hcju/isabelleps huggingface dataset:
s0g_dom_def: shows "s0g_dom \<equiv> {s. Domain (az s \<union> lkr s) \<subseteq> Domain (kn s)}"
HR_H: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "coding_hf x" shows "HR x \<lbrakk>Goedel_I.HF x\<rbrakk>e"
walk_tl: fixes G :: "('e, 'f) Graph_scheme"   and xs :: "'e list" assumes "Graph G"   and "Graph.walk G xs" shows "Graph.walk G (tl xs)"
fps_unit_factor_zero_iff: fixes f :: "'a fps" shows "(unit_factor f = 0) = (f = 0)"
ord_inv: fixes x :: "'a" assumes "x \<in> carrier G" shows "ord (inv x) = ord x"
rec_t_eq_step: fixes tp :: "(action \<times> nat) list"   and c :: "nat \<times> cell list \<times> cell list" assumes "case c of (s, l, r) \<Rightarrow> s \<le> length tp div 2" shows "trpl_code (step0 c tp) = rec_exec rec_newconf [code tp, trpl_code c]"
inter_empty2: fixes i :: "nat_int" shows "bot \<sqinter> i = bot"
inj_idom_hom_axioms: shows "inj_idom_hom to_fract"
effect_bindI: fixes f :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'b"   and g :: "'b \<Rightarrow> 'c Heap"   and h'' :: "heap"   and r' :: "'c" assumes "effect f h h' r"   and "effect (g r) h' h'' r'" shows "effect (f \<bind> g) h h'' r'"
assms_1: shows "(a, b) \<in> set (arcs i j (x # xs))"
chine_in_hom: shows "\<guillemotleft>\<mu>.chine : \<mu>.dom.apex \<rightarrow> \<mu>.cod.apex\<guillemotright>"
c_d_4: shows "\<nexists>p. p tagged_division_of cbox c d \<and> g fine p"
sin_periodic: fixes x :: "real" shows "sin (x + 2 * pi) = sin x"
last_appendR: fixes ys :: "'g list"   and xs :: "'g list" assumes "ys \<noteq> []" shows "last (xs @ ys) = last ys"
STL4: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> \<box>F \<longrightarrow> \<box>G"
vec1I_def: fixes ze :: "'a"   and on :: "'a"   and n :: "nat"   and i :: "nat" shows "vec1I ze on n i \<equiv> replicate i ze @ on # replicate (n - 1 - i) ze"
canon_ang_def: fixes \<alpha> :: "real" shows "\<downharpoonright>\<alpha>\<downharpoonleft> = (THE \<alpha>''. canon_ang_P \<alpha> \<alpha>'')"
mono: fixes A :: "'a set"   and x :: "'a"   and B :: "'a set" assumes "local.MaxR.MaxR_opt A = Some x" shows "\<exists>y. local.MaxR.MaxR_opt (A \<union> B) = Some y \<and> (x, y) \<in> r"
j_nonabort: shows "chaos \<sqsubseteq> (j::'a::refinement_lattice)"
subst_closed: fixes f :: "'form"   and s :: "'subst" assumes "closed f" shows "subst s f = f"
safe_hd_rep: fixes q :: "'a queue"   and \<alpha> :: "'a option"   and q' :: "'a queue" assumes "safe_hd q = (\<alpha>, q')" shows "linearize q = linearize q' \<and> (case \<alpha> of None \<Rightarrow> linearize q = [] | Some a \<Rightarrow> linearize q \<noteq> [] \<and> a = hd (linearize q))"
C: shows "set (p\<^sub>2 # ps) = PS\<^sub>L \<union> PS\<^sub>R"
has_no_zeros_def: fixes l :: "rat list" shows "has_no_zeros l = (0 \<notin> set l)"
map_of_Cons_code_2: fixes l :: "'d"   and v :: "'c"   and ps :: "('d \<times> 'c) list"   and k :: "'d" shows "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
domrangefix: fixes x :: "'a" shows "(d x = x) = (r x = x)"
BLP_m_spec_def: shows "BLP_m_spec \<equiv> new_configured_SecurityInvariant (SINVAR_BLPtrusted.sinvar, SINVAR_BLPtrusted.default_node_properties, SINVAR_BLPtrusted.receiver_violation, BLPtrusted.node_props \<lparr>node_properties = BLP_security_levels\<rparr>)"
lossless_return_pmf_None: shows "\<not> lossless_spmf (return_pmf None)"
simps_5: fixes b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "Language.normalize (Cond b c\<^sub>1 c\<^sub>2) = Cond b (Language.normalize c\<^sub>1) (Language.normalize c\<^sub>2)"
fixp_preserves_mcont2: fixes F :: "('c \<Rightarrow> 'd \<Rightarrow> 'e llist) \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e llist"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e llist"   and lubb :: "('c \<times> 'd) set \<Rightarrow> 'c \<times> 'd"   and ordb :: "'c \<times> 'd \<Rightarrow> 'c \<times> 'd \<Rightarrow> bool" assumes "\<And>x. mono_llist (\<lambda>f. case x of (x, xa) \<Rightarrow> F (curry f) x xa)"   and "f \<equiv> curry (llist.fixp_fun (\<lambda>f (x, y). F (curry f) x y))"   and "\<And>f. mcont lubb ordb lSup (\<sqsubseteq>) (\<lambda>(x, y). f x y) \<Longrightarrow> mcont lubb ordb lSup (\<sqsubseteq>) (\<lambda>(x, y). F f x y)" shows "mcont lubb ordb lSup (\<sqsubseteq>) (\<lambda>(x, y). f x y)"
jkbpTn_length: fixes t :: "'s Trace"   and n :: "nat"   and T :: "'s Trace set" assumes "t \<in> jkbpT\<^bsub>n\<^esub> T" shows "tLength t = n"
hyps_2: shows "poly (poly_lift p) [:x:] = [:poly p x:]"
conv_radius_shift: fixes f :: "nat \<Rightarrow> 'a"   and m :: "nat" shows "conv_radius (\<lambda>n. f (n + m)) = conv_radius f"
jump_gnz_2: fixes x :: "real" assumes "a < x"   and "x < b"   and "g x \<noteq> 0" shows "jumpF h (at_right x) = 0"
maximum_in: fixes i :: "nat_int" shows "i \<noteq> bot \<longrightarrow> maximum i \<^bold>\<in> i"
norm_add_leD: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm (plus a b) \<le> c" shows "norm b \<le> norm a + c"
nth_superfluous_append: fixes i :: "nat"   and a :: "'a list"   and b :: "'a list" assumes "i < length a" shows "(a @ b) ! i = a ! i"
LLL_D_pos: fixes fs :: "int vec list" assumes "LLL_invariant_weak fs" shows "0 < D fs"
assms_3: shows "(state0, state) \<in> transitionRelation decisionVars"
same_level_path_valid_path: fixes as :: "'edge list" assumes "same_level_path as" shows "valid_path as"
simps_3: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>c = (\<lambda>\<I>. \<lbrakk>insert (t \<cdot> \<I>) M; S\<rbrakk>\<^sub>c \<I>)"
assert_all_def: fixes assert_all_state :: "(nat \<times> linear_poly) list \<Rightarrow> ('a \<times> 'b atom) list \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list"   and as :: "('a \<times> 'b atom) list" assumes "AssertAllState assert_all_state" shows "AssertAllState.assert_all assert_all_state t as \<equiv> let s = assert_all_state t as in if \<U> s then Inl (the (\<U>\<^sub>c s)) else Inr (\<V> s)"
IH: fixes a :: "int"   and a' :: "int" assumes "[a = a'] (mod 2 ^ p - 1)" shows "[gen_lucas_lehmer_sequence'' p a n = gen_lucas_lehmer_sequence a' n] (mod 2 ^ p - 1)"
coerce_sinl: fixes x :: "'c" shows "COERCE('c \<oplus> 'd, 'a \<oplus> 'b)\<cdot>(sinl\<cdot>x) = sinl\<cdot>(COERCE('c, 'a)\<cdot>x)"
less_option_None_Some_code: fixes x :: "'b" shows "(None < Some x) = True"
Rep_formula_inverse: fixes x :: "('a, 'b, 'c) Formula.formula" shows "Formula.formula.Abs_formula (Formula.formula.Rep_formula x) = x"
pdivmod_via_divmod_list: fixes f :: "'b poly"   and g :: "'b poly" shows "(f div g, f mod g) = (let cg = coeffs g in if cg = [] then (0, f) else let cf = coeffs f; ilc = inverse (last cg); ch = map ((*) ilc) cg; (q, r) = divmod_poly_one_main_list [] (rev cf) (rev ch) (1 + length cf - length cg) in (poly_of_list (map ((*) ilc) q), poly_of_list (rev r)))"
inset: shows "p \<in> grid b (ds' \<union> ds)"
map_of_delete_aux: fixes xs :: "('c \<times> 'd) list"   and k :: "'c"   and k' :: "'c" assumes "distinct (map fst xs)" shows "map_of (AList.delete_aux k xs) k' = ((map_of xs)(k := None)) k'"
finlsts_rec_LNil: fixes c :: "'b"   and d :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b \<Rightarrow> 'b" shows "finlsts_rec c d LNil = c"
has_codomain_iff_arr: fixes f :: "'a \<times> 'a \<times> 'a" shows "(B.VVV.codomains f \<noteq> {}) = B.VVV.arr f"
weakSimI: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and C :: "'e" assumes "eqvt Rel"   and "\<And>\<Psi>'' \<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; bn \<alpha> \<sharp>* subject \<alpha>; bn \<alpha> \<sharp>* C; distinct (bn \<alpha>); \<alpha> \<noteq> \<tau>\<rbrakk> \<Longrightarrow> \<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>'' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>'', P', Q') \<in> Rel)"   and "\<And>Q'. \<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q' \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto><Rel> Q"
image_subsetI: fixes A :: "'g set"   and f :: "'g \<Rightarrow> 'h"   and B :: "'h set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f ` A \<subseteq> B"
C_def: shows "C = (\<lambda>F. interior (F \<union> - t)) ` {F. finite F \<and> F \<subseteq> t}"
col_permutation_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.Col Bet B C A"
strategy_def: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" shows "G'.strategy p \<sigma> \<equiv> \<forall>v\<in>G'.VV p. \<not> G'.deadend v \<longrightarrow> v \<rightarrow>\<^bsub>G'\<^esub> \<sigma> v"
da_elim_cases_26: fixes Env :: "env"   and B :: "lname set"   and T :: "ty"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l (New T[e])\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
oalist_inv_rawD2: fixes xs :: "('a \<times> 'd) list" assumes "oalist_inv_raw xs" shows "sorted_wrt lt (map fst xs)"
distinct_2: shows "1\<^sub>r \<noteq> 0\<^sub>r"
st_le: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HFinite"   and "y \<in> HFinite"   and "x \<le> y" shows "st x \<le> st y"
br_comp_alt: fixes \<alpha> :: "'c \<Rightarrow> 'e"   and I :: "'c \<Rightarrow> bool"   and R :: "('e \<times> 'd) set" shows "br \<alpha> I O R = {(c, a). I c \<and> (\<alpha> c, a) \<in> R}"
qequiv_sym: fixes a :: "'e['f, 'g]"   and b :: "'e['h, 'g]" assumes "a \<cong>\<^sub>Q b" shows "b \<cong>\<^sub>Q a"
length_addition: fixes a :: "nat"   and b :: "nat"   and c :: "nat" shows "length (addition a b c) = 7"
assms: shows "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<Rel> Q"
lin_dep_crit: fixes A :: "'c set"   and S :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "finite A"   and "A \<subseteq> S"   and "a \<in> A \<rightarrow> carrier R"   and "v \<in> A"   and "a v \<noteq> \<zero>"   and "lincomb a A = \<zero>\<^bsub>M\<^esub>" shows "lin_dep S"
atMost_iff: fixes i :: "'a"   and k :: "'a" shows "(i \<in> {..k}) = (i \<le> k)"
is_derivation_derivation: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "c_manifold charts k"   and "c_manifold.is_derivation charts k X p"   and "f \<in> c_manifold.diff_fun_space charts k"   and "g \<in> c_manifold.diff_fun_space charts k" shows "X (f * g) = f p * X g + g p * X f"
Casts_Cons: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and T :: "ty"   and v :: "val"   and v' :: "val"   and Ts :: "ty list"   and vs :: "val list"   and vs' :: "val list" assumes "P \<turnstile> T casts v to v' "   and "P \<turnstile> Ts Casts vs to vs' " shows "P \<turnstile> T # Ts Casts v # vs to v' # vs' "
one_real_def: shows "1 \<equiv> Real.Real (\<lambda>n. 1)"
ksup_idem: fixes f :: "'a \<Rightarrow> 'b set" shows "f \<squnion> f = f"
step_3_similar: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat (??.Jordan_Normal_Form_Existence.step_3 A) A"
prob0_fst_qubits_eq: fixes n :: "nat"   and v :: "complex Matrix.mat" shows "prob0_fst_qubits n v = (cmod (v $$ (0, 0)))\<^sup>2 + (cmod (v $$ (1, 0)))\<^sup>2"
iIN_add: fixes n :: "nat"   and d :: "nat"   and k :: "nat" shows "[n\<dots>,d] \<oplus> k = [n + k\<dots>,d]"
assms_4: shows "d \<le> forest_components f * e\<^sup>T * top"
2: shows "prv (imp \<phi> (PP (encF N \<langle>\<phi>\<rangle>)))"
unfold: shows "the_and = foldr\<cdot>trand\<cdot>TT"
rq_trans: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<le>s v"   and "v \<le>s w" shows "w\<^sup><\<inverse>v  \<cdot> v\<^sup><\<inverse>u  = w\<^sup><\<inverse>u "
prems_2: shows "\<Gamma> \<turnstile> t is u : T\<^sub>1\<rightarrow>T\<^sub>2"
DDH_bound2: fixes \<G> :: "'b cyclic_group"   and \<A> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf" assumes "ddh_ext \<G>" shows "\<bar>spmf (ddh.DDH0 \<G> \<A>) True - spmf (ddh.DDH1 \<G> \<A>) True\<bar> \<le> ddh.advantage \<G> \<A> + \<bar>spmf (ddh.ddh_1 \<G> \<A>) True - spmf (ddh.DDH1 \<G> \<A>) True\<bar>"
sorted_wrt_filter: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "sorted_wrt f xs" shows "sorted_wrt f (filter P xs)"
SReal_hcmod_hcomplex_of_complex: fixes r :: "complex" shows "hcmod (hcomplex_of_complex r) \<in> \<real>"
wellformed_inc_item: fixes x :: "item"   and s :: "symbol"   and k :: "nat" assumes "wellformed_item x"   and "next_symbol x = Some s"   and "k \<le> length Doc"   and "item_end x \<le> k" shows "wellformed_item (inc_item x k)"
Gmult_neg_left: fixes g :: "'g"   and v :: "'v" assumes "g \<in> G"   and "v \<in> V" shows "- g *\<cdot> g *\<cdot> v = v"
norm_eff_def: fixes i :: "instr"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list" shows "norm_eff i P pc \<tau> = map (\<lambda>pc'. (pc', \<lfloor>eff\<^sub>i (i, P, \<tau>)\<rfloor>)) (succs i \<tau> pc)"
foldD_commute: fixes A :: "'b set"   and x :: "'b"   and e :: "'a" assumes "finite A"   and "x \<in> B"   and "e \<in> D"   and "A \<subseteq> B" shows "x \<cdot> foldD D (\<cdot>) e A = foldD D (\<cdot>) (x \<cdot> e) A"
ev_at_imp_snth: fixes P :: "'a stream \<Rightarrow> bool"   and n :: "nat"   and \<omega> :: "'a stream" assumes "ev_at P n \<omega>" shows "P (sdrop n \<omega>)"
assms: shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>h h''"
Gd: fixes fs :: "int vec list"   and k :: "nat" assumes "LLL_invariant_weak fs"   and "k \<le> m" shows "rat_of_int (gs.Gramian_determinant fs k) = (\<Prod>j<k. \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) j\<parallel>\<^sup>2)"
MC_T_not_sconst_strong: fixes f :: "'s \<Rightarrow> 'd"   and u :: "'d"   and Y :: "'s cfg set"   and X :: "'s cfg set"   and r :: "ennreal"   and cfg :: "'s cfg"   and x :: "'s" assumes "\<forall>l. \<forall>cfg\<in>\<Union> (cfg_on ` f -` {u}) \<inter> Y \<inter> X. emeasure (measure_pmf (action cfg)) (f -` {u}) \<le> r"   and "r < 1"   and "cfg \<in> cfg_on x"   and "almost_everywhere (MC.T cfg) (pred_stream (\<lambda>x. x \<in> X))"   and "almost_everywhere (MC.T cfg) (alw (\<lambda>\<omega>. HLD (f -` {u}) (smap state \<omega>) \<longrightarrow> ev (HLD Y) \<omega>))" shows "AE \<omega> in MC.T cfg. smap (f \<circ> state) \<omega> \<noteq> sconst u"
poly_subst_monom_mult: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and c :: "'c"   and t :: "'d \<Rightarrow>\<^sub>0 nat"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f (MPoly_Type_Class.punit.monom_mult c t p) = MPoly_Type_Class.punit.monom_mult c 0 (subst_pp f t * poly_subst f p)"
rotate_append: fixes l :: "'b list"   and q :: "'b list" shows "rotate (length l) (l @ q) = q @ l"
elem_set_of: fixes x :: "ref"   and st :: "dag"   and t :: "dag" assumes "x \<in> set_of st"   and "set_of st \<subseteq> set_of t" shows "x \<in> set_of t"
n1_def: shows "n1 \<equiv> alpha1 \<or> beta1 \<or> gamma1"
second_summand_in: fixes y :: "i"   and x :: "i" shows "P y (x \<oplus> y)"
nhdE: fixes U :: "'a set"   and x :: "'a" assumes "U \<in> nhds x" obtains m :: "'a set" where "U \<subseteq> carrier" and "m open" and "x \<in> m" and "m \<subseteq> U"
assms: shows "n -as\<rightarrow>\<^sub>\<surd>* (_Exit_)"
simps: fixes fstate :: "('a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a) discr" shows "\<aF>\<cdot>fstate = (case undiscr fstate of (PC (Lambda lab vs c, \<beta>), as, ve, b) \<Rightarrow> if length vs = length as then let \<beta>' = \<beta>(lab \<mapsto> b); ve' = ve \<union>. \<Union>.map2 (\<lambda>v. smap_singleton (v, b)) vs as in \<aC>\<cdot>(Discr (c, \<beta>', ve', b)) else \<bottom> | (PP (Plus c), [], bb) \<Rightarrow> \<bottom> | (PP (Plus c), [x], bb) \<Rightarrow> \<bottom> | (PP (Plus c), [x, xa], bb) \<Rightarrow> \<bottom> | (PP (Plus c), [x, xa, cnts], ve, b) \<Rightarrow> let b' = \<anb> b c; \<beta> = [c \<mapsto> b] in (\<Union>cnt\<in>cnts. \<aF>\<cdot>(Discr (cnt, [{}], ve, b'))) \<union> {((c, \<beta>), cont) |cont. cont \<in> cnts} | (PP (Plus c), x # xa # cnts # ac # lista, bb) \<Rightarrow> \<bottom> | (PP (prim.If ct cf), [], bb) \<Rightarrow> \<bottom> | (PP (prim.If ct cf), [x], bb) \<Rightarrow> \<bottom> | (PP (prim.If ct cf), [x, cntts], bb) \<Rightarrow> \<bottom> | (PP (prim.If ct cf), [x, cntts, cntfs], ve, b) \<Rightarrow> (let b' = \<anb> b ct; \<beta> = [ct \<mapsto> b] in (\<Union>cnt\<in>cntts. \<aF>\<cdot>(Discr (cnt, [], ve, b'))) \<union> {((ct, \<beta>), cnt) |cnt. cnt \<in> cntts}) \<union> (let b' = \<anb> b cf; \<beta> = [cf \<mapsto> b] in (\<Union>cnt\<in>cntfs. \<aF>\<cdot>(Discr (cnt, [], ve, b'))) \<union> {((cf, \<beta>), cnt) |cnt. cnt \<in> cntfs}) | (PP (prim.If ct cf), x # cntts # cntfs # ac # lista, bb) \<Rightarrow> \<bottom> | (AStop, [], bb) \<Rightarrow> \<bottom> | (AStop, [x], xa, xb) \<Rightarrow> {} | (AStop, x # a # lista, bb) \<Rightarrow> \<bottom>)"
distinct_1: fixes x2 :: "'d" shows "Done \<noteq> Skip x2"
augment_cf_refine: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and p :: "(nat \<times> nat) list" assumes "(cf, f) \<in> cfi_rel"   and "NPreflow.isAugmentingPath c s t f p" shows "(Graph.augment_cf cf (set p) (resCap_cf cf p), NFlow.augment_with_path c f p) \<in> cfi_rel"
toFull_iff_fromFull: fixes \<sigma> :: "'tp"   and a :: "univ"   and b :: "univ" assumes "intT \<sigma> a" shows "(toFull \<sigma> a = b) = (a = fromFull \<sigma> b)"
atoms_norm: fixes r :: "'a rexp" shows "atoms (norm r) \<subseteq> atoms r"
pres_typeD: fixes step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and n :: "nat"   and A :: "'b set"   and s :: "'b"   and p :: "nat"   and q :: "nat"   and s' :: "'b" assumes "pres_type step n A"   and "s \<in> A"   and "p < n"   and "(q, s') \<in> set (step p s)" shows "s' \<in> A"
exec_def: fixes G :: "jvm_prog"   and maxs :: "nat"   and rT :: "ty"   and et :: "ex_table"   and bs :: "bytecode" shows "exec G maxs rT et bs \<equiv> err_step (length bs) (\<lambda>pc. app (bs ! pc) G maxs rT pc (length bs) et) (\<lambda>pc. eff (bs ! pc) G pc et)"
commutative_square_def: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" shows "commutative_square f g h k \<equiv> cospan f g \<and> span h k \<and> local.dom f = cod h \<and> f \<cdot> h = g \<cdot> k"
rec1_def: fixes s1 :: "('d, 'd IF1 \<times> 'e, 'd IF2 \<times> 'f) F1 \<Rightarrow> 'e"   and s2 :: "('d, 'd IF1 \<times> 'e, 'd IF2 \<times> 'f) F2 \<Rightarrow> 'f" shows "rec1 s1 s2 = snd \<circ> fold1 <ctor1 \<circ> F1map id fst fst , s1> <ctor2 \<circ> F2map id fst fst , s2>"
hoare_choice: fixes p :: "'e"   and S :: "'f \<Rightarrow> 'e"   and T :: "'f \<Rightarrow> 'e"   and q :: "'f" shows "\<Turnstile> p{| S \<sqinter> T |}q = (\<Turnstile> p{| S |}q \<and> \<Turnstile> p{| T |}q)"
mat_adj_def: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_adj A = mat_cnj (mat_transpose A)"
lapp_infT: fixes r :: "'b llist"   and s :: "'b llist"   and A :: "'b set" shows "(r @@ s \<in> A\<^sup>\<omega>) = (r \<in> A\<^sup>\<star> \<and> s \<in> A\<^sup>\<omega> \<or> r \<in> A\<^sup>\<omega>)"
int_of_integer_integer_of_int: fixes k :: "int" shows "int_of_integer (integer_of_int k) = k"
integrable_squareD: fixes X :: "'a \<Rightarrow> real" assumes "integrable M (\<lambda>x. (X x)\<^sup>2)"   and "random_variable borel X" shows "integrable M X"
consistent_def: fixes S :: "('a, 'b) fm set" shows "consistent S \<equiv> \<nexists>S'. set S' \<subseteq> S \<and> (S' \<turnstile> \<^bold>\<bottom>)"
elims: fixes x :: "'fsym trm"   and y :: "'tp" assumes "Ik.tpOf x = y" obtains xa :: "Preliminaries.var" where "x = Var xa" and "y = TE.tpOfV xa"   | f :: "'fsym" and Tl :: "'fsym trm list" where "x = Fn f Tl" and "y = resOf f"
ras_pop_def: fixes s :: "'c array \<times> nat" shows "ras_pop s \<equiv> let (a, n) = s in ras_shrink (a, n - 1)"
eq: shows "f (a /\<^sub>R norm a) = b /\<^sub>R norm b"
adjoint_pair_def: fixes f :: "'b"   and g :: "'b" shows "D.adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D f g \<eta> \<epsilon>"
inject_4: fixes x51 :: "'b ptrm"   and x52 :: "'b ptrm"   and y51 :: "'b ptrm"   and y52 :: "'b ptrm" shows "(PPair x51 x52 = PPair y51 y52) = (x51 = y51 \<and> x52 = y52)"
set_mset_remdups_mset: fixes A :: "'a multiset" shows "set_mset (remdups_mset A) = set_mset A"
dvd_mult: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a dvd c" shows "a dvd b * c"
equiv_order_trans: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "semiattractive A less_eq"   and "equivpartp less_eq x y"   and "less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"
skip_eval\<^sub>w: fixes mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat" shows "eval_abv\<^sub>C \<langle>Skip, mds, mem\<rangle>\<^sub>C \<langle>Stop, mds, mem\<rangle>\<^sub>C"
test_bit_rsplit: fixes sw :: "'c word list"   and w :: "'d word"   and m :: "nat"   and k :: "nat" assumes "sw = word_rsplit w"   and "m < size (hd sw)"   and "k < length sw" shows "rev sw ! k !! m = w !! (k * size (hd sw) + m)"
arbitrage_processI: fixes Mkt :: "(bool stream, 'b) discrete_market"   and p :: "'b \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real" assumes "\<exists>m. self_financing Mkt p \<and> trading_strategy p \<and> (\<forall>w\<in>space M. cls_val_process Mkt p 0 w = 0) \<and> (AE w in M. 0 \<le> cls_val_process Mkt p m w) \<and> 0 < prob {w \<in> space M. 0 < cls_val_process Mkt p m w}" shows "arbitrage_process Mkt p"
lrev_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "xs \<in> UNIV\<^sup>\<star>"   and "ys \<in> UNIV\<^sup>\<star>" shows "lrev (xs @@ ys) = lrev ys @@ lrev xs"
Star_eq_map_rexp_iff_1: fixes r :: "'d rexp"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e rexp" shows "(Star r = map_rexp f x) = (\<exists>r'. x = Star r' \<and> map_rexp f r' = r)"
ipPartition_def: fixes A :: "'a set set"   and B :: "'a set set" shows "ipPartition A B \<equiv> \<forall>a\<in>A. \<forall>b\<in>B. a \<inter> b = {} \<or> b \<subseteq> a"
multiset_prime_factorization_nat_correct: fixes n :: "nat" shows "prime_factorization n = mset (prime_factorization_nat n)"
assert_all_state_def: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list" shows "assert_all_state t as \<equiv> assert_loop as (init t)"