hcju/isabelleps huggingface dataset:
rel_simps_28: fixes n :: "num" shows "\<not> numeral n \<le> (0::'b)"
fractrel_iff: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b \<times> 'b"   and y :: "'b \<times> 'b" assumes "class.idom times one plus zero minus uminus" shows "idom.fractrel times zero x y = (snd x \<noteq> zero \<and> snd y \<noteq> zero \<and> times (fst x) (snd y) = times (fst y) (snd x))"
max_D_in: shows "Max_mset \<delta> \<in># \<delta>"
subadditive_converges_unbounded': fixes u :: "nat \<Rightarrow> real"   and N0 :: "nat" assumes "eventually_subadditive u N0"   and "\<not> bdd_below {u n / real n |n. N0 < n}" shows "(\<lambda>n. ereal (u n / real n)) \<longlonglongrightarrow> - \<infinity>"
bst_join2: fixes l :: "('a \<times> 'b) tree"   and r :: "('a \<times> 'b) tree" assumes "Tree2.bst l"   and "Tree2.bst r"   and "\<forall>x\<in>Tree2.set_tree l. \<forall>y\<in>Tree2.set_tree r. x < y" shows "Tree2.bst (join2 l r)"
ci: shows "Ci = select_min_weight_clause P0 (P1 # P)"
wf_app: fixes r :: "('a \<times> 'a) set"   and f :: "'b \<Rightarrow> 'a" assumes "wf r" shows "wf {(x, y). (f x, f y) \<in> r}"
UP_car_memE_1: fixes g :: "nat \<Rightarrow> 'a"   and x :: "nat" assumes "g \<in> carrier (UP R)" shows "g x \<in> carrier R"
finish_impl_def: fixes u :: "'e"   and s :: "('e, 'f) simple_state_scheme" shows "finish_impl u s \<equiv> ASSERT (ss_stack s \<noteq> [] \<and> u \<in> on_stack s) \<bind> (\<lambda>_. let s = s\<lparr>ss_stack := tl (ss_stack s)\<rparr> in Let (s\<lparr>on_stack := on_stack s - {u}\<rparr>) RETURN)"
rtos_hom5: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0"   and "i div s = r - Suc 0" shows "Suc (rtos r s i) div r = Suc (i mod s)"
Resolution_LSC1: fixes S :: "'b literal set set" assumes "S \<turnstile> \<box>" shows "\<exists>F. is_nnf F \<and> cnf F \<subseteq> S \<and> F, {#} \<Rightarrow>\<^sub>n"
size_of_lhs: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list" shows "dim_vec (w_vec p qs signs) = length signs"
same0: shows "(prod_list (map ((!) (map_sgas sign)) xa) = 1) = (\<forall>p\<in>set (retrieve_polys qs xa). poly p x = 0)"
Chi_HAStates: fixes T :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "T \<in> HAStates A" shows "Chi A T \<subseteq> HAStates A"
hom_dvd: fixes p :: "'b poly"   and q :: "'b poly"   and a :: "'b" assumes "p dvd q" shows "poly p a dvd poly q a"
rep_eq: shows "cblinfun_apply 0 = (\<lambda>x. 0::'b)"
corthogonal_distinct: fixes us :: "'b vec list" assumes "corthogonal us" shows "distinct us"
sgn_if: fixes x :: "'b" shows "sgn x = (if x = (0::'b) then 0::'b else if (0::'b) < x then 1::'b else - (1::'b))"
assms: shows "\<exists>x u. initial_arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y x u"
integrable_spike_set: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'a set" assumes "f integrable_on S"   and "negligible {x \<in> S - T. f x \<noteq> (0::'b)}"   and "negligible {x \<in> T - S. f x \<noteq> (0::'b)}" shows "f integrable_on T"
set_intros_2: fixes y :: "'c"   and x22 :: "'c llist"   and x21 :: "'c" assumes "y \<in> lset x22" shows "y \<in> lset (LCons x21 x22)"
conjugI: fixes r :: "'b list"   and s :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "r \<cdot> s = u"   and "s \<cdot> r = v" shows "u \<sim> v"
assms_2: fixes x :: "'a" assumes "x \<notin> carrier T" shows "f x = undefined"
igSwapBinp_termMOD: shows "igSwapBinp termMOD = swapBinp"
fun_upd_idem_iff: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" shows "(f(x := y) = f) = (f x = y)"
rep_eq: shows "bounds_of_interval 1 = (1::'a, 1::'a)"
elims_3: fixes x :: "'c \<Rightarrow> nat"   and xa :: "nat \<times> nat list \<times> 'c list" assumes "\<not> count_inv x xa" obtains f :: "'c \<Rightarrow> nat" and u :: "nat" and ns :: "nat list" and xs :: "'c list" where "x = f" and "xa = (u, ns, xs)" and "\<not> (\<forall>x. count (mset xs) x = f x)"
hyps: fixes thesis :: "bool" assumes "\<And>ub. \<lbrakk>\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> x < ub; \<forall>x\<ge>ub. \<forall>p\<in>set ps. sgn (poly p x) = sgn_pos_inf p; a < ub\<rbrakk> \<Longrightarrow> thesis"   and "0 \<notin> set ps" shows "thesis"
update_sqn: fixes dsn :: "nat"   and dsk :: "k"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and rip :: "nat"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "(dsn = 0) = (dsk = unk)" shows "sqn rt dip \<le> sqn (update rt rip (dsn, dsk, val, hops, nhip, pre)) dip"
mreq_end2: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and x :: "'c"   and b :: "'b list"   and c :: "'b list"   and a :: "'b" assumes "applied_rule_rev C x b = applied_rule_rev C x c" shows "applied_rule_rev C x (a # b) = applied_rule_rev C x (a # c)"
step: fixes i :: "Integer" assumes "defined i"   and "0 \<le> \<lbrakk>i\<rbrakk>"   and "\<And>j. \<lbrakk>defined j; \<lbrakk>j\<rbrakk> < \<lbrakk>i\<rbrakk>\<rbrakk> \<Longrightarrow> P j" shows "P i"
infinite_imp_bij_betw: fixes A :: "'c set"   and a :: "'c" assumes "infinite A" shows "\<exists>h. bij_betw h A (A - {a})"
SUP_apply: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'c"   and A :: "'e set"   and x :: "'d" shows "Sup (f ` A) x = (SUP y\<in>A. f y x)"
same_snapshot_state_implies_same_result_snapshot_state: fixes c :: "('a, 'b, 'c) configuration"   and p :: "'a"   and d :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration" assumes "ps c p = ps d p"   and "states c p = states d p"   and "c \<turnstile> ev \<mapsto> c'"   and "d \<turnstile> ev \<mapsto> d'" shows "ps d' p = ps c' p"
extensional_push_forward: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real" shows "extensional0 dest.diff_fun_space (push_forward X)"
right_angled_def: fixes l :: "triangle" shows "right_angled l = (\<exists>x\<in>set (Rep_triangle l). x \<sim> 90)"
restrict_top_subset: fixes x :: "'a"   and i :: "'a ifex"   and vr :: "'a"   and vl :: "bool" assumes "x \<in> ifex_var_set (restrict_top i vr vl)" shows "x \<in> ifex_var_set i"
zero_not_in_nonzero: fixes R :: "('a, 'b) ring_scheme" shows "\<zero>\<^bsub>R\<^esub> \<notin> nonzero R"
s1_2: shows "g1 = s1 \<otimes> r1"
cases: fixes x :: "atomUni" obtains p :: "real \<times> real \<times> real" where "x = EqUni pa__"   | p :: "real \<times> real \<times> real" where "x = LeqUni pb__"   | p :: "real \<times> real \<times> real" where "x = NeqUni pc__"   | p :: "real \<times> real \<times> real" where "x = LessUni p"
measurable_nxt: fixes P :: "'a stream \<Rightarrow> bool"   and M :: "'a measure" assumes "Measurable.pred (stream_space M) P" shows "Measurable.pred (stream_space M) (nxt P)"
sum_zero_eq_neg: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier S"   and "y \<in> carrier S"   and "x \<oplus>\<^bsub>S\<^esub> y = \<zero>\<^bsub>S\<^esub>" shows "x = \<ominus>\<^bsub>S\<^esub> y"
mon_ah_def: fixes h :: "'b \<Rightarrow> 'b set" shows "mon_ah h \<equiv> \<Union> {h m |m. True}"
ce_rel_lm_15: fixes x :: "nat"   and y :: "nat"   and A :: "nat set" shows "((x, y) \<in> ce_set_to_rel A) = (c_pair x y \<in> A)"
step: shows "\<Gamma>\<turnstile> ([], (pcs_normal, pcs_abrupt) # pcss @ css, s) \<rightarrow> (cs', pcss @ css, t)"
minimalEnabled_def: fixes msg :: "('p, 'v) message" shows "minimalEnabled msg \<equiv> (\<exists>p. isReceiverOf p msg) \<and> enabled (last exec) msg \<and> (\<exists>n<length exec. enabled (execConf n) msg \<and> (\<forall>n'\<ge>n. n' < length trace \<longrightarrow> msg \<noteq> execMsg n') \<and> (\<forall>n' msg'. (\<exists>p. isReceiverOf p msg') \<and> enabled (last exec) msg' \<and> n' < length trace \<and> enabled (execConf n') msg' \<and> (\<forall>n''\<ge>n'. n'' < length trace \<longrightarrow> msg' \<noteq> execMsg n'') \<longrightarrow> n \<le> n'))"
simps: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec" shows "optimize_no_cond A b c = (case create_optimal_solutions A b c of Inl x \<Rightarrow> Inl x | Inr X \<Rightarrow> Inr (fst (split_n_m_x (dim_vec c) (dim_vec b) X)))"
in_carrier_2: shows "set (map h Us) \<subseteq> carrier S"
init_same_loc: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap" assumes "P \<turnstile> \<langle>INIT C (Cs,b) \<leftarrow> unit,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',(h', l', sh')\<rangle>" shows "l = l'"
ws_prog_cdeclD: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "class G C = Some c"   and "C \<noteq> Object"   and "ws_prog G" shows "is_class G (super c) \<and> (super c, C) \<notin> (subcls1 G)\<^sup>+"
baseE_2: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and ds :: "nat set" assumes "p \<in> sparsegrid' dm" shows "p \<in> grid (base ds p) ds"
res: shows "LLL_Impl.basis_reduction_add_rows n upw i state = state'"
deg_sect_empty: fixes d :: "nat" shows "deg_sect {} d = (if d = 0 then {0} else {})"
swap_msgs_Trans_RecvMarker: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecvMarker ev"   and "isTrans ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e' i = msgs e i"
assms_4: shows "\<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>,P'\<rangle>"
healthy_equivI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_trans t u"   and "healthy t" shows "healthy u"
Some_le: fixes x :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b option" shows "(\<lfloor>x\<rfloor> \<sqsubseteq>\<^bsub>Opt.le r\<^esub> z) = (\<exists>y. z = \<lfloor>y\<rfloor> \<and> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
that: fixes e :: "'a \<Rightarrow> nat" assumes "bij_betw e S UNIV" shows "thesis"
l\<delta>: fixes aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list"   and X :: "(mregex, event_data option list set) mapping"   and ms :: "mregex" assumes "st = (aux, X)"   and "ms \<in> LPDs mr" shows "qtable n (fv_regex r) (mem_restr R) (\<lambda>v. Regex.match (Formula.sat \<sigma> V (map the v)) (from_mregex ms \<phi>s) i (i + length aux)) (l\<delta> (\<lambda>x. x) X rels ms)"
assms_2: fixes k :: "int" assumes "0 \<le> k"   and "k < b"   and "n mod b = k" shows "P n"
simps_1: fixes uu :: "('d \<times> 'd) set"   and uv :: "'c \<Rightarrow> 'd"   and uw :: "'d" shows "ipurge_tr_rev uu uv uw [] = []"
intro: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b" assumes "bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B" shows "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"
nth_prime_le_iff: shows "(nth_prime 0 \<le> nth_prime n) = (0 \<le> n)"
Diff_Compl: fixes A :: "'c set"   and B :: "'c set" shows "A - - B = A \<inter> B"
IMP: assumes "TERM If" shows "\<Gamma>2b \<or>\<^sub>A \<Gamma>2c \<Longrightarrow>\<^sub>t \<Gamma>'"
fun_diff_def: fixes A :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c" shows "A - B = (\<lambda>x. A x - B x)"
\<A>_2: fixes a :: "'a set" assumes "a \<in> \<A>" shows "fst x \<in> a"
ct_suffix_take: fixes ts :: "ct list"   and ts' :: "ct list"   and n :: "nat" assumes "ct_suffix ts ts'" shows "ct_suffix (take (length ts - n) ts) (take (length ts' - n) ts')"
sum_mono: fixes K :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset"   and g :: "'c \<Rightarrow> 'd multiset" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<subseteq># g i" shows "comm_monoid_add.sum (+) {#} f K \<subseteq># comm_monoid_add.sum (+) {#} g K"
induct: fixes P1 :: "('d, 'c) Expr \<Rightarrow> bool"   and P2 :: "('d, 'c) Expr list \<Rightarrow> bool"   and Expr :: "('d, 'c) Expr" assumes "\<And>x. P1 (Const x)"   and "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (Op x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 Expr"
list_ball_nth: fixes n :: "nat"   and xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "n < length xs"   and "\<forall>x\<in>set xs. P x" shows "P (xs ! n)"
alg_select: shows "\<forall>i\<in>II. alg (fst (fst (Rep_IIT i))) (snd (fst (Rep_IIT i))) (fst (snd (Rep_IIT i))) (snd (snd (Rep_IIT i)))"
rcosI: fixes G :: "('a, 'b) monoid_scheme"   and h :: "'a"   and H :: "'a set"   and x :: "'a" assumes "Group.group G"   and "h \<in> H"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "h \<otimes>\<^bsub>G\<^esub> x \<in> H #>\<^bsub>G\<^esub> x"
seqr_right_zero: fixes P :: "('b, 'd) urel" shows "P ;; false = false"
trivial_limit_def: fixes F :: "'b filter" shows "(F = bot) = (\<forall>\<^sub>F x in F. False)"
ord_resolve_ground_inst_sound: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and \<sigma> :: "'s"   and E :: "'a literal multiset"   and I :: "'a set"   and \<eta> :: "'s" assumes "ord_resolve CAs DA AAs As \<sigma> E"   and "I \<Turnstile>m mset CAs \<cdot>cm \<sigma> \<cdot>cm \<eta>"   and "I \<Turnstile> DA \<cdot> \<sigma> \<cdot> \<eta>"   and "is_ground_subst \<eta>" shows "I \<Turnstile> E \<cdot> \<eta>"
summable_Dirichlet_L': fixes s :: "complex" assumes "\<chi> \<noteq> \<chi>\<^sub>0"   and "0 < Re s" shows "summable (\<lambda>n. \<chi> n * of_nat n powr - s)"
keys_entries: fixes k :: "'a"   and t :: "('a, 'b) rbt" shows "(k \<in> set (keys t)) = (\<exists>v. (k, v) \<in> set (entries t))"
intros_1: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "(\<pi> k, \<pi> i) \<in> pdg"
Tw_def: fixes D :: "('c, 'e) Order_scheme"   and T :: "('d, 'f) Order_scheme" shows "Tw\<^bsub>D,T\<^esub> = (\<lambda>a\<in>carrier D. SOME x. x \<in> carrier T \<and> ord_equiv (Iod D (segment D a)) (Iod T (segment T x)))"
ltl_ltake: fixes n :: "enat"   and xs :: "'c llist" shows "ltl (ltake n xs) = ltake (epred n) (ltl xs)"
fixing_arr_gives_natural_transformation_2: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'b \<Rightarrow> 'd"   and f2 :: "'b" assumes "binary_functor A1 A2 B F"   and "partial_magma.arr A2 f2" shows "natural_transformation A1 B (\<lambda>f1. F (f1, partial_magma.dom A2 f2)) (\<lambda>f1. F (f1, partial_magma.cod A2 f2)) (\<lambda>f1. F (f1, f2))"
expand: fixes hd :: "('a, 'b) hd"   and hd' :: "('a, 'b) hd" assumes "is_Var hd = is_Var hd'"   and "\<lbrakk>is_Var hd; is_Var hd'\<rbrakk> \<Longrightarrow> var hd = var hd'"   and "\<lbrakk>is_Sym hd; is_Sym hd'\<rbrakk> \<Longrightarrow> hd.sym hd = hd.sym hd'" shows "hd = hd'"
empty_path_nodes: fixes prog :: "cmd"   and n :: "w_node"   and n' :: "w_node" assumes "While_CFG.path prog n [] n'" shows "n = n'"
simps_1: fixes f :: "'a" shows "list_comb f [] = f"
Un_absorb2: fixes B :: "'e set"   and A :: "'e set" assumes "B \<subseteq> A" shows "A \<union> B = A"
carT1_def: shows "carT1 = {(Kl, lab) |Kl lab. isTree Kl lab \<and> isNode1 Kl lab []}"
QR_lemma_06: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.b p q + QR.c p q = QR.r p q"
sum_list_closed: fixes R :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and as :: "'b list" assumes "Ring1 R"   and "set (map f as) \<subseteq> R" shows "sum_list (map f as) \<in> R"
mono: shows "mono (\<lambda>p x1 x2. \<exists>\<omega> s. x1 = s \<and> x2 = \<omega> \<and> p (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K' s))"
power_down_nonneg_iff: fixes prec :: "nat"   and b :: "real"   and n :: "nat" shows "(0 \<le> power_down prec b n) = (even n \<or> 0 \<le> b)"
Diag_TensorE_1: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "\<^bold>\<langle>un_Prim t\<^bold>\<rangle> = t"
bind_try_Done_Fail: fixes f :: "'d \<Rightarrow> ('a, 'b, 'c) gpv"   and x :: "'d"   and gpv :: "('d, 'b, 'c) gpv" assumes "f x = Fail" shows "bind_gpv (TRY gpv ELSE Done x) f = bind_gpv gpv f"
nodes_add_node: fixes v :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "nodes (add_node v g) = insert v (nodes g)"
Pow_singleton_iff: fixes X :: "'a set"   and Y :: "'a set" shows "(Pow X = {Y}) = (X = {} \<and> Y = {})"
ind_squarefree: shows "ind squarefree = dirichlet_prod g f"
lt_eq_min_term_monomial: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "i.lpp p = i.punit.min_term" shows "monomial (i.lcf p) i.punit.min_term = p"
distinct_3: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x13 :: "floatarith"   and x14 :: "form"   and x31 :: "floatarith"   and x32 :: "floatarith" shows "Bound x11 x12 x13 x14 \<noteq> Less x31 x32"
simps_3: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.rank_skew_invar (t # t' # bq) = (SkewBinomialHeapStruc.rank t \<le> SkewBinomialHeapStruc.rank t' \<and> SkewBinomialHeapStruc.rank_invar (t' # bq))"
hseqI: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a"   and \<nu> :: "'a"   and c :: "'a" assumes "B.in_hhom \<mu> a b"   and "B.in_hhom \<nu> b c" shows "B.hseq \<nu> \<mu>"
T_F: fixes s :: "'c event list"   and P :: "'c process" assumes "s \<in> T P" shows "(s, {}) \<in> F P"
filter_diff_mset: fixes P :: "'k \<Rightarrow> bool"   and M :: "'k multiset"   and N :: "'k multiset" shows "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
WCFG_deterministic: fixes prog :: "cmd"   and n\<^sub>1 :: "label"   and et\<^sub>1 :: "p_edge_kind"   and n\<^sub>1' :: "label"   and n\<^sub>2 :: "label"   and et\<^sub>2 :: "p_edge_kind"   and n\<^sub>2' :: "label" assumes "prog \<turnstile> n\<^sub>1 -et\<^sub>1\<rightarrow>\<^sub>p n\<^sub>1'"   and "prog \<turnstile> n\<^sub>2 -et\<^sub>2\<rightarrow>\<^sub>p n\<^sub>2'"   and "n\<^sub>1 = n\<^sub>2"   and "n\<^sub>1' \<noteq> n\<^sub>2'" shows "\<exists>Q Q'. et\<^sub>1 = IEdge (Q)\<^sub>\<surd> \<and> et\<^sub>2 = IEdge (Q')\<^sub>\<surd> \<and> (\<forall>s. (Q s \<longrightarrow> \<not> Q' s) \<and> (Q' s \<longrightarrow> \<not> Q s))"
coeff_diff: fixes p :: "'a poly"   and q :: "'a poly"   and n :: "nat" shows "poly.coeff (p - q) n = poly.coeff p n - poly.coeff q n"
beta_cfun: fixes f :: "'b \<Rightarrow> 'c"   and u :: "'b" assumes "cont f" shows "(\<Lambda> x. f x)\<cdot>u = f u"
simps_18: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and l :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Floor l) env = None"
distinct_23: shows "IndexOfOp \<noteq> string_binop.GreaterOp"
span_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ('b, 'c) vec"   and S :: "('b, 'c) vec set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> vec.span S" shows "sum f A \<in> vec.span S"
a_get_disconnected_nodes_def: fixes document_ptr :: "(_) document_ptr" shows "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_nodes document_ptr = get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes"
isLub_le_isUb: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "isLub R S x"   and "isUb R S y" shows "x \<le> y"
ide_in_hom: fixes a :: "'c" shows "Cop.ide a = Cop.in_hom a a a"
obtain_t_subset_points: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and grouping :: "int"   and index :: "int" assumes "t_wise_balance point_set block_collection grouping index" obtains T :: "'b set" where "T \<subseteq> point_set" and "int (card T) = grouping" and "finite T"
dim_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "dim_vec (vec n f) = n"
no_zero_height_Step_derivs: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, 0, ll, t) \<in> Step" shows "False"
ra1: shows "fps_inv a $ 1 \<noteq> (0::'a)"
p: shows "p = (\<Sum>f\<in>A. q f * f)"
summable_mult2: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "summable f" shows "summable (\<lambda>n. f n * c)"
assms_2: fixes i :: "'a" assumes "i \<in> A" shows "0 \<le> f i"
strict_coboundedI1: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c < a" shows "c < sup a b"
access_empty_lookup: fixes root :: "('a att_scheme \<times> 'b, 'a att_scheme, 'c) env"   and path :: "'c list"   and uid :: "nat" shows "access root path uid {} = lookup root path"
intro: fixes R :: "('d, 'e) ring_scheme"   and S :: "('f, 'g) ring_scheme"   and h :: "'d \<Rightarrow> 'f" assumes "ring R"   and "ring S"   and "ring_hom_ring_axioms R S h" shows "ring_hom_ring R S h"
wset_thread_ok_upd_ws: fixes ws :: "'d \<Rightarrow> 'e wait_set_status option"   and ts :: "'d \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'d"   and xln :: "'f \<times> 'g \<Rightarrow>f nat"   and w :: "'e wait_set_status option" assumes "wset_thread_ok ws ts"   and "ts t = \<lfloor>xln\<rfloor>" shows "wset_thread_ok (ws(t := w)) ts"
measurable_convolution1: fixes A :: "'a measure"   and M :: "'b measure"   and N :: "'b measure" shows "A \<rightarrow>\<^sub>M (M \<star> N) = borel_measurable A"
PsiInv_stutter: shows "\<turnstile> unchanged (x, y, sem, pc1, pc2) \<and> $PsiInv \<longrightarrow> PsiInv$"
assms_3: shows "\<forall>B\<in>H. atom i \<sharp> B"
rep_eq: shows "blinfun_apply no_real_inner.zerosnd = (\<lambda>(x, y). (0, y))"
\<tau>exec_moves_append: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and es' :: "'addr expr1 list" assumes "\<tau>exec_moves ci P t es h s s'" shows "\<tau>exec_moves ci P t (es @ es') h s s'"
leg1_simps_4: shows "trg (tab\<^sub>1 f) = trg f"
aux: fixes i :: "'a" assumes "weight_spmf p = 0" shows "spmf p i = 0"
seqI': fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>S c\<guillemotright>" shows "S.seq g f"
dependent_finite: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set" assumes "module scale"   and "finite S" shows "module.dependent scale S = (\<exists>u. (\<exists>v\<in>S. u v \<noteq> (0::'b)) \<and> (\<Sum>v\<in>S. scale (u v) v) = (0::'c))"
mult_sm_mm: fixes k :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "k *\<^sub>s\<^sub>m A *\<^sub>m\<^sub>m B = k *\<^sub>s\<^sub>m (A *\<^sub>m\<^sub>m B)"