hcju/isabelleps huggingface dataset:
r_invk: fixes CT :: "nat \<Rightarrow> classDef option"   and m :: "nat"   and C :: "nat"   and xs :: "nat list"   and e0 :: "exp"   and ds :: "exp list"   and es :: "exp list"   and e0' :: "exp" assumes "mbody(CT,m,C) = xs . e0"   and "substs [xs [\<mapsto>] ds, this \<mapsto> New C es] e0 = e0'" shows "CT \<turnstile> MethodInvk (New C es) m ds \<rightarrow> e0'"
assms_2: shows "a \<notin> {uv, rev_G uv}"
of_nat_mono: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and i :: "nat"   and j :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "i \<le> j" shows "less_eq (semiring_1.of_nat one plus zero i) (semiring_1.of_nat one plus zero j)"
takeWhile_dropWhile_real_prefix: fixes x :: "'b"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> set xs"   and "\<not> P x" shows "\<exists>y ys. xs = takeWhile P xs @ y # ys \<and> \<not> P y \<and> dropWhile P xs = y # ys"
first_countable_basis_Int_stableE: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.first_countable_topology open" obtains \<A> :: "'b set set" where "countable \<A>" and "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" and "\<And>A. A \<in> \<A> \<Longrightarrow> open A" and "\<And>S. \<lbrakk>open S; x \<in> S\<rbrakk> \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S" and "\<And>A B. \<lbrakk>A \<in> \<A>; B \<in> \<A>\<rbrakk> \<Longrightarrow> A \<inter> B \<in> \<A>"
borel_measurable_simple_function: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "simple_function M f" shows "f \<in> borel_measurable M"
q_Succ_def: fixes x :: "hf" shows "q_Succ x \<equiv> q_Eats x x"
Max_mono: fixes M :: "'a word set"   and N :: "'a word set" assumes "M \<subseteq> N"   and "M \<noteq> {}"   and "finite N" shows "signed.Max M \<le>s signed.Max N"
ik\<^sub>s\<^sub>s\<^sub>t_def: fixes A :: "('a, 'b) stateful_strand_step list" shows "ik\<^sub>s\<^sub>s\<^sub>t A \<equiv> {t. receive\<langle>t\<rangle> \<in> set A}"
reachable_def: shows "reachable \<equiv> \<lambda>g \<phi>. {x. reachablep g \<phi> x}"
step_r_complete_aux: fixes \<R> :: "('d \<Rightarrow> real) set set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and A :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)"   and u :: "'d \<Rightarrow> real"   and R :: "('d \<Rightarrow> real) set"   and l :: "'e"   and g :: "('d, real) cconstraint"   and a :: "'f"   and r :: "'d list"   and l' :: "'e" assumes "\<R> = {region X I r |I r. valid_region X k I r}"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "u \<in> R"   and "R \<in> \<R>"   and "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'"   and "u \<turnstile> g"   and "[r\<rightarrow>0]u \<turnstile> inv_of A l'" shows "R = R \<inter> {u. u \<turnstile> g} \<and> region_set' (R \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> inv_of A l'} = region_set' R r 0 \<and> region_set' (R \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> inv_of A l'} \<in> \<R>"
Gp_carrier: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H " shows "carrier (\<natural>\<^bsub>G\<^esub>H) = H"
list_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and A :: "'a set" shows "(map f xs \<in> list (length xs) A) = (f ` set xs \<subseteq> A)"
calcChainAtomRev: fixes xvec :: "name list"   and yvec :: "name list"   and x :: "name" assumes "length xvec = length yvec"   and "x \<sharp> xvec"   and "x \<sharp> yvec" shows "[xvec yvec]\<^sup>- \<bullet>\<^sub>v x = x"
smult_mult2: fixes a :: "'c"   and A :: "('c, 'd) sq_matrix"   and B :: "('c, 'd) sq_matrix" shows "a *\<^sub>S (A * B) = A * a *\<^sub>S B"
cos_converges: fixes x :: "'a" shows "(\<lambda>n. cos_coeff n *\<^sub>R x ^ n) sums cos x"
Call_\<tau>red1r_obj: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and M :: "String.literal"   and ps :: "'addr expr1 list" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (e\<bullet>M(ps), xs) (e'\<bullet>M(ps), xs')"
star_of_0_less: fixes y :: "'a" shows "(0 < star_of y) = ((0::'a) < y)"
preserves_cod: fixes f :: "'a" assumes "arr f" shows "cod (CCC.prod f b) = CCC.prod (cod f) b"
cos_minus_converges: fixes x :: "'a" shows "(\<lambda>n. cos_coeff n *\<^sub>R (- x) ^ n) sums cos x"
rep_eq: fixes x :: "'d" shows "\<lbrakk>\<guillemotleft>x\<guillemotright>\<rbrakk>\<^sub>e = (\<lambda>b. x)"
f1: shows "\<forall>m ma. m \<rightarrow>\<^sub>M ma = {f \<in> space m \<rightarrow> space ma. \<forall>B. B \<in> sets ma \<longrightarrow> f -` B \<inter> space m \<in> sets m}"
SUP_mult_right_ennreal: fixes f :: "'a \<Rightarrow> ennreal"   and I :: "'a set"   and c :: "ennreal" shows "Sup (f ` I) * c = (SUP i\<in>I. f i * c)"
completely_multiplicative_function_axioms: shows "completely_multiplicative_function \<chi>"
ring_distribs_1: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a \<cdot> (b + c) = a \<cdot> b + a \<cdot> c"
distinct_1: fixes x1 :: "int"   and x2 :: "int" shows "Const' x1 \<noteq> Intv' x2"
SpecUnfolded: shows "new_configured_SecurityInvariant (sinvar_spec, nm_default m, nm_receiver_violation m, nP) = Some \<lparr>c_sinvar = \<lambda>G. sinvar_spec G nP, c_offending_flows = \<lambda>G. SecurityInvariant_withOffendingFlows.set_offending_flows sinvar_spec G nP, c_isIFS = nm_receiver_violation m\<rparr>"
sf_aux3: fixes z :: "nat"   and m :: "nat" assumes "sf z + m < sf (z + 1)" shows "m \<le> z"
div: assumes "n div 2 < n"   and "0 < n" shows "R"
that: fixes g :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "\<And>n. continuous_on UNIV (g n)"   and "AE x in lebesgue_on S. (\<lambda>n. g n x) \<longlonglongrightarrow> h x" shows "thesis"
red1_hext_incr: fixes uf :: "bool"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "hp s \<unlhd> hp s'"
MP: fixes H :: "'b pl set"   and p :: "'b pl"   and q :: "'b pl" assumes "H |- p -> q"   and "H |- p" shows "H |- q"
ltrm_deg_0: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "deg R p = 0" shows "monom (UP R) (p (deg R p)) (deg R p) = p"
is_top_sorted_distinct: fixes R :: "('a \<times> 'a) set"   and l :: "'a list" assumes "is_top_sorted R l" shows "distinct l"
word_clz_max: shows "word_clz w \<le> size w"
maxim_equality: fixes A :: "'b set"   and x :: "'b" assumes "s.isMaxim A x" shows "s.maxim A = x"
cakeml_static_env_axioms: shows "cakeml_static_env static_cenv"
right_uniqueI: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>A x y; A x z\<rbrakk> \<Longrightarrow> y = z" shows "right_unique A"
1: assumes "is_left_adjoint f" shows "ide f \<and> C.iso (Leg0 (Dom f))"
Expr: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e :: "expr"   and Q :: "'b assn" assumes "G,A\<turnstile>{Normal P} e-\<succ> {Q\<leftarrow>\<diamondsuit>}" shows "G,A\<turnstile>{Normal P} .Expr e. {Q}"
simps_3: fixes uv :: "'a list" shows "local.newton_coefficients_main [] uv = []"
suminf_exist_split: fixes r :: "real"   and f :: "nat \<Rightarrow> 'b" assumes "0 < r"   and "summable f" shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
swapEnvDom_preserves_good: fixes rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option"   and xs :: "'d"   and x :: "'e"   and y :: "'e" assumes "goodEnv rho" shows "goodEnv (swapEnvDom xs x y rho)"
modaliseMultiset_def: fixes a :: "'b"   and \<Gamma> :: "('a, 'b) form multiset" shows "a \<cdot> \<Gamma> \<equiv> (\<lambda>p. Modal a [p]) \<cdot>\<cdot> \<Gamma>"
cases: fixes a :: "'var"   and S :: "'node SDG_node set"   and n :: "'node SDG_node" assumes "a \<in> rv S n" obtains as :: "'edge list" and n' :: "'node SDG_node" and V :: "'var" where "a = V" and "parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n'" and "n' \<in> HRB_slice S" and "V \<in> Use\<^bsub>SDG\<^esub> n'" and "\<forall>n''. valid_SDG_node n'' \<and> parent_node n'' \<in> set (sourcenodes as) \<longrightarrow> V \<notin> Def\<^bsub>SDG\<^esub> n''"
HeronConf_interp_stepwise_strictly_precedes_cases: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
monoid_axioms: shows "monoid transformation.Sym (compose G) (identity G)"
le_of_comp_set_less_eq: fixes le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "le_of_comp (comp_of_ords (ord.set_less_eq le) (ord.set_less le)) = ord.set_less_eq le"
drop_Nil: fixes n :: "nat" shows "drop n \<epsilon> = \<epsilon>"
fresh_Some: fixes a :: "atom"   and x :: "'a" shows "a \<sharp> Some x = a \<sharp> x"
fr_refl: fixes A :: "assn"   and B :: "assn"   and C :: "assn" assumes "A \<Longrightarrow>\<^sub>A B" shows "A * C \<Longrightarrow>\<^sub>A B * C"
i_take_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Down> n \<equiv> map f [0..<n]"
hassoc_update_def: fixes k :: "'a"   and v :: "'b"   and ht :: "('a \<times> 'b) list" shows "hassoc_update k v ht = return ((k, v) # ht)"
implode_explode: fixes x :: "ZF" shows "implode (explode x) = x"
rel_SIod: fixes D :: "'c Order"   and E :: "('c, 'd) Order_scheme" assumes "Order D"   and "Order E"   and "carrier E \<subseteq> carrier D"   and "\<forall>a\<in>carrier E. \<forall>b\<in>carrier E. a \<preceq>\<^bsub>E\<^esub> b = a \<preceq>\<^bsub>D\<^esub> b" shows "rel E = rel (SIod D (carrier E))"
vfsequence_vcard_vcons: fixes xs :: "V"   and x :: "V" assumes "vfsequence xs" shows "vcard (xs #\<^sub>\<circ> x) = ZFC_in_HOL.succ (vcard xs)"
lsl_rpc_galois: fixes x :: "'b"   and z :: "'b"   and y :: "'b" shows "(x \<sqinter> \<nu> z \<le> y) = (z \<le> rpc x y)"
select_convs_1: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "inst \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = inst"
ordinal_0_plus: fixes x :: "ordinal" shows "0 + x = x"
reduced_run_invar_1: fixes q :: "'state"   and v\<^sub>1 :: "'action list"   and v\<^sub>2 :: "'action stream"   and l :: "'action list"   and w :: "'action list"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and u :: "'action stream" assumes "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "v\<^sub>1 @ l =\<^sub>F w\<^sub>1"
alphaInp_imp_asInp_equal: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option"   and qinp' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" assumes "qGoodInp qinp"   and "(qinp, qinp') \<in> alphaInp" shows "asInp qinp = asInp qinp'"
enum_less: fixes P :: "'b \<Rightarrow> bool"   and i :: "nat"   and j :: "nat" assumes "infinitely_many1 P"   and "i < j" shows "infinitely_many1.enum P i < infinitely_many1.enum P j"
finsum_empty: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "comm_ring R addition multiplication zero unit" shows "monoid.finprod R addition zero {} f = zero"
norm_cauchy_schwarz_eq: fixes x :: "'b"   and y :: "'b" shows "(x \<bullet> y = norm x * norm y) = (norm x *\<^sub>R y = norm y *\<^sub>R x)"
simps_1: fixes p :: "nat"   and x :: "real" shows "power_down p x 0 = 1"
equivP_pick_preserves: fixes P :: "'a \<Rightarrow> bool"   and \<phi> :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a set" assumes "equivP P \<phi>"   and "(P /// \<phi>) X" shows "P (pick X)"
l8_3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "A \<noteq> B"   and "Tarski_neutral_dimensionless.Col Bet B A A'" shows "Tarski_neutral_dimensionless.Per Bet Cong A' B C"
SemPut: fixes s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and x :: "Var"   and l :: "Location"   and C :: "Class"   and Flds :: "(Field \<times> Val) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and F :: "Field"   and e :: "Expr" assumes "fst s x = RVal (Loc l)"   and "lookup (snd s) l = Some (C, Flds)"   and "t = (fst s, (l, C, (F, evalE e (fst s)) # Flds) # snd s)" shows " s , Put x F e \<rightarrow>\<^sub>1  t "
assms_2: shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>)"
word_neq_0_conv: fixes w :: "'b word" shows "(w \<noteq> 0) = (0 < w)"
prems: shows "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)"
eventually_disj_iff: fixes P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(\<forall>\<^sub>F x in \<U>. P x \<or> Q x) = (eventually P \<U> \<or> eventually Q \<U>)"
points_index_alt_def: fixes ps :: "'a set" shows "\<B> index ps = size (filter_mset ((\<subseteq>) ps) \<B>)"
assms_2: shows "b_e = Unop_i t iop \<or> b_e = Unop_f t fop \<or> b_e = Testop t testop"
simps: fixes x :: "nat" shows "addr_of_ref (Ref x) = x"
is_weak_ranking_Cons: fixes x :: "'b set"   and xs :: "'b set list" shows "is_weak_ranking (x # xs) = (x \<noteq> {} \<and> is_weak_ranking xs \<and> x \<inter> \<Union> (set xs) = {})"
simps_3: fixes \<A> :: "'a \<Rightarrow> bool"   and F :: "'a formula" shows "\<A> \<Turnstile> \<^bold>\<not> F = (\<not> \<A> \<Turnstile> F)"
strong_induct: fixes x1 :: "(name \<times> ty) list"   and x2 :: "lam"   and x3 :: "ty"   and P :: "'a \<Rightarrow> (name \<times> ty) list \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<turnstile> x2 : x3"   and "\<And>\<Gamma> x T y. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P y \<Gamma> (Var x) T"   and "\<And>\<Gamma> t1 T1 T2 t2 y. \<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<And>z. P z \<Gamma> t1 (T1 \<rightarrow> T2); \<Gamma> \<turnstile> t2 : T1; \<And>z. P z \<Gamma> t2 T1\<rbrakk> \<Longrightarrow> P y \<Gamma> (App t1 t2) T2"   and "\<And>x \<Gamma> T1 t T2 y. \<lbrakk>x \<sharp> y; x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2; \<And>z. P z ((x, T1) # \<Gamma>) t T2\<rbrakk> \<Longrightarrow> P y \<Gamma> (Lam [x].t) (T1 \<rightarrow> T2)" shows "P z x1 x2 x3"
chain: fixes C :: "'b set" assumes "set_chain C"   and "\<forall>x\<in>C. P x" shows "P (Sup C)"
rw_rat_is_rewrite_ord: shows "is_rewrite_ord rw_rat"
prems: fixes c :: "'a" assumes "poly (pCons a p) (x + y) = poly (pCons a p) x + c * y" shows "thesis"
DESTRimpl_Some: fixes s :: "'a"   and ni :: "'b"   and i :: "'c ifex" assumes "I s"   and "(ni, i) \<in> R s" shows "ospec (DESTRimpl ni s) bf_True"
z_3: shows "f y - f x = (y - x) * f' z"
invertible_inverse_inverse: fixes u :: "'a" assumes "invertible u"   and "u \<in> M" shows "inverse (inverse u) = u"
cong_exp_mod_2: fixes c :: "'c"   and a :: "'c"   and n :: "'c"   and b :: "nat" shows "[c = (a mod n) ^ b] (mod n) = [c = a ^ b] (mod n)"
Fst: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<langle>X, Y\<rangle> \<in> parts H" shows "X \<in> parts H"
topological_basis_prod: fixes A :: "'c set set"   and B :: "'d set set" assumes "topological_basis A"   and "topological_basis B" shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
assms_2: assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "F \<alpha> P'"
assms: fixes i :: "nat" assumes "i < n" shows "u i \<in> space\<^sub>N N"
adapt: shows "P \<subseteq> {s. \<exists>Z. init s \<in> P' s Z \<and> (\<forall>t. t \<in> Q' s Z \<longrightarrow> return s t \<in> R s t) \<and> (\<forall>t. t \<in> A' s Z \<longrightarrow> return s t \<in> A)}"
conversion_conversion_idemp: fixes A :: "('b \<times> 'b) set" shows "(A\<^sup>\<leftrightarrow>\<^sup>*)\<^sup>\<leftrightarrow>\<^sup>* = A\<^sup>\<leftrightarrow>\<^sup>*"
cinner_scaleC_left: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and r :: "complex"   and x :: "'c"   and y :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (scaleC r x) y = cnj r * cinner x y"
convex_hull_list_mono: fixes Ws :: "'b vec list"   and n :: "nat"   and Vs :: "'b vec list" assumes "set Ws \<subseteq> carrier_vec n"   and "set Vs \<subseteq> set Ws" shows "gram_schmidt.convex_hull_list n Vs \<subseteq> gram_schmidt.convex_hull_list n Ws"
assms_2: fixes c :: "complex"   and x :: "'a" shows "f (c *\<^sub>C x) = cnj c *\<^sub>C f x"
kronecker_of_mult: fixes A :: "'b mat"   and C :: "'b mat"   and B :: "'b mat"   and D :: "'b mat" assumes "dim_col A = dim_row C"   and "dim_col B = dim_row D" shows "kronecker_product A B * kronecker_product C D = kronecker_product (A * C) (B * D)"
mapl_G_cong: fixes x :: "('p, 'q, 'r, 's, 't, 'u, 'v) G"   and l1 :: "'p \<Rightarrow> 'w"   and l1' :: "'p \<Rightarrow> 'w"   and l2 :: "'q \<Rightarrow> 'x"   and l2' :: "'q \<Rightarrow> 'x" assumes "\<And>z. z \<in> set1_G x \<Longrightarrow> l1 z = l1' z"   and "\<And>z. z \<in> set2_G x \<Longrightarrow> l2 z = l2' z" shows "mapl_G l1 l2 x = mapl_G l1' l2' x"
all_union_tsI1: fixes c :: "'a"   and A :: "'b \<Rightarrow> 'a set"   and k :: "'b"   and B :: "'a set" assumes "c \<in> A k" shows "c \<in> (A \<Rightarrow>\<union>\<^sub>\<forall> B) k"
Compl_Un: fixes A :: "'c set"   and B :: "'c set" shows "- (A \<union> B) = - A \<inter> - B"
simps_2: fixes ux :: "String.literal \<Rightarrow> 'a"   and uy :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and uz :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool" shows "semantics ux uy uz Falsity = False"
simps: fixes i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and Q :: "(('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Diagonal_to_Smith_row_i_PQ i bezout (P, A, Q) = diagonal_to_Smith_i_PQ [i + 1..<min (nrows A) (ncols A)] i bezout (P, A, Q)"
prems: fixes t :: "real" assumes "a < t"   and "t < b" shows "Q t = P t"
abelian_relative_homology_group: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set" shows "comm_group (relative_homology_group p X S)"
lub_upper: fixes S :: "'a set"   and x :: "'a" assumes "S \<subseteq> A"   and "x \<in> S" shows "(x, lub S cl) \<in> r"
IH: shows "(oid \<in> set (inserted_ids ops)) = (\<exists>ref val. (oid, Insert ref val) \<in> set ops)"
gkerQmp: fixes H :: "'a set"   and N :: "'a set" assumes "G \<triangleright> H"   and "G \<triangleright> N"   and "H \<subseteq> N" shows "gker\<^bsub>(G / H),(G / N)\<^esub> (Qm\<^bsub>G H,N\<^esub>) = carrier (\<natural>N / H)"
assms_3: shows "\<forall>(t, t')\<in>set D. fv t \<union> fv t' \<subseteq> X"
input2state_cons_shift: fixes i :: "value list"   and x1 :: "nat"   and a :: "value"   and b :: "value" assumes "input2state i $ x1 = Some a" shows "input2state (b # i) $ (Suc x1) = Some a"
usubstappt_antimon: fixes V :: "variable set"   and U :: "variable set"   and \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and \<theta> :: "trm" assumes "V \<subseteq> U"   and "usubstappt \<sigma> U \<theta> \<noteq> undeft" shows "usubstappt \<sigma> U \<theta> = usubstappt \<sigma> V \<theta>"
dim_gen_eigenspace: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and ev :: "'a"   and k :: "nat" assumes "jordan_nf A n_as" shows "local.dim_gen_eigenspace A ev k = sum_list (map (min k) (map fst (filter (\<lambda>(n, e). e = ev) n_as)))"
hom: shows "if atends then homotopic_paths S g h else homotopic_loops S g h"
ccINF_lower: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and i :: "'c"   and f :: "'c \<Rightarrow> 'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "i \<in> A" shows "less_eq (Inf (f ` A)) (f i)"
pair_lessI2: fixes a :: "nat"   and b :: "nat"   and s :: "nat"   and t :: "nat" assumes "a \<le> b"   and "s < t" shows "((a, s), b, t) \<in> pair_less"
assms_3: shows "distributed M P Y (\<lambda>x. ennreal (Py x))"
k_def: shows "k = (LEAST k. i < k \<and> snd (r k) \<noteq> n)"
subterm_of_a_subterm_is_a_subterm: fixes u :: "'b trm"   and q :: "indices list"   and v :: "'b trm"   and t :: "'b trm"   and p :: "indices list" assumes "subterm u q v"   and "subterm t p u" shows "subterm t (p @ q) v"
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)"
non_exec_assign_local: fixes \<sigma> :: "'a control_state_scheme"   and upd :: "('b list \<Rightarrow> 'b list) \<Rightarrow> 'a control_state_scheme \<Rightarrow> 'a control_state_scheme"   and rhs :: "'a control_state_scheme \<Rightarrow> 'b"   and M :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "\<not> exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> _bind_SE _ (assign_local upd rhs) M) = (upd (map_hd (\<lambda>_. rhs \<sigma>)) \<sigma> \<Turnstile> M)"
obs_singleton: fixes n :: "'a LDCFG_node"   and S :: "'a LDCFG_node set" shows "(\<exists>m. CFGExit_wf.obs n (CFGExit_wf.wod_backward_slice S) = {m}) \<or> CFGExit_wf.obs n (CFGExit_wf.wod_backward_slice S) = {}"
simps_1: fixes f :: "'b \<Rightarrow> 'b" shows "f ^^ 0 = id"
integral_cexp_minus_to_infinity: fixes a :: "complex"   and c :: "real" assumes "0 < Re a" shows "integral {c..} (\<lambda>x. exp (x *\<^sub>R - a)) = exp (c *\<^sub>R - a) / a"
f2: shows "(flow0 x \<circ> r) \<longlonglongrightarrow> flow0 p t"
assms_5: shows "(\<xi>, p) \<in> reachable A I"