hcju/isabelleps huggingface dataset:
1: fixes u :: "real" assumes "min (g a) (g b) \<le> u"   and "u \<le> max (g a) (g b)" shows "\<exists>x\<in>{a..b}. u = g x"
pw_filter_nb_1: fixes b :: "'c \<Rightarrow> bool"   and I :: "'c nres" shows "nofail (filter_nb b I) = nofail I"
diff_union_cancelR: fixes M :: "'a multiset"   and N :: "'a multiset" shows "M + N - N = M"
finite_set_of_finite_maps: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "finite {m. dom m = A \<and> ran m \<subseteq> B}"
Gromov_converging_at_boundary_imp_not_constant': fixes u :: "nat \<Rightarrow> 'b" assumes "Gromov_converging_at_boundary u" shows "\<not> (\<forall>m n. u m = u n)"
assms_1: shows "partial_saturation S E2 (S \<union> R)"
lookup_cacheD: fixes g :: "'g"   and f :: "'node \<Rightarrow> 'a"   and x :: "'node"   and y :: "'a" assumes "Mapping.lookup (cache g f) x = Some y" shows "y = f x"
exhaust: fixes y :: "init_call_status" obtains "y = No_ics"   | x21 :: "char list" and x22 :: "char list list" where "y = Calling x21 x22"   | x3 :: "char list list" where "y = Called x3"   | x41 :: "char list list" and x42 :: "nat" where "y = Throwing x41 x42"
cones_map_pointwise: fixes x :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and \<chi> :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and x' :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and \<chi>' :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "local.cone x \<chi>"   and "local.cone x' \<chi>'"   and "f \<in> A_B.hom x' x" shows "(cones_map f \<chi> = \<chi>') = (\<forall>a. A.ide a \<longrightarrow> (\<lambda>\<chi>\<in>{\<chi>. Limit.cone (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (at a D) (B.cod (A_B.Map f a)) \<chi>}. \<lambda>j. if J.arr j then \<chi> j \<cdot>\<^sub>B A_B.Map f a else B.null) (at a \<chi>) = at a \<chi>')"
pochhammer_Gamma: fixes z :: "'a"   and n :: "nat" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "pochhammer z n = Gamma (z + of_nat n) / Gamma z"
set_disconnected_nodes_get_host: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_host_locs. r h h')"
surj_id_the_inv: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "top_pres ((`) f)"   and "g \<circ> f = id" shows "g = the_inv f"
S3ClerkUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [MClkNext memCh crCh cst p]_(c p) \<and> $S3 rmhist p \<longrightarrow> unchanged c p"
orient_lit_inst_coincide: fixes L1 :: "'a literal"   and t :: "'a trm"   and s :: "'a trm"   and polarity :: "sign"   and \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list" assumes "orient_lit_inst L1 t s polarity \<sigma>"   and "coincide_on \<sigma> \<eta> (vars_of_lit L1)" shows "orient_lit_inst L1 t s polarity \<eta>"
constant_rules_1: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph"   and c :: "'b" assumes "standard' C G"   and "c \<in> C" shows "maintained (const_exists c) G"
\<phi>_\<psi>: fixes x :: "'a"   and g :: "'b"   and y :: "'b" assumes "Adj.C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G' x\<guillemotright>" shows "\<phi>' y (\<psi>' x g) = g"
map_indets_inverseE: fixes f :: "'b \<Rightarrow> 'c" assumes "inj f" obtains g :: "'c \<Rightarrow> 'b" where "g = the_inv f" and "g \<circ> f = id" and "map_indets g \<circ> map_indets f = id"
eventually_nhds: fixes "open" :: "'b set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "class.topological_space open" shows "eventually P (topological_space.nhds open a) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
Acts: fixes act :: "('d \<times> 'd) set"   and F :: "'d program"   and s :: "'d"   and s' :: "'d" assumes "act \<in> Acts F"   and "s \<in> reachable F"   and "(s, s') \<in> act" shows "s' \<in> reachable F"
domD: fixes a :: "'g"   and m :: "'g \<Rightarrow> 'h option" assumes "a \<in> dom m" shows "\<exists>b. m a = Some b"
less_multiset\<^sub>D\<^sub>M_def: fixes M :: "'a multiset"   and N :: "'a multiset" shows "less_multiset\<^sub>D\<^sub>M M N = (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> k < a)))"
that: fixes f :: "'a \<Rightarrow> nat" assumes "\<forall>x\<in>X. \<forall>y\<in>X. (f x \<le> f y) = ((x, y) \<in> r)" shows "thesis"
prob_sum: fixes I :: "'a set"   and s :: "'s"   and P :: "'a \<Rightarrow> 's stream \<Rightarrow> bool"   and Q :: "'s stream \<Rightarrow> bool" assumes "finite I"   and "\<And>n. n \<in> I \<Longrightarrow> {x \<in> space (T s). P n x} \<in> T.events s"   and "{x \<in> space (T s). Q x} \<in> T.events s"   and "AE x in T s. (\<forall>n\<in>I. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. n \<in> I \<and> P n x))" shows "T.prob s {x \<in> space (T s). Q x} = (\<Sum>n\<in>I. T.prob s {x \<in> space (T s). P n x})"
vfinite_induct: fixes F :: "V"   and P :: "V \<Rightarrow> bool" assumes "vfinite F"   and "P []\<^sub>\<circ>"   and "\<And>x F. \<lbrakk>vfinite F; x \<notin>\<^sub>\<circ> F; P F\<rbrakk> \<Longrightarrow> P (vinsert x F)" shows "P F"
special_subgroup_word_length: fixes T :: "'w set"   and w :: "'w" assumes "T \<in> Pow S"   and "w \<in> \<langle>T\<rangle>" shows "word_length T w = S_length w"
hyps: fixes nc :: "nat"   and c :: "'a list" assumes "mat nr nc m"   and "vec nc c" shows "scalar_prod (map (\<lambda>k. scalar_prod r (m ! k)) [0..<nc]) c = scalar_prod r (map (\<lambda>k. scalar_prod (row m k) c) [0..<nr])"
that: fixes i :: "nat" assumes "i < length xs"   and "i < length ys"   and "take i xs = take i ys"   and "(xs ! i, ys ! i) \<in> r" shows "thesis"
nBnd_def: shows "nBnd = of_nat (n + 1) * DBnd"
winding_eq: shows "complex_of_real (2 * pi) * \<i> * winding_number g 0 = Ln fi - Ln st"
subset_singleton_iff: fixes X :: "'a set"   and a :: "'a" shows "(X \<subseteq> {a}) = (X = {} \<or> X = {a})"
intros_1: fixes P :: "pi" shows "P \<equiv>\<^sub>e P"
simps: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and s :: "'a" shows "terminates_within g s = (let (has_next, next) = g in if has_next s then map_option (\<lambda>n. n + 1) (terminates_within g (snd (next s))) else Some 0)"
Max_ge: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "x \<le> Max A"
select_convs_3: fixes hosts :: "'a set"   and flows_fix :: "('a \<times> 'a) set"   and flows_state :: "('a \<times> 'a) set"   and more :: "'b" shows "flows_state \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = flows_state"
simps_13: fixes b :: "nat"   and I :: "\<I>"   and \<phi> :: "formula" shows "fvi b (Next I \<phi>) = fvi b \<phi>"
prefix_fininf_item: fixes a :: "'a"   and u :: "'a list"   and v :: "'a stream" shows "(a # u \<le>\<^sub>F\<^sub>I a ## v) = (u \<le>\<^sub>F\<^sub>I v)"
spanningForest_eq_basis: shows "spanningForest = s.SpanningForest"
Suc_1: fixes l :: "nat" assumes "lk = l - k"   and "k \<le> l"   and "\<forall>j\<in>{k..<l}. v \<notin> local.writes (path \<sigma> j)" shows "(\<sigma>\<^bsup>l\<^esup>) v = (\<sigma>\<^bsup>k\<^esup>) v"
finfun_All_except_const: fixes A :: "'a list"   and b :: "bool" shows "finfun_All_except A (finfun_const b) = (b \<or> set A = UNIV)"
t1_space_subtopology: fixes X :: "'a topology"   and U :: "'a set" assumes "t1_space X" shows "t1_space (subtopology X U)"
induct: fixes P :: "dag \<Rightarrow> bool"   and a0 :: "dag" assumes "P Tip"   and "\<And>vref. P (Node Tip vref Tip)"   and "\<And>vref l vref1 r. P (Node Tip vref (Node l vref1 r))"   and "\<And>l vref1 r vref. P (Node (Node l vref1 r) vref Tip)"   and "\<And>l1 vref1 r1 vref l2 vref2 r2. \<lbrakk>\<And>x. \<not> (x vref = 0 \<or> x vref = 1) \<Longrightarrow> P (Node l1 vref1 r1); \<And>x x2. \<lbrakk>\<not> (x vref = 0 \<or> x vref = 1); bdt (Node l1 vref1 r1) x = Some x2\<rbrakk> \<Longrightarrow> P (Node l2 vref2 r2)\<rbrakk> \<Longrightarrow> P (Node (Node l1 vref1 r1) vref (Node l2 vref2 r2))" shows "P a0"
distinct_226: fixes x14 :: "Syntax.term"   and x91 :: "Syntax.term"   and x92 :: "Syntax.term" shows "Unroll x14 \<noteq> App x91 x92"
T0_eqI: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier"   and "\<And>u. u \<in> nhds x \<Longrightarrow> y \<in> u"   and "\<And>v. v \<in> nhds y \<Longrightarrow> x \<in> v" shows "x = y"
sim_def: fixes M :: "('d, 'e, 'f) KripkeStructure"   and M' :: "('d, 'e, 'g) KripkeStructure"   and f :: "'f \<Rightarrow> 'g" shows "sim M M' f \<equiv> sim_range M M' f \<and> sim_val M M' f \<and> sim_f M M' f \<and> sim_r M M' f"
MST_validn_valid: assumes "\<forall>K. MST_validn K" shows "MST_valid"
ccpo_Sup_upper_inv: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set"   and x :: "'b" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq C"   and "less (Sup C) x" shows "x \<notin> C"
lformula_RESTRICT_VARS: fixes \<phi> :: "('a, 'k) aformula"   and vs :: "'k \<Rightarrow> nat list"   and ks :: "'k list" assumes "lformula \<phi>" shows "lformula (RESTRICT_VARS ks vs \<phi>)"
keys_def: fixes t :: "('b, 'c) rbt" shows "RBT_Impl.keys t = map fst (RBT_Impl.entries t)"
mono_F_syn: fixes S :: "'a process"   and A :: "'a set"   and Q :: "'a process" shows "F (S \<lbrakk>A\<rbrakk> Q) = F (Q \<lbrakk>A\<rbrakk> S)"
h2: shows "aff_dim (affine hull S) \<le> aff_dim UNIV"
SDG_param_in_edge: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and ins :: "'var list"   and outs :: "'var list"   and V :: "'var"   and x :: "nat"   and n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs"   and "V = ins ! x"   and "x < length ins"   and "n = Actual_in (sourcenode a, x)"   and "n' = Formal_in (targetnode a, x)" shows "n -p:V\<rightarrow>\<^bsub>in\<^esub> n'"
trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "g \<in> O[F](h)" shows "f \<in> O[F](h)"
asList_def: fixes A :: "'b set" shows "asList A \<equiv> SOME as. distinct as \<and> set as = A"
mono_step: fixes S :: "(char list \<Rightarrow> int) set" shows "mono (step S)"
distance_attains_sup: fixes s :: "'b set"   and a :: "'b" assumes "compact s"   and "s \<noteq> {}" shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"
sds_lowering_neutral_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf"   and agents' :: "'a set"   and alts' :: "'b set" shows "sds_lowering_neutral agents alts sds agents' alts' \<equiv> neutral_sds agents alts sds \<and> sds_lowering agents alts sds agents' alts'"
Compl_UN: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "- \<Union> (B ` A) = (\<Inter>x\<in>A. - B x)"
discriminant: fixes a :: "real"   and x :: "real"   and b :: "real"   and c :: "real" assumes "a * x\<^sup>2 + b * x + c = 0" shows "0 \<le> b\<^sup>2 - 4 * a * c"
pochhammer_of_int: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "int"   and n :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "comm_semiring_1.pochhammer times one plus zero (ring_1.of_int one plus zero minus x) n = ring_1.of_int one plus zero minus (pochhammer x n)"
wfP_induct: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "wfP r"   and "\<And>x. \<forall>y. r y x \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
pair_spmf_alt_def: fixes p :: "'h spmf"   and q :: "'i spmf" shows "pair_spmf p q = p \<bind> (\<lambda>x. q \<bind> (\<lambda>y. return_spmf (x, y)))"
Cinfinite_limit_finite: fixes X :: "'d set"   and r :: "('d \<times> 'd) set" assumes "finite X"   and "X \<subseteq> Field r"   and "Cinfinite r" shows "\<exists>y\<in>Field r. \<forall>x\<in>X. x \<noteq> y \<and> (x, y) \<in> r"
P_def: shows "P = (\<lambda>p. p \<in> local.\<P> \<longrightarrow> return_pmf B \<succeq>[\<R>] p)"
image_vector_to_cblinfun: fixes x :: "'a" shows "vector_to_cblinfun x *\<^sub>S \<top> = ccspan {x}"
step_invariant_induction: fixes c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and c' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and I :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> bool"   and P :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> bool" assumes "c \<Rightarrow>\<^sup>* c'"   and "\<not> boring_step c'"   and "invariant (\<Rightarrow>) I"   and "I c"   and "\<And>\<Gamma> e x S. I (\<Gamma>, App e x, S) \<Longrightarrow> P (\<Gamma>, App e x, S) (\<Gamma>, e, Arg x # S)"   and "\<And>\<Gamma> y e x S. I (\<Gamma>, Lam [y]. e, Arg x # S) \<Longrightarrow> P (\<Gamma>, Lam [y]. e, Arg x # S) (\<Gamma>, e[y::=x], S)"   and "\<And>\<Gamma> x e S. \<lbrakk>map_of \<Gamma> x = Some e; \<not> isVal e; I (\<Gamma>, Var x, S)\<rbrakk> \<Longrightarrow> P (\<Gamma>, Var x, S) (delete x \<Gamma>, e, Upd x # S)"   and "\<And>\<Gamma> x e S. \<lbrakk>map_of \<Gamma> x = Some e; isVal e; I (\<Gamma>, Var x, S)\<rbrakk> \<Longrightarrow> P (\<Gamma>, Var x, S) ((x, e) # delete x \<Gamma>, e, S)"   and "\<And>\<Gamma> x e S. \<lbrakk>x \<notin> domA \<Gamma>; isVal e; I (\<Gamma>, e, Upd x # S)\<rbrakk> \<Longrightarrow> P (\<Gamma>, e, Upd x # S) ((x, e) # \<Gamma>, e, S)"   and "\<And>\<Delta> \<Gamma> e S. \<lbrakk>atom ` domA \<Delta> \<sharp>* \<Gamma>; atom ` domA \<Delta> \<sharp>* S; I (\<Gamma>, Let \<Delta> e, S)\<rbrakk> \<Longrightarrow> P (\<Gamma>, Let \<Delta> e, S) (\<Delta> @ \<Gamma>, e, S)"   and "\<And>\<Gamma> scrut e1 e2 S. I (\<Gamma>, scrut ? e1 : e2, S) \<Longrightarrow> P (\<Gamma>, scrut ? e1 : e2, S) (\<Gamma>, scrut, Alts e1 e2 # S)"   and "\<And>\<Gamma> b e1 e2 S. I (\<Gamma>, Bool b, Alts e1 e2 # S) \<Longrightarrow> P (\<Gamma>, Bool b, Alts e1 e2 # S) (\<Gamma>, if b then e1 else e2, S)"   and "\<And>c. P c c"   and "\<And>c c' c''. \<lbrakk>c \<Rightarrow>\<^sup>* c'; c' \<Rightarrow>\<^sup>* c''; P c c'; P c' c''\<rbrakk> \<Longrightarrow> P c c''" shows "P c c'"
rreq_rrep_sn_simps_4: fixes d :: "nat"   and dip :: "nat" shows "rreq_rrep_sn (Newpkt d dip) = True"
83: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> (y \<squnion> top) = x \<sqinter> (y \<squnion> top)"
leg1_in_hom_1: shows "\<guillemotleft>tab\<^sub>1 f \<star> f\<^sub>0g\<^sub>1.p\<^sub>1 : src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<rightarrow> trg (f \<star> g)\<guillemotright>"
V_imp_Nil: fixes trn :: "('state, 'act, 'out) trans"   and tr :: "('state, 'act, 'out) trans list" assumes "V (trn # tr) = []" shows "V tr = []"
representation_ne_zero: fixes basis :: "'b set"   and v :: "'b"   and b :: "'b" assumes "representation basis v b \<noteq> (0::'a)" shows "b \<in> basis"
na: shows "(norm a)\<^sup>2 = 1 - b\<^sup>2"
order_tendstoD_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and y :: "'a"   and F :: "'b filter"   and a :: "'a" assumes "class.order_topology less_eq less open"   and "filterlim f (topological_space.nhds open y) F"   and "less a y" shows "\<forall>\<^sub>F x in F. less a (f x)"
Union_comp: fixes B :: "'a set set"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>A\<in>B. finite A"   and "\<And>A1 A2 x. \<lbrakk>A1 \<in> B; A2 \<in> B; A1 \<noteq> A2; x \<in> A1; x \<in> A2\<rbrakk> \<Longrightarrow> g x = (0::'b)" shows "sum g (\<Union> B) = (sum \<circ> sum) g B"
dim_eq_full: fixes S :: "('b, 'c) vec set" shows "(vec.dim S = finite_dimensional_vector_space.dimension cart_basis) = (vec.span S = UNIV)"
finite: shows "finite {b. f b \<noteq> (0::'a)}"
A: fixes n :: "nat" shows "A n = (\<Prod>(a, i)\<in>as \<inter> UNIV \<times> {n}. a)"
pirreducible_imp_not_splitted: fixes p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "p \<in> carrier (poly_ring M)"   and "pirreducible\<^bsub>M\<^esub> (carrier M) p"   and "degree p \<noteq> 1" shows "\<not> M.splitted p"
Pow_Int_eq: fixes A :: "'b set"   and B :: "'b set" shows "Pow (A \<inter> B) = Pow A \<inter> Pow B"
simps_7: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "subc c s (And p q) = And (subc c s p) (subc c s q)"
B_sub: shows "fst ` B \<subseteq> dgrad_p_set d m"
iterate_add_distrib: fixes m :: "nat"   and n :: "nat"   and a :: "'b" shows "iterate_add (m + n) a = iterate_add m a + iterate_add n a"
ln_approx_aux3_def: fixes x' :: "real"   and e :: "real"   and n :: "nat"   and y :: "real"   and d :: "real" shows "ln_approx_aux3 x' e n y d = (\<bar>2 * y * (\<Sum>k<n. inverse (real (2 * k + 1)) * y\<^sup>2 ^ k) + d - x'\<bar> \<le> e - d)"
rtranclp_reflclp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "R\<^sup>=\<^sup>=\<^sup>*\<^sup>* = R\<^sup>*\<^sup>*"
iso_tuple_UNIV_I: fixes x :: "'c" shows "x \<in> UNIV \<equiv> True"
hint: fixes n :: "nat" shows "h n integrable_on g ` S \<and> integral (g ` S) (h n) \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * h n (g x))"
sums: shows "(\<lambda>m. k / real (p ^ Suc m)) sums (k / real (p - 1))"
step_invariant_def: fixes A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<times> 'd \<times> 'c \<Rightarrow> bool" shows "(A \<TTurnstile>\<^sub>A (I \<rightarrow>) P) = (\<forall>a. I a \<longrightarrow> (\<forall>s\<in>reachable A I. \<forall>s'. (s, a, s') \<in> automaton.trans A \<longrightarrow> P (s, a, s')))"
simps_7: fixes t :: "'c local.term" shows "Can \<^bold>\<r>\<^bold>[t\<^bold>] = Can t"
sol_eq_r: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "x \<in> carrier G"   and "x \<cdot> a = b" shows "x = b \<cdot> \<rho> a"
nres_order_simps_21: fixes x :: "'g"   and Y :: "'g set" shows "(RETURN x \<le> RES Y) = (x \<in> Y)"
that: fixes c5 :: "real" assumes "\<And>x. x\<^sub>0 \<le> x \<Longrightarrow> c5 * f_approx x \<le> f x"   and "0 < c5" shows "thesis"
partial_deriv_one_d_integrable: fixes xc :: "real" assumes "xc \<in> cbox a b" shows "((\<lambda>y. partial_vector_derivative (\<lambda>a. F a \<bullet> j) i (y, xc)) has_integral F (g1 xc, xc) \<bullet> j - F (g2 xc, xc) \<bullet> j) (cbox (g2 xc) (g1 xc))"
Vagree_univ: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real" shows "Vagree \<nu> \<nu>' Denotational_Semantics.allvars = (\<nu> = \<nu>')"
Plus_2: shows "\<guillemotleft>ACI_norm_alt s\<guillemotright> = \<guillemotleft>s\<guillemotright>"
eq_snd_iff: fixes b :: "'a"   and p :: "'c \<times> 'a" shows "(b = snd p) = (\<exists>a. p = (a, b))"
star_rtc1: fixes x :: "'a" shows "(1::'a) + x + x\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
SUBSET_PROJ_DOM_DISJ: fixes vs :: "'d set"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and vs' :: "'d set" assumes "vs \<subseteq> prob_dom (prob_proj PROB (prob_dom PROB - vs'))" shows "disjnt vs vs'"
power_up_simp: fixes p :: "nat"   and x :: "real"   and n :: "nat" shows "power_up p x (Suc n) = (if odd n then truncate_up p ((power_up p x (Suc n div 2))\<^sup>2) else truncate_up p (x * power_up p x n))"
upper_closed: fixes y :: "'a" assumes "y \<in> carrier \<Y>\<^bsub>F\<^esub>" shows "\<pi>\<^sub>*\<^bsub>F\<^esub> y \<in> carrier \<X>\<^bsub>F\<^esub>"
Quotient_alt_def5: fixes R :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Abs :: "'f \<Rightarrow> 'g"   and Rep :: "'g \<Rightarrow> 'f"   and T :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "Quotient R Abs Rep T = (T \<le> BNF_Def.Grp UNIV Abs \<and> BNF_Def.Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>)"
Nml_Nmlize_1: fixes t :: "'a E.term" assumes "E.Arr t" shows "E.Nml \<^bold>\<lfloor>t\<^bold>\<rfloor>"
closed_Times: fixes S :: "'b set"   and T :: "'c set" assumes "closed S"   and "closed T" shows "closed (S \<times> T)"
ii: shows "{#mset_lit (subst_lit x \<eta>). x \<in># mset_set ((\<lambda>x. subst_lit x \<theta>) ` D)#} \<subseteq># {#mset_lit (subst_lit x \<eta>). x \<in># {#subst_lit x \<theta>. x \<in># mset_set D#}#}"
asig_comp_def: fixes a1 :: "'d set \<times> 'd set \<times> 'd set"   and a2 :: "'d set \<times> 'd set \<times> 'd set" shows "asig_comp a1 a2 \<equiv> (inputs a1 \<union> inputs a2 - (outputs a1 \<union> outputs a2), outputs a1 \<union> outputs a2, internals a1 \<union> internals a2)"
convergent_powser_sin_series_stream: shows "convergent_powser (msllist_of_msstream sin_series_stream)"
irbb_irfb: shows "bb\<^sup>-\<^sub>\<R> = \<smile> \<circ> fb\<^sup>-\<^sub>\<R>"
decr_grading_p_monom_mult: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "hom_grading d" shows "decr_grading_p d n (monom_mult c t p) = monom_mult c (decr_grading d n t) (decr_grading_p d n p)"
order_on_defs_3: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
subset_dfs_start: fixes q0 :: "bool list" assumes "nfa_is_node A q0" shows "bdd_lookup (fst (subset_dfs A q0)) q0 = Some 0"
all_RED: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "lam"   and \<tau> :: "ty"   and \<theta> :: "(name \<times> lam) list" assumes "\<Gamma> \<turnstile> t : \<tau>"   and "\<theta> closes \<Gamma>" shows "\<theta><t> \<in> RED \<tau>"
assms: shows "(v, v') \<in> subs\<^sup>+"
lem: fixes p :: "bool list"   and q :: "bool list"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, q) \<in> eps R" shows "(False # p, False # q) \<in> eps (RegExp2NAe.conc L R)"
modular: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<squnion> y \<sqinter> z = y \<sqinter> (x \<squnion> z)"
is_glb_bin: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "{x, y} >>| x"
insert_sub: fixes A :: "'c set"   and B :: "'c set"   and a :: "'c" assumes "A \<subseteq> B"   and "a \<in> B" shows "insert a A \<subseteq> B"
wfI_min: fixes R :: "('c \<times> 'c) set" assumes "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q" shows "wf R"
strict_trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "strict_prefix b a"   and "strict_prefix c b" shows "strict_prefix c a"