hcju/isabelleps huggingface dataset:
vpra_empty_valid_call_list_rev: fixes cs :: "'edge list"   and n :: "'node" assumes "valid_call_list cs n" shows "valid_path_rev_aux [] (rev cs)"
synlr_def: shows "synlr \<equiv> {R. \<forall>a. {d. (d, a) \<in> R} \<in> admS}"
l_distributive_def: fixes fmul :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and fadd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "l_distributive fmul fadd \<equiv> \<forall>a u v. fmul (fadd u v) a = fadd (fmul u a) (fmul v a)"
RETURN_SPEC_refine: fixes x :: "'f"   and R :: "('f \<times> 'e) set"   and \<Phi> :: "'e \<Rightarrow> bool" assumes "\<exists>x'. (x, x') \<in> R \<and> \<Phi> x'" shows "RETURN x \<le> \<Down> R (SPEC \<Phi>)"
Deriv_inter: fixes a :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "Deriv a (A \<inter> B) = Deriv a A \<inter> Deriv a B"
worec_fixpoint: fixes H :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" assumes "local.adm_wo H" shows "local.worec H = H (local.worec H)"
is_unit_div_mult2_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one"   and "dvd.dvd times c one" shows "divide a (times b c) = divide (divide a b) c"
nprv_eql_symI: fixes F :: "'fmla set"   and t1 :: "'trm"   and t2 :: "'trm" assumes "nprv F (eql t1 t2)"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "F \<subseteq> fmla"   and "finite F" shows "nprv F (eql t2 t1)"
IH: fixes gpv :: "('c, 'a, 'b) gpv" assumes "\<And>x. x \<in> results_gpv \<I> gpv \<Longrightarrow> f x = g x" shows "expectation_gpv' gpv = expectation_gpv'' gpv"
min_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" shows "ord.min less_eq a b = (if less_eq a b then a else b)"
sat_RESTRICT: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "sat \<AA> \<phi> = (\<AA> \<Turnstile> RESTRICT \<phi>)"
abc_ex_path_unique: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c" shows "\<exists>!Q. Q \<in> \<P> \<and> a \<in> Q \<and> b \<in> Q \<and> c \<in> Q"
ideD_1: fixes a :: "'a" assumes "D.J.ide a" shows "D.J.arr a"
boundedI: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite A"   and "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> a \<preceq> x" shows "ordered_powerprod_lin.Max A \<preceq> x"
fst_initmissing_netgmap_pair_fst: fixes s :: "((state \<times> 'a) \<times> 'b) net_state" shows "fst (initmissing (netgmap (\<lambda>(p, q). (fst (id p), snd (id p), q)) s)) = fst (initmissing (netgmap fst s))"
read_value_typeable_justifed: fixes f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'md \<Rightarrow> 'addr val list \<Rightarrow> 'x"   and C :: "String.literal"   and M :: "String.literal"   and vs :: "'addr val list"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('addr, 'thread_id) justifying_execution"   and status :: "status"   and a :: "nat"   and ad :: "'addr"   and al :: "addr_loc"   and v :: "'addr val" assumes "ts_ok wfx (thr (h.start_state f P C M vs)) h.start_heap"   and "wf_syscls P"   and "P \<turnstile> (E, ws) weakly_justified_by J"   and "range (justifying_exec \<circ> J) \<subseteq> h.\<E>_start f P C M vs status"   and "E \<in> h.\<E>_start f P C M vs status"   and "P \<turnstile> (E, ws) \<surd>"   and "enat a < llength E"   and "action_obs E a = NormalAction (ReadMem ad al v)" shows "\<exists>T. P \<turnstile> ad@al : T \<and> P \<turnstile> v :\<le> T"
Associativity: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and \<Psi>'' :: "'d" assumes "assertion SCompose SImp SBottom SChanEq" shows "assertionAux.AssertionStatEq SImp (SCompose (SCompose \<Psi> \<Psi>') \<Psi>'') (SCompose \<Psi> (SCompose \<Psi>' \<Psi>''))"
fresh_list_elem: fixes a :: "atom"   and \<Gamma> :: "'a list"   and e :: "'a" assumes "a \<sharp> \<Gamma>"   and "e \<in> set \<Gamma>" shows "a \<sharp> e"
map_of_update_with_aux': fixes v :: "'c"   and k :: "'d"   and f :: "'c \<Rightarrow> 'c"   and ps :: "('d \<times> 'c) list"   and k' :: "'d" shows "map_of (update_with_aux v k f ps) k' = (map_of ps(k \<mapsto> case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v)) k'"
smult_monom: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "Polynomial.smult a (Polynomial.monom b n) = Polynomial.monom (a * b) n"
prems_1: fixes i :: "nat" assumes "i < n"   and "i < length (x # xs)" shows "P ((x # xs) ! i)"
simps_36: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Stmt.If x51 x52 x53 \<noteq> x31@[x32]"
qtables: shows "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) \<phi>s rels"
nolb_And: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm"   and xs :: "'a list"   and l :: "'a"   and x :: "'a" shows "nolb (And \<phi>\<^sub>1 \<phi>\<^sub>2) xs l x = (nolb \<phi>\<^sub>1 xs l x \<and> nolb \<phi>\<^sub>2 xs l x)"
fin: fixes A :: "'a vec set" assumes "A.Ker.fin_dim"   and "A.lin_indpt A"   and "A \<subseteq> mat_kernel A" shows "finite A"
SUP_cong: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e \<Rightarrow> 'f"   and D :: "'e \<Rightarrow> 'f"   and Sup :: "'f set \<Rightarrow> 'f" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
is_functor: shows "functor (\<cdot>\<^sub>J) (\<cdot>) \<chi>'.A.map"
simps_6: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and k :: "nat"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Power f k) env = map_option (\<lambda>t. tm_pow prec ord I a t k) (approx_tm prec ord I a f env)"
functional_converse_relation: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "left_unique R"   and "right_total R" shows "\<forall>y. \<exists>!x. R x y"
cases: fixes x :: "acom" obtains C\<^sub>1 :: "acom" and C\<^sub>2 :: "acom" where "x = C\<^sub>1__;; C\<^sub>2__"   | b :: "bexp" and C\<^sub>1 :: "acom" and C\<^sub>2 :: "acom" where "x = IF b__ THEN C\<^sub>1 ELSE C\<^sub>2"   | uu_ :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set" and uv_ :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set" and uw_ :: "((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set" and C :: "acom" where "x = {uu_/uv_/uw_} CONSEQ Ca__"   | I :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set" and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int" and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat" and Es :: "char list set \<times> (char list \<Rightarrow> char list set)" and b :: "bexp" and C :: "acom" where "x = {(I, S, E, Es)} WHILE b DO C"   | "x = SKIP"   | v :: "char list" and va :: "aexp" where "x = v ::= va"
fref_mono: fixes P' :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and R' :: "('b \<times> 'a) set"   and R :: "('b \<times> 'a) set"   and S :: "('c \<times> 'd) set"   and S' :: "('c \<times> 'd) set" assumes "\<And>x. P' x \<Longrightarrow> P x"   and "R' \<subseteq> R"   and "S \<subseteq> S'" shows "[P]\<^sub>f R \<rightarrow> S \<subseteq> [P']\<^sub>f R' \<rightarrow> S'"
iIN_sub: fixes k :: "nat"   and n :: "nat"   and d :: "nat" shows "k \<ominus> [n\<dots>,d] = (if n + d \<le> k then [k - (n + d)\<dots>,d] else if n \<le> k then [\<dots>k - n] else {})"
proj_def: fixes r :: "('d \<times> 'c) set"   and x :: "'d" shows "Equiv_Relations.proj r x = r `` {x}"
open_subset_interior: fixes S :: "'b set"   and T :: "'b set" assumes "open S" shows "(S \<subseteq> interior T) = (S \<subseteq> T)"
eq1: shows "fls_of_fpxs f = fls_shift (- fls_subdegree (fls_of_fpxs f)) (fps_to_fls f')"
ss_hdw_nodes: shows "set (ss ! hd w) \<subseteq> set nodes"
bool_taut_test_def: shows "bool_taut_test \<equiv> taut_test Boolean_Expression_Checkers.ifex_of"
xy: fixes y :: "'a"   and x :: "'a" assumes "y \<in> space_as_set M" shows "is_orthogonal (x - P *\<^sub>V x) y"
simps_1: fixes valuation :: "Literal list" shows "formulaFalse [] valuation = False"
add_indep_normal: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> real"   and \<sigma> :: "real"   and \<tau> :: "real"   and \<mu> :: "real"   and \<nu> :: "real" assumes "indep_var borel X borel Y"   and "0 < \<sigma>"   and "0 < \<tau>"   and "distributed M lborel X (\<lambda>x. ennreal (normal_density \<mu> \<sigma> x))"   and "distributed M lborel Y (\<lambda>x. ennreal (normal_density \<nu> \<tau> x))" shows "distributed M lborel (\<lambda>x. X x + Y x) (\<lambda>x. ennreal (normal_density (\<mu> + \<nu>) (sqrt (\<sigma>\<^sup>2 + \<tau>\<^sup>2)) x))"
prod_list_dvd: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "x dvd prod_list xs"
Dom_disj: fixes P :: "'b hrel"   and Q :: "'b hrel" shows "Dom (P \<or> Q) = (Dom P \<or> Dom Q)"
inj_on_fst_map_to_set: fixes m :: "'f \<Rightarrow> 'g option" shows "inj_on fst (map_to_set m)"
prems_2: fixes i :: "nat"   and j :: "nat" assumes "i \<le> Suc n"   and "j \<le> Suc n"   and "i \<noteq> j" shows "I i <+> I j = carrier R"
formulaEntailsFormulaIffEntailsAllItsClauses: fixes formula :: "Literal list list"   and formula' :: "Literal list list" shows "formulaEntailsFormula formula formula' = (\<forall>clause. clause el formula' \<longrightarrow> formulaEntailsClause formula clause)"
suminf_cong: fixes f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c" assumes "\<And>n. f n = g n" shows "suminf f = suminf g"
add_0_right: fixes a :: "'c" shows "a \<sqinter> top = a"
mult_delta_left: fixes b :: "bool"   and x :: "'d"   and y :: "'d" shows "(if b then x else (0::'d)) * y = (if b then x * y else (0::'d))"
wts_wts': fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and es :: "expr list"   and Ts :: "ty list" assumes "P,E,h \<turnstile> es [:] Ts" shows "P,E,h \<turnstile> es [:'] Ts"
invimage_vsingleton_in: fixes b :: "V"   and A :: "V"   and a :: "V" assumes "b \<in>\<^sub>\<circ> A" shows "set {\<langle>a, b\<rangle>} -`\<^sub>\<circ> A = set {a}"
Point: shows "p\<^sub>1 = Point x\<^sub>1 y\<^sub>1"
monad_altc_axioms: shows "monad_altc return bind altc"
a_rcos_sum: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme"   and a :: "'a"   and b :: "'a" assumes "ideal I R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "I +>\<^bsub>R\<^esub> a <+>\<^bsub>R\<^esub> (I +>\<^bsub>R\<^esub> b) = I +>\<^bsub>R\<^esub> a \<oplus>\<^bsub>R\<^esub> b"
max_ipv6_addr_max_word: shows "max_ipv6_addr = - 1"
f: shows "integrable (Pi\<^sub>M (I \<union> J) M) f"
finite_borel_measure_def: fixes M :: "real measure" shows "finite_borel_measure M \<equiv> finite_measure M \<and> finite_borel_measure_axioms M"
card_of_Func_mono: fixes A1 :: "'a set"   and A2 :: "'a set"   and B :: "'b set" assumes "A1 \<subseteq> A2"   and "B \<noteq> {}" shows "|Func A1 B| \<le>o |Func A2 B|"
lm031: fixes N :: "'c set"   and G :: "'d set" shows "{a. Domain a \<subseteq> N \<and> Range a \<in> all_partitions G} = Domain -` Pow N \<inter> Range -` all_partitions G"
not_numeral_less_zero: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (numeral.numeral one plus n) zero"
lt_punit_def: fixes cmp_term :: "'a nat_term_order" shows "lt_punit cmp_term \<equiv> ordered_term.lt fst (le_of_nat_term_order cmp_term)"
subset_closed_segment: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "({a--b} \<subseteq> {c--d}) = (a \<in> {c--d} \<and> b \<in> {c--d})"
ivl_disj_un_singleton_1: fixes l :: "'b" shows "{l} \<union> {l<..} = {l..}"
openin_open_eq: fixes s :: "'a set"   and t :: "'a set" assumes "open s" shows "openin (top_of_set s) t = (open t \<and> t \<subseteq> s)"
produces_imp_atms_leq: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset"   and A :: "'b"   and B :: "'b" assumes "production N C = {A}"   and "B \<in> atms_of C" shows "B \<le> A"
assms_1: fixes i :: "nat" shows "0 \<le> f i"
valid_W_inv_mo_co_lock: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state"   and p :: "'d process_name" assumes "valid_W_inv s"   and "sys_mem_lock s = None" shows "valid_W_inv (s(sys := s sys\<lparr>mem_lock := Some p\<rparr>))"
trivial_group_alt: fixes G :: "('b, 'c) monoid_scheme" assumes "Group.group G" shows "trivial_group G = (\<exists>a. carrier G \<subseteq> {a})"
php_def: fixes mr :: "'i simple_match" shows "php mr = (proto mr \<in> Proto ` {TCP, UDP, L4_Protocol.SCTP})"
fplus_fzero_right: fixes m :: "'c\<cdot>'d" shows "fplus\<cdot>m\<cdot>fzero = m"
simps_1: fixes a :: "'a" shows "\<oo> (Atom a) = False"
openI: fixes m :: "'a set" assumes "m \<in> T" shows "m open\<^bsub>T\<^esub>"
enabledI: fixes A :: "state \<times> state \<Rightarrow> bool" shows "\<turnstile> A \<longrightarrow> $Enabled A"
g_bigtheta: shows "g \<in> \<Theta>(\<lambda>x. real x powr p * ln (real x) powr p')"
infT_simp: fixes r :: "'b llist"   and A :: "'b set" assumes "r \<in> A\<^sup>\<omega>" shows "r \<in> UNIV\<^sup>\<omega>"
l0f2f1e: shows "f1 / (f2 - f1 * e) \<le> real l0"
intros_8: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs\<^sub>1 :: "'c rule list"   and m' :: "'c match_expr"   and rs\<^sub>2 :: "'c rule list"   and rrs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some (rs\<^sub>1 @ Rule m' Return # rs\<^sub>2)"   and "matches \<gamma> m' p"   and "\<Gamma>,\<gamma>,p\<turnstile> rs\<^sub>1 \<Rightarrow>\<^sub>z Undecided"   and "\<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m (Call chain) # rrs \<Rightarrow>\<^sub>z t"
calc_spread_spec: fixes k :: "'a"   and ps :: "(real, 'a) vec list" shows "calc_spread k ps = spread k (set ps)"
respects_cut': fixes A :: "('v \<times> 'v) set" shows "A \<subseteq> S A \<times> S A"
less_numeral_extra_1: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero one"
intros_10: fixes F :: "'a formula"   and \<Gamma> :: "'a formula multiset"   and G :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "F, \<Gamma> \<Rightarrow> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta>"
continuous_on_arcosh: fixes A :: "real set" assumes "A \<subseteq> {1..}" shows "continuous_on A arcosh"
assms_2: assumes "QD('u\<^sub>1) = QD('u\<^sub>2)" shows "\<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q"
convex_on_inverse: fixes A :: "real set" assumes "A \<subseteq> {0<..}" shows "convex_on A inverse"
s3: shows "s3 = init_lvars G invDeclC \<lparr>name = mn, parTs = pTs'\<rparr> (invmode statM e) a vs s2"
Keys_def: fixes F :: "('e \<Rightarrow>\<^sub>0 'f) set" shows "Keys F = \<Union> (keys ` F)"
exhaust_sel: fixes prot_atom :: "'a prot_atom" obtains "prot_atom = prot_atom.Atom (prot_atom.un_Atom prot_atom)"   | "prot_atom = Value"   | "prot_atom = SetType"   | "prot_atom = AttackType"   | "prot_atom = Bottom"   | "prot_atom = OccursSecType"
seqE: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, s\<rangle> \<Rightarrow> t" obtains ti :: "state" where "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> ti" and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, ti\<rangle> \<Rightarrow> t"
hyps_3: shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> Normal s' \<and> \<not> AlternativeSmallStep.inf \<Gamma> [c2] [] (Normal s')"
assms: shows "x \<in>\<^sub>\<circ> vpairs r"
ys: shows "({x} \<union> set xs) \<inter> set ys = {}"
bind_return': fixes M :: "'b measure" shows "M \<bind> return M = M"
bi_total_MF_Rel: shows "bi_total MF_Rel"
initState_Friend: fixes i :: "nat" shows "initState (Friend i) = {Key (shrK (Friend i))}"
strict_sorted_list_of_set: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less (linorder.sorted_list_of_set less_eq A)"
finsum_cong2: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c" assumes "A = B"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i"   and "f \<in> B \<rightarrow> carrier M" shows "finsum M f A = finsum M g B"
linear_independent_injective_image: fixes f :: "'d \<Rightarrow> 'e"   and s :: "'d set" assumes "clinear f"   and "cindependent s"   and "inj_on f (cspan s)" shows "cindependent (f ` s)"
a: shows "x \<cdot> d z \<le> d z \<cdot> y"
ring_hom_one: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
simps_1: fixes ok :: "nat"   and x :: "'b"   and f :: "'b list"   and f' :: "'b list"   and y :: "'b"   and r :: "'b list"   and r' :: "'b list" shows "exec (Rev ok (x # f) f' (y # r) r') = Rev (ok + 1) f (x # f') r (y # r')"
simps_3: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and l :: "floatarith"   and r :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Add l r) env = (case (approx_tm prec ord I a l env, approx_tm prec ord I a r env) of (None, b) \<Rightarrow> None | (Some t1, None) \<Rightarrow> None | (Some t1, Some t2) \<Rightarrow> Some (tm_add t1 t2))"
invert_opposite_sides_of_square: fixes f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "\<phi>.B.seq f g"   and "f \<cdot>\<^sub>B g = h \<cdot>\<^sub>B k"   and "\<phi>.B.iso f"   and "\<phi>.B.iso k" shows "\<phi>.B.seq g (\<phi>.B.inv k) \<and> \<phi>.B.seq (\<phi>.B.inv f) h \<and> g \<cdot>\<^sub>B \<phi>.B.inv k = \<phi>.B.inv f \<cdot>\<^sub>B h"
simps_1: fixes x :: "'a" shows "pp_last (Node x) = x"
Solve_exec_ns'_def: fixes preprocess :: "('a \<times> 'b ns_constraint) list \<Rightarrow> (nat \<times> linear_poly) list \<times> ('a \<times> 'b atom) list \<times> ((nat, 'b) mapping \<Rightarrow> (nat, 'b) mapping) \<times> 'a list"   and assert_all :: "(nat \<times> linear_poly) list \<Rightarrow> ('a \<times> 'b atom) list \<Rightarrow> 'a list + (nat, 'b) mapping" shows "Solve_exec_ns' preprocess assert_all \<equiv> Preprocess preprocess \<and> AssertAll assert_all"
rtrancl_induct: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and P :: "'d \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P a"   and "\<And>y z. \<lbrakk>(a, y) \<in> r\<^sup>*; (y, z) \<in> r; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
map_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "vertical_composite A B F G H \<sigma> \<tau>" shows "vertical_composite.map A B \<sigma> \<tau> f = (if partial_magma.arr A f then B (\<tau> (partial_magma.cod A f)) (\<sigma> f) else partial_magma.null B)"
fundfoldpairs_induced_autormorph_bij: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "(f, g) \<in> fundfoldpairs" shows "bij (induced_automorph f g)"
post_del_adjL: fixes l :: "('c \<times> nat) tree"   and a :: "'c"   and lv :: "nat"   and r :: "('c \<times> nat) tree"   and l' :: "('c \<times> nat) tree"   and b :: "'c" assumes "invar \<langle>l, (a, lv), r\<rangle>"   and "pre_adjust \<langle>l', (b, lv), r\<rangle>" shows "post_del \<langle>l, (a, lv), r\<rangle> (adjust \<langle>l', (b, lv), r\<rangle>)"
u3_4_1: shows "u \<le>\<^sub>v ys"
5: shows "minarc (v * - v\<^sup>T \<sqinter> g) \<le> v * - v\<^sup>T \<sqinter> - - g"
simps_4: fixes v :: "nat \<Rightarrow> 'a"   and l :: "linear_poly"   and r :: "rat" shows "v \<Turnstile>\<^sub>c GEQ l r = (r *R (1::'a) \<le> l \<lbrace> v \<rbrace>)"
ar_c_top: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "'c"   and b :: "'c" assumes "Ring R"   and "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "c_top (b_ag R) I (a \<uplus>\<^bsub>R\<^esub> I) (b \<uplus>\<^bsub>R\<^esub> I) = (a \<plusminus>\<^bsub>R\<^esub> b) \<uplus>\<^bsub>R\<^esub> I"
simps: fixes k :: "nat"   and xs :: "'b list" shows "fast_select k xs = (if length xs \<le> 20 then sort xs ! k else let x = fast_select (((length xs + 4) div 5 - 1) div 2) (map median_le_5 (chop 5 xs)); (ls, es, gs) = threeway_partition x xs in if k < length ls then fast_select k ls else if k < length ls + length es then x else fast_select (k - length ls - length es) gs)"
seq_char': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b local.term set"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.comp C g f) = (free_monoidal_category.ARR C f \<and> free_monoidal_category.ARR C g \<and> monoidal_language.Dom C (free_monoidal_category.rep C g) = monoidal_language.Cod C (free_monoidal_category.rep C f))"
lhs_def: shows "lhs \<equiv> \<lambda>plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1. vector_space_pair_ow (Collect (Domainp B\<^sub>1)) plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 (Collect (Domainp B\<^sub>2))"
Arg_inverse: fixes z :: "complex" shows "Arg (inverse z) = (if z \<in> \<real> then Arg z else - Arg z)"
row_space_eq_col_space_transpose: fixes A :: "'a mat" shows "row_space A = col_space A\<^sup>T"
d: fixes p :: "real \<Rightarrow> complex" assumes "path p"   and "pathfinish p = pathstart p"   and "\<forall>t\<in>{0..1}. cmod (p t - g t) < d" shows "homotopic_loops (- {z}) g p"
character_double_dual_iso: fixes x :: "'a \<Rightarrow> complex" assumes "x \<in> carrier (Characters G)" shows "character (Characters (Characters G)) (G'.double_dual_iso x)"