes x :: "'a"   and xa :: "'b"   and P :: "bool" assumes "\<not> applies_Yes x xa" shows "P"
simps_2: fixes cfg :: "('p, 'v, 's) configuration"   and fStepCfg :: "('p, 'v, 's) configuration list \<Rightarrow> ('p, 'v) message list \<Rightarrow> ('p, 'v, 's) configuration list"   and fStepMsg :: "('p, 'v, 's) configuration list \<Rightarrow> ('p, 'v) message list \<Rightarrow> ('p, 'v) message list"   and n :: "nat" shows "infiniteExecutionMsg cfg fStepCfg fStepMsg (Suc n) = fStepMsg (infiniteExecutionCfg cfg fStepCfg fStepMsg n) (infiniteExecutionMsg cfg fStepCfg fStepMsg n)"
True: shows "a = (0::'a) \<or> f = 0"
gr_transl_rules: fixes x :: "('b, nat) labeled_graph \<times> ('b, nat) labeled_graph"   and T :: "('b allegorical_term \<times> 'b allegorical_term) set" assumes "x \<in> transl_rules T" shows "graph_rule x"
simple_match_to_of_match_def: fixes m :: "32 simple_match"   and ifs :: "char list list" shows "simple_match_to_of_match m ifs \<equiv> let npm = \<lambda>p. fst p = 0 \<and> snd p = - 1; sb = \<lambda>p. if npm p then [None] else if fst p \<le> snd p then map (Some \<circ> (\<lambda>pfx. (pfxm_prefix pfx, ~~ pfxm_mask pfx))) (wordinterval_CIDR_split_prefixmatch (WordInterval (fst p) (snd p))) else [] in concat (map (\<lambda>iif. concat (map (\<lambda>sport. map (simple_match_to_of_match_single m iif (proto m) sport) (sb (dports m))) (sb (sports m)))) (if iiface m = ifaceAny then [None] else concat (map (\<lambda>i. if match_iface (iiface m) i then [Some i] else []) ifs)))"
winding_number_pos_le: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> 0 \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z))" shows "0 \<le> Re (winding_number \<gamma> z)"
contractible_Euclidean_space: fixes n :: "nat" shows "contractible_space (Euclidean_space n)"
simps_2: fixes r :: "real" shows "ereal r * \<infinity> = (if r = 0 then 0 else if 0 < r then \<infinity> else - \<infinity>)"
inject: fixes x1 :: "'a"   and x2 :: "'a stream"   and y1 :: "'a"   and y2 :: "'a stream" shows "(x1 ## x2 = y1 ## y2) = (x1 = y1 \<and> x2 = y2)"
filter_cong: fixes xs :: "'g list"   and ys :: "'g list"   and P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" shows "filter P xs = filter Q ys"
prv_exi_commute: fixes x :: "'var"   and y :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "y \<in> var"   and "\<phi> \<in> fmla" shows "prv (imp (exi x (exi y \<phi>)) (exi y (exi x \<phi>)))"
prob_space_axioms: fixes s :: "'a cfg" shows "prob_space (MDP.MC.T s)"
update_simps_11: fixes up :: "(nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and s :: "('a, 'b) state" shows "\<U> (\<B>\<^sub>i\<^sub>l_update up s) = \<U> s"
acyclic: fixes xa :: "'v"   and y :: "'v" assumes "xa \<notin> X"   and "y \<in> X" shows "h_diff i (xa, y) = 0"
prems_4: shows "eqvt_at transform_sumC (PropLetBody (AnalLet as' body' a'), body')"
compose_def: fixes A :: "'c set"   and g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" shows "compose A g f = (\<lambda>x\<in>A. g (f x))"
ext_inject: fixes dpm :: "(nat \<times> nat \<times> nat) pointermap"   and dcl :: "nat \<times> nat \<times> nat \<Rightarrow> nat option"   and more :: "'a"   and dpm' :: "(nat \<times> nat \<times> nat) pointermap"   and dcl' :: "nat \<times> nat \<times> nat \<Rightarrow> nat option"   and more' :: "'a" shows "(\<lparr>dpm = dpm, dcl = dcl, \<dots> = more\<rparr> = \<lparr>dpm = dpm', dcl = dcl', \<dots> = more'\<rparr>) = (dpm = dpm' \<and> dcl = dcl' \<and> more = more')"
Ide_in_Hom: fixes t :: "'c local.term" assumes "Ide t" shows "t \<in> Hom t t"
convex_on_diff_2: fixes I :: "real set"   and f :: "real \<Rightarrow> real"   and x :: "real"   and y :: "real"   and t :: "real" assumes "convex_on I f"   and "x \<in> I"   and "y \<in> I"   and "x < t"   and "t < y" shows "(f x - f y) / (x - y) \<le> (f t - f y) / (t - y)"
jordan_block_index_3: fixes n :: "nat"   and k :: "'b" shows "dim_col (jordan_block n k) = n"
split_min_inv: fixes t :: "('a \<times> 'b) tree"   and m :: "'a"   and t' :: "('a \<times> 'b) tree" assumes "split_min t = (m, t')"   and "inv t"   and "t \<noteq> \<langle>\<rangle>" shows "inv t'"
a_get_tag_name_def: fixes element_ptr :: "(_) element_ptr" shows "CD.a_get_tag_name element_ptr = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name"
comp_inv_arr: fixes f :: "'a"   and g :: "'a" assumes "A.inverse_arrows f g" shows "g \<cdot>\<^sub>A f = A.dom f"
ipurge_tr_rev_aux_first_offset: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a"   and zs :: "'a list"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set" assumes "xs = ys @ x # zs \<and> ipurge_tr_rev_aux I D (sources_aux I D U (x # zs)) ys = [] \<and> (\<exists>v\<in>sources_aux I D U zs. (D x, v) \<in> I)" shows "ys = take (the (offset 0 x xs)) xs"
mul_eq_def: fixes r :: "('a \<times> 'a) set" shows "mul_eq r = {(M, N). \<exists>I J K. M = I + K \<and> N = I + J \<and> set_mset K \<subseteq> r \<down>m J}"
has_integral_implies_lebesgue_measurable: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'b"   and \<Omega> :: "'a set" assumes "(f has_integral I) \<Omega>" shows "(\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x) \<in> borel_measurable lebesgue"
trivial_imp_no_distinct: fixes X :: "'c set"   and x :: "'c"   and y :: "'c" assumes "trivial X"   and "x \<in> X"   and "y \<in> X" shows "x = y"
emp_gen_set: shows "\<langle>{}\<rangle> = {\<epsilon>}"
Rep_uexpr_inject: fixes x :: "('b, 'a) uexpr"   and y :: "('b, 'a) uexpr" shows "(\<lbrakk>x\<rbrakk>\<^sub>e = \<lbrakk>y\<rbrakk>\<^sub>e) = (x = y)"
pf: shows "pf \<equiv> rev (if n < 2 then [] else case next_candidates 0 of (j, is) \<Rightarrow> prime_factorization_nat_main n j is [])"
distinct_subtract_entries_raw: fixes xs :: "('b \<times> 'c) list"   and ys :: "('b \<times> 'c) list" assumes "distinct (map fst xs)" shows "distinct (map fst (subtract_entries_raw xs ys))"
simps_16: fixes x2 :: "'c" shows "set1_list_op (Delete x2) = {x2}"
iMin_gr_iff: fixes I :: "'a set"   and n :: "'a" assumes "I \<noteq> {}" shows "(n < iMin I) = (\<forall>a\<in>I. n < a)"
mono_mono_def: fixes F :: "('c \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'f" shows "mono_mono F = (mono F \<and> (\<forall>f. mono f \<longrightarrow> mono (F f)))"
d_idem2: fixes x :: "'a" shows "d x \<cdot> d x = d x"
e2: fixes \<sigma> :: "nat \<Rightarrow> val" assumes "\<sigma> \<in> space (state_measure (shift_var_set V) (case_nat t1 \<Gamma>))" shows "return_val (expr_sem_rf \<sigma> e2) = expr_sem \<sigma> e2"
bij_betwI: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'c" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> A"   and "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x"   and "\<And>y. y \<in> B \<Longrightarrow> f (g y) = y" shows "bij_betw f A B"
chain_ocrit_def: fixes data :: "nat \<times> 'e"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" shows "chain_ocrit data hs ps p q = (let v = lt (fst p); l = term_of_pair (lcs (pp_of_term v) (lp (fst q)), component_of_term v); i = fst (snd p); j = fst (snd q) in \<exists>h\<in>set hs. let k = fst (snd h) in k \<noteq> i \<and> k \<noteq> j \<and> lt (fst h) adds\<^sub>t l \<and> pair_in_list ps i k \<and> pair_in_list ps j k \<and> fst h \<noteq> 0)"
list_set_union: fixes X :: "'c set"   and Y :: "'c set"   and xs :: "'d list"   and A :: "('d \<times> 'c) set"   and ys :: "'d list" assumes "X \<inter> Y = {}"   and "(xs, X) \<in> \<langle>A\<rangle>list_set_rel"   and "(ys, Y) \<in> \<langle>A\<rangle>list_set_rel" shows "(xs @ ys, X \<union> Y) \<in> \<langle>A\<rangle>list_set_rel"
rep_eq: fixes x :: "('b, nat) alist" shows "is_empty_Bag_impl x = list_all (\<lambda>x. snd x = 0) (alist.impl_of x)"
size_annos: fixes C :: "'a acom" shows "length (annos C) = asize (strip C)"
fold_fun_left_comm: fixes A :: "'c set"   and x :: "'c"   and z :: "('c, real) mapping" assumes "finite A" shows "Mapping.update x p (Finite_Set.fold (\<lambda>x. Mapping.update x p) z A) = Finite_Set.fold (\<lambda>x. Mapping.update x p) (Mapping.update x p z) A"
y_eq: shows "Sup (integral\<^sup>N M ` G) = (SUP i. integral\<^sup>N M (gs i))"
p_preserves_inv_implications_nonneg: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t" assumes "next_propagate' c0 c1 loc t"   and "inv_implications_nonneg c0"   and "inv_imp_plus_work_nonneg c0" shows "inv_implications_nonneg c1"
LeftDerivationIntros_def: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list" shows "LeftDerivationIntros a D L = (\<forall>index. 1 \<le> index \<and> index < length L \<longrightarrow> LeftDerivationIntrosAt a D L index)"
tl_subset: fixes xs :: "'b list"   and A :: "'b set" assumes "xs \<noteq> []"   and "set xs \<subseteq> A" shows "set (tl xs) \<subseteq> A"
dim_update_mat_1: fixes A :: "'b mat"   and ij :: "nat \<times> nat"   and a :: "'b" shows "dim_row (A |\<^sub>m ij \<mapsto> a) = dim_row A"
fenum_def: shows "fenum \<equiv> flat (smap (\<lambda>n. stake n rules) (fromN 1))"
ID1: fixes x :: "'a" shows "ID\<cdot>x = x"
hyps: fixes m :: "nat \<Rightarrow> nat" assumes "\<forall>j<length ts. let b = ts ! j in case b of (uu_, uua_, uub_, sb\<^sub>j, uuc_, uud_, uue_) \<Rightarrow> a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j)" shows "flush_all_until_volatile_write ts (m(a := v)) = (flush_all_until_volatile_write ts m)(a := v)"
pt_swap_bij'': fixes a :: "'b"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "[(a, a)] \<bullet> x = x"
hrabs_le_Infinitesimal: fixes e :: "real star"   and x :: "real star" assumes "e \<in> Infinitesimal"   and "\<bar>x\<bar> \<le> e" shows "x \<in> Infinitesimal"
simps_1: fixes s :: "'s" shows "lowest_tops_impl [] s = Some (None, s)"
bex_triv_one_point2: fixes A :: "'c set"   and a :: "'c" shows "(\<exists>x\<in>A. a = x) = (a \<in> A)"
a_merge: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "real_of_int (int (T_merge t1 t2) + \<Phi> (merge t1 t2) - \<Phi> t1 - \<Phi> t2) \<le> 3 * log 2 (real (size1 t1 + size1 t2)) + 1"
M_eq: shows "M = measure_of \<Omega> (G ` I) (Eps P)"
insert_node_writes: fixes ptr :: "(_) object_ptr"   and new_child :: "(_) node_ptr"   and reference_child_opt :: "(_) node_ptr option"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_child_nodes_locs ptr) (local.a_insert_node ptr new_child reference_child_opt) h h'"
min_tminus_distrib: fixes a :: "real"   and b :: "real"   and c :: "real" shows "min a b \<ominus> c = min (a \<ominus> c) (b \<ominus> c)"
if1: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e1 :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and e2 :: "exp0"   and e3 :: "exp0"   and e' :: "exp0"   and bv :: "'b state \<times> (v, v) result" assumes "evaluate ck env s1 e1 (s2, Rval v1) \<and> do_if v1 e2 e3 = Some e' \<and> evaluate ck env s2 e' bv" shows "evaluate ck env s1 (exp0.If e1 e2 e3) bv"
hidings_entails_def: fixes t :: "'c \<Rightarrow> 'd set"   and s :: "'c \<Rightarrow> 'e set"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(t hidings s entails R) = (\<forall>k. \<forall>x\<in>t k. \<forall>y\<in>s k. R x y)"