hcju/isabelleps huggingface dataset:
distributed_unique: fixes S :: "'d measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> ennreal"   and Py :: "'d \<Rightarrow> ennreal" assumes "distributed M S X Px"   and "distributed M S X Py" shows "AE x in S. Px x = Py x"
case_eq_if: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and vertex :: "'b vertex" shows "(case vertex of SOURCE \<Rightarrow> f1 | vertex.SINK \<Rightarrow> f2 | \<langle>x\<rangle> \<Rightarrow> f3 x) = (if vertex = SOURCE then f1 else if vertex = vertex.SINK then f2 else f3 (vertex.inner vertex))"
infdist_singleton: fixes x :: "'c"   and y :: "'c" shows "infdist x {y} = dist x y"
hyps: assumes "M \<noteq> {#}"   and "M \<in> multisets A" shows "mulex_on P A {#} M"
per2__sams: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> E"   and "E \<noteq> F"   and "Per A B C"   and "Per D E F" shows "SAMS A B C D E F"
nn_integral_const: fixes M :: "'b measure"   and c :: "ennreal" shows "\<integral>\<^sup>+ x. c \<partial>M = c * emeasure M (space M)"
insert_length': fixes l :: "'c list"   and i :: "nat"   and x :: "'c" shows "length (list_insert l i x) = length (x # l)"
a: shows "map_of (map (\<lambda>x. (x, f x)) (a # xs)) = map_of (map (\<lambda>x. (x, f x)) xs)(a \<mapsto> f a)"
sum_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and K :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>i. i \<in> K \<Longrightarrow> less_eq (f i) (g i)" shows "less_eq (comm_monoid_add.sum plus zero f K) (comm_monoid_add.sum plus zero g K)"
f0_1: shows "x * f $ 0 = (1::'a)"
assms_3: fixes x :: "complex" assumes "x \<in> X" shows "algebraic x"
absorb2: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "max a b = b"
assms_4: shows "local.my_U q \<le> local.my_U p"
gfp_fixpoint: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "f (\<mu> f) = \<mu> f"
size_multiset_add_mset: fixes f :: "'k \<Rightarrow> nat"   and a :: "'k"   and M :: "'k multiset" shows "size_multiset f (add_mset a M) = Suc (f a) + size_multiset f M"
piecewise_continuous_on_insert_rightI: fixes a :: "'a"   and b :: "'a"   and I :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "piecewise_continuous_on a b I f" shows "piecewise_continuous_on a b (insert b I) f"
minimal_unsat_state_core_def: fixes s :: "('a, 'b) state" shows "minimal_unsat_state_core s = (unsat_state_core s \<and> (distinct_indices_state s \<longrightarrow> subsets_sat_core s))"
decr_sorted: fixes xs :: "'a list" assumes "decr xs" shows "sorted (rev xs)"
lpfpc_pfp: fixes f :: "'b option acom \<Rightarrow> 'b option acom"   and c0 :: "com"   and c :: "'b option acom" assumes "lpfp\<^sub>c f c0 = Some c" shows "f c \<sqsubseteq> c"
is_derivation_def: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c" shows "src.manifold_eucl.dest.is_derivation X p = (linear_on src.manifold_eucl.dest.diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X \<and> (\<forall>f g. f \<in> src.manifold_eucl.dest.diff_fun_space \<longrightarrow> g \<in> src.manifold_eucl.dest.diff_fun_space \<longrightarrow> X (f * g) = f p * X g + g p * X f))"
induct: fixes P :: "('b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres) \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres"   and a1 :: "'b"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat"   and a5 :: "nat" assumes "\<And>f m n. P f m n 0 0 0"   and "\<And>f m n k. P f m n k n n \<Longrightarrow> P f m n (Suc k) 0 0"   and "\<And>f m n k i. P f m n k i n \<Longrightarrow> P f m n k (Suc i) 0"   and "\<And>f m n k i j. P f m n k i j \<Longrightarrow> P f m n k i (Suc j)" shows "P a0 a1 a2 a3 a4 a5"
inj_on_finite: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "inj_on f A"   and "order.greater_eq B (f ` A)"   and "finite B" shows "finite A"
infdist_pos_not_in_closed: fixes S :: "'c set"   and x :: "'c" assumes "closed S"   and "S \<noteq> {}"   and "x \<notin> S" shows "0 < infdist x S"
simps_3: fixes x1 :: "'b"   and x2 :: "'b"   and xs :: "'b list" shows "changes (x1 # x2 # xs) = (if x1 * x2 < (0::'b) then 1 + changes (x2 # xs) else if x2 = (0::'b) then changes (x1 # xs) else changes (x2 # xs))"
while: fixes b :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val" shows "eval\<^sub>w_simple_abv (While b c, mem) (Stmt.If b (c ;; While b c) Stop, mem)"
Chn_triangle_eq_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and \<eta> :: "'b arrow_of_spans_data"   and \<epsilon> :: "'b arrow_of_spans_data" assumes "adjunction_data_in_span_bicategory C prj0 prj1 f g \<eta> \<epsilon>" shows "Chn (span_vertical_category.vcomp C (bicategory.lunit (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_bicategory.unit C prj0 prj1) (span_vertical_category.src C) (span_vertical_category.trg C) f) (span_vertical_category.vcomp C (span_bicategory.hcomp C prj0 prj1 \<epsilon> f) (span_vertical_category.vcomp C (bicategory.\<a>' (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_vertical_category.src C) (span_vertical_category.trg C) f g f) (span_vertical_category.vcomp C (span_bicategory.hcomp C prj0 prj1 f \<eta>) (category.inv (span_vertical_category.vcomp C) (bicategory.runit (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1) (span_bicategory.assoc C prj0 prj1) (span_bicategory.unit C prj0 prj1) (span_vertical_category.src C) f)))))) = C (prj0 (Leg0 (Dom g)) (Leg1 (Dom f))) (C (Chn \<eta>) (Leg0 (Dom f)))"
cont: shows "continuous_on (set_of X \<times> set_of Y) (\<lambda>(x, y). x * y)"
filter_hd_P_rep_indep: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and xs :: "'d list" assumes "\<forall>x. P x x"   and "\<forall>a b. P x a \<longrightarrow> P a b \<longrightarrow> P x b"   and "filter (P x) xs \<noteq> []" shows "hd (filter (P (hd (filter (P x) xs))) xs) = hd (filter (P x) xs)"
root_rat_ceiling_def: fixes p :: "nat"   and x :: "rat" shows "root_rat_ceiling p x \<equiv> - root_rat_floor p (- x)"
lea_in_angle: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and C :: "'p" assumes "A B P LeA A B C"   and "A B OS C P" shows "P InAngle A B C"
isolate_variable_sparse_monom_mult: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a"   and q :: "'a mpoly"   and v :: "nat"   and n :: "nat" shows "isolate_variable_sparse (MPoly_Type.monom m a * q) v n = (if lookup m v \<le> n then MPoly_Type.monom (Poly_Mapping.update v 0 m) a * isolate_variable_sparse q v (n - lookup m v) else 0)"
transitive: fixes P :: "ccs"   and Q :: "ccs"   and R :: "ccs" assumes "P \<approx> Q"   and "Q \<approx> R" shows "P \<approx> R"
m_inv_closed: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "subgroup H G"   and "x \<in> H" shows "inv\<^bsub>G\<^esub> x \<in> H"
countable_SIGMA: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (A i)" shows "countable (Sigma I A)"
less_le: fixes x :: "'c word"   and y :: "'c word" shows "(x <s y) = (x \<le>s y \<and> x \<noteq> y)"
ListMem_iff: fixes x :: "'b"   and xs :: "'b list" shows "ListMem x xs = (x \<in> ran\<^sub>l xs)"
conv_dist_inf: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y)\<^sup>T = x\<^sup>T \<sqinter> y\<^sup>T"
eventually_at_topological: fixes "open" :: "'e set \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool"   and a :: "'e"   and s :: "'e set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
valid_strategy_updates: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and w0 :: "'a" assumes "G'.strategy p \<sigma>"   and "v0 \<rightarrow>\<^bsub>G'\<^esub> w0" shows "G'.strategy p (\<sigma>(v0 := w0))"
left_idem: fixes a :: "'b"   and b :: "'b" shows "max a (max a b) = max a b"
simps_3: fixes a :: "'b"   and b :: "'b" shows "Num a + Num b = Num (a + b)"
card_order_infinite_isLimOrd: fixes r :: "('a \<times> 'a) set" assumes "Card_order r"   and "infinite (Field r)" shows "isLimOrd r"
BAD_iff: fixes f :: "nat \<Rightarrow> 'a list"   and P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" shows "(f \<in> BAD P) = ((\<forall>i. f i \<in> lists A) \<and> bad P f)"
infsetsum_cmult_right: fixes c :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "c \<noteq> (0::'c) \<Longrightarrow> f abs_summable_on A" shows "(\<Sum>\<^sub>ax\<in>A. c * f x) = c * infsetsum f A"
lfp_fusion: fixes g :: "'d \<rightarrow> 'c"   and f :: "'d \<rightarrow> 'd"   and h :: "'c \<rightarrow> 'c" assumes "g\<cdot>\<bottom> = \<bottom>"   and "g oo f = h oo g" shows "g\<cdot>(fix\<cdot>f) = fix\<cdot>h"
simps_1: fixes l :: "'a \<Rightarrow> 'a"   and r :: "'a \<Rightarrow> 'a" shows "traverse_dir l r L = l"
connectedI: fixes g :: "'b ugraph" assumes "\<And>u v. \<lbrakk>u \<in> nodes g; v \<in> nodes g\<rbrakk> \<Longrightarrow> (u, v) \<in> (edges g)\<^sup>*" shows "Undirected_Graph.connected g"
abs_conv_abscissa_completely_multiplicative_log_deriv: fixes f :: "'b fds" assumes "completely_multiplicative_function (fds_nth f)"   and "fds_nth f 1 \<noteq> (0::'b)" shows "abs_conv_abscissa (fds_deriv f / f) \<le> abs_conv_abscissa f"
some: fixes a :: "'a"   and b :: "'b" assumes "fmlookup m x = Some a"   and "fmlookup n x = Some b"   and "P a b" shows "thesis"
continuous_at_Inf_mono: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "mono f"   and "continuous (at_right (Inf S)) f"   and "S \<noteq> {}"   and "bdd_below S" shows "f (Inf S) = Inf (f ` S)"
point_chainI: fixes M :: "('e \<Rightarrow> 'f) set"   and x :: "'e" assumes "is_chain M" shows "is_chain ((\<lambda>f. f x) ` M)"
Red_I_to_Inf: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and N :: "'a set" assumes "calculus Bot Inf entails Red_I Red_F" shows "Red_I N \<subseteq> Inf"
bind_ask: fixes f :: "'b \<Rightarrow> 'm"   and g :: "'a list \<Rightarrow> 'm" shows "bind (ask f) g = ask (\<lambda>r. bind (f r) g)"
inf_assoc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
sel: fixes x :: "unit + 'c \<times> 'c list_R1.list_R1" shows "list_R1.unR (list_R1.list_R1 x) = x"
cmp_simps_3: shows "src cmp = src \<rho>\<sigma>.tab"
cblinfun_apply_inject: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and y :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "((*\<^sub>V) x = (*\<^sub>V) y) = (x = y)"
parts_used: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> parts H"
length_update_until: fixes pos :: "\<I>"   and I :: "bool"   and rel1 :: "'a option list set"   and rel2 :: "'a option list set"   and nt :: "nat"   and aux :: "(nat \<times> 'a option list set \<times> 'a option list set) list" shows "length (update_until pos I rel1 rel2 nt aux) = Suc (length aux)"
element_ptr_casts_commute3: fixes node_ptr :: "(_) node_ptr" assumes "is_element_ptr_kind node_ptr" shows "cast (the (cast node_ptr)) = node_ptr"
set_up: shows "pgallery (C0 # Cs @ [C])"
\<alpha>01: shows "\<alpha> # \<alpha>s = \<alpha>0 # \<alpha>1 # \<alpha>s'"
p': shows "prec c1 (\<lambda>s. \<down>\<^sub>t (c2, s)) s = p'"
assms: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a"   and y :: "'a" assumes "js = xs @ ys"   and "x \<in> set xs"   and "y \<in> set ys" shows "x \<le> y"
normalize_sort_code: fixes cs :: "(String.literal \<times> String.literal) list"   and s :: "String.literal set" shows "normalize_sort (set cs) s = exenormalize_sort cs s"
inner_simps_5: fixes r :: "real"   and x :: "'b"   and y :: "'b" shows "r *\<^sub>R x \<bullet> y = r * (x \<bullet> y)"
comp: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'a" shows "(-`) f \<circ> (-`) g = (-`) (g \<circ> f)"
H: shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"
assms: shows "(a, h') = alloc (map f [0..<n]) h"
index_add_vec_2: fixes v\<^sub>1 :: "'c vec"   and v\<^sub>2 :: "'c vec" shows "dim_vec (v\<^sub>1 + v\<^sub>2) = dim_vec v\<^sub>2"
interval_lowerbound_def: fixes s :: "'b set" shows "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x\<in>s. x \<bullet> i) *\<^sub>R i)"
case1: assumes "0 \<le> x"   and "0 \<le> y" shows "- 4611686018427387904 \<le> x * y"
singleton_inject: fixes a :: "'x"   and b :: "'x" assumes "{a} = {b}" shows "a = b"
Times_empty: fixes A :: "'b set"   and B :: "'c set" shows "(A \<times> B = {}) = (A = {} \<or> B = {})"
igSubstAllIPresIGWlsAll_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSubstAllIPresIGWlsAll MOD \<equiv> igSubstIPresIGWls MOD \<and> igSubstAbsIPresIGWlsAbs MOD"
once_none_nxt_always_none: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and t :: "(String.literal \<times> value list) stream" shows "alw (nxt (state_eq None)) (make_full_observation e None r p t)"
finvimage_vunion_right: fixes r :: "V"   and A :: "V"   and B :: "V" shows "r -`\<^sub>\<bullet> (A \<union>\<^sub>\<circ> B) = r -`\<^sub>\<bullet> A \<union>\<^sub>\<circ> r -`\<^sub>\<bullet> B"
iwlsFSwSTR_qSwapAll_igSwapAll_qInitAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm"   and qA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs" assumes "iwlsFSwSTR MOD" shows "qInit MOD (qX #[[z1 \<and> z2]]_zs) = igSwap MOD zs z1 z2 (qInit MOD qX) \<and> qInitAbs MOD (qA $[[z1 \<and> z2]]_zs) = igSwapAbs MOD zs z1 z2 (qInitAbs MOD qA)"
reduction_P1: fixes R1_12_0T :: "nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'v_OT121 spmf"   and n :: "nat"   and M :: "bool \<times> bool \<times> bool \<times> bool"   and c0 :: "bool"   and c1 :: "bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf" shows "\<bar>spmf (R1_14 R1_12_0T n M (c0, c1) \<bind> D) True - spmf (funct_OT_14 M (c0, c1) \<bind> (\<lambda>(out1, out2). S1_14 n M out1 \<bind> D)) True\<bar> \<le> 3 * adv_OT12 n"
integral_nonneg_eq_0_iff_AE: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f"   and "AE x in M. 0 \<le> f x" shows "(integral\<^sup>L M f = 0) = (AE x in M. f x = 0)"
a2: fixes \<theta> :: "('b, 'a) trm" assumes "\<theta> \<in> range args" shows "(sterm_sem I \<theta> has_derivative frechet I \<theta> \<nu>) (at \<nu>)"
lcm_proj1_if_dvd: fixes b :: "'a"   and a :: "'a" assumes "b dvd a" shows "lcm a b = normalize a"
that: fixes s :: "complex set" assumes "open s"   and "0 \<in> s"   and "f holomorphic_on s"   and "\<And>z. z \<in> s \<Longrightarrow> f z = eval_fps F z" shows "thesis"
fun_type_inv_wf: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and t :: "('a, 'b) Term.term"   and f :: "'a"   and T :: "('a, 'c) Term.term list" assumes "typed_model arity public Ana \<Gamma>"   and "\<Gamma> t = Fun f T"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t" shows "arity f = length T"
w_simps_5: shows "local.dom TTfgh_TfTgh.chine = TTfgh_TfTgh.chine"
xs: shows "set xs \<subseteq> outs_\<I> \<I> \<times> UNIV"
prob_space_distr: fixes f :: "'b \<Rightarrow> 'c"   and M' :: "'c measure" assumes "X.random_variable M' f" shows "prob_space (distr (distr M MX X) M' f)"
not_ffs_2: fixes x :: "f" shows "(x \<noteq> Aodv_Basic.inv) = (x = val)"
less_add_Suc1: fixes i :: "nat"   and m :: "nat" shows "i < Suc (i + m)"
hyps_3: shows "almost_full_on (adds) {x. supp_fun x \<subseteq> V}"
A: shows "{\<cdot> p } = {\<cdot> q }"
nret: shows "\<forall>n'. \<pi>' n' \<noteq> local.return"
inv_induced_id_sum_list: fixes ss :: "'w list" assumes "ss \<in> lists S" shows "the_inv_into (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>) (GroupByPresentationInducedFun.induced_hom S (map (charpair S) ` R) id) (sum_list ss) = BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>) (FreeGroup S) `` {Abs_freelist ss})"
xs: shows "filter (\<lambda>(i, Ail). Ail \<noteq> (0::'a)) (map (\<lambda>i. (i, A $$ (i, l))) [Suc l..<dim_row A]) \<noteq> []"
set_remove1_eq: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs" shows "set (remove1 x xs) = set xs - {x}"
assms_4: shows "hs = fst (add_indices (compl gs bs (ps -- sps) sps data) data)"
step: fixes i :: "nat" shows "(S i, S (i + Suc n)) \<in> A ^^ Suc n"
l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel"
map_prod_simp: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and a :: "'g"   and b :: "'h" shows "map_prod f g (a, b) = (f a, g b)"
le_inf_iff: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "less_eq x (inf y z) = (less_eq x y \<and> less_eq x z)"
f_inv_f: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "surjective f" shows "f (inv f x) = x"
bind_bind: fixes m :: "'d\<cdot>'b"   and f :: "'d \<rightarrow> 'c\<cdot>'b"   and g :: "'c \<rightarrow> 'a\<cdot>'b" shows "m \<bind> f \<bind> g = m \<bind> (\<Lambda> x. f\<cdot>x \<bind> g)"
ins_ins2_3: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real"   and n :: "nat"   and d :: "nat"   and x :: "'b"   and t :: "'b tree"   and t' :: "'b tree"   and n' :: "nat"   and h' :: "nat" assumes "RBTi2 TYPE('b) bal_i e"   and "RBTi2.ins2 bal_i n d x t = Unbal2 t' n' h'" shows "RBTi1.ins bal_i n d x t = Unbal t' \<and> n' = size t' \<and> h' = height t'"
map\<^sub>0_def: fixes a :: "'c" shows "G.map\<^sub>0 a \<equiv> src\<^sub>D (G a)"
d_IN_cmult: fixes c :: "ennreal"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "d_IN (\<lambda>e. c * f e) x = c * d_IN f x"
elims: fixes x :: "Vars.var"   and xa :: "Vars.var"   and xb :: "Vars.var"   and y :: "Vars.var" assumes "x[xa::v=xb] = y" obtains xc :: "Vars.var" and ya :: "Vars.var" and z :: "Vars.var" where "x = xc" and "xa = ya" and "xb = z" and "y = (if xc = ya then z else xc)"
dg_Set_Hom_vsubset_dg_Par_Hom: fixes a :: "V"   and \<alpha> :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> dg_Set \<alpha>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> dg_Set \<alpha>\<lparr>Obj\<rparr>" shows "Hom (dg_Set \<alpha>) a b \<subseteq>\<^sub>\<circ> Hom (dg_Par \<alpha>) a b"
R: fixes c :: "'a sequent"   and F :: "'a"   and Fs :: "'a form list"   and ps :: "'b list" assumes "c \<equiv>  \<Empt> \<Rightarrow>* Compound F Fs"   and "ps \<noteq> []" shows "(ps, c) \<in> upRules"
angle_def: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "<A>_v \<equiv> A \<and> \<not> unchanged v"
hf_induct_ax: fixes P :: "hf \<Rightarrow> bool"   and x :: "hf" assumes "P 0"   and "\<forall>x. P x \<longrightarrow> (\<forall>y. P y \<longrightarrow> P (x \<triangleleft> y))" shows "P x"
E25: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<diamond>\<langle>A\<rangle>_v \<longrightarrow> \<diamond>A"
safe_distance_3r_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and \<delta> :: "real" shows "safe_distance_3r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = v\<^sub>e * \<delta> - v\<^sub>e\<^sup>2 / 2 / a\<^sub>e - v\<^sub>o * \<delta> - 1 / 2 * a\<^sub>o * \<delta>\<^sup>2"
hyps_2: shows "\<Gamma> \<turnstile> cfg'' \<rightarrow>\<^sup>* cfg\<^sub>2"
ot_base_def: fixes \<G> :: "'d cyclic_group" shows "ot_base \<G> \<equiv> finite (carrier \<G>) \<and> 0 < order \<G> \<and> prime (order \<G>)"
subset_entailed: fixes N2 :: "'g set"   and N1 :: "'g set" assumes "N2 \<subseteq> N1" shows "N1 \<Turnstile>G N2"
union_def: shows "DList_Set.union = DList_Set.fold DList_Set.insert"
simps_4: fixes MI :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and e :: "nat \<Rightarrow> U"   and f :: "form"   and g :: "form" shows "FEval MI e (FDisj f g) = (FEval MI e f \<or> FEval MI e g)"
ivl_disj_un_two_8: fixes l :: "'c"   and m :: "'c"   and u :: "'c" assumes "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m<..u} = {l..u}"
transpose_mat_of_rows: fixes n :: "nat"   and vs :: "'b vec list" shows "(mat_of_rows n vs)\<^sup>T = mat_of_cols n vs"
reachable_def: fixes xs :: "'a set" shows "reachable xs = {(x, y). y \<in> set (succs x)}\<^sup>* `` xs"
points_index_ps_nin: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and ps :: "'b set" assumes "incidence_system point_set block_collection"   and "\<not> ps \<subseteq> point_set" shows "block_collection index ps = 0"
step': fixes vsa :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option"   and obsa :: "('a, 'b) obs_event action llist"   and aa :: "'c" assumes "X vsa obsa aa" shows "obsa = LNil \<or> (\<exists>obs' obs''. obsa = lappend obs' obs'' \<and> obs' \<noteq> LNil \<and> ta_seq_consist P vsa obs' \<and> (lfinite obs' \<longrightarrow> (\<exists>a. X (mrw_values P vsa (list_of obs')) obs'' a) \<or> ta_seq_consist P (mrw_values P vsa (list_of obs')) obs''))"
distinct_2: fixes x2 :: "final_decision" shows "Decision x2 \<noteq> Undecided"
intros: fixes C :: "'a literal multiset"   and n :: "nat"   and A :: "'a"   and D :: "'a literal multiset" shows "unord_resolve (C + replicate_mset (Suc n) (Pos A)) (add_mset (Neg A) D) (C + D)"
assms_2: shows "strict_subtuple (\<le>) \<pi> \<xi>"
af_identity: fixes x :: "'d af" shows "Applicative.ap (pure id) x = x"
assms_2: shows "\<forall>c\<in>\<X>. \<not> (\<exists>d\<le>k c. u c = real d)"
simps_4: fixes x :: "quat"   and y :: "quat" shows "Im3 (x - y) = Im3 x - Im3 y"
lm057: fixes pair :: "'c \<times> 'd" shows "fst pair = snd (flip pair) \<and> snd pair = fst (flip pair)"
Keys_to_list_Nil: shows "punit.Keys_to_list [] = []"