hcju/isabelleps huggingface dataset:
a_App: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e\<^sub>1 :: "Syntax.term"   and eP\<^sub>1 :: "Syntax.term"   and eV\<^sub>1 :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty"   and e\<^sub>2 :: "Syntax.term"   and eP\<^sub>2 :: "Syntax.term"   and eV\<^sub>2 :: "Syntax.term" assumes "\<Gamma> \<turnstile> e\<^sub>1, eP\<^sub>1, eV\<^sub>1 : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"   and "\<Gamma> \<turnstile> e\<^sub>2, eP\<^sub>2, eV\<^sub>2 : \<tau>\<^sub>1" shows "\<Gamma> \<turnstile> App e\<^sub>1 e\<^sub>2, App eP\<^sub>1 eP\<^sub>2, App eV\<^sub>1 eV\<^sub>2 : \<tau>\<^sub>2"
intro: fixes M1 :: "'c measure"   and M2 :: "'d measure" assumes "sigma_finite_measure M1"   and "sigma_finite_measure M2" shows "pair_sigma_finite M1 M2"
ev_at_HLD_single_imp_snth: fixes x :: "'a"   and n :: "nat"   and \<omega> :: "'a stream" assumes "ev_at (HLD {x}) n \<omega>" shows "\<omega> !! n = x"
prems_8: shows "mk_eqcl' (y # xs) ii jj l T ! i = Some m"
components_full_pmdl: fixes K :: "'k set" shows "component_of_term ` Keys (full_pmdl K) = K"
df_system_axioms: shows "df_system cs\<^sub>0 clstep"
aform_of_ivl_def: fixes l :: "'b"   and u :: "'b" shows "aform_of_ivl l u = ((l + u) /\<^sub>R 2, pdevs_of_ivl l u)"
in_mono: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
simps_4: fixes \<A> :: "'c ltl set"   and \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "\<A> \<Turnstile>\<^sub>P \<phi> or \<psi> = (\<A> \<Turnstile>\<^sub>P \<phi> \<or> \<A> \<Turnstile>\<^sub>P \<psi>)"
inj_imp_image_group_is_group: fixes G :: "('c, 'd) monoid_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "inj_on f (carrier G)" shows "Group.group (image_group f G)"
Cont: fixes generat :: "('b, 'a, ('b, 'a, 'c) rpv) generat"   and gpv :: "('b, 'a, 'c) gpv"   and c :: "('b, 'a, 'c) rpv"   and input :: "'c" assumes "generat \<in> set_spmf (the_gpv gpv)"   and "c \<in> generat_conts generat"   and "x \<in> outs'_gpv (c input)"   and "P (c input)" shows "P gpv"
cong_iff_lin: fixes a :: "'a"   and b :: "'a"   and m :: "'a" shows "[a = b] (mod m) = (\<exists>k. b = a + m * k)"
list_gcd_greatest: fixes xs :: "'a list"   and y :: "'a" assumes "\<And>x. x \<in> set xs \<Longrightarrow> y dvd x" shows "y dvd gcd_list xs"
lemma_2_10_29: fixes b :: "'a"   and a :: "'a" shows "b l\<rightarrow> (a l\<rightarrow> b) r\<rightarrow> a = b l\<rightarrow> a"
b: shows "\<forall>x. pi \<bullet> f x = f (pi \<bullet> x)"
merge_emb_step_at: fixes p :: "dir list"   and d1 :: "dir"   and q :: "dir list"   and d2 :: "dir"   and t :: "('c, 'd) tm" shows "emb_step_at p d1 (emb_step_at (p @ [opp d1] @ q) d2 t) = emb_step_at p d1 t"
image_Int: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "inj f" shows "f ` (A \<inter> B) = f ` A \<inter> f ` B"
le_imp_less_or_eq: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "y < x \<or> x = y"
summable_minus_iff: fixes f :: "nat \<Rightarrow> 'a" shows "summable (\<lambda>n. - f n) = summable f"
mono_gamma: fixes a :: "'av"   and b :: "'av" assumes "a \<sqsubseteq> b" shows "\<gamma> a \<subseteq> \<gamma> b"
h_len: assumes "length (pivot_positions A) \<le> dim_row A" shows "dim_col (take_cols A (map snd (pivot_positions A))) = dim_col (1\<^sub>m (length (pivot_positions A)) @\<^sub>r 0\<^sub>m (dim_row A - length (pivot_positions A)) (length (pivot_positions A)))"
cnj_add: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and cnj :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.cnj divide inverse times one plus zero minus uminus cnj" shows "cnj (plus a b) = plus (cnj a) (cnj b)"
elims: fixes x :: "v sem_env"   and xa :: "(v sem_env, 'b) result"   and y :: "(v sem_env, 'b) result" assumes "combine_dec_result x xa = y" obtains env :: "v sem_env" and e :: "'b error_result" where "x = enva__" and "xa = Rerr e" and "y = Rerr e"   | env :: "v sem_env" and env' :: "v sem_env" where "x = env" and "xa = Rval env'" and "y = Rval (make_sem_env (nsAppend (sem_env.v env') (sem_env.v env)) (nsAppend (c env') (c env)))"
invariant_start: fixes r :: "'s"   and s :: "'s" assumes "wf_state r"   and "wf_state s" shows "invariant r s ([([], r, s)], [], {(post r, post s)})"
characteristic_iso_2: fixes f :: "'a" assumes "Left.ide f" shows "Left.in_hom (\<i>[trg f] \<star> f) ((trg f \<star> trg f) \<star> f) (trg f \<star> f)"
assms: fixes x :: "'a pdevs" assumes "x \<in> set X" shows "degree x \<le> K"
map\<^sub>1_in_hom_2: fixes f :: "'a" assumes "C.ide f" shows "\<guillemotleft>\<tau>\<^sub>1 f : G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f) \<Rightarrow>\<^sub>D \<tau>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D F f\<guillemotright>"
converse_subset_swap: fixes r :: "('d \<times> 'e) set"   and s :: "('e \<times> 'd) set" shows "(r \<subseteq> s\<inverse>) = (r\<inverse> \<subseteq> s)"
s_strict_mono: fixes u :: "real"   and t :: "real" assumes "u < t"   and "t \<le> delayed_safe_distance.other.t_stop"   and "0 \<le> u" shows "delayed_safe_distance.other.s u < delayed_safe_distance.other.s t"
join_idem: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<squnion> x = x"
timpl_closure'_timpls_trancl_eq: fixes c :: "('c set \<times> 'c set) set" shows "timpl_closure' (c\<^sup>+) = timpl_closure' c"
notin_range_Some: fixes x :: "'e option" shows "(x \<notin> range Some) = (x = None)"
le_reflexivity: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Le Bet Cong A B A B"
k_def: fixes G1 :: "'a poly"   and G2 :: "'a poly" shows "??.Subresultant.k G1 G2 = (LEAST i. ??.Subresultant.F G1 G2 (Suc i) = 0)"
M_def: shows "\<forall>m\<ge>M. \<forall>n\<ge>M. \<parallel>X m - X n\<parallel> < \<epsilon>"
map_poly_hom_monom: fixes ia :: "nat"   and a :: "'a poly"   and i :: "nat" shows "map_poly (\<lambda>p. coeff p ia) (monom a i) = monom (coeff a ia) i"
MIV: shows "IntVal \<in> count_space UNIV \<rightarrow>\<^sub>M count_space (range IntVal)"
ang_vec_c_def: fixes z1 :: "complex"   and z2 :: "complex" shows "\<angle>c z1 z2 \<equiv> \<bar>\<angle> z1 z2\<bar>"
dsf: shows "matrix_sum d f n \<in> carrier_mat d d"
transfer_FOREACHoci_plain: fixes iterate :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'d set"   and ordR :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and fi :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool" assumes "set_iterator_genord iterate s ordR"   and "\<And>x \<sigma>. RETURN (fi x \<sigma>) \<le> f x \<sigma>" shows "RETURN (iterate c fi \<sigma>) \<le> FOREACH\<^sub>O\<^sub>C\<^bsup>ordR,I\<^esup> s c f \<sigma>"
h_plus_mono': fixes i :: "nat"   and j :: "nat"   and e :: "'v \<times> 'v" assumes "i \<le> j" shows "h_plus i e \<le> h_plus j e"
assms: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> set xys" shows "y < x"
wf_sum_def: fixes divisor :: "'c \<Rightarrow> bool"   and r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set" shows "wf_sum divisor r r' \<equiv> {(x, y). \<not> divisor x \<and> \<not> divisor y} \<inter> r' \<union> {(x, y). \<not> divisor x \<and> divisor y} \<union> {(x, y). divisor x \<and> divisor y} \<inter> r"
get_impl_def: fixes M :: "('a SecurityInvariant \<times> 'a SecurityInvariant_configured) list" shows "get_impl M \<equiv> map fst M"
h: shows "i < length (pivot_positions A) \<longrightarrow> fst (pivot_positions A ! i) = i"
lm05: fixes P :: "('a \<times> 'b) set" assumes "inj_on fst P" shows "runiq P"
leI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less"   and "\<not> less x y" shows "less_eq y x"
sum_eq: shows "integral\<^sup>N (count_space X) f = integral\<^sup>N (count_space (\<^bold>E `` X)) g"
BinOp\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>2 :: "nat exp"   and v\<^sub>2 :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and bop :: "bop"   and v :: "val" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v\<^sub>1,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v\<^sub>2,s\<^sub>2\<rangle>"   and "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>"
that_2: assumes "\<not> c"   and "run e h h' r" shows "thesis"
assms_3: shows "T \<inter> rel_interior u \<noteq> {}"
int_times_rat_def: fixes i :: "int"   and x :: "rat" shows "int_times_rat i x = rat_of_int i * x"
psubst_Var_simp1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "goodEnv rho"   and "rho xs x = None" shows "Var xs x #[rho] = Var xs x"
p_1: shows "(\<^bold>\<not> \<^bold>\<forall> p) \<in> Extend S f"
sumcondlim: shows "\<forall>x. (\<lambda>i. expl_cond_expect M Y X x * indpre i x) sums (expl_cond_expect M Y X x * indicat_real (Y -` (A \<inter> range Y) \<inter> space M) x)"
interval_bij_def: shows "interval_bij = (\<lambda>(a, b) (u, v) x. \<Sum>i\<in>Basis. (u \<bullet> i + (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i) * (v \<bullet> i - u \<bullet> i)) *\<^sub>R i)"
inext_less_conv: fixes n :: "nat"   and I :: "nat set" shows "(n \<in> I \<and> (n < Max I \<or> infinite I)) = (n < inext n I)"
assms_2: shows "\<D>\<^sub>\<circ> a = I"
diff_eq1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "x - y = x + - y"
qSwap_preserves_qGood2: fixes zs :: "'e"   and x :: "'d"   and y :: "'d"   and X :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood (X #[[x \<and> y]]_zs)" shows "FixVars.qGood X"
set_elem_equalityI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B"   and "\<And>x. x \<in> B \<Longrightarrow> x \<in> A" shows "A = B"
l10: shows "Fract (fst p ^ 3) (snd p ^ 3) - Fract (3 * fst p * (snd p)\<^sup>2) (snd p ^ 3) = 1"
tendsto_norm: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. \<parallel>f x\<parallel>) \<longlongrightarrow> \<parallel>a\<parallel>) F"
a0: fixes m :: "nat"   and i :: "nat" assumes "m < 2 ^ n"   and "i \<le> n"   and "0 \<le> m" shows "bin_rep_aux n m ! i = 0 \<or> bin_rep_aux n m ! i = 1"
filter_mset_union: fixes A :: "'d multiset"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x. \<lbrakk>x \<in># A; P x; Q x\<rbrakk> \<Longrightarrow> False" shows "filter_mset P A + filter_mset Q A = {#x \<in># A. P x \<or> Q x#}"
support_adapt_def: fixes Mkt :: "('a, 'c) discrete_market"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" shows "support_adapt Mkt pf = (\<forall>asset\<in>support_set pf. borel_adapt_stoch_proc F (prices Mkt asset))"
isomorphic_transitive: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c"   and h :: "'c \<times> 'c" assumes "C.VV.isomorphic f g"   and "C.VV.isomorphic g h" shows "C.VV.isomorphic f h"
redundant_def: fixes C :: "'a Literal set"   and S :: "'a Literal set set" shows "redundant C S = (tautology C \<or> (\<exists>D. D \<in> S \<and> subsumes D C))"
canonical_embedding_is_hom: fixes K :: "('c multiset \<Rightarrow> 'a) set" assumes "subring K L" shows "L.poly_of_const \<in> ring_hom (L\<lparr>carrier := K\<rparr>) (K [X]\<^bsub>L\<^esub>)"
proj_arcs_G_simps_3: fixes a :: "'b" assumes "a \<notin> {uv, rev_G uv}" shows "proj_arcs_G a = a"
simps_4: fixes f1 :: "'b"   and f2 :: "'b" shows "(case False of True \<Rightarrow> f1 | False \<Rightarrow> f2) = f2"
outside_canonical_retraction_chamber_images_3: shows "canonical_retraction A D ` E = C"
points_to_Mutate: fixes y :: "'c"   and x :: "'c"   and s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state"   and p :: "'d process_name"   and ws :: "('a, 'b, 'c) mem_store_action list"   and r :: "'c"   and f :: "'a"   and opt_r' :: "'c option" shows "(x points_to y) (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := opt_r')\<rparr>) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>)) = (r \<noteq> x \<and> (x points_to y) s \<or> r = x \<and> obj_at (\<lambda>s. True) r s \<and> (opt_r' = Some y \<or> (x points_to y) s \<and> obj_at (\<lambda>obj. \<exists>f'. obj_fields obj f' = Some y \<and> f \<noteq> f') r s))"
split: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c"   and unit :: "unit" shows "P (case unit of () \<Rightarrow> f) = (unit = () \<longrightarrow> P f)"
ladder_last_j_of_cut: fixes L :: "(nat \<times> nat \<times> nat) list"   and n :: "nat" assumes "L \<noteq> []" shows "ladder_last_j (ladder_cut L n) = ladder_last_j L"
le_fin_def: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind" shows "traces.le_fin ind w\<^sub>1 w\<^sub>2 \<equiv> \<exists>v\<^sub>1. (traces.eq_swap ind)\<^sup>*\<^sup>* (w\<^sub>1 @ v\<^sub>1) w\<^sub>2"
hyps: fixes q :: "'a poly"   and r :: "'a poly"   and dr :: "nat"   and q' :: "'a poly" shows "snd (pseudo_divmod_main (x * lc) q r (Polynomial.smult x d) dr n) = snd (pseudo_divmod_main lc q' (Polynomial.smult (x ^ n) r) d dr n)"
map_id: fixes t :: "'b tree" shows "map_tree id t = t"
prod_mset_delta: fixes y :: "'c"   and c :: "'a"   and A :: "'c multiset" shows "prod_mset {#if x = y then c else (1::'a). x \<in># A#} = c ^ count A y"
set_map_1: fixes f1 :: "'d \<Rightarrow> 'a"   and f2 :: "'e \<Rightarrow> 'b"   and f3 :: "'f \<Rightarrow> 'c"   and v :: "('d, 'e, 'f) expr" shows "RID\<^sub>E (map_expr f1 f2 f3 v) = f1 ` RID\<^sub>E v"
lnth_lappend: fixes xs :: "'a llist"   and ys :: "'a llist"   and n :: "nat" shows "lnth (xs @\<^sub>l ys) n = (if enat n < llength xs then lnth xs n else lnth ys (n - the_enat (llength xs)))"
card_eq_Suc_0_ex1: fixes A :: "'d set" shows "(card A = Suc 0) = (\<exists>!x. x \<in> A)"
trancl_id: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "r\<^sup>+ = r"
ptrm: fixes \<sigma> :: "'fmla"   and n :: "nat" assumes "\<sigma> \<in> ptrm n" shows "\<sigma> \<in> fmla"
false_rule: fixes c :: "'d Heap"   and Q :: "'d \<Rightarrow> assn" shows "<false> c <Q>"
A': shows "A' \<equiv> mult_row (interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)) i ((1::'a) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j)"
noDAsortQ: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "noDenyAll1 p" shows "noDenyAll1 (qsort p l)"
ordinal_order_less_le_not_le: fixes x :: "ordinal"   and y :: "ordinal" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
block_complement_size: fixes b :: "'a set" assumes "b \<subseteq> bl" shows "card (derived_incidence_sys.block_complement b) = card bl - card b"
set_sort: fixes f :: "'a \<Rightarrow> 'b word"   and xs :: "'a list" shows "set (signed.sort_key f xs) = set xs"
liouville_lambda_prime_power: fixes p :: "nat"   and k :: "nat" assumes "prime p" shows "liouville_lambda (p ^ k) = (- (1::'b)) ^ k"
set0: shows "{w \<in> space M. f w \<noteq> g w} \<in> sets (F t) \<and> emeasure M {w \<in> space M. f w \<noteq> g w} = 0"
induct: fixes P :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> 'd"   and a1 :: "'c list" assumes "\<And>f. P f []"   and "\<And>f x xs. P f (filter (\<lambda>y. f x \<noteq> f y) xs) \<Longrightarrow> P f (x # xs)" shows "P a0 a1"
relabel_action_bound': fixes fxl :: "(nat \<times> nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> nat)"   and p :: "pr_operation' list"   and fxl' :: "(nat \<times> nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> nat)" assumes "(fxl, p, fxl') \<in> trcl pr_algo_lts'" shows "length (filter is_RELABEL' p) \<le> 2 * (card V)\<^sup>2"
fls_decompose_reg_pr_parts_2: fixes f :: "'a fls" shows "f = fls_regpart_as_fls f + fls_prpart_as_fls f"
init_segment_iff_initial_segment: fixes xs :: "nat list"   and ys :: "nat list" assumes "strict_sorted xs"   and "strict_sorted ys" shows "init_segment (list.set xs) (list.set ys) = initial_segment xs ys"
simps: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" shows "(a \<in> nnfis f M) = (\<exists>u x y. a = y \<and> u\<up>f \<and> (\<forall>xa. x xa \<in> sfis (u xa) M) \<and> x\<up>y)"
orthogonal_complement_zero: shows "orthogonal_complement {0::'b} = UNIV"
xor_left_commute: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<oplus> y \<oplus> z = y \<oplus> x \<oplus> z"
in_mono: fixes A :: "'f set"   and B :: "'f set"   and x :: "'f" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
param_in_slice1: fixes n'' :: "'node SDG_node"   and V :: "'var"   and p :: "'pname"   and n' :: "'node SDG_node"   and n :: "'node SDG_node" assumes "n'' s-p:V\<rightarrow>\<^bsub>in\<^esub> n'"   and "n' \<in> sum_SDG_slice1 n" shows "n'' \<in> sum_SDG_slice1 n"
pcr_cr_eq: shows "pcr_ltl_prop_equiv_quotient (=) = cr_ltl_prop_equiv_quotient"
mgu_finite_subst_domain: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu s t = Some \<sigma>" shows "finite (subst_domain \<sigma>)"
is_constant_constructors_simps_3: fixes n :: "nat"   and x :: "'a ltln" shows "is_constant (X\<^sub>n' n x) = is_constant x"
image_set: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "f ` set xs = set (map f xs)"
dtor1_def: shows "dtor1 = fold1 (F1map id ctor1 ctor2) (F2map id ctor1 ctor2)"
has_integral_empty: fixes f :: "'c \<Rightarrow> 'd" shows "(f has_integral (0::'d)) {}"
simps_1: fixes A :: "specID"   and m :: "Keys" shows "know A (kKS m) = (ine A (kE m) \<or> kKS m \<in> LocalSecrets A)"
is_iam_def: fixes m :: "nat \<Rightarrow> 'a option"   and a :: "'a option array" shows "is_iam m a \<equiv> \<exists>\<^sub>Al. a \<mapsto>\<^sub>a l * \<up> (m = iam_of_list l)"
indep_in_imp_subset_basis_in: fixes X :: "'a set" assumes "indep_in \<E> X" shows "\<exists>B. basis_in \<E> B \<and> X \<subseteq> B"
ENR_closed_Un: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "closed T"   and "ENR S"   and "ENR T"   and "ENR (S \<inter> T)" shows "ENR (S \<union> T)"
g'f: shows "internal_equivalence g' f \<phi> \<psi>"
kern_comp_gHom: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and N :: "'g set" assumes "Group F"   and "Group G"   and "gsurj\<^bsub>F,G\<^esub> f"   and "G \<triangleright> N" shows "gker\<^bsub>F,(G / N)\<^esub> (Pj G N \<circ>\<^bsub>F\<^esub> f) = iim F G f N"
b: shows "Sup_pres (\<lambda>x. \<bottom>)"
path_split_set': fixes v :: "'V"   and p :: "('V \<times> 'W \<times> 'V) list"   and v' :: "'V"   and W :: "'V set" assumes "is_path v p v'"   and "v' \<in> W" obtains p1 :: "('V \<times> 'W \<times> 'V) list" and p2 :: "('V \<times> 'W \<times> 'V) list" and u :: "'V" where "p = p1 @ p2" and "is_path v p1 u" and "is_path u p2 v'" and "int_vertices p1 \<subseteq> - W" and "u \<in> W"
until_ev_suntil: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "(\<phi> until \<psi>) \<omega>"   and "ev \<psi> \<omega>" shows "(\<phi> suntil \<psi>) \<omega>"
quicksort_correct_case1: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and xs0 :: "'c list"   and lo :: "nat"   and hi :: "nat"   and xs :: "'c list"   and xs' :: "'c list"   and p :: "nat" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. x \<noteq> y \<Longrightarrow> R (h x) (h y) \<or> R (h y) (h x)"   and "quicksort_pre R h xs0 lo hi xs"   and "partition_spec R h xs lo hi xs' p"   and "p - 1 \<le> lo"   and "hi \<le> p + 1" shows "quicksort_post R h lo hi xs xs'"
conjugate_cancel_iff: fixes conjugate :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.conjugate conjugate" shows "(conjugate a = conjugate b) = (a = b)"
R_def: fixes n :: "nat" shows "R n = map_prod Some id ` (\<^bold>E \<inter> A_n n \<times> Y n) \<union> {None} \<times> Y n"
mod_dvd_simp: fixes x :: "nat"   and y :: "nat" shows "(x mod y = 0) = (y dvd x)"
list_all2_update_cong: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and x :: "'c"   and y :: "'d"   and i :: "nat" assumes "list_all2 P xs ys"   and "P x y" shows "list_all2 P (xs[i := x]) (ys[i := y])"
runit_in_vhom: fixes f :: "'a"   and a :: "'a" assumes "B.ide f"   and "src\<^sub>B f = a" shows "\<guillemotleft>B.runit f : f \<star>\<^sub>B a \<rightarrow>\<^sub>B f\<guillemotright>"
of_complex_of_int_eq: fixes z :: "int" shows "of_complex (of_int z) = of_int z"
exhaust_sel: fixes SkewBinomialTree :: "('c, 'd) SkewBinomialTree" obtains "SkewBinomialTree = Node (val SkewBinomialTree) (prio SkewBinomialTree) (rank SkewBinomialTree) (children SkewBinomialTree)"
rel_Grp: fixes A1 :: "'f set"   and f1 :: "'f \<Rightarrow> 'i"   and A2 :: "'g set"   and f2 :: "'g \<Rightarrow> 'j" shows "Resumption.resumption.rel_resumption (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. results x \<subseteq> A1 \<and> outputs x \<subseteq> A2} (Resumption.resumption.map_resumption f1 f2)"
set_intros_4: fixes yc :: "('a, 'b) Term.term"   and x22 :: "('a, 'b) Term.term list"   and ye :: "'b"   and x21 :: "'a" assumes "yc \<in> set x22"   and "ye \<in> fv yc" shows "ye \<in> fv (Fun x21 x22)"
foldr_disj_map_rlfm: fixes f :: "'a \<Rightarrow> MIR.num \<Rightarrow> fm"   and xs :: "(fm \<times> 'a \<times> MIR.num) list" assumes "\<forall>n s. numbound0 s \<longrightarrow> isrlfm (f n s)"   and "\<forall>(\<phi>, n, s)\<in>set xs. numbound0 s \<and> isrlfm \<phi>" shows "isrlfm (foldr MIR.disj (map (\<lambda>(\<phi>, n, s). MIR.conj \<phi> (f n s)) xs) F)"
hseq_char: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "hseq \<nu> \<mu> = (arr \<mu> \<and> arr \<nu> \<and> src \<nu> = trg \<mu>)"
add_0: fixes a :: "'d" shows "(0::'d) + a = a"
nth_eq_iff_index_eq: fixes xs :: "'e list"   and i :: "nat"   and j :: "nat" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "(xs ! i = xs ! j) = (i = j)"
hyps_11: shows "A\<^sub>P \<sharp>* P"
to_metric_completion_def: fixes x :: "'b" shows "to_metric_completion x = abs_metric_completion (\<lambda>n. x)"
measure_sum: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and R :: "nat set"   and B :: "nat \<Rightarrow> 'b set" assumes "measure_space M"   and "\<forall>j1\<in>R. \<forall>j2\<in>R. j1 \<noteq> j2 \<longrightarrow> B j1 \<inter> B j2 = {}"   and "\<forall>j\<in>R. B j \<in> measurable_sets M"   and "finite R" shows "Measure.measure M (\<Union> (B ` R)) = (\<Sum>j\<in>R. Measure.measure M (B j))"