hcju/isabelleps huggingface dataset:
map_seq: fixes f :: "'a" assumes "A.arr f" shows "B.seq (\<tau> (A.cod f)) (\<sigma> f)"
oth_class_taut_6_b: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<or> \<psi> \<^bold>\<equiv> \<^bold>\<not>(\<^bold>\<not>\<phi> \<^bold>& \<^bold>\<not>\<psi>) in v]"
assms_1: shows "a \<noteq> \<infinity> \<or> b \<noteq> \<infinity>"
subgroup_finite: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G"   and "finite (carrier G)" shows "finite H"
term_okD1: 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 t :: "Core.term" assumes "term_ok \<Theta> t" shows "term_ok' (sig \<Theta>) t"
is_valid_operator_sas_plus_then_1: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "\<forall>(v, a)\<in>set (sas_plus_operator.precondition_of op). v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)"
suc_empty: fixes s :: "nat \<Rightarrow> 'b"   and m :: "nat" assumes "emptyseq (s |\<^sub>s m)" shows "emptyseq (s |\<^sub>s Suc m)"
Infinitesimal_diff: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Infinitesimal"   and "y \<in> Infinitesimal" shows "x - y \<in> Infinitesimal"
select_convs_12: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getCl \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getCl"
prodag_sameTr2: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and B :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)"   and "\<forall>k\<in>I. A k = B k" shows "prod_pOp I A = prod_pOp I B"
classes_consistent: fixes S :: "('a, 'b) eq_object_scheme"   and A :: "'a set" assumes "equivalence S"   and "A \<subseteq> carrier S" shows "is_closed\<^bsub>S\<^esub> (closure_of\<^bsub>S\<^esub> A)"
exp_cnj: fixes z :: "complex" shows "cnj (exp z) = exp (cnj z)"
mexp: shows "m = 10 * (m div 10) + c \<and> c < 10"
U_def: shows "U = {v \<in> V. T v = e \<sharp>\<cdot> v}"
is_element_ptr_kind_none: fixes node_ptr :: "(_) node_ptr" assumes "\<not> is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr" shows "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr = None"
commute: fixes a :: "'c"   and b :: "'c" shows "sup a b = sup b a"
vector_vector_star: fixes v :: "'a" assumes "vector v" shows "(v * v\<^sup>T)\<^sup>\<star> = Rf v (v\<^sup>T)"
MatchId: fixes a :: "name"   and P :: "pi" shows "[a\<frown>a]P \<equiv>\<^sub>s P"
assms_1: shows "\<exists>tid\<in>actHn (t n). length (bc (\<sigma>\<^bsub>tid\<^esub>t n)) = x"
atS_un: fixes p :: "'e"   and l :: "'f set"   and l' :: "'f set"   and s :: "('g, 'f, 'e, 'h, 'i) system_state" shows "atS p (l \<union> l') s = (atS p l s \<or> atS p l' s)"
simps_3: fixes ys :: "'a list" shows "sublist_at [] ys 0 = True"
primfun_dominates: fixes f :: "primfun"   and g :: "primfun" assumes "f < g" shows "dominates at_top (eval_primfun' f) (eval_primfun' g)"
SUP_le_iff: 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 f :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and u :: "'e" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (f ` A)) u = (\<forall>i\<in>A. less_eq (f i) u)"
assms: shows "(red F)\<^sup>*\<^sup>* p 0"
i: shows "ground_clause (subst_cl (subst_cl (subst_cl (cl_ecl C) \<sigma>) \<theta>) \<eta>)"
bzero: shows "\<beta> fs i \<noteq> (0::'a)"
actions_ok'_iff: fixes s :: "('g, 'h) locks \<times> (('h \<Rightarrow> ('x \<times> 'g \<Rightarrow>f nat) option) \<times> 'i) \<times> ('h \<Rightarrow> 'j wait_set_status option) \<times> 'h set"   and t :: "'h"   and ta :: "('g, 'h, 'k, 'i, 'j, 'l) thread_action" shows "actions_ok' s t ta = (lock_ok_las' (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub> \<and> thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub> \<and> cond_action_oks' s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub> \<and> wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> \<and> interrupt_actions_ok' (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>)"
Weierstrass_m_test'_ev: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and M :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>A. norm (f n x) \<le> M n"   and "summable M" shows "uniformly_convergent_on A (\<lambda>n x. \<Sum>i<n. f i x)"
hcomp_in_vhomE: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<nu> \<star> \<mu> : f \<Rightarrow>\<^sub>B g\<guillemotright>" obtains "B.arr \<mu>" and "B.arr \<nu>" and "src \<nu> = trg \<mu>" and "src \<mu> = src f" and "src \<mu> = src g" and "trg \<nu> = trg f" and "trg \<nu> = trg g"
polynate_norm: fixes p :: "'b poly" shows "isnpoly (polynate p)"
inner_add_right: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (plus y z) = inner x y + inner x z"
cong_modulus_mult: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and division_segment :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and m :: "'b"   and n :: "'b" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment"   and "unique_euclidean_semiring.cong modulo x y (times m n)" shows "unique_euclidean_semiring.cong modulo x y m"
not_zero: shows "\<forall>i\<in>{i. i < n}. m i \<noteq> 0"
reachable_mono: fixes H :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c"   and G :: "('c, 'd) pre_digraph" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"   and "subgraph H G" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
dummy_Lcm_def: fixes x :: "'b set" shows "dummy_Lcm x = Lcm x"
hyps_1: shows "liftAll (\<lambda>a. qAFresh ys (y @ys[z1 \<and> z2]_zs) (a #[[z1 \<and> z2]]_zs) = qAFresh ys y a) inp"
punit_monom_mult_monomial: fixes c :: "'b"   and s :: "'a"   and v :: "'a"   and d :: "'b" shows "MPoly_Type_Class.punit.monom_mult c s (monomial d v) = monomial (c * d) (s + v)"
that: fixes c :: "'a" assumes "poly p (x + y) = poly p x + c * y" shows "thesis"
one_minus_one_minus_q_x_n_integral_2: fixes q :: "real"   and n :: "nat" assumes "q \<in> {0<..<1}" shows "LBINT x=0..\<infinity>. 1 - (1 - q powr x) ^ n = - harm n / ln q"
update'_def: fixes v :: "'V"   and \<sigma> :: "'V set \<times> ('V \<Rightarrow> ('V \<times> 'W \<times> 'V) list option)" shows "update' v \<sigma> \<equiv> ASSERT (update_pre v \<sigma>) \<bind> (\<lambda>_. let (wl, res) = \<sigma>; wv = path_weight' (res v); pv = res v in FOREACH\<^bsup>uinvar v wl res\<^esup> (succ G v) (\<lambda>(w', v') (wl, res). if wv + Num w' < path_weight' (res v') then ASSERT (v' \<in> wl \<and> pv \<noteq> None) \<bind> (\<lambda>_. RETURN (wl, res(v' \<mapsto> the pv @ [(v, w', v')]))) else RETURN (wl, res)) (wl, res))"
prv_eqv_rawpsubst: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "prv (eqv \<phi> \<psi>)" shows "prv (eqv (rawpsubst \<phi> txs) (rawpsubst \<psi> txs))"
cod_cod: fixes f :: "'b" assumes "D.arr f" shows "D.cod (D.cod f) = D.cod f"
simps_3: fixes states :: "hf set"   and init :: "hf"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf"   and more :: "'c" shows "dfa.final \<lparr>dfa.states = states, init = init, final = final, nxt = nxt, \<dots> = more\<rparr> = final"
dist: shows "distinct (map fst (step (hd w) (ss ! hd w)))"
isState_hd_purgeIdle: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list" assumes "GM_sec_model.wffp do sl"   and "isState (hd sl)" shows "GM_sec_model.purgeIdle sl \<noteq> [] \<and> hd (GM_sec_model.purgeIdle sl) = hd sl"
R: fixes b1 :: "'a"   and c1 :: "'b" assumes "P b1" shows "hn_refine (R1 b1 c1 * \<Gamma>) (c c1) (R1p b1 c1 * \<Gamma>') R (b b1)"
cong_int: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "[a = b] (mod m)" shows "[int a = int b] (mod int m)"
pell_valuation_pos_pos: fixes x :: "int"   and y :: "int" assumes "0 < x"   and "0 < y"   and "solution (x, y)" shows "1 < pell_valuation (x, y)"
cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_some: fixes obj :: "(_) Object"   and shadow_root :: "('m, 'n, 'o, 'p option) RShadowRoot_scheme" shows "(cast obj = Some shadow_root) = (cast shadow_root = obj)"
exec_move_BinOpI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and bop :: "bop"   and e' :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e \<guillemotleft>bop\<guillemotright> e') h s ta h' s'"
poly_map_poly_cnj: fixes p :: "complex poly"   and x :: "complex" shows "poly (map_poly cnj p) x = cnj (poly p (cnj x))"
valof_def: shows "valof \<equiv> map_fun Rep_float id (\<lambda>(s, e, f). let x = TYPE(('a, 'b) IEEE.float) in if e = 0 then (- 1) ^ unat s * (2 / 2 ^ bias x) * (real_of_word f / 2 ^ LENGTH('b)) else (- 1) ^ unat s * (2 ^ unat e / 2 ^ bias x) * (1 + real_of_word f / 2 ^ LENGTH('b)))"
continuous_map_canonical_const: fixes X :: "'b topology"   and c :: "'c" shows "continuous_map X euclidean (\<lambda>x. c)"
prj_simps_6: shows "C.cod \<mu>_\<nu>\<pi>.prj\<^sub>0 = \<nu>\<pi>.apex"
BisT_transC: fixes indis :: "('a \<times> 'a) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a"   and c' :: "('b, 'c) com"   and s' :: "'a" assumes "PL_Indis indis"   and "(c, d) \<in> PL_Indis.BisT tval aval indis"   and "PL.mustT tval aval c s"   and "PL.mustT tval aval d t"   and "(s, t) \<in> indis"   and "PL.transC tval aval (c, s) (c', s')" obtains d' :: "('b, 'c) com" and t' :: "'a" where "PL.MtransC tval aval (d, t) (d', t')" and "(s', t') \<in> indis" and "(c', d') \<in> PL_Indis.BisT tval aval indis"
sum_upto_altdef: fixes f :: "nat \<Rightarrow> 'a"   and x :: "real" shows "sum_upto f x = sum f {0<..nat \<lfloor>x\<rfloor>}"
ffact_add_diff_assoc: fixes a :: "'b"   and n :: "nat" shows "(a - of_nat n) * ffact n a + of_nat n * ffact n a = a * ffact n a"
fun_upds_apply: fixes m :: "'d \<Rightarrow> 'c option"   and xs :: "'d list"   and ys :: "'c list"   and x :: "'d" shows "(m(xs [\<mapsto>] ys)) x = (let xs' = take (length ys) xs in if x \<in> set xs' then Some (ys ! last_index xs' x) else m x)"
list_all2_antisym: fixes P :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'e list" assumes "\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y"   and "list_all2 P xs ys"   and "list_all2 Q ys xs" shows "xs = ys"
eq_fold: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "F g A = Finite_Set.fold ((\<^bold>*) \<circ> g) \<^bold>1 A"
discrete_Markov_process_axioms: shows "discrete_Markov_process M K"
barb: shows "(\<exists>S. S \<in>S Q \<and> S\<down><SWB>a) \<or> (\<exists>T. T \<in>T Q \<and> T\<down><TWB>a)"
le_infE: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x (inf a b)" obtains "less_eq x a" and "less_eq x b"
sin_series: fixes z :: "'a" shows "(\<lambda>n. ((- 1) ^ n / fact (2 * n + 1)) *\<^sub>R z ^ (2 * n + 1)) sums sin z"
distinct_map_filterI: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "distinct (list.map f xs)" shows "distinct (list.map f (List.filter P xs))"
take_update: fixes n :: "nat"   and l :: "'a list"   and i :: "nat"   and x :: "'a" shows "take n (l[i := x]) = (take n l)[i := x]"
in_gen_check: fixes xs :: "nat list"   and s :: "nat"   and as :: "nat list" assumes "(xs, s) \<in> set (c2.gen_check as)" shows "length xs = length as \<and> s = as \<bullet> xs"
iwlsFSbSwTR_iterSTR_termFSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSbSwTR MOD" shows "termFSbImorph (iterSTR MOD) (iterAbsSTR MOD) MOD"
all_distinct_left: fixes l :: "'c tree"   and x :: "'c"   and b :: "bool"   and r :: "'c tree" assumes "all_distinct (Node l x b r)" shows "all_distinct l"
length_Cons: fixes x :: "'k"   and xs :: "'k list" shows "length (x # xs) = Suc (length xs)"
disjoint_sets_def: fixes X :: "'c set"   and Y :: "'c set" shows "disjoint_sets X Y = SPEC (\<lambda>b. b \<longrightarrow> X \<inter> Y = {})"
synth_analz_fst: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> synth (analz H)" shows "X \<in> synth (analz H)"
prob: shows "(\<Sum>\<^sup>+ x. ennreal (case x of (x, y) \<Rightarrow> enn2real (h x y))) = 1"
differentiable_on_comp: fixes g :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and f :: "'d \<Rightarrow> 'e" assumes "g differentiable_on S"   and "f differentiable_on g ` S" shows "f \<circ> g differentiable_on S"
eq_minus_cnj_iff_imag: fixes z :: "complex" shows "(cnj z = - z) = is_imag z"
Wax_satPB: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "GE.satPB \<xi> Wax"
card_of_ordLeq: fixes A :: "'d set"   and B :: "'e set" shows "(\<exists>f. inj_on f A \<and> f ` A \<subseteq> B) = (|A| \<le>o |B|)"
wf_asym: fixes r :: "('g \<times> 'g) set"   and a :: "'g"   and x :: "'g" assumes "wf r"   and "(a, x) \<in> r" obtains "(x, a) \<notin> r"
simps: fixes a :: "'d"   and r :: "('e \<times> 'd) set" shows "(a \<in> Range r) = (\<exists>aa b. a = b \<and> (aa, b) \<in> r)"
Q: shows "four_block_mat Q1 (0\<^sub>m n m) (0\<^sub>m m n) Q2 \<in> carrier_mat (n + m) (n + m)"
ord_def: fixes n :: "'a"   and a :: "'a" shows "ord n a = (if coprime n a then LEAST d. 0 < d \<and> [a ^ d = 1::'a] (mod n) else 0)"
csa_in_hyp: shows "(csa \<in> set (find_consistent_signs_at_roots (coprime_r (cast_rat_list fs)) (cast_rat_list fs))) = (csa \<in> set (characterize_consistent_signs_at_roots (coprime_r (cast_rat_list fs)) (cast_rat_list fs)))"
elims_3: fixes x :: "real_max_bot"   and xa :: "real_max_bot" assumes "\<not> x < xa" obtains uu_ :: "real_max_bot" where "xa__ = uu_" and "xa = MInfty"   | x :: "real" and y :: "real" where "xa__ = R x" and "xa = R y" and "\<not> x < y"
tendsto_cong_limit: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter"   and k :: "'c" assumes "(f \<longlongrightarrow> l) F"   and "k = l" shows "(f \<longlongrightarrow> k) F"
det_rank_iff: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n" shows "(det A \<noteq> (0::'a)) = (rank A = n)"
partn_lst_VWF_degenerate: fixes k :: "nat"   and n :: "nat"   and \<alpha>s :: "V list" assumes "k < n" shows "partn_lst_VWF \<omega> (ord_of_nat k # \<alpha>s) n"
Diff_empty: fixes A :: "'b set" shows "(A::'b set) - {} = A"
uminus_moebius_def: shows "uminus_class.uminus = moebius_inv"
hom_dvd: fixes p :: "'a"   and q :: "'a" assumes "p dvd q" shows "hom p dvd hom q"
rev: fixes xs :: "'a list" shows "sum_list (rev xs) = sum_list xs"
HSem_reorder: fixes \<Gamma> :: "('a \<times> 'b) list"   and \<Delta> :: "('a \<times> 'b) list"   and ESem :: "'b \<Rightarrow> ('a \<Rightarrow> 'c) \<rightarrow> 'c"   and \<rho> :: "'a \<Rightarrow> 'c" assumes "map_of \<Gamma> = map_of \<Delta>" shows "has_ESem.HSem ESem \<Gamma>\<cdot>\<rho> = has_ESem.HSem ESem \<Delta>\<cdot>\<rho>"
frame_rem4: fixes P :: "assn" shows "P \<Longrightarrow>\<^sub>t emp"
A: fixes x :: "'a" assumes "x \<in> A" shows "\<exists>y. y \<in> A \<and> R x y"
comb_def: shows "comb = a *\<^sub>R p + b *\<^sub>R q + c *\<^sub>R r"
sets_count_space: fixes \<Omega> :: "'a set" shows "sets (count_space \<Omega>) = Pow \<Omega>"
path_split_second: fixes n :: "'node"   and as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as @ a # as'\<rightarrow>* n'" shows "sourcenode a -a # as'\<rightarrow>* n'"
upd_invs_Some_eq: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and tas :: "('f, 'i, 'j) new_thread_action list"   and t :: "'f"   and x :: "'g \<times> 'h \<Rightarrow>f nat"   and I :: "'f \<Rightarrow> 'k option"   and Q :: "'k \<Rightarrow> 'f \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> bool" assumes "thread_oks ts tas"   and "ts t = \<lfloor>x\<rfloor>" shows "upd_invs I Q tas t = I t"
dc2c_c2dc: fixes \<chi> :: "nat \<Rightarrow> complex" assumes "character G \<chi>" shows "dc2c (c2dc \<chi>) = \<chi>"
assms: shows "\<not> rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2"
C_card_eq_D_plus_E: shows "card C = card D + card E"
4: shows "w \<sqinter> top * prim_E w v e \<le> prim_E w v e"
nsets_Pi_contra: fixes A' :: "'a set"   and A :: "'a set"   and n :: "nat"   and B :: "'a set \<Rightarrow> 'b set" assumes "A' \<subseteq> A" shows "Pi ([A]\<^bsup>n\<^esup>) B \<subseteq> Pi ([A']\<^bsup>n\<^esup>) B"
upset_to_downset2: fixes dual :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.ord_with_dual dual less_eq" shows "ord.upset less_eq = (`) dual \<circ> ord.downset less_eq \<circ> dual"
orderE: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" obtains "a = inf a b"
Inf_superset_mono: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and B :: "'d set"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "B \<subseteq> A" shows "less_eq (Inf A) (Inf B)"
identify_blocks_def: fixes A :: "'b mat"   and i :: "nat" shows "identify_blocks A i = identify_blocks_main A i []"
subsums_def: fixes c\<^sub>2 :: "('c, 'd) conf"   and c\<^sub>1 :: "('c, 'd) conf" shows "c\<^sub>2 \<sqsubseteq> c\<^sub>1 \<equiv> states c\<^sub>2 \<subseteq> states c\<^sub>1"
solution_n0: fixes s :: "'s" assumes "s \<in> S"   and "n s = 0" shows "x s = 0"
simps_2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "PAC_Checker_Init.merge f xs [] = xs"
less_le_not_le: fixes x :: "'k multiset"   and y :: "'k multiset" shows "(x \<subset># y) = (x \<subseteq># y \<and> \<not> y \<subseteq># x)"
presheaf_of_rings_axioms_def: fixes is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd" shows "presheaf_of_rings_axioms is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> ((\<forall>U V. is_open U \<longrightarrow> is_open V \<longrightarrow> V \<subseteq> U \<longrightarrow> ring_homomorphism (\<rho> U V) (\<FF> U) (add_str U) (mult_str U) (zero_str U) (one_str U) (\<FF> V) (add_str V) (mult_str V) (zero_str V) (one_str V)) \<and> \<FF> {} = {b}) \<and> (\<forall>U. is_open U \<longrightarrow> (\<forall>x. x \<in> \<FF> U \<longrightarrow> \<rho> U U x = x)) \<and> (\<forall>U V W. is_open U \<longrightarrow> is_open V \<longrightarrow> is_open W \<longrightarrow> V \<subseteq> U \<longrightarrow> W \<subseteq> V \<longrightarrow> (\<forall>x. x \<in> \<FF> U \<longrightarrow> \<rho> U W x = (\<rho> V W \<circ> \<rho> U V) x))"
assms_2: assumes "\<Phi>"   and "\<Psi>" shows "m' \<le> \<Down> R c"
hf_to_ide_def: fixes x :: "hf" shows "hf_to_ide x = mkIde (UP ` hfset x)"
override_D_def: fixes m1 :: "'c \<Rightarrow> 'd decision option"   and m2 :: "'c \<Rightarrow> 'd decision option" shows "m1 \<Oplus>\<^sub>D m2 = (\<lambda>x. case m2 x of \<bottom> \<Rightarrow> m1 x | \<lfloor>allow a\<rfloor> \<Rightarrow> case m1 x of \<bottom> \<Rightarrow> \<lfloor>allow a\<rfloor> | \<lfloor>allow \<alpha>\<rfloor> \<Rightarrow> \<lfloor>allow a\<rfloor> | \<lfloor>deny b\<rfloor> \<Rightarrow> \<lfloor>deny b\<rfloor> | \<lfloor>deny a\<rfloor> \<Rightarrow> \<lfloor>deny a\<rfloor>)"
f: shows "\<guillemotleft>f : trg\<^sub>C g \<rightarrow>\<^sub>C src\<^sub>C g\<guillemotright> \<and> C.ide f \<and> D.isomorphic (F f) f'"
Units_inv_Units: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> Units G" shows "inv\<^bsub>G\<^esub> x \<in> Units G"
fps_mult_fps_X_power_nonzero_1: fixes f :: "'a fps"   and n :: "nat" assumes "f \<noteq> 0" shows "fps_X ^ n * f \<noteq> 0"
ind2: shows "lin_indpt (set (((adjuster n w us + w) # us) @ ws))"
ag_r_inv1: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c" assumes "aGroup A"   and "x \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> x = \<zero>\<^bsub>A\<^esub>"
sel_4: fixes x4 :: "'a set" shows "the_Abs (Abs x4) = x4"
vdomain_atE: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> xs" obtains b :: "V" where "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> xs" and "xs\<lparr>a\<rparr> = b"
dH_equal_def: fixes E :: "'c \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and DA :: "'d \<Rightarrow> 'f"   and d :: "'f"   and H :: "('f \<times> 'c) set"   and m :: "'d \<Rightarrow> 'e"   and m' :: "'d \<Rightarrow> 'e" shows "WHATWHERE.dH_equal E DA d H m m' \<equiv> WHATWHERE.d_equal DA d m m' \<and> (\<forall>(d', e)\<in>H. d' \<le> d \<longrightarrow> E e m = E e m')"