hcju/isabelleps huggingface dataset:
obtain_fresh: fixes V :: "'var set"   and XS :: "('a, 'b, 'varSort, 'var, 'c) qTerm set set"   and AS :: "('a, 'b, 'varSort, 'var, 'c) qAbs set set"   and Rho :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option) set"   and xs :: "'varSort" assumes "|V| <o |UNIV| \<or> finite V"   and "|XS| <o |UNIV| \<or> finite XS"   and "|AS| <o |UNIV| \<or> finite AS"   and "|Rho| <o |UNIV| \<or> finite Rho"   and "Ball XS good"   and "Ball AS goodAbs"   and "Ball Rho goodEnv" shows "\<exists>z. z \<notin> V \<and> (\<forall>X\<in>XS. fresh xs z X) \<and> (\<forall>A\<in>AS. freshAbs xs z A) \<and> (\<forall>rho\<in>Rho. freshEnv xs z rho)"
domA_append: fixes a :: "('a \<times> 'b) list"   and b :: "('a \<times> 'b) list" shows "domA (a @ b) = domA a \<union> domA b"
mkarr_memb_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "t \<in> f"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "monoidal_language.Arr C t"
weakBisimCoinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>\<^sup>^<(X \<union> weakBisim)> Q"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> (Q, P) \<in> X" shows "P \<approx> Q"
top_left_normalize: fixes a :: "'a" shows "gcd (0::'a) a = normalize a"
Teq: shows "T = \<Union> p \<union> \<Union> (r1 - p) \<inter> \<Union> q"
in_zmset_notin_mset_pos: fixes x :: "'a"   and M :: "'a zmultiset" assumes "x \<in>#\<^sub>z M"   and "x \<notin># mset_pos M" shows "x \<in># mset_neg M"
simps_17: fixes h :: "hash"   and t' :: "Syntax.term"   and x :: "Syntax.var" shows "Hash h[t' / x] = Hash h"
s_def: shows "s = (if even (degree p) then 1::'a else - (1::'a))"
map_rbt_sorted: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and t :: "('a, 'c) rbt" shows "ord.rbt_sorted less (RBT_Impl.map f t) = ord.rbt_sorted less t"
diff_between_chartsE: fixes d1 :: "('a, 'c) chart"   and d2 :: "('b, 'b) chart"   and y :: "'a" assumes "d1 \<in> src.atlas"   and "d2 \<in> src.manifold_eucl.dest.atlas"   and "y \<in> domain d1"   and "y \<in> src.carrier"   and "f y \<in> domain d2" obtains X :: "'c set" where "k-smooth_on X (apply_chart d2 \<circ> f \<circ> inv_chart d1)" and "apply_chart d1 y \<in> X" and "open X" and "X = codomain d1 \<inter> inv_chart d1 -` (src.carrier \<inter> f -` domain d2)"
simple_path_subpath_eq: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'a" shows "simple_path (subpath u v g) = (path (subpath u v g) \<and> u \<noteq> v \<and> (\<forall>x y. x \<in> closed_segment u v \<and> y \<in> closed_segment u v \<and> g x = g y \<longrightarrow> x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u))"
weak_GB_is_strong_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "G \<subseteq> punit.dgrad_p_set d m"   and "\<And>f. \<lbrakk>f \<in> local.punit.pmdl G; f \<in> punit.dgrad_p_set d m\<rbrakk> \<Longrightarrow> (punit.red G)\<^sup>*\<^sup>* f 0" shows "punit.is_Groebner_basis G"
bvxorxor_elim: fixes emBits :: "nat"   and a :: "bit list"   and b :: "bit list" assumes "roundup emBits 8 * 8 - emBits \<le> length a"   and "length a = length b" shows "maskedDB_zero (bvxor (maskedDB_zero (bvxor a b) emBits) b) emBits = bv_prepend (roundup emBits 8 * 8 - emBits) \<zero> (drop (roundup emBits 8 * 8 - emBits) a)"
order_iff_strict: fixes x :: "'o"   and b :: "'a"   and a :: "'a" shows "le x b a = (lt x b a \<or> a = b)"
fmadd_fmupd: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f"   and a :: "'e"   and b :: "'f"   and n :: "'e \<rightharpoonup>\<^sub>f 'f" shows "m ++\<^sub>f n(a \<mapsto>\<^sub>f b) = (m ++\<^sub>f n)(a \<mapsto>\<^sub>f b)"
Infinitesimal_add_approx: fixes y :: "'a star"   and x :: "'a star"   and z :: "'a star" assumes "y \<in> Infinitesimal"   and "x + y = z" shows "x \<approx> z"
is_heap_children_larger: fixes t :: "('c, 'd) bintree" assumes "is_heap_list (priority t) (children t)" shows "\<forall>x\<in>set (children t). priority t \<le> priority x"
conv_radius_conv_Sup: fixes f :: "nat \<Rightarrow> 'a" shows "conv_radius f = Sup {r. \<forall>z. ereal (norm z) < r \<longrightarrow> summable (\<lambda>n. f n * z ^ n)}"
one_dim_apply_is_times: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "A o\<^sub>C\<^sub>L B = A * B"
project_constrains_D: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "F \<squnion> project h UNIV G \<in> A co B" shows "extend h F \<squnion> G \<in> extend_set h A co extend_set h B"
ball_cong_simp: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x \<in> B =simp=> P x = Q x" shows "(\<forall>x\<in>A. P x) = (\<forall>x\<in>B. Q x)"
G_def: shows "G = (\<lambda>x w. \<forall>xa. (P xa m\<rightarrow> xa x) w)"
hyps_2: assumes "H = {}" shows "Pf \<lbrakk>\<guillemotleft>A IMP B\<guillemotright>\<rbrakk>e"
idiff_eint_0_right: fixes n :: "eint" shows "n - eint 0 = n"
simps_2: fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" shows "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
lookup_minus: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and g :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" shows "lookup (f - g) k = lookup f k - lookup g k"
simps_5: fixes f1 :: "fdecl \<Rightarrow> 'a"   and f2 :: "mdecl \<Rightarrow> 'a"   and x1 :: "fdecl" shows "(case fdecl x1 of fdecl x \<Rightarrow> f1 x | mdecl x \<Rightarrow> f2 x) = f1 x1"
cobounded1: fixes a :: "'c"   and b :: "'c" shows "inf a b \<le> a"
tensor0_def: fixes d :: "nat list" shows "tensor0 d = tensor_from_vec d (vec0 (prod_list d))"
distinct_189: fixes x7 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Abs x7 \<noteq> Power x151 x152"
alphas_2: fixes bs :: "atom set"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'a" shows "(bs, x) \<approx>res R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y)"
h1: shows "lose t = [True] \<or> lose t = [False]"
Cauchy_def: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'd" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
bypass': fixes z :: "'v \<times> 'v" assumes "z \<in> set q" shows "z \<notin> RF S"
deltafun_def: fixes b :: "'b"   and a :: "'a" shows "b \<delta> a = (\<lambda>x. if x = a then b else (0::'b))"
subgroup_conjugation_is_surj0: fixes g :: "'a"   and H :: "'a set" assumes "g \<in> carrier G"   and "H \<subseteq> carrier G" shows "g <# (inv g <# H #> g) #> inv g = H"
non_inf_delta_gt: fixes \<delta> :: "'c" assumes "(0::'c) < \<delta>" shows "non_inf {(a, b). delta_gt \<delta> a b}"
ordinal_rec_oLimit: fixes s :: "ordinal \<Rightarrow> ordinal \<Rightarrow> ordinal"   and z :: "ordinal"   and f :: "nat \<Rightarrow> ordinal" assumes "\<forall>p x. x \<le> s p x" shows "ordinal_rec z s (oLimit f) = oLimit (\<lambda>n. ordinal_rec z s (f n))"
subdiv_ends_2: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d" assumes "subdiv_step G rev_G H rev_H u v w uv uw vw" shows "head G uv = v"
widen_rules_4: fixes P :: "'a prog"   and h :: "heap"   and loc :: "val list"   and LT :: "ty err list"   and LT' :: "ty err list" assumes "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT"   and "P \<turnstile> LT [\<le>\<^sub>\<top>] LT'" shows "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT'"
enabled_iff: fixes s :: "'a cfg"   and \<omega> :: "'a cfg stream" shows "MDP.MC.enabled s \<omega> = (MDP.MC.enabled (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K_cfg s))"
ex_un_\<xi>': shows "\<exists>!\<xi>'. adjoint_equivalence_in_bicategory (\<cdot>\<^sub>S) (\<star>\<^sub>S) S.\<a> S.\<i> S.src S.trg (S.UP f) (S.UP g) (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f)) \<xi>'"
pgeq: fixes t :: "real" assumes "t \<in> {0..1}" shows "p (g t) = f t"
inj_on_map_add_Un: fixes m :: "'a \<Rightarrow> 'b option"   and m' :: "'a \<Rightarrow> 'b option"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "inj_on m (dom m)"   and "inj_on m' (dom m')"   and "m \<subseteq>\<^sub>m Some \<circ> f"   and "m' \<subseteq>\<^sub>m Some \<circ> f"   and "inj_on f (dom m' \<union> dom m)"   and "A = dom m'"   and "B = dom m" shows "inj_on (m ++ m') (A \<union> B)"
simps_10: fixes y :: "name"   and I :: "('d, 'e, 'f) input" shows "guarded' (\<nu>yI) = False"
compute_bound_tm_correct: fixes t :: "taylor_model"   and I :: "float interval list"   and e :: "nat \<Rightarrow> real"   and a :: "float interval list"   and x0 :: "real"   and prec :: "nat" assumes "num_params (tm_poly t) \<le> length I"   and "develops_at_within e a I"   and "x0 \<in>\<^sub>i range_tm e t" shows "x0 \<in>\<^sub>r compute_bound_tm prec I a t"
ccpo: shows "class.ccpo lub_fun le_fun (mk_less le_fun)"
Unf_abs_def: shows "\<up>Unf \<equiv> map_fun Rep Abs Unf"
in_\<R>_i: shows "((cms\<^sub>A ! i, mem\<^sub>A), cms\<^sub>C ! i, mem\<^sub>C) \<in> \<R>_rel (cms ! i)"
exp\<omega>_addO: fixes n :: "Ord"   and m :: "Ord" shows "exp\<omega> (addO n m) = mulO (exp\<omega> n) (exp\<omega> m)"
subu: shows "I.subcocycle (\<lambda>n x. u n ((Tinv ^^ n) x))"
madd_def: fixes m :: "int"   and x :: "int"   and y :: "int" shows "x ++\<^bsub>m\<^esub> y = (x + y) mod m"
g3mod2_modRules2: shows "g3mod2 \<subseteq> modRules2"
diff_union_swap: fixes a :: "'a"   and b :: "'a"   and M :: "'a multiset" assumes "a \<noteq> b" shows "b, M - (a, {#}) = (b, M) - (a, {#})"
Limsup_greatest: fixes F :: "'b filter"   and x :: "'d"   and f :: "'b \<Rightarrow> 'd" assumes "\<And>P. eventually P F \<Longrightarrow> x \<le> Sup (f ` Collect P)" shows "x \<le> Limsup F f"
fresh_eqvt: fixes p :: "perm"   and a :: "atom"   and x :: "'a" shows "p \<bullet> (a \<sharp> x) = p \<bullet> a \<sharp> p \<bullet> x"
convex_connected_collinear: fixes S :: "'b set" assumes "collinear S" shows "convex S = connected S"
extreme_point_of_convex_hull_eq: fixes S :: "'b set"   and x :: "'b" assumes "compact S"   and "\<And>T. T \<subset> S \<Longrightarrow> convex hull T \<noteq> convex hull S" shows "(x extreme_point_of convex hull S) = (x \<in> S)"
prv_imp_dsjEE: fixes \<phi> :: "fm"   and \<chi> :: "fm"   and \<psi> :: "fm" assumes "\<phi> \<in> UNIV"   and "\<chi> \<in> UNIV"   and "\<psi> \<in> UNIV"   and "{} \<turnstile> \<phi> IMP \<psi>"   and "{} \<turnstile> \<chi> IMP \<psi>" shows "{} \<turnstile> \<phi> OR \<chi> IMP \<psi>"
data_dependency_consistent_instrs_append: fixes T :: "nat set"   and xs :: "instr list"   and ys :: "instr list" shows "data_dependency_consistent_instrs T (xs @ ys) = (data_dependency_consistent_instrs T xs \<and> data_dependency_consistent_instrs (T \<union> load_tmps xs) ys \<and> load_tmps ys \<inter> \<Union> (fst ` store_sops xs) = {})"
assms_2: fixes k :: "'a"   and x :: "'b" shows "f k (0::'b) x = x"
Suci_mod_s_2: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0"   and "i mod s < s - Suc 0" shows "Suc i mod s = Suc (i mod s)"
intros_7: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and G :: "'b formula" assumes "F, \<Gamma> \<Rightarrow> \<Delta> \<down> n"   and "G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F \<^bold>\<or> G, \<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
max_top: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and x :: "'a" assumes "class.order_top less_eq less top" shows "ord.max less_eq top x = top"
phl_skip: fixes x :: "'a list set" shows "p_prod x p_one \<subseteq> p_prod p_one x"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (load_sub3 instr curr_win rd 10 address s1)) \<and> t2 = snd (fst (load_sub3 instr curr_win rd 10 address s2))"
final: shows "dfa.final M \<subseteq> dfa.states M"
zip_append2: fixes xs :: "'c list"   and ys :: "'d list"   and zs :: "'d list" shows "zip xs (ys @ zs) = zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
p_prod_onel: fixes X :: "'a list set" shows "p_prod p_one X = X"
simps_6: fixes v :: "'a" shows "(dFAILi \<le> dRETURN v) = False"
split_conv: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and a :: "'b"   and b :: "'c" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
ring_mat_def: fixes ty :: "'b itself"   and n :: "nat"   and b :: "'c" shows "ring_mat ty n b \<equiv> \<lparr>carrier = carrier_mat n n, mult = (*), one = 1\<^sub>m n, zero = 0\<^sub>m n n, add = (+), \<dots> = b\<rparr>"
delete_ran_def: fixes p :: "'b \<Rightarrow> nat"   and j :: "nat" shows "delete_ran p j \<equiv> \<lambda>i. delete_index j (p i)"
wp_hoare_link: fixes p :: "'b upred"   and Q :: "'b hrel"   and r :: "'b upred" shows "\<lbrace>p\<rbrace> Q \<lbrace>r\<rbrace>\<^sub>u = (Q wp r \<sqsubseteq> p)"
streams_iff_snth: fixes s :: "'d stream"   and X :: "'d set" shows "(s \<in> streams X) = (\<forall>n. s !! n \<in> X)"
adv_not_know2: fixes P :: "specID"   and A :: "specID"   and m :: "Secrets" assumes "out P \<subseteq> ins A"   and "\<not> know A (sKS m)" shows "\<not> eout P (sE m)"
add_less_same_cancel1: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and a :: "'d" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus b a) b = less a zero"
tot: shows "\<forall>y z. y \<in> {y. x \<noteq> y \<and> y pd\<rightarrow> x} \<and> z \<in> {y. x \<noteq> y \<and> y pd\<rightarrow> x} \<longrightarrow> z pd\<rightarrow> y \<or> y pd\<rightarrow> z"
comp_def: fixes g :: "'a"   and f :: "'a" shows "local.comp g f = (if (C.arr f \<and> Ide (C.dom f) \<and> Ide (C.cod f)) \<and> (C.arr g \<and> Ide (C.dom g) \<and> Ide (C.cod g)) \<and> C.cod f = C.dom g then C g f else C.null)"
simps_6: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "fol fm"   and q :: "fol fm" shows "semantics e f g (And p q) = (semantics e f g p \<and> semantics e f g q)"
fixespointwise_restrict1: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "fixespointwise f A" shows "fixespointwise (restrict1 f B) A"
elims: fixes x :: "'e \<Rightarrow> 'd"   and xa :: "'e list"   and xb :: "'d list"   and y :: "'d list" assumes "map_tailrec_rev x xa xb = y" obtains f :: "'e \<Rightarrow> 'd" and bs :: "'d list" where "x = fa__" and "xa = []" and "xb = bs__" and "y = bs__"   | f :: "'e \<Rightarrow> 'd" and a :: "'e" and as :: "'e list" and bs :: "'d list" where "x = f" and "xa = a # as" and "xb = bs" and "y = map_tailrec_rev f as (f a # bs)"
B: shows "prv (imp (ldsj \<phi>1s') (ldsj \<phi>2s))"
nodes_states: fixes p :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme"   and r :: "(('e \<times> 'f) \<times> 'd) list" assumes "p \<in> nodes A"   and "path A r p" shows "set (states r p) \<subseteq> nodes A"
PPs_insertI: fixes tx :: "'a \<Rightarrow>\<^sub>0 nat"   and X :: "'a set"   and t :: "'a \<Rightarrow>\<^sub>0 nat"   and x :: "'a"   and e :: "nat" assumes "tx \<in> .[X]"   and "t = monomial e x + tx" shows "t \<in> .[insert x X]"
plus_mono: fixes x\<^sub>2 :: "'b"   and x\<^sub>1 :: "'b"   and y\<^sub>2 :: "'b"   and y\<^sub>1 :: "'b" assumes "x\<^sub>2 \<le> x\<^sub>1"   and "y\<^sub>2 \<le> y\<^sub>1" shows "x\<^sub>2 + y\<^sub>2 \<le> x\<^sub>1 + y\<^sub>1"
homeomorphism_on_cong: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and X' :: "'c set"   and f' :: "'c \<Rightarrow> 'd" assumes "homeomorphism_on X f"   and "X' = X"   and "\<And>x. x \<in> X \<Longrightarrow> f' x = f x" shows "homeomorphism_on X' f'"
hyps_3: assumes "f \<in> F \<rightarrow> UNIV"   and "F \<subseteq> carrier_vec n" shows "dim_vec (module.lincomb (module_cpx_vec n) f F) = n"
set_\<Phi>o: fixes a :: "'t" assumes "S'.ide a" shows "S.set (\<Psi>o a) = \<psi> ` S'.set a"
L_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "L i \<le> L j"
mult_right_sub_dist_sup_right: fixes y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd" shows "y \<odot> z \<preceq> (x \<oplus> y) \<odot> z"
zero_le_mult_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a b) = (less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero)"
connected_component_set_empty: fixes x :: "'a" shows "connected_component_set {} x = {}"
set_child_nodes_get_shadow_root: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
min_galleryI_betw_compare: fixes X :: "'b set set"   and x :: "'b set"   and y :: "'b set"   and xs :: "'b set list"   and ys :: "'b set list" assumes "ChamberComplex X"   and "x \<noteq> y"   and "SimplicialComplex.maxsimpchain X (x # xs @ [y])"   and "SimplicialComplex.min_maxsimpchain X (x # ys @ [y])"   and "length xs = length ys" shows "SimplicialComplex.min_maxsimpchain X (x # xs @ [y])"
arl_append_def: shows "arl_append \<equiv> \<lambda>(a, n) x. Array.len a \<bind> (\<lambda>len. if n < len then Array.upd n x a \<bind> (\<lambda>a. return (a, n + 1)) else let newcap = 2 * len in array_grow a newcap default \<bind> (\<lambda>a. Array.upd n x a \<bind> (\<lambda>a. return (a, n + 1))))"
set_2: fixes x21 :: "'b"   and x22 :: "'b list" shows "set (x21 # x22) = insert x21 (set x22)"
case_prod_beta': fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (cc x) (snd x))"
finite_vars: fixes s :: "('d, 'c) tm" shows "finite (vars s)"
rel_interior_aff_dim: fixes S :: "'c set" assumes "convex S" shows "aff_dim (rel_interior S) = aff_dim S"
order_0_monom: fixes c :: "'a"   and n :: "nat" assumes "c \<noteq> (0::'a)" shows "Polynomial.order (0::'a) (monom c n) = n"
cond_uu: shows "continuous_on (U \<times> U) (\<lambda>(x, y). F x y)"
strand_vars_split_2: fixes S :: "('d, 'c) strand_step list"   and S' :: "('d, 'c) strand_step list" shows "wfrestrictedvars\<^sub>s\<^sub>t (S @ S') = wfrestrictedvars\<^sub>s\<^sub>t S \<union> wfrestrictedvars\<^sub>s\<^sub>t S'"
ce_rel_lm_22: fixes r :: "(nat \<times> nat) set" assumes "r \<in> ce_rels" shows "\<exists>p\<in>PrimRec3. r = {(x, y). \<exists>u. p x y u = 0}"
tmneg_blt: fixes n :: "nat"   and t :: "tm" assumes "tmboundslt n t" shows "tmboundslt n (tmneg t)"
gt_sym_imp_hd: fixes gt_sym :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and ground_heads_var :: "'b \<Rightarrow> 'a set"   and g :: "'a"   and f :: "'a" assumes "Lambda_Free_Term.ground_heads gt_sym arity_sym arity_var ground_heads_var"   and "gt_sym g f" shows "ground_heads.gt_hd gt_sym ground_heads_var (Sym g) (Sym f)"
zero_cap_simp: fixes u :: "nat"   and v :: "nat"   and c :: "nat \<times> nat \<Rightarrow> 'b" assumes "(u, v) \<notin> Graph.E c" shows "c (u, v) = (0::'b)"
A: shows "\<forall>x\<in>set ns. case x of 0 \<Rightarrow> True | Suc 0 \<Rightarrow> True | Suc (Suc m) \<Rightarrow> bn_valid m (p - bn_count ns) (foldl (+) 0 ns - bn_count ns)"
fr_eq_msubst: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list"   and x :: "'b" assumes "islin p" shows "(\<exists>x. Ifm vs (x # bs) p) = (Ifm vs (x # bs) (minusinf p) \<or> Ifm vs (x # bs) (plusinf p) \<or> (\<exists>(c, t)\<in>set (uset p). \<exists>(d, s)\<in>set (uset p). Ifm vs (x # bs) (msubst p ((c, t), d, s))))"
poly_subst_id: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and f :: "'c \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "\<And>x. x \<in> indets p \<Longrightarrow> f x = monomial (1::'d) (monomial 1 x)" shows "poly_subst f p = p"
glb_singleton: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.po below" shows "po.glb below {x} = x"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_same a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "wt t \<ge>\<^sub>p wt s" and "head t = head s" and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)"
ComSecTyping_single_is_sound: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool"   and DA :: "'b \<Rightarrow> 'e"   and AssignSideCondition :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and WhileSideCondition :: "'a \<Rightarrow> bool"   and IfSideCondition :: "'a \<Rightarrow> ('a, 'b) MWLfCom \<Rightarrow> ('a, 'b) MWLfCom \<Rightarrow> bool"   and c :: "('a, 'b) MWLfCom" assumes "Type_System E BMap DA AssignSideCondition WhileSideCondition IfSideCondition"   and "Type_System.ComSecTyping AssignSideCondition WhileSideCondition IfSideCondition c" shows "Strong_Security.Strongly_Secure (MWLf_semantics.MWLfSteps_det E BMap) DA [c]"
inf_eq: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = - (- x \<squnion> - y)"
TR_deMorgan: fixes x :: "tr"   and y :: "tr" shows "neg\<cdot>(x orelse y) = (neg\<cdot>x andalso neg\<cdot>y)"
list_strip_comb: fixes t :: "'a" shows "list_comb (fst (strip_comb t)) (snd (strip_comb t)) = t"
cauchyD: fixes X :: "nat \<Rightarrow> rat"   and r :: "rat" assumes "Real.cauchy X"   and "0 < r" shows "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
intro: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" assumes "df_system s\<^sub>0 lstep"   and "run_axioms s\<^sub>0 lstep s" shows "run s\<^sub>0 lstep s"
is_arg_minD2: fixes m :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "is_arg_min m P x"   and "P y" shows "\<not> m y < m x"
elem_def: fixes S :: "('c, 'd) eq_object_scheme"   and x :: "'c"   and A :: "'c set" shows "(x .\<in>\<^bsub>S\<^esub> A) = (\<exists>y\<in>A. x .=\<^bsub>S\<^esub> y)"
bin_rep_aux_neq_nil: fixes n :: "nat"   and m :: "nat" shows "bin_rep_aux n m \<noteq> []"
S_def: fixes A :: "('v \<times> 'v) set" shows "S A \<equiv> {r} \<union> fst ` A \<union> snd ` A"
assms_3: shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"
vconverse_vinsert: fixes a :: "V"   and b :: "V"   and r :: "V" shows "(vinsert \<langle>a, b\<rangle> r)\<inverse>\<^sub>\<circ> = vinsert \<langle>b, a\<rangle> (r\<inverse>\<^sub>\<circ>)"
AddSA_CompFun: fixes SA :: "('d, 'e, 'f) seqauto"   and HA :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates HA = {}"   and "S \<in> HAStates HA" shows "CompFun (HA [++] (S, SA)) = (CompFun HA [f+] (S, SA))"