hcju/isabelleps huggingface dataset:
concI: fixes u :: "'a list"   and A :: "'a list set"   and v :: "'a list"   and B :: "'a list set" assumes "u \<in> A"   and "v \<in> B" shows "u @ v \<in> A @@ B"
set: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and l :: "'l"   and v :: "('r, 'l, 'v) val" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Assign (VE (Loc l)) (VE v)])"   and "l \<in> dom (\<sigma>;;\<tau>)" shows "revision_step r s (s(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (CV Unit)])))"
strictly_exactD: fixes P :: "'a \<Rightarrow> bool"   and h :: "'a"   and h' :: "'a" assumes "strictly_exact P"   and "P h"   and "P h'" shows "h = h'"
sum_closed: fixes A :: "'b set"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "ring_closed A"   and "\<And>x. x \<in> X \<Longrightarrow> f x \<in> A" shows "sum f X \<in> A"
insert_disjoint_2: fixes a :: "'c"   and A :: "'c set"   and B :: "'c set" shows "({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)"
tendsto_cot: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F"   and "sin a \<noteq> (0::'a)" shows "((\<lambda>x. cot (f x)) \<longlongrightarrow> cot a) F"
scaleR_matrix_vector_assoc: fixes k :: "real"   and A :: "((real, 'c) vec, 'a) vec"   and v :: "(real, 'c) vec" shows "k *\<^sub>R (A *v v) = k *\<^sub>R A *v v"
cases: fixes x :: "('c, 'd) ioa \<times> 'c \<times> ('d \<times> 'c) list" obtains A :: "('c, 'd) ioa" and s :: "'c" and ps :: "('d \<times> 'c) list" and p' :: "'d \<times> 'c" and p :: "'d \<times> 'c" where "x = (Aa__, s__, Cons p__ (Cons p' ps))"   | A :: "('c, 'd) ioa" and s :: "'c" and p :: "'d \<times> 'c" where "x = (Ab__, sa__, [p])"   | A :: "('c, 'd) ioa" and s :: "'c" where "x = (A, s, [])"
f1_Two: fixes x :: "three"   and y :: "three" assumes "x \<noteq> Zero \<and> y \<noteq> Zero" shows "f1 x y = Two"
is_linorder_rel_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "is_linorder_rel R \<equiv> (\<forall>x y. R x y \<or> R y x) \<and> (\<forall>x y z. R x y \<longrightarrow> R y z \<longrightarrow> R x z)"
dvd_div_eq_1: fixes c :: "nat"   and a :: "nat"   and b :: "nat" assumes "c dvd a"   and "c dvd b"   and "a div c = b div c" shows "a = b"
Alexander_subbase: fixes \<B> :: "'c set set"   and X :: "'c topology" assumes "topology (arbitrary union_of (finite intersection_of (\<lambda>x. x \<in> \<B>) relative_to \<Union> \<B>)) = X"   and "\<And>C. \<lbrakk>C \<subseteq> \<B>; \<Union> C = topspace X\<rbrakk> \<Longrightarrow> \<exists>C'. finite C' \<and> C' \<subseteq> C \<and> \<Union> C' = topspace X" shows "compact_space X"
is_RB_upt_def: fixes d :: "'a \<Rightarrow> nat"   and rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t" shows "is_RB_upt d rword G u = (G \<subseteq> dgrad_sig_set' (length fs) d \<and> (\<forall>v. v \<prec>\<^sub>t u \<longrightarrow> d (pp_of_term v) \<le> dgrad_max d \<longrightarrow> component_of_term v < length fs \<longrightarrow> is_RB_in d rword G v))"
less_eq_Def_def: fixes x :: "Def"   and y :: "Def" shows "(x \<le> y) = (x = y \<or> x < y)"
mono_on_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "mono_on f A \<equiv> \<forall>r s. r \<in> A \<and> s \<in> A \<and> r \<le> s \<longrightarrow> f r \<le> f s"
Sup_least: fixes A :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> z \<le> x" shows "z \<le> \<Sqinter> A"
mult_nonneg_nonneg: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (times a b)"
subcocycle_integrable_2: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "subcocycle u" shows "u n \<in> borel_measurable M"
independent_inj_image: fixes S :: "'b set" assumes "independent S"   and "inj f" shows "independent (f ` S)"
lt_plus_lessE: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "ord_term_strict (ordered_term.lt term_of_pair ord_term p) (ordered_term.lt term_of_pair ord_term (p + q))" shows "ord_term_strict (ordered_term.lt term_of_pair ord_term p) (ordered_term.lt term_of_pair ord_term q)"
poly_monom: fixes a :: "'a"   and n :: "nat"   and x :: "'a" shows "poly (monom a n) x = a * x ^ n"
product_topology_open_contains_basis: fixes T :: "'a \<Rightarrow> 'b topology"   and I :: "'a set"   and U :: "('a \<Rightarrow> 'b) set"   and x :: "'a \<Rightarrow> 'b" assumes "openin (product_topology T I) U"   and "x \<in> U" shows "\<exists>X. x \<in> Pi\<^sub>E I X \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> Pi\<^sub>E I X \<subseteq> U"
distinct_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and x21 :: "'c"   and x22 :: "'c"   and x1 :: "'c" assumes "antiparallel_edges \<Delta>" shows "antiparallel_edges.Edge x21 x22 \<noteq> antiparallel_edges.Vertex x1"
rep_eq: fixes x :: "'f" shows "\<lbrakk>\<guillemotleft>x\<guillemotright>\<rbrakk>\<^sub>e = (\<lambda>b. x)"
simps_1: shows "stails\<cdot>[::] = [:[::]:]"
bdd_below_image_mono: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "mono f"   and "bdd_below A" shows "bdd_below (f ` A)"
assms_2: fixes x :: "'a"   and x' :: "'b" assumes "(x, x') \<in> R'" shows "f x \<le> \<Down> R (f' x')"
d2: shows "H \<turnstile> All2 z' r (SyntaxN.Ex x (SyntaxN.Ex y (SyntaxN.Ex x' (SyntaxN.Ex y' (HPair a b EQ HPair (Var x) (Var y) AND Var z' EQ HPair (Var x') (Var y') AND OrdP (Var x) AND OrdP (Var x') AND (Var x EQ Var x' IMP Var y EQ Var y'))))))"
mset_add: fixes a :: "'a"   and A :: "'a multiset" assumes "a \<in># A" obtains B :: "'a multiset" where "A = add_mset a B"
rebalance_last_tree_order: fixes ts :: "('a btree \<times> 'a) list"   and ls :: "('a btree \<times> 'a) list"   and sub :: "'a btree"   and sep :: "'a"   and k :: "nat"   and t :: "'a btree" assumes "ts = ls @ [(sub, sep)]"   and "\<forall>s\<in>set (subtrees ts). order k s"   and "almost_order k t"   and "length ts \<le> 2 * k"   and "height sub = height t" shows "almost_order k (rebalance_last_tree k ts t)"
get_empty: fixes empty :: "'a"   and get :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and add :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and to_list :: "'a \<Rightarrow> ('b \<times> 'c) list"   and x :: "'b" assumes "env empty get add to_list" shows "get empty x = None"
hom_max: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "(a \<le> b) = (f a \<le> f b)" shows "f (max a b) = max (f a) (f b)"
that: fixes M' :: "real" assumes "exponential_order M' c f" shows "thesis"
least_min_alg1: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e" assumes "alg B1 B2 s1 s2" shows "min_alg1 s1 s2 \<subseteq> B1"
max_ge_x: fixes x :: "'a"   and y :: "'a" shows "x \<le> max x y"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (src f) = B.arr f"
If_tE: fixes b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and s :: "char list \<Rightarrow> int"   and x :: "nat"   and t :: "char list \<Rightarrow> int" assumes "(IF b THEN c1 ELSE c2, s) \<Rightarrow> x \<Down> t" obtains xa :: "nat" where "x = Suc xa__" and "bval b s" and "(c1, s) \<Rightarrow> xa__ \<Down> t"   | xa :: "nat" where "x = Suc xa" and "\<not> bval b s" and "(c2, s) \<Rightarrow> xa \<Down> t"
pw_eq_iff: fixes S :: "'b nres"   and S' :: "'b nres" shows "(S = S') = (nofail S = nofail S' \<and> (\<forall>x. inres S x = inres S' x))"
induct: fixes P :: "'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b up\<^sub>d \<Rightarrow> bool"   and a0 :: "'b tree234"   and a1 :: "'b"   and a2 :: "'b up\<^sub>d" assumes "\<And>l a r. P l a (T\<^sub>d r)"   and "\<And>ll b rl a r. P \<langle>ll, b, rl\<rangle> a (Up\<^sub>d r)"   and "\<And>ll b ml c rl a r. P \<langle>ll, b, ml, c, rl\<rangle> a (Up\<^sub>d r)"   and "\<And>t1 a t2 b t3 c t4 d t5. P \<langle>t1, a, t2, b, t3, c, t4\<rangle> d (Up\<^sub>d t5)"   and "\<And>b v. P \<langle>\<rangle> b (Up\<^sub>d v)" shows "P a0 a1 a2"
lt_in_keys: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p \<noteq> 0" shows "ordered_term.lt term_of_pair ord_term p \<in> keys p"
is_closed_subst_bv2_unchanged: fixes t :: "Core.term"   and n :: "nat"   and u :: "Core.term" assumes "is_closed t" shows "subst_bv2 t n u = t"
ennreal_inverse_mult: fixes a :: "ennreal"   and b :: "ennreal" assumes "a < top"   and "b < top" shows "inverse (a * b) = inverse a * inverse b"
RN_deriv_unique: fixes f :: "'b \<Rightarrow> ennreal"   and N :: "'b measure" assumes "f \<in> borel_measurable (distr M M' T)"   and "density (distr M M' T) f = N" shows "AE x in distr M M' T. f x = RN_deriv (distr M M' T) N x"
configuration_step_non_empty: fixes q' :: "'b"   and n :: "nat"   and q :: "'b" assumes "configuration q' n \<noteq> {}"   and "q = \<delta> q' (w n)" shows "configuration q (Suc n) \<noteq> {}"
nan_not_less: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x \<or> IEEE.is_nan y" shows "\<not> x < y"
bind_assoc_pmf: fixes A :: "'e pmf"   and B :: "'e \<Rightarrow> 'd pmf"   and C :: "'d \<Rightarrow> 'c pmf" shows "A \<bind> B \<bind> C = A \<bind> (\<lambda>x. B x \<bind> C)"
map_scale_zero_right: fixes k :: "'b" shows "k \<cdot> 0 = 0"
P_Ana_key: shows "\<forall>t K T k. M \<turnstile> t \<longrightarrow> P t \<longrightarrow> Ana t = (K, T) \<longrightarrow> M \<turnstile> k \<longrightarrow> k \<in> set K \<longrightarrow> P k"
G_map: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and g :: "'a"   and v :: "'c" assumes "FGModuleHom G smult V smult' T"   and "g \<in> G"   and "v \<in> V" shows "T (aezfun_scalar_mult.Gmult smult g v) = aezfun_scalar_mult.Gmult smult' g (T v)"
gt_sub_arg: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "App s t >\<^sub>t t"
split_beta: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and prod :: "'b \<times> 'c" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
hyps: fixes xs :: "'a llist" assumes "enat n < llength xs" shows "lnth (lmap f xs) n = f (lnth xs n)"
deriv: fixes y :: "real" assumes "real_of_int a \<le> y" shows "(G has_real_derivative g y) (at y within {real_of_int a..})"
deriv_ix_def: fixes d :: "nat \<times> nat \<times> nat" shows "deriv_ix d = snd (snd d)"
ide_char: fixes a :: "'a" shows "S.ide a = (S.arr a \<and> ide a)"
c1: fixes n :: "nat" assumes "x = 2 * n" shows "P"
st: shows "s \<in> \<S>\<^sub>a \<and> t \<in> \<S>\<^sub>a \<and> t \<le>\<^sub>\<S> \<lambda> (s, A) \<and> \<not> (t \<circ> B \<le>\<^sub>\<K> B + C \<and> t \<circ> C \<le>\<^sub>\<K> B + C)"
proj_unproj: fixes k :: "'b"   and e :: "'b \<Rightarrow> 'c" assumes "k \<in> basis" shows "unproj e proj k = e k"
subst_cls_lists_id_subst: fixes Cs :: "'a literal multiset list" shows "Cs \<cdot>\<cdot>cl replicate (length Cs) id_subst = Cs"
onl_def: fixes \<Gamma> :: "'j \<Rightarrow> ('h, 'i, 'j, 'k) seqp"   and P :: "'g \<times> 'k \<Rightarrow> bool" shows "onl \<Gamma> P \<equiv> \<lambda>(\<xi>, p). \<forall>l\<in>labels \<Gamma> p. P (\<xi>, l)"
fresh_star_Un: fixes as :: "atom set"   and bs :: "atom set"   and x :: "'e" shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
simps_1: fixes x :: "three" shows "f2 x Zero = x"
w_in_hom_1: shows "\<guillemotleft>w' : src f' \<rightarrow> src f\<guillemotright>"
Cd_singleton: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" shows "(x \<in> ManiPref.Cd d X) = (ManiPref.Cd d X = {x})"
rbt_min_opt_induct: fixes P :: "('b, unit) RBT_Impl.rbt \<Rightarrow> bool"   and t :: "('b, unit) RBT_Impl.rbt" assumes "P rbt.Empty"   and "\<And>color t1 a b t2. \<lbrakk>P t1; P t2; t1 = rbt.Empty\<rbrakk> \<Longrightarrow> P (Branch color t1 a b t2)"   and "\<And>color t1 a b t2. \<lbrakk>P t1; P t2; t1 \<noteq> rbt.Empty\<rbrakk> \<Longrightarrow> P (Branch color t1 a b t2)" shows "P t"
psiCasesInject: fixes CsP :: "('f \<times> ('d, 'e, 'f) psi) list"   and CsQ :: "('f \<times> ('d, 'e, 'f) psi) list" shows "(psiCases CsP = psiCases CsQ) = (CsP = CsQ)"
right_adjoint_is_ide: fixes f :: "'a" assumes "C.is_right_adjoint f" shows "C.ide f"
si_empty1: fixes X :: "'a event set" shows "setinterleaving ([], X, []) = {[]}"
length_add1: fixes x :: "real"   and y :: "real" shows "\<forall>ts v. (\<exists>va u. v=va\<parallel>u \<and> \<parallel>ext va\<parallel> = x \<and> \<parallel>ext u\<parallel> = y) \<longrightarrow> \<parallel>ext v\<parallel> = x + y"
at_supp: fixes x :: "'a" assumes "at TYPE('a)" shows "supp x = {x}"
select_convs_1: fixes edge :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and capacity :: "'a \<times> 'a \<Rightarrow> ennreal"   and source :: "'a"   and sink :: "'a"   and more :: "'b" shows "capacity \<lparr>edge = edge, capacity = capacity, source = source, sink = sink, \<dots> = more\<rparr> = capacity"
r_findr: fixes e :: "nat" shows "eval r_findr [e] = findr e"
SpanI: fixes ms :: "'v list" assumes "set ms \<subseteq> V"   and "V \<subseteq> RSpan ms" shows "V = RSpan ms"
ind: fixes s' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps0 (s, l, r) B stp = (s', l', r')"   and "s' \<noteq> 0" shows "steps0 (s + off, l, r) (A @ shift B off) stp = (s' + off, l', r')"
CatExpDom: fixes f :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme" assumes "f \<in> mor\<^bsub>CatExp A B\<^esub>" shows "dom\<^bsub>CatExp A B\<^esub> f = NTDom f"
wm_to_state_eq: fixes M :: "object atom formula set"   and p :: "predicate"   and as :: "object list" shows "wm_to_state M (p, as) = (Atom (predAtm p as) \<in> M)"
prefs_wf: fixes i :: "'a" assumes "i \<in> agents" shows "finite_total_preorder_on alts (R i)"
cases: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b llist" assumes "chain R a" obtains x :: "'b" where "a = LCons x__ LNil"   | xs :: "'b llist" and x :: "'b" where "a = LCons x xs" and "chain R xs" and "R x (lhd xs)"
rep_eq: fixes x :: "'a set"   and xa :: "'a \<Rightarrow> 'b" shows "Rep_pfun (pfun_entries x xa) = (\<lambda>xb. if xb \<in> x then Some (xa xb) else None)"
conj_len: fixes p :: "'b list"   and x :: "'b list"   and s :: "'b list" assumes "p \<cdot> x = x \<cdot> s" shows "\<^bold>|p\<^bold>| = \<^bold>|s\<^bold>|"
app\<^sub>i_Invoke: fixes M :: "char list"   and n :: "nat"   and P :: "'b prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Invoke M n, P, pc, mxs, T\<^sub>r, ST, LT) = (n < length ST \<and> (ST ! n \<noteq> NT \<longrightarrow> (\<exists>C D Ts T m. ST ! n = Class C \<and> P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = m in D \<and> P \<turnstile> rev (take n ST) [\<le>] Ts)))"
swap_fresh_fresh: fixes a :: "atom"   and x :: "'b"   and b :: "atom" assumes "a \<sharp> x"   and "b \<sharp> x" shows "(a \<rightleftharpoons> b) \<bullet> x = x"
sym_lens_def: fixes S :: "<'d, 'e> \<Longleftrightarrow> 'f" shows "sym_lens S \<equiv> (vwb_lens \<V>\<^bsub>S\<^esub> \<and> vwb_lens \<C>\<^bsub>S\<^esub>) \<and> \<V>\<^bsub>S\<^esub> \<bowtie> \<C>\<^bsub>S\<^esub> \<and> bij_lens (\<V>\<^bsub>S\<^esub> +\<^sub>L \<C>\<^bsub>S\<^esub>)"
eq_sym: fixes x :: "'a"   and y :: "'a" shows "eq\<cdot>x\<cdot>y = eq\<cdot>y\<cdot>x"
map_iterator_genord_rule_insert_P: assumes "map_iterator_genord it m (\<lambda>_ _. True)"   and "I {} \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>True; k \<in> dom m - it; m k = Some v; it \<subseteq> dom m; I it \<sigma>; \<forall>k' v'. k' \<in> dom m - it - {k} \<and> m k' = Some v' \<longrightarrow> True; \<forall>k' v'. k' \<in> it \<and> m k' = Some v' \<longrightarrow> True\<rbrakk> \<Longrightarrow> I (insert k it) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I (dom m) \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> dom m; \<not> True; I it \<sigma>; \<forall>k v k' v'. k \<in> it \<and> m k = Some v \<and> k' \<notin> it \<and> m k' = Some v' \<longrightarrow> True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
SourcesS11_L2: shows "Sources level2 sS11 = {sS9}"
minus_floatarith_def: fixes f :: "floatarith"   and g :: "floatarith" shows "f - g = Add f (Minus g)"
assms_8: shows "b \<cdot> l \<le> l \<cdot> b"
mat_def: fixes k :: "'d" shows "mat k = (\<chi>i j. if i = j then k else (0::'d))"
insertI: fixes A :: "'b set"   and a :: "'b" assumes "finite A" shows "finite (a \<triangleright> A)"
lm034: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G" shows "a\<inverse> \<in> injections (Range a) N \<and> Range a \<in> all_partitions G"
qe: fixes as :: "'a list" assumes "\<forall>a\<in>set as. depends\<^sub>0 a \<and> anormal a" shows "is_dnf_qe qe as"
cases: fixes x :: "'a list" obtains "xa__ = []"   | x :: "'a" and xs :: "'a list" where "xa__ = x # xs"
lexord_asym: fixes R :: "('a \<times> 'a) set" assumes "asym R" shows "asym (lexord R)"
Sup_exp_sound: fixes S :: "('a \<Rightarrow> real) set"   and b :: "real" assumes "\<And>P. P \<in> S \<Longrightarrow> sound P"   and "\<And>P. P \<in> S \<Longrightarrow> bounded_by b P" shows "sound (Sup_exp S)"
util_def_conf: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "ordinal_utility carrier relation u"   and "x \<in> carrier"   and "y \<in> carrier" shows "(u y \<le> u x) = x \<succeq>[relation] y"
ends_in_int: fixes a :: "'a"   and b :: "'a" shows "a \<in> interval a b \<and> b \<in> interval a b"
AxSelf: fixes m :: "Body"   and x :: "'a Point" assumes "IOb m" shows "m sees m at x \<longrightarrow> onAxisT x"
INF_greatest: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and u :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq u (f i)" shows "less_eq u (Inf (f ` A))"
path_components_of_maximal: fixes C :: "'b set"   and X :: "'b topology"   and S :: "'b set" assumes "C \<in> path_components_of X"   and "path_connectedin X S"   and "\<not> disjnt C S" shows "S \<subseteq> C"
Rep_Arity_inject: fixes x :: "Arity"   and y :: "Arity" shows "(Rep_Arity x = Rep_Arity y) = (x = y)"
zero_neq_neg_one: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_char_0 one times plus zero minus uminus" shows "zero \<noteq> uminus one"
HC_intros_1: fixes F :: "'b formula"   and G :: "'b formula" shows "AX0 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F"
quat_unfold: fixes q :: "quat" shows "q = quat_of_real (quat.Re q) + \<i> * quat_of_real (Im1 q) + \<j> * quat_of_real (Im2 q) + \<k> * quat_of_real (Im3 q)"
add_mono_thms_linordered_field_2: fixes i :: "'g"   and j :: "'g"   and k :: "'g"   and l :: "'g" assumes "i = j \<and> k < l" shows "i + k < j + l"
xd_diff_2: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j \<le> n" shows "local.xd j i \<noteq> (0::'a)"
SeqHaddP_Zero_E: fixes s :: "tm"   and w :: "tm"   and z :: "tm" shows "{SeqHaddP s w Zero z} \<turnstile> w EQ z"
remdups_map_remdups: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "remdups (map f (remdups xs)) = remdups (map f xs)"
case_1: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'c" shows "(case Infinity of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = f1"
nested_subalg_is_sigma_finite: fixes G :: "'a measure" assumes "subalgebra M G"   and "subalgebra G G" shows "sigma_finite_subalgebra M G"
1: shows "coreflexive ((v \<sqinter> - i \<sqinter> - i\<^sup>T) * (v\<^sup>T \<sqinter> - i \<sqinter> - i\<^sup>T))"
generate_DB_def: fixes PS :: "bit list" shows "generate_DB PS = PS @ [\<zero>, \<zero>, \<zero>, \<zero>, \<zero>, \<zero>, \<zero>, \<one>] @ salt"
simps_3: fixes SUC :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and wf0 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'b"   and \<phi> :: "('c, 'a) aformula" shows "Formula_Operations.wf SUC wf0 n (FNot \<phi>) = Formula_Operations.wf SUC wf0 n \<phi>"
normalize_quot_in_normalized_fracts: fixes x :: "'b \<times> 'b" shows "normalize_quot x \<in> normalized_fracts"
lim_imp_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'd"   and f0 :: "'d" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> f0) F" shows "Limsup F f = f0"
fresher_trans: fixes x :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and y :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and z :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
assms: shows "\<Psi> \<rhd> M\<langle>N\<rangle>.P \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N'\<rangle> \<prec> P'"
run_one_step_basic_call_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x12 :: "nat"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Call x12) = (s', vs', res)" shows "\<exists>r. res = RSNormal r"
null_space_def: fixes A :: "(('c, 'd) vec, 'e) vec" shows "null_space A = {x. A *v x = 0}"
K0_def: shows "K0 = K \<inter> null_sets M"
X'_def: fixes i :: "nat"   and x :: "'a" shows "X' i x \<equiv> X i x - m"
uno_simps_2: fixes z :: "'a word"   and n :: "nat" shows "unat (word_of_nat (unat z div n)) = unat z div n"
axioms_1: fixes x :: "'a \<Longrightarrow> 'b" assumes "mwb_lens x" shows "weak_lens x"
feasible_sound: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "feasible t"   and "sound P" shows "sound (t P)"
card_Un_disjoint: fixes A :: "'e set"   and B :: "'e set" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "card (A \<union> B) = card A + card B"
rel_distinct_2: fixes R :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and y2 :: "'l" shows "\<not> rel_option R (Some y2) None"
ex_less_of_int: fixes x :: "'b" shows "\<exists>z. x < of_int z"
boundedI: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "c \<le> a" shows "b + c \<le> a"