hcju/isabelleps huggingface dataset:
set_new_pairs_sorted: fixes rel :: "nat \<times> 'g \<Rightarrow> bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and gs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "set (punit.new_pairs_sorted rel gs bs hs data) = Pair True ` (set hs \<times> (set gs \<union> set bs)) \<union> set (pairs (punit.add_pairs_single_sorted (rel data)) False hs)"
ord_resolve_sound: fixes CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and E :: "'a literal multiset"   and I :: "'a set" assumes "ord_resolve CAs DA AAs As E"   and "I \<Turnstile>m mset CAs"   and "I \<Turnstile> DA" shows "I \<Turnstile> E"
kuhn_simplex_pair_def: fixes p :: "nat"   and n :: "nat"   and b_s :: "nat \<Rightarrow> nat"   and u_s :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and b_t :: "nat \<Rightarrow> nat"   and u_t :: "nat \<Rightarrow> nat"   and t :: "(nat \<Rightarrow> nat) set" shows "kuhn_simplex_pair p n b_s u_s s b_t u_t t \<equiv> kuhn_simplex p n b_s u_s s \<and> kuhn_simplex p n b_t u_t t"
cases: fixes f :: "nat \<Rightarrow> 'd"   and g :: "nat \<Rightarrow> 'd" assumes "aeseq_eq f g" obtains "finite {n. f n \<noteq> g n}"
sets_Collect_finite_Ex: fixes S :: "'c set"   and M :: "'d measure"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>i. i \<in> S \<Longrightarrow> {x \<in> space M. P i x} \<in> sets M"   and "finite S" shows "{x \<in> space M. \<exists>i\<in>S. P i x} \<in> sets M"
clinear_scale_complex: fixes f :: "complex \<Rightarrow> complex"   and c :: "complex"   and b :: "complex" assumes "clinear f" shows "f (c * b) = c * f b"
ARSsecure_def: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and R :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and S :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c :: "IMP" shows "ARSsecure A R S c = ( \<Turnstile> c : A  \<and> RSsecure R S c)"
Rep_seqauto_inverse: fixes x :: "('a, 'b, 'c) seqauto" shows "Abs_seqauto (Rep_seqauto x) = x"
always_eventually: fixes P :: "'i \<Rightarrow> bool"   and F :: "'i filter" assumes "\<forall>x. P x" shows "eventually P F"
mult_matrix_def: fixes fmul :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and fadd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'c matrix"   and B :: "'d matrix" shows "mult_matrix fmul fadd A B \<equiv> mult_matrix_n (max (ncols A) (nrows B)) fmul fadd A B"
as_proj_pair_1: fixes p :: "('d, 'e) fmap"   and e :: "('d, 'f) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) list"   and vs :: "'d set" shows "as_proj ((p, e) # as) vs = (if fmdom' (fmrestrict_set vs e) \<noteq> {} then action_proj (p, e) vs # as_proj as vs else as_proj as vs)"
of_nat_simps_3: fixes m :: "nat" shows "of_nat (Suc m) = (1::'a) + of_nat m"
mk_mop_rl0_np: fixes mf :: "'e nres"   and f :: "'e" assumes "mf \<equiv> RETURN f" shows "(RETURN f, mf) \<in> \<langle>Id\<rangle>nres_rel"
inj_on_pullback_facet: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b set"   and z :: "'c set" assumes "inj_on f x"   and "z \<lhd> f ` x" obtains y :: "'b set" where "y \<lhd> x" and "f ` y = z"
take_0: fixes xs :: "'d list" shows "take 0 xs = []"
finfun_single_def: fixes x :: "'a" shows "finfun_single x = {}\<^sub>f(x $:= True)"
HMA_vec_elements: shows "rel_fun HMA_V (=) vec_elements vec_elements_h"
left_coset_equality: fixes a :: "'a"   and b :: "'a" assumes "a \<cdot>| L = b \<cdot>| L"   and "a \<in> G"   and "b \<in> G" shows "\<eta> a = \<eta> b"
SUP_eqI: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) x"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) y) \<Longrightarrow> less_eq x y" shows "Sup (f ` A) = x"
ANDaux2: fixes x :: "('b set set, 'c) Combinators"   and xs :: "('b set set, 'c) Combinators list"   and y :: "('b set set, 'c) Combinators" assumes "allNetsDistinct (x # xs)"   and "y \<in> set xs" shows "allNetsDistinct [x, y]"
a: fixes x :: "'a" shows "d x \<cdot> x = x"
int_or_lt0: fixes x :: "int"   and y :: "int" shows "(x OR y < 0) = (x < 0 \<or> y < 0)"
cop_F_cases: fixes x :: "'a"   and ds :: "'b set"   and fp :: "'a set" assumes "x \<in> cop_F ds fp" obtains "x \<in> fp"   | "x \<in> CD_on ds (- Rf CH fp) - fp"
intros: fixes a :: "'d"   and b :: "'e"   and r :: "('d \<times> 'e) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
lbrules_1: shows "\<forall>x\<in>{1..m}. \<forall>y\<in>{1..m}. x \<noteq> y \<longrightarrow> A x \<inter> A y = {}"
elims: fixes x :: "('fsym, 'psym) lit"   and y :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit set" assumes "glitOfL x = y" obtains at :: "('fsym, 'psym) atm" where "x = Pos at__" and "y = {grdLit x |x. x \<in> varsA at__ \<and> (prot (GE.tpOfV x) \<or> protFw (GE.tpOfV x) \<and> x \<in> nvA at__)}"   | at :: "('fsym, 'psym) atm" where "x = Neg at" and "y = {grdLit x |x. x \<in> varsA at \<and> prot (GE.tpOfV x)}"
iIN_iMODb_subset_conv: fixes d' :: "nat"   and n' :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "0 < d'" shows "([n'\<dots>,d'] \<subseteq> [ r, mod m, c ]) = (r \<le> n' \<and> m = Suc 0 \<and> n' + d' \<le> r + m * c)"
hoare_fixpoint: fixes F :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and p :: "'c \<Rightarrow> 'b"   and y :: "'a" assumes "mono_mono F"   and "\<And>w f. mono f \<and> \<Turnstile> Sup_less p w{| f |}y \<Longrightarrow> \<Turnstile> p w{| F f |}y" shows "\<Turnstile> \<Squnion>range p{| lfp F |}y"
up_dist_sup: fixes x :: "'a"   and y :: "'a" shows "\<up>(x \<squnion> y) = \<up>x \<inter> \<up>y"
eq: shows "S \<union> \<Union> c = U - \<Union> (components (U - S) - c)"
step_def: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and s :: "'a"   and oa :: "'b option" shows "Kernel.step kstep s oa \<equiv> case oa of None \<Rightarrow> s | Some a \<Rightarrow> kstep s a"
cone_add_cone: fixes C :: "'a vec set" assumes "C \<subseteq> carrier_vec n" shows "cone C + cone C = cone C"
simps: fixes a :: "'b set" shows "fsigma a = (\<exists>F. a = \<Union> (range F) \<and> (\<forall>x. closed (F x)))"
errMOD_igWlsDisj: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igWlsDisj MOD" shows "igWlsDisj (errMOD MOD)"
measure_UNION_le: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" shows "Sigma_Algebra.measure M (\<Union> (F ` I)) \<le> (\<Sum>i\<in>I. Sigma_Algebra.measure M (F i))"
a_n_sqrt: shows "\<forall>\<^sub>F n in sequentially. ((real (n - 1) / 2) powr (3 / 2) - real (n - 1)) / 2 < real_of_int (a n)"
assms_2: shows "term_ok \<Theta> (Abs T t)"
elims: fixes x :: "nterm"   and y :: "name option" assumes "unfree x = y" obtains name :: "name" where "x = Nvar name" and "y = Some name"   | v :: "name" where "x = Nconst v__" and "y = None"   | v :: "name" and va :: "nterm" where "x = (\<Lambda>\<^sub>n va__. vaa__)" and "y = None"   | v :: "nterm" and va :: "nterm" where "x = v $\<^sub>n va" and "y = None"
ExtCategory_def: fixes C :: "('d, 'e, 'f) Category_scheme" shows "ExtCategory C \<equiv> (Dom C \<in> extensional (mor\<^bsub>C\<^esub>) \<and> Cod C \<in> extensional (mor\<^bsub>C\<^esub>)) \<and> Category.Id C \<in> extensional (obj\<^bsub>C\<^esub>) \<and> (\<lambda>(x, y). x ;;\<^bsub>C\<^esub> y) \<in> extensional {(f, g) |f g. f \<approx>>\<^bsub>C\<^esub> g}"
sum_negf: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "(\<Sum>x\<in>A. - f x) = - sum f A"
stern_brocot_conv: shows "stern_brocot_recurse = tree_recurse (recip \<circ> succ \<circ> recip) succ (1, 1)"
invfun_r: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and b :: "'d" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B"   and "b \<in> B" shows "f ((f\<inverse>\<^bsub>B,A\<^esub>) b) = b"
axioms_1: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option" assumes "complete_tree_automaton TA A" shows "det_tree_automaton TA A"
lt_in_keys: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "lpp p \<in> keys p"
size_bin_to_bl: fixes n :: "nat"   and w :: "int" shows "length (bin_to_bl n w) = n"
d_repr_def: fixes n :: "nat"   and m :: "nat"   and ds :: "int iarray"   and fs :: "int vec list" shows "LLL.d_repr n m ds fs = (ds = IArray.of_fun (LLL.d n fs) (Suc m))"
Ide_implies_Can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Ide C t" shows "monoidal_language.Can C t"
Gamma_euler'_aux1: fixes n :: "nat"   and z :: "'b" assumes "0 < n" shows "exp (z * of_real (ln (real n + 1))) = (\<Prod>k = 1..n. exp (z * of_real (ln (1 + 1 / real k))))"
fdom_fdom_res: fixes A :: "'a set"   and f :: "('a, 'b) ffun" shows "fdom (A \<lhd>\<^sub>f f) = A \<inter> fdom f"
stutter_extend: shows "bisimulation R (stutter_extend (stutter_extend A)) (stutter_extend (stutter_extend B))"
ordertype_mono: fixes X :: "'a set"   and Y :: "'a set"   and r :: "('a \<times> 'a) set" assumes "X \<subseteq> Y"   and "wf r"   and "trans r"   and "small Y" shows "ordertype X r \<le> ordertype Y r"
simps: fixes a1 :: "'a set"   and a2 :: "'a \<Rightarrow> nat"   and a3 :: "'a \<Rightarrow> Regions.intv"   and a4 :: "('a \<times> 'a) set" shows "Regions.valid_region a1 a2 a3 a4 = (\<exists>X\<^sub>0 X I r k. a1 = X \<and> a2 = k \<and> a3 = I \<and> a4 = r \<and> X\<^sub>0 = {x \<in> X. \<exists>d. I x = Regions.intv.Intv d} \<and> refl_on X\<^sub>0 r \<and> trans r \<and> total_on X\<^sub>0 r \<and> (\<forall>x\<in>X. Regions.valid_intv (k x) (I x)))"
iii: shows "{#mset_lit (subst_lit x \<eta>). x \<in># mset_set ((\<lambda>x. subst_lit x \<theta>) ` D)#} \<subseteq># {#mset_lit (subst_lit (subst_lit x \<theta>) \<eta>). x \<in># mset_set D#}"
periodic_simps_3: fixes x :: "real"   and n :: "int" shows "fds_perzeta (x + real_of_int n) = fds_perzeta x"
sum_squares_eq_zero_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "(plus (times x x) (times y y) = zero) = (x = zero \<and> y = zero)"
weakResPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and x :: "name" assumes "\<Psi> \<rhd> P \<leadsto><Rel> Q"   and "eqvt Rel'"   and "x \<sharp> \<Psi>"   and "Rel \<subseteq> Rel'"   and "\<And>\<Psi>' R S y. \<lbrakk>(\<Psi>', R, S) \<in> Rel; y \<sharp> \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>S) \<in> Rel'" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<leadsto><Rel'> \<lparr>\<nu>x\<rparr>Q"
fin: fixes t :: "'t" assumes "r1.final_thread s1 t" shows "r2.final_thread (ls2, (ts2, m2), ws2, is2) t"
rcos_disjoint: fixes H :: "'a set" assumes "subgroup H G" shows "pairwise disjnt (rcosets H)"
lemma_3_10_iii_i: fixes H :: "'a set" assumes "H \<in> filters"   and "cong_r H = cong_l H" shows "H \<in> normalfilters"
private_const_deduct: fixes c :: "'fun"   and M :: "('fun, 'var) Term.term set" assumes "\<not> public c"   and "M \<turnstile> Fun c []" shows "Fun c [] \<in> M \<or> (\<exists>m\<in>subterms\<^sub>s\<^sub>e\<^sub>t M. M \<turnstile> m \<and> (\<forall>k\<in>set (fst (Ana m)). M \<turnstile> m) \<and> Fun c [] \<in> set (snd (Ana m)))"
X_minus_plus: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R" shows "X_poly_minus R a = X_poly_plus R (\<ominus>\<^bsub>R\<^esub> a)"
rtc2: fixes x :: "'a"   and y :: "'a" assumes "(1::'a) + x + y \<cdot> y \<le> y" shows "x\<^sup>\<star> \<le> y"
invariant: fixes y :: "'ret"   and s' :: "'s"   and s :: "'s"   and x :: "'call" assumes "(y, s') \<in> results'_gpv (callee s x)"   and "I s" shows "I s'"
N2_live: shows "\<turnstile> \<box>[(N1 \<or> N2) \<and> \<not> beta1]_(x, y, sem, pc1, pc2) \<and> SF(N2)_(x, y, sem, pc1, pc2) \<longrightarrow> \<diamond>pc2 = #a"
simps_1: fixes mag :: "'a"   and dim :: "(int, 'b) dimvec"   and more :: "'c" shows "mag \<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> = mag"
is_natural_2: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (\<tau> (partial_magma.cod A f)) (F f) = \<tau> f"
upcarr: shows "up \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))"
game2c_2d: shows "\<bar>spmf (map_spmf fst game2_c) True - spmf (map_spmf fst game2_d) True\<bar> \<le> spmf (map_spmf snd game2_c) True"
disc_iff_1: fixes xs :: "'b llist" shows "lnull (lalternate xs) = lnull xs"
irreducible\<^sub>d_mD_1: fixes m :: "int"   and f :: "int poly" assumes "poly_mod.irreducible\<^sub>d_m m f" shows "0 < degree (poly_mod.Mp m f)"
eq_iff: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(x = y) = (x \<preceq> y \<and> y \<preceq> x)"
least_power_props_1: fixes a :: "'a" shows "(p ^^ least_power p a) a = a"
phl_while: fixes p :: "'a"   and s :: "'a"   and w :: "'a"   and x :: "'a" assumes "p \<cdot> s \<le> s \<cdot> p \<cdot> s"   and "p \<cdot> w \<le> w \<cdot> p \<cdot> w"   and "\<lbrace>p \<cdot> s\<rbrace>x\<lbrace>p\<rbrace>" shows "\<lbrace>p\<rbrace>it (s \<cdot> x) \<cdot> w\<lbrace>p \<cdot> w\<rbrace>"
br'_algo_def: fixes \<delta> :: "('a, 'b) ta_rule set" shows "br'_algo \<delta> \<equiv> \<lparr>wa_cond = br'_cond, wa_step = br'_step \<delta>, wa_initial = br'_initial \<delta>, wa_invar = br'_invar \<delta>\<rparr>"
injectivity: fixes L :: "'b \<Rightarrow> nat"   and E :: "('b \<times> 'b) set"   and V :: "'b set"   and M :: "('b \<times> 'b) set"   and e1 :: "'b \<times> 'b"   and e2 :: "'b \<times> 'b" assumes "OSC L E"   and "matching V E M"   and "e1 \<in> matching_i 1 V E M L"   and "e2 \<in> matching_i 1 V E M L"   and "e1 \<noteq> e2" shows "endpoint_inV {v \<in> V. L v = 1} e1 \<noteq> endpoint_inV {v \<in> V. L v = 1} e2"
MVOKBARF_def: fixes v :: "Vals" shows "MVOKBARF v = (v \<in> MemVal \<or> v = OK \<or> v = BadArg \<or> v = RPCFailure)"
card_idx_L_transform_state: shows "|UNIV| <o |UNIV|"
Span_contains_Spans_Cons_right: fixes m :: "'v"   and ms :: "'v list" assumes "m \<in> V" shows "RSpan ms \<subseteq> RSpan (m # ms)"
Cast1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and C :: "ty"   and a :: "'addr"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Cast C (Throw a),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
has_as_pullback_def: fixes f0 :: "'a"   and f1 :: "'a"   and p0 :: "'a"   and p1 :: "'a" shows "has_as_pullback f0 f1 p0 p1 \<equiv> cospan f0 f1 \<and> limit_cone cospan_shape.comp (\<cdot>) (cospan_diagram.map (\<cdot>) f0 f1) (local.dom p0) (cospan_diagram.mkCone (\<cdot>) f0 f1 p0 p1)"
rename_preserves: fixes h :: "'c \<Rightarrow> 'd"   and G :: "'c program"   and v :: "'d \<Rightarrow> 'e" assumes "bij h" shows "(rename h G \<in> preserves v) = (G \<in> preserves (v \<circ> h))"
vcompE: fixes ac :: "V"   and r :: "V"   and s :: "V" assumes "ac \<in>\<^sub>\<circ> r \<circ>\<^sub>\<circ> s" obtains a :: "V" and b :: "V" and c :: "V" where "ac = \<langle>a, c\<rangle>" and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> s" and "\<langle>b, c\<rangle> \<in>\<^sub>\<circ> r"
ereal_of_enat_def: fixes n :: "enat" shows "ereal_of_enat n = (case n of enat n \<Rightarrow> ereal (real n) | \<infinity> \<Rightarrow> \<infinity>)"
assms_2: shows "integrable M (\<lambda>x. f (T x))"
invariant_run_index: fixes P :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat"   and p :: "'c"   and A :: "('d, 'c) ngba"   and Q :: "nat \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'c \<Rightarrow> bool" assumes "P n p"   and "\<And>n p. P n p \<Longrightarrow> \<exists>a. (fst a \<in> alphabet A \<and> snd a \<in> transition A (fst a) p) \<and> P (Suc n) (snd a) \<and> Q n p a" obtains r :: "('d \<times> 'c) stream" where "run A r p" and "\<And>i. P (n + i) (target (stake i r) p)" and "\<And>i. Q (n + i) (target (stake i r) p) (r !! i)"
RP_complete_if_fair: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "chain (\<leadsto>) Sts"   and "fair_state_seq Sts"   and "Q_of_state (lhd Sts) = {}"   and "\<not> satisfiable (grounding_of_clss (N_of_state (lhd Sts) \<union> P_of_state (lhd Sts) \<union> Q_of_state (lhd Sts)))" shows "{#} \<in> Q_of_state (Liminf_state Sts)"
simps_5: fixes \<sigma> :: "'a basic_type" shows "Real \<squnion> \<sigma> = (if Real \<le> \<sigma> then \<sigma> else if \<sigma> \<le> Real then Real else OclAny)"
axioms_1: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e" assumes "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f" shows "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
k_def: shows "I (s ! k) = i \<and> k < length s"
secure_equals_c_secure: fixes I :: "('a \<times> 'a) set"   and step :: "'d \<Rightarrow> 'b \<Rightarrow> 'd"   and out :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and s\<^sub>0 :: "'d"   and D :: "'b \<Rightarrow> 'a" assumes "refl I" shows "secure (c_process step out s\<^sub>0) I (c_dom D) = c_secure step out s\<^sub>0 I D"
scount_eq_0D: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" assumes "scount P \<omega> = 0" shows "alw (\<lambda>xs. \<not> P xs) \<omega>"
r_P2: fixes b :: "nat"   and x :: "nat"   and y :: "nat"   and r :: "nat"   and a :: "'a" assumes "b < q"   and "x < q"   and "y < q"   and "r < q" shows "((x, a, r, (y + q - b) mod q), (x * ((y + q - b) mod q) + q - r) mod q, (x * y + q - (x * ((y + q - b) mod q) + q - r) mod q) mod q) = ((x, a, (x * ((y + q - b) mod q) + q - (x * ((y + q - b) mod q) + q - r) mod q) mod q, (y + q - b) mod q), (x * ((y + q - b) mod q) + q - r) mod q, (x * y + q - (x * ((y + q - b) mod q) + q - r) mod q) mod q)"
seqs_def: fixes n :: "nat" shows "seqs n = {xs. length xs = n}"
bnd_1: shows "v x \<ge>\<^sub>l\<^sub>b \<B>\<^sub>l s x"
pmaxsimpchainI_maxsimpchain: fixes xs :: "'a set list" assumes "maxsimpchain xs"   and "distinct xs" shows "pmaxsimpchain xs"
min_pconj: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" shows "min a b .& min c d \<le> min (a .& c) (b .& d)"
aNDSetsEq: fixes a :: "('a set set, 'b) Combinators list"   and b :: "('a set set, 'b) Combinators list" assumes "singleCombinators a"   and "singleCombinators b"   and "set a = set b"   and "allNetsDistinct b" shows "allNetsDistinct a"
add_scalar_prod_distrib: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec"   and v\<^sub>3 :: "'c vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "(v\<^sub>1 + v\<^sub>2) \<bullet> v\<^sub>3 = v\<^sub>1 \<bullet> v\<^sub>3 + v\<^sub>2 \<bullet> v\<^sub>3"
minus_diff_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "uminus (minus a b) = minus b a"
comp_assoc: fixes h :: "'d"   and g :: "'d"   and f :: "'d" shows "(h \<cdot>\<^sub>D g) \<cdot>\<^sub>D f = h \<cdot>\<^sub>D g \<cdot>\<^sub>D f"
pred_succ: fixes r :: "('c \<times> 'c) set"   and j :: "'c" assumes "wo_rel r"   and "aboveS r j \<noteq> {}" shows "pred r (succ r j) = j"
2: shows "Left_a.ide (a \<star> f) \<and> Left_a.ide (a \<star> g)"
fdj: shows "\<forall>p q. f (Or p q) = Or (f p) (f q)"
submoduleE_4: fixes H :: "'c set"   and a :: "'a"   and b :: "'c" assumes "Module.submodule H R M"   and "a \<in> carrier R"   and "b \<in> H" shows "a \<odot>\<^bsub>M\<^esub> b \<in> H"
get_ancestors_di_def: shows "get_ancestors_di \<equiv> l_get_ancestors_di\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_di Shadow_DOM.get_parent get_host get_disconnected_document"
remove_max_max_lemma: fixes m :: "'a"   and t :: "'a list"   and l :: "'a list" shows "fst (foldl f (m, t) l) = Max (SelectionSort_Functional.set (m # l))"
r_right_minus_eq: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "(a \<ominus> b = \<zero>) = (a = b)"
ide_char: fixes a :: "'a \<times> 'a \<times> 'a" shows "Cop.VVV.ide a = (Cop.VVV.arr a \<and> VxVxV.ide a)"
kuratowski_planar_def: fixes G :: "('c, 'd) pre_digraph" shows "kuratowski_planar G \<equiv> \<nexists>H. subgraph H G \<and> (\<exists>K rev_K rev_H. subdivision (K, rev_K) (H, rev_H) \<and> (K\<^bsub>3,3\<^esub> K \<or> K\<^bsub>5\<^esub> K))"
hyps: shows "ack i (Suc j) \<le> ack (Suc i) j"
conga_right_comm: 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"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C F E D"
DLOe_code_lemmas_5: shows "??.QEdlo.lift_dnfeq_qe = DLO\<^sub>e.lift_dnfeq_qe"
power_up_eq_zero_iff: fixes prec :: "nat"   and b :: "real"   and n :: "nat" shows "(power_up prec b n = 0) = (b = 0 \<and> n \<noteq> 0)"
mod_diff_right_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (minus a (modulo b c)) c = modulo (minus a b) c"
listrel_Nil1: fixes xs :: "'c list"   and r :: "('b \<times> 'c) set" assumes "([], xs) \<in> listrel r" obtains "xs = []"