hcju/isabelleps huggingface dataset:
P''Chain: shows "P''[x::=u] \<Longrightarrow>\<^sub>\<tau> P'"
prems_2: shows "(x' # xvec) \<sharp>* (y # yvec)"
has_bochner_integral_even_function: fixes f :: "real \<Rightarrow> 'b"   and x :: "'b" assumes "has_bochner_integral lborel (\<lambda>x. indicat_real {0..} x *\<^sub>R f x) x"   and "\<And>x. f (- x) = f x" shows "has_bochner_integral lborel f (2 *\<^sub>R x)"
extends_def: fixes S :: "'a list"   and S' :: "'a list" shows "(S \<lesssim> S') = (\<exists>S''. S' = S'' @ S)"
abs_Integer_def: shows "Num_syn_class.abs = (\<Lambda> (MkI\<cdot>x). MkI\<cdot>\<bar>x\<bar>)"
deriv_sat_preserving: fixes Ns :: "'a literal multiset set llist" assumes "chain (\<triangleright>) Ns"   and "satisfiable (lhd Ns)" shows "satisfiable (Sup_llist Ns)"
assms_1: shows "(\<box>(\<lambda>s. P s \<longrightarrow> (\<diamond>R) s)) \<sigma>"
bhv2cnf_ge_lActive: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<and> t\<rangle> \<le> \<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n"
poincare_circle_def: fixes z :: "complex_homo"   and r :: "real" shows "poincare_circle z r = {z' \<in> unit_disc. poincare_distance z z' = r}"
af_congruentness: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and xs :: "'a set list" assumes "\<phi> \<sim> \<psi>" shows "af \<phi> xs \<sim> af \<psi> xs"
linear_dependent_inj_imageD: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and s :: "('a, 'b) vec set" assumes "Vector_Spaces.linear (*s) (*s) f"   and "vec.dependent (f ` s)"   and "inj_on f (vec.span s)" shows "vec.dependent s"
frameImpResCommLeft': fixes x :: "name"   and xvec :: "name list"   and F :: "'b frame" shows "\<lparr>\<nu>x\<rparr>\<lparr>\<nu>*xvec\<rparr>F \<hookrightarrow>\<^sub>F \<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>x\<rparr>F"
is_valid_operator_sas_plus_then_4: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and op :: "('a, 'b) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "\<forall>(v, a)\<in>set (effect_of op). \<R>\<^sub>+ \<Psi> v \<noteq> {} \<and> a \<in> \<R>\<^sub>+ \<Psi> v"
prj_simps_3: shows "C.arr \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0"
ordLeq3_finite_infinite: fixes A :: "'d set"   and B :: "'e set" assumes "finite A"   and "infinite B" shows "ordLeq3 (card_of A) (card_of B)"
hom_0: fixes hom :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "zero_hom_0 hom"   and "hom x = (0::'c)" shows "x = (0::'b)"
lincomb_cong: fixes A :: "'a vec set"   and B :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a"   and b :: "'a vec \<Rightarrow> 'a" assumes "A = B"   and "A \<subseteq> carrier_vec n"   and "\<And>v. v \<in> A \<Longrightarrow> a v = b v" shows "lincomb a A = lincomb b B"
add_self_mod_2: fixes m :: "nat" shows "(m + m) mod 2 = 0"
Rep_mat_impl_cases: fixes y :: "nat \<times> nat \<times> 'a iarray iarray" assumes "y \<in> {(nr, nc, m). IArray.length m = nr \<and> IArray.all (\<lambda>r. IArray.length r = nc) m}" obtains x :: "'a mat_impl" where "y = Rep_mat_impl x"
fun_strict_order_def: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set" shows "f <. g \<equiv> f \<le> g \<and> f \<noteq> g"
simps_2: fixes \<O> :: "nat set"   and x :: "'b memref"   and xs :: "'b memref list" shows "read_only_reads \<O> (x # xs) = (case x of Write\<^sub>s\<^sub>b volatile x xa xb A L R W \<Rightarrow> if volatile then read_only_reads (\<O> \<union> A - R) xs else read_only_reads \<O> xs | Read\<^sub>s\<^sub>b volatile a t v \<Rightarrow> if \<not> volatile \<and> a \<notin> \<O> then insert a (read_only_reads \<O> xs) else read_only_reads \<O> xs | Prog\<^sub>s\<^sub>b p1 p2 list \<Rightarrow> read_only_reads \<O> xs | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> read_only_reads (\<O> \<union> A - R) xs)"
WPO_NS_subst: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('f, 'b) Term.term" assumes "(s, t) \<in> WPO_NS" shows "(s \<cdot> \<sigma>, t \<cdot> \<sigma>) \<in> WPO_NS"
potential: fixes p :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b"   and u :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "p 0 = (0::'b)"   and "\<And>n. (0::'b) \<le> p n"   and "\<And>n. t n + p (n + 1) - p n \<le> u n" shows "sum t {..<n} \<le> sum u {..<n}"
b2: fixes x :: "'a" assumes "x \<in> M" shows "x \<in> (id - projection M) -` {0::'a}"
coboundedI2: fixes b :: "'b upred"   and c :: "'b upred"   and a :: "'b upred" assumes "c \<sqsubseteq> b" shows "c \<sqsubseteq> (a \<and> b)"
finsert_iff: fixes a :: "'c"   and b :: "'c"   and A :: "'c fset" shows "(a |\<in>| finsert b A) = (a = b \<or> a |\<in>| A)"
hom_mult: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e"   and x :: "'c"   and y :: "'c" assumes "group_hom G H h"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
eventually_const: fixes F :: "'c filter"   and P :: "bool" assumes "F \<noteq> bot" shows "(\<forall>\<^sub>F x in F. P) = P"
insert_def: fixes a :: "'b"   and B :: "'b set" shows "insert a B = {x. x = a} \<union> B"
assms: fixes i :: "nat" assumes "i < n" shows "p A i m"
c: shows "\<not> (cnf (nnf (\<^bold>\<not> F)) \<turnstile> \<box>)"
simps: fixes a1 :: "'b com option list"   and a2 :: "'b"   and a3 :: "'b com option list"   and a4 :: "'b" shows "(a1, a2) -pc\<rightarrow> (a3, a4) = (\<exists>i Ps s r t. a1 = Ps \<and> a2 = s \<and> a3 = Ps[i := r] \<and> a4 = t \<and> i < length Ps \<and> (Ps ! i, s) -c\<rightarrow> (r, t))"
simps: fixes A :: "'d atMost \<Rightarrow> 'd atMost \<Rightarrow> 'c" shows "sqmatrix_of_matrix (Matrix A) = SqMatrix A"
1: shows "r\<^sup>T \<le> v\<^sup>T"
pw_leof_iff: fixes m :: "'c nres"   and m' :: "'c nres" shows "(m \<le>\<^sub>n m') = (nofail m \<longrightarrow> (\<forall>x. inres m x \<longrightarrow> inres m' x))"
simps: fixes m :: "nat"   and r :: "nat"   and t :: "nat" shows "nonstop m r t = nstd (conf m r t)"
single_comp_portfolio: fixes asset :: "'c"   and qty :: "nat \<Rightarrow> 'd \<Rightarrow> real" shows "portfolio (qty_single asset qty)"
ereal_PInfty_eq_plus: fixes a :: "ereal"   and b :: "ereal" shows "(\<infinity> = a + b) = (a = \<infinity> \<or> b = \<infinity>)"
Atm_Sbis: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and atm :: "'c"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "PL_Indis.compatAtm aval indis atm" shows "(Atm atm, Atm atm) \<in> PL_Indis.Sbis aval tval cval indis"
diff_cancel: fixes a :: "'y" shows "a - a = (0::'y)"
normalize: shows "(R ===> R) (arith_ops_record.normalize ops) normalization_semidom_class.normalize"
Standard_minus: fixes x :: "'a star" assumes "x \<in> Standard" shows "- x \<in> Standard"
cases: fixes a :: "msg"   and bad :: "agent set"   and IK :: "msg set"   and H :: "chan set" assumes "a \<in> extr bad IK H" obtains M :: "msg" where "a = M__" and "M__ \<in> IK"   | c :: "tag" and A :: "agent" and B :: "agent" and M :: "msg" where "a = M" and "Chan c A B M \<in> H" and "c = insec \<or> c = auth \<or> A \<in> bad \<or> B \<in> bad"
right_diff_distrib': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times a (minus b c) = minus (times a b) (times a c)"
pmdl_induct: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and P :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool" assumes "p \<in> pmdl B"   and "P 0"   and "\<And>a p c t. \<lbrakk>a \<in> pmdl B; P a; p \<in> B; c \<noteq> (0::'c)\<rbrakk> \<Longrightarrow> P (a + monom_mult c t p)" shows "P p"
distinct_31: shows "EqualsIgnoreCaseOp \<noteq> string_binop.LessEqOp"
convert_substInfinitesimalQuadratic_less: fixes var :: "nat"   and p :: "real mpoly"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and p' :: "real \<times> real \<times> real"   and a :: "real mpoly"   and A :: "real"   and b :: "real mpoly"   and B :: "real"   and c :: "real mpoly"   and C :: "real"   and d :: "real mpoly"   and D :: "real" assumes "convert_poly var p (xs' @ x # xs) = Some p'"   and "insertion (nth_default 0 (xs' @ x # xs)) a = A"   and "insertion (nth_default 0 (xs' @ x # xs)) b = B"   and "insertion (nth_default 0 (xs' @ x # xs)) c = C"   and "insertion (nth_default 0 (xs' @ x # xs)) d = D"   and "D \<noteq> 0"   and "0 \<le> C"   and "var \<notin> vars a"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "var \<notin> vars d"   and "length xs' = var" shows "eval (quadratic_sub_fm var a b c d (convertDerivative var p)) (xs' @ x # xs) = evalUni (map_atomUni (quadraticSubUni A B C D) (convertDerivativeUni p')) x"
length_sorted_list_of_multiset: fixes A :: "'b multiset" shows "length (sorted_list_of_multiset A) = size A"
assms_1: shows "\<exists>f. h \<stileturn> f"
open_ball: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and d :: "real" assumes "class.metric_space dist uniformity open" shows "open {y. dist x y < d}"
simps_4: fixes M :: "('b, 'c, 'd) KripkeStructure"   and w :: "'d"   and a :: "'b"   and \<phi> :: "('b, 'c) Kform" shows "M, w \<Turnstile> \<^bold>K\<^sub>a \<phi> = (\<forall>w'\<in>relations M a `` {w}. M, w' \<Turnstile> \<phi>)"
arith_special_3: shows "(1::'c) + (1::'c) = (2::'c)"
moebius_decomposition_similarity: fixes a :: "complex"   and b :: "complex"   and d :: "complex" assumes "a \<noteq> 0" shows "mk_moebius a b 0 d = moebius_similarity (a / d) (b / d)"
rec_1: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> 'a" shows "rec_list f1 f2 [] = f1"
abs_ereal_ge0: fixes x :: "ereal" assumes "0 \<le> x" shows "\<bar>x\<bar> = x"
ideD_1: fixes a :: "'a1" assumes "C1.ide a" shows "C1.arr a"
simply_connected_eq_homotopic_circlemaps: fixes S :: "'a set" shows "simply_connected S = (\<forall>f g. continuous_on (sphere 0 1) f \<and> f ` sphere 0 1 \<subseteq> S \<and> continuous_on (sphere 0 1) g \<and> g ` sphere 0 1 \<subseteq> S \<longrightarrow> homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f g)"
odd_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "num" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (numeral.numeral one plus (num.Bit1 n))"
sound_monitor: fixes nfv :: "nat"   and fv :: "nat set"   and sat :: "'c trace \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool"   and M :: "'c prefix \<Rightarrow> (nat \<times> 'd option list) set"   and i :: "nat"   and v :: "'d option list"   and \<pi> :: "'c prefix"   and \<sigma> :: "'c trace" assumes "monitor nfv fv sat M"   and "(i, v) \<in> M \<pi>"   and "prefix_of \<pi> \<sigma>" shows "sat \<sigma> (map the v) i"
Sup_sat_def: shows "Sup = semilattice_neutr_set.F max bot"
sum_eq_0_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "finite F" shows "(comm_monoid_add.sum plus zero f F = zero) = (\<forall>a\<in>F. f a = zero)"
closure_eq_empty: fixes S :: "'a set" shows "(closure S = {}) = (S = {})"
poly_deg_times: fixes p :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'd"   and q :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'd" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "poly_deg (p * q) = poly_deg p + poly_deg q"
left_gpv_inline: fixes callee :: "'b \<Rightarrow> 'g \<Rightarrow> ('h \<times> 'b, 'c, 'e) gpv"   and gpv :: "('a, 'g, 'h) gpv"   and s :: "'b" shows "left_gpv (inline callee gpv s) = inline (\<lambda>s q. left_gpv (callee s q)) gpv s"
psubst_Var_simp2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "goodEnv rho"   and "rho xs x = Some X" shows "Var xs x #[rho] = X"
vwb_lens_mwb: fixes x :: "'a \<Longrightarrow> 'b" assumes "vwb_lens x" shows "mwb_lens x"
right_diff_distrib: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.ring plus zero minus uminus times" shows "times a (minus b c) = minus (times a b) (times a c)"
induct: fixes x :: "'b list"   and G :: "'b list set"   and P :: "'b list \<Rightarrow> bool" assumes "x \<in> \<langle>G\<rangle>\<^sub>F"   and "P \<epsilon>"   and "\<And>w. w \<in> G \<Longrightarrow> P w"   and "\<And>w1 w2. \<lbrakk>w1 \<in> \<langle>G\<rangle>\<^sub>F; P w1; w2 \<in> \<langle>G\<rangle>\<^sub>F; P w2\<rbrakk> \<Longrightarrow> P (w1 \<cdot> w2)"   and "\<And>p q w. \<lbrakk>p \<in> \<langle>G\<rangle>\<^sub>F; P p; q \<in> \<langle>G\<rangle>\<^sub>F; P q; p \<cdot> w \<in> \<langle>G\<rangle>\<^sub>F; P (p \<cdot> w); w \<cdot> q \<in> \<langle>G\<rangle>\<^sub>F; P (w \<cdot> q)\<rbrakk> \<Longrightarrow> P w" shows "P x"
uminus_p64_def: fixes p :: "uint64"   and x :: "uint64" shows "uminus_p64 p x = (if x = 0 then 0 else p - x)"
substitution_code_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val" shows "substitution_code g next = the (the_trivial (snd next) (the (phis g next)))"
Limit_ftimes_in_VsetI: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V" assumes "Limit \<alpha>"   and "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>" shows "A \<times>\<^sub>\<bullet> B \<in>\<^sub>\<circ> Vset \<alpha>"
anycast_simps_2: fixes P :: "'a \<Rightarrow> bool"   and ips :: "nat set"   and m :: "'a" shows "anycast P (groupcast ips m) = P m"
collapse_impl_def_opt: fixes x :: "'b"   and s :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list" shows "collapse_impl x s \<equiv> case s of (S, B, I, P) \<Rightarrow> GS.idx_of_impl (S, B, I, P) x \<bind> (\<lambda>i. ASSERT (i + 1 \<le> length B) \<bind> (\<lambda>_. let B = take (i + 1) B in RETURN (S, B, I, P)))"
sup_inf_distrib1: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x \<union># (y \<inter># z) = x \<union># y \<inter># (x \<union># z)"
Ax_canonical: fixes k :: "'a"   and \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" shows "Atom k, \<Gamma> \<Rightarrow> Atom k, \<Delta>"
g_def: shows "g \<equiv> \<lambda>x y u i. (1 - u) * x i + u * y i"
category_with_terminal_object_axioms: shows "category_with_terminal_object (\<cdot>)"
supp_def: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.pt permute" shows "pt.supp permute x = {a. infinite {b. permute (a \<rightleftharpoons> b) x \<noteq> x}}"
A7: shows "\<forall>TQ SR. TQ \<in>T Q \<and> SR \<in>S R \<longrightarrow> (TQ, \<lbrakk>SR\<rbrakk>) \<in> TRel"
length_prefixes: fixes xs :: "'a list" shows "length (prefixes xs) = length xs + 1"
starlike_imp_simply_connected: fixes S :: "'b set" assumes "starlike S" shows "simply_connected S"
Agent_Pair: fixes X :: "agent"   and X' :: "msg"   and Y' :: "msg" shows "Messages.Agent X \<noteq> \<langle>X', Y'\<rangle>"
case_6: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> 'd"   and f5 :: "'e \<Rightarrow> 'd"   and f6 :: "nat \<Rightarrow> 'd"   and f7 :: "'e \<Rightarrow> 'd"   and f8 :: "'d"   and x6 :: "nat" shows "(case deliver x6 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f6 x6"
leg1_composite: shows "\<mu>\<nu>.leg1 = \<mu>.leg1 \<cdot> \<mu>\<nu>.prj\<^sub>1"
cross7_refl: fixes x :: "(real, 7) vec" shows "x \<times>\<^sub>7 x = 0"
complex_cnj_complex_of_real: fixes x :: "real" shows "cnj (cor x) = cor x"
hom_graph_union_inf: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b"   and T :: "'a set"   and y :: "'a formula"   and b :: "'b" assumes "hom_graph f S x a"   and "hom_graph f T y b" shows "hom_graph f (S \<union> T) (x \<sqinter> y) (a \<sqinter> b)"
imult_infinity: fixes n :: "enat" assumes "0 < n" shows "\<infinity> * n = \<infinity>"
read_icache2_mod_state: fixes s :: "'b sparc_state"   and addr :: "word32"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "fst (case read_instr_cache s addr of None \<Rightarrow> (None, s) | Some w \<Rightarrow> (Some w, s)) = fst (case read_instr_cache (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) addr of None \<Rightarrow> (None, s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>) | Some w \<Rightarrow> (Some w, s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>))"
sup_aci_1: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
sorted_many: fixes x :: "'a"   and y :: "'a"   and zs :: "'a list" assumes "x \<le> y"   and "Specialisation_Examples.sorted (y \<Colon> zs)" shows "Specialisation_Examples.sorted (x \<Colon> y \<Colon> zs)"
clock_numbering_3: shows "\<forall>c\<in>X. v c \<le> n"
lnull_lconcat: fixes xss :: "'c llist llist" shows "lnull (lconcat xss) = (lset xss \<subseteq> {xs. lnull xs})"
Goal: fixes xveca :: "name list"   and Na :: "'a"   and P'a :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xveca\<rparr>\<langle>Na\<rangle> \<prec> P'a"   and "xveca \<sharp>* M"   and "xveca \<sharp>* yvec"   and "xveca \<sharp>* zvec" shows "p \<bullet> \<Psi> \<rhd> P \<longmapsto> (p \<bullet> M)\<lparr>\<nu>*xveca\<rparr>\<langle>Na\<rangle> \<prec> P'a"
integral_indicator: fixes M :: "'b measure"   and A :: "'b set" shows "integral\<^sup>L M (rind A) = Sigma_Algebra.measure M (A \<inter> space M)"
rep_plus: fixes r :: "'c \<Rightarrow>\<^sub>0 'c"   and s :: "'c \<Rightarrow>\<^sub>0 'c" shows "ideal.rep (r + s) = ideal.rep r + ideal.rep s"
r_nth: fixes e :: "nat"   and n :: "nat" shows "eval r_nth [e, n] \<down>= e_nth e n"
less_irrefl: fixes x :: "nat list" shows "\<not> x <\<^sub>v x"
simps_2: fixes f :: "'d \<Rightarrow> 'd option" shows "option_comp_pow f (Suc 0) = f"
abs_eq: shows "min_sls = Abs_semilattice_set min"
times_divide_eq_left: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "times (divide b c) a = divide (times b a) c"
pderiv_pcompose: fixes p :: "'a poly"   and q :: "'a poly" shows "pderiv (p \<circ>\<^sub>p q) = pderiv p \<circ>\<^sub>p q * pderiv q"
completeness_countable: fixes Cs :: "fterm literal set set" assumes "unification"   and "infinite UNIV"   and "finite Cs"   and "\<forall>C\<in>Cs. finite C"   and "\<forall>F G. \<not> eval\<^sub>c\<^sub>s F G Cs" shows "\<exists>Cs'. resolution_deriv Cs Cs' \<and> {} \<in> Cs'"
a_lcomm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
gba_impl_rel_eext_internal_def: fixes Rm :: "('i \<times> 'l) set"   and Rv :: "('g \<times> 'j) set"   and Rl :: "('h \<times> 'k) set" shows "gba_impl_rel_eext Rm Rv Rl \<equiv> \<langle>Rm, Rv \<rightarrow> Rl \<rightarrow> bool_rel\<rangle>gen_gba_impl_rel_eext"
esssup_zero_measure: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" shows "emeasure M {x \<in> space M. esssup M f < f x} = 0"
welltyped_unique: fixes t :: "expr"   and T :: "Unary_PCF.type"   and U :: "Unary_PCF.type" assumes "t ::: T"   and "t ::: U" shows "T = U"
update_assignment_alt_def: fixes u :: "('a \<times> 'b) list"   and as :: "'a \<Rightarrow> 'b" shows "update_assignment_alt u as = (\<lambda>v. case map_of u v of None \<Rightarrow> as v | Some n \<Rightarrow> n)"
prop_rules_def: shows "prop_rules = cycle [andI, andE, impI, impE]"
update_arg_wf_def: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat" shows "update_arg_wf r \<equiv> \<pi>\<^sub>4 r = val \<and> (\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk) \<and> (\<pi>\<^sub>3 r = unk \<longrightarrow> \<pi>\<^sub>5 r = 1)"
kgdd1_points_index_group_block: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set"   and groups :: "'b set set"   and ps :: "'b set" assumes "K_GDD point_set block_collection sizes groups"   and "ps \<subseteq> point_set"   and "card ps = 2" shows "(block_collection + mset_set groups) index ps = 1"
natural_transformation_axioms: shows "natural_transformation B C K L \<tau>'"
matchPres: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "P \<approx> Q" shows "[a\<frown>b]P \<approx> [a\<frown>b]Q"
list_all2_Cons: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'b"   and ys :: "'b list" shows "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
lang_ENC_formula: fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.wf_formula \<Sigma> n \<phi>" shows "project.lang (set \<circ> \<sigma> \<Sigma>) \<pi> lookup n (formula.ENC n (FOV \<phi>)) = \<Union> {formula.enc \<Sigma> (w, I) |w I. length I = n \<and> formula.wf_interp_for_formula \<Sigma> (w, I) \<phi>}"
induct: fixes P :: "('b, 'a) Term.term \<Rightarrow> bool"   and a0 :: "('b, 'a) Term.term" assumes "\<And>x. P (Var x)"   and "\<And>f T. (\<And>x. x \<in> set T \<Longrightarrow> P x) \<Longrightarrow> P (Fun f T)" shows "P a0"
dI: shows "diff_invariant (\<lambda>s. \<nu> s \<noteq> \<mu> s) f U S t\<^sub>0 G"
smult_lincomb: fixes rs :: "'r list"   and s :: "'r"   and ms :: "'m list" assumes "set rs \<subseteq> R"   and "s \<in> R"   and "set ms \<subseteq> M" shows "s \<cdot> rs \<bullet>\<cdot> ms = map ((*) s) rs \<bullet>\<cdot> ms"
snd2: fixes B :: "'a trm"   and B' :: "'a trm"   and A :: "'a trm" assumes "B >>> B'" shows "Snd (SimplyTyped.Pair A B) >>> B'"
components_are_iso: fixes a :: "'a" assumes "C.ide a" shows "D.iso (F a)"
polynomial_function_sum: fixes I :: "'c set"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'e" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> polynomial_function (\<lambda>x. f x i)" shows "polynomial_function (\<lambda>x. sum (f x) I)"