hcju/isabelleps huggingface dataset:
Basis_list: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Basis_list :: "'a list"   and eucl_down :: "int \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_down :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_up :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "set Basis_list = Basis"
cinner_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "\<langle>x, y\<rangle> = \<langle>fst x, fst y\<rangle> + \<langle>snd x, snd y\<rangle>"
right: assumes "\<exists>x. eval (elimVar var L F (Eq p)) (xs @ x # \<Gamma>)" shows "(\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) \<or> (\<forall>x. aEval (Eq p) (xs @ x # \<Gamma>))"
simps_5: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'g"   and x2 :: "'h" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
br_def: fixes \<alpha> :: "'a \<Rightarrow> 'c"   and I :: "'a \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
eSuc_enat_iff: fixes x :: "enat"   and y :: "nat" shows "(eSuc x = enat y) = (\<exists>n. y = Suc n \<and> x = enat n)"
invKey_eq: fixes K :: "nat"   and K' :: "nat" shows "(invKey K = invKey K') = (K = K')"
union_disjoint: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "prod g (A \<union> B) = prod g A * prod g B"
assms_2: shows "constructive_security real2 ideal2 sim2 \<I>_real2 \<I>_inner2 \<I>_common2 bound2 lossless2 w2"
lambda_system_caratheodory: fixes f :: "'a set \<Rightarrow> ennreal"   and A :: "nat \<Rightarrow> 'a set" assumes "outer_measure_space M f"   and "range A \<subseteq> lambda_system \<Omega> M f"   and "disjoint_family A" shows "\<Union> (range A) \<in> lambda_system \<Omega> M f \<and> (\<Sum>i. f (A i)) = f (\<Union> (range A))"
removeLeaf_multiset: fixes v' :: "'c"   and t' :: "'c Tree"   and t :: "'c Tree" assumes "(v', t') = removeLeaf t"   and "t \<noteq> E" shows "{#v'#} + Heap.multiset t' = Heap.multiset t"
f0: assumes "reset_trap_val s = False \<and> get_ET (cpu_reg_val PSR s) = 0" shows "False"
card_eq: shows "card {inversion -` {x} \<inter> dc_crypto |x. x \<in> inversion ` dc_crypto} = card (inversion ` dc_crypto)"
cast_rat_list_def: fixes qs :: "rat poly list" shows "cast_rat_list qs = map real_of_rat_poly qs"
ltl_const_equiv_const_1: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>C true\<^sub>n = (eval \<phi> = Yes)"
permute_minus_cancel_2: fixes p :: "perm"   and x :: "'c" shows "- p \<bullet> p \<bullet> x = x"
vector_minus_component: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec"   and i :: "'b" shows "(x - y) $ i = x $ i - y $ i"
finfun_Diag_const_code_1: fixes b :: "'b"   and c :: "'c" shows "finfun_Diag (finfun_const b) (finfun_const c) = finfun_const (b, c)"
wls_diff_fresh_vsubst: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and y :: "'var"   and y1 :: "'var"   and ys :: "'varSort" assumes "wls s X"   and "y \<noteq> y1" shows "fresh ys y (X #[y1 // y]_ys)"
project_enc_extend: fixes x :: "('a \<times> bool list) list"   and w :: "'a list"   and I :: "(nat + nat set) list" assumes "map \<pi> x = enc (w, I)"   and "\<forall>(uu_, x)\<in>set x. x \<noteq> []" shows "x = enc (w, Inr (positions_in_row x 0) # I)"
inj_def: fixes f :: "'a \<Rightarrow> 'b" shows "inj f = (\<forall>x y. f x = f y \<longrightarrow> x = y)"
HEndPhase0_HInv5_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p" shows "HInv5_inner s' p"
Plus_def: fixes A :: "'a set"   and B :: "'b set" shows "A <+> B = Inl ` A \<union> Inr ` B"
induct: fixes x1 :: "'a basic_type"   and x2 :: "'a basic_type"   and P :: "'a basic_type \<Rightarrow> 'a basic_type \<Rightarrow> bool" assumes "x1 \<sqsubset>\<^sub>B x2"   and "P OclVoid Boolean"   and "P OclVoid UnlimitedNatural"   and "P OclVoid String"   and "\<And>\<C>. P OclVoid \<langle>\<C>\<rangle>\<^sub>\<T>"   and "\<And>\<E>. P OclVoid (Enum \<E>)"   and "P UnlimitedNatural Integer"   and "P Integer Real"   and "\<And>\<C> \<D>. \<C> < \<D> \<Longrightarrow> P \<langle>\<C>\<rangle>\<^sub>\<T> \<langle>\<D>\<rangle>\<^sub>\<T>"   and "P Boolean OclAny"   and "P Real OclAny"   and "P String OclAny"   and "\<And>\<C>. P \<langle>\<C>\<rangle>\<^sub>\<T> OclAny"   and "\<And>\<E>. P (Enum \<E>) OclAny" shows "P x1 x2"
ofilter_def: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wo_rel r" shows "ofilter r A \<equiv> A \<subseteq> Field r \<and> (\<forall>a\<in>A. under r a \<subseteq> A)"
h4d: shows "aEval (Neq (A + B * Var var)) (L[var := sqrt Cv]) = (\<not> eval (And (fm.Atom (Leq (A * B))) (fm.Atom (Eq (A\<^sup>2 - B\<^sup>2 * c)))) (L[var := sqrt Cv]))"
bij_betw_index: fixes xs :: "'b list"   and X :: "'b set"   and l :: "nat" assumes "distinct xs"   and "X = set xs"   and "l = length xs" shows "bij_betw (index xs) X {0..<l}"
list_appendE: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and x :: "'a" assumes "xs = ys @ zs"   and "x \<in> set xs" obtains "x \<in> set ys"   | "x \<in> set zs"
divisor_set_mult: fixes m :: "nat"   and n :: "nat" shows "divisor_set (m * n) = {i * j |i j. i \<in> divisor_set m \<and> j \<in> divisor_set n}"
prems_1: fixes x :: "'a" assumes "x \<in> insert x__ A" shows "f x \<noteq> 0"
ide_dom: fixes f :: "'a" assumes "A.arr f" shows "A.ide (A.dom f)"
pnet_lift: fixes onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool"   and p :: "net_tree" assumes "\<And>ii R\<^sub>i. \<langle>ii : onp ii : R\<^sub>i\<rangle>\<^sub>o \<Turnstile> (otherwith S {ii} (oarrivemsg I), other U {ii} \<rightarrow>) global (P ii)"   and "\<And>\<xi>. S \<xi> \<xi>"   and "\<And>\<xi>. U \<xi> \<xi>"   and "\<And>i R. \<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, uu_). castmsg (I \<sigma>) a)"   and "\<And>i R. \<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a \<noteq> \<tau> \<and> (\<forall>d. a \<noteq> i:deliver(d)) \<longrightarrow> S (\<sigma> i) (\<sigma>' i))"   and "\<And>i R. \<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a = \<tau> \<or> (\<exists>d. a = i:deliver(d)) \<longrightarrow> U (\<sigma> i) (\<sigma>' i))" shows "opnet onp p \<Turnstile> (otherwith S (net_tree_ips p) (oarrivemsg I), other U (net_tree_ips p) \<rightarrow>) global (\<lambda>\<sigma>. \<forall>i\<in>net_tree_ips p. P i \<sigma>)"
2: shows "Fun (local.comp (eval b c) (HF'.prod h b)) = restrict (Fun (eval b c) \<circ> Fun (HF'.prod h b)) (local.set (HF'.prod a b))"
F: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> s" shows "False"
d2_def: fixes x :: "'a" shows "d2 x = d' x / 2"
arrI: fixes f :: "'a" assumes "left (trg f) f" shows "Left.arr f"
from_decomp_def: fixes x :: "'a"   and y :: "'a"   and n :: "nat" shows "from_decomp x y n = lift x div lift y ^ n"
eq: fixes i :: "real"   and j :: "real" assumes "i \<in> I"   and "j \<in> I" shows "im_sheaf_morphisms (V i) (V i \<inter> V j) (s i) = im_sheaf_morphisms (V j) (V i \<inter> V j) (s j)"
Cons: fixes v :: "Vertices"   and w :: "Vertices"   and vs :: "Vertices list" assumes "v \<rightarrow>\<^bsub>T3\<^esub> w"   and "walk (w # vs)" shows "walk (v # w # vs)"
conj_distrib1: fixes c :: "'a::refinement_lattice"   and d\<^sub>0 :: "'a::refinement_lattice"   and d\<^sub>1 :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice) \<iinter> ((d\<^sub>0::'a::refinement_lattice) \<iinter> (d\<^sub>1::'a::refinement_lattice)) = c \<iinter> d\<^sub>0 \<iinter> (c \<iinter> d\<^sub>1)"
minus_ltrm_degree_drop: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P"   and "deg R f = Suc n" shows "deg R (f \<ominus>\<^bsub>P\<^esub> monom P (f (deg R f)) (deg R f)) \<le> n"
comb_repr: fixes \<psi> :: "'b" assumes "\<psi> \<in> span B" shows "comb (repr \<psi>) = \<psi>"
nths_eq_ConsD: fixes xs :: "'b list"   and I :: "nat set"   and x :: "'b"   and as :: "'b list" assumes "nths xs I = x # as" shows "\<exists>ys zs. xs = ys @ x # zs \<and> length ys \<in> I \<and> (\<forall>i\<in>I. length ys \<le> i) \<and> nths zs {i - length ys - 1 |i. i \<in> I \<and> length ys < i} = as"
in_Gr: fixes x :: "'a"   and y :: "'b"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "((x, y) \<in> BNF_Def.Gr A f) = (x \<in> A \<and> f x = y)"
ab_3: shows "\<forall>i\<le>N. fpxs_root_order (coeff p i) dvd b"
reflect_tranclp: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "\<And>x y. S (f x) (f y) \<Longrightarrow> R x y"   and "bij_on_trancl S f"   and "S\<^sup>+\<^sup>+ (f x) (f y)" shows "R\<^sup>+\<^sup>+ x y"
A3': fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "norm_metric.smC a b c c" shows "a = b"
hnr_ASSERT: fixes \<Phi> :: "bool"   and \<Gamma> :: "assn"   and c :: "'c Heap"   and \<Gamma>' :: "assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and c' :: "'d nres" assumes "\<Phi> \<Longrightarrow> hn_refine \<Gamma> c \<Gamma>' R c'" shows "hn_refine \<Gamma> c \<Gamma>' R (ASSERT \<Phi> \<bind> (\<lambda>_. c'))"
bisimilar_def: fixes a :: "('d, 'e) Stream"   and b :: "('d, 'f) Stream" shows "(a \<approx> b) = (unstream\<cdot>a = unstream\<cdot>b \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"
step_3_a_dims_main: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_row (step_3_a i j A) = dim_row A \<and> dim_col (step_3_a i j A) = dim_col A"
eq_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(a = divide b c) = (if c \<noteq> zero then times a c = b else a = zero)"
cond_hiding_entails_def: fixes t :: "'c \<Rightarrow> 'd option"   and s :: "'c \<Rightarrow> 'e option"   and C :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(t hiding s under C entails R) = (\<forall>k. ! x:t k: ! y:s k: C x y \<longrightarrow> R x y)"
widen_refl: fixes P :: "'a prog"   and T :: "ty" shows "P \<turnstile> T \<le> T"
vars1'_def: shows "st2.vars1' = ind_in_set st2.vars0 ` vars2"
no_prefix_targets_FAIL: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and PM :: "('d, 'e, 'f \<times> 'g) FSM"   and p :: "(('d \<times> 'e) \<times> 'f \<times> 'g) list"   and q :: "'f \<times> 'g"   and k :: "nat" assumes "productF M2 M1 FAIL PM"   and "path PM p q"   and "k < length p" shows "target (take k p) q \<noteq> FAIL"
exec_bind_SE_failure2: fixes A :: "'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e"   and M :: "'e \<Rightarrow> (bool \<times> 'e) option" assumes "A \<sigma> = None" shows "\<not> (\<sigma> \<Turnstile> A ;- M)"
sigma_finite_measure_axioms: fixes n :: "nat" shows "sigma_finite_measure (M n)"
Tinv_fmpt: assumes "invertible_qmpt" shows "fmpt M Tinv"
funcsetI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f \<in> funcset A B"
componentsI: fixes x :: "'a"   and U :: "'a set" assumes "x \<in> U" shows "connected_component_set U x \<in> components U"
set1_S_def: shows "set1_S = set1_G \<circ> Rep_S"
one_element_vec_def: fixes n :: "nat"   and e :: "'b" shows "one_element_vec n e = vec n (\<lambda>i. e)"
rT_def: fixes x :: "'s" shows "rT x = restrict_space (T x) {\<omega>. enabled x \<omega>}"
assms_4: fixes j :: "'b" assumes "j \<in> J" shows "Group.group (Hs j)"
prems: fixes x :: "'a" assumes "x \<in> insert x__ A" shows "f x \<in> \<real>"
card_fset_Suc: fixes S :: "'b fset"   and n :: "nat" assumes "card_fset S = Suc n" shows "\<exists>x T. x |\<notin>| T \<and> S = insert_fset x T \<and> card_fset T = n"
dotprod_1_right: fixes xs :: "nat list" shows "xs \<bullet> replicate (length xs) 1 = sum_list xs"
is_finite_nonempty: shows "{a. is_finite a} \<noteq> {}"
rev_induct: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (xs @ [x])" shows "P xs"
mem_filter: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a list" assumes "P x"   and "List.member A x" shows "List.member (filter P A) x"
cases: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and order :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'b" assumes "inf_wf r order a1 a2" obtains n :: "'c" and m :: "'c" and x :: "'b" where "a1 = m__" and "a2 = xa__" and "order n__ m__" and "inf_wf r order n__ xa__"   | x :: "'b" and y :: "'b" and n :: "'c" and m :: "'c" where "a1 = m" and "a2 = x" and "r\<^sup>+\<^sup>+ x y" and "inf_wf r order n y"
elims: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a"   and xb :: "'a list"   and y :: "'a list" assumes "insert_sort_insert_by x xa xb = y" obtains cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and e :: "'a" where "x = cmp__" and "xa = e__" and "xb = []" and "y = [e__]"   | cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and e :: "'a" and xc :: "'a" and xs :: "'a list" where "x = cmp" and "xa = e" and "xb = xc # xs" and "y = (if cmp e xc then e # xc # xs else xc # insert_sort_insert_by cmp e xs)"
correctness: fixes m0 :: "bool"   and m1 :: "bool"   and c :: "bool" shows "protocol (m0, m1) c = funct_OT_12 (m0, m1) c"
ev: shows "A *v (\<chi>i. complex_of_real (v $h i)) = complex_of_real ev *s (\<chi>i. complex_of_real (v $h i))"
sylow_greater_zero: shows "0 < card (subgroups_of_size (p ^ a))"
set_val_ok: fixes h :: "(_) heap"   and character_data_ptr :: "(_) character_data_ptr"   and tag :: "char list" assumes "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "character_data_ptr |\<in>| character_data_ptr_kinds h" shows "h \<turnstile> ok set_val character_data_ptr tag"
terminate_halt: fixes l :: "nat"   and lm :: "'b"   and tp :: "(action \<times> nat) list"   and stp :: "nat"   and m :: "nat"   and rs :: "nat"   and n :: "nat" assumes "steps0 (Suc 0, Bk \<up> l, <lm>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"   and "tm_wf0 tp"   and "0 < rs" shows "terminate rec_halt [code tp, bl2wc (<lm>)]"
initial_refine: fixes v0 :: "'v"   and D0 :: "'v set"   and I :: "'v \<Rightarrow> node_state option"   and v0i :: "'v" assumes "v0 \<notin> D0"   and "(I, D0) \<in> oGS_rel"   and "(v0i, v0) \<in> Id" shows "(initial_impl v0i I, initial v0 D0) \<in> GS_rel"
first: shows "(\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0) \<longrightarrow> (\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<or> (\<exists>(a', b', c')\<in>set les. a' = 0 \<and> b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or> a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> ((\<forall>(d, e, f)\<in>set les. \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a'). \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<or> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' - sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)))"
power_numeral_reduce: fixes a :: "'a"   and n :: "num" shows "a ^ numeral n = a * a ^ pred_numeral n"
a_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> a_kernel R S h"   and "y \<in> a_kernel R S h" shows "x \<oplus> y \<in> a_kernel R S h"
typeof_conf: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P,h \<turnstile> v :\<le> T"
satisfiable_def: fixes formula :: "Literal list list" shows "satisfiable formula \<equiv> \<exists>valuation. model valuation formula"
impossible_Cons: fixes xs :: "'g list"   and ys :: "'g list"   and x :: "'g" assumes "length xs \<le> length ys" shows "(xs = x # ys) = False"
ntr_sl_move_left: fixes p :: "'b"   and ww :: "('b, 'c) label list list"   and u :: "'a"   and r :: "'a list"   and ce :: "'a list multiset"   and w :: "('b, 'c) label list"   and r' :: "'a list"   and ce' :: "'a list multiset" assumes "({#[entry fg p]#}, ww, {#u # r#} + ce) \<in> trcl (ntr fg)"   and "(([u], ce), w, r', ce') \<in> trcl (trss fg)"   and "initialproc fg p"   and "length r' \<le> 1"   and "w \<noteq> []" shows "\<exists>ww'. ({#[entry fg p]#}, ww', {#r' @ r#} + ce') \<in> trcl (ntr fg)"
assms_1: shows "s \<in> init (closed (pnet np n))"
IH2_4: assumes "\<not> M \<Turnstile> G" shows "\<not> M(p := False) \<Turnstile> G"
t_eq_imp_bij_func: fixes xs :: "'observation list"   and ys :: "'observation list" assumes "t xs = t ys" shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! f i)"
liftAll2_def: fixes phi :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and inp1 :: "'f \<Rightarrow> 'd option"   and inp2 :: "'f \<Rightarrow> 'e option" shows "liftAll2 phi inp1 inp2 \<equiv> \<forall>i v1 v2. inp1 i = Some v1 \<and> inp2 i = Some v2 \<longrightarrow> phi v1 v2"
andfold_MatchExp_not_disc_mapMatch: fixes disc :: "'c \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'c"   and ls :: "'b list" assumes "\<forall>a. \<not> disc (C a)" shows "\<not> has_disc disc (andfold_MatchExp (map (Match \<circ> C) ls))"
False: shows "\<not> wt (App s t) >\<^sub>p wt s"
o4: shows "h = 3\<^sup>2 * 7 * 13"
BusInterface_def: fixes activation :: "nat \<Rightarrow> nat list"   and return :: "nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and store :: "nat \<Rightarrow> 'a Frame list"   and send :: "nat \<Rightarrow> 'a Frame list"   and get :: "nat \<Rightarrow> nat list" shows "BusInterface activation return recv store send get \<equiv> Receive recv store activation \<and> Send return send get activation"
assms_6: shows "w \<le> (r\<^sup>T * g\<^sup>\<star>)\<^sup>T * (r\<^sup>T * g\<^sup>\<star>) \<sqinter> g"
lookup_Mapping_code_1: fixes al :: "('a, 'b) alist" shows "Mapping.lookup (Assoc_List_Mapping al) = DAList.lookup al"
bottom_def: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.pcpo below" shows "pcpo.bottom below = (THE x. \<forall>y. below x y)"
a_is_weakly_dom_component_safe_def: fixes S\<^sub>a\<^sub>r\<^sub>g :: "(_) object_ptr set"   and S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t :: "(_) object_ptr set"   and h :: "(_) heap"   and h' :: "(_) heap" shows "a_is_weakly_dom_component_safe S\<^sub>a\<^sub>r\<^sub>g S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t h h' = (let removed_pointers = fset (object_ptr_kinds h) - fset (object_ptr_kinds h'); added_pointers = fset (object_ptr_kinds h') - fset (object_ptr_kinds h); arg_components = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_dom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> a_get_dom_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_dom_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> a_get_dom_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<union> added_pointers \<and> (\<forall>outside_ptr\<in>fset (object_ptr_kinds h) \<inter> fset (object_ptr_kinds h') - (\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> a_get_dom_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
dbproj_Cons_1: fixes i :: "'a"   and d :: "'b"   and D :: "('a \<times> 'b) list" shows "dbproj i ((i, d) # D) = (i, d) # dbproj i D"
confluentp_eq_semiconfluentp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "confluentp r = semiconfluentp r"
uminus_all: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "ccw' (- a) (- b) (- c) = ccw' a b c"
finite_comm_group_adjoin_axioms_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" shows "finite_comm_group_adjoin_axioms G H a \<equiv> a \<in> carrier G \<and> a \<notin> H"
ucast_revcast: fixes w :: "'c word" shows "ucast w = word_reverse (revcast (word_reverse w))"
vsv_vempty: shows "vsv []\<^sub>\<circ>"
mkdisjoint_un: fixes A :: "nat \<Rightarrow> 'b set"   and n :: "nat" assumes "\<And>n. A n \<subseteq> A (Suc n)" shows "A n = \<Union> (mkdisjoint A ` {..n})"
rt: shows "y ^ 3 + cor e * y + cor f = 0"
induct: fixes P :: "'c list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c list"   and a1 :: "'c list" assumes "\<And>bs. P [] bs"   and "\<And>a as bs. \<lbrakk>a \<in> set bs \<Longrightarrow> P as bs; a \<notin> set bs \<Longrightarrow> P as bs\<rbrakk> \<Longrightarrow> P (a # as) bs" shows "P a0 a1"
idiff_ile_mono2: fixes m :: "enat"   and n :: "enat"   and l :: "enat" assumes "m \<le> n" shows "l - n \<le> l - m"
distinct_fst_inj: fixes ps :: "('c \<times> 'd) list"   and f :: "'c \<Rightarrow> 'e" assumes "distinct (map fst ps)"   and "inj f" shows "distinct (map fst (map (\<lambda>(k, v). (f k, v)) ps))"
f'_def: shows "f' = - inverse (m\<^sup>2)"
valid_deep_model': fixes r :: "nat"   and rs :: "nat list" shows "valid_net (deep_model' r rs)"
INF_cong: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d \<Rightarrow> 'e"   and D :: "'d \<Rightarrow> 'e"   and Inf :: "'e set \<Rightarrow> 'e" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
indep_vars_def2: fixes M' :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'a \<Rightarrow> 'e"   and I :: "'d set" shows "indep_vars M' X I = ((\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> indep_sets (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I)"
\<eta>_p_CAs0': shows "\<forall>i<n. CAs0' ! i \<cdot> \<eta>s0' ! i = CAs0' ! i \<cdot> \<eta>"
zero_not_in_Basis: shows "(0::'a) \<notin> {1::'a}"