hcju/isabelleps huggingface dataset:
compl_less_swap1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and y :: "'a"   and x :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less y (uminus x)" shows "less x (uminus y)"
class_fieldsI: fixes dcl :: "char list"   and cl :: "cl"   and fds :: "fd list"   and meth_defs :: "meth_def list" shows "class_fields (cld_def dcl cl fds meth_defs) fds"
upd_cs_intra_path: fixes as :: "'edge list"   and cs :: "'edge list" assumes "\<forall>a\<in>set as. intra_kind (kind a)" shows "upd_cs cs as = cs"
Flow_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "Flow c s t f \<equiv> Preflow c s t f \<and> Flow_axioms c s t f"
permutes_univ: fixes p :: "'c \<Rightarrow> 'c" shows "(p permutes UNIV) = (\<forall>y. \<exists>!x. p x = y)"
X: shows "simple_distributed M X (\<lambda>x. prob (X -` {x} \<inter> space M))"
gSubstAbsPresGWlsAbs_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) Recursion.model_scheme" shows "gSubstAbsPresGWlsAbs MOD \<equiv> \<forall>ys Y' Y y us s A' A. isInBar (us, s) \<and> wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<and> wlsAbs (us, s) A' \<and> gWlsAbs MOD (us, s) A \<longrightarrow> gWlsAbs MOD (us, s) (gSubstAbs MOD ys Y' Y y A' A)"
reach6: fixes x :: "Value"   and s :: "Store"   and l :: "Location" assumes "isprimitive \<tau> x" shows "\<not> s\<turnstile> l reachable_from x"
bet__le1213: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C" shows "A B Le A C"
simps_3: fixes S :: "nat \<Rightarrow> type_scheme"   and sch1 :: "type_scheme"   and sch2 :: "type_scheme" shows "bound_scheme_inst S (sch1 =-> sch2) = bound_scheme_inst S sch1 =-> bound_scheme_inst S sch2"
genideal_self': fixes i :: "'a" assumes "i \<in> carrier R" shows "i \<in> Idl {i}"
assms_2: shows "\<exists>n'. latestAct_cond tid t n n'"
hom_0: fixes x :: "real" assumes "of_real x = (0::'a)" shows "x = 0"
notSpecKeysSecretsExpr_def: fixes P :: "specID"   and e :: "Expression list" shows "notSpecKeysSecretsExpr P e \<equiv> (\<forall>x. kE x mem e \<longrightarrow> kKS x \<notin> specKeysSecrets P) \<and> (\<forall>y. sE y mem e \<longrightarrow> sKS y \<notin> specKeysSecrets P)"
eq: shows "MPoly_Type.coeff (Var i * mpoly_of_poly i p) mon = (\<Sum>a. MPoly_Type.coeff (mpoly_of_poly i p) a when mon = moni + a)"
arr_def: fixes f :: "'a arrow_of_spans_data" shows "V.arr f \<equiv> V.domains f \<noteq> {} \<or> V.codomains f \<noteq> {}"
distr_convolution_real: fixes M :: "(real \<times> real) measure"   and f :: "real \<times> real \<Rightarrow> ennreal" assumes "has_density M lborel f" shows "has_density (distr M borel (\<lambda>(x, y). x + y)) lborel (\<lambda>z. \<integral>\<^sup>+ x. f (x, z - x) \<partial>lborel)"
empty_not_add_mset: fixes a :: "'a"   and A :: "'a multiset" shows "\<Empt> \<noteq> add_mset a A"
get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_type_wf: fixes h :: "(_) heap"   and object_ptr :: "(_) object_ptr" assumes "NodeClass.type_wf h" shows "(object_ptr |\<in>| object_ptr_kinds h) = (get object_ptr h \<noteq> None)"
wmax_lemma: fixes w1 :: "32 Word.word"   and r1 :: "real"   and w2 :: "32 Word.word"   and r2 :: "real" assumes "w1 \<equiv>\<^sub>E r1"   and "w2 \<equiv>\<^sub>E r2" shows "wmax w1 w2 \<equiv>\<^sub>E max r1 r2"
r_step_Mn_prim: shows "prim_recfn 1 r_step_Mn"
hyps: fixes p :: "'a poly" assumes "degree p < degree pa__"   and "p \<noteq> 0" shows "\<exists>A. size A = degree p \<and> p = smult (lead_coeff p) (\<Prod>x\<in>#A. [:- x, 1::'a:])"
le_neq_trans: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># b"   and "a \<noteq> b" shows "a \<subset># b"
out: shows "outs_gpv \<I> d \<subseteq> A"
assms_1: fixes \<sigma> :: "'a" shows "P \<sigma> \<longrightarrow> P' \<sigma>"
mask_zero: fixes x :: "'b word"   and a :: "nat" assumes "is_aligned x a" shows "x AND mask a = 0"
is_ring_hom_ring: shows "ring_hom_ring R S h"
zcf_monom_sub': fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "p \<in> carrier P"   and "a \<in> carrier R" shows "zcf (Cring_Poly.compose R (monom P a (Suc n)) p) = a \<otimes> zcf p [^] Suc n"
setdist_gt_0_closedin: fixes U :: "'c set"   and S :: "'c set"   and x :: "'c" assumes "closedin (top_of_set U) S"   and "x \<in> U"   and "S \<noteq> {}"   and "x \<notin> S" shows "0 < setdist {x} S"
distinct_25: fixes x21 :: "POPLmarkRecord.type"   and x22 :: "trm"   and x81 :: "pat"   and x82 :: "trm"   and x83 :: "trm" shows "(\<lambda>:x21. x22) \<noteq> (LET x81 = x82 IN x83)"
couSim: shows "weak_reduction_coupled_simulation (indRelTEQ TRel) (STCal Source Target)"
eq_fin_length: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "length w\<^sub>1 = length w\<^sub>2"
DynCom: fixes r :: "'a set"   and a :: "('a, 'b, 'c) ann"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and F :: "'c set"   and c :: "'a \<Rightarrow> ('a, 'b, 'c) com"   and Q :: "'a set"   and A :: "'a set" assumes "r \<subseteq> pre a"   and "\<forall>s\<in>r. \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> a (c s) Q, A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnRec r a) (DynCom c) Q, A"
resolution_class_blocks_constant_size: fixes S :: "'a set multiset"   and bl :: "'a set" assumes "S \<in># \<P>"   and "bl \<in># S" shows "int (card bl) = \<k>"
augment_list_prefix: fixes n :: "nat"   and m :: "nat"   and f :: "'c list \<Rightarrow> 'c"   and p :: "'c list" assumes "n \<le> m" shows "prefix ((augment_list f ^^ n) p) ((augment_list f ^^ m) p)"
trimmed_pos_MSLCons: fixes x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" shows "trimmed_pos (MS (MSLCons x xs) f) = trimmed_pos (fst x)"
plus_nd_fun_def: fixes f :: "'a nd_fun"   and g :: "'a nd_fun" shows "f + g = (f\<^sub>\<bullet> \<squnion> g\<^sub>\<bullet>)\<^sup>\<bullet>"
some_geodesic_closed: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" shows "closed {x--S--y}"
exec_step_ind_Return_Last: fixes M\<^sub>0 :: "char list"   and P :: "jvm_prog"   and h :: "heap"   and stk\<^sub>0 :: "val list"   and loc\<^sub>0 :: "val list"   and C\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and sh :: "sheap" assumes "M\<^sub>0 \<noteq> clinit" shows "exec_step_ind (StepI Return) P h stk\<^sub>0 loc\<^sub>0 C\<^sub>0 M\<^sub>0 pc ics [] sh (None, h, [], sh)"
env_len_def: fixes env :: "'a list set"   and l :: "nat" shows "env_len env l = (\<forall>xs\<in>env. length xs = l)"
RedCallNull: fixes M :: "char list"   and vs :: "val list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>null\<bullet>M(map Val vs),s\<rangle> \<rightarrow> \<langle>THROW NullPointer,s\<rangle>"
conga_cop__or_out_ts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and C' :: "'p" assumes "Coplanar A B C C'"   and "A B C CongA A B C'" shows "B Out C C' \<or> A B TS C C'"
set_map: fixes f :: "'d \<Rightarrow> 'b"   and v :: "'c \<Rightarrow> 'd" shows "range (f \<circ> v) = f ` range v"
hd_def: shows "ODList.hd \<equiv> list.hd \<circ> toList"
bdl_star_def: fixes x :: "'a" shows "bdl_star x = top"
CollectI: fixes P :: "'e \<Rightarrow> bool"   and a :: "'e" assumes "P a" shows "a \<in> {x. P x}"
mem_permute_set: fixes x :: "'c"   and p :: "perm"   and S :: "'c set" shows "(x \<in> p \<bullet> S) = (- p \<bullet> x \<in> S)"
lift_maximal_literal: fixes L :: "'a literal"   and \<sigma> :: "('a \<times> 'a trm) list"   and C :: "'a literal set" assumes "maximal_literal (subst_lit L \<sigma>) (subst_cl C \<sigma>)" shows "maximal_literal L C"
prems_2: shows "basis_reduction_main (upw, i, fs) = fs'"
axioms_2: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subring H R" shows "submonoid H R"
hoare_r_def: fixes p :: "'b upred"   and Q :: "'b hrel"   and r :: "'b upred" shows "\<lbrace>p\<rbrace> Q \<lbrace>r\<rbrace>\<^sub>u = ((\<lceil>p\<rceil>\<^sub>< \<Rightarrow> \<lceil>r\<rceil>\<^sub>>) \<sqsubseteq> Q)"
fractrel_normalize_quot_left: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a" assumes "snd x \<noteq> (0::'a)" shows "fractrel (normalize_quot x) y = fractrel x y"
singular_chain_extend: fixes x :: "'c \<Rightarrow>\<^sub>0 int"   and p :: "nat"   and X :: "'d topology"   and f :: "'c \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'd) \<Rightarrow>\<^sub>0 int" assumes "\<And>c. c \<in> Poly_Mapping.keys x \<Longrightarrow> singular_chain p X (f c)" shows "singular_chain p X (frag_extend f x)"
parts_msg_mpair: fixes X :: "msg"   and Y :: "msg" shows "parts_msg \<lbrace>X, Y\<rbrace> = insert \<lbrace>X, Y\<rbrace> (parts_msg X \<union> parts_msg Y)"
less_eq_mreal_def: fixes x :: "mreal"   and y :: "mreal" shows "x \<le> y \<equiv> x + y = y"
pw_leI: fixes S' :: "'c nres"   and S :: "'c nres" assumes "nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x)" shows "S \<le> S'"
Prim_Invar1_loc_axioms: shows "Prim_Invar1_loc w g r (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u))"
subcls_irrefl: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal" assumes "wf_prog wf_md P"   and "(subcls1 P)\<^sup>+\<^sup>+ C D" shows "C \<noteq> D"
sum_Un: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B" shows "sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
le_diff_conv2: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "(c \<subseteq># b - a) = (c + a \<subseteq># b)"
indRelp_indRel_eq: shows "indRelp = indRelInfix"
add_neg_neg: fixes a :: "'a"   and b :: "'a" assumes "a < (0::'a)"   and "b < (0::'a)" shows "a + b < (0::'a)"
id_o: fixes g :: "'a \<Rightarrow> 'b" shows "id \<cdot> g = g"
assms: fixes a :: "'a"   and list :: "'a list" assumes "y = a # list" shows "P"
keys_of_Ltk: fixes A :: "agent" shows "keys_of A \<subseteq> range LtK"
f2: fixes x :: "'a discr \<rightarrow> 'b" shows "cont (F x)"
ffb_fbd_dual: fixes f :: "'d \<Rightarrow> 'c set" shows "\<partial> \<circ> fb\<^sub>\<F> f = bd\<^sub>\<F> (op\<^sub>K f) \<circ> \<partial>"
wp_R1: fixes p :: "'a"   and q :: "'a" shows "d p \<le> wp (R p q) q"
arg_min_inj_eq: fixes f :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "inj_on f {x. P x}"   and "P a"   and "\<forall>y. P y \<longrightarrow> f a \<le> f y" shows "arg_min f P = a"
shift1_inv_some_snd_1: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and B :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "shift1_inv (qs, B)"   and "1 < card B"   and "(h, U) = (SOME b. b \<in> B \<and> card (snd b) = Suc m)" shows "(h, U) \<in> B"
null_eqI: fixes n :: "'s" assumes "\<And>f. n \<cdot> f = n \<and> f \<cdot> n = n" shows "n = S.null"
total_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "Relation.total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
zero_prod_def: shows "0 = (0::'b, 0::'c)"
DAnotTL: fixes xs :: "('b, 'c) Combinators list" assumes "xs \<noteq> []"   and "wellformed_policy1 (xs @ [DenyAll])" shows "False"
lc1aux: shows "{l. length l = 1 \<and> valid l} = {[B]}"
fls_prpart_as_fls_nth: fixes f :: "'a fls"   and n :: "int" shows "fls_nth (fls_prpart_as_fls f) n = (if n < 0 then fls_nth f n else (0::'a))"
nested_include_restrict: fixes I :: "nat set"   and z :: "'b option list"   and t :: "'b option list"   and A :: "nat set" assumes "restrict I z = t"   and "A \<subseteq> I" shows "restrict A z = restrict A t"
simps_1: fixes uu :: "'c \<Rightarrow> bool" shows "find uu [] = None"
all_val: fixes p :: "linear_poly" assumes "\<forall>v. \<exists>v'. (\<forall>x\<in>vars p. v' x = v x) \<and> p \<lbrace> v' \<rbrace> = (0::'a)" shows "p = 0"
Nmlize_Dom: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Dom t\<^bold>\<rfloor> = Dom \<^bold>\<lfloor>t\<^bold>\<rfloor>"
pos_at_zero: shows "(0::'a) \<le> eval_poly (\<lambda>w. 0::'a) p"
plus_iarray_def: fixes A :: "'a iarray"   and B :: "'a iarray" shows "A + B = (let length_A = IArray.length A; length_B = IArray.length B; n = max length_A length_B; A' = IArray.of_fun (\<lambda>a. if a < length_A then A !! a else (0::'a)) n; B' = IArray.of_fun (\<lambda>a. if a < length_B then B !! a else (0::'a)) n in IArray.of_fun (\<lambda>a. A' !! a + B' !! a) n)"
continuous_within_compose3: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and g :: "'a \<Rightarrow> 'b"   and s :: "'c set" assumes "isCont g (f x)"   and "continuous (at x within s) f" shows "continuous (at x within s) (\<lambda>x. g (f x))"
eq_set_eq: fixes xs :: "'d list"   and ys :: "'d list" assumes "equivclp cancel1 xs ys" shows "set xs = set ys"
bdd_above_iff_small: fixes X :: "V set" shows "bdd_above X = small X"
evContainsDisj: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and A0 :: "formula"   and A1 :: "formula"   and gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" assumes "EV (contains f (i, FConj Neg A0 A1))"   and "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))" shows "EV (contains f (0, A0)) \<and> EV (contains f (0, A1))"
3: shows "\<guillemotleft>S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f) : UP.map\<^sub>0 (src f) \<Rightarrow>\<^sub>S S.UP g \<star>\<^sub>S S.UP f\<guillemotright> \<and> S.iso (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f))"
oracle_encrypt_def: fixes k :: "'prf_key \<times> 'upf_key"   and b :: "bool"   and L :: "(bool list \<times> bool list \<times> 'hash) set" shows "ind_cca'.oracle_encrypt k b L = (\<lambda>(msg1, msg0). case length msg1 = prf_clen \<and> length msg0 = prf_clen of True \<Rightarrow> encrypt k (if b then msg1 else msg0) \<bind> (\<lambda>c. return_spmf (Some c, {c} \<union> L)) | False \<Rightarrow> return_spmf (None, L))"
lead_coeff_factor_2: fixes u :: "'a poly"   and v :: "'a poly"   and w :: "'a poly" assumes "u = v * w" shows "lead_coeff (Polynomial.smult (lead_coeff w) v) = lead_coeff u"
strong_delete_point_groups_index_one: fixes G1 :: "'a set"   and x :: "'a"   and G2 :: "'a set"   and xa :: "'a"   and y :: "'a" assumes "G1 \<in> {b - {x} |b. b \<in># \<B> \<and> x \<in> b}"   and "G2 \<in> {b - {x} |b. b \<in># \<B> \<and> x \<in> b}"   and "G1 \<noteq> G2"   and "xa \<in> G1"   and "y \<in> G2" shows "str_del_point_blocks x index {xa, y} = 1"
subst_fv_dom_ground_if_ground_img: fixes t :: "('d, 'c) Term.term"   and s :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "fv t \<subseteq> subst_domain s"   and "ground (subst_range s)" shows "fv (t \<cdot> s) = {}"
comp_arr_inv: fixes f :: "'c"   and g :: "'c" assumes "C.inverse_arrows f g" shows "f \<cdot>\<^sub>C g = C.dom g"
is_monomial_setI: fixes A :: "('a \<Rightarrow>\<^sub>0 'b) set" assumes "\<And>p. p \<in> A \<Longrightarrow> is_monomial p" shows "is_monomial_set A"
no_plus_overflow_neg: fixes x :: "'b word"   and y :: "'b word" assumes "x < - y" shows "x \<le> x + y"
runE: fixes f :: "'c Heap"   and g :: "'c \<Rightarrow> 'b Heap"   and \<sigma> :: "heap option"   and \<sigma>'' :: "heap option"   and r :: "'b" assumes "run (f \<bind> g) \<sigma> \<sigma>'' r" obtains \<sigma>' :: "heap option" and r' :: "'c" where "run f \<sigma> \<sigma>' r'" and "run (g r') \<sigma>' \<sigma>'' r"
e_app_cong: fixes e1 :: "exp"   and e1' :: "exp"   and e2 :: "exp"   and e2' :: "exp" assumes "E e1 = E e1'"   and "E e2 = E e2'" shows "E (EApp e1 e2) = E (EApp e1' e2')"
prems_1: shows "P,E \<turnstile> es [::] Ts'"
in_rel: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a pmf"   and b :: "'b pmf" shows "rel_pmf R a b = (\<exists>z. z \<in> {x. set_pmf x \<subseteq> {(x, y). R x y}} \<and> map_pmf fst z = a \<and> map_pmf snd z = b)"
norm_pths_1: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (norm (x - y) \<le> 0)"
distinct_17: fixes x3 :: "bool"   and x5 :: "'b" shows "Bool x3 \<noteq> Addr x5"
ID_ceq: fixes eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "ID CEQ('b) = Some eq" shows "eq = (=)"
mkArr_eqI: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's"   and A' :: "'s set"   and B' :: "'s set"   and F' :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B F)"   and "A = A'"   and "B = B'"   and "\<And>x. x \<in> A \<Longrightarrow> F x = F' x" shows "mkArr A B F = mkArr A' B' F'"
psubstt_lift_2: fixes f :: "'b \<Rightarrow> 'a"   and ts :: "'b FOL_Fitting.term list" shows "psubstts f (liftts ts) = liftts (psubstts f ts)"
perron_frobenius_axioms_def: fixes A :: "((real, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "perron_frobenius_axioms A \<equiv> fixed_mat.irreducible A"
l: shows "trace (A - B) = (\<Sum>i = 0..<n. A $$ (i, i)) - (\<Sum>i = 0..<n. B $$ (i, i))"
V_def: shows "V \<equiv> {v. \<forall>x\<in>X. (0::'a) \<le> v x}"
path_conforms_with_strategy_start: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and v :: "'c"   and w :: "'c"   and P :: "'c llist"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "ParityGame.path_conforms_with_strategy G p (LCons v (LCons w P)) \<sigma>"   and "v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)" shows "\<sigma> v = w"
CompositionSupport_def: fixes ESi :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>i :: "'b V_rec" shows "CompositionSupport ESi \<V> \<V>i \<equiv> (ES_valid ESi \<and> isViewOn \<V>i E\<^bsub>ESi\<^esub>) \<and> V\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ESi\<^esub> = V\<^bsub>\<V>i\<^esub> \<and> C\<^bsub>\<V>\<^esub> \<inter> E\<^bsub>ESi\<^esub> \<subseteq> C\<^bsub>\<V>i\<^esub>"
r_zero: fixes x :: "rat vec" assumes "x \<in> Rn" shows "x + 0\<^sub>v n = x"
assms_2: fixes i :: "'i" assumes "i \<in> I" shows "open (U i)"
assms_3: fixes k :: "'k" assumes "k \<in> component_of_term ` Keys B" shows "\<exists>b\<in>B. b \<noteq> 0 \<and> component_of_term (lt b) = k \<and> lp b = (0::'a)"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set" assumes "v0_v1_Digraph G v0 v1"   and "DisjointPaths_axioms G v0 v1 paths" shows "DisjointPaths G v0 v1 paths"
HLD_mono: fixes R :: "'a set"   and \<omega> :: "'a stream"   and S :: "'a set" assumes "HLD R \<omega>"   and "R \<subseteq> S" shows "HLD S \<omega>"
det_cols_sum: fixes M :: "('c, 'd) sq_matrix"   and T :: "'d set"   and a :: "'d \<Rightarrow> 'e \<Rightarrow> ('c, 'd) vec"   and S :: "'e set" shows "Square_Matrix.det (upd_cols M T (\<lambda>i. sum (a i) S)) = (\<Sum>f\<in>T \<rightarrow>\<^sub>E S. Square_Matrix.det (upd_cols M T (\<lambda>i. a i (f i))))"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>) (\<cdot>) src local.map \<phi>" shows "equivalence_functor (\<cdot>) (\<cdot>) src"
nths_replicate: fixes n :: "nat"   and x :: "'a"   and A :: "nat set" shows "nths (replicate n x) A = replicate (card {i. i < n \<and> i \<in> A}) x"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) (\<lambda>f. f \<otimes>\<^sub>D \<I>\<^sub>D) D.map D.\<rho>"
l_coset_def: fixes G :: "('d, 'e) monoid_scheme"   and a :: "'d"   and H :: "'d set" shows "a <#\<^bsub>G\<^esub> H = (\<Union>h\<in>H. {a \<otimes>\<^bsub>G\<^esub> h})"
fct: shows "(if stk_strict_subs S then S else {}, A) \<in> {(s, a). stk_strict_subs s}"
impl_of: fixes x :: "('a, 'b) alist" shows "impl_of x \<in> {xs. (distinct \<circ> list.map fst) xs}"
rel_with_target_impl_TRel_reflects_barbs: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_reflects_binary_pred Rel (\<lambda>b. hasBarb b (STCalWB SWB TWB))"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel" shows "rel_reflects_binary_pred TRel (\<lambda>b. hasBarb b TWB)"
corec_sel_1: fixes g1 :: "'b \<Rightarrow> 'a pmf"   and g2 :: "'b \<Rightarrow> 'a \<Rightarrow> 'a scheduler + 'b"   and a :: "'b" shows "action_sch (corec_scheduler g1 g2 a) = g1 a"
preserves_comp_1: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'b \<Rightarrow> 'd"   and f1' :: "'a"   and f1 :: "'a" assumes "binary_functor A1 A2 B F"   and "partial_magma.arr A1 (A1 f1' f1)" shows "(\<lambda>f2. F (A1 f1' f1, f2)) = vertical_composite.map A2 B (\<lambda>f2. F (f1, f2)) (\<lambda>f2. F (f1', f2))"
order_r: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a r\<rightarrow> b = (1::'a))"
keys_add: fixes f :: "'b \<Rightarrow>\<^sub>0 'c"   and g :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (f + g) \<subseteq> keys f \<union> keys g"
axis_inverse: fixes v :: "(real, 'a) vec" assumes "v \<in> Basis" shows "\<exists>i. v = axis i 1"
best_division_abs: fixes n :: "int"   and a :: "int" assumes "0 < n" shows "\<exists>k. 2 * \<bar>a - k * n\<bar> \<le> n"
contracted_no_degree2_simp: fixes G :: "'b pair_pre_digraph"   and H :: "'b pair_pre_digraph" assumes "pair_graph G"   and "subdivision_pair G H"   and "verts3 (with_proj G) = pverts G" shows "gen_contr_graph (with_proj H) (verts3 (with_proj H)) = G"
assms: shows "cont lub ord \<Union> (\<subseteq>) f"
swap_id_common': fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<noteq> b"   and "a \<noteq> c" shows "Fun.swap a c id \<circ> Fun.swap b c id = Fun.swap b c id \<circ> Fun.swap a b id"
oexp_less: fixes \<beta> :: "V"   and \<gamma> :: "V"   and \<alpha> :: "V" assumes "\<beta> \<in> elts \<gamma>"   and "Ord \<gamma>"   and "1 < \<alpha>"   and "Ord \<alpha>" shows "\<alpha> \<up> \<beta> < \<alpha> \<up> \<gamma>"
msetext_huet_compat_cons: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "msetext_huet gt ys xs" shows "msetext_huet gt (x # ys) (x # xs)"
sumUnionDisjoint1: fixes C :: "'c set set"   and f :: "'c \<Rightarrow> 'd" assumes "\<forall>A\<in>C. finite A"   and "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}" shows "sum f (\<Union> C) = sum (sum f) C"
assms_2: shows "p\<^sub>1 \<in> set ps"
nth_Cons_pos: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" assumes "0 < n" shows "(x # xs) ! n = xs ! (n - 1)"
surj_fin_dim: assumes "V.fin_dim"   and "T ` carrier V = carrier W" shows "W.fin_dim"
subrelI: fixes r :: "('b \<times> 'c) set"   and s :: "('b \<times> 'c) set" assumes "\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s" shows "r \<subseteq> s"
f_nth: fixes x :: "nat"   and n :: "nat" assumes "x < Suc n" shows "f x \<down>= e_nth (f \<triangleright> n) x"
span_neg: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "x \<in> phull S" shows "- x \<in> phull S"
matrix_inv_right: fixes M :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" assumes "invertible M" shows "M ** matrix_inv M = Finite_Cartesian_Product.mat (1::'a)"