hcju/isabelleps huggingface dataset:
norm_eq_zero: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(norm x = 0) = (x = zero)"
g: fixes x :: "'a" assumes "x \<in> carrier G" shows "h x = g (N #>\<^bsub>G\<^esub> x)"
simps_5: fixes P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" shows "guarded (P \<parallel> Q) = (guarded P \<and> guarded Q)"
ordering_top_axioms: shows "ordering_top (\<lambda>x y. prefix y x) (\<lambda>x y. strict_prefix y x) []"
get_tag_name_locs_def: shows "get_tag_name_locs = (\<lambda>element_ptr. {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name)})"
mayDiverge_transC: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "('a, 'c) com"   and s :: "'b" assumes "PL.mayDiverge tval aval c s" shows "\<exists>c' s'. PL.transC tval aval (c, s) (c', s') \<and> PL.mayDiverge tval aval c' s'"
inv_unique': fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "x \<otimes> y = \<one>"   and "y \<otimes> x = \<one>" shows "y = inv x"
AE_completion: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure" assumes "almost_everywhere M P" shows "almost_everywhere (completion M) P"
ps_chain_to_int_chain_simp: fixes V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget"   and E :: "(node fset \<times> command_gadget \<times> node fset) list"   and \<Pi> :: "(node \<rightharpoonup>\<^sub>f topbot, node + node fset \<times> command_gadget \<times> node fset) chain" shows "ps_chain_to_int_chain (Graph V \<Lambda> E) \<Pi> = chainmap (ps_to_int (Graph V \<Lambda> E)) (case_sum (Inl \<circ> \<Lambda>) (Inr \<circ> snd3)) \<Pi>"
Sup_upper: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "x \<le> \<Squnion> A"
irrefl: fixes a :: "'d DBMEntry" shows "\<not> a \<prec> a"
negligible_finite: fixes s :: "'a set" assumes "finite s" shows "negligible s"
sources_aux_append: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and U :: "'a set"   and xs :: "'b list"   and ys :: "'b list" shows "sources_aux I D U (xs @ ys) = sources_aux I D (sources_aux I D U ys) xs"
spair_sigs_spp_def: fixes p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and q :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "spair_sigs_spp p q = (let t1 = punit.lt (snd p); t2 = punit.lt (snd q); l = lcs t1 t2 in ((l - t1) \<oplus> fst p, (l - t2) \<oplus> fst q))"
open_ereal: fixes S :: "real set" assumes "open S" shows "open (ereal ` S)"
sp_arraycpy_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and y :: "char list"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "sp \<pi> P (x[] ::= y) t = (\<exists>vx. let s = t(x := vx) in t x = s y \<and> P s)"
While_le: fixes Inv :: "'b"   and b :: "bexp"   and c :: "'b acom"   and P :: "'b"   and w :: "'b acom" shows "({Inv} WHILE b DO c {P} \<le> w) = (\<exists>Inv' c' P'. w = {Inv'} WHILE b DO c' {P'} \<and> c \<le> c' \<and> Inv \<le> Inv' \<and> P \<le> P')"
divide_nonpos_nonneg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less_eq zero y" shows "less_eq (divide x y) zero"
flincomb_concat: fixes rss :: "'f list list"   and mss :: "'w list list" assumes "list_all2 (\<lambda>rs ms. length rs = length ms) rss mss" shows "concat rss \<bullet>\<sharp>\<star> concat mss = sum_list (map2 (\<bullet>\<sharp>\<star>) rss mss)"
zero_le_power: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a" shows "less_eq zero (power.power one times a n)"
null_eqI: fixes n :: "'c \<times> 'c \<times> 'c" assumes "\<And>f. C.VxVxV.comp n f = n \<and> C.VxVxV.comp f n = n" shows "n = C.VxVxV.null"
correct_def: shows "ped_commit.correct \<equiv> \<forall>m. valid_msg m \<longrightarrow> spmf (ped_commit.correct_game m) True = 1"
rplusinf_inf: fixes p :: "fm"   and bs :: "real list" assumes "isrlfm p" shows "\<exists>z. \<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p"
errMOD_igSubstIGAbsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igAbsIPresIGWls MOD"   and "igWlsDisj MOD"   and "igSubstIPresIGWls MOD"   and "igSubstIGAbs MOD" shows "igSubstIGAbsSTR (errMOD MOD)"
Diff_triv: fixes A :: "'a set"   and B :: "'a set" assumes "A \<inter> B = {}" shows "A - B = A"
evenE: fixes a :: "'a" assumes "even a" obtains b :: "'a" where "a = (2::'a) * b"
round_down_shift: fixes p :: "int"   and x :: "real"   and k :: "int" shows "round_down p (x * 2 powr real_of_int k) = 2 powr real_of_int k * round_down (p + k) x"
Transset_succ: fixes x :: "V" assumes "Transset x" shows "Transset (ZFC_in_HOL.succ x)"
simps_1: fixes a :: "'a" shows "a +^ 0 = (0::'a)"
eq_elimR: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "eq \<phi> \<psi> \<pi>l"   and "\<psi> \<pi>l" obtains "\<phi> \<pi>l"
HPhase1or2Write_HInv5: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2Write s s' p d"   and "HInv5_inner s q"   and "HInv4c s p"   and "HInv4a s p" shows "HInv5_inner s' q"
all_def: fixes p :: "'a \<Rightarrow> bool"   and as :: "'a iarray" shows "IArray.all p as = (\<forall>a\<in>set (IArray.list_of as). p a)"
rep_eq: fixes x :: "uint32"   and xa :: "uint32" shows "Rep_uint32 (x + xa) = Rep_uint32 x + Rep_uint32 xa"
le_utrans_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "le_utrans t u = (\<forall>P. unitary P \<longrightarrow> t P \<tturnstile> u P)"
minus_div_mult_eq_mod: fixes a :: "'a"   and b :: "'a" shows "a - a div b * b = a mod b"
atLeastLessThan_empty: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "class.order less_eq less"   and "less_eq b a" shows "ord.atLeastLessThan less_eq less a b = {}"
take_0: fixes xs :: "'a list" shows "take 0 xs = []"
antisym: fixes x :: "'a"   and y :: "'a" assumes "boa x y"   and "boa y x"   and "x \<in> A" shows "x = y"
sorted_wrt_rev: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "sorted_wrt P (rev xs) = sorted_wrt (\<lambda>x y. P y x) xs"
fmran'E: fixes y :: "'d"   and m :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "y \<in> fmran' m" obtains x :: "'e" where "fmlookup m x = Some y"
distr_pair_swap: shows "stock_measure t1 \<Otimes>\<^sub>M N' = distr (N' \<Otimes>\<^sub>M stock_measure t1) (stock_measure t1 \<Otimes>\<^sub>M N') (\<lambda>(x, y). (y, x))"
simps_1: fixes L :: "(real mpoly \<times> nat) list" shows "chophelper [] L = (L, [])"
o_def: fixes f :: "'ai \<Rightarrow> 'ah"   and g :: "'ag \<Rightarrow> 'ai" shows "f \<circ> g = (\<lambda>x. f (g x))"
enum_pos_injective: fixes x :: "'c"   and y :: "'c" assumes "enum_pos x = enum_pos y" shows "x = y"
max1: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> getRel idt G" shows "x = y"
count_mset_set_2: fixes A :: "'b set"   and x :: "'b" assumes "infinite A" shows "count (mset_set A) x = 0"
v0_lset_P: shows "P $ n \<in> lset P'"
cc_restr_eq_subset: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and G :: "CoCalls"   and G2 :: "CoCalls" assumes "S \<subseteq> S'"   and "G G|` S' = G2 G|` S'" shows "G G|` S = G2 G|` S"
VectorSpaceI: fixes V :: "'a set"   and smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" assumes "Group V"   and "RModule_axioms UNIV smult V" shows "VectorSpace smult V"
good_item_simps_48: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and xa :: "'var" assumes "good Y" shows "Var xsa xa #[Y / xa]_xsa = Y"
padr_padl_eq: fixes v :: "'b vec"   and n :: "nat"   and m :: "nat"   and u :: "'b vec" assumes "v \<in> carrier_vec n" shows "(v @\<^sub>v 0\<^sub>v m = 0\<^sub>v n @\<^sub>v u) = (v = 0\<^sub>v n \<and> u = 0\<^sub>v m)"
conjugate_exponent_def: fixes p :: "'b" shows "conjugate_exponent p = (1::'b) + (1::'b) / (p - (1::'b))"
distinct_takeWhile: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "distinct xs" shows "distinct (takeWhile P xs)"
f_Exec_Stream_Acc_LocalState_append: fixes k :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and ys :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun (xs @ ys) c = f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs c @ f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun ys (f_Exec_Comp trans_fun (xs \<odot> k) c)"
f: fixes S :: "'a set" assumes "S \<in> \<A>" shows "f S \<subseteq> S"
kripke_def: fixes M :: "('b, 'c, 'd) KripkeStructure" shows "kripke M \<equiv> \<forall>a. relations M a \<subseteq> worlds M \<times> worlds M"
mset_subset_eq_add_right: fixes B :: "'b multiset"   and A :: "'b multiset" shows "B \<subseteq># A + B"
absorb2: fixes a :: "'a"   and b :: "'a" assumes "a \<preceq> b" shows "ordered_powerprod_lin.max a b = b"
awalk_verts_conv: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list" shows "pre_digraph.awalk_verts G u p = (if p = [] then [u] else map (tail G) p @ [head G (last p)])"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (write_cpu w cr s1)) \<and> t2 = snd (fst (write_cpu w cr s2))"
thetaSAtm_def: fixes atm :: "'b" shows "thetaSAtm atm \<equiv> {(Atm atm, Atm atm), (Done, Done)}"
norm_less: fixes z :: "complex" assumes "cmod (\<xi> - z) = r" shows "cmod (f \<xi>) < cmod (f z)"
sub_num_simps_4: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (num.Bit0 k) num.One = numeral.numeral one plus (Num.BitM k)"
vsubset_vinsert_rightI: fixes A :: "V"   and B :: "V"   and b :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "A \<subseteq>\<^sub>\<circ> vinsert b B"
rel_mono: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R1a :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a" shows "rel_prod R1 R2 \<le> rel_prod R1a R2a"
assms_4: shows "G \<psi> \<in> \<G> \<inter> {\<chi>. w \<Turnstile> F \<chi>}"
vpairs_iff_pairs: fixes a :: "V"   and b :: "V"   and r :: "V" shows "(\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vpairs r) = ((a, b) \<in> pairs r)"
sint_greater_eq: fixes w :: "'b word" shows "- (2 ^ (LENGTH('b) - Suc 0)) \<le> sint w"
apex_def: shows "S.apex \<equiv> C.dom S.leg0"
matrix_inv_P: shows "matrix_change_of_basis Y Y' = matrix_inv P"
invert_side_of_triangle_2: fixes h :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.arr h"   and "VV.comp f g = h"   and "VV.iso g" shows "VV.seq h (VV.inv g) \<and> f = VV.comp h (VV.inv g)"
carrier_submanifold: shows "sub.carrier = codomain c \<inter> manifold_eucl.dest.carrier"
fls_lr_inverse_X_inv_power_2: fixes n :: "nat"   and y :: "'a" shows "fls_right_inverse (fls_X_inv ^ n) y = fls_shift (- int n) (fls_const y)"
list_to_poly_Nil: fixes cs :: "'e list" shows "list_to_poly [] cs = 0"
vec_first_last_append: fixes v :: "'a vec"   and n :: "nat"   and m :: "nat" assumes "v \<in> carrier_vec (n + m)" shows "vec_first v n @\<^sub>v vec_last v m = v"
C_eq_iD: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "separated p"   and "noDenyAll1 p"   and "wellformed_policy1_strong p" shows "C (list2FWpolicy (insertDenies p)) = C (list2FWpolicy p)"
lower_lift_monom: fixes f :: "nat \<Rightarrow> real"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "real"   and i :: "nat"   and j :: "nat" shows "insertion f (MPoly_Type.monom m a) = insertion f (lowerPoly i j (liftPoly i j (MPoly_Type.monom m a)))"
normal_mult_x: fixes f :: "real poly" assumes "normal_poly f" shows "normal_poly (f * [:0, 1:])"
comp2_1: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec2"   and "g \<in> PrimRec1"   and "h \<in> PrimRec1" shows "(\<lambda>x. f (g x) (h x)) \<in> PrimRec1"
porder_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "porder le \<equiv> (\<forall>x. le x x) \<and> (\<forall>x y z. le x y \<longrightarrow> le y z \<longrightarrow> le x z) \<and> (\<forall>x y. le x y \<longrightarrow> le y x \<longrightarrow> x = y)"
walk_array_append: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list"   and x :: "'c"   and b :: "'b" shows "walk_array f (Array (xs @ [x])) b = f (walk_array f (Array xs) b) x"
cases: fixes a :: "'a"   and r :: "('a \<times> 'a) set" assumes "a \<in> Wellfounded.acc r" obtains x :: "'a" where "a = x" and "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r"
seqseq_fold_reduce: fixes n :: "nat" shows "seqseq n = fold_reduce 0 n"
not_digit: fixes n :: "nat" assumes "\<not> is_digit n"   and "P (n div 10)"   and "P (n mod 10)" shows "P n"
Min_big_d'_le: fixes i :: "nat"   and y :: "nat list" assumes "i < length a"   and "big_d' a b y i \<noteq> []"   and "length y \<le> length b" shows "Min (set (big_d' a b y i)) \<le> Max (set b)"
csmall_nstep_Suc_nend: fixes o' :: "'state \<times> 'coll"   and P :: "'prog"   and \<sigma> :: "'state"   and n1 :: "nat" assumes "o' \<in> csmall_nstep P \<sigma> (Suc n1)" shows "\<sigma> \<notin> endset"
zero_smult: fixes m :: "'v" assumes "m \<in> V" shows "(0::'f) \<cdot> m = (0::'v)"
halve_stepsizes_spec_def: shows "halve_stepsizes_spec = SPEC (\<lambda>n. True)"
retract1_ab: fixes x :: "'Fab"   and y :: "'Fb" assumes "R (\<pi>_Fabb x) y" shows "\<exists>z. Eab x z \<and> y = \<pi>_Fabb z"
EF_lem1: fixes x :: "'b"   and f :: "'b set" assumes "x \<in> f \<or> x \<in> EX' (EF f)" shows "x \<in> EF f"
irreducible_primitive_connect: fixes f :: "'b poly" assumes "primitive f" shows "irreducible\<^sub>d f = irreducible f"
le_cases3: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f"   and z :: "'f" assumes "class.linorder less_eq less" obtains "less_eq x y" and "less_eq y z"   | "less_eq y x" and "less_eq x z"   | "less_eq x z" and "less_eq z y"   | "less_eq z y" and "less_eq y x"   | "less_eq y z" and "less_eq z x"   | "less_eq z x" and "less_eq x y"
int_cases: fixes z :: "int" obtains n :: "nat" where "z = int n__"   | n :: "nat" where "z = - int (Suc n)"
round_nil: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and q :: "nat"   and r :: "nat"   and t :: "nat \<times> nat list \<times> 'c list" shows "(fst (snd (round index key p q r t)) \<noteq> []) = (\<exists>n\<in>set (fst (snd t)). 0 < n)"
vcomp_in_hhom: fixes \<nu> :: "'a"   and \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "B.seq \<nu> \<mu>"   and "src \<nu> = a"   and "trg \<nu> = b" shows "\<guillemotleft>\<nu> \<cdot> \<mu> : a \<rightarrow>\<^sub>B b\<guillemotright>"
psi_at_0: fixes j :: "nat" shows "\<psi> j 0 \<down>= j"
restrict0_apply_out: fixes x :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "x \<notin> A" shows "restrict0 A f x = (0::'d)"
simps_2: shows "channel.recv_oracle Void () = return_spmf (None, Void)"
TC_DEFL: shows "TC('a)\<cdot>DEFL('b) = DEFL('b\<cdot>'a)"
P_prop: shows "[\<lbrace>x\<^sup>P,P\<rbrace> \<^bold>& \<^bold>\<not>\<lbrace>y\<^sup>P,P\<rbrace> in v]"
tabulate_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "tabulate n f = IArray.of_fun f n"
Dflow_def: fixes x0 :: "'c" shows "Dflow x0 = mvar.flow x0 0 1\<^sub>L"
sub1: fixes A :: "'c"   and k :: "'c"   and a :: "'c"   and b :: "'c" assumes "A \<equiv> k + a" shows "A - b \<equiv> k + (a - b)"
pell_mul_assoc: fixes z1 :: "'b \<times> 'b"   and z2 :: "'b \<times> 'b"   and z3 :: "'b \<times> 'b" shows "pell_mul z1 (pell_mul z2 z3) = pell_mul (pell_mul z1 z2) z3"
5: shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w\<^sub>f \<rightarrow> w\<^sub>f'\<guillemotright> \<and> \<beta>\<^sub>f = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta>\<^sub>f = \<theta>\<^sub>f' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)"
vV: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and q_ty :: "'c itself" assumes "Knuth_ex_4_6_2_22_main a b u v w" shows "v = #(#v + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q q_ty)) (#Knuth_ex_4_6_2_22_main.v' q_ty b u v w))"
has_codomain_iff_arr: fixes f :: "'c" shows "(J'.codomains f \<noteq> {}) = J'.arr f"
module_hom_ow_axioms_def: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f" shows "module_hom_ow_axioms U\<^sub>M\<^sub>_\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 plus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 f \<equiv> f ` U\<^sub>M\<^sub>_\<^sub>1 \<subseteq> U\<^sub>M\<^sub>_\<^sub>2 \<and> (\<forall>b1 b2. b1 \<in> U\<^sub>M\<^sub>_\<^sub>1 \<longrightarrow> b2 \<in> U\<^sub>M\<^sub>_\<^sub>1 \<longrightarrow> f (plus\<^sub>M\<^sub>_\<^sub>1 b1 b2) = plus\<^sub>M\<^sub>_\<^sub>2 (f b1) (f b2)) \<and> (\<forall>r U\<^sub>C\<^sub>R\<^sub>1 b. r \<in> U\<^sub>C\<^sub>R\<^sub>1 \<longrightarrow> b \<in> U\<^sub>M\<^sub>_\<^sub>1 \<longrightarrow> f (scale\<^sub>1 r b) = scale\<^sub>2 r (f b))"
mult_minus1_right: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "'b" assumes "class.ring_1 one times plus zero minus uminus" shows "times z (uminus one) = uminus z"
rem_implicit_pres_ops_wf_partial_state: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and s :: "(nat \<times> nat) list" shows "ast_problem.wf_partial_state (rem_implicit_pres_ops prob) s = ast_problem.wf_partial_state prob s"
eq2: shows "\<forall>p\<in>set Ps'. \<exists>\<Phi>' \<Psi>'. p = ( \<Phi>' + \<Gamma>' \<Rightarrow>* \<Psi>' + \<Delta>')"
has_integral_iff: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and S :: "'a set" shows "(f has_integral i) S = (f integrable_on S \<and> integral S f = i)"
unitary_is_length_preserving_bis: fixes U :: "complex mat"   and v :: "complex mat" assumes "unitary U"   and "dim_row v = dim_col U"   and "dim_col v = 1" shows "\<parallel>col (U * v) 0\<parallel> = \<parallel>col v 0\<parallel>"
dsg4: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> d y = d y \<cdot> d x"
mult_onel: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b" assumes "class.ab_near_semiring_one plus times one" shows "times one x = x"
inv_char: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> x = y"
assms: shows "b \<^bold>\<in> \<Squnion>RepFun A B"
10: shows "spanning_forest (w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) g"
inner_zero_left: fixes x :: "'a" shows "inner (0::'a) x = 0"
explainApplicableToEachNonDecision: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and conflictFlag :: "bool"   and C :: "Literal list"   and literal :: "Literal" assumes "InvariantReasonClauses F M"   and "InvariantCFalse conflictFlag M C"   and "conflictFlag = True"   and "opposite literal el C"   and "\<not> literal el decisions M" shows "\<exists>clause. formulaEntailsClause F clause \<and> isReason clause literal (elements M)"
CodF: shows "A_B.Cod (fst Fg) = A_B.Map (A_B.cod (fst Fg))"
f_def: shows "f \<equiv> \<lambda>x. (4 * ln x + 3) / (ln 2 * x)"
relpow_image: fixes r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b"   and r' :: "('b \<times> 'b) set"   and s :: "'a"   and t :: "'a"   and n :: "nat" assumes "\<And>s t. (s, t) \<in> r \<Longrightarrow> (f s, f t) \<in> r'"   and "(s, t) \<in> r ^^ n" shows "(f s, f t) \<in> r' ^^ n"
sum_distribution_cut: fixes X :: "'key \<times> 'message list \<Rightarrow> 'a"   and x :: "'a"   and Y :: "'key \<times> 'message list \<Rightarrow> 'b" shows "prob (X -` {x} \<inter> msgs) = (\<Sum>y\<in>Y ` space \<mu>. prob ((\<lambda>x. (X x, Y x)) -` {(x, y)} \<inter> msgs))"
vreal_of_real_impl_def: shows "vreal_of_real_impl = (SOME V_of. inj V_of)"
assms: shows "\<exists>b e. terminal_arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F b a e"
sSum_cont_H: fixes H :: "'a set"   and K :: "'a set" assumes "submodule R M H"   and "submodule R M K" shows "H \<subseteq> H \<minusplus> K"
equal_linear_poly_def: fixes x :: "linear_poly"   and y :: "linear_poly" shows "equal_class.equal x y = (linear_poly_map x = linear_poly_map y)"
r_enumdom_def: shows "r_enumdom \<equiv> Cn 2 r_pdec1 [Mn 2 r_dovedelay]"
weak_pareto_def: fixes scf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and Is :: "'a set"   and Pcond :: "'b set \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> bool" shows "weak_pareto scf A Is Pcond \<equiv> \<forall>P x y. Pcond A Is P \<and> x \<in> A \<and> y \<in> A \<and> (\<forall>i\<in>Is. x \<^bsub>(P i)\<^esub>\<prec> y) \<longrightarrow> x \<^bsub>(scf P)\<^esub>\<prec> y"