hcju/isabelleps huggingface dataset:
gr_implies_not_zero: fixes m :: "'a multiset"   and n :: "'a multiset" assumes "m \<subset># n" shows "n \<noteq> {#}"
l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs \<and> l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs"
gbinomial_of_nat_symmetric: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "of_nat n gchoose k = of_nat n gchoose (n - k)"
cSym: fixes R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "(\<Psi>, R, S) \<in> Rel" shows "(\<Psi>, S, R) \<in> Rel"
abs_eq: fixes xa :: "i"   and x :: "j \<Rightarrow> i \<Rightarrow> bool" shows "(xa \<Turnstile> make\<o> x) = x dj xa"
fset_induct: fixes P :: "'b fset \<Rightarrow> bool"   and S :: "'b fset" assumes "P {||}"   and "\<And>x S. P S \<Longrightarrow> P (finsert x S)" shows "P S"
IIface: fixes \<beta> :: "'a common_primitive \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> ternaryvalue" assumes "primitive_matcher_generic \<beta>" shows "\<forall>p i. \<beta> (IIface i) p = bool_to_ternary (match_iface i (p_iiface p))"
tuple_decode_nonzero: fixes n :: "nat"   and i :: "nat" assumes "0 < n" shows "tuple_decode n i = pdec1 i # tuple_decode (n - 1) (pdec2 i)"
SD_pref_profileD_1: fixes i :: "'a"   and q :: "'b pmf"   and p :: "'b pmf" assumes "i \<in> agents"   and "SD (R i) q p" shows "p \<in> lotteries_on alts"
RPCidle: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and p :: "PrIds"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState" shows "\<turnstile> \<not> $Calling send p \<longrightarrow> \<not> RPCNext send rcv rst p"
one_int_def: shows "1 \<equiv> abs_int (1, 0)"
some_gcd_0_1: fixes a :: "'a" shows "is_gcd (some_gcd a (0::'a)) a (0::'a)"
i_shrink_last_nth: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat" assumes "0 < k" shows "(f \<div>\<^sub>l k) n = f (n * k + k - Suc 0)"
int_transE_proj: fixes T :: "'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wt T" shows "Ik.int (transE \<xi>) T = proj (tpOf T) (I.int \<xi> T)"
indep_subset: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set"   and Y :: "'b set" assumes "indep_system carrier indep"   and "indep X"   and "Y \<subseteq> X" shows "indep Y"
operative_realI: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "real set \<Rightarrow> 'a" assumes "operative f z g" shows "operative_real f z g"
coeffs_def: fixes p :: "'a poly" shows "coeffs p = (if p = 0 then [] else map (coeff p) [0..<Suc (degree p)])"
spmf_rel_mono_strong: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c spmf"   and g :: "'d spmf"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. \<lbrakk>x \<in> set_spmf f; y \<in> set_spmf g; A x y\<rbrakk> \<Longrightarrow> B x y" shows "rel_spmf B f g"
omega_cases: fixes x :: "'b"   and z :: "'b"   and R :: "('b \<times> 'b) set" assumes "(x, z) \<in> Omega_Algebra_Models.omega R" obtains y :: "'b" where "(x, y) \<in> R" and "(y, z) \<in> Omega_Algebra_Models.omega R"
fc_equiv_def: fixes G :: "('e, 'f) pre_digraph"   and M :: "'f pre_map"   and a :: "'f"   and b :: "'f" assumes "digraph_map G M" shows "digraph_map.fc_equiv M a b \<equiv> a \<in> pre_digraph_map.face_cycle_set M b"
dims_Aw: shows "dims Aw = replicate (2 * N_half) (last rs)"
lim_inverse_n: shows "(\<lambda>n. inverse (of_nat n)) \<longlonglongrightarrow> (0::'b)"
sff_2: shows "square_free (prod_list (map fst bs))"
NotActualDisjD: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "\<not> [\<^bold>\<A>(\<phi> \<^bold>\<or> \<psi>) in v]" shows "\<not> [\<^bold>\<A>\<phi> \<^bold>\<or> \<^bold>\<A>\<psi> in v]"
B_rw: shows "(\<Prod>i = 0..<n. B $$ (g i, \<sigma> i)) = (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))"
mult_not_zero: fixes a :: "'a"   and b :: "'a" assumes "a \<sqinter> b \<noteq> bot" shows "a \<noteq> bot \<and> b \<noteq> bot"
subdg_objD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
greaterThanAtMost_empty_iff: fixes k :: "complex"   and l :: "complex" shows "(complex_lex.greaterThanAtMost k l = {}) = (\<not> k <\<^sub>\<complex> l)"
real_complete: fixes A :: "real set" assumes "\<exists>a. a \<in> A"   and "\<exists>y. \<forall>a\<in>A. a \<le> y" shows "\<exists>x. lub A x"
eval_rec_models: fixes X :: "'s odlist"   and Y :: "'s odlist"   and s :: "'s"   and \<phi> :: "('a, 'p) Kform" assumes "toSet X \<subseteq> toSet Y"   and "s \<in> toSet X" shows "(s \<in> toSet (eval_rec (clock_evalProp envVal) (clock_knowledge envObs Y) (clock_commonKnowledge envObs Y) X \<phi>)) = clock_repMC (toSet Y), s \<Turnstile> \<phi>"
Poly_cCons_eq_pCons_Poly: fixes a :: "'b"   and p :: "'b list" shows "Poly (a ## p) = pCons a (Poly p)"
invar_ins_tree: fixes t :: "'b tree"   and ts :: "'b tree list" assumes "invar_tree t"   and "invar ts"   and "\<forall>t'\<in>set ts. rank t \<le> rank t'" shows "invar (ins_tree t ts)"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Cod (\<nu>' \<star> \<mu>'))"
C1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<Psi> \<simeq> \<Psi>'" shows "(\<Psi>', P, Q) \<in> Rel'"
holomorphic_derivI: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and x :: "complex"   and T :: "complex set" assumes "f holomorphic_on S"   and "open S"   and "x \<in> S" shows "(f has_field_derivative deriv f x) (at x within T)"
that: fixes h :: "('c \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'a" assumes "h \<in> hom (free_Abelian_group S) G"   and "\<And>x. x \<in> S \<Longrightarrow> h (frag_of x) = f x" shows "thesis"
intros: fixes base :: "nat"   and M :: "hmultiset"   and e :: "nat" assumes "2 \<le> base"   and "\<forall>m\<in>#hmsetmset M. e \<le> decode base 0 m" shows "aligned\<^sub>d base e M"
nvC_varsC: fixes x :: "Preliminaries.var"   and c :: "('a, 'b) lit list" assumes "x \<in> nvC c" shows "x \<in> varsC c"
ps: shows "ps \<subseteq> bl \<and> int (card ps) = \<t>"
simps_7: fixes x1 :: "'b"   and x2 :: "'a"   and x3 :: "'b list" shows "set2_ta_rule x1 \<rightarrow> x2 x3 = {x2}"
saturated_def: fixes N :: "'a set" shows "saturated N = (Inf_from N \<subseteq> Red_I N)"
out_arcs_del_arc_iff: fixes a :: "'b"   and u :: "'a" shows "out_arcs (del_arc a) u = (if tail G a = u then out_arcs G u - {a} else out_arcs G u)"
division_filter_not_empty: fixes a :: "'a"   and b :: "'a" shows "division_filter (cbox a b) \<noteq> bot"
pow_eq_div2: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier R"   and "[m] \<cdot> x = [n] \<cdot> x" shows "[(m - n)] \<cdot> x = \<zero>"
ptensor_mat_assoc: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and vars3 :: "nat set"   and m1 :: "'b mat"   and m2 :: "'b mat"   and m3 :: "'b mat" assumes "vars1 \<inter> vars2 = {}"   and "(vars1 \<union> vars2) \<inter> vars3 = {}"   and "vars1 \<union> vars2 \<union> vars3 \<subseteq> {0..<length dims}" shows "partial_state2.ptensor_mat dims (vars1 \<union> vars2) vars3 (partial_state2.ptensor_mat dims vars1 vars2 m1 m2) m3 = partial_state2.ptensor_mat dims vars1 (vars2 \<union> vars3) m1 (partial_state2.ptensor_mat dims vars2 vars3 m2 m3)"
largest_contained_prefix_subset: fixes a :: "'b word"   and r :: "'b wordinterval"   and p :: "'b prefix_match" assumes "largest_contained_prefix a r = Some p" shows "wordinterval_to_set (prefix_to_wordinterval p) \<subseteq> wordinterval_to_set r"
algebra_iff_Int: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "algebra \<Omega> M = (M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<Omega> - a \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<inter> b \<in> M))"
scaleR_nonpos_nonneg: fixes a :: "real"   and x :: "'a" assumes "a \<le> 0"   and "(0::'a) \<le> x" shows "a *\<^sub>R x \<le> (0::'a)"
AR_imp_ANR: fixes S :: "'b set" assumes "AR S" shows "ANR S"
member_filter: fixes P :: "'b \<Rightarrow> bool"   and L :: "'b list"   and x :: "'b" assumes "List.member (filter P L) x" shows "List.member L x"
STA_drop_block': fixes A :: "'b set"   and n :: "nat"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list" assumes "A, n \<turnstile> (ps, a) # branch"   and "(ps, a) \<in>. branch" shows "A, Suc n \<turnstile> branch"
ipurge_tr_rev_trace: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and u :: "'d" assumes "secure P I D"   and "xs \<in> traces P" shows "ipurge_tr_rev I D u xs \<in> traces P"
Union_iff: fixes A :: "'a"   and C :: "'a set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
ssubst_sopen_distrib: fixes t' :: "sterm"   and x :: "char list"   and n :: "nat"   and s :: "sterm"   and p :: "sterm"   and t :: "sterm" assumes "lc t'" shows "[x \<rightarrow> t'] {n \<rightarrow> [s,p]} t = {n \<rightarrow> [[x \<rightarrow> t'] s,[x \<rightarrow> t'] p]} [x \<rightarrow> t'] t"
sign_simps_51: fixes ak :: "'d"   and bk :: "'d" shows "(ak * bk < (0::'d)) = ((0::'d) < ak \<and> bk < (0::'d) \<or> ak < (0::'d) \<and> (0::'d) < bk)"
insert_nth_take_drop: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "insert_nth n x xs = take n xs @ [x] @ drop n xs"
prop_atoms_subfrmlsn: fixes \<phi> :: "'b ltln" shows "prop_atoms \<phi> \<subseteq> subfrmlsn \<phi>"
exp_of_derived_in_carrier: fixes H :: "'a set"   and n :: "nat" assumes "H \<subseteq> carrier G" shows "(derived G ^^ n) H \<subseteq> carrier G"
insert_mono: fixes C :: "'q set"   and D :: "'q set"   and a :: "'q" assumes "C \<subseteq> D" shows "insert a C \<subseteq> insert a D"
distinct_5: fixes x1 :: "b_e"   and x41 :: "nat"   and x42 :: "e list"   and x43 :: "e list" shows "$x1 \<noteq> Label x41 x42 x43"
sop_assoc_def: fixes R :: "('e, 'f) Ring_scheme"   and s :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and A :: "'g set" shows "sop_assoc R s A = (\<forall>a\<in>carrier R. \<forall>b\<in>carrier R. \<forall>x\<in>A. a \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<^sub>s\<cdot> x = a \<^sub>s\<cdot> (b \<^sub>s\<cdot> x))"
objI_trg: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr a"   and "Span.trg a = a" shows "Span.obj a"
ereal_uminus_complement: fixes S :: "ereal set" shows "uminus ` (- S) = - uminus ` S"
ACI_sym: fixes r :: "'b rexp"   and s :: "'b rexp" assumes "ACI r s" shows "ACI s r"
add_flip_cancel: fixes a :: "'c"   and b :: "'c" shows "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0"
djf_def: fixes f :: "'a \<Rightarrow> fm"   and p :: "'a"   and q :: "fm" shows "djf f p q = (if q = T then T else if q = F then f p else let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)"
fromQ: fixes x :: "'a['c, 'b]" shows "fromQ x \<in> {x. Quantity.dim x = QD('c)}"
weight_stable_le: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('f, 'd) Term.term" assumes "weight s \<le> weight t"   and "vars_term_ms (SCF s) \<subseteq># vars_term_ms (SCF t)" shows "weight (s \<cdot> \<sigma>) \<le> weight (t \<cdot> \<sigma>)"
cont_image: fixes f :: "'b \<Rightarrow> 'c" shows "cont \<Union> (\<subseteq>) \<Union> (\<subseteq>) ((`) f)"
assms: shows "wlp \<pi> (WHILE b DO c) Q s\<^sub>0"
eucl_truncate_down: fixes e :: "nat"   and x :: "'b" shows "eucl_truncate_down e x \<le> x"
swap_commute: fixes n :: "nat"   and m :: "nat"   and p :: "'b poly" shows "swap n m p = swap m n p"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) f"
star_inductl: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and star :: "'a \<Rightarrow> 'a"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.left_near_kleene_algebra plus times one less_eq less star"   and "less_eq (plus z (times x y)) y" shows "less_eq (times (star x) z) y"
S2: shows "{x. \<exists>y. abs_of_diff x a + y = 0} = {a}"
ResF: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and x :: "name" assumes "P \<Longrightarrow>\<alpha> \<prec> P'"   and "x \<sharp> \<alpha>" shows "<\<nu>x>P \<Longrightarrow>\<alpha> \<prec> <\<nu>x>P'"
multiplicity_eqI: fixes p :: "'a"   and n :: "nat"   and x :: "'a" assumes "p ^ n dvd x"   and "\<not> p ^ Suc n dvd x" shows "multiplicity p x = n"
r_into_rtrancl: fixes p :: "'x \<times> 'x"   and r :: "('x \<times> 'x) set" assumes "p \<in> r" shows "p \<in> r\<^sup>*"
a_translate_def: shows "a_translate = (\<lambda>r\<in>carrier R. restrict ((\<oplus>) r) (carrier R))"
2: shows "\<guillemotleft>\<Psi>o yx : Hom_DopxG.map yx \<rightarrow>\<^sub>S Hom_FopxC.map yx\<guillemotright>"
map_1: fixes f :: "'d \<Rightarrow> 'c" shows "map f [] = []"
baEA: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b"   and \<phi> :: "'b \<Rightarrow> bool" assumes "\<exists>i\<ge>n. active c (t i)"   and "dynamic_component.eval tCMP active c t t' n (dynamic_component.ba \<phi>)" shows "\<phi> (tCMP c (t (dynamic_component.nxtAct active c t n)))"
run_one_step_label_break_imp_break: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and ln :: "nat"   and les :: "e list"   and es :: "e list"   and s' :: "s"   and vs' :: "v list"   and n :: "nat"   and res :: "v list" assumes "run_one_step d i (s, vs, ves, Label ln les es) = (s', vs', RSBreak n res)" shows "run_step d i (s, vs, es) = (s', vs', RSBreak (Suc n) res)"
test_inf_left_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "test x"   and "test y"   and "test z" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
qFresh_qPsubst_commute_qAbs: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and xs :: "'varSort"   and x :: "'var" assumes "qGood X"   and "qGoodEnv rho"   and "qFreshEnv xs x rho" shows "qAbs xs x X $[[rho]] $= qAbs xs x X #[[rho]]"
Int_atLeastAtMostL2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.linorder less_eq less" shows "ord.atLeastAtMost less_eq a b \<inter> ord.atLeast less_eq c = ord.atLeastAtMost less_eq (ord.max less_eq a c) b"
BF: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) in v]"
pair_measure_def: fixes A :: "'c measure"   and B :: "'d measure" shows "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B) {a \<times> b |a b. a \<in> sets A \<and> b \<in> sets B} (\<lambda>X. \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>B \<partial>A)"
ipresSwap_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'd, 'i, 'j) model_scheme" shows "ipresSwap h MOD \<equiv> \<forall>zs z1 z2 s X. wls s X \<longrightarrow> h (X #[z1 \<and> z2]_zs) = igSwap MOD zs z1 z2 (h X)"
ln_upper_11_def: fixes x :: "real" shows "ln_upper_11 x \<equiv> (5 * x ^ 5 + 647 * x ^ 4 + 4397 * x ^ 3 + 6397 * x\<^sup>2 + 2272 * x + 142) * (x - 1) / (30 * (6 * x ^ 5 + 75 * x ^ 4 + 200 * x ^ 3 + 150 * x\<^sup>2 + 30 * x + 1))"
Ball_mem_monad_gt_zero: fixes x :: "real star"   and u :: "real star" assumes "0 < x"   and "x \<notin> Infinitesimal"   and "u \<in> monad x" shows "0 < u"
generate_subset_eqI: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier G"   and "B \<subseteq> A"   and "A - B \<subseteq> generate G B" shows "generate G A = generate G B"
convex_box_1: fixes a :: "'a"   and b :: "'a" shows "convex (cbox a b)"
linorder_neqE: fixes x :: "'c"   and y :: "'c" assumes "x \<noteq> y" obtains "x < y"   | "y < x"
alpha_tConj: fixes tset1 :: "('a, 'b, 'c) Tree set['a]"   and tset2 :: "('a, 'b, 'c) Tree set['a]" shows "(tConj tset1 =\<^sub>\<alpha> tConj tset2) = rel_bset (=\<^sub>\<alpha>) tset1 tset2"
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_inject: fixes x :: "(_) shadow_root_ptr"   and y :: "(_) shadow_root_ptr" shows "(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 x = 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 y) = (x = y)"
takeWhile_eq_all_conv: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(takeWhile P xs = xs) = (\<forall>x\<in>set xs. P x)"
second_isomorphism_grp_def: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and S :: "'c set" shows "second_isomorphism_grp H G S \<equiv> H \<lhd> G \<and> second_isomorphism_grp_axioms G S"
size_tabulate: fixes ks :: "'c list"   and f :: "'c \<Rightarrow> 'd" shows "Mapping.size (Mapping.tabulate ks f) = length (remdups ks)"
simps_2: fixes f :: "'c \<rightarrow> 'b"   and x :: "'c"   and xs :: "'c llist" shows "lmap\<cdot>f\<cdot>(x :@ xs) = f\<cdot>x :@ lmap\<cdot>f\<cdot>xs"
Aexp_fresh_bot: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and v :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "EdomArityAnalysis Aexp"   and "atom v \<sharp> e" shows "(Aexp e\<cdot>a) v = \<bottom>"
Says_Tgs_message_form: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "B \<noteq> Tgs \<and> authK \<notin> range shrK \<and> authK \<in> authKeys evs \<and> authK \<in> symKeys \<and> servK \<notin> range shrK \<and> servK \<notin> authKeys evs \<and> servK \<in> symKeys \<and> servTicket = Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<and> Key servK \<notin> used (before Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) on evs) \<and> Ts = CT (before Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) on evs)"
assms_1: shows "lower_on \<sigma> \<eta> (vars_of_lit L)"
nat_intv_not_int: fixes c :: "nat"   and d :: "real" assumes "real c < d"   and "d < real (c + 1)" shows "d \<notin> \<int>"
xs: shows "\<not> (\<exists>x\<in>lset xs. \<not> P x)"
assms_16: shows "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"
mono: fixes F :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "is_filter F"   and "\<forall>x. P x \<longrightarrow> Q x"   and "F P" shows "F Q"
var_disjoint_def: fixes Cs :: "'a literal multiset list" shows "var_disjoint Cs = (\<forall>\<sigma>s. length \<sigma>s = length Cs \<longrightarrow> (\<exists>\<tau>. \<forall>i<length Cs. \<forall>S. S \<subseteq># Cs ! i \<longrightarrow> S \<cdot> \<sigma>s ! i = S \<cdot> \<tau>))"
preserves_dom: fixes f :: "'b \<times> 'b" assumes "B.VV.arr f" shows "C.VV.dom (F.FF f) = F.FF (B.VV.dom f)"
assms_1: fixes x :: "'oid" assumes "x \<in> set xs" shows "x < oid"
Call1ThrowParams: fixes es :: "'addr expr1 list"   and vs :: "'addr val list"   and a :: "'addr"   and es' :: "'addr expr1 list"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and v :: "'addr val"   and M :: "String.literal"   and s :: "'heap \<times> 'addr val list" assumes "es = map Val vs @ Throw a # es'" shows "uf,P,t \<turnstile>1 \<langle>Val v\<bullet>M(es),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
V'_to_V: fixes v :: "'v" assumes "v \<in> G'.V" shows "fi v \<in> V"
some_betw_xor: fixes Q :: "'a set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "Q \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "c \<in> Q"   and "a \<noteq> b"   and "a \<noteq> c"   and "b \<noteq> c" shows "[[a b c]] \<and> \<not> [[b c a]] \<and> \<not> [[c a b]] \<or> [[b c a]] \<and> \<not> [[a b c]] \<and> \<not> [[c a b]] \<or> [[c a b]] \<and> \<not> [[a b c]] \<and> \<not> [[b c a]]"
assign_local_def: fixes upd :: "('c list \<Rightarrow> 'c list) \<Rightarrow> 'b control_state_scheme \<Rightarrow> 'b control_state_scheme"   and rhs :: "'b control_state_scheme \<Rightarrow> 'c" shows "assign_local upd rhs = assign (\<lambda>\<sigma>. (upd \<circ> map_hd) (\<lambda>_. rhs \<sigma>) \<sigma>)"
disjoint_insert_1: fixes B :: "'g set"   and a :: "'g"   and A :: "'g set" shows "(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})"
empty_sem_env_def: shows "cake_static_env.empty_sem_env = make_sem_env nsEmpty as_static_cenv"
linorder_ow_def: fixes U :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ls :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "linorder_ow U le ls \<equiv> order_ow U le ls \<and> linorder_ow_axioms U le"
abs_le_zero_iff: fixes abs :: "'i \<Rightarrow> 'i"   and plus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and zero :: "'i"   and minus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and uminus :: "'i \<Rightarrow> 'i"   and less_eq :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and less :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "'i" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs a) zero = (a = zero)"
wordinterval_setminus_def: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_setminus r1 r2 = wordinterval_compress (??.WordInterval.wordinterval_setminus' r1 r2)"
ntrp_stack_decomp: fixes u :: "'a"   and s :: "'a list"   and r :: "'a list"   and c :: "'a list multiset"   and ww :: "('b, 'c) label list el_step list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((u # s @ r, c), ww, s', c') \<in> trcl (ntrp fg)" obtains v :: "'a" and rr :: "'a list" where "s' = v # rr @ r" and "((u # s, c), ww, v # rr, c') \<in> trcl (ntrp fg)"
Reals_numeral: fixes w :: "num" shows "numeral w \<in> \<real>"
proj_def: fixes v :: "'b"   and u :: "'b" shows "proj v u = scale (inner v u / inner u u) u"
pt_Collect_eqvt: fixes pi :: "('b \<times> 'b) list"   and P :: "'a \<Rightarrow> bool" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> {x. P x} = {x. P (rev pi \<bullet> x)}"