hcju/isabelleps huggingface dataset:
fmrel_addI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('e, 'c) fmap"   and n :: "('e, 'd) fmap"   and a :: "('e, 'c) fmap"   and b :: "('e, 'd) fmap" assumes "fmrel P m n"   and "fmrel P a b" shows "fmrel P (m ++\<^sub>f a) (n ++\<^sub>f b)"
hRe_hcomplex_polar: fixes r :: "real star"   and a :: "real star" shows "hRe (hcomplex_of_hypreal r * HComplex ((*f* cos) a) ((*f* sin) a)) = r * (*f* cos) a"
assms_2: shows "rat_of_int i * r \<in> \<int>"
distinct_path_intro': fixes p :: "('v \<times> 'w \<times> 'v) list"   and v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "valid_unMultigraph.is_trail (rem_unPath p G) v ps v'" shows "is_trail v ps v'"
"1_4": shows "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) (A_pos @ A_neg) L"
s_top_assoc: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and K :: "'e set"   and L :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<guillemotright> K "   and "G \<guillemotright> L " shows "H \<diamondop>\<^bsub>G\<^esub> K \<diamondop>\<^bsub>G\<^esub> L = H \<diamondop>\<^bsub>G\<^esub> (K \<diamondop>\<^bsub>G\<^esub> L)"
stable: fixes i :: "'b" assumes "i \<in> I" shows "Int_stable (F i)"
ipresSwap_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'h, 'm, 'n) model_scheme" shows "ipresSwap h MOD \<equiv> \<forall>zs z1 z2 s X. wls s X \<longrightarrow> h (X #[z1 \<and> z2]_zs) = igSwap MOD zs z1 z2 (h X)"
mem_appR: fixes B :: "'a list"   and x :: "'a"   and A :: "'a list" assumes "List.member B x" shows "List.member (A @ B) x"
intros_8: fixes X :: "cmsg"   and Y :: "cmsg" assumes "X \<in> cpayload"   and "Y \<in> cpayload" shows "cAenc X Y \<in> cpayload"
comp_arr_dom: fixes f :: "'b"   and a :: "'b" assumes "B.arr f"   and "B.dom f = a" shows "f \<cdot>\<^sub>B a = f"
add_anno_def: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and upd :: "Eg1.var ModeUpd" shows "add_anno \<Gamma> \<S> upd = (\<lambda>x. Some (to_total \<Gamma> x)) |` add_anno_dom \<Gamma> \<S> upd"
coprime4: shows "coprime (A div radical A * (B div radical B)) (C div radical C)"
integral: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "field R"   and "a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>"
isLub_upper: fixes S :: "'a set"   and L :: "'a"   and y :: "'a" assumes "isLub S cl L"   and "y \<in> S" shows "(y, L) \<in> r"
that: fixes u :: "'a" assumes "is_unit u"   and "a = u * b" shows "thesis"
word_not_simps_2: fixes y :: "'a word" shows "\<not> word_pred 0 < y"
complex_of_real_integrable_eq: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real" shows "complex_integrable M (\<lambda>x. complex_of_real (f x)) = integrable M f"
nn_integral_has_integral_lebesgue: fixes \<Omega> :: "'b set"   and f :: "'b \<Rightarrow> real"   and I :: "real" assumes "\<And>x. x \<in> \<Omega> \<Longrightarrow> 0 \<le> f x"   and "(f has_integral I) \<Omega>" shows "\<integral>\<^sup>+ x. ennreal (indicat_real \<Omega> x * f x) \<partial>lborel = ennreal I"
simps_6: fixes k :: "nat"   and C :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "vc ({k} Ab C) Q = (vc C (\<lambda>s. enat k * Q s) \<and> 0 < k)"
w'_def: shows "w' \<equiv> a * f + t * #w"
Arr_in_mkarr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C t" shows "t \<in> free_monoidal_category.mkarr C t"
topspace_euclidean_subtopology: fixes S :: "'a set" shows "topspace (top_of_set S) = S"
RecvMarker_implies_Marker_in_set: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and cid :: "nat"   and p :: "'p"   and q :: "'p" assumes "c \<turnstile> ev \<mapsto> c'"   and "ev = RecvMarker cid p q" shows "Marker \<in> set (msgs c cid)"
apex_def: shows "\<mu>\<nu>.apex \<equiv> C.dom \<mu>\<nu>.leg0"
hfref_weaken_pre_nofail': fixes f :: "'a \<Rightarrow> 'b Heap"   and g :: "'c \<Rightarrow> 'd nres"   and P :: "'c \<Rightarrow> bool"   and R :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and S :: "'d \<Rightarrow> 'b \<Rightarrow> assn"   and Q :: "'c \<Rightarrow> bool" assumes "(f, g) \<in> [P]\<^sub>a R \<rightarrow> S"   and "\<And>x. \<lbrakk>nofail (g x); Q x\<rbrakk> \<Longrightarrow> P x" shows "(f, g) \<in> [Q]\<^sub>a R \<rightarrow> S"
rep_eq: fixes x :: "'a['c, 'b]" shows "fromQ (- x) = \<lparr>mag = - mag (fromQ x), dim = Quantity.dim (fromQ x), unit_sys = unit\<rparr>"
max_degree_def: fixes A :: "('c poly, 'd) sq_matrix" shows "max_degree A = (MAX (i, j). degree (to_fun A i j))"
is_dghm_axioms: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
coll_ex_scaling: fixes b :: "real \<times> real"   and c :: "real \<times> real"   and a :: "real \<times> real" assumes "b \<noteq> c"   and "coll a b c" shows "\<exists>r. a = b + r *\<^sub>R (c - b)"
indexed_weak_nominal_ts_axioms: shows "indexed_weak_nominal_ts TYPE('idx) (S_action.Act \<tau>) (\<turnstile>\<^sub>S) (\<rightarrow>\<^sub>S)"
Body_triple_valid_Suc: fixes n :: "nat"   and P :: "'b assn"   and pn :: "pname"   and Q :: "'b assn" shows "|=n:{P}. the (body pn) .{Q} = |=Suc n:{P}. BODY pn .{Q}"
Gets_Server_message_form': fixes B :: "agent"   and M :: "nat"   and X :: "msg"   and Nb :: "nat"   and K :: "nat"   and evs :: "event list" assumes "Gets B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>X, Nonce Nb, Key K\<rbrace>\<rbrace> \<in> set evs"   and "B \<notin> bad"   and "evs \<in> orb" shows "K \<notin> range shrK \<and> (\<exists>A Na. X = Crypt (shrK A) \<lbrace>Nonce Na, Key K\<rbrace>)"
Hoare_halt_def: fixes P :: "cell list \<times> cell list \<Rightarrow> bool"   and p :: "(action \<times> nat) list"   and Q :: "cell list \<times> cell list \<Rightarrow> bool" shows "{P} p {Q} \<equiv> \<forall>tp. P tp \<longrightarrow> (\<exists>n. is_final (steps0 (1, tp) p n) \<and> Q holds_for steps0 (1, tp) p n)"
poly_as_sum: fixes p :: "'b poly"   and x :: "'b" shows "poly p x = (\<Sum>i\<le>degree p. x ^ i * coeff p i)"
simps_1: fixes X :: "'c event set" shows "setinterleaving ([], X, []) = {[]}"
g_def: shows "g = (\<lambda>x. f x - real_cond_exp M Invariants f x)"
asubg_nsubg: fixes A :: "('e, 'f) aGroup_scheme"   and H :: "'e set" assumes "aGroup A"   and "A +> H" shows "b_ag A \<triangleright> H"
rep_eq: fixes x :: "real\<^sup>+" shows "real_of_pos x = Rep_pos x"
disj_DW2: shows "disjoint_family (\<lambda>n. DW (n + 1))"
strict_mono_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "class.order less_eq less" shows "order.strict_mono less f = (\<forall>x y. less x y \<longrightarrow> f x < f y)"
init_index_valid: fixes as :: "('i \<times> 'a atom) set"   and t :: "(nat \<times> linear_poly) list" shows "index_valid as (init t)"
partition_factors_main: fixes s :: "nat"   and xs :: "('a \<times> nat) list"   and a :: "('a \<times> nat) list"   and b :: "('a \<times> nat) list" assumes "partition_factors_main s xs = (a, b)" shows "mset xs = mset a + mset b"
distinct_1: fixes x11 :: "real"   and x12 :: "real list"   and x21 :: "real"   and x22 :: "real list" shows "Less x11 x12 \<noteq> Eq x21 x22"
partition_aux_def: fixes r :: "'a \<times> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "partition_aux r xs \<equiv> while (Not \<circ> List.null \<circ> fst) (partition_aux_body r) (xs, [])"
closedin_closed_subset: fixes U :: "'a set"   and V :: "'a set"   and T :: "'a set"   and S :: "'a set" assumes "closedin (top_of_set U) V"   and "T \<subseteq> U"   and "S = V \<inter> T" shows "closedin (top_of_set T) S"
independent_backwards_2: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and u :: "'a"   and Us :: "'a list" assumes "ring R"   and "ring.independent R K (u # Us)" shows "ring.independent R K Us"
is_GB_cofactor_boundI: fixes G :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and F :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and b :: "nat" assumes "extended_ord.punit.is_Groebner_basis G"   and "ideal G = ideal F"   and "\<Union> (indets ` G) \<subseteq> \<Union> (indets ` F)"   and "\<And>g. g \<in> G \<Longrightarrow> \<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f\<in>F'. poly_deg (q f * f) \<le> b)" shows "extended_ord.is_GB_cofactor_bound F b"
extended_predist_commute: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_predist x y = extended_predist y x"
inv'_2: shows "length (map of_int_hom.vec_hom fs) = m"
otherwith_syncD: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'b set"   and P :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a"   and a :: "'c" assumes "otherwith Q I P \<sigma> \<sigma>' a" shows "\<forall>j. j \<notin> I \<longrightarrow> Q (\<sigma> j) (\<sigma>' j)"
12370: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (- - x \<sqinter> y) = - x"
dist_cancel_add1: fixes t0 :: "'b"   and et :: "'b" shows "dist (t0 + et) t0 = norm et"
Induction_ax_imp_hfthm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "x \<in> Induction_ax" obtains A :: "fm" where "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e" and "{} \<turnstile> A"
coin_coin': fixes S0 :: "bool"   and m1 :: "bool" shows "map_spmf (\<lambda>S3. S0 \<oplus> S3 \<oplus> m1) coin_spmf = coin_spmf"
elims_3: fixes x :: "('b \<times> 'c) tree" assumes "\<not> Leftist_Heap.linorder_class.heap x" obtains l :: "('b \<times> 'c) tree" and m :: "'b" and uu_ :: "'c" and r :: "('b \<times> 'c) tree" where "x = \<langle>l, (m, uu_), r\<rangle>" and "\<not> (Ball (Tree2.set_tree l \<union> Tree2.set_tree r) ((\<le>) m) \<and> Leftist_Heap.linorder_class.heap l \<and> Leftist_Heap.linorder_class.heap r)"
iprefix_eq_iprefix_take_ex: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "(xs \<sqsubseteq> f) = (\<exists>n. f \<Down> n = xs)"
RECT_rule: fixes body :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and V :: "('c \<times> 'c) set"   and pre :: "'c \<Rightarrow> bool"   and x :: "'c"   and M :: "'c \<Rightarrow> 'd" assumes "trimono body"   and "wf V"   and "pre x"   and "\<And>f x. \<lbrakk>\<And>x'. \<lbrakk>pre x'; (x', x) \<in> V\<rbrakk> \<Longrightarrow> f x' \<le> M x'; pre x; REC\<^sub>T body = f\<rbrakk> \<Longrightarrow> body f x \<le> M x" shows "REC\<^sub>T body x \<le> M x"
bind_nondet_def: fixes m :: "('a, 'm) nondetT"   and f :: "'a \<Rightarrow> ('a, 'm) nondetT" shows "bind_nondet m f = NondetT (bind (run_nondet m) (\<lambda>A. merge A (run_nondet \<circ> f)))"
add_less_same_cancel2: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a + b \<subset># b) = (a \<subset># {#})"
uy_1: fixes x :: "'a" assumes "x \<in> S" shows "0 \<le> uy x"
hyps: shows "filter p xs @ filter (Not \<circ> p) xs <~~> xs"
Gamma_semilattice_def: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set" shows "Gamma_semilattice num' plus' \<gamma> \<equiv> Val_semilattice \<gamma> num' plus'"
tape_of_nl_cons_app2: fixes c :: "nat"   and xs :: "nat list"   and b :: "nat" shows "<c # xs @ [b]> = <c # xs> @ Bk # Oc \<up> Suc b"
perp_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D" shows "B A Perp C D"
code: fixes P :: "'d \<Rightarrow> bool"   and s :: "'d stream" shows "sfilter P s = shd (sdrop_while (Not \<circ> P) s) ## sfilter P (stl (sdrop_while (Not \<circ> P) s))"
a_get_child_nodes_def: fixes ptr :: "(_) object_ptr" shows "local.a_get_child_nodes ptr = invoke a_get_child_nodes_tups ptr ()"
addr_loc_type_fun: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr"   and al :: "addr_loc"   and T :: "ty"   and T' :: "ty" assumes "P,h \<turnstile> a@al : T"   and "P,h \<turnstile> a@al : T'" shows "T = T'"
min_less_iff_disj: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "(signed.min x y <s z) = (x <s z \<or> y <s z)"
continuous_def: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "continuous F f = (f \<longlongrightarrow> f (netlimit F)) F"
fls_regpart_to_fls_trivial: fixes f :: "'a fls" assumes "0 \<le> fls_subdegree f" shows "fls_regpart_as_fls f = f"
ideal_ele_sumTr1: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<plusminus> b \<in> I"   and "a \<in> I" shows "b \<in> I"
sign_swap_id: fixes a :: "'a"   and b :: "'a" shows "sign (Fun.swap a b id) = (if a = b then 1 else - 1)"
rfb_def: shows "fb\<^sub>\<R> = fb\<^sub>\<F> \<circ> \<F>"
ZY_coord: shows "\<forall>i. coord X (Z $ i) = coord X (Y $ i)"
equalityE: fixes A :: "'f set"   and B :: "'f set" assumes "A = B" obtains "A \<subseteq> B" and "B \<subseteq> A"
count_inverse: fixes x :: "'b multiset" shows "Abs_multiset (count x) = x"
preserves_reflects_arr: fixes f :: "'d" shows "D.arr (\<eta> f) = D.arr f"
abs_eq: fixes x :: "nat" shows "pi_float_interval x = Interval (lb_pi x, ub_pi x)"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and i :: "nat"   and v :: "'a" shows "(x # xs)[i := v] = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # xs[j := v])"
spmf_rel_conversep: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "rel_spmf R\<inverse>\<inverse> = (rel_spmf R)\<inverse>\<inverse>"
6: shows "\<forall>i j. i < row_length M1 * row_length M2 * row_length M3 \<and> j < length M1 * length M2 * length M3 \<longrightarrow> (M1 \<otimes> M2 \<otimes> M3) ! j ! i = M1 ! (j div length M3 div length M2) ! (i div row_length M3 div row_length M2) * M2 ! (j div length M3 mod length M2) ! (i div row_length M3 mod row_length M2) * M3 ! (j mod length M3) ! (i mod row_length M3)"
vars_apps: fixes s :: "('b, 'a) tm"   and ss :: "('b, 'a) tm list" shows "vars (apps s ss) = vars s \<union> \<Union> (vars ` set ss)"
mojmir_accept_to_Acc: fixes \<psi> :: "'a ltl"   and \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<phi> :: "'a ltl" assumes "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "mojmir_def.accept \<up>af\<^sub>G (Abs \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"   and "\<pi> (G \<psi>) = mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G (Abs \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}" shows "accepting_pair\<^sub>R (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) (Acc\<^sub>\<A> \<Sigma> \<pi> (G \<psi>)) w"
finite_subset_image: fixes B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "finite B"   and "B \<subseteq> f ` A" shows "\<exists>C\<subseteq>A. finite C \<and> B = f ` C"
simps: fixes i :: "int"   and j :: "int" shows "opt_bst2 i j = (if j < i then \<langle>\<rangle> else if i = j then \<langle>\<langle>\<rangle>, i, \<langle>\<rangle>\<rangle> else let left = root (opt_bst2 i (j - 1)); right = root (opt_bst2 (i + 1) j) in argmin (local.wpl i j) (map (\<lambda>k. \<langle>opt_bst2 i (k - 1), k, opt_bst2 (k + 1) j\<rangle>) [left..right]))"
prime_elem_multiplicity_prod_distrib: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "zero \<notin> f ` A"   and "finite A" shows "factorial_semiring.multiplicity times one p (comm_monoid_mult.prod times one f A) = (\<Sum>x\<in>A. factorial_semiring.multiplicity times one p (f x))"
assert_fun_def: fixes p :: "'b"   and q :: "'b" shows "assert_fun p q = p \<sqinter> q"
ordinal_oLog_monoR: fixes x :: "ordinal"   and y :: "ordinal"   and b :: "ordinal" assumes "x \<le> y" shows "oLog b x \<le> oLog b y"
hyps_1: shows "l \<in> UNIV\<^sup>\<star>"
assms: fixes x :: "'a" shows "\<lbrace>P x\<rbrace> p \<lbrace>Q x\<rbrace>"
left_inverse: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "inverse a * a = (1::'a)"
g_1: shows "eval g (y # hd (trace n f g xs y) # xs) \<down>"
viewL_correct_1: fixes t :: "('a, 'b) FingerTree" assumes "t = FingerTree.empty" shows "FingerTree.viewL t = None"
ereal_less_3: fixes r :: "real" shows "(ereal r < 1) = (r < 1)"
sorted_wrt_filter: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and Q :: "'a \<Rightarrow> bool" assumes "sorted_wrt P xs" shows "sorted_wrt P (filter Q xs)"
ps_ge_zero: shows "\<forall>c. 0 < ps c"
Span_strict_incl: fixes Us :: "'a list"   and Vs :: "'a list" assumes "set Us \<subseteq> carrier R"   and "set Vs \<subseteq> carrier R"   and "Span K Us \<subset> Span K Vs" shows "\<exists>v\<in>set Vs. v \<notin> Span K Us"
Inf_if_Inf_from: fixes \<iota> :: "'a inference"   and Inf :: "'a inference set"   and N :: "'a set" assumes "\<iota> \<in> inference_system.Inf_from Inf N" shows "\<iota> \<in> Inf"
lexordp_eq_simps_3: fixes less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and xs :: "'d list" shows "ord.lexordp_eq less (x # xs) [] = False"
elims_3: fixes x :: "Core.term \<Rightarrow> bool"   and xa :: "Core.term" assumes "\<not> exists_subterm x xa" obtains P :: "Core.term \<Rightarrow> bool" and t :: "Core.term" where "x = P" and "xa = t" and "\<not> (P t \<or> (case t of Abs ty x \<Rightarrow> exists_subterm P x | t $ u \<Rightarrow> exists_subterm P t \<or> exists_subterm P u | _ \<Rightarrow> False))"
perm_swap_1: fixes a :: "name"   and b :: "name"   and x :: "'a" shows "[(a, b)] \<bullet> [(a, b)] \<bullet> x = x"
sums_iff_shift: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and s :: "'b" shows "(\<lambda>i. f (i + n)) sums s = f sums (s + sum f {..<n})"
NN_neg: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (NN \<langle>\<phi>\<rangle> \<langle>neg \<phi>\<rangle>)"
sympD: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "symp r"   and "r b a" shows "r a b"
prems_5: fixes x :: "Vars.var"   and ea :: "exp" assumes "(x, ea) \<in> set as" shows "True"
assms_3: shows "x + y + sum_list xs = (0::'a)"
isomorphic_def: fixes a :: "'a \<times> 'a"   and a' :: "'a \<times> 'a" shows "B.VV.isomorphic a a' = (\<exists>f. B.VV.in_hom f a a' \<and> B.VV.iso f)"
atU_def: fixes U :: "'f set"   and c :: "'f list multiset" shows "atU U c \<equiv> \<exists>s. s \<in># c \<and> atU_s U s"
case_prodE': fixes c :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and p :: "'d \<times> 'e"   and z :: "'f" assumes "(case p of (a, b) \<Rightarrow> c a b) z" obtains x :: "'d" and y :: "'e" where "p = (x, y)" and "c x y z"
vec_nth_inverse: fixes x :: "('b, 'c) vec" shows "vec_lambda (($) x) = x"
lconjby_uminus: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.group_add minus plus zero uminus" shows "minus (plus x (uminus y)) x = uminus (minus (plus x y) x)"
"4_3": shows "\<forall>i<length (y # ys). (x # xs) ! i = (y # ys) ! i"
sigma_sets_subset: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and a :: "'c set set" assumes "sigma_algebra \<Omega> M"   and "a \<subseteq> M" shows "sigma_sets \<Omega> a \<subseteq> M"
assms_1: shows "\<forall>(P, Q)\<in>RelA. CondA (P, Q)"
gen_model_rels_rev: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w' :: "'g"   and w :: "'g"   and w'' :: "'g"   and a :: "'e" assumes "kripke M"   and "w' \<in> worlds (gen_model M w)"   and "(w', w'') \<in> relations M a" shows "(w', w'') \<in> relations (gen_model M w) a"
discr0_MtransC_indis: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "discr0 c"   and "mustT c s"   and "(c, s) \<rightarrow>*c (c', s')" shows "s \<approx> s'"
nth_map_eq': fixes xs :: "'f list"   and ys :: "'g list"   and f :: "'f \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h"   and i :: "nat" assumes "length xs = length ys"   and "map f xs = map g ys"   and "i < length xs" shows "f (xs ! i) = g (ys ! i)"
jumpF_add: fixes p :: "real poly"   and q :: "real poly" assumes "poly p x = 0"   and "coprime p q" shows "jumpF_polyR q p x = jumpF_polyR 1 (q * p) x"
Derivation_take_derive: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list"   and n :: "nat" assumes "Derivation a D b" shows "Derivation a (take n D) (Derive a (take n D))"
edge_Some2E: fixes x :: "'v option"   and y :: "'v" assumes "edge' x \<langle>y\<rangle>" obtains "x = None" and "y \<in> \<^bold>V" and "y \<notin> A \<Gamma>"   | z :: "'v" where "x = \<langle>z\<rangle>" and "edge \<Gamma> z y \<or> edge \<Gamma> y z"
disj_sparse_row_singleton: fixes i :: "nat"   and j :: "nat"   and y :: "'a"   and v :: "(nat \<times> 'a) list"   and x :: "'a" assumes "i \<le> j"   and "sorted_spvec ((j, y) # v)" shows "disj_matrices (sparse_row_vector v) (singleton_matrix 0 i x)"
subcocycle_lim_ereal_def: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "subcocycle_lim_ereal u = (if \<exists>g. g \<in> borel_measurable Invariants \<and> (\<forall>x. g x < \<infinity>) \<and> (AE x in M. (\<lambda>xa. ereal (u xa x / real xa)) \<longlonglongrightarrow> g x) then SOME g. g \<in> borel_measurable Invariants \<and> (\<forall>x. g x < \<infinity>) \<and> (AE x in M. (\<lambda>xa. ereal (u xa x / real xa)) \<longlonglongrightarrow> g x) else (\<lambda>_. 0))"
inject_6: fixes x81 :: "'b ltlr"   and x82 :: "'b ltlr"   and y81 :: "'b ltlr"   and y82 :: "'b ltlr" shows "(x81 U\<^sub>r x82 = y81 U\<^sub>r y82) = (x81 = y81 \<and> x82 = y82)"
measurable_density_eq1: fixes g :: "'a \<Rightarrow> 'b"   and Mg :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and Mg' :: "'b measure" shows "(g \<in> density Mg f \<rightarrow>\<^sub>M Mg') = (g \<in> Mg \<rightarrow>\<^sub>M Mg')"
bet3__bet: 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"   and E :: "'a"   and D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B E"   and "Bet A D E"   and "Bet B C D" shows "Bet A C E"
rel_simps_28: fixes n :: "num" shows "\<not> numeral n \<le> (0::'a)"
nth_mapWithIndex: fixes i :: "nat"   and xs :: "'d list"   and f :: "nat \<Rightarrow> 'd \<Rightarrow> 'e" assumes "i < length xs" shows "mapWithIndex f xs ! i = f i (xs ! i)"
IH: shows "canonical_proj_ext (\<lambda>i. I (Suc i)) n ` carrier R = carrier (RDirProd_list (map (\<lambda>i. R Quot I (Suc i)) [0..<Suc n]))"
entails_def: fixes t :: "'a \<Rightarrow> 'b option"   and P :: "'b \<Rightarrow> bool" shows "(t entails P) = (\<forall>k. ! x:t k: P x)"
leg1_commutes: shows "\<mu>.cod.leg1 \<cdot> \<mu>.chine = \<mu>.leg1"
simps_1: fixes x :: "nat"   and y :: "nat" shows "eval_AddNumUbx [OpUbx1 x, OpUbx1 y] = Some (OpUbx1 (x + y))"
mult_le_less_imp_less: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "a \<le> b"   and "c < d"   and "(0::'c) < a"   and "(0::'c) \<le> c" shows "a * c < b * d"
xy: shows "y = x @\<^sub>v vec_of_scal (0::'a)"
sums: shows "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. (- 1) ^ k / fact (2 * k + 1) * x ^ (2 * k + 1)) sums sin x"
Suc_reindex_ivl: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "sum g {m..n} + g (Suc n) = g m + (\<Sum>i = m..n. g (Suc i))"
preserves_arr: fixes f :: "'a" assumes "F.A.arr f" shows "F.B.arr (F f)"