hcju/isabelleps huggingface dataset:
sees_method_fun: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and TS :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and b' :: "staticb"   and TS' :: "ty list"   and T' :: "ty"   and m' :: "'a"   and D' :: "char list" assumes "P \<turnstile> C sees M, b :  TS\<rightarrow>T = m in D"   and "P \<turnstile> C sees M, b' :  TS'\<rightarrow>T' = m' in D'" shows "b = b' \<and> TS' = TS \<and> T' = T \<and> m' = m \<and> D' = D"
derives1_is_sentence: fixes \<alpha> :: "symbol list"   and \<beta> :: "symbol list" assumes "derives1 \<alpha> \<beta>" shows "is_sentence \<alpha> \<and> is_sentence \<beta>"
closed_orbit_in_domain: fixes x :: "'a" assumes "closed_orbit x" shows "x \<in> X"
differentiable_transform_within: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set"   and d :: "real"   and g :: "'b \<Rightarrow> 'c" assumes "f differentiable at x within s"   and "0 < d"   and "x \<in> s"   and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "g differentiable at x within s"
\<G>_elements: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl" assumes "\<G> \<subseteq> \<^bold>G \<phi>" shows "Only_G \<G>"
propagate: shows "if index + 1 < length L then \<exists>\<beta>. LeftDerivation \<alpha> (take n D) \<beta> \<and> ladder_\<gamma> (\<alpha> @ []) D L index = \<beta> @ [] \<and> ladder_j L index < length \<beta> else \<exists>n' \<beta> \<delta>'. (index = 0 \<or> ladder_prev_n L index < n') \<and> n' \<le> n \<and> LeftDerivation \<alpha> (take n' D) \<beta> \<and> LeftDerivation (\<alpha> @ []) (take n' D) (\<beta> @ []) \<and> derivation_ge (drop n' D) (length \<beta>) \<and> LeftDerivation [] (derivation_shift (drop n' D) (length \<beta>) 0) \<delta>' \<and> ladder_\<gamma> (\<alpha> @ []) D L index = \<beta> @ \<delta>' \<and> ladder_j L index < length \<beta>"
HPhase1or2Write_valueChosen2: fixes s :: "state"   and s' :: "state"   and pp :: "Proc"   and d :: "Disk"   and D :: "Disk set"   and b :: "nat"   and p :: "Proc" assumes "HPhase1or2Write s s' pp d"   and "D \<in> MajoritySet"   and "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))"   and "HInv4a s pp" shows "\<forall>d\<in>D. b \<le> bal (disk s' d p) \<and> (\<forall>q. phase s' q = 1 \<and> b \<le> mbal (dblock s' q) \<and> hasRead s' q d p \<longrightarrow> (\<exists>br\<in>blocksRead s' q d. b \<le> bal (block br)))"
map_scale_eq_times: fixes c :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'b" shows "c \<cdot> p = monomial c (0::'a) * p"
quotient_is_cring: assumes "cring R" shows "cring (R Quot I)"
ComplI: fixes c :: "'e"   and A :: "'e set" assumes "c \<in> A \<Longrightarrow> False" shows "c \<in> - A"
bisim: shows "t \<turnstile> s1 \<approx> s2"
assms_2: fixes x :: "nat" shows "(x \<in> A) = (f x \<in> B)"
defined_neg: fixes b :: "tr" shows "defined (neg\<cdot>b) = defined b"
wf_jvm_prog_phi_def: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list" shows "wf_jvm_prog\<^bsub>\<Phi>\<^esub> \<equiv> wf_prog (\<lambda>P C (M, b, Ts, T\<^sub>r, mxs, mxl\<^sub>0, is, xt). wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt (\<Phi> C M))"
nop_vdomain: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop A n f" shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"
dynmethdSomeD: fixes G :: "prog"   and statC :: "qtname"   and dynC :: "qtname"   and sig :: "sig"   and dynM :: "qtname \<times> methd" assumes "dynmethd G statC dynC sig = Some dynM"   and "is_class G dynC"   and "ws_prog G" shows "G\<turnstile>dynC\<preceq>\<^sub>C statC \<and> (\<exists>statM. methd G statC sig = Some statM)"
iffs: fixes iiface :: "iface"   and oiface :: "iface"   and src :: "'a word \<times> nat"   and dst :: "'a word \<times> nat"   and proto :: "protocol"   and sports :: "16 word \<times> 16 word"   and dports :: "16 word \<times> 16 word"   and more :: "'b"   and iiface' :: "iface"   and oiface' :: "iface"   and src' :: "'a word \<times> nat"   and dst' :: "'a word \<times> nat"   and proto' :: "protocol"   and sports' :: "16 word \<times> 16 word"   and dports' :: "16 word \<times> 16 word"   and more' :: "'b" shows "(\<lparr>iiface = iiface, oiface = oiface, src = src, dst = dst, proto = proto, sports = sports, dports = dports, \<dots> = more\<rparr> = \<lparr>iiface = iiface', oiface = oiface', src = src', dst = dst', proto = proto', sports = sports', dports = dports', \<dots> = more'\<rparr>) = (iiface = iiface' \<and> oiface = oiface' \<and> src = src' \<and> dst = dst' \<and> proto = proto' \<and> sports = sports' \<and> dports = dports' \<and> more = more')"
openin_subset: fixes U :: "'a topology"   and S :: "'a set" assumes "openin U S" shows "S \<subseteq> topspace U"
ffold_finsert_fremove: fixes z :: "bool"   and x :: "val \<times> val"   and A :: "(val \<times> val) fset" shows "ffold (good_entry xa__ e \<rho> \<circ> good_prod) z (finsert x A) = (good_entry xa__ e \<rho> \<circ> good_prod) x (ffold (good_entry xa__ e \<rho> \<circ> good_prod) z (A |-| {|x|}))"
cring_simprules_26: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "cring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
distinct_Keys_to_list: fixes ps :: "(('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "distinct (pp_pm.extended_ord.punit.Keys_to_list ps)"
assms_3: fixes L :: "('a, 'b) Term.term literal"   and K :: "('a, 'b) Term.term literal"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b) Term.term option" assumes "match_term_list [(atm_of L, atm_of K)] \<sigma> = Some \<tau>"   and "is_pos L = is_pos K" shows "leq L K"
polyfun_polynomial: shows "polyfun {..<weight_space_dim} polynomial_f"
poly_map_iterateoi_axioms: shows "poly_map_iterateoi \<alpha> invar list_ordered_it"
simps_12: shows "get_axiom AdMult = diff_times_axiom"
single_is_union: fixes a :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" shows "({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
monotone_def: fixes orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g" shows "monotone orda ordb f = (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
cont_case_prod_iff: fixes orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and luba :: "'d set \<Rightarrow> 'd"   and lubb :: "'e set \<Rightarrow> 'e" assumes "monotone (rel_prod orda ordb) leq (\<lambda>(x, y). f x y)"   and "class.preorder orda (mk_less orda)"   and "lub_singleton luba"   and "class.preorder ordb (mk_less ordb)"   and "lub_singleton lubb" shows "cont (prod_lub luba lubb) (rel_prod orda ordb) lub leq (\<lambda>(x, y). f x y) = ((\<forall>x. cont lubb ordb lub leq (f x)) \<and> (\<forall>y. cont luba orda lub leq (\<lambda>x. f x y)))"
states_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and mxs :: "nat"   and mxl :: "nat" shows "states P mxs mxl \<equiv> fst (JVM_SemiType.sl P mxs mxl)"
orthogonal_mat_of_cols: fixes ws :: "'c vec list"   and n :: "nat" assumes "set ws \<subseteq> carrier_vec n"   and "corthogonal ws"   and "length ws = n" shows "corthogonal_mat (mat_of_cols n ws)"
c_assoc_have_key_is_pr: shows "c_assoc_have_key \<in> PrimRec2"
integrable_ccontinuous_explicit: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>a b. f integrable_on cbox a b" obtains N :: "'a set" where "negligible N" and "\<And>x e. \<lbrakk>x \<notin> N; 0 < e\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>h. 0 < h \<and> h < d \<longrightarrow> norm (integral (cbox x (x + h *\<^sub>R One)) f /\<^sub>R h ^ DIM('a) - f x) < e"
vertical_chop_own_trans: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u--w \<longrightarrow> own u = own w"
G_perm_apply_identity: fixes G :: "'a set"   and binop :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and e :: "'a"   and g :: "'a" assumes "BinOpSetGroup G binop e"   and "g \<in> G" shows "BinOpSetGroup.Abs_G_perm G binop g \<rightarrow> e = g"
permut_def: fixes as :: "'c list"   and f :: "nat \<Rightarrow> nat" shows "permut as f = permut_aux as f as"
FinalAllow_approximating_in_doubt_allow: fixes \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<beta> :: "'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue"   and rs :: "'a rule list"   and \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and p :: "'b" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalAllow" shows "(\<beta>, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow"
Red1AAss: fixes h :: "'heap"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and w :: "'addr val"   and U :: "ty"   and P :: "'addr J1_prog"   and h' :: "'heap"   and uf :: "bool"   and t :: "'thread_id"   and l :: "'addr val list" assumes "typeof_addr h a = \<lfloor>Array_type T n\<rfloor>"   and "0 \<le>s i"   and "sint i < int n"   and "typeof\<^bsub>h\<^esub> w = \<lfloor>U\<rfloor>"   and "P \<turnstile> U \<le> T"   and "heap_write h a (ACell (nat (sint i))) w h'" shows "uf,P,t \<turnstile>1 \<langle>addr a\<lfloor>Val (Intg i)\<rceil> := Val w,(h, l)\<rangle> -\<lbrace>WriteMem a (ACell (nat (sint i))) w\<rbrace>\<rightarrow> \<langle>unit,(h', l)\<rangle>"
collinear_def: fixes A :: "point"   and B :: "point"   and C :: "point" shows "collinear A B C = parallel A B A C"
index_take_if_set: fixes x :: "'b"   and n :: "nat"   and xs :: "'b list" assumes "x \<in> set (take n xs)" shows "index (take n xs) x = index xs x"
listrel1_def: fixes r :: "('g \<times> 'g) set" shows "listrel1 r = {(xs, ys). \<exists>us z z' vs. xs = us @ z # vs \<and> (z, z') \<in> r \<and> ys = us @ z' # vs}"
update_conv': fixes k :: "'a"   and v :: "'b"   and al :: "('a \<times> 'b) list" shows "map_of (AList.update k v al) = map_of al(k \<mapsto> v)"
lc_eq_zero_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'e" shows "(lc p = (0::'e)) = (p = 0)"
xor_def2: fixes x :: "'a"   and y :: "'a" shows "x \<oplus> y = (x \<^bold>\<squnion> y) \<^bold>\<sqinter> (\<sim> x \<^bold>\<squnion> \<sim> y)"
seq_char: fixes g :: "'a"   and f :: "'a" shows "L.seq g f = (L.arr f \<and> L.arr g \<and> seq g f)"
subset_\<P>Suc: fixes k :: "nat"   and u :: "nat" shows "\<P> k u \<subseteq> \<P> k (Suc u)"
mk_disjoint_insert: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "\<exists>B. A = a \<triangleright> B \<and> a \<notin> B"
gexp_max_input_In: fixes v :: "vname"   and l :: "value list" shows "GExp.max_input (In v l) = AExp.max_input (V v)"
rep_eq: shows "mapping_rbt.impl_of RBT_Mapping2.empty = rbt.Empty"
iter_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "iter MOD \<equiv> check \<circ> iterSTR (errMOD MOD)"
goedel_first_classic_strong: assumes "consistent" shows "\<not> {} \<turnstile> \<phi>G \<and> \<not> {} \<turnstile> neg \<phi>G \<and> eval_fm e0 \<phi>G"
has_fields_Object: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and D :: "char list"   and fs :: "(char list \<times> ty) list"   and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list"   and FDTs :: "((char list \<times> char list) \<times> ty) list" assumes "class P Object = \<lfloor>(D, fs, ms)\<rfloor>"   and "FDTs = map (\<lambda>(F, T). ((F, Object), T)) fs" shows "P \<turnstile> Object has_fields FDTs"
C_eq_normalize: fixes p :: "((int \<times> int) set set, int) Combinators" assumes "DenyAll \<in> set (policy2list p)"   and "allNetsDistinct (policy2list p)"   and "all_in_list (policy2list p) (Nets_List p)" shows "C (list2FWpolicy (FWNormalisationCore.normalize p)) = C p"
assms: shows "\<forall>I. finitely_generated_ideal I \<longrightarrow> principal_ideal I"
simps_1: fixes T :: "ty" shows "is_Array (T\<lfloor>\<rceil>) = True"
GB_adds_lt: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "i.punit.is_Groebner_basis G"   and "f \<in> punit.pmdl G"   and "f \<noteq> 0" obtains g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where "g \<in> G" and "g \<noteq> 0" and "punit.adds_term (i.lpp g) (i.lpp f)"
invertible_det_nz: fixes A :: "(('b, 'c) vec, 'c) vec" shows "invertible A = (det A \<noteq> (0::'b))"
ignore: fixes s :: "'s"   and x :: "'call" assumes "I s"   and "\<not> consider' x" shows "interaction_bounded_by consider (callee s x) 0"
1: shows "1 \<le> Re (winding_number \<gamma> z)"
kat_cond_def: fixes p :: "'a"   and x :: "'a"   and y :: "'a" shows "if p then x else y = \<tt>\<tt> p \<cdot> x + !p \<cdot> y"
fst_add_indices: fixes ns :: "(('f \<Rightarrow>\<^sub>0 'g) \<times> 'h) list \<times> 'i"   and data' :: "nat \<times> 'i" shows "map fst (fst (add_indices ns data')) = map fst (fst ns)"
stable_onD: fixes CX :: "'d set"   and trap :: "'d set"   and x0 :: "'d"   and t :: "real" assumes "stable_on CX trap"   and "flow0 x0 t \<in> trap"   and "t \<in> existence_ivl0 x0"   and "0 < t"   and "\<And>s. \<lbrakk>0 < s; s \<le> t\<rbrakk> \<Longrightarrow> flow0 x0 s \<in> CX" shows "x0 \<in> trap"
index_zero_vec_1: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "0\<^sub>v n $ i = (0::'a)"
discounted_integrable: fixes M :: "'c measure"   and N :: "'c measure"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and n :: "nat"   and r :: "real" assumes "prob_space M"   and "integrable N (X n)"   and "- 1 < r"   and "space N = space M" shows "integrable N (discounted_value r X n)"
cross3_simps_17: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a - (b - c) = a + c - b"
scale_minus_right: fixes a :: "real"   and x :: "'a" shows "a *\<^sub>R - x = - (a *\<^sub>R x)"
divide_nonneg_nonneg: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less_eq zero y" shows "less_eq zero (divide x y)"
get_empty: fixes empty :: "'c"   and get :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and add :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'c"   and to_list :: "'c \<Rightarrow> ('d \<times> 'e) list"   and x :: "'d" assumes "env empty get add to_list" shows "get empty x = None"
simps_10: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'g"   and f3 :: "'i \<Rightarrow> 'g"   and x2 :: "'h" shows "(case sum_bot.Left x2 of Bottom \<Rightarrow> f1 | sum_bot.Left x \<Rightarrow> f2 x | sum_bot.Right x \<Rightarrow> f3 x) = f2 x2"
weight_def: fixes f :: "nat \<Rightarrow> nat"   and \<pi> :: "nat \<Rightarrow> nat" shows "local.weight f \<pi> = signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i) * B $$ (f i, \<pi> i))"
integral_diff: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "integrable M f"   and "integrable M g" shows "LINT x|M. f x - g x = integral\<^sup>L M f - integral\<^sup>L M g"
assms: shows "P \<turnstile> \<langle>INIT C ([C],b) \<leftarrow> unit,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"
alice_state: fixes \<phi> :: "complex Matrix.mat" assumes "state 1 \<phi>" shows "state 3 (alice \<phi>)"
tube_lemma: fixes S :: "'a set"   and a :: "'b"   and U :: "('a \<times> 'b) set"   and T :: "'b set" assumes "compact S"   and "S \<noteq> {}"   and "(\<lambda>x. (x, a)) ` S \<subseteq> U"   and "openin (top_of_set (S \<times> T)) U" obtains V :: "'b set" where "openin (top_of_set T) V" and "a \<in> V" and "S \<times> V \<subseteq> U"
map_simp_ide: fixes a :: "'c" assumes "C.ide a" shows "G\<epsilon>o\<eta>G.map a = (G \<circ> \<epsilon>.map) a \<cdot>\<^sub>D (\<eta>.map \<circ> G) a"
1: fixes m :: "nat" assumes "N < m" shows "s m n = r"
encode_interfering_operator_pair_exclusion_def: fixes \<Pi> :: "'a strips_problem"   and k :: "nat"   and op\<^sub>1 :: "'a strips_operator"   and op\<^sub>2 :: "'a strips_operator" shows "encode_interfering_operator_pair_exclusion \<Pi> k op\<^sub>1 op\<^sub>2 \<equiv> let ops = \<Pi>\<^sub>\<O> in \<^bold>\<not> (Atom (Operator k (index ops op\<^sub>1))) \<^bold>\<or> \<^bold>\<not> (Atom (Operator k (index ops op\<^sub>2)))"
prc: shows "prc (g, length ts) (f, length ss) = (s, True)"
existence_ivl_eq_rev: fixes x0 :: "'a" shows "existence_ivl0 x0 = preflect 0 ` ll_on_open.existence_ivl (range (preflect 0)) (\<lambda>t. - f) X 0 x0"
var_not_in_aEval: fixes var :: "nat"   and \<phi> :: "atom"   and L :: "real list" assumes "freeIn var (fm.Atom \<phi>)" shows "(\<exists>x. aEval \<phi> (L[var := x])) = (\<forall>x. aEval \<phi> (L[var := x]))"
in_charts_in_atlas: fixes x :: "('b, 'b) chart" assumes "x \<in> manifold_eucl.dest.charts_submanifold (codomain c)" shows "x \<in> a.dest.atlas"
axioms_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "category C"
degree_m_eq_lead_coeff: fixes f :: "int poly" assumes "q.degree_m f = degree f" shows "lead_coeff (q.Mp f) = q.M (lead_coeff f)"
lNactGe: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "dynamic_component.lNAct active c t n \<le> n'"   and "active c (t n')" shows "n \<le> n'"
left_commute: fixes b :: "'k multiset"   and a :: "'k multiset"   and c :: "'k multiset" shows "b \<union># (a \<union># c) = a \<union># (b \<union># c)"
drop_tl: fixes n :: "nat"   and xs :: "'d list" shows "drop n (tl xs) = tl (drop n xs)"
prems_3: shows "p\<^sub>4 = local.add a p\<^sub>1 p\<^sub>3"
offs_pred_next_cons_less: fixes ns :: "nat list"   and ub :: "nat"   and x :: "'c"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and j :: "nat" assumes "offs_pred ns ub (x # xs) index key mi ma"   and "index key x (length ns) mi ma = i"   and "offs_set_prev ns (x # xs) index key mi ma i \<noteq> {}"   and "Max (offs_set_prev ns (x # xs) index key mi ma i) = j" shows "offs_next ns ub (x # xs) index key mi ma j < offs_next (ns[i := Suc (ns ! i)]) ub xs index key mi ma j"
self_domain_iff_ide: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "partial_magma C" shows "(a \<in> partial_magma.domains C a) = partial_magma.ide C a"
assms_1: shows "igSubstAllIPresIGWlsAll MOD"
is_ascii_butlast: fixes str :: "char list" assumes "is_ascii str" shows "is_ascii (butlast str)"
sum_nonneg: 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 A :: "'f set"   and f :: "'f \<Rightarrow> 'e" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x)" shows "less_eq zero (comm_monoid_add.sum plus zero f A)"
strict_boundedE: fixes b :: "'b multiset"   and c :: "'b multiset"   and a :: "'b multiset" assumes "b \<union># c \<subset># a" obtains "b \<subset># a" and "c \<subset># a"
verts_in_translation: fixes X :: "'b allegorical_term" shows "inv_translation (vertices (translation X))"
il_sub_inf_left_isotone_var: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "y \<sqinter> x \<squnion> z \<sqinter> x \<le> (y \<squnion> z) \<sqinter> x"
smult_conv_mult: fixes s :: "'b"   and p :: "'b mpoly" shows "smult s p = monom 0 s * p"
closed_sum_comm: fixes A :: "'d set"   and B :: "'d set" shows "A +\<^sub>M B = B +\<^sub>M A"
Min_le_iff: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'e set"   and x :: "'e" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq (linorder.Min less_eq A) x = (\<exists>a\<in>A. less_eq a x)"
UNIV_eq_I: fixes A :: "'b set" assumes "\<And>x. x \<in> A" shows "UNIV = A"
fresh_subst: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y" shows "fresh zs z (X #[Y / y]_ys) = ((zs = ys \<and> z = y \<or> fresh zs z X) \<and> (fresh ys y X \<or> fresh zs z Y))"
no_small_separation: fixes S :: "'a set" assumes "Separation G v0 v1 S" shows "Suc n \<le> card S"
intro: fixes P :: "'c \<Rightarrow> bool"   and E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>P a; E a b\<rbrakk> \<Longrightarrow> Q b"   and "\<And>a. Q a \<Longrightarrow> P a" shows "Graph_Invariants E P Q"
assms: shows "x \<in> carrier (reduced_homology_group (int p) (nsphere p))"
nth_list_update_eq: fixes i :: "nat"   and xs :: "'k list"   and x :: "'k" assumes "i < length xs" shows "xs[i := x] ! i = x"
simps_7: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm" shows "Ifm vs bs (NOT p) = (\<not> Ifm vs bs p)"
prems_3: shows "a \<in> read_only (share (x # sb) \<S>)"
I2: shows "set_integrable lborel {1 / 2..1} (\<lambda>t. t powr (a - 1) * (1 - t) powr (b - 1))"
fresh_def: fixes a :: "atom"   and x :: "'d" shows "a \<sharp> x \<equiv> a \<notin> supp x"
poly_eval_map_indets: fixes a :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "poly_eval a (map_indets g p) = poly_eval (a \<circ> g) p"
get_addr2_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "'c \<times> inst_operand list" assumes "low_equal s1 s2" shows "get_addr (snd instr) (snd (fst (get_curr_win () s1))) = get_addr (snd instr) (snd (fst (get_curr_win () s2)))"