hcju/isabelleps huggingface dataset:
simps_6: fixes prec :: "nat"   and bs :: "float interval option list" shows "approx prec Pi bs = Some (pi_float_interval prec)"
IH: shows "\<Union> {propos y |y. y = i \<or> y \<in> set xs} = propos (foldl f i xs)"
1: fixes a :: "'c" assumes "C.obj a" shows "D.ide (F.map\<^sub>0 a)"
power_minus_is_div: fixes b :: "nat"   and a :: "nat" assumes "b \<le> a" shows "2 ^ (a - b) = 2 ^ a div 2 ^ b"
induct: fixes x1 :: "comb"   and x2 :: "comb"   and P :: "comb \<Rightarrow> comb \<Rightarrow> bool" assumes "x1 \<Rrightarrow>\<^sup>1 x2"   and "\<And>x. P x x"   and "\<And>x y. P (K \<bullet> x \<bullet> y) x"   and "\<And>x y z. P (S \<bullet> x \<bullet> y \<bullet> z) (x \<bullet> z \<bullet> (y \<bullet> z))"   and "\<And>x y z w. \<lbrakk>x \<Rrightarrow>\<^sup>1 y; P x y; z \<Rrightarrow>\<^sup>1 w; P z w\<rbrakk> \<Longrightarrow> P (x \<bullet> z) (y \<bullet> w)" shows "P x1 x2"
asymp_equiv_sym: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "(f \<sim>[F] g) = (g \<sim>[F] f)"
isom_Worder: fixes T :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order T"   and "ord_isom D T f" shows "Worder T"
simps_13: fixes v :: "('b, 'c) Combinators"   and va :: "('b, 'c) Combinators"   and vb :: "('b, 'c) Combinators"   and vc :: "('b, 'c) Combinators"   and l :: "'b set list" shows "smaller (v \<oplus> va) (vb \<oplus> vc) l = (v \<oplus> va = vb \<oplus> vc \<or> (if bothNet (v \<oplus> va) = bothNet (vb \<oplus> vc) then case vb \<oplus> vc of DenyAllFromTo a b \<Rightarrow> v \<oplus> va = DenyAllFromTo b a | _ \<Rightarrow> True else position (bothNet (v \<oplus> va)) l \<le> position (bothNet (vb \<oplus> vc)) l))"
rsd_winners_manipulation: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and Ri' :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a"   and agents' :: "'a list" assumes "pref_profile_wf agents alts R"   and "total_preorder_on alts Ri'"   and "i \<in> agents"   and "set agents' = agents"   and "finite agents"   and "finite alts"   and "alts \<noteq> {}" shows "\<forall>x\<in>rsd_winners (R(i := Ri')) alts agents'. \<forall>y\<in>rsd_winners R alts agents'. R i x y"
widen_refl: fixes P :: "'b prog"   and T :: "ty" shows "P \<turnstile> T \<le> T"
runiq_basic: fixes R :: "('a \<times> 'b) set" shows "runiq R = (\<forall>x y y'. (x, y) \<in> R \<and> (x, y') \<in> R \<longrightarrow> y = y')"
lt_i: shows "lt (fs ! i \<odot> monomial (1::'b) (term_of_pair (0::'a, j))) = term_of_pair (punit.lt (fs ! i), j)"
simps: fixes i :: "nat" shows "local.F i = (if i = 0 then 1 else if i = 1 then s.ffp G1 else if i = 2 then s.ffp G2 else let G = pseudo_mod (local.F (i - 2)) (local.F (i - 1)) in if local.F (i - 1) = 0 \<or> G = 0 then 0 else Polynomial.smult (inverse (local.b i)) G)"
pair_COCOUNT: fixes A :: "(real \<times> 'a) set"   and M :: "'a measure" assumes "A \<in> sets (COCOUNT \<Otimes>\<^sub>M M)" shows "\<exists>J F X. X \<in> sets M \<and> F \<in> J \<rightarrow> sets M \<and> countable J \<and> A = (UNIV - J) \<times> X \<union> Sigma J F"
f2: shows "(\<Lambda> * (\<v> - 1) div (\<k> - 1) - 1 \<le> (\<v> * (\<Lambda> * (\<v> - 1) div (\<k> - 1)) - \<v> * \<k>) div \<k>) = (\<k> * (\<Lambda> * (\<v> - 1) div (\<k> - 1) - 1) \<le> \<v> * (\<Lambda> * (\<v> - 1) div (\<k> - 1) - \<k>))"
bex_triv_one_point1: fixes A :: "'e set"   and a :: "'e" shows "(\<exists>x\<in>A. x = a) = (a \<in> A)"
SN_imp_minimal: fixes A :: "('b \<times> 'b) set" assumes "SN A" shows "\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. (z, y) \<in> A \<longrightarrow> y \<notin> Q)"
downset_set_idem: shows "ord.downset_set (\<lambda>x y. y \<le> x) \<circ> ord.downset_set (\<lambda>x y. y \<le> x) = ord.downset_set (\<lambda>x y. y \<le> x)"
incrIndexList_def: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" shows "incrIndexList ls m nmax \<equiv> 1 < m \<and> 1 < nmax \<and> hd ls = 0 \<and> last ls = nmax - 1 \<and> |ls| = m \<and> last (butlast ls) < last ls \<and> increasing ls"
contour_integral_integral: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" shows "contour_integral g f = integral {0..1} (\<lambda>x. f (g x) * vector_derivative g (at x))"
lub_sum_iff: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'d" shows "(\<forall>k. f k \<le> x) = (sup_monoid.sum f {k. True} \<le> x)"
intros_2: fixes p :: "nat spmf"   and S :: "'a spmf \<Rightarrow> 'b spmf \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a spmf"   and g :: "nat \<Rightarrow> 'b spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> S (f x) (g x)" shows "trace_eq_simcl S (p \<bind> f) (p \<bind> g)"
basisV: shows "set vs \<subseteq> V \<and> V = RSpan vs \<and> R_lin_independent vs"
gt: shows "mat_gt_arc (0\<^sub>m nr nc) A"
bs_sub: shows "set bs \<subseteq> dgrad_sig_set' (length fs) dgrad"
F_hiding: fixes P :: "'a process"   and A :: "'a set" shows "F (P \ A) = {(s, X). \<exists>t. s = trace_hide t (ev ` A) \<and> (t, X \<union> ev ` A) \<in> F P} \<union> {(s, X). s \<in> div_hide P A}"
less_eq_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.transitive less_eq less" shows "less_eq x y = (x = y \<or> less x y)"
mult_comp_support_set: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and qty :: "nat \<Rightarrow> 'd \<Rightarrow> real" shows "support_set (qty_mult_comp pf1 qty) \<subseteq> support_set pf1"
dagger_slide: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (y \<cdot> x)\<^sup>\<star> = (x \<cdot> y)\<^sup>\<star> \<cdot> x"
attach_callee_fuse: fixes cnv_adv_core :: "'r \<Rightarrow> 'x \<Rightarrow> ('ab \<times> 'r, 'af, 'aj) gpv"   and cnv_adv_rest :: "'s \<Rightarrow> 'y \<Rightarrow> ('ac \<times> 's, 'ag, 'ak) gpv"   and cnv_usr_core :: "'t \<Rightarrow> 'z \<Rightarrow> ('ad \<times> 't, 'ah, 'al) gpv"   and cnv_usr_rest :: "'u \<Rightarrow> 'aa \<Rightarrow> ('ae \<times> 'u, 'ai, 'am) gpv"   and core :: "('v, 'an, 'af, 'ah, 'aj, 'al) core"   and rest :: "('w, 'an, 'ag, 'ai, 'ak, 'am, 'ao) rest_scheme"   and f_init :: "'ao \<Rightarrow> 'ap" shows "attach_callee ((cnv_adv_core \<ddagger>\<^sub>I cnv_adv_rest) \<ddagger>\<^sub>I cnv_usr_core \<ddagger>\<^sub>I cnv_usr_rest) (fused_resource.fuse core rest) = apply_state_iso iso_trisplit (fused_resource.fuse (attach_core cnv_adv_core cnv_usr_core core) (attach_rest cnv_adv_rest cnv_usr_rest f_init rest))"
seq_power_front: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice"   and n :: "nat" assumes "seq_distrib_right (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (seq_distrib_right.seq_power seq (nil::'b::refinement_lattice) (a::'b::refinement_lattice) (n::nat)) a = seq a (seq_distrib_right.seq_power seq nil a n)"
permute_Lam: fixes f :: "'c \<Rightarrow> 'd"   and p :: "perm" assumes "cont f" shows "p \<bullet> (\<Lambda> x. f x) = (\<Lambda> x. (p \<bullet> f) x)"
associative: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> G"   and "b \<in> G"   and "c \<in> G" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
prv_cnjI: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv \<phi>"   and "prv \<chi>" shows "prv (cnj \<phi> \<chi>)"
ent_pure_post_iff_sng: fixes P :: "assn"   and b :: "bool" shows "(P \<Longrightarrow>\<^sub>A \<up> b) = ((\<forall>h. h \<Turnstile> P \<longrightarrow> b) \<and> (P \<Longrightarrow>\<^sub>A emp))"
borel_predict_stoch_proc_borel_measurable: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> 'c" assumes "borel_predict_stoch_proc F X" shows "\<forall>n. random_variable borel (X n)"
odeliver_completeTE: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and i :: "nat"   and d :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" assumes "((\<sigma>, s), i:deliver(d), \<sigma>', s') \<in> ocnet_sos S" obtains "((\<sigma>, s), i:deliver(d), \<sigma>', s') \<in> S" and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j"
IT_K': fixes x :: "'a" assumes "x \<in> space M" shows "Ionescu_Tulcea (K' x) (\<lambda>_. M)"
simps_2: fixes x2 :: "'h" shows "Fault \<noteq> OK x2"
wo_rel_def: fixes r :: "('d \<times> 'd) set" shows "wo_rel r \<equiv> Well_order r"
RR_def2: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "xx \<notin> FvarsT t1" shows "RR t1 t2 = subst (subst R t1 yy) t2 xx"
less_eq_bot_def: fixes x :: "'a bot"   and y :: "'a bot" shows "(x \<le> y) = (case x of Value x \<Rightarrow> case y of Value y \<Rightarrow> x \<le> y | Bot \<Rightarrow> False | Bot \<Rightarrow> True)"
atom_def: fixes a :: "'b" shows "atom a = ([True], \<lambda>b s. if s = [True] \<and> b = a then {[False]} else {}, \<lambda>s. s = [False])"
rbl_add: fixes n :: "nat"   and bina :: "int"   and binb :: "int" shows "rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) = rev (bin_to_bl n (bina + binb))"
set_tree_inter: fixes t1 :: "('a \<times> color) tree"   and t2 :: "('a \<times> color) tree" assumes "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.set_tree (inter_rbt t1 t2) = Tree2.set_tree t1 \<inter> Tree2.set_tree t2"
sum_in_spanI: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and q :: "'c \<Rightarrow> 'b"   and B :: "'c set" assumes "module scale" shows "(\<Sum>b\<in>B. scale (q b) b) \<in> module.span scale B"
leq_zero_imp: fixes x :: "'b" assumes "(x, s.zero) \<in> s" shows "x = s.zero"
that: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "punit.is_Groebner_basis G"   and "ideal G = ideal F"   and "G \<subseteq> P[X]"   and "\<And>g. g \<in> G \<Longrightarrow> \<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f. q f \<in> P[X] \<and> poly_deg (q f * f) \<le> b \<and> (f \<notin> F' \<longrightarrow> q f = 0))" shows "thesis"
fis_singleton_alt: fixes f :: "'b fset" shows "fis_singleton f = (\<exists>e. f = {|e|})"
subtuple_antisym: fixes ym :: "'d \<rightharpoonup>\<^sub>f 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xm :: "'d \<rightharpoonup>\<^sub>f 'e" assumes "subtuple (\<lambda>x y. f x y \<or> x = y) xm ym"   and "subtuple (\<lambda>x y. f x y \<and> \<not> f y x \<or> x = y) ym xm" shows "xm = ym"
distinct_3: fixes x1 :: "'a"   and x31 :: "'a acom"   and x32 :: "'a acom" shows "SKIP {x1} \<noteq> x31;; x32"
h_3: shows "subgraph (map_graph_fn G' h) G'"
that: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> 0 = x"   and "\<pi> n = local.return"   and "\<forall>k\<le>n. \<pi> k \<noteq> y" shows "thesis"
not_zero_iarray: shows "\<not> is_zero_iarray (vec_to_iarray (row i (Gauss_Jordan A)))"
card_UNIV_set_def: shows "card_UNIV_class.card_UNIV = Phantom('a set) (let c = of_phantom card_UNIV_class.card_UNIV in if c = 0 then 0 else 2 ^ c)"
iso_wf: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "wf r"   and "iso r r' f" shows "wf r'"
carrier_polynomial_shell: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "p \<in> carrier (poly_ring (K [X]))"
trans: fixes a :: "'h"   and b :: "'h"   and c :: "'h" assumes "a \<le> b"   and "b \<le> c" shows "a \<le> c"
hindrance_by: fixes a :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<epsilon> :: "ennreal" assumes "a \<in> A \<Gamma>"   and "a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)"   and "d_OUT f a < weight \<Gamma> a"   and "\<epsilon> < weight \<Gamma> a - d_OUT f a" shows "hindrance_by \<Gamma> f \<epsilon>"
horner_sum_eq_sum: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'b list" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_0.horner_sum plus zero times f a xs = comm_monoid_add.sum plus zero (\<lambda>n. times (f (xs ! n)) (power.power one times a n)) {0..<length xs}"
simps: fixes A :: "'c mat"   and B :: "'c mat"   and i :: "nat"   and j :: "nat" shows "gauss_jordan_main A B i j = (let nr = dim_row A; nc = dim_col A in if i < nr \<and> j < nc then let aij = A $$ (i, j) in if aij = (0::'c) then case concat (map (\<lambda>i'. if A $$ (i', j) \<noteq> (0::'c) then [i'] else []) [Suc i..<nr]) of [] \<Rightarrow> gauss_jordan_main A B i (Suc j) | i' # x \<Rightarrow> gauss_jordan_main (swaprows i i' A) (swaprows i i' B) i j else if aij = (1::'c) then let v = \<lambda>i. A $$ (i, j) in gauss_jordan_main (eliminate_entries v A i j) (eliminate_entries v B i j) (Suc i) (Suc j) else let iaij = inverse aij in gauss_jordan_main (multrow i iaij A) (multrow i iaij B) i j else (A, B))"
PX5d_range: fixes d :: "D2" shows "Field (PX5d d) \<subseteq> {x. X5d x = d}"
evaldjf_qf: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> fm" assumes "\<forall>x\<in>set xs. qfree (f x)" shows "qfree (evaldjf f xs)"
Gp_smult_sg_nsg: fixes H :: "'a set"   and N :: "'a set" assumes "G \<guillemotright> H "   and "G \<triangleright> N" shows "Group (\<natural>H \<diamondop> N)"
cobounded2: fixes a :: "nat"   and b :: "nat" shows "gcd a b dvd b"
Sup_upto_llist_mono: fixes j :: "enat"   and k :: "enat"   and Xs :: "'b set llist" assumes "j \<le> k" shows "Sup_upto_llist Xs j \<subseteq> Sup_upto_llist Xs k"
less_eq_integer_def: shows "(\<le>) \<equiv> map_fun int_of_integer (map_fun int_of_integer id) (\<le>)"
assms_4: shows "DD ars r (\<upsilon>, \<sigma>', \<sigma>'', \<upsilon>')"
csum_com: fixes p1 :: "('d \<times> 'd) set"   and p2 :: "('e \<times> 'e) set" shows "p1 +c p2 =o p2 +c p1"
r_inv_ex: fixes x :: "rat vec" assumes "x \<in> Rn" shows "\<exists>y\<in>Rn. x + y = 0\<^sub>v n"
vcons_3: shows "vdistinct (xs #\<^sub>\<circ> x)"
pHom_memTr: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g"   and n :: "nat" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y" shows "\<forall>c. pol_coeff S (n, c) \<longrightarrow> f (polyn_expr R X n (n, c)) = polyn_expr A Y n (n, cmp f c)"
f_expand_nth_mult: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" assumes "n < length xs"   and "0 < k" shows "xs \<odot> k ! (n * k) = xs ! n"
assms: shows "(\<exists>v. t = Var v) \<or> (\<exists>f. t = Fun f [])"
simps_17: fixes x :: "nat"   and a :: "floatarith"   and n :: "nat"   and vs :: "real list" shows "isDERIV x (a ^\<^sub>e Suc n) vs = isDERIV x a vs"
wt_app: fixes \<Gamma> :: "(nat \<times> ty) list"   and e1 :: "exp"   and A :: "ty"   and B :: "ty"   and e2 :: "exp" assumes "\<Gamma> \<turnstile> e1 : TFun (A \<rightarrow> B)"   and "\<Gamma> \<turnstile> e2 : A" shows "\<Gamma> \<turnstile> EApp e1 e2 : B"
iterateoi_correct: fixes S :: "'s" assumes "invar S" shows "set_iterator_linord (iterateoi S) (\<alpha> S)"
of_nat_eq_iff: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and m :: "nat"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = semiring_1.of_nat one plus zero n) = (m = n)"
correctCompositionDiffLevelsS9: shows "correctCompositionDiffLevels sS9"
FW_cyc_free_preservation: fixes n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'b) \<le> len m i i xs" shows "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> (0::'b) \<le> len (FW m n) i i xs"
can_occur_def: fixes ev :: "('a, 'b, 'c) event"   and c :: "('a, 'b, 'c) configuration" shows "can_occur ev c \<equiv> case ev of Trans p s s' \<Rightarrow> states c p = s \<and> trans p s s' | Send i p q s s' msg \<Rightarrow> states c p = s \<and> channel i = Some (p, q) \<and> send i p q s s' msg | Recv i p q s s' msg \<Rightarrow> states c p = s \<and> channel i = Some (q, p) \<and> 0 < length (msgs c i) \<and> hd (msgs c i) = Msg msg \<and> recv i p q s s' msg | Snapshot p \<Rightarrow> \<not> ps c p \<noteq> None | RecvMarker i p q \<Rightarrow> channel i = Some (q, p) \<and> 0 < length (msgs c i) \<and> hd (msgs c i) = Marker"
lower_power_zero: fixes a :: "nat"   and b :: "nat" shows "lowerPowers a b 0 = 0"
state_repcs: fixes s :: "'s \<times> ('c \<Rightarrow> real)"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" shows "state (repcs s cfg) = s"
single_valued_relcomp: fixes r :: "('a \<times> 'b) set"   and s :: "('b \<times> 'c) set" assumes "single_valued r"   and "single_valued s" shows "single_valued (r O s)"
coeff_monom_mult: fixes c :: "'a"   and n :: "nat"   and p :: "'a poly"   and k :: "nat" shows "poly.coeff (Polynomial.monom c n * p) k = (if k < n then 0::'a else c * poly.coeff p (k - n))"
Cond1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and b :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and b' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>b,s\<rangle> -ta\<rightarrow> \<langle>b',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>if (b) e1 else e2,s\<rangle> -ta\<rightarrow> \<langle>if (b') e1 else e2,s'\<rangle>"
genby_order2: fixes s :: "'a" assumes "s + s = (0::'a)" shows "\<langle>{s}\<rangle> = {0::'a, s}"
source_neq_sink: shows "source (residual_network f) \<noteq> sink (residual_network f)"
assms_2: fixes y :: "'b" shows "cont (\<lambda>x. g x y)"
assms_1: shows "invariantsHoldInState stateA F0 decisionVars"
Polys_closed_times: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<in> P[X]"   and "q \<in> P[X]" shows "p * q \<in> P[X]"
pseudo_proj_False_stake: fixes n :: "nat"   and w :: "bool stream" shows "stake n (pseudo_proj_False n w) = stake n w"
map_mmupd_def: fixes m :: "'b \<Rightarrow> 'a option"   and K :: "'b set"   and v :: "'a"   and k :: "'b" shows "map_mmupd m K v k \<equiv> if k \<in> K then Some v else m k"
cont_repcs1: fixes s :: "'s \<times> ('c \<Rightarrow> real)"   and s' :: "'s \<times> ('c \<Rightarrow> real)"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "abss s \<in> \<S>"   and "s' \<in> set_pmf (action (repcs s cfg))"   and "cfg \<in> R_G.cfg_on (abss s)" shows "cont (repcs s cfg) s' = repcs s' (cont cfg (abss s'))"
fsubset_finsert: fixes x :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "x |\<notin>| A" shows "(A |\<subseteq>| finsert x B) = (A |\<subseteq>| B)"
select_convs_3: fixes Pol :: "('c \<times> 'c) set"   and Map :: "'d \<Rightarrow> 'c"   and Dom :: "'c"   and In :: "'d list"   and Out :: "'d list"   and more :: "'e" shows "Dom \<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = Dom"
star_inductl_var_equiv: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.left_near_kleene_algebra plus times one less_eq less star" shows "less_eq (times (star x) y) y = less_eq (times x y) y"
not_isFault_iff: fixes t :: "('d, 'e) xstate" shows "(\<not> isFault t) = (\<forall>f. t \<noteq> Fault f)"
sum_Un_nat: fixes A :: "'k set"   and B :: "'k set"   and f :: "'k \<Rightarrow> nat" assumes "finite A"   and "finite B" shows "sum f (A \<union> B) = sum f A + sum f B - sum f (A \<inter> B)"
strict_assoc: fixes f :: "('a E.term, 'a) arr"   and g :: "('a E.term, 'a) arr"   and h :: "('a E.term, 'a) arr" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "ide (\<a> f g h)"
e_type_comp_conc1: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and es :: "e list"   and es' :: "e list"   and ts :: "t list"   and ts' :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> es @ es' : ts _> ts'" shows "\<exists>ts''. \<S>\<bullet>\<C> \<turnstile> es : ts _> ts'' \<and> \<S>\<bullet>\<C> \<turnstile> es' : ts'' _> ts'"
conj_zero_left: fixes x :: "'a" shows "(0::'a) AND x = (0::'a)"
L_transform_Tree\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) FL_Formula.Tree\<^sub>\<alpha>" shows "p \<bullet> L_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha> = L_transform_Tree\<^sub>\<alpha> (p \<bullet> t\<^sub>\<alpha>)"
f2: fixes R' :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and gpv :: "('d, 'a, 'c) gpv"   and gpv' :: "('f, 'b, 'e) gpv"   and C :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "('d, 'a, ('d, 'a, 'c) rpv) generat"   and y :: "('f, 'b, ('f, 'b, 'e) rpv) generat"   and a :: "('d, 'a, 'c) rpv"   and b :: "('f, 'b, 'e) rpv"   and a' :: "'c"   and \<alpha> :: "'e"   and \<beta> :: "'c"   and b' :: "'e" assumes "??.Generative_Probabilistic_Value.foo R' outs_gpv outs_gpv gpv gpv' C C'"   and "x \<in> set_spmf (the_gpv gpv)"   and "y \<in> set_spmf (the_gpv gpv')"   and "a \<in> generat_conts x"   and "b \<in> generat_conts y"   and "R' a' \<alpha>"   and "R' \<beta> b'" shows "??.Generative_Probabilistic_Value.foo R' outs_gpv outs_gpv (a a') (b b') C C'"
rep_eq: fixes x :: "'c" shows "Rep_ell2 (ket x) = (\<lambda>y. if x = y then 1 else 0)"
bounded_sesquilinear_add: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "bounded_sesquilinear A"   and "bounded_sesquilinear B" shows "bounded_sesquilinear (\<lambda>x y. A x y + B x y)"
wf_minit0: fixes \<phi> :: "Formula.formula"   and n :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set" assumes "safe_formula \<phi>"   and "\<forall>x\<in>fv \<phi>. x < n"   and "pred_mapping (\<lambda>x. x = 0) P" shows "wf_mformula \<sigma> 0 P V n R (minit0 n \<phi>) \<phi>"
bottom_closed: shows "\<bottom>\<^bsub>L\<^esub> \<in> carrier L"
Eq_Var2: fixes x :: "nat"   and c :: "event_data" shows "P (formula.Eq (Var x) (Const c))"
sel_rews_2: fixes x :: "'e\<cdot>('g, 'f) writerT" shows "(runWriterT\<cdot>x = \<bottom>) = (x = \<bottom>)"
relcompI: fixes a :: "'e"   and b :: "'f"   and r :: "('e \<times> 'f) set"   and c :: "'g"   and s :: "('f \<times> 'g) set" assumes "(a, b) \<in> r"   and "(b, c) \<in> s" shows "(a, c) \<in> r O s"
nonzero_mult_divide_mult_cancel_left2: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<noteq> (0::'a)" shows "c * a / (b * c) = a / b"
flow_initial_time: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> S"   and "x0 \<in> Y" shows "Y.flow t0 x0 t0 = x0"
akra_bazzi_light: fixes a :: "real"   and c :: "real"   and C\<^sub>1 :: "real"   and n\<^sub>0 :: "nat"   and f :: "nat \<Rightarrow> real"   and b :: "real"   and d :: "real"   and C\<^sub>2 :: "real" assumes "0 < a"   and "0 < c"   and "a + c < 1"   and "0 \<le> C\<^sub>1"   and "\<forall>n>n\<^sub>0. f n = f (nat \<lceil>a * real n + b\<rceil>) + f (nat \<lceil>c * real n + d\<rceil>) + C\<^sub>1 * real n + C\<^sub>2" shows "\<exists>C\<^sub>3 C\<^sub>4. \<forall>n. f n \<le> C\<^sub>3 * real n + C\<^sub>4"