hcju/isabelleps huggingface dataset:
of_nat_0_less_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (semiring_1.of_nat one plus zero n) = (0 < n)"
x_y_nonzero: shows "x_y \<noteq> 0"
ext_inject: fixes ok :: "bool"   and more :: "'b"   and ok' :: "bool"   and more' :: "'b" shows "(\<lparr>ok = ok, \<dots> = more\<rparr> = \<lparr>ok = ok', \<dots> = more'\<rparr>) = (ok = ok' \<and> more = more')"
unwinding_implies_NI_indirect_sources: shows "NI_indirect_sources"
sup_commute: fixes x :: "'d"   and y :: "'d" shows "x \<sqinter> y = y \<sqinter> x"
simps_4: fixes v :: "bool" shows "is_Intg (Bool v) = False"
diagonal_step_code: fixes A :: "(('b, 'a) vec, 'c) vec"   and i :: "nat"   and j :: "nat"   and c :: "'b"   and v :: "'b"   and a :: "'c" shows "($) (diagonal_step_row A i j c v a) = (\<lambda>b. if a = from_nat i \<and> b = from_nat i then c else if a = from_nat j \<and> b = from_nat j then v * A $ from_nat j $ from_nat j else A $ a $ b)"
isOpair: fixes x :: "ZF"   and y :: "ZF" shows "isOpair (Opair x y) = True"
shadow_root_ptr_kinds_def: fixes heap :: "(_) heap" shows "shadow_root_ptr_kinds heap = the |`| cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |`| ffilter is_shadow_root_ptr_kind\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (document_ptr_kinds heap)"
cyclic_on_orbit: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" assumes "f permutes S"   and "finite S" shows "cyclic_on f (orbit f x)"
Ker_memI: fixes a :: "'a" assumes "\<eta> a = \<one>'"   and "a \<in> G" shows "a \<in> Ker"
simps_2: fixes v1 :: "'d"   and e :: "'d edge"   and es :: "'d edge list"   and v2 :: "'d" shows "Graph.ces v1 (e # es) v2 = (src e = v1 \<and> Graph.ces (tgt e) es v2)"
root_2: shows "[:- a, 1:] * p \<noteq> 0"
strictly_preserves_T: fixes f :: "'a"   and g :: "'a" assumes "F.C.arr f"   and "F.C.arr g" shows "F (F.C.tensor f g) = F.D.tensor (F f) (F g)"
gt_compat_arg: fixes s' :: "('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary (App s' t)"   and "local.kbo_std.gt s' s" shows "local.kbo_std.gt (App s' t) (App s t)"
countable_finite: fixes S :: "'b set" assumes "finite S" shows "countable S"
IdFtor'FtorM: fixes C :: "('a, 'b, 'c) Category_scheme" assumes "Category C" shows "FunctorM FId' C"
mat_complement_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "f\<^sup>c = (\<lambda>i j. - f i j)"
Val_inv_axioms: shows "Val_inv num_ivl (+) \<gamma>_ivl in_ivl inv_plus_ivl inv_less_ivl"
D: shows "D (ys ! n) \<notin> sinks I D u (take (Suc n) ys)"
ring_simprules_11: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z)"
valid_Not: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and x :: "('d, 'b, 'c) formula" assumes "nominal_ts satisfies transition" shows "nominal_ts.valid satisfies transition P (Formula.Not x) = (\<not> nominal_ts.valid satisfies transition P x)"
asymp_equiv_power: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "f \<sim>[F] g" shows "(\<lambda>x. f x ^ n) \<sim>[F] (\<lambda>x. g x ^ n)"
sel_3: fixes x1 :: "'a extended"   and x2 :: "'a"   and x3 :: "'a extended"   and x4 :: "'b" shows "S (InsertMessage x1 x2 x3 x4) = x3"
cong__le3412: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D" shows "Tarski_neutral_dimensionless.Le Bet Cong C D A B"
assms_1: shows "x \<in> \<langle>\<C>\<rangle>"
filterlim_mono: fixes f :: "'c \<Rightarrow> 'd"   and F2 :: "'d filter"   and F1 :: "'c filter"   and F2' :: "'d filter"   and F1' :: "'c filter" assumes "filterlim f F2 F1"   and "F2 \<le> F2'"   and "F1' \<le> F1" shows "filterlim f F2' F1'"
arctan_eq_Re_Arctan: fixes x :: "real" shows "arctan x = Re (Arctan (complex_of_real x))"
to_bot: shows "\<G>_Fset N - Red_F_G (\<G>_Fset N) \<Turnstile>G \<G>_F B"
ipurge_tr_length: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "length (ipurge_tr I D u xs) \<le> length xs"
box_eq_empty_1: fixes a :: "'a"   and b :: "'a" shows "(box a b = {}) = (\<exists>i\<in>Basis. b \<bullet> i \<le> a \<bullet> i)"
Red_I_without_red_F: fixes N :: "'f set" shows "Red_I (N - Red_F N) = Red_I N"
abs_ccache_def: fixes cc :: "((nat \<times> (nat \<Rightarrow> contour option)) \<times> d) set" shows "|cc| = (\<Union>((c, \<beta>), d)\<in>cc. {((c, |\<beta>|), p) |p. p \<in> |d|})"
qSwapAbs_preserves_alphaAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "qGoodAbs A \<or> qGoodAbs A'"   and "A $= A'" shows "A $[[z1 \<and> z2]]_zs $= A' $[[z1 \<and> z2]]_zs"
monad_reader_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and ask :: "('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_reader return bind ask" shows "monad_reader (return_state return) (bind_state bind) (ask_state ask)"
image_group_carrier: fixes f :: "'d \<Rightarrow> 'c"   and G :: "('d, 'e) monoid_scheme" shows "carrier (image_group f G) = f ` carrier G"
mem_affine_3_minus2: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and v :: "real" assumes "affine S"   and "x \<in> S"   and "y \<in> S"   and "z \<in> S" shows "x - v *\<^sub>R (y - z) \<in> S"
Abs_bd_type_F_cases: fixes x :: "bd_type_F" obtains y :: "bd_type_F1 + bd_type_F2" where "x = Abs_bd_type_F y" and "y \<in> UNIV"
normal_form_minus_normal_form: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "F \<subseteq> P[X]" shows "normal_form F (normal_form F p - normal_form F q) = normal_form F p - normal_form F q"
WT_coreD_cpoke: fixes \<I>_adv :: "('a, 'b) \<I>"   and \<I>_usr :: "('c, 'd) \<I>"   and I :: "'e \<Rightarrow> bool"   and core :: "('e, 'f, 'a, 'c, 'b, 'd) core"   and s' :: "'e"   and s :: "'e"   and e :: "'f" assumes "WT_core \<I>_adv \<I>_usr I core"   and "s' \<in> set_spmf (cpoke core s e)"   and "I s" shows "I s'"
err_semilat_eslI: fixes L :: "'b set \<times> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'b \<Rightarrow> 'b)" assumes "semilat L" shows "err_semilat (Err.esl L)"
exact_sequence_sum_lemma: fixes G :: "'a monoid"   and h :: "'a \<Rightarrow> 'a"   and A :: "'a monoid"   and C :: "'a monoid"   and k :: "'a \<Rightarrow> 'a"   and B :: "'a monoid"   and D :: "'a monoid"   and g :: "'a \<Rightarrow> 'a"   and i :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a"   and j :: "'a \<Rightarrow> 'a" assumes "comm_group G"   and "h \<in> Group.iso A C"   and "k \<in> Group.iso B D"   and "exact_seq ([D, G, A], [g, i])"   and "exact_seq ([C, G, B], [f, j])"   and "\<And>x. x \<in> carrier A \<Longrightarrow> f (i x) = h x"   and "\<And>x. x \<in> carrier B \<Longrightarrow> g (j x) = k x" shows "(\<lambda>(x, y). i x \<otimes>\<^bsub>G\<^esub> j y) \<in> Group.iso (A \<times>\<times> B) G \<and> (\<lambda>z. (f z, g z)) \<in> Group.iso G (C \<times>\<times> D)"
final: shows "dfa.final \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr> \<subseteq> dfa.states \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr>"
fvi_trm_iff_fv_trm: fixes x :: "nat"   and b :: "nat"   and t :: "trm" shows "(x \<in> fvi_trm b t) = (x + b \<in> fv_trm t)"
order_iff: fixes a :: "'d upred"   and b :: "'d upred" shows "(b \<sqsubseteq> a) = (a = (a \<and> b))"
T_def: shows "T = cball \<xi> e \<inter> U"
unique_factorization_m_imp_factorization: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" assumes "p.unique_factorization_m f cfs" shows "p.factorization_m f cfs"
lexordp_eq_refl: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" shows "ord.lexordp_eq less xs xs"
assms_10: shows "sparse_row_matrix A1 \<le> A"
make_vert_equiv_mat: fixes n :: "nat" shows "make_vert_equiv n = mat1 (2 ^ n)"
REP_in_Map: fixes A :: "('a, 'a set) arr" assumes "arr A" shows "REP A \<in> Map A"
trss_stack_decomp_s: fixes s :: "'a list"   and r :: "'a list"   and c :: "'a list multiset"   and e :: "('b, 'c) label"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s @ r, c), e, s', c') \<in> trss fg"   and "s \<noteq> []" shows "\<exists>sp'. s' = sp' @ r \<and> ((s, c), e, sp', c') \<in> trss fg"
asm: shows "\<Gamma> \<turnstile> s \<equiv> t : T"
dgrad_set_zero: fixes m :: "nat" shows "dgrad_set (\<lambda>_. 0) m = UNIV"
rel_fun_refl_eq_onp: fixes f :: "'b \<Rightarrow> 'a"   and X :: "'b set"   and A :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>z. z \<in> f ` X \<Longrightarrow> A z z" shows "rel_fun (eq_on X) A f f"
smap_Union_map_rev: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and l :: "'c list" shows "\<Union>.map f (rev l) = \<Union>.map f l"
compress_top_def: fixes x :: "wall"   and y :: "wall" shows "compress_top x y \<equiv> \<exists>B. x = basic (make_vert_block (nat (domain_wall B))) \<circ> B \<and> y = B \<circ> basic [] \<and> codomain_wall B = 0 \<and> is_tangle_diagram B"
sorted_list_of_multiset_empty: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_multiset less_eq {#} = []"
y_min: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list"   and y' :: "'c" assumes "ProofStepInduct_NonTrivial_P_k_pre G v0 v1 paths P_new sep_size P_k_pre y P_k_post"   and "y' \<in> set P_k_pre" shows "\<not> ProofStepInduct_NonTrivial.hitting_Q_or_new_last G v0 v1 paths P_new sep_size y'"
pigeonhole: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "\<And>i. i \<le> Suc n \<Longrightarrow> f i \<le> n" shows "\<exists>i j. i \<le> Suc n \<and> j < i \<and> f i = f j"
ge: shows "0 \<le> inner_prod v (B *\<^sub>v v)"
Nml_HcompD_4: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Nml V src trg (t \<^bold>\<star> u)" shows "bicategorical_language.Nml V src trg u"
real_of_card: fixes A :: "'a set" shows "real (card A) = (\<Sum>x\<in>A. 1)"
uconnected_mono: fixes A :: "'b uprod set"   and A' :: "'b uprod set" assumes "A \<subseteq> A'" shows "uconnected A \<subseteq> uconnected A'"
prems: shows "P,E,s \<turnstile> e : T \<surd>"
sup_derivable_qfp: fixes p :: "'a" assumes "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"   and "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x"   and "extreme_bound A (\<sqsubseteq>) (Collect derivable) p" shows "f p \<sim> p"
numeral_Bit1_div_2: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and n :: "num" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 num.One)) = numeral.numeral one plus n"
that: fixes t :: "'a list" assumes "x \<in> t*"   and "y \<in> t*" shows "thesis"
fls_base_factor_X_inv: shows "fls_base_factor fls_X_inv = 1"
intro: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" assumes "galois_connection G"   and "coretract_axioms G" shows "coretract G"
finite_set_intervals2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and x :: "'b"   and l :: "'b"   and u :: "'b"   and S :: "'b set" assumes "class.linorder less_eq less"   and "P x"   and "less_eq l x"   and "less_eq x u"   and "l \<in> S"   and "u \<in> S"   and "finite S"   and "\<forall>x\<in>S. less_eq l x"   and "\<forall>x\<in>S. less_eq x u" shows "(\<exists>s\<in>S. P s) \<or> (\<exists>a\<in>S. \<exists>b\<in>S. (\<forall>y. less a y \<and> less y b \<longrightarrow> y \<notin> S) \<and> less a x \<and> less x b \<and> P x)"
convex_affine_hull: fixes S :: "'c set" shows "convex (affine hull S)"
infinite_iff_countable_subset: fixes S :: "'a set" shows "infinite S = (\<exists>f. inj f \<and> range f \<subseteq> S)"
long_ch_Y: shows "[f[a\<^sub>1 .. a ..  a\<^sub>n]Y]"
take_cols_pivot_eq: fixes A :: "'b mat" assumes "row_echelon_form A"   and "dim_row A \<le> dim_col A" shows "take_cols A (map snd (pivot_positions A)) = 1\<^sub>m (length (pivot_positions A)) @\<^sub>r 0\<^sub>m (dim_row A - length (pivot_positions A)) (length (pivot_positions A))"
g3: shows "|~ F \<longrightarrow> \<circle>F"
null_sets_completionI: fixes N :: "'a set"   and M :: "'a measure" assumes "N \<in> null_sets M" shows "N \<in> null_sets (completion M)"
replay''_code: fixes thy :: "exetheory"   and vs :: "(variable \<times> typ) list"   and ns :: "variable set"   and Hs :: "Core.term list"   and P :: "proofterm" assumes "exe_wf_theory' thy" shows "replay'' (translate_theory thy) vs ns Hs P = exereplay'' thy vs ns Hs P"
f_1: fixes epsilon :: "real" assumes "0 < epsilon" shows "f epsilon \<in> sets M"
foo_axioms: shows "foo f a b c project_'a inject_'a project_Nat_nat_Nat_nat_fun inject_Nat_nat_Nat_nat_fun project_Int_int inject_Int_int project_Nat_nat inject_Nat_nat"
enat_ex_split: fixes P :: "enat \<Rightarrow> bool" shows "(\<exists>c. P c) = (P \<infinity> \<or> (\<exists>x. P (enat x)))"
partial_order_onD_3: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "partial_order_on A r" shows "antisym r"
IH_2: fixes y :: "'a ifex" assumes "ro_ifex xb2"   and "ro_ifex y"   and "\<And>ass. val_ifex xb2 ass = val_ifex y ass" shows "xb2 = y"
simps_1: fixes ivl1 :: "float interval"   and ivl2 :: "float interval"   and f :: "float interval \<Rightarrow> float interval \<Rightarrow> float interval option" shows "lift_bin (Some ivl1) (Some ivl2) f = f ivl1 ivl2"
simps_15: fixes x21 :: "'a"   and x22 :: "'a list" shows "list.set (x21 # x22) = insert x21 (list.set x22)"
not_in_affected_iff_orbit_eq_singleton: fixes a :: "'a"   and f :: "'a perm" shows "(a \<notin> affected f) = (orbit f a = {a})"
\<iota>_in_hom': shows "\<guillemotleft>\<iota>\<^sub>D : D.cod \<iota>\<^sub>D \<otimes>\<^sub>D D.cod \<iota>\<^sub>D \<rightarrow>\<^sub>D D.cod \<iota>\<^sub>D\<guillemotright>"
prec: shows "- log 2 x \<le> real_of_int p"
LLL_f_bound: fixes i :: "nat"   and j :: "nat" assumes "i < m"   and "j < n" shows "\<bar>fs ! i $ j\<bar> \<le> int (N ^ m * 2 ^ (m - 1) * m)"
induct: fixes P1 :: "tm \<Rightarrow> bool"   and P2 :: "tm list \<Rightarrow> bool"   and tm :: "tm" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (Fn x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 tm"
b_not0: shows "gcd a1 a2 \<noteq> 0"
wf_marking_def: fixes t :: "dag"   and mark_old :: "ref \<Rightarrow> bool"   and mark_new :: "ref \<Rightarrow> bool"   and marked :: "bool" shows "wf_marking t mark_old mark_new marked = (case t of Tip \<Rightarrow> mark_new = mark_old | Node lt p rt \<Rightarrow> (\<forall>n. n \<notin> set_of t \<longrightarrow> mark_new n = mark_old n) \<and> (\<forall>n. n \<in> set_of t \<longrightarrow> mark_new n = marked))"
equivalence_pseudofunctor_axioms: shows "equivalence_pseudofunctor (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B local.map cmp"
less_eq_prod_def: fixes p1 :: "'a \<times> 'b"   and p2 :: "'a \<times> 'b" shows "(p1 \<le> p2) = (fst p1 \<le> fst p2 \<and> snd p1 \<le> snd p2)"
sublmm_4: fixes a :: "'lbl strand_label \<times> ('fun, 'var) stateful_strand_step"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (unlabel (a # A))"   and "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # A) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # A) = {}"   and "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # A) = {}"   and "\<forall>t\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel (a # A)) \<union> pair ` snd ` set D. \<forall>t'\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel (a # A)) \<union> pair ` snd ` set D. (\<exists>\<delta>. Unifier \<delta> t t') \<longrightarrow> \<Gamma> t = \<Gamma> t'" shows "\<forall>t\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel A) \<union> pair ` snd ` set D. \<forall>t'\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel A) \<union> pair ` snd ` set D. (\<exists>\<delta>. Unifier \<delta> t t') \<longrightarrow> \<Gamma> t = \<Gamma> t'"
assms_2: fixes x :: "'a" assumes "x \<in> space N" shows "f x = g x"
ordertype_eq_iff: fixes \<alpha> :: "V"   and r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "Ord \<alpha>"   and "wf r"   and "small A"   and "total_on A r"   and "trans r" shows "(ordertype A r = \<alpha>) = (\<exists>f. bij_betw f A (elts \<alpha>) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. (f x < f y) = ((x, y) \<in> r)))"
consistent_Extend: fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set"   and f :: "nat \<Rightarrow> ('c, 'b) fm list \<times> 'b" assumes "infinite UNIV"   and "finite A"   and "consistent A S"   and "finite (\<Union> (block_nominals ` S))" shows "consistent A (Extend A S f)"
simps: fixes s :: "nat set"   and tab :: "'a option list set"   and st :: "nat set"   and tup :: "'a option list" shows "semiJoin (s, tab) (st, tup) = (s, Set.filter (isSameIntersection tup (s \<inter> st)) tab)"
inject: fixes x21 :: "'b"   and x22 :: "'b list"   and y21 :: "'b"   and y22 :: "'b list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
locally_connected_open_connected_component: fixes S :: "'b set" shows "locally connected S = (\<forall>t x. openin (top_of_set S) t \<and> x \<in> t \<longrightarrow> openin (top_of_set S) (connected_component_set t x))"
msig_def: fixes m :: "qtname \<times> mdecl" shows "msig m = fst (snd m)"
link2: fixes x :: "'b"   and lx :: "'b hp list"   and h :: "'b hp" shows "link (Hp x lx) h = (case h of Hp y ly \<Rightarrow> if x < y then Hp x (Hp y ly # lx) else Hp y (Hp x lx # ly))"
refl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "star r x x"
null_sets_subset: fixes B :: "'c set"   and M :: "'c measure"   and A :: "'c set" assumes "B \<in> null_sets M"   and "A \<in> sets M"   and "A \<subseteq> B" shows "A \<in> null_sets M"
rrep_1_update_changes: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l = PRrep-:1 \<longrightarrow> rt \<xi> \<noteq> update (rt \<xi>) (dip \<xi>) (dsn \<xi>, kno, val, hops \<xi> + 1, sip \<xi>, {}))"
pmatch_list_typerr: fixes vs :: "v list"   and ps :: "pat list"   and cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and s :: "v store_v list" assumes "length vs = length ps" shows "fold2 (\<lambda>p v m. case m of No_match \<Rightarrow> No_match | Match_type_error \<Rightarrow> Match_type_error | Match env \<Rightarrow> pmatch_single cenv s p v env) Match_type_error ps vs Match_type_error = Match_type_error"
Class_widen: fixes P :: "'a prog"   and C :: "char list"   and T :: "ty" assumes "subtype P (Class C) T" shows "\<exists>D. T = Class D"
WT_fv: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "P,E \<turnstile> e :: T" shows "fv e \<subseteq> dom E"
has_fps_expansion_imp_continuous: fixes f :: "'a \<Rightarrow> 'a"   and F :: "'a fps"   and A :: "'a set" assumes "f has_fps_expansion F" shows "continuous (at (0::'a) within A) f"
fun_apps_Cons: fixes s :: "('c, 'd) tm"   and sa :: "('c, 'd) tm"   and ss :: "('c, 'd) tm list" shows "fun (apps (App s sa) ss) = apps s (butlast (sa # ss))"
equal_pp_def: fixes p :: "('a, 'b) pp"   and q :: "('a, 'b) pp" shows "equal_class.equal p q \<equiv> \<forall>t. lookup_pp p t = lookup_pp q t"
a_remove_child_locs_def: fixes set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" shows "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child_locs set_child_nodes_locs set_disconnected_nodes_locs ptr owner_document = set_child_nodes_locs ptr \<union> set_disconnected_nodes_locs owner_document"
exec_step_ind_Checkcast: fixes P :: "jvm_prog"   and C :: "char list"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "cast_ok P C h (hd stk)" shows "exec_step_ind (StepI (Checkcast C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
is_ground_atm_def: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and A :: "'a" shows "substitution_ops.is_ground_atm subst_atm A = (\<forall>\<sigma>. A = subst_atm A \<sigma>)"
eq0: shows "exH_k (Suc k) = st.pmat_extension (H_k k) * st2.pmat_extension hadamard"
inner_minus_left: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (uminus x) y = - inner x y"