hcju/isabelleps huggingface dataset:
prod_principal_idealTr0: fixes a :: "'a"   and b :: "'a"   and z :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "z \<in> set_mult R (R \<diamondsuit>\<^sub>p a) (R \<diamondsuit>\<^sub>p b)" shows "z \<in> R \<diamondsuit>\<^sub>p (a \<cdot>\<^sub>r b)"
ownership_distinct: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>i \<union> all_acquired sb\<^sub>i) \<inter> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) = {}"
birkhoff_sum_def: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat"   and x :: "'a" shows "birkhoff_sum f n x = (\<Sum>i<n. f ((T ^^ i) x))"
ParityGame_def: fixes G :: "('c, 'd) ParityGame_scheme" shows "ParityGame G \<equiv> Digraph G \<and> ParityGame_axioms G"
rts: shows "{x. \<exists>q ri. (q, ri) \<in> orig \<and> poly (real_of_int_poly q) x = 0} = (\<Union>(q, ri)\<in>set (todo @ old). {x. poly (real_of_int_poly q) x = 0})"
isos_compose: fixes f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "B.VV.iso f"   and "B.VV.iso f'"   and "B.VV.seq f' f" shows "B.VV.iso (B.VV.comp f' f)"
takeWhile_idem: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "takeWhile P (takeWhile P xs) = takeWhile P xs"
bounded_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 "bounded (S 0)" obtains a :: "'a" where "\<And>n. a \<in> S n"
butterfly_def: fixes s :: "'c"   and t :: "'b" shows "butterfly s t = vector_to_cblinfun s o\<^sub>C\<^sub>L vector_to_cblinfun t*"
isPath_append: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and p1 :: "(nat \<times> nat) list"   and p2 :: "(nat \<times> nat) list"   and v :: "nat" shows "Graph.isPath c u (p1 @ p2) v = (\<exists>w. Graph.isPath c u p1 w \<and> Graph.isPath c w p2 v)"
comp_null_2: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "f \<cdot> null = null"
finite_comm_group_Characters: shows "finite_comm_group (Characters (Characters G))"
add_mset_not_empty: fixes a :: "'a"   and A :: "'a multiset" shows "add_mset a A \<noteq> \<Empt>"
cases: fixes a1 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a2 :: "nat \<Rightarrow> nat"   and a3 :: "pr_operation'"   and a4 :: "nat \<times> nat \<Rightarrow> 'capacity"   and a5 :: "nat \<Rightarrow> nat" assumes "((a1, a2), a3, a4, a5) \<in> pr_algo_lts'" obtains f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fa__" and "a2 = la__" and "a3 = NONSAT_PUSH'" and "a4 = push_effect fa__ ea__" and "a5 = la__" and "Height_Bounded_Labeling c s t fa__ la__" and "nonsat_push_precond fa__ la__ ea__"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and e :: "nat \<times> nat" where "a1 = fb__" and "a2 = lb__" and "a3 = SAT_PUSH' e" and "a4 = push_effect fb__ e" and "a5 = lb__" and "Height_Bounded_Labeling c s t fb__ lb__" and "sat_push_precond fb__ lb__ e"   | f :: "nat \<times> nat \<Rightarrow> 'capacity" and l :: "nat \<Rightarrow> nat" and u :: "nat" where "a1 = f" and "a2 = l" and "a3 = RELABEL'" and "a4 = f" and "a5 = relabel_effect f l u" and "Height_Bounded_Labeling c s t f l" and "relabel_precond f l u"
hd_rev: fixes xs :: "'b list" assumes "xs \<noteq> \<epsilon>" shows "hd (rev xs) = last xs"
h7: shows "\<not> 4 * k + 3 dvd a'"
sig_regular_top_reduced_lt_unique: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "is_sig_GB_upt d G (lt q)"   and "p \<in> dgrad_sig_set' (length fs) d"   and "q \<in> dgrad_sig_set' (length fs) d"   and "lt p = lt q"   and "p \<noteq> 0"   and "q \<noteq> 0"   and "\<not> is_sig_red (\<prec>\<^sub>t) (=) G p"   and "\<not> is_sig_red (\<prec>\<^sub>t) (=) G q" shows "punit.lt (rep_list p) = punit.lt (rep_list q)"
gcd_proj2_if_dvd_int: fixes y :: "int"   and x :: "int" assumes "y dvd x" shows "gcd x y = \<bar>y\<bar>"
simps_43: fixes x71 :: "nat"   and x72 :: "msg"   and x4 :: "nat" shows "Crypt x71 x72 \<noteq> Key x4"
traces_prefixclosed_def: fixes ES :: "'a ES_rec" shows "traces_prefixclosed ES \<equiv> prefixclosed Tr\<^bsub>ES\<^esub>"
shext_refl: fixes sh :: "sheap" shows "sh \<unlhd>\<^sub>s sh"
mk_less_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "mk_less R = (\<lambda>x y. R x y \<and> \<not> R y x)"
Cons: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and v :: "val"   and s1 :: "state"   and es :: "expr list"   and vs :: "val list"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>es\<doteq>\<succ>vs\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>e # es\<doteq>\<succ>v # vs\<rightarrow> s2"
minus_add_cancel: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "plus (uminus a) (plus a b) = b"
adj_path_app: fixes v :: "'v"   and ps :: "'v list"   and u :: "'v" assumes "adj_path v ps"   and "ps \<noteq> []"   and "adjacent (last ps) u" shows "adj_path v (ps @ [u])"
perm_swap_2: fixes a :: "vrs"   and b :: "vrs"   and x :: "'a" shows "[(a, b)] \<bullet> [(a, b)] \<bullet> x = x"
covering_space_locally_path_connected_eq: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "covering_space C p S" shows "locally path_connected S = locally path_connected C"
ex_mset: fixes X :: "'a multiset" shows "\<exists>xs. mset xs = X"
ccpo_monoI: fixes f :: "'g \<Rightarrow> 'h" assumes "mono f" shows "monotone (\<le>) (\<le>) f"
sep_add_assoc: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.pre_sep_algebra plus zero sep_disj"   and "sep_disj x y"   and "sep_disj y z"   and "sep_disj x z" shows "plus (plus x y) z = plus x (plus y z)"
null_sets_UN: fixes N :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "\<And>i. N i \<in> null_sets M" shows "\<Union> (range N) \<in> null_sets M"
simps_33: fixes vd :: "'a"   and ve :: "bexp"   and vf :: "'a acom"   and vg :: "'a"   and v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a" shows "({vd} WHILE ve DO vf {vg} \<sqsubseteq> IF v THEN va ELSE vb {vc}) = False"
Inf_top_conv_1: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Inf A = top) = (\<forall>x\<in>A. x = top)"
lookup: fixes x :: "'b \<Rightarrow>\<^sub>0 'c" shows "poly_mapping.lookup x \<in> {f. finite {x. f x \<noteq> (0::'c)}}"
return_time_eqI: fixes f :: "'e \<Rightarrow> 'e"   and X :: "'e set"   and P :: "'e set"   and t :: "real"   and x :: "'e" assumes "auto_ll_on_open f X"   and "closed P"   and "0 < t"   and "t \<in> auto_ll_on_open.existence_ivl0 f X x"   and "auto_ll_on_open.flow0 f X x t \<in> P"   and "\<And>s. \<lbrakk>0 < s; s < t\<rbrakk> \<Longrightarrow> auto_ll_on_open.flow0 f X x s \<notin> P" shows "auto_ll_on_open.return_time f X P x = t"
p_nth: fixes i :: "nat" assumes "i < length p" shows "p ! i = (face_cycle_succ ^^ i) a0"
dim_row_mat_1: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" shows "dim_row (Matrix.mat nr nc f) = nr"
outs: shows "outs_\<I> \<I>1 = UNIV"
l2_inv2_def: shows "l2_inv2 \<equiv> {s. \<forall>Ra A B gny. Auth B A \<langle>Number 0, gny, Exp Gen (NonceF (Ra $ nx))\<rangle> \<in> chan s \<longrightarrow> A \<notin> bad s \<and> B \<notin> bad s \<longrightarrow> (\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress s Rb) {xny, xgnx, xgny, xsk} \<and> gny = Exp Gen (NonceF (Rb $ ny)) \<and> guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx))))}"
Number_Exp: fixes X :: "nat"   and X' :: "msg"   and Y' :: "msg" shows "Number X \<noteq> Exp X' Y'"
wt_elim_cases_15: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and e :: "expr"   and mn :: "mname"   and pT' :: "ty list"   and p :: "expr list"   and T :: "tys" assumes "E,dt\<Turnstile>In1l ({accC,statT,mode}e\<cdot>mn( {pT'}p))\<Colon>T" obtains pTs :: "ty list" and statDeclT :: "ref_ty" and m :: "mhead" where "T = Inl (resTy m)" and "E,dt\<Turnstile>e\<Colon>-RefT statT" and "E,dt\<Turnstile>p\<Colon>\<doteq>pTs" and "max_spec (prg E) (cls E) statT \<lparr>name = mn, parTs = pTs\<rparr> = {((statDeclT, m), pT')}" and "accC = cls E" and "mode = invmode m e"
cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "('a, 'c, 'e) Vardi.var"   and a2 :: "('b, 'd, 'e) Vardi.var" assumes "rel_var R S a1 a2" obtains x :: "('a \<times> 'b, 'c \<times> 'd, 'e) Vardi.var" where "a1 = map_var fst fst x" and "a2 = map_var snd snd x" and "set1_var x \<subseteq> {(x, y). R x y}" and "set2_var x \<subseteq> {(x, y). S x y}"
split_aforms_def: fixes xs :: "('a \<times> 'a pdevs) list"   and i :: "nat" shows "split_aforms xs i = (let splits = map (\<lambda>x. split_aform x i) xs in (map fst splits, map snd splits))"
simps_3: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real" shows "aEvalUni (LeqUni (a, b, c)) x = (a * x\<^sup>2 + b * x + c \<le> 0)"
bounded_imp_convergent_subsequence: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b" assumes "class.heine_borel dist uniformity open"   and "metric_space.bounded dist (range f)" shows "\<exists>l r. strict_mono r \<and> filterlim (f \<circ> r) (topological_space.nhds open l) sequentially"
l_inv_ex: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "\<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
is_offending_flows_def: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and f :: "('c \<times> 'c) set"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" shows "SecurityInvariant_withOffendingFlows.is_offending_flows sinvar f G nP \<equiv> \<not> sinvar G nP \<and> sinvar (delete_edges G f) nP"
AssignIdx_simp: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and i :: "aexp"   and a :: "aexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (x[i] ::= a, s) \<Rightarrow> t = (t = s(x := (s x)(aval i s := aval a s)))"
bezw_aux: fixes x :: "nat"   and y :: "nat" shows "int (gcd x y) = fst (bezw x y) * int x + snd (bezw x y) * int y"
field_differentiable_within_exp: fixes z :: "'a"   and s :: "'a set" shows "exp field_differentiable at z within s"
select_convs_2: fixes nodes :: "'a set"   and edges :: "('a \<times> 'a) set"   and more :: "'b" shows "edges \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = edges"
simps_2: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'b list"   and x :: "'b" shows "ipurge_tr I D u (xs @ [x]) = (if D x \<in> sinks I D u (xs @ [x]) then ipurge_tr I D u xs else ipurge_tr I D u xs @ [x])"
cod_simps_3: shows "doms.\<nu>\<pi>.cod.leg1 = doms.\<nu>\<pi>.leg1"
integral_0_1_ln_times_powr_1: fixes e :: "real" assumes "- 1 < e" shows "LBINT x=0..1. - ln x * x powr e = 1 / (e + 1)\<^sup>2"
size1_height_if_incomplete: fixes t :: "'b tree" assumes "\<not> complete t" shows "size1 t < 2 ^ height t"
polysub_same_0: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "p -\<^sub>p p = 0\<^sub>p"
pwd_spy_id_sesk_def: fixes n :: "nat" shows "pwd_spy_id_sesk n \<equiv> let (s, S, A, B, C, D) = pwd_spy_id_pubk n; SK = (Some S, {A, B}, {C, D}) in (insert (Spy, \<langle>n, SesKey SK\<rangle>) s, S, A, B, C, D)"
TS_inv'_det2: fixes s :: "'a list"   and h :: "'a list"   and x :: "'a"   and i :: "'a list" shows "TS_inv' (s, h) x i = (\<exists>hs. (s, h) = (if x = hd i then i else rev i, [x, x] @ hs)) \<or> (s, h) = (if x = hd i then i else rev i, [])"
fst_set_full_gb: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "fst ` set (full_gb bs) = (\<lambda>v. monomial (1::'e) (term_of_pair (0::'a, component_of_term v))) ` Keys (fst ` set bs)"
t'_def: shows "t' \<equiv> SOME t'. t' |\<in>| cont t \<and> fst s'a' = fst (tree.root t')"
comp_null_2: fixes f :: "'a" shows "f \<cdot>\<^sub>S S.null = S.null"
abs_sgn: fixes k :: "'a" shows "\<bar>k\<bar> = k * sgn k"
cong_refl: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b b a"
clear_Bit_eq: fixes w :: "'b word"   and n :: "nat" shows "clearBit w n = unset_bit n w"
fresh_fun_def: fixes h :: "'b \<Rightarrow> 'a" shows "fresh_fun h \<equiv> THE fr. \<forall>a. a \<sharp> h \<longrightarrow> h a = fr"
ev_neq_start_implies_ev_neq: fixes y :: "'a"   and xs :: "'a stream" assumes "ev (Not \<circ> HLD {y}) (y ## xs)" shows "ev (\<lambda>xs. shd xs \<noteq> shd (stl xs)) (y ## xs)"
f0: shows "\<phi> \<Otimes> |\<beta>\<^sub>0\<^sub>0\<rangle> = Tensor.mat_of_cols_list 8 [[\<alpha> / complex_of_real (sqrt 2), 0, 0, \<alpha> / complex_of_real (sqrt 2), \<beta> / complex_of_real (sqrt 2), 0, 0, \<beta> / complex_of_real (sqrt 2)]]"
coboundedI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<preceq> c" shows "a \<sqinter> b \<preceq> c"
P'''Trans: shows "P''' \<longmapsto> a<x> \<prec> P''"
unit_simps_5: shows "trg\<^sub>C \<eta> = src\<^sub>C f"
stoch_proc_filt_def: fixes M :: "'d measure"   and X :: "nat \<Rightarrow> 'd \<Rightarrow> 'e"   and N :: "'e measure"   and n :: "nat" shows "stoch_proc_filt M X N n = gen_subalgebra M (sigma_sets (space M) (\<Union>i\<in>{m. m \<le> n}. {X i -` A \<inter> space M |A. A \<in> sets N}))"
inject_4: fixes x2 :: "trm"   and x1 :: "trm"   and y2 :: "trm"   and y1 :: "trm" shows "(x2 \<cdot> x1 = y2 \<cdot> y1) = (x2 = y2 \<and> x1 = y1)"
Uvariation_refl: fixes \<nu> :: "variable \<Rightarrow> real"   and V :: "variable set" shows "Uvariation \<nu> \<nu> V"
not_Leaf_if_not_balanced1: fixes l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree" assumes "\<not> balanced1 l r" shows "r \<noteq> \<langle>\<rangle>"
simps_3: fixes \<phi> :: "'a MFOTL.formula" shows "MFOTL.future_reach (formula.Neg \<phi>) = MFOTL.future_reach \<phi>"
V_3: shows "\<forall>v\<in>set V. \<exists>x. v = Var x"
intro: fixes effect_apply :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and L :: "'c \<times> 'a fs_set \<times> 'a \<Rightarrow> 'a fs_set" assumes "eqvt effect_apply"   and "eqvt L" shows "effect_nominal_ts_axioms effect_apply L"
bij_betw_imp_surj_on: fixes f :: "'q \<Rightarrow> 'r"   and A :: "'q set"   and B :: "'r set" assumes "bij_betw f A B" shows "f ` A = B"
set_category_given_img_axioms: shows "set_category_given_img (\<cdot>) S.some_img \<AA>"
simps_6: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list" shows "eval_wellf (Pr n f g) (0 # xs) = eval_wellf f xs"
B'_altdef: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>" shows "B' \<epsilon> = (*\<^sub>R) (inverse (1 + \<epsilon>)) -` B"
zero: fixes i :: "nat"   and q :: "nat \<Rightarrow> nat" assumes "i \<in> {0..<n}"   and "q permutes {0..<n}" shows "prod' (Fun.swap n i id \<circ> q) = (0::'a)"
c_cons_pos: fixes x :: "nat"   and u :: "nat" shows "0 < c_cons x u"
tendsto_at_iff_tendsto_nhds_within: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a"   and s :: "'b set" assumes "f x = y" shows "(f \<longlongrightarrow> y) (at x within s) = (f \<longlongrightarrow> y) (inf (nhds x) (principal s))"
select_convs_5: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cTemperatureF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cTemperatureF"
subset_imageE: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "B \<subseteq> f ` A" obtains C :: "'c set" where "C \<subseteq> A" and "B = f ` C"
finite_varsL: fixes l :: "('a, 'b) lit" shows "finite (varsL l)"
iso_class_memb_is_ide: fixes F :: "'a set"   and f :: "'a" assumes "B.is_iso_class F"   and "f \<in> F" shows "B.ide f"
normalize_quotE': fixes x :: "'b \<times> 'b" assumes "snd x \<noteq> (0::'b)" obtains d :: "'b" where "fst x = fst (normalize_quot x) * d" and "snd x = snd (normalize_quot x) * d" and "d dvd fst x" and "d dvd snd x" and "d \<noteq> (0::'b)"
dghm_eqI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<GG> :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<FF> : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"   and "\<GG>\<lparr>ObjMap\<rparr> = \<FF>\<lparr>ObjMap\<rparr>"   and "\<GG>\<lparr>ArrMap\<rparr> = \<FF>\<lparr>ArrMap\<rparr>"   and "\<AA> = \<CC>"   and "\<BB> = \<DD>" shows "\<GG> = \<FF>"
openin_Int: fixes U :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin U S"   and "openin U T" shows "openin U (S \<inter> T)"
content_dvd_contentI: fixes p :: "'b poly"   and q :: "'b poly" assumes "p dvd q" shows "content p dvd content q"
foldri_def: fixes l :: "'d list" shows "foldri l = foldli (rev l)"
update_convs_8: fixes cross_edges' :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and counter :: "nat"   and discovered :: "'a \<Rightarrow> nat option"   and finished :: "'a \<Rightarrow> nat option"   and pending :: "('a \<times> 'a) set"   and stack :: "'a list"   and tree_edges :: "('a \<times> 'a) set"   and back_edges :: "('a \<times> 'a) set"   and cross_edges :: "('a \<times> 'a) set"   and more :: "'b" shows "cross_edges_update cross_edges' \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges' cross_edges, \<dots> = more\<rparr>"
assms_5: shows "(r, Expunge e2 mo2 r) \<in> set (node_deliver_messages xs)"
h_nonneg: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x \<le> n + 1"   and "y \<le> n + 1" shows "0 \<le> h n x y"
inj_image_filter_cp_loc_def: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('d, 'e, 'f) set_ops_scheme"   and ops3 :: "('g, 'h, 'i) set_ops_scheme" shows "inj_image_filter_cp_loc ops1 ops2 ops3 \<equiv> StdSet ops1 \<and> StdSet ops2 \<and> StdSet ops3"
insert_mset: fixes q :: "('c, 'd) BinomialTree list"   and e :: "'c"   and a :: "'d" assumes "queue_invar q" shows "queue_to_multiset (local.insert e a q) = queue_to_multiset q + {#(e, a)#}"
nths_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and A :: "nat set" shows "nths (map f xs) A = map f (nths xs A)"
take_Suc_Cons: fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" shows "take (Suc n) (x # xs) = x # take n xs"
replace_comp_self_financing: fixes Mkt :: "('c, 'd) discrete_market"   and x :: "'d"   and pf2 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "\<forall>n w. prices Mkt x n w = cls_val_process Mkt pf2 n w"   and "portfolio pf1"   and "portfolio pf2"   and "self_financing Mkt pf1"   and "self_financing Mkt pf2" shows "self_financing Mkt (qty_replace_comp pf1 x pf2)"
invar_r_ge_k: fixes k :: "'b"   and v :: "real"   and l :: "'b kdt"   and r :: "'b kdt" assumes "invar (Node k v l r)" shows "\<forall>p\<in>set_kdt r. v < p $ k"
tail_monomial: fixes v :: "'a"   and c :: "'c" shows "punit.tail (monomial c v) = 0"
exec_move_Throw: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (throw e) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
HInfinite_add_ge_zero2: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HInfinite"   and "0 \<le> y"   and "0 \<le> x" shows "y + x \<in> HInfinite"
le_cases: fixes x :: "'a"   and y :: "'a" obtains "le x y"   | "le y x"
VectorSpaceHom: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T" shows "VectorSpaceHom smult V smult T"
unique_factorization_m_smultD: fixes d :: "int"   and f :: "int poly"   and c :: "int"   and fs :: "int poly multiset"   and di :: "int" assumes "p.unique_factorization_m (Polynomial.smult d f) (c, fs)"   and "p.M (di * d) = 1" shows "p.unique_factorization_m f (c * di, fs)"
affine_no_rel_frontier: fixes S :: "'b set" assumes "affine S" shows "rel_frontier S = {}"
Lcm_nat_abs_eq: fixes K :: "int set" shows "(LCM k\<in>K. nat \<bar>k\<bar>) = nat (Lcm K)"
compact_space_topspace_empty: fixes X :: "'d topology" assumes "topspace X = {}" shows "compact_space X"
linear_normal_iff: fixes x :: "real" shows "normal_poly [:- x, 1:] = (x \<le> 0)"
EM_remainder_converges_cnj_iff: fixes a :: "int"   and f :: "real \<Rightarrow> complex"   and n :: "nat" assumes "continuous_on {real_of_int a..} f" shows "EM_remainder_converges n (\<lambda>x. cnj (f x)) a = EM_remainder_converges n f a"
third_mertens_const_pos: shows "0 < third_mertens_const"
fls_times_fps_to_fls: fixes f :: "'b fps"   and g :: "'b fps" shows "fps_to_fls (f * g) = fps_to_fls f * fps_to_fls g"
closure_V_int: fixes Z :: "('c \<Rightarrow> real) set" shows "Closure\<^sub>\<alpha> Z = Closure\<^sub>\<alpha> (Z \<inter> V)"
simps_2: fixes a :: "'b" shows "\<dd> (Atom a) = (\<lambda>b. if a = b then One else Zero)"
add_0: fixes a :: "'b" shows "(0::'b) + a = a"
abs_eq: fixes xa :: "'c llist"   and x :: "'c llist \<times> 'd" shows "lappendt xa (case x of (xs, a) \<Rightarrow> tllist_of_llist a xs) = (case apfst (lappend xa) x of (xs, a) \<Rightarrow> tllist_of_llist a xs)"