hcju/isabelleps huggingface dataset:
0: fixes b :: "'a \<Rightarrow>\<^sub>0 'b" shows "q b \<in> punit.dgrad_p_set d m"
rbt_interwk_is_rbt: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "rbt_sorted t1"   and "rbt_sorted t2" shows "is_rbt (rbt_inter_with_key f t1 t2)"
induced_automorphism_selfcomp_halfmorphism_f: shows "ChamberComplexMorphism (f \<turnstile> X) (f \<turnstile> X) (\
 \<circ> \
)"
wo_rel2_def: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" shows "wo_rel2 r s \<equiv> Well_order r \<and> Well_order s"
assms: shows "GEN_OP ie is_empty (A \<rightarrow> bool_rel)"
independece_dest_alt: fixes C :: "'b pmf set"   and P :: "('b pmf \<times> 'b pmf) set" assumes "independent_vnm C P" shows "\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. \<forall>\<alpha>\<in>{0<..1}. p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r"
weak_reduction_correspondence_simulation_and_closures_3: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_correspondence_simulation Rel Cal" shows "weak_reduction_correspondence_simulation (Rel\<^sup>*) Cal"
U_le_f: fixes i :: "nat"   and x :: "'a" shows "U i x \<le> f x"
on_curvep_infinity: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and b :: "'a"   and x :: "'a"   and y :: "'a" assumes "cring R" shows "cring.on_curvep R a b (x, y, \<zero>\<^bsub>R\<^esub>) = (x \<in> carrier R \<and> y \<in> carrier R)"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'a" shows "deg_pm x = deg_fun (lookup x)"
card_Collect_less_nat: fixes n :: "nat" shows "card {i. i < n} = n"
induct: fixes x :: "'b formula multiset"   and P :: "'b formula multiset \<Rightarrow> bool" assumes "x \<Rightarrow>\<^sub>n"   and "\<And>k \<Gamma>. P (\<^bold>\<not> (Atom k), Atom k, \<Gamma>)"   and "\<And>\<Gamma>. P (\<bottom>, \<Gamma>)"   and "\<And>F G \<Gamma>. \<lbrakk>F, G, \<Gamma> \<Rightarrow>\<^sub>n; P (F, G, \<Gamma>)\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G, \<Gamma>)"   and "\<And>F \<Gamma> G. \<lbrakk>F, \<Gamma> \<Rightarrow>\<^sub>n; P (F, \<Gamma>); G, \<Gamma> \<Rightarrow>\<^sub>n; P (G, \<Gamma>)\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G, \<Gamma>)"   and "\<And>F G \<Gamma>. \<lbrakk>\<^bold>\<not> F, \<^bold>\<not> G, \<Gamma> \<Rightarrow>\<^sub>n; P (\<^bold>\<not> F, \<^bold>\<not> G, \<Gamma>)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (F \<^bold>\<or> G), \<Gamma>)"   and "\<And>F \<Gamma> G. \<lbrakk>\<^bold>\<not> F, \<Gamma> \<Rightarrow>\<^sub>n; P (\<^bold>\<not> F, \<Gamma>); \<^bold>\<not> G, \<Gamma> \<Rightarrow>\<^sub>n; P (\<^bold>\<not> G, \<Gamma>)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (F \<^bold>\<and> G), \<Gamma>)"   and "\<And>F \<Gamma> G. \<lbrakk>\<^bold>\<not> F, \<Gamma> \<Rightarrow>\<^sub>n; P (\<^bold>\<not> F, \<Gamma>); G, \<Gamma> \<Rightarrow>\<^sub>n; P (G, \<Gamma>)\<rbrakk> \<Longrightarrow> P (F \<^bold>\<rightarrow> G, \<Gamma>)"   and "\<And>F G \<Gamma>. \<lbrakk>F, \<^bold>\<not> G, \<Gamma> \<Rightarrow>\<^sub>n; P (F, \<^bold>\<not> G, \<Gamma>)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (F \<^bold>\<rightarrow> G), \<Gamma>)"   and "\<And>F \<Gamma>. \<lbrakk>F, \<Gamma> \<Rightarrow>\<^sub>n; P (F, \<Gamma>)\<rbrakk> \<Longrightarrow> P (\<^bold>\<not> (\<^bold>\<not> F), \<Gamma>)" shows "P x"
antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
st2_def: shows "st2 = [Frame f pc \<Sigma>2]"
s_prod_def: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "R \<cdot> S = {(a, A). \<exists>B. (a, B) \<in> R \<and> (\<exists>f. (\<forall>b\<in>B. (b, f b) \<in> S) \<and> A = \<Union> {f b |b. b \<in> B})}"
assms_3: shows "q \<in> sterms \<Gamma> p"
closed_segment_subset_convex_hull: fixes x :: "'a"   and S :: "'a set"   and y :: "'a" assumes "x \<in> convex hull S"   and "y \<in> convex hull S" shows "closed_segment x y \<subseteq> convex hull S"
SBHWriteVolatile: fixes a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'c memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Write True a (D, f) A L R W # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>, sb @ [Write\<^sub>s\<^sub>b True a (D, f) (f \<theta>) A L R W], m, True, \<O>, \<R>, \<S>)"
simps_4: fixes l :: "dag"   and vref1 :: "ref"   and r :: "dag"   and vref :: "ref" shows "bdt (Node (Node l vref1 r) vref Tip) = Map.empty"
nonempty_ex: fixes A :: "'d set" assumes "A \<noteq> {}" shows "\<exists>x. x \<in> A"
remdups_remdups: fixes xs :: "'c list" shows "remdups (remdups xs) = remdups xs"
startsymbol_dom: shows "\<SS> \<in> \<NN>"
fileq: shows "filter (\<lambda>y. y < dim_row A) [0..<length (pivot_positions A)] = [0..<length (pivot_positions A)]"
augment_aux: fixes X :: "'a set"   and Y :: "'a set" assumes "indep X"   and "indep Y"   and "card X = Suc (card Y)" shows "\<exists>x\<in>X - Y. indep (insert x Y)"
top_widen_top: shows "\<top> \<nabla> \<top> = \<top>"
14: shows "(w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> selected_edge h j g = bot"
assms_1: shows "interaction_bounded_by' (\<lambda>_. True) \<A> q"
m_inv_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "subgroup H G"   and "x \<in> H" shows "inv\<^bsub>G\<^esub> x \<in> H"
arg_croot: fixes n :: "nat"   and x :: "complex" shows "arg (croot n x) = arg x / real n"
equiv_iff_eq_norm: fixes t :: "'c local.term"   and u :: "'c local.term" shows "local.equiv t u = (Arr t \<and> Arr u \<and> \<^bold>\<parallel>t\<^bold>\<parallel> = \<^bold>\<parallel>u\<^bold>\<parallel>)"
deg_pp_alt: fixes s :: "('b, 'a) pp" shows "deg_pp s = sum (lookup_pp s) (keys_pp s)"
e_altdef: fixes k :: "nat" shows "e k = insertion Root (sym_mpoly {..<n} k)"
FVF: shows "{Inl y, Inr y, Inr x} \<inter> FVF \<phi> = {}"
pivot_funD_2: fixes A :: "'b mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat"   and j :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr"   and "j < f i" shows "A $$ (i, j) = (0::'b)"
assms_2: shows "Pi\<^sub>E A' B \<noteq> {}"
induces_equivalence: shows "\<exists>F \<eta> \<epsilon>. equivalence_of_categories (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<eta> \<epsilon>"
set_scanl1_is_One: shows "set (scanl1 (\<lambda>x y. if x = One \<or> y = One then One else Zero) (map (\<lambda>xs. case xs of [] \<Rightarrow> Zero | x # xa \<Rightarrow> One) (map wrap [0..<n + 1]))) = {One}"
iNextWeak_iAll_conv: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNextWeak t0 I P = (\<box> t {inext t0 I} \<down>> t0. P t)"
exhaust: fixes y :: "('a, 'b, 'c) sum3" obtains x1 :: "'a" where "y = In1 x1"   | x2 :: "'b" where "y = In2 x2"   | x3 :: "'c" where "y = In3 x3"
dbl_dec_simps_5: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one (numeral.numeral one plus k) = numeral.numeral one plus (Num.BitM k)"
B: shows "finite {y. [(x, y)] \<bullet> xvec \<noteq> xvec}"
subcategory_axioms: shows "subcategory (\<cdot>) (right a)"
simps: fixes k :: "int"   and l :: "int" shows "k AND l = (if k \<in> {0, - 1} \<and> l \<in> {0, - 1} then - of_bool (odd k \<and> odd l) else of_bool (odd k \<and> odd l) + 2 * (k div 2 AND l div 2))"
g1_path_continuous: shows "continuous_on (cbox a b) (\<lambda>x. (g1 x, x))"
G'_def: shows "G' = \<GG> ! (length \<GG> - 2)"
simps_5: fixes Inv :: "'a"   and b :: "bexp"   and c :: "'a acom"   and P :: "'a" shows "strip ({Inv} WHILE b DO c {P}) = WHILE b DO strip c"
Int_UNIV: fixes A :: "'a set"   and B :: "'a set" shows "(A \<inter> B = UNIV) = (A = UNIV \<and> B = UNIV)"
prefix_le_fininf_transp: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 \<le> w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>F\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
map_is_Nil_conv: fixes f :: "'l \<Rightarrow> 'k"   and xs :: "'l list" shows "(map f xs = []) = (xs = [])"
intros_1: fixes c :: "('c, 'd) conf" shows "se_star c [] c"
prefix_cases: fixes xs :: "'b list"   and ys :: "'b list" obtains "xs \<le>p ys"   | "ys <p xs"   | "xs \<parallel> ys"
surjective: fixes r :: "('b, 'c, 'd) ioa_scheme" shows "r = \<lparr>ioa.asig = ioa.asig r, start = start r, trans = ioa.trans r, \<dots> = ioa.more r\<rparr>"
antisym: fixes x :: "'t"   and y :: "'t" assumes "x \<preceq>\<^sub>t y"   and "y \<preceq>\<^sub>t x" shows "x = y"
insert_rule: fixes k :: "nat"   and t :: "'a btree"   and ti :: "'a btnode ref option"   and x :: "'a" assumes "0 < k"   and "sorted_less (inorder t)" shows "<btree_assn k t ti> local.insert k x ti <btree_assn k (abs_split.insert k x t)>\<^sub>t"
to_total_type_wellformed: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and x :: "'Var" assumes "types_wellformed \<Gamma>" shows "type_wellformed (to_total \<Gamma> x)"
measure_pmf_posI: fixes x :: "'a"   and p :: "'a pmf"   and A :: "'a set" assumes "x \<in> set_pmf p"   and "x \<in> A" shows "0 < measure_pmf.prob p A"
effect_run: fixes c :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'b" assumes "effect c h h' r" shows "run c (Some h) (Some h') r"
intros: fixes f :: "'L"   and ss :: "'Q set list"   and s :: "'Q set" assumes "A f = Some (length ss)"   and "listsp (\<lambda>s. s \<subseteq> Q) ss"   and "s = \<delta>ss_lhs f ss" shows "s \<rightarrow> f ss \<in> \<delta>ss"
eq_Min_iff: assumes "finite F"   and "F \<noteq> {}" shows "(Min F = Min F) = (Min F \<in> F \<and> (\<forall>a\<in>F. Min F \<le> a))"
simps_16: fixes V :: "'e"   and T :: "ty"   and vo :: "'g val option"   and e :: "('e, 'f, 'g) exp" shows "fv {V:T=vo; e} = fv e - {V}"
simps_26: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x11 :: "nat"   and x12 :: "nat"   and x13 :: "nat"   and x14 :: "nat"   and x15 :: "k"   and x16 :: "nat"   and x17 :: "nat"   and x18 :: "nat"   and x19 :: "bool" shows "(case Rreq x11 x12 x13 x14 x15 x16 x17 x18 x19 of Rreq x xa xb xc xd xe xf xg xh \<Rightarrow> f1 x xa xb xc xd xe xf xg xh | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f1 x11 x12 x13 x14 x15 x16 x17 x18 x19"
fmdom'_fmrestrict_set: fixes X :: "'c set"   and f :: "('c, 'd) fmap" shows "fmdom' (fmrestrict_set X f) = X \<inter> fmdom' f"
g_min_impl: shows "map_min \<alpha> invar g_min"
deg: shows "Polynomial_Expression.degree p \<noteq> Polynomial_Expression.degree q"
these_image_Some_eq: fixes A :: "'b set" shows "Option.these (Some ` A) = A"
mprodR_Suc: fixes e :: "nat \<Rightarrow> nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "e \<in> {j. j \<le> Suc n} \<rightarrow> {j. 0 \<le> j}"   and "f \<in> {j. j \<le> Suc n} \<rightarrow> carrier R" shows "mprod_expR R e f (Suc n) = mprod_expR R e f n \<cdot>\<^sub>r f (Suc n)^\<^bsup>R e (Suc n)\<^esup>"
length: fixes A :: "'b llist set"   and N :: "enat" assumes "GSgame A N" shows "\<forall>e\<in>A. llength e = 2 * N"
inj_on_opp_chambers': fixes C :: "'a set"   and D :: "'a set" assumes "folding_g.chamber C"   and "C \<notin> g \<turnstile> X"   and "folding_g.chamber D"   and "D \<notin> g \<turnstile> X"   and "g ` C = g ` D" shows "C = D"
bijI': fixes f :: "'h \<Rightarrow> 'i" assumes "\<And>x y. (f x = f y) = (x = y)"   and "\<And>y. \<exists>x. y = f x" shows "bij f"
cls: shows "class P C = \<lfloor>(D, fs, ms)\<rfloor>"
filterlim_split_at: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'c filter"   and x :: "'b" assumes "filterlim f F (at_left x)"   and "filterlim f F (at_right x)" shows "filterlim f F (at x)"
Gamma_reflection_aux: obtains h' :: "complex \<Rightarrow> complex" where "continuous_on UNIV h'" and "\<And>z. ((\<lambda>z. if z \<in> \<int> then 0 else complex_of_real pi * cot (complex_of_real pi * z) + Digamma z - Digamma (1 - z)) has_field_derivative h' z) (at z)"
submonoid_nonempty: fixes G :: "('c, 'd) monoid_scheme" shows "\<not> submonoid {} G"
big_clocked_unclocked_equiv: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r1 :: "(v, v) result" shows "BigStep.evaluate False env s e (s', r1) = (\<exists>c. BigStep.evaluate True env (update_clock (\<lambda>_. c) s) e (update_clock (\<lambda>_. 0) s', r1) \<and> r1 \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock s')"
mem_diff_2: shows "\<exists>c\<^sub>A' mds\<^sub>A' mem\<^sub>A'. (\<langle>c\<^sub>A', mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>c\<^sub>C', mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C) \<in> \<R> \<and> \<langle>c\<^sub>A', mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A \<in> abs.loc_reach \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A"
Hd_head_id: fixes s :: "('a, 'b) tm" assumes "num_args s = 0" shows "Hd (head s) = s"
i_shrink_def: fixes f :: "nat \<Rightarrow> 'b message_af"   and k :: "nat" shows "f \<div> k \<equiv> i_aggregate f k last_message"
upd_def: shows "upd = tighten_poly_bounds_binary p1 p2"
inj_decode: fixes base :: "nat"   and e :: "nat" assumes "2 \<le> base" shows "inj_on (decode base e) {M. well_base\<^sub>h base M \<and> aligned\<^sub>d base e M}"
get_min_mset: fixes ts :: "'b tree list"   and x :: "'b" assumes "ts \<noteq> []"   and "invar ts"   and "x \<in># mset_heap ts" shows "get_min ts \<le> x"
nth_default_Cons: fixes dflt :: "'f"   and x :: "'f"   and xs :: "'f list"   and n :: "nat" shows "nth_default dflt (x # xs) n = (case n of 0 \<Rightarrow> x | Suc n' \<Rightarrow> nth_default dflt xs n')"
arcs_blocks_elem: fixes bl :: "'a set" assumes "bl \<in># arcs_blocks" shows "\<exists>x y. bl = {x, y}"
T_delete_def: fixes x :: "'b"   and t :: "'b tree" shows "T_delete x t = 1 + (if t = \<langle>\<rangle> then 0 else T_splay x t + (case splay x t of \<langle>l, a, r\<rangle> \<Rightarrow> if x \<noteq> a then 0 else if l = \<langle>\<rangle> then 0 else T_splay_max l))"
lfilter_llist_of: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "lfilter P (llist_of xs) = llist_of (filter P xs)"
field_differentiable_at_cos: fixes z :: "'a" shows "cos field_differentiable at z"
nonzero_Complexs_inverse: fixes a :: "'b" assumes "a \<in> \<complex>"   and "a \<noteq> (0::'b)" shows "inverse a \<in> \<complex>"
add_block_alt: fixes b :: "'a set" shows "add_block b = add_mset b \<B>"
ah_cases: fixes h :: "'b \<Rightarrow> 'b set"   and m :: "'b" assumes "h \<in> ah" obtains "h m = {}"   | "m \<in> h m"
simp_num_pair_ci: fixes bs :: "real list"   and t :: "Ferrack.num"   and n :: "int" shows "(case simp_num_pair (t, n) of (t, n) \<Rightarrow> Inum bs t / real_of_int n) = (case (t, n) of (t, n) \<Rightarrow> Inum bs t / real_of_int n)"
merge_preserves: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list" assumes "merge as bs us vs" shows "concat as = concat us \<and> concat bs = concat vs"
Rats_abs_iff: fixes x :: "real" shows "(\<bar>x\<bar> \<in> \<rat>) = (x \<in> \<rat>)"
abs_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b set \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b set"   and x :: "'b" assumes "quot_type R Abs Rep" shows "abs R Abs x = Abs (Collect (R x))"
prems_1: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"
intro: assumes "CARD('a) = CARD('b) * CARD('c)" shows "rebase_mult TYPE('a) TYPE('b) TYPE('c)"
a: shows " t \<longrightarrow>\<^sub>1\<^sup>* t2"
distinct_1: fixes x1 :: "'a" shows "prot_atom.Atom x1 \<noteq> Value"
slice2_param_out_slice2: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and V :: "'var"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice2 n"   and "n s-p:V\<rightarrow>\<^bsub>out\<^esub> n'" shows "nx \<in> sum_SDG_slice2 n'"
lt_monomial: fixes c :: "'c"   and t :: "'t" assumes "c \<noteq> (0::'c)" shows "lt (monomial c t) = t"
Greater_def: fixes \<theta> :: "('d, 'f) trm"   and \<theta>' :: "('d, 'f) trm" shows "Greater \<theta> \<theta>' = (Geq \<theta> \<theta>' && ! (Geq \<theta>' \<theta>))"
mult_closed: fixes R :: "'a set"   and r :: "'a"   and s :: "'a" assumes "Ring1 R"   and "r \<in> R"   and "s \<in> R" shows "r * s \<in> R"
ereal_dense2: fixes x :: "ereal"   and y :: "ereal" assumes "x < y" shows "\<exists>z. x < ereal z \<and> ereal z < y"
Gauss_Jordan_PA_eq: fixes A :: "(('d, 'e) vec, 'f) vec" shows "snd (Gauss_Jordan_PA A) = Gauss_Jordan A"
not_\<omega>_le_1: shows "\<not> \<omega> \<le> 1"
sublist_rev_left: fixes xs :: "'c list"   and ys :: "'c list" shows "sublist (rev xs) ys = sublist xs (rev ys)"
select: fixes xs :: "'b list"   and s :: "natural \<times> natural" assumes "xs \<noteq> []" shows "fst (Random.select xs s) \<in> set xs"
HF: fixes A :: "fm"   and H :: "fm set" assumes "A \<in> HF_axioms" shows "H \<turnstile> A"
comp_assoc: fixes h :: "'d"   and g :: "'d"   and f :: "'d" shows "(h \<cdot> g) \<cdot> f = h \<cdot> g \<cdot> f"
lt_eq_min_term_monomial: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt p = min_term" shows "monomial (lc p) min_term = p"
simps_2: fixes g1 :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)"   and g2 :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'b \<times> 'c)"   and s2 :: "'c" shows "append_has_next g1 g2 (Inr s2) = fst g2 s2"
intros_3: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and t :: "('a, 'b) Term.term"   and M :: "('a, 'b) Term.term set"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "typed_model arity public Ana \<Gamma>"   and "t \<in> typed_model.SMP arity Ana \<Gamma> M"   and "typed_model.wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<Gamma> \<delta>"   and "\<forall>t\<in>subst_range \<delta>. intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity t" shows "t \<cdot> \<delta> \<in> typed_model.SMP arity Ana \<Gamma> M"
simps_1: fixes nat :: "nat"   and nat' :: "nat" shows "(Suc nat = Suc nat') = (nat = nat')"
finished_ss_sccs_tj_stack: shows "dom (finished s) \<subseteq> \<Union> (sccs s) \<union> set (tj_stack s)"
bexE: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>A. P x" obtains x :: "'b" where "x \<in> A" and "P x"
simps_1: fixes f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "transfer \<Up>f (cf # cfs) = (f (fst cf), snd cf) # cfs"
summable_mult: fixes f :: "nat \<Rightarrow> 'c"   and c :: "'c" assumes "summable f" shows "summable (\<lambda>n. c * f n)"
snd_conv: fixes x1 :: "'c"   and x2 :: "'b" shows "time (x1, x2) = x2"
f01: fixes z :: "complex" shows "f z \<noteq> 0 \<and> f z \<noteq> 1"
coset_join2: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier G"   and "subgroup H G"   and "x \<in> H" shows "H #> x = H"
simps_6: fixes P :: "'b ptrm" shows "ptrm_fvs (PFst P) = ptrm_fvs P"
out_arcs_empty: fixes x :: "'a" assumes "x \<in> verts G" shows "(out_arcs G x = {}) = (out_arcs H x = {})"
eventually_always_next_stable: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "(\<diamond>P) \<sigma>"   and "(\<box>(\<lambda>s. P s \<longrightarrow> (\<circle>P) s)) \<sigma>" shows "(\<diamond>\<box>P) \<sigma>"
SUP_upper: 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 i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
lift_preserves_beta: fixes r :: "Core.term"   and s :: "Core.term"   and i :: "nat" assumes "r \<rightarrow>\<^sub>\<beta> s" shows "lift r i \<rightarrow>\<^sub>\<beta> lift s i"
map_seq: fixes f :: "'a" assumes "D.J.arr f" shows "D.C.seq (\<tau> (D.J.cod f)) (\<chi> f)"
YFtorNTCatDom: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" shows "NTCatDom (YFtorNT C f) = Op C"
subset_Worder: fixes T :: "'a set" assumes "T \<subseteq> carrier D" shows "Worder (Iod D T)"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "hom.ide a"   and "hom.ide a'"   and "hom'.in_hom g (trg f \<star> a' \<star> e) (trg f \<star> a \<star> e)" shows "\<exists>fa. hom.in_hom fa a' a \<and> trg f \<star> fa \<star> e = g"
Mx: fixes F :: "'a set" assumes "finite F" shows "L2_set (cmod \<circ> x) F \<le> Mx"
take_rewrite_swap: fixes m :: "nat"   and xs :: "'a list"   and n :: "nat"   and x :: "'a" shows "take m (rewrite xs n x) = rewrite (take m xs) n x"
coreflexive_bot_closed: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b" assumes "class.non_associative_left_semiring one times sup less_eq less bot" shows "less_eq bot one"
conc_fun_chain: fixes R :: "('a \<times> 'b) set"   and S :: "('b \<times> 'c) set"   and M :: "'c nres" shows "\<Down> R (\<Down> S M) = \<Down> (R O S) M"
expand: fixes inference :: "'b inference"   and inference' :: "'b inference" assumes "prems_of inference = prems_of inference' \<and> concl_of inference = concl_of inference'" shows "inference = inference'"
A2: fixes x :: "'a list"   and y :: "'a list" assumes "x \<noteq> []"   and "y \<noteq> []" shows "f x y \<noteq> []"
CallRedsThrowParams: fixes e :: "char list exp"   and s0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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 es :: "char list exp list"   and vs\<^sub>1 :: "val list"   and a :: "char list exp"   and es\<^sub>2 :: "char list exp list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs\<^sub>1 @ throw a # es\<^sub>2,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>2\<rangle>"