hcju/isabelleps huggingface dataset:
antisym_conv: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "(y \<le> x) = (x = y)"
MinPredicate2: fixes P :: "nat \<Rightarrow> bool" assumes "\<exists>n. P n" shows "\<exists>n0. P n0 \<and> (n0 = 0 \<or> \<not> P (n0 - 1))"
assms: fixes y :: "'a" assumes "y \<in> keys t" shows "f y = monomial (c y) (s y)"
tllist_all2_tfinite2_terminalD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 P Q xs ys"   and "tfinite ys" shows "Q (terminal xs) (terminal ys)"
closed_segment_in_Reals: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> closed_segment a b"   and "a \<in> \<real>"   and "b \<in> \<real>" shows "x \<in> \<real>"
elims_2: fixes x :: "'a dres"   and xa :: "'a dres" assumes "x \<le> xa" obtains uu_ :: "'a dres" where "x = dSUCCEEDi" and "xa = uu_"   | "x = dFAILi" and "xa = dFAILi"   | v :: "'a" where "x = dRETURN v" and "xa = dFAILi"   | a :: "'a" and b :: "'a" where "x = dRETURN a" and "xa = dRETURN b" and "a = b"
nth_last_index: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "xs ! last_index xs x = x"
listsetD_1: fixes ys :: "'b list"   and xs :: "'b set list" assumes "ys \<in> listset xs" shows "length ys = length xs"
hom_induced_id_gen: fixes X :: "'b topology"   and f :: "'b \<Rightarrow> 'b"   and c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set"   and p :: "int"   and S :: "'b set" assumes "continuous_map X X f"   and "\<And>x. x \<in> topspace X \<Longrightarrow> f x = x"   and "c \<in> carrier (relative_homology_group p X S)" shows "hom_induced p X S X S f c = c"
Field_ofilter: fixes r :: "('c \<times> 'c) set" assumes "wo_rel r" shows "ofilter r (Field r)"
ReachCfg1Cfg2: shows "reachable cfg1 cfg2"
cross_ratio_swap_13_24: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and s :: "proj2" assumes "cross_ratio_correct p q r s"   and "r \<noteq> s" shows "cross_ratio r s p q = cross_ratio p q r s"
sums_le: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "\<And>n. f n \<le> g n"   and "f sums s"   and "g sums t" shows "s \<le> t"
ideD_2: fixes a :: "'a \<times> 'a" assumes "B.VV.ide a" shows "B.VV.dom a = a"
has_integral_f: fixes k :: "nat" assumes "c \<le> real k" shows "(f k has_integral exp (- a * c) / a - exp (- a * real k) / a) {c..}"
bdd_aboveI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and M :: "'c" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x M" shows "preorder.bdd_above less_eq A"
oalist_tc_of_list_def: shows "oalist_tc_of_list = OAlist_tc"
swap_simp: fixes x :: "'l"   and y :: "'k" shows "prod.swap (x, y) = (y, x)"
com_Unique: fixes c :: "com"   and s :: "loc \<Rightarrow> nat"   and s1 :: "loc \<Rightarrow> nat" assumes "(c, s) -[eval \<inter> {((e, s), n, t). \<forall>nt'. (e, s) -|-> nt' \<longrightarrow> (n, t) = nt'}]-> s1" shows "\<forall>s2. (c, s) -[eval]-> s2 \<longrightarrow> s2 = s1"
fst_set_map_charpair_un_uminus: fixes ss :: "'a list"   and S :: "'a set" assumes "ss \<in> lists (S \<union> uminus ` S)" shows "order.greater_eq S (fst ` set (map (charpair S) ss))"
spans_image: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and V :: "'e set"   and B :: "'e set" assumes "module_hom s1 s2 f"   and "V \<subseteq> module.span s1 B" shows "f ` V \<subseteq> module.span s2 (f ` B)"
wgraph_specs_3: shows "\<alpha> empty_graph = (\<lambda>_. \<infinity>)"
scaleR_sum_right: fixes a :: "real"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "a *\<^sub>R sum f A = (\<Sum>x\<in>A. a *\<^sub>R f x)"
ipresSubst_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'd, 'i, 'j) model_scheme" shows "ipresSubst h MOD \<equiv> \<forall>ys Y y s X. wls (asSort ys) Y \<and> wls s X \<longrightarrow> h (X #[Y / y]_ys) = igSubst MOD ys (h Y) y (h X)"
p_decomp: shows "p = p1 @ (x, y) # p2"
bind_map_pmf: fixes f :: "'e \<Rightarrow> 'd"   and M :: "'e pmf"   and g :: "'d \<Rightarrow> 'c pmf" shows "map_pmf f M \<bind> g = M \<bind> (\<lambda>x. g (f x))"
us'_def: shows "us' \<equiv> gram_schmidt0 n ws"
hom_mult: fixes x :: "'a poly"   and y :: "'a poly"   and p :: "'a poly" shows "(x * y) \<circ>\<^sub>p p = x \<circ>\<^sub>p p * y \<circ>\<^sub>p p"
Enc_Aenc: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Enc X Y \<noteq> Aenc X' Y'"
antisymPI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>r x y; r y x\<rbrakk> \<Longrightarrow> x = y" shows "antisymp r"
pr_symk_analz: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat" assumes "(evs, S, A, U) \<in> protocol" shows "(Key (symK n) \<in> analz (A \<union> spies evs)) = (n \<in> bad)"
choose_pivot3_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and lo :: "nat"   and hi :: "nat" shows "choose_pivot3 R h xs lo hi = ASSERT (lo < length xs) \<bind> (\<lambda>_. ASSERT (hi < length xs) \<bind> (\<lambda>_. let k' = (hi - lo) div 2; k = lo + k' in ASSERT (k < length xs) \<bind> (\<lambda>_. let start = h (xs ! lo); mid = h (xs ! k); end = h (xs ! hi) in if R start mid \<and> R mid end \<or> R end mid \<and> R mid start then RETURN k else if R start end \<and> R end mid \<or> R mid end \<and> R end start then RETURN hi else RETURN lo)))"
a4: shows "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"
restrict_ifex_ordered_invar: fixes b :: "'a ifex"   and var :: "'a"   and val :: "bool" assumes "ifex_ordered b" shows "ifex_ordered (restrict b var val)"
min_MInf2: fixes x :: "ereal" shows "min x (- \<infinity>) = - \<infinity>"
tensor_mat_carrier: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "tensor_mat m1 m2 \<in> carrier_mat d d"
hyps: assumes "is_subobj P (D, D # rev Cs')" shows "is_subobj P (C, D # rev Cs')"
Class_closed: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "equivalence S E"   and "a \<in> equivalence.Class S E b"   and "b \<in> S" shows "a \<in> S"
prems_1: shows "|filter P (x # xs)| = |x # xs|"
distinct_10: fixes x61 :: "('a, 'b, 'c) expr"   and x62 :: "('a, 'b, 'c) expr"   and x1 :: "('a, 'b, 'c) val" shows "Assign x61 x62 \<noteq> VE x1"
\<epsilon>_in_terms_of_\<psi>: fixes x :: "'a" assumes "Adj.C.ide x" shows "Adj'.\<epsilon> x = \<psi>' x (G' x)"
chine_assoc_in_hom: shows "\<guillemotleft>\<mu>_\<nu>\<pi>_\<rho>.chine_assoc : HH\<mu>H\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C H\<mu>HH\<nu>\<pi>\<rho>.chine\<guillemotright>"
divides_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p1 :: "'b list"   and p2 :: "'b list" assumes "class.semiring_0 plus zero times" shows "semiring_0.divides plus zero times p1 p2 = (\<exists>q. semiring_0.poly plus zero times p2 = semiring_0.poly plus zero times (semiring_0.pmult plus zero times p1 q))"
sprod_one: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and m :: "'e" assumes "R module M"   and "m \<in> carrier M" shows "1\<^sub>r\<^bsub>R\<^esub> \<cdot>\<^sub>s\<^bsub>M\<^esub> m = m"
lprodr_simps: fixes a :: "'c"   and b :: "'d"   and c :: "'e" shows "lprodr (a, b, c) = ((a, b), c)"
pfinite_converterD: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter"   and a :: "'h" assumes "pfinite_converter \<I> \<I>' conv"   and "a \<in> outs_\<I> \<I>" shows "pfinite_gpv \<I>' (run_converter conv a) \<and> (\<forall>(b, conv')\<in>results_gpv \<I>' (run_converter conv a). pfinite_converter \<I> \<I>' conv')"
eqD_18: fixes y :: "cmsg"   and N :: "tag" assumes "eq y (cTag N)" shows "y = cTag N"
sorted_enumerate: fixes n :: "nat"   and xs :: "'b list" shows "sorted (map fst (enumerate n xs))"
perfect_hiding: fixes \<A> :: "(('a \<times> 'a) \<times> nat \<Rightarrow> ((nat \<times> nat) \<times> 'c) spmf) \<times> ('a \<times> 'a \<Rightarrow> 'c \<Rightarrow> bool spmf)" shows "chaum_ped_\<Sigma>_commit.abstract_com.perfect_hiding_ind_cpa \<A>"
multpw_local: fixes X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set" assumes "(X, Y) \<in> multpw ns" shows "(X, Y) \<in> multpw (ns \<inter> set_mset X \<times> set_mset Y)"
tc_not_0: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p \<noteq> 0" shows "ordered_term.tc term_of_pair ord_term p \<noteq> (0::'f)"
span_superset: fixes Sa :: "('a \<Rightarrow> real) set" assumes "Sa \<subseteq> diff_fun_space" shows "Sa \<subseteq> sub.diff_fun_space.m1.span Sa"
dvd_add_times_triv_left_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus (times c a) b) = dvd.dvd times a b"
get_shadow_root_ptr_simp6: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and character_data_ptr :: "(_) character_data_ptr"   and f :: "'u option RCharacterData_scheme"   and h :: "(_) heap" shows "get shadow_root_ptr (put character_data_ptr f h) = get shadow_root_ptr h"
f11: shows "\<not> (\<exists>(a', b', c')\<in>set d. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
Inr_eq_map_sum_iff: fixes x :: "'e"   and f :: "'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'e"   and y :: "'f + 'g" shows "(Inr x = map_sum f g y) = (\<exists>z. y = Inr z \<and> x = g z)"
FvarsT_num: fixes n :: "'b" assumes "n \<in> num" shows "FvarsT n = {}"
in_set_tlD: fixes x :: "'k"   and xs :: "'k list" assumes "x \<in> set (tl xs)" shows "x \<in> set xs"
lt2_def: shows "lt2 \<equiv> less_than <*lex*> less_than"
unbounded_nat_set_infinite: fixes A :: "nat set" assumes "\<forall>i. \<exists>j\<ge>i. j \<in> A" shows "infinite A"
2: assumes "u \<in> FvarsT (rawpsubstT r (zip (map Var us) (map snd txs)))" shows "False"
cring_simprules_27: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<otimes> \<zero> = \<zero>"
silent_moves_nodestack_notempty: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s')"   and "ms \<noteq> []" shows "ms' \<noteq> []"
g'_primepow: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "g' (p ^ k) = (1::'a) - f p"
right_unit: fixes a :: "'b" assumes "a \<in> M'" shows "a \<cdot>' \<one>' = a"
summableI: fixes f :: "nat \<Rightarrow> 'a" shows "summable f"
sum_inner_Basis_scaleR: fixes b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "b \<in> Basis" shows "(\<Sum>i\<in>Basis. (i \<bullet> b) *\<^sub>R f i) = f b"
cases: fixes a :: "fm" assumes "a \<in> induction_axioms" obtains j :: "name" and i :: "name" and A :: "fm" where "a = A(i::=Zero) IMP SyntaxN.All i (SyntaxN.All j (A IMP A(i::=Var j) IMP A(i::=Eats (Var i) (Var j)))) IMP SyntaxN.All i A" and "atom j \<sharp> (i, A)"
case_nat_unfold: fixes x :: "'b"   and f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "case n of 0 \<Rightarrow> x | Suc x \<Rightarrow> f x \<equiv> if n = 0 then x else f (n - 1)"
ladder_every_stage_5: fixes index :: "nat" assumes "index < length L"   and "0 < index" shows "ladder_i L' index < length (ladder_\<alpha> \<alpha> E L' index)"
Aexp_Var: fixes n :: "Arity"   and x :: "Vars.var" shows "up\<cdot>n \<sqsubseteq> \<A>\<^bsub>n\<^esub> (Var x) x"
bisim1NewArrayFail: fixes P :: "'b prog"   and h :: "'heap"   and T :: "ty"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and v :: "'addr val" shows "P,newA T\<lfloor>e\<rceil>,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([v], xs, length (compE2 e), \<lfloor>a\<rfloor>)"
wellformed2_sorted: fixes p :: "((int \<times> int) set set, int) Combinators list"   and l :: "(int \<times> int) set set set list" assumes "all_in_list p l"   and "distinct p"   and "allNetsDistinct p"   and "singleCombinators p" shows "wellformed_policy2 (FWNormalisationCore.sort p l)"
NC_disjoint_def: fixes v :: "'a V_rec" shows "NC_disjoint v \<equiv> N\<^bsub>v\<^esub> \<inter> C\<^bsub>v\<^esub> = {}"
strengthen_step_6: shows "\<forall>e\<in>pred (init_conf prb'). finite (Bexp.vars e)"
simulation1_inf_step: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and s1 :: "'g"   and tls1 :: "'h llist"   and s2 :: "'i" assumes "bisimulation trsys1 trsys2 bisim tlsim"   and "trsys.inf_step trsys1 s1 tls1"   and "bisim s1 s2" shows "\<exists>tls2. trsys.inf_step trsys2 s2 tls2 \<and> llist_all2 tlsim tls1 tls2"
\<tau>move2_Try1D: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e1 :: "'addr expr1"   and C' :: "String.literal"   and V :: "nat"   and e2 :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (try e1 catch(C' V) e2) pc xcp"   and "pc < length (compE2 e1)" shows "\<tau>move2 P h stk e1 pc xcp"
pseudominimal_ne_bottom: fixes P :: "'b set"   and x :: "'b" assumes "codomain.pseudominimal_in P x" shows "x \<noteq> codomain.bottom P"
mset_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "mset (map f xs) = f `# mset xs"
trg_closed: fixes f :: "'a" assumes "Arr f" shows "Arr (trg\<^sub>B f)"
E_1: shows "E \<subseteq> {F. finite F \<and> F \<subseteq> t}"
simps_1: fixes b :: "'a list" shows "revg [] b = b"
prime_power_eq_one_iff: fixes p :: "'a"   and n :: "nat" assumes "prime p" shows "(p ^ n = (1::'a)) = (n = 0)"
ide_right: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "partial_magma.ide V g"
vsingleton_insert_inj_eq: fixes y :: "V"   and x :: "V"   and A :: "V" shows "(set {y} = vinsert x A) = (x = y \<and> A \<subseteq>\<^sub>\<circ> set {y})"
poly_roots_prod_mset: fixes A :: "'b poly multiset" assumes "0 \<notin># A" shows "poly_roots (\<Prod>\<^sub># A) = \<Sum>\<^sub># (image_mset poly_roots A)"
the_inv_into_f_f: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a" assumes "inj_on f A"   and "x \<in> A" shows "the_inv_into A f (f x) = x"
bilinear_r_add: fixes M1 :: "('g, 'a, 'h) Module_scheme"   and M2 :: "('i, 'a, 'h) Module_scheme"   and N :: "('j, 'a, 'h) Module_scheme"   and m :: "'g"   and m21 :: "'i"   and m22 :: "'i"   and f :: "'g \<times> 'i \<Rightarrow> 'j" assumes "R module M1"   and "R module M2"   and "R module N"   and "m \<in> carrier M1"   and "m21 \<in> carrier M2"   and "m22 \<in> carrier M2"   and "bilinear_map f R M1 M2 N" shows "f (m, m21 \<plusminus>\<^bsub>M2\<^esub> m22) = f (m, m21) \<plusminus>\<^bsub>N\<^esub> f (m, m22)"
LNot: fixes y :: "name"   and M :: "trm"   and N :: "trm"   and x :: "name"   and a :: "coname"   and b :: "coname" assumes "y \<sharp> (M, N)"   and "x \<sharp> (N, y)"   and "a \<sharp> (M, N, b)"   and "b \<sharp> M"   and "y \<noteq> x"   and "b \<noteq> a" shows "Cut <a>.NotR x.M a y.NotL <b>.N y \<longrightarrow>\<^sub>l Cut <b>.N x.M"
SFAccInitThrow\<^sub>1: fixes P :: "nat exp prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and h :: "heap"   and ls :: "val list"   and a :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile>\<^sub>1 \<langle>INIT D ([D],False) \<leftarrow> unit,(h, ls, sh)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sF{D},(h, ls, sh)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"
Abs_fin_string_inject: fixes x :: "char list"   and y :: "char list" assumes "x \<in> {s. length s \<le> MAX_STR}"   and "y \<in> {s. length s \<le> MAX_STR}" shows "(Abs_fin_string x = Abs_fin_string y) = (x = y)"
\<beta>0: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "\<beta> i \<noteq> (0::'b)"
PseudoHA_def: fixes SA :: "('d, 'e, 'f) seqauto"   and D :: "'f data" shows "PseudoHA SA D = Abs_hierauto (D, {SA}, SAEvents SA, EmptyMap (States SA))"
cfg_on_div_absc: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and st :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> cfg_on_div st"   and "st \<in> S" shows "absc cfg \<in> R_G_cfg_on_div (abss st)"
ab_spec_def: fixes ab :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" shows "ab_spec ab = ((\<forall>gs bs ns data. ns \<noteq> [] \<longrightarrow> set (ab gs bs ns data) = set bs \<union> set ns) \<and> (\<forall>gs bs data. ab gs bs [] data = bs))"
prog_unclocked_ignore: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and prog :: "top0 list"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_prog ck env s prog res" shows "\<forall>cnt s' r. res = (s', r) \<and> r \<noteq> Rerr (Rabort Rtimeout_error) \<longrightarrow> evaluate_prog False env (update_clock (\<lambda>_. cnt) s) prog (update_clock (\<lambda>_. cnt) s', r)"
vmc_path_lset_induction_simple: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "v0 \<in> S"   and "\<And>P v0. \<lbrakk>vmc_path_no_deadend G P v0 p \<sigma>; v0 \<in> S\<rbrakk> \<Longrightarrow> vmc_path_no_deadend.w0 P \<in> S" shows "lset P \<subseteq> S"
until_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<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 minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and at :: "'a \<Rightarrow> nat \<Rightarrow> 'a"   and p :: "'a"   and q :: "'a" assumes "class.temporal Inf Sup inf less_eq less sup bot top minus uminus at" shows "temporal.until Inf Sup inf at p q = Sup (range (\<lambda>n. inf (Inf (at p ` {i. i < n})) (at q n)))"
cases: fixes S :: "'a spmf \<Rightarrow> 'b spmf \<Rightarrow> bool"   and a1 :: "'a spmf"   and a2 :: "'b spmf" assumes "trace_eq_simcl S a1 a2" obtains p :: "'a spmf" and q :: "'b spmf" where "a1 = p__" and "a2 = q" and "S p__ q"   | p :: "nat spmf" and f :: "nat \<Rightarrow> 'a spmf" and g :: "nat \<Rightarrow> 'b spmf" where "a1 = p \<bind> f" and "a2 = p \<bind> g" and "\<And>x. x \<in> set_spmf p \<Longrightarrow> S (f x) (g x)"
assert_disch1: fixes P :: "'d \<Rightarrow> bool"   and \<sigma> :: "'d"   and M :: "bool \<Rightarrow> 'd \<Rightarrow> (bool \<times> 'd) option" assumes "P \<sigma>" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (assert\<^sub>S\<^sub>E P) M) = (\<sigma> \<Turnstile> M True)"
powr_minus_divide: fixes x :: "'b"   and a :: "'b" shows "x powr - a = (1::'b) / x powr a"
eliminate_entries_gen_zero_def: fixes minu :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and time :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and v :: "integer \<Rightarrow> 'b"   and A :: "'b mat"   and I :: "nat"   and J :: "nat" shows "eliminate_entries_gen_zero minu time z v A I J = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if v (integer_of_nat i) \<noteq> z \<and> i \<noteq> I then minu (A $$ (i, j)) (time (v (integer_of_nat i)) (A $$ (I, j))) else A $$ (i, j))"
set_iterator_linord_rule_P: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and it :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'c set"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.set_iterator_linord less_eq it S0"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; x \<in> S; I S \<sigma>; S \<subseteq> S0; \<And>x'. x' \<in> S0 - S \<Longrightarrow> less_eq x' x; \<And>x'. x' \<in> S \<Longrightarrow> less_eq x x'\<rbrakk> \<Longrightarrow> I (S - {x}) (f x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<And>x x'. \<lbrakk>x \<in> S; x' \<in> S0 - S\<rbrakk> \<Longrightarrow> less_eq x' x; \<not> c \<sigma>; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
t_def: shows "t = (\<lambda>x. x + a) ` S"
natural_isomorphism_axioms: shows "natural_isomorphism B.VV.comp (\<cdot>\<^sub>B) (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>) (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>) (\<lambda>\<mu>\<nu>. fst \<mu>\<nu> \<star>\<^sub>B snd \<mu>\<nu>)"
assms: fixes t :: "real" assumes "a < t"   and "t < b" shows "P t = c"
e: shows "((\<lambda>x. F (\<gamma>2 (\<phi> x)) \<bullet> b * (vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1}) \<bullet> b)) has_integral integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))) {0..1}"
a: shows "\<not> x \<sharp> idn \<Gamma> b"