hcju/isabelleps huggingface dataset:
aHom_compos: fixes L :: "('e, 'f) aGroup_scheme"   and M :: "('g, 'h) aGroup_scheme"   and N :: "('i, 'j) aGroup_scheme"   and f :: "'e \<Rightarrow> 'g"   and g :: "'g \<Rightarrow> 'i" assumes "aGroup L"   and "aGroup M"   and "aGroup N"   and "f \<in> aHom L M"   and "g \<in> aHom M N" shows "compos L g f \<in> aHom L N"
closest_pair_bf_code_eq: fixes ps :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and \<delta>' :: "int"   and c\<^sub>0' :: "int \<times> int"   and c\<^sub>1' :: "int \<times> int" assumes "1 < length ps"   and "(c\<^sub>0, c\<^sub>1) = closest_pair_bf ps"   and "(\<delta>', c\<^sub>0', c\<^sub>1') = closest_pair_bf_code ps" shows "c\<^sub>0 = c\<^sub>0' \<and> c\<^sub>1 = c\<^sub>1'"
circ_zero: shows "bot\<^sup>\<circ> = (1::'a)"
last_eval2: fixes u :: "'a"   and v :: "'a" shows "last [u, v] = v"
that: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j < length bs"   and "v = ord_term_lin.max (term_of_pair (punit.lt (bs ! i), k + j)) (term_of_pair (punit.lt (bs ! j), k + i))" shows "thesis"
permutations_of_multisetD: fixes xs :: "'b list"   and A :: "'b multiset" assumes "xs \<in> permutations_of_multiset A" shows "mset xs = A"
plus_period: fixes x :: "'a" shows "f (x + period) = f x"
commute: fixes a :: "'d multiset"   and b :: "'d multiset" shows "a \<inter># b = b \<inter># a"
path_component_of_eq_empty: fixes X :: "'b topology"   and x :: "'b" shows "(path_component_of_set X x = {}) = (x \<notin> topspace X)"
lt_monomial: fixes c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "c \<noteq> (0::'b)" shows "i.lpp (monomial c t) = t"
mset_le_not_refl: fixes M :: "'a multiset" shows "\<not> M < M"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c' Q,A"
allAllocationsFinite: fixes N :: "'c set"   and G :: "'d set" assumes "finite N"   and "finite G" shows "finite (allAllocations N G)"
PTrans: shows "P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"
split_list_last: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
valid_dbm_def: fixes M :: "'d \<Rightarrow> 'd \<Rightarrow> 'e DBMEntry"   and n :: "'d" shows "valid_dbm M n \<equiv> dbm_int M n \<and> (\<forall>i\<le>n. M (0::'d) i \<le> \<one>)"
irreflI: fixes R :: "('g \<times> 'g) set" assumes "\<And>a. (a, a) \<notin> R" shows "irrefl R"
M_carrier: shows "make_mat n k B \<in> carrier_mat n n"
prime_factorization_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> (0::'a)"   and "y \<noteq> (0::'a)" shows "prime_factorization (x * y) = prime_factorization x + prime_factorization y"
uncurry_simp: fixes f :: "'a1 \<times> 'a2"   and \<tau> :: "'a1 \<Rightarrow> ('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" assumes "A1xA2.arr f" shows "uncurry \<tau> f = E.map (\<tau> (fst f), snd f)"
cases: fixes \<mu> :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"   and m :: "'b"   and ts :: "'c \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option"   and ts' :: "'c \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option" assumes "multithreaded_base_measure.m\<mu>t \<mu> m ts ts'" obtains t :: "'c" and x :: "'a" and x' :: "'a" and ln' :: "'d \<Rightarrow>f nat" and ln :: "'d \<Rightarrow>f nat" where "finite (dom ts)" and "ts t = \<lfloor>(x, ln)\<rfloor>" and "ts' t = \<lfloor>(x', ln')\<rfloor>" and "\<mu> (x, m) (x', m)" and "\<And>t'. t' \<noteq> t \<Longrightarrow> ts t' = ts' t'"
ground_sub: fixes t :: "fterm"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "ground\<^sub>t t" shows "t \<cdot>\<^sub>t \<sigma> = t"
simps_15: fixes v :: "fm"   and va :: "fm" shows "conjuncts (Iff v va) = [Iff v va]"
sym: fixes s :: "'b event list"   and X :: "'b event set"   and t :: "'b event list" shows "setinterleaving (s, X, t) = setinterleaving (t, X, s)"
Option_bind_eq_Some_conv: fixes x :: "'b option"   and y :: "'b \<Rightarrow> 'a option"   and z :: "'a" shows "(x \<bind> y = \<lfloor>z\<rfloor>) = (\<exists>x'. x = \<lfloor>x'\<rfloor> \<and> y x' = \<lfloor>z\<rfloor>)"
deriv_strip: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P c'' UNIV,UNIV"
conjugate_square_ge_0_vec: fixes v :: "'c vec" shows "(0::'c) \<le> inner_prod v v"
simps_36: fixes vd :: "'a acom"   and ve :: "'a acom"   and v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a" shows "(vd;; ve \<sqsubseteq> {v} WHILE va DO vb {vc}) = False"
simps_3: fixes g :: "char list \<Rightarrow> bool"   and h :: "'a pfm \<Rightarrow> bool"   and p :: "'a pfm"   and q :: "'a pfm" shows "peval g h (p \<^bold>\<or>\<^sub>! q) = (peval g h p \<or> peval g h q)"
simps_4: fixes n :: "nat"   and a :: "'b" shows "local.wf n (Atom a) = wf_atom n a"
simps_2: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and ys :: "'c list" shows "sorted_wrt P (x # ys) = (Ball (set ys) (P x) \<and> sorted_wrt P ys)"
kat_2_equiv: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(y \<cdot> a z \<le> a x \<cdot> y) = (d x \<cdot> y \<cdot> a z = \<bottom>)"
take_bit_word_eq_self_iff: fixes n :: "nat"   and w :: "'b word" shows "(take_bit n w = w) = (LENGTH('b) \<le> n \<or> w < 2 ^ n)"
convergent_stirling_integral: fixes n :: "nat" assumes "0 < n" shows "convergent (\<lambda>N. integral {0..real N} (\<lambda>x. complex_of_real (pbernpoly n x) / (complex_of_real x + s) ^ n))"
elims: fixes x :: "pr_operation'"   and y :: "pr_operation" assumes "project_operation x = y" obtains "x = RELABEL'" and "y = RELABEL"   | "x = NONSAT_PUSH'" and "y = PUSH"   | uu_ :: "nat \<times> nat" where "x = SAT_PUSH' uu_" and "y = PUSH"
pred_def: shows "pred \<equiv> \<Lambda> (ValN\<cdot>n). case n of 0 \<Rightarrow> \<bottom> | Suc n \<Rightarrow> ValN\<cdot>n"
simps_7: fixes \<phi> :: "atom fm" shows "negation_free (AllQ \<phi>) = negation_free \<phi>"
n_subid: fixes x :: "'a" shows "n x \<sqsubseteq> (1::'a)"
least_simps_2: fixes x :: "'c option" shows "least x None = x"
euclidean_size_of_nat: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size (semiring_1.of_nat one plus zero n) = n"
simps: fixes I :: "'a set"   and v :: "nat \<Rightarrow> rat"   and cs :: "('a \<times> constraint) set" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s cs = v \<Turnstile>\<^sub>c\<^sub>s (snd ` (cs \<inter> I \<times> UNIV))"
rimg_ag: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "aGroup (rimg A R f)"
hyps: fixes child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child"   and "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent"   and "h \<turnstile> get_component parent \<rightarrow>\<^sub>r c" shows "parent \<in> set c"
simps_1: fixes e1 :: "'a \<times> 'a \<times> 'a poly"   and e2 :: "'a \<times> 'a \<times> 'a poly"   and es :: "('a \<times> 'a \<times> 'a poly) list" shows "local.neville_aitken_impl_main (e1 # e2 # es) = local.neville_aitken_impl_main (local.neville_aitken_merge (e1 # e2 # es))"
expr_typing_cong: fixes e :: "expr"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<Gamma>' :: "nat \<Rightarrow> pdf_type"   and t :: "pdf_type" assumes "\<And>x. x \<in> free_vars e \<Longrightarrow> \<Gamma> x = \<Gamma>' x" shows "(\<Gamma> \<turnstile> e : t) = (\<Gamma>' \<turnstile> e : t)"
standard_lifting_family: shows "\<forall>q\<in>Q. tiebreaker_lifting Bot_F Inf_F Bot_G (entails_q q) (Inf_G_q q) (Red_I_q q) (Red_F_q q) (\<G>_F_q q) (\<G>_I_q q) Prec_F_g"
dom_simp: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.VVV.dom f = B.VxVxV.dom f"
order_refl: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "order r" shows "x \<sqsubseteq>\<^bsub>r\<^esub> x"
filter_insort_key_neg: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and f :: "'c \<Rightarrow> 'd"   and xs :: "'c list" assumes "\<not> P x" shows "filter P (insort_key f x xs) = filter P xs"
crdt_ops_independent_suf: fixes xs :: "('a \<times> 'b) list"   and oid :: "'a"   and oper :: "'b"   and deps :: "'b \<Rightarrow> 'a set"   and ys :: "('a \<times> 'b) list"   and zs :: "('a \<times> 'b) list"   and op2 :: "'b"   and r :: "'a" assumes "spec_ops (xs @ [(oid, oper)]) deps"   and "crdt_ops (ys @ [(oid, oper)] @ zs) deps"   and "set (xs @ [(oid, oper)]) = set (ys @ [(oid, oper)] @ zs)"   and "op2 \<in> snd ` set zs"   and "r \<in> deps op2" shows "r \<noteq> oid"
finite_set_less_eq_aux''_finite: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "(A \<sqsubseteq>''' B) = (A \<sqsubseteq>'' B)"
RETURN_def: fixes x :: "'z" shows "RETURN x \<equiv> RES {x}"
cases: fixes a1 :: "('p, 's, 'm) configuration"   and a2 :: "('p, 's, 'm) event list"   and a3 :: "('p, 's, 'm) configuration" assumes "trace a1 a2 a3" obtains c :: "('p, 's, 'm) configuration" where "a1 = cb__" and "a2 = []" and "a3 = cb__"   | c :: "('p, 's, 'm) configuration" and ev :: "('p, 's, 'm) event" and c' :: "('p, 's, 'm) configuration" and t :: "('p, 's, 'm) event list" and c'' :: "('p, 's, 'm) configuration" where "a1 = c" and "a2 = ev # t" and "a3 = c''" and "c \<turnstile> ev \<mapsto> c'" and "trace c' t c''"
plus_one_non_zero: fixes n :: "nat" shows "n + 1 \<noteq> 0"
isomorphic_implies_hpar_1: fixes f :: "'b"   and f' :: "'b" assumes "f \<cong>\<^sub>D f'" shows "D.ide f"
that: fixes v :: "nat \<times> 'b" assumes "v \<in> V"   and "\<forall>u\<in>greachable A w V v. \<not> gaccepting A u" shows "thesis"
sets_PiM_I: fixes J :: "'c set"   and I :: "'c set"   and E :: "'c \<Rightarrow> 'd set"   and M :: "'c \<Rightarrow> 'd measure" assumes "finite J"   and "J \<subseteq> I"   and "\<forall>i\<in>J. E i \<in> sets (M i)" shows "prod_emb I M J (Pi\<^sub>E J E) \<in> sets (Pi\<^sub>M I M)"
resolutionD2: fixes S :: "'a set multiset" assumes "S \<in># \<P>" shows "distinct_mset S"
dim_eq_card_independent: fixes B :: "'c set" assumes "vs2.independent B" shows "vs2.dim B = card B"
alignUp_def3: fixes a :: "'b word"   and sz :: "nat" shows "alignUp a sz = 2 ^ sz + (a - 1 AND NOT (mask sz))"
prod_dvd_prod: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.comm_monoid_mult times one"   and "\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times (f a) (g a)" shows "dvd.dvd times (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
poly_y_minus_x_def: fixes p :: "'b poly" shows "poly_y_minus_x p \<equiv> poly_lift p \<circ>\<^sub>p y_x"
set_intros_1: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" shows "x21 \<in> funs_term (Fun x21 x22)"
vertexI1: fixes \<Gamma> :: "('a, 'b) graph_scheme"   and x :: "'a"   and y :: "'a" assumes "edge \<Gamma> x y" shows "vertex \<Gamma> x"
dom_cond_override: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'd option"   and t :: "'c \<Rightarrow> 'd option" shows "dom (cond_override C s t) \<subseteq> dom s \<union> dom t"
inj_idom_divide_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_idom_divide_hom hom \<equiv> idom_divide_hom hom \<and> inj_idom_hom hom"
match_implicit_pres_correct: fixes effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and s :: "nat \<Rightarrow> nat option" assumes "distinct (map (\<lambda>(uu_, v, uu_, uu_). v) effs)"   and "s \<in> valid_states" shows "match_implicit_pres effs s = (s \<in> subsuming_states (map_of (implicit_pres effs)))"
chine_assoc'_def: shows "fgh.chine_assoc' \<equiv> \<langle>\<langle>fgh.Prj\<^sub>1 \<lbrakk>fgh.\<mu>.leg0, fgh.\<nu>.leg1\<rbrakk> fgh.Prj\<^sub>1\<^sub>0\<rangle> \<lbrakk>fgh.\<nu>.leg0 \<cdot> fgh.\<mu>\<nu>.prj\<^sub>0, fgh.\<pi>.leg1\<rbrakk> fgh.Prj\<^sub>0\<^sub>0\<rangle>"
check_ex_unique: fixes eX :: "'c withERR" assumes "eX \<noteq> ERR" shows "\<exists>!X. eX = OK X"
eucl_abs: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and 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 inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and abs :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "abs x = comm_monoid_add.sum plus zero (\<lambda>i. scaleR \<bar>inner x i\<bar> i) Basis"
Ri_seq: fixes x :: "'b"   and y :: "'c"   and z :: "'c"   and w :: "'d" shows "Ri x y z \<le> Ri x w (Ri w y z)"
singleCombinatorsConc: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list" assumes "singleCombinators (x # xs)" shows "singleCombinators xs"
to_nat_plus_1_set: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and a :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.to_nat Rep a + 1 \<in> {1..<CARD('b) + 1}"
scc_ofI_reachable: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>* v" shows "u \<in> scc_of v"
g_to_rev_list_def: fixes m :: "'s" shows "g_to_rev_list m \<equiv> iterateo m (#) []"
dom_mkArr: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and img :: "'a \<Rightarrow> 'a"   and \<AA> :: "('b \<times> 'b) set"   and A :: "'a set"   and B :: "'a set"   and F :: "'a \<Rightarrow> 'a" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)" shows "partial_magma.dom S (set_category_given_img.mkArr S img \<AA> A B F) = set_category_given_img.mkIde S img \<AA> A"
vec_nth_real_1_iff_cbox: fixes S :: "(real, 1) vec set"   and a :: "real"   and b :: "real" shows "((\<lambda>x. x $ 1) ` S = {a..b}) = (S = cbox (vec a) (vec b))"
encR: fixes TRel :: "'procT \<Rightarrow> 'procT \<Rightarrow> bool"   and S :: "'procS" shows "indRelRTp TRel (SourceTerm S) (TargetTerm (\<lbrakk>S\<rbrakk>))"
HStartBallot_Inv2b: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "Inv2b s"   and "HStartBallot s s' p" shows "Inv2b s'"
FAccNone: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l, sh)\<rangle>"   and "h a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NoSuchFieldError,(h, l, sh)\<rangle>"
V: shows "UNpart01 cl dl P - {BrnFT cl dl} = UNpart1 cl dl P"
vintersection_vempty: fixes A :: "V" shows "[]\<^sub>\<circ> \<subseteq>\<^sub>\<circ> A"
arr_dom: fixes f :: "'s" assumes "arr f" shows "arr (local.dom f)"
h2: shows "h \<turnstile> new_element \<rightarrow>\<^sub>h h2"
simps_1: shows "\<Phi> \<langle>\<rangle> = 0"
transp_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "transp_on P A = (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. P x y \<and> P y z \<longrightarrow> P x z)"
osum_embedL: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "embed r (r +o r') Inl"
zhmset_of_nonneg: fixes M :: "hmultiset" shows "0 \<le> zhmset_of M"
simps: fixes P :: "(('b \<times> 'c \<times> 'b) set \<times> ('b \<times> 'c \<times> 'b) set) set" shows "combine_pairs P = (\<Union> (fst ` P), snd ` P)"
hs_iterate_impl: shows "imp_set_iterate is_hashset hs_is_it hs_it_init hs_it_has_next hs_it_next"
IH: fixes i :: "nat" assumes "i < length T"   and "\<forall>f. \<forall>a\<in>set (P f). f = a" shows "T ! i = S ! i"
disj1: fixes x :: "Vars.var" shows "carrier ((FBinds \<Gamma>\<cdot>ae) x) \<inter> domA \<Delta> = {}"
gt_fun_imp: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "fun t >\<^sub>t s" shows "t >\<^sub>t s"
pairwiseD: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "pairwise R S"   and "x \<in> S"   and "y \<in> S"   and "x \<noteq> y" shows "R x y"
T_simp: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and tensor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unity :: "'a"   and lunit :: "'a \<Rightarrow> 'a"   and runit :: "'a \<Rightarrow> 'a"   and assoc :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_monoidal_category C tensor unity lunit runit assoc"   and "partial_magma.arr C f"   and "partial_magma.arr C g" shows "elementary_monoidal_category.T C tensor (f, g) = tensor f g"
isin: fixes t :: "'b tree"   and x :: "'b" assumes "True \<and> Sorted_Less.sorted (inorder t)" shows "isin t x = (x \<in> set (inorder t))"
A: shows "range A \<subseteq> lambda_system \<Omega> M f"
tendsto_Zfun_iff: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter" shows "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
refl: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "star r x x"
hyps: fixes thesis :: "bool" assumes "\<And>B. \<lbrakk>card B = n; B \<subseteq> A\<rbrakk> \<Longrightarrow> thesis"   and "n \<le> card A" shows "thesis"
count_list_map_ge: fixes xs :: "'a list"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "count_list xs x \<le> count_list (map f xs) (f x)"
typeof_lit_neq_Class: fixes v :: "'a val"   and C :: "String.literal" shows "typeof v \<noteq> \<lfloor>Class C\<rfloor>"
simps_2: shows "\<lbrace>\<^bold>\<I>\<rbrace> = \<I>\<^sub>D"
split_list_def: fixes xs :: "'b list" shows "split_list xs = (let n = length xs div 2 in (take n xs, drop n xs))"
SN_Ret: fixes u :: "trm" assumes "SN u" shows "SN [u]"
finite_Atleast_Atmost_nat: fixes f :: "'c \<Rightarrow> 'b" shows "finite {f x |x. x \<in> UNIV}"
ennreal_one_neq_top: shows "1 \<noteq> top"
D_SKIP_H: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" shows "(G, Skip, HighSec) \<in> Deriv"
back_subst: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "P a"   and "a = b" shows "P b"
retract_of_subset: fixes T :: "'c set"   and S :: "'c set"   and s' :: "'c set" assumes "T retract_of S"   and "T \<subseteq> s'"   and "s' \<subseteq> S" shows "T retract_of s'"
iT_Div_iEx_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<diamond> t I \<oslash> k. P t) = (\<diamond> t I. P (t div k))"
welltypedE_5: fixes e1 :: "expr"   and e2 :: "expr"   and T :: "Unary_PCF.type" assumes "e1 \<cdot> e2 ::: T" obtains T :: "Unary_PCF.type" where "e1 ::: T \<rightarrow> Tb__" and "e2 ::: T"
dense_accessible_frontier_points_connected: fixes S :: "'b set"   and x :: "'b"   and V :: "'b set" assumes "open S"   and "connected S"   and "x \<in> S"   and "V \<noteq> {}"   and "openin (top_of_set (frontier S)) V" obtains g :: "real \<Rightarrow> 'b" where "arc g" and "g ` {0..<1} \<subseteq> S" and "pathstart g = x" and "pathfinish g \<in> V"
perp_foot_hyp2: fixes a :: "proj2"   and l :: "proj2_line"   and b :: "proj2" assumes "a \<in> hyp2"   and "proj2_incident a l"   and "b \<in> hyp2" shows "perp_foot b l \<in> hyp2"
card_mono: fixes B :: "'a set"   and A :: "'a set" assumes "finite B"   and "A \<subseteq> B" shows "card A \<le> card B"
nat_pow_closed: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and n :: "nat" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> n \<in> carrier G"
pw_Sup_inres: fixes X :: "'b nres set"   and r :: "'b" shows "inres (Sup X) r = (\<exists>M\<in>X. inres M r)"
synth_mono: fixes G :: "msg set"   and H :: "msg set" assumes "G \<subseteq> H" shows "synth G \<subseteq> synth H"
ceiling_altdef: fixes x :: "'a" shows "\<lceil>x\<rceil> = (if x = of_int \<lfloor>x\<rfloor> then \<lfloor>x\<rfloor> else \<lfloor>x\<rfloor> + 1)"
some_geodesic_segment_description_2: fixes x :: "'b"   and y :: "'b"   and S :: "'b set" assumes "\<nexists>G. geodesic_segment_between G x y \<and> G \<subseteq> S" shows "{x--S--y} = {x, y}"
assms: fixes Tvec :: "'a list"   and \<Psi>' :: "'b" assumes "length xvec = length Tvec" shows "(\<Psi> \<otimes> \<Psi>', P[xvec::=Tvec], Q[xvec::=Tvec]) \<in> Rel"
mgp_facesAt_no_loop: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "minGraphProps g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)" shows "f \<bullet> v \<noteq> v"
AbstrLevels_S7_A42: fixes i :: "AbstrLevelsID" assumes "sS7 \<in> AbstrLevel i" shows "sA42 \<notin> AbstrLevel i"
lim_sequentially: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
derive_size_1: fixes i :: "'b"   and x :: "'b sort_key extended \<times> 'b sort_key extended" assumes "\<lbrakk>Final i\<rbrakk> \<le> fst x \<and> is_interval x" shows "int_size (derive_left x) < int_size x"
homeomorphism_prod: fixes a :: "'e set"   and c :: "'f set"   and f :: "'e \<Rightarrow> 'f"   and f' :: "'f \<Rightarrow> 'e"   and b :: "'g set"   and d :: "'h set"   and g :: "'g \<Rightarrow> 'h"   and g' :: "'h \<Rightarrow> 'g" assumes "homeomorphism a c f f'"   and "homeomorphism b d g g'" shows "homeomorphism (a \<times> b) (c \<times> d) (\<lambda>(x, y). (f x, g y)) (\<lambda>(x, y). (f' x, g' y))"
sig_red_rtrancl_lt_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q" shows "punit.lt (rep_list q) \<preceq> punit.lt (rep_list p)"
norm_char: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and a :: "'c" assumes "character G \<chi>" shows "cmod (\<chi> a) = (if a \<in> carrier G then 1 else 0)"
seqll_onll_swap: fixes i :: "'a"   and \<Gamma> :: "'e \<Rightarrow> ('c, 'd, 'e, 'f) seqp"   and P :: "('b \<times> 'f) \<times> 'g \<times> 'b \<times> 'f \<Rightarrow> bool" shows "seqll i (onll \<Gamma> P) = onll \<Gamma> (seqll i P)"
sums_Suc: fixes f :: "nat \<Rightarrow> 'b"   and l :: "'b" assumes "(\<lambda>n. f (Suc n)) sums l" shows "f sums (l + f 0)"