hcju/isabelleps huggingface dataset:
merge_on_F: fixes A :: "'e set"   and rha :: "'e \<Rightarrow> 'f"   and boa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and ma :: "'e \<Rightarrow> 'e \<Rightarrow> 'e option"   and B :: "'g set"   and rhb :: "'g \<Rightarrow> 'h"   and bob :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and mb :: "'g \<Rightarrow> 'g \<Rightarrow> 'g option" assumes "merge_on A rha boa ma"   and "merge_on B rhb bob mb" shows "merge_on {x. set1_F\<^sub>m x \<subseteq> A \<and> set3_F\<^sub>m x \<subseteq> B} (root_hash_F rha rhb) (blinding_of_F rha boa rhb bob) (merge_F rha ma rhb mb)"
simps_8: fixes v1 :: "'a"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (V v1) (Plus v va) = True"
trivial_limit_eventually: fixes net :: "'b filter"   and P :: "'b \<Rightarrow> bool" assumes "net = bot" shows "eventually P net"
RSpan_single_nonzero: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and m :: "'b" assumes "RModule R smult M"   and "m \<in> M"   and "m \<noteq> (0::'b)" shows "R_scalar_mult.RSpan R smult [m] \<noteq> 0"
mechanism_stable_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and \<phi> :: "('e \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('f \<Rightarrow> 'd set \<Rightarrow> 'd set) \<Rightarrow> 'e set \<Rightarrow> 'd set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.mechanism_stable Xd Xh ds \<phi> = (\<forall>Pd Ch. ContractsWithBilateralSubstitutesAndIRC Xd Xh Pd Ch \<longrightarrow> Contracts.stable_on Pd Ch ds (\<phi> Pd Ch ds))"
fin_digraphI: assumes "finite (verts G)"   and "finite (arcs G)" shows "fin_digraph G"
phis_measure_update_le: fixes k :: "'node \<times> 'b"   and a :: "'a"   and p :: "('node \<times> 'b, 'a) mapping"   and v :: "'b"   and g :: "'g" shows "phis_measure g v (Mapping.update k a p) \<le> phis_measure g v p"
max_prob_T_state_Nil: fixes s :: "'b"   and o :: "'a" shows "Max {emeasure (T (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as [] @- \<omega>'} |as. length as = length [] \<and> set as \<subseteq> \<S>} = 1"
bv_mult_def: fixes w1 :: "bit list"   and w2 :: "bit list" shows "bv_mult w1 w2 = nat_to_bv (bv_to_nat w1 * bv_to_nat w2)"
lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "sum g {..<Suc n} = g 0 + (\<Sum>i<n. g (Suc i))"
abs_eq: fixes x :: "'a" shows "mk_pos x = Abs_pos (if (0::'a) \<le> x then x else (0::'a))"
simps_10: shows "pratt1_max \<top>\<^sub>1 \<infinity>\<^sub>1 = \<top>\<^sub>1"
of_hypnat_0_le_iff: fixes n :: "nat star" shows "0 \<le> of_hypnat n"
Partitions_def: fixes x :: "'b list" shows "Partitions x \<equiv> {(x\<^sub>p, x\<^sub>s). x\<^sub>p @ x\<^sub>s = x}"
wf_simple_strand_first_Send_var_split: fixes S :: "('b, 'a) strand_step list"   and t :: "('c, 'a) Term.term"   and \<I> :: "'a \<Rightarrow> ('c, 'd) Term.term" assumes "wf\<^sub>s\<^sub>t {} S"   and "simple S"   and "\<exists>v\<in>wfrestrictedvars\<^sub>s\<^sub>t S. t \<cdot> \<I> = \<I> v" shows "\<exists>v S\<^sub>p\<^sub>r\<^sub>e S\<^sub>s\<^sub>u\<^sub>f. S = S\<^sub>p\<^sub>r\<^sub>e @ send\<langle>Var v\<rangle>\<^sub>s\<^sub>t # S\<^sub>s\<^sub>u\<^sub>f \<and> t \<cdot> \<I> = \<I> v \<and> \<not> (\<exists>w\<in>wfrestrictedvars\<^sub>s\<^sub>t S\<^sub>p\<^sub>r\<^sub>e. t \<cdot> \<I> = \<I> w)"
lunionM_assoc: fixes x :: "'m"   and y :: "'m"   and z :: "'m" shows "lunionM (lunionM x y) z = lunionM x (lunionM y z)"
replicate_empty: fixes n :: "nat"   and x :: "'a" shows "(replicate n x = []) = (n = 0)"
that: fixes a :: "real" assumes "0 < a"   and "\<And>x. x \<in> C \<Longrightarrow> {- a..a} \<subseteq> existence_ivl0 x" shows "thesis"
h: shows "bij_betw h (UNIV // R) (hfset (ord_of n))"
eventually_mem_existence_ivl: fixes b :: "real"   and x0 :: "'a" assumes "b \<in> existence_ivl0 x0" shows "\<forall>\<^sub>F x in at x0. b \<in> existence_ivl0 x"
range_prefix_match_sm: fixes pfx :: "'b prefix_match"   and rg :: "'b wordinterval" shows "wordinterval_to_set (fst (range_prefix_match pfx rg)) = fst (ipset_prefix_match pfx (wordinterval_to_set rg))"
empty_eq_union: fixes M :: "'a multiset"   and N :: "'a multiset" shows "({#} = M + N) = (M = {#} \<and> N = {#})"
not_learn_lim_V: fixes i :: "nat" shows "\<not> learn_lim \<phi> (V_bclim i) (\<sigma> i)"
finite_lattice_complete_Inf_lower: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "Inf A \<le> x"
atLeastLessThan_iff: fixes i :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and l :: "'g"   and u :: "'g" shows "(i \<in> ord.atLeastLessThan less_eq less l u) = (less_eq l i \<and> less i u)"
f2: shows "\<langle>|v\<rangle>| * |w\<rangle> \<Otimes> \<langle>|v\<rangle>| * |w\<rangle> = \<langle>|v\<rangle>| * |w\<rangle> \<Otimes> \<langle>|s\<rangle>| * |s\<rangle>"
Var: fixes rs :: "dB list"   and n :: "nat" assumes "listsp WN rs" shows "WN (Var n \<degree>\<degree> rs)"
IH: fixes S :: "'a set" assumes "finite S"   and "pairwise orthogonal S" shows "\<exists>U. pairwise orthogonal (S \<union> U) \<and> span (S \<union> U) = span (S \<union> T)"
30: fixes x :: "'a" shows "bot \<squnion> x = x"
finite_Lim_measure_incseq: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sets N"   and "incseq A" shows "(\<lambda>i. Sigma_Algebra.measure N (A i)) \<longlonglongrightarrow> Sigma_Algebra.measure N (\<Union> (range A))"
proper_interval_set_Compl_aux: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "finite UNIV"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "proper_intrvl.proper_interval_set_Compl_aux less proper_interval None 0 xs ys = (\<exists>A. ord.set_less_aux less_eq (set xs) A \<and> ord.set_less_aux less_eq A (- set ys))"
scan_last: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list" shows "last (a # scan f xs a) = fold f xs a"
collapse_2: fixes dining_event :: "dining_event" assumes "\<not> is_picks dining_event" shows "putsdown (phil dining_event) (fork dining_event) = dining_event"
revpoly: shows "\<forall>x. - 1 * poly [:c, b, a:] x = poly [:- c, - b, - a:] x"
Mp_smult_1: fixes a :: "int"   and f :: "int poly" shows "q.eq_m (Polynomial.smult (q.M a) f) (Polynomial.smult a f)"
P: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and lambda :: "nat"   and mu :: "nat" assumes "Basis.properties f x0 lambda mu" shows "fx0.properties f x0 lambda mu"
option_assn_simps_1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and v' :: "'b option" shows "option_assn P None v' = \<up> (v' = None)"
seq_comp_prop_1: fixes P :: "'c option process"   and Q :: "'c option process" shows "([], {}) \<in> seq_comp_failures P Q"
count_mset_0_iff: fixes xs :: "'c list"   and x :: "'c" shows "(count (mset xs) x = 0) = (x \<notin> set xs)"
ccTTree_mono2: fixes G :: "CoCalls"   and G' :: "CoCalls"   and S :: "Vars.var set" assumes "G \<sqsubseteq> G'" shows "ccTTree S G \<sqsubseteq> ccTTree S G'"
f_nonneg: fixes x :: "nat" shows "0 \<le> f x"
graph_rel_def: fixes Rv :: "('a \<times> 'c) set"   and Rw :: "('b \<times> 'd) set" shows "\<langle>Rv, Rw\<rangle>graph_rel \<equiv> {(g, g'). (nodes g, nodes g') \<in> \<langle>Rv\<rangle>set_rel \<and> (edges g, edges g') \<in> \<langle>Rv \<times>\<^sub>r Rw \<times>\<^sub>r Rv\<rangle>set_rel}"
subst_eq_if_disjoint_vars_ground: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<xi> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" assumes "subst_domain \<delta> \<inter> subst_domain \<xi> = {}"   and "ground (subst_range \<xi>)"   and "ground (subst_range \<delta>)" shows "t \<cdot> \<delta> \<cdot> \<xi> = t \<cdot> \<xi> \<cdot> \<delta>"
assms_3: shows "t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x'', m''\<rangle>"
do_store_action_def: fixes wact :: "('d, 'f, 'g) mem_store_action" shows "do_store_action wact = (\<lambda>s. case wact of mw_Mark r gc_mark \<Rightarrow> s\<lparr>heap := (heap s)(r := map_option (obj_mark_update (\<lambda>_. gc_mark)) (heap s r))\<rparr> | mw_Mutate r f new_r \<Rightarrow> s\<lparr>heap := (heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := new_r)\<rparr>) (heap s r))\<rparr> | mw_Mutate_Payload r f pl \<Rightarrow> s\<lparr>heap := (heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_payload := (obj_payload obj)(f := pl)\<rparr>) (heap s r))\<rparr> | mw_fA gc_mark \<Rightarrow> s\<lparr>fA := gc_mark\<rparr> | mw_fM gc_mark \<Rightarrow> s\<lparr>fM := gc_mark\<rparr> | mw_Phase gc_phase \<Rightarrow> s\<lparr>phase := gc_phase\<rparr>)"
wf_rexp_of_alt: fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.wf_formula \<Sigma> n \<phi>" shows "alphabet.wf (wf_atom \<Sigma>) n (formula.rexp_of_alt \<Sigma> n \<phi>)"
square_eq_1_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> '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 :: "'b" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(times x x = one) = (x = one \<or> x = uminus one)"
map_remdups': fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map f (remdups' f xs) = remdups (map f xs)"
cod_eqI': fixes a :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "a \<in> V.codomains f" shows "a = V.cod f"
assms_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "d x \<cdot> (y \<cdot> z) = d x \<cdot> y \<cdot> z"
summable_scaleR_right: fixes X :: "nat \<Rightarrow> 'a"   and r :: "real" assumes "summable X" shows "summable (\<lambda>n. r *\<^sub>R X n)"
simps: fixes f :: "heap \<Rightarrow> ('b \<times> heap) option" shows "execute (Heap.Heap f) = f"
simps: fixes s :: "'a list"   and l :: "nat"   and u :: "nat" shows "substr s l u = take (u - Suc l) (drop l s)"
compute_flow_val_correct: fixes am :: "nat \<Rightarrow> nat list"   and z :: "int nres" assumes "is_adj_map am"   and "(spec v. v = f.val) \<le> z" shows "compute_flow_val am cf \<le> z"
future_bounded: shows "MFOTL.future_reach \<phi> \<noteq> \<infinity>"
unit_mult_div_div: fixes a :: "'a"   and b :: "'a" assumes "is_unit a" shows "b * ((1::'a) div a) = b div a"
new_actions_for_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and adal :: "'c \<times> addr_loc" shows "new_actions_for P E adal = {a \<in> actions E. adal \<in> action_loc P E a \<and> is_new_action (action_obs E a)}"
x_2: shows "sum_list ys = freeword_funlift f x"
one_plus_numeral: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.numeral plus" shows "plus one (numeral.numeral one plus n) = numeral.numeral one plus (num.One + n)"
coboundedI1: fixes a :: "'a::semilattice_inf"   and c :: "'a::semilattice_inf"   and b :: "'a::semilattice_inf" assumes "(a::'a::semilattice_inf) \<le> (c::'a::semilattice_inf)" shows "(a::'a::semilattice_inf) \<sqinter> (b::'a::semilattice_inf) \<le> (c::'a::semilattice_inf)"
delete_block_subset: fixes b :: "'a set" shows "del_block b \<subseteq># \<B>"
solution_iff_nth_solution: fixes z :: "nat \<times> nat" shows "solution z = (z \<in> range nth_solution)"
alg_dvd_uminus_right_iff: fixes x :: "'b"   and y :: "'b" shows "(x alg_dvd - y) = (x alg_dvd y)"
assert_bound_tableau: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and a :: "'a \<times> 'b atom"   and s' :: "('a, 'b) state" assumes "AssertBound assert_bound"   and "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "assert_bound a s = s'" shows "\<T> s' = \<T> s \<and> \<nabla> s'"
map_snd_zip_take: fixes xs :: "'f list"   and ys :: "'e list" shows "map snd (zip xs ys) = take (min (length xs) (length ys)) ys"
shiftl_of_bl: fixes bl :: "bool list"   and n :: "nat" shows "of_bl bl << n = of_bl (bl @ replicate n False)"
aiVN: shows "approx_floatarith p (inner_floatariths (map floatarith.Var [0..<length a]) (map floatarith.Var [length a..<length a + length b])) (map (\<lambda>x. (x, 0)) a @ map (\<lambda>x. (x, 0)) b) \<noteq> None"
Abs_star_induct: fixes P :: "'b star \<Rightarrow> bool"   and x :: "'b star" assumes "\<And>y. y \<in> star \<Longrightarrow> P (Abs_star y)" shows "P x"
inject_1: fixes x2 :: "'b"   and y2 :: "'b" shows "(PVar x2 = PVar y2) = (x2 = y2)"
part_fn_subset: fixes r :: "nat"   and s :: "nat"   and YY :: "'b set"   and f :: "'b set \<Rightarrow> nat"   and Y :: "'b set" assumes "part_fn r s YY f"   and "Y \<subseteq> YY" shows "part_fn r s Y f"
refl: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l, Z\<rangle>"
hiding_advantage_ind_cpa_def: fixes \<A> :: "('a \<Rightarrow> ((nat \<times> nat) \<times> 'c) spmf) \<times> ('a \<Rightarrow> 'c \<Rightarrow> bool spmf)" shows "ped_commit.hiding_advantage_ind_cpa \<A> \<equiv> \<bar>spmf (ped_commit.hiding_game_ind_cpa \<A>) True - 1 / 2\<bar>"
intro: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R"   and "cring R"   and "primeideal_axioms I R" shows "primeideal I R"
all_tuples_def: shows "all_tuples \<equiv> \<Union> (snd ` (set (linearize data_prev) \<union> set (linearize data_in)))"
insert_before_removes_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ptr' :: "(_) object_ptr"   and node :: "(_) node_ptr"   and child :: "(_) node_ptr option"   and h' :: "(_) heap"   and children :: "(_) node_ptr list" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "ptr \<noteq> ptr'"   and "h \<turnstile> Shadow_DOM.insert_before ptr node child \<rightarrow>\<^sub>h h'"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr' \<rightarrow>\<^sub>r node # children" shows "h' \<turnstile> Shadow_DOM.get_child_nodes ptr' \<rightarrow>\<^sub>r children"
simps_2: fixes t :: "('c, 'd) BinomialTree" shows "rank_invar [t] = True"
elims_2: fixes x :: "'a"   and xa :: "'b node_action" assumes "inoclosed x xa" obtains uu_ :: "'a" and ii :: "nat set" and ni :: "nat set" and m :: "'b" where "x = uu_" and "xa = ii\<not>ni:arrive(m)" and "eq_newpkt m"   | uw_ :: "'a" and v :: "nat set" and va :: "'b" where "x = uw___" and "xa = v__:*cast(va__)"   | uw_ :: "'a" and v :: "nat" and va :: "nat" where "x = uwa___" and "xa = vb__:deliver(vaa__)"   | uw_ :: "'a" and v :: "nat" and va :: "nat" where "x = uwb___" and "xa = connect(vc__, vab__)"   | uw_ :: "'a" and v :: "nat" and va :: "nat" where "x = uwc___" and "xa = disconnect(v, va)"   | uw_ :: "'a" where "x = uw_" and "xa = \<tau>\<^sub>n"
hd_exp_plus: fixes xs :: "('a \<times> real) msllist"   and ys :: "('a \<times> real) msllist" shows "ms_aux_hd_exp (plus_ms_aux xs ys) = combine_options max (ms_aux_hd_exp xs) (ms_aux_hd_exp ys)"
cod_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VoV.seq g f" shows "VoV.cod (VoV.comp g f) = VoV.cod g"
distinct_36: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Stop \<noteq> x31@[x32]"
div_pCons_eq: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "pCons a p div q = (if q = 0 then 0 else pCons (coeff (pCons a (p mod q)) (degree q) / lead_coeff q) (p div q))"
lift_start_obs_def: fixes t :: "'a"   and obs :: "'b list" shows "lift_start_obs t obs = (t, InitialThreadAction) # map (\<lambda>ob. (t, NormalAction ob)) obs"
REC_rule_arb: fixes body :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and pre :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and arb :: "'d"   and x :: "'b"   and M :: "'d \<Rightarrow> 'b \<Rightarrow> 'c" assumes "trimono body"   and "pre arb x"   and "\<And>f arb x. \<lbrakk>\<And>arb' x. pre arb' x \<Longrightarrow> f x \<le> M arb' x; pre arb x; f \<le> REC body\<rbrakk> \<Longrightarrow> body f x \<le> M arb x" shows "REC body x \<le> M arb x"
shows_paren_def: fixes s :: "char list \<Rightarrow> char list" shows "shows_paren s = shows CHR ''('' \<circ> s \<circ> shows CHR '')''"
lift_keys_def: fixes i :: "nat"   and xs :: "(('a, 'b) pp \<times> nat, 'c) oalist_ntm" shows "lift_keys_pprod i xs = oalist_of_list_ntm (map_raw (\<lambda>kv. (map_component_pprod ((+) i) (fst kv), snd kv)) (list_of_oalist_ntm xs))"
ex231c: fixes x :: "'a" shows "x \<le> x \<sqinter> id x \<sqinter> x"
wia_eq: shows "wrap_iso_arcs hom f x = wrap_iso_arcs hom f y"
replicate_eq_list_of_eucl_zero: shows "replicate DIM('a) 0 = list_of_eucl (0::'a)"
map_indets_map_indets: fixes f :: "'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'f"   and p :: "('g \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'e" shows "map_indets f (map_indets g p) = map_indets (f \<circ> g) p"
lower_le_upper: fixes i :: "'a interval" shows "lower i \<le> upper i"
hyps_15: fixes thesis :: "bool" assumes "\<And>V. \<lbrakk>V \<subseteq> U - {x}; (monomial (1::'a) t, V) \<in> set (snd (ps0, qs0))\<rbrakk> \<Longrightarrow> thesis"   and "0 \<notin> S" shows "thesis"
strict_iff_order: fixes a :: "'a"   and b :: "'a" shows "(a \<prec> b) = (a \<preceq> b \<and> a \<noteq> b)"
pirreducible_degree: fixes K :: "'a set"   and p :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "pirreducible K p" shows "1 \<le> degree p"
indep_vars_iff_distr_eq_PiM: fixes I :: "'d set"   and X :: "'d \<Rightarrow> 'b \<Rightarrow> 'e"   and i :: "'i"   and M' :: "'d \<Rightarrow> 'e measure" assumes "I \<noteq> {}"   and "\<And>i. Y.random_variable i (M' i) (X i)" shows "Y.indep_vars i M' X I = (distr (distr M borel (Y i)) (Pi\<^sub>M I M') (\<lambda>x. \<lambda>i\<in>I. X i x) = Pi\<^sub>M I (\<lambda>i. distr (distr M borel (Y i)) (M' i) (X i)))"
push_forward_def: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real" shows "push_forward X = restrict0 dest.diff_fun_space (\<lambda>g. X (restrict0 src.carrier (g \<circ> f)))"
assms_3: shows "a11 * k2\<^sup>2 - 2 * a12 * k1 * k2 + a22 * k1\<^sup>2 \<noteq> 0"
a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes node_ptr :: "(_) node_ptr"   and uu :: "unit" shows "a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr uu = Heap_Error_Monad.bind (get_root_node (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 node_ptr)) (\<lambda>root. case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r root of None \<Rightarrow> Heap_Error_Monad.bind document_ptr_kinds_M (\<lambda>ptrs. Heap_Error_Monad.bind (filter_M (\<lambda>document_ptr. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disconnected_nodes. return (root \<in> 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 ` set disconnected_nodes))) ptrs) (\<lambda>candidates. return (hd candidates))) | Some document_ptr \<Rightarrow> return document_ptr)"
maxbidEquivalence: fixes N :: "'c set"   and \<Omega> :: "'d set"   and a :: "('c \<times> 'd set) set" shows "summedBidVectorRel (pseudoAllocation a <| (N \<times> finestpart \<Omega>)) N \<Omega> = summedBidVectorRel (maxbidAlg a N \<Omega>) N \<Omega>"
set_scf_list: fixes xs :: "'a list"   and scf :: "nat \<Rightarrow> nat" assumes "\<forall>i<length xs. 0 < scf i" shows "set (scf_list scf xs) = set xs"
l1_inv4I: fixes x :: "l1_state" assumes "\<And>Rb A B gnx. 0 < signalsResp x (Commit A B (Exp gnx (NonceF (Rb $ ny)))) \<Longrightarrow> guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress x Rb = Some {xny, xgnx, xgny, xsk, xEnd} \<and> guessed_frame Rb xgnx = Some gnx" shows "x \<in> l1_inv4"
of_nat_le_0_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) zero = (m = 0)"
assms_4: fixes k :: "'a" assumes "k \<in> K" shows "h [k] = k"
diffprop: fixes v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" assumes "\<And>G \<phi>. IsProperInX (\<lambda>x. \<lparr>G,\<^bold>\<iota>y. \<phi> y x\<rparr>)"   and "[\<lparr>A!,x\<^sup>P\<rparr> in v]"   and "[\<lparr>A!,y\<^sup>P\<rparr> in v]"   and "[x \<^bold>\<noteq> y in v]" shows "[\<^bold>\<exists>F. \<^bold>\<not>(\<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>F,y\<^sup>P\<rparr>) in v]"
lin_subspace: fixes x :: "'a" assumes "x \<in> E" shows "Subspace.subspace (lin x) E"
Blinfun_inject: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> {f. bounded_linear f}"   and "y \<in> {f. bounded_linear f}" shows "(Blinfun x = Blinfun y) = (x = y)"
not_numeral_less_zero: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (numeral.numeral one plus n) zero"
res_li1: fixes x :: "'a"   and y :: "'a" shows "(x \<leftarrow> y) \<cdot> y \<le> x"
conjugate_exponent_def: fixes p :: "'a" shows "conjugate_exponent p = (1::'a) + (1::'a) / (p - (1::'a))"
not_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" shows "(\<not> less x y) = less_eq y x"
triangle_in_hom_1: shows "\<guillemotleft>(\<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g) \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) : f \<star> src f \<Rightarrow> trg f \<star> f\<guillemotright>"
fls_right_inverse_idempotent_ring1: fixes f :: "'a fls"   and x :: "'a"   and y :: "'a" assumes "fls_nth f (fls_subdegree f) * x = (1::'a)"   and "x * y = (1::'a)" shows "fls_right_inverse (fls_right_inverse f x) y = f"
assms_1: shows "sourcenode a \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
sup_continuous_SUP: fixes I :: "'a set"   and M :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> sup_continuous (M i)" shows "sup_continuous (Sup (M ` I))"
g'_def: shows "g' = fps_shift (subdegree g) g"
info_real_alg_1: fixes x :: "real_alg"   and p :: "int poly"   and n :: "nat" assumes "info_real_alg x = Inr (p, n)" shows "p represents real_of x \<and> card {y. y \<le> real_of x \<and> poly (real_of_int_poly p) y = 0} = n \<and> irreducible p"
N: shows "(map (inv q) (filter (\<lambda>x. x \<in> range q) (xs @ zs)), inv q ` S) \<in> failures Q"
sorted_upto: fixes m :: "int"   and n :: "int" shows "sorted [m..n]"
attract: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<sqsubseteq> x"   and "x \<sqsubseteq> y"   and "z \<sqsubseteq> y"   and "x \<in> Collect derivable"   and "y \<in> Collect derivable"   and "z \<in> Collect derivable" shows "z \<sqsubseteq> x"