hcju/isabelleps huggingface dataset:
that: fixes b :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<or> [b, a]\<^sub>\<circ> \<in>\<^sub>\<circ> r" shows "thesis"
1: shows "\<forall>x. n (S1 x \<triangle> S2 x) \<le> n (S1 x)"
prems: shows "vars (p1 + p2) \<subseteq> {..<n}"
lex_prod_def: fixes ra :: "('a \<times> 'a) set"   and rb :: "('b \<times> 'b) set" shows "ra <*lex*> rb = {((a, b), a', b'). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
fresh_1: fixes a :: "name"   and x1 :: "'d" shows "a \<sharp> \<langle>\<epsilon>, x1\<rangle> = a \<sharp> x1"
I_vals_LBset: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I" shows "v ` I \<subseteq> LBset 0"
1: shows "\<forall>x\<in>MPLS PROB. x < k"
is_sig_redD2: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and sing_reg :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top_tail :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and F :: "('c \<Rightarrow>\<^sub>0 'e) set"   and p :: "'c \<Rightarrow>\<^sub>0 'e" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.is_sig_red pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs sing_reg top_tail F p" shows "ord.is_le_rel ord ord_strict top_tail"
gcd_zero_iff: fixes a :: "'a"   and b :: "'a" shows "(gcd a b = (0::'a)) = (a = (0::'a) \<and> b = (0::'a))"
antisym: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "antisymmetric A less_eq"   and "less_eq x y"   and "less_eq y x"   and "x \<in> A"   and "y \<in> A" shows "x = y"
gamma4_x_const: shows "\<forall>x. gamma4 x \<bullet> i = a"
less_aexp_antisym: fixes x :: "'b aexp"   and y :: "'b aexp" shows "(x < y) = (\<not> y < x \<and> x \<noteq> y)"
alw_discrCf: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and \<omega> :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" assumes "enabled cf \<omega>"   and "discrCf cf" shows "alw (holds discrCf) \<omega>"
collect_locks'_subset_collect_locks: fixes las :: "'a lock_actions" shows "collect_locks' las \<subseteq> collect_locks las"
permutations_of_set_empty_iff: fixes A :: "'b set" shows "(permutations_of_set A = {}) = infinite A"
get_err_mode_def: fixes v :: "sparc_state_var" shows "get_err_mode v \<equiv> err v"
assms_3: shows "\<forall>y. y \<in> {length p - 1 |p. valid_path Pi p \<and> distinct p} \<longrightarrow> y \<le> x"
not_essential_times: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and g :: "complex \<Rightarrow> complex" assumes "not_essential f z"   and "not_essential g z"   and "isolated_singularity_at f z"   and "isolated_singularity_at g z" shows "not_essential (\<lambda>w. f w * g w) z"
fpsrel_iff: fixes f :: "'a fps"   and g :: "'a fps"   and p :: "'a" shows "((f, g) \<in> fpsmodrel p) = (\<forall>n. f $ n mod p = g $ n mod p)"
the_NF_step: fixes A :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "SN A"   and "CR A"   and "(a, b) \<in> A" shows "the_NF A a = the_NF A b"
know2knows_neg_s: fixes A :: "specID"   and m :: "Secrets" assumes "\<not> know A (sKS m)" shows "\<not> knows A [sE m]"
Reachability_Problem_axioms: shows "Reachability_Problem K S S1 S2"
sub2_det: fixes A :: "'a mat"   and d :: "nat"   and l :: "nat"   and r :: "'a"   and r' :: "'a"   and A' :: "'a mat" assumes "A \<in> carrier_mat d d"   and "local.sub2 d l (r, A) = (r', A')"   and "r \<noteq> (0::'a)"   and "l < d" shows "r * det A' = r' * det A"
lsbisE2: fixes xa :: "'d"   and x :: "'d"   and B1 :: "'e set"   and B2 :: "'d set"   and s1 :: "'e \<Rightarrow> ('f, 'e, 'd) F1"   and s2 :: "'d \<Rightarrow> ('g, 'e, 'd) F2" assumes "(xa, x) \<in> lsbis2 B1 B2 s1 s2" shows "\<exists>z\<in>F2in UNIV (lsbis1 B1 B2 s1 s2) (lsbis2 B1 B2 s1 s2). F2map id fst fst z = s2 xa \<and> F2map id snd snd z = s2 x"
has_field_derivative_rGamma_nonpos_int: fixes n :: "nat"   and A :: "'a set" shows "(rGamma has_field_derivative (- (1::'a)) ^ n * fact n) (at (- of_nat n) within A)"
Sup_bot_conv_2: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup A) = (\<forall>x\<in>A. x = bot)"
Step1_5_3c: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> \<box>(\<forall>l. $MemInv mm l) \<and> \<box>$ImpInv rmhist p \<and> ImpLive p \<longrightarrow> WF(MemReturn memCh (resbar rmhist) p)_(rtrner memCh!p, resbar rmhist!p)"
atLeast_Suc_atMost: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "prod g {m..n} = g m * prod g {Suc m..n}"
distinct_map_filter: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "distinct (list.map f xs)" shows "distinct (list.map f (List.filter P xs))"
assms_4: shows "List_Spec.list_order some_ops x y"
Saturated_Graph_axioms: shows "Saturated_Graph task.nodes task.inPorts task.outPorts vertices nodeOf task_edges"
seq_power_front: fixes a :: "'a::refinement_lattice"   and n :: "nat" shows "((a::'a::refinement_lattice) \<^sup>;^ (n::nat)) ; a = a ; (a \<^sup>;^ n)"
assms_1: shows "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"
hyps_4: fixes x :: "nat" assumes "x < count_weights True (remove_weights m1)" shows "extract_weights True m1 x = extract_weights True m2 x"
d1: fixes x :: "real"   and y :: "real" assumes "g x = g y"   and "u \<le> x"   and "x \<le> v"   and "u \<le> y"   and "y \<le> v" shows "x = y \<or> x = u \<and> y = v \<or> x = v \<and> y = u"
topological_basis_closed: fixes B :: "'at set set" assumes "on U with \<tau> : \<guillemotleft>topological_basis\<guillemotright> B" shows "B \<subseteq> Pow U"
subst_elimD: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and v :: "'c"   and t :: "('b, 'a) Term.term" assumes "subst_elim \<sigma> v" shows "v \<notin> fv (t \<cdot> \<sigma>)"
B: shows "offs_num (length ns) xs index key mi ma i = 0"
elims_3: fixes x :: "('a \<times> ('b \<Rightarrow> 'c) \<times> 'd) list"   and xa :: "'a \<Rightarrow> 'c"   and xb :: "'b" assumes "\<not> postList x xa xb" obtains xc :: "'a" and e :: "'b \<Rightarrow> 'c" and v :: "'d" and xs :: "('a \<times> ('b \<Rightarrow> 'c) \<times> 'd) list" and l :: "'a \<Rightarrow> 'c" and s :: "'b" where "x = (xc, e, v) # xs" and "xa = l" and "xb = s" and "\<not> (l xc = e s \<and> postList xs l s)"
that: fixes c :: "complex" assumes "f = (\<lambda>x. c)" shows "thesis"
int_simps_24: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (#True \<or> P) = #True"
less_dual_def: shows "(<) = rel_dual (\<lambda>x y. y < x)"
tan_mono_le_eq: fixes x :: "real"   and y :: "real" assumes "- (pi / 2) < x"   and "x < pi / 2"   and "- (pi / 2) < y"   and "y < pi / 2" shows "(tan x \<le> tan y) = (x \<le> y)"
has_vector_derivative_at_within: fixes f :: "real \<Rightarrow> 'b"   and f' :: "'b"   and x :: "real"   and S :: "real set" assumes "(f has_vector_derivative f') (at x)" shows "(f has_vector_derivative f') (at x within S)"
set_mset_def: fixes M :: "'f multiset" shows "set_mset M = {x. 0 < count M x}"
cpu_reg_mod_mem: fixes asi :: "word8"   and addr :: "word32"   and s :: "'b sparc_state"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "fst (memory_read asi addr s) = fst (memory_read asi addr (s\<lparr>cpu_reg := new_cpu_reg\<rparr>))"
sub_to_poly: fixes g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "g \<in> carrier P"   and "a \<in> carrier R" shows "Cring_Poly.compose R (to_polynomial R a) g = to_polynomial R a"
boundedI: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "c \<le> a" shows "b \<squnion> c \<le> a"
a_3: fixes i :: "'a" assumes "i \<in> domain x" shows "open (a i)"
conf_def: fixes P :: "'b prog"   and h :: "heap"   and v :: "val"   and T :: "ty" shows "P,h \<turnstile> v :\<le> T \<equiv> \<exists>T'. typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor> \<and> P \<turnstile> T' \<le> T"
functor_abbrev_def: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and A :: "('a, 'c, 'e) Category_scheme"   and B :: "('b, 'd, 'f) Category_scheme" shows "Ftor F : A \<longrightarrow> B \<equiv> Functor F \<and> CatDom F = A \<and> CatCod F = B"
pure_stream_numeral: fixes n :: "num" shows "numeral n = pure (numeral n)"
intro: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set" assumes "Group G"   and "RModule (aezfun_setspan G) smult V" shows "FGModule G smult V"
subsumption_and_semantics: fixes C :: "'at Literal set"   and D :: "'at Literal set"   and I :: "'at set" assumes "subsumes C D"   and "I \<Turnstile> C" shows "I \<Turnstile> D"
open_continuous_vimage: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "open s"   and "continuous_on s f"   and "open B" shows "open (f -` B \<inter> s)"
circ_zero: shows "bot\<^sup>\<star> = (1::'a)"
GPicard2: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T"   and "connected T"   and "S \<noteq> {}"   and "open S"   and "\<And>x. \<lbrakk>x islimpt S; x \<in> T\<rbrakk> \<Longrightarrow> x \<in> S" shows "S = T"
distinct_map: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "distinct (map f xs) = (distinct xs \<and> inj_on f (set xs))"
assms_2: fixes \<xi> :: "'a" shows "U \<xi> \<xi>"
nonneg: fixes x :: "'b"   and y :: "'b" shows "0 \<le> norm_dist x y"
Q_hit_pre_edges: shows "H.edges_of_walk (Q_hit_pre @ [y]) \<inter> B \<noteq> {}"
nth_filter: fixes j :: "nat"   and P :: "'b \<Rightarrow> bool"   and ls :: "'b list" assumes "j < length (filter P ls)" shows "P (filter P ls ! j)"
r_init: fixes s :: "'a"   and T :: "('a, 'b) TS_scheme" assumes "s \<in> init T" shows "s \<in> reach T"
comp_arr_ide: fixes a :: "'c"   and f :: "'c" assumes "Fop.Bop.ide a"   and "Fop.Bop.seq f a" shows "Fop.Bop.comp f a = f"
cases: fixes a :: "('state \<times> 'rule) stream" assumes "epath a" obtains steps :: "('state \<times> 'rule) stream" and sl :: "'state fset" where "a = steps" and "snd (shd steps) \<in> R" and "fst (shd (stl steps)) |\<in>| sl" and "effStep (shd steps) sl" and "epath (stl steps)"
CardinalityPrognosisVar_axioms: shows "CardinalityPrognosisVar prognosis"
B: shows "(Asset m, Token m (Auth_PriK m) B' C' SK) \<in> s"
intro: fixes M :: "'c pre_map"   and G :: "('d, 'c) pre_digraph" assumes "edge_succ M permutes arcs G"   and "\<And>v. \<lbrakk>v \<in> verts G; out_arcs G v \<noteq> {}\<rbrakk> \<Longrightarrow> cyclic_on (edge_succ M) (out_arcs G v)" shows "digraph_map_axioms G M"
swap_atom: fixes a :: "nat"   and L :: "real list"   and b :: "nat"   and F :: "atom" assumes "a < length L"   and "b < length L" shows "aEval F L = aEval (swap_atom a b F) (swap_list a b L)"
ground_ns_in_ground_limit_st: shows "Liminf_llist Gs - sr.Rf (Liminf_llist Gs) \<subseteq> grounding_of_clss (Q_of_state (Liminf_state Sts))"
simps: fixes a :: "'c"   and A :: "('d, 'c) nba"   and p :: "'c" shows "(a \<in> reachable A p) = (a = p \<or> (\<exists>q aa. a = snd aa \<and> q \<in> reachable A p \<and> fst aa \<in> alphabet A \<and> snd aa \<in> transition A (fst aa) q))"
cont_const: fixes c :: "'h" shows "cont (\<lambda>x. c)"
presVar_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)" shows "presVar (fromIMor h) MOD"
polyn_minus_nsum: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k \<le> fst c" shows "-\<^sub>a polyn_expr R X k c = \<Sigma>\<^sub>e R (\<lambda>j. (-\<^sub>a\<^bsub>S\<^esub> snd c j) \<cdot>\<^sub>r X^\<^bsup>R j\<^esup>) k"
tagged_division_of_def: fixes s :: "('b \<times> 'b set) set"   and i :: "'b set" shows "(s tagged_division_of i) = (s tagged_partial_division_of i \<and> \<Union> {K. \<exists>x. (x, K) \<in> s} = i)"
simps_8: fixes x1 :: "'a"   and x2 :: "'a stream" shows "sset (x1 ## x2) = insert x1 (sset x2)"
M: shows "M = add_mset (Max_mset M) (M - {#Max_mset M#})"
schroeder_5_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> - z) = (y * z\<^sup>T \<le> - x\<^sup>T)"
interior_closed_segment_ge2: fixes a :: "'a"   and b :: "'a" assumes "2 \<le> DIM('a)" shows "interior (closed_segment a b) = {}"
analz_Un_implConfidSet_aux_3: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implSecureSet_aux Ag G \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
hyps_3: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val" assumes "y = (cms ! k, mem)"   and "k < length cms" shows "map snd (cms[k := fst z]) \<in> reachable_mode_states (cms, mem)"
face_cycle_set_def: fixes M :: "'c pre_map"   and a :: "'c" shows "pre_digraph_map.face_cycle_set M a = orbit (pre_digraph_map.face_cycle_succ M) a"
homh: shows "rupture_surj A P \<circ> poly_of_const \<in> ring_hom (L\<lparr>carrier := A\<rparr>) (Rupt A P)"
additive_implies_homogenous: fixes f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. f (x + y) = f x + f y" shows "f (0::'c) = (0::'d)"
flush_all_until_volatile_write_Prog\<^sub>s\<^sub>b_commute: fixes i :: "nat"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and p\<^sub>1 :: "'b"   and p\<^sub>2 :: "'b"   and mis :: "instr list"   and sb :: "'b memref list"   and \<D> :: "'c"   and \<O> :: "'d"   and \<R> :: "'e"   and \<D>' :: "'c"   and \<R>' :: "'e"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis # sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p, is, \<theta>, sb, \<D>', \<O>, \<R>')]) m = flush_all_until_volatile_write ts m"
is_node_ptr_kind_cast: fixes node_ptr :: "(_) node_ptr" shows "is_node_ptr_kind (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 node_ptr)"
ccSup_insert: fixes A :: "'a set"   and a :: "'a" assumes "countable A" shows "Sup (insert a A) = sup a (Sup A)"
linear_scale: fixes f :: "'b \<Rightarrow> 'c"   and r :: "complex"   and b :: "'b" assumes "clinear f" shows "f (r *\<^sub>C b) = r *\<^sub>C f b"
ochoiceT2: fixes \<sigma> :: "nat \<Rightarrow> 'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'e"   and q' :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat"   and p :: "('e, 'f, 'g, 'h) seqp" assumes "((\<sigma>, q), a, \<sigma>', q') \<in> oseqp_sos \<Gamma> i" shows "((\<sigma>, p \<oplus> q), a, \<sigma>', q') \<in> oseqp_sos \<Gamma> i"
subset_replicate: fixes n :: "nat"   and x :: "'a" shows "set (replicate n x) \<subseteq> {x}"
freeOrd: fixes h :: "'b \<Rightarrow> 'c"   and s :: "'b \<Rightarrow> 'b"   and s' :: "'c \<Rightarrow> 'c"   and l :: "(nat \<Rightarrow> 'b) \<Rightarrow> 'b"   and l' :: "(nat \<Rightarrow> 'c) \<Rightarrow> 'c"   and n :: "Ord"   and z :: "'b" assumes "\<And>n. h (s n) = s' (h n)"   and "\<And>f. h (l f) = l' (\<lambda>i. h (f i))" shows "h (\<langle>n\<rangle>\<^sub>O z s l) = \<langle>n\<rangle>\<^sub>O (h z) s' l'"
Plus_FullR: fixes r :: "'a language" shows "Plus r \<Sigma>\<^sup>* = \<Sigma>\<^sup>*"
con2: shows "continuous_on UNIV (\<lambda>x. \<chi>i. sterm_sem I (args i) x)"
MkIde_Cod: fixes a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.arr a" shows "A_B.MkIde (A_B.Cod a) = A_B.cod a"
maximal_wrt_def: fixes A :: "'a"   and C :: "'a literal multiset" shows "maximal_wrt A C = (\<forall>B\<in>atms_of C. \<not> less_atm A B)"
path_concatenation_start_points: fixes x :: "'a"   and y :: "'a" assumes "end_points x = start_points y"   and "x ; 1 \<cdot> y\<^sup>T ; 1 = 0" shows "start_points (x + y) = start_points x"
map_undefined: fixes a :: "'a" assumes "a \<notin> S" shows "\<alpha> a = undefined"
length_lv_rel: fixes ys :: "real list"   and y :: "'b" assumes "(ys, y) \<in> lv_rel" shows "length ys = DIM('b)"
order_trans_rules_23: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
iT_iff_4: fixes x :: "nat"   and r :: "nat"   and m :: "nat" shows "(x \<in> [ r, mod m ]) = (x mod m = r mod m \<and> r \<le> x)"
D_in_Sts_Suc: fixes l :: "nat" assumes "i \<le> l"   and "enat (Suc l) < llength Sts" shows "D \<in> Q_of_state (lnth Sts (Suc l))"
hom_graph_exists: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b formula" shows "\<exists>a S. hom_graph f S x a"
simps: fixes xs :: "'b list"   and i :: "nat" shows "nth_of xs i = (if length xs \<le> i then None else Some (xs ! i))"
wtc_pres: fixes n :: "nat"   and c :: "'a list"   and pc :: "nat"   and s :: "'a" assumes "pres_type step n A"   and "c ! pc \<in> A"   and "c ! (pc + 1) \<in> A"   and "s \<in> A"   and "pc < n" shows "wtc c pc s \<in> A"
bool_of_val_def: fixes v :: "32 word" shows "bool_of_val v \<equiv> v \<noteq> 0"
subset_Un_eq: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subseteq> B) = (A \<union> B = B)"
lalternate_finite_helper: fixes xs :: "'b llist" assumes "lfinite (lalternate xs)" shows "lfinite xs"
rSim: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto>\<^sup>^<(bisim O X O bisim)> Q"
wtl_append: fixes a :: "'b list"   and b :: "'b list"   and c :: "'a list"   and pc :: "nat"   and s :: "'a" shows "wtl (a @ b) c pc s = wtl b c (pc + length a) (wtl a c pc s)"
lfinite_ltakeWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lfinite (ltakeWhile P xs) = (lfinite xs \<or> (\<exists>x\<in>lset xs. \<not> P x))"
row_def: fixes A :: "'c mat"   and i :: "nat" shows "row A i = vec (dim_col A) (\<lambda>j. A $$ (i, j))"
parts_init: shows "parts (used s\<^sub>0) = used s\<^sub>0 \<union> range (Hash \<circ> Agent) \<union> range (Hash \<circ> Auth_PubKey) \<union> range (\<lambda>n. \<lbrace>Hash (Agent n), Hash (Auth_PubKey n)\<rbrace>)"
valid_W_invE_3: fixes r :: "'e"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and p :: "'g process_name" assumes "r \<in> W (s p)"   and "valid_W_inv s" shows "obj_at (\<lambda>s. True) r s"
genideal_zero: shows "Idl\<^bsub>K [X]\<^esub> {\<zero>\<^bsub>K [X]\<^esub>} = {\<zero>\<^bsub>K [X]\<^esub>}"
plus_int_code_2: fixes l :: "int" shows "0 + l = l"
inverse_unique: fixes f :: "'a"   and g :: "'a" assumes "B.inverse_arrows f g" shows "B.inv f = g"
wi2l: fixes r :: "'b wordinterval" shows "(\<Union>(i, j)\<in>set (wi2l r). {i..j}) = wordinterval_to_set r"
psubseteq_ran_subset: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" assumes "f \<subseteq>\<^sub>p g" shows "pran f \<subseteq> pran g"
induct: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'b list"   and x2 :: "'b list"   and Pa :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "list_emb P x1 x2"   and "\<And>ys. Pa [] ys"   and "\<And>xs ys y. \<lbrakk>list_emb P xs ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa xs (y # ys)"   and "\<And>x y xs ys. \<lbrakk>P x y; list_emb P xs ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa (x # xs) (y # ys)" shows "Pa x1 x2"
RP_model: fixes St :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and St' :: "'a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set"   and I :: "'a set" assumes "St \<leadsto>RP St'" shows "(I \<TTurnstile>s \<G>_Fset (N_of_state St' \<union> P_of_state St' \<union> Q_of_state St')) = (I \<TTurnstile>s \<G>_Fset (N_of_state St \<union> P_of_state St \<union> Q_of_state St))"
either_af_def: fixes B :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "either_af B \<equiv> reflp B"
map_of_reinsert_SomeD: fixes D :: "'g"   and fs :: "('f \<times> 'e) list"   and F :: "'f"   and T :: "'e" assumes "map_of (map (\<lambda>(F, y). ((F, D), y)) fs) (F, D) = \<lfloor>T\<rfloor>" shows "map_of fs F = \<lfloor>T\<rfloor>"
dom_simp: fixes f :: "'a" assumes "S.arr f" shows "S.dom f = local.dom f"
s'a'_def: shows "s'a' \<equiv> SOME (s', a'). P (s', a')"
poly_subst_def: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "poly_subst f p = (\<Sum>t\<in>keys p. punit.monom_mult (lookup p t) 0 (subst_pp f t))"
extensible_chain_sup: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph"   and R :: "('e, 'g) labeled_graph \<times> ('e, 'g) labeled_graph"   and j :: "nat"   and f :: "('g \<times> 'f) set" assumes "chain S"   and "extensible R (S j) f" shows "extensible R (chain_sup S) f"
enn2real_less_iff: fixes x :: "ennreal"   and c :: "real" assumes "x < \<top>" shows "(enn2real x < c) = (x < ennreal c)"
term_variants_pred_param: fixes P :: "'a \<Rightarrow> 'a list"   and t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and f :: "'a"   and g :: "'a"   and S :: "('a, 'b) Term.term list"   and T :: "('a, 'b) Term.term list" assumes "term_variants_pred P t s"   and "f = g \<or> g \<in> set (P f)" shows "term_variants_pred P (Fun f (S @ t # T)) (Fun g (S @ s # T))"
step_def: fixes g :: "'g" shows "step g \<equiv> \<lambda>(u, p). (CFG_SSA_Transformed_notriv_base.uses' \<alpha>n defs (uses(g := u)) (phis(g := p)) chooseNext_all g, CFG_SSA_Transformed_notriv_base.phis' \<alpha>n defs (uses(g := u)) (phis(g := p)) chooseNext_all g)"
\<alpha>_\<gamma>: shows "LeftDerivation \<alpha> (take v D) \<gamma>"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "Right_a.seq g f" shows "Right_a.comp g f = g \<cdot> f"
Const\<^sub>0_one: shows "Const\<^sub>0 (1::'b) = 1"
get_ancestors_def: fixes ptr :: "(_) object_ptr" shows "get_ancestors ptr = Heap_Error_Monad.bind (check_in_heap ptr) (\<lambda>_. 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>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> return [] | Some node_ptr \<Rightarrow> Heap_Error_Monad.bind (get_parent node_ptr) (\<lambda>parent_ptr_opt. case parent_ptr_opt of None \<Rightarrow> return [] | Some parent_ptr \<Rightarrow> get_ancestors parent_ptr)) (\<lambda>ancestors. return (ptr # ancestors)))"
map_of_map_to_set: fixes l :: "('d \<times> 'e) list"   and m :: "'d \<Rightarrow> 'e option" assumes "distinct (map fst l)" shows "(map_of l = m) = (set l = map_to_set m)"
inv_fg_cancel2: fixes l :: "(bool \<times> 'd) list" shows "Cancelation.normalize (inv_fg l @ l) = []"
indeps_1: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a" assumes "myss x y" shows "x \<bowtie> y"
assms: fixes e :: "ereal" assumes "0 < e" shows "x \<le> y + e"
vertextype_def: fixes g :: "graph"   and v :: "nat" shows "vertextype g v \<equiv> (tri g v, quad g v, except g v)"
nontrivial: shows "\<^bold>V - {source \<Delta>, sink \<Delta>} \<noteq> {}"
ide_in_hom: fixes a :: "'d \<times> 'c" shows "DopxC.ide a = DopxC.in_hom a a a"