hcju/isabelleps huggingface dataset:
bval_eq_if_eq_le: fixes l :: "nat"   and s\<^sub>1 :: "char list \<Rightarrow> int"   and s\<^sub>2 :: "char list \<Rightarrow> int"   and b :: "bexp" assumes "s\<^sub>1 = s\<^sub>2 (\<le> l)"   and "sec b \<le> l" shows "bval b s\<^sub>1 = bval b s\<^sub>2"
prems_3: shows "causal_program_history is\<^sub>s\<^sub>b ((r # sb) @ sb')"
disj_def: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "P \<or> Q \<equiv> \<lambda>A. P A \<or> Q A"
ord_divides: fixes a :: "nat"   and d :: "nat"   and n :: "nat" shows "[a ^ d = 1] (mod n) = (ord n a dvd d)"
1: shows "Can (Inv (u\<^bold>\<down>) \<^bold>\<cdot> t\<^bold>\<down>) \<and> Dom (Inv (u\<^bold>\<down>) \<^bold>\<cdot> t\<^bold>\<down>) = t \<and> Cod (Inv (u\<^bold>\<down>) \<^bold>\<cdot> t\<^bold>\<down>) = u"
g_def: shows "g = (\<lambda>x. 1 / of_nat n * (\<Sum>m<n. \<chi> m * unity_root n (- int x * int m)))"
ord0_prec_def: shows "ord0_prec = (\<Union>f i. {(f i, ord0_Lim f)})"
meas_gim: fixes T :: "(real, 'n) vec set" assumes "T \<subseteq> g ` S"   and "T \<in> sets lebesgue" shows "{x \<in> S. g x \<in> T} \<in> sets lebesgue"
ring_distrib1: fixes R :: "('d, 'e) Ring_scheme"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> (y \<plusminus>\<^bsub>R\<^esub> z) = x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<plusminus>\<^bsub>R\<^esub> x \<cdot>\<^sub>r\<^bsub>R\<^esub> z"
wt_start_def: fixes P :: "'a prog"   and C :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and mxl\<^sub>0 :: "nat"   and \<tau>s :: "(ty list \<times> ty err list) option list" shows "wt_start P C b Ts mxl\<^sub>0 \<tau>s \<equiv> case b of Static \<Rightarrow> P \<turnstile> \<lfloor>([], map OK Ts @ replicate mxl\<^sub>0 Err)\<rfloor> \<le>' \<tau>s ! 0 | NonStatic \<Rightarrow> P \<turnstile> \<lfloor>([], OK (Class C) # map OK Ts @ replicate mxl\<^sub>0 Err)\<rfloor> \<le>' \<tau>s ! 0"
interaction_bounded_by_restrict_gpvI: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat"   and \<I> :: "('d, 'f) \<I>" assumes "interaction_bounded_by consider gpv n" shows "interaction_bounded_by consider (restrict_gpv \<I> gpv) n"
minus_add: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> (x \<oplus>\<^bsub>G\<^esub> y) = \<ominus>\<^bsub>G\<^esub> x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> y"
diagonal_step_dvd1: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and j :: "nat" assumes "is_bezout_ext bezout" shows "diagonal_step A i j (case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> d) (case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> v) $ from_nat i $ from_nat i dvd A $ from_nat i $ from_nat i"
dep_def: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and v1 :: "'d"   and v2 :: "'d" shows "dep PROB v1 v2 \<equiv> (\<exists>a. a \<in> PROB \<and> (v1 \<in> fmdom' (fst a) \<and> v2 \<in> fmdom' (snd a) \<or> v1 \<in> fmdom' (snd a) \<and> v2 \<in> fmdom' (snd a))) \<or> v1 = v2"
sumE: fixes s :: "'m + 'n" obtains x :: "'m" where "s = Inl x"   | y :: "'n" where "s = Inr y"
rx_le_Ax: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "x \<in> X" shows "le_vec (r x *s x) (A *v x)"
simps_17: shows "OUTfromCh data17 = {data15}"
a_star_Cut: fixes M :: "trm"   and M' :: "trm"   and N :: "trm"   and N' :: "trm"   and a :: "coname"   and x :: "name" assumes "M \<longrightarrow>\<^sub>a* M'"   and "N \<longrightarrow>\<^sub>a* N'" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>a* Cut <a>.M' x.N'"
filter_inf_closed: fixes F :: "'a set"   and x :: "'a"   and y :: "'a" assumes "ord_class.filter F"   and "x \<in> F"   and "y \<in> F" shows "x \<sqinter> y \<in> F"
ereal_diff_add_eq_diff_diff_swap: fixes y :: "ereal"   and x :: "ereal"   and z :: "ereal" assumes "\<bar>y\<bar> \<noteq> \<infinity>" shows "x - (y + z) = x - y - z"
eq_iff: fixes x :: "'c list"   and y :: "'c list" shows "(x = y) = (prefix x y \<and> prefix y x)"
normalize_field: fixes x :: "'b" shows "normalize x = (if x = (0::'b) then 0::'b else (1::'b))"
create_element_def: fixes document_ptr :: "(_) document_ptr"   and tag :: "char list" shows "create_element document_ptr tag = Heap_Error_Monad.bind new_element (\<lambda>new_element_ptr. Heap_Error_Monad.bind (set_tag_name new_element_ptr tag) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes document_ptr (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr # disc_nodes)) (\<lambda>_. return new_element_ptr))))"
index_smult_mat_3: fixes a :: "'b"   and A :: "'b mat" shows "dim_col (a \<cdot>\<^sub>m A) = dim_col A"
parZero: fixes P :: "pi" shows "P \<parallel> \<zero> \<simeq> P"
simps_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "not_shadows_vconsts (Vabs cs \<Gamma>) = (pred_fmap id (fmmap not_shadows_vconsts \<Gamma>) \<and> list_all (\<lambda>(pat, t). fdisjnt all_consts (frees pat) \<and> \<not> shadows_consts t) cs)"
one: shows "\<forall>i\<in>A. [y = u i] (mod m i)"
wf_bounded_supset: fixes S :: "'b set" assumes "finite S" shows "wf {(Q', Q). Q \<subset> Q' \<and> Q' \<subseteq> S}"
hyps: fixes x0 :: "real \<times> real"   and x :: "real \<times> real" assumes "ccw'.sortedP x0 (map ((+) x \<circ> g) xs)" shows "ccw'.sortedP (x0 - x) (map g xs)"
assms: shows "fold (\<lambda>(T, U) subs. Option.bind subs (assoc_match T U)) (zip (x # xs) (y # ys)) (Some subs) = Some subs'"
UN_extend_simps_10: fixes B :: "'f \<Rightarrow> 'd set"   and f :: "'e \<Rightarrow> 'f"   and A :: "'e set" shows "(\<Union>a\<in>A. B (f a)) = \<Union> (B ` f ` A)"
itrm_id: fixes y :: "'a itrm" shows "Pure \<I> \<diamondop> y \<simeq> y"
INT_decseq_offset: fixes F :: "nat \<Rightarrow> 'c set"   and n :: "nat" assumes "decseq F" shows "\<Inter> (range F) = \<Inter> (F ` {n..})"
all_nat: fixes P :: "nat \<Rightarrow> bool" shows "(\<forall>x. P x) = (\<forall>x\<ge>0. P (nat x))"
is_cparserE: fixes p :: "char list \<Rightarrow> char list + 'a \<times> char list" assumes "is_cparser p" obtains "\<And>s r x. p s = Inr (x, r) \<Longrightarrow> length r < length s"
interaction_bound_Done: fixes "consider" :: "'d \<Rightarrow> bool"   and x :: "'e" shows "interaction_bound consider (Generative_Probabilistic_Value.Done x) = 0"
tests_def: shows "tests = concat_lists (map (\<lambda>(v, j). map (Pair j) (if j = 0 then dp v else df v)) (map (\<lambda>j. (poly P j, j)) js))"
scaleR_cont: fixes f :: "'a \<Rightarrow> 'b"   and a :: "real" assumes "f \<in> bcontfun" shows "(\<lambda>x. a *\<^sub>R f x) \<in> bcontfun"
PO_l2_inv3: shows "reach l2 \<subseteq> l2_inv3"
G_completeness: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) G.efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "('b, 'c) epsym \<Rightarrow> univ list \<Rightarrow> bool" assumes "ProblemIkTpartG_GEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl eintT eintF eintP" shows "CM.Model wtFsym wtPsym arOf resOf parOf \<Phi> (ProblemIkTpartG_GEModel.intT prot protFw eintT eintP) (ProblemIkTpartG_GEModel.intF eintF) (ProblemIkTpartG_GEModel.intP eintP)"
Sup_set_fold: 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 xs :: "'b list" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (set xs) = fold sup xs bot"
disconnected_nodes_eq2_h2: fixes doc_ptr :: "(_) document_ptr" shows "|h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h' \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
check_tableau: fixes s :: "('i, 'a) state"   and v :: "nat \<Rightarrow> 'a" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "let s' = check s in v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> s' \<and> \<triangle> (\<T> s') \<and> \<nabla> s'"
cart_eq_inner_axis: fixes a :: "(real, 'a) Finite_Cartesian_Product.vec"   and i :: "'a" shows "a $h i = a \<bullet> axis i 1"
pdevs_val_pdevs_of_list_inlE: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and X :: "(real \<times> real) pdevs" assumes "e \<in> UNIV \<rightarrow> I"   and "uminus ` I = I"   and "0 \<in> I" obtains e' :: "nat \<Rightarrow> real" where "pdevs_val e (pdevs_of_list (inl X)) = pdevs_val e' X" and "e' \<in> UNIV \<rightarrow> I"
1: shows "finite {f. dom f \<subseteq> nodes A \<and> ran f \<subseteq> {0..2 * card (nodes A)}}"
space_measure_of_conv: fixes \<Omega> :: "'b set"   and A :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal" shows "space (measure_of \<Omega> A \<mu>) = \<Omega>"
lapprox_rat: fixes prec :: "nat"   and x :: "int"   and y :: "int" shows "real_of_float (lapprox_rat prec x y) \<le> real_of_int x / real_of_int y"
th: shows "(\<exists>x. Ifm (x # bs) p) = (\<exists>x. Ifm (x # bs) (rlfm (simpfm p)))"
f_def: fixes b :: "'b set" shows "f b = (b \<subseteq> U\<^sub>V\<^sub>S \<and> \<not> local.dependent b \<and> local.span b = local.span V)"
bdd_above_def: fixes A :: "'b set" shows "bdd_above A = (\<exists>M. \<forall>x\<in>A. x \<le> M)"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>local.inv f : local.inv a \<rightarrow>\<^sub>A local.inv b\<guillemotright>"
sw_ident: fixes ys :: "'b"   and y :: "'a"   and xs :: "'b"   and x :: "'a" shows "x @xs[y \<and> y]_ys = x"
INF_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'d set"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. less_eq (f n) (g m)" shows "less_eq (Inf (f ` A)) (Inf (g ` B))"
inv_bval'_correct: fixes s :: "char list \<Rightarrow> int"   and S :: "'av st option"   and bv :: "bool"   and b :: "bexp" assumes "s \<in> \<gamma>\<^sub>o S"   and "bv = bval b s" shows "s \<in> \<gamma>\<^sub>o (inv_bval' b bv S)"
xor_list_assoc: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs [\<oplus>] ys) [\<oplus>] zs = xs [\<oplus>] ys [\<oplus>] zs"
card_insert_if: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "card (insert x A) = (if x \<in> A then card A else Suc (card A))"
linear_compose: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and s3 :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "Vector_Spaces.linear s1 s2 f"   and "Vector_Spaces.linear s2 s3 g" shows "Vector_Spaces.linear s1 s3 (g \<circ> f)"
assms_1: shows "A \<turnstile>\<^sub>C callee1(s1) \<approx> callee2(s2)"
le_supI: fixes a :: "'d multiset"   and x :: "'d multiset"   and b :: "'d multiset" assumes "a \<subseteq># x"   and "b \<subseteq># x" shows "a \<union># b \<subseteq># x"
auto_split_lemma: fixes A :: "'d \<times> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<times> ('d \<Rightarrow> bool)"   and ps :: "'c list"   and q :: "'d"   and res :: "'c list \<times> 'c list"   and xs :: "'c list" shows "auto_split A (delta A ps q) res ps xs = maxsplit (\<lambda>ys. fin A (delta A ys q)) res ps xs"
simps_2: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c Trace"   and x :: "'c" shows "tMap f (xs \<leadsto> x) = tMap f xs \<leadsto> f x"
lex_pp_strict_alt: fixes s :: "('a, 'b) pp"   and t :: "('a, 'b) pp" shows "lex_pp_strict s t = (\<not> lex_pp t s)"
vec_eq: fixes m :: "'a"   and n :: "'a" shows "(vec m = vec n) = (m = n)"
elims_2: fixes x :: "('a \<times> constraint) list"   and xa :: "'a set"   and xb :: "'a simplex_state" assumes "checked_simplex x xa xb" obtains cs :: "('a \<times> constraint) list" and J :: "'a set" and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" where "x = cs" and "xa = J" and "xb = Simplex_State s" and "checked_simplex' cs J s"
respectfulness_4: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl"   and \<nu> :: "'b set" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "af_G_letter_opt \<phi> \<nu> \<equiv>\<^sub>P af_G_letter_opt \<psi> \<nu>"
stateful_policy_to_network_graph_def: fixes \<T> :: "'a stateful_policy" shows "stateful_policy_to_network_graph \<T> = \<lparr>nodes = hosts \<T>, edges = all_flows \<T>\<rparr>"
plus_Mp_1: fixes f :: "int poly"   and g :: "int poly" shows "r.eq_m (r.Mp f + g) (f + g)"
epsclo_nextl: fixes Q :: "hf set"   and xs :: "'a list" shows "ST.epsclo (ST.nextl Q xs) = ST.nextl Q xs"
cccFix_nonrec_eq: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and x :: "Vars.var"   and e :: "exp"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "CoCallArityAnalysis.cccFix_nonrec cccExp x e\<cdot>i = (CoCallArityAnalysis.Afix_nonrec cccExp x e\<cdot>i, CoCallArityAnalysis.CCfix_nonrec cccExp x e\<cdot>i)"
S5n_def: fixes M :: "('e, 'f, 'g) KripkeStructure" shows "S5n M \<equiv> \<forall>a. equiv (worlds M) (relations M a)"
bij_to_nat: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "bij_betw (mod_type.to_nat Rep) UNIV {0..<CARD('d)}"
fls_inverse_def': fixes f :: "'a fls" shows "inverse f = fls_right_inverse f (inverse (fls_nth f (fls_subdegree f)))"
arev_eq: fixes G :: "('b, 'c) pre_digraph"   and arev :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "bidirected_digraph G arev"   and "a \<notin> arcs G" shows "arev a = a"
x2_def: shows "x2 = Min (insert (x + 1) ((\<lambda>X. x + (Sup (Re ` X) - x) / 2) ` T))"
not_conga_sym: 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 D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "\<not> Tarski_neutral_dimensionless.CongA Bet Cong D E F A B C"
scc_of_def: fixes u :: "'a" shows "G.scc_of u \<equiv> {v. u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v \<and> v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> u}"
f_nonneg: fixes i :: "nat" shows "0 \<le> int (aa i) gchoose (i - j) + (int (bb i) gchoose (i - j))"
prems_1: shows "v \<notin> ifex_var_set (IF v1 t e)"
iimg: fixes a :: "'a"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "a \<in> B" shows "iimg B f (f a) = f a"
Crypt_imp_invKey_keysFor: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> H" shows "invKey K \<in> keysFor H"
nchotomy_relcomppE: fixes f :: "'e \<Rightarrow> 'd"   and r :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'f"   and c :: "'g" assumes "\<And>y. \<exists>x. y = f x"   and "(r OO s) a c" obtains b :: "'e" where "r a (f b)" and "s (f b) c"
pratt_primeI: fixes xs :: "pratt list"   and p :: "nat" assumes "valid_cert xs"   and "Prime p \<in> set xs" shows "prime p"
axioms_1: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.comm_monoid G" shows "Group.monoid G"
neutr_eq_iff: fixes a :: "'c"   and b :: "'c" shows "(top = inf a b) = (a = top \<and> b = top)"
pderiv_monom: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<in> carrier Zp" shows "pderiv (monom (UP Zp) a n) = monom (UP Zp) ([n] \<cdot> a) (n - 1)"
list_all2_last: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b"   and ys :: "'c list"   and y :: "'c" shows "list_all2 A (xs @ [x]) (ys @ [y]) = (list_all2 A xs ys \<and> A x y)"
openin_clauses_1: fixes U :: "'c topology" shows "openin U {}"
lm026: fixes P1 :: "'b set set"   and P2 :: "'b set set"   and X :: "'b set"   and Y :: "'b set" assumes "\<Union> P1 \<inter> \<Union> P2 = {}"   and "is_non_overlapping P1"   and "is_non_overlapping P2"   and "X \<in> P1 \<union> P2"   and "Y \<in> P1 \<union> P2"   and "X = Y" shows "X \<inter> Y \<noteq> {}"
arr_cod: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.VV.arr (C.VV.cod f)"
oracle_def: fixes len :: "nat"   and \<sigma> :: "'a \<Rightarrow> bool list option"   and x :: "'a" shows "hash_oracle.oracle len \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV len) \<bind> (\<lambda>bs. return_spmf (bs, \<sigma>(x \<mapsto> bs))) | Some bs \<Rightarrow> return_spmf (bs, \<sigma>))"
PO_l3_inv4: shows "reach l3 \<subseteq> l3_inv4"
spmf_bind: fixes p :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf"   and y :: "'c" shows "spmf (p \<bind> f) y = LINT x|measure_spmf p. spmf (f x) y"
mat_mult: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'a list list"   and nc :: "nat"   and m2 :: "'a list list"   and ze :: "'a"   and pl :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and ti :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "Matrix_Legacy.mat nr n m1"   and "Matrix_Legacy.mat n nc m2" shows "Matrix_Legacy.mat nr nc (mat_multI ze pl ti nr m1 m2)"
P9: shows "Bet C E C' \<and> Bet D E D'"
assm: shows "v=v1--v2 \<and> restrict v1 (res ts) c \<noteq> bot \<and> consec (lan v1) (lan v2) \<and> \<not> consec (restrict v1 (res ts) c) (restrict v2 (res ts) c)"
subgraph_trans: fixes G'' :: "nat set \<times> nat set set"   and G' :: "nat set \<times> nat set set"   and G :: "nat set \<times> nat set set" assumes "subgraph G'' G'"   and "subgraph G' G" shows "subgraph G'' G"
isEmpty_correct: fixes s :: "'queue" assumes "queue_invar s" shows "queue_isEmpty s = (queue_\<alpha> s = [])"
assms_2: shows "inj_on (\<lambda>b. x * b mod q) ({..<q} - {0})"
noread: fixes v :: "'Val" shows "\<langle>c, mds, mem(x := v)\<rangle> \<leadsto> \<langle>c', mds', mem'(x := v)\<rangle>"
qSwap_compose: fixes zs' :: "'c"   and x' :: "'d"   and y' :: "'d"   and zs :: "'c"   and x :: "'d"   and y :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "X #[[x \<and> y]]_zs #[[x' \<and> y']]_zs' = X #[[x' \<and> y']]_zs' #[[x @zs[x' \<and> y']_zs' \<and> y @zs[x' \<and> y']_zs']]_zs"
in_sets_embed_measure: fixes A :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "A \<in> sets M" shows "f ` A \<in> sets (embed_measure M f)"
le_unat_uoi: fixes y :: "nat"   and z :: "'a word" assumes "y \<le> unat z" shows "unat (word_of_nat y) = y"
s_def: fixes t :: "real" shows "other.s t = (if t \<le> 0 then s\<^sub>o else if t \<le> other.t_stop then other.p t else other.p_max)"
sym_symcl: fixes Rel :: "('a \<times> 'a) set" shows "sym (symcl Rel)"
isSubobj_isClass: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and R :: "char list \<times> char list list" assumes "is_subobj P R" shows "is_class P (mdc R)"
finiteHom_def: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme"   and f :: "'h \<Rightarrow> 'i" shows "finiteHom I A f = (f \<in> carr_prodag I A \<and> (\<exists>H\<subseteq>I. finite H \<and> (\<forall>j\<in>I - H. f j = \<zero>\<^bsub>A j\<^esub>)))"
plus_right_distributivity: fixes a :: "int poly fract"   and b :: "int poly fract"   and c :: "int poly fract" shows "rat_poly_times (rat_poly_plus a b) c = rat_poly_plus (rat_poly_times a c) (rat_poly_times b c)"
link_mset: fixes t1 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and t2 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.tree_to_multiset (SkewBinomialHeapStruc.link t1 t2) = SkewBinomialHeapStruc.tree_to_multiset t1 + SkewBinomialHeapStruc.tree_to_multiset t2"
MergeGuardsD_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P merge_guards c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
pos_single: fixes x :: "real" shows "pos [x] = (0 < x)"
assms_1: shows "za \<noteq> \<infinity>\<^sub>h"
assms_6: fixes j :: "nat" assumes "j < m" shows "fs ! j \<in> carrier_vec n"
rg_l_unit: fixes a :: "'a" assumes "a \<in> carrier R" shows "1\<^sub>r \<cdot>\<^sub>r a = a"
bisim_must_sync_preserve1: fixes t :: "'t"   and x1 :: "'x1"   and m1 :: "'m1"   and x2 :: "'x2"   and m2 :: "'m2" assumes "t \<turnstile> (x1, m1) \<approx> (x2, m2)"   and "t \<turnstile> \<langle>x1, m1\<rangle> \<wrong>1" shows "t \<turnstile> \<langle>x2, m2\<rangle> \<wrong>2"
cs: shows "(c, s) \<rightarrow>*c (c'', s'')"
clop_idem_var: fixes x :: "'b" shows "cl_op (cl_op x) = cl_op x"
bv_chop_length_fst: fixes w :: "bit list"   and i :: "nat" shows "length (fst (bv_chop w i)) = length w - i"
1: shows "subst \<rho> (chop s) = emb_step_at (replicate (num_args (fun s)) dir.Left) dir.Right (subst \<rho> s)"
prems: fixes z :: "'a" assumes "z \<in> set (y # xs)" shows "f z = f x"
integrable_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integrable (restrict_space M \<Omega>) f = integrable M (\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x)"
contour_integral_bound_part_circlepath_strong: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real"   and k :: "complex set"   and B :: "real" assumes "f contour_integrable_on part_circlepath z r s t"   and "finite k"   and "0 \<le> B"   and "0 < r"   and "s \<le> t"   and "\<And>x. x \<in> path_image (part_circlepath z r s t) - k \<Longrightarrow> cmod (f x) \<le> B" shows "cmod (contour_integral (part_circlepath z r s t) f) \<le> B * r * (t - s)"
smult_smult: fixes a :: "'b"   and b :: "'b"   and p :: "'b poly" shows "Polynomial.smult a (Polynomial.smult b p) = Polynomial.smult (a * b) p"
list_ex_simps_1: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "list_ex P (x # xs) = (P x \<or> list_ex P xs)"