hcju/isabelleps huggingface dataset:
bij_betw_respects_domain_permutation: fixes A :: "'a set"   and B :: "'b set" shows "(\<lambda>f. bij_betw f A B) respects domain_permutation A B"
cp: shows "char_poly A = [:- e, 1::'a:] * (\<Prod>e\<leftarrow>es. [:- e, 1::'a:])"
code: fixes x :: "gauss_int"   and y :: "gauss_int" shows "x * y = Gauss_Int (ReZ x * ReZ y - ImZ x * ImZ y) (ReZ x * ImZ y + ImZ x * ReZ y)"
dens: shows "has_parametrized_subprob_density (state_measure (V \<union> V') \<Gamma>) F (stock_measure t) f"
nat_pow_zero: fixes n :: "nat" assumes "n \<noteq> 0" shows "\<zero>\<^bsub>poly_ring R\<^esub> [^]\<^bsub>poly_ring R\<^esub> n = \<zero>\<^bsub>poly_ring R\<^esub>"
implies_dist_inf: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<squnion> z - x = y - x \<squnion> (z - x)"
p'_pos_iff: fixes x :: "real" shows "(0 < p' x) = (if 0 < a then - v / a < x else if a < 0 then x < - v / a else 0 < v)"
intros: fixes l :: "'d"   and g :: "('e, 'f) cconstraint"   and a :: "'g"   and r :: "'e list"   and l' :: "'d"   and A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and u :: "'e \<Rightarrow> 'f"   and u' :: "'e \<Rightarrow> 'f" assumes "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" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsub>a\<^esub> \<langle>l', u'\<rangle>"
lea_asym: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.LeA Bet Cong D E F A B C" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F"
colossless_gpv_coinduct: fixes X :: "('d, 'e, 'f) gpv \<Rightarrow> bool"   and gpv :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>" assumes "X gpv"   and "\<And>gpv. X gpv \<Longrightarrow> lossless_spmf (the_gpv gpv) \<and> (\<forall>out c input. IO out c \<in> set_spmf (the_gpv gpv) \<longrightarrow> input \<in> responses_\<I> \<I> out \<longrightarrow> X (c input) \<or> colossless_gpv \<I> (c input))" shows "colossless_gpv \<I> gpv"
full_elims_3: fixes n :: "nat"   and l :: "'a tree23"   and p :: "int \<times> 'a"   and m :: "'a tree23"   and q :: "int \<times> 'a"   and r :: "'a tree23" assumes "full n (Branch3 l p m q r)" obtains na :: "nat" where "n = Suc na" and "full na l" and "full na m" and "full na r"
events_eq_borel: fixes M :: "real measure" assumes "real_distribution M" shows "sets M = sets borel"
rcosets_non_empty: fixes R :: "nat set" assumes "R \<in> rcosets\<^bsub>G\<^esub> H" shows "R \<noteq> {}"
WTrt_binop_widen_mono: fixes P :: "'c prog"   and T1 :: "ty"   and bop :: "bop"   and T2 :: "ty"   and T :: "ty"   and T1' :: "ty"   and T2' :: "ty" assumes "P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T"   and "P \<turnstile> T1' \<le> T1"   and "P \<turnstile> T2' \<le> T2" shows "\<exists>T'. P \<turnstile> T1'\<guillemotleft>bop\<guillemotright>T2' : T' \<and> P \<turnstile> T' \<le> T"
Un: fixes a :: "'b set"   and M :: "'b measure"   and b :: "'b set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a \<union> b \<in> sets M"
fold_insert2: fixes A :: "'a set"   and x :: "'a"   and z :: "'a option" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold MaxR_f z (insert x A) = Finite_Set.fold MaxR_f (MaxR_f x z) A"
SP_post_def: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool" shows "SP_post M l B = (\<lambda>s0 r t. \<forall>s l1 n. (M, l, s, n, l1, r) \<in> Step \<longrightarrow> B s0 s t)"
slide_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "slide w1 w2" shows "kauff_mat w1 = kauff_mat w2"
pr1_tuple: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "local.comp (hfsetcat.pr1 (cod f) (cod g)) (hfsetcat.tuple f g) = f"
opnet_net_ips_net_tree_ips_init: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and ns :: "'e net_state"   and np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p :: "net_tree" assumes "(\<sigma>, ns) \<in> init (opnet np p)" shows "net_ips ns = net_tree_ips p"
eexI: fixes F :: "(state \<Rightarrow> 'b) \<Rightarrow> (nat \<Rightarrow> state) \<Rightarrow> bool"   and x :: "state \<Rightarrow> 'b" shows "\<turnstile> F x \<longrightarrow> (\<exists>\<exists> x. F x)"
set_del_list: fixes xs :: "'b list"   and x :: "'b" assumes "sorted xs" shows "list.set (del_list x xs) = list.set xs - {x}"
arrI: fixes f1 :: "'a"   and f2 :: "'a" assumes "B.arr f1"   and "B.arr f2" shows "B.VxV.arr (f1, f2)"
one_def: fixes one :: "'c"   and times :: "'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 bot :: "'c"   and top :: "'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c" assumes "class.linorder_stone_relation_algebra_expansion one times uminus less_eq less bot top inf sup conv" shows "one = top"
to_rat_rat_def: fixes x :: "rat" shows "to_rat x = x"
vc_transl: fixes C :: "Nielson_VCG.acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "Nielson_VCG.vc C Q" shows "Nielson_VCGi.vc (transl C) Q UNIV UNIV"
axioms_1: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_clinear f" shows "clinear f"
card_extensional_funcset_inj_on: fixes S :: "'c set"   and T :: "'d set" assumes "finite S"   and "finite T"   and "card S \<le> card T" shows "card {f \<in> S \<rightarrow>\<^sub>E T. inj_on f S} = fact (card T) div fact (card T - card S)"
s0_\<beta>_s': shows "s0\<^bsub>SES\<^esub> \<beta>\<Longrightarrow>\<^bsub>SES\<^esub> s'"
bv_to_nat_upper_range: fixes w :: "bit list" shows "bv_to_nat w < 2 ^ length w"
less_diff_eq: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a < c - b) = (a + b < c)"
add_0_left: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g" assumes "class.monoid_add plus zero" shows "plus zero a = a"
R\<^sub>0_in_hom: shows "Maps.in_hom R\<^sub>0 (Maps.MkIde (src (tab\<^sub>0 r))) (Maps.MkIde (src r))"
l11_4_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B"   and "D \<noteq> E"   and "F \<noteq> E"   and "\<forall>A' C' D' F'. B Out A' A \<and> B Out C' C \<and> E Out D' D \<and> E Out F' F \<and> Cong B A' E D' \<and> Cong B C' E F' \<longrightarrow> Cong A' C' D' F'" shows "A B C CongA D E F"
subgroupE_4: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "subgroup H (add_monoid R)"   and "a \<in> H"   and "b \<in> H" shows "a \<oplus> b \<in> H"
nullhomotopic_into_contractible: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "contractible T" obtains c :: "'b" where "homotopic_with_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
order_trans: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
simps_5: fixes p :: "'a poly" shows "polynate (Neg p) = ~\<^sub>p (polynate p)"
Pd_linear': fixes d :: "'b" shows "Linear_order (Pd' d)"
axioms_1: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ext_compat_list ext" shows "ext ext"
min_galleryD_min_betw: fixes x :: "'a set"   and xs :: "'a set list"   and y :: "'a set"   and ys :: "'a set list" assumes "min_gallery (x # xs @ [y])"   and "gallery (x # ys @ [y])" shows "order.greater_eq (length ys) (length xs)"
G_plus1: fixes x :: "real" assumes "0 < x" shows "G (x + 1) = x * G x"
a_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> y = y \<oplus> x"
is_ring_morphism: fixes U :: "'c set" assumes "is_open\<^sub>Y U" shows "ring_homomorphism (\<phi>\<^sub>f U) (\<O>\<^sub>Y U) (add_str\<^sub>Y U) (mult_str\<^sub>Y U) (zero_str\<^sub>Y U) (one_str\<^sub>Y U) (local.im_sheaf U) (add_im_sheaf U) (mult_im_sheaf U) (zero_im_sheaf U) (one_im_sheaf U)"
church_rosser_equiv: fixes y :: "'a"   and x :: "'a" shows "(y\<^sup>\<star> ; x\<^sup>\<star> \<le> x\<^sup>\<star> ; y\<^sup>\<star>) = ((x + y)\<^sup>\<star> = x\<^sup>\<star> ; y\<^sup>\<star>)"
spr_simAbs_agent_abs: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "toSet (snd ec) = agent_abs a t"
\<alpha>_limit_point_def: fixes x :: "'a"   and p :: "'a" shows "rev.\<alpha>_limit_point x p = ({..0} \<subseteq> rev.existence_ivl0 x \<and> (\<exists>s. s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity> \<and> (rev.flow0 x \<circ> s) \<longlonglongrightarrow> p))"
sorted_rec: fixes cmp :: "'b comparator"   and y :: "'b"   and x :: "'b"   and xs :: "'b list" shows "Sorting_Algorithms.sorted cmp (y # x # xs) = (compare cmp y x \<noteq> Greater \<and> Sorting_Algorithms.sorted cmp (x # xs))"
wp_while: fixes b :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "state \<Rightarrow> bool" shows "wp (WHILE b DO c) Q = (\<lambda>s. if b s then wp (c; WHILE b DO c) Q s else Q s)"
nsqnr_def': fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat" shows "nsqn\<^sub>r r = (if \<pi>\<^sub>4 r = Aodv_Basic.inv then \<pi>\<^sub>2 r - 1 else \<pi>\<^sub>2 r)"
CardinalityPrognosisLam_axioms: shows "CardinalityPrognosisLam prognosis"
strong_del_point_design: fixes p :: "'a" shows "design (del_point p) (str_del_point_blocks p)"
assms_2: fixes i :: "'b" assumes "i \<in> I" shows "A i \<in> M"
between_equality: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C"   and "Bet B A C" shows "A = B"
bddh_subsetbdd: fixes l :: "nat"   and A :: "bool list bdd list \<times> bool list"   and bdd' :: "bool list bdd"   and q :: "bool list" assumes "list_all (bddh l) (fst A)"   and "bddh l bdd'"   and "nfa_is_node A q" shows "bddh l (subsetbdd (fst A) q bdd')"
tss'_not_empty: shows "\<not> Optimized_MTL.is_empty tss'"
bij_betw_function_of: fixes enum :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw enum {0..<card A} A" shows "bij_betw (function_of A enum) {xs. set xs \<subseteq> B \<and> length xs = card A} (A \<rightarrow>\<^sub>E B)"
ccpo_Sup_upper: fixes A :: "'b set"   and x :: "'b" assumes "Complete_Partial_Order.chain ordb A"   and "x \<in> A" shows "ordb x (lubb A)"
list_nth_append4: fixes i :: "nat"   and x :: "'a list"   and y :: "'b list" assumes "i < Suc (length x + length y)"   and "\<not> i - Suc (length x) < Suc (length y)" shows "False"
good_item_simps_74: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and xsa :: "'varSort"   and xa :: "'var" assumes "good X"   and "goodAbs A"   and "freshAbs xsa xa A" shows "A $[X / xa]_xsa = A"
downset_hdE: fixes xss :: "'b list set"   and xs :: "'b list" assumes "downset xss"   and "xs \<in> xss"   and "xs \<noteq> []" shows "[hd xs] \<in> xss"
usubstr_def: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and R :: "fml list \<times> fml" shows "usubstr \<sigma> R \<equiv> if usubstappf \<sigma> Denotational_Semantics.allvars (snd R) \<noteq> undeff \<and> (\<forall>\<phi>\<in>set (fst R). usubstappf \<sigma> Denotational_Semantics.allvars \<phi> \<noteq> undeff) then Some (map (the \<circ> usubstappf \<sigma> Denotational_Semantics.allvars) (fst R), the (usubstappf \<sigma> Denotational_Semantics.allvars (snd R))) else None"
assms_4: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
efficient_funpow_correct: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b"   and n :: "nat" assumes "\<And>x z. f x (f x z) = f (f x x) z" shows "efficient_funpow f y x n = (f x ^^ n) y"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (x XOR xa) = int_of_integer x XOR int_of_integer xa"
assms_2: shows "(V', as', as) \<in> dependent_live_vars n''"
simps_6: fixes I :: "interp"   and \<alpha> :: "game"   and \<phi> :: "fml" shows "fml_sem I (\<langle> \<alpha> \<rangle> \<phi>) = game_sem I \<alpha> (fml_sem I \<phi>)"
chain_iterate: fixes F :: "'a \<rightarrow> 'a" shows "chain (\<lambda>i. iterate i\<cdot>F\<cdot>\<bottom>)"
hyps: assumes "\<forall>a\<in>set neq. case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0" shows "\<exists>y'>q. \<forall>a\<in>set neq. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0"
0: shows "pick {a. a < m \<and> a \<in> I} i = pick I i"
fls_deriv_X_intpow: fixes i :: "int" shows "fls_deriv (fls_X_intpow i) = of_int i * fls_X_intpow (i - 1)"
coboundedI2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "semilattice_order f less_eq less"   and "less_eq b c" shows "less_eq (f a b) c"
hOfFax_def: fixes f :: "'fsym" shows "hOfFax f = map2 (Pr \<circ> Guard) (arOf f) (map singl (GE.getTvars (arOf f)))"
sum_integral_diff_series_nonneg: fixes n :: "nat" shows "0 \<le> sum_integral_diff_series n"
fr: shows "(f \<longlongrightarrow> r) (at t within {t..})"
lookup_hom_component: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" shows "lookup (hom_component p n) = (\<lambda>t. lookup p t when deg_pm t = n)"
monotone_gfp_eadd1: fixes y :: "enat" shows "monotone (\<lambda>x y. y \<le> x) (\<lambda>x y. y \<le> x) (\<lambda>x. x + y)"
eval_tm_SNOC: fixes "is" :: "int list"   and \<AA> :: "interp"   and x :: "bool list" assumes "length is \<le> #\<^sub>V \<AA>"   and "#\<^sub>V \<AA> = length x" shows "eval_tm (SNOC x \<AA>) is = eval_tm \<AA> is + 2 ^ Length \<AA> * scalar_product (map of_bool x) is"
dsg3: fixes x :: "'a"   and y :: "'a" shows "ad (ad (ad (ad x) \<cdot> y)) = ad (ad x) \<cdot> ad (ad y)"
obtain_lower_bound_finite: fixes Us :: "'a set set" assumes "finite Us"   and "Us \<noteq> {}"   and "Us \<subseteq> I" obtains W :: "'a set" where "W \<in> I" and "\<forall>U\<in>Us. W \<subseteq> U"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(SKIP {x1} = SKIP {y1}) = (x1 = y1)"
Variable: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and i :: "nat" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst" shows "Generic_Syntax.Variable var i \<in> var"
sexp_case_Leaf: fixes c :: "'d \<Rightarrow> 'c"   and d :: "nat \<Rightarrow> 'c"   and e :: "('d, unit) Old_Datatype.node set \<Rightarrow> ('d, unit) Old_Datatype.node set \<Rightarrow> 'c"   and a :: "'d" shows "sexp_case c d e (Leaf a) = c a"
exec_step_ind_Getfield_NoField: fixes v :: "val"   and stk :: "val list"   and D :: "char list"   and fs :: "Objects.fields"   and h :: "heap"   and P :: "jvm_prog"   and F :: "char list"   and C :: "char list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "v = hd stk"   and "(D, fs) = the (h (the_Addr v))"   and "v \<noteq> Null"   and "\<nexists>t b. P \<turnstile> D has F,b:t in C" shows "exec_step_ind (StepI (Getfield F C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt NoSuchFieldError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
real_of_nat_eq_numeral_power_cancel_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and y :: "nat"   and x :: "num"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero y = power.power one times (numeral.numeral one plus x) n) = (y = numeral x ^ n)"
new_scheme_list_le: fixes n :: "nat"   and m :: "nat"   and A :: "type_scheme list" assumes "n \<le> m"   and "new_tv n A" shows "new_tv m A"
map_ident: fixes t :: "'b multiset" shows "{#x. x \<in># t#} = t"
inverse_aform'_def: fixes p :: "nat"   and X :: "real \<times> real pdevs" shows "inverse_aform' p X = (let l = Inf_aform' p X; u = Sup_aform' p X; a = min \<bar>l\<bar> \<bar>u\<bar>; b = max \<bar>l\<bar> \<bar>u\<bar>; sq = truncate_up p (b * b); alpha = - real_divl p 1 sq; dmax = truncate_up p (real_divr p 1 a - alpha * a); dmin = truncate_down p (real_divl p 1 b - alpha * b); zeta' = truncate_up p ((dmin + dmax) / 2); zeta = if l < 0 then - zeta' else zeta'; delta = truncate_up p (zeta - dmin); res1 = trunc_bound_eucl p (alpha * fst X); res2 = trunc_bound_eucl p (fst res1 + zeta); zs = trunc_bound_pdevs p (scaleR_pdevs alpha (snd X)) in ((fst res2, fst zs), sum_list' p [delta, snd res1, snd res2, snd zs]))"
iffs: fixes nodes :: "'a set"   and edges :: "('a \<times> 'a) set"   and more :: "'b"   and nodes' :: "'a set"   and edges' :: "('a \<times> 'a) set"   and more' :: "'b" shows "(\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = \<lparr>nodes = nodes', edges = edges', \<dots> = more'\<rparr>) = (nodes = nodes' \<and> edges = edges' \<and> more = more')"
inject_8: fixes x9 :: "(String.literal \<times> 'a OCL_Types.type option \<times> 'a expr) list"   and y9 :: "(String.literal \<times> 'a OCL_Types.type option \<times> 'a expr) list" shows "(TupleLiteral x9 = TupleLiteral y9) = (x9 = y9)"
simps_2: fixes m :: "'b match_expr"   and a :: "action"   and rs :: "'b rule list" shows "normalize_rules_dnf (Rule m a # rs) = map (\<lambda>m. Rule m a) (normalize_match m) @ normalize_rules_dnf rs"
assms: shows "finite {t. Re (g t - z) = 0 \<and> 0 \<le> t \<and> t \<le> 1}"
convex_translation_iff_1: fixes f :: "'c \<Rightarrow> 'b"   and y :: "'b"   and S :: "'c set" shows "convex ((\<lambda>x. f x + y) ` S) = convex (f ` S)"
inorder_insert: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (Weight_Balanced_Trees.insert x t) = ins_list x (Tree2.inorder t)"
A_FactGroup_iso_set: assumes "h ` carrier G = carrier H" shows "(\<lambda>X. the_elem (h ` X)) \<in> Group.iso (G A_Mod a_kernel G H h) (add_monoid H)"
emeasure_lim_emb: fixes J :: "int set"   and X :: "(int \<Rightarrow> 'a) set" assumes "J \<subseteq> UNIV"   and "finite J"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>i. borel))" shows "emeasure local.lim (emb UNIV J X) = emeasure (P J) X"
assms_2: shows "fin_make_untimed (inf_truncate z t) \<noteq> []"
infinite_chain_above: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set"   and x :: "'b" assumes "Complete_Partial_Order.chain ord C"   and "infinite C"   and "x \<in> C"   and "finite {y \<in> C. ord y x}" shows "infinite {y \<in> C. ord x y}"
rotate_rev: fixes n :: "nat"   and xs :: "'a list" shows "rotate n (rev xs) = rev (rotate (length xs - n mod length xs) xs)"
trm_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category_with_terminal_object C"   and "partial_magma.arr C f" shows "partial_magma.in_hom C (category_with_terminal_object.trm C f) (partial_magma.dom C f) (category_with_terminal_object.some_terminal C)"
isMaxim_unique: fixes r :: "('d \<times> 'd) set"   and A :: "'d set"   and x :: "'d"   and y :: "'d" assumes "wo_rel r"   and "wo_rel.isMaxim r A x"   and "wo_rel.isMaxim r A y" shows "x = y"
select: shows "l = STR ''select'' \<longrightarrow> P (Some 1, [], <1 $:= Some (hd i), 2 $:= Some (Num 0)>)"
lessE: fixes a :: "'c"   and b :: "'c" assumes "a < b" obtains c :: "'c" where "b = a + c" and "c \<noteq> (0::'c)"
antisym: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat" assumes "b \<preceq> a"   and "a \<preceq> b" shows "a = b"
label_wrt_special_coset1': fixes \<phi> :: "'a \<Rightarrow> 'a"   and w0 :: "'a permutation"   and v :: "'a" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0"   and "w0 \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>"   and "v \<in> C0" shows "{w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> (w0 \<rightarrow> v) = w0 \<rightarrow> v} = w0 +o \<langle>(\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) - {the_inv_into (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) fundantivertex v}\<rangle>"
Domainp_Domain_eq: fixes r :: "('a \<times> 'b) set" shows "Domainp (\<lambda>x xa. (x, xa) \<in> r) = (\<lambda>x. x \<in> Domain r)"
sign_simps_4: fixes ae :: "'b"   and be :: "'b"   and ce :: "'b" shows "ae * be * ce = ae * (be * ce)"
ZSeq: fixes P\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c\<^sub>1 :: "com"   and P\<^sub>2 :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c\<^sub>2 :: "com"   and P\<^sub>3 :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>Z {P\<^sub>1} c\<^sub>1 {P\<^sub>2}"   and "\<turnstile>\<^sub>Z {P\<^sub>2} c\<^sub>2 {P\<^sub>3}" shows "\<turnstile>\<^sub>Z {P\<^sub>1} c\<^sub>1;; c\<^sub>2 {P\<^sub>3}"
empty: shows "GEN_OP emp {} (\<langle>B\<rangle>Rs3)"
part_circlepath_half_finite_inter: fixes st :: "real"   and tt :: "real"   and r :: "real"   and c :: "complex"   and z0 :: "complex"   and d :: "real" assumes "st \<noteq> tt"   and "r \<noteq> 0"   and "c \<noteq> 0" shows "finite {t. part_circlepath z0 r st tt t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}"
cond_override_None: fixes old :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and c :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and new :: "'d \<Rightarrow> 'c option" assumes "old k = None" shows "cond_override c old new k = new k"
Basis_zero: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "zero \<notin> Basis"