hcju/isabelleps huggingface dataset:
updates_conv': fixes ks :: "'c list"   and vs :: "'d list"   and al :: "('c \<times> 'd) list" shows "map_of (updates ks vs al) = map_of al(ks [\<mapsto>] vs)"
op_map_ball_def: fixes m :: "'g \<Rightarrow> 'h option"   and P :: "'g \<times> 'h \<Rightarrow> bool" shows "op_map_ball m P \<equiv> Ball (map_to_set m) P"
cone_axioms: shows "D.cone (C.dom e) (D.mkCone e)"
interaction_any_bounded_converterD: fixes conv :: "('a, 'b, 'c, 'd) converter"   and n :: "enat"   and a :: "'a" assumes "interaction_any_bounded_converter conv n" shows "interaction_any_bounded_by (run_converter conv a) n \<and> (\<forall>(b, conv')\<in>results'_gpv (run_converter conv a). interaction_any_bounded_converter conv' n)"
FSbImorph_def: fixes h :: "'k \<Rightarrow> 'l"   and hA :: "'m \<Rightarrow> 'n"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'k, 'm, 'p) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'l, 'n, 'q) model_scheme" shows "FSbImorph h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> ipresIGCons h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSubstAll h hA MOD MOD'"
fds_nth_plus: fixes f :: "'b fds"   and g :: "'b fds" shows "fds_nth (f + g) = (\<lambda>n. fds_nth f n + fds_nth g n)"
adjunction_unit_determines_counit: 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"   and \<epsilon>' :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>"   and "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>'" shows "\<epsilon> = \<epsilon>'"
simps_8: fixes f :: "'d \<Rightarrow> 'c" shows "map_option f \<bottom> = \<bottom>"
11: shows "prv (RR (encPf prf) \<langle>\<phi>R\<rangle>)"
function_smult_eval: fixes a :: "'a"   and f :: "'c \<Rightarrow> 'a"   and b :: "'c" assumes "a \<in> carrier R"   and "f \<in> carrier F"   and "b \<in> S" shows "(a \<odot>\<^bsub>F\<^esub> f) b = a \<otimes> f b"
unannotated: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val"   and E :: "('Var, 'AExp, 'BExp) Stmt list"   and mds :: "Mode \<Rightarrow> 'Var set" assumes "(c, mem) \<leadsto>\<^sub>s (c', mem')" shows "\<langle>cxt_to_stmt E c, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>cxt_to_stmt E c', mds, mem'\<rangle>\<^sub>w"
IH: fixes x :: "real"   and I :: "float interval" assumes "x \<in>\<^sub>r I" shows "list_ex (in_real_interval x) (subdivide_interval n I)"
power_poly_f_mod_binary: fixes m :: "'a poly"   and a :: "'a poly"   and n :: "nat" shows "power_poly_f_mod m a n = (if n = 0 then 1 mod m else let (d, r) = divmod_nat n 2; rec = power_poly_f_mod m (a * a mod m) d in if r = 0 then rec else rec * a mod m)"
constant_functor_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c" shows "constant_functor A B b \<equiv> category A \<and> category B \<and> constant_functor_axioms B b"
Domainp_iff: fixes T :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c" shows "Domainp T x = (\<exists>y. T x y)"
HSigmaE: fixes c :: "hf"   and A :: "hf"   and B :: "hf \<Rightarrow> hf" assumes "c \<^bold>\<in> HSigma A B" obtains x :: "hf" and y :: "hf" where "x \<^bold>\<in> A" and "y \<^bold>\<in> B x" and "c = \<langle>x, y\<rangle>"
ket_def: shows "ket \<equiv> map_fun id Abs_ell2 (\<lambda>x y. if x = y then 1 else 0)"
rel_chain_def: fixes l :: "'b list"   and r :: "('b \<times> 'b) set" shows "rel_chain l r = (\<forall>k<length l - 1. (l ! k, l ! (k + 1)) \<in> r)"
map_ide_simp: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.ide a" shows "\<alpha>' a = local.inv \<a>[fst a, fst (snd a), snd (snd a)]"
round_num_denom_def: fixes n :: "int"   and d :: "int" shows "round_num_denom n d = (2 * n + d) div (2 * d)"
exec_blockAbrupt: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and bdy :: "('e, 'd, 'f) com"   and init :: "'e \<Rightarrow> 'e"   and s :: "'e"   and t :: "'e"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Abrupt t" shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> \<Rightarrow> Abrupt (return s t)"
residue_mult_closed: fixes k :: "nat"   and x :: "int"   and y :: "int" shows "x \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> y \<in> carrier (residue_ring (p ^ k))"
Cons_both_possible_leftE: fixes t :: "'a ttree"   and x :: "'a"   and xs :: "'a list"   and t' :: "'a ttree" assumes "possible t x"   and "xs \<in> paths (nxt t x \<otimes>\<otimes> t')" shows "x # xs \<in> paths (t \<otimes>\<otimes> t')"
pos_less_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less a (divide b c) = less (times a c) b"
ProofStepInduct_NonTrivial_P_k_pre_axioms_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" shows "ProofStepInduct_NonTrivial_P_k_pre_axioms G v0 v1 paths P_new sep_size P_k_pre y P_k_post \<equiv> ProofStepInduct_NonTrivial.P_k G v0 v1 paths P_new sep_size = P_k_pre @ y # P_k_post \<and> ProofStepInduct_NonTrivial.hitting_Q_or_new_last G v0 v1 paths P_new sep_size y \<and> (\<forall>y'. y' \<in> set P_k_pre \<longrightarrow> \<not> ProofStepInduct_NonTrivial.hitting_Q_or_new_last G v0 v1 paths P_new sep_size y')"
intro: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and Basis :: "'b set" assumes "vector_space scale"   and "finite_dimensional_vector_space_axioms scale Basis" shows "finite_dimensional_vector_space scale Basis"
inv_dining_def: fixes ps :: "'a list"   and fs :: "'b list" shows "inv_dining ps fs \<equiv> (\<forall>i. Suc i < N \<longrightarrow> (fs ! Suc i = (1::'b)) = (ps ! Suc i \<noteq> (0::'a))) \<and> (fs ! (N - 1) = (2::'b)) = (ps ! 0 \<noteq> (0::'a)) \<and> (\<forall>i<N - 1. (fs ! i = (2::'b)) = (ps ! Suc i = (2::'a))) \<and> (fs ! 0 = (1::'b)) = (ps ! 0 = (2::'a)) \<and> (\<forall>i<N. fs ! i = (0::'b) \<or> fs ! i = (1::'b) \<or> fs ! i = (2::'b)) \<and> (\<forall>i<N. ps ! i = (0::'a) \<or> ps ! i = (1::'a) \<or> ps ! i = (2::'a) \<or> ps ! i = (3::'a)) \<and> length fs = N \<and> length ps = N"
Collect_eq_comp': fixes P :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" shows "{(x, y). P x y} O {(c, a). c = f a} = {(x, a). P x (f a)}"
linorder_rank_le: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and x :: "'b" assumes "finite A" shows "linorder_rank R A x \<le> card A"
assms: shows "\<forall>k\<in>set ks. k < l"
symm_closure_of_preorder_is_trans: fixes Rel :: "('a \<times> 'a) set" assumes "\<forall>a b c. (a, b) \<in> Rel \<and> (c, b) \<in> Rel \<and> (a, c) \<notin> Rel \<longrightarrow> (b, a) \<in> Rel \<or> (b, c) \<in> Rel"   and "\<forall>a b c. (a, b) \<in> Rel \<and> (a, c) \<in> Rel \<and> (b, c) \<notin> Rel \<longrightarrow> (b, a) \<in> Rel \<or> (c, a) \<in> Rel"   and "refl Rel"   and "trans Rel" shows "trans (symcl Rel)"
HInit_Inv2b: fixes s :: "state" shows "HInit s \<longrightarrow> Inv2b s"
prefix_same_cases: fixes xs\<^sub>1 :: "'a list"   and ys :: "'a list"   and xs\<^sub>2 :: "'a list" assumes "prefix xs\<^sub>1 ys"   and "prefix xs\<^sub>2 ys" shows "prefix xs\<^sub>1 xs\<^sub>2 \<or> prefix xs\<^sub>2 xs\<^sub>1"
web_of_network_sel_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and e :: "'a \<times> 'a" shows "weight (web_of_network \<Delta>) e = capacity \<Delta> e"
arr_iff_in_hom: fixes f :: "'a" shows "A.arr f = \<guillemotleft>f : A.dom f \<rightarrow>\<^sub>A A.cod f\<guillemotright>"
bisimulation_def: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j) graph_rec_scheme"   and B :: "('i, 'k) graph_rec_scheme" shows "bisimulation R A B \<equiv> (graph A \<and> graph B) \<and> simulation R A B \<and> simulation (R\<inverse>) B A"
pivot_valuation_id: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot :: "nat \<Rightarrow> nat \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "Pivot eq_idx_for_lvar pivot"   and "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "\<V> (pivot x\<^sub>i x\<^sub>j s) = \<V> s"
osendT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, {l}send(s\<^sub>m\<^sub>s\<^sub>g) . p), send (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i)), \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
13: shows "(w \<sqinter> - F) * (F \<sqinter> w\<^sup>T) = bot"
circ_decompose_11: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> = (x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> * x\<^sup>\<star>"
in_next_args: fixes v :: "'val" assumes "v \<in> set next_args" shows "v = snd next \<or> v = uninst_code.substitution_code phis g next"
in_s_set: fixes x :: "'d"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set" assumes "x \<in> s_set R s A" shows "(\<exists>r\<in>carrier R. \<exists>a\<in>A. x = r \<^sub>s\<cdot> a) \<or> x \<in> A"
r_cancel_one': fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and a :: "'d" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(x = a \<otimes>\<^bsub>G\<^esub> x) = (a = \<one>\<^bsub>G\<^esub>)"
Op_preserves_good: fixes inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs set option"   and delta :: "'f" assumes "goodInp inp"   and "goodBinp binp" shows "good (Op delta inp binp)"
c0: shows "\<forall>i<length Vs + 1. (0::'a) \<le> c i"
insort_is_Cons: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "\<forall>x\<in>set xs. f a \<le> f x" shows "insort_key f a xs = a # xs"
regular_bot_top: fixes x :: "'a" shows "regular x = (x = bot \<or> x = top)"
disj: shows "{l. length l = m \<and> sum_list l = N \<and> hd l = 0} \<inter> {l. length l = m \<and> sum_list l = N \<and> hd l \<noteq> 0} = {}"
distinct_sort_on: fixes r :: "('a \<times> 'a) set"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "Linear_order r" shows "distinct (sort_key_on r f xs) = distinct xs"
LLL_invI: fixes fs :: "int vec list"   and i :: "nat"   and upw :: "bool" assumes "set fs \<subseteq> carrier_vec n"   and "length fs = m"   and "lattice_of fs = L"   and "i \<le> m"   and "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"   and "gram_schmidt_fs.reduced n (map of_int_hom.vec_hom fs) \<alpha> i"   and "upw \<or> \<mu>_small fs i" shows "LLL_invariant upw i fs"
Same_Vote_Refines: shows "PO_refines Id v_TS sv_TS"
ordered_semiring_record_simps_5: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
dagger_unfoldr_distr: fixes y :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" shows "y \<union> (x\<^sup>* O x) O y = x\<^sup>* O y"
in_small_setE: fixes a :: "V"   and A :: "V set" assumes "a \<in>\<^sub>\<circ> set A"   and "small A" obtains "a \<in> A"
cases: fixes a1 :: "real"   and a2 :: "(real \<times> 'a) stream" assumes "wf_times a1 a2" obtains t :: "real" and t' :: "real" and \<omega> :: "(real \<times> 'a) stream" and s' :: "'a" where "a1 = t" and "a2 = (t', s') ## \<omega>" and "t < t'" and "wf_times t' \<omega>"
bl_drop_leading_zeros: fixes bs :: "bool list" shows "of_bl (dropWhile Not bs) = of_bl bs"
phis'_def: fixes g :: "'g"   and x :: "'node \<times> 'val" shows "phis' g x \<equiv> case x of (n, v) \<Rightarrow> if v = chooseNext g then None else map_option (map (substNext g)) (phis g (n, v))"
scc_root_exists: fixes scc :: "'v set" assumes "is_scc E scc"   and "scc \<inter> dom (discovered s) \<noteq> {}" shows "\<exists>r. scc_root s r scc"
new_shadow_root_ptr_new: fixes h :: "(_) heap" shows "shadow_root_ptr.Ref (Suc (fMax (finsert 0 (shadow_root_ptr.the_ref |`| shadow_root_ptrs h)))) |\<notin>| shadow_root_ptrs h"
ipresIGFresh_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and MOD :: "('h, 'i, 'd, 'j, 'k, 'a, 'g, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "ipresIGFresh h (FixSyn.termMOD Delta) MOD = FixSyn.ipresFresh Delta h MOD"
hoare_partial_correct_def: fixes P :: "complex mat"   and S :: "com"   and Q :: "complex mat" shows "(\<Turnstile>\<^sub>p {P} S {Q}) = (\<forall>\<rho>\<in>density_states. trace (P * \<rho>) \<le> trace (Q * denote S \<rho>) + (trace \<rho> - trace (denote S \<rho>)))"
fun_upd_upd: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b"   and z :: "'b" shows "f(x := y, x := z) = f(x := z)"
nontriv_solution_imp_solution: fixes D :: "nat"   and z :: "'a \<times> 'a" assumes "pell D"   and "pell.nontriv_solution D z" shows "pell.solution D z"
while_lemma: fixes s :: "state"   and w :: "com"   and t :: "state"   and b :: "state \<Rightarrow> bool"   and c :: "com"   and P :: "state \<Rightarrow> bool" assumes "s -w\<rightarrow> t"   and "w = WHILE b DO c \<and> P s \<and> (\<forall>s s'. P s \<and> b s \<and> s -c\<rightarrow> s' \<longrightarrow> P s')" shows "P t \<and> \<not> b t"
rcosets_non_empty: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and R :: "'c set" assumes "subgroup H G"   and "R \<in> rcosets\<^bsub>G\<^esub> H" shows "R \<noteq> {}"
simps_3: fixes m :: "nat"   and n :: "nat" shows "count_Mn (recf.Id m n) = 0"
span_superset: fixes S :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "S \<subseteq> phull S"
intro: fixes d :: "'a \<rightarrow> 'a" assumes "\<And>x. d\<cdot>(d\<cdot>x) = d\<cdot>x"   and "\<And>x. d\<cdot>x \<sqsubseteq> x" shows "deflation d"
antisym_conv2: fixes x :: "nat list"   and y :: "nat list" assumes "x \<le>\<^sub>v y" shows "(\<not> x <\<^sub>v y) = (x = y)"
f: shows "\<guillemotleft>f : x \<rightarrow>\<^sub>C x'\<guillemotright>"
out__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A Out B C" shows "Coplanar A B C D"
quot_setI: fixes x :: "'a"   and a :: "'a"   and A :: "'a set" assumes "x * a \<in> A" shows "a \<in> A \<div> x"
cp_OclNot: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" shows "not X \<tau> = not (\<lambda>_. X \<tau>) \<tau>"
intros_10: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula" assumes "\<Gamma> \<turnstile> F \<^bold>\<or> G"   and "F \<triangleright> \<Gamma> \<turnstile> H"   and "G \<triangleright> \<Gamma> \<turnstile> H" shows "\<Gamma> \<turnstile> H"
eq_iff: fixes a :: "'b Word.word"   and b :: "'b Word.word" shows "(a = b) = (b \<le>s a \<and> a \<le>s b)"
coefficients_isconst: fixes p :: "poly"   and n :: "nat" assumes "isnpolyh p n" shows "\<forall>q\<in>set (coefficients p). isconstant q"
plossless_converter_3: shows "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"
verticesFrom_pres_nodes: fixes v :: "nat"   and f :: "face"   and w :: "nat" assumes "v \<in> \<V> f"   and "w \<in> \<V> f" shows "w \<in> set (verticesFrom f v)"
Stewart: fixes B :: "'a"   and C :: "'a"   and D :: "'a"   and a :: "real"   and b :: "real"   and A :: "'a"   and c :: "real"   and d :: "real"   and m :: "real"   and n :: "real" assumes "between (B, C) D"   and "a = dist B C"   and "b = dist A C"   and "c = dist B A"   and "d = dist A D"   and "m = dist B D"   and "n = dist C D" shows "b\<^sup>2 * m + c\<^sup>2 * n = a * (d\<^sup>2 + m * n)"
union_inter: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "finite A"   and "finite B" shows "sum g (A \<union> B) + sum g (A \<inter> B) = sum g A + sum g B"
convex_on_sum: fixes s :: "'c set"   and C :: "'d set"   and f :: "'d \<Rightarrow> real"   and a :: "'c \<Rightarrow> real"   and y :: "'c \<Rightarrow> 'd" assumes "finite s"   and "s \<noteq> {}"   and "convex_on C f"   and "convex C"   and "sum a s = 1"   and "\<And>i. i \<in> s \<Longrightarrow> 0 \<le> a i"   and "\<And>i. i \<in> s \<Longrightarrow> y i \<in> C" shows "f (\<Sum>i\<in>s. a i *\<^sub>R y i) \<le> (\<Sum>i\<in>s. a i * f (y i))"
to_tree_order_child2: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<noteq> ptr"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set nodes" obtains parent :: "(_) object_ptr" where "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some parent" and "parent \<in> set nodes"
valid_mD: fixes m :: "'a SecurityInvariant_configured" assumes "m \<in> set M" shows "configured_SecurityInvariant m"
WT_swap_lassocr: fixes \<I>1 :: "('i, 'l) \<I>"   and \<I>2 :: "('j, 'm) \<I>"   and \<I>3 :: "('k, 'n) \<I>" shows "\<I>1 \<oplus>\<^sub>\<I> (\<I>2 \<oplus>\<^sub>\<I> \<I>3), \<I>2 \<oplus>\<^sub>\<I> (\<I>1 \<oplus>\<^sub>\<I> \<I>3) \<turnstile>\<^sub>C swap_lassocr \<surd>"
fold_insert: fixes A :: "'a set"   and x :: "'a"   and z :: "'a option" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold MaxR_f z (insert x A) = MaxR_f x (Finite_Set.fold MaxR_f z A)"
assms_2: shows "\<exists>!Q. Q \<in> subgroups_of_size q"
fmap_add_ltr_def: fixes m1 :: "('d, 'e) fmap"   and m2 :: "('d, 'e) fmap" shows "m1 ++ m2 \<equiv> m2 ++\<^sub>f m1"
length_greater_0_conv: fixes xs :: "'c list" shows "(0 < length xs) = (xs \<noteq> [])"
\<psi>_in_terms_of_\<epsilon>o: fixes x :: "'d"   and g :: "'c"   and y :: "'c" assumes "D.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>C G x\<guillemotright>" shows "\<psi> x g = Adj.\<epsilon>o x \<cdot>\<^sub>D F g"
RootState_def: fixes ST :: "('d, 'e, 'f) status" shows "RootState ST \<equiv> SOME S. S \<in> Conf ST \<and> S \<in> States (HARoot (HA ST))"
cdf_nondecreasing: fixes x :: "real"   and y :: "real"   and k :: "nat" assumes "x \<le> y" shows "cdf (\<mu> k) x \<le> cdf (\<mu> k) y"
upd_def: fixes L :: "('a \<times> 'b) list"   and a :: "'a"   and b :: "'b" shows "L[a\<mapsto>b] = (a, b) # delete L a"
Fract_eq_zero_iff: fixes a :: "'c"   and b :: "'c" shows "(Fraction_Field.Fract a b = 0) = (a = (0::'c) \<or> b = (0::'c))"
usubstappt_det: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm"   and V :: "variable set" assumes "usubstappt \<sigma> U \<theta> \<noteq> undeft"   and "usubstappt \<sigma> V \<theta> \<noteq> undeft" shows "usubstappt \<sigma> U \<theta> = usubstappt \<sigma> V \<theta>"
fpxs_nth_below_val: fixes n :: "rat"   and f :: "'b fpxs" assumes "n < fpxs_val f" shows "fpxs_nth f n = (0::'b)"
rpr_le_less_trans: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and z :: "'a"   and A :: "'a set" assumes "x \<^bsub>r\<^esub>\<preceq> y"   and "y \<^bsub>r\<^esub>\<prec> z"   and "rpr A r" shows "x \<^bsub>r\<^esub>\<prec> z"
not_\<S>_reps: fixes l :: "'s"   and R :: "('c \<Rightarrow> real) set" assumes "(l, R) \<notin> \<S>" shows "reps (l, R) \<notin> S"
subspace_diff: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "m1.subspace S"   and "x \<in> S"   and "y \<in> S" shows "x - y \<in> S"
paste_subrel: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" assumes "P || Domain Q \<subseteq> Q" shows "P +< Q = P \<union> Q"
2547: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<squnion> (- - x \<sqinter> - y \<squnion> z) = - x \<squnion> (- y \<squnion> z)"
assms_5: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
quantifiable_T8: fixes w :: "i"   and \<psi> :: "'b \<Rightarrow> \<o>" shows "[\<^bold>\<forall>x. \<psi> x in w] = (\<forall>x. [\<psi> x in w])"
real_cond_exp_pos: fixes f :: "'a \<Rightarrow> real" assumes "AE x in M. 0 \<le> f x"   and "f \<in> borel_measurable M" shows "AE x in M. 0 \<le> real_cond_exp M F f x"
map_acom_SKIP: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c acom"   and S' :: "'b" shows "(ACom.map_acom f c = SKIP {S'}) = (\<exists>S. c = SKIP {S} \<and> S' = f S)"
GH_def: shows "GH = G\<lparr>carrier := H\<rparr>"
Table_axioms: shows "Table f1 f2 f1' f2' e c l0"
dims1_def: shows "st.dims1 = nths dims0 vars1'"
encL: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<R>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> SourceTerm S"
clock_sim_r: shows "sim_r Clock.MC clock_simMC clock_sim"
stable_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "stable_on ds X = (individually_rational_on ds X \<and> stable_no_blocking_on ds X)"
ex_sinks_sources_aux: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'d list"   and V :: "'c set" shows "(\<exists>u\<in>sinks_aux I D U xs. \<exists>v\<in>V. (u, v) \<in> I) = (\<exists>u\<in>U. \<exists>v\<in>sources_aux I D V xs. (u, v) \<in> I)"
\<tau>move2While3: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and c :: "'addr expr1"   and e :: "'addr expr1" shows "\<tau>move2 P h stk (while (c) e) (Suc (Suc (length (compE2 c) + length (compE2 e)))) None"
ring_tOp_closed: fixes R :: "('d, 'e) Ring_scheme"   and x :: "'d"   and y :: "'d" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<in> carrier R"
fv_subterms: fixes t :: "('b, 'a) Term.term" shows "fv\<^sub>s\<^sub>e\<^sub>t (subterms t) = fv t"
iffalse: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<iota> :: "nat"   and c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom" assumes "BMap (E b m) = False" shows "SRabbr (if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi) m [] (Some c2) m"
simps_1: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "equal_base.list_distinct equal [] = True"
mset_nondetMs_3: fixes return :: "'a multiset \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a multiset \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_commute return bind" shows "monad_alt (nondetM_base.return_nondet return (\<lambda>x. {#x#})) (nondetM_base.bind_nondet bind (mmerge return bind)) (nondetM_base.alt_nondet return bind (+))"