hcju/isabelleps huggingface dataset:
hyp_wrong_at_last: fixes j :: "nat"   and t :: "nat" shows "\<phi> (the (s (e_butlast (local.prefixes j (Suc t))))) (e_length (local.prefixes j (Suc t)) - 1) \<noteq> \<psi> j (e_length (local.prefixes j (Suc t)) - 1)"
integral_vec1_eq: fixes S :: "(real, 1) vec set"   and f :: "(real, 1) vec \<Rightarrow> 'a" shows "integral S f = integral ((\<lambda>x. x $ 1) ` S) (f \<circ> vec)"
llist_all2_lmap1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and xs :: "'c llist"   and ys :: "'b llist" shows "llist_all2 P (lmap f xs) ys = llist_all2 (\<lambda>x. P (f x)) xs ys"
xs: shows "(xs, X) \<in> \<langle>\<langle>A\<rangle>default_rel d\<rangle>list_wset_rel"
gg_def: fixes mrk :: "'node set"   and atom :: "'a \<Rightarrow> bool"   and ptr :: "'a \<Rightarrow> 'node"   and x :: "'a" shows "gg mrk atom ptr x \<equiv> ptr x \<noteq> nil \<and> ptr x \<notin> mrk \<and> \<not> atom x"
Wa_def: fixes D :: "('c, 'd) Order_scheme"   and W :: "'c set"   and g :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "Wa D W g a = (W \<subseteq> carrier D \<and> Worder (Iod D W) \<and> a \<in> W \<and> (\<forall>x\<in>W. a \<preceq>\<^bsub>D\<^esub> x) \<and> (\<forall>x\<in>W. if ExPre (Iod D W) x then g (Pre (Iod D W) x) = x else if a \<noteq> x then Algebra2.Sup D (segment (Iod D W) x) = x else a = a))"
sys_block_sizes_def: shows "combine_sys.sys_block_sizes \<equiv> {int (card bl) |bl. bl \<in># \<B>\<^sup>+}"
set_iterator_I: fixes l0 :: "'a list"   and S :: "'a set"   and iti :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b" assumes "distinct l0"   and "S = set l0"   and "iti = foldli l0" shows "set_iterator iti S"
mplus_fail2: fixes m :: "'a match" shows "m +++ Fixrec.fail = m"
FormP_Mem: fixes x :: "tm"   and y :: "tm" shows "{TermP x, TermP y} \<turnstile> FormP (Q_Mem x y)"
simps_1: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and u :: "nat"   and v :: "nat" shows "linordered_monoid.len m u v [] = m u v"
norm_diff_triangle_le: 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 x :: "'c"   and y :: "'c"   and e1 :: "real"   and z :: "'c"   and e2 :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm (minus x y) \<le> e1"   and "norm (minus y z) \<le> e2" shows "norm (minus x z) \<le> e1 + e2"
"with": shows "module.span \<equiv> span.with (0::'e) (+)"
chamber_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexEndomorphism X f"   and "SimplicialComplex.maxsimp X C" shows "SimplicialComplex.maxsimp X (f ` C)"
removen_same: fixes xs :: "'b list"   and n :: "nat" assumes "length xs \<le> n" shows "removen n xs = xs"
longest_common_prefix_prefix1: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<and>\<^sub>p ys \<le>p xs"
Par_ZObis_discrL1: fixes c1 :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "c1 \<approx>01 d"   and "discr c2" shows "c1 | c2 \<approx>01 d"
vec_nth_inner_scaleR_craziness: fixes f :: "real \<Rightarrow> real"   and x :: "('a, 'b) vec"   and i :: "'b"   and j :: "'a" shows "f (x $ i \<bullet> j) *\<^sub>R j = (\<Sum>xa\<in>UNIV. f (x $ xa \<bullet> j) *\<^sub>R axis xa j) $ i"
linorder_cases': fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less y x"
assms: shows "gen_lossless_gpv co \<I> (gpv \<bind> f)"
adds_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and s :: "'c" assumes "class.ninv_comm_monoid_add plus zero" shows "plus.adds plus s zero = (s = zero)"
face_cycle_set_def: fixes M :: "'e pre_map"   and a :: "'e" shows "pre_digraph_map.face_cycle_set M a = orbit (pre_digraph_map.face_cycle_succ M) a"
s: shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) qs = type0 init x y"
wlsAbs_inversion: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsAbs (xs, s) A" shows "isInBar (xs, s) \<and> (\<forall>x X. good X \<and> A = Abs xs x X \<longrightarrow> wls s X)"
lec_poly_def: fixes le_constraint :: "'a le_constraint" shows "lec_poly le_constraint = (case le_constraint of Le_Constraint x1 x2 x3 \<Rightarrow> x2)"
rep_eq: fixes x :: "('a, 'b) RBT.rbt" shows "mapping.rep (RBT_Mapping.Mapping x) = RBT.lookup x"
assms_7: shows "coeff q 0 \<noteq> (0::'a)"
IH: fixes c :: "'a" assumes "u \<turnstile>\<^bsub>v,n\<^esub> m"   and "v c \<le> n"   and "\<forall>c\<in>set cs. v c \<le> n" shows "dbm_entry_val u (Some c) None (len m (v c) 0 (map v cs))"
map_poly_pCons_hom: fixes a :: "real"   and p :: "real poly" shows "map_poly of_real (pCons a p) = pCons (of_real a) (map_poly of_real p)"
squarefree_asymptotics'': shows "(\<lambda>x. real (card {n. real n \<le> x \<and> squarefree n})) \<sim>[at_top] (*) (6 / pi\<^sup>2)"
cong_col_mid: fixes A :: "'p"   and C :: "'p"   and B :: "'p" assumes "A \<noteq> C"   and "Col A B C"   and "Cong A B B C" shows "B Midpoint A C"
polyadd_norm: fixes p :: "poly"   and q :: "poly" assumes "isnpoly p"   and "isnpoly q" shows "isnpoly (p +\<^sub>p q)"
assms_3: fixes x :: "V" assumes "x \<in> elts A" shows "app f x = app g x"
span_insertI: fixes p :: "'b"   and B :: "'b set"   and r :: "'b" assumes "p \<in> local.span B" shows "p \<in> local.span (insert r B)"
d3_const_def: shows "d3_const \<equiv> \<d> 3"
many_strongly_connected_iff_7: fixes x :: "'a" shows "many_strongly_connected x = (x\<^sup>T\<^sup>+ = x\<^sup>+)"
process_rule_1_futures: fixes xs :: "'e list"   and P :: "'e process" assumes "xs \<in> traces P" shows "([], {}) \<in> futures P xs"
relhoare_refl: fixes R :: "('e \<times> 'e) set" shows "{Id} R, R {> Id}"
is_coext_def: fixes x' :: "'d"   and f :: "'c"   and g :: "'d" shows "is_coext x' f g \<equiv> \<guillemotleft>g : x' \<rightarrow>\<^sub>D Go x\<guillemotright> \<and> f = \<epsilon>o x \<cdot>\<^sub>C F g"
fold_ohearn1: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "ir_valid P c SKIP Q = (\<forall>t'. ohearn (\<lambda>t. P t t') c (\<lambda>t. Q t t'))"
initial_ps2_def: fixes S :: "node fset"   and G :: "diagram" shows "initial_ps2 S G \<equiv> [ initials G |-| S |=> Top ] ++\<^sub>f [ S |=> Bot ]"
prime_normalize_iff: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalization_semidom.prime zero times one normalize (normalize p) = comm_semiring_1.prime_elem times one zero p"
oreachable_local: fixes s :: "'a \<times> 'b"   and A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'c"   and s' :: "'a \<times> 'b" assumes "s \<in> oreachable A S U"   and "(s, a, s') \<in> automaton.trans A"   and "S (fst s) (fst s') a" shows "s' \<in> oreachable A S U"
a_coset_join1: fixes G :: "('e, 'f) ring_scheme"   and H :: "'e set"   and x :: "'e" assumes "abelian_group G"   and "H +>\<^bsub>G\<^esub> x = H"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "x \<in> H"
minus_mult_commute: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = times a (uminus b)"
simps_34: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x41 ;; x42 \<noteq> x31@[x32]"
Aij: shows "A $h i $h j \<noteq> 0"
ANR_retract_of_ANR: fixes T :: "'a set"   and S :: "'a set" assumes "ANR T"   and "S retract_of T" shows "ANR S"
fid_on_eqI: fixes a :: "V"   and b :: "V"   and A :: "V" assumes "a = b"   and "a \<in>\<^sub>\<circ> A" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> fid_on A"
model_to_dimacs_model_sound: fixes n_vars :: "nat"   and f :: "nat formula"   and M :: "nat \<Rightarrow> bool" assumes "1 < n_vars"   and "\<forall>v\<in>atoms f. 0 < v \<and> v < n_vars"   and "is_cnf f"   and "M \<Turnstile> f" shows "dimacs_model (model_to_dimacs_model M [1..<n_vars]) (cnf_to_dimacs f)"
entropy_partition: fixes X :: "'a \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'e" assumes "simple_function M X" shows "\<H>(X) = \<H>(f \<circ> X) + \<H>(X | f \<circ> X)"
reverse_phase_sorted: fixes xs :: "int list" shows "\<forall>ys\<in>set (map (\<lambda>ys. if decr ys then rev ys else ys) (cuts xs)). sorted ys"
ideal_game_1_def: fixes x :: "'grp \<times> 'grp"   and y :: "bool"   and z :: "'e"   and A :: "('grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> unit \<Rightarrow> 'f \<Rightarrow> 'd spmf)" shows "mal_def.ideal_game_1 x y z A = (let (A1, A2) = A in A1 x z \<bind> (\<lambda>(x', aux_out). mal_def.trusted_party x' y \<bind> (\<lambda>(out1, out2). A2 x' z out1 aux_out \<bind> (\<lambda>out1'. return_spmf (out1', out2)))))"
simps_5: fixes y :: "parity" shows "plus_parity Either y = Either"
matrix_transpose_mul: fixes A :: "(('b, 'e) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and B :: "(('b, 'd) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose (A ** B) = Finite_Cartesian_Product.transpose B ** Finite_Cartesian_Product.transpose A"
dcd_7: shows "\<forall>j'\<in>{LEAST i'. k' < i' \<and> (\<exists>i. cs\<^bsup>path \<sigma>\<^esup> i = cs\<^bsup>path \<sigma>'\<^esup> i')..<n'}. v \<notin> local.writes (path \<sigma>' j')"
exhaust_sel: fixes msllist :: "'a msllist" obtains "msllist = MSLNil"   | "msllist = MSLCons (un_MSLCons1 msllist) (un_MSLCons2 msllist)"
nfa_accepting_def: fixes A :: "bool list bdd list \<times> bool list" shows "nfa_accepting A = nfa_accepting' (snd A)"
lists_accI: fixes xs :: "'g list"   and r :: "('g \<times> 'g) set" assumes "xs \<in> Wellfounded.acc (listrel1 r)" shows "xs \<in> lists (Wellfounded.acc r)"
iterate_sound: fixes w :: "nat \<Rightarrow> 'a set"   and f :: "'a ltln \<Rightarrow> 'a ltln"   and \<phi> :: "'a ltln"   and n :: "nat" assumes "\<And>\<phi>. w \<Turnstile>\<^sub>n f \<phi> = w \<Turnstile>\<^sub>n \<phi>" shows "w \<Turnstile>\<^sub>n iterate f \<phi> n = w \<Turnstile>\<^sub>n \<phi>"
st_vec_nonneg: fixes x :: "'a st_vec"   and i :: "'a" shows "0 \<le> st_vec x $h i"
default_unique_ACS: fixes otherbot :: "'a" assumes "\<forall>G f nP i. wf_graph G \<and> f \<in> set_offending_flows G nP \<and> i \<in> fst ` f \<longrightarrow> \<not> sinvar G (nP(i := otherbot))" shows "otherbot = \<bottom>"
at_vchop_dist1: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and \<psi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<forall>v'. v = c > v' \<longrightarrow> (\<exists>v u. v'=v--u \<and> \<phi> ts v \<and> \<psi> ts u)) \<longrightarrow> (\<exists>va u. v=va--u \<and> (\<forall>v'. va = c > v' \<longrightarrow> \<phi> ts v') \<and> (\<forall>v'. u = c > v' \<longrightarrow> \<psi> ts v'))"
vc_sound'': fixes C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" assumes "vc C Q"   and "\<exists>k>0. \<forall>l s. P l s \<longrightarrow> pre C Q l s \<and> time C s \<le> k * e s"   and "finite (support Q)"   and "finite (varacom C)" shows "\<turnstile>\<^sub>1 {P} strip C { e \<Down> Q}"
purge_Cons_notIn: fixes u :: "'U"   and G :: "'U set"   and c :: "'C"   and ucl :: "('U \<times> 'C) list" assumes "u \<in> G" shows "purge G ((u, c) # ucl) = purge G ucl"
mult_commute_abs: fixes c :: "'i" shows "(\<lambda>x. x * c) = (*) c"
lnth_ltake: fixes m :: "nat"   and n :: "enat"   and xs :: "'a llist" assumes "enat m < n" shows "lnth (ltake n xs) m = lnth xs m"
powerset_lift_UNION: fixes g :: "'e \<rightarrow> 'c set"   and A :: "'d \<Rightarrow> 'e set"   and S :: "'d set" shows "(\<Union>x\<in>S. \<^ps> g\<cdot>(A x)) = \<^ps> g\<cdot>(\<Union> (A ` S))"
elims: fixes x :: "'a list"   and y :: "'a list" assumes "psums x = y" obtains "x = []" and "y = []"   | xa :: "'a" where "x = [xa__]" and "y = [xa__]"   | xa :: "'a" and ya :: "'a" and xs :: "'a list" where "x = xa # ya # xs" and "y = xa # psums ((xa + ya) # xs)"
rceil_def: fixes x :: "real" shows "rceil x = real_of_int \<lceil>x\<rceil>"
hyps: fixes \<Gamma> :: "'a constr list"   and \<Phi> :: "'a TESL_atomic list" assumes "\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g" shows "\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
cong_3: fixes x :: "'a"   and t :: "real" assumes "x \<in> Y"   and "t \<in> S" shows "f t x = g t x"
simps_1: fixes d :: "int" shows "intv'_const (Smaller' d) = d"
fls_mult_distrib_subdegree_ge_0_2: fixes f :: "'a fls"   and g :: "'a fls"   and h :: "'a fls" assumes "0 \<le> fls_subdegree f"   and "0 \<le> fls_subdegree g"   and "0 \<le> fls_subdegree h" shows "h * (f + g) = h * f + h * g"
select_convs_1: fixes V :: "'b set"   and N :: "'b set"   and C :: "'b set"   and more :: "'c" shows "V \<lparr>V = V, N = N, C = C, \<dots> = more\<rparr> = V"
RU_sound: fixes u :: "'n"   and w :: "('p, 'ba) label list el_step list"   and s' :: "'n list"   and c' :: "'n list multiset"   and U :: "'n set" assumes "(([u], {#}), w, s', c') \<in> trcl (ntrp fg)"   and "atU U (add_mset s' c')" shows "\<exists>Ml Me h. (u, Ml, Me, h) \<in> RU_cs fg U \<and> Ml \<subseteq> mon_loc fg w \<and> Me \<subseteq> mon_env fg w \<and> h \<le> \<alpha>ah (map (\<alpha>nl fg) w)"
Minimal_Solutions_def: shows "local.Minimal_Solutions = {(x, y). (x, y) \<in> local.Solutions \<and> nonzero x \<and> \<not> (\<exists>(u, v)\<in>local.Solutions. nonzero u \<and> u @ v <\<^sub>v x @ y)}"
mset_subset_eqI: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "\<And>a. count A a \<le> count B a" shows "A \<subseteq># B"
count_greater_zero_iff: fixes M :: "'a multiset"   and x :: "'a" shows "(0 < count M x) = (x \<in># M)"
PrimRec1'_def: shows "PrimRec1' \<equiv> {x. PrimRec1'p x}"
select_convs_3: 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 "getM \<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> = getM"
permutation_compose: fixes p :: "'e \<Rightarrow> 'e"   and q :: "'e \<Rightarrow> 'e" assumes "permutation p"   and "permutation q" shows "permutation (p \<circ> q)"
oprod_Well_order: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "Well_order (r *o r')"
ipurge_tr_rev_idem: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "ipurge_tr_rev I D u (ipurge_tr_rev I D u xs) = ipurge_tr_rev I D u xs"
dijkstra'_refines: shows "dijkstra' \<le> \<Down> Id dijkstra"
length_tl_list_minus_butlast_gt_zero: fixes s :: "'b list"   and t :: "'b list" assumes "length s < length t"   and "strict_prefix (butlast s) t"   and "0 < length s" shows "0 < length (tl (t - butlast s))"
fg: fixes n :: "nat" assumes "N \<le> n" shows "norm (f n) \<le> cmod (g n)"
distr: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and c :: "('a \<times> 'a) set" assumes "valid_frac a"   and "valid_frac b"   and "valid_frac c" shows "mult_rel a (add_rel b c) = add_rel (mult_rel a b) (mult_rel a c)"
linord_of_list_total_on: fixes xs :: "'a list" shows "total_on (set xs) (linord_of_list xs)"
continuousE: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" assumes "continuous f S T" obtains "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>" and "\<forall>m. m open\<^bsub>T\<^esub> \<longrightarrow> carrier\<^bsub>S\<^esub> \<inter> f -` m open\<^bsub>S\<^esub>"
substitution: fixes B :: "val"   and e :: "exp"   and \<Gamma>' :: "(nat \<times> val) list"   and A :: "val"   and v :: "exp"   and x :: "nat"   and \<Gamma> :: "(nat \<times> val) list" assumes "B \<in> E e \<Gamma>'"   and "A \<in> E v []"   and "\<Gamma>' \<approx> (x, A) # \<Gamma>"   and "is_val v" shows "B \<in> E (subst x v e) \<Gamma>"
val_Zp_p_int_unit: fixes n :: "int" assumes "n mod p \<noteq> 0" shows "val_Zp ([n] \<cdot> \<one>) = 0"
trancl_S: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "pre_order_pair S NS" shows "S\<^sup>+ = S"
simps: fixes A :: "'a mat"   and lhs_vec :: "rat vec" shows "reduce_mat_cols A lhs_vec = take_cols_from_matrix A (find_nonzeros_from_input_vec lhs_vec)"
empty_rep: fixes t :: "('a, 'b) FingerTree" shows "(t = FingerTree.empty) = (Rep_FingerTree t = Empty)"
intros_1: fixes t1 :: "('c, 'd) tm"   and t2 :: "('c, 'd) tm" shows "App t1 t2 \<rightarrow>\<^sub>e\<^sub>m\<^sub>b t1"
inverse: fixes x :: "nat"   and q :: "nat" assumes "gcd x q = 1"   and "0 < q" shows "[int x * fst (bezw x q) = 1] (mod int q)"
case_1: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Eq of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f1"
NotifyWatchesLoopQEffect: fixes state :: "State"   and M :: "(Literal \<times> bool) list"   and literal :: "Literal"   and decision :: "bool"   and Wl :: "nat list"   and newWl :: "nat list" assumes "getM state = M @ [(opposite literal, decision)]"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "InvariantConsistent (getM state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"   and "uniq Wl"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) M" shows "let state' = notifyWatches_loop literal Wl newWl state in (\<forall>l. l \<in> set (getQ state') - set (getQ state) \<longrightarrow> (\<exists>clause. clause el getF state \<and> literal el clause \<and> isUnitClause clause l (elements (getM state)))) \<and> (\<forall>clause. clause \<in> set Wl \<longrightarrow> (\<forall>l. isUnitClause (getF state ! clause) l (elements (getM state)) \<longrightarrow> l \<in> set (getQ state')))"
has_integral_sum: fixes T :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and i :: "'a \<Rightarrow> 'c"   and S :: "'b set" assumes "finite T"   and "\<And>a. a \<in> T \<Longrightarrow> (f a has_integral i a) S" shows "((\<lambda>x. \<Sum>a\<in>T. f a x) has_integral sum i T) S"
rQuots_def: fixes xs :: "'b list"   and A :: "'b list set" shows "rQuots xs A = {ys. ys @ rev xs \<in> A}"
image_TimesA: fixes f :: "'i \<Rightarrow> 'g"   and g :: "'j \<Rightarrow> 'h"   and A :: "'i set"   and B :: "'j set" shows "(\<lambda>(x, y). (f x, g y)) ` (A \<times> B) = f ` A \<times> g ` B"
basis_null_space_iarrays_def: fixes A :: "'d iarray iarray" shows "basis_null_space_iarrays A = (let GJ = Gauss_Jordan_iarrays_PA (transpose_iarray A); rank_A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of (snd GJ))) in set (map (\<lambda>i. row_iarray i (fst GJ)) [rank_A..<ncols_iarray A]))"
in_lset_conv_lnth: fixes x :: "'c"   and xs :: "'c llist" shows "(x \<in> lset xs) = (\<exists>n. enat n < llength xs \<and> xs $ n = x)"
is_monoid: fixes G :: "('d, 'e) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
derived_consistent: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H"   and "subgroup H G" shows "derived (G\<lparr>carrier := H\<rparr>) K = derived G K"
ai: fixes i :: "nat" assumes "i \<in> S" shows "a i \<in> sigma_sets sp A"
simps_8: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly" shows "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
comp_char: fixes g :: "'c"   and f :: "'c" shows "C'.comp g f = (if C'.arr f \<and> C'.arr g \<and> C.seq g f then g \<cdot>\<^sub>C f else C.null)"
simps_2: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and ts :: "dB list"   and Ts :: "LambdaType.type list" shows "(e \<tturnstile> t # ts : Ts) = (case Ts of [] \<Rightarrow> False | T # Ts \<Rightarrow> e \<turnstile> t : T \<and> e \<tturnstile> ts : Ts)"
join_comm: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" shows "x \<squnion>\<^bsub>L\<^esub> y = y \<squnion>\<^bsub>L\<^esub> x"
int_extend_disjoint: fixes K1 :: "'a set"   and K2 :: "'a set" assumes "K1 \<in> \<D>"   and "K2 \<in> \<D>"   and "K1 \<noteq> K2" shows "interior (extend K1) \<inter> interior (extend K2) = {}"
assms_3: shows "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>)"
comp_ipresIGSwapAll: fixes h :: "'k \<Rightarrow> 'l"   and hA :: "'m \<Rightarrow> 'n"   and MOD :: "('o, 'p, 'q, 'r, 's, 't, 'k, 'm, 'u) model_scheme"   and MOD' :: "('v, 'w, 'q, 'r, 'x, 't, 'l, 'n, 'y) model_scheme"   and h' :: "'l \<Rightarrow> 'z"   and hA' :: "'n \<Rightarrow> 'aa"   and MOD'' :: "('ab, 'ac, 'q, 'ad, 'ae, 't, 'z, 'aa, 'af) model_scheme" assumes "ipresIGWlsAll h hA MOD MOD'"   and "ipresIGSwapAll h hA MOD MOD'"   and "ipresIGSwapAll h' hA' MOD' MOD''" shows "ipresIGSwapAll (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
rev_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "rev (filter P xs) = filter P (rev xs)"
list_order_equiv: fixes ops :: "('c \<times> ('c, 'd) list_op) list"   and x :: "'c"   and y :: "'c" shows "List_Spec.list_order ops x y = Insert_Spec.list_order (insertions ops) x y"
tuple_in'_def: shows "tuple_in' \<equiv> upd_set tuple_in (\<lambda>_. nt) X"
Collect_cong: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x::'b. (P::'b \<Rightarrow> bool) x = (Q::'b \<Rightarrow> bool) x" shows "{x::'b. (P::'b \<Rightarrow> bool) x} = {x::'b. (Q::'b \<Rightarrow> bool) x}"