hcju/isabelleps huggingface dataset:
int_pow_closed: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier (G\<lparr>carrier := generate G {a}\<rparr>)" shows "x [^]\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> i \<in> carrier (G\<lparr>carrier := generate G {a}\<rparr>)"
that: fixes H :: "'a set"   and a :: "'a" assumes "H0 \<subseteq> H"   and "subgroup H G"   and "a \<in> carrier G - H"   and "carrier G = adjoin G H a" shows "thesis"
Proc_CFG_EntryD: fixes prog :: "cmd"   and et :: "p_edge_kind"   and n' :: "label" assumes "prog \<turnstile> Entry -et\<rightarrow>\<^sub>p n'" shows "n' = Exit \<and> et = IEdge (\<lambda>s. False)\<^sub>\<surd> \<or> n' = Label 0 \<and> et = IEdge (\<lambda>s. True)\<^sub>\<surd>"
c_def: shows "c \<in> carrier Zp \<and> f \<bullet> a \<ominus> f \<bullet> b = (a \<ominus> b) \<otimes> c"
LD_1'_def: fixes r :: "('a \<times> 'a) set"   and \<beta> :: "'a"   and \<alpha> :: "'a"   and \<sigma>1 :: "'a list"   and \<sigma>2 :: "'a list"   and \<sigma>3 :: "'a list" shows "LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 = (set \<sigma>1 \<subseteq> r \<down>s {\<beta>} \<and> length \<sigma>2 \<le> 1 \<and> set \<sigma>2 \<subseteq> {\<alpha>} \<and> set \<sigma>3 \<subseteq> r \<down>s {\<alpha>, \<beta>})"
div_add_self1: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus b a) b = plus (divide a b) one"
h4: shows "satisfy_equation_R p qs [([], []), ([0], []), ([], [0])] [[1], [0], [- 1]]"
case_y_not_in_range: shows "\<forall>x. x \<notin> cbox a b \<longrightarrow> integral UNIV (\<lambda>y. f (y, x)) = (0::'c)"
comp_right_dist_sum: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "class.idempotent_semiring one times sup less_eq less bot" shows "comm_monoid_add.sum sup bot (\<lambda>k. times (f k) x) {k. True} = times (comm_monoid_add.sum sup bot f {k. True}) x"
sorted_filter_greater_subset_drop: fixes xs :: "'a list"   and i :: "nat" assumes "sorted xs"   and "i < length xs" shows "filter_mset ((<) (xs ! i)) (mset xs) \<subseteq># mset (drop (Suc i) xs)"
diag_coeff: fixes x :: "'b poly"   and i :: "nat" shows "diag (coeff x i) = map_sq_matrix (\<lambda>x. coeff x i) (diag x)"
spec_ops_remove1: fixes xs :: "('b \<times> 'c) list"   and deps :: "'c \<Rightarrow> 'b set"   and x :: "'b \<times> 'c" assumes "spec_ops xs deps" shows "spec_ops (remove1 x xs) deps"
op_dghm_components_1: fixes \<FF> :: "V" shows "op_dghm \<FF>\<lparr>ObjMap\<rparr> = \<FF>\<lparr>ObjMap\<rparr>"
cases: fixes x :: "'b \<times> 'b tree" obtains xa :: "'b" and t :: "'b tree" where "x = (xa, t)"
intros_1: fixes G :: "prog"   and membr :: "qtname \<times> memberdecl"   and "class" :: "qtname"   and accclass :: "qtname" assumes "G \<turnstile> membr member_of class"   and "G \<turnstile> Class class accessible_in pid accclass"   and "G \<turnstile> membr in class permits_acc_from accclass" shows "G \<turnstile> membr of class accessible_from accclass"
subst_introduces_no_rids: fixes e :: "('r, 'l, 'v) expr"   and x :: "'v"   and e' :: "('r, 'l, 'v) expr" shows "RID\<^sub>E (subst e x e') \<subseteq> RID\<^sub>E e \<union> RID\<^sub>E e'"
distinct_1: fixes x1 :: "res_crash"   and x21 :: "nat"   and x22 :: "v list" shows "RSCrash x1 \<noteq> RSBreak x21 x22"
nonempty_bind_const: fixes A :: "'b set"   and B :: "'c set" assumes "A \<noteq> {}" shows "A \<bind> (\<lambda>_. B) = B"
inv_into_into: fixes x :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "x \<in> f ` A" shows "inv_into A f x \<in> A"
inner_minus_right: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (uminus y) = - inner x y"
map_prod_sym: fixes A :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b" assumes "sym A" shows "sym (map_prod f f ` A)"
rTau: assumes "\<Psi> \<rhd> P \<sim> P'" shows "Prop pTau P'"
setops\<^sub>s\<^sub>s\<^sub>t_cons_subset: fixes A :: "('d, 'e) stateful_strand_step list"   and a :: "('d, 'e) stateful_strand_step" shows "setops\<^sub>s\<^sub>s\<^sub>t A \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t (a # A)"
tan_monotone': fixes y :: "real"   and x :: "real" assumes "- (pi / 2) < y"   and "y < pi / 2"   and "- (pi / 2) < x"   and "x < pi / 2" shows "(y < x) = (tan y < tan x)"
crdt_ops_reorder_spec: fixes xs :: "('c \<times> 'd) list"   and x :: "'c \<times> 'd"   and deps :: "'d \<Rightarrow> 'c set"   and ys :: "('c \<times> 'd) list"   and zs :: "('c \<times> 'd) list" assumes "spec_ops (xs @ [x]) deps"   and "crdt_ops (ys @ [x] @ zs) deps"   and "set (xs @ [x]) = set (ys @ [x] @ zs)" shows "crdt_ops (ys @ zs @ [x]) deps"
abs_matrix_def: fixes A :: "'b matrix" shows "\<bar>A\<bar> = sup A (- A)"
unop_v11_def: fixes A :: "V"   and f :: "V" shows "unop_v11 A f \<equiv> nop_v11 A (1\<^sub>\<nat>) f"
hyps: fixes xc :: "char list list"   and x :: "char list list"   and xa :: "char list list"   and xb :: "cmd" assumes "ps' = px # xc"   and "(px, x, xa, xb) \<in> set procs"   and "containsCall procs xb xc p'"   and "containsCall procs c ps p"   and "(p', ins, outs, c) \<in> set procs" shows "containsCall procs xb (xc @ p' # ps) p"
surjective: fixes r :: "('d, 'e, 'f, 'g) pre_RedBlack_scheme" shows "r = \<lparr>red = red r, black = black r, subs = subs r, init_conf = init_conf r, confs = confs r, marked = marked r, strengthenings = strengthenings r, \<dots> = pre_RedBlack.more r\<rparr>"
constructive_security_aux_obsf_axioms: shows "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv"
L'_def: fixes F' :: "'c filter"   and g' :: "'c \<Rightarrow> 'b" shows "L' F' g' = {f. \<exists>c>0. \<forall>\<^sub>F x in F'. R (norm (f x)) (c * norm (g' x))}"
h_5: shows "steps0 (st, l', r) tp stp = (aa, ba, ca)"
sorted_wrt_map: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "sorted_wrt R (map f xs) = sorted_wrt (\<lambda>x y. R (f x) (f y)) xs"
step_z_mono: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and Z :: "('e \<Rightarrow> 'f) set"   and l' :: "'d"   and Z' :: "('e \<Rightarrow> 'f) set"   and W :: "('e \<Rightarrow> 'f) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>"   and "Z \<subseteq> W" shows "\<exists>W'. A \<turnstile> \<langle>l, W\<rangle> \<leadsto> \<langle>l', W'\<rangle> \<and> Z' \<subseteq> W'"
f1: shows "rk {A', B', P} \<le> 3"
roofedD: fixes x :: "'a"   and G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set"   and p :: "'a list"   and y :: "'a" assumes "x \<in> roofed_gen G B S"   and "path G x p y"   and "y \<in> B" shows "(\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S"
qTermQSwappedLess_wf: shows "wf qTermQSwappedLess"
iTILL_union': fixes n :: "nat"   and n' :: "nat" shows "[\<dots>n] \<union> [\<dots>n'] = [\<dots>max n n']"
Fvars_fm_simps_4: fixes A :: "fm"   and B :: "fm" shows "Fvars (A AND B) = Fvars A \<union> Fvars B"
scast_up_scast_id: fixes w :: "'b word" assumes "is_up SCAST('b \<rightarrow> 'c)" shows "SCAST('c \<rightarrow> 'b) (SCAST('b \<rightarrow> 'c) w) = w"
bin_rest_numeral_simps_8: fixes w :: "num" shows "- numeral (num.Bit1 w) div 2 = - numeral (w + num.One)"
the_left_def: fixes xs :: "int list"   and j :: "nat" shows "the_left xs j = (GREATEST i. i < j \<and> xs ! i < xs ! j)"
e: fixes y :: "real" assumes "dist y (real_of_ereal x) < e" shows "ereal y \<in> S"
take_all_iff: fixes n :: "nat"   and xs :: "'b list" shows "(xs \<down> n = xs) = (length xs \<le> n)"
Impl_nvalid_Suc: fixes n :: "nat"   and P :: "assn"   and M :: "cname \<times> mname"   and Q :: "assn" assumes "\<Turnstile>n: (P, body M, Q)" shows "\<Turnstile>Suc n: (P, Impl M, Q)"
extremum_uniqueI: fixes a :: "'a" assumes "a \<le> \<bottom>" shows "a = \<bottom>"
Ptl_conforms: shows "path_conforms_with_strategy p (ltl P) \<sigma>"
that: fixes g :: "complex \<Rightarrow> complex" assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)" shows "thesis"
cmp_surj: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and g :: "'f \<Rightarrow> 'g"   and C :: "'g set" assumes "f \<in> A \<rightarrow> B"   and "surj_to f A B"   and "g \<in> B \<rightarrow> C"   and "surj_to g B C" shows "surj_to (cmp g f) A C"
A3b: fixes y :: "i" shows "cod y \<cdot> y \<cong> y"
tminus_def: fixes x :: "real"   and y :: "real" shows "x \<ominus> y = max (x - y) 0"
assms_1: shows "vs @ xs \<in> TS M2 M1 \<Omega> V m i"
simps_2: fixes literal :: "Literal"   and valuation :: "Literal list" shows "inconsistent (literal # valuation) = (opposite literal el valuation \<or> inconsistent valuation)"
IH_4: assumes "insert\<langle>t,s\<rangle> \<in> set A" shows "insert\<langle>t \<cdot> \<theta>,s \<cdot> \<theta>\<rangle> \<in> set (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>)"
IH_1: fixes n :: "'node"   and u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" assumes "n \<in> set ns"   and "set ns \<subseteq> set (\<alpha>n g)"   and "distinct ns"   and "foldr (aux_2 g) ns (Mapping.empty, Mapping.empty) = (u', p')" shows "uses' g n = lookup_multimap u' n"
rtrancl_subset_rtrancl: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "r \<subseteq> s\<^sup>*" shows "r\<^sup>* \<subseteq> s\<^sup>*"
cupdate_def: fixes g :: "'G"   and v :: "'V"   and wv :: "'W infty"   and \<sigma> :: "'qw \<times> 'mr" shows "cupdate g v wv \<sigma> = ASSERT (dinvarC_add \<sigma>) \<bind> (\<lambda>_. let (wl, res) = \<sigma>; pv = mpath' (mr.lookup v res) in FOREACH (succ (g.\<alpha> g) v) (\<lambda>(w', v') (wl, res). if wv + Num w' < mpath_weight' (mr.lookup v' res) then RETURN (qw.insert wl v' (wv + Num w'), mr.update v' ((v, w', v') # the pv, val wv + w') res) else RETURN (wl, res)) (wl, res))"
assms: shows "\<langle>r, s\<rangle> \<in>\<^sub>\<circ> vfield A"
1: shows "finsum M g (insert a A) = g a \<oplus>\<^bsub>M\<^esub> finsum M g A"
topo1: shows "top_on\<^sub>o (annos C1 ! i) (- vars (strip C2))"
compact_space_imp_Bolzano_Weierstrass: fixes X :: "'c topology"   and S :: "'c set" assumes "compact_space X"   and "infinite S"   and "S \<subseteq> topspace X" shows "X derived_set_of S \<noteq> {}"
nat_pow_distrib: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "[n] \<cdot> (x \<oplus> y) = [n] \<cdot> x \<oplus> [n] \<cdot> y"
map_shuffle: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b" assumes "length xs = length ys" shows "[xs [\<mapsto>] ys, x \<mapsto> y] = [xs @ [x] [\<mapsto>] ys @ [y]]"
VLambda_eq_D1: fixes A :: "V"   and f :: "V \<Rightarrow> V"   and B :: "V"   and g :: "V \<Rightarrow> V" assumes "VLambda A f = VLambda B g" shows "A = B"
simple_functionD_1: fixes M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "simple_function M g" shows "finite (g ` space M)"
cInf_lower: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and X :: "'c set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> X"   and "preorder.bdd_below less_eq X" shows "less_eq (Inf X) x"
P2_OT_14_FT: fixes m0 :: "bool"   and m1 :: "bool"   and m2 :: "bool"   and m3 :: "bool" shows "R2_14 (m0, m1, m2, m3) (False, True) = funct_OT_14 (m0, m1, m2, m3) (False, True) \<bind> (\<lambda>(out1, out2). S2_14 (False, True) out2)"
shift2_invD_4: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and d0 :: "nat" assumes "shift2_inv qs"   and "d0 < d" shows "card {q \<in> set qs. poly_deg (fst q) = d0 \<and> m < card (snd q)} \<le> 1"
min_minus': fixes m :: "nat"   and k :: "nat" shows "min (m - k) m = m - k"
senior_token_run: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "(semi_mojmir_def.senior \<delta> q\<^sub>0 w x n = semi_mojmir_def.senior \<delta> q\<^sub>0 w y n) = (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = semi_mojmir_def.token_run \<delta> q\<^sub>0 w y n)"
Nil_neq_Cons: fixes x :: "'b"   and xs :: "'b SList.list"   and R :: "bool" assumes "[] = x # xs" shows "R"
equiv_order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<simeq> y"   and "y \<sqsubseteq> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubseteq> z"
oreturn_bind: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'b \<Rightarrow> 'c option" shows "oreturn x |>> f = f x"
has_fields_decl_above: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and F :: "char list"   and D :: "char list"   and b :: "staticb"   and T :: "ty" assumes "P \<turnstile> C has_fields FDTs"   and "((F, D), b, T) \<in> set FDTs" shows "P \<turnstile> C \<preceq>\<^sup>* D"
conv_dist_comp: fixes x :: "'a"   and y :: "'a" shows "id (x \<sqinter> y) = id y \<sqinter> id x"
chain_iterates: fixes f :: "'a \<Rightarrow> 'a" assumes "monotone (\<le>) (\<le>) f" shows "Complete_Partial_Order.chain (\<le>) (ccpo_class.iterates f)"
pair_in: shows "(wp (body ;; Embed (Sup_trans (fst ` {x. P (fst x) (snd x) \<and> feasible (fst x) \<and> (\<forall>Q. unitary Q \<longrightarrow> unitary (snd x Q)) \<and> le_trans (fst x) (lfp_trans (\<lambda>a. wp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) \<and> le_utrans (gfp_trans (\<lambda>a. wlp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) (snd x)})) \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip), wlp (body ;; Embed (Inf_utrans (snd ` {x. P (fst x) (snd x) \<and> feasible (fst x) \<and> (\<forall>Q. unitary Q \<longrightarrow> unitary (snd x Q)) \<and> le_trans (fst x) (lfp_trans (\<lambda>a. wp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) \<and> le_utrans (gfp_trans (\<lambda>a. wlp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) (snd x)})) \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) \<in> {x. P (fst x) (snd x) \<and> feasible (fst x) \<and> (\<forall>Q. unitary Q \<longrightarrow> unitary (snd x Q)) \<and> le_trans (fst x) (lfp_trans (\<lambda>a. wp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) \<and> le_utrans (gfp_trans (\<lambda>a. wlp (body ;; Embed a \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip))) (snd x)}"
compE_def: fixes \<rho> :: "'b \<Rightarrow> 'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and x :: "'b" shows "compE \<rho> \<xi> x \<equiv> int \<xi> (\<rho> x)"
drop: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Drop], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
comp_converter_parametric: shows "rel_fun (rel_converter (BNF_Def.Grp UNIV id) (BNF_Def.Grp UNIV id)\<inverse>\<inverse> (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g)) (rel_fun (rel_converter (BNF_Def.Grp UNIV f)\<inverse>\<inverse> (BNF_Def.Grp UNIV g) (BNF_Def.Grp UNIV h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse>) (rel_converter (BNF_Def.Grp UNIV id) (BNF_Def.Grp UNIV id)\<inverse>\<inverse> (BNF_Def.Grp UNIV h) (BNF_Def.Grp UNIV k)\<inverse>\<inverse>)) comp_converter comp_converter"
os_is_empty_def: fixes b :: "'b node ref option" shows "os_is_empty b \<equiv> return (b = None)"
N: shows "\<forall>n>N. f $$ - int n = (0::'a)"
TE_resOf_def: shows "TE_resOf \<equiv> ProblemIkTpart.TE_resOf resOf"
map_def: fixes f :: "'a" shows "\<rho>.map f = (if arr f then \<r>[cod f] \<cdot> (f \<otimes> \<I>) else null)"
update_inv: fixes k :: "'a"   and v :: "'c" shows "lift_p P (update k v)"
nat_bit_induct: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "\<And>n. \<lbrakk>P n; 0 < n\<rbrakk> \<Longrightarrow> P (2 * n)"   and "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))" shows "P n"
is_sublist_trans: fixes as :: "'a list"   and bs :: "'a list"   and cs :: "'a list" assumes "is_sublist as bs"   and "is_sublist bs cs" shows "is_sublist as cs"
plossless_converterI: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter" assumes "\<And>a. a \<in> outs_\<I> \<I> \<Longrightarrow> plossless_gpv \<I>' (run_converter conv a)"   and "\<And>a b conv'. \<lbrakk>a \<in> outs_\<I> \<I>; (b, conv') \<in> results_gpv \<I>' (run_converter conv a)\<rbrakk> \<Longrightarrow> plossless_converter \<I> \<I>' conv'" shows "plossless_converter \<I> \<I>' conv"
last_el_greatest: fixes x :: "'a" shows "x \<le> last_el"
ind_topology_axioms_def: fixes X :: "'c set"   and S :: "'c set" shows "ind_topology_axioms X S \<equiv> S \<subseteq> X"
less_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y < x"   and "z < y" shows "z < x"
filter_leD: fixes F :: "'a filter"   and F' :: "'a filter"   and P :: "'a \<Rightarrow> bool" assumes "F \<le> F'"   and "eventually P F'" shows "eventually P F"
update_convs_4: fixes getC' :: "Literal list \<Rightarrow> Literal list"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a" shows "getC_update getC' \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> = \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC' getC, \<dots> = more\<rparr>"
map_commutes_b: fixes Q :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "\<forall>x xs. concat (map (\<lambda>n. if Q (f n) then [f n] else []) xs) = filter Q (map f xs) \<longrightarrow> concat (map (\<lambda>n. if Q (f n) then [f n] else []) (x # xs)) = filter Q (map f (x # xs))"
set: fixes x1 :: "'b literal multiset multiset"   and x2 :: "'b literal multiset"   and x3 :: "'b literal multiset" shows "set_inference (Infer x1 x2 x3) = (\<Union>uu\<in>set_mset x1. \<Union> (set_literal ` set_mset uu)) \<union> \<Union> (set_literal ` set_mset x2) \<union> \<Union> (set_literal ` set_mset x3)"
i_expand_nth_mod_neq_0: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < n mod k" shows "(f \<odot> k) n = NoMsg"
rep_eq: fixes x :: "'e"   and xa :: "'e fset" shows "(x |\<in>| xa) = (x \<in> fset xa)"
comp_lunit_lunit'_1: fixes a :: "'c" assumes "C.ide a" shows "\<l>\<^sub>C[a] \<cdot>\<^sub>C C.lunit' a = a"
simps_12: fixes num' :: "nat \<Rightarrow> nat"   and id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "num_update num' \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num' num, sid = sid, \<dots> = more\<rparr>"
that_2: fixes y :: "'a"   and ys :: "'a list" assumes "xs = y # ys" shows "compare cmp x y \<noteq> Greater"
l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf DocumentClass.known_ptr Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs"
assms_4: fixes v :: "nat \<times> 'b" assumes "v \<in> V" shows "gsuccessors A w V v \<noteq> {}"
map_eq_conv: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and g :: "'e \<Rightarrow> 'd" shows "(map f xs = map g xs) = (\<forall>x\<in>set xs. f x = g x)"
numeral_power_le_of_nat_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and i :: "num"   and n :: "nat"   and x :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (power.power one times (numeral.numeral one plus i) n) (semiring_1.of_nat one plus zero x) = (numeral i ^ n \<le> x)"
map_of_eq_None_iff: fixes xys :: "('b \<times> 'a) list"   and x :: "'b" shows "(Map.map_of xys x = None) = (x \<notin> fst ` set xys)"
vertical_composite_axioms: shows "vertical_composite A B F G K \<rho> \<sigma>\<tau>.map"
must_syncE: fixes t :: "'t"   and x :: "'x"   and m :: "'m" assumes "t \<turnstile> \<langle>x, m\<rangle> \<wrong>" obtains ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" and "actions_ok s t ta" and "m = shr s"
is_terminal: fixes x' :: "'a"   and f :: "'a" assumes "arrow_from_functor C C (\<lambda>x. local.prod x b) x' c f" shows "\<exists>!g. e.is_coext x' f g"
ie1: shows "te q (i + 1) \<le> te p (i + 1)"
unique_connecting_path_set_2: fixes v :: "'a"   and w :: "'a" assumes "v \<in> V"   and "w \<in> V" shows "w \<in> set (v \<leadsto> w)"
dom_comp_is_union: shows "subst_domain (\<sigma>1 \<circ>\<^sub>s \<theta>2) = subst_domain \<sigma>1 \<union> subst_domain \<theta>2"
bi_unique_rel_set: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "bi_unique A" shows "bi_unique (rel_set A)"
wf_subst_fsubst: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>" shows "\<delta> \<in> fsubst"
v11_app_vconverse_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_uminus" shows "-\<^sub>\<real> vreal_uminus\<inverse>\<^sub>\<circ>\<lparr>a\<rparr> = a"
focus_minus: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "focus X (p - q) = focus X p - focus X q"
id_closes: fixes \<Gamma> :: "(name \<times> ty) list" shows "SN.id \<Gamma> closes \<Gamma>"
mset_ecl_and_mset_lit: fixes L :: "'a literal"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "L \<in> cl_ecl C"   and "finite (cl_ecl C)" shows "mset_lit (subst_lit L \<sigma>) \<in># mset_ecl (C, \<sigma>)"