hcju/isabelleps huggingface dataset:
simps_6: fixes S :: "'a formula \<Rightarrow> 'a formula"   and F :: "'a formula"   and G :: "'a formula" shows "freshnats.tseytin_tran1 S (F \<^bold>\<rightarrow> G) = [S (F \<^bold>\<rightarrow> G) \<^bold>\<leftrightarrow> S F \<^bold>\<rightarrow> S G] @ freshnats.tseytin_tran1 S F @ freshnats.tseytin_tran1 S G"
vsubst_eq_swap: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and y1 :: "'var"   and y2 :: "'var"   and ys :: "'varSort" assumes "good X"   and "y1 = y2 \<or> fresh ys y1 X" shows "X #[y1 // y2]_ys = X #[y1 \<and> y2]_ys"
f1f2'c: shows "f1 \<le> f2' / c"
cut_less_0_empty: fixes I :: "nat set" shows "I \<down>< 0 = {}"
rev_morph_def: fixes f :: "'c list \<Rightarrow> 'd list" shows "rev_morph f = rev \<circ> f \<circ> rev"
assms_4: shows "f \<in> UNIV \<rightarrow> {- 1<..<1}"
le_infE: 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 x :: "'b::type"   and a :: "'b::type"   and b :: "'b::type" assumes "class.semilattice_inf (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)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type))" obtains "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (a::'b::type)" and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (b::'b::type)"
subK: fixes X :: "complex set" assumes "compact X"   and "X \<subseteq> S" shows "\<exists>N. \<forall>n\<ge>N. X \<subseteq> K n"
bottom_right_bottom: fixes a :: "'a" shows "lcm a (0::'a) = (0::'a)"
assms_1: shows "map fst vs = mcp' (map fst (vs @ xs)) V"
walk_dropWhile: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and f :: "'c \<Rightarrow> bool" assumes "Graph G"   and "Graph.walk G xs" shows "Graph.walk G (dropWhile f xs)"
hyps_2: fixes xs :: "['a]" assumes "distinct xs"   and "\<forall>a\<in>set ys. \<forall>x\<in>set xs. eq\<cdot>a\<cdot>x \<noteq> \<bottom>" shows "xs \\ ys = filter\<cdot>(\<Lambda> x. neg\<cdot>(elem\<cdot>x\<cdot>ys))\<cdot>xs"
matrix'_matrix_change_of_basis: fixes B :: "(('a, 'b) vec, 'b) vec"   and B' :: "(('a, 'b) vec, 'b) vec"   and C :: "(('a, 'c) vec, 'c) vec"   and C' :: "(('a, 'c) vec, 'c) vec"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "is_basis (set_of_vector B)"   and "is_basis (set_of_vector B')"   and "is_basis (set_of_vector C)"   and "is_basis (set_of_vector C')"   and "linear (*s) (*s) f" shows "matrix' B' C' f = matrix_change_of_basis C C' ** matrix' B C f ** matrix_change_of_basis B' B"
span_span: fixes A :: "'b set" shows "vs1.span (vs1.span A) = vs1.span A"
distinct_49: fixes x51 :: "'b poly"   and x52 :: "'b poly"   and x81 :: "'b poly"   and x82 :: "nat"   and x83 :: "'b poly" shows "Mul x51 x52 \<noteq> CN x81 x82 x83"
l_inv: fixes x :: "'a" assumes "x \<in> carrier H'" shows "inv\<^bsub>H'\<^esub> x \<otimes>\<^bsub>H'\<^esub> x = \<one>\<^bsub>H'\<^esub>"
a: shows "d (d x \<cdot> y) \<le> d x \<cdot> d y"
abstractness_essential: shows "\<lfloor>\<lambda>w. \<forall>x. (Abstract x \<^bold>\<rightarrow> \<^bold>\<box>Abstract x) w\<rfloor>"
wls_swap_involutive: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "wls s X" shows "X #[x \<and> y]_zs #[x \<and> y]_zs = X"
intro: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'h" assumes "\<And>y x. f y \<circ> f x = f x \<circ> f y" shows "folding f"
ChamberSubcomplexI: fixes Y :: "'b set set" assumes "order.greater_eq Y Y"   and "ChamberComplex Y"   and "\<And>y. SimplicialComplex.maxsimp Y y \<Longrightarrow> codomain.chamber y" shows "codomain.ChamberSubcomplex Y"
eq_iff: fixes a :: "'e list"   and b :: "'e list" shows "(a = b) = (prefix b a \<and> prefix a b)"
dropWhile_hd_tl: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "xs \<noteq> []" shows "dropWhile P xs = (if P (hd xs) then dropWhile P (tl xs) else xs)"
foldr_fold: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> (f y \<circ>\<circ> f) x = (f x \<circ>\<circ> f) y" shows "foldr f xs = fold f xs"
semIntAll_prWlsAll: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prWlsAll (semInt SEM) (semIntAbs SEM) SEM"
characteriseAx: fixes r :: "'a list \<times> sequent" assumes "r \<in> Ax" shows "r = ([],  \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) \<or> (\<exists>i xs. r = ([],  \<LM> At i xs  \<RM> \<Rightarrow>* \<LM> At i xs  \<RM>))"
strict_implies_not_eq: fixes b :: "('d, 'e) Term.term"   and a :: "('d, 'e) Term.term" assumes "b \<rhd> a" shows "a \<noteq> b"
greater_iff_sym_less: fixes b :: "'a"   and a :: "'a" shows "(cmp b a = Greater) = (cmp a b = Less)"
concat_inf_truncate_nonempty_el: fixes z :: "nat \<Rightarrow> 'd list"   and i :: "nat" assumes "z i \<noteq> []" shows "concat (inf_truncate z i) \<noteq> []"
predicate2_eqD: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" assumes "A = B" shows "A a b = B a b"
continuous_at: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "isCont f x = f \<midarrow>x\<rightarrow> f x"
g: fixes F :: "'h filter"   and G :: "('h \<Rightarrow> real) set"   and H :: "('h \<Rightarrow> real) set"   and g :: "'h \<Rightarrow> real" assumes "landau_function_family_pair F G H g" shows "g \<in> G"
Arcsin_1: shows "Arcsin 1 = complex_of_real (pi / 2)"
lift_def: fixes x :: "V"   and y :: "V" shows "lift x y \<equiv> set ((+) x ` elts y)"
Max_pos_neg_less_multiset: fixes C :: "'b literal multiset"   and A :: "'b"   and D :: "'b literal multiset" assumes "Max_mset C = Pos A"   and "Neg A \<in># D" shows "C < D"
u: shows "u = #v * #w + Polynomial.smult (of_nat q) (#f)"
inject_3: fixes x41 :: "('b, 'c) Combinators"   and x42 :: "('b, 'c) Combinators"   and y41 :: "('b, 'c) Combinators"   and y42 :: "('b, 'c) Combinators" shows "(x41 \<oplus> x42 = y41 \<oplus> y42) = (x41 = y41 \<and> x42 = y42)"
normalize_primitive_extract_preserves_nnf_normalized: fixes m :: "'c match_expr"   and disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'd"   and C :: "'d \<Rightarrow> 'c"   and f :: "'d negation_type list \<Rightarrow> 'd list" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C" shows "\<forall>mn\<in>set (normalize_primitive_extract (disc, sel) C f m). normalized_nnf_match mn"
OclNot_defargs: fixes \<tau> :: "'a state \<times> 'a state"   and P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> not P" shows "\<tau> \<Turnstile> \<delta> P"
element_ptr_casts_commute2: fixes element_ptr :: "(_) element_ptr" shows "cast (cast element_ptr) = Some element_ptr"
diff_self_eq_0: fixes m :: "nat" shows "m - m = zero_class.zero"
residuated_multl: fixes y :: "'a" shows "residuated (\<lambda>x. x \<cdot> y)"
InvariantUniqAfterApplyBackjump: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "getConflictFlag state"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantUniqC (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "0 < currentLevel (getM state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)" shows "let state' = applyBackjump state in InvariantUniq (getM state')"
eventually_at_bot_not_equal: fixes c :: "'a" shows "\<forall>\<^sub>F x in at_bot. x \<noteq> c"
degree_cong: fixes x :: "'b pdevs"   and y :: "'c pdevs" assumes "\<And>i. (pdevs_apply x i = (0::'b)) = (pdevs_apply y i = (0::'c))" shows "degree x = degree y"
pred_subset_eq2: fixes R :: "('b \<times> 'c) set"   and S :: "('b \<times> 'c) set" shows "((\<lambda>x y. (x, y) \<in> R) \<le> (\<lambda>x y. (x, y) \<in> S)) = (R \<subseteq> S)"
alt_ex1E': fixes P :: "'b \<Rightarrow> bool" assumes "\<exists>!x. P x" obtains x :: "'b" where "P x" and "\<exists>\<^sub>\<le>\<^sub>1 x. P x"
min_def: fixes a :: "'c word"   and b :: "'c word" shows "signed.min a b = (if a \<le>s b then a else b)"
exhaust: fixes y :: "'b ext" obtains "y = ext.Bot"   | x2 :: "'b" where "y = ext.Val x2"   | "y = ext.Top"
image_subset_iff: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and B :: "'d set" shows "(f ` A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
simps_3: fixes n :: "nat"   and y :: "'a"   and x :: "'a" assumes "n \<noteq> 0"   and "even n" shows "local.efficient_power y x n = local.efficient_power y (f x x) (n div 2)"
price_structure_init: fixes pyf :: "'a \<Rightarrow> real"   and T :: "nat"   and \<pi> :: "real"   and pr :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "price_structure pyf T \<pi> pr" shows "\<forall>w\<in>space M. pr 0 w = \<pi>"
analz_monotonic: fixes c :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "c \<in> analz G"   and "G \<subseteq> H" shows "c \<in> analz H"
perzeta_conv_hurwitz_zeta_multiplication': fixes k :: "nat"   and s :: "complex"   and a :: "int" assumes "0 < k"   and "s \<noteq> 1" shows "perzeta (real_of_int a / real k) s = of_nat k powr - s * (\<Sum>n = 1..k. exp (complex_of_real (2 * pi * real n * real_of_int a / real k) * \<i>) * hurwitz_zeta (real n / real k) s)"
igAbsRen_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igAbsRen MOD \<equiv> \<forall>xs y x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igFresh MOD xs y X \<longrightarrow> igAbs MOD xs y (igSubst MOD xs (igVar MOD xs y) x X) = igAbs MOD xs x X"
th: fixes n :: "nat"   and a :: "nat"   and e :: "nat"   and d :: "nat" assumes "coprime n a"   and "e \<le> d" shows "[a ^ d = a ^ e] (mod n) = [d = e] (mod ord n a)"
map_add_Some_iff: fixes m :: "'b \<Rightarrow> 'a option"   and n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a" shows "(Map.map_add m n k = Some x) = (n k = Some x \<or> n k = None \<and> m k = Some x)"
successors_def: fixes \<alpha>n :: "'b \<Rightarrow> 'a list"   and inEdges' :: "'b \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'c) list"   and g :: "'b"   and m :: "'a" shows "graph_path_base.successors \<alpha>n inEdges' g m \<equiv> concat (map (\<lambda>n. if m \<in> set (graph_path_base.predecessors inEdges' g n) then [n] else []) (\<alpha>n g))"
take_bit_nat_less_eq_self: fixes n :: "nat"   and m :: "nat" shows "take_bit n m \<le> m"
eq_abstract_rule_ax_def: shows "eq_abstract_rule_ax \<equiv> Ct STR ''Pure.all'' ((Core.aT \<rightarrow> propT) \<rightarrow> propT) $ Abs Core.aT (mk_eq' Core.bT (Core.f $ Bv 0) (Core.g $ Bv 0)) \<longmapsto> mk_eq (Abs Core.aT (Core.f $ Bv 0)) (Abs Core.aT (Core.g $ Bv 0))"
select_convs_3: fixes name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "new \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = new"
composite_simps_1: shows "src (f \<star> src f) = src (src f)"
ground_flit_of_hlit: fixes l :: "hterm literal" shows "ground\<^sub>l (flit_of_hlit l)"
inverse_arrowsE: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" obtains "C.ide (g \<cdot> f)" and "C.ide (f \<cdot> g)"
pj_mem: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set"   and x :: "'d" assumes "Ring R"   and "ideal R I"   and "x \<in> carrier R" shows "pj R I x = x \<uplus>\<^bsub>R\<^esub> I"
hd_list_of: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "hd (list_of A) = Min A"
ap_state_rev_rev_def: shows "(\<diamondop>) = ap_dual pure (\<diamondop>) (\<diamondop>)"
divide_strict_left_mono: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and b :: "'e"   and a :: "'e"   and c :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less b a"   and "less zero c"   and "less zero (times a b)" shows "less (divide c a) (divide c b)"
merge_sorted_map_partitions: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat"   and p :: "nat" assumes "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "isPartition_map R h xs lo hi p"   and "sorted_sublist_map R h xs lo (p - Suc 0)"   and "sorted_sublist_map R h xs (Suc p) hi"   and "lo \<le> hi"   and "lo \<le> p"   and "p \<le> hi"   and "hi < length xs" shows "sorted_sublist_map R h xs lo hi"
cut_ge_mem_iff: fixes x :: "'b"   and I :: "'b set"   and t :: "'b" shows "(x \<in> I \<down>\<ge> t) = (x \<in> I \<and> t \<le> x)"
assms: fixes X :: "nat \<Rightarrow> real" assumes "filterlim X at_bot sequentially" shows "(\<lambda>n. f (X n)) \<longlonglongrightarrow> y"
span_eq: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set"   and T :: "'d set" assumes "module scale" shows "(module.span scale S = module.span scale T) = (S \<subseteq> module.span scale T \<and> T \<subseteq> module.span scale S)"
sup_inf_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<sqinter> z \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
no_spoofing_iface_def: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and rs :: "'b common_primitive rule list" shows "no_spoofing_iface iface ipassmt rs \<equiv> no_spoofing_algorithm iface ipassmt rs {} {}"
con_comp_traces: fixes P :: "'f process"   and Q :: "'g process"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e" shows "traces (P \<parallel> Q <p, q>) = Domain (con_comp_failures P Q p q)"
\<mu>'_1: shows "\<forall>s\<in>G ` I. \<mu>' s = \<mu> (sel s)"
Int_empty_right: fixes A :: "'d set" shows "A \<inter> {} = {}"
outs_\<I>_uniform: fixes A :: "'h set"   and B :: "'i set" shows "outs_\<I> (\<I>_uniform A B) = (if B = {} then {} else A)"
new_element_ptr: shows "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"
powsum_ext: fixes x :: "'a"   and n :: "nat" shows "x \<le> x\<^bsub>0\<^esub>\<^bsup>Suc n\<^esup>"
T2: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> \<box>\<box>[P]_v = \<box>[P]_v"
asm: fixes ts :: "'b triple set"   and G :: "'b triple set" assumes "ts \<subseteq> G" shows "G||-ts"
l_m: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c" assumes "Ring R"   and "a \<in> carrier R" shows "-\<^sub>a\<^bsub>R\<^esub> a \<plusminus>\<^bsub>R\<^esub> a = \<zero>\<^bsub>R\<^esub>"
select_result_E: fixes P :: "'ap \<Rightarrow> bool"   and h :: "'ar"   and f :: "('ar, 'aq, 'ap) prog" assumes "P |h \<turnstile> f|\<^sub>r"   and "h \<turnstile> ok f" obtains x :: "'ap" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "P x"
ord_less_eq_trans: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b < a"   and "b = c" shows "c < a"
finite_INT: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and I :: "'d set"   and A :: "'d \<Rightarrow> 'c set" assumes "semiring_of_sets \<Omega> M"   and "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" shows "\<Inter> (A ` I) \<in> M"
eq_iff: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "(a = b) = (b \<preceq> a \<and> a \<preceq> b)"
size_eq_ex_count_lt: fixes M :: "'b multiset"   and N :: "'b multiset" assumes "size M = size N"   and "M \<noteq> N" shows "\<exists>x. count M x < count N x"
sub_block_upd_fresh: fixes i :: "'b"   and block :: "('c, 'b) fm list \<times> 'b"   and f :: "'b \<Rightarrow> 'd"   and j :: "'d" assumes "i \<notin> block_nominals block" shows "sub_block (f(i := j)) block = sub_block f block"
set_integral_mono: fixes M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "set_integrable M A f"   and "set_integrable M A g"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x" shows "set_lebesgue_integral M A f \<le> set_lebesgue_integral M A g"
counit_in_vhom: shows "\<guillemotleft>\<psi> : f \<star> g \<Rightarrow> src g\<guillemotright>"
fmrel_trans: fixes xm :: "'h \<rightharpoonup>\<^sub>f 'e"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and ym :: "'h \<rightharpoonup>\<^sub>f 'f"   and zm :: "'h \<rightharpoonup>\<^sub>f 'g" assumes "\<And>x y z. \<lbrakk>x \<in> fmran' xm; P x y; Q y z\<rbrakk> \<Longrightarrow> R x z"   and "fmrel P xm ym"   and "fmrel Q ym zm" shows "fmrel R xm zm"
rhs_def: shows "rhs \<equiv> expectation_gpv 0 \<I> (\<lambda>_. y)"
tagged_division_ofI: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set" assumes "finite s"   and "\<And>x K. (x, K) \<in> s \<Longrightarrow> x \<in> K"   and "\<And>x K. (x, K) \<in> s \<Longrightarrow> K \<subseteq> i"   and "\<And>x K. (x, K) \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b"   and "\<And>x1 K1 x2 K2. \<lbrakk>(x1, K1) \<in> s; (x2, K2) \<in> s; (x1, K1) \<noteq> (x2, K2)\<rbrakk> \<Longrightarrow> interior K1 \<inter> interior K2 = {}"   and "\<Union> {K. \<exists>x. (x, K) \<in> s} = i" shows "s tagged_division_of i"
weight_0: fixes w :: "'c \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'c \<times> nat \<Rightarrow> 'c \<times> nat \<Rightarrow> bool"   and pr_weak :: "'c \<times> nat \<Rightarrow> 'c \<times> nat \<Rightarrow> bool"   and least :: "'c \<Rightarrow> bool"   and scf :: "'c \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and t :: "('c, 'd) Term.term" assumes "admissible_kbo w w0 pr_strict pr_weak least scf" shows "(weight_fun.weight w w0 scf t = 0) = False"
refl_relcl: fixes L :: "'b set set" assumes "lattice L" shows "refl (relcl L)"
eval_elim_cases_1: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and t :: "Term.term"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>(Some xc, s) \<midarrow>t\<succ>\<rightarrow> (v, s')" obtains "v = undefined3 t" and "s' = (Some xc, s)"
that: fixes y :: "'a"   and z :: "'a" assumes "y \<in> span S"   and "\<And>w. w \<in> span S \<Longrightarrow> orthogonal z w"   and "x = y + z" shows "thesis"
returns_to_congI1: fixes A :: "'a set"   and x :: "'a" assumes "returns_to A x" shows "auto_ll_on_open.returns_to g Y A x"
transient_def: fixes A :: "'c set" shows "transient A \<equiv> {F. \<exists>act\<in>Acts F. act `` A \<subseteq> - A}"
l11_44_2_b: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and A :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LtA Bet Cong B A C B C A" shows "Tarski_neutral_dimensionless.Lt Bet Cong B C B A"
surjective_mk_program: fixes F :: "'b program" shows "mk_program (Init F, Acts F, AllowedActs F) = F"
less_le_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x < y"   and "y \<le> z" shows "x < z"
induct: fixes P :: "(int \<times> int) list \<Rightarrow> bool"   and a0 :: "(int \<times> int) list" assumes "P []"   and "\<And>uu_. P [uu_]"   and "\<And>p\<^sub>0 p\<^sub>1. P [p\<^sub>0, p\<^sub>1]"   and "\<And>p\<^sub>0 v vb vc. P (v # vb # vc) \<Longrightarrow> P (p\<^sub>0 # v # vb # vc)" shows "P a0"
k1_2: fixes k' :: "'a" assumes "k' < k1" shows "f k' = g k'"
Rep_data_tuple: fixes D :: "'b data" shows "Rep_data D = (DataValue D, Data.DataSpace D)"
absolutely_integrable_periodic_offset_2: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and c :: "real" assumes "f absolutely_integrable_on {a..b}"   and "\<And>x. f (x + (b - a)) = f x" shows "(\<lambda>x. f (c + x)) absolutely_integrable_on {a..b}"
step_invariantI: fixes A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool"   and P :: "'a \<times> 'b \<times> 'a \<Rightarrow> bool" assumes "\<And>s a s'. \<lbrakk>s \<in> reachable A I; (s, a, s') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P (s, a, s')" shows "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"
Disj_Neg_1: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A OR B"   and "H \<turnstile> Neg B" shows "H \<turnstile> A"
simps: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a" shows "Domainp r a = (\<exists>aa b. a = aa \<and> r aa b)"
interaction_bound_inline1': fixes q :: "enat"   and out' :: "'call"   and out :: "'call'"   and c' :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s"   and p :: "enat"   and input :: "'ret'"   and x :: "'ret" assumes "\<And>s x. consider' x \<Longrightarrow> interaction_bound consider (callee s x) \<le> q"   and "\<And>s x. \<not> consider' x \<Longrightarrow> interaction_bound consider (callee s x) = 0"   and "Inr (out', out, c', rpv) \<in> set_spmf (local.inline1' gpv s)"   and "interaction_bound consider' gpv \<le> p" shows "if consider' out' then (if consider out then eSuc (interaction_bound consider (c' input)) else interaction_bound consider (c' input)) \<le> q \<and> eSuc (interaction_bound consider' (rpv x)) \<le> p else \<not> consider out \<and> interaction_bound consider (c' input) = 0 \<and> interaction_bound consider' (rpv x) \<le> p"
col_mult2: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and B :: "'a mat"   and nc :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc"   and "j < nc" shows "col (A * B) j = A *\<^sub>v col B j"
concurrent_add_remove_independent_technical: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e :: "'b"   and ir :: "'a" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Add i e) \<in> set (node_deliver_messages xs)"   and "(ir, Rem is e) \<in> set (node_deliver_messages xs)" shows "hb (i, Add i e) (ir, Rem is e)"
inorder_update: fixes t :: "('c \<times> 'd) tree"   and a :: "'c"   and b :: "'d" assumes "sorted1 (inorder t)" shows "inorder (update a b t) = upd_list a b (inorder t)"
abs_of_pos: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less zero a" shows "abs a = a"
Limsup_lessD: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'b"   and y :: "'b" assumes "Limsup F f < y" shows "\<forall>\<^sub>F x in F. f x < y"
hyps_5: shows "local.general_U q = measure_pmf.expectation q local.ocU"
FieldI2: fixes i :: "'c"   and j :: "'c"   and R :: "('c \<times> 'c) set" assumes "(i, j) \<in> R" shows "j \<in> Field R"