hcju/isabelleps huggingface dataset:
updown': fixes d :: "nat"   and dm :: "nat"   and p :: "(nat \<times> int) list"   and lm :: "nat"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "d \<le> dm"   and "p \<in> sparsegrid dm lm" shows "updown' dm lm d \<alpha> p = (\<Sum>p'\<in>lgrid (base {0..<d} p) {0..<d} lm. \<alpha> p' * (\<Prod>d' = 0..<d. l2_\<phi> (p' ! d') (p ! d')))"
vec_to_iarray_norm: fixes A :: "(real, 'c) vec" shows "norm A = norm_iarray (vec_to_iarray A)"
exists_base: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set" assumes "subfield K R"   and "dimension n K E" shows "\<exists>Vs. set Vs \<subseteq> carrier R \<and> local.independent K Vs \<and> length Vs = n \<and> Span K Vs = E"
cases: fixes x :: "nat \<times> cell list \<times> cell list" obtains s :: "nat" and l :: "cell list" and r :: "cell list" where "x = (s, l, r)"
interval_union_mono1: fixes a :: "'b interval"   and A :: "'b interval" shows "set_of a \<subseteq> set_of (sup a A)"
map_spmf_of_set_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "map_spmf f (spmf_of_set A) = spmf_of_set (f ` A)"
replicate_app_Cons_same: fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" shows "replicate n x @ x # xs = x # replicate n x @ xs"
wf'_3: shows "nonneg_cexpr (shift_var_set (set (vs @ vs'))) (case_nat BOOL \<Gamma>) f"
cindex_pathE_def: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" shows "cindex_pathE g z = cindexE 0 1 (\<lambda>t. Im (g t - z) / Re (g t - z))"
eq: shows "(\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) = (\<lambda>x. \<Sum>i\<in>Basis. ((\<lambda>y. \<Sum>j\<in>Basis. if j = i then y *\<^sub>R j else (0::'b)) \<circ> (\<lambda>x. norm (\<Sum>j\<in>Basis. if j = i then (x \<bullet> i) *\<^sub>R j else (0::'b))) \<circ> f) x)"
impI_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and q :: "'b"   and A :: "'b set"   and p :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "q \<in> Cn (A \<union> {p})" shows "imp_PL p q \<in> Cn A"
not_empty_K: shows "{k. k \<le> n \<and> \<not> cycle_free_up_to m k n} \<noteq> {}"
rbd_irbd_inv1: shows "bd\<^sup>-\<^sub>\<R> \<circ> bd\<^sub>\<R> = id"
elims: fixes x :: "('b \<times> color) tree"   and y :: "nat" assumes "bheight x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "('b \<times> color) tree" and xa :: "'b" and c :: "color" and r :: "('b \<times> color) tree" where "x = \<langle>l, (xa, c), r\<rangle>" and "y = (if c = Black then bheight l + 1 else bheight l)"
flip_at_base_simps_3: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a \<noteq> c"   and "b \<noteq> c" shows "(a \<leftrightarrow> b) \<bullet> c = c"
WEAK: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and H :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" assumes "G \<rhd> c : A"   and "G \<subseteq> H" shows "H \<rhd> c : A"
negative_zle: fixes n :: "nat"   and m :: "nat" shows "- int n \<le> int m"
distrib_lincomb: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and rs :: "'a list"   and ms :: "'b list" assumes "VectorSpaceHom smult V smult' T"   and "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "T (scalar_mult.lincomb smult rs ms) = scalar_mult.lincomb smult' rs (map T ms)"
mult_zero_right: fixes times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and a :: "'m" assumes "class.mult_zero times zero" shows "times a zero = zero"
simps_3: fixes Chn :: "'b"   and Dom :: "'b span_data"   and Cod :: "'b span_data"   and more :: "'c" shows "Cod \<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr> = Cod"
subset_insert_iff: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "order.greater_eq (insert x B) A = (if x \<in> A then order.greater_eq B (A - {x}) else order.greater_eq B A)"
chamber_singleton: fixes S :: "'b set"   and w :: "'b" assumes "CoxeterComplex S"   and "w \<in> \<langle>S\<rangle>" shows "SimplicialComplex.maxsimp (CoxeterComplex.TheComplex S) (dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w})"
countable_preimages_def: fixes B :: "'d set"   and Y :: "'c \<Rightarrow> 'd" shows "countable_preimages B Y = (\<lambda>n. if infinite B \<or> finite B \<and> n < card B then Y -` {from_nat_into B n} else {})"
rreq_dip_kD_oip_sqn: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRreq-:16..PRreq-:17} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>) \<and> (osn \<xi> < sqn (rt \<xi>) (oip \<xi>) \<or> sqn (rt \<xi>) (oip \<xi>) = osn \<xi> \<and> the (dhops (rt \<xi>) (oip \<xi>)) \<le> Suc (hops \<xi>) \<and> the (flag (rt \<xi>) (oip \<xi>)) = val))"
convergent_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'a"   and m :: "nat" shows "convergent (\<lambda>n. f (n + m)) = convergent f"
coeff_gt: fixes p :: "nat"   and j :: "nat" assumes "p < j" shows "Bernstein_Poly_01 j p = 0"
do_store_action_simps_1: fixes xa :: "'h"   and x :: "bool"   and xb :: "('e, 'f, 'g, 'h) Model.local_state" shows "do_store_action (mw_Mark xa x) xb = xb\<lparr>heap := (heap xb)(xa := map_option (obj_mark_update (\<lambda>_. x)) (heap xb xa))\<rparr>"
continuous_open_vimage: fixes S :: "'b set"   and f :: "'c \<Rightarrow> 'b" assumes "open S"   and "\<And>x. isCont f x" shows "open (f -` S)"
map_member_plus: fixes x :: "'a"   and y :: "'b"   and f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "((x, y) \<in>\<^sub>m f ++ g) = (x \<notin> dom g \<and> (x, y) \<in>\<^sub>m f \<or> (x, y) \<in>\<^sub>m g)"
as0_15: shows "DA0 = D0 + negs (mset As0)"
tx_range: shows "x \<in>\<^sub>i range_tm e tg"
Pareto_iff: fixes x :: "'b"   and y :: "'b" shows "Pareto R' x y = (\<forall>i\<in>agents'. R' i x y)"
ia_less_length: shows "ia < length (rev [0..<IArray.length (vec_to_iarray A)])"
unconst_trm_def: fixes t :: "'b trm" shows "unconst t = (case t of Var x \<Rightarrow> Map.empty x | trm.Const a \<Rightarrow> Some (to_name a) | trm1 \<cdot> x \<Rightarrow> Map.empty x)"
that: fixes as :: "'edge list" assumes "parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n'" shows "thesis"
IH: fixes VS :: "taylor_model list"   and X :: "taylor_model list"   and vs :: "real list" assumes "approx_slp' p ord I i ras VS = Some X"   and "vs all_in\<^sub>i range_tms e VS"   and "\<And>tm. tm \<in> set VS \<Longrightarrow> num_params (tm_poly tm) \<le> length I" shows "interpret_slp ras vs all_in\<^sub>i range_tms e X"
simps_2: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and c :: "('d, 'e, 'f) configuration"   and i :: "nat"   and p :: "'d"   and q :: "'d"   and c' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel" shows "distributed_system.next channel trans send recv c (RecvMarker i p q) c' = (distributed_system.can_occur channel trans send recv (RecvMarker i p q) c \<and> (\<forall>r. states c r = states c' r) \<and> (\<forall>r. r \<noteq> p \<longrightarrow> ps c r = ps c' r) \<and> Marker # msgs c' i = msgs c i \<and> cs c' i = (fst (cs c i), Done) \<and> (if ps c p \<noteq> None then ps c p = ps c' p \<and> (\<forall>i'. i' \<noteq> i \<longrightarrow> msgs c' i' = msgs c i') \<and> (\<forall>i'. i' \<noteq> i \<longrightarrow> cs c i' = cs c' i') else ps c' p = Some (states c p) \<and> (\<forall>i'. i' \<noteq> i \<and> (\<exists>r. channel i' = Some (p, r)) \<longrightarrow> msgs c' i' = msgs c i' @ [Marker]) \<and> (\<forall>i'. i' \<noteq> i \<and> (\<exists>r. channel i' = Some (r, p)) \<longrightarrow> cs c' i' = (fst (cs c i'), Recording)) \<and> (\<forall>i'. i' \<noteq> i \<and> (\<nexists>r. channel i' = Some (p, r)) \<longrightarrow> msgs c' i' = msgs c i') \<and> (\<forall>i'. i' \<noteq> i \<and> (\<nexists>r. channel i' = Some (r, p)) \<longrightarrow> cs c' i' = cs c i')))"
nth_default_coeffs_eq: fixes p :: "'b poly" shows "nth_default (0::'b) (coeffs p) = coeff p"
BAD_iff: fixes f :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(f \<in> BAD P) = ((\<forall>i. f i \<in> A) \<and> bad P f)"
supset_pseudominimal_inD2: fixes P :: "'b set set"   and x :: "'b set"   and z :: "'b set" assumes "dual_order.pseudominimal_in P x"   and "z \<in> P"   and "order.greater z x" shows "z = dual_order.bottom P"
strict_weak_trans: fixes a :: "'b pmf"   and b :: "'b pmf"   and c :: "'b pmf" assumes "a \<prec>[Pareto (SD \<circ> R')] b"   and "Pareto (SD \<circ> R') b c" shows "a \<prec>[Pareto (SD \<circ> R')] c"
t1_space_closedin_finite: fixes X :: "'c topology" shows "t1_space X = (\<forall>S. finite S \<and> S \<subseteq> topspace X \<longrightarrow> closedin X S)"
assms_2: fixes x :: "complex" assumes "x \<in> A" shows "f x \<notin> \<int>\<^sub>\<le>\<^sub>0"
insert_sort_mset: fixes xs :: "'c list" shows "mset (insert_sort xs) = mset xs"
simps_6: fixes f1 :: "('b \<times> nat, 'c) message \<Rightarrow> 'a"   and f2 :: "'b \<times> nat \<Rightarrow> ('b \<times> nat, 'c) message \<Rightarrow> 'a"   and x21 :: "'b \<times> nat"   and x22 :: "('b \<times> nat, 'c) message" shows "(case Receive x21 x22 of Send x \<Rightarrow> f1 x | Receive x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
OF_priority_match_def: fixes \<gamma> :: "'e set \<Rightarrow> 'f \<Rightarrow> bool"   and flow_entries :: "('e, 'd) flow_entry_match list"   and packet :: "'f" shows "OF_priority_match \<gamma> flow_entries packet \<equiv> let m = filter (\<lambda>f. \<gamma> (ofe_fields f) packet) flow_entries; m' = filter (\<lambda>f. \<forall>fo\<in>set m. ofe_prio fo \<le> ofe_prio f) m in case m' of [] \<Rightarrow> NoAction | [s] \<Rightarrow> Action (ofe_action s) | s # aa # lista \<Rightarrow> Undefined"
spectrum_def: shows "local.local.spectrum \<equiv> {I. pr_ideal local.carrier_quotient_ring I local.add_rel local.mult_rel local.zero_rel local.one_rel}"
comp_left_one: fixes x :: "'a" shows "(1::'a) * x = x"
ord_pres_dual: fixes f :: "'b \<Rightarrow> 'c" assumes "ord_pres f" shows "ord_pres (\<partial>\<^sub>F f)"
elims_2: fixes x :: "exetheory" assumes "exe_wf_theory x" obtains cto :: "(String.literal \<times> typ) list" and tao :: "(String.literal \<times> nat) list" and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" and ax :: "Core.term list" where "x = ExeTheory (ExeSignature cto tao sa) ax" and "exe_sig_conds (ExeSignature cto tao sa) \<and> (\<forall>p\<in>set ax. term_ok (translate_theory (ExeTheory (ExeSignature cto tao sa) ax)) p \<and> typ_of p = Some propT) \<and> is_std_sig (translate_signature (ExeSignature cto tao sa)) \<and> exe_wf_sig (ExeSignature cto tao sa) \<and> eq_axs \<subseteq> set ax"
minimal_up: fixes z :: "'a\<^sub>\<bottom>" shows "Ibottom \<sqsubseteq> z"
Rep_Tangle_Diagram: fixes x :: "Tangle_Diagram" shows "Rep_Tangle_Diagram x \<in> {x. is_tangle_diagram x}"
is_extensional: fixes f :: "'a2" assumes "\<not> A2.arr f" shows "\<tau>'o\<tau>.map f = B.null"
prj_chine_assoc'_2: shows "trg_trg_f.Prj\<^sub>0\<^sub>1 \<cdot> trg_trg_f.chine_assoc' = trg_trg_f.Prj\<^sub>1\<^sub>0"
refl_on_Int: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and B :: "'b set"   and s :: "('b \<times> 'b) set" assumes "refl_on A r"   and "refl_on B s" shows "refl_on (A \<inter> B) (r \<inter> s)"
unreachStatic: fixes s :: "Store"   and f :: "AttId"   and x :: "Value" shows "\<not> s\<turnstile> staticLoc f reachable_from x"
range_eq_singletonD: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and x :: "'b" assumes "range f = {a}" shows "f x = a"
correctly_synchronized_def: fixes P :: "'a prog"   and \<E> :: "('b \<times> ('c, 'b) obs_event action) llist set" shows "correctly_synchronized P \<E> = (\<forall>E\<in>\<E>. \<forall>ws. P \<turnstile> (E, ws) \<surd> \<longrightarrow> sequentially_consistent P (E, ws) \<longrightarrow> (\<forall>a\<in>actions E. \<forall>a'\<in>actions E. P,E \<turnstile>a\<dagger>a' \<longrightarrow> P,E \<turnstile> a \<le>hb a' \<or> P,E \<turnstile> a' \<le>hb a))"
Synchronized1Throw2: fixes xs :: "'addr val list"   and V :: "nat"   and a' :: "'addr"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and ad :: "'addr"   and h :: "'heap" assumes "xs ! V = Addr a'"   and "V < length xs" shows "uf,P,t \<turnstile>1 \<langle>insync\<^bsub>V\<^esub> (a) Throw ad,(h, xs)\<rangle> -\<lbrace>(Unlock, a'), SyncUnlock a'\<rbrace>\<rightarrow> \<langle>Throw ad,(h, xs)\<rangle>"
nontriv_solution_altdef: fixes D :: "nat"   and z :: "'a \<times> 'a" assumes "pell D" shows "pell.nontriv_solution D z = (pell.solution D z \<and> z \<noteq> (1::'a, 0::'a))"
intensional_simps_10: fixes y :: "'a \<Rightarrow> bool" shows "#True \<longrightarrow> y \<equiv> y"
Suc_2: shows "u \<in> graph A w (Suc k)"
aval'_sound: fixes s :: "char list \<Rightarrow> int"   and S :: "'av Abs_State.st"   and a :: "aexp" assumes "s \<in> \<gamma>\<^sub>f S" shows "aval a s \<in> \<gamma> (aval' a S)"
order_less_subst1: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and b :: "'b"   and c :: "'b" assumes "a < f b"   and "b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "a < f c"
cr_fset_def: shows "cr_fset \<equiv> \<lambda>x y. x = fset y"
chooset_def: fixes p :: "fm" shows "chooset p \<equiv> let q = zlfm p; d = \<delta> q; B = remdups (map (\<lambda>(t, k). (simpnum t, k)) (\<rho> q)); a = remdups (map (\<lambda>(t, k). (simpnum t, k)) (\<alpha>_\<rho> q)) in if length B \<le> length a then (q, B, d) else (mirror q, a, d)"
Iagree_sub: fixes A :: "('d + 'e + 'f) set"   and B :: "('d + 'e + 'f) set"   and I :: "('d, 'e, 'f) interp"   and J :: "('d, 'e, 'f) interp" assumes "A \<subseteq> B"   and "Iagree I J B" shows "Iagree I J A"
antisymmetric_one_closed: shows "antisymmetric (1::'a)"
minus_right_nil: fixes xs :: "'b list" shows "xs - [] = xs"
scc_ofI_reachable: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "u \<in> G.scc_of v"
cobounded2: fixes a :: "'d"   and b :: "'d" shows "min a b \<le> b"
fun_rel_mono: fixes R1 :: "('d \<times> 'e) set"   and R2 :: "('d \<times> 'e) set"   and R3 :: "('f \<times> 'g) set"   and R4 :: "('f \<times> 'g) set" assumes "R1 \<subseteq> R2"   and "R3 \<subseteq> R4" shows "R2 \<rightarrow> R3 \<subseteq> R1 \<rightarrow> R4"
is_gcdI: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x dvd a"   and "x dvd b"   and "\<And>y. \<lbrakk>y dvd a; y dvd b\<rbrakk> \<Longrightarrow> y dvd x" shows "is_gcd x a b"
compl_le_swap1: fixes y :: "'a"   and x :: "'a" assumes "y \<le> - x" shows "x \<le> - y"
pred_inject: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and aa :: "'a stream" shows "pred_stream P (a ## aa) = (P a \<and> pred_stream P aa)"
k_def: shows "k = \<bar>p * (p - 1)\<bar> * max ((1 / 2) powr (p - 2)) ((3 / 2) powr (p - 2)) / 2"
cases: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and a :: "'b"   and n :: "'b"   and S :: "'b set" assumes "CFG sourcenode targetnode valid_edge Entry"   and "a \<in> CFG.obs sourcenode targetnode valid_edge n S" obtains as :: "'a list" and n' :: "'b" where "a = n'" and "CFG.path sourcenode targetnode valid_edge n as n'" and "\<forall>nx\<in>set (CFG.sourcenodes sourcenode as). nx \<notin> S" and "n' \<in> S"
dom_eqI: fixes a :: "'a"   and f :: "'a" assumes "R.ide a"   and "R.seq f a" shows "R.dom f = a"
iso_iff: fixes h :: "'a \<Rightarrow> 'b"   and G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme" shows "(h \<in> Group.iso G H) = (h \<in> hom G H \<and> h ` carrier G = carrier H \<and> inj_on h (carrier G))"
res: shows "roots_of_rat_poly p = concat (map (roots_of_rat_poly_main \<circ> fst) pis)"
chain_fun_ordD: fixes le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "('b \<Rightarrow> 'c) set"   and x :: "'b" assumes "Complete_Partial_Order.chain (fun_ord le) Y" shows "Complete_Partial_Order.chain le ((\<lambda>f. f x) ` Y)"
add_set: fixes x :: "'node"   and y :: "'node"   and S :: "'node set"   and mrk :: "'node set" assumes "x \<noteq> y"   and "x \<in> path S mrk" shows "x \<in> path (insert y S) (insert y mrk)"
beta_cases_1: fixes i :: "nat"   and t :: "Core.term"   and P :: "bool" assumes "Bv i \<rightarrow>\<^sub>\<beta> t" shows "P"
dominates_trans': fixes g :: "'g"   and n :: "'node"   and n' :: "'node"   and n'' :: "'node" assumes "old.dominates g n n'"   and "old.dominates g n' n''" shows "old.dominates g n n''"
simps_4: fixes p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "FOL_Fitting.size_form (And p q) = FOL_Fitting.size_form p + FOL_Fitting.size_form q + 1"
finite_G: fixes G :: "('c, 'd) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" assumes "sylow G p a m" shows "finite (carrier G)"
mcp_unique: fixes z :: "'b list"   and W :: "'b list set"   and p :: "'b list"   and p' :: "'b list" assumes "mcp z W p"   and "mcp z W p'" shows "p = p'"
asIMOD_igSwapCls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSwapCls (asIMOD SEM)"
trivial_bounds_sin: shows "\<forall>x. sin x \<in> {- 1..1}"
LeastI2_wellorder: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and a :: "'d"   and Q :: "'d \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "P a"   and "\<And>a. \<lbrakk>P a; \<forall>b. P b \<longrightarrow> less_eq a b\<rbrakk> \<Longrightarrow> Q a" shows "Q (ord.Least less_eq P)"
pp_idempotent: fixes x :: "'a" shows "- - - - x = - - x"
fmdomI: fixes m :: "('n, 'm) fmap"   and x :: "'n"   and y :: "'m" assumes "fmlookup m x = Some y" shows "x |\<in>| fmdom m"
measurable_shd: fixes M :: "'b measure" shows "shd \<in> stream_space M \<rightarrow>\<^sub>M M"
le_iff_lt_or_eq: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a"   and y :: "'a" assumes "class.Ord_linear eq compare" shows "(Ord.le compare\<cdot>x\<cdot>y = TT) = (Ord.lt compare\<cdot>x\<cdot>y = TT \<or> eq\<cdot>x\<cdot>y = TT)"
red_lcl_incr: fixes e :: "char list exp"   and h\<^sub>0 :: "heap"   and l\<^sub>0 :: "sfields"   and sh\<^sub>0 :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>e,(h\<^sub>0, l\<^sub>0, sh\<^sub>0),b\<rangle> \<rightarrow> \<langle>e',(h\<^sub>1, l\<^sub>1, sh\<^sub>1),b'\<rangle>" shows "dom l\<^sub>0 \<subseteq> dom l\<^sub>1"
monad_reader_stateT': assumes "monad_reader Monad_Overloading.return Monad_Overloading.bind ask" shows "monad_reader Monad_Overloading.return Monad_Overloading.bind ask"
that_1: assumes "\<paragraph>''Invar pres''" shows "I s'"
subsetI: fixes A :: "'b set"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
map_of_SomeD: fixes xs :: "('b \<times> 'a) list"   and k :: "'b"   and y :: "'a" assumes "map_of xs k = \<lfloor>y\<rfloor>" shows "(k, y) \<in> set xs"
right_neutral: fixes a :: "'b" shows "a * (1::'b) = a"
in_pairsI: fixes aps :: "bool \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> (bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and flag :: "bool"   and a :: "('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and b :: "('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and xs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" assumes "\<And>gs bs h ps. set (aps flag gs bs h ps) = set ps \<union> Pair flag ` ({h} \<times> (set gs \<union> set bs))"   and "a \<noteq> b"   and "a \<in> set xs"   and "b \<in> set xs" shows "(flag, a, b) \<in> set (pairs aps flag xs) \<or> (flag, b, a) \<in> set (pairs aps flag xs)"
red_single_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" shows "red_single p q f t = (f \<noteq> 0 \<and> lookup p (t \<oplus> lt f) \<noteq> (0::'c) \<and> q = p - monom_mult (lookup p (t \<oplus> lt f) / lc f) t f)"
prj0_ext: fixes f :: "'a"   and g :: "'a" assumes "\<not> cospan f g" shows "\<p>\<^sub>0[f, g] = null"
full_functor_axioms: shows "full_functor S.comp C S.map"
n_o_def: fixes n :: "'b \<Rightarrow> 'a"   and opt :: "'b option" shows "n_o n opt = (case opt of None \<Rightarrow> 0::'a | Some x \<Rightarrow> n x + (1::'a))"
Neville_Aitken: shows "alg = Neville_Aitken"
sqmatrix_mult_0_right: fixes A :: "('c, 'd) sqmatrix" shows "A \<cdot> 0 = 0"
snd_bf_subset: fixes j' :: "nat"   and i' :: "nat"   and j :: "nat"   and i :: "nat"   and k :: "nat"   and M :: "nat set"   and r :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "((j', i'), j, i) \<in> pair_less"   and "(j', i') \<in> IJ k" shows "snd (bf M r (j, i)) \<subseteq> snd (bf M r (j', i'))"
finite_indets: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "finite (indets p)"
op_defs_7: fixes F :: "('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "exi F \<pi>l \<equiv> \<exists>\<pi>. wfp AP' \<pi> \<and> stateOf \<pi> = (if \<pi>l \<noteq> [] then stateOf (last \<pi>l) else s0) \<and> F \<pi> \<pi>l"
sets_pair_measure_cong: fixes M1 :: "'b measure"   and M1' :: "'b measure"   and M2 :: "'c measure"   and M2' :: "'c measure" assumes "sets M1 = sets M1'"   and "sets M2 = sets M2'" shows "sets (M1 \<Otimes>\<^sub>M M2) = sets (M1' \<Otimes>\<^sub>M M2')"
hd_rand: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and dst :: "pdf_dist" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Random dst e \<Rightarrow> apply_dist_to_dens dst f"
non_atomic_Exp: fixes x :: "msg"   and y :: "msg" shows "composed (Exp x y)"
solve_for_lhs_R_def: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "(nat list \<times> nat list) list"   and matr :: "rat mat" shows "solve_for_lhs_R p qs subsets matr = matr_option (dim_row matr) (mat_inverse_var matr) *\<^sub>v v_vec_R p qs subsets"
label_1: shows "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
terminates_elim_cases_10: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>Throw \<down> s" obtains f :: "'f" where "sb__ = Fault f"   | "sb__ = Stuck"   | s :: "'e" where "sb__ = Normal sc__"   | s :: "'e" where "sb__ = Abrupt s"