hcju/isabelleps huggingface dataset:
KCKCKCK_KCK: shows "K \<circ> C \<circ> K \<circ> C \<circ> K \<circ> C \<circ> K = K \<circ> C \<circ> K"
new_def: fixes A :: "ref set" shows "new A = (SOME a. a \<notin> {Null} \<union> A)"
t_below: shows "t \<sqsubseteq> DEFL('a)"
awalk_Cons_iff: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and w :: "'a" shows "S.awalk u (e # es) w = (e \<in> parcs sG \<and> u = fst e \<and> S.awalk (snd e) es w)"
expr_typing_failE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and t :: "pdf_type"   and t' :: "pdf_type" assumes "\<Gamma> \<turnstile> Fail t : t'" obtains "t' = t"
monom_vars_list_subst: fixes m :: "'c monom"   and f :: "'c \<Rightarrow> ('d monom \<times> 'e) list"   and g :: "'c \<Rightarrow> ('d monom \<times> 'e) list" assumes "\<And>w. w \<in> set (monom_vars_list m) \<Longrightarrow> f w = g w" shows "monom_subst f m = monom_subst g m"
before_snd: fixes r2 :: "'a"   and r1 :: "'a"   and vs :: "'a list" assumes "r2 \<in> set (snd (splitAt r1 vs))" shows "before vs r1 r2"
Graph13: fixes a :: "nat set"   and M :: "node list"   and j :: "nat" assumes "a \<subset> Blacks M"   and "j < length M" shows "a \<subset> Blacks (M[j := Black])"
axioms_4: fixes \<V> :: "'b set"   and \<B> :: "'b set multiset"   and \<k> :: "int"   and \<B>' :: "'b set multiset"   and \<t> :: "int"   and \<Lambda>\<^sub>t :: "int"   and \<Lambda>\<^sub>t' :: "int" assumes "two_t_designs_eq_points \<V> \<B> \<k> \<B>' \<t> \<Lambda>\<^sub>t \<Lambda>\<^sub>t'" shows "t_design \<V> \<B>' \<k> \<t> \<Lambda>\<^sub>t'"
intro: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d"   and M :: "'d pre_map" assumes "tail G a \<notin> pre_digraph.isolated_verts (bidel_arc.H G M a)" shows "bidel_arc_tail_conn_axioms G M a"
colossless_restrict_gpvI: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" assumes "colossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>" shows "colossless_gpv \<I> (restrict_gpv \<I> gpv)"
islimptE: fixes x :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "x islimpt S"   and "x \<in> T"   and "open T" obtains y :: "'a" where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
\<R>\<^sub>3_elim': fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and c\<^sub>1 :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set"   and c\<^sub>2 :: "('a, 'c, 'd) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'a set"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "sifum_types.\<R>\<^sub>3_aux ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type ((c\<^sub>1, mds), mem\<^sub>1) \<Gamma> \<S> P ((c\<^sub>2, mds\<^sub>2), mem\<^sub>2)" obtains c\<^sub>1' :: "('a, 'c, 'd) Stmt" and \<Gamma>' :: "'a \<Rightarrow> 'd set option" and \<S>' :: "'a set \<times> 'a set" and Pa :: "'d set" and c\<^sub>2' :: "('a, 'c, 'd) Stmt" and c :: "('a, 'c, 'd) Stmt" where "c\<^sub>1 = c\<^sub>1'__ ;; c__" and "(((c\<^sub>1'__, mds), mem\<^sub>1), (c\<^sub>2'__, mds), mem\<^sub>2) \<in> sifum_types.\<R>\<^sub>1 ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type \<Gamma>'__ \<S>'__ Pa__" and "sifum_types.has_type ev\<^sub>B aexp_vars bexp_vars \<C>_vars bexp_neg assign_post dma_type \<Gamma>'__ \<S>'__ Pa__ c__ \<Gamma> \<S> P" and "c\<^sub>2 = c\<^sub>2'__ ;; c__" and "mds\<^sub>2 = mds"   | c\<^sub>1' :: "('a, 'c, 'd) Stmt" and \<Gamma>' :: "'a \<Rightarrow> 'd set option" and \<S>' :: "'a set \<times> 'a set" and Pa :: "'d set" and c\<^sub>2' :: "('a, 'c, 'd) Stmt" and c :: "('a, 'c, 'd) Stmt" where "c\<^sub>1 = c\<^sub>1' ;; c" and "sifum_types.\<R>\<^sub>3_aux ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type ((c\<^sub>1', mds), mem\<^sub>1) \<Gamma>' \<S>' Pa ((c\<^sub>2', mds), mem\<^sub>2)" and "sifum_types.has_type ev\<^sub>B aexp_vars bexp_vars \<C>_vars bexp_neg assign_post dma_type \<Gamma>' \<S>' Pa c \<Gamma> \<S> P" and "c\<^sub>2 = c\<^sub>2' ;; c" and "mds\<^sub>2 = mds"
wf_tuple_Suc_fviD: fixes n :: "nat"   and b :: "nat"   and \<phi> :: "'b MFOTL.formula"   and v :: "'a option list" assumes "wf_tuple (Suc n) (MFOTL.fvi b \<phi>) v" shows "wf_tuple n (MFOTL.fvi (Suc b) \<phi>) (tl v)"
is_Inf_greatest: fixes A :: "'b set"   and inf :: "'b"   and z :: "'b" assumes "is_Inf A inf"   and "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x" shows "z \<sqsubseteq> inf"
blinfun_of_matrix_minus: fixes x :: "'e \<Rightarrow> 'd \<Rightarrow> real"   and y :: "'e \<Rightarrow> 'd \<Rightarrow> real" shows "blinfun_of_matrix x - blinfun_of_matrix y = blinfun_of_matrix (x - y)"
\<Z>_srng_\<Z>_sgrp_vmult: shows "\<Z>_sgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vmult\<rparr>]\<^sub>\<circ>"
compl_pmdlD: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> 'g) list \<times> 'h"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" assumes "compl_pmdl compl"   and "is_Groebner_basis (fst ` set gs)"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps"   and "unique_idx (gs @ bs) data" shows "fst ` set (fst (compl gs bs (ps -- sps) sps data)) \<subseteq> pmdl (args_to_set (gs, bs, ps))"
operator_encoding_subset_encoding: shows "cnf (encode_all_frame_axioms \<Pi> t) \<subseteq> cnf (\<Phi> \<Pi> t)"
current_bipartite_web_finite: fixes f :: "('v + 'v) \<times> ('v + 'v) \<Rightarrow> ennreal"   and e :: "('v + 'v) \<times> ('v + 'v)" assumes "current (bipartite_web_of \<Gamma>) f" shows "f e \<noteq> \<top>"
fold1_def: fixes s1 :: "('d, 'e, 'f) F1 \<Rightarrow> 'e"   and s2 :: "('d, 'e, 'f) F2 \<Rightarrow> 'f" shows "fold1 s1 s2 = fst (LFP.fold s1 s2)"
intros: assumes "\<I>,\<I> \<turnstile>\<^sub>C 1\<^sub>C \<sim> map_converter id id id id 1\<^sub>C"   and "\<I>, \<I> \<turnstile>\<^sub>C 1\<^sub>C \<surd>" shows "wiring \<I> \<I> 1\<^sub>C (id, id)"
open_segment_line_hyperplanes: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "{a<--<b} = range (\<lambda>u. a + u *\<^sub>R (b - a)) \<inter> {x. a \<bullet> (b - a) < x \<bullet> (b - a) \<and> x \<bullet> (b - a) < b \<bullet> (b - a)}"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes> y \<in> carrier G"
ucast_and_mask: fixes x :: "'c word"   and n :: "nat" shows "UCAST('c \<rightarrow> 'b) (x AND mask n) = UCAST('c \<rightarrow> 'b) x AND mask n"
rename_subst: fixes p :: "'var \<Rightarrow> 'var"   and s :: "'subst"   and f :: "'form" shows "renameLCs p (subst s f) = subst (subst_renameLCs p s) (renameLCs p f)"
qtop_ueval: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'a"   and x :: "('c, 'b) uexpr"   and y :: "('d, 'b) uexpr"   and z :: "('e, 'b) uexpr"   and w :: "('f, 'b) uexpr"   and b :: "'b" shows "\<lbrakk>qtop f x y z w\<rbrakk>\<^sub>e b = f (\<lbrakk>x\<rbrakk>\<^sub>e b) (\<lbrakk>y\<rbrakk>\<^sub>e b) (\<lbrakk>z\<rbrakk>\<^sub>e b) (\<lbrakk>w\<rbrakk>\<^sub>e b)"
cong_iff_lin_poly: fixes a :: "'c poly"   and b :: "'c poly"   and m :: "'c poly" shows "[a = b] (mod m) = (\<exists>k. b = a + m * k)"
le: fixes x :: "'n"   and k :: "nat" assumes "x \<in> S" shows "f (Suc k) x \<le> f k x"
system_if_mat_kernel: fixes Q :: "'b mat"   and v :: "'b vec" assumes "dim_row Q = dim_col Q"   and "dim_row Q = dim_vec v" shows "(Q\<^sup>T *\<^sub>v v = v) = (v \<in> mat_kernel (Q - 1\<^sub>m (dim_row Q))\<^sup>T)"
AndAssign: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and x :: "nat"   and t :: "Formula.trm"   and conf :: "nat \<times> Formula.trm"   and \<psi>' :: "Formula.formula" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "x < n"   and "x \<notin> fv \<phi>'"   and "fv_trm t \<subseteq> fv \<phi>'"   and "(x, t) = conf"   and "\<psi>' = formula.Eq (trm.Var x) t \<or> \<psi>' = formula.Eq t (trm.Var x)" shows "wf_mformula \<sigma> j P V n R (MAndAssign \<phi> conf) (formula.And \<phi>' \<psi>')"
D_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "D x y = (fst x = fst y \<and> D (snd x) (snd y))"
that: fixes n :: "nat" assumes "x = - of_nat n" shows "thesis"
Max_eq_iff: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and m :: "'d" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(linorder.Max less_eq A = m) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq a m))"
GSubspaces_2: shows "\<forall>U\<in>set Us. RModule FG (\<cdot>) U \<and> U \<subseteq> V"
lc_Var: fixes xT :: "char list \<times> Unary_PCF.type" shows "lc \<langle>xT\<rangle>"
t1: shows "(SUP x. cmod r * norm (f x) / norm x) = cmod r * (SUP x. norm (f x) / norm x)"
isoE: fixes f :: "'b" assumes "B.iso f" obtains g :: "'b" where "B.inverse_arrows f g"
p_imp_false: fixes P :: "'b upred" shows "(P \<Rightarrow> false) = (\<not> P)"
distinct_19: shows "hp_Mark \<noteq> hp_IdleMarkSweep"
mult_left_mono: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times c a) (times c b)"
that: fixes r :: "real" assumes "0 < r"   and "\<forall>y\<in>cball x r. 0 < g y" shows "thesis"
g1: shows "0 \<le> Sup {norm (f x) / norm x |x. x \<noteq> (0::'a)}"
lim_1_over_n: shows "(\<lambda>n. (1::'b) / of_nat n) \<longlonglongrightarrow> (0::'b)"
\<epsilon>o_in_hom: fixes x :: "'c" assumes "C.ide x" shows "\<guillemotleft>\<epsilon>o x : F (G x) \<rightarrow>\<^sub>C x\<guillemotright>"
finite_V: fixes V :: "name set" assumes "V \<subseteq> Va__" shows "finite V"
index_Cons: fixes x :: "'a"   and xs :: "'a list"   and a :: "'a" shows "index (x # xs) a = (if x = a then 0 else index xs a + 1)"
dis: shows "p = 0 \<or> (let v1 = lt p; v2 = lt q in v1 \<prec>\<^sub>t v2 \<or> v1 = v2 \<and> lookup p v1 = lookup q v2 \<and> lower p v1 \<prec>\<^sub>p lower q v2)"
fork: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr"   and r' :: "'a" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Rfork e])"   and "r' \<notin> RID\<^sub>G s" shows "substitution.revision_step subst r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid r')]), r' \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e)))"
E_hd: fixes t :: "trm"   and t' :: "trm"   and l :: "char list"   and fs :: "(char list \<times> trm) list" assumes "t \<longmapsto> t'" shows "(l, t) \<Colon> fs [\<longmapsto>] (l, t') \<Colon> fs"
image_mset_mset_set: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "image_mset f (mset_set A) = mset_set (f ` A)"
base_in_hom_2: shows "\<guillemotleft>s : s \<Rightarrow> s\<guillemotright>"
halfspace_simps_5: fixes sctns :: "'c sctn set" shows "below_halfspaces sctns = \<Inter> (below_halfspace ` sctns)"
distinct_220: fixes x11 :: "Syntax.term"   and x91 :: "Syntax.term"   and x92 :: "Syntax.term" shows "Prj1 x11 \<noteq> App x91 x92"
one_le_power: fixes a :: "'d"   and n :: "nat" assumes "(1::'d) \<le> a" shows "(1::'d) \<le> a ^ n"
content_def: fixes p :: "'a poly" shows "content p = gcd_list (Polynomial.coeffs p)"
arith_simps_12: fixes m :: "num"   and n :: "num" shows "num.Bit0 m * num.Bit0 n = num.Bit0 (num.Bit0 (m * n))"
index_Basis_list_axis2: fixes j :: "'b"   and i :: "'a" shows "index Basis_list (axis j (axis i 1)) = index enum_class.enum j * CARD('a) + index enum_class.enum i"
app3: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and es :: "exp0 list"   and s2 :: "'b state"   and vs :: "v list" assumes "evaluate_list ck env s1 (rev es) (s2, Rval vs) \<and> do_opapp (rev vs) = None" shows "evaluate ck env s1 (App Opapp es) (s2, Rerr (Rabort Rtype_error))"
vsv_vdomain_vrange_vsingleton: fixes a :: "V"   and b :: "V" assumes "\<D>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>) = ZFC_in_HOL.set {a}"   and "\<R>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>) = ZFC_in_HOL.set {b}" shows "\<BB>\<lparr>Cod\<rparr> = ZFC_in_HOL.set {\<langle>a, b\<rangle>}"
lookup_higher_when: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and u :: "'t"   and v :: "'t" shows "lookup (higher p u) v = (lookup p v when u \<prec>\<^sub>t v)"
rank_transpose: fixes A :: "(('d, 'f) vec, 'e) vec" shows "rank (Finite_Cartesian_Product.transpose A) = rank A"
DOM_tensor: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "partial_magma.arr (free_monoidal_category.comp C) g" shows "monoidal_language.Dom C (free_monoidal_category.rep C (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C f g)) = monoidal_language.Dom C (free_monoidal_category.rep C f) \<^bold>\<otimes> monoidal_language.Dom C (free_monoidal_category.rep C g)"
mirror: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and x :: "int" assumes "iszlfm p (a # bs)" shows "Ifm (real_of_int x # bs) (mirror p) = Ifm (real_of_int (- x) # bs) p"
Inr_inject: fixes x :: "'h"   and y :: "'h" assumes "Inr x = Inr y" shows "x = y"
kripke_rels_worlds: fixes w :: "'b"   and w' :: "'b"   and M :: "('c, 'd, 'b) KripkeStructure"   and a :: "'c" assumes "(w, w') \<in> relations M a"   and "kripke M" shows "w \<in> worlds M \<and> w' \<in> worlds M"
comp_ide_arr: fixes b :: "'c"   and f :: "'c" assumes "Cop.ide b"   and "Cop.seq b f" shows "b \<cdot>\<^sup>o\<^sup>p f = f"
keys_eq_empty: fixes c :: "'b \<Rightarrow>\<^sub>0 'c" shows "(Poly_Mapping.keys c = {}) = (c = 0)"
resl1: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<cdot> (x \<leftarrow> y) \<le> z \<cdot> x \<leftarrow> y"
not_less_Least: fixes less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and k :: "'h"   and P :: "'h \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "less k (ord.Least less_eq P)" shows "\<not> P k"
fxt: shows "(\<lambda>x. f (x + t)) absolutely_integrable_on {- pi..pi}"
apply_operations_Snoc: fixes xs :: "('msgid \<times> 'op) list"   and x :: "'msgid \<times> 'op" shows "hb.apply_operations (xs @ [x]) = hb.apply_operations xs \<rhd> interp_msg x"
trans_F: fixes P :: "'d process"   and Q :: "'d process"   and S :: "'d process" assumes "P \<sqsubseteq>\<^sub>F Q"   and "Q \<sqsubseteq>\<^sub>F S" shows "P \<sqsubseteq>\<^sub>F S"
primify_contains: fixes x :: "'c + 'c" shows "x \<in> primify x"
extremum_uniqueI: fixes a :: "'e list" assumes "prefix a []" shows "a = []"
adm_Q: shows "adm (\<lambda>x. Q (fst x) (snd x))"
divide_formula: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "x \<in> carrier (padic_int p)"   and "y \<in> carrier (padic_int p)"   and "y \<noteq> \<zero>\<^bsub>padic_int p\<^esub>"   and "padic_integers.val_Zp p y \<le> padic_integers.val_Zp p x" shows "y \<otimes>\<^bsub>padic_int p\<^esub> padic_integers.divide p x y = x"
div_0: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and a :: "'e" assumes "class.semidom_divide divide plus minus zero times one" shows "divide zero a = zero"
vector_space_ow_axioms: shows "vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 (+\<^sub>V\<^sub>S\<^sub>_\<^sub>1) 0\<^sub>V\<^sub>S\<^sub>_\<^sub>1 (-\<^sub>V\<^sub>S\<^sub>_\<^sub>1) uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 (*s\<^sub>o\<^sub>w\<^sub>_\<^sub>1)"
bigthetaI': fixes c1 :: "real"   and c2 :: "real"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "0 < c1"   and "0 < c2"   and "\<forall>\<^sub>F x in F. c1 * norm (g x) \<le> norm (f x) \<and> norm (f x) \<le> c2 * norm (g x)" shows "f \<in> \<Theta>[F](g)"
at_simp: fixes a :: "'c"   and j :: "'j"   and \<tau> :: "'j \<Rightarrow> ('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop.arr a"   and "J.arr j"   and "Cop_S.arr (\<tau> j)" shows "YoD.at a \<tau> j = \<chi>.\<Phi>.Ya.Cop_S.Map (\<tau> j) a"
hIm_hcomplex_polar: fixes r :: "real star"   and a :: "real star" shows "hIm (hcomplex_of_hypreal r * HComplex ((*f* cos) a) ((*f* sin) a)) = r * (*f* sin) a"
incoming_def: fixes v :: "nat" shows "cf.incoming v \<equiv> {(u, v) |u. (u, v) \<in> cf.E}"
ell2_norm_smult_2: fixes x :: "'c \<Rightarrow> complex"   and c :: "complex" assumes "has_ell2_norm x" shows "ell2_norm (\<lambda>i. c * x i) = cmod c * ell2_norm x"
fminus_finsert2: fixes A :: "'a fset"   and a :: "'a"   and B :: "'a fset" shows "A |-| finsert a B = A |-| {|a|} |-| B"
h': shows "h2 \<turnstile> (if owner_document \<noteq> old_document then Heap_Error_Monad.bind (get_disconnected_nodes old_document) (\<lambda>old_disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes old_document (remove1 node old_disc_nodes)) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes owner_document) (\<lambda>disc_nodes. set_disconnected_nodes owner_document (node # disc_nodes)))) else return ()) \<rightarrow>\<^sub>h h'"
is_red_indI1: fixes f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F"   and "f \<noteq> 0"   and "p \<noteq> 0"   and "local.punit.adds_term (lpp f) (lpp p)" shows "punit.is_red F p"
rep_nat_term_prod_def: fixes u :: "'b \<times> 'c" shows "rep_nat_term u = (rep_nat_pp (fst u), rep_nat (snd u))"
reach_wv: shows "w \<rightarrow>\<^sup>*\<^bsub>Digraph_Component.union G H\<^esub> v"
univ_poly_carrier_units_incl: shows "Units (poly_ring R) \<subseteq> {[k] |k. k \<in> carrier R - {\<zero>}}"
sums: fixes f :: "'a \<Rightarrow> 'b"   and X :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "bounded_linear f"   and "X sums a" shows "(\<lambda>n. f (X n)) sums f a"
JN_stable: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'c program"   and A :: "'c set" shows "(JOIN I F \<in> stable A) = (\<forall>i\<in>I. F i \<in> stable A)"
cofactor_def: shows "Square_Matrix.cofactor \<equiv> map_fun id of_fun (\<lambda>A i j. Square_Matrix.det (minor A i j))"
1: assumes "\<forall>m n. x [^] m = x [^] n \<longrightarrow> m = n" shows "infinite (carrier (subgroup_generated G {x}))"
Entry_target: fixes prog :: "cmd"   and a :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node" assumes "valid_edge prog a"   and "targetnode a = (_Entry_)" shows "False"
\<pi>_\<theta>_bigo: shows "(\<lambda>x. primes_pi x - primes_theta x / ln x) \<in> O(\<lambda>x. x / (ln x)\<^sup>2)"
s2r_inter_hom: fixes A :: "'a set"   and B :: "'a set" shows "s2r (A \<inter> B) = s2r A \<inter> s2r B"
htai_empty_def: shows "htai_empty \<equiv> hta_empty"
dim_1: shows "dim_col (char_matrix A ev ^\<^sub>m k) = n"
contra3: fixes s :: "real \<times> real"   and p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "distinct4 s p q r" shows "\<not> (Counterclockwise_2D_Arbitrary.lex p r \<and> Counterclockwise_2D_Arbitrary.lex p s \<and> Counterclockwise_2D_Arbitrary.lex q r \<and> Counterclockwise_2D_Arbitrary.lex q s \<and> insquare p r q s)"
Rep_cpx_sqr_mat: fixes x :: "cpx_sqr_mat" shows "Rep_cpx_sqr_mat x \<in> {M |M. square_mat M}"
pw_bind_inres: fixes M :: "'f nres"   and f :: "'f \<Rightarrow> 'e nres" shows "inres (M \<bind> f) = (\<lambda>x. nofail M \<longrightarrow> (\<exists>y. inres M y \<and> inres (f y) x))"
wellformed: fixes point_set :: "'c set"   and block_collection :: "'c set multiset"   and b :: "'c set" assumes "incidence_system point_set block_collection"   and "b \<in># block_collection" shows "b \<subseteq> point_set"
rep_eq: fixes x :: "'e \<Rightarrow> 'g"   and xa :: "'f \<Longrightarrow> 'g" shows "\<lbrakk>\<langle>x\<rangle>\<^sub>s xa\<rbrakk>\<^sub>e = (\<lambda>b. get\<^bsub>xa\<^esub> (x b))"
INF_mult_left_ennreal: fixes I :: "'a set"   and c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal" assumes "I = {} \<Longrightarrow> c \<noteq> 0"   and "\<lbrakk>c = \<top>; \<exists>i\<in>I. 0 < f i\<rbrakk> \<Longrightarrow> \<exists>p>0. \<forall>i\<in>I. p \<le> f i" shows "c * \<Sqinter> (f ` I) = (\<Sqinter>i\<in>I. c * f i)"
equivalence_functor_axioms: shows "equivalence_functor hom.comp hom'.comp (\<lambda>\<mu>. e\<^sub>1 \<star> \<mu> \<star> d\<^sub>0)"
set_zero_plus: fixes C :: "'a set" shows "(0::'a) +o C = C"
removeAll_def: shows "removeAll \<equiv> \<lambda>uu uua. rec_list (\<lambda>x. []) (\<lambda>y xs xsa x. if x = y then xsa x else y # xsa x) uua uu"
Hi_def: shows "Hi = (embed1_blinfun o\<^sub>L fst_blinfun) + (embed2_blinfun o\<^sub>L T o\<^sub>L snd_blinfun - (f' (x, y) o\<^sub>L embed1_blinfun o\<^sub>L fst_blinfun))"
know2knowsPQ_k: fixes P :: "specID"   and m :: "Keys"   and Q :: "specID" assumes "know P (kKS m) \<or> know Q (kKS m)" shows "knows P [kE m] \<or> knows Q [kE m]"
ide_dom: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "ide (local.dom f)"
that_2: fixes sh :: "'state"   and a :: "'act"   and ou :: "'out" assumes "reach sh"   and "step sh a = (ou, s)"   and "\<not> T (Trans sh a ou s)" shows "thesis"
orderE: fixes b :: "'b multiset"   and a :: "'b multiset" assumes "b \<subseteq># a" obtains "a = a \<union># b"
extension_valid_maximal_conforming: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and v' :: "'c" assumes "vmc_path G P v0 p \<sigma>"   and "v' \<rightarrow>\<^bsub>G\<^esub> v0"   and "v' \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<Longrightarrow> \<sigma> v' = v0" shows "vmc_path G (LCons v' P) v' p \<sigma>"