hcju/isabelleps huggingface dataset:
StaticCallRedsFinal: 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 e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and vs :: "val list"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and C :: "char list"   and Cs'' :: "char list list"   and Ds :: "char list list"   and Cs' :: "char list list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "expr"   and vs' :: "val list"   and l\<^sub>2' :: "char list \<Rightarrow> val option"   and ef :: "expr"   and h\<^sub>3 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>3 :: "char list \<Rightarrow> val option" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>ref (a, Cs),s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "P \<turnstile> Path last Cs to C unique"   and "P \<turnstile> Path last Cs to C via Cs'' "   and "Ds = (Cs @\<^sub>p Cs'') @\<^sub>p Cs'"   and "P \<turnstile> C has least M = (Ts, T, pns, body) via Cs'"   and "length vs = length pns"   and "P \<turnstile> Ts Casts vs to vs' "   and "l\<^sub>2' = [this \<mapsto> Ref (a, Ds), pns [\<mapsto>] vs']"   and "P,E(this \<mapsto> Class (last Ds), pns [\<mapsto>] Ts) \<turnstile> \<langle>body,(h\<^sub>2, l\<^sub>2')\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>3)\<rangle>"   and "final ef" shows "P,E \<turnstile> \<langle>Call e \<lfloor>C\<rfloor> M es,s\<^sub>0\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, l\<^sub>2)\<rangle>"
OUTfromVCorrect1_data23: shows "OUTfromVCorrect1 data23"
lens_comp_def: fixes Y :: "'b \<Longrightarrow> 'd"   and X :: "'d \<Longrightarrow> 'c" shows "Y ;\<^sub>L X = \<lparr>lens_get = get\<^bsub>Y\<^esub> \<circ> get\<^bsub>X\<^esub>, lens_put = \<lambda>\<sigma> v. put\<^bsub>X\<^esub> \<sigma> (put\<^bsub>Y\<^esub> (get\<^bsub>X\<^esub> \<sigma>) v)\<rparr>"
distinct_31: fixes x4 :: "nat"   and x5 :: "msg" shows "Key x4 \<noteq> Hash x5"
exhaust: fixes y :: "'b literal" obtains x1 :: "'b" where "y = x1\<^sup>+"   | x2 :: "'b" where "y = x2\<inverse>"
relcomp_unfold: fixes r :: "('a \<times> 'c) set"   and s :: "('c \<times> 'b) set" shows "r ; s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>f\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
D_base_finite'': fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "finite {len m i j xs |xs. set xs \<subseteq> {0..k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs}"
fps_sub_nth: fixes f :: "'b fps"   and g :: "'b fps"   and n :: "nat" shows "(f - g) $ n = f $ n - g $ n"
simps: fixes a :: "msg"   and H :: "msg set" shows "(a \<in> parts H) = ((\<exists>X. a = X \<and> X \<in> H) \<or> (\<exists>X Y. a = X \<and> \<langle>X, Y\<rangle> \<in> parts H) \<or> (\<exists>X Y. a = Y \<and> \<langle>X, Y\<rangle> \<in> parts H) \<or> (\<exists>X Y. a = X \<and> Enc X Y \<in> parts H) \<or> (\<exists>X Y. a = X \<and> Aenc X Y \<in> parts H) \<or> (\<exists>X Y. a = X \<and> Sign X Y \<in> parts H))"
univ_poly_def: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" shows "K [X]\<^bsub>R\<^esub> = \<lparr>carrier = {p. polynomial\<^bsub>R\<^esub> K p}, monoid.mult = ring.poly_mult R, one = [\<one>\<^bsub>R\<^esub>], zero = [], add = ring.poly_add R\<rparr>"
splitFace_edge_disj: fixes g :: "graph"   and u :: "nat"   and v :: "nat"   and f :: "face"   and vs :: "nat list"   and f\<^sub>1 :: "face"   and f\<^sub>2 :: "face"   and g' :: "graph" assumes "minGraphProps g"   and "pre_splitFace g u v f vs"   and "(f\<^sub>1, f\<^sub>2, g') = splitFace g u v f vs" shows "edges_disj g'"
simps_4: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'h list \<Rightarrow> 'g" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
psubst_eval: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'b"   and p :: "('d, 'c) form" shows "eval e f g (psubst h p) = eval e (\<lambda>p. f (h p)) g p"
U: shows "(map_mat int_of_rat A)\<^sup>T \<in> carrier_mat n n"
compact_components: fixes s :: "'a set"   and c :: "'a set" assumes "compact s"   and "c \<in> components s" shows "compact c"
finite_subset: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subseteq> B"   and "finite B" shows "finite A"
gc2_def: shows "gc2 \<equiv> SOME gc2. 0 < gc2 \<and> (\<forall>\<^sub>F x in sequentially. gc2 * g' (real x) \<le> g x)"
pairs_phi_less_eq: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter" assumes "triple phi"   and "(x, y) \<in> triple.pairs phi" shows "phi (- x) \<le> y"
finite_imageD: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "finite (f ` A)"   and "inj_on f A" shows "finite A"
closest_point_def: fixes S :: "'b set"   and a :: "'b" shows "closest_point S a = (SOME x. x \<in> S \<and> (\<forall>y\<in>S. dist a x \<le> dist a y))"
rejects_prefix: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list"   and t' :: "(String.literal \<times> value list) list" assumes "\<not> recognises_execution e s r t" shows "\<not> recognises_execution e s r (t @ t')"
min_eq_arg_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a"   and n :: "'a" assumes "class.linorder less_eq less" shows "(ord.min less_eq m n = m) = less_eq m n"
holomorphic_perzeta: fixes q :: "real"   and A :: "complex set" assumes "q \<notin> \<int> \<or> 1 \<notin> A" shows "perzeta q holomorphic_on A"
simps: fixes cms :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)"   and mem' :: "'Var \<Rightarrow> 'Val"   and X :: "'Var set" shows "change_respecting (cms, mem) (cms', mem') X = ((cms, mem) \<leadsto> (cms', mem') \<and> (\<forall>\<sigma>. dom \<sigma> = vars_and_\<C> X \<longrightarrow> (cms, subst \<sigma> mem) \<leadsto> (cms', subst \<sigma> mem')))"
E\<^sub>i: fixes x :: "i"   and y :: "i" shows "E (x \<cdot> y) \<^bold>\<leftarrow> E x \<^bold>\<and> E y \<^bold>\<and> \<^bold>\<not> (\<^bold>\<forall>xa. \<^bold>\<not> (xa \<cdot> xa \<cong> xa \<^bold>\<and> x \<cdot> xa \<cong> x \<^bold>\<and> xa \<cdot> y \<cong> y))"
select_convs_1: fixes getM :: "(Literal \<times> bool) list"   and more :: "'a" shows "getM \<lparr>getM = getM, \<dots> = more\<rparr> = getM"
typ_of1_weaken_Ts: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ"   and T :: "typ" assumes "typ_of1 Ts t = Some ty" shows "typ_of1 (Ts @ [T]) t = Some ty"
cbelow_on_induct: fixes A :: "'c set"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "cbelow_on A ord z rel a b"   and "\<lbrakk>a \<in> A; ord a z\<rbrakk> \<Longrightarrow> P a"   and "\<And>b c. \<lbrakk>cbelow_on A ord z rel a b; rel b c; c \<in> A; ord c z; P b\<rbrakk> \<Longrightarrow> P c" shows "P b"
map_fst_map_const_snd: fixes b :: "'b"   and xs :: "'a list" shows "map fst (map (\<lambda>s. (s, b)) xs) = xs"
mem_existence_ivl_iv_defined_1: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t - t0 \<in> existence_ivl0 x0" shows "t0 \<in> UNIV"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(extB.Val x2 = extB.Val y2) = (x2 = y2)"
unifier\<^sub>l\<^sub>s_unifier\<^sub>t\<^sub>s: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" assumes "unifier\<^sub>l\<^sub>s \<sigma> L" shows "unifier\<^sub>t\<^sub>s \<sigma> (literal_to_term ` L)"
enc_reflects_pred_iff_indRelL_preserves_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_reflects_pred Pred = rel_preserves_pred indRelL Pred"
map_option_case: fixes f :: "'b \<Rightarrow> 'a"   and y :: "'b option" shows "map_option f y = (case y of None \<Rightarrow> None | \<lfloor>x\<rfloor> \<Rightarrow> \<lfloor>f x\<rfloor>)"
finite_fv_set: fixes M :: "('a, 'b) Term.term set" assumes "finite M" shows "finite (fv\<^sub>s\<^sub>e\<^sub>t M)"
normalize_mult: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize" shows "normalize (times a b) = times (normalize a) (normalize b)"
\<pi>0: shows "\<pi> 0 = \<pi>' 0"
assms_2: shows "mi \<le> \<Down>\<^sub>E E R m"
simps_1: fixes x21 :: "'a"   and x22 :: "'a"   and x23 :: "'b"   and y21a :: "'a"   and y22a :: "'a"   and y23a :: "'b" shows "(MkArr x21 x22 x23 = MkArr y21a y22a y23a) = (x21 = y21a \<and> x22 = y22a \<and> x23 = y23a)"
uint_mask_eq: fixes n :: "nat" shows "uint (mask n) = mask (min LENGTH('a) n)"
simps_2: fixes m :: "nat"   and n :: "nat" shows "(Suc m \<le> n) = (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
Node_5: shows "well_valued0 as (Node i j a t1 t2)"
Elementary_divisor_ring_imp_Hermite_ring: assumes "OFCLASS('a, elementary_divisor_ring_class)" shows "OFCLASS('a, Hermite_ring_class)"
hconf_start: fixes wf_mb :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'a \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list" assumes "wf_prog wf_mb P" shows "P \<turnstile> start_heap P \<surd>"
oddE: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" obtains b :: "'c" where "a = plus (times (numeral.numeral one plus (num.Bit0 num.One)) b) one"
not_is_unit_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "\<not> dvd.dvd times zero one"
complex_mult: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" shows "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
s_is_partial_execution: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "distributed_system channel" shows "distributed_system.s channel trans send recv c t i = distributed_system.run_trace channel trans send recv (c, take i t)"
has_integral_cbox: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'b"   and a :: "'a"   and b :: "'a" shows "(f has_integral I) (cbox a b) = (sum (\<lambda>(x, k). content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter (cbox a b))"
op_set_npick_remove_def: fixes X :: "'e set" shows "op_set_npick_remove X = SPEC (\<lambda>(x, X'). x \<in> X \<and> (X' = X - {x} \<or> X' = X))"
nn_integral_measure_spmf_conv_measure_pmf: fixes f :: "'c \<Rightarrow> ennreal"   and p :: "'c spmf" assumes "f \<in> borel_measurable (count_space UNIV)" shows "integral\<^sup>N (measure_spmf p) f = integral\<^sup>N (restrict_space (measure_pmf p) (range Some)) (f \<circ> the)"
sees_method_mono2: fixes C' :: "char list"   and C :: "char list"   and P :: "'a prog"   and wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list"   and b' :: "staticb"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "'a"   and D' :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D"   and "P \<turnstile> C' sees M, b' :  Ts'\<rightarrow>T' = m' in D'" shows "b = b' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> subtype P T' T"
fin_maintained_def: fixes R :: "('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph"   and G :: "('d, 'f) labeled_graph" shows "fin_maintained R G \<equiv> \<forall>F f. finite_graph F \<longrightarrow> subgraph F (fst R) \<longrightarrow> extensible (F, fst R) G f \<longrightarrow> graph_homomorphism F G f \<longrightarrow> extensible (F, snd R) G f"
le0: fixes xs :: "nat list" shows "zeroes (length xs) \<le>\<^sub>v xs"
tauChainCons: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" obtains P'' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> \<tau>.P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P''" and "\<Psi> \<rhd> P' \<sim> P''"
assms_5: shows "K1 \<inter> {x. c \<le> x \<bullet> k} = K2 \<inter> {x. c \<le> x \<bullet> k}"
sor: shows "(\<sigma>, s) \<in> oreachable (\<langle>i : \<zeta>\<^sub>i : R\<^sub>i\<rangle>\<^sub>o) S U"
nth_take_prefix: fixes ys :: "'c list"   and xs :: "'c list" assumes "length ys \<le> length xs"   and "\<forall>i<length ys. xs ! i = ys ! i" shows "take (length ys) xs = ys"
prems_1: shows "dims (subtensor_combine ds As) \<noteq> []"
restrict_top_impl_def: fixes e :: "nat"   and vr :: "nat"   and vl :: "bool"   and s :: "bdd" shows "mi.restrict_top_impl e vr vl s = mi.case_ifexi (\<lambda>s. Some (e, s)) (\<lambda>s. Some (e, s)) (\<lambda>v te ee s. Some (if v = vr then if vl then te else ee else e, s)) e s"
ask_option_parametric: fixes R :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and M :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "(((R ===> M) ===> M) ===> (R ===> rel_optionT M) ===> rel_optionT M) ask_option ask_option"
gcd_left_idem: fixes a :: "'a"   and b :: "'a" shows "gcd a (gcd a b) = gcd a b"
P: shows "extreme {q \<in> Q. F q = q} (\<lambda>x y. y \<sqsubseteq>\<^sup>s x) P"
exhaust: fixes y :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" obtains x1 :: "'a" and x2 :: "'b" and x3 :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" where "y = BsSkewBinomialHeapStruc.Element x1 x2 x3"
AsSeqExpr1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a2 :: "('e, 'd, 'f) ann"   and A :: "'e set"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com"   and a :: "'e set"   and Q :: "'e set"   and c2 :: "('e, 'd, 'f) com" assumes "assertionsR \<Gamma> \<Theta> (pre a2) A a1 c1 a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnComp a1 a2) (Seq c1 c2) a"
filter_IFS_no_violations_subseteq_input: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and Es :: "('b \<times> 'b) list" shows "set (filter_IFS_no_violations G M Es) \<subseteq> set Es"
alphas_abs_eqvt_3: fixes ds :: "atom list"   and x :: "'a"   and es :: "atom list"   and y :: "'a"   and p :: "perm" assumes "(ds, x) \<approx>abs_lst (es, y)" shows "(p \<bullet> ds, p \<bullet> x) \<approx>abs_lst (p \<bullet> es, p \<bullet> y)"
add_cancel_left_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus b a = a) = (b = zero)"
dis_log2_def: fixes \<A> :: "'a \<Rightarrow> (nat \<times> nat) spmf" shows "dis_log2 \<A> = TRY sample_uniform (order \<G>) \<bind> (\<lambda>w. let h = \<^bold>g [^] w in \<A> h \<bind> (\<lambda>(w1', w2'). return_spmf [w = w1' + x * w2'] (mod order \<G>))) ELSE return_spmf False"
skelAbs_simp: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "skelAbs (Abs xs x X) = Branch (\<lambda>i. Some (skel X)) Map.empty"
yun_rel_pow: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly"   and n :: "nat" assumes "yun_rel F c f" shows "yun_rel (F ^ n) (c ^ n) (f ^ n)"
cases: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and a :: "('a, 'b) tm" assumes "arity.wary_fo arity_sym arity_var a" obtains s :: "('a, 'b) tm" where "a = s" and "is_Hd s \<or> is_Sym (head s)" and "enat (num_args s) = arity.arity_hd arity_sym arity_var (head s)" and "\<forall>t\<in>set (args s). arity.wary_fo arity_sym arity_var t"
norm_mult_ineq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.real_normed_algebra plus zero minus uminus scaleR times dist norm sgn uniformity open" shows "norm (times x y) \<le> norm x * norm y"
p: shows "\<Turnstile>\<^sub>p {P} S {Q}"
axioms_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu>" shows "identity_arrow_of_spans C \<nu>"
P: shows "P \<in> carrier_mat (dim_row A) (dim_row A)"
poly_mod_add_left: fixes x :: "'b poly"   and y :: "'b poly"   and z :: "'b poly" shows "(x + y) mod z = x mod z + y mod z"
Const_zero: fixes c :: "'a"   and X :: "'a set"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set" assumes "c \<in> X"   and "I c = Regions.intv.Const 0" shows "zero c (Regions.region X I r)"
eqOnUC_0_Suc0: fixes AP' :: "('U, 'C, 'Out) Noninterference.aprop set"   and \<pi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream"   and \<pi>' :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream"   and u :: "'U"   and c :: "'C" assumes "wfp AP' \<pi>"   and "wfp AP' \<pi>'" shows "eqOnUC 0 (Suc 0) u c [\<pi>, \<pi>'] = ((\<exists>st. stateOf \<pi> = State st u c) = (\<exists>st'. stateOf \<pi>' = State st' u c))"
smult_scalar_prod_distrib: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec"   and a :: "'c" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "(a \<cdot>\<^sub>v v\<^sub>1) \<bullet> v\<^sub>2 = a * (v\<^sub>1 \<bullet> v\<^sub>2)"
getEnv_vsubstEnv4: fixes ys :: "'varSort"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and z :: "'var" assumes "freshEnv ys y rho" shows "(rho &[z // y]_ys) ys y = Some (Var ys z)"
assms_2: shows "Rs \<noteq> \<tau> \<prec> P \<parallel> !P"
wt_err_step_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a err \<Rightarrow> (nat \<times> 'a err) list"   and \<tau>s :: "'a err list" shows "wt_err_step r step \<tau>s = wt_step (Err.le r) Err step \<tau>s"
box: shows "\<forall>a ba. (\<exists>sola. sol 0 = sola 0 \<and> (\<exists>t. (a, ba) = mk_v I (OVar vid1) (sola 0, b) (sola t) \<and> 0 \<le> t \<and> (sola solves_ode (\<lambda>a. ODEs I vid1)) {0..t} {x. Predicates I vid1 (\<chi>i. dterm_sem I (local.empty i) (mk_v I (OVar vid1) (sola 0, b) x))})) \<longrightarrow> directional_derivative I (f1 fid2 vid1) (a, ba) \<le> directional_derivative I (f1 fid1 vid1) (a, ba)"
is_direct_product_def: fixes x :: "'a"   and y :: "'a" shows "is_direct_product x y \<equiv> x\<^sup>\<smile> ; x = 1' \<and> y\<^sup>\<smile> ; y = 1' \<and> x ; x\<^sup>\<smile> \<cdot> y ; y\<^sup>\<smile> = 1' \<and> x\<^sup>\<smile> ; y = 1"
flow_initial_time_if: fixes x0 :: "real \<times> real" shows "c.flow0 x0 0 = (if 0 \<in> UNIV \<and> x0 \<in> UNIV then x0 else 0)"
dg_Arr_vsubset_Vset: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<CC>\<lparr>Arr\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
walls_betw_subset_wall_crossings: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "folding_g.gallery (C # Cs @ [D])" shows "order.greater_eq (set (wall_crossings (C # Cs @ [D]))) (walls_betw C D)"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x xs. (\<And>xa y. (xa, y) = partition (\<lambda>y. f y = f x) xs \<Longrightarrow> P y) \<Longrightarrow> P (x # xs)" shows "P a0"
numer_cf3_def: shows "numer_cf3 \<equiv> \<lambda>x. x ^ 3 + 12 * x\<^sup>2 + 60 * x + 120"
height_btree_order: fixes ls :: "('b btree \<times> 'b) list"   and a :: "'b btree \<times> 'b"   and t :: "'b btree" shows "height (Node (ls @ [a]) t) = height (Node (a # ls) t)"
lookup_mult_vec: fixes p :: "'b \<Rightarrow>\<^sub>0 'a"   and q :: "'b \<Rightarrow>\<^sub>0 'a"   and v :: "'b" shows "lookup (punit.mult_vec p q) v = lookup (punit.proj_poly (punit.component_of_term v) p * punit.proj_poly (punit.component_of_term v) q) (punit.pp_of_term v)"
map2_val_compat'D1: fixes f :: "('a \<times> 'd) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o"   and zs :: "('a \<times> 'd) list \<times> 'o" assumes "map2_val_compat' f"   and "oalist_inv zs" shows "oalist_inv (f zs)"
elims: fixes x :: "'a list \<times> 'a list"   and y :: "'a option \<times> 'a list \<times> 'a list" assumes "safe_hd_t x = y" obtains "x = ([], [])" and "y = (None, [], [])"   | l :: "'a" where "x = ([], [l__])" and "y = (Some l__, [], [l__])"   | l :: "'a" and v :: "'a" and va :: "'a list" where "x = ([], la__ # v # va)" and "y = (let fs = rev (v # va) in (Some (hd fs), fs, [la__]))"   | f :: "'a" and fs :: "'a list" and l :: "'a" and ls :: "'a list" where "x = (f__ # fs__, l # ls)" and "y = (Some f__, f__ # fs__, l # ls)"   | f :: "'a" and fs :: "'a list" where "x = (f # fs, [])" and "y = undefined"