hcju/isabelleps huggingface dataset:
simps_3: fixes n :: "nat"   and i :: "int"   and a :: "Cooper.num" shows "zsplit0 (CN n i a) = (let (i', a') = zsplit0 a in if n = 0 then (i + i', a') else (i', CN n i a'))"
flow0_transversal_segment_monotone_step: fixes a :: "'a"   and b :: "'a"   and t1 :: "real"   and t2 :: "real"   and x :: "'a"   and t :: "real" assumes "transversal_segment a b"   and "t1 \<le> t2"   and "{t1..t2} \<subseteq> existence_ivl0 x"   and "flow0 x t1 \<in> {a<--<b}"   and "flow0 x t2 \<in> {flow0 x t1<--<b}"   and "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"   and "t2 < t"   and "t \<in> existence_ivl0 x" shows "flow0 x t \<notin> {a<--<flow0 x t2}"
Unify: fixes S :: "('fun, 'var) strand_step list"   and f :: "'fun"   and T' :: "('fun, 'var) Term.term list"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and T :: "('fun, 'var) Term.term list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "simple S"   and "Fun f T' \<in> ik\<^sub>s\<^sub>t S"   and "Some \<delta> = mgu (Fun f T) (Fun f T')" shows "(S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S', \<theta>) \<leadsto> (S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>, \<theta> \<circ>\<^sub>s \<delta>)"
set_cases: fixes e :: "'d"   and a :: "'d list" assumes "e \<in> set a" obtains z2 :: "'d list" where "a = e # z2__"   | z1 :: "'d" and z2 :: "'d list" where "a = z1 # z2" and "e \<in> set z2"
IH: fixes b :: "nat" assumes "local.prefixes t j \<down>= b" shows "t < e_length b"
least_squares_approximation4: fixes S :: "'a set"   and v :: "'a" assumes "subspace S" shows "\<exists>!p. p \<in> S \<and> (\<forall>y\<in>S - {p}. norm (v - p) < norm (v - y))"
elims: fixes x :: "char list set"   and y :: "char list" assumes "fresh_name x = y" obtains S :: "char list set" where "x = S" and "y = (if S = {} then ''a'' else replicate (Max (length ` S) + 1) CHR ''a'')"
flatten_times: fixes p :: "'c \<Rightarrow>\<^sub>0 'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'c \<Rightarrow>\<^sub>0 'd" shows "flatten (p * q) = flatten p * flatten q"
axioms_1: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('d, 'e, 'f) set_ops_scheme" assumes "g_set_xy_defs_loc ops1 ops2" shows "StdSet ops1"
cond_action_ok_init_fin_descend_stateI: fixes final :: "'c \<Rightarrow> bool"   and s :: "('d, 'e) locks \<times> (('e \<Rightarrow> ((status \<times> 'c) \<times> 'd \<Rightarrow>f nat) option) \<times> 'f) \<times> ('e \<Rightarrow> 'g wait_set_status option) \<times> 'e set"   and t :: "'e"   and ct :: "'e conditional_action" assumes "final_thread.cond_action_ok (final_thread.init_fin_final final) s t ct" shows "final_thread.cond_action_ok final (init_fin_descend_state s) t ct"
special_points_on_y_axis''_1: shows "0\<^sub>h \<in> circline_set y_axis"
konigDown_def: shows "konigDown \<equiv> konigDtree (\<lambda>(s, S). S \<in> structure \<and> \<not> sat S s) descent"
subset: fixes m :: "nat" assumes "0 < m" shows "S m \<subseteq> box (complex_of_real (- real m) - complex_of_real (real (2 * m + 1) * pi) * \<i>) (complex_of_real (real m) + complex_of_real (real (2 * m + 1) * pi) * \<i>)"
that: fixes c :: "'c" assumes "homotopic_with_canon (\<lambda>x. True) T U f (\<lambda>x. c)" shows "thesis"
confluent_quotient_ACI: shows "confluent_quotient (~) (~~) (~~) (~~) (~~) (~~) (map_rexp fst) (map_rexp snd) (map_rexp fst) (map_rexp snd) rel_rexp rel_rexp rel_rexp set_rexp set_rexp"
strategy_extends_backwards_VVp: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and w :: "'a" assumes "v0 \<in> G'.VV p"   and "\<sigma> v0 = w"   and "v0 \<rightarrow>\<^bsub>G'\<^esub> w"   and "G'.strategy p \<sigma>"   and "G'.winning_strategy p \<sigma> w" shows "G'.winning_strategy p \<sigma> v0"
upd_cols_insert: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and I :: "'d set"   and f :: "'d \<Rightarrow> ('c, 'd) vec" shows "upd_cols M (insert i I) f = upd_col (upd_cols M I f) i (f i)"
igba_impl_rel_eext_internal_def: fixes Rm :: "('c \<times> 'f) set"   and Rv :: "('a \<times> 'd) set"   and Rl :: "('b \<times> 'e) set" shows "igba_impl_rel_eext Rm Rv Rl \<equiv> \<langle>Rm, Rv \<rightarrow> Rl \<rightarrow> bool_rel\<rangle>gen_igba_impl_rel_eext"
add_divide_eq_if_simps_4: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and z :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "minus a (divide b z) = (if z = zero then a else divide (minus (times a z) b) z)"
Deriv_epsilon: fixes a :: "'b" shows "Deriv a {[]} = {}"
rref_condition2: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "reduced_row_echelon_form A" shows "\<forall>i. \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'a)) = (1::'a)"
AsCondExpr2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and Q :: "'e set"   and A :: "'e set"   and a2 :: "('e, 'd, 'f) ann"   and c2 :: "('e, 'd, 'f) com"   and a :: "'e set"   and r :: "'e set"   and a1 :: "('e, 'd, 'f) ann"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com" assumes "assertionsR \<Gamma> \<Theta> Q A a2 c2 a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnBin r a1 a2) (Cond b c1 c2) a"
squarefree_part_unit: fixes n :: "'b" assumes "is_unit n" shows "squarefree_part n = n"
all: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'a"   and \<phi> :: "'c" assumes "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi"   and "x \<in> var"   and "\<phi> \<in> fmla" shows "all x \<phi> \<in> fmla"
r_take': fixes n :: "nat"   and x :: "nat" shows "eval r_take [n, x] \<down>= e_take n x"
suminf_pos2: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "summable f"   and "\<And>n. (0::'a) \<le> f n"   and "(0::'a) < f i" shows "(0::'a) < suminf f"
abs_eq: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and x :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus (Word.Word x) = ring_1.of_int one plus zero minus (signed_take_bit (LENGTH('l) - Suc 0) x)"
step_rfunc_adv: fixes p :: "'s_rest spmf"   and q :: "'s_rest' spmf"   and ia :: "'iadv_rest" assumes "S p q"   and "ia \<in> IA" shows "p \<bind> (\<lambda>s1. map_spmf fst (rfunc_adv rest1 s1 ia)) = q \<bind> (\<lambda>s2. map_spmf fst (rfunc_adv rest2 s2 ia))"
eq_\<I>_gpv_relcompp: fixes A :: "'a \<Rightarrow> 'e \<Rightarrow> bool"   and A' :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>" shows "eq_\<I>_gpv (A OO A') \<I> = eq_\<I>_gpv A \<I> OO eq_\<I>_gpv A' \<I>"
HVZK_unfold1: assumes "\<Sigma>_protocol" shows "\<forall>h w e. (h, w) \<in> Rel \<longrightarrow> e \<in> challenge_space \<longrightarrow> R h w e = S h e"
emeasure_count_space_infinite: fixes X :: "'c set"   and A :: "'c set" assumes "X \<subseteq> A"   and "infinite X" shows "emeasure (count_space A) X = \<infinity>"
control_state_ext_def: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'b" shows "\<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons control_state_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple break_status return_status) more"
ZObis_sym: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "c \<approx>01 d" shows "d \<approx>01 c"
card_bijections_domain_and_range_permutation_eq_0: fixes A :: "'c set"   and B :: "'d set" assumes "card A \<noteq> card B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 0"
assert_def: fixes p :: "'e"   and q :: "'e" shows "{. p .} q \<equiv> p \<sqinter> q"
f_ind: fixes ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and anything :: "nat list" assumes "rec_ci f = (ap, arity, fp)" shows "{\<lambda>nl. nl = xs @ r # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ r # 0 # 0 \<up> (fp - Suc arity) @ anything}"
wod_backward_slice_valid_node: fixes n :: "'a LDCFG_node"   and S :: "'a LDCFG_node set" assumes "n \<in> CFGExit_wf.wod_backward_slice S" shows "CFGExit_wf.valid_node n"
as_S_not_empty: shows "{x \<in> I. \<nexists>n. I \<rightarrow> n = x} \<noteq> {}"
mdecl_qmdecl_simp: fixes sig :: "sig"   and new :: "qtname \<times> methd" shows "mdecl (mthd (qmdecl sig new)) = mdecl (sig, mthd new)"
par_comp_constr: fixes \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "par_comp \<A> Sec"   and "typing_cond (unlabel \<A>)"   and "\<I> \<Turnstile> \<langle>unlabel \<A>\<rangle>"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" shows "\<exists>\<I>\<^sub>\<tau>. (subst_domain \<I>\<^sub>\<tau> = \<V> \<and> ground (subst_range \<I>\<^sub>\<tau>)) \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>\<^sub>\<tau> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>) \<and> \<I>\<^sub>\<tau> \<Turnstile> \<langle>unlabel \<A>\<rangle> \<and> ((\<forall>l. \<I>\<^sub>\<tau> \<Turnstile> \<langle>proj_unl l \<A>\<rangle>) \<or> (\<exists>\<A>'. prefix \<A>' \<A> \<and> strand_leaks\<^sub>l\<^sub>s\<^sub>t \<A>' Sec \<I>\<^sub>\<tau>))"
dist_triangle_half_l: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x1 :: "'a"   and y :: "'a"   and e :: "real"   and x2 :: "'a" assumes "class.metric_space dist uniformity open"   and "dist x1 y < e / 2"   and "dist x2 y < e / 2" shows "dist x1 x2 < e"
p'_def: fixes t :: "real" shows "ego_other.p' t = v\<^sub>o - v\<^sub>e + (a\<^sub>o - a\<^sub>e) * t"
impl_of_cases: fixes y :: "('a \<times> 'b) list" assumes "y \<in> {xs. (distinct \<circ> map fst) xs}" obtains x :: "('a, 'b) alist" where "y = alist.impl_of x"
arr_assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c" shows "D.arr \<a>\<^sub>D[a, b, c]"
sum_gt_neutral_dest: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "\<one> < a + b" shows "\<exists>d. Le d \<le> a \<and> Le (- d) \<le> b"
finfun_apply: fixes x :: "('a, 'b) finfun" shows "finfun_apply x \<in> finfun"
select_convs_5: fixes ob :: "'b set"   and ar :: "'c set"   and dom :: "'c \<Rightarrow> 'b"   and cod :: "'c \<Rightarrow> 'b"   and id :: "'b \<Rightarrow> 'c"   and comp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "category.id \<lparr>ob = ob, ar = ar, dom = dom, cod = cod, id = id, comp = comp, \<dots> = more\<rparr> = id"
Rep_IF2_inverse: fixes x :: "'d IF2" shows "Abs_IF2 (Rep_IF2 x) = x"
pos_pp_np_help: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "(0::'b) \<le> f x" shows "pp f x = f x \<and> np f x = (0::'b)"
poly_lift_hom_sub: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "p \<in> carrier P"   and "q \<in> carrier P" shows "poly_lift_hom R S \<phi> (Cring_Poly.compose R p q) = Cring_Poly.compose S (poly_lift_hom R S \<phi> p) (poly_lift_hom R S \<phi> q)"
real_cond_exp_gr_c: fixes f :: "'a \<Rightarrow> real"   and c :: "real" assumes "integrable M f"   and "AE x in M. c < f x" shows "AE x in M. c < real_cond_exp M F f x"
cba_bres1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> z) = (x \<le> proto_near_quantale.bres Sup (\<le>) (\<sqinter>) y z)"
oseq_choiceTE: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and p1 :: "('e, 'f, 'g, 'h) seqp"   and p2 :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat" assumes "((\<sigma>, p1 \<oplus> p2), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i" obtains "((\<sigma>, p1), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"   | "((\<sigma>, p2), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"
r_right_minus_eq: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier P"   and "b \<in> carrier P" shows "(a \<ominus>\<^bsub>P\<^esub> b = \<zero>\<^bsub>P\<^esub>) = (a = b)"
loc_srj_0: shows "nat_to_sch (c_pair 1 0) = Base_suc"
columns_transpose: fixes A :: "(('a, 'b) vec, 'c) vec" shows "columns (Finite_Cartesian_Product.transpose A) = rows A"
finalNonConflictState_1: fixes state :: "State"   and decisionVars :: "nat set" assumes "getConflictFlag state = False"   and "\<not> applicableDecide state decisionVars"   and "\<not> applicableConflict state" shows "\<not> formulaFalse (getF state) (elements (getM state))"
intro: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>a b. \<lbrakk>a \<in> S; b \<in> S\<rbrakk> \<Longrightarrow> pls a b = pls b a" shows "ab_semigroup_add_ow_axioms S pls"
symmetric_mpoly_def: fixes A :: "nat set"   and p :: "'a mpoly" shows "symmetric_mpoly A p = (\<forall>\<pi>. \<pi> permutes A \<longrightarrow> mpoly_map_vars \<pi> p = p)"
sel_2: fixes x :: "quat" shows "Im1 (inverse x) = - Im1 x / ((quat.Re x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2)"
rb_sp_append_one: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv1 :: "'d \<times> nat"   and res :: "('d \<times> nat) edge list"   and re :: "('d \<times> nat) edge"   and rv2 :: "'d \<times> nat" assumes "RedBlack prb" shows "SubRel.subpath (red prb) rv1 (res @ [re]) rv2 (subs prb) = (SubRel.subpath (red prb) rv1 res (src re) (subs prb) \<and> re \<in> edges (red prb) \<and> (tgt re = rv2 \<or> (tgt re, rv2) \<in> subs prb))"
hseq_char: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" shows "hseq \<nu> \<mu> = (arr \<mu> \<and> arr \<nu> \<and> src \<nu> = trg \<mu>)"
llength_eq_0: fixes xs :: "'a llist" shows "(llength xs = 0) = lnull xs"
simps_2: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and s :: "'a"   and q :: "'b"   and qs :: "'b list"   and a :: "'c"   and as :: "'c list" shows "On_Off.steps step s (q # qs) (a # as) = On_Off.steps step (step s q a) qs as"
ln_2_ge': shows "2 / 3 \<le> ln 2"
spaceN_sum: fixes I :: "'c set"   and x :: "'c \<Rightarrow> 'd"   and N :: "'d quasinorm" assumes "\<And>i. i \<in> I \<Longrightarrow> x i \<in> space\<^sub>N N" shows "sum x I \<in> space\<^sub>N N"
prime_imp_coprime_nat: fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "\<not> p dvd n" shows "coprime p n"
B: shows "Y = (\<Union>n\<le>length xs. \<Union>W\<in>Ry n. W) \<and> (\<forall>W\<in>Ry 0. xs \<notin> sentences P \<and> None \<notin> set xs \<and> (xs, W) \<in> failures P \<or> xs \<in> sentences P \<and> (\<exists>U V. (xs, U) \<in> failures P \<and> ([], V) \<in> failures Q \<and> W = insert None U \<inter> V)) \<and> (\<forall>n\<in>{0<..length xs}. \<forall>W\<in>Ry n. take (length xs - n) xs \<in> sentences P \<and> (drop (length xs - n) xs, W) \<in> failures Q) \<and> (\<exists>n\<in>{..length xs}. \<exists>W. W \<in> Ry n)"
key_sets_crypt_1: fixes X :: "msg"   and K :: "key"   and Y :: "msg" assumes "X \<noteq> Crypt K Y" shows "key_sets X {Crypt K Y} \<subseteq> insert (InvKey K) ` key_sets X {Y}"
vector_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and y :: "'b"   and x :: "'b"   and z :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit"   and "relation_algebra.is_vector top composition y" shows "composition (inf x (converse y)) z = composition x (inf y z)"
1: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> {a} \<rightarrow>\<^sub>E B"   and "y \<in> {a} \<rightarrow>\<^sub>E B"   and "x a = y a" shows "x = y"
lconcat_eq_LNil: fixes xss :: "'a llist llist" shows "(lconcat xss = LNil) = (lset xss \<subseteq> {LNil})"
assms_3: shows "\<forall>r\<in>R. Well_order r"
restrict_top_eq: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex"   and val :: "bool" assumes "ifex_ordered (IF v t e)" shows "restrict (IF v t e) v val = restrict_top (IF v t e) v val"
imp: fixes \<phi>1 :: "fm"   and \<phi>2 :: "fm" assumes "\<phi>1 \<in> UNIV"   and "\<phi>2 \<in> UNIV" shows "\<phi>1 IMP \<phi>2 \<in> UNIV"
eucl_of_list_mem_lv_rel: fixes x :: "real list" assumes "length x = DIM('a)" shows "(x, eucl_of_list x) \<in> lv_rel"
matrix_vector_mult_hom: fixes A :: "((real, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "(real, 'c) Finite_Cartesian_Product.vec" shows "HMA_Connect.map_matrix of_real A *v map_vector of_real v = map_vector of_real (A *v v)"
x'_def: shows "x' = nextElem xs (hd xs) x"
is_unit_gauss_int_iff: fixes x :: "gauss_int" shows "is_unit x = (x \<in> {1, - 1, \<i>\<^sub>\<int>, - \<i>\<^sub>\<int>})"
fract_collapse_2: shows "Fraction_Field.Fract (1::'b) (1::'b) = 1"
subobj_aux_rev: 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"   and C :: "char list"   and C' :: "char list"   and Cs :: "char list list"   and C'' :: "char list" assumes "is_subobj P (C, C' # rev Cs @ [C''])" shows "is_subobj P (C, C' # rev Cs)"
finite_nat_set_iff_bounded: fixes N :: "nat set" shows "finite N = (\<exists>m. \<forall>n\<in>N. n < m)"
wfLearnLess: fixes C :: "Literal list" shows "wf (learnLess C)"
final_finished: fixes F_empty :: "'a"   and F_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) instr fundef option"   and F_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) instr fundef \<Rightarrow> 'a"   and F_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e) instr fundef) list"   and heap_empty :: "'f"   and heap_get :: "'f \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'f \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'f"   and heap_to_list :: "'f \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and s :: "('a, 'f, ('b, 'c) frame) state" assumes "std F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy>"   and "std.final F_get s" shows "finished (std.step F_get heap_get heap_add is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy>) s"
L_delta: fixes a :: "'a"   and s :: "'s" shows "L (delta a s) = Deriv a (L s)"
not_and_lt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "\<not> (A B Lt C D \<and> C D Lt A B)"
constT_ss_objT: fixes P :: "ast_problem" shows "ast_domain.constT (domain P) \<subseteq>\<^sub>m ast_problem.objT P"
regular_closed_sup: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.stone_algebra inf less_eq less sup bot top uminus"   and "x = uminus (uminus x)"   and "y = uminus (uminus y)" shows "sup x y = uminus (uminus (sup x y))"
distinct_clearjunk: fixes al :: "('c \<times> 'd) list" shows "distinct (map fst (clearjunk al))"
density_density_divide: fixes f :: "'d \<Rightarrow> real"   and M :: "'d measure"   and g :: "'d \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> g x"   and "AE x in M. f x = 0 \<longrightarrow> g x = 0" shows "density (density M (\<lambda>x. ennreal (f x))) (\<lambda>x. ennreal (g x / f x)) = density M (\<lambda>x. ennreal (g x))"
measures_def: fixes fs :: "('b \<Rightarrow> nat) list" shows "measures fs = inv_image (lex less_than) (\<lambda>a. map (\<lambda>f. f a) fs)"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_end2 n (l, r) = (\<exists>i j. i + j = Suc n \<and> 0 < n \<and> l = Oc \<up> i @ [Bk] \<and> r = Bk \<up> j @ Oc \<up> n)"
d\<mu>: fixes j :: "nat"   and ii :: "nat" assumes "j < length fs"   and "ii < length fs" shows "rat_of_int (fs.d\<mu> ii j) = rat_of_int (fs'.d fs (Suc j)) * fs.gs.\<mu> ii j"
meet_le: fixes L :: "('c, 'd) gorder_scheme"   and z :: "'c"   and x :: "'c"   and y :: "'c" assumes "weak_lower_semilattice L"   and "z \<sqsubseteq>\<^bsub>L\<^esub> x"   and "z \<sqsubseteq>\<^bsub>L\<^esub> y"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "z \<sqsubseteq>\<^bsub>L\<^esub> x \<sqinter>\<^bsub>L\<^esub> y"
simps_1: shows "Expr2ZF Vx = ZFTriple (nat2Nat 0) (nat2Nat 0) Empty"
SCSubset: fixes b :: "('b, 'c) Combinators list"   and a :: "('b, 'c) Combinators list" assumes "singleCombinators b"   and "set a \<subseteq> set b" shows "singleCombinators a"
antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x"   and "x \<in> B"   and "y \<in> B" shows "x = y"
ordinal_mod_greater: fixes x :: "ordinal"   and y :: "ordinal" assumes "x < y" shows "x mod y = x"
simps: fixes a1 :: "'b rexp"   and a2 :: "'b rexp" shows "ACI a1 a2 = ((\<exists>r. a1 = r \<and> a2 = r) \<or> (\<exists>r s. a1 = s \<and> a2 = r \<and> ACI r s) \<or> (\<exists>r s t. a1 = r \<and> a2 = t \<and> ACI r s \<and> ACI s t) \<or> (\<exists>r1 s1 r2 s2. a1 = Plus r1 r2 \<and> a2 = Plus s1 s2 \<and> ACI r1 s1 \<and> ACI r2 s2) \<or> (\<exists>r1 s1 r2 s2. a1 = Times r1 r2 \<and> a2 = Times s1 s2 \<and> ACI r1 s1 \<and> ACI r2 s2) \<or> (\<exists>r s. a1 = Star r \<and> a2 = Star s \<and> ACI r s) \<or> (\<exists>r s t. a1 = Plus (Plus r s) t \<and> a2 = Plus r (Plus s t)) \<or> (\<exists>r s. a1 = Plus r s \<and> a2 = Plus s r) \<or> (\<exists>r. a1 = Plus r r \<and> a2 = r))"
fv_snd_rcv_strand_subset_5: fixes S :: "('b, 'a) strand_step list" shows "wfvarsoccs\<^sub>s\<^sub>t S \<subseteq> fv\<^sub>s\<^sub>t S"
in_set_conv_decomp_first: fixes x :: "'e"   and xs :: "'e list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
simps_3: fixes f :: "('d, 'e) bintree \<Rightarrow> 'c"   and t :: "('d, 'e) bintree"   and xs :: "('d, 'e) bintree option list" shows "dfs f (Some t # xs) = bt_dfs f t @ dfs f xs"
retract_of_space_empty: fixes X :: "'b topology" shows "({} retract_of_space X) = (topspace X = {})"
while_option_stop: fixes b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c" assumes "while_option b c s = Some t" shows "\<not> b t"