hcju/isabelleps huggingface dataset:
fact_ge_Suc_0_nat: fixes n :: "nat" shows "Suc 0 \<le> fact n"
Lambert_W_lessI: fixes x :: "real"   and y :: "real" assumes "x < y * exp y"   and "- 1 \<le> y"   and "- exp (- 1) \<le> x" shows "Lambert_W x < y"
sup_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> B = Collect (s2p A \<squnion> s2p B)"
weighted_majority_nonagent1: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and x :: "'alt"   and y :: "'alt" assumes "is_pref_profile R"   and "x \<notin> alts" shows "weighted_majority R x y = 0"
assms_5: shows "qs \<in> lang (seq [Atom x, Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])"
mat_def: fixes k :: "'b" shows "mat k = (\<chi>i j. if i = j then k else (0::'b))"
that: assumes "\<And>p. p \<in> P \<Longrightarrow> p \<noteq> {}"   and "\<Union> P = A"   and "\<And>p p'. \<lbrakk>p \<in> P; p' \<in> P; p \<noteq> p'\<rbrakk> \<Longrightarrow> p \<inter> p' = {}" shows "thesis"
set_zip_rightD: fixes x :: "'g"   and y :: "'h"   and xs :: "'g list"   and ys :: "'h list" assumes "(x, y) \<in> set (zip xs ys)" shows "y \<in> set ys"
term_induct_fresh: fixes P :: "('a, 'b, 'varSort, 'var, 'c) param"   and phi :: "('a, 'b, 'varSort, 'var, 'c) qTerm set \<Rightarrow> bool"   and phiAbs :: "('a, 'b, 'varSort, 'var, 'c) qAbs set \<Rightarrow> bool"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "goodPar P"   and "\<And>xs x. phi (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>goodInp inp; goodBinp binp; liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (Op delta inp binp)"   and "\<And>xs x X. \<lbrakk>good X; x \<notin> varsOf P; \<And>Y. Y \<in> termsOf P \<Longrightarrow> fresh xs x Y; \<And>A. A \<in> absOf P \<Longrightarrow> freshAbs xs x A; \<And>rho. rho \<in> envsOf P \<Longrightarrow> freshEnv xs x rho; phi X\<rbrakk> \<Longrightarrow> phiAbs (Abs xs x X)" shows "(good X \<longrightarrow> phi X) \<and> (goodAbs A \<longrightarrow> phiAbs A)"
swap_5: fixes ii :: "nat"   and j :: "nat" assumes "ii < m"   and "j < ii" shows "fs'.gs.\<mu> ii j = (if ii = k - 1 then fs.gs.\<mu> k j else if ii = k then if j = k - 1 then fs.gs.\<mu> k (k - 1) * \<parallel>fs.gs.gso (k - 1)\<parallel>\<^sup>2 / \<parallel>fs'.gs.gso (k - 1)\<parallel>\<^sup>2 else fs.gs.\<mu> (k - 1) j else if k < ii \<and> j = k then fs.gs.\<mu> ii (k - 1) - fs.gs.\<mu> k (k - 1) * fs.gs.\<mu> ii k else if k < ii \<and> j = k - 1 then fs.gs.\<mu> ii (k - 1) * fs'.gs.\<mu> k (k - 1) + fs.gs.\<mu> ii k * \<parallel>fs.gs.gso k\<parallel>\<^sup>2 / \<parallel>fs'.gs.gso (k - 1)\<parallel>\<^sup>2 else fs.gs.\<mu> ii j)"
r: shows "is_arg_min norm (\<lambda>t. t \<in> M) r"
inj_on_diff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f A" shows "inj_on f (A - B)"
iterate_to_list_map_linord_correct: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and it :: "(('a \<times> 'b) list \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'b \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list"   and m :: "'a \<Rightarrow> 'b option" assumes "class.linorder less_eq less"   and "linorder.set_iterator_map_linord less_eq it (map_to_set m)" shows "map_of (iterate_to_list it) = m \<and> distinct (map fst (iterate_to_list it)) \<and> linorder.sorted less_eq (map fst (rev (iterate_to_list it)))"
fresh_1: fixes a :: "atom"   and tm1 :: "tm"   and tm2 :: "tm" shows "a \<sharp> tm1 IN tm2 = (a \<sharp> tm1 \<and> a \<sharp> tm2)"
cut_greater_not_empty_imp: fixes I :: "'a set"   and t :: "'a" assumes "I \<down>> t \<noteq> {}" shows "I \<noteq> {}"
simps_2: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and e :: "'c"   and v :: "'d"   and xs :: "('a \<times> 'c \<times> 'd) list"   and q :: "'c \<Rightarrow> 'a \<Rightarrow> 'b" shows "ListUpdate f ((x, e, v) # xs) q = (ListUpdate f xs q)(x := q e x)"
G_ctor_o_fold: fixes s :: "('e, 'f F) G0 \<Rightarrow> 'f" shows "ctor_fold_G s \<circ> ctor_G = s \<circ> map_pre_G id (ctor_fold_G s)"
eAbs_simp2: fixes xs :: "'varSort"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and X :: "'a"   and x :: "'var" assumes "\<forall>s. \<not> (isInBar (xs, s) \<and> igWls MOD s X)" shows "eAbs MOD xs x (OK X) = ERR"
assms: shows "intersect_segment_xline prec y xl = Some (m0, M0)"
comp_char: fixes g :: "'a"   and f :: "'a" shows "local.comp g f = (if arr f \<and> arr g \<and> C.seq g f then C g f else C.null)"
set_sum: fixes H :: "'a set"   and K :: "'a set" assumes "H \<subseteq> carrier M"   and "K \<subseteq> carrier M" shows "H \<minusplus> K = {x. \<exists>h\<in>H. \<exists>k\<in>K. x = h \<plusminus> k}"
cases: fixes x :: "'c + 'd + 'e" obtains a :: "'c" where "x = Inl a"   | b :: "'d" where "x = Inr (Inl b)"   | c :: "'e" where "x = Inr (Inr c)"
distinct_5: fixes x1 :: "name"   and x4 :: "nat" shows "Const x1 \<noteq> Bound x4"
intro: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and empty :: "'b" assumes "dp_consistency lookup update P"   and "mem_correct_empty lookup update P empty" shows "dp_consistency_empty lookup update P empty"
calc_x_axis_intersection: fixes H :: "circline" assumes "is_poincare_line H"   and "intersects_x_axis H" shows "calc_x_axis_intersection H \<in> circline_set H \<inter> circline_set x_axis"
i_Exec_Stream_Acc_LocalState_1: fixes localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" shows "i_Exec_Comp_Stream_Acc_LocalState (Suc 0) localState trans_fun input c = localState \<circ> i_Exec_Comp_Stream trans_fun input c"
auxx8: fixes x :: "('b, 'c) Combinators" shows "removeShadowRules1_alternative_rev [x] = [x]"
f: shows "f = Polynomial.smult c (\<Prod>a\<leftarrow>as. [:- a, 1:])"
assms: shows "scut\<^sup>+\<^sup>+ p q"
no_oif_match_def: shows "no_oif_match \<equiv> list_all (\<lambda>m. oiface (match_sel m) = ifaceAny)"
r: fixes b :: "'a" shows "Itm vs (b # bs) r = Itm vs (x # bs) r"
normalize_positive_ports_step_def: fixes disc_sel :: "('b common_primitive \<Rightarrow> bool) \<times> ('b common_primitive \<Rightarrow> ipt_l4_ports)"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and m :: "'b common_primitive match_expr" shows "normalize_positive_ports_step disc_sel C m \<equiv> let (spts, rst) = primitive_extractor disc_sel m in case (getPos spts, getNeg spts) of (pspts, []) \<Rightarrow> case l4_ports_compress pspts of CannotMatch \<Rightarrow> [] | MatchesAll \<Rightarrow> [rst] | MatchExpr m \<Rightarrow> map (\<lambda>spt. MatchAnd (Match (C spt)) rst) (singletonize_L4Ports m)"
assms_1: fixes i :: "'a" assumes "i \<in> dom" shows "R i x y"
rprI: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "complete A r"   and "refl_on A r"   and "trans r" shows "rpr A r"
qGoodBinp_def: fixes qbinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs option" shows "qGoodBinp qbinp \<equiv> liftAll qGoodAbs qbinp \<and> |{i. qbinp i \<noteq> None}| <o |UNIV|"
relImage_relInvImage: fixes R :: "('d \<times> 'd) set"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "R \<subseteq> f ` A \<times> f ` A" shows "BNF_Greatest_Fixpoint.relImage (BNF_Greatest_Fixpoint.relInvImage A R f) f = R"
scope_cases: fixes v :: "'form \<times> nat list"   and i :: "('form, 'var) antecedent"   and v' :: "'form \<times> nat list" assumes "scope' v i v'" obtains c :: "'form" and is' :: "nat list" and "is" :: "nat list" and i :: "nat" where "v = (c, is)" and "ia__ = in_port_at (c, is) i" and "v' = (c, is')" and "c \<in> set conclusions" and "is' \<in> (#) 0 ` it_paths (it' c)" and "prefix (is @ [i]) is'"
projection_split_first: fixes \<tau> :: "'b list"   and X :: "'b set"   and x :: "'b"   and xs :: "'b list" assumes "\<tau> \<upharpoonleft> X = x # xs" shows "\<exists>\<alpha> \<beta>. \<tau> = \<alpha> @ [x] @ \<beta> \<and> \<alpha> \<upharpoonleft> X = []"
lsum_append: fixes f :: "'a \<Rightarrow> nat"   and al :: "'a list"   and bl :: "'a list" shows "lsum f (al @ bl) = lsum f al + lsum f bl"
col_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points" shows "col A B C \<equiv> \<exists>l. incid A l \<and> incid B l \<and> incid C l"
degree_frame: fixes G :: "('a, 'b) graph"   and x :: "'a"   and v :: "'a"   and v' :: "'a"   and w :: "'b" assumes "valid_unMultigraph G"   and "finite (edges G)"   and "x \<notin> {v, v'}" shows "degree x (del_unEdge v w v' G) = degree x G"
slist_eq_TT_snil_1: fixes xs :: "[:'a:]" shows "(eq\<cdot>xs\<cdot>[::] = TT) = (xs = [::])"
le_iff_inf: fixes x :: "'a multiset"   and y :: "'a multiset" shows "(x \<subseteq># y) = (x \<inter># y = x)"
map\<^sub>0_in_hom_1: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<tau>\<^sub>0 a : \<tau>.F.map\<^sub>0 a \<rightarrow>\<^sub>D \<tau>.G.map\<^sub>0 a\<guillemotright>"
rv_def: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)" shows "rv M = {f. measure_space M \<and> f \<in> measurable (measurable_sets M) \<bool>}"
g_impl_rel_ext_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "\<langle>Rm, Rv\<rangle>g_impl_rel_ext \<equiv> \<langle>Rm, \<langle>Rv\<rangle>fun_set_rel, \<langle>Rv\<rangle>slg_rel, \<langle>Rv\<rangle>list_set_rel\<rangle>gen_g_impl_rel_ext"
pivotandupdate_nolhs: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<B>\<^sub>l s x\<^sub>i = Some c \<or> \<B>\<^sub>u s x\<^sub>i = Some c" shows "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s (pivot_and_update x\<^sub>i x\<^sub>j c s)"
map_disc_iff: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c tree" shows "(map_tree f a = Leaf) = (a = Leaf)"
polychain_of_ccw_conjunction: fixes Ps :: "(real \<times> real) list"   and z :: "(real \<times> real) \<times> real \<times> real"   and Pc :: "real \<times> real" assumes "ccw'.sortedP 0 Ps"   and "z \<in> set (polychain_of Pc Ps)" shows "list_all (\<lambda>(xi, xj). ccw xi xj (fst z) \<and> ccw xi xj (snd z)) (polychain_of Pc Ps)"
tab_simps_4: shows "local.dom f.tab = tab\<^sub>1 f"
span_span: fixes A :: "'b set" assumes "A \<subseteq> U\<^sub>V\<^sub>S" shows "local.span (local.span A) = local.span A"
finite_domain: fixes P :: "'d \<Rightarrow>\<^sub>F 'e" shows "finite (domain P)"
evaln_elim_cases_12: fixes G :: "prog"   and s :: "st"   and e :: "expr"   and T :: "ref_ty"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (e InstOf T)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains va :: "val" where "v = In1 (Bool (va \<noteq> Null \<and> G,snd s'\<turnstile>va fits RefT T))" and "G\<turnstile>Norm s \<midarrow>e-\<succ>va\<midarrow>n\<rightarrow> s'"
uminus_Pos: fixes A :: "'a" shows "- Pos A = Neg A"
run_asset_i_rel: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "s \<Turnstile> fst (run_asset_i n s)"
CFGExit_axioms: fixes prog :: "cmd" shows "CFGExit sourcenode targetnode kind (valid_edge prog) (_Entry_) (_Exit_)"
dup_simps_1: fixes f :: "'a" assumes "arr f" shows "arr (ECC.dup f)"
diag_mat_zero: fixes n :: "nat" shows "diag_mat (0\<^sub>m n n) = replicate n (0::'b)"
intro: fixes n :: "nat" assumes "1 < n" shows "residues_nat n"
intro: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "ring_ops ops R" shows "idom_ops ops R"
iUntil_iWeakUntil_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t'. t' \<U> t I. Q t) = ((P t'. t' \<W> t I. Q t) \<and> (\<diamond> t I. Q t))"
last_message_last_message_hold: fixes xs :: "'b message_af list" shows "last_message (last_message_hold xs) = last_message xs"
lemTime2Sym: fixes x :: "'a Point"   and y :: "'a Point" shows "time2 x y = time2 y x"
less_le_not_le_basic_type: fixes \<tau> :: "'a basic_type"   and \<sigma> :: "'a basic_type" shows "(\<tau> < \<sigma>) = (\<tau> \<le> \<sigma> \<and> \<not> \<sigma> \<le> \<tau>)"
closed_euler1: assumes "Digraph_Component.connected G"   and "\<And>u. u \<in> verts G \<Longrightarrow> in_degree G u = out_degree G u" shows "\<exists>u p. euler_trail u p u"
interp_ops_distinct: fixes ops :: "('c \<times> ('c, 'd) list_op) list" assumes "list_ops ops" shows "distinct (fst (interp_ops ops))"
cases: fixes x :: "'b status" obtains ok :: "nat" and xa :: "'b" and f :: "'b list" and f' :: "'b list" and y :: "'b" and r :: "'b list" and r' :: "'b list" where "x = Rev ok__ (xa__ # f) f'__ (y__ # r) r'__"   | ok :: "nat" and f' :: "'b list" and y :: "'b" and r' :: "'b list" where "x = Rev ok [] f'a__ [y] r'a__"   | f' :: "'b list" and r' :: "'b list" where "x = App 0 f'b__ r'b__"   | v :: "nat" and xa :: "'b" and f' :: "'b list" and r' :: "'b list" where "x = App (Suc v__) (xa # f') r'"   | "x = Idle"   | v :: "nat" and vb :: "'b list" and vd :: "'b list" where "x = Rev va__ [] vb__ [] vd__"   | v :: "nat" and vb :: "'b list" and va :: "'b" and vf :: "'b" and vg :: "'b list" and vd :: "'b list" where "x = Rev vc__ [] vba__ (vaa__ # vf # vg) vda__"   | v :: "nat" and va :: "'b list" and vb :: "'b list" and vd :: "'b list" where "x = Rev ve__ va vbb__ [] vd"   | vc :: "nat" and vb :: "'b list" where "x = App (Suc vc) [] vb"   | v :: "'b list" where "x = Done v"
Max_in: fixes A :: "'a DBMEntry set" assumes "finite A"   and "A \<noteq> {}" shows "linordered_monoid.Max A \<in> A"
digraph_map: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map" assumes "digraph_map G M" shows "digraph_map G M"
pivot_positions_3: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat nr nc"   and "pivot_fun A f nc" shows "distinct (map snd (pivot_positions A))"
f_def: fixes l :: "'a vec" shows "f l \<equiv> \<Sum>i = 0..<length lst. if l = lst ! i then v $ i else (0::'a)"
cl_subset: shows "cl C (T \<inter> wens F act X) \<subseteq> T \<inter> wens F act X"
p3: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All li (SyntaxN.All j (SyntaxN.All j' (SyntaxN.All sj (SyntaxN.All lj (SeqQuoteP (Var i) (Var i') (Var si) (Var li) IMP SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP HaddP (Var li) (Var lj) (Var k) IMP (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))))))))(si::=Var si)"
Disjunctive_fun_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "Disjunctive_fun R = (\<forall>i. R i \<in> Apply.Disjunctive)"
vector_derivative_cnj_within: fixes x :: "real"   and A :: "real set"   and f :: "real \<Rightarrow> complex" assumes "at x within A \<noteq> bot"   and "f differentiable at x within A" shows "vector_derivative (\<lambda>z. cnj (f z)) (at x within A) = cnj (vector_derivative f (at x within A))"
arr_Par_arr_RelD_1: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Par \<alpha> T" shows "arr_Rel \<alpha> T"
Abs_enat_cases: fixes x :: "enat" obtains y :: "nat option" where "x = Abs_enat y" and "y \<in> UNIV"
assms_2: fixes x :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "P x"
F_def: shows "F \<equiv> \<lambda>x. \<Sum>i\<in>Basis. f i (x \<bullet> i) *\<^sub>R i"
power_strict_increasing_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and x :: "nat"   and y :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one b" shows "less (power.power one times b x) (power.power one times b y) = (x < y)"
member_bind: fixes x :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a set" shows "(x \<in> Set.bind A f) = (x \<in> \<Union> (f ` A))"
separable_bound_def: fixes f :: "int poly" shows "separable_bound f = max \<bar>resultant f (pderiv f)\<bar> (max \<bar>lead_coeff f\<bar> \<bar>lead_coeff (pderiv f)\<bar>)"
aux'_eq: shows "aux' = (case aux0 of [] \<Rightarrow> [(\<tau> \<sigma> ne, mrtabulate mrs (r\<epsilon>_strict n rels))] | x # aux' \<Rightarrow> if fst x = \<tau> \<sigma> ne then x # aux' else (\<tau> \<sigma> ne, mrtabulate mrs (r\<epsilon>_strict n rels)) # x # aux')"
interleave_ccTTree: fixes S :: "Vars.var set"   and G :: "CoCalls"   and S' :: "Vars.var set"   and G' :: "CoCalls" shows "ccTTree S G \<otimes>\<otimes> ccTTree S' G' \<sqsubseteq> ccTTree (S \<union> S') (G \<squnion> G' \<squnion> S G\<times> S')"
intros: fixes a :: "'d"   and b :: "'e"   and r :: "('d \<times> 'e) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
a_in_ar_coset: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> carrier R" shows "a \<in> a \<uplus>\<^bsub>R\<^esub> I"
simps_1: fixes prec :: "nat"   and n :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and f :: "form"   and bs :: "float interval option list"   and ss :: "nat list" shows "approx_form prec (Bound (Var n) a b f) bs ss = (case (approx prec a bs, approx prec b bs) of (None, b) \<Rightarrow> False | (Some ivl1, None) \<Rightarrow> False | (Some ivl1, Some ivl2) \<Rightarrow> approx_form' prec f (ss ! n) n (sup ivl1 ivl2) bs ss)"
map_try_gpv: fixes f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e"   and gpv :: "('g, 'h, 'f) gpv"   and gpv' :: "('g, 'h, 'f) gpv" shows "map_gpv f g (TRY gpv ELSE gpv') = TRY map_gpv f g gpv ELSE map_gpv f g gpv'"
galois_bijection_axioms_def: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "galois_bijection_axioms G \<equiv> (isotone \<X>\<^bsub>G\<^esub> \<Y>\<^bsub>G\<^esub> \<pi>\<^sup>*\<^bsub>G\<^esub> \<and> isotone \<Y>\<^bsub>G\<^esub> \<X>\<^bsub>G\<^esub> \<pi>\<^sub>*\<^bsub>G\<^esub>) \<and> (\<forall>x. x \<in> carrier \<X>\<^bsub>G\<^esub> \<longrightarrow> \<pi>\<^sub>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>G\<^esub> x) = x) \<and> (\<forall>y. y \<in> carrier \<Y>\<^bsub>G\<^esub> \<longrightarrow> \<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y) = y)"
Cons_less_Cons: fixes a :: "'b"   and x :: "'b list"   and b :: "'b"   and y :: "'b list" shows "(a # x < b # y) = (length x < length y \<or> length x = length y \<and> (a < b \<or> a = b \<and> x < y))"
scene_compat_self_compl: fixes A :: "'a scene" shows "A ##\<^sub>S - A"
expansion_gamma_onI: fixes As :: "'a set set"   and A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<lbrakk>\<Union> As \<subseteq> A; As \<noteq> {}\<rbrakk> \<Longrightarrow> \<Inter> (f ` As) \<subseteq> f (\<Union> As)" shows "expansion_gamma_on A As f"
that: fixes C :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite C"   and "disjnt C S"   and "continuous_on (\<Union> \<F> - C) g"   and "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
behaviour_refinement: fixes R :: "('e \<times> 'f) set"   and Ta :: "('e, 'g) TS_scheme"   and Tc :: "('f, 'h) TS_scheme"   and bc :: "'f list" assumes "PO_refines R Ta Tc"   and "bc \<in> beh Tc" shows "\<exists>ba\<in>beh Ta. (ba, bc) \<in> seq_lift R"
nextElem_is_nextElem: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "distinct xs"   and "x \<in> set xs" shows "is_nextElem xs x y = (nextElem xs (hd xs) x = y)"
map_spmf_bind_spmf: fixes f :: "'f \<Rightarrow> 'e"   and p :: "'g spmf"   and g :: "'g \<Rightarrow> 'f spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
true_clss_set_mset: fixes I :: "'b set"   and CC :: "'b literal multiset multiset" shows "(I \<TTurnstile>s set_mset CC) = (I \<TTurnstile>m CC)"
sm_QF: fixes k :: "nat" shows "strict_mono_sets ({..k} \<times> {..<m}) (QF k)"
reflect_poly_eq_zero: fixes f :: "'a poly" shows "(reflect_poly f = 0) = (f = 0)"
individually_rational_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "TruePref_tax.individually_rational_on ds X = (TruePref_tax.CD_on ds X = X \<and> Singleton_for_d.CH X = X)"
ahm_\<alpha>_def: fixes m :: "uint32 \<Rightarrow> ('b \<Rightarrow> 'a option) option"   and k :: "'b" shows "ahm_\<alpha> m k \<equiv> case m (hashcode k) of None \<Rightarrow> None | Some cm \<Rightarrow> cm k"
divide_float_def: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "a / b = fdiv To_nearest a b"
correctCompositionDiffLevelsS12: shows "correctCompositionDiffLevels sS12"
wf_lderiv0: fixes idx :: "'n"   and a :: "'a"   and x :: "'x" assumes "wf0 idx a"   and "lformula0 a" shows "local.wf idx (lderiv0 x a)"
F1: shows "R1 \<Longrightarrow>\<^sub>A P x * F1"
bigo_lesseq4: fixes f :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(h)"   and "\<forall>x. (0::'d) \<le> g x"   and "\<forall>x. g x \<le> \<bar>f x\<bar>" shows "g \<in> O(h)"
exhaust: fixes y :: "accent" obtains "y = Acute"   | "y = Grave"   | "y = Macron"
obroadcastT: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, {l}broadcast(s\<^sub>m\<^sub>s\<^sub>g) . p), broadcast (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i)), \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
STEPFMT_4: shows "((uh # rh, ceh), LCall q, entry fg q # uh' # rh, ceh) \<in> trss fg"
step'_def: shows "step' = Step (\<lambda>x e S. case S of None \<Rightarrow> None | Some S \<Rightarrow> Some (update S x (aval' e S))) (\<lambda>b S. S)"
classical_operator_exists_finite: fixes \<pi> :: "'c \<Rightarrow> 'd option" shows "classical_operator_exists \<pi>"
normalize_mult: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom_multiplicative divide plus minus zero times one unit_factor normalize" shows "normalize (times a b) = times (normalize a) (normalize b)"
at_neq_bot: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b" assumes "class.perfect_space open" shows "topological_space.at_within open a UNIV \<noteq> bot"
coboundedI: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "a \<in> A" shows "linorder.Min (\<sqsubseteq>) A \<sqsubseteq> a"
closed_Collect_le_within: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and R :: "'c set" assumes "continuous_on UNIV f"   and "continuous_on UNIV g"   and "closed R" shows "closed {x \<in> R. f x \<le> g x}"
simps_9: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Arr \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = (Arr t \<and> Arr u \<and> Arr v \<and> Src t = Trg u \<and> Src u = Trg v)"
self_append_conv2: fixes ys :: "'e list"   and xs :: "'e list" shows "(ys = append xs ys) = (xs = [])"
inf_sup_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> (x \<sqinter> z)"
"7_1": shows "t @- y @- r = fromN (Suc 0) ||| s"
exists_gen: fixes I :: "'a set" assumes "ideal I R" shows "\<exists>a\<in>carrier R. I = PIdl a"
tendsto_mono: fixes "open" :: "'a set \<Rightarrow> bool"   and F :: "'b filter"   and F' :: "'b filter"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a" assumes "class.topological_space open"   and "F \<le> F'"   and "filterlim f (topological_space.nhds open l) F'" shows "filterlim f (topological_space.nhds open l) F"
drop: fixes \<C> :: "t_context"   and t :: "t" shows "\<C> \<turnstile> [Drop] : [t] _> []"