hcju/isabelleps huggingface dataset:
div_mult_swap: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c b" shows "times a (divide b c) = divide (times a b) c"
set_takeWhile: fixes as :: "'a list list"   and y :: "'a list" assumes "strict_sorted (concat as)"   and "as \<in> lists (- {[]})" shows "list.set (takeWhile (\<lambda>x. x < y) as) = {x \<in> list.set as. x < y}"
HLD_Stream: fixes X :: "'c set"   and x :: "'c"   and \<omega> :: "'c stream" shows "HLD X (x ## \<omega>) = (x \<in> X)"
A: shows "\<forall>s\<in>set (toList (snd ec)). envObs a s = envObs a (tLast t)"
ode_set_def: fixes ode_ops :: "ode_ops"   and X :: "'c set" shows "ode_set ode_ops X = mk_safe ode_ops X \<bind> (\<lambda>_. approx_slp_appr (ode_e ode_ops) (ode_slp ode_ops) (list_of_eucl ` X))"
reaching_ret: fixes x :: "'n" assumes "x \<in> local.nodes" shows "\<exists>\<pi> n. is_path \<pi> \<and> \<pi> 0 = x \<and> \<pi> n = local.return"
weakly_preserves_src: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "C.isomorphic (F (src\<^sub>B \<mu>)) (src\<^sub>C (F \<mu>))"
elims_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'd list \<Rightarrow> 'd"   and intP :: "'b \<Rightarrow> 'd list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'd"   and xa :: "('a, 'b) lit" assumes "M.Struct wtFsym wtPsym arOf resOf intT intF intP"   and "Struct.satL intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and a :: "('a, 'b) atm" where "x = \<xi>__" and "xa = Pos a__" and "Struct.satA intF intP \<xi>__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and a :: "('a, 'b) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> Struct.satA intF intP \<xi> a"
"1_2": fixes x :: "'a option list set list"   and xa :: "'a option list set list"   and x2 :: "(nat set list \<times> nat set \<times> nat set list \<times> nat set \<times> nat set list) \<times> 'a option list set list \<times> 'a option list set \<times> 'a option list set list \<times> 'a option list set \<times> 'a option list set list"   and xb :: "nat set list \<times> nat set \<times> nat set list \<times> nat set \<times> nat set list"   and y :: "'a option list set list \<times> 'a option list set \<times> 'a option list set list \<times> 'a option list set \<times> 'a option list set list"   and xc :: "nat set list"   and ya :: "nat set \<times> nat set list \<times> nat set \<times> nat set list"   and xd :: "nat set"   and yb :: "nat set list \<times> nat set \<times> nat set list"   and xe :: "nat set list"   and yaa :: "nat set \<times> nat set list"   and xf :: "nat set"   and yab :: "nat set list"   and xg :: "'a option list set list"   and yac :: "'a option list set \<times> 'a option list set list \<times> 'a option list set \<times> 'a option list set list"   and xh :: "'a option list set"   and yc :: "'a option list set list \<times> 'a option list set \<times> 'a option list set list"   and xi :: "'a option list set list"   and yd :: "'a option list set \<times> 'a option list set list"   and xj :: "'a option list set"   and ye :: "'a option list set list" assumes "\<not> length A_pos + length A_neg \<noteq> length L"   and "x = take (length A_pos) L"   and "xa = drop (length A_pos) L"   and "dominate_True A_pos x = Some x2"   and "(xb, y) = x2"   and "(xc, ya) = xb"   and "(xd, yb) = ya"   and "(xe, yaa) = yb"   and "(xf, yab) = yaa"   and "(xg, yac) = y"   and "(xh, yc) = yac"   and "(xi, yd) = yc"   and "(xj, ye) = yd"   and "xc @ (xd \<union> xf) # xe @ yab \<noteq> []"   and "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((xc @ (xd \<union> xf) # xe @ yab) @ A_neg) ((xg @ bin_join n xd xh True xf xj # xi @ ye) @ xa)" shows "mmulti_join n (xc @ (xd \<union> xf) # xe @ yab) A_neg ((xg @ bin_join n xd xh True xf xj # xi @ ye) @ xa) = mmulti_join' (xc @ (xd \<union> xf) # xe @ yab) A_neg ((xg @ bin_join n xd xh True xf xj # xi @ ye) @ xa)"
lmult_0: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and m :: "'d" assumes "Module.module R M"   and "m \<in> carrier M" shows "\<zero>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> m = \<zero>\<^bsub>M\<^esub>"
ipresCons_imp_ipresSubstAll_aux: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme"   and P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and us :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "ipresCons h hA MOD"   and "igSubstCls MOD"   and "igConsIPresIGWls MOD"   and "igFreshCls MOD"   and "wlsPar P" shows "(wls s X \<longrightarrow> (\<forall>ys y Y. y \<in> varsOfS P ys \<and> Y \<in> termsOfS P (asSort ys) \<longrightarrow> h (X #[Y / y]_ys) = igSubst MOD ys (h Y) y (h X))) \<and> (wlsAbs (us, s') A \<longrightarrow> (\<forall>ys y Y. y \<in> varsOfS P ys \<and> Y \<in> termsOfS P (asSort ys) \<longrightarrow> hA (A $[Y / y]_ys) = igSubstAbs MOD ys (h Y) y (hA A)))"
simps_2: fixes I :: "('e, 'f, 'd) interp"   and P :: "'d"   and terms :: "'d \<Rightarrow> ('e, 'd) trm" shows "fml_sem I ($\<phi> P terms) = {\<nu>. Predicates I P (\<chi>i. dterm_sem I (terms i) \<nu>)}"
inner_vec_def: fixes x :: "('d, 'e) vec"   and y :: "('d, 'e) vec" shows "x \<bullet> y = (\<Sum>i\<in>UNIV. x $ i \<bullet> y $ i)"
singleton_mem_set_subseqs: fixes x :: "'c"   and xs :: "'c list" shows "([x] \<in> set (subseqs xs)) = (x \<in> set xs)"
induct: fixes P :: "'c list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c list"   and a1 :: "'c list" assumes "\<And>x xs y ys. \<lbrakk>x = y \<Longrightarrow> P xs (y # ys); \<lbrakk>x \<noteq> y; x < y\<rbrakk> \<Longrightarrow> P xs (y # ys); \<lbrakk>x \<noteq> y; \<not> x < y\<rbrakk> \<Longrightarrow> P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []" shows "P a0 a1"
primepow_iff_even_sqr: fixes n :: "nat" shows "primepow n = primepow_even (n\<^sup>2)"
partial_order_def: fixes L :: "('a, 'b) gorder_scheme" shows "partial_order L \<equiv> weak_partial_order L \<and> partial_order_axioms L"
poly_mapping_plus_induct: fixes P :: "('c \<Rightarrow>\<^sub>0 'd) \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'd" assumes "P 0"   and "\<And>p c t. \<lbrakk>c \<noteq> (0::'d); t \<notin> keys p; P p\<rbrakk> \<Longrightarrow> P (monomial c t + p)" shows "P p"
simple_cipher_axioms: shows "simple_cipher prf_key_gen prf_fun prf_domain prf_dlen prf_clen upf_key_gen"
wf_nFNot: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" assumes "local.wf idx (FNot \<phi>)" shows "local.wf idx (nFNot \<phi>)"
scale_right_imp_eq: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<noteq> (0::'a)"   and "a * x = b * x" shows "a = b"
grd_def: fixes x :: "'b" shows "grd x = - wp x \<bottom>"
InvariantEquivalentZLAfterApplyDecide: fixes state :: "State"   and F0 :: "Literal list list"   and Vbl :: "nat set" assumes "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantEquivalentZL (getF state) (getM state) F0" shows "let state' = applyDecide state Vbl in InvariantEquivalentZL (getF state') (getM state') F0"
scale_zero_right: fixes a :: "'c" shows "a * (0::'c) = (0::'c)"
next_state_invariant: shows "\<forall>s execs. invariant s \<longrightarrow> invariant (next_state s execs)"
vector_7_6: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b"   and x4 :: "'b"   and x5 :: "'b"   and x6 :: "'b"   and x7 :: "'b" shows "vector [x1, x2, x3, x4, x5, x6, x7] $ 6 = x6"
strongAppend: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs"   and Rel'' :: "(ccs \<times> ccs) set" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "Q \<leadsto>[Rel'] R"   and "Rel O Rel' \<subseteq> Rel''" shows "P \<leadsto>\<^sup>^<Rel''> R"
almost_full_on_subset: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<subseteq> B"   and "almost_full_on P B" shows "almost_full_on P A"
positively_responsive_def: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set" shows "positively_responsive scf A Is \<equiv> \<forall>P P' x y. profile A Is P \<and> profile A Is P' \<and> x \<in> A \<and> y \<in> A \<and> (\<forall>i\<in>Is. (x \<^bsub>(P i)\<^esub>\<prec> y \<longrightarrow> x \<^bsub>(P' i)\<^esub>\<prec> y) \<and> (x \<^bsub>(P i)\<^esub>\<approx> y \<longrightarrow> x \<^bsub>(P' i)\<^esub>\<preceq> y)) \<and> (\<exists>k\<in>Is. x \<^bsub>(P k)\<^esub>\<approx> y \<and> x \<^bsub>(P' k)\<^esub>\<prec> y \<or> y \<^bsub>(P k)\<^esub>\<prec> x \<and> x \<^bsub>(P' k)\<^esub>\<preceq> y) \<longrightarrow> x \<^bsub>(scf P)\<^esub>\<preceq> y \<longrightarrow> x \<^bsub>(scf P')\<^esub>\<prec> y"
simps_4: fixes orderA :: "('e, 'f) gorder_scheme"   and orderB :: "('d, 'g) gorder_scheme"   and lower :: "'e \<Rightarrow> 'd"   and upper :: "'d \<Rightarrow> 'e"   and more :: "'h" shows "\<pi>\<^sub>*\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = upper"
assms_1: shows "\<forall>x. x \<in> S \<longrightarrow> trms_ecl x = {}"
vec_simps_18: fixes ac :: "'a"   and bc :: "'a"   and cb :: "'a" shows "ac + (bc - cb) = ac + bc - cb"
LT28: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "|~ P \<longrightarrow> \<circle>P \<or> \<circle>Q" shows "|~ (P \<longrightarrow> \<circle>P) \<or> \<diamond>Q"
sup_left_idem: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<union># (x \<union># y) = x \<union># y"
interaction_bound_IO_consider: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and "consider" :: "'e \<Rightarrow> bool"   and input :: "'f" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "consider out" shows "eSuc (interaction_bound consider (c input)) \<le> interaction_bound consider gpv"
2: shows "\<forall>d\<in>set D - set Di. (t, s) \<cdot>\<^sub>p \<I> \<noteq> d \<cdot>\<^sub>p \<I>"
connected_component_of_trans: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "connected_component_of X x y"   and "connected_component_of X y z" shows "connected_component_of X x z"
hyps_3: shows "reciprocal_poly p (sum P F) = (\<Sum>k\<in>F. reciprocal_poly p (P k))"
bisimulation_def: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "bisimulation trsys1 trsys2 bisim tlsim \<equiv> (\<forall>s1 s2 tl1 s1'. bisim s1 s2 \<longrightarrow> trsys1 s1 tl1 s1' \<longrightarrow> (\<exists>s2' tl2. trsys2 s2 tl2 s2' \<and> bisim s1' s2' \<and> tlsim tl1 tl2)) \<and> (\<forall>s1 s2 tl2 s2'. bisim s1 s2 \<longrightarrow> trsys2 s2 tl2 s2' \<longrightarrow> (\<exists>s1' tl1. trsys1 s1 tl1 s1' \<and> bisim s1' s2' \<and> tlsim tl1 tl2))"
consec_un_not_elem1: fixes i :: "nat_int"   and j :: "nat_int"   and n :: "nat" shows "consec i j \<and> n \<^bold>\<notin> i \<squnion> j \<longrightarrow> n \<^bold>\<notin> i"
of_nat_add: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m + n) = plus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
flatten_sigma_2_sf_subset: fixes \<phi> :: "'b ltln"   and M :: "'b ltln set" shows "sf \<phi>[M]\<^sub>\<Sigma>\<^sub>2 \<subseteq> (\<Union>\<psi>\<in>sf \<phi>. sf \<psi>[M]\<^sub>\<Sigma>\<^sub>2)"
L_fst_simp: fixes r :: "'b rexp" shows "lang r = lang (fst (simp r))"
nth_derivative_exists: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and i :: "nat"   and x :: "'b"   and h :: "'b" assumes "higher_differentiable_on X f (Suc i)"   and "open X"   and "x \<in> X" shows "\<exists>f'. ((\<lambda>x. nth_derivative i f x h) has_derivative f') (at x) \<and> f' h = nth_derivative (Suc i) f x h"
of_nat_le_of_nat_power_cancel_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "nat"   and w :: "nat"   and x :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (power.power one times (semiring_1.of_nat one plus zero b) w) (semiring_1.of_nat one plus zero x) = (b ^ w \<le> x)"
filter_insort: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and P :: "'h \<Rightarrow> bool"   and x :: "'h" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map f xs)"   and "P x" shows "filter P (linorder.insort_key less_eq f x xs) = linorder.insort_key less_eq f x (filter P xs)"
induct: fixes P :: "('a option list set \<Rightarrow> 'a option list set \<Rightarrow> 'b) \<Rightarrow> 'a option list set list \<times> 'a option list set list \<Rightarrow> bool"   and a0 :: "'a option list set \<Rightarrow> 'a option list set \<Rightarrow> 'b"   and a1 :: "'a option list set list \<times> 'a option list set list" assumes "\<And>f x xs y ys. P f (xs, ys) \<Longrightarrow> P f (x # xs, y # ys)"   and "\<And>f ys. P f ([], ys)"   and "\<And>f xs. P f (xs, [])" shows "P a0 a1"
Collect_mem_eq: fixes A :: "'g set" shows "{x. x \<in> A} = A"
power2_less_imp_less: fixes x :: "'a"   and y :: "'a" assumes "x\<^sup>2 < y\<^sup>2"   and "(0::'a) \<le> y" shows "x < y"
RSpan_lincomb_closed: fixes rs :: "'f list"   and ms :: "'v list"   and ns :: "'v list" assumes "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V"   and "set ns \<subseteq> RSpan ms" shows "rs \<bullet>\<cdot> ns \<in> RSpan ms"
map_project_Int_ENC: fixes X :: "nat set"   and n :: "nat"   and Z :: "('a \<times> bool list) list set" assumes "0 \<notin> X"   and "X \<subseteq> {0..<n + 1}"   and "Z \<subseteq> lists ((set \<circ> \<sigma> \<Sigma>) (n + 1))" shows "map \<pi> ` (Z \<inter> lang (n + 1) (ENC (n + 1) X) - {[]}) = map \<pi> ` Z \<inter> lang n (ENC n ((\<lambda>x. x - 1) ` X)) - {[]}"
greaterThanAtMost_iff: fixes i :: "'t"   and l :: "'t"   and u :: "'t" shows "(i \<in> ord_term_lin.greaterThanAtMost l u) = (l \<prec>\<^sub>t i \<and> i \<preceq>\<^sub>t u)"
CAs0_in_M: shows "\<forall>CA0\<in>set CAs0. CA0 \<in> M"
DefaultPData_def: fixes D :: "'b dataspace" shows "DefaultPData D = Abs_pdata (replicate (PartNum D) None, D)"
finite_set_ins_swap: fixes A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "finite A" shows "max a (Max (Set.insert b A)) = max b (Max (Set.insert a A))"
nat_term_comp_lex_comp: shows "nat_term_comp lex_comp"
Id_on_iff: fixes x :: "'c"   and y :: "'c"   and A :: "'c set" shows "((x, y) \<in> Id_on A) = (x = y \<and> x \<in> A)"
inject_5: fixes x2 :: "trm"   and x1 :: "ty"   and y2 :: "trm"   and y1 :: "ty" shows "(x2 \<cdot>\<^sub>\<tau> x1 = y2 \<cdot>\<^sub>\<tau> y1) = (x2 = y2 \<and> x1 = y1)"
iso_verts_eq_iff: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and u :: "'c"   and v :: "'c" assumes "pre_digraph.digraph_isomorphism G hom"   and "u \<in> verts G"   and "v \<in> verts G" shows "(iso_verts hom u = iso_verts hom v) = (u = v)"
arity_generalize_fundef: fixes F_empty :: "'a"   and F_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'g, 'h, 'i) Ubx.instr fundef option"   and F_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'g, 'h, 'i) Ubx.instr fundef \<Rightarrow> 'a"   and F_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'f, 'g, 'h, 'i) Ubx.instr fundef) list"   and heap_empty :: "'j"   and heap_get :: "'j \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'j \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'j"   and heap_to_list :: "'j \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and box_ubx1 :: "'h \<Rightarrow> 'c"   and unbox_ubx1 :: "'c \<Rightarrow> 'h option"   and box_ubx2 :: "'i \<Rightarrow> 'c"   and unbox_ubx2 :: "'c \<Rightarrow> 'i option"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'f \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'f option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'e"   and \<UU>\<bb>\<xx>\<OO>\<pp> :: "'g \<Rightarrow> ('c, 'h, 'i) unboxed list \<Rightarrow> ('c, 'h, 'i) unboxed option"   and \<UU>\<bb>\<xx> :: "'f \<Rightarrow> Unboxed.type option list \<Rightarrow> 'g option"   and \<BB>\<oo>\<xx> :: "'g \<Rightarrow> 'f"   and \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> :: "'g \<Rightarrow> Unboxed.type option list \<times> Unboxed.type option"   and fd2 :: "('c, 'k, 'l, 'm, 'f, 'g, 'h, 'i) Ubx.instr fundef" assumes "ubx F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>" shows "arity (ubx.generalize_fundef box_ubx1 box_ubx2 \<BB>\<oo>\<xx> fd2) = arity fd2"
filtermap_ident: fixes F :: "'a filter" shows "filtermap (\<lambda>x. x) F = F"
assms_4: shows "rel_set T xl (a' \<inter> b')"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "x \<le> y"
phi_universal: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> \<P>" obtains i :: "nat" where "\<phi> i = f"
j_gt_m_pigeonhole: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "lb T A j"   and "m < j" shows "\<exists>x\<in>{1..j}. \<exists>y\<in>{1..j}. \<exists>z\<in>{1..m}. x \<noteq> y \<and> x \<in> A z \<and> y \<in> A z"
s: shows "(flow0 y \<circ> s) \<longlonglongrightarrow> z"
is_dghm_axioms: shows "\<FF> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
UP_map\<^sub>0_obj: fixes a :: "'a" assumes "obj a" shows "UP.map\<^sub>0 a = S.UP\<^sub>0 a"
matrix_mul_sum_alt: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "A ** B = (\<chi>i. \<Sum>k\<in>UNIV. A $h i $h k *s B $h k)"
aXb_old: shows "Derives1 \<delta> i r (a @ [X] @ b)"
accept_def: fixes w :: "nat \<Rightarrow> 'b" shows "accept w \<equiv> \<exists>r. is_run r \<and> w = L \<circ> r"
bst_mrbst_inter_diff: fixes t' :: "'a tree"   and b :: "bool"   and t1 :: "'a tree"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_inter_diff b t1 t2)"   and "bst t1"   and "bst t2" shows "bst t'"
f_def: shows "f = (\<lambda>x. if x \<in> space M then g x else 1)"
renaming: shows "\<R>\<^sub>G (id(r' := r'', r'' := r')) id s\<^sub>2 = s\<^sub>2'"
GreatestI_ex: fixes P :: "'b \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (Greatest P)"
Red_I_def: fixes N :: "'f set" shows "Red_I N = {\<iota> \<in> Inf. redundant_infer N \<iota>}"
subprob_space_return: fixes x :: "'a"   and M :: "'a measure" assumes "x \<in> space M" shows "subprob_space (return M x)"
sets_Collect_finite_All: fixes S :: "'c set"   and M :: "'d measure"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>i. i \<in> S \<Longrightarrow> {x \<in> space M. P i x} \<in> sets M"   and "finite S" shows "{x \<in> space M. \<forall>i\<in>S. P i x} \<in> sets M"
integral_density: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> g x" shows "integral\<^sup>L (density M (\<lambda>x. ennreal (g x))) f = LINT x|M. g x *\<^sub>R f x"
hyps_4: shows "sh'' = sh'(C \<mapsto> (sfs, Error))"
ordertype_\<omega>\<omega>: fixes F :: "nat \<Rightarrow> nat list set" assumes "\<And>j. ordertype (F j) (lenlex less_than) = \<omega> \<up> ord_of_nat j" shows "ordertype (\<Union> (range F)) (lenlex less_than) = \<omega> \<up> \<omega>"
AppR: fixes u :: "trm"   and u' :: "trm"   and t :: "trm" assumes "u \<longmapsto> u'" shows "t \<cdot> u \<longmapsto> t \<cdot> u'"
io_target_from_path: fixes M :: "('d, 'e, 'f) FSM"   and w :: "('d \<times> 'e) list"   and tr :: "'f list"   and q :: "'f" assumes "path M (w || tr) q"   and "length w = length tr" shows "target (w || tr) q \<in> io_targets M q w"
that: fixes a :: "'a \<times> 'b"   and s :: "('a, 'b) FingerTree" assumes "viewL t = Some (a, s)"   and "toList t = a # toList s" shows "thesis"
code: fixes a :: "real" shows "cis a = Complex (cos a) (sin a)"
intro: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor"   and G :: "('d, 'e, 'g, 'h) functor"   and u :: "'d \<Rightarrow> 'h" assumes "two_cats AA BB"   and "natural_transformation_axioms AA BB F G u" shows "u : F \<Rightarrow> G in Func (AA , BB )"
prems: shows "local.create_subseqs base elements (Suc n) = (out, state)"
count_Sigma_mset: fixes A :: "'a multiset"   and B :: "'a \<Rightarrow> 'b multiset"   and a :: "'a"   and b :: "'b" shows "count (Sigma_mset A B) (a, b) = count A a * count (B a) b"
monoE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b"   and f' :: "'b" assumes "category C"   and "category.mono C g"   and "partial_magma.arr C (C g f)"   and "partial_magma.arr C (C g f')"   and "C g f = C g f'" shows "f' = f"
sum_list_lec: fixes ls :: "'a le_constraint list" shows "sum_list ls = Le_Constraint (sum_list (map lec_rel ls)) (sum_list (map lec_poly ls)) (sum_list (map lec_const ls))"
WT_callee: fixes s :: "'d" shows "\<I> \<turnstile>c callee s \<surd>"
difference_list_opt_lem0: fixes u :: "'a set" assumes "\<forall>t\<in>set ts. u \<inter> t = {}" shows "difference_list_opt s ts = difference_list_opt (s - u) ts"
sfis_intro: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and S :: "nat set"   and A :: "nat \<Rightarrow> 'b set"   and x :: "nat \<Rightarrow> real" assumes "measure_space M"   and "\<forall>i\<in>S. A i \<in> measurable_sets M"   and "nonnegative x"   and "finite S" shows "(\<Sum>i\<in>S. x i * Measure.measure M (A i)) \<in> sfis (\<lambda>t. \<Sum>i\<in>S. x i * \<chi> (A i) t) M"
Smith_normal_form_upt_k_Suc_imp_k: fixes A :: "(('a, 'd) vec, 'e) vec"   and k :: "nat" assumes "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<Suc k] bezout) k"   and "k < min (nrows A) (ncols A)" shows "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<k] bezout) k"
VARusefulSYSTEM_def: shows "VARusefulSYSTEM \<equiv> \<forall>v. VARto v \<noteq> {}"
pw_le_iff: fixes S :: "'c nres"   and S' :: "'c nres" shows "(S \<le> S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x))"