hcju/isabelleps huggingface dataset:
approx_form_approx_form': fixes prec :: "nat"   and f :: "form"   and s :: "nat"   and n :: "nat"   and ivl :: "float interval"   and bs :: "float interval option list"   and ss :: "nat list"   and x :: "real" assumes "approx_form' prec f s n ivl bs ss"   and "x \<in>\<^sub>r ivl" obtains ivl' :: "float interval" where "x \<in>\<^sub>r ivl'" and "approx_form prec f (bs[n := Some ivl']) ss"
subgroup_generated_subset_carrier_subset: fixes S :: "'c set" assumes "S \<subseteq> carrier H" shows "S \<subseteq> carrier (subgroup_generated H S)"
union_of_def: fixes P :: "'b set set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> bool" shows "P union_of Q \<equiv> \<lambda>S. \<exists>\<U>. P \<U> \<and> \<U> \<subseteq> Collect Q \<and> \<Union> \<U> = S"
simps: fixes v :: "'vertex"   and p :: "('form, 'var) out_port" shows "valid_out_port (v, p) = (v |\<in>| vertices \<and> p |\<in>| outPorts (nodeOf v))"
rev_subsetD: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
support_single2: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "support (\<lambda>l. P) = {}"
rev_image_eqI: fixes x :: "'x"   and A :: "'x set"   and b :: "'y"   and f :: "'x \<Rightarrow> 'y" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
le_iff_sup: fixes x :: "'c multiset"   and y :: "'c multiset" shows "(x \<subseteq># y) = (x \<union># y = y)"
eq_refl: fixes x :: "nat list"   and y :: "nat list" assumes "x = y" shows "x \<le>\<^sub>v y"
le_approx_implies_le_ref: fixes P :: "'c process"   and Q :: "'c process" assumes "P \<sqsubseteq> Q" shows "P \<le> Q"
scaleC_minus1_left_vec: fixes v :: "'b vec" shows "- (1::'b) \<cdot>\<^sub>v v = - v"
is_strong_nat_string_simps_1: shows "\<not> is_strong_nat_string []"
star_plus_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.B2_algebra plus times one zero less_eq less star" shows "plus one (star x) = star x"
SAT: fixes n :: "nat" shows "B \<Gamma> \<inter> \<^bold>V - {b} \<subseteq> SAT \<Gamma> (h n)"
gen_preserved1: fixes Xs :: "W.tvar list"   and \<theta> :: "(W.tvar \<times> ty) list"   and T :: "ty" assumes "Xs \<sharp>* \<theta>" shows "\<theta><gen T Xs> = gen (\<theta><T>) Xs"
invertible_column_add: fixes a :: "'b"   and b :: "'b"   and q :: "'c" assumes "a \<noteq> b" shows "invertible (column_add (Finite_Cartesian_Product.mat (1::'c)) a b q)"
f: shows "continuous_on (Sigma A B) (\<lambda>(a, b). f a b)"
toplevel_inters_wf: fixes n :: "nat"   and s :: "'b rexp" shows "local.wf n s = (\<forall>r\<in>toplevel_inters s. local.wf n r)"
ax1: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> F"
lex_pm_trans: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd"   and u :: "'c \<Rightarrow>\<^sub>0 'd" assumes "lex_pm s t"   and "lex_pm t u" shows "lex_pm s u"
FAssRedsThrow2: fixes e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(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"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "expr"   and r :: "nat \<times> char list list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Throw r,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>1\<bullet>F{Cs} := e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw r,s\<^sub>2\<rangle>"
strip_while_Nil: fixes P :: "'a \<Rightarrow> bool" shows "strip_while P [] = []"
locally_compact_homeomorphism_projection_closed: fixes S :: "'a set" assumes "locally compact S" obtains T :: "('a \<times> 'b) set" and f :: "'a \<Rightarrow> 'a \<times> 'b" where "closed T" and "homeomorphism S T f fst"
3: shows "\<r>[Right.cod \<mu>] \<in> Right.hom (cod \<mu> \<star> src \<mu>) (cod \<mu>)"
in_hhomE: fixes \<mu> :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>" obtains "D.arr \<mu>" and "D.obj a" and "D.obj b" and "src\<^sub>D \<mu> = a" and "trg\<^sub>D \<mu> = b"
ediv_enat: fixes a :: "nat"   and k :: "nat" shows "ediv (enat a) k = enat (mydiv a k)"
finite_Image: fixes A :: "'b set"   and R :: "('b \<times> 'c) set" assumes "finite A" shows "finite (R `` A) = (\<forall>x\<in>A. finite {y. (x, y) \<in> R})"
linear_surjective_isomorphism: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "surj f"   and "vec.dim UNIV = vec.dim UNIV" shows "\<exists>f'. Vector_Spaces.linear (*s) (*s) f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
field_differentiable_linear: fixes c :: "'a"   and F :: "'a filter" shows "(*) c field_differentiable F"
D_ASSIGN: fixes e :: "Expr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var" assumes "Expr_low e" shows "(G, Assign x e, Sec (\<lambda>(s, t, \<beta>). \<exists>r. s = (update (fst r) x (evalE e (fst r)), snd r) \<and> r \<equiv>\<^sub>\<beta> t)) \<in> Deriv"
rel_alt: shows "rel = (\<Union>(t, rel)\<in>set aux'. if left I \<le> \<tau> \<sigma> ne - t then lookup rel mr else empty_table)"
pdivmod_monic_i_def: fixes ops :: "'b arith_ops_record"   and cf :: "'b list"   and cg :: "'b list" shows "pdivmod_monic_i ops cf cg \<equiv> case divmod_poly_one_main_i ops [] (rev cf) (rev cg) (1 + length cf - length cg) of (q, r) \<Rightarrow> (poly_of_list_i ops q, poly_of_list_i ops (rev r))"
simps_2: fixes x :: "'a"   and l :: "'a tree"   and r :: "'a tree"   and h :: "nat" shows "erase (MKT x l r h) = MKT\<^sub>0 x (erase l) (erase r)"
Goal: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q"   and "Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> R"   and "P \<approx> Q" shows "P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> R"
mat_on_ket: fixes A :: "complex mat"   and v :: "complex vec" assumes "dim_col A = dim_vec v" shows "A * |v\<rangle> = |col (A * |v\<rangle>) 0\<rangle>"
rep_eq: fixes x :: "\<Pi>\<^sub>1"   and xa :: "\<kappa>" shows "eval\<o> \<lparr>x,xa\<rparr> = (\<lambda>s w. proper xa \<and> eval\<Pi>\<^sub>1 x (\<nu>\<upsilon> (rep xa)) s w)"
vwb_region: fixes S :: "<'d, 'e> \<Longleftrightarrow> 'f" assumes "sym_lens S" shows "vwb_lens \<V>\<^bsub>S\<^esub>"
eq_iff: fixes x :: "'a poly poly"   and y :: "'a poly poly" shows "(poly_y_x x = poly_y_x y) = (x = y)"
absolutely_integrable_mult_Fejer_kernel_reflected_part2: fixes f :: "real \<Rightarrow> real"   and d :: "real"   and n :: "nat"   and t :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x"   and "d \<le> pi" shows "(\<lambda>x. Fejer_kernel n x * f (t - x)) absolutely_integrable_on {0..d}"
star_prod_unfold_le: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y)\<^sup>\<star> \<le> (1::'a) + x \<cdot> (y \<cdot> x)\<^sup>\<star> \<cdot> y"
fps_mult_fps_X_plus_1_nth: fixes a :: "'b fps"   and n :: "nat" shows "((1 + fps_X) * a) $ n = (if n = 0 then a $ n else a $ n + a $ (n - 1))"
le_refl_Integer: fixes x :: "Integer" assumes "x \<noteq> \<bottom>" shows "le\<cdot>x\<cdot>x = TT"
b2: shows "q2 = 0 \<or> b2 \<in> sqrt_irrat"
abs_lcm_int: fixes i :: "int"   and j :: "int" shows "\<bar>lcm i j\<bar> = lcm i j"
le_supI1: fixes x :: "'c multiset"   and a :: "'c multiset"   and b :: "'c multiset" assumes "x \<subseteq># a" shows "x \<subseteq># a \<union># b"
sq: fixes p :: "rat poly" assumes "p \<in> set fs" shows "square_free p"
comm_semiring_0_cancel_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.comm_semiring_0_cancel plus minus zero times \<equiv> class.cancel_comm_monoid_add plus minus zero \<and> class.comm_semiring plus times"
case_k: shows "\<forall>u<d * k. \<forall>v<d * k. u * v \<noteq> d * k"
lt_lower: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lt (lower p v) \<preceq>\<^sub>t lt p"
elims: fixes x :: "('b \<times> nat) option"   and xa :: "'b"   and y :: "('b \<times> nat) option" assumes "lock_lock x xa = y" obtains t :: "'b" where "x = None" and "xa = tb__" and "y = \<lfloor>(tb__, 0)\<rfloor>"   | t' :: "'b" and n :: "nat" and t :: "'b" where "x = \<lfloor>(t', n)\<rfloor>" and "xa = t" and "y = \<lfloor>(t', Suc n)\<rfloor>"
simps_1: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and q :: "'a"   and l :: "nat"   and rA :: "'a \<times> 'a mat" shows "sub1 mf q 0 l rA = rA"
norm1_def: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec" shows "norm1 v = (\<Sum>i\<in>UNIV. norm (v $h i))"
GF_advice_minus_subformulas: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and X :: "'b ltln set" assumes "\<psi> \<notin> subformulas\<^sub>\<mu> \<phi>" shows "\<phi>[X - {\<psi>}]\<^sub>\<nu> = \<phi>[X]\<^sub>\<nu>"
a': shows "( \<Gamma> \<Rightarrow>* Compound F Fs, n) \<in> derivable R*"
division_segment_euclidean_size: 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 division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "times (division_segment a) (semiring_1.of_nat one plus zero (euclidean_size a)) = a"
rep_eq: fixes x :: "CoCalls set" shows "Rep_CoCall (coCallsLub x) = \<Union> (Rep_CoCall ` x)"
algebraic_int_times: fixes x :: "'b"   and y :: "'b" assumes "algebraic_int x"   and "algebraic_int y" shows "algebraic_int (x * y)"
assms: shows "is_longest_prefix_routing (r1 # rn # rss)"
inner_eq_zero_abs_BasisI: fixes y :: "'c"   and b :: "'c" assumes "\<bar>y\<bar> \<in> Basis"   and "b \<in> Basis"   and "\<bar>y\<bar> \<noteq> b" shows "y \<bullet> b = 0"
\<Psi>o_in_hom: fixes yx :: "'d \<times> 'c" assumes "DopxC.ide yx" shows "\<guillemotleft>\<Psi>o yx : Hom_DopxG.map yx \<rightarrow>\<^sub>S Hom_FopxC.map yx\<guillemotright>"
prime_divisor_exists: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero"   and "\<not> dvd.dvd times a one" shows "\<exists>b. dvd.dvd times b a \<and> normalization_semidom.prime zero times one normalize b"
num_funs_subst: fixes t :: "('a, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term" shows "num_funs t \<le> num_funs (t \<cdot> \<sigma>)"
hom_closed: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c"   and x :: "'a" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h x \<in> carrier H"
sym_block_mult_one: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "multiplicity bl = 1"
"4_1": fixes x :: "bool \<times> bool"   and xa :: "bool"   and y :: "bool" assumes "x = P x__ y__"   and "(xa, y) = x"   and "\<not> xa"   and "y"   and "\<And>i. \<lbrakk>i < length xs; i < length ys; fst (P (xs ! i) (ys ! i))\<rbrakk> \<Longrightarrow> fst (P' (xs ! i) (ys ! i))"   and "\<And>i. \<lbrakk>i < length xs; i < length ys; snd (P (xs ! i) (ys ! i))\<rbrakk> \<Longrightarrow> snd (P' (xs ! i) (ys ! i))" shows "(fst (lex_ext_unbounded P xs ys) \<longrightarrow> fst (lex_ext_unbounded P' xs ys)) \<and> (snd (lex_ext_unbounded P xs ys) \<longrightarrow> snd (lex_ext_unbounded P' xs ys))"
sign_simps_6: fixes bh :: "'a"   and ah :: "'a"   and cg :: "'a" shows "bh * (ah * cg) = ah * (bh * cg)"
wf_L_Prec: shows "wfP (\<sqsubset>l)"
n'_def: shows "n' = (\<lambda>x l v r _ _. n x l v r)"
execute_preserves_valid: fixes s :: "nat \<Rightarrow> nat option"   and name :: "char list" assumes "s \<in> valid_states"   and "enabled name s" shows "execute name s \<in> valid_states"
assms: shows "class.bezout_ring (*) (1::'a) (+) (0::'a) (-) uminus"
Bseq_NSBseq: fixes X :: "nat \<Rightarrow> 'a" assumes "Bseq X" shows "NSBseq X"
fInf_distr_var: fixes F :: "('d \<Rightarrow> 'c) set"   and g :: "'b \<Rightarrow> 'd" shows "\<Sqinter> F \<circ> g = (\<Sqinter>f\<in>F. f \<circ> g)"
cases: fixes a1 :: "bool stream"   and a2 :: "'marker stream"   and a3 :: "('sequent \<times> 'rule) stream" assumes "follow a1 a2 a3" obtains M' :: "'marker" and Ms :: "'marker stream" and s' :: "'sequent" and steps :: "('sequent \<times> 'rule) stream" and M :: "'marker" and b :: "bool" and s :: "'sequent" and r :: "'rule" and bs :: "bool stream" where "a1 = b ## bs" and "a2 = M ## Ms" and "a3 = (s, r) ## steps" and "M' = shd Ms" and "s' = fst (shd steps)" and "(M, b, M') \<in> \<delta> s r s'" and "follow bs Ms steps"
bisimilar_def: fixes a :: "('b, 'c) Stream"   and b :: "('b, 'd) Stream" shows "(a \<approx> b) = (unstream\<cdot>a = unstream\<cdot>b \<and> a \<noteq> \<bottom> \<and> b \<noteq> \<bottom>)"
Sup_atMost: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and y :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (ord.atMost less_eq y) = y"
refl_onPD: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "refl_onP A r"   and "a \<in> A" shows "r a a"
ImpL: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula"   and \<Delta> :: "'a formula multiset"   and G :: "'a formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "G, \<Gamma> \<Rightarrow> \<Delta>" shows "F \<^bold>\<rightarrow> G, \<Gamma> \<Rightarrow> \<Delta>"
cSUP_least: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and M :: "'d" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) M" shows "less_eq (Sup (f ` A)) M"
Pivot_axioms: shows "Pivot eq_idx_for_lvar pivot_code"
e: shows "on_triple f `` E \<subseteq> edges (S (max i j))"
eq1: shows "eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2"
Int_Union2: fixes B :: "'b set set"   and A :: "'b set" shows "\<Union> B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
add_diff_cancel_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a c) (plus b c) = minus a b"
index_conv_size_if_notin: fixes x :: "'c"   and xs :: "'c list" assumes "x \<notin> set xs" shows "index xs x = length xs"
simps_3: fixes \<phi> :: "'b formula" shows "future_reach (Neg \<phi>) = future_reach \<phi>"
holomorphic_on_id: fixes s :: "complex set" shows "id holomorphic_on s"
NOTLEFT_elim: fixes x :: "name"   and M :: "trm"   and B :: "ty" assumes "x:M \<in> NOTLEFT (NOT B) (\<parallel><B>\<parallel>)" obtains a' :: "coname" and M' :: "trm" where "M = NotL <a'>.M' x" and "fin (NotL <a'>.M' x) x" and "<a'>:M' \<in> \<parallel><B>\<parallel>"
AE_iff_null_sets: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure" shows "almost_everywhere (completion M) P = ({x \<in> space (completion M). \<not> P x} \<in> null_sets (completion M))"
graph_antimono: fixes A :: "('d, 'c) nba"   and w :: "'d stream" shows "decseq (graph A w)"
fresh_Next_def: shows "fresh_Next \<equiv> fresh.Next next default"
card_Inv_mtf2: fixes xs :: "'b list"   and j :: "nat"   and ys :: "'b list"   and i :: "nat" assumes "xs ! j = ys ! 0"   and "j < length xs"   and "dist_perm xs ys" shows "int (card (Inv (swaps [i..<j] xs) ys)) = int (card (Inv xs ys)) - int (j - i)"
inf_tl_def: fixes s :: "nat \<Rightarrow> 'd" shows "inf_tl s \<equiv> \<lambda>i. s (Suc i)"
a_rcos_self: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<in> I +> x"
postdominate_antisym: fixes n' :: "'node"   and n :: "'node" assumes "n' postdominates n"   and "n postdominates n'" shows "n = n'"
span_add_eq2: fixes y :: "'a"   and S :: "'a set"   and x :: "'a" assumes "y \<in> ideal S" shows "(x + y \<in> ideal S) = (x \<in> ideal S)"
ordinal_rec0_mono: fixes s :: "ordinal \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "ordinal"   and y :: "ordinal"   and z :: "'a" assumes "\<forall>p x. x << s p x"   and "x \<le> y" shows "ordinal_rec0 z s lub x << ordinal_rec0 z s lub y"
inorder_bal_tree: fixes n :: "nat"   and t :: "'a tree" assumes "n \<le> size t" shows "inorder (bal_tree n t) = take n (inorder t)"
eq: shows "K1 \<inter> {x. c \<le> x \<bullet> k} = K2 \<inter> {x. c \<le> x \<bullet> k}"
ToTC_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "T.ToTC f \<equiv> if CCC.arr f then T (T (fst f, fst (snd f)), snd (snd f)) else null"
t_wcode_main_first_part_def: shows "t_wcode_main_first_part \<equiv> [(L, 1), (L, 2), (L, 7), (R, 3), (R, 4), (W0, 3), (R, 4), (R, 5), (W1, 6), (R, 5), (R, 13), (L, 6), (R, 0), (R, 8), (R, 9), (Nop, 8), (R, 10), (W0, 9), (R, 10), (R, 11), (W1, 12), (R, 11), (R, t_twice_len + 14), (L, 12)]"
CsafeI: fixes t :: "real"   and x :: "'c" assumes "t \<in> existence_ivl0 x" shows "x \<in> Csafe"
Bal: shows "ins n d x \<langle>l, y, r\<rangle> = Bal t'"
select_convs_1: fixes pverts :: "'a set"   and parcs :: "('a \<times> 'a) set"   and more :: "'b" shows "pverts \<lparr>pverts = pverts, parcs = parcs, \<dots> = more\<rparr> = pverts"
lessThan_Suc_atMost: fixes k :: "nat" shows "{..<Suc k} = {..k}"
mono: shows "\<forall>n m. n < m \<longrightarrow> f n < f m"
distinct_3: fixes x11 :: "trm"   and x12 :: "trm"   and x3 :: "fmla" shows "x11 EQ x12 \<noteq> neg x3"
ins_inorder_pairs: fixes t :: "('a, 'b) rbt"   and x :: "'a"   and v :: "'b" assumes "rbt_sorted t" shows "rbt_in_traverse_pairs (ins x v t) = ordered_insert_pairs x v (rbt_in_traverse_pairs t)"
partial_saturation_def: fixes S :: "'at Literal set set"   and A :: "'at"   and R :: "'at Literal set set" shows "partial_saturation S A R = (\<forall>P1 P2. P1 \<in> S \<longrightarrow> P2 \<in> S \<longrightarrow> redundant (resolvent_upon P1 P2 A) R)"
assms_5: shows "prv (eqv \<phi>1 \<phi>2)"
Unity_in_unity: shows "\<^bold>\<I> \<in> \<I>"
CondTrue: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "s \<in> b"   and "\<Gamma>\<turnstile>c\<^sub>1 \<down> Normal s" shows "\<Gamma>\<turnstile>Cond b c\<^sub>1 c\<^sub>2 \<down> Normal s"
join_compl_top: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "\<^bold>-\<^bsub>L\<^esub> x \<squnion>\<^bsub>L\<^esub> x = \<top>\<^bsub>L\<^esub>"
N_def: shows "N = {s \<in> S. p s = 0} \<union> S2"
hb_consistent_prefix: fixes xs :: "('a \<times> operation) event list"   and i :: "nat" assumes "xs prefix of i" shows "hb.hb_consistent (node_deliver_messages xs)"
hdb_subset_n: shows "set (hd b # a ! hd b) \<subseteq> {0..<n}"
h2c: shows "all_list_constr_R (subsets_smash_R (length qs1) subsets1 subsets2) (length (qs1 @ qs2))"
exec_move_AAssI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e2 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e1 :: "'addr expr1"   and e3 :: "'addr expr1"   and v :: "'addr val" assumes "exec_move ci P t e2 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (e1\<lfloor>e2\<rceil> := e3) h (stk @ [v], loc, length (compE2 e1) + pc, xcp) ta h' (stk' @ [v], loc', length (compE2 e1) + pc', xcp')"
reduction_language_stuttering: fixes u :: "'interpretation stream" assumes "u \<in> S.language" obtains v :: "'interpretation stream" where "v \<in> R.language" and "(!!) u \<approx> (!!) v"
either_belowI: fixes t :: "'b ttree"   and t'' :: "'b ttree"   and t' :: "'b ttree" assumes "t \<sqsubseteq> t''"   and "t' \<sqsubseteq> t''" shows "t \<oplus>\<oplus> t' \<sqsubseteq> t''"
segment_as_ball_2: fixes a :: "'b"   and b :: "'b" shows "{a<--<b} = affine hull {a, b} \<inter> ball ((a + b) /\<^sub>R 2) (norm (b - a) / 2)"
mcont_const: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and luba :: "'d set \<Rightarrow> 'd"   and orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'c" assumes "class.ccpo Sup less_eq less" shows "mcont luba orda Sup less_eq (\<lambda>x. c)"
sets_PiM_I_countable: fixes I :: "'d set"   and E :: "'d \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)"
WT_implies_WTrt: fixes P :: "'addr J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and T :: "ty"   and h :: "'heap" assumes "P,E \<turnstile> e :: T" shows "P,E,h \<turnstile> e : T"
res: shows "local.quadratic_hensel_loop j = (q', S', T', D', H')"
reduction_inv_7: fixes t' :: "db"   and P :: "bool" assumes "DBff \<rightarrow>\<^sub>v t'" shows "P"
simps: fixes x :: "real \<times> real"   and a :: "(real \<times> real) list" shows "ccw'.sortedP x a = (a = [] \<or> (\<exists>ys xa. a = xa # ys \<and> (\<forall>y\<in>set ys. ccw' x xa y) \<and> ccw'.sortedP x ys))"
id_hom: fixes CC :: "('d, 'e, 'f) category_scheme"   and A :: "'d" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>" shows "Id\<^bsub>CC\<^esub> A \<in> Hom\<^bsub>CC\<^esub> A A"
in_enum: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "'a" assumes "class.enum enum enum_all enum_ex" shows "x \<in> set enum"
atoms_list_conjE: fixes a :: "'c"   and fs :: "'c fm list" assumes "a \<in> atoms (list_conj fs)" shows "a \<in> \<Union> (atoms ` set fs)"