hcju/isabelleps huggingface dataset:
neighbourhood_base_of_def: fixes P :: "'b set \<Rightarrow> bool"   and X :: "'b topology" shows "neighbourhood_base_of P X \<equiv> \<forall>x\<in>topspace X. neighbourhood_base_at x P X"
below_antisym: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.po below"   and "below x y"   and "below y x" shows "x = y"
le_plus: fixes c :: "'b word"   and b :: "'b word"   and a :: "'b word" assumes "c \<le> b - a"   and "a \<le> b" shows "a + c \<le> b"
left_adjoint_functor_axioms: shows "left_adjoint_functor (\<cdot>) (\<cdot>) (\<lambda>x. x \<otimes> b)"
assms: shows "x \<in> fv\<^sub>s\<^sub>e\<^sub>t (rm_vars X \<theta> ` Y)"
simps_1: fixes ws :: "'b stream list" shows "shd (stranspose ws) = map shd ws"
x: shows "x \<in> rat_of_int ` {y. y * y = i}"
Suc: shows "psi (Discrete.sqrt n) = psi_even n"
adm_def: fixes P :: "'a \<Rightarrow> bool" shows "adm P = (\<forall>Y. chain Y \<longrightarrow> (\<forall>i. P (Y i)) \<longrightarrow> P (\<Squnion>i. Y i))"
BinPacking_Complete_axioms: shows "BinPacking_Complete U w c"
set_same: fixes r :: "'a array"   and x :: "'a list"   and y :: "'a list"   and h :: "heap" shows "Array.set r x (Array.set r y h) = Array.set r x h"
FIXME_refl_on_insert_absorb: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a" assumes "refl_on A r"   and "x \<in> A" shows "insert x (r `` {x}) = r `` {x}"
nth_list_update: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a"   and j :: "nat" assumes "i \<sqsubset> length xs" shows "xs[i := x] ! j = (if i = j then x else xs ! j)"
simps_1: fixes uu :: "('b \<times> 'b) set"   and uv :: "'a \<Rightarrow> 'b"   and uw :: "'b set" shows "ipurge_tr_rev_aux uu uv uw [] = []"
add_neg_numeral_simps_2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and m :: "num"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus (numeral.numeral one plus m)) (numeral.numeral one plus n) = neg_numeral.sub minus plus one n m"
closeI_valid_sound: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat" assumes "sound (SG, C)"   and "seq_valid (SG ! i)" shows "sound (closeI SG i, C)"
listsp_lists_eq: fixes A :: "'a set" shows "listsp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> lists A)"
Complexs_minus: fixes a :: "'b" assumes "a \<in> \<complex>" shows "- a \<in> \<complex>"
SystemDECut_def: shows "SystemDECut = ([[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) (TT && Geq (f0 fid1) (Const 0))]]Geq (Differential (Var vid1)) (Differential (Const 0)) \<leftrightarrow> ([[EvolveODE (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) (TT && Geq (f0 fid1) (Const 0))]][[DiffAssign vid1 (f0 fid1)]]Geq (Differential (Var vid1)) (Differential (Const 0))))"
iT_Minus_Int: fixes k :: "nat"   and A :: "nat set"   and B :: "nat set" shows "k \<ominus> A \<inter> B = (k \<ominus> A) \<inter> (k \<ominus> B)"
eq_neutr_iff: fixes a :: "'a"   and b :: "'a" shows "(inf a b = top) = (a = top \<and> b = top)"
tendsto_Zfun_iff: fixes f :: "'b \<Rightarrow> 'd"   and a :: "'d"   and F :: "'b filter" shows "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
lens_get_put_quasi_commute: fixes Y :: "'b \<Longrightarrow> 'c"   and X :: "'d \<Longrightarrow> 'c"   and s :: "'c"   and v :: "'d" assumes "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> s v) = put\<^bsub>X /\<^sub>L Y\<^esub> (get\<^bsub>Y\<^esub> s) v"
inputPres: fixes P :: "pi"   and x :: "name"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and a :: "name" assumes "\<forall>y. (P[x::=y], Q[x::=y]) \<in> Rel"   and "eqvt Rel" shows "a<x>.P \<leadsto>\<guillemotleft>Rel\<guillemotright> a<x>.Q"
mprefix_singl: fixes a :: "'d"   and P :: "'d \<Rightarrow> 'd process" shows "Mprefix {a} P = (a \<rightarrow> P a)"
fMTaux: fixes x :: "('b, 'c) Combinators" assumes "\<not> member DenyAll x" shows "first_bothNet x \<noteq> {}"
wp_skip_r: fixes r :: "'b upred" shows "II wp r = r"
tc: fixes x :: "'a" assumes "transitive x" shows "x\<^sup>\<star> ; x = x"
fmpred_map: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and m :: "('c, 'e) fmap" shows "fmpred P (fmmap f m) = fmpred (\<lambda>k v. P k (f v)) m"
Conj\<^sub>\<alpha>: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c) Formula.Tree\<^sub>\<alpha> set['a]" assumes "finite (supp tset\<^sub>\<alpha>)"   and "\<And>t\<^sub>\<alpha>. t\<^sub>\<alpha> \<in> set_bset tset\<^sub>\<alpha> \<Longrightarrow> Formula.hereditarily_fs t\<^sub>\<alpha>" shows "Formula.hereditarily_fs (Formula.Conj\<^sub>\<alpha> tset\<^sub>\<alpha>)"
is_natural_1: fixes f :: "'a \<times> 'a \<times> 'a" assumes "CCC.arr f" shows "T.ToTC f \<cdot> \<alpha>' (CCC.dom f) = \<alpha>' f"
weak_ranking_Union: shows "\<Union> (set (weak_ranking (map_relation (inv f) le))) = inv f -` carrier"
flow_eq_rev_2: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "flow0 x0 t = ll_on_open.flow (range (preflect 0)) (\<lambda>t. - f) X 0 x0 (preflect 0 t)"
geq: shows "norm A - (1 - \<epsilon>) * norm A \<le> norm (A *\<^sub>V \<psi>)"
f11: fixes A :: "'a set" shows "- A - - (fmask1 - - fmask2) = c2 \<or> fmask1 - A \<noteq> c1"
strip_guards_simps_11: fixes F :: "'f set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "strip_guards F (Catch c\<^sub>1 c\<^sub>2) = Catch (strip_guards F c\<^sub>1) (strip_guards F c\<^sub>2)"
strong_induct: fixes P :: "'a \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a"   and ty :: "ty" assumes "\<And>list z. P z (PR list)"   and "\<And>ty z. (\<And>z. P z ty) \<Longrightarrow> P z (NOT ty)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 AND ty2)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 OR ty2)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 IMP ty2)" shows "P z ty"
distinct_product: fixes xs :: "'a list"   and ys :: "'b list" assumes "distinct xs"   and "distinct ys" shows "distinct (List.product xs ys)"
\<phi>_in_hom_2: shows "\<guillemotleft>f\<^sub>0g\<^sub>1.\<phi> : tab\<^sub>0 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 \<rightarrow> tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0\<guillemotright>"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs"   and "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs" shows "l_set_child_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs get_tag_name get_tag_name_locs"
apartment_face_distance_eq_chamber_distance_compare_other_chamber: fixes A :: "'a set set"   and C :: "'a set"   and D :: "'a set"   and E :: "'a set"   and z :: "'a set" assumes "A \<in> \<A>"   and "chamber C"   and "chamber D"   and "chamber E"   and "C \<in> A"   and "D \<in> A"   and "E \<in> A"   and "z \<lhd> C"   and "z \<lhd> D"   and "C \<noteq> D"   and "order.greater_eq (chamber_distance D E) (chamber_distance C E)" shows "face_distance z E = chamber_distance C E"
Trans: fixes Ax :: "('a, 'b) Axioms"   and A :: "'a"   and e1 :: "('a, 'b) Expression"   and e2 :: "('a, 'b) Expression"   and B :: "'a"   and e3 :: "('a, 'b) Expression" assumes "Axioms Ax"   and "Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Axioms.Theory Ax"   and "Vx : A \<turnstile> e2\<equiv>e3 : B \<in> Axioms.Theory Ax" shows "Vx : A \<turnstile> e1\<equiv>e3 : B \<in> Axioms.Theory Ax"
euclidean_eq_iff: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and x :: "'e"   and y :: "'e" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(x = y) = (\<forall>b\<in>Basis. inner x b = inner y b)"
filterlim_minus_nat_at_top: fixes k :: "nat" shows "LIM n sequentially. n - k :> sequentially"
subtuple_acyclic': fixes ym :: "'e \<rightharpoonup>\<^sub>f 'd"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xm :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "acyclicP_on (fmran' ym) P"   and "subtuple (\<lambda>x y. P x y \<or> x = y)\<^sup>+\<^sup>+ xm ym"   and "subtuple (\<lambda>x y. P x y \<or> x = y) ym xm" shows "xm = ym"
before_def: fixes f :: "'b \<Rightarrow> 'c" shows "$f \<equiv> %s. f (first s)"
exhaust: fixes y :: "bop" obtains "y = Eq"   | "y = Add"
index_outer_prod: fixes v :: "'c vec"   and n :: "nat"   and w :: "'c vec"   and m :: "nat"   and i :: "nat"   and j :: "nat" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec m"   and "i < n"   and "j < m" shows "outer_prod v w $$ (i, j) = v $ i * conjugate (w $ j)"
sumlist_nf: shows "sum_list (map fst n_as) = n"
lspasl_cs_der: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a"   and h4 :: "'a" assumes "h1,h2\<triangleright>h0"   and "h3,h4\<triangleright>h0" shows "\<exists>h5 h6 h7 h8. (h5,h6\<triangleright>h1) \<and> (h7,h8\<triangleright>h2) \<and> (h5,h7\<triangleright>h3) \<and> (h6,h8\<triangleright>h4) \<and> (h1,h2\<triangleright>h0) \<and> (h3,h4\<triangleright>h0)"
wt_subst: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "legal_zpassign A"   and "wary_subst \<rho>"   and "wary s" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt (subst \<rho> s))) = eval_tpoly (subst_zpassign \<rho> A) (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
\<gamma>_def: shows "\<gamma> = radiant_of (C - center)"
cut_le_less_inext_conv: fixes t :: "nat"   and I :: "nat set" assumes "t \<in> I"   and "t \<noteq> Max I \<or> infinite I" shows "I \<down>\<le> t = I \<down>< inext t I"
3: shows "AE x in lborel. (\<lambda>i. indicat_real {l i..u i} x *\<^sub>R f x) \<longlonglongrightarrow> indicat_real (einterval a b) x *\<^sub>R f x"
fresh_erase_fresh: fixes x :: "'a"   and \<tau> :: "ty" assumes "atom x \<sharp> \<tau>" shows "atom x \<sharp> erase \<tau>"
wf_interp_for_formula_any_Inl: fixes w :: "'a list"   and p :: "nat"   and I :: "(nat + nat set) list"   and \<phi> :: "'a formula" assumes "wf_interp_for_formula (w, Inl p # I) \<phi>" shows "\<forall>p. wf_interp_for_formula (w, Inl p # I) \<phi>"
simps_3: fixes x21 :: "'c tree"   and x22 :: "'c"   and x23 :: "'c tree" shows "\<langle>x21, x22, x23\<rangle> \<noteq> \<langle>\<rangle>"
tendsto_eventually: fixes "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and l :: "'c"   and net :: "'d filter" assumes "class.topological_space open"   and "\<forall>\<^sub>F x in net. f x = l" shows "filterlim f (topological_space.nhds open l) net"
valid_sops_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "valid_sops ts \<equiv> valid_write_sops ts \<and> valid_store_sops ts"
l_card_cosets_equal: fixes c :: "'a set"   and H :: "'a set" assumes "c \<in> lcosets H"   and "H \<subseteq> carrier G"   and "finite (carrier G)" shows "card H = card c"
decompose_equation_vars: fixes e :: "'b equation"   and t :: "'b trm"   and s :: "'b trm" assumes "decompose_equation e t s" shows "vars_of t \<subseteq> vars_of_eq e"
tree2_cases: fixes y :: "('b \<times> 'c) tree" obtains "y = \<langle>\<rangle>"   | x21 :: "('b \<times> 'c) tree" and a :: "'b" and b :: "'c" and x23 :: "('b \<times> 'c) tree" where "y = \<langle>x21, (a, b), x23\<rangle>"
invol_dual_var: fixes x :: "'c" shows "- (- x) = x"
order_sym_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sim> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubseteq> z"
diamond_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "diamond R = commute R R"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat" assumes "senior x xa = y" obtains x :: "nat" and n :: "nat" where "xa__ = x" and "xa = n" and "y = the (oldest_token (token_run x n) n)"
IH: fixes x0 :: "real \<times> real" assumes "list_all (\<lambda>x. x \<noteq> 0) xs"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs" shows "distinct (map snd (polychain_of x0 xs))"
orci_def: fixes e1 :: "nat"   and e2 :: "nat"   and s :: "bddi" shows "orci e1 e2 s \<equiv> tci s \<bind> (\<lambda>(t, s). iteci_lu e1 t e2 s)"
keys_uminus: fixes p :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (- p) = keys p"
assms_10: shows "isUIP (opposite (getCl state)) (getC state) (getM state)"
coeffs_map_poly_hom: fixes p :: "int poly" shows "coeffs (of_int_poly p) = map of_int (coeffs p)"
order_trans_rules_27: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a < b"   and "b = c" shows "a < c"
multiset_inter_def: fixes A :: "'k multiset"   and B :: "'k multiset" shows "A \<inter># B = A - (A - B)"
M_def: shows "M \<equiv> dens_ctxt_measure \<Y> \<rho>"
grid_union_eq: fixes ds' :: "nat set"   and b :: "(nat \<times> int) list"   and ds :: "nat set" shows "(\<Union>p\<in>grid b ds. grid p ds') = grid b (ds' \<union> ds)"
hh_2: shows "hermitean H2 \<and> H2 \<noteq> mat_zero"
Refl_under_in: fixes r :: "('c \<times> 'c) set"   and a :: "'c" assumes "Refl r"   and "a \<in> Field r" shows "a \<in> under r a"
grid_subset_alldim: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set" assumes "p \<in> grid b ds" shows "p \<in> grid b {0..<length b}"
scale_left_diff_distrib: fixes a :: "complex"   and b :: "complex"   and x :: "'b" shows "(a - b) *\<^sub>C x = a *\<^sub>C x - b *\<^sub>C x"
Delta_CDPlayer_CTRL: shows "Delta CDPlayer_CTRL = CDPlayer_CTRL_Delta"
has_ivl_integral_def: fixes f :: "real \<Rightarrow> 'b"   and y :: "'b"   and a :: "real"   and b :: "real" shows "(f has_ivl_integral y) a b = (if a \<le> b then (f has_integral y) {a..b} else (f has_integral - y) {b..a})"
nhid_eq_id: fixes i :: "nat" shows "ptoy i \<TTurnstile> onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>(\<xi>, l). l \<in> {PToy-:2..PToy-:8} \<longrightarrow> nhid \<xi> = state.id \<xi>)"
equals_RESUMPTION_corec: fixes h :: "'f \<Rightarrow> ('g + 'h + 'i, 'j) Old_Datatype.node set"   and f :: "'f \<Rightarrow> ('i, 'h, 'g, 'j, 'f) resumption_corec" assumes "\<And>x. h x = (case f x of Terminal_corec a \<Rightarrow> TERMINAL a | Linear_corec b x' \<Rightarrow> LINEAR b (h x') | Branch_corec c xs \<Rightarrow> BRANCH c (\<lambda>d. h (xs d)) | Resumption_corec r \<Rightarrow> Rep_resumption r)" shows "h = RESUMPTION_corec f"
nsets_eq_empty_iff: fixes A :: "'a set"   and r :: "nat" shows "([A]\<^bsup>r\<^esup> = {}) = (finite A \<and> card A < r)"
dmu_ij_state: fixes j :: "nat"   and ii :: "nat" assumes "j < ii"   and "ii < m" shows "dmu_ij_state state ii j = d\<mu> fs ii j"
mult_assoc: fixes c :: "'b ppath set"   and b :: "'b ppath set"   and a :: "'b ppath set" shows "c ; (b ; a) = c ; b ; a"
induct_step: fixes st :: "'a"   and P :: "'b hrel" assumes "\<lbrace>p \<and> (e, \<guillemotleft>st\<guillemotright>)\<^sub>u \<in>\<^sub>u \<guillemotleft>R\<guillemotright>\<rbrace> P \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p \<and> e =\<^sub>u \<guillemotleft>st\<guillemotright>\<rbrace> F P \<lbrace>q\<rbrace>\<^sub>u"
that: fixes b :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> g"   and "[b, c]\<^sub>\<circ> \<in>\<^sub>\<circ> f" shows "thesis"
Hyp: fixes A :: "fmla"   and H :: "fmla set" assumes "A \<in> H" shows "H \<turnstile> A"
wary_AppE\<^sub>h: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "arity.wary arity_sym arity_var (App s t)" obtains "arity.wary arity_sym arity_var s" and "arity.wary arity_sym arity_var t" and "of_nat (num_args s) < hmset_of_enat (arity.arity_hd arity_sym arity_var (head s))"
simps_9: fixes \<Sigma> :: "nat \<Rightarrow> 'b set"   and project :: "'b \<Rightarrow> 'b"   and lookup :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'c rexp"   and s :: "'c rexp" assumes "project \<Sigma> project" shows "project.lang \<Sigma> project lookup n (rexp.Inter r s) = project.lang \<Sigma> project lookup n r \<inter> project.lang \<Sigma> project lookup n s"
order_gt_0_iff_finite: shows "(0 < order (mult_of R)) = finite (carrier (mult_of R))"
a2: shows "\<forall>p\<in>set Ps. \<exists>\<Phi>' \<Psi>' m. m \<le> n' \<and> ( \<Phi>' + \<Gamma>' \<Rightarrow>* \<Psi>' + \<Delta>', m) \<in> derivable  R* \<and> p = ( \<Phi>' \<Rightarrow>* \<Psi>' \<oplus> F \<nabla> [x].A)"
minus: fixes x :: "real"   and xa :: "'c" shows "- x *\<^sub>R xa = - (x *\<^sub>R xa)"
int_pow_1: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> 1 = x"
hypnat_le_zero_cancel: fixes n :: "nat star" shows "(n \<le> 0) = (n = 0)"
part_insert: fixes A :: "'a set"   and P :: "'a set set"   and X :: "'a set" assumes "part A P"   and "X \<inter> A = {}" shows "part (A \<union> X) (insert X P)"
successively_Cons: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list" shows "successively P (x # xs) = (xs = [] \<or> P x (hd xs) \<and> successively P xs)"
hyps_1: shows "p2 = MPoly_Type.monom m a"
zero_le_square: fixes a :: "'a" shows "(0::'a) \<le> a * a"
parallel_reduction_pairE: fixes A :: "'a trm"   and B :: "'a trm"   and X :: "'a trm" assumes "SimplyTyped.Pair A B >> X" shows "\<exists>A' B'. A >> A' \<and> B >> B' \<and> X = SimplyTyped.Pair A' B'"
3: shows "wf'\<^sub>s\<^sub>s\<^sub>t X T1"
in_homE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and b :: "'b" assumes "category C"   and "partial_magma.in_hom C f a b" obtains "partial_magma.arr C f" and "partial_magma.dom C f = a" and "partial_magma.cod C f = b"
is_category: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "span_vertical_category C" shows "category (span_vertical_category.vcomp C)"
splitAt_simpC: fixes ram :: "'a"   and a :: "'a list"   and b :: "'a list" assumes "ram \<notin> set a" shows "fst (splitAt ram (a @ b)) = a @ fst (splitAt ram b)"
vec_setI: fixes v :: "'b Matrix.vec"   and i :: "nat"   and a :: "'b" assumes "v $v i = a"   and "i < dim_vec v" shows "a \<in> set\<^sub>v v"
Int_lower1: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> B \<subseteq> A"
assms_1: shows "x1 \<in> {a<--<b}"
mono_inf: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h"   and A :: "'g"   and B :: "'g" assumes "class.semilattice_inf inf less_eq less"   and "order.mono less_eq f" shows "f (inf A B) \<le> f A \<sqinter> f B"
inject_3: fixes x31 :: "'b E.term"   and x32 :: "'b E.term"   and y31 :: "'b E.term"   and y32 :: "'b E.term" shows "(x31 \<^bold>\<star> x32 = y31 \<^bold>\<star> y32) = (x31 = y31 \<and> x32 = y32)"
preserving: fixes u :: "complex_homo"   and a :: "complex" assumes "u \<in> unit_disc"   and "is_real a"   and "cmod a < 1"   and "P (moebius_pt (blaschke a) u)" shows "P u"
"1_3": shows "length as' \<le> 2 ^ card (prob_dom PROB) - 1"
k: shows "godel_code' ps j = godel_code' (butlast ps) j * Pi (length ps + j - 1) ^ last ps"
non_speculative_ltake: fixes P :: "'a prog"   and vs :: "'b \<times> addr_loc \<Rightarrow> 'b val set"   and obs :: "('b, 'c) obs_event action llist"   and n :: "enat" assumes "non_speculative P vs obs" shows "non_speculative P vs (ltake n obs)"
\<theta>_bound: shows "0 < \<theta> \<and> \<theta> < pi"
Rats_eq_range_of_rat_o_nat_to_rat_surj: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.field_char_0 divide inverse times one plus zero minus uminus" shows "field_char_0.Rats divide one plus zero minus = range (field_char_0.of_rat divide one plus zero minus \<circ> nat_to_rat_surj)"
apreservation: fixes \<Gamma> :: "char list \<Rightarrow> ty"   and a :: "aexp"   and \<tau> :: "ty"   and s :: "char list \<Rightarrow> val"   and v :: "val" assumes "\<Gamma> \<turnstile> a : \<tau>"   and "taval a s v"   and "\<Gamma> \<turnstile> s" shows "type v = \<tau>"
abs_summable_on_Sigma_project2: fixes f :: "'d \<times> 'e \<Rightarrow> 'f"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and x :: "'d" assumes "f abs_summable_on Sigma A B"   and "x \<in> A" shows "(\<lambda>y. f (x, y)) abs_summable_on B x"
writeFailCorrect: fixes cfile1 :: "CFile"   and index :: "nat"   and "value" :: "byte" assumes "cfWrite cfile1 index value = None" shows "afWrite (abstFn cfile1) index value = None"
Impossible\<^sub>1_def: shows "Impossible \<equiv> \<lambda>F. \<^bold>\<box>(\<^bold>\<forall>x. \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>)"
prj_in_hom_1: shows "\<guillemotleft>\<mu>\<nu>.prj\<^sub>1 : \<mu>.leg0 \<down>\<down> \<nu>.leg1 \<rightarrow>\<^sub>C \<mu>.apex\<guillemotright>"
inj_on_the_inv_into: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" assumes "inj_on f A" shows "inj_on (the_inv_into A f) (f ` A)"
map_charpair_uniform: fixes ss :: "'b list"   and S :: "'b set" assumes "ss \<in> lists S" shows "map (charpair S) ss = map pairtrue ss"
simps: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat" shows "interaction_bounded_by consider gpv n = (interaction_bound consider gpv \<le> n)"
sorted_wrt_nth_less: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list"   and i :: "nat"   and j :: "nat" assumes "sorted_wrt P xs"   and "i < j"   and "j < length xs" shows "P (xs ! i) (xs ! j)"
partial_magma_axioms: shows "partial_magma (\<cdot>\<^sub>J)"
transitive: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<sim> Q"   and "Q \<sim> R" shows "P \<sim> R"
strict_trans2: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry"   and c :: "'d DBMEntry" assumes "b \<prec> a"   and "c \<preceq> b" shows "c \<prec> a"
join_imp_conversion: fixes r :: "('b \<times> 'b) set" shows "r\<^sup>\<down> \<subseteq> r\<^sup>\<leftrightarrow>\<^sup>*"
pair_return_pmf2: fixes x :: "'b pmf"   and y :: "'c" shows "pair_pmf x (return_pmf y) = map_pmf (\<lambda>x. (x, y)) x"