hcju/isabelleps huggingface dataset:
strategyproofness_2: shows "(pmf (sds R10) a < pmf (sds R36) a \<or> pmf (sds R10) a + pmf (sds R10) b < pmf (sds R36) a + pmf (sds R36) b) \<or> pmf (sds R10) a = pmf (sds R36) a \<and> pmf (sds R10) a + pmf (sds R10) b = pmf (sds R36) a + pmf (sds R36) b"
sum_list_map_unique_vars: fixes f :: "'a"   and ts :: "('a, 'b) Term.term list" shows "num_unique_vars (Fun f ts) \<le> sum_list (map num_unique_vars ts)"
simps_1: fixes i :: "nat" shows "a i 0 = fs ! i"
add_pow_rdistr: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and k :: "nat" assumes "semiring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<otimes>\<^bsub>R\<^esub> [k] \<cdot>\<^bsub>R\<^esub> b = [k] \<cdot>\<^bsub>R\<^esub> (a \<otimes>\<^bsub>R\<^esub> b)"
assms_3: fixes \<E> :: "'a set set" assumes "\<E> \<subseteq> \<D>"   and "finite \<E>" shows "Sigma_Algebra.measure M (\<Union> \<E>) \<le> B"
that_2: fixes X :: "'a set" assumes "M = RES X" shows "thesis"
qSkel_qSwap: fixes zs :: "'f"   and x :: "'g"   and y :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "qSkel (X #[[x \<and> y]]_zs) = qSkel X"
mod_starD: fixes A :: "assn"   and B :: "assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> A * B" shows "\<exists>h1 h2. h1 \<Turnstile> A \<and> h2 \<Turnstile> B"
intros_1: fixes C :: "'a eclause"   and S :: "'a eclause set" assumes "C \<in> S" shows "derivable_ecl C S"
vec_of_basis_enum_minus: fixes b1 :: "'b"   and b2 :: "'b" shows "vec_of_basis_enum (b1 - b2) = vec_of_basis_enum b1 - vec_of_basis_enum b2"
valuate_def: fixes x :: "(nat \<Rightarrow> real) \<Rightarrow> 'b" shows "valuate x = x ` (UNIV \<rightarrow> {- 1..1})"
fold_insert_remove: fixes A :: "'b fset set set"   and z :: "'b fset set"   and x :: "'b fset set" assumes "finite A" shows "Finite_Set.fold (\<otimes>) z (insert x A) = x \<otimes> Finite_Set.fold (\<otimes>) z (A - {x})"
assms_2: shows "stab s i' (nat_of_int c) = Some cl"
total_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and i :: "'c" assumes "matrix_stone_relation_algebra.total_var f" shows "\<exists>j. f (i, j) = top"
h: shows "mat (length subsets) (length signs) (\<lambda>(i, j). z_R (subsets ! i) (signs ! j)) = mat (length subsets) (length signs) (\<lambda>(i, y). map (\<lambda>index_list. map (z_R index_list) signs) subsets ! i ! y)"
generalized: fixes jmps :: "jump set"   and Env :: "env"   and T :: "tys" assumes "\<forall>j. fst s0 = Some (Jump j) \<longrightarrow> j \<in> jmps"   and "jumpNestingOk jmps t"   and "Env\<turnstile>t\<Colon>T"   and "prg Env = G" shows "(\<forall>j. fst s1 = Some (Jump j) \<longrightarrow> j \<in> jmps) \<and> (normal s1 \<longrightarrow> (\<forall>w upd. v = In2 (w, upd) \<longrightarrow> (\<forall>s j val. fst s \<noteq> Some (Jump j) \<longrightarrow> fst (upd val s) \<noteq> Some (Jump j))))"
simps_1: fixes t :: "'a tree23" shows "trees (T t) = {t}"
simp3: shows "flows_state (generate_valid_stateful_policy_IFSACS G M edgesList) \<subseteq> edges G"
binomial_pmf_altdef': fixes A :: "'a set"   and n :: "nat"   and p :: "real"   and dflt :: "bool" assumes "finite A"   and "card A = n"   and "p \<in> {0..1}" shows "binomial_pmf n p = map_pmf (\<lambda>f. card {x \<in> A. f x}) (Pi_pmf A dflt (\<lambda>_. bernoulli_pmf p))"
Ana_subterm': fixes s :: "('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "s \<in> set (snd (Ana t))" shows "s \<sqsubseteq> t"
that: fixes a :: "'a \<times> 'b"   and s :: "('a, 'b) FingerTree" assumes "viewR t = Some (a, s)"   and "toList t = toList s @ [a]" shows "thesis"
abs_sgn: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and 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 a :: "'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "sgn (abs a) = zero_neq_one.of_bool one zero (a \<noteq> zero)"
DESTRimpl_rule3: fixes s :: "'s"   and ni :: "'ni"   and v :: "'a"   and n1 :: "'a ifex"   and n2 :: "'a ifex" assumes "I s"   and "(ni, IF v n1 n2) \<in> R s" shows "ospec (DESTRimpl ni s) (\<lambda>r. \<exists>ni1 ni2. r = IFD v ni1 ni2 \<and> (ni1, n1) \<in> R s \<and> (ni2, n2) \<in> R s)"
independence1: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y = (0::'a)" shows "x\<^sup>\<star> \<cdot> y = y"
Q: shows "all_subfrmls (fst (n, ns)) \<subseteq> B \<and> (\<forall>nd\<in>snd (n, ns). all_subfrmls nd \<subseteq> B)"
map_poly_def: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" shows "map_poly f p = Poly (map f (coeffs p))"
unfold_congs_6: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "'c"   and f :: "'c \<Rightarrow> 'c"   and f' :: "'c \<Rightarrow> 'c" assumes "r = r'"   and "more r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "more_update f r = more_update f' r'"
coeff_sum: fixes p :: "'c \<Rightarrow> 'b poly"   and A :: "'c set"   and i :: "nat" shows "coeff (sum p A) i = (\<Sum>x\<in>A. coeff (p x) i)"
gcd_dvd2: fixes a :: "'d"   and b :: "'d" shows "gcd a b dvd b"
induct: fixes x1 :: "Var list \<times> (int \<times> Instr) list \<times> int"   and x2 :: "int"   and x3 :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and x4 :: "nat"   and x5 :: "int"   and x6 :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and P :: "Var list \<times> (int \<times> Instr) list \<times> int \<Rightarrow> int \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes "(x1, x2, x3, x4, x5, x6) \<in> MStep"   and "\<And>k t s ll l M. \<lbrakk>k = 0; t = s; ll = l\<rbrakk> \<Longrightarrow> P M l s k ll t"   and "\<And>M l s n l1 r k l2 t m. \<lbrakk>(M, l, s, n, l1, r) \<in> Step; (M, l1, r, k, l2, t) \<in> MStep; P M l1 r k l2 t; m = Suc k + n\<rbrakk> \<Longrightarrow> P M l s m l2 t" shows "P x1 x2 x3 x4 x5 x6"
rep_eq: fixes x :: "'b fset"   and xa :: "'b fset" shows "(x |\<subseteq>| xa) = (fset x \<subseteq> fset xa)"
redp_red_eq: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "redp P = red' P"
intro: fixes x :: "'c"   and X :: "'c set" assumes "x \<in> X" shows "ind_mor_btw_stalks_axioms X x"
le_floor_iff: fixes z :: "int"   and x :: "'a" shows "(z \<le> \<lfloor>x\<rfloor>) = (of_int z \<le> x)"
cong_3_swap: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C Cong3 A' B' C'" shows "B A C Cong3 B' A' C'"
obs_neq_ret_contradicts: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c"   and n :: "nat"   and \<sigma>' :: "'b \<Rightarrow> 'c"   and i :: "nat" assumes "IFC prob"   and "IFC_def.path prob \<sigma> n = IFC_def.return prob"   and "IFC_def.obs prob \<sigma>' i \<noteq> IFC_def.obs prob \<sigma> i"   and "IFC_def.obs prob \<sigma>' i \<noteq> None" shows "\<exists>k k'. IFC_def.contradicts prob (\<sigma>', k') (\<sigma>, k) \<and> IFC_def.path prob \<sigma> k \<in> dom (IFC_def.att prob) \<or> IFC_def.contradicts prob (\<sigma>, k) (\<sigma>', k') \<and> IFC_def.path prob \<sigma>' k' \<in> dom (IFC_def.att prob)"
fls_X_inv_power_times_conv_shift_2: fixes f :: "'a fls"   and n :: "nat" shows "f * fls_X_inv ^ n = fls_shift (int n) f"
assms_1: shows "io_targets PM (initial PM) io = {FAIL}"
eucl_truncate_up_real_def: fixes prec :: "nat"   and b :: "real" shows "eucl_truncate_up prec b = truncate_up prec b"
rank_Gauss_Jordan_eq: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A = (let A' = Gauss_Jordan A in card {row i A' |i. row i A' \<noteq> 0})"
Ideal2_def: fixes m2 :: "nat"   and m1 :: "nat"   and out2 :: "nat" shows "local.sim_non_det_def.Ideal2 m2 m1 out2 = S2 m2 out2 \<bind> (\<lambda>view2. Out2 m2 m1 out2 \<bind> (\<lambda>out2. return_spmf (view2, out2)))"
resultant_impl_generic_def: fixes div_exp :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> 'b"   and f :: "'b poly"   and g :: "'b poly" shows "resultant_impl_generic div_exp f g = (if length (coeffs g) \<le> length (coeffs f) then resultant_impl_main div_exp f g else let res = resultant_impl_main div_exp g f in if even (degree f) \<or> even (degree g) then res else - res)"
lenB: fixes i :: "'a set"   and j :: "'a" assumes "i \<in> \<I>"   and "j \<in> J i" shows "length (B j) = c"
fns2: fixes fr :: "'b \<Rightarrow> 'b"   and fa :: "'a \<Rightarrow> 'a" assumes "Abs \<circ> fr \<circ> Rep = fa" shows "(norm \<circ> fr \<circ> norm = fr \<circ> norm) = (Rep \<circ> fa = fr \<circ> Rep)"
simps_4: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lnull xs \<or> \<not> lnull ys" shows "ltl (lappend xs ys) = (case xs of LNil \<Rightarrow> case ys of LCons uu uua \<Rightarrow> uua | LCons x xs' \<Rightarrow> lappend xs' ys)"
sb: shows "\<Union> (left_lang ` qs1) \<subseteq> \<Union> (left_lang ` qs2)"
subst_lsst_unlabel_member: fixes x :: "('e, 'f) stateful_strand_step"   and A :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" assumes "x \<in> set (unlabel A)" shows "x \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<in> set (unlabel (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>))"
prolong_betw2: fixes Q :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "a \<noteq> b" shows "\<exists>c\<in>Q. [[a b c]]"
irreducible_imp_square_free: fixes p :: "'a poly" assumes "irreducible p" shows "square_free p"
bernoulli_stream_pref_prob_neq_zero: fixes n :: "nat"   and x :: "bool stream" assumes "0 < p"   and "p < 1" shows "emeasure M {w \<in> space M. stake n w = stake n x} \<noteq> 0"
IH_2: shows "chain_boundary p (chain_map p g b) = chain_map (p - Suc 0) g (chain_boundary p b)"
is_maxsplitter_maxsplit: fixes P :: "'c list \<Rightarrow> bool" shows "is_maxsplitter P (\<lambda>xs. maxsplit P ([], xs) [] xs)"
tauChainInduct: fixes P :: "ccs"   and P' :: "ccs"   and Prop :: "ccs \<Rightarrow> bool" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "Prop P"   and "\<And>P' P''. \<lbrakk>P \<Longrightarrow>\<^sub>\<tau> P'; P' \<longmapsto>\<tau> \<prec> P''; Prop P'\<rbrakk> \<Longrightarrow> Prop P''" shows "Prop P'"
has_derivative_eq_rhs: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and F :: "'c filter"   and g' :: "'c \<Rightarrow> 'd" assumes "(f has_derivative f') F"   and "f' = g'" shows "(f has_derivative g') F"
db\<^sub>s\<^sub>s\<^sub>t_in_cases: fixes t :: "('c, 'd) Term.term"   and s :: "('c, 'd) Term.term"   and A :: "('c, 'd) stateful_strand_step list"   and I :: "'d \<Rightarrow> ('c, 'd) Term.term"   and D :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" assumes "(t, s) \<in> set (db'\<^sub>s\<^sub>s\<^sub>t A I D)" shows "(t, s) \<in> set D \<or> (\<exists>t' s'. insert\<langle>t',s'\<rangle> \<in> set A \<and> t = t' \<cdot> I \<and> s = s' \<cdot> I)"
convex3: fixes u :: "real"   and v :: "real"   and a :: "real \<times> real"   and b :: "real \<times> real"   and d :: "real \<times> real"   and c :: "real \<times> real" assumes "0 \<le> u"   and "0 \<le> v"   and "u + v = 1"   and "ccw' a b d"   and "ccw' a b c" shows "ccw' a b (u *\<^sub>R c + v *\<^sub>R d)"
zero_in_Field: assumes "Field oexp \<noteq> {}" shows "o.zero \<in> Field oexp"
groups_of_size_def: fixes g :: "nat" shows "groups_of_size g \<equiv> card {G \<in> \<G>. card G = g}"
in_set_least_squares_approximation_eq_full_rank: fixes A :: "((real, 'a) vec, 'b) vec"   and x :: "(real, 'a) vec"   and b :: "(real, 'b) vec" assumes "rank A = ncols A" shows "(x \<in> set_least_squares_approximation A b) = (x = matrix_inv (Finite_Cartesian_Product.transpose A ** A) ** Finite_Cartesian_Product.transpose A *v b)"
LN_6: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [Label n es' lfilledk] @ es'' : t1s _> t2s"
saction_simps_2: fixes P :: "'a prog"   and a :: "'b"   and al :: "addr_loc"   and v :: "'b val" shows "saction P (NormalAction (ReadMem a al v)) = is_volatile P al"
assms_3: shows "term_ok \<Theta> (mk_eq t u)"
less_acc_def: fixes a :: "acc_modi"   and b :: "acc_modi" shows "(a < b) = (case a of Private \<Rightarrow> b = Package \<or> b = Protected \<or> b = Public | Package \<Rightarrow> b = Protected \<or> b = Public | Protected \<Rightarrow> b = Public | Public \<Rightarrow> False)"
rep_eq: fixes x :: "ocircline" shows "pos_oriented x = pos_oriented_clmat (rep_ocircline x)"
assms_3: shows "\<forall>i\<le>d. v \<notin> Vd i"
Upair_nonEmpty: fixes a :: "ZF"   and b :: "ZF" shows "Upair a b \<noteq> Empty"
norm_triangle_ineq2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm a - norm b \<le> norm (minus a b)"
in_rtrancl_closure_iff_in_rtrancl_fun: fixes a :: "'b"   and b :: "'b"   and TI :: "('b \<times> 'b) list" shows "((a, b) \<in> (set TI)\<^sup>*) = in_rtrancl TI a b"
E_def: shows "E = mat_of_cols (dim_row A) (map (col C) [2..<dim_col A])"
filter_sorted_list_of_set: fixes xs :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "finite xs" shows "filter P (sorted_list_of_set xs) = sorted_list_of_set {x \<in> xs. P x}"
assms_2: shows "max_Var_floatariths ea \<le> n"
is_none_def: fixes x :: "'b option" shows "Option.is_none x = (x = None)"
Im1_def: fixes quat :: "quat" shows "Im1 quat = (case quat of Quat x1 x2 x3 x4 \<Rightarrow> x2)"
partition_main_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and lo :: "nat"   and hi :: "nat"   and xs0 :: "'c list" shows "partition_main R h lo hi xs0 = ASSERT (hi < length xs0) \<bind> (\<lambda>_. RETURN (h (xs0 ! hi)) \<bind> (\<lambda>pivot. WHILE\<^sub>T\<^bsup>partition_main_inv R h lo hi xs0\<^esup> (\<lambda>(i, j, xs). j < hi) (\<lambda>(i, j, xs). ASSERT (i < length xs \<and> j < length xs) \<bind> (\<lambda>_. if R (h (xs ! j)) pivot then RETURN (i + 1, j + 1, swap xs i j) else RETURN (i, j + 1, xs))) (lo, lo, xs0) \<bind> (\<lambda>(i, j, xs). ASSERT (i < length xs \<and> j = hi \<and> lo \<le> i \<and> hi < length xs \<and> mset xs = mset xs0) \<bind> (\<lambda>_. RETURN (swap xs i hi, i)))))"
poly_power_card_as_sum_of_monoms: fixes h :: "'b mod_ring poly" shows "h ^ CARD('b) = (\<Sum>i\<le>degree h. monom (coeff h i) (CARD('b) * i))"
intros_18: fixes c1 :: "f64"   and c2 :: "f64"   and fop :: "relop_f" shows "\<lparr>[$C ConstFloat64 c1, $C ConstFloat64 c2, $Relop_f T_f64 fop]\<rparr> \<leadsto> \<lparr>[$C ConstInt32 (wasm_bool (app_relop_f fop c1 c2))]\<rparr>"
rref_condition2_explicit: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d" assumes "reduced_row_echelon_form A"   and "\<not> is_zero_row i A" shows "A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'b)) = (1::'b)"
asym: fixes b :: "'a word"   and a :: "'a word" assumes "b <s a"   and "a <s b" shows "False"
sum_mset_mono_complex_lex: fixes A :: "complex multiset"   and B :: "complex multiset" assumes "rel_mset (\<le>\<^sub>\<complex>) A B" shows "\<Sum>\<^sub># A \<le>\<^sub>\<complex> \<Sum>\<^sub># B"
assms_6: shows "has_least_prefixpoint (h \<circ> g)"
extend2: fixes f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "f \<in> sub.diff_fun_space"   and "x \<in> B" shows "extend f x = f x"
assms: shows "s' = snd (fst (call_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
mem_mod_def: fixes asi :: "word8"   and addr :: "word36"   and val :: "word8"   and state :: "'b sparc_state" shows "mem_mod asi addr val state \<equiv> let state1 = state\<lparr>mem := (mem state)(asi := mem state asi(addr \<mapsto> val))\<rparr> in if uint asi = 8 \<or> uint asi = 10 then let asi2 = word_of_int (uint asi + 1) in state1\<lparr>mem := (mem state1)(asi2 := (mem state1 asi2)(addr := None))\<rparr> else if uint asi = 9 \<or> uint asi = 11 then let asi2 = word_of_int (uint asi - 1) in state1\<lparr>mem := (mem state1)(asi2 := (mem state1 asi2)(addr := None))\<rparr> else state1"
po_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "po_on P A = (irreflp_on P A \<and> transp_on P A)"
build_ip_partition_same_fw: fixes V :: "'a wordinterval"   and c :: "parts_connection"   and rs :: "'a simple_rule list" assumes "V \<in> set (build_ip_partition c rs)" shows "\<forall>ip1\<in>wordinterval_to_set V. \<forall>ip2\<in>wordinterval_to_set V. same_fw_behaviour_one ip1 ip2 c rs"
Resolution_LSC_pre_nodisj: fixes S :: "'b literal set set"   and R :: "'b literal set" assumes "S \<turnstile> R"   and "finite R"   and "finite S"   and "Ball S finite" shows "\<exists>S' R'. \<forall>\<Gamma>. is_nnf_mset \<Gamma> \<longrightarrow> is_disj R' \<and> is_nnf S' \<and> cnf R' = {R} \<and> cnf S' \<subseteq> S \<and> (R', S', \<Gamma> \<Rightarrow>\<^sub>n \<longrightarrow> S', \<Gamma> \<Rightarrow>\<^sub>n)"
ck_unifier_thm: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and t :: "'c trm"   and s :: "'c trm"   and \<sigma> :: "('c \<times> 'c trm) list"   and k :: "inferences" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "ck_unifier t s \<sigma> k" shows "t \<lhd> \<sigma> = s \<lhd> \<sigma>"
polyfun_roots_finite: fixes c :: "nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat" assumes "c k \<noteq> (0::'a)"   and "k \<le> n" shows "finite {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'a)}"
1: shows "gallery (fst (Spair s) \<Turnstile> (fold fst (map Spair ss) \<Turnstile> Cs))"
proper_it'_def: fixes it :: "'a \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and it' :: "'a \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "proper_it' it it' \<equiv> \<forall>s. proper_it (it s) (it' s)"
finite_imp_card_less: fixes A :: "'b set" assumes "finite A" shows "|A| <o \<AA>"
preserves_section_retraction: fixes e :: "'j"   and m :: "'j" assumes "J.ide (e \<cdot>\<^sub>J m)" shows "C.ide (D e \<cdot> D m)"
concat_parts_leq: fixes a :: "nat"   and d :: "nat"   and v :: "'b list" assumes "a * d \<le> length v" shows "concat (map (fixed_length_sublist v d) [0..<a]) = take (a * d) v"
IH: fixes t :: "('a, 'b) tm" assumes "\<not> list_all (\<lambda>x. x = dir.Left) p" shows "num_args (emb_step_at p d t) = num_args t"
monotone_fun_ord_apply: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "monotone orda (fun_ord ordb) f = (\<forall>x. monotone orda ordb (\<lambda>y. f y x))"
subseq_of_length_def: fixes n :: "nat"   and xs :: "'c list"   and ys :: "'c list" shows "subseq_of_length n xs ys \<equiv> ys \<in> set (subseqs xs) \<and> length ys = n"
assms: shows "\<DD>\<ee>\<II>\<nn>\<ll> opinl = op"
prems_2: shows "length ys = card {a \<in> - A. a < length xs + length ys}"
that: fixes e' :: "nat \<Rightarrow> real" assumes "pdevs_vals e X = pdevs_vals e' (summarize_pdevs_list p I d X)"   and "\<And>i. i < d \<Longrightarrow> e i = e' i"   and "e' \<in> UNIV \<rightarrow> {- 1..1}" shows "thesis"
order_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
A_4: shows "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'"
fold_graph_insertE_aux: fixes z :: "'a"   and B :: "'a set"   and y :: "'a"   and a :: "'a" assumes "fold_graph f z B y"   and "a \<in> B"   and "z \<in> A"   and "B \<subseteq> A" shows "\<exists>y'. y = f a y' \<and> fold_graph f z (B - {a}) y' \<and> y' \<in> A"
of_bl_length: fixes xs :: "bool list" assumes "length xs < LENGTH('b)" shows "of_bl xs < 2 ^ length xs"
p_not_D: shows "\<Union> (set p) \<inter> D = {}"
Units_l_inv: fixes x :: "'a" assumes "x \<in> Units (units_group K)" shows "inv\<^bsub>units_group K\<^esub> x \<otimes>\<^bsub>units_group K\<^esub> x = \<one>\<^bsub>units_group K\<^esub>"
simps_2: fixes p :: "'b \<times> 'c"   and ps :: "('b \<times> 'c) list" shows "map_of (p # ps) = map_of ps(fst p \<mapsto> snd p)"
vsv_ex1_app1: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> xs'" shows "(b = xs'\<lparr>a\<rparr>) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> xs')"
vimage2p_mono: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'f"   and R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'c"   and y :: "'e"   and S :: "'d \<Rightarrow> 'f \<Rightarrow> bool" assumes "BNF_Def.vimage2p f g R x y"   and "R \<le> S" shows "BNF_Def.vimage2p f g S x y"
WHILEIT_rule: fixes R :: "('a \<times> 'a) set"   and I :: "'a \<Rightarrow> bool"   and s :: "'a"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "wf R"   and "I s"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> f s \<le> SPEC (\<lambda>s'. I s' \<and> (s', s) \<in> R)"   and "\<And>s. \<lbrakk>I s; \<not> b s\<rbrakk> \<Longrightarrow> \<Phi> s" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f s \<le> SPEC \<Phi>"
set_mode_get_shadow_root: fixes ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
simps_5: fixes uw :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'd)"   and ux :: "'d \<Rightarrow> bool"   and uy :: "'c match_expr"   and uz :: "'c match_expr" shows "normalized_n_primitive uw ux (MatchNot (MatchAnd uy uz)) = False"
iso_dghm_ObjMap_inv_dghm_ObjMap_app: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>inv_dghm \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<rparr> = a"
smooth_on_times_fun: fixes k :: "enat"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (f * g)"
tab_simps_2: shows "src \<tau> = src t\<^sub>0"
Trg_HcompNml: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Trg (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Trg t"
set_tag_name_get_shadow_root: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
rel_F_eq_refl: fixes x :: "('n, 'o, 'p, 'q, 'r, 's, 't, 'u, 'v, 'w) F" shows "rel_F (=) (=) (=) (=) (=) (=) (=) (=) (=) x x"
Act_eqvt: fixes p :: "perm"   and \<alpha> :: "'g"   and x :: "('e, 'f, 'g) formula" shows "p \<bullet> Act \<alpha> x = Act (p \<bullet> \<alpha>) (p \<bullet> x)"
entry_valid: fixes p :: "'p" shows "proc_of fg (entry fg p) = p"