hcju/isabelleps huggingface dataset:
vrrestriction_vsubset_vrange: fixes B :: "V" assumes "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> B" shows "r \<restriction>\<^sup>r\<^sub>\<circ> B = r"
assms: shows "inj_on f (Field r1 \<union> Field r2)"
dom_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.dom (g \<cdot>\<^sub>C f) = C.dom f"
StepRel_def: shows "StepRel \<equiv> fst \<circ> snd \<circ> snd \<circ> Rep_kripke"
simps_2: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and as :: "'b list"   and bs :: "'a list" shows "map_tailrec_rev f (a # as) bs = map_tailrec_rev f as (f a # bs)"
RModuleHom_axioms: shows "RModuleHom R (\<cdot>) M (\<star>) T"
rep_eq: fixes x :: "'d pdevs"   and xa :: "nat"   and xb :: "'d" shows "pdevs_apply (pdev_upd x xa xb) = (pdevs_apply x)(xa := xb)"
to_fun_Fun_zero: shows "to_fun \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>function_ring (carrier R) R\<^esub>"
prob_algebra_def: fixes K :: "'c measure" shows "prob_algebra K = restrict_space (subprob_algebra K) {M. prob_space M}"
simps_5: fixes h :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and \<phi>\<^sub>1 :: "'c fm"   and \<phi>\<^sub>2 :: "'c fm"   and xs :: "'d list" shows "interpret h (Or \<phi>\<^sub>1 \<phi>\<^sub>2) xs = (interpret h \<phi>\<^sub>1 xs \<or> interpret h \<phi>\<^sub>2 xs)"
primeidealCE: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "ideal I R"   and "cring R"   and "\<not> primeideal I R" obtains "carrier R = I"   | "\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes>\<^bsub>R\<^esub> b \<in> I \<and> a \<notin> I \<and> b \<notin> I"
F2a: shows "\<forall>x\<^sub>2 x\<^sub>1. x\<^sub>1 \<subseteq> x\<^sub>2 \<longrightarrow> x\<^sub>2 = x\<^sub>2 \<union> x\<^sub>1"
runit'_simps_5: fixes f :: "'d" assumes "D.ide f" shows "D.cod \<r>\<^sub>D\<^sup>-\<^sup>1[f] = f \<star>\<^sub>D src\<^sub>D f"
possible_traces_def: fixes SES :: "('b, 'a) SES_rec" shows "possible_traces SES \<equiv> {t. enabled SES s0\<^bsub>SES\<^esub> t}"
coll_convex: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real"   and d :: "real \<times> real"   and u :: "real"   and v :: "real" assumes "coll a b c"   and "coll a b d"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "coll a b (u *\<^sub>R c + v *\<^sub>R d)"
colossless_gpv_1: shows "colossless_gpv \<I> gpv"
validI: fixes A :: "'b triple set"   and G :: "prog"   and t :: "Term.term"   and P :: "'b assn"   and Q :: "'b assn" assumes "\<And>n s0 L accC T C v s1 Y Z. \<lbrakk>\<forall>t\<in>A. G\<Turnstile>n\<Colon>t; s0\<Colon>\<preceq>(G, L); normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>t\<Colon>T; normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> C; G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1); P Y s0 Z\<rbrakk> \<Longrightarrow> Q v s1 Z \<and> s1\<Colon>\<preceq>(G, L)" shows "G,A|\<Turnstile>\<Colon>{{P} t\<succ> {Q}}"
set_def: fixes ba :: "'c \<times> 'c" shows "Hom.set ba \<equiv> \<phi> (fst ba, snd ba) ` C.hom (fst ba) (snd ba)"
simps_4: fixes sys :: "'a \<Rightarrow> ('b, 'c) process_term"   and T1 :: "('d, 'a) process_term"   and T2 :: "('d, 'a) process_term" shows "isACT sys (CH T1 T2) = False"
rel_fundefs_None1: fixes f :: "'k \<Rightarrow> ('b, 'l, 'm, 'n, 'e) Inca.instr fundef option"   and g :: "'k \<Rightarrow> ('b, 'l, 'm, 'n, 'e, 'g, 'h, 'i) Ubx.instr fundef option"   and x :: "'k" assumes "rel_fundefs f g"   and "f x = None" shows "g x = None"
Liminf_mono: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x \<le> g x" shows "Liminf F f \<le> Liminf F g"
node_in_scc_of_node: fixes v :: "'c"   and E :: "('c \<times> 'c) set" shows "v \<in> scc_of E v"
n: shows "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n \<A>))"
span_span: fixes A :: "'a set" shows "ideal (ideal A) = ideal A"
MergeGuardsI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P merge_guards c Q,A"
BT_eq_30': fixes i :: "nat"   and j :: "nat" assumes "3 \<le> i"   and "i \<le> k + 1"   and "j < n (i - 1)" shows "subresultant j (F 1) (F 2) = Polynomial.smult ((- 1) ^ (\<Sum>l\<leftarrow>[3..<i]. (n (l - 2) - j) * (n (l - 1) - j)) * (\<Prod>l\<leftarrow>[3..<i]. (\<beta> l / \<alpha> l) ^ (n (l - 1) - j)) * (\<Prod>l\<leftarrow>[3..<i]. f (l - 1) ^ (\<delta> (l - 2) + \<delta> (l - 1)))) (subresultant j (F (i - 2)) (F (i - 1)))"
Inf_real_def: fixes X :: "real set" shows "Inf X = - Sup (uminus ` X)"
h2: shows "\<Gamma> \<turnstile> t is u : T\<^sub>1\<rightarrow>T\<^sub>2"
internal_inv_Branching: fixes p :: "bool list"   and l :: "tree"   and r :: "tree" shows "internal p (Branching l r) = (p = [] \<or> (\<exists>a p'. p = a # p' \<and> (a \<longrightarrow> internal p' l) \<and> (\<not> a \<longrightarrow> internal p' r)))"
Pi_eq_empty: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" shows "(Pi A B = {}) = (\<exists>x\<in>A. B x = {})"
mapping_eqI': fixes m :: "('b, 'c) mapping"   and d :: "'c"   and m' :: "('b, 'c) mapping" assumes "\<And>x. x \<in> Mapping.keys m \<Longrightarrow> Mapping.lookup_default d m x = Mapping.lookup_default d m' x"   and "Mapping.keys m = Mapping.keys m'" shows "m = m'"
SC_SCp_eq: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" shows "(\<exists>n. \<Gamma> \<Rightarrow> \<Delta> \<down> n) = \<Gamma> \<Rightarrow> \<Delta>"
permute_swap_cancel2: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "atom"   and b :: "atom"   and x :: "'e" assumes "class.pt permute" shows "permute (a \<rightleftharpoons> b) (permute (b \<rightleftharpoons> a) x) = x"
space_closed: shows "{{}, X, S - X, S} \<subseteq> Pow S"
elims: fixes x :: "Preliminaries.var \<Rightarrow> 'c trm"   and xa :: "('c, 'd) atm"   and y :: "('c, 'd) atm" assumes "substA x xa = y" obtains \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm" and T1 :: "'c trm" and T2 :: "'c trm" where "x = \<pi>__" and "xa = Eq T1 T2" and "y = Eq (subst \<pi>__ T1) (subst \<pi>__ T2)"   | \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm" and p :: "'d" and Tl :: "'c trm list" where "x = \<pi>" and "xa = Pr p Tl" and "y = Pr p (map (subst \<pi>) Tl)"
Cauchy_Schwarz_complex_vec: fixes x :: "complex vec"   and n :: "nat"   and y :: "complex vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "inner_prod x y * inner_prod y x \<le> inner_prod x x * inner_prod y y"
Eps_subset_hd: fixes n :: "nat" shows "Eps (\<Phi> (F n)) \<subseteq> hd (F n)"
ok_rename_iff: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and G :: "'c program" assumes "bij h" shows "rename h F ok rename h G = F ok G"
galleryD_chamber: fixes xs :: "'a set list"   and x :: "'a set" assumes "folding.gallery xs"   and "x \<in> set xs" shows "folding.chamber x"
t_1: shows "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
lossless_spmfD_set_spmf_nonempty: fixes p :: "'c spmf" assumes "lossless_spmf p" shows "set_spmf p \<noteq> {}"
assms: shows "set \<Gamma>\<^sub>2 \<subseteq> set \<Gamma>\<^sub>1"
r0: shows "r (Suc k) (b $ 0) ^ Suc k = b $ 0"
MD_k1: shows "mat_delete (make_mat n k B) 0 k $$ (k - 1, 0) = B $$ (1, 0)"
finite_sorted_distinct_unique: fixes A :: "'a set" assumes "finite A" shows "\<exists>!xs. set xs = A \<and> sorted xs \<and> distinct xs"
ipresWlsAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'varSort, 'sort, 'l, 'm, 'h, 'i, 'n) model_scheme" shows "ipresWlsAll h hA MOD \<equiv> ipresWls h MOD \<and> ipresWlsAbs hA MOD"
dependent_nat_choice: fixes P :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<exists>x. P 0 x"   and "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y" shows "\<exists>f. \<forall>n. P n (f n) \<and> Q n (f n) (f (Suc n))"
P_1_lemma: fixes p :: "nat"   and q :: "nat"   and res :: "int \<times> int" assumes "QR p q"   and "0 \<le> fst res"   and "fst res < int p"   and "0 \<le> snd res"   and "snd res < int q" shows "\<exists>!x. QR.P_1 p q res x"
cases: fixes a1 :: "('e, 'f, 'g) Tree"   and a2 :: "('e, 'f, 'g) Tree" assumes "(a1, a2) \<in> Tree_wf" obtains t :: "('e, 'f, 'g) Tree" and tset :: "('e, 'f, 'g) Tree set['e]" where "a1 = t__" and "a2 = tConj tset" and "t__ \<in> set_bset tset"   | t :: "('e, 'f, 'g) Tree" where "a1 = ta__" and "a2 = tNot ta__"   | t :: "('e, 'f, 'g) Tree" and \<alpha> :: "'g" where "a1 = t" and "a2 = tAct \<alpha> t"
len: shows "length (transpose (transpose xs)) = length (takeWhile (\<lambda>x. x \<noteq> []) xs)"
sublens_put_put: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'f"   and u :: "'d" assumes "mwb_lens X"   and "Y \<subseteq>\<^sub>L X" shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>X\<^esub> \<sigma> u"
sparse_row_vector_pprt: fixes v :: "(nat \<times> 'a) list" assumes "sorted_spvec v" shows "sparse_row_vector (pprt_spvec v) = pprt (sparse_row_vector v)"
f_in_K: fixes n :: "nat" shows "K n \<noteq> {}"
l_adjoint_def: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and x :: "'b::refinement_lattice" shows "((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)\<^sup>\<flat>) (x::'b::refinement_lattice) \<equiv> \<Sqinter> {y::'b::refinement_lattice. x \<sqsubseteq> F y}"
consistency_onD: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set"   and C :: "'a set" assumes "consistency_on A f"   and "B \<subseteq> A"   and "C \<subseteq> B"   and "f B \<subseteq> C" shows "f B = f C"
simps_4: fixes sh :: "sheap"   and e\<^sub>1 :: "'a exp"   and bop :: "bop"   and e\<^sub>2 :: "'a exp" shows "iconf sh (e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2) = (case val_of e\<^sub>1 of None \<Rightarrow> iconf sh e\<^sub>1 \<and> \<not> sub_RI e\<^sub>2 | \<lfloor>v\<rfloor> \<Rightarrow> iconf sh e\<^sub>2)"
simps_4: fixes aneg :: "'c \<Rightarrow> 'c fm"   and anormal :: "'c \<Rightarrow> bool"   and I\<^sub>a :: "'c \<Rightarrow> 'd list \<Rightarrow> bool"   and depends\<^sub>0 :: "'c \<Rightarrow> bool"   and decr :: "'c \<Rightarrow> 'c"   and qe :: "'c fm \<Rightarrow> 'c fm"   and \<phi> :: "'c fm" assumes "ATOM aneg anormal I\<^sub>a depends\<^sub>0 decr" shows "ATOM.lift_nnf_qe aneg qe (ExQ \<phi>) = qe (ATOM.nnf aneg (ATOM.lift_nnf_qe aneg qe \<phi>))"
chamber_system_into: shows "order.greater_eq folding_g.\<C> (f \<turnstile> folding_g.\<C>)"
lesser_con: shows "\<forall>n m. n \<^bold>\<in> lan v1 \<and> m \<^bold>\<in> lan v2 \<longrightarrow> n < m"
simps_1: shows "eq\<cdot>[::]\<cdot>[::] = TT"
simps_4: fixes x2 :: "'e"   and x1 :: "'d" shows "Inr x2 \<noteq> Inl x1"
lcm_div_le: fixes a :: "nat"   and b :: "nat" shows "lcm a b div b \<le> a"
Inter_greatest: fixes A :: "'c set set"   and C :: "'c set" assumes "\<And>X. X \<in> A \<Longrightarrow> C \<subseteq> X" shows "C \<subseteq> \<Inter> A"
Increasing_Un: fixes F :: "'b program"   and f :: "'b \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c set" assumes "F \<in> Increasing f"   and "F \<in> Increasing g" shows "F \<in> Increasing (\<lambda>s. f s \<union> g s)"
size_def: fixes m :: "('a, 'b) mapping" shows "Mapping.size m = (if finite (Mapping.keys m) then card (Mapping.keys m) else 0)"
f'_def: shows "f' = fls_base_factor_to_fps (fls_of_fpxs f)"
interval_bisection_step: fixes P :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "P {}"   and "\<And>S T. \<lbrakk>P S; P T; interior S \<inter> interior T = {}\<rbrakk> \<Longrightarrow> P (S \<union> T)"   and "\<not> P (cbox a b)" obtains c :: "'a" and d :: "'a" where "\<not> P (cbox c d)" and "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> c \<bullet> i \<and> c \<bullet> i \<le> d \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i \<and> 2 * (d \<bullet> i - c \<bullet> i) \<le> b \<bullet> i - a \<bullet> i"
mset_subset_eq_add_right: fixes B :: "'k multiset"   and A :: "'k multiset" shows "B \<subseteq># A + B"
Nonce_Sign: fixes X :: "nonce_t"   and X' :: "msg"   and Y' :: "msg" shows "Nonce X \<noteq> Sign X' Y'"
le_env_free_tv: fixes A :: "type_scheme list"   and B :: "type_scheme list" assumes "A \<le> B" shows "free_tv B \<subseteq> free_tv A"
prems_1: shows "ws'' t' = \<lfloor>InWS w\<rfloor>"
tdev_nonneg: fixes X :: "'b pdevs" shows "(0::'b) \<le> tdev X"
ag_l_inv1: fixes x :: "'a" assumes "x \<in> carrier A" shows "-\<^sub>a x \<plusminus> x = \<zero>"
lem_6_1_2'_3: fixes X :: "'x set"   and k :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and i :: "nat"   and d :: "nat" assumes "finite X"   and "standard_decomp k ps"   and "exact_decomp X 0 ps"   and "Suc 0 \<le> i"   and "i \<le> card X"   and "\<b> ps (Suc i) \<le> d"   and "d < \<b> ps i" shows "card {(h', U'). (h', U') \<in> set (ps\<^sub>+) \<and> poly_deg h' = d \<and> card U' = i} = 1"
span_eq_iff: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s :: "'d set" assumes "module scale" shows "(module.span scale s = s) = module.subspace scale s"
ln2_le_25_over_36: shows "ln 2 \<le> 25 / 36"
wt0_eq: shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt0 \<zeta>)) = eval_tpoly B (map_tpoly (\<lambda>x. x) zhmset_of (wt0 \<zeta>))"
f_mult: fixes n :: "nat"   and ra :: "real" assumes "n \<noteq> 0" shows "f (real_of_int (int n) * ra + real_of_int (int (n - 1))) = real_of_int (int n) * f ra"
pm_of_idx_pm_take: fixes f :: "nat \<Rightarrow>\<^sub>0 'c"   and j :: "nat"   and xs :: "'d list" assumes "keys f \<subseteq> {0..<j}" shows "pm_of_idx_pm (take j xs) f = pm_of_idx_pm xs f"
eom: shows "exponential_order (2 * L) c (\<lambda>x. f x - g x)"
simps_2: fixes s :: "('c, 'd) tm"   and t :: "('c, 'd) tm"   and ts :: "('c, 'd) tm list" shows "apps s (t # ts) = apps (App s t) ts"
partizan_l: shows "\<forall>e. Elem e l \<longrightarrow> e \<in> games_lfp"
primepow_not_unit: fixes p :: "'a" assumes "primepow p" shows "\<not> is_unit p"
sim_valI: fixes M :: "('b, 'c, 'a) KripkeStructure"   and M' :: "('b, 'c, 'd) KripkeStructure"   and f :: "'a \<Rightarrow> 'd" assumes "\<And>u. u \<in> worlds M \<Longrightarrow> valuation M u = valuation M' (f u)" shows "sim_val M M' f"
smult_assoc: fixes r :: "('f, 'g) aezfun"   and s :: "('f, 'g) aezfun"   and m :: "'v" assumes "r \<in> FG"   and "s \<in> FG"   and "m \<in> V" shows "r \<cdot> (s \<cdot> m) = r * s \<cdot> m"
fps_deriv_power': fixes a :: "'b fps"   and n :: "nat" shows "fps_deriv (a ^ n) = of_nat n * fps_deriv a * a ^ (n - 1)"
ssubst_Zero: fixes V :: "name set" assumes "finite V" shows "ssubst Zero V F = Zero"
cv_hom_par: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> U \<parallel> y \<parallel> U = x \<parallel> y \<parallel> U"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and v :: "'d list" shows "set (map f v) = f ` set v"
mu'_mu_large_row_i: fixes ii :: "nat" assumes "i < ii"   and "ii < m" shows "gs2.\<mu> ii i = fs.gs.\<mu> ii (i - 1) - fs.gs.\<mu> i (i - 1) * fs.gs.\<mu> ii i"
P2_real_model'_def: fixes M :: "'a \<times> 'a"   and \<sigma> :: "bool"   and z :: "'f"   and \<A> :: "(bool \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a \<times> 'a) \<times> nat) \<times> 'g) spmf) \<times> ('a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> 'e spmf)" shows "P2_real_model' M \<sigma> z \<A> = (let (x0, x1) = M; (\<A>1, \<A>2, \<A>3) = \<A> in \<A>1 \<sigma> z \<bind> (\<lambda>((h0, h1, a, b0, b1), s). assert_spmf (h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> a \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>) \<bind> (\<lambda>_. \<A>2 (h0, h1, a, b0, b1) s \<bind> (\<lambda>(((in1, in2, in3), r), s'). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in funct_DH_ZK (h, a, b) ((in1, in2, in3), r) \<bind> (\<lambda>(out_zk_funct, uu_). assert_spmf out_zk_funct \<bind> (\<lambda>_. P2_real_model_end M ((h0, h1, a, b0, b1), s') \<A>3))))))"
nonneg_lincomb_def: fixes n :: "nat"   and c :: "'b vec \<Rightarrow> 'b"   and Vs :: "'b vec set"   and b :: "'b vec" shows "gram_schmidt.nonneg_lincomb n c Vs b = (module.lincomb (module_vec TYPE('b) n) c Vs = b \<and> c ` Vs \<subseteq> {x. (0::'b) \<le> x})"
exhaust: fixes y :: "('a, 'b, 'c) state" obtains x1 :: "'a" where "y = Idle x1"   | x21 :: "'a" and x22 :: "'b" and x23 :: "'c" where "y = State x21 x22 x23"
assms_1: shows "(f has_field_derivative f') (at a within S)"
Conf_ChiPlus: fixes T :: "'d"   and S :: "'d"   and ST :: "('d, 'e, 'f) status" assumes "(T, S) \<in> ChiPlus (HA ST)" shows "S \<in> Conf ST \<longrightarrow> T \<in> Conf ST"
spr_ii_sim: shows "sim SPR.MC spr_ii_simMC spr_ii_sim"
Inter_eq: fixes A :: "'c set set" shows "\<Inter> A = {x. \<forall>B\<in>A. x \<in> B}"
aform_err_lemma1: fixes err :: "real"   and X1 :: "real"   and A :: "real"   and e :: "'a \<Rightarrow> real"   and d :: "'a"   and B :: "real"   and e1 :: "real"   and x :: "real"   and p :: "nat" assumes "- 1 \<le> err"   and "err \<le> 1"   and "X1 + (A - e d * B + err * B) - e1 \<le> x" shows "X1 + (A - e d * B) - truncate_up p (\<bar>B\<bar> + e1) \<le> x"
widen_ArrayRefT: fixes G :: "prog"   and t :: "ref_ty"   and T :: "ty" assumes "G\<turnstile>RefT t.[]\<preceq>T" shows "T = Class Object \<or> (\<exists>s. T = RefT s.[] \<and> G\<turnstile>RefT t\<preceq>RefT s)"
Sbis_sym: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "c \<approx>s d" shows "d \<approx>s c"
use_premise: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "healthy t"   and "\<And>s. P s \<Longrightarrow> 1 \<le> t \<guillemotleft> Q \<guillemotright> s" shows "\<guillemotleft> P \<guillemotright> \<tturnstile> t \<guillemotleft> Q \<guillemotright>"
noninert_gauss_int_prime_def: fixes p :: "nat" shows "noninert_gauss_int_prime p \<equiv> prime p \<and> [p = 1] (mod 4)"
translation_assoc: fixes b :: "'a"   and a :: "'a"   and S :: "'a set" shows "(+) b ` (+) a ` S = (+) (a + b) ` S"
GuardFault: fixes s :: "'a"   and g :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and f :: "'c"   and c :: "('a, 'b, 'c) com" assumes "s \<notin> g" shows "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> \<Rightarrow> Fault f"
symp_def: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "symp r = (\<forall>x y. r x y \<longrightarrow> r y x)"
line_suitable_junctors_4: fixes \<A> :: "'b set"   and F :: "'b formula"   and G :: "'b formula" shows "line_suitable \<A> (F \<^bold>\<rightarrow> G) = (line_suitable \<A> F \<and> line_suitable \<A> G)"
proots_within_union: fixes p :: "'a poly"   and A :: "'a set"   and B :: "'a set" shows "proots_within p A \<union> proots_within p B = proots_within p (A \<union> B)"
BangRelRel': shows "bangRel Rel \<subseteq> Rel'"
is_iso_dghm_axioms_def: fixes \<BB> :: "V"   and \<FF> :: "V" shows "is_iso_dghm_axioms \<BB> \<FF> \<equiv> (v11 (\<FF>\<lparr>ObjMap\<rparr>) \<and> v11 (\<FF>\<lparr>ArrMap\<rparr>)) \<and> \<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) = \<BB>\<lparr>Obj\<rparr> \<and> \<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) = \<BB>\<lparr>Arr\<rparr>"
Abs_fin_string_inverse: fixes y :: "char list" assumes "y \<in> {s. length s \<le> MAX_STR}" shows "Rep_fin_string (Abs_fin_string y) = y"
RedThrowNull: fixes E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P,E \<turnstile> \<langle>throw null,s\<rangle> \<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<rangle>"
data_in'_def: shows "data_in' \<equiv> fold (\<lambda>(t, X). append_queue (t, X)) move data_in"
affine_hull_convex_Int_nonempty_interior: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "S \<inter> interior T \<noteq> {}" shows "affine hull (S \<inter> T) = affine hull S"
order_less: assumes "\<not> prime p" shows "order G < p - 1"
m2ZF_def: fixes t :: "(ZF, ZF) TermEquivClT" shows "m2ZF t \<equiv> ZFTriple (TDomain t) (implode (Expr2ZF ` TExprSet t)) (TCodomain t)"
fac_two_conjug_prim_root: fixes u :: "'b list"   and r :: "'b list"   and k :: "nat"   and s :: "'b list"   and l :: "nat" assumes "u \<le>f r \<^sup>@ k"   and "u \<le>f s \<^sup>@ l"   and "r \<noteq> \<epsilon>"   and "s \<noteq> \<epsilon>"   and "\<^bold>|r\<^bold>| + \<^bold>|s\<^bold>| - gcd \<^bold>|r\<^bold>| \<^bold>|s\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "\<rho> r \<sim> \<rho> s"
assms_1: shows "exact_seq ([C, B, A], [g, f])"
closed_segment_def: fixes a :: "'c"   and b :: "'c" shows "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b |u. 0 \<le> u \<and> u \<le> 1}"
set_ofI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "P x y" shows "(x, y) \<in> set_of P"
relabel_increase_u: fixes u :: "nat" assumes "relabel_precond f (relabel_effect f l u) u" shows "relabel_effect f l u u < relabel_effect f (relabel_effect f l u) u u"
nth_least_nth_least: fixes n :: "nat"   and A :: "nat set"   and B :: "'b set" assumes "enat n < esize A"   and "\<And>i. i \<in> A \<Longrightarrow> enat i < esize B" shows "nth_least B (nth_least A n) = nth_least (nth_least B ` A) n"
domainOfGraph: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "runiq (graph X f) \<and> Domain (graph X f) = X"
assms_2: shows "e = Binop_i t iop \<and> is_int_t t \<or> e = Binop_f t fop \<and> is_float_t t \<or> e = Relop_i t irop \<and> is_int_t t \<or> e = Relop_f t frop \<and> is_float_t t"
suminf_mult: fixes f :: "nat \<Rightarrow> 'c"   and c :: "'c" assumes "summable f" shows "(\<Sum>n. c * f n) = c * suminf f"
green_typeII_chain_axioms_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" shows "green_typeII_chain_axioms i j F two_chain s \<equiv> valid_typeII_division s two_chain \<and> (\<forall>twoC\<in>two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> j) i)"
S12: shows "fn_to_set (\<lambda>x y. p_a x y * p_b x y) = A \<union> B"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('a, 'b) lit" assumes "\<not> U.satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('a, 'b) atm" where "x = \<xi>'__" and "xa = Pos a__" and "\<not> U.satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('a, 'b) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> U.satA \<xi> a"
argmaxProperty: fixes X :: "'a set"   and mx :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "finite X \<and> mx \<in> X \<and> (\<forall>aa\<in>X - {mx}. f aa < f mx) \<longrightarrow> argmax f X = {mx}"
add_assoc: fixes a :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> bool" shows "((a \<and>* b) \<and>* c) = (a \<and>* b \<and>* c)"