hcju/isabelleps huggingface dataset:
assms: shows "{a..b} \<subseteq> existence_ivl t0 x"
univ_poly_is_abelian_group: shows "abelian_group (K [X])"
pairD: fixes a :: "'a"   and b :: "'b"   and p :: "'a \<times> 'b" assumes "(a, b) = p" shows "a = fst p \<and> b = snd p"
body_split_def: fixes p :: "prog"   and s\<^sub>L :: "stmt"   and s\<^sub>H :: "stmt" shows "body_split p s\<^sub>L s\<^sub>H \<equiv> body p = Seq s\<^sub>L s\<^sub>H"
remove_duplicate_premise: fixes P :: "prop"   and Q :: "prop" shows "(\<lbrakk>PROP P; PROP P\<rbrakk> \<Longrightarrow> PROP Q) \<equiv> (PROP P \<Longrightarrow> PROP Q)"
the_unique_root_eqI_2: fixes plr :: "int poly \<times> rat \<times> rat"   and y :: "real" assumes "\<exists>!x. root_cond plr x"   and "root_cond plr y" shows "real_of_1 plr = y"
mod_ordinal_def: fixes x :: "ordinal"   and y :: "ordinal" shows "x mod y = x - y * (x div y)"
rs: shows "r @@ s \<in> A\<^sup>\<infinity>"
assms_1: shows "H \<turnstile> SeqHaddP s j (SUCC k) y"
DERIV_zero_connected_constant: fixes S :: "'a set"   and K :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "connected S"   and "open S"   and "finite K"   and "continuous_on S f"   and "\<forall>x\<in>S - K. (f has_field_derivative (0::'a)) (at x)" obtains c :: "'a" where "\<And>x. x \<in> S \<Longrightarrow> f x = c"
wt_ge_\<epsilon>\<^sub>h: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and s :: "('s, 'v) tm" assumes "legal_zpassign A" shows "zhmset_of \<epsilon>\<^sub>h \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
converse_ordermap_mono: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and y :: "'c"   and x :: "'c" assumes "ordermap A r y \<in> elts (ordermap A r x)"   and "wf r"   and "total_on A r"   and "x \<in> A"   and "y \<in> A"   and "small A" shows "(y, x) \<in> r"
c_3: shows "\<forall>i x. eval psi [i, x] = eval r_phi [the (eval c [i]), x]"
lapp_suff_llength: fixes r :: "'b llist"   and A :: "'b set"   and s :: "'b llist" assumes "r \<in> A\<^sup>\<star>" shows "(r @@ s) \<up> LList2.llength r = s"
Abs_key_order_inject: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and y :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "x \<in> {compare. comparator compare}"   and "y \<in> {compare. comparator compare}" shows "(Abs_key_order x = Abs_key_order y) = (x = y)"
Join_transient_I1: fixes F :: "'b program"   and A :: "'b set"   and G :: "'b program" assumes "F \<in> transient A" shows "F \<squnion> G \<in> transient A"
BLP_m_spec: fixes nP :: "nat \<Rightarrow> SINVAR_BLPtrusted.node_config" assumes "nP = (\<lambda>v. case BLP_security_levels v of None \<Rightarrow> SINVAR_BLPtrusted.default_node_properties | Some c \<Rightarrow> c)" shows "BLP_m_spec = Some \<lparr>c_sinvar = \<lambda>G. SINVAR_BLPtrusted.sinvar G nP, c_offending_flows = \<lambda>G. SINVAR_BLPtrusted.set_offending_flows G nP, c_isIFS = SINVAR_BLPtrusted.receiver_violation\<rparr>"
neq_LNil_conv: fixes xs :: "'c llist" shows "(xs \<noteq> LNil) = (\<exists>x xs'. xs = LCons x xs')"
1: shows "finite (\<Delta> ` (S \<union> N))"
basic_subseqs_impl_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" shows "basic_subseqs_impl f = Sublists_Impl (\<lambda>a xs n. (subseqs_length f a n xs, a, xs, n)) (\<lambda>(a, xs, n). (subseqs_length f a (Suc n) xs, a, xs, Suc n))"
update_convs_1: fixes zc' :: "int \<Rightarrow> int"   and xc :: "int"   and yc :: "int"   and zc :: "int"   and more :: "'a" shows "zc_update zc' \<lparr>xc = xc, yc = yc, zc = zc, \<dots> = more\<rparr> = \<lparr>xc = xc, yc = yc, zc = zc' zc, \<dots> = more\<rparr>"
preim_of_lideal_is_lideal: fixes I :: "'c set"   and B :: "'c set"   and addB :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multB :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zeroB :: "'c"   and oneB :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and addA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroA :: "'d"   and oneA :: "'d" assumes "lideal I B addB multB zeroB oneB"   and "ring_homomorphism f A addA multA zeroA oneA B addB multB zeroB oneB" shows "lideal (f \<^sup>\<inverse> A I) (f \<^sup>\<inverse> A B) addA multA zeroA oneA"
fps_const_nonzero_eq_nonzero: fixes c :: "'a" assumes "c \<noteq> (0::'a)" shows "fps_const c \<noteq> 0"
timpls_Un_mono: fixes FP :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" shows "mono (\<lambda>X. FP \<union> (\<Union>x\<in>X. \<Union>(a, b)\<in>TI. set \<langle>a --\<guillemotright> b\<rangle>\<langle>x\<rangle>))"
is_red_implies_0_red_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "B \<subseteq> dgrad_p_set d m"   and "pmdl B \<subseteq> pmdl A"   and "\<And>q. \<lbrakk>q \<in> pmdl A; q \<in> dgrad_p_set d m; q \<noteq> 0\<rbrakk> \<Longrightarrow> is_red B q"   and "p \<in> pmdl A"   and "p \<in> dgrad_p_set d m" shows "(red B)\<^sup>*\<^sup>* p 0"
b: shows "rep_list (monomial (1::'b) (term_of_pair (0::'a, j))) = fs ! j"
E3: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> F \<longrightarrow> \<diamond>F"
sinks_aux_subset: fixes U :: "'b set"   and I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "U \<subseteq> sinks_aux I D U xs"
A1: shows "\<forall>SP SQ. SP \<in>S P \<and> SQ \<in>S Q \<longrightarrow> (SP, SQ) \<in> SRel\<^sup>+"
assms_4: shows "((\<lambda>x. exp (ln (f x) * g x)) expands_to E) basis"
embedI: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "Well_order r"   and "Well_order s"   and "\<And>a. a \<in> Field r \<Longrightarrow> f a \<in> Field s \<and> f ` underS r a \<subseteq> underS s (f a)" shows "\<exists>g. embed r s g"
assms_1: shows "u \<cdot> r \<bowtie> v \<cdot> s"
lcm_nat_abs_left_eq: fixes k :: "int"   and n :: "nat" shows "lcm (nat \<bar>k\<bar>) n = nat (lcm k (int n))"
cases: fixes x :: "'c tree" obtains "x = \<langle>\<rangle>"   | l :: "'c tree" and a :: "'c" and r :: "'c tree" where "x = \<langle>l, a, r\<rangle>"
app_vimage_iff: fixes b :: "V"   and r :: "V"   and A :: "V" shows "(b \<in>\<^sub>\<circ> r `\<^sub>\<circ> A) = (\<exists>a\<in>elts A. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r)"
\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c_not_Unknown: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr" shows "ternary_ternary_eval (map_match_tac (\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c \<gamma>) p m) \<noteq> TernaryUnknown"
select_convs_4: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cCurrentF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cCurrentF"
subgroup_inverse_iff: fixes G :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and x :: "'c" assumes "subgroup G M composition unit"   and "monoid.invertible M composition unit x"   and "x \<in> M" shows "(monoid.inverse M composition unit x \<in> G) = (x \<in> G)"
measures_less: fixes f :: "'a \<Rightarrow> nat"   and x :: "'a"   and y :: "'a"   and fs :: "('a \<Rightarrow> nat) list" assumes "f x < f y" shows "(x, y) \<in> measures (f # fs)"
finter_iff: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" shows "(c |\<in>| A |\<inter>| B) = (c |\<in>| A \<and> c |\<in>| B)"
inject_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(Pure x1 = Pure y1) = (x1 = y1)"
map_simp: fixes f :: "'d" assumes "arr f" shows "\<ll>' f = local.inv (\<ll> (cod f)) \<cdot> local.map f"
\<theta>: shows "\<theta> = \<theta>\<^sub>1 @ \<theta>\<^sub>2"
only_fork_introduces_rids': fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and r :: "'a"   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 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" assumes "substitution subst"   and "substitution.revision_step subst r s s'"   and "\<And>\<sigma> \<tau> \<E> e. s r \<noteq> Some (\<sigma>, \<tau>, \<E> [Rfork e])"   and "r' \<notin> RID\<^sub>G s" shows "r' \<notin> RID\<^sub>G s'"
arg_correct: fixes z :: "complex" assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> - pi < arg z \<and> arg z \<le> pi"
set_ConsD: fixes y :: "'g"   and x :: "'g"   and xs :: "'g list" assumes "y \<in> set (x # xs)" shows "y = x \<or> y \<in> set xs"
indVfbasis_indV: fixes hfvs :: "(('f, 'g) aezfun \<Rightarrow> 'v) list" assumes "hfvs \<in> set indVfbasis" shows "set hfvs \<subseteq> indV"
leq_antisym: fixes leq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.partial_order leq"   and "leq x y"   and "leq y x" shows "x = y"
fmap_rel_in_dom_iff: fixes aa :: "('a, 'b) fmap"   and a'a :: "('c, 'd) fmap"   and K :: "('a \<times> 'c) set"   and V :: "('b \<times> 'd) set"   and a :: "'a"   and a' :: "'c" assumes "(aa, a'a) \<in> \<langle>K, V\<rangle>fmap_rel"   and "(a, a') \<in> K" shows "(a' \<in># dom_m a'a) = (a \<in># dom_m aa)"
intersect_list'_nodes_finite: fixes AA :: "('c, 'd) nba list" assumes "list_all (finite \<circ> NBA.nodes) AA" shows "finite (NGBA.nodes (intersect_list' AA))"
filter_map: fixes P :: "'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "filter P (map f xs) = map f (filter (P \<circ> f) xs)"
list_of_oalist_tc_of_list: fixes xs :: "('a \<times> 'b) list" shows "list_of_oalist_tc (oalist_tc_of_list xs) = sort_oalist_tc xs"
elims: fixes P :: "ast_problem"   and x :: "plan_action"   and y :: "ground_action" assumes "ast_problem.resolve_instantiate P x = y" obtains n :: "char list" and args :: "object list" where "x = PAction n args" and "y = instantiate_action_schema (the (ast_domain.resolve_action_schema (domain P) n)) args"
simps: fixes n :: "'a"   and m :: "'b" shows "tape_of_nat_prod (n, m) = <n> @ [Bk] @ <m>"
open_right: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "class.linorder_topology less_eq less open"   and "open S"   and "x \<in> S"   and "less x y" shows "\<exists>b. less x b \<and> ord.atLeastLessThan less_eq less x b \<subseteq> S"
well_com_hadamard_k: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat"   and k :: "nat" assumes "grover_state_sig n f dims R K"   and "k \<le> n" shows "state_sig.well_com dims (grover_state_sig.hadamard_n n dims K k)"
405: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> (y \<sqinter> z)) \<sqinter> - (x \<sqinter> y) = - (x \<sqinter> y)"
adjugate_mult_det: fixes A :: "('c, 'd) sq_matrix" shows "Square_Matrix.adjugate A * A = diag (Square_Matrix.det A)"
subgraph_union_iff: fixes H1 :: "('c, 'd) pre_digraph"   and H2 :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "wf_digraph H1"   and "wf_digraph H2"   and "compatible H1 H2" shows "subgraph (Digraph_Component.union H1 H2) G = (subgraph H1 G \<and> subgraph H2 G)"
d_def: shows "d = sign_changes (coeffs p) - count_roots_with ((<) 0) p"
type_update_empty: fixes ts :: "checker_type"   and cons :: "ct list" shows "type_update ts cons (Type []) = consume ts cons"
simps_3: fixes n :: "nat" shows "\<d> n \<equiv> if n = 3 then 0 else if n = 4 then 2058 else if n = 5 then 4819 else if n = 6 then 7120 else squanderTarget"
pochhammer_Gamma: fixes z :: "'b"   and n :: "nat" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "pochhammer z n = Gamma (z + of_nat n) / Gamma z"
Int_subset_iff: fixes C :: "'e set"   and A :: "'e set"   and B :: "'e set" shows "(C \<subseteq> A \<inter> B) = (C \<subseteq> A \<and> C \<subseteq> B)"
is_functor: shows "functor (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) A_B.map"
cd_trans: fixes j :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and k :: "nat" assumes "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> i"   and "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j" shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i"
simps_3: fixes d :: "int" shows "intv'_const (Intv' d) = d"
lmeasurable_compact: fixes S :: "'c set" assumes "compact S" shows "S \<in> lmeasurable"
quotient_web_sel_4: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "B (quotient_web \<Gamma> f) = B \<Gamma>"
map_to_set_map_of: fixes l :: "('a \<times> 'b) list" assumes "distinct (map fst l)" shows "map_to_set (map_of l) = set l"
att_andD2: fixes x1 :: "'a attree"   and x2 :: "'a attree list"   and s :: "'a set \<times> 'a set" assumes "\<turnstile>x1 # x2 \<oplus>\<^sub>\<and>\<^bsup>s\<^esup>" shows "\<turnstile>x2 \<oplus>\<^sub>\<and>\<^bsup>(snd (attack x1), snd s)\<^esup>"
elims: fixes x :: "'b literal set"   and xa :: "('b \<times> 'b trm) list"   and y :: "'b literal set" assumes "subst_cl x xa = y" obtains C :: "'b literal set" and s :: "('b \<times> 'b trm) list" where "x = C" and "xa = s" and "y = {L. \<exists>L'. L' \<in> C \<and> L = subst_lit L' s}"
invalid: shows "\<not> \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
mult_assoc_5: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex"   and C :: "complex \<times> complex \<times> complex \<times> complex"   and D :: "complex \<times> complex \<times> complex \<times> complex"   and E :: "complex \<times> complex \<times> complex \<times> complex" shows "A *\<^sub>m\<^sub>m (B *\<^sub>m\<^sub>m C *\<^sub>m\<^sub>m D) *\<^sub>m\<^sub>m E = A *\<^sub>m\<^sub>m B *\<^sub>m\<^sub>m C *\<^sub>m\<^sub>m (D *\<^sub>m\<^sub>m E)"
cindependent_inter_scaleC_cindependent: fixes B :: "'d set"   and c :: "complex" assumes "cindependent B"   and "c \<noteq> 1" shows "B \<inter> (*\<^sub>C) c ` B = {}"
less_eq_pred_def: fixes P :: "'b Predicate.pred"   and Q :: "'b Predicate.pred" shows "(P \<le> Q) = (pred.eval P \<le> pred.eval Q)"
Scons_UN1_y: fixes M :: "('a, 'b) node set"   and f :: "'c \<Rightarrow> ('a, 'b) node set" shows "Scons M (\<Union> (range f)) = (\<Union>x. Scons M (f x))"
Pd_shuffle_to_top_Field: fixes Y :: "'a set"   and ds' :: "'b set"   and d :: "'b" assumes "inj_on Xd Y" shows "Field (Pd_shuffle_to_top ds' Y d) = Field (Pd d) \<union> (if d \<in> ds' then Pdds'.dX Y d else {})"
S1_def: fixes p :: "nat"   and a :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)" shows "??.Euler_Criterion.S1 p = {0<..int p - 1}"
assms_3: assumes "ip \<in> iD rt" shows "P rt ip"
comp_arr_dom: fixes f :: "(bool, bool list) concrete_category.arr"   and a :: "(bool, bool list) concrete_category.arr" assumes "J.arr f"   and "J.dom f = a" shows "f \<cdot>\<^sub>J a = f"
measurable_abs_UNIV: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and M :: "'e measure"   and N :: "'d \<Rightarrow> 'f measure" assumes "\<And>n. f n \<in> M \<rightarrow>\<^sub>M N n" shows "(\<lambda>\<omega> n. f n \<omega>) \<in> M \<rightarrow>\<^sub>M Pi\<^sub>M UNIV N"
echelon_form_of_upt_k_det_def: fixes bezout :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<times> 'e \<times> 'e \<times> 'e \<times> 'e"   and A' :: "'h \<times> (('e, 'f) vec, 'g) vec"   and k :: "nat" shows "echelon_form_of_upt_k_det bezout A' k = (let A = snd A'; f = foldl (echelon_form_of_column_k_det bezout) (1::'e, A, 0) [0..<Suc k] in (fst f, fst (snd f)))"
iT_Minus_inext2: fixes k :: "nat"   and I :: "nat set"   and n :: "nat" assumes "k \<ominus> I \<noteq> {}"   and "n \<le> k" shows "inext n (k \<ominus> I) = k - iprev (k - n) I"
lprefix_code_2: fixes x :: "'c"   and xs :: "'c llist" shows "lprefix (LCons x xs) LNil = False"
simps_1: fixes m :: "'b match_expr" shows "compress_normalize_primitive_monad [] m = Some m"
prems: shows "Keys (set fs) = set ks \<union> set vs"
has_vector_derivative_cong_ev: fixes S :: "real set"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and x :: "real"   and f' :: "'a" assumes "\<forall>\<^sub>F x in nhds x. x \<in> S \<longrightarrow> f x = g x"   and "f x = g x" shows "(f has_vector_derivative f') (at x within S) = (g has_vector_derivative f') (at x within S)"
a: shows "\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. \<forall>\<alpha>\<in>{0<..<1}. p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r"
insertE: fixes a :: "'c"   and b :: "'c"   and A :: "'c set" assumes "a \<in> {b} \<union> A" obtains "a = b"   | "a \<in> A"
call_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i = call_type CALL" shows "snd (call_instr i s) = False"
r_parallel': fixes i :: "nat"   and j :: "nat"   and x :: "nat" shows "eval r_parallel [i, j, x] = parallel i j x"
del_UnEdge_even': fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and v' :: "'a" assumes "valid_unMultigraph G"   and "(v, w, v') \<in> edges G"   and "finite (edges G)" shows "(v' \<in> odd_nodes_set (del_unEdge v w v' G)) = even (degree v' G)"
proj_exec_proj_eq_exec_proj': fixes vs :: "'a set"   and s :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" shows "fmrestrict_set vs (exec_plan (fmrestrict_set vs s) (as_proj as vs)) = fmrestrict_set vs (exec_plan s (as_proj as vs))"
fun_left_comm: fixes x :: "'ag"   and y :: "'ag"   and z :: "'ag" assumes "x \<in> U"   and "y \<in> U"   and "z \<in> U" shows "y +\<^sub>o\<^sub>w (x +\<^sub>o\<^sub>w z) = x +\<^sub>o\<^sub>w (y +\<^sub>o\<^sub>w z)"
simps: fixes a :: "atom"   and T :: "atom set"   and H :: "chmsg set" shows "(a \<in> extr T H) = ((\<exists>At. a = At \<and> At \<in> T) \<or> (\<exists>c A B M At. a = At \<and> StatCh c A B (Msg M) \<in> H \<and> At \<in> set M \<and> (confid \<notin> c \<or> A \<in> bad \<or> B \<in> bad)) \<or> (\<exists>c K M At. a = At \<and> DynCh c K (Msg M) \<in> H \<and> At \<in> set M \<and> (confid \<notin> c \<or> aKey K \<in> extr T H)))"
homogenize_id_iff: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "(homogenize x p = p) = homogeneous p"
foundation20: fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "\<tau> \<Turnstile> \<delta> X" shows "\<tau> \<Turnstile> \<upsilon> X"
corthogonalI: fixes vs :: "'a vec list" assumes "\<And>i j. \<lbrakk>i < length vs; j < length vs\<rbrakk> \<Longrightarrow> (vs ! i \<bullet>c vs ! j = (0::'a)) = (i \<noteq> j)" shows "corthogonal vs"
sat_primal_def: fixes A :: "'b mat"   and b :: "'b vec" shows "sat_primal A b = {x. [A*\<^sub>vx]\<le>b}"
f_lesbegue_integrable: shows "integrable lborel (\<lambda>x. if x \<in> Dx_pair then partial_vector_derivative (\<lambda>a. F a \<bullet> j) i x else 0)"
replace_update_2: fixes k :: "'a"   and m :: "('a, 'b) mapping"   and v :: "'b" assumes "k \<in> Mapping.keys m" shows "Mapping.replace k v m = Mapping.update k v m"
sup_assoc: fixes sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.semilattice_sup sup less_eq less" shows "sup (sup x y) z = sup x (sup y z)"
Un_empty_right: fixes A :: "'c set" shows "A \<union> {} = A"
Abs_fset_induct: fixes P :: "'b fset \<Rightarrow> bool"   and x :: "'b fset" assumes "\<And>y. y \<in> {A. finite A} \<Longrightarrow> P (Abs_fset y)" shows "P x"
eq: fixes z :: "complex" assumes "z \<in> T - {a}" shows "g z = (z - a) * f z"
S: fixes s :: "'a"   and s' :: "'b" assumes "(s, s') \<in> R"   and "rwof m c f s"   and "rwof m' c' f' s'"   and "c s"   and "c' s'" shows "f s \<le>\<^sub>n \<Down> R (f' s')"
simps_983: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x291 :: "t"   and x292 :: "relop_i" shows "(case Relop_i x291 x292 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f29 x291 x292"
exhaust: fixes y :: "('a, 'b) state" obtains x :: "'a \<Rightarrow> 'b \<times> 'a" where "y = State x"
arr_cod_iff_arr: fixes f :: "'a \<times> 'a" shows "VoV.arr (VoV.cod f) = VoV.arr f"
coeff_Abs_poly_If_le: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "coeff (Abs_poly (\<lambda>i. if i \<le> n then f i else (0::'b))) = (\<lambda>i. if i \<le> n then f i else (0::'b))"
s_def: shows "s = (if conv_abscissa f = - \<infinity> then \<sigma> - 1 else (\<sigma> + real_of_ereal (conv_abscissa f)) / 2)"
subst_comp_te: fixes R :: "nat \<Rightarrow> typ"   and S :: "nat \<Rightarrow> typ"   and t :: "typ" shows "$ R ($ S t) = $ (\<lambda>x. $ R (S x)) t"
simps_2: fixes x2 :: "'b nmultiset multiset"   and y2 :: "'b nmultiset multiset" shows "(MSet x2 = MSet y2) = (x2 = y2)"
gen_eq_gen: fixes I :: "'a set"   and J :: "'a set"   and P :: "'a set set" assumes "{I, J} \<subseteq> P"   and "gen P I \<inter> gen P J \<noteq> {}" shows "gen P I = gen P J"
add: fixes S1 :: "'c set"   and S2 :: "'d set"   and s1 :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and s2 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and b1 :: "'c"   and b2 :: "'c" assumes "module_hom_on S1 S2 s1 s2 f"   and "b1 \<in> S1"   and "b2 \<in> S1" shows "f (b1 + b2) = f b1 + f b2"
dimension_independent: fixes K :: "'a set"   and Us :: "'a list" assumes "R.independent K Us" shows "R.dimension (length Us) K (R.Span K Us)"