hcju/isabelleps huggingface dataset:
prefix_append_subsequence: fixes i :: "nat"   and j :: "nat"   and w :: "nat \<Rightarrow> 'b" assumes "i \<le> j" shows "prefix i w @ w [i \<rightarrow> j] = prefix j w"
extend_SKIP: fixes h :: "'c \<times> 'd \<Rightarrow> 'e" assumes "Extend h" shows "extend h \<bottom> = \<bottom>"
succ_5: shows "ord_rec_Seq T G s' (succ k) y'"
lookup_default_empty: fixes d :: "'b"   and k :: "'c" shows "lookup_default d Mapping.empty k = d"
lspasl_p_der: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a" assumes "h1,h2\<triangleright>h0"   and "h1,h2\<triangleright>h3" shows "(h1,h2\<triangleright>h0) \<and> h0 = h3"
size_repr: shows "groups_of_size g = size {#x \<in># image_mset card (mset_set \<G>). x = g#}"
augment_edge_impl_def: fixes cf :: "nat \<times> nat \<Rightarrow> int"   and e :: "nat \<times> nat"   and cap :: "int" shows "augment_edge_impl cf e cap \<equiv> cf_get cf e \<bind> (\<lambda>v. cf_set cf e (v - cap) \<bind> (\<lambda>cf. let e = prod.swap e in cf_get cf e \<bind> (\<lambda>v. cf_set cf e (v + cap) \<bind> return)))"
wrap_iso_arcs_iso_arcs: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and x :: "'d"   and f :: "'d \<Rightarrow> 'd" assumes "pre_digraph.digraph_isomorphism G hom"   and "x \<in> arcs G" shows "pre_digraph.wrap_iso_arcs G hom f (iso_arcs hom x) = iso_arcs hom (f x)"
af_equiv: fixes \<phi> :: "'b ltl"   and w :: "'b set list"   and \<nu> :: "'b set" shows "af \<phi> (w @ [\<nu>]) = step (af\<^sub>\<UU> (Unf \<phi>) w) \<nu>"
SLin_axioms: shows "SLin (\<bullet>) \<gamma> \<bottom>"
zcount_ps: shows "0 < zcount (c_pts c loc1) a"
butlast_conv_take: fixes xs :: "'c list" shows "butlast xs = take (length xs - 1) xs"
reduction_def: fixes \<alpha> :: "rat" shows "LLL.reduction \<alpha> = (4 + \<alpha>) / (4 * \<alpha>)"
mult_isor: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.near_dioid plus times less_eq less"   and "less_eq x y" shows "less_eq (times x z) (times y z)"
all_coarser_partitions_with_list_def: fixes elem :: "'b"   and Ps :: "'b set list list" shows "all_coarser_partitions_with_list elem Ps = concat (map (coarser_partitions_with_list elem) Ps)"
assms: shows "\<forall>s. P s \<longrightarrow> Q s"
cont2cont_Let: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "cont f"   and "\<And>y. cont (\<lambda>x. g x y)"   and "\<And>x. cont (g x)" shows "cont (\<lambda>x. Let (f x) (g x))"
Liminf_le: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "F \<noteq> bot"   and "\<forall>\<^sub>F x in F. f x \<le> l" shows "Liminf F f \<le> l"
Hash_in_parts_respond: fixes B :: "agent"   and M :: "msg"   and RB :: "msg"   and H :: "msg set"   and PB :: "msg"   and K :: "nat"   and evs :: "event list" assumes "Hash \<lbrace>Key (shrK B), M\<rbrace> \<in> parts (insert RB H)"   and "(PB, RB, K) \<in> respond evs" shows "Hash \<lbrace>Key (shrK B), M\<rbrace> \<in> parts H"
module_hom_with_def: fixes plus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and plus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>M\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>M\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f" shows "module_hom_with plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 f \<equiv> module_with plus\<^sub>M\<^sub>_\<^sub>1 zero\<^sub>M\<^sub>_\<^sub>1 minus\<^sub>M\<^sub>_\<^sub>1 uminus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 \<and> module_with plus\<^sub>M\<^sub>_\<^sub>2 zero\<^sub>M\<^sub>_\<^sub>2 minus\<^sub>M\<^sub>_\<^sub>2 uminus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 \<and> module_hom_with_axioms plus\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 f"
hyps: assumes "\<alpha> \<in># r|as|" shows "\<exists>\<sigma>1 \<sigma>3. as = \<sigma>1 @ [\<alpha>] @ \<sigma>3 \<and> \<alpha> \<notin> r \<down>l \<sigma>1"
B_gets_good_key: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and NB :: "nat"   and evs :: "event list"   and NA :: "nat" assumes "Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>, Crypt K (Nonce NB)\<rbrace> \<in> set evs"   and "Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> \<in> set evs"   and "\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "Key K \<notin> analz (knows Spy evs)"
prfOf_def: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and eql :: "'e \<Rightarrow> 'e \<Rightarrow> 'f"   and cnj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and imp :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and all :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and exi :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and fls :: "'f"   and dsj :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and num :: "'e set"   and enc :: "'f \<Rightarrow> 'e"   and prv :: "'f \<Rightarrow> bool"   and bprv :: "'f \<Rightarrow> bool"   and P :: "'f"   and isTrue :: "'f \<Rightarrow> bool"   and Pf :: "'f"   and n :: "'e"   and \<phi> :: "'f" assumes "Minimal_Truth_Soundness_HBL1iff_Compl_Pf var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num enc prv bprv P isTrue Pf" shows "Minimal_Truth_Soundness_HBL1iff_Compl_Pf.prfOf var Var FvarsT Fvars subst enc bprv Pf n \<phi> \<equiv> bprv (Minimal_Truth_Soundness_HBL1iff_Compl_Pf.PPf var Var FvarsT Fvars subst Pf n (enc \<phi>))"
hyps_13: shows "(ps1, qs1) = split (monomial 1 x + t) U ((\<lambda>f. f - monomial 1 x) ` S)"
assign_def: fixes f :: "'a control_state_scheme \<Rightarrow> 'a control_state_scheme" shows "assign f = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some ((), \<sigma>) else Some ((), f \<sigma>))"
aff_dim_negative_iff: fixes S :: "'b set" shows "(aff_dim S < 0) = (S = {})"
supto_defined: fixes i :: "int"   and j :: "int" shows "supto\<cdot>(MkI\<cdot>i)\<cdot>(MkI\<cdot>j) \<noteq> \<bottom>"
a_star_trans: fixes M1 :: "trm"   and M2 :: "trm"   and M3 :: "trm" assumes "M1 \<longrightarrow>\<^sub>a* M2"   and "M2 \<longrightarrow>\<^sub>a* M3" shows "M1 \<longrightarrow>\<^sub>a* M3"
group_action_axioms_def: fixes G :: "('d, 'e) monoid_scheme"   and \<phi> :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and M :: "'f set" shows "group_action_axioms G \<phi> M \<equiv> group_hom G (BijGroup M) \<phi>"
eq_sum: fixes I :: "'b set"   and p :: "'b \<Rightarrow> 'c" assumes "finite I" shows "sum' p I = sum p I"
ALG_Pxy_def: fixes Strat :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and x :: "'c"   and y :: "'c" shows "ALG_Pxy Strat qs init x y = (\<Sum>i<length qs. ALG_P' Strat qs init i x y)"
AF_fp: fixes p :: "'b set" shows "\<^bold>A\<^bold>F p = p \<union> \<^bold>A\<^bold>X \<^bold>A\<^bold>F p"
lborel_integral_real_affine: fixes c :: "real"   and f :: "real \<Rightarrow> 'b"   and t :: "real" assumes "c \<noteq> 0" shows "integral\<^sup>L lborel f = \<bar>c\<bar> *\<^sub>R (LINT x|lborel. f (t + c * x))"
\<psi>_\<phi>: fixes f :: "'c"   and b :: "'c"   and a :: "'c" assumes "\<guillemotleft>f : b \<rightarrow> a\<guillemotright>" shows "R.\<psi> (b, a) ((S.UP \<circ> inl) f) = f"
simps_1: fixes P :: "'state"   and tset :: "('b, 'pred, 'act) Tree set['b]" shows "valid_Tree P (tConj tset) = Ball (set_bset tset) (valid_Tree P)"
ccFromList_def: shows "ccFromList \<equiv> (id ---> Abs_CoCall) (\<lambda>xs. {(x, y). list_pairs xs (x, y) \<or> list_pairs xs (y, x)})"
Rep_group: fixes x :: "'d group" shows "Rep_group x \<in> {G. Group.group G}"
e_nth: fixes e :: "nat"   and n :: "nat" shows "e_nth e n = (if n < e_length e then Partial_Recursive.list_decode e ! n else 0)"
ffact_Suc: fixes n :: "nat"   and a :: "'a" shows "ffact (Suc n) a = a * ffact n (a - (1::'a))"
carrier_is_subfield: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "subfield (carrier R) R"
compact_space_prod_topology: fixes X :: "'d topology"   and Y :: "'e topology" shows "compact_space (prod_topology X Y) = (topspace (prod_topology X Y) = {} \<or> compact_space X \<and> compact_space Y)"
Fvars_PP: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and P :: "'c"   and t :: "'b" assumes "HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv enc P"   and "t \<in> trm" shows "Fvars (HBL1.PP var subst P t) = FvarsT t"
extremum_unique: fixes a :: "'b" shows "(a \<le> bot) = (a = bot)"
scaleR_vec_def: shows "(*\<^sub>R) \<equiv> \<lambda>r x. \<chi>i. r *\<^sub>R x $h i"
Abs_type: fixes \<Gamma> :: "binding list"   and S :: "POPLmarkRecord.type"   and s :: "trm"   and T :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type"   and U' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> (\<lambda>:S. s) : T"   and "\<Gamma> \<turnstile> T <: U \<rightarrow> U'" obtains S' :: "POPLmarkRecord.type" where "\<Gamma> \<turnstile> U <: S" and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'" and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'"
cyclic_on_def: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set" shows "cyclic_on f S = (\<exists>s\<in>S. S = orbit f s)"
finsert_fminus1: fixes x :: "'a"   and B :: "'a fset"   and A :: "'a fset" assumes "x |\<in>| B" shows "finsert x A |-| B = A |-| B"
rep_unity: shows "\<F>C.rep \<I> = \<^bold>\<parallel>\<^bold>\<I>\<^bold>\<parallel>"
wffp_last_purgeIdle: fixes do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and sl :: "('a, 'b, 'c) state list" assumes "GM_sec_model.wffp do sl"   and "GM_sec_model.purgeIdle sl \<noteq> []" shows "local.getGMState (last (GM_sec_model.purgeIdle sl)) = local.getGMState (last sl)"
map_index_id: fixes xs :: "'b list" shows "map_index (curry snd) xs = xs"
affine_dependent_iff_dependent: fixes a :: "'a"   and S :: "'a set" assumes "a \<notin> S" shows "affine_dependent (insert a S) = dependent ((+) (- a) ` S)"
nuc_comp_assoc: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "nucleus f" shows "f (x \<cdot> f (y \<cdot> z)) = f (f (x \<cdot> y) \<cdot> z)"
maxsimpchainD_maxsimp: fixes X :: "'b set set"   and xs :: "'b set list"   and x :: "'b set" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimpchain X xs"   and "x \<in> set xs" shows "SimplicialComplex.maxsimp X x"
cases: fixes G :: "('c, 'd) Graph_scheme"   and a :: "'c llist" assumes "Digraph G"   and "Digraph.valid_path G a" obtains "a = LNil"   | v :: "'c" where "a = LCons v__ LNil" and "v__ \<in> V\<^bsub>G\<^esub>"   | v :: "'c" and w :: "'c" and Ps :: "'c llist" where "a = LCons v Ps" and "v \<in> V\<^bsub>G\<^esub>" and "w \<in> V\<^bsub>G\<^esub>" and "v \<rightarrow>\<^bsub>G\<^esub> w" and "Digraph.valid_path G Ps" and "\<not> lnull Ps" and "lhd Ps = w"
abs_eq: fixes xa :: "'b \<Rightarrow> 'c"   and x :: "'b \<Rightarrow> 'c" assumes "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'c)}) xa xa"   and "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'c)}) x x" shows "Abs_poly_mapping xa + Abs_poly_mapping x = Abs_poly_mapping (\<lambda>k. xa k + x k)"
\<beta>': fixes p :: "fm"   and d :: "int"   and bbs :: "bool list"   and a :: "int"   and bs :: "int list" assumes "iszlfm p"   and "d_\<beta> p 1"   and "d_\<delta> p d"   and "0 < d" shows "\<forall>x. \<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>set (\<beta> p). Ifm bbs ((Inum (a # bs) b + j) # bs) p) \<longrightarrow> Ifm bbs (x # bs) p \<longrightarrow> Ifm bbs ((x - d) # bs) p"
add_mset_remove_trivial_If: fixes a :: "'a"   and N :: "'a multiset" shows "add_mset a (N - {#a#}) = (if a \<in># N then N else add_mset a N)"
arith_simps_62: fixes a :: "'b" shows "(0::'b) * a = (0::'b)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) Dom_f.map Dom_f.map Dom_f.map"
AAss_\<tau>red1t_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and i :: "'addr expr1"   and xs :: "'addr val list"   and i' :: "'addr expr1"   and xs' :: "'addr val list"   and a :: "'addr val"   and e :: "'addr expr1" assumes "\<tau>red1gt uf P t h (i, xs) (i', xs')" shows "\<tau>red1gt uf P t h (Val a\<lfloor>i\<rceil> := e, xs) (Val a\<lfloor>i'\<rceil> := e, xs')"
split_sel: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd llist \<Rightarrow> 'c"   and llist :: "'d llist" shows "P (case llist of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = ((llist = LNil \<longrightarrow> P f1) \<and> (llist = LCons (lhd llist) (ltl llist) \<longrightarrow> P (f2 (lhd llist) (ltl llist))))"
index_mult_mat_3: fixes A :: "'d mat"   and B :: "'d mat" shows "dim_col (A * B) = dim_col B"
image_eqI: fixes b :: "'ap"   and f :: "'aq \<Rightarrow> 'ap"   and x :: "'aq"   and A :: "'aq set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
setVar'_gState_progress_rel: fixes prog :: "program"   and g :: "'a gState_scheme"   and gl :: "bool"   and v :: "String.literal"   and idx :: "integer option"   and val :: "integer"   and p :: "pState" assumes "gState_inv prog g" shows "(g, fst (setVar' gl v idx val g p)) \<in> gState_progress_rel prog"
finfun_upd_apply_same: fixes f :: "'b \<Rightarrow>f 'a"   and a :: "'b"   and b :: "'a" shows "(f(a $:= b)) $ a = b"
comp_apply: fixes f :: "'e \<Rightarrow> 'b"   and g :: "'f \<Rightarrow> 'e"   and x :: "'f" shows "(f \<circ> g) x = f (g x)"
n_restrictl: fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> y \<le> y"
distinct_1: fixes x1 :: "'b equation"   and x2 :: "'b equation" shows "Pos x1 \<noteq> Neg x2"
eq_minus_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(a = uminus (divide b c)) = (if c \<noteq> zero then times a c = uminus b else a = zero)"
trans: fixes S :: "('c, 'd) eq_object_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "equivalence S"   and "x .=\<^bsub>S\<^esub> y"   and "y .=\<^bsub>S\<^esub> z"   and "x \<in> carrier S"   and "y \<in> carrier S"   and "z \<in> carrier S" shows "x .=\<^bsub>S\<^esub> z"
Osum_Linear_order: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "Linear_order r"   and "Linear_order r'" shows "Linear_order (r \<union>o r')"
analz_insert_Agent: fixes A :: "agent"   and H :: "msg set" shows "analz (insert (Messages.Agent A) H) = insert (Messages.Agent A) (analz H)"
path_invar: fixes g :: "'g"   and ns :: "'node list" assumes "local.path g ns" shows "invar g"
primes_dvd_imp_eq: fixes p :: "'a"   and q :: "'a" assumes "prime p"   and "prime q"   and "p dvd q" shows "p = q"
f1: shows "(ad x \<oplus> ad y) \<cdot> d x = ad (ad (ad x) \<cdot> ad (ad y)) \<cdot> ad (ad x) \<cdot> ad (ad (ad y))"
hyp: assumes "finals es" shows "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es,s\<rangle>"
reduced_head_imp_exchange: fixes w :: "'w"   and s :: "'w"   and as :: "'w list"   and cs :: "'w list" assumes "S_reduced_for w (s # as)"   and "S_reduced_for w cs" shows "\<exists>a ds es. cs = ds @ [a] @ es \<and> S_reduced_for w (s # ds @ es)"
assert_atoms_equiv_bounds: fixes s :: "('i, 'a) state"   and ats :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "(snd ` ats) \<doteq> \<B> s" shows "(snd ` (ats \<union> {a})) \<doteq> \<B> (assert a s)"
set_drop_conv: fixes n :: "nat"   and l :: "'a list" shows "set (drop n l) = {l ! i |i. n \<le> i \<and> i < length l}"
fmap_add_ltr_def: fixes m1 :: "('c, 'd) fmap"   and m2 :: "('c, 'd) fmap" shows "m1 ++ m2 \<equiv> m2 ++\<^sub>f m1"
iprev_UNIV: fixes n :: "nat" shows "iprev n UNIV = n - Suc 0"
inj_on_chamber: fixes C :: "'a set" assumes "domain.chamber C" shows "inj_on f C"
arr_Set_ArrDom_in_Vset: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Set \<alpha> T" shows "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
balance_trace_axioms: shows "balance_trace (\<Rightarrow>) stack"
\<beta>: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and d :: "int"   and x :: "int" assumes "iszlfm p (a # bs)"   and "d_\<beta> p 1"   and "d_\<delta> p d"   and "0 < d"   and "\<not> (\<exists>j\<in>{1..d}. \<exists>b\<in>Inum (a # bs) ` set (\<beta> p). real_of_int x = b + real_of_int j)"   and "Ifm (real_of_int x # bs) p" shows "Ifm (real_of_int (x - d) # bs) p"
pfa_extend_grow_def: shows "pfa_extend_grow \<equiv> \<lambda>(a, n) (b, m). array_ensure a (n + m) default \<bind> (\<lambda>a'. blit b 0 a' n m \<bind> (\<lambda>_. return (a', n + m)))"
induct: fixes P :: "iface \<Rightarrow> (iface \<Rightarrow> ('b word \<times> nat) list option) \<Rightarrow> 'b common_primitive rule list \<Rightarrow> 'b wordinterval \<Rightarrow> 'b wordinterval \<Rightarrow> bool"   and a0 :: "iface"   and a1 :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and a2 :: "'b common_primitive rule list"   and a3 :: "'b wordinterval"   and a4 :: "'b wordinterval" assumes "\<And>iface ipassmt allowed denied1. P iface ipassmt [] allowed denied1"   and "\<And>iface ipassmt m rs allowed denied1. P iface ipassmt rs (wordinterval_union allowed (get_exists_matching_src_ips_executable iface m)) denied1 \<Longrightarrow> P iface ipassmt (Rule m Accept # rs) allowed denied1"   and "\<And>iface ipassmt m rs allowed denied1. P iface ipassmt rs allowed (wordinterval_union denied1 (wordinterval_setminus (get_all_matching_src_ips_executable iface m) allowed)) \<Longrightarrow> P iface ipassmt (Rule m Drop # rs) allowed denied1"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Log # va) ux_ uy_"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Reject # va) ux_ uy_"   and "\<And>uu_ uv_ vb vd va ux_ uy_. P uu_ uv_ (Rule vb (Call vd) # va) ux_ uy_"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Return # va) ux_ uy_"   and "\<And>uu_ uv_ vb vd va ux_ uy_. P uu_ uv_ (Rule vb (Goto vd) # va) ux_ uy_"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Empty # va) ux_ uy_"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Unknown # va) ux_ uy_" shows "P a0 a1 a2 a3 a4"
vec_last_def: fixes v :: "'b Matrix.vec"   and n :: "nat" shows "vec_last v n \<equiv> Matrix.vec n (\<lambda>i. v $v (dim_vec v - n + i))"
sin_ge_minus_one: fixes x :: "real" shows "- 1 \<le> sin x"
tNN_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" shows "TE.int \<xi> (tNN T) = Ik.int \<xi> T"
urel_integer_eq: fixes x :: "integer"   and y :: "int"   and x' :: "integer"   and y' :: "int" assumes "urel_integer x y"   and "urel_integer x' y'" shows "(x = x') = (y = y')"
biQuot_compl: fixes a :: "'b"   and b :: "'b"   and A :: "'b list set" shows "biQuot a b (- A) = - biQuot a b A"
eq_game_def: fixes G :: "game"   and H :: "game" shows "eq_game G H \<equiv> ge_game (G, H) \<and> ge_game (H, G)"
action_of_inject: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "x \<in> {p. is_CSP_process p}"   and "y \<in> {p. is_CSP_process p}" shows "(action_of x = action_of y) = (x = y)"
sorted_wrt_map_mono: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "sorted_wrt Q xs"   and "\<And>x y. Q x y \<Longrightarrow> P (f x) (f y)" shows "sorted_wrt P (map f xs)"
oa_Gx: shows "out_arcs G x = (uw \<rightleftharpoons>\<^sub>F uv) ` (vw \<rightleftharpoons>\<^sub>F rev_G uv) ` (out_arcs H x - {rev_H uw} - {rev_H vw})"
CHECK_dres_def: fixes f :: "unit \<Rightarrow> 'a"   and P :: "bool" shows "CHECK_dres f P = (if P then dRETURN () else let _ = f () in dSUCCEED)"
cinner_CBasis: fixes u :: "'a"   and v :: "'a" assumes "u \<in> CBasis"   and "v \<in> CBasis" shows "\<langle>u, v\<rangle> = (if u = v then 1 else 0)"
less_eint_def: fixes m :: "eint"   and n :: "eint" shows "(m < n) = (case m of eint m1 \<Rightarrow> case n of eint n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True | \<infinity> \<Rightarrow> False)"
aext_id: fixes P :: "('b, 'c) uexpr" shows "P \<oplus>\<^sub>p \<Sigma> = P"
subst_eq: fixes x :: "name"   and u :: "lam" shows "Var x[x::=u] = u"
arr_cod: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.arr C (partial_magma.cod C f)"
inj_on_map_addI1: fixes m :: "'a \<Rightarrow> 'b option"   and A :: "'a set"   and m' :: "'a \<Rightarrow> 'b option" assumes "inj_on m A"   and "m \<subseteq>\<^sub>m m ++ m'"   and "A \<subseteq> dom m" shows "inj_on (m ++ m') A"
prod_emb_def: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and K :: "'d set"   and X :: "('d \<Rightarrow> 'e) set" shows "prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (\<Pi>\<^sub>E i\<in>I. space (M i))"
unreachable_bounded_path_only: fixes d' :: "'a"   and ab :: "'a set"   and e :: "'a" assumes "d' \<notin> \<emptyset> ab e"   and "d' \<in> ab"   and "d' \<noteq> e"   and "e \<in> \<E>"   and "ab \<in> \<P>"   and "e \<notin> ab" shows "\<exists>d'e. path d'e d' e"
VLambda_vsingleton: fixes a :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda (ZFC_in_HOL.set {a}) f = ZFC_in_HOL.set {\<langle>a, f a\<rangle>}"
\<G>\<F>_semantics': fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<G>\<F> \<phi> w = subformulas\<^sub>\<mu> \<phi> \<inter> {\<psi>. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)}"