hcju/isabelleps huggingface dataset:
p_prod_def: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "R \<parallel> S = {(a, A). \<exists>B C. A = B \<union> C \<and> (a, B) \<in> R \<and> (a, C) \<in> S}"
small_by_big: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"   and "iconf (shp s) e"   and "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>"   and "final e'" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
su3_4: shows "dist (flow0 x su3) u < duv"
r_rev_aux: fixes e :: "nat"   and xs :: "nat list"   and i :: "nat" assumes "Partial_Recursive.list_decode e = xs"   and "0 < length xs"   and "i < length xs" shows "eval r_rev_aux [i, e] \<down>= tuple_encode i (rev (take (Suc i) xs))"
test_add_closed: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "test (p + q)"
ParCptnEnv: fixes P :: "'b com option list"   and t :: "'b"   and xs :: "('b com option list \<times> 'b) list"   and s :: "'b" assumes "(P, t) # xs \<in> par_cptn" shows "(P, s) # (P, t) # xs \<in> par_cptn"
bpE_3: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and S :: "'b set"   and L :: "'b set"   and P :: "'b set set" assumes "BinPacking U w c S L"   and "BinPacking.bp U w c P" shows "\<Union> P = U"
leg0_uniquely_isomorphic_1: shows "tab\<^sub>0 (r \<star> s) \<star> cmp.chine \<cong> tab\<^sub>0 s \<star> p\<^sub>0"
proj_poly_syz_plus: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "proj_poly_syz n (p + q) = proj_poly_syz n p + proj_poly_syz n q"
ssubst_Eats: fixes V :: "name set"   and t :: "tm"   and u :: "tm" assumes "finite V" shows "ssubst (Eats t u) V F = Eats (ssubst t V F) (ssubst u V F)"
rcosets_assoc: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and M1 :: "'c set"   and M2 :: "'c set"   and M3 :: "'c set" assumes "H \<lhd> G"   and "M1 \<in> rcosets\<^bsub>G\<^esub> H"   and "M2 \<in> rcosets\<^bsub>G\<^esub> H"   and "M3 \<in> rcosets\<^bsub>G\<^esub> H" shows "M1 <#>\<^bsub>G\<^esub> M2 <#>\<^bsub>G\<^esub> M3 = M1 <#>\<^bsub>G\<^esub> (M2 <#>\<^bsub>G\<^esub> M3)"
landau_primfuns_iff_2: fixes fs :: "(primfun \<times> real) list" shows "((\<lambda>_. 1) \<in> o(eval_primfuns fs)) = pos_primfun_list (group_primfuns fs)"
bit_take_bit_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "nat"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit (take_bit m a) n = (n < m \<and> bit a n)"
Gets_imp_Says: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> set_pur" shows "\<exists>A. Says A B X \<in> set evs"
lsteps_smaller_card_Max_fun': fixes xs :: "'a list"   and l :: "nat"   and f :: "'a \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "\<exists>x\<in>set xs. l \<le> f x" shows "lsteps xs f l u v + l * u \<le> v * length xs + u * Max (f ` set xs \<union> {0})"
gfinprod_closed: fixes G :: "('b, 'c) monoid_scheme"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "Group.comm_monoid G"   and "f \<in> A \<rightarrow> carrier G" shows "gfinprod G f A \<in> carrier G"
comp_null_2: fixes f :: "'a" shows "Right_a.comp f Right_a.null = Right_a.null"
in_mono: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "order.greater_eq B A" shows "x \<in> A \<longrightarrow> x \<in> B"
ps_1: shows "x\<^sub>3 = x\<^sub>2"
map_is_empty: fixes m :: "'q" assumes "Q_invar m" shows "Q_is_empty m = (Q_lookup m = Map.empty)"
cases: fixes x :: "'b constr list \<times> nat \<times> 'b TESL_atomic list \<times> 'b TESL_atomic list" obtains \<Gamma> :: "'b constr list" and n :: "nat" and \<Psi> :: "'b TESL_atomic list" and \<Phi> :: "'b TESL_atomic list" where "x = \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>"
irreducible_connect_rev: fixes p :: "'a poly" assumes "irreducible p"   and "0 < degree p" shows "irreducible\<^sub>d p"
measurement_dim: fixes d :: "nat"   and n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and k :: "nat" assumes "measurement d n M"   and "k < n" shows "M k \<in> carrier_mat d d"
start_class_def: fixes C :: "char list"   and M :: "char list" shows "start_class C M = (Start, Object, [], [start_method C M, start_clinit])"
Pd_above_irrelevant: fixes Pd' :: "'b \<Rightarrow> ('a \<times> 'a) set"   and d' :: "'b"   and X :: "'a set"   and x :: "'a"   and ds :: "'b set" assumes "Linear_order (Pd' d')"   and "Field (Pd' d') \<subseteq> {y. Xd y = d'}"   and "\<forall>d. d \<noteq> d' \<longrightarrow> Pd' d = Pd'_tax d"   and "Singleton_for_d.dX X d' \<subseteq> Field (Pd' d')"   and "Above (Pd' d') (Singleton_for_d.dX X d') \<subseteq> Above (Pd'_tax d') (Singleton_for_d.dX X d')"   and "x \<in> X"   and "TruePref_tax.stable_on ds X" shows "\<exists>y\<in>Singleton_for_d.CH (Contracts.fp_cop_F Pd' Ch ds). (x, y) \<in> Pd' (Xd x)"
intro: fixes S :: "'c set"   and 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" assumes "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "sheaf_of_rings_axioms S is_open \<FF> \<rho> zero_str" shows "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
less_eq_float_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "(a \<le> b) = fle a b"
merge_correct: fixes l1 :: "'a list"   and l2 :: "'a list" assumes "sorted l1"   and "sorted l2" shows "sorted (merge l1 l2) \<and> mset (merge l1 l2) = mset l1 + mset l2 \<and> set (merge l1 l2) = set l1 \<union> set l2"
dg_SemiCAT_is_arrI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<^bsub>dg_SemiCAT \<alpha>\<^esub> \<BB>"
tensor_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and u :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t"   and "monoidal_language.Arr C u" shows "free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C (free_monoidal_category.mkarr C t) (free_monoidal_category.mkarr C u) = free_monoidal_category.mkarr C (t \<^bold>\<otimes> u)"
abs_convergent_prod_imp_LIMSEQ: fixes f :: "nat \<Rightarrow> 'e" assumes "abs_convergent_prod f" shows "f \<longlonglongrightarrow> (1::'e)"
bconf_Cons: fixes P :: "'a prog"   and sh :: "sheap"   and e :: "'b exp"   and es :: "'b exp list"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (e # es,b) \<surd>) = (case val_of e of None \<Rightarrow> P,sh \<turnstile>\<^sub>b (e,b) \<surd> | \<lfloor>v\<rfloor> \<Rightarrow> P,sh \<turnstile>\<^sub>b (es,b) \<surd>)"
simps_1: fixes x :: "'b" shows "remove1 x [] = []"
distinct_85: fixes x8 :: "'b ltl"   and x9 :: "'b ltl" shows "G x8 \<noteq> F x9"
fconverse_vunion: fixes r :: "V"   and s :: "V" shows "(r \<union>\<^sub>\<circ> s)\<inverse>\<^sub>\<bullet> = r\<inverse>\<^sub>\<bullet> \<union>\<^sub>\<circ> s\<inverse>\<^sub>\<bullet>"
simps_2: fixes n :: "num" shows "numeral (num.Bit0 n) = numeral n + numeral n"
assms: shows "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' y f"
simps_10: fixes x41 :: "expr"   and x42 :: "com"   and x43 :: "com" shows "if (x41) x42 else x43 \<noteq> Skip"
Lp_Banach_lattice_1: fixes f :: "'b \<Rightarrow> real"   and p :: "ennreal"   and M :: "'b measure" assumes "f \<in> space\<^sub>N (\<LL> p M)" shows "(\<lambda>x. \<bar>f x\<bar>) \<in> space\<^sub>N (\<LL> p M)"
that: assumes "x \<notin> sset xs"   and "sdistinct xs" shows "thesis"
pterm_to_sterm_frees: fixes t :: "pterm" assumes "pre_strong_term_class.wellformed t" shows "frees (pterm_to_sterm t) = frees t"
CALL_cor: fixes t :: "state" shows "{(\<lambda>z s. (z = s \<and> body \<down> s) \<and> (s, t) \<in> termi_call_steps, CALL, \<lambda>z. exec z body)} \<turnstile>\<^sub>t {\<lambda>z s. (z = s \<and> body \<down> s) \<and> s = t} body {\<lambda>z. exec z body}"
P_continuous_on: fixes t :: "real set" assumes "t \<subseteq> {0<..}" shows "continuous_on t P"
applicableBackjumpCharacterization: fixes stateA :: "State" shows "applicableBackjump stateA = (\<exists>bc bl level. isUnitClause bc bl (elements (prefixToLevel level (getM stateA))) \<and> formulaEntailsClause (getF stateA) bc \<and> var bl \<in> vars (getF stateA) \<union> vars (elements (getM stateA)) \<and> 0 \<le> level \<and> level < currentLevel (getM stateA))"
cardSuc_Card_order: fixes r :: "('b \<times> 'b) set" assumes "Card_order r" shows "Card_order (cardSuc r)"
minGraphProps_def: fixes g :: "graph" shows "minGraphProps g \<equiv> minGraphProps' g \<and> facesAt_eq g \<and> faceListAt_len g \<and> facesAt_distinct g \<and> faces_distinct g \<and> faces_subset g \<and> edges_sym g \<and> edges_disj g \<and> face_face_op g"
assms_1: shows "mcont luba orda Sup (\<le>) f"
equiv_up_toI: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com" assumes "\<And>s s'. P s \<Longrightarrow> (c, s) \<Rightarrow> s' = (c', s) \<Rightarrow> s'" shows "P \<Turnstile> c \<sim> c'"
gcd_dvd1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) a"
jumpNestingOk_expr_simp: fixes jmps :: "jump set"   and e :: "expr" shows "jumpNestingOk jmps (In1l e) = True"
SD_inefficient_support_subset: fixes p' :: "'alt pmf"   and p :: "'alt pmf" assumes "\<not> SD_efficient R p'"   and "set_pmf p' \<subseteq> set_pmf p"   and "p \<in> lotteries_on alts" shows "\<not> SD_efficient R p"
E_2: assumes "\<not> QuadRes (int p) a" shows "[a ^ ((p - 1) div 2) = - 1] (mod int p)"
rel_fundef_rewrite_both: fixes fd1 :: "('dyn, 'a, 'b, 'c, 'opinl) Inca.instr fundef"   and fd2 :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and y :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr"   and x :: "('dyn, 'a, 'b, 'c, 'opinl) Inca.instr"   and pc :: "nat" assumes "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"   and "norm_instr y = x" shows "rel_fundef (\<lambda>x y. x = norm_instr y) (rewrite_fundef_body fd1 pc x) (rewrite_fundef_body fd2 pc y)"
nxt: fixes q :: "hf"   and x :: "'a" assumes "q \<in> dfa.states Canon.DFA" shows "dfa.nxt Canon.DFA q x \<in> dfa.states Canon.DFA"
op_set_wcard_def: fixes X :: "'c set" shows "op_set_wcard X = SPEC ((\<le>) (card X))"
monom_one_mult_comm: fixes n :: "nat"   and m :: "nat" shows "monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m = monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n"
distinct_1: fixes x2 :: "name" shows "Zero \<noteq> Var x2"
star_inductr_var: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> y" shows "y \<cdot> x\<^sup>\<star> \<le> y"
UUZ: shows "closedin (top_of_set U) (U - Z)"
asprod_ge: fixes b :: "ant"   and N :: "nat" assumes "0 < b"   and "N \<noteq> 0" shows "an N \<le> int N *\<^sub>a b"
\<theta>_twopow_diff: fixes k :: "nat" shows "\<theta> (2 ^ Suc k) - \<theta> (2 ^ k) < 2 ^ Suc k * ln 2"
memo_list_trancl: fixes r :: "('b \<times> 'b) list"   and a :: "'b" shows "set (memo_list_trancl r a) = rel_ext (\<lambda>b. (a, b) \<in> (set r)\<^sup>+)"
upset_to_downset2: shows "\<up> = (`) \<partial> \<circ> \<down> \<circ> \<partial>"
partial_term_of_anything: fixes x :: "'a itself"   and nt :: "??.Quickcheck_Narrowing.narrowing_term"   and t :: "term" shows "partial_term_of x nt \<equiv> t"
triangular_to_jnf_steps_dims_4: fixes A :: "'b mat" shows "dim_col (??.Jordan_Normal_Form_Existence.step_2 A) = dim_col A"
red2: shows "r2.r_syntax t x2 m2 ta2' x2' m2'"
poly_DERIV: fixes p :: "'a poly"   and x :: "'a" shows "(poly p has_field_derivative poly (pderiv p) x) (at x)"
IDirProds_def: fixes G :: "('d, 'e) monoid_scheme"   and S :: "'f \<Rightarrow> 'd set"   and I :: "'f set" shows "IDirProds G S I = generate G (\<Union> (S ` I))"
distinct_5: fixes x1 :: "nat"   and x41 :: "pdf_operator"   and x42 :: "expr" shows "Var x1 \<noteq> x41 $$ x42"
h: shows "d x \<cdot> y \<cdot> ad z = (0::'a)"
ans1: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.antidomain_near_semiring antidomain_op plus times one zero less_eq less" shows "times (antidomain_op x) x = zero"
prob_component_def: fixes p :: "real"   and w :: "bool stream"   and n :: "nat" shows "prob_component p w n = (if w !! n then p else 1 - p)"
chain_all_nthtriple: fixes \<phi> :: "'a \<times> 'b \<times> 'a \<Rightarrow> bool"   and \<Pi> :: "('a, 'b) chain" shows "chain_all \<phi> \<Pi> = (\<forall>i<chainlen \<Pi>. \<phi> (nthtriple \<Pi> i))"
the_least_prefixpoint_def: fixes f :: "'a \<Rightarrow> 'a" shows "p\<mu> f = (THE x. is_least_prefixpoint f x)"
simps_28: shows "J.BT \<noteq> J.TT"
chine_composite_1: shows "HH\<mu>H\<nu>\<pi>\<rho>.chine = \<nu>\<pi>.leg0 \<cdot> \<mu>_\<nu>\<pi>.prj\<^sub>0 \<down>\<down> \<rho>.leg1"
axioms_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "tarski_absolute_space C B" shows "tarski_absolute_space_axioms B"
offs_pred_asc: fixes ns :: "nat list"   and ub :: "nat"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and j :: "nat" assumes "offs_pred ns ub xs index key mi ma"   and "i < j"   and "j < length ns"   and "0 < offs_num (length ns) xs index key mi ma i"   and "0 < offs_num (length ns) xs index key mi ma j" shows "ns ! i + offs_num (length ns) xs index key mi ma i \<le> ns ! j"
icrit_spec_def: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool" shows "icrit_spec crit = (\<forall>d m data gs bs hs p q. dickson_grading d \<longrightarrow> fst ` (set gs \<union> set bs \<union> set hs) \<subseteq> dgrad_p_set d m \<longrightarrow> unique_idx (gs @ bs @ hs) data \<longrightarrow> is_Groebner_basis (fst ` set gs) \<longrightarrow> p \<in> set hs \<longrightarrow> q \<in> set gs \<union> set bs \<union> set hs \<longrightarrow> fst p \<noteq> 0 \<longrightarrow> fst q \<noteq> 0 \<longrightarrow> crit data gs bs hs p q \<longrightarrow> crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs \<union> set hs)) (fst p) (fst q))"
gamma_def: shows "\<gamma> = (\<lambda>x. (b, y2 + y1 * x))"
is_locally_frac_def: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and V :: "'a set set" shows "is_locally_frac s V \<equiv> \<exists>r f. r \<in> R \<and> f \<in> R \<and> (\<forall>\<qq>\<in>V. f \<notin> \<qq> \<and> s \<qq> = quotient_ring.frac (R\<setminus>\<qq>) R (+) (\<cdot>) \<zero> r f)"
Rep_run: fixes x :: "'b run" shows "Rep_run x \<in> {\<rho>. \<forall>c. mono (\<lambda>n. time (\<rho> n c))}"
to_bl_unfold: fixes w :: "'b word" shows "to_bl w = rev (map (bit w) [0..<LENGTH('b)])"
characteristic_function_def: fixes A :: "'b set"   and x :: "'b" shows "\<chi> A x \<equiv> if x \<in> A then 1 else 0"
simps_1: fixes x :: "'a"   and a :: "('a, 'c) rbt"   and y :: "'a"   and s :: "'c"   and lt :: "('a, 'c) rbt"   and z :: "'a"   and v :: "'c"   and rt :: "('a, 'c) rbt" shows "rbt_del_from_right x a y s (Branch B lt z v rt) = balance_right a y s (rbt_del x (Branch B lt z v rt))"
isDERIV_approx: fixes xs :: "real list"   and vs :: "float interval option list"   and prec :: "nat"   and x :: "nat"   and f :: "floatarith" assumes "bounded_by xs vs"   and "isDERIV_approx prec x f vs" shows "isDERIV x f xs"
finite_nodes_gsuccessors: fixes A :: "('b, 'a) nba"   and v :: "nat \<times> 'a"   and w :: "'b stream" assumes "finite (nodes A)"   and "v \<in> gunodes A w" shows "finite (gusuccessors A w v)"
igFreshCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "igFreshCls MOD \<equiv> igFreshIGVar MOD \<and> igFreshIGAbs1 MOD \<and> igFreshIGAbs2 MOD \<and> igFreshIGOp MOD"
tso_store_refs_simps_4: fixes m :: "'b"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state"   and r' :: "'a" shows "mut_m.tso_store_refs m (s(sys := s sys\<lparr>heap := (sys_heap s)(r' := None)\<rparr>)) = mut_m.tso_store_refs m s"
inf_le1: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y \<le> x"
rule_nonterminal_type: fixes N :: "symbol"   and \<alpha> :: "symbol list" assumes "(N, \<alpha>) \<in> \<RR>" shows "is_nonterminal N"
last_nns_mono: fixes kdt :: "'b kdt"   and p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list"   and n :: "nat" assumes "invar kdt"   and "sorted_wrt_dist p ps"   and "n \<le> length ps"   and "0 < n" shows "dist (last (nearest_nbors n ps p kdt)) p \<le> dist (last ps) p"
surjective: fixes r :: "('d, 'e, 'f) Category_scheme" shows "r = \<lparr>Obj = obj\<^bsub>r\<^esub>, Mor = mor\<^bsub>r\<^esub>, Dom = Dom r, Cod = Cod r, Id = Category.Id r, Comp = (;;\<^bsub>r\<^esub>), \<dots> = more r\<rparr>"
induced_hom_equality: fixes w :: "'a freeword" assumes "w \<in> FreeGroup S" shows "F (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {w})) = lift_f w"
independent_explicit_finite_subsets: fixes A :: "'d set" shows "cindependent A = (\<forall>S\<subseteq>A. finite S \<longrightarrow> (\<forall>u. (\<Sum>v\<in>S. u v *\<^sub>C v) = (0::'d) \<longrightarrow> (\<forall>v\<in>S. u v = 0)))"
tauActionChain_2: fixes P :: "pi"   and P' :: "pi" assumes "P \<noteq> P'"   and "P \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'"
CSP1_R_commute: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "CSP1 (R P) = R (CSP1 P)"
assms_4: shows "A\<^sub>P\<^sub>Q \<sharp>* \<Psi>"
distinct_adj_append_iff: fixes xs :: "'g list"   and ys :: "'g list" shows "distinct_adj (xs @ ys) = (distinct_adj xs \<and> distinct_adj ys \<and> (xs = [] \<or> ys = [] \<or> last xs \<noteq> hd ys))"
in_arcs_imp_in_arcs_ends: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs G" shows "(fst e, snd e) \<in> parcs G"
expand_set: fixes g :: "'c \<Rightarrow> 'a" shows "G g = F.F g {a. g a \<noteq> \<^bold>1}"
Sup_iter_unfold_var: fixes x :: "'a" shows "(1::'a) \<squnion> (\<Squnion>n. x \<cdot> x ^ n) = \<Squnion> range ((^) x)"
CatDom: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme" assumes "PreFunctor F" shows "Category (CatDom F)"
list_updates_nth_ge: fixes vs :: "'a list"   and x :: "nat"   and xs :: "nat list"   and ys :: "'a list" assumes "length vs \<le> x"   and "length xs = length ys" shows "list_updates xs ys vs ! x = vs ! x"
gf: fixes z :: "complex" assumes "z \<in> S" shows "g (f z) = z"
orthogonal_currentD_in: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set"   and v :: "'a"   and u :: "'a" assumes "orthogonal_current \<Gamma> f S"   and "v \<in> RF\<^bsub>\<Gamma>\<^esub> S"   and "u \<notin> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S" shows "f (u, v) = 0"
T_on_on': fixes s0 :: "'state"   and qs :: "'request list"   and A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)"   and x :: "'b" shows "T s0 qs (off2 A (s0, x) qs) = T_on' A (s0, x) qs"
boundOutputFreshSet_1: fixes Xs :: "name set"   and xvec :: "name list"   and B :: "('d, 'e, 'f) boundOutput" shows "Xs \<sharp>* (\<lparr>\<nu>*xvec\<rparr>B) = (\<forall>x\<in>Xs. x mem xvec \<or> x \<sharp> B)"
Compl_disjoint: fixes A :: "'c set" shows "A \<inter> - A = {}"
jumpF_def: fixes f :: "real \<Rightarrow> real"   and F :: "real filter" shows "jumpF f F \<equiv> if filterlim f at_top F then 1 / 2 else if filterlim f at_bot F then - 1 / 2 else 0"
sigma_finite_measure_axioms: shows "sigma_finite_measure (count_space X)"
ord_pp_punit_alt: shows "ord_pp_punit = le_of_nat_term_order"
square_integrable_imp_lebesgue: fixes f :: "real \<Rightarrow> real"   and S :: "real set" assumes "f square_integrable S" shows "S \<in> sets lebesgue"
conc_subset_lists: fixes A :: "'a list set"   and S :: "'a set"   and B :: "'a list set" assumes "A \<subseteq> lists S"   and "B \<subseteq> lists S" shows "A @@ B \<subseteq> lists S"
action_order_def: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat"   and a' :: "nat" shows "(E \<turnstile> a \<le>a a') = (a \<in> actions E \<and> a' \<in> actions E \<and> (if is_new_action (action_obs E a) then is_new_action (action_obs E a') \<longrightarrow> a \<le> a' else \<not> is_new_action (action_obs E a') \<and> a \<le> a'))"
lcm_def: fixes a :: "'a"   and b :: "'a" shows "normalization_euclidean_semiring_class.lcm a b = normalize (a * b div normalization_euclidean_semiring_class.gcd a b)"
known_ptrs_new_ptr: fixes h' :: "(_) heap"   and h :: "(_) heap"   and new_ptr :: "(_) object_ptr" assumes "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|new_ptr|}"   and "DocumentClass.known_ptr new_ptr"   and "DocumentClass.known_ptrs h" shows "DocumentClass.known_ptrs h'"
that: fixes j :: "nat"   and step :: "nat"   and start :: "int" assumes "j < k"   and "0 < step"   and "is_arith_prog_on l start step a b"   and "arith_prog start step ` {..<l} \<subseteq> col -` {j} \<inter> {a..b}" shows "thesis"
le: fixes X :: "nat set" assumes "X \<subseteq> {..n}" shows "sum f X \<le> sum g (R `` X)"
univalent_fn: fixes R :: "('b \<times> 'c) set" assumes "univalent R" shows "BNF_Def.Gr (Domain R) (\<lambda>x. SOME y. (x, y) \<in> R) = R"
Nml_HcompD_2: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml (t \<^bold>\<star> u)" shows "C.arr (un_Prim t)"
const: fixes k :: "nat"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. enat k * P s} c {\<lambda>s. enat k * Q s}"   and "0 < k" shows "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}"