hcju/isabelleps huggingface dataset:
env_restr_perm': fixes p :: "perm"   and S :: "'a set"   and \<rho> :: "'a \<Rightarrow> 'b" assumes "supp p \<sharp>* S"   and "finite S" shows "p \<bullet> \<rho> f|` S = \<rho> f|` S"
P_lnull_ltl_deadend_v0: assumes "lnull (ltl P)" shows "deadend v"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_loop_goon m lm (l, r) = (wprepare_loop_goon_in_middle m lm (l, r) \<or> wprepare_loop_goon_on_rightmost m lm (l, r))"
contI: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>Y. chain Y \<Longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i)" shows "cont f"
cfg_onD_cont: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real) set"   and t :: "'s \<times> ('c \<Rightarrow> real) set" assumes "cfg \<in> R_G.cfg_on s"   and "t \<in> set_pmf (action cfg)" shows "cont cfg t \<in> R_G.cfg_on t"
fz: shows "f (0::'a) (0::'a) = (0::'a)"
MapsToObj_2: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and X :: "'g"   and Y :: "'g" assumes "Category C"   and "f maps\<^bsub>C\<^esub> X to Y" shows "Y \<in> obj\<^bsub>C\<^esub>"
sba2_commutative: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
assms: shows "s' = snd (fst (sub_instr_sub1 instr_name result rs1_val operand2 s)) \<and> get_S (cpu_reg_val PSR s) = 0"
wf_k: fixes x :: "'v"   and y :: "'v"   and i :: "nat" assumes "(x, y) \<in> \<^bold>E" shows "k_i i (y, x) \<le> f_i i (x, y) + k_i i (x, y)"
field_cont_on_bot_edge_image: shows "continuous_on {p. \<exists>x. x \<in> cbox a b \<and> p = (g2 x, x)} F"
sumPres: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<simeq> Q" shows "P \<oplus> R \<simeq> Q \<oplus> R"
cf_no_self_loop: fixes u :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "(u, u) \<notin> cfE_of f"
map_spmf_cong: fixes p :: "'c spmf"   and q :: "'c spmf"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "map_spmf f p = map_spmf g q"
reflexive_mult_closed: fixes x :: "'a"   and y :: "'a" assumes "top \<le> x"   and "top \<le> y" shows "top \<le> x \<sqinter> y"
fdemodalisation2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "( |x\<rangle> y \<le> d z) = (x \<cdot> d y \<le> d z \<cdot> x)"
assms_2: fixes \<epsilon> :: "real" assumes "\<epsilon> < u"   and "inverse C < \<epsilon>" shows "(\<lambda>n. norm (f n)) \<in> o((^) \<epsilon>)"
reflectl__conga: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and P' :: "'p" assumes "A \<noteq> B"   and "B \<noteq> P"   and "P P' ReflectL A B" shows "A B P CongA A B P'"
qAFreshAll_simps_2: fixes xs :: "'d"   and x :: "'e"   and delta :: "'h"   and inp :: "'f \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qTerm option"   and binp :: "'g \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qAbs option" shows "qAFresh xs x (qOp delta inp binp) = (liftAll (qAFresh xs x) inp \<and> liftAll (qAFreshAbs xs x) binp)"
assms_2: shows "nprv (insert \<phi> F) \<chi>"
sym_design_vk_gt_kl: assumes "\<Lambda> + 2 \<le> \<k>" shows "\<k> - \<Lambda> < \<v> - \<k>"
node_def: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and r :: "('b \<times> nat) tree" shows "node l a r = (let rl = mht l; rr = mht r in if rr \<le> rl then \<langle>l, (a, rr + 1), r\<rangle> else \<langle>r, (a, rl + 1), l\<rangle>)"
add_minus_cancel: fixes a :: "'a"   and b :: "'a" shows "a + (- a + b) = b"
interaction_scheme_unique_aux: fixes as :: "'a list list"   and as' :: "'a list list"   and bs :: "'a list list"   and bs' :: "'a list list" assumes "concat as = concat as'"   and "concat bs = concat bs'"   and "as \<in> lists (- {[]})"   and "bs \<in> lists (- {[]})"   and "strict_sorted (interact as bs)"   and "length bs \<le> length as"   and "length as \<le> Suc (length bs)"   and "as' \<in> lists (- {[]})"   and "bs' \<in> lists (- {[]})"   and "strict_sorted (interact as' bs')"   and "length bs' \<le> length as'"   and "length as' \<le> Suc (length bs')"   and "length as = length as'"   and "length bs = length bs'" shows "as = as' \<and> bs = bs'"
assms_5: shows "c \<turnstile> ev' \<mapsto> d'"
eq_onE: fixes X :: "'d set"   and a :: "'d"   and b :: "'d" assumes "eq_on X a b" obtains "b \<in> X" and "a = b"
S_def: shows "S = - {z. Re z = 0 \<and> Im z \<le> 0} - (sngs - sngs')"
l1s_refines_s0: shows "refines R01s med01s s0 l1"
inf_comp: shows "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (i + 1)"
fds_nth_logderiv_completely_multiplicative: fixes f :: "'b fds"   and n :: "nat" assumes "completely_multiplicative_function (fds_nth f)"   and "fds_nth f 1 \<noteq> (0::'b)" shows "fds_nth (fds_deriv f / f) n = - fds_nth f n * mangoldt n"
ag_inc_zero: fixes A :: "('h, 'i) aGroup_scheme" assumes "aGroup A" shows "\<zero>\<^bsub>A\<^esub> \<in> carrier A"
has_complex_derivative_locally_invertible: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex" assumes "f holomorphic_on S"   and "\<xi> \<in> S"   and "open S"   and "deriv f \<xi> \<noteq> 0" obtains r :: "real" where "0 < r" and "ball \<xi> r \<subseteq> S" and "open (f ` ball \<xi> r)" and "inj_on f (ball \<xi> r)"
mkPinj_cn: fixes y :: "nat"   and Q :: "pol" assumes "y \<noteq> 0"   and "isnorm Q" shows "isnorm (mkPinj y Q)"
scaleRat_right_distrib: fixes a :: "rat"   and x :: "'a"   and y :: "'a" shows "a *R (x + y) = a *R x + a *R y"
inject_3: fixes x5 :: "nat"   and y5 :: "nat" shows "(Addr x5 = Addr y5) = (x5 = y5)"
continuous_map_into_fulltopology: fixes X :: "'b topology"   and X' :: "'c topology"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_map X (subtopology X' T) f" shows "continuous_map X X' f"
abs_eq: fixes x :: "nat \<Rightarrow> 'b" shows "of_seq x = stream_of_llist (inf_llist x)"
append_eq_conv_conj: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "(xs @ ys = zs) = (xs = take |xs| zs \<and> ys = drop |xs| zs)"
netgmap_netgmap_rhs: fixes i :: "nat"   and s2 :: "'d net_state"   and sr :: "'d \<Rightarrow> 'e \<times> 'f"   and s1 :: "'d net_state" assumes "i \<in> net_ips s2" shows "(fst (netgmap sr s1) ++ fst (netgmap sr s2)) i = fst (netgmap sr s2) i"
delete_edges_as_filter: fixes G :: "'b list_graph"   and l :: "('b \<times> 'b) list" shows "FiniteListGraph.delete_edges G l = \<lparr>nodesL = nodesL G, edgesL = filter (\<lambda>x. x \<notin> set l) (edgesL G)\<rparr>"
opt_less_simps_1: fixes y :: "int option" shows "opt_less None y = True"
split_elimVar: fixes L :: "atomUni list"   and eq :: "(real \<times> real \<times> real) list"   and les :: "(real \<times> real \<times> real) list"   and leq :: "(real \<times> real \<times> real) list"   and neq :: "(real \<times> real \<times> real) list"   and L' :: "atomUni list"   and x :: "real" assumes "separateAtoms L = (eq, les, leq, neq)" shows "(\<exists>l\<in>set L. evalUni (elimVarUni_atom L' l) x) = ((\<exists>(a', b', c')\<in>set eq. evalUni (elimVarUni_atom L' (EqUni (a', b', c'))) x) \<or> (\<exists>(a', b', c')\<in>set les. evalUni (elimVarUni_atom L' (LessUni (a', b', c'))) x) \<or> (\<exists>(a', b', c')\<in>set leq. evalUni (elimVarUni_atom L' (LeqUni (a', b', c'))) x) \<or> (\<exists>(a', b', c')\<in>set neq. evalUni (elimVarUni_atom L' (NeqUni (a', b', c'))) x))"
simps: fixes x :: "complex"   and y :: "complex"   and a :: "complex"   and b :: "complex" shows "(x, y) *\<^sub>v\<^sub>v (a, b) = x * a + y * b"
Nfunc_LeastTr2_7: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set"   and k :: "nat" assumes "\<forall>l\<le>n. G \<guillemotright> f l "   and "\<forall>l\<le>n. G \<guillemotright> g l "   and "inj_on g {i. i \<le> n}"   and "\<exists>i\<le>n. f i \<noteq> g i"   and "f k = g (LEAST k. f k \<noteq> g k)" shows "(LEAST k. f k \<noteq> g k) < k"
coprimeI: 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 a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "\<And>c. \<lbrakk>dvd.dvd times c a; dvd.dvd times c b\<rbrakk> \<Longrightarrow> dvd.dvd times c one" shows "algebraic_semidom.coprime times one a b"
map_fst_pair_spmf: fixes p :: "'c spmf"   and q :: "'d spmf" shows "map_spmf fst (pair_spmf p q) = scale_spmf (weight_spmf q) p"
prappend_signed_list_single_Cons: fixes x :: "'a \<times> bool"   and y :: "'a \<times> bool"   and ys :: "('a \<times> bool) list" shows "prappend_signed_list [x] (y # ys) = (if y = flip_signed x then ys else x # y # ys)"
to_language_in_language: fixes L :: "'a language" shows "to_language (Collect (in_language L)) = L"
ex_toFull: fixes \<sigma> :: "'tp" shows "\<exists>F. bij_betw F {a. intT \<sigma> a} UNIV"
cast\<^sub>N\<^sub>o\<^sub>d\<^sub>e\<^sub>2\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_def: fixes node :: "'a RNode_scheme" shows "cast node = RObject.extend (RObject.truncate node) (Inl (RObject.more node))"
elims: fixes x :: "nat \<Rightarrow> nat"   and xa :: "nat set \<times> nat set set"   and y :: "nat set \<times> nat set set" assumes "map_ugraph x xa = y" obtains f :: "nat \<Rightarrow> nat" and V :: "nat set" and E :: "nat set set" where "x = f" and "xa = (V, E)" and "y = (f ` V, (`) f ` E)"
a: shows "<a>:M \<in> NOTRIGHT (NOT B) (\<parallel>(B)\<parallel>)"
trivial_add_mset_remove_iff: fixes a :: "'a"   and N :: "'a multiset"   and b :: "'a" shows "(add_mset a (remove1_mset b N) = N) = (a \<in># N \<and> a = b)"
left_neutral: fixes a :: "'a list set" shows "p_prod p_one a = a"
simps_5: fixes f :: "atom \<Rightarrow> nat \<Rightarrow> atom"   and \<phi> :: "atom fm"   and \<psi> :: "atom fm"   and n :: "nat" shows "map_fm_binders f (Or \<phi> \<psi>) n = Or (map_fm_binders f \<phi> n) (map_fm_binders f \<psi> n)"
permute_list_distinct: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'c list" assumes "f ` {..<length xs} \<subseteq> {..<length xs}"   and "distinct xs" shows "permute_list f xs = map (\<lambda>x. xs ! f (index xs x)) xs"
PO_a0i_inv1_iagree: shows "reach a0i \<subseteq> a0i_inv1_iagree"
steps_reachable: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and s\<^sub>0 :: "'c"   and b :: "'c" assumes "Graph_Defs.steps E xs"   and "hd xs = s\<^sub>0"   and "last xs = b" shows "Graph_Start_Defs.reachable E s\<^sub>0 b"
add_new_vmuaux_def: shows "add_new_vmuaux = (\<lambda>args rel1 rel2 nt (t, auxlist). (nt, update_until args rel1 rel2 nt auxlist))"
elims_2: fixes x :: "'a atom set"   and xa :: "(nat \<Rightarrow> 'a option) \<times> (nat \<Rightarrow> 'a option)" assumes "x \<doteq> xa" obtains as :: "'a atom set" and lb :: "nat \<Rightarrow> 'a option" and ub :: "nat \<Rightarrow> 'a option" where "x = as" and "xa = (lb, ub)" and "\<forall>v. v \<Turnstile>\<^sub>a\<^sub>s as = v \<Turnstile>\<^sub>b (lb, ub)"
l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.to_tree_order Shadow_DOM.first_in_tree_order"
bell10_index_3: shows "|\<beta>\<^sub>1\<^sub>0\<rangle> $$ (2, 0) = 0"
splitAt_simpA: fixes ram :: "'c"   and b :: "'c list" shows "fst (splitAt ram (ram # b)) = []"
listrel_rtrancl_eq_rtrancl_listrel1: fixes r :: "('g \<times> 'g) set" shows "listrel (r\<^sup>*) = (listrel1 r)\<^sup>*"
A: shows "filter P xs = filter P (takeWhile P xs @ dropWhile P xs)"
U_2: shows "\<forall>R\<in>U. R \<in> \<R>"
N: shows "prob {x. (1 / 2) ^ N < \<bar>x\<bar>} \<noteq> 0"
ss_422iii: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "is_p_fun y" shows "(x \<cdot> z ; y\<^sup>T) ; y = x ; y \<cdot> z"
Abs_BIS_cases: fixes x :: "BIS" obtains y :: "unit" where "x = Abs_BIS y" and "y \<in> UNIV"
Rep_FingerTree: fixes x :: "('a, 'b) FingerTree" shows "Rep_FingerTree x \<in> {t. FingerTreeStruc.ft_invar t}"
fun_upd_meet_simp: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" shows "(f \<squnion> g)(x := y) = f(x := y) \<squnion> g(x := y)"
hom_induced_chain_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T"   and "singular_relcycle p X S c" shows "hom_induced (int p) X S Y T f (homologous_rel_set p X S c) = homologous_rel_set p Y T (chain_map p f c)"
rel_cong: fixes x :: "'c set['d]"   and ya :: "'c set['d]"   and y :: "'e set['d]"   and xa :: "'e set['d]"   and R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'e \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set_bset ya; yb \<in> set_bset xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "rel_bset R x y = rel_bset Ra ya xa"
SC_tl: fixes p :: "('a, 'b) Combinators list" assumes "singleCombinators p" shows "singleCombinators (tl p)"
convex_def: fixes s :: "'d set" shows "convex s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
decreasing_closed_nest: fixes S :: "nat \<Rightarrow> 'a set" assumes "\<And>n. closed (S n)"   and "\<And>n. S n \<noteq> {}"   and "\<And>m n. m \<le> n \<Longrightarrow> S n \<subseteq> S m"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>S n. \<forall>y\<in>S n. dist x y < e" obtains a :: "'a" where "\<And>n. a \<in> S n"
MakeCatComp: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "f \<approx>>\<^bsub>C\<^esub> g" shows "f ;;\<^bsub>MakeCat C\<^esub> g = f ;;\<^bsub>C\<^esub> g"
insert_before_list_distinct: fixes x :: "'m"   and xs :: "'m list"   and ref :: "'m option" assumes "x \<notin> set xs"   and "distinct xs" shows "distinct (insert_before_list x ref xs)"
eventually_mp: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x"   and "eventually P F" shows "eventually Q F"
naturality: fixes f :: "'c" assumes "C.arr f" shows "C (C.map (C.cod f)) (C.map f) = C (C.map f) (C.map (C.dom f))"
inj_map_strong: fixes x :: "'d option"   and xa :: "'d option"   and f :: "'d \<Rightarrow> 'e"   and fa :: "'d \<Rightarrow> 'e" assumes "\<And>z za. \<lbrakk>z \<in> set_option x; za \<in> set_option xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_option f x = map_option fa xa" shows "x = xa"
in_rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d rexp"   and b :: "'e rexp" shows "rel_rexp R a b = (\<exists>z. z \<in> {x. set_rexp x \<subseteq> {(x, y). R x y}} \<and> map_rexp fst z = a \<and> map_rexp snd z = b)"
fds_converges_iff: fixes f :: "'a fds"   and s :: "'a" shows "fds_converges f s = (\<lambda>n. fds_nth f n / nat_power n s) sums eval_fds f s"
tranclp_induct: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ a b"   and "\<And>y. r a y \<Longrightarrow> P y"   and "\<And>y z. \<lbrakk>r\<^sup>+\<^sup>+ a y; r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
simps_6: fixes f1 :: "char list option \<Rightarrow> 'd \<Rightarrow> 'c"   and f2 :: "char list option \<Rightarrow> 'e \<Rightarrow> 'c"   and x21 :: "char list option"   and x22 :: "'e" shows "(case sum.Inr x21 x22 of sum.Inl x xa \<Rightarrow> f1 x xa | sum.Inr x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
Suc_1: fixes i :: "nat" assumes "k = j - i"   and "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "P i j"
fds_deriv_times: fixes f :: "'b fds"   and g :: "'b fds" shows "fds_deriv (f * g) = fds_deriv f * g + f * fds_deriv g"
swap_different_sorts: fixes a :: "atom"   and b :: "atom" assumes "sort_of a \<noteq> sort_of b" shows "(a \<rightleftharpoons> b) = 0"
x: shows "is_Read\<^sub>s\<^sub>b x \<or> is_Prog\<^sub>s\<^sub>b x \<or> is_Ghost\<^sub>s\<^sub>b x"
graph_def: fixes \<alpha> :: "'d \<Rightarrow> ('e, 'f) graph"   and invar :: "'d \<Rightarrow> bool" shows "graph \<alpha> invar \<equiv> (\<forall>g. invar g \<longrightarrow> finite (nodes (\<alpha> g))) \<and> (\<forall>g. invar g \<longrightarrow> finite (edges (\<alpha> g))) \<and> (\<forall>g. invar g \<longrightarrow> valid_graph (\<alpha> g))"
assms_1: shows "N = bernoulli_stream ((1 + r - d) / (u - d))"
that_1: assumes "is_exn \<sigma>'"   and "\<sigma>' = \<tau>" shows "thesis"
route2match_def: fixes r :: "('b, 'c) routing_rule_scheme" shows "route2match r = \<lparr>iiface = ifaceAny, oiface = ifaceAny, src = (0, 0), dst = (pfxm_prefix (routing_match r), routing_prefix r), proto = ProtoAny, sports = (0, - 1), dports = (0, - 1)\<rparr>"
image2_eqI: fixes b :: "'b"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and c :: "'d"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "b = f x"   and "c = g x"   and "x \<in> A" shows "(b, c) \<in> BNF_Greatest_Fixpoint.image2 A f g"
HInfinite_HFinite_add_cancel: fixes x :: "'a star"   and y :: "'a star" assumes "x + y \<in> HInfinite"   and "y \<in> HFinite" shows "x \<in> HInfinite"
map_simp_ide: 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>.map a = \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 (fst a) (fst (snd a)) (snd (snd a)), Dom = Dom ((fst a \<star> fst (snd a)) \<star> snd (snd a)), Cod = Cod (fst a \<star> fst (snd a) \<star> snd (snd a))\<rparr>"
foundation8: fixes \<tau> :: "'b state \<times> 'b state"   and x :: "'b state \<times> 'b state \<Rightarrow> 'c" shows "\<tau> \<Turnstile> \<delta> x \<or> \<tau> \<Turnstile> x \<triangleq> invalid \<or> \<tau> \<Turnstile> x \<triangleq> null"
neighbourhood_base_at_def: fixes x :: "'c"   and P :: "'c set \<Rightarrow> bool"   and X :: "'c topology" shows "neighbourhood_base_at x P X \<equiv> \<forall>W. openin X W \<and> x \<in> W \<longrightarrow> (\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> W)"
lim: shows "(f \<longlongrightarrow> flim) at_bot"
continuous_map_product_projection: fixes k :: "'b"   and I :: "'b set"   and X :: "'b \<Rightarrow> 'c topology" assumes "k \<in> I" shows "continuous_map (product_topology X I) (X k) (\<lambda>x. x k)"
elims: fixes x :: "'a"   and xa :: "'a simplex_state"   and y :: "'a list + 'a simplex_state" assumes "assert_simplex x xa = y" obtains i :: "'a" and cs :: "QDelta ns_constraint list" and asi :: "('a, ('a \<times> QDelta atom) list) mapping" and tv :: "(nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping" and ui :: "'a list" and s :: "('a, QDelta) state" where "x = i" and "xa = Simplex_State (cs, (asi, tv, ui), s)" and "y = (if i \<in> set ui then Inl [i] else case assert_all_s (list_map_to_fun asi i) s of Inl x \<Rightarrow> Inl x | Inr s' \<Rightarrow> Inr (Simplex_State (cs, (asi, tv, ui), s')))"
Rep_S_inject: fixes x :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) S"   and y :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) S" shows "(Rep_S x = Rep_S y) = (x = y)"
Complex_eq_i: fixes x :: "real"   and y :: "real" shows "(Complex x y = \<i>) = (x = 0 \<and> y = 1)"
simps_1: fixes C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e\<^sub>i :: "'a exp"   and e :: "'a exp" shows "init_switch (INIT C (Cs,b) \<leftarrow> e\<^sub>i) e = INIT C (Cs,b) \<leftarrow> e"
i_join_i_expand_iMOD: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af"   and n :: "nat" assumes "0 < k" shows "f \<odot> k \<Join>\<^sub> [ n * k, mod k ] = f \<Join>\<^sub> [n\<dots>]"
Throw_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (throw e, xs) (throw e', xs')"
induct: fixes P :: "'a list \<Rightarrow> bool"   and list :: "'a list" assumes "P []"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (x1 \<Colon> x2)" shows "P list"
assms: shows "sig_red_single sing_reg top_tail p q f t"
accepts_def: fixes A :: "'a \<times> ('b \<Rightarrow> 'a \<Rightarrow> 'a set) \<times> ('a \<Rightarrow> bool)"   and w :: "'b list" shows "accepts A w = (\<exists>q\<in>delta A w (start A). fin A q)"
assms_2: shows "- (3 \<times> x \<squnion> y) = - y"
nrm_dot: fixes x :: "'a"   and y :: "'a" shows "(x - y) \<bullet> rot (x - y) = 0"
measure_eqI: fixes M :: "'a measure"   and N :: "'a measure" assumes "sets M = sets N"   and "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A" shows "M = N"
eucl_of_list_nth: fixes xs :: "real list" assumes "length xs = DIM('a)" shows "eucl_of_list xs = (\<Sum>i<DIM('a). xs ! i *\<^sub>R Basis_list ! i)"
assms: shows "X \<inter> Domain P \<subseteq> Domain Q"
\<psi>_mapsto: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "\<psi> (b, a) \<in> local.set (b, a) \<rightarrow> C.hom b a"
mon_pl_ileq: fixes w :: "('b set \<times> 'b set) list"   and w' :: "('b set \<times> 'b set) list" assumes "w \<preceq> w'" shows "mon_pl w \<subseteq> mon_pl w'"
m: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and x :: "'a" assumes "R N M" shows "m N M x \<in> space (prob_algebra (stream_space (count_space UNIV)))"
kripke_rels_worlds: fixes w :: "'a"   and w' :: "'a"   and M :: "('b, 'c, 'a) KripkeStructure"   and a :: "'b" assumes "(w, w') \<in> relations M a"   and "kripke M" shows "w \<in> worlds M \<and> w' \<in> worlds M"
fr_eq_\<upsilon>: fixes p :: "fm"   and bs :: "real list"   and x :: "real" assumes "isrlfm p" shows "(\<exists>x. Ifm (x # bs) p) = (Ifm (x # bs) (minusinf p) \<or> Ifm (x # bs) (plusinf p) \<or> (\<exists>(t, k)\<in>set (\<Upsilon> p). \<exists>(s, l)\<in>set (\<Upsilon> p). Ifm (x # bs) (\<upsilon> p (Add (Mul l t) (Mul k s), 2 * k * l))))"
ConjI: fixes A :: "fm"   and B :: "fm" assumes "ss_fm A"   and "ss_fm B" shows "ss_fm (A AND B)"
boundedFlipNegQuantifier: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "(\<not> (\<forall>x\<in>A. \<not> P x)) = (\<exists>x\<in>A. P x)"
mat_ge_refl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat" assumes "class.non_strict_order less_eq" shows "ord.mat_ge less_eq A A"
inf_absorb1: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y" shows "inf x y = x"
assms_2: assumes "\<forall>a\<in>A. a .\<in> B"   and "\<forall>b\<in>B. b .\<in> A" shows "P"
sum_list_scf_list: fixes ts :: "('f, 'b) Term.term list"   and f :: "nat \<Rightarrow> nat" assumes "\<And>i. i < length ts \<Longrightarrow> 0 < f i" shows "sum_list (map weight ts) \<le> sum_list (map weight (scf_list f ts))"
det_row_add: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and a :: "('c, 'd) vec"   and b :: "('c, 'd) vec" shows "Square_Matrix.det (upd_row M i (a + b)) = Square_Matrix.det (upd_row M i a) + Square_Matrix.det (upd_row M i b)"
span_superset: fixes S :: "'c set" shows "S \<subseteq> span S"
elims_1: fixes x :: "'a extB"   and xa :: "'a extB"   and y :: "bool" assumes "(x \<le> xa) = y" obtains uu_ :: "'a extB" where "xa__ = extB.Bot" and "xa = uu_" and "ya__"   | uv_ :: "'a" where "xa__ = extB.Val uv_" and "xa = extB.Bot" and "\<not> ya__"   | x :: "'a" and y :: "'a" where "xa__ = extB.Val x" and "xa = extB.Val y" and "ya__ = (x \<le> y)"