hcju/isabelleps huggingface dataset:
prime_factorization_mult: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "factorial_semiring.prime_factorization zero times one normalize (times x y) = factorial_semiring.prime_factorization zero times one normalize x + factorial_semiring.prime_factorization zero times one normalize y"
UP_m_comm: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "UP_cring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "p \<otimes>\<^bsub>UP R\<^esub> q = q \<otimes>\<^bsub>UP R\<^esub> p"
Equality_def: fixes A :: "'b set"   and B :: "'b set" shows "A <==> B = (- A \<union> B) \<inter> (A \<union> - B)"
diff_compose: fixes k :: "enat"   and M1 :: "('c, 'd) chart set"   and M2 :: "('e, 'f) chart set"   and f :: "'c \<Rightarrow> 'e"   and M3 :: "('g, 'h) chart set"   and g :: "'e \<Rightarrow> 'g" assumes "diff k M1 M2 f"   and "diff k M2 M3 g" shows "diff k M1 M3 (g \<circ> f)"
in_write_sops_conv: fixes sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and xs :: "'b memref list" shows "(sop \<in> write_sops xs) = (\<exists>volatile a v A L R W. Write\<^sub>s\<^sub>b volatile a sop v A L R W \<in> set xs)"
inducts: fixes x1 :: "'d"   and x2 :: "'d"   and x3 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> (('d \<times> 'c) \<times> 'd \<times> 'b) list \<Rightarrow> bool" assumes "terminal_path x1 x2 x3"   and "\<And>v. terminal_vertex v \<Longrightarrow> P v v []"   and "\<And>v\<^sub>1 p\<^sub>1 v\<^sub>2 p\<^sub>2 v' pth. \<lbrakk>((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) \<in> edges; terminal_path v\<^sub>2 v' pth; P v\<^sub>2 v' pth; hyps (nodeOf v\<^sub>1) p\<^sub>1 = None\<rbrakk> \<Longrightarrow> P v\<^sub>1 v' (((v\<^sub>1, p\<^sub>1), v\<^sub>2, p\<^sub>2) # pth)" shows "P x1 x2 x3"
total_onD: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "total_on A r"   and "x \<in> A"   and "y \<in> A" shows "(x, y) \<in> r \<or> x = y \<or> (y, x) \<in> r"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. P (y # xs) \<Longrightarrow> P (x # y # xs)" shows "P a0"
simps_2: fixes onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'c) \<times> 'd, 'e seq_action) automaton"   and p\<^sub>1 :: "net_tree"   and p\<^sub>2 :: "net_tree" shows "opnet onp (p\<^sub>1 \<parallel> p\<^sub>2) = \<lparr>init = {(\<sigma>, SubnetS s\<^sub>1 s\<^sub>2) |\<sigma> s\<^sub>1 s\<^sub>2. (\<sigma>, s\<^sub>1) \<in> init (opnet onp p\<^sub>1) \<and> (\<sigma>, s\<^sub>2) \<in> init (opnet onp p\<^sub>2) \<and> net_ips s\<^sub>1 \<inter> net_ips s\<^sub>2 = {}}, trans = opnet_sos (automaton.trans (opnet onp p\<^sub>1)) (automaton.trans (opnet onp p\<^sub>2))\<rparr>"
Equivalence_def: shows "Equivalence = {(a, b). \<exists>A\<in>Partition. a \<in> A \<and> b \<in> A}"
ExitBlockNormal: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and nrms :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and abrs :: "('e, 'd, 'f) com list" assumes "\<Gamma>\<turnstile>nrms,css \<Down> Normal s" shows "\<Gamma>\<turnstile>[],(nrms, abrs) # css \<Down> Normal s"
simps_15: fixes v :: "fm"   and va :: "fm" shows "not (Iff v va) = NOT (Iff v va)"
pre_adj_if_postR: fixes lv :: "('c \<times> nat) tree"   and l :: "'c"   and a :: "nat"   and r :: "('c \<times> nat) tree"   and r' :: "('c \<times> nat) tree" assumes "invar \<langle>lv, (l, a), r\<rangle>"   and "post_del r r'" shows "pre_adjust \<langle>lv, (l, a), r'\<rangle>"
InitializePhase_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "InitializePhase s s' p = (disksWritten s' = (disksWritten s)(p := {}) \<and> blocksRead s' = (blocksRead s)(p := \<lambda>d. {}))"
integral_add: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f integrable_on S"   and "g integrable_on S" shows "integral S (\<lambda>x. f x + g x) = integral S f + integral S g"
simps: fixes a1 :: "((nat \<times> nat) \<times> transition) fset"   and a2 :: "nat"   and a3 :: "nat \<Rightarrow>f value option"   and a4 :: "(String.literal \<times> value list) list" shows "recognises_execution a1 a2 a3 a4 = ((\<exists>e s r. a1 = e \<and> a2 = s \<and> a3 = r \<and> a4 = []) \<or> (\<exists>e s r l i t. a1 = e \<and> a2 = s \<and> a3 = r \<and> a4 = (l, i) # t \<and> (\<exists>(s', T)|\<in>|possible_steps e s r l i. recognises_execution e s' (evaluate_updates T i r) t)))"
multiset_inter_commute: fixes A :: "'b multiset"   and B :: "'b multiset" shows "A \<inter># B = B \<inter># A"
oexp_sup: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V" assumes "\<alpha> \<noteq> 0"   and "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "\<alpha> \<up> (\<beta> \<squnion> \<gamma>) = \<alpha> \<up> \<beta> \<squnion> \<alpha> \<up> \<gamma>"
vec_to_iarray_row': fixes i :: "'d"   and A :: "(('b, 'c) vec, 'd) vec" shows "vec_to_iarray (row i A) = matrix_to_iarray A !! mod_type_class.to_nat i"
less_eqE: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less_eq a b" obtains c :: "'d" where "b = plus a c"
simps_1: fixes \<mu> :: "'c" shows "\<^bold>\<lfloor>\<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0\<^bold>\<rfloor> = \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0"
length_takefill: fixes fill :: "'a"   and n :: "nat"   and l :: "'a list" shows "length (takefill fill n l) = n"
exec_move_Seq1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e;; e') h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
inner_iarray_def: fixes A :: "real iarray"   and B :: "real iarray" shows "A \<bullet>i B = (\<Sum>n = 0..<IArray.length A. A !! n * B !! n)"
l_neg: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x \<oplus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"
uminus_fun_def: fixes A :: "'a \<Rightarrow> 'b" shows "- A = (\<lambda>x. - A x)"
support_wpw_Un: fixes b :: "bexp"   and c :: "com"   and x :: "char list"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (\<lambda>l. wpw b c (l x) Q l) \<subseteq> insert x (\<Union>n. support (wpw b c n Q))"
contour_integral_def: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" shows "contour_integral g f \<equiv> SOME i. (f has_contour_integral i) g \<or> \<not> f contour_integrable_on g \<and> i = 0"
simple_ruleset_tail: fixes r :: "'a rule"   and rs :: "'a rule list" assumes "simple_ruleset (r # rs)" shows "simple_ruleset rs"
plus_ratfps_code: fixes x :: "'a ratfps"   and y :: "'a ratfps" shows "local.quot_of_ratfps (x + y) = (let (a, b) = local.quot_of_ratfps x; (c, d) = local.quot_of_ratfps y in normalize_quot (a * d + b * c, b * d))"
inj_on_opp_chambers': fixes C :: "'a set"   and D :: "'a set" assumes "folding.chamber C"   and "C \<notin> f \<turnstile> X"   and "folding.chamber D"   and "D \<notin> f \<turnstile> X"   and "f ` C = f ` D" shows "C = D"
xs_min: shows "\<forall>x. x \<in> set (j # xs) \<longrightarrow> x < min (nrows A) (ncols A)"
C: shows "C \<in> carrier_mat (m - 1) (m - 1)"
res_seq_subseq: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "is_subseq_of (res_seq s k) (res_seq s (Suc k))"
merge_def: fixes fs :: "('c \<Rightarrow> 'd option) set" shows "merge fs = (\<lambda>x. if \<exists>f\<in>fs. x \<in> dom f then THE y. \<forall>f\<in>fs. x \<in> dom f \<longrightarrow> f x = y else None)"
c0: fixes y :: "'a vec" assumes "y \<in> Y" shows "(0::'a) \<le> c y"
ereal_mult_m1: fixes x :: "ereal" shows "x * ereal (- 1) = - x"
A2': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and p :: "'b"   and q :: "'b"   and r :: "'b"   and s :: "'b" assumes "tarski_first3 C"   and "C a b p q"   and "C a b r s" shows "C p q r s"
unfoldI: fixes P :: "pi"   and Q :: "pi" assumes "P \<leadsto><weakBisim> Q"   and "Q \<leadsto><weakBisim> P" shows "P \<simeq> Q"
Suc: shows "(x \<oplus> y) [^] n = (\<Oplus>i\<in>{..n}. [(n choose i)] \<cdot> (x [^] i \<otimes> y [^] (n - i)))"
F2: shows "R2 \<Longrightarrow>\<^sub>A P y * F2"
axioms_1: fixes return :: "'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and altc :: "'e cset \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_state_altc return bind get put altc" shows "monad_state return bind get put"
simps_26: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "nat"   and x13 :: "nat"   and x14 :: "k"   and x15 :: "nat"   and x16 :: "nat"   and x17 :: "nat"   and x18 :: "bool" shows "(case Rreq x11 x12 x13 x14 x15 x16 x17 x18 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f1 x11 x12 x13 x14 x15 x16 x17 x18"
ide_is_iso: fixes a :: "'a" assumes "ide a" shows "local.iso a"
no: fixes I :: "'c"   and a :: "'b"   and b :: "'b"   and J :: "'c"   and c :: "'b"   and d :: "'b" assumes "Q I a b"   and "Q J c d"   and "a \<noteq> b \<and> b \<noteq> c \<and> c \<noteq> d \<and> a = d \<or> a \<noteq> b \<and> b = c \<and> c \<noteq> d \<and> a = d" shows "P I J"
intro: fixes Ch :: "'d \<Rightarrow> 'e set \<Rightarrow> 'e set" assumes "\<forall>h. irc (Ch h)" shows "ContractsWithIRC_axioms Ch"
hta_empty_def: fixes u :: "unit" shows "hta_empty u \<equiv> init_hta (hs.empty ()) (ls.empty ())"
order_Rep_N: fixes a :: "'b N"   and b :: "'b N" shows "(a \<le> b) = (Rep_N a \<le> Rep_N b)"
mkarr_memb_1: fixes t :: "'c local.term"   and f :: "'c local.term set" assumes "t \<in> f"   and "\<F>C.arr f" shows "Arr t"
has_integral_cmult_real: fixes c :: "real"   and f :: "'c \<Rightarrow> real"   and x :: "real"   and A :: "'c set" assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A" shows "((\<lambda>x. c * f x) has_integral c * x) A"
subst_support_comp_split_1: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<I> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "\<theta> \<circ>\<^sub>s \<delta> supports \<I>"   and "subst_domain \<theta> \<inter> range_vars \<theta> = {}" shows "\<theta> supports \<I>"
continuous_on_codomain: fixes c :: "('d, 'c) chart" shows "continuous_on (codomain c) (inv_chart c)"
intro: fixes U\<^sub>1 :: "'a set"   and sup\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and U\<^sub>2 :: "'b set"   and sup\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "semilattice_sup_ow U\<^sub>1 sup\<^sub>1 le\<^sub>1 ls\<^sub>1"   and "semilattice_sup_ow U\<^sub>2 sup\<^sub>2 le\<^sub>2 ls\<^sub>2" shows "semilattice_sup_pair_ow U\<^sub>1 sup\<^sub>1 le\<^sub>1 ls\<^sub>1 U\<^sub>2 sup\<^sub>2 le\<^sub>2 ls\<^sub>2"
h1c2: shows "Const (real (i + 1)) \<noteq> 0"
rk2_fas'_def: fixes ode_ops :: "ode_ops"   and D :: "nat" shows "ode_operations.rk2_fas' ode_ops D = map fold_const_fa (ode_operations.rk2_fas ode_ops (floatarith.Var (2 * D)) (map floatarith.Var [0..<D]) (floatarith.Var (2 * D + 1)) (map floatarith.Var [D..<2 * D]) (floatarith.Var (2 * D + 2)))"
card_bijections_range_permutation_eq_1: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B"   and "card A = card B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // range_permutation A B) = 1"
brace90shannon: fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and G :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and H :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and ass :: "'a \<Rightarrow> bool"   and i :: "'a" shows "bf_ite F G H ass = bf_ite (\<lambda>l. l i) (bf_ite (bf_restrict i True F) (bf_restrict i True G) (bf_restrict i True H)) (bf_ite (bf_restrict i False F) (bf_restrict i False G) (bf_restrict i False H)) ass"
simps: fixes p :: "int"   and pm :: "int"   and m :: "nat"   and bnd :: "int" shows "find_exponent_main p pm m bnd = (if bnd < pm then m else find_exponent_main p (pm * p) (Suc m) bnd)"
rcoset_mult_add: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ideal I R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "[mod I:] (I +>\<^bsub>R\<^esub> x) \<Otimes>\<^bsub>R\<^esub> (I +>\<^bsub>R\<^esub> y) = I +>\<^bsub>R\<^esub> x \<otimes>\<^bsub>R\<^esub> y"
in_set_simps_3: fixes x :: "'c" shows "(x \<in> set []) = False"
Gamma_def: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set" shows "Gamma num' plus' \<gamma> \<equiv> Val_abs \<gamma> num' plus'"
numeral_tree_conv_pure: fixes n :: "num" shows "numeral n = pure (numeral n)"
HEndPhase0_maxBalInp: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and b :: "nat"   and v :: "InputsOrNi" assumes "HEndPhase0 s s' q"   and "maxBalInp s b v"   and "Inv1 s" shows "maxBalInp s' b v"
subs_wf_sub_rel_of: fixes prb :: "('d, 'e, 'f) pre_RedBlack" assumes "RedBlack prb" shows "wf_sub_rel_of (red prb) (subs prb)"
split_extended_all: fixes P :: "'c \<Rightarrow> prop" shows "(\<And>z. PROP P z) \<equiv> (\<And>u y. PROP P (h (u, y)))"
winningAllocationPossible: fixes b :: "'d \<times> 'e set \<Rightarrow> 'f"   and G :: "'e set"   and N :: "'d set" shows "winningAllocationRel N G (\<subseteq>) b (allAllocations N G)"
vcard_vunion: fixes A :: "V"   and B :: "V" assumes "vdisjnt A B" shows "vcard (A \<union>\<^sub>\<circ> B) = vcard A \<oplus> vcard B"
gs: shows "gs = fs[i := fs ! i + of_int l \<cdot>\<^sub>v fs ! j]"
some_equality: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(SOME x. P x) = a"
is_strongly_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 "is_strongly_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> local.a_get_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h). set |h \<turnstile> local.a_get_component ptr|\<^sub>r; arg_components' = \<Union>ptr\<in>(\<Union>ptr\<in>S\<^sub>a\<^sub>r\<^sub>g. set |h \<turnstile> local.a_get_component ptr|\<^sub>r) \<inter> fset (object_ptr_kinds h'). set |h' \<turnstile> local.a_get_component ptr|\<^sub>r in removed_pointers \<subseteq> arg_components \<and> added_pointers \<subseteq> arg_components' \<and> S\<^sub>r\<^sub>e\<^sub>s\<^sub>u\<^sub>l\<^sub>t \<subseteq> arg_components' \<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> local.a_get_component ptr|\<^sub>r). preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t outside_ptr id) h h'))"
f: shows "\<forall>i. simple_function lebesgue (f i)"
assms_3: shows "B = A - {a} \<union> {b}"
cones_map_mapsto: fixes f :: "'b" assumes "D.C.arr f" shows "D.cones_map f \<in> extensional (D.cones (D.C.cod f)) \<inter> (D.cones (D.C.cod f) \<rightarrow> D.cones (D.C.dom f))"
wf_trm_subst_rm_vars: fixes t :: "('fun, 'b) Term.term"   and \<delta> :: "'b \<Rightarrow> ('fun, 'b) Term.term"   and X :: "'b set" assumes "wf\<^sub>t\<^sub>r\<^sub>m (t \<cdot> \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m (t \<cdot> rm_vars X \<delta>)"
simps_3: fixes "is" :: "'b set"   and t :: "'b"   and b :: "bool" shows "interrupt_action_ok is (IsInterrupted t b) = (b = (t \<in> is))"
to_tree_order_si_def: fixes ptr :: "(_) object_ptr" shows "to_tree_order_si ptr = Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. Heap_Error_Monad.bind (case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> return [] | Some element_ptr \<Rightarrow> Heap_Error_Monad.bind (get_shadow_root element_ptr) (\<lambda>shadow_root_opt. case shadow_root_opt of None \<Rightarrow> return [] | Some shadow_root_ptr \<Rightarrow> return [cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr])) (\<lambda>shadow_root_part. Heap_Error_Monad.bind (map_M to_tree_order_si (map cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r children @ shadow_root_part)) (\<lambda>treeorders. return (ptr # concat treeorders))))"
repc_preds: fixes I :: "interp"   and f :: "char"   and d :: "real" shows "Preds (repc I f d) = Preds I"
is_natural_2: fixes f :: "'a" assumes "S.arr f" shows "L (S.cod f) \<cdot>\<^sub>S L f = L f"
degree3_coeffs: fixes p :: "'a poly" assumes "degree p = 3" shows "\<exists>a b c d. p = [:d, c, b, a:] \<and> a \<noteq> (0::'a)"
sum_augmenting_alt: fixes A :: "(nat \<times> nat) set"   and p :: "(nat \<times> nat) list" assumes "finite A" shows "sum (augmentingFlow p) A = resCap p * of_nat (card (A \<inter> set p))"
inverse_arrowsE: fixes f :: "'d"   and g :: "'d" assumes "D.inverse_arrows f g" obtains "D.ide (g \<cdot>\<^sub>D f)" and "D.ide (f \<cdot>\<^sub>D g)"
Nil_in_conc: fixes A :: "'b list set"   and B :: "'b list set" shows "([] \<in> A @@ B) = ([] \<in> A \<and> [] \<in> B)"
admits_triangular_reduction_intro: fixes U :: "'b mat"   and A :: "'b mat" assumes "invertible_mat U"   and "U \<in> carrier_mat (dim_col A) (dim_col A)"   and "lower_triangular (A * U)" shows "admits_triangular_reduction A"
to_bl_of_bin: fixes bin :: "int" shows "to_bl (word_of_int bin) = bin_to_bl LENGTH('a) bin"
wf_subcls1: fixes G :: "prog" assumes "ws_prog G" shows "wf ((subcls1 G)\<inverse>)"
collision_within_eq: assumes "s\<^sub>o \<le> ego.s_stop"   and "ego.s_stop < other.s_stop" shows "collision {0..} = collision {0..<min ego.t_stop other.t_stop}"
simps_10: fixes p :: "fm"   and q :: "fm" shows "bound0 (And p q) = (bound0 p \<and> bound0 q)"
result_return_resumption: fixes x :: "'d" shows "result (return_resumption x) = x"
sum_le_included: 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 s :: "'d set"   and t :: "'e set"   and g :: "'e \<Rightarrow> 'c"   and i :: "'e \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite s"   and "finite t"   and "\<forall>y\<in>t. less_eq zero (g y)"   and "\<forall>x\<in>s. \<exists>y\<in>t. i y = x \<and> less_eq (f x) (g y)" shows "less_eq (comm_monoid_add.sum plus zero f s) (comm_monoid_add.sum plus zero g t)"
ignores_old_vars_conc_meval_sched: fixes P :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list \<Rightarrow> bool"   and sched :: "nat list"   and gc\<^sub>C :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and gc\<^sub>C' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" assumes "ignores_old_vars P"   and "conc.meval_sched_abv gc\<^sub>C sched gc\<^sub>C'"   and "length (fst gc\<^sub>C) = length cms" shows "P (map snd (fst gc\<^sub>C)) = P (map snd (fst gc\<^sub>C'))"
Exists: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and i :: "name" assumes "H \<turnstile> A IMP B"   and "atom i \<sharp> B"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> SyntaxN.Ex i A IMP B"
eq_imp_below: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.po below"   and "x = y" shows "below x y"
Units_def: fixes G :: "('c, 'd) monoid_scheme" shows "Units G = {y \<in> carrier G. \<exists>x\<in>carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>}"
split_indicator: fixes P :: "'b \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" shows "P (indicator S x) = ((x \<in> S \<longrightarrow> P (1::'b)) \<and> (x \<notin> S \<longrightarrow> P (0::'b)))"
test_compl_three: fixes x :: "'a" shows "tc (tc (tc x)) = tc x"
strictly_generalizes_atm_def: fixes A :: "('a, 'b) Term.term"   and B :: "('a, 'b) Term.term" shows "strictly_generalizes_atm A B = strict generalizes_atm A B"
apred_def: shows "apred \<equiv> \<lambda>a (pc, s, stk) (pc', s', stk'). pc' = pc + size (acomp a) \<and> s' = s \<and> stk' = aval a s # stk"
step: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "\<phi> xs"   and "(\<phi> until \<psi>) (stl xs)" shows "(\<phi> until \<psi>) xs"
hassoc_lookup_def: fixes k :: "'b"   and ht :: "('b \<times> 'a) list" shows "hassoc_lookup k ht = return (map_of ht k)"
Let: fixes p :: "pat"   and t :: "trm"   and \<Gamma> :: "(name \<times> ty) list"   and T :: "ty"   and \<Delta> :: "(name \<times> ty) list"   and u :: "trm"   and U :: "ty" assumes "supp p \<sharp>* t"   and "\<Gamma> \<turnstile> t : T"   and "\<turnstile> p : T \<Rightarrow> \<Delta>"   and "\<Delta> @ \<Gamma> \<turnstile> u : U" shows "\<Gamma> \<turnstile> (LET p = t IN u) : U"
elims: fixes x :: "'b list"   and xa :: "'b \<times> 'b option"   and y :: "'b list" assumes "insert_spec x xa = y" obtains xs :: "'b list" and oid :: "'b" where "xb__ = xsb__" and "xa = (oida__, None)" and "y = oida__ # xsb__"   | oid :: "'b" and v :: "'b" where "xb__ = []" and "xa = (oidb__, Some v)" and "y = []"   | x :: "'b" and xs :: "'b list" and oid :: "'b" and ref :: "'b" where "xb__ = x # xs" and "xa = (oid, Some ref)" and "y = (if x = ref then x # oid # xs else x # insert_spec xs (oid, Some ref))"
is_extensional: fixes f :: "'a" assumes "\<not> B.arr f" shows "UP f = null"