hcju/isabelleps huggingface dataset:
2: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" shows "(hom_induced p X {} X S id \<circ> hom_induced p (subtopology X T) {} X {} id) x = hom_induced p (subtopology X T) {} X S id x"
part_equivp_symp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "part_equivp R"   and "R x y" shows "R y x"
J_rel_coind_ind: fixes R2 :: "'d JF1 \<Rightarrow> 'e JF1 \<Rightarrow> bool"   and f :: "'d JF1 \<Rightarrow> 'e JF1 \<Rightarrow> ('d \<times> 'e, 'd JF1 \<times> 'e JF1, 'd JF2 \<times> 'e JF2) F1"   and R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R3 :: "'d JF2 \<Rightarrow> 'e JF2 \<Rightarrow> bool"   and g :: "'d JF2 \<Rightarrow> 'e JF2 \<Rightarrow> ('d \<times> 'e, 'd JF1 \<times> 'e JF1, 'd JF2 \<times> 'e JF2) F2"   and z1 :: "('d \<times> 'e) JF1"   and z2 :: "('d \<times> 'e) JF2" assumes "\<forall>x y. R2 x y \<longrightarrow> f x y \<in> F1in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F1map fst fst fst (f x y) = dtor1 x \<and> F1map snd snd snd (f x y) = dtor1 y"   and "\<forall>x y. R3 x y \<longrightarrow> g x y \<in> F2in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F2map fst fst fst (g x y) = dtor2 x \<and> F2map snd snd snd (g x y) = dtor2 y" shows "(\<forall>a\<in>JF1set z1. \<forall>x y. R2 x y \<and> z1 = unfold1 (\<lambda>(x, y). f x y) (\<lambda>(x, y). g x y) (x, y) \<longrightarrow> R1 (fst a) (snd a)) \<and> (\<forall>a\<in>JF2set z2. \<forall>x y. R3 x y \<and> z2 = unfold2 (\<lambda>(x, y). f x y) (\<lambda>(x, y). g x y) (x, y) \<longrightarrow> R1 (fst a) (snd a))"
assms_5: shows "Prj\<^sub>0\<^sub>0 \<cdot> h = Prj\<^sub>0\<^sub>0 \<cdot> h'"
RELATES_def: fixes R :: "('a \<times> 'b) set" shows "RELATES R \<equiv> True"
assms_2: shows "\<not> R\<^sup>+\<^sup>+ x y"
e1_no_c: shows "e1 \<in> E\<^bsub>SES\<^esub> - C\<^bsub>\<V>\<^esub>"
"2_4": shows "D' = And (reset' (And D (abstr g (\<lambda>i j. \<infinity>) v)) n r v (0::'c)) (abstr (inv_of A l') (\<lambda>i j. \<infinity>) v)"
n_def: shows "res_seq s (Suc k) 0 = res_seq s k n"
tendsto_mono: fixes "open" :: "'b set \<Rightarrow> bool"   and F :: "'c filter"   and F' :: "'c filter"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b" assumes "class.topological_space open"   and "F \<le> F'"   and "filterlim f (topological_space.nhds open l) F'" shows "filterlim f (topological_space.nhds open l) F"
sorted_map: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (map f xs) = sorted_wrt (\<lambda>x y. less_eq (f x) (f y)) xs"
at_begin_fst_awtn_Bk: fixes as :: "nat"   and am :: "nat list"   and q :: "nat"   and aaa :: "cell list"   and ires :: "cell list" assumes "at_begin_fst_awtn (as, am) (q, aaa, []) ires" shows "at_begin_fst_awtn (as, am) (q, aaa, [Bk]) ires"
assms_1: shows "a \<in>\<^sub>\<circ> \<Union>\<^sub>\<circ> A"
insertion_pow: fixes f :: "nat \<Rightarrow> 'a"   and p :: "'a mpoly"   and i :: "nat" shows "insertion f (p ^ i) = insertion f p ^ i"
Fvars_cnj: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "Fvars (cnj \<phi> \<chi>) = Fvars \<phi> \<union> Fvars \<chi>"
le_setdistI: fixes s :: "'b set"   and t :: "'b set"   and d :: "real" assumes "s \<noteq> {}"   and "t \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> s; y \<in> t\<rbrakk> \<Longrightarrow> d \<le> dist x y" shows "d \<le> setdist s t"
mset_runs: fixes xs :: "'b list" shows "mset_mset (runs xs) = mset xs"
d\<^sub>\<kappa>_def: shows "Semantics.d\<^sub>\<kappa> \<equiv> map_fun eval\<kappa> id id"
SFImplWF: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> SF(A)_v \<longrightarrow> WF(A)_v"
unfold: shows "endswith = (\<Lambda> pat. selem\<cdot>pat oo stails)"
IH: assumes "n \<in> A"   and "n < length xs" shows "nths xs A ! card {i. i < n \<and> i \<in> A} = xs ! n"
no_0_in_smods_ext: fixes p :: "real poly"   and q :: "real poly" shows "0 \<notin> set (smods_ext p q)"
leof_lift: fixes m :: "'b nres"   and F :: "'b nres" assumes "m \<le> F" shows "m \<le>\<^sub>n F"
sub_inc_sysI: fixes \<U> :: "'a set"   and \<A> :: "'a set multiset" assumes "incidence_system \<U> \<A>"   and "\<U> \<subseteq> \<V>\<^sup>+"   and "\<A> \<subseteq># \<B>\<^sup>+" shows "sub_incidence_system \<U> \<A> \<V>\<^sup>+ \<B>\<^sup>+"
sup_cong: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier L"   and "B \<subseteq> carrier L"   and "A {.=} B" shows "\<Squnion>A .= \<Squnion>B"
WTrt_binop_THE: fixes P :: "'b prog"   and T1 :: "ty"   and bop :: "bop"   and T2 :: "ty"   and T :: "ty" assumes "P \<turnstile> T1\<guillemotleft>bop\<guillemotright>T2 : T" shows "The (WTrt_binop P T1 bop T2) = T"
map_id: fixes t :: "'c dual" shows "map_dual id t = t"
Sup_aform_err_def: fixes p :: "nat"   and Rd :: "(real \<times> real pdevs) \<times> real" shows "Sup_aform_err p Rd = float_of (truncate_up p (Sup_aform' p (fst Rd) + \<bar>snd Rd\<bar>))"
decrnum: fixes t :: "Ferrack.num"   and x :: "real"   and bs :: "real list" assumes "numbound0 t" shows "Inum (x # bs) t = Inum bs (decrnum t)"
assms_1: shows "constructive_security_aux_obsf real_resource ideal_resource sim (\<I>_real1 \<oplus>\<^sub>\<I> \<I>_real2) (\<I>_ideal1 \<oplus>\<^sub>\<I> \<I>_ideal2) \<I>_common adv"
class_ordered_semiring: fixes gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'c"   and b :: "'d" assumes "order_pair gt d" shows "ordered_semiring (class_ordered_semiring TYPE('c) gt b)"
preserves_hseq: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (F \<mu>) \<and> D.arr (F \<nu>) \<and> src\<^sub>D (F \<mu>) = trg\<^sub>D (F \<nu>)"
simps_1: fixes dims :: "nat list"   and P :: "complex mat" shows "state_sig.wlp dims SKIP P = P"
Uniq_def: fixes P :: "'g \<Rightarrow> bool" shows "Uniq P \<equiv> \<forall>x y. P x \<longrightarrow> P y \<longrightarrow> y = x"
ran_CompFun: fixes HA :: "('a, 'b, 'c) hierauto" shows "\<Union> (ran (CompFun HA)) = SAs HA - {Root (SAs HA) (CompFun HA)}"
acc_acom: assumes "acc {(a, a'). a \<sqsubseteq> a'}" shows "acc {(c, c'). c \<sqsubseteq> c'}"
Restrp_well_founded: shows "well_founded UNIV ((\<sqsubset>) \<restriction> A \<restriction> UNIV)"
E23: fixes F :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool" shows "|~ \<circle>F \<longrightarrow> \<diamond>F"
antipar_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "trg g = src f"
convex_cone_hull: fixes S :: "'a set" assumes "convex S" shows "convex (cone hull S)"
P_disjoint: fixes xs :: "'a list"   and v :: "'a" assumes "xs \<in> paths"   and "xs \<noteq> P"   and "v \<in> set xs"   and "v \<in> set P" shows "v = v0 \<or> v = v1"
simps_2: fixes l :: "dag"   and a :: "ref"   and r :: "dag" shows "root (Node l a r) = a"
map_sum_sel_2: fixes s :: "'a + 'b"   and f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c" assumes "\<not> isl s" shows "projr (map_sum f g s) = g (projr s)"
Polys_closed_normal_form: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "F \<subseteq> P[X]"   and "p \<in> P[X]" shows "normal_form F p \<in> P[X]"
ide_cod: fixes f :: "'d \<times> 'c" assumes "DopxC.arr f" shows "DopxC.ide (DopxC.cod f)"
suicide_aux_r: fixes Y :: "'a list set"   and X :: "'a list set"   and n :: "nat" assumes "\<forall>w\<in>Y. 0 \<le> |w|"   and "\<forall>w\<in>X\<^bsup>Suc n\<^esup>. n \<le> |w|" shows "\<forall>w\<in>Y \<cdot> X\<^bsup>Suc n\<^esup>. n \<le> |w|"
pset_deriv: fixes a :: "'a"   and r :: "'a rexp" shows "pset (deriv a r) = pderiv a r"
carrier_matD_2: fixes A :: "'d mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "dim_col A = nc"
pref_cancel': fixes u :: "'b list"   and v :: "'b list"   and z :: "'b list" assumes "u \<le>p v" shows "z \<cdot> u \<le>p z \<cdot> v"
find_zip_1: fixes cond :: "'a \<Rightarrow> bool"   and p :: "'a"   and v :: "'a list"   and s :: "'a"   and x :: "'a"   and y :: "'a" assumes "find (cond \<circ> snd) (zip (p # v) (v @ [s])) = Some (x, y)"   and "v \<noteq> []" shows "cond y"
bibd_block_transformations_axioms: shows "bibd_block_transformations \<V> \<B> \<k> \<Lambda> bl"
circuit_in_min_dep_in: fixes \<E> :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "\<E> \<subseteq> carrier"   and "circuit_in \<E> X"   and "x \<in> X" shows "indep_in \<E> (X - {x})"
of_int_add: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w + z) = plus (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
label_stab_map_W_fundchamber_image: fixes \<phi> :: "'a \<Rightarrow> 'a"   and w0 :: "'a permutation" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0"   and "w0 \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>" shows "(\<lambda>v. {w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> v = v}) ` w0 `\<rightarrow> C0 = dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})) {w0}"
validFrom_def: fixes s :: "'state"   and tr :: "('state, 'act, 'out) trans list" shows "validFrom s tr \<equiv> tr = [] \<or> valid tr \<and> srcOf (hd tr) = s"
identity_\<nu>_def: shows "(\<^bold>=) \<equiv> \<lambda>x y. x\<^sup>P \<^bold>= y\<^sup>P"
ordIso_reflexive: fixes r :: "('a \<times> 'a) set" assumes "Well_order r" shows "r =o r"
\<psi>_in_hom: fixes f :: "'c" assumes "C.ide f" shows "\<guillemotleft>\<tau>.\<psi> f : F f \<Rightarrow>\<^sub>D \<sigma>'.map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f)\<guillemotright>"
F: fixes n :: "nat" shows "F n \<noteq> [] \<and> sorted_wrt (\<subseteq>) (F n) \<and> set (F n) \<subseteq> {S. infinite S} \<and> set (F n) \<subseteq> Pow M \<and> Inf ` set (F n) \<subseteq> M"
eq: shows "(\<Prod>i\<in>UNIV - {k}. (if i = k then c *s a i else b i) $ p i) = (\<Prod>i\<in>UNIV - {k}. (if i = k then a i else b i) $ p i)"
isin_set_inorder: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and t :: "'b btree"   and x :: "'b" assumes "split split"   and "sorted_less (inorder t)" shows "split.isin split t x = (x \<in> set (inorder t))"
costBarrierDecreases': fixes a :: "(signs \<times> predicate \<times> vbl list) list"   and num :: "nat"   and fm :: "formula"   and list :: "(nat \<times> formula) list"   and iA :: "nat \<times> formula"   and fSucn :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list" shows "\<not> SATAxiom (sequent (a, (num, fm) # list)) \<longrightarrow> iA \<noteq> (num, fm) \<longrightarrow> \<not> proofTree (tree subs (a, (num, fm) # list)) \<longrightarrow> fSucn \<in> subs (a, (num, fm) # list) \<longrightarrow> iA \<in> set list \<longrightarrow> costBarrier iA fSucn < costBarrier iA (a, (num, fm) # list)"
sprod_mHom_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'j) Module_scheme"   and a :: "'g"   and f :: "'e \<Rightarrow> 'f" shows "sprod_mHom R M N a f = (\<lambda>x\<in>carrier M. a \<cdot>\<^sub>s\<^bsub>N\<^esub> f x)"
assms_1: shows "\<rho>\<^sub>1 \<in> carrier_mat d d"
mem_Joints_msum_aforms'I: fixes a :: "real list"   and x :: "(real \<times> real pdevs) list"   and b :: "real list"   and y :: "(real \<times> real pdevs) list" assumes "a \<in> Joints x"   and "b \<in> Joints y" shows "map2 (+) a b \<in> Joints (msum_aforms' x y)"
diverge_lfilter_LNil: fixes xs :: "'b llist"   and P :: "'b \<Rightarrow> bool" assumes "\<forall>x\<in>lset xs. \<not> P x" shows "lfilter P xs = LNil"
inj_semigroup_add_hom_axioms: shows "inj_semigroup_add_hom hom"
cases: fixes a1 :: "nat"   and a2 :: "'a set"   and a3 :: "'a set" assumes "dimension a1 a2 a3" obtains K :: "'a set" where "a1 = 0" and "a2 = Kb__" and "a3 = {\<zero>}"   | v :: "'a" and E :: "'a set" and n :: "nat" and K :: "'a set" where "a1 = Suc n" and "a2 = K" and "a3 = line_extension K v E" and "v \<in> carrier R" and "v \<notin> E" and "dimension n K E"
ordinal_oLog_exp: fixes b :: "ordinal"   and x :: "ordinal" assumes "oSuc 0 < b" shows "oLog b (b ** x) = x"
weakly_logically_equivalent_def: fixes P :: "'state"   and Q :: "'state" shows "P \<equiv>\<cdot> Q \<equiv> \<forall>x. weak_formula x \<longrightarrow> P \<Turnstile> x = Q \<Turnstile> x"
assms_2: shows "g \<in> S \<rightarrow> T"
DDH_bound2: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "\<bar>spmf (DDH0 \<A>) True - spmf (DDH1 \<A>) True\<bar> \<le> advantage \<A> + \<bar>spmf (ddh_1 \<A>) True - spmf (DDH1 \<A>) True\<bar>"
is_aligned_no_overflow': fixes p :: "'b word"   and n :: "nat" assumes "is_aligned p n" shows "p \<le> p + (2 ^ n - 1)"
take_bin2bl_lem1: fixes m :: "nat"   and w :: "int"   and bs :: "bool list" shows "take m (bin_to_bl_aux m w bs) = bin_to_bl m w"
alg_mult_gt_zero_iff_is_root: fixes p :: "'a list"   and x :: "'a" assumes "p \<in> carrier (poly_ring R)" shows "(0 < alg_mult p x) = is_root p x"
Pg_def: fixes K :: "('c, 'd) Ring_scheme"   and v :: "'c \<Rightarrow> ant" shows "Pg K v = (SOME x. x \<in> carrier K - {\<zero>\<^bsub>K\<^esub>} \<and> v x = Lv K v)"
\<psi>_norm: shows "vec_norm \<psi> = 1"
fresh_ty_dom_cons: fixes X :: "tyvrs"   and Y :: "binding"   and \<Gamma> :: "binding list" shows "X \<sharp> ty_dom (Y # \<Gamma>) = (X \<sharp> tyvrs_of Y \<and> X \<sharp> ty_dom \<Gamma>)"
converse_rtrancl_induct2: fixes ax :: "'a"   and ay :: "'b"   and bx :: "'a"   and "by" :: "'b"   and r :: "(('a \<times> 'b) \<times> 'a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "((ax, ay), bx, by) \<in> r\<^sup>*"   and "P bx by"   and "\<And>a b aa ba. \<lbrakk>((a, b), aa, ba) \<in> r; ((aa, ba), bx, by) \<in> r\<^sup>*; P aa ba\<rbrakk> \<Longrightarrow> P a b" shows "P ax ay"
cases: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'a option"   and a2 :: "'b option" assumes "ord_option ord a1 a2" obtains x :: "'b option" where "a1 = None" and "a2 = x__"   | x :: "'a" and y :: "'b" where "a1 = Some x" and "a2 = Some y" and "ord x y"
mset_extend: fixes S :: "('a, 'b) sequent"   and c :: "('a, 'b) sequent" shows "ModalSequents.mset (extend S c) = ModalSequents.mset S + ModalSequents.mset c"
le_iff_add: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq a b = (\<exists>c. b = plus a c)"
inv_comm: fixes x :: "'b"   and y :: "'b" assumes "x \<otimes>\<^bsub>G i\<^esub> y = \<one>\<^bsub>G i\<^esub>"   and "x \<in> carrier (G i)"   and "y \<in> carrier (G i)" shows "y \<otimes>\<^bsub>G i\<^esub> x = \<one>\<^bsub>G i\<^esub>"
uniq: fixes x' :: "real" assumes "0 \<le> x'"   and "x' \<le> pi"   and "cos x' = y" shows "x' = x"
adjugate_mult_det: fixes A :: "('c, 'd) sq_matrix" shows "adjugate A * A = diag (Square_Matrix.det A)"
lexord_intro_elems: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'b list"   and R :: "('b \<times> 'b) set" assumes "i < length xs"   and "i < length ys"   and "(xs ! i, ys ! i) \<in> R"   and "\<forall>j<i. xs ! j = ys ! j" shows "(xs, ys) \<in> lexord R"
simps_2: fixes d :: "int"   and i :: "int"   and ks :: "int list" shows "divisor (Dvd d i ks) = d"
poly_mapping_induct: fixes P :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and f :: "'a \<Rightarrow>\<^sub>0 'b" assumes "\<And>k v. P (Poly_Mapping.single k v)"   and "\<And>f g k v. \<lbrakk>P f; P g; g = Poly_Mapping.single k v; k \<notin> keys f\<rbrakk> \<Longrightarrow> P (f + g)" shows "P f"
fold_empty: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c" shows "DList_Set.fold f DList_Set.empty b = b"
simps_2: fixes m :: "nat" shows "ack (Suc m) 0 = ack m 1"
eqci_def: fixes a :: "'a"   and b :: "'a" shows "eqci a b \<equiv> return (a = b)"
rhsnot0: shows "\<Lambda> * (\<v> - 1) \<noteq> 0"
lookup_pair_inj: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys"   and "lookup_pair_tc xs = lookup_pair_tc ys" shows "xs = ys"
diff_gt_0_iff_gt_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "(0 < a - b) = (a = \<top> \<and> b = \<top> \<or> b < a)"
less_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f < g) = (f \<le> g \<and> \<not> g \<le> f)"
AndE2: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F \<^bold>\<and> G" shows "\<Gamma> \<turnstile> G"
terminates_Normal_Call_body: fixes p :: "'d"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e" assumes "p \<in> dom \<Gamma>" shows "\<Gamma>\<turnstile>Call p \<down> Normal s = \<Gamma>\<turnstile>the (\<Gamma> p) \<down> Normal s"
natural_eqI: fixes m :: "natural"   and n :: "natural" assumes "nat_of_natural m = nat_of_natural n" shows "m = n"
Int_absorb2: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "A \<inter> B = A"
g1: shows "L\<lbrakk>Vx : A \<turnstile> e1 : B\<rbrakk> \<rightarrow> IMor g1"
transE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "(a, b) \<in> r"   and "(b, c) \<in> r" shows "(a, c) \<in> r"
prod_relE: fixes p :: "'a \<times> 'b"   and p' :: "'c \<times> 'd"   and R1 :: "('a \<times> 'c) set"   and R2 :: "('b \<times> 'd) set" assumes "(p, p') \<in> R1 \<times>\<^sub>r R2" obtains a :: "'a" and b :: "'b" and a' :: "'c" and b' :: "'d" where "p = (a, b)" and "p' = (a', b')" and "(a, a') \<in> R1" and "(b, b') \<in> R2"
indep_inI: fixes X :: "'a set"   and \<E> :: "'a set" assumes "X \<subseteq> \<E>"   and "indep_in \<E> X" shows "\<E>.indep_in \<E> X"
dual: shows "class.linorder (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x)"
ord_le_eq_trans: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a \<preceq> b"   and "b = c" shows "a \<preceq> c"
elementLastTrailIsLastElementsTrail: fixes M :: "('b \<times> bool) list" assumes "M \<noteq> []" shows "element (last M) = last (elements M)"
CS: shows "- S = {x. dist a x \<le> r \<and> x \<notin> S} \<union> {x. r \<le> dist a x \<and> x \<notin> S}"
Var_def: shows "Var \<equiv> map_fun id MPoly Var\<^sub>0"
visits_via_def: fixes P :: "'c llist"   and A :: "'c set"   and W :: "'c set" shows "visits_via P A W \<equiv> \<exists>n. enat n < llength P \<and> P $ n \<in> W \<and> lset (ltake (enat n) P) \<subseteq> A"
type_fresh_subst: fixes x :: "Syntax.tvar"   and t :: "ty"   and s :: "ty"   and y :: "Syntax.tvar" assumes "atom x \<sharp> t"   and "atom x \<sharp> s" shows "atom x \<sharp> subst_type t s y"
alts'_def: shows "alts' = {a, b, c, d}"
b_dest: assumes "- b \<in> Basis" shows "b = - b' \<and> b' \<in> Basis"
l_comb_mem: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and A :: "'f set"   and H :: "'e set"   and s :: "nat \<Rightarrow> 'f"   and n :: "nat"   and m :: "nat \<Rightarrow> 'e" assumes "R module M"   and "ideal R A"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "m \<in> {j. j \<le> n} \<rightarrow> H" shows "l_comb R M n s m \<in> carrier M"
lm008: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" shows "P +< Q = (P outside Domain Q) +< Q"
SeqSubstFormP_Ex: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and k1 :: "tm"   and x :: "tm"   and x' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, s1, k1, x, x', v, i)"   and "atom k \<sharp> (s1, k1, x, x', v, i)" shows "{SeqSubstFormP v i x x' s1 k1, TermP i, VarP v} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqSubstFormP v i (Q_Ex x) (Q_Ex x') (Var s) (Var k)))"
conj_ac_2: fixes a :: "'c"   and b :: "'c" shows "a AND b = b AND a"
TC_lift: fixes y :: "V"   and x :: "V" assumes "y \<noteq> 0" shows "TC (lift x y) = TC x \<squnion> lift x (TC y)"
fmdom_drop: fixes a :: "'c"   and m :: "('c, 'd) fmap" shows "fmdom (fmdrop a m) = fmdom m |-| {|a|}"
ESem_fresh_cong: fixes e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> 'Value"   and \<rho>' :: "Vars.var \<Rightarrow> 'Value" assumes "\<rho> f|` fv e = \<rho>' f|` fv e" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub>"
chain_def: fixes w :: "nat \<Rightarrow> 'a list" shows "chain w \<equiv> incseq w \<and> (\<forall>k. \<exists>l. k < length (w l))"
less_le_not_le: fixes xa :: "'o"   and x :: "'a"   and y :: "'a" shows "lt xa x y = (le xa x y \<and> \<not> le xa y x)"
has_integral_affinity: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and a :: "'b"   and b :: "'b"   and m :: "real"   and c :: "'b" assumes "(f has_integral i) (cbox a b)"   and "m \<noteq> 0" shows "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral (1 / \<bar>m\<bar> ^ DIM('b)) *\<^sub>R i) ((\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b)"
param_Let: fixes Ra :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set" shows "(Let, Let) \<in> Ra \<rightarrow> (Ra \<rightarrow> Rr) \<rightarrow> Rr"
quasi_inversesI: fixes f :: "'b"   and g :: "'b" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f \<cong>\<^sub>D g \<star>\<^sub>D f"   and "f \<star>\<^sub>D g \<cong>\<^sub>D trg\<^sub>D f" shows "D.quasi_inverses f g"
distinct_1: fixes x1 :: "rat"   and x21 :: "rat"   and x22 :: "rat" shows "Rat_Info x1 \<noteq> Sqrt_Info x21 x22"
f_Exec_Stream_Acc_LocalState_one: fixes k :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'f message_af"   and c :: "'e" assumes "0 < k" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun [x] c = [localState (f_Exec_Comp trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) c)]"
iso_assoc': fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "local.iso \<a>\<^sup>-\<^sup>1[f, g, h]"
dual_transaction_strand: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" assumes "wellformed_transaction T" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_strand T) = dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T) @ transaction_selects T @ transaction_checks T @ transaction_updates T @ dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T)"
finfun_apply_inject: fixes x :: "'c \<Rightarrow>f 'd"   and y :: "'c \<Rightarrow>f 'd" shows "(finfun_apply x = finfun_apply y) = (x = y)"
legs_form_cospan_1: shows "C.cospan f.leg0 src.leg1"
list_all2_NilL: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "list_all2 R [] ys = (ys = [])"
map_fst_zip: fixes xs :: "'e list"   and ys :: "'f list" assumes "length xs = length ys" shows "map fst (zip xs ys) = xs"
fresh: shows "dom (share_all_until_volatile_write ts (share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<S>)) - dom (share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<S>) \<subseteq> (\<Union>(uu_, uu_, uu_, uu_, uu_, \<O>, \<R>)\<in>set ts. \<O>)"