hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and a2 :: "'d"   and a3 :: "'e \<Rightarrow> 'f"   and a4 :: "'g"   and a5 :: "'d"   and a6 :: "'e \<Rightarrow> 'f" assumes "a1 \<turnstile> \<langle>a2, a3\<rangle> \<rightarrow>\<^bsub>a4\<^esub> \<langle>a5, a6\<rangle>" obtains A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)" and l :: "'d" and g :: "('e, 'f) cconstraint" and a :: "'g" and r :: "'e list" and l' :: "'d" and u :: "'e \<Rightarrow> 'f" and u' :: "'e \<Rightarrow> 'f" where "a1 = A" and "a2 = l" and "a3 = u" and "a4 = a" and "a5 = l'" and "a6 = u'" and "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" and "u \<turnstile> g" and "u' \<turnstile> inv_of A l'" and "u' = [r\<rightarrow>(0::'f)]u"
rtrancl_into_rtrancl2: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> r\<^sup>*"
rel_refl_strong: fixes x :: "'c dual"   and Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z. z \<in> set_dual x \<Longrightarrow> Ra z z" shows "rel_dual Ra x x"
add_minus_left_eq_diff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "- x + y = y - x"
rel_transp: fixes R1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "transp R1"   and "transp R2" shows "transp (tllist_all2 R1 R2)"
spanning_tree_def: fixes F :: "('c, 'd) graph"   and G :: "('c, 'd) graph" shows "spanning_tree F G \<equiv> tree F \<and> subgraph F G"
cong_2_pow: fixes a :: "nat"   and b :: "nat" assumes "[a = b] (mod 2)" shows "(- 1) ^ a = (- 1) ^ b"
ld_\<beta>: shows "LeftDerives1 (\<alpha> @ \<delta>) i r \<beta>"
simps_9: fixes cs :: "instr list" shows "unwind cs [] = []"
psubset_eq: fixes A :: "'f set"   and B :: "'f set" shows "(A \<subset> B) = (A \<subseteq> B \<and> A \<noteq> B)"
confluent_quotient_fim: shows "confluent_quotient cancel1\<inverse>\<inverse> (equivclp cancel1) (equivclp cancel1) (equivclp cancel1) (equivclp cancel1) (equivclp cancel1) (map fst) (map snd) (map fst) (map snd) list_all2 list_all2 list_all2 set set"
Units_inv_Units: fixes x :: "'a" assumes "x \<in> Units R" shows "inv x \<in> Units R"
inv1: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], rev u @ h) v = ([x, y], rev v @ rev u @ h)"
spr_jview_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'f"   and a :: "'b" assumes "Environment jkbp" shows "Environment.spr_jview envObs a = tMap (envObs a)"
ie1: fixes k :: "nat"   and i :: "nat" assumes "0 < k" shows "int (aa i) gchoose k + (int (bb i) gchoose k) \<le> int (aa i + bb i) gchoose k"
Collect_not_less_iMin: fixes k :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "k < iMin {x. P x}" shows "\<not> P k"
bind_get: fixes f :: "'s \<Rightarrow> 'm"   and g :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind (get f) g = get (\<lambda>s. bind (f s) g)"
do_log_exp: fixes op :: "lop"   and v :: "v"   and e :: "exp0"   and e' :: "exp0" assumes "do_log op v e = Some (Exp e')" shows "e = e'"
borel_measurable_vimage_open_interval: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>a b. {x \<in> S. f x \<in> box a b} \<in> sets (lebesgue_on S))"
Exp: fixes a :: "cmsg"   and b :: "cmsg"   and c :: "cmsg"   and d :: "cmsg" assumes "eq a b"   and "eq c d" shows "eq (cExp a c) (cExp b d)"
divide_cancel_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide c a = divide c b) = (c = zero \<or> a = b)"
simps_3: fixes x :: "real"   and y :: "real" shows "(R x \<le> R y) = (x \<le> y)"
SUP_bot_conv_2: 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 B :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup (B ` A)) = (\<forall>x\<in>A. B x = bot)"
bpE_1: fixes P :: "'a set set" assumes "bp P" shows "disjoint P"
that: fixes z :: "'a"   and Z :: "'a multiset" assumes "Y = add_mset z Z"   and "(x, z) \<in> ns"   and "(X, Z) \<in> multpw ns" shows "thesis"
exp_strong_exhaust: fixes y :: "exp"   and c :: "'a" obtains var :: "Vars.var" where "y = Var var__"   | exp :: "exp" and var :: "Vars.var" where "y = App exp__ vara__"   | \<Gamma> :: "(Vars.var \<times> exp) list" and exp :: "exp" where "atom ` domA \<Gamma> \<sharp>* c" and "y = Terms.Let \<Gamma> expa__"   | var :: "Vars.var" and exp :: "exp" where "{atom var} \<sharp>* c" and "y = Lam [var]. exp"   | b :: "bool" where "y = Bool b"   | scrut :: "exp" and e1 :: "exp" and e2 :: "exp" where "y = (scrut ? e1 : e2)"
prefix_Nil: fixes xs :: "'a list" shows "(xs \<le> []) = (xs = [])"
uminus_INF: 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 minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "uminus (Inf (B ` A)) = Sup ((\<lambda>x. uminus (B x)) ` A)"
fiter_unfoldl: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "mono f"   and "mono g" shows "f \<sqinter> (g \<circ> fiter f g) = fiter f g"
simps_2: shows "\<dd> Zero = (\<lambda>_. Zero)"
app: shows "approx prec (DERIV_floatarith n f) vs = Some ivl"
iso_unit\<^sub>P\<^sub>B\<^sub>U: fixes a :: "'a" assumes "weak_unit a" shows "local.iso \<i>[a]"
wti_less: fixes pc :: "nat" assumes "stable r step \<tau>s pc"   and "Suc pc < length \<tau>s" shows "wti c pc (\<tau>s ! pc) \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! Suc pc"
eqpoll_trans: fixes A :: "'b set"   and B :: "'c set"   and C :: "'d set" assumes "A \<approx> B"   and "B \<approx> C" shows "A \<approx> C"
Domain_unfold: fixes r :: "('d \<times> 'e) set" shows "Domain r = {x. \<exists>y. (x, y) \<in> r}"
a': shows "( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Modal M Ms, n) \<in> derivable (ext R R2 M1 M2)"
iT_Mult_cut_less2: fixes k :: "nat"   and I :: "nat set"   and t :: "nat" assumes "0 < k" shows "(I \<otimes> k) \<down>< (t * k) = I \<down>< t \<otimes> k"
bound: fixes \<eta> :: "nat" shows "interaction_any_bounded_converter (conv \<eta>) (bound_conv \<eta>)"
left_unique_OO: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "left_unique R"   and "left_unique S" shows "left_unique (R OO S)"
elem_exact: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> carrier L"   and "a \<in> A" shows "a .\<in> A"
id_subst_comp_substs: fixes \<sigma>s :: "'s list" shows "replicate (length \<sigma>s) id_subst \<odot>s \<sigma>s = \<sigma>s"
mult_less_cancel_right_disj: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a c) (times b c) = (less zero c \<and> less a b \<or> less c zero \<and> less b a)"
abs: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma>\<^sub>1 :: "(name, value) fmap"   and \<Gamma>\<^sub>2 :: "(name, value) fmap" assumes "fmrel_on_fset (ids (Sabs cs)) erelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2" shows "Vabs cs \<Gamma>\<^sub>1 \<approx>\<^sub>e Vabs cs \<Gamma>\<^sub>2"
r_ifz_prim: shows "prim_recfn 3 r_ifz"
measurable_cong: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and M' :: "'c measure" assumes "\<And>w. w \<in> space M \<Longrightarrow> f w = g w" shows "(f \<in> M \<rightarrow>\<^sub>M M') = (g \<in> M \<rightarrow>\<^sub>M M')"
preorder_one_closed: shows "coreflexive (1::'a) \<and> transitive (1::'a)"
dnso4: fixes x :: "'a"   and y :: "'a" shows "d (x + y) = d x + d y"
direct_decomp_repeated_eq_zero: fixes A :: "'a set"   and ss :: "'a set list"   and X :: "'a set" assumes "direct_decomp A ss"   and "1 < count_list ss X"   and "\<And>s. s \<in> set ss \<Longrightarrow> (0::'a) \<in> s" shows "X = {0::'a}"
intros: fixes xs :: "'a list"   and ys :: "'a list" shows "initial_segment xs (xs @ ys)"
det_int_fg: shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) integrable_on S"
le_lang_equiv: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool" assumes "ltl_equivalence eq" shows "eq \<le> (\<sim>\<^sub>L)"
AE: fixes k :: "real" shows "AE x\<in>{0..k} in lebesgue. cmod (f x) \<le> B k"
Fvars_softSubst_subst_in: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "x \<notin> FvarsT t"   and "x \<in> Fvars \<phi>" shows "Fvars (softSubst \<phi> t x) = Fvars (subst \<phi> t x)"
is_lubD2: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a"   and u :: "'a" assumes "class.po below"   and "po.is_lub below S x"   and "po.is_ub below S u" shows "below x u"
rabin_accept_to_Acc: assumes "accept\<^sub>R (\<MM>.\<delta>\<^sub>\<R>, \<MM>.q\<^sub>\<R>, {(\<MM>.fail\<^sub>R \<union> \<MM>.merge\<^sub>R j, \<MM>.succeed\<^sub>R j) |j. j < \<MM>.max_rank}) w"   and "\<pi> (G \<psi>) = \<MM>.smallest_accepting_rank" shows "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (Acc \<Sigma> \<pi> (G \<psi>)) w"
unit_inv_comm: fixes x :: "'a"   and y :: "'a" assumes "x ** y = one"   and "unit x"   and "unit y" shows "y ** x = one"
simps_1: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option" shows "redT_updTs ts [] = ts"
vector_uminus_component: fixes x :: "('a, 'b) vec"   and i :: "'b" shows "(- x) $ i = - x $ i"
maxBound_def: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "enum.maxBound enum \<equiv> last enum"
ext_inject: fixes Obj :: "'g set"   and Mor :: "'h set"   and Dom :: "'h \<Rightarrow> 'g"   and Cod :: "'h \<Rightarrow> 'g"   and Id :: "'g \<Rightarrow> 'h"   and Comp :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and more :: "'i"   and Obj' :: "'g set"   and Mor' :: "'h set"   and Dom' :: "'h \<Rightarrow> 'g"   and Cod' :: "'h \<Rightarrow> 'g"   and Id' :: "'g \<Rightarrow> 'h"   and Comp' :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and more' :: "'i" shows "(\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = \<lparr>Obj = Obj', Mor = Mor', Dom = Dom', Cod = Cod', Id = Id', Comp = Comp', \<dots> = more'\<rparr>) = (Obj = Obj' \<and> Mor = Mor' \<and> Dom = Dom' \<and> Cod = Cod' \<and> Id = Id' \<and> Comp = Comp' \<and> more = more')"
f: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
unique_ctran_Basic: fixes x :: "('b com option \<times> 'b) list"   and f :: "'b \<Rightarrow> 'b"   and s :: "'b"   and i :: "nat"   and j :: "nat" assumes "x \<in> cptn"   and "x ! 0 = (Some (Basic f), s)"   and "Suc i < length x"   and "x ! i -c\<rightarrow> x ! Suc i"   and "Suc j < length x"   and "i \<noteq> j" shows "x ! j -e\<rightarrow> x ! Suc j"
dvd_imp_order_le: fixes q :: "'b poly"   and p :: "'b poly"   and a :: "'b" assumes "q \<noteq> 0"   and "p dvd q" shows "order a p \<le> order a q"
rfr_base: fixes r :: "real"   and w :: "'c" shows "disc_rfr_proc r 0 w = 1"
inf_sup_aci_1: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type"   and y :: "'b::type" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type) = inf y x"
initQ_def: fixes r :: "'a" shows "Prim2.initQ r \<equiv> (\<lambda>_. \<infinity>)(r := 0)"
comp: fixes M1 :: "moebius"   and M2 :: "moebius" assumes "unit_disc_fix M1"   and "P M1"   and "unit_disc_fix M2"   and "P M2" shows "P (M1 + M2)"
aType_empty: fixes a :: "nat \<Rightarrow> aType list"   and t :: "nat" assumes "msg (Suc 0) a"   and "a t \<noteq> [sc_ack]" shows "a t = []"
before_conv_take: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "before x xs = set (take (index xs x) xs)"
check_eqv_def: shows "check_eqv \<equiv> Presb.check_eqv"
map_sq_matrix_diag: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c" assumes "f (0::'c) = (0::'b)" shows "map_sq_matrix f (diag c) = diag (f c)"
flow_network_def: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" shows "flow_network \<Delta> g \<equiv> flow_attainability \<Delta> \<and> flow_network_axioms \<Delta> g"
solve_def: fixes a :: "nat list"   and b :: "nat list" shows "solve a b = special_solutions a b @ non_special_solutions a b"
rtc_implies_srtc: fixes a :: "'a"   and b :: "'a" assumes "a \<rightarrow>\<^sup>* b" shows "a \<leftrightarrow>\<^sup>* b"
irreducibleI: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "p \<noteq> zero"   and "\<not> dvd.dvd times p one"   and "\<And>a b. p = times a b \<Longrightarrow> dvd.dvd times a one \<or> dvd.dvd times b one" shows "comm_semiring_1.irreducible times one zero p"
a_mult_idem: fixes x :: "'a" shows "ad x \<cdot> ad x = ad x"
fst_poly_pfd_simple: fixes x :: "'a poly"   and cs :: "('a \<times> nat) list" shows "fst (poly_pfd_simple x cs) = fst (partial_fraction_decomposition x (map (\<lambda>(c, n). ([:1::'a, - c:], n)) cs))"
C: shows "IntAgrK (S (Card n, n, run)) = Some d"
Derives1_nonterminal: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "CFG.Derives1 \<NN> \<TT> \<RR> a i r b"   and "splits_at a i \<alpha> N \<beta>" shows "fst r = N \<and> CFG.is_nonterminal \<NN> N"
integral_restrict_Int: fixes S :: "'b set"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "S \<in> sets lebesgue"   and "T \<in> sets lebesgue" shows "LINT x|lebesgue_on T. (if x \<in> S then f x else (0::'c)) = integral\<^sup>L (lebesgue_on (S \<inter> T)) f"
exhaust: fixes y :: "'b bot" obtains x1 :: "'b" where "y = bot.Value x1"   | "y = bot.Bot"
IH: shows "prod (prob_component pr w) {0..<n} = lprob_comp pr (stake n w)"
lift_pref_profile_wf: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a set"   and alts' :: "'b set" assumes "pref_profile_wf agents alts R"   and "agents \<subseteq> agents'"   and "alts \<subseteq> alts'"   and "finite alts'" shows "pref_profile_wf agents' alts' (lift_pref_profile agents alts agents' alts' R)"
inv_preserv: fixes S :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "inv_iv S X"   and "x \<in> V - X" shows "inv_iv (insert x S) (X \<union> neighbors x \<union> {x})"
transP_def: fixes \<phi> :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "transP \<phi> \<equiv> \<forall>x y z. \<phi> x y \<and> \<phi> y z \<longrightarrow> \<phi> x z"
assms: fixes p :: "('a, 'b, 'c, 'd) seqp" assumes "p \<in> Range \<sigma>" shows "\<exists>pn. p = \<Gamma> pn"
relH_subset: fixes bs :: "nat set"   and h :: "heap"   and h' :: "heap"   and as :: "nat set" assumes "relH bs h h'"   and "as \<subseteq> bs" shows "relH as h h'"
multpw_cancel: fixes ns :: "('b \<times> 'b) set"   and X :: "'b multiset"   and Z :: "'b multiset"   and Y :: "'b multiset" assumes "refl ns"   and "trans ns" shows "((X + Z, Y + Z) \<in> multpw ns) = ((X, Y) \<in> multpw ns)"
closed_segment_commute: fixes a :: "'a"   and b :: "'a" shows "{a--b} = {b--a}"
edka2_refine: shows "edka2 \<le> \<Down> Id edka"
open_generated_order: shows "open = generate_topology (range lessThan \<union> range greaterThan)"
infinity_evalUni_LeqUni: fixes p :: "real \<times> real \<times> real"   and x :: "real" shows "(\<exists>x. \<forall>y<x. aEvalUni (LeqUni p) y) = evalUni (substNegInfinityUni (LeqUni p)) x"
simps_2: fixes t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "formula_of_constraint (t1, True, MLess, t2) = formula.Less t1 t2"
type_conf_NT: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> NT) = (P,E,h \<turnstile> e : NT)"
closed_closedin: fixes S :: "'b set" shows "closed S = closedin euclidean S"
lthird_gt0: shows "0 < real_of_float (lapprox_rat prec 1 3) + 1"
ccw'_sortedP_mirror: fixes x0 :: "real \<times> real"   and p0 :: "real \<times> real"   and xs :: "(real \<times> real) list" shows "ccw'.sortedP x0 (map (mirror_point p0) xs) = ccw'.sortedP (mirror_point p0 x0) xs"
smallest_closed_cdi1: fixes M :: "'b measure" shows "sets M \<subseteq> smallest_ccdi_sets (space M) (sets M)"
cases: fixes x :: "real_alg_2" obtains n :: "nat" and p :: "int poly" and l :: "rat" and r :: "rat" where "x = Irrational n (p, l, r__)"   | r :: "rat" where "x = Rational r"
lem1: shows "\<forall>e>0. \<exists>d>0. \<forall>z. norm (z - y) < d \<longrightarrow> norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
it_to_list_def: fixes it :: "'e \<Rightarrow> ('f \<Rightarrow> bool) \<Rightarrow> ('g \<Rightarrow> 'g list \<Rightarrow> 'g list) \<Rightarrow> 'h list \<Rightarrow> 'd"   and s :: "'e" shows "it_to_list it s \<equiv> it s (\<lambda>_. True) (\<lambda>x l. l @ [x]) []"
Max_le_iMin_imp_singleton: fixes I :: "'a set" assumes "finite I"   and "I \<noteq> {}"   and "Max I \<le> iMin I" shows "I = {iMin I}"
sum_wt_ParT_sub_WtFT_notPickFT_0: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I :: "nat set"   and s :: "'state" assumes "properL cl"   and "WtFT cl = 1"   and "n < length cl"   and "I \<subseteq> {..<brn (cl ! n)}"   and "n = pickFT cl \<longrightarrow> 0 \<notin> I" shows "sum (wt (ParT cl) s) ((+) (brnL cl n) ` I) = 0"
index_f_def: shows "index_f \<equiv> SOME h. bij_betw h (UNIV // eq_app_right language) (hfset (ord_of min_states))"
frac_eq_Ex: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> R \<times> (R\<setminus>I)"   and "(r', s') \<in> R \<times> (R\<setminus>I)"   and "local.frac r s = local.frac r' s'" obtains s1 :: "'a" where "s1 \<in> R\<setminus>I" and "s1 \<cdot> (s' \<cdot> r - s \<cdot> r') = \<zero>"
det_bind: fixes m :: "'a dres"   and M :: "'a nres"   and f :: "'a \<Rightarrow> 'b dres"   and F :: "'a \<Rightarrow> 'b nres" assumes "nres_of m \<le> M"   and "\<And>x. nres_of (f x) \<le> F x" shows "nres_of (m \<bind> f) \<le> M \<bind> F"
in_pair_in_set: fixes A :: "'b"   and B :: "'b"   and As :: "'b list" assumes "(A, B) \<in> set (pairs As)" shows "A \<in> set As \<and> B \<in> set As"
distinct_27: fixes x3 :: "nat"   and x4 :: "nat" shows "Nonce x3 \<noteq> Pan x4"
diff_union_cancelL: fixes N :: "'a multiset"   and M :: "'a multiset" shows "N + M - N = M"
quad_part_2_deg: fixes var :: "nat"   and sq :: "real mpoly"   and p :: "real mpoly" assumes "var \<notin> vars sq" shows "MPoly_Type.degree (quadratic_part_2 var sq p) var \<le> 1"
ran_upds: fixes ks :: "'a list"   and vs :: "'b list"   and m :: "'a \<Rightarrow> 'b option" assumes "length ks = length vs" shows "ran (m(ks [\<mapsto>] vs)) \<subseteq> ran m \<union> set vs"
main: fixes i :: "nat" assumes "i < n"   and "dim_vec x = n" shows "(vec\<^sub>h x $ i = (0::'b)) = (x $ i = (0::'a))"
distinct_23: fixes x3 :: "fresh_t"   and x4 :: "key" shows "Nonce x3 \<noteq> Key x4"
add_poly_p'_def: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset" shows "add_poly_p' p q = SPEC (\<lambda>r. add_poly_p\<^sup>*\<^sup>* (p, q, {#}) ({#}, {#}, r))"
pprt_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "lattice_ab_group_add.pprt zero sup (uminus x) = uminus (lattice_ab_group_add.nprt zero inf x)"
lunit_char_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "prebicategory_with_homs_and_units V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "H (trg f) (prebicategory_with_homs_and_units.lunit V H \<a> \<i> trg f) = V (H (\<i> (trg f)) f) (category.inv V (\<a> (trg f) (trg f) f))"
ins3_def: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real"   and n :: "nat"   and d :: "nat"   and x :: "'b"   and t :: "'b tree" assumes "RBTi2 TYPE('b) bal_i e" shows "RBTi2.ins3 bal_i n d x t = Time_Monad.val (RBTi2.ins3_tm bal_i n d x t)"
integrable_Re: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> complex" assumes "complex_integrable M f" shows "integrable M (\<lambda>x. Re (f x))"
lipschitz_on_closed_Union: fixes I :: "'b set"   and M :: "real"   and U :: "'b \<Rightarrow> real set"   and f :: "real \<Rightarrow> 'c"   and u :: "real"   and v :: "real" assumes "\<And>i. i \<in> I \<Longrightarrow> M-lipschitz_on (U i) f"   and "\<And>i. i \<in> I \<Longrightarrow> closed (U i)"   and "finite I"   and "0 \<le> M"   and "{u..v} \<subseteq> \<Union> (U ` I)" shows "M-lipschitz_on {u..v} f"
subset_refl: fixes A :: "'b set" shows "order.greater_eq A A"