hcju/isabelleps huggingface dataset:
infplays_augment_one_joint_1: fixes p :: "'c list"   and \<sigma>\<^sub>e :: "'c list \<Rightarrow> 'c"   and \<sigma>\<^sub>o :: "'c list \<Rightarrow> 'c" assumes "even (length p)" shows "strategy_progression (joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o) (augment_list \<sigma>\<^sub>e p) = strategy_progression (joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o) p"
valid_in_def: shows "(\<Turnstile>) \<equiv> map_fun id (map_fun eval\<o> id) (\<lambda>v \<phi>. \<phi> dj v)"
intro: fixes r :: "('d \<times> 'd) set"   and linear_cf :: "'d set \<Rightarrow> 'd set" assumes "Linear_order r"   and "\<And>X. linear_cf X \<equiv> set_option (MaxR.MaxR_opt r X)" shows "linear_cf r linear_cf"
ivl_disj_un_two_3: fixes l :: "'c"   and m :: "'c"   and u :: "'c" assumes "l \<le> m"   and "m \<le> u" shows "{l..<m} \<union> {m..<u} = {l..<u}"
l9_31: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and X :: "'a"   and Y :: "'a"   and Z :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.OS Bet A X Y Z"   and "Tarski_neutral_dimensionless.OS Bet A Z Y X" shows "Tarski_neutral_dimensionless.TS Bet A Y X Z"
fold_fusion_def: fixes g :: "('b, 'c) generator"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'c" shows "fold_fusion g f s = fold f (list.unfoldr g s)"
is_extensional: fixes f :: "'c \<times> 'c" assumes "\<not> C.VV.arr f" shows "\<Phi>\<^sub>G f = D.null"
ide_compE: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>C.ide (g \<cdot> f)" obtains "\<F>C.seq g f" and "\<F>C.seq f g" and "g \<cdot> f = \<F>C.dom f" and "g \<cdot> f = \<F>C.cod g"
size_neq: fixes x :: "'b com" shows "size x \<noteq> 0"
linear_axioms: shows "Vector_Spaces.linear (*b) local.scale (Pair (0::'b))"
pmdl_row_echelon: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ts :: "'t list" assumes "Keys (set ps) \<subseteq> set ts"   and "distinct ts" shows "pmdl (set (mat_to_polys ts (row_echelon (polys_to_mat ts ps)))) = pmdl (set ps)"
lcm_proj1_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and m :: "'a"   and n :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(lcm m n = normalize m) = dvd.dvd times n m"
set_times_mono3_b: fixes a :: "'b"   and C :: "'b set"   and x :: "'b"   and D :: "'b set" assumes "a \<in> C"   and "x \<in> a *o D" shows "x \<in> C * D"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "semilattice_neutr f z" shows "semilattice_neutr_set f z"
sep_substate_disj_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj x y" shows "sep_algebra.sep_substate plus sep_disj x (plus x y)"
ns_mul_ext_def: fixes ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" shows "ns_mul_ext ns s \<equiv> (mult2_alt_ns (ns\<inverse>) (s\<inverse>))\<inverse>"
block_design_axioms_def: fixes block_collection :: "'b set multiset"   and u_block_size :: "int" shows "block_design_axioms block_collection u_block_size \<equiv> \<forall>bl. bl \<in># block_collection \<longrightarrow> int (card bl) = u_block_size"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and ternaryvalue :: "ternaryvalue" shows "P (case ternaryvalue of TernaryTrue \<Rightarrow> f1 | TernaryFalse \<Rightarrow> f2 | TernaryUnknown \<Rightarrow> f3) = ((ternaryvalue = TernaryTrue \<longrightarrow> P f1) \<and> (ternaryvalue = TernaryFalse \<longrightarrow> P f2) \<and> (ternaryvalue = TernaryUnknown \<longrightarrow> P f3))"
block_set_nempty_imp_block_ex: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "incidence_system point_set block_collection"   and "block_collection \<noteq> {#}" shows "\<exists>bl. bl \<in># block_collection"
coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b \<sqsubseteq> c" shows "a \<sqinter> b \<sqsubseteq> c"
insert_spec_none: fixes xs :: "'b list"   and oid :: "'b" shows "set (insert_spec xs (oid, None)) = set xs \<union> {oid}"
lift_PDG_wcd: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and nx :: "'g"   and H :: "'i set"   and L :: "'i set" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit (StrongPostdomination.weak_control_dependence sourcenode targetnode valid_edge Exit)"   and "StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit nx" shows "PDG src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_Def Def Entry Exit H L) (lift_Use Use Entry Exit H L) state_val NewExit (StrongPostdomination.weak_control_dependence src trg (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewExit)"
Abs_unit_cases: fixes x :: "unit" obtains y :: "bool" where "x = Abs_unit y" and "y \<in> {True}"
a: shows " t \<longrightarrow>\<^sub>1\<^sup>* t1"
subresultant_prs: fixes F :: "nat \<Rightarrow> 'a fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a fract"   and G1 :: "'a poly"   and G2 :: "'a poly"   and div_exp :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and Gk :: "'a poly"   and hk :: "'a"   and i :: "nat" assumes "subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2"   and "\<And>x y n. to_fract x ^ n / to_fract y ^ (n - 1) \<in> range to_fract \<Longrightarrow> to_fract (div_exp x y n) = to_fract x ^ n / to_fract y ^ (n - 1)"   and "subresultant_prs div_exp G1 G2 = (Gk, hk)" shows "F k = map_poly to_fract Gk \<and> subresultant_prs_locale.h \<delta> f k = to_fract hk \<and> (i \<noteq> 0 \<longrightarrow> F i \<in> range (map_poly to_fract)) \<and> (3 \<le> i \<longrightarrow> i \<le> Suc k \<longrightarrow> \<beta> i \<in> range to_fract)"
min_maxsimpchainI_betw: fixes x :: "'a set"   and y :: "'a set"   and xs :: "'a set list" assumes "x \<noteq> y"   and "gallery (x # xs @ [y])"   and "\<And>ys. gallery (x # ys @ [y]) \<Longrightarrow> order.greater_eq (length ys) (length xs)" shows "min_gallery (x # xs @ [y])"
index_mat_swapcols_2: fixes k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "dim_row (swapcols k l A) = dim_row A"
constant_degree: fixes p :: "'a poly" shows "constant (poly p) = (Polynomial.degree p = 0)"
poly_lift_hom_smult: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and p :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "p \<in> carrier (UP R)"   and "a \<in> carrier R" shows "poly_lift_hom R S \<phi> (a \<odot>\<^bsub>UP R\<^esub> p) = \<phi> a \<odot>\<^bsub>UP S\<^esub> poly_lift_hom R S \<phi> p"
ND0aux3: fixes x :: "'a"   and y :: "'a"   and p :: "'b"   and b :: "('a, 'b) Combinators list" assumes "AllowPortFromTo x y p \<in> set b" shows "x \<in> set (net_list_aux b)"
eqvt_at_def: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "eqvt_at f x \<equiv> \<forall>p. p \<bullet> f x = f (p \<bullet> x)"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "g \<cdot> f = g \<cdot>\<^sub>C f"
subgroupE_1: fixes H :: "('b \<Rightarrow>\<^sub>0 int) set" assumes "subgroup H (free_Abelian_group (\<Union> (S ` I)))" shows "H \<subseteq> carrier (free_Abelian_group (\<Union> (S ` I)))"
Range_Id_on: fixes A :: "'e set" shows "Range (Id_on A) = A"
inverse_arrowsE: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and g :: "'d" assumes "category C"   and "category.inverse_arrows C f g" obtains "partial_magma.ide C (C g f)" and "partial_magma.ide C (C f g)"
limproc_is_lub: fixes S :: "nat \<Rightarrow> 'a process" assumes "chain S" shows "range S <<| lim_proc (range S)"
max_Var_floatarithimage_Var: fixes X :: "nat set" shows "max_Var_floatarith ` Var ` X = Suc ` X"
Sigma_Int_distrib1: fixes I :: "'b set"   and J :: "'b set"   and C :: "'b \<Rightarrow> 'c set" shows "Sigma (I \<inter> J) C = Sigma I C \<inter> Sigma J C"
assms: shows "k < length (trace_parallel_plan_strips I \<pi>) - 1"
field_differentiable_add_const: fixes c :: "'a"   and F :: "'a filter" shows "(+) c field_differentiable F"
FINFUNCD: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "f \<in> FINFUNC"   and "x \<in> Field s" shows "f x \<in> Field r"
eval_rs: fixes s :: "nat \<Rightarrow> nat option"   and x :: "nat" assumes "total_cons s" shows "s x = eval (total_cons.r_s s) [x]"
sphere_empty: fixes r :: "real"   and a :: "'a" assumes "r < 0" shows "sphere a r = {}"
simps: fixes a :: "msg"   and H :: "msg set" shows "(a \<in> synth H) = ((\<exists>X. a = X \<and> X \<in> H) \<or> (\<exists>agt. a = msg.Agent agt) \<or> (\<exists>n. a = Number n) \<or> (\<exists>X. a = Hash X \<and> X \<in> synth H) \<or> (\<exists>X Y. a = \<lbrace>X, Y\<rbrace> \<and> X \<in> synth H \<and> Y \<in> synth H) \<or> (\<exists>X K. a = Crypt K X \<and> X \<in> synth H \<and> Key K \<in> H))"
order_mult: fixes p :: "'a poly"   and q :: "'a poly"   and a :: "'a" assumes "p * q \<noteq> 0" shows "Polynomial.order a (p * q) = Polynomial.order a p + Polynomial.order a q"
alpha_Tree_tAct: fixes t1 :: "('e, 'f, 'g) Tree"   and t2 :: "('e, 'f, 'g) Tree"   and \<alpha> :: "'g" assumes "t1 =\<^sub>\<alpha> t2" shows "tAct \<alpha> t1 =\<^sub>\<alpha> tAct \<alpha> t2"
t_Lam_inversion: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and t :: "lam"   and T :: "ty" assumes "\<Gamma> \<turnstile> Lam [x].t : T"   and "x \<sharp> \<Gamma>" shows "\<exists>T\<^sub>1 T\<^sub>2. T = T\<^sub>1 \<rightarrow> T\<^sub>2 \<and> (x, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2"
collapse_2: fixes step :: "('c, 'd) step" assumes "is_Yield step" shows "Yield (un_Yield1 step) (un_Yield2 step) = step"
radicals_expr_subset: fixes r :: "expr"   and e :: "expr" assumes "r \<in> radicals e" shows "radicals r \<subset> radicals e"
last_message_hold_def: fixes xs :: "'b message_af list" shows "last_message_hold xs \<equiv> last_message_hold_init xs NoMsg"
left_cancel: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "(a + b = a + c) = (b = c)"
g_def: fixes x :: "'a" shows "g x = (\<Sum>i. b i * (x - \<xi>) ^ i)"
if_rel_def: fixes b :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'd"   and h :: "'d \<Rightarrow> 'd" shows "if_rel b f g h = {(s, t). if b s then t = f s else t = g s \<or> t = h s}"
f'_def: shows "f' \<equiv> \<lambda>w. f w / (w - z)"
mk_release_semantics: fixes w :: "nat \<Rightarrow> 'a set"   and x :: "'a ltln"   and y :: "'a ltln" shows "w \<Turnstile>\<^sub>n mk_release x y = w \<Turnstile>\<^sub>n x R\<^sub>n y"
\<Delta>'''_sel_3: shows "source \<Delta>''' = source \<Delta>"
remove_invalid_point_block: fixes p :: "'a"   and bl :: "'a set" assumes "p \<notin> \<V>"   and "bl \<in># \<B>" shows "bl - {p} = bl"
runit'_in_vhom: fixes f :: "'a"   and a :: "'a" assumes "ide f"   and "src f = a" shows "\<guillemotleft>\<r>\<^sup>-\<^sup>1[f] : f \<Rightarrow> f \<star> a\<guillemotright>"
nf_invar_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat" shows "nf_invar c src dst PRED C N d \<equiv> nf_invar' c src dst PRED C N d \<and> nf_invar_axioms C N"
distrib_imp1: fixes x :: "'a upred"   and y :: "'a upred"   and z :: "'a upred" assumes "\<And>x y z. (x \<and> (y \<or> z)) = (x \<and> y \<or> x \<and> z)" shows "(x \<or> y \<and> z) = ((x \<or> y) \<and> (x \<or> z))"
terminal_def: fixes subs :: "'a \<Rightarrow> 'a set"   and delta :: "'a" shows "terminal subs delta = (subs delta = {})"
\<psi>_2: shows "Fvars \<psi> = {inp}"
norm1_ge_0: fixes f :: "'a poly" shows "(0::'a) \<le> norm1 f"
rep_eq: fixes x :: "'f"   and xa :: "('f, 'e) generator" shows "generator (remove1_trans x xa) = remove1_raw x (generator xa)"
full_multiset_induct: fixes P :: "'a multiset \<Rightarrow> bool"   and B :: "'a multiset" assumes "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B" shows "P B"
l10_10: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P' :: "'a"   and P :: "'a"   and A :: "'a"   and B :: "'a"   and Q' :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Reflect Bet Cong P' P A B"   and "Tarski_neutral_dimensionless.Reflect Bet Cong Q' Q A B" shows "Cong P Q P' Q'"
abs_convergent_prod_imp_convergent_prod: fixes f :: "nat \<Rightarrow> 'a" assumes "abs_convergent_prod f" shows "convergent_prod f"
Healthy_if: fixes P :: "'a upred"   and H :: "'a upred \<Rightarrow> 'a upred" assumes "P is H" shows "H P = P"
stepsST_refl: fixes P :: "('procS, 'procT) Proc" shows "P \<longmapsto>ST* P"
Seq_transC_invert: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c1 ;; c2, s) \<rightarrow>c (c', s')" obtains c1' :: "('test, 'atom) com" where "(c1, s) \<rightarrow>c (c1', s')" and "c' = c1' ;; c2"   | "(c1, s) \<rightarrow>t s'" and "c' = c2"
skl1_trans_def: shows "skl1_trans \<equiv> \<Union>m Ra Rb A B x y. (skl1_step1 Ra A B \<union> skl1_step2 Rb A B x y \<union> skl1_step3 Ra A B x y \<union> skl1_step4 Rb A B x y \<union> l1_learn m)\<^sup>="
il_inf_right_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) \<sqinter> z = x \<sqinter> z \<squnion> y \<sqinter> z"
monoAux: fixes A :: "(pi \<times> pi) set"   and B :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi" assumes "A \<subseteq> B" shows "P \<leadsto><A> Q \<longrightarrow> P \<leadsto><B> Q"
AX10_def: shows "AX10 \<equiv> {x. AX10p x}"
that_2: fixes n :: "nat"   and b :: "bool discr" assumes "Y = (\<lambda>m. if m < n then \<bottom> else B\<cdot>b)" shows "thesis"
card_as_sums: fixes A :: "'d set"   and B :: "'d set"   and P :: "'d \<Rightarrow> bool" assumes "A = {x \<in> B. P x}"   and "finite B" shows "card A = (\<Sum>x\<in>B. if P x then 1 else 0)"
sub_monom: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat"   and g :: "nat \<Rightarrow> 'c" assumes "UP_domain R"   and "a \<in> carrier R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>"   and "f = monom (UP R) a n"   and "g \<in> carrier (UP R)" shows "deg R (Cring_Poly.compose R f g) = n * deg R g"
b: shows "b = Determinant.det (submatrix (P * A * Q) I J)"
Rep_measure: fixes x :: "'c measure" shows "Rep_measure x \<in> {(\<Omega>, A, \<mu>). (\<forall>a\<in>- A. \<mu> a = 0) \<and> measure_space \<Omega> A \<mu>}"
sylvester_mat_sub_index: fixes i :: "nat"   and m :: "nat"   and n :: "nat"   and j :: "nat"   and p :: "'b poly"   and q :: "'b poly" assumes "i < m + n"   and "j < m + n" shows "sylvester_mat_sub m n p q $$ (i, j) = (if i < n then if i \<le> j \<and> j - i \<le> m then coeff p (m + i - j) else (0::'b) else if i - n \<le> j \<and> j \<le> i then coeff q (i - j) else (0::'b))"
eventually_imp_distrib: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<diamond>(\<lambda>s. P s \<longrightarrow> Q s) = (\<lambda>s. (\<box>P) s \<longrightarrow> (\<diamond>Q) s)"
downset_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ord.downset less_eq = ord.downset_set less_eq \<circ> \<eta>"
tm_comp_final: fixes A :: "(action \<times> nat) list"   and l :: "cell list"   and r :: "cell list"   and B :: "(action \<times> nat) list"   and stp :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "tm_wf0 A"   and "steps0 (1, l, r) B stp = (0, l', r')" shows "steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (0, l', r')"
keys_bulkload: fixes xs :: "'b list" shows "keys (bulkload xs) = {0..<length xs}"
sw_involutive: fixes xs :: "'b"   and x :: "'a"   and y :: "'a"   and zs :: "'b"   and z :: "'a" shows "z @zs[x \<and> y]_xs @zs[x \<and> y]_xs = z"
isometry_on_lipschitz: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f" shows "1-lipschitz_on X f"
inducts: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "lam \<Rightarrow> lam \<Rightarrow> bool" assumes " x1 \<longrightarrow>\<^sub>\<beta>\<^sup>* x2"   and "\<And>M. P M M"   and "\<And>M1 M2 M3. \<lbrakk> M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M2; P M1 M2;  M2 \<longrightarrow>\<^sub>\<beta> M3\<rbrakk> \<Longrightarrow> P M1 M3" shows "P x1 x2"
DBSpec_impl_FullSpec: shows "\<turnstile> DBSpec \<longrightarrow> FullSpec"
config_rand_COMB: fixes h :: "nat list"   and init :: "nat list"   and qs :: "nat list" shows "Partial_Cost_Model.config'_rand (COMB h) (fst (COMB h) init \<bind> (\<lambda>is. return_pmf (init, is))) qs = bernoulli_pmf (8 / 10) \<bind> (\<lambda>b. Partial_Cost_Model.config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs \<bind> (\<lambda>(b1, b2). Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h)) (fst (Partial_Cost_Model.embed (rTS h)) init \<bind> (\<lambda>is. return_pmf (init, is))) qs \<bind> (\<lambda>(t1, t2). return_pmf (if b then (b1, Inl b2) else (t1, Inr t2)))))"
infinite_set_less_eq_Complement: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "infinite UNIV" shows "A \<sqsubseteq> - B"
LIMSEQ_binaryset: fixes f :: "'c set \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" assumes "f {} = (0::'b)" shows "(\<lambda>n. \<Sum>i<n. f (binaryset A B i)) \<longlonglongrightarrow> f A + f B"
lenq: shows "\<^bold>|take q w\<^bold>| = q"
wf_type_conj: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f \<and> \<Gamma> \<turnstile>\<^sub>w\<^sub>f T"
x1_inL: shows "x1 \<in> \<Sigma>1.L"
accs_mono: fixes \<delta> :: "('a, 'b) ta_rule set"   and n :: "'b tree"   and q :: "'a"   and \<delta>' :: "('a, 'b) ta_rule set" assumes "accs \<delta> n q"   and "\<delta> \<subseteq> \<delta>'" shows "accs \<delta>' n q"
intro: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "f ` A \<subseteq> A" shows "fixed_point_proof A f"
iso_inv_block_in_alt: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd"   and bl :: "'c set" assumes "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map"   and "bij_map ` bl \<in># \<B>'"   and "bl \<subseteq> \<V>" shows "bl \<in># \<B>"
rexp_empty_iff: fixes r :: "'a rexp" shows "rexp_empty r = (lang r = {})"
qequiv_def: shows "(\<cong>\<^sub>Q) \<equiv> map_fun fromQ (map_fun fromQ id) (=)"
list_update_nonempty: fixes xs :: "'b list"   and k :: "nat"   and x :: "'b" shows "(xs[k := x] = []) = (xs = [])"
bind_map_spmf: fixes f :: "'j \<Rightarrow> 'i"   and p :: "'j spmf"   and g :: "'i \<Rightarrow> 'h spmf" shows "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
simps: fixes p1 :: "'a list"   and p2 :: "'a list" shows "poly_add p1 p2 = (if length p2 \<le> length p1 then local.normalize (map2 (\<oplus>) p1 (replicate (length p1 - length p2) \<zero> @ p2)) else poly_add p2 p1)"
3: fixes u :: "'b" shows "sorted_wrt lt (map fst (update_by_pair (k, u) xs))"
list_min_def: fixes l :: "'a list" shows "list_min l = fold min l"
word32_bounds_1: fixes x :: "32 word" shows "- (2 ^ (size x - 1)) = - 2147483648"
suminf_sq_mtx_diag: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "\<forall>i. (\<lambda>n. f n i) sums (\<Sum>n. f n i)" shows "(\<Sum>n. \<d>\<i>\<a>\<g> i. f n i) = (\<d>\<i>\<a>\<g> i. \<Sum>n. f n i)"
declassified_subterms_secrets_subset: fixes A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and I :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<forall>n m. n \<noteq> m \<longrightarrow> GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj n A)) (trms\<^sub>l\<^sub>s\<^sub>t (proj m A)) Sec"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t I"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range I)"   and "subst_domain I = \<V> \<and> ground (subst_range I)" shows "subterms\<^sub>s\<^sub>e\<^sub>t (declassified\<^sub>l\<^sub>s\<^sub>t A I) \<subseteq> Sec \<union> {m. {} \<turnstile>\<^sub>c m}"
trelPS: shows "rel_weakly_preserves_barbs TRel TWB"
simps_4: fixes disc :: "'b \<Rightarrow> bool"   and neg :: "bool"   and m1 :: "'b match_expr"   and m2 :: "'b match_expr" shows "has_disc_negated disc neg (MatchAnd m1 m2) = (has_disc_negated disc neg m1 \<or> has_disc_negated disc neg m2)"
IH: fixes B :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "B \<subset> S"   and "f permutes B" shows "\<exists>C. (\<forall>a\<in>C. cyclic_on f a) \<and> \<Union> C = B \<and> (\<forall>c1\<in>C. \<forall>c2\<in>C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {})"
inducts: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat" assumes "interaction_bounded_by consider gpv n" obtains "interaction_bound consider gpv \<le> n"
commute: fixes a :: "'b multiset"   and b :: "'b multiset" shows "a \<inter># b = b \<inter># a"
b: fixes x :: "'b" assumes "x \<in> S" shows "b \<le> norm (m x)"
R3: shows "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"
wp_Loop1: fixes P :: "'a \<Rightarrow> real"   and body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "sound P"   and "healthy (wp body)" shows "wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P = lfp_exp (\<lambda>Q s. \<guillemotleft> G \<guillemotright> s * wp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s)"
assms_1: shows "A \<turnstile>\<^sub>R res \<sim> res'"
select_convs_1: fixes role :: "role_t"   and owner :: "agent"   and partner :: "agent"   and more :: "'a" shows "role \<lparr>role = role, owner = owner, partner = partner, \<dots> = more\<rparr> = role"
plus_sup: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y) \<sqinter> top = top \<sqinter> (x \<sqinter> top) \<squnion> top \<sqinter> y \<sqinter> top"
nonempty_I: shows "{i. postrecording_event t i \<and> 0 < card {j \<in> {i + 1..<length t}. prerecording_event t j}} \<noteq> {}"
lookup_minus: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and g :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" shows "poly_mapping.lookup (f - g) k = poly_mapping.lookup f k - poly_mapping.lookup g k"
finite: fixes A :: "'e set" assumes "class.finite TYPE('e)" shows "finite A"
reflp: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" assumes "merkle_interface h bo m" shows "reflp bo"
cancel_scalar: fixes a :: "'f"   and u :: "'v"   and v :: "'v" assumes "a \<noteq> (0::'f)"   and "u \<in> V"   and "v \<in> V"   and "a \<cdot> u = a \<cdot> v" shows "u = v"
in_sets_Sup: fixes M :: "'c measure set"   and X :: "'c set"   and m :: "'c measure"   and A :: "'c set" assumes "\<And>m. m \<in> M \<Longrightarrow> space m = X"   and "m \<in> M"   and "A \<in> sets m" shows "A \<in> sets (Sup M)"