hcju/isabelleps huggingface dataset:
Abs_ainj_ex: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "good X"   and "good X'" shows "(Abs xs x X = Abs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> fresh xs y X \<and> fresh xs y X' \<and> X #[y \<and> x]_xs = X' #[y \<and> x']_xs))"
gt_ge_tpoly_trans: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and r :: "('a pvar, hmultiset) tpoly"   and q :: "('a pvar, hmultiset) tpoly"   and p :: "('a pvar, hmultiset) tpoly" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.gt_tpoly ground_heads_var \<delta> arity_sym wt_sym r q"   and "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym q p" shows "tkbo_coefs.gt_tpoly ground_heads_var \<delta> arity_sym wt_sym r p"
intro: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "ab_semigroup_add_ow S pls"   and "comm_monoid_add_ow_axioms S pls z" shows "comm_monoid_add_ow S pls z"
length_getVars: fixes \<sigma>l :: "'tp list" shows "length (getVars \<sigma>l) = length \<sigma>l"
add_def: fixes x :: "'a"   and y :: "'a" shows "x + y = Abs ((Rep x + Rep y) mod int CARD('a))"
H_on_ket_one_is_\<psi>\<^sub>1\<^sub>1: shows "H * |Deutsch.one\<rangle> = Matrix.mat 2 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if i = 0 then 1 / sqrt 2 else - 1 / sqrt 2))"
Lcm_complex_def: shows "Lcm = normalization_euclidean_semiring_class.Lcm"
bounded_iff: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(Max A \<le> x) = (\<forall>a\<in>A. a \<le> x)"
subtype_sound: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and t :: "'d set"   and P :: "'d set"   and t' :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "sifum_types.subtype ev\<^sub>B t P t'" shows "\<forall>mem. sifum_lang_no_dma.pred ev\<^sub>B P mem \<longrightarrow> sifum_types.type_max ev\<^sub>B t mem \<le> sifum_types.type_max ev\<^sub>B t' mem"
simps_2: fixes l :: "('b ivl \<times> 'b) tree"   and a :: "'b ivl"   and m :: "'b"   and r :: "('b ivl \<times> 'b) tree"   and x :: "'b ivl" shows "search \<langle>l, (a, m), r\<rangle> x = (if overlap x a then True else if l \<noteq> \<langle>\<rangle> \<and> low x \<le> max_hi l then search l x else search r x)"
rep_eq: fixes x :: "('c, 'd) fmap" shows "fset (fmdom x) = dom (fmlookup x)"
assms_1: shows "\<forall>i\<in>I. subgroup (S i) G"
tuple_since'_def: shows "tuple_since' \<equiv> upd_set tuple_since (\<lambda>_. nt) (X - Mapping.keys tuple_since)"
real_gauss_int_norm: fixes z :: "gauss_int" shows "real (gauss_int_norm z) = (cmod (gauss2complex z))\<^sup>2"
mkarr_rep: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "free_monoidal_category.mkarr C (free_monoidal_category.rep C f) = f"
i2: shows "\<forall>D\<in>MajoritySet. \<forall>q. \<exists>d\<in>D. bal (dblock s' p) < mbal (disk s' d q) \<longrightarrow> hasRead s' p d q"
ext_trans_before_irrefl_axioms_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_trans_before_irrefl_axioms ext \<equiv> \<forall>xs gt. (\<forall>z\<in>set xs. \<forall>y\<in>set xs. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x) \<longrightarrow> (\<forall>x\<in>set xs. \<not> gt x x) \<longrightarrow> \<not> ext gt xs xs"
has_bochner_integral_cong_AE: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "has_bochner_integral M f x = has_bochner_integral M g x"
assms_3: shows "ring_hom_ring (K [X]) A h"
set_spmf_map_pmf: fixes f :: "'d \<Rightarrow> 'c option"   and p :: "'d pmf" shows "set_spmf (map_pmf f p) = (\<Union>x\<in>set_pmf p. set_option (f x))"
distinct_17: fixes x61 :: "'b formula"   and x62 :: "'b formula" shows "\<bottom> \<noteq> x61 \<^bold>\<rightarrow> x62"
inject_1: fixes x1 :: "variable"   and y1 :: "variable" shows "(term.VAR x1 = term.VAR y1) = (x1 = y1)"
l_child_lower_bound: fixes p :: "int"   and l :: "int" shows "(p < l_child l p) = (l \<le> p)"
d: shows "\<Sum>\<^sub># (image_mset size (\<Gamma> + \<Delta>)) = sum_list (map size (\<Gamma>' @ \<Delta>'))"
sum_inner_Basis_eq: fixes b :: "'c"   and f :: "'c \<Rightarrow> real" assumes "b \<in> Basis" shows "(\<Sum>i\<in>Basis. i \<bullet> b * f i) = f b"
first_map_add_submap: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option"   and x :: "'a \<Rightarrow> 'b option" assumes "\<And>x. x \<in> dom m1 \<and> x \<in> dom m2 \<Longrightarrow> m1 x = m2 x"   and "m1 ++ m2 \<subseteq>\<^sub>m x" shows "m1 \<subseteq>\<^sub>m x"
elims_2: fixes x :: "('d, 'e, 'f) FSM"   and xa :: "'d list set"   and xb :: "('d, 'e) ATC set"   and xc :: "('d, 'e, 'g) FSM" assumes "x \<preceq>\<lbrakk>xa.xb\<rbrakk> xc" obtains M1 :: "('d, 'e, 'f) FSM" and TS :: "'d list set" and \<Omega> :: "('d, 'e) ATC set" and M2 :: "('d, 'e, 'g) FSM" where "x = M1" and "xa = TS" and "xb = \<Omega>" and "xc = M2" and "\<forall>iseq\<in>TS. atc_io_reduction_on M1 M2 iseq \<Omega>"
span_base: fixes Sa :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a"   and a :: "(('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real" assumes "Sa \<subseteq> cotangent_space p"   and "a \<in> Sa" shows "a \<in> span Sa"
IH1_1: fixes f :: "floatarith" assumes "f \<in> subterms fa1" shows "f \<in> Mapping.keys M1"
seq_6: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel = ennreal r"
h2: shows "well_def_signs (length qs) (get_signs (reduce_system p (qs, matrix, subsets, signs)))"
5495: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- - (x \<sqinter> y) \<sqinter> - (x \<squnion> z) = bot"
lsl_iso_var: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "\<nu> x \<le> \<nu> y"
"2_3": shows "poly (poly_of_pm x p) a = poly_eval (\<lambda>y. a when y = x) p"
assms: shows "run_one_step d i (s, vs, ves, Label x41 x42 x43) = (s', vs', res)"
actsUpdate_def: fixes a :: "'a"   and ec :: "'s odlist"   and A :: "(('s, 'aAct list) trie, ('s, ('obs, 's odlist) mapping) trie) AlgState" shows "SPRsingle.KBP.actsUpdate a ec A \<equiv> update trie_odlist_MapOps ec (local.spr_simAction a ec) (aActs A)"
exp_first_two_terms: fixes x :: "'a" shows "exp x = (1::'a) + x + (\<Sum>n. x ^ (n + 2) /\<^sub>R fact (n + 2))"
IO: shows "IO out c \<in> set_spmf (the_gpv gpv)"
assms: shows "(i_start, i_end) \<in> set (identify_blocks B k)"
list_lex_less_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs <\<^sub>l ys) = ((xs, ys) \<in> lexord {(u, v). u < v})"
simps_177: fixes x121 :: "expr"   and x122 :: "expr"   and x81 :: "expr"   and x82 :: "char list"   and x83 :: "char list list" shows "x121;; x122 \<noteq> x81\<bullet>x82{x83}"
t2: shows "term_ok \<Theta> (Abs \<tau>' s $ Fv x \<tau>')"
pivot_positions_gen_transfer: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "(R ===> R ===> (=)) (=) (=)" shows "(R ===> mat_rel R ===> (=)) pivot_positions_gen pivot_positions_gen"
addr_in_ipset_from_cidr_code: fixes addr :: "'b word"   and pre :: "'b word"   and len :: "nat" shows "(addr \<in> ipset_from_cidr pre len) = (pre && (mask len << LENGTH('b) - len) \<le> addr \<and> addr \<le> pre || mask (LENGTH('b) - len))"
restrict_map_upds: fixes xs :: "'c list"   and ys :: "'d list"   and D :: "'c set"   and m :: "'c \<Rightarrow> 'd option" assumes "length xs = length ys"   and "set xs \<subseteq> D" shows "m(xs [\<mapsto>] ys) |` D = (m |` (D - set xs))(xs [\<mapsto>] ys)"
R_lin_independent_ConsD2: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b"   and ms :: "'b list"   and r :: "'a"   and rs :: "'a list" assumes "R_scalar_mult R"   and "R_scalar_mult.R_lin_independent R smult (m # ms)"   and "set (r # rs) \<subseteq> R"   and "scalar_mult.lincomb smult (r # rs) (m # ms) = (0::'b)" shows "r = (0::'a)"
chamber_in_apartment: fixes Y :: "'a set set"   and C :: "'a set" assumes "Y \<in> \<A>"   and "C \<in> Y"   and "chamber C" shows "SimplicialComplex.maxsimp Y C"
abs_eq: fixes x :: "32 word" shows "uint8_of_uint32 (Abs_uint32 x) = Abs_uint8 (ucast x)"
isometry_iterates: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "isometry f" shows "isometry (f ^^ n)"
gext_lupd1: fixes vn :: "lname"   and v :: "val"   and s1 :: "st"   and s2 :: "st" shows "lupd(vn\<mapsto>v) s1\<le>|s2 = s1\<le>|s2"
subspace_tangent_space: fixes p :: "'c" shows "subspace (manifold_eucl.dest.tangent_space p)"
aux_bnd_mono: shows "N * m \<le> 4 ^ (m - 1) * N ^ m * m * m"
converges: fixes x :: "real" assumes "x \<in> {- R<..<R}" shows "summable (\<lambda>n. f n * real (Suc n) * x ^ n)"
simps_1: fixes a :: "nat \<Rightarrow> complex" shows "IFFT 0 a = a"
iso_inv_iso: fixes f :: "'d" assumes "D.iso f" shows "D.iso (D.inv f)"
Q: shows "Q = (\<lambda>((s, l, r), n). findnth_inv ly n (as, lm) (s, l, r) ires)"
in_closed_iff_infdist_zero: fixes A :: "'b set"   and x :: "'b" assumes "closed A"   and "A \<noteq> {}" shows "(x \<in> A) = (infdist x A = 0)"
zero_order_1: fixes x :: "'a multiset" shows "{#} \<subseteq># x"
unitary_def: fixes M :: "complex Matrix.mat" shows "unitary M \<equiv> M\<^sup>\<dagger> * M = 1\<^sub>m (dim_col M) \<and> M * M\<^sup>\<dagger> = 1\<^sub>m (dim_row M)"
that: fixes n :: "nat" assumes "\<forall>a\<in>sset (smap f (gtrace (sdrop n r) (gtarget (stake n r) v))). odd a" shows "thesis"
Abs_bit1'_def: fixes x :: "int" shows "Abs_bit1' x = Abs_bit1 (x mod int CARD('a bit1))"
pr_simps_3: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "cod \<pp>\<^sub>0[a, b] = b"
uncurry: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" assumes "P \<longrightarrow> Q \<longrightarrow> R" shows "P \<and> Q \<longrightarrow> R"
map_add_eval_right': fixes x :: "'c"   and h :: "'c \<Rightarrow> 'd option"   and h' :: "'c \<Rightarrow> 'd option" assumes "x \<notin> dom h"   and "h \<bottom> h'" shows "(h ++ h') x = h' x"
Scope_simp: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (SCOPE c, s) \<Rightarrow> t = (\<exists>s'. t = <s|s'> \<and> \<pi>: (c, <<>|s>) \<Rightarrow> s')"
sbintrunc_If: fixes n :: "nat"   and x :: "int" assumes "- 3 * 2 ^ n \<le> x \<and> x < 3 * 2 ^ n" shows "signed_take_bit n x = (if x < - (2 ^ n) then x + 2 * 2 ^ n else if 2 ^ n \<le> x then x - 2 * 2 ^ n else x)"
ideD_2: fixes a :: "('obj, 'obj) concrete_category.arr" assumes "ide a" shows "local.dom a = a"
absorb1: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "a \<sqinter> b = a"
distrib_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b) \<cdot> c = a \<cdot> c + b \<cdot> c"
list_trisect: fixes x :: "nat"   and lst :: "'b list" assumes "x < length lst" shows "[0..<length lst] = [0..<x] @ x # [Suc x..<length lst]"
coreflexive_symmetric: fixes x :: "'a" assumes "coreflexive x" shows "symmetric x"
Inf_prod_def: fixes A :: "('c \<times> 'd) set" shows "Inf A = (Inf (fst ` A), Inf (snd ` A))"
name_of_nat_def: fixes n :: "nat" shows "name_of_nat n \<equiv> Abs_name (Atom (Sort ''SyntaxN.name'' []) n)"
antisym: fixes y :: "'a"   and x :: "'a" assumes "y \<sqsubseteq> x"   and "x \<sqsubseteq> y"   and "x \<in> P"   and "y \<in> P" shows "x = y"
unit_disc_fix_moebius_preserve_poincare_between: fixes M :: "moebius"   and u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "unit_disc_fix M"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc" shows "poincare_between (moebius_pt M u) (moebius_pt M v) (moebius_pt M w) = poincare_between u v w"
equal_refl: fixes equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.equal equal" shows "equal x x = True"
comm_monoid_add_isom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "comm_monoid_add_isom hom \<equiv> monoid_add_isom hom"
sylow_eq: fixes G :: "('c, 'd) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" shows "sylow G p a m = (Group.group G \<and> sylow_axioms G p a m)"
rep_eq: fixes x :: "Linear_Aggregation_Algebras.unit"   and xa :: "Linear_Aggregation_Algebras.unit" shows "(x < xa) = (Linear_Aggregation_Algebras.unit.Rep_unit x < Linear_Aggregation_Algebras.unit.Rep_unit xa)"
ex1_poincare_line_general: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<noteq> v"   and "u \<noteq> inversion v" shows "\<exists>!l. is_poincare_line l \<and> u \<in> circline_set l \<and> v \<in> circline_set l"
Nmlize_red: fixes t :: "'a local.term" assumes "Ide t" shows "\<^bold>\<lfloor>t\<^bold>\<down>\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
sum_unique_nonzero: fixes i :: "'b"   and f :: "'b \<Rightarrow> real" assumes "\<And>j. \<lbrakk>j \<in> UNIV; j \<noteq> i\<rbrakk> \<Longrightarrow> f j = 0" shows "sum f UNIV = f i"
distinct_1: fixes x2 :: "'f" shows "Done \<noteq> Skip x2"
witnessed_def: fixes p :: "('b, 'c) fm"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list" shows "witnessed p a branch \<equiv> \<exists>i. (\<^bold>@ i p) at a in' set branch \<and> (\<^bold>\<diamond> Nom i) at a in' set branch"
hom_decomp_append_iff: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "hom_decomp (ps @ qs) = (hom_decomp ps \<and> hom_decomp qs)"
MGF_simult_Methd_lemma: fixes ms :: "(qtname \<times> sig) set"   and G :: "prog"   and A :: "state triple set" assumes "finite ms"   and "G,A \<union> (\<lambda>(C, sig). {Normal \<doteq>} In1l (Methd C sig)\<succ> {G\<rightarrow>}) ` ms|\<turnstile>(\<lambda>(C, sig). {Normal \<doteq>} In1l (body G C sig)\<succ> {G\<rightarrow>}) ` ms" shows "G,A|\<turnstile>(\<lambda>(C, sig). {Normal \<doteq>} In1l (Methd C sig)\<succ> {G\<rightarrow>}) ` ms"
prob_compl: fixes A :: "'b set"   and M :: "'b pmf" assumes "A \<in> measure_pmf.events M" shows "measure_pmf.prob M (space (measure_pmf M) - A) = 1 - measure_pmf.prob M A"
ok_def: fixes F :: "'b program"   and G :: "'b program" shows "F ok G \<equiv> Acts F \<subseteq> AllowedActs G \<and> Acts G \<subseteq> AllowedActs F"
degree_map_poly: fixes f :: "'c \<Rightarrow> 'd"   and p :: "'c poly" assumes "\<And>x. x \<noteq> (0::'c) \<Longrightarrow> f x \<noteq> (0::'d)" shows "degree (map_poly f p) = degree p"
OUTfromChCorrect_data4: shows "OUTfromChCorrect data4"
subspace_vec_eq: fixes X :: "(real, 'a) vec set" shows "vec.subspace X = subspace X"
subst_typ'_simulates_tsubst_gen': fixes pairs :: "(variable \<times> String.literal set) list"   and t :: "Core.term"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" assumes "distinct pairs"   and "tvs t \<subseteq> set pairs" shows "tsubst t \<rho> = subst_typ' (map (\<lambda>(x, y). ((x, y), \<rho> x y)) pairs) t"
N_3: shows "invar \<langle>l, (x, n), r\<rangle>"
card_ge_dim_independent: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "independent B"   and "dim V \<le> card B" shows "V \<subseteq> span B"
cases: fixes x :: "('c, 'd, 'varSort, 'var, 'e) qTerm" obtains xs :: "'varSort" and x :: "'var" where "xa__ = qVar xs x"   | delta :: "'e" and inp :: "'c \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qTerm option" and binp :: "'d \<Rightarrow> ('c, 'd, 'varSort, 'var, 'e) qAbs option" where "xa__ = qOp delta inp binp"
eq: fixes x :: "'a" assumes "x \<in> space M" shows "indicator A (T x) = indicator (T -` A \<inter> space M) x"
fv_def: fixes e :: "'d" shows "fv e = {v. atom v \<in> supp e}"
assoc_oo: fixes f :: "'c \<rightarrow> 'b"   and g :: "'d \<rightarrow> 'c"   and h :: "'a \<rightarrow> 'd" shows "f oo g oo h = (f oo g) oo h"
shift_mono: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I1 :: "nat set"   and I2 :: "nat set" shows "(local.shift cl n I1 \<subseteq> local.shift cl n I2) = (I1 \<subseteq> I2)"
gauss_int_norm_def: fixes z :: "gauss_int" shows "gauss_int_norm z = nat ((ReZ z)\<^sup>2 + (ImZ z)\<^sup>2)"
\<rr>_ide_simp: fixes f :: "'a" assumes "B.ide f" shows "B.\<rr> f = B.runit f"
f_def: shows "f \<equiv> \<lambda>x. ln (b * x + hb * x / ln x powr (1 + e)) powr (- e / 2)"
erlang_density_def: fixes k :: "nat"   and l :: "real"   and x :: "real" shows "erlang_density k l x = (if x < 0 then 0 else l ^ Suc k * x ^ k * exp (- l * x) / fact k)"
is_monomial_monomial_ordered: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "is_monomial p" obtains c :: "'c" and v :: "'a" where "c \<noteq> (0::'c)" and "punit.lc p = c" and "punit.lt p = v" and "p = monomial c v"
return_pmf_if: fixes P :: "bool"   and a :: "'a"   and b :: "'a" shows "return_pmf (if P then a else b) = (if P then return_pmf a else return_pmf b)"
some_in_eq: fixes A :: "'a set" shows "((SOME x. x \<in> A) \<in> A) = (A \<noteq> {})"
cs_recording_implies_snapshot: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "snd (cs (distributed_system.s channel trans send recv init t i) cid) = Recording"   and "channel cid = Some (p, q)" shows "ps (distributed_system.s channel trans send recv init t i) q \<noteq> None"
is_solution_def: fixes x :: "('d, 'e) vec"   and A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "is_solution x A b = (A *v x = b)"
I1: fixes S :: "'b" assumes "invariant S" shows "P S []"
neq_zero_conv_eSuc: fixes n :: "enat" shows "(n \<noteq> 0) = (\<exists>n'. n = eSuc n')"
int_pow_1: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> 1 = x"
ex_snap: shows "\<exists>i. \<not> ps (S t i) p \<noteq> None \<and> ps (S t (i + 1)) p \<noteq> None"
simps_1: fixes u0 :: "'a" shows "prep_wit_blue u0 NO_CYC = NO_CYC"
has_vector_derivative_transform: fixes x :: "real"   and S :: "real set"   and g :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a"   and f' :: "'a" assumes "x \<in> S"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"   and "(f has_vector_derivative f') (at x within S)" shows "(g has_vector_derivative f') (at x within S)"
P1_security_inf_the: fixes m1 :: "bool \<times> bool"   and m2 :: "bool" shows "etp.OT_12.perfect_sec_P1 m1 m2"
len_le_n: fixes i :: "nat" shows "length (worst_chain (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) (\<lambda>ys xs. length xs < length ys) i) \<le> n"
ReachableStepFirst: fixes cfg :: "('p, 'v, 's) configuration"   and cfg' :: "('p, 'v, 's) configuration" assumes "reachable cfg cfg'" shows "cfg = cfg' \<or> (\<exists>cfg1 msg p. cfg \<turnstile> msg \<mapsto> cfg1 \<and> enabled cfg msg \<and> isReceiverOf p msg \<and> reachable cfg1 cfg')"
span_insert_idI: fixes r :: "'c"   and B :: "'c set" assumes "r \<in> ideal B" shows "ideal (insert r B) = ideal B"
queue_invar_simps_2: fixes t :: "('b, 'c) SkewBinomialTree"   and q :: "('b, 'c) SkewBinomialTree list" shows "queue_invar (t # q) = (tree_invar t \<and> queue_invar q)"
release_all_shared_exchange: fixes sb :: "'b memref list"   and S' :: "nat set"   and S :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "\<forall>a\<in>all_shared sb. (a \<in> S') = (a \<in> S)" shows "release sb S' \<R> = release sb S \<R>"
wp_def: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "local.wp c Q s \<equiv> \<exists>t. \<pi>: (c, s) \<Rightarrow> t \<and> Q t"
finfun_upd_apply: fixes f :: "'b \<Rightarrow>f 'a"   and a :: "'b"   and b :: "'a"   and a' :: "'b" shows "f(a $:= b) $ a' = (if a = a' then b else f $ a')"
CP: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v] \<Longrightarrow> [\<psi> in v]" shows "[\<phi> \<^bold>\<rightarrow> \<psi> in v]"
order_trans_rules_26: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a = b"   and "b \<le> c" shows "a \<le> c"
range_ereal: shows "range ereal = UNIV - {\<infinity>, - \<infinity>}"
strip_widen_acom: fixes c' :: "'a acom"   and c :: "'a acom" assumes "strip c' = strip c" shows "strip (c \<nabla>\<^sub>c c') = strip c"
addfeqstut: fixes t :: "nat \<Rightarrow> 'b" shows "stutstep (first t ## t) 0"
inj_map_inv_map: fixes \<pi> :: "'d \<Rightarrow> 'c option" shows "inj_map (inv_map \<pi>)"
good_fun_def: fixes f :: "(val \<times> val) fset"   and x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> bval) list" shows "good_fun f x e \<rho> \<equiv> ffold (good_entry x e \<rho>) True (good_prod |`| f)"
Bossy_CH_range: fixes X :: "'a set"   and X' :: "'a set" assumes "X \<subseteq> Bossy.CH X'" shows "X \<subseteq> set [d1h1, d1h3, d2h1, d2h3, d3h1, d3h3]"
constrains_Un: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A co A'"   and "F \<in> B co B'" shows "F \<in> A \<union> B co A' \<union> B'"
simps_1: fixes k :: "nat" shows "\<Z> k 0 = {}"
in_dom_in_ran_m: fixes i :: "'a"   and N :: "('a, 'b) fmap" assumes "i \<in># dom_m N" shows "the (fmlookup N i) \<in># ran_m N"
exhaust: fixes y :: "'c iarray" obtains x :: "'c list" where "y = IArray x"
coprime_0_2: 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" shows "comm_monoid_mult.coprime times one zero p = dvd.dvd times p one"
bounded_iff: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" shows "(a \<preceq> linordered_monoid.min b c) = (a \<preceq> b \<and> a \<preceq> c)"
r_inv: fixes x :: "cltn2" assumes "x \<in> carrier \<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "x \<otimes>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> inv\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x = \<one>\<^bsub>\<lparr>carrier = Collect is_K2_isometry, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub>"
le_bot: fixes a :: "'a" assumes "a \<le> \<bottom>" shows "a = \<bottom>"