hcju/isabelleps huggingface dataset:
finprod_Suc3: fixes f :: "nat \<Rightarrow> 'a vec"   and n :: "nat" assumes "f \<in> {..n} \<rightarrow> carrier_vec n" shows "finsum V f {..n} = f n + finsum V f {..<n}"
eq1: shows "tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K) * exexH_k k = exexH_k (Suc k)"
nat_term_order_eq_POT_ps': fixes ps :: "bool"   and to1 :: "'a nat_term_order"   and to2 :: "'a nat_term_order"   and dg :: "bool" assumes "ps \<or> is_scalar TYPE('a)" shows "nat_term_order_eq to1 (POT to2) dg ps = nat_term_order_eq to1 to2 dg ps"
MI: shows "indicator X \<in> borel_measurable (state_measure (shift_var_set (V \<union> V')) (case_nat t \<Gamma>))"
comp_apply: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'f \<Rightarrow> 'd"   and x :: "'f" shows "(f \<circ> g) x = f (g x)"
permutes_inverses_2: fixes p :: "'e \<Rightarrow> 'e"   and S :: "'e set"   and x :: "'e" assumes "p permutes S" shows "inv p (p x) = x"
sublens_greatest: fixes X :: "'a \<Longrightarrow> 'b" assumes "vwb_lens X" shows "X \<subseteq>\<^sub>L \<Sigma>"
supp_8: fixes x2 :: "tyvrs"   and x3 :: "ty"   and x1 :: "ty" shows "supp (\<forall>x2<:x1. x3) = supp ([x2].x3) \<union> supp x1"
sigma_finite: fixes i :: "nat" obtains A :: "nat \<Rightarrow> val set" where "range A \<subseteq> sets (stock_measure (\<Gamma> i))" and "\<Union> (range A) = space (stock_measure (\<Gamma> i))" and "\<And>ia. emeasure (stock_measure (\<Gamma> i)) (A ia) \<noteq> \<infinity>"
simps: fixes a1 :: "'p"   and a2 :: "'p" shows "rel_12_21 a1 a2 = (a1 = \<^bold>1 \<and> a2 = \<^bold>2 \<or> a1 = \<^bold>2 \<and> a2 = \<^bold>1)"
nofail_RES_conv: fixes m :: "'a nres" shows "nofail m = (\<exists>M. m = RES M)"
psi_converg_imp_prefix': fixes j :: "nat"   and x :: "nat" assumes "\<psi> j x \<down>" shows "\<exists>t b. local.prefixes t j \<down>= b \<and> x < e_length b \<and> \<psi> j x \<down>= e_nth b x"
integral_iff: fixes R :: "('b, 'c) ring_scheme"   and a :: "'b"   and b :: "'b" assumes "domain R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "(a \<otimes>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>) = (a = \<zero>\<^bsub>R\<^esub> \<or> b = \<zero>\<^bsub>R\<^esub>)"
simps_2: fixes V :: "'b set"   and N :: "'b set"   and C :: "'b set"   and more :: "'c" shows "N \<lparr>V = V, N = N, C = C, \<dots> = more\<rparr> = N"
pmf_None_eq_weight_spmf: fixes p :: "'h spmf" shows "pmf p None = 1 - weight_spmf p"
and_mask_0_iff_le_mask: fixes w :: "'b word"   and n :: "nat" shows "(w && ~~ mask n = 0) = (w \<le> mask n)"
sf_at_Suc: fixes x :: "nat" shows "sf (x + 1) = sf x + x + 1"
irrefl_def: fixes r :: "('d \<times> 'd) set" shows "irrefl r = (\<forall>a. (a, a) \<notin> r)"
imp_map_lookup_axioms: shows "imp_map_lookup hassoc_map_assn hassoc_lookup"
simps_3: fixes f :: "'a itrm"   and x :: "'a itrm" shows "opaque (f \<diamondop> x) = opaque f @ opaque x"
assms: shows "(x, y) \<in> set (alls2 (Max (set b)) (Max (set a)) a b)"
H: fixes a :: "char list list \<times> int"   and aa :: "char list list"   and b :: "int"   and pq :: "(char list list \<times> int) list"   and r :: "(char list multiset \<times> int) multiset"   and p' :: "(char list multiset \<times> int) multiset"   and p :: "(char list list \<times> 'a) list" assumes "(q, q') \<in> sorted_poly_rel"   and "a = (aa, b)"   and "(pq, r) \<in> unsorted_poly_rel"   and "p' = add_mset (mset aa, b) A"   and "\<forall>x\<in>set p. term_order aa (fst x)"   and "sorted_wrt var_order aa"   and "distinct aa"   and "b \<noteq> 0"   and "\<And>aaa. (aaa, 0) \<notin># q'" shows "(pq @ map (mult_monomials (aa, b)) q, {#case x of (ys, n) \<Rightarrow> (remdups_mset (mset aa + ys), n * b). x \<in># q'#} + r) \<in> unsorted_poly_rel"
assms_2: shows "edges G1 \<inter> edges G2 = {}"
no_relabel_adm_edges: assumes "lo u = l u" shows "adm_edges f l \<subseteq> adm_edges fo lo"
const_in_Prim1: fixes c :: "nat" shows "(\<lambda>_. Some c) \<in> Prim1"
delta_hedging_eq_derivative_price: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat"   and w :: "bool stream" assumes "N = bernoulli_stream ((1 + r - d) / (u - d))"   and "der \<in> borel_measurable (G matur)"   and "n \<le> matur" shows "val_process Mkt (delta_hedging N der matur) n w = rn_price N der matur n w"
filterlim_inverse_at_iff: fixes g :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "(LIM x F. inverse (g x) :> at (0::'d)) = filterlim g at_infinity F"
Sup: fixes M :: "'a set"   and a :: "'a" assumes "M \<in> Chains leq"   and "M \<noteq> {}"   and "\<And>x. x \<in> M \<Longrightarrow> x \<in> iterates_above a" shows "lub M \<in> iterates_above a"
pref_mod_list: fixes u :: "'b list"   and ws :: "'b list list" assumes "u <p concat ws" obtains j :: "nat" and r :: "'b list" where "j < \<^bold>|ws\<^bold>|" and "r <p ws ! j" and "concat (take j ws) \<cdot> r = u"
Rep_star: fixes x :: "'b star" shows "Rep_star x \<in> star"
suffixes_not_Nil: fixes xs :: "'b list" shows "suffixes xs \<noteq> []"
intro: fixes y :: "'c"   and v1 :: "'c"   and P_new :: "'c list"   and G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and sep_size :: "nat" assumes "y \<noteq> v1"   and "y = DisjointPathsPlusOne.new_last P_new"   and "\<And>paths' P_new'. ProofStepInduct G v0 v1 paths' P_new' sep_size \<Longrightarrow> Digraph.distance (ProofStepInduct_NonTrivial_P_k_pre.H G v0) (last P_new) v1 \<le> Digraph.distance (ProofStepInduct_NonTrivial_P_k_pre.H G v0) (last P_new') v1" shows "ProofStepInduct_y_eq_new_last_axioms G v0 v1 P_new sep_size y"
Bounded_vec_def: fixes b :: "'b" shows "Bounded_vec b = {x. \<forall>i<dim_vec x. \<bar>x $ i\<bar> \<le> b}"
CondRed: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>if (e') e\<^sub>1 else e\<^sub>2,s',b'\<rangle>"
1: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on {a..b} f" shows "P a b {} f"
f_aggregate_eq_f_aggregate_take: fixes xs :: "'b list"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" shows "f_aggregate (xs \<down> (length xs div k * k)) k ag = f_aggregate xs k ag"
additive: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and g :: "'b"   and g' :: "'b" assumes "VectorSpaceHom smult V smult' T"   and "g \<in> V"   and "g' \<in> V" shows "T (g + g') = T g + T g'"
iT_arith_iAll_conv_2: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t I \<otimes> k. P t) = (\<box> t I. P (t * k))"
sub_of_denot_equiv_ground: fixes C :: "fterm literal set"   and E :: "char list \<Rightarrow> hterm"   and G :: "char list \<Rightarrow> hterm list \<Rightarrow> bool" shows "(\<exists>l\<in>C. eval\<^sub>l E HFun G l) = (\<exists>l\<in>C \<cdot>\<^sub>l\<^sub>s sub_of_denot E. eval\<^sub>l E HFun G l) \<and> ground\<^sub>l\<^sub>s (C \<cdot>\<^sub>l\<^sub>s sub_of_denot E)"
awalk_verts_non_Nil: fixes u :: "'a"   and p :: "'b list" shows "G.awalk_verts u p \<noteq> []"
simps: fixes a1 :: "'b formula"   and a2 :: "'b formula"   and a3 :: "'b formula" shows "Con (a1::'b formula) (a2::'b formula) (a3::'b formula) = ((\<exists>(F::'b formula) G::'b formula. a1 = F \<^bold>\<and> G \<and> a2 = F \<and> a3 = G) \<or> (\<exists>(F::'b formula) G::'b formula. a1 = \<^bold>\<not> (F \<^bold>\<or> G) \<and> a2 = \<^bold>\<not> F \<and> a3 = \<^bold>\<not> G) \<or> (\<exists>(F::'b formula) G::'b formula. a1 = \<^bold>\<not> (F \<^bold>\<rightarrow> G) \<and> a2 = F \<and> a3 = \<^bold>\<not> G) \<or> (\<exists>F::'b formula. a1 = \<^bold>\<not> (\<^bold>\<not> F) \<and> a2 = F \<and> a3 = F))"
hashcode_array_def: fixes a :: "'a array" shows "hashcode a = foldl (\<lambda>h v. h * 33 + hashcode v) 0 (list_of_array a)"
mult_strict_mono': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less c d"   and "less_eq zero a"   and "less_eq zero c" shows "less (times a c) (times b d)"
n_notin_Nset_pred: fixes n :: "nat" assumes "0 < n" shows "\<not> n \<le> n - Suc 0"
assigns_if_good_approx: fixes Env :: "env"   and s0 :: "state"   and e :: "expr"   and b :: "val"   and s1 :: "state" assumes "prg Env\<turnstile>s0 \<midarrow>e-\<succ>b\<rightarrow> s1"   and "normal s1"   and "Env\<turnstile>e\<Colon>-PrimT Boolean" shows "assigns_if (the_Bool b) e \<subseteq> dom (locals (snd s1))"
set_child_nodes_get_tag_name: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
set_eq_insert: fixes A :: "'a set"   and x :: "'a"   and x' :: "'a" assumes "A \<subseteq> carrier S"   and "x \<in> carrier S"   and "x' \<in> carrier S"   and "x .= x'" shows "insert x A {.=} insert x' A"
state_rank_step_foldl: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "(\<lambda>q. semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n) = foldl (semi_mojmir_def.step \<Sigma> \<delta> q\<^sub>0) (semi_mojmir_def.initial q\<^sub>0) (map w [0..<n])"
greaterThan_iff: fixes i :: "'a"   and k :: "'a" shows "(i \<in> {k<..}) = (k < i)"
the_prog_def: fixes prog :: "('a, 'b, 'c) prog" shows "the_prog prog = (case prog of Prog x \<Rightarrow> x)"
simps_1: fixes ys :: "'f list" shows "[] @ ys = ys"
fbind_funion: fixes M :: "'d fset"   and N :: "'d fset"   and f :: "'d \<Rightarrow> 'c fset" shows "fbind (M |\<union>| N) f = fbind M f |\<union>| fbind N f"
has_as_product_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> 's"   and a :: "'s" shows "has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J (\<cdot>) D a \<pi>"
oalist_inv_rawD1: fixes x :: "'o"   and xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw x xs" shows "(0::'c) \<notin> snd ` set xs"
insert_eqpoll_cong: fixes A :: "'a set"   and B :: "'b set"   and a :: "'a"   and b :: "'b" assumes "A \<approx> B"   and "a \<notin> A"   and "b \<notin> B" shows "insert a A \<approx> insert b B"
simps_1: fixes n :: "nat"   and c :: "'a" shows "split_by_degree n (c)\<^sub>p = ((c)\<^sub>p, 0\<^sub>p)"
non: shows "\<not> (a \<unlhd> c \<or> c \<unlhd> a)"
closure_aff_dim: fixes S :: "'b set" shows "aff_dim (closure S) = aff_dim S"
implied_frontier_alt_in_pointstamps: fixes b :: "'t"   and c :: "('loc, 't, 'a) configuration_scheme"   and loc2 :: "'loc" assumes "b \<in>\<^sub>A implied_frontier_alt c loc2" obtains a :: "'t" and s :: "'sum" and loc1 :: "'loc" where "a \<in>\<^sub>A frontier (c_pts c loc1)" and "s \<in>\<^sub>A path_summary loc1 loc2" and "results_in a s = b"
Spy_analz_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> kerbIV_gets" shows "(Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
swap_def: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" shows "Fun.swap a b f = f(a := f b, b := f a)"
filternew_subseteq_flows_state: fixes \<T> :: "'a stateful_policy" shows "filternew_flows_state \<T> \<subseteq> flows_state \<T>"
of_nat_neq_0: fixes n :: "nat" shows "of_nat (Suc n) \<noteq> (0::'b)"
downset_elim: fixes K :: "'a multiset"   and r :: "('a \<times> 'a) set"   and J :: "'a multiset" assumes "set_mset K \<subseteq> r \<down>m J" shows "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r"
euclidean_size_lcm_le1: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "euclidean_size a \<le> euclidean_size (lcm a b)"
ccw_translate3_aux: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and x :: "real" assumes "det3 0 a b \<noteq> 0"   and "x < 1"   and "ccw 0 (a - x *\<^sub>R a) (b - x *\<^sub>R a)" shows "ccw 0 a b"
set_mset_mset: fixes xs :: "'c list" shows "set_mset (mset xs) = set xs"
orbit_not_empty: fixes f :: "'a perm"   and a :: "'a" shows "orbit f a \<noteq> {}"
forbidden_path_prefix_is_hyp_free: fixes v :: "'vertex"   and es :: "(('vertex \<times> ('form, 'var) out_port) \<times> 'vertex \<times> ('form, 'var) antecedent) stream"   and n :: "nat" assumes "forbidden_path v es" shows "hyps_free (rev (stake n es))"
id: fixes xa :: "int"   and qa :: "int" shows "(xa \<in> {0..<qa}) = (0 \<le> xa \<and> xa < qa)"
gen_wf_defs_def: shows "gen_wf_defs \<equiv> map_fun Rep_gen_cfg_wf id gen_defs"
tj_stack_discovered: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan_invar G s" shows "set (tj_stack s) \<subseteq> dom (discovered s)"
f: shows "v \<notin> dom (map_of as\<^sub>1)"
simps_12: fixes \<Phi> :: "'c"   and P :: "('a, 'b, 'c) psi"   and C :: "('a, 'b, 'c) psiCase" shows "extractFrame'' (\<box> \<Phi> \<Rightarrow> P C ) = \<bottom>\<^sub>F"
div_galois: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "(x \<le> z \<leftarrow> y) = (y \<le> x \<rightarrow> z)"
lm070: fixes R :: "('c \<times> 'd) set"   and x :: "'c"   and A :: "'d set" shows "{R \<union> {(x, y)} |y. y \<in> A} \<subseteq> totalRels ({x} \<union> Domain R) (A \<union> Range R)"
continuous_on_max: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on A f"   and "continuous_on A g" shows "continuous_on A (\<lambda>x. max (f x) (g x))"
product_simps_1: fixes A :: "('f, 'g, 'd) FSM"   and B :: "('f, 'g, 'e) FSM"   and a :: "'f \<times> 'g"   and p\<^sub>1 :: "'d"   and p\<^sub>2 :: "'e" shows "succ (product A B) a (p\<^sub>1, p\<^sub>2) = succ A a p\<^sub>1 \<times> succ B a p\<^sub>2"
local: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<zeta> :: "'b"   and R :: "nat set"   and a :: "'c node_action"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and \<zeta>' :: "'b"   and R' :: "nat set" assumes "(\<sigma>, NodeS ii \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) S U"   and "((\<sigma>, NodeS ii \<zeta> R), a, \<sigma>', NodeS ii \<zeta>' R') \<in> automaton.trans (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o)"   and "S \<sigma> \<sigma>' a"   and "P \<sigma> \<zeta> R" shows "P \<sigma>' \<zeta>' R'"
rt_fresh_as_fresherI: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt1"   and "dip \<in> kD rt2"   and "the (rt1 dip) \<sqsubseteq> the (rt2 dip)"   and "the (rt2 dip) \<sqsubseteq> the (rt1 dip)" shows "rt1 \<approx>\<^bsub>dip\<^esub> rt2"
cancels_to_1_def: fixes l1 :: "(bool \<times> 'e) list"   and l2 :: "(bool \<times> 'e) list" shows "cancels_to_1 l1 l2 = (\<exists>i. cancels_to_1_at i l1 l2)"
post_meas0_def: fixes n :: "nat"   and v :: "complex Matrix.mat"   and i :: "nat" shows "post_meas0 n v i \<equiv> complex_of_real (1 / sqrt (prob0 n v i)) \<cdot>\<^sub>m |Matrix.vec (2 ^ n) (\<lambda>j. if \<not> select_index n i j then v $$ (j, 0) else 0)\<rangle>"
arr_mset_remove_first: fixes i :: "int"   and r :: "int"   and arr :: "int \<Rightarrow> 'a" assumes "i \<le> r" shows "arr_mset arr (i - 1) r = arr_mset arr i r + {#arr (i - 1)#}"
insert_ident: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "x \<notin> A"   and "x \<notin> B" shows "(insert x A = insert x B) = (A = B)"
qSwapAll_preserves_qAFreshAll: fixes ys :: "'a"   and zs :: "'a"   and z1 :: "'b"   and z2 :: "'b"   and y :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm"   and A :: "('c, 'd, 'a, 'b, 'e) qAbs" shows "qAFresh ys (y @ys[z1 \<and> z2]_zs) (X #[[z1 \<and> z2]]_zs) = qAFresh ys y X \<and> qAFreshAbs ys (y @ys[z1 \<and> z2]_zs) (A $[[z1 \<and> z2]]_zs) = qAFreshAbs ys y A"
simps_1: fixes m :: "nat"   and i :: "'b"   and j :: "'b"   and p :: "'a"   and xs :: "('b \<times> 'a) list" shows "local.select_min_main m i ((j, p) # xs) = (let n = measure p in if n < m then local.select_min_main n j xs else local.select_min_main m i xs)"
ultrafilters_def: shows "ultrafilters = properfilters \<inter> maximalfilters"
ipurge_ref_subset: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and u :: "'f"   and xs :: "'e list"   and X :: "'e set" shows "ipurge_ref I D u xs X \<subseteq> X"
mod_le_divisor: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "m mod n \<le> n"
resTauCases: fixes \<Psi> :: "'b"   and x :: "name"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<longmapsto> \<tau> \<prec> P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> P'"   and "\<And>P'. \<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P' \<Longrightarrow> Prop (\<lparr>\<nu>x\<rparr>P')" shows "Prop P'"
f_expand_f_shrink_id_take: fixes xs :: "'b message_af list"   and k :: "nat" assumes "\<And>i. \<lbrakk>i < length xs; 0 < i mod k\<rbrakk> \<Longrightarrow> xs ! i = NoMsg" shows "xs \<div> k \<odot> k = xs \<down> (length xs div k * k)"
n_val_in_aug_inf: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "n_val K v x \<in> Z\<^sub>\<infinity>"
Stop_def: shows "Stop \<equiv> action_of (R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> wait A'))"
span_char: shows "local.span D = {x \<in> carrier_vec n. x \<bullet> c = (0::'a)}"
cInf_mono: fixes B :: "'b set"   and A :: "'b set" assumes "B \<noteq> {}"   and "bdd_below A"   and "\<And>b. b \<in> B \<Longrightarrow> \<exists>a\<in>A. a \<le> b" shows "Inf A \<le> Inf B"
comp_min_basis_is_red: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red (set xs) f" shows "is_red (set (comp_min_basis xs)) f"
roots_of_2_irr_3: fixes p :: "int poly" assumes "poly_cond p"   and "0 < degree p" shows "distinct (map real_of_2 (roots_of_2_irr p))"
sup_principal: fixes A :: "'a set"   and B :: "'a set" shows "sup (principal A) (principal B) = principal (A \<union> B)"
head_nz: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "(head p = 0\<^sub>p) = (p = 0\<^sub>p)"
lookup_append_some: fixes env :: "('f, 'g, 'h) env"   and xs :: "'h list"   and e :: "('f, 'g, 'h) env"   and ys :: "'h list" assumes "lookup env xs = Some e" shows "lookup env (xs @ ys) = lookup e ys"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop2 n (l, r) = (\<exists>i j any. i + j = n \<and> 0 < n \<and> 0 < i \<and> 0 < j \<and> (l, r) = (Oc \<up> i, any # Bk \<up> j @ Oc \<up> j))"
derivation_ge_LeftDerivationLadder: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and d :: "nat"   and \<alpha> :: "symbol list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list"   and index :: "nat" assumes "derivation_ge D d"   and "LeftDerivationLadder \<alpha> D L \<gamma>"   and "d \<le> ladder_i L 0"   and "index < length L" shows "d \<le> ladder_i L index \<and> d \<le> ladder_j L index"
nth_append_first: fixes i :: "nat"   and l :: "'b list"   and l' :: "'b list" assumes "i < length l" shows "(l @ l') ! i = l ! i"
hh_1: shows "hermitean H \<and> H \<noteq> mat_zero"
trans: fixes x :: "('a, 'b) rose_tree\<^sub>m"   and y :: "('a, 'b) rose_tree\<^sub>m"   and z :: "('a, 'b) rose_tree\<^sub>m" assumes "blinding_of_tree h bo x y"   and "blinding_of_tree h bo y z"   and "x \<in> UNIV" shows "blinding_of_tree h bo x z"
iterate_add_eq_add_iff2: fixes i :: "nat"   and j :: "nat"   and u :: "'a"   and m :: "'a"   and n :: "'a" assumes "i \<le> j" shows "(iterate_add i u + m = iterate_add j u + n) = (m = iterate_add (j - i) u + n)"
simps_4: fixes instT :: "((variable \<times> String.literal set) \<times> typ) list"   and insts :: "((variable \<times> typ) \<times> Core.term) list"   and T :: "typ"   and t :: "Core.term" shows "subst_type_term instT insts (Abs T t) = Abs (subst_typ instT T) (subst_type_term instT insts t)"
ex_has_least_nat: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a"   and m :: "'a \<Rightarrow> nat" assumes "P k" shows "\<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> m x \<le> m y)"
asymp_equiv_compose': fixes f :: "'a \<Rightarrow> 'b"   and G :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "f \<sim>[G] g"   and "filterlim h G F" shows "(\<lambda>x. f (h x)) \<sim>[F] (\<lambda>x. g (h x))"
rel_simps_51: fixes n :: "num" shows "(0::'b) < numeral n"
value_plus_symmetry: fixes x :: "value option"   and y :: "value option" shows "value_plus x y = value_plus y x"
simp2: fixes x :: "brick list"   and ys :: "wall"   and y :: "brick list" assumes "codomain_block x \<noteq> 0"   and "is_tangle_diagram (basic (make_vert_block (nat (codomain_block x)))) \<and> is_tangle_diagram ys \<longrightarrow> is_tangle_diagram (basic (make_vert_block (nat (codomain_block x))) \<otimes> ys)" shows "is_tangle_diagram (basic x) \<and> is_tangle_diagram (y * ys) \<longrightarrow> is_tangle_diagram (basic x \<otimes> y * ys)"
decomp2: shows "C0 # Cs @ [C] = Es @ E # B # Bs"
nonempty_subseteq_mset_iff_singleton: fixes M :: "'b multiset"   and x :: "'b"   and P :: "bool" shows "(M \<noteq> {#} \<and> M \<subseteq># {#x#} \<and> P) = (M = {#x#} \<and> P)"
notin_remove_fset: fixes x :: "'b"   and S :: "'b fset" shows "x |\<notin>| remove_fset x S"
set_relator_eq_onp: fixes P :: "'a \<Rightarrow> bool" shows "rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"
proper_it'_def: fixes it :: "'e \<Rightarrow> ('f \<Rightarrow> bool) \<Rightarrow> ('g \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f"   and it' :: "'e \<Rightarrow> ('h \<Rightarrow> bool) \<Rightarrow> ('g \<Rightarrow> 'h \<Rightarrow> 'h) \<Rightarrow> 'h \<Rightarrow> 'h" shows "proper_it' it it' \<equiv> \<forall>s. proper_it (it s) (it' s)"
remove_small_job: fixes T :: "nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "makespan (T(x := T x + y)) \<noteq> T x + y" shows "makespan (T(x := T x + y)) = makespan T"
vs: shows "vs = getFrN (map snd txs) (map fst txs) [subst \<phi> s y] (length txs)"
lift_IT: fixes t :: "dB"   and i :: "nat" assumes "IT t" shows "IT (lift t i)"
\<psi>D_im_single: fixes T :: "'v \<Rightarrow> 'w"   and h :: "'g"   and v :: "'v" assumes "T \<in> GRepHomSet (\<star>) W"   and "h \<in> set H_rcoset_reps"   and "v \<in> set Vfbasis" shows "\<psi> T (- h *\<currency> induced_vector v) = - h *\<star> T v"
assms: shows "fresh_idn \<notin> fst ` tvsT \<tau>"
distinct_3: fixes x1 :: "agent"   and x3 :: "nat" shows "Agent x1 \<noteq> Nonce x3"
B_mod_2': fixes X :: "int"   and w :: "int" assumes "X = 2" shows "w BIT True mod X = 1 \<and> w BIT False mod X = 0"
assms: fixes i :: "nat" assumes "i \<in> t" shows "enat i < esize s"
ipPartitionUnion: fixes As :: "'a set set"   and Cs :: "'a set set"   and Bs :: "'a set set" shows "(ipPartition As Cs \<and> ipPartition Bs Cs) = ipPartition (As \<union> Bs) Cs"
enum_n_top: fixes a :: "nat \<Rightarrow> nat" assumes "a \<in> t" shows "(a = t.enum n) = (\<forall>a'\<in>t. a' \<le> a)"
CP_not_closed_subseteq: shows "\<exists>V U. V \<subseteq> U \<and> U \<in> CP \<and> V \<notin> CP"
and_mask_shiftr_comm: fixes m :: "nat"   and w :: "'b word"   and n :: "nat" assumes "m \<le> size w" shows "w AND mask m >> n = (w >> n) AND mask (m - n)"
sameDom_def: fixes inp1 :: "'d \<Rightarrow> 'e option"   and inp2 :: "'d \<Rightarrow> 'f option" shows "sameDom inp1 inp2 \<equiv> \<forall>i. (inp1 i = None) = (inp2 i = None)"