hcju/isabelleps huggingface dataset:
eventually_at_top_linorderI: fixes c :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "\<And>x. c \<le> x \<Longrightarrow> P x" shows "eventually P at_top"
consistent_iarrays_def: fixes A :: "'d iarray iarray"   and b :: "'d iarray" shows "consistent_iarrays A b = (let GJ = Gauss_Jordan_iarrays_PA A; rank_A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of (snd GJ))); P_mult_b = fst GJ *iv b in (if \<not> is_zero_iarray P_mult_b then greatest_not_zero P_mult_b + 1 else 0) \<le> rank_A)"
k_def: shows "k = min 1 (r / (2 * dist x y))"
simps_6: fixes uv :: "poscheckvariant"   and t :: "('c, 'd) Term.term"   and t' :: "('c, 'd) Term.term" shows "trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>uv: t \<in> t'\<rangle> = {t, t'}"
box_phi_a_1: fixes v :: "i"   and \<phi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>"   and x :: "\<nu>" assumes "[\<^bold>\<box>(\<^bold>\<forall>F. \<phi> F \<^bold>\<rightarrow> \<^bold>\<box>\<phi> F) in v]" shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F) \<^bold>\<rightarrow> \<^bold>\<box>(\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F)) in v]"
Sigma_Diff_distrib2: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and B :: "'c \<Rightarrow> 'd set" shows "(SIGMA i:I. A i - B i) = Sigma I A - Sigma I B"
dom_kil: shows "dom_kildall start = kildall r f step start"
smult_0_left: fixes p :: "'c poly" shows "smult (0::'c) p = 0"
closure_Un_frontier: fixes S :: "'b set" shows "closure S = S \<union> frontier S"
simps: fixes a1 :: "bool stream"   and a2 :: "'marker stream"   and a3 :: "('sequent \<times> 'rule) stream" shows "follow a1 a2 a3 = (\<exists>M' Ms s' steps M b s r bs. 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)"
span_mem: fixes K :: "('e, 'f) ring_scheme"   and V :: "('e, 'g, 'h) module_scheme"   and E :: "'g set"   and u :: "'g" assumes "vectorspace K V"   and "E \<subseteq> carrier V"   and "u \<in> E" shows "u \<in> module.span K V E"
can_implies_arr: fixes f :: "'c local.term set" assumes "can f" shows "arr f"
prv_cnj_mono: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c"   and \<chi>1 :: "'c"   and \<chi>2 :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla"   and "prv (imp \<phi>1 \<chi>1)"   and "prv (imp \<phi>2 \<chi>2)" shows "prv (imp (cnj \<phi>1 \<phi>2) (cnj \<chi>1 \<chi>2))"
prob_space_distrD: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "prob_space (distr M N f)" shows "prob_space M"
cases: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" obtains "x = []"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # q"
f_eq_g_measurable: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and g :: "'b \<Rightarrow> real" assumes "f \<in> rv M"   and "g \<in> rv M" shows "{w. f w = g w} \<in> measurable_sets M"
square_free_m_cong: fixes f :: "int poly"   and g :: "int poly" assumes "square_free_m f"   and "f =m g" shows "square_free_m g"
gb_schema_dummy_induct: fixes P :: "nat \<times> nat \<times> 'd \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list \<times> ((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set \<Rightarrow> bool"   and data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "struct_spec sel ap ab compl"   and "\<And>bs data D. P data D bs [] (gs @ bs, D)"   and "\<And>bs ps sps data D. \<lbrakk>ps \<noteq> []; sps = sel gs bs ps (snd data); fst data \<le> count_const_lt_components (fst (compl gs bs (ps -- sps) sps (snd data)))\<rbrakk> \<Longrightarrow> P data D bs ps (full_gb (gs @ bs), D)"   and "\<And>bs ps sps aux hs rc data data' D D'. \<lbrakk>ps \<noteq> []; sps = sel gs bs ps (snd data); aux = compl gs bs (ps -- sps) sps (snd data); (hs, data') = add_indices aux (snd data); rc = fst data - count_const_lt_components (fst aux); 0 < rc; D' = D \<union> (set hs \<times> (set gs \<union> set bs \<union> set hs) \<union> set (ps -- sps) -\<^sub>p set (ap gs bs (ps -- sps) hs data')); P (rc, data') D' (ab gs bs hs data') (ap gs bs (ps -- sps) hs data') (gb_schema_dummy (rc, data') D' (ab gs bs hs data') (ap gs bs (ps -- sps) hs data'))\<rbrakk> \<Longrightarrow> P data D bs ps (gb_schema_dummy (rc, data') D' (ab gs bs hs data') (ap gs bs (ps -- sps) hs data'))" shows "P data D bs ps (gb_schema_dummy data D bs ps)"
to_language_in_language: fixes L :: "'b language" shows "to_language (Collect (in_language L)) = L"
Ferio: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A E B"   and "B I C" shows "A Z C"
measurable_pair_measure: fixes f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and g :: "'c \<Rightarrow> 'e measure"   and L :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "g \<in> M \<rightarrow>\<^sub>M subprob_algebra L" shows "(\<lambda>x. f x \<Otimes>\<^sub>M g x) \<in> M \<rightarrow>\<^sub>M subprob_algebra (N \<Otimes>\<^sub>M L)"
subgraph_slim: shows "subgraph (with_proj sG.slim) (with_proj slim)"
type_is_class: fixes G :: "prog"   and C :: "qtname" assumes "is_type G (Class C)" shows "is_class G C"
multi_psub_of_add_self: fixes A :: "'k multiset"   and x :: "'k" shows "A \<subset># add_mset x A"
nat_power_0_right: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.nat_power one times plus zero nat_power"   and "0 < n" shows "nat_power n zero = one"
subset: shows "{x. pmf q x < pmf p x} \<subseteq> set_pmf p"
combine_values: fixes v :: "exp"   and v1 :: "val"   and \<rho> :: "(nat \<times> val) list"   and v2 :: "val" assumes "isval v"   and "v1 \<in> E v \<rho>"   and "v2 \<in> E v \<rho>" shows "\<exists>v3. v3 \<in> E v \<rho> \<and> v1 \<squnion> v2 = Some v3"
sappend_def: fixes Al :: "'a list set"   and Bl :: "'a list set" shows "Al @@ Bl = {al @ bl |al bl. al \<in> Al \<and> bl \<in> Bl}"
homotopy_eqv_contractibility: fixes S :: "'a set"   and T :: "'b set" assumes "S homotopy_eqv T" shows "contractible S = contractible T"
continuous_on_add: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x + g x)"
nat_uint_eq: fixes w :: "'b word" shows "nat (uint w) = unat w"
of_nat_max: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "nat"   and y :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "semiring_1.of_nat one plus zero (max x y) = ord.max less_eq (semiring_1.of_nat one plus zero x) (semiring_1.of_nat one plus zero y)"
hyps_1: shows "liftAll (\<lambda>a. qFresh ys (y @ys[z1 \<and> z2]_zs) (a #[[z1 \<and> z2]]_zs) = qFresh ys y a) inp"
squarefree_factorial_semiring'': fixes n :: "'a" assumes "n \<noteq> (0::'a)" shows "squarefree n = (\<forall>p. prime p \<longrightarrow> multiplicity p n \<le> 1)"
induct: fixes x1 :: "('a, 'b) form list"   and x2 :: "('a, 'b) form"   and P :: "('a, 'b) form list \<Rightarrow> ('a, 'b) form \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>a G. a \<in> set G \<Longrightarrow> P G a"   and "\<And>G. P G TT"   and "\<And>G a. \<lbrakk>G \<turnstile> FF; P G FF\<rbrakk> \<Longrightarrow> P G a"   and "\<And>a G. \<lbrakk>a # G \<turnstile> FF; P (a # G) FF\<rbrakk> \<Longrightarrow> P G (Neg a)"   and "\<And>G a. \<lbrakk>G \<turnstile> Neg a; P G (Neg a); G \<turnstile> a; P G a\<rbrakk> \<Longrightarrow> P G FF"   and "\<And>a G. \<lbrakk>Neg a # G \<turnstile> FF; P (Neg a # G) FF\<rbrakk> \<Longrightarrow> P G a"   and "\<And>G a b. \<lbrakk>G \<turnstile> a; P G a; G \<turnstile> b; P G b\<rbrakk> \<Longrightarrow> P G (And a b)"   and "\<And>G a b. \<lbrakk>G \<turnstile> And a b; P G (And a b)\<rbrakk> \<Longrightarrow> P G a"   and "\<And>G a b. \<lbrakk>G \<turnstile> And a b; P G (And a b)\<rbrakk> \<Longrightarrow> P G b"   and "\<And>G a b. \<lbrakk>G \<turnstile> a; P G a\<rbrakk> \<Longrightarrow> P G (Or a b)"   and "\<And>G b a. \<lbrakk>G \<turnstile> b; P G b\<rbrakk> \<Longrightarrow> P G (Or a b)"   and "\<And>G a b c. \<lbrakk>G \<turnstile> Or a b; P G (Or a b); a # G \<turnstile> c; P (a # G) c; b # G \<turnstile> c; P (b # G) c\<rbrakk> \<Longrightarrow> P G c"   and "\<And>a G b. \<lbrakk>a # G \<turnstile> b; P (a # G) b\<rbrakk> \<Longrightarrow> P G (Impl a b)"   and "\<And>G a b. \<lbrakk>G \<turnstile> Impl a b; P G (Impl a b); G \<turnstile> a; P G a\<rbrakk> \<Longrightarrow> P G b"   and "\<And>G a n. \<lbrakk>G \<turnstile> a[App n []/0]; P G (a[App n []/0]); news n G; new n a\<rbrakk> \<Longrightarrow> P G (Forall a)"   and "\<And>G a t. \<lbrakk>G \<turnstile> Forall a; P G (Forall a)\<rbrakk> \<Longrightarrow> P G (a[t/0])"   and "\<And>G a t. \<lbrakk>G \<turnstile> a[t/0]; P G (a[t/0])\<rbrakk> \<Longrightarrow> P G (Exists a)"   and "\<And>G a n b. \<lbrakk>G \<turnstile> Exists a; P G (Exists a); a[App n []/0] # G \<turnstile> b; P (a[App n []/0] # G) b; news n G; new n a; new n b\<rbrakk> \<Longrightarrow> P G b" shows "P x1 x2"
sel_4: fixes x1 :: "'c"   and x2 :: "'d"   and x3 :: "nat"   and x4 :: "('c, 'd) SkewBinomialTree list" shows "children (Node x1 x2 x3 x4) = x4"
subset_eq_imp_le_multiset: fixes M :: "'b multiset"   and N :: "'b multiset" assumes "M \<subseteq># N" shows "M \<le> N"
rel_3: shows "0 \<le> int_of_uint32 x'"
indRelLTPO_refl: fixes TRel :: "('procT \<times> 'procT) set" assumes "refl TRel" shows "refl (indRelLTPO TRel)"
unat_div_ten: shows "unat (n div 10) = unat n div 10"
indoc_Cons: fixes a :: "'item"   and b :: "'item"   and v :: "'item list" shows "indoc a (b # v) = (a = b \<or> a \<noteq> b \<and> ind a b \<and> indoc a v)"
eFreshAbs_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and A :: "'d"   and ys :: "'varSort"   and y :: "'var" assumes "\<forall>us s. \<not> igWlsAbs MOD (us, s) A" shows "eFreshAbs MOD ys y (OK A)"
strong_exhaust: fixes y :: "ty"   and c :: "'a" obtains "y = One"   | x21 :: "ty" and x22 :: "ty" where "y = Fun x21 x22"   | x31 :: "ty" and x32 :: "ty" where "y = Syntax.Sum x31 x32"   | x41 :: "ty" and x42 :: "ty" where "y = Syntax.Prod x41 x42"   | x51 :: "Syntax.tvar" and x52 :: "ty" where "{atom x51} \<sharp>* c" and "y = Mu x51 x52"   | x6 :: "Syntax.tvar" where "y = Alpha x6"   | x7 :: "ty" where "y = AuthT x7"
c_def: shows "c = (1 / (E * real DIM('a))) ^ DIM('a)"
New: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and h :: "heap"   and a :: "nat"   and P :: "(char list list \<times> char list exp) prog"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and h' :: "heap"   and l :: "sfields" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "new_Addr h = \<lfloor>a\<rfloor>"   and "P \<turnstile> C has_fields FDTs"   and "h' = h(a \<mapsto> blank P C)" shows "P \<turnstile> \<langle>new C,(h, l, sh)\<rangle> \<Rightarrow> \<langle>addr a,(h', l, sh)\<rangle>"
Longest_common_prefix_ex: fixes L :: "'c list set" assumes "L \<noteq> {}" shows "\<exists>ps. (\<forall>xs\<in>L. prefix ps xs) \<and> (\<forall>qs. (\<forall>xs\<in>L. prefix qs xs) \<longrightarrow> length qs \<le> length ps)"
prv_dsj_com_imp: fixes \<phi> :: "fm"   and \<chi> :: "fm" assumes "\<phi> \<in> UNIV"   and "\<chi> \<in> UNIV" shows "{} \<turnstile> \<phi> OR \<chi> IMP \<chi> OR \<phi>"
bcontfun\<^sub>N_1: fixes f :: "'c \<Rightarrow> 'd" shows "eNorm bcontfun\<^sub>N f = (if f \<in> bcontfun then ennreal (norm (Bcontfun f)) else \<infinity>)"
unique_connecting_path_rev: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "v \<leadsto>\<^bsub>G\<^esub> w = rev (w \<leadsto>\<^bsub>G\<^esub> v)"
lit_ord_dominating_term: fixes s1 :: "'a trm"   and s2 :: "'a trm"   and t2 :: "'a trm"   and x1 :: "'a literal"   and t1 :: "'a trm"   and p1 :: "sign"   and x2 :: "'a literal"   and p2 :: "sign" assumes "(s1, s2) \<in> trm_ord \<or> (s1, t2) \<in> trm_ord"   and "orient_lit x1 s1 t1 p1"   and "orient_lit x2 s2 t2 p2"   and "vars_of_lit x1 = {}"   and "vars_of_lit x2 = {}" shows "(x1, x2) \<in> lit_ord"
fresh_Pair: fixes a :: "atom"   and x :: "'e"   and y :: "'f" shows "a \<sharp> (x, y) = (a \<sharp> x \<and> a \<sharp> y)"
trunc_ell2_lim_at_UNIV: fixes \<psi> :: "'c ell2" shows "((\<lambda>S. trunc_ell2 S \<psi>) \<longlongrightarrow> \<psi>) (finite_subsets_at_top UNIV)"
stepC_sound: fixes n :: "C"   and c :: "nat" assumes "n \<in> hbase_ext (Suc (Suc c))" shows "stepC c n \<in> hbase (Suc (Suc c))"
closed_segment_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "(closed_segment a b = closed_segment c d) = ({a, b} = {c, d})"
K: shows "(fill xs (ns[index key y (length ns) mi ma := Suc (ns ! index key y (length ns) mi ma)]) index key ub mi ma)[ns ! index key y (length ns) mi ma := Some y] ! j = Some x"
intros_3: fixes p :: "('a, 'b) fm"   and q :: "('a, 'b) fm" assumes "\<turnstile> p \<^bold>\<longrightarrow> q"   and "\<turnstile> p" shows "\<turnstile> q"
is_terminal: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' x f" shows "\<exists>!g. \<epsilon>x.is_coext x' f g"
incl_def: fixes f :: "'t" shows "S'.incl f = (S'.arr f \<and> S'.set (S'.dom f) \<subseteq> S'.set (S'.cod f) \<and> f = S'.mkArr (S'.set (S'.dom f)) (S'.set (S'.cod f)) (\<lambda>x. x))"
invoke_ptr_in_heap: fixes h :: "(_) heap"   and xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> 'o \<Rightarrow> ((_) heap, exception, 'n) prog)) list"   and ptr :: "(_) object_ptr"   and args :: "'o" assumes "h \<turnstile> ok invoke xs ptr args" shows "ptr |\<in>| object_ptr_kinds h"
set_intros_1: fixes y :: "'b btree \<times> 'b"   and x21 :: "('b btree \<times> 'b) list"   and ya :: "'b btree"   and yb :: "'b"   and x22 :: "'b btree" assumes "y \<in> set x21"   and "ya \<in> Basic_BNFs.fsts y"   and "yb \<in> set_btree ya" shows "yb \<in> set_btree (Node x21 x22)"
second_vertex_def: shows "second_vertex \<equiv> \<lambda>xs. hd (tl xs)"
isomorphic_implies_equicomposable_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<tau> :: "'b" assumes "weak_composition V H"   and "category.isomorphic V f g" shows "(H \<tau> f \<noteq> partial_magma.null V) = (H \<tau> g \<noteq> partial_magma.null V)"
surjective: fixes r :: "('a, 'b) edge_scheme" shows "r = \<lparr>src = src r, tgt = tgt r, \<dots> = edge.more r\<rparr>"
iT_Plus_commute: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" shows "I \<oplus> a \<oplus> b = I \<oplus> b \<oplus> a"
invc2I: fixes t :: "('b \<times> color) tree" assumes "invc t" shows "invc2 t"
zdvd_mult_cancel: fixes k :: "int"   and m :: "int"   and n :: "int" assumes "k * m dvd k * n"   and "k \<noteq> 0" shows "m dvd n"
sum_of_prideals3: fixes I :: "'a set"   and n :: "nat" assumes "ideal R I" shows "\<forall>f. (\<forall>l\<le>n. f l \<in> carrier R) \<and> f ` {i. i \<le> n} \<subseteq> I \<longrightarrow> sum_pr_ideals R f n \<subseteq> I"
l\<^sub>2': shows "l\<^sub>2' = [this \<mapsto> Addr a, pns [\<mapsto>] vs]"
set_map_2: fixes f1 :: "'f \<Rightarrow> 'd"   and f2 :: "'g \<Rightarrow> 'c"   and f3 :: "'h \<Rightarrow> 'e"   and v :: "('f, 'g, 'h) F2" shows "F2set2 (F2map f1 f2 f3 v) = f2 ` F2set2 v"
rep_eq: fixes x :: "'b list" shows "lookup (Poly_Mapping.nth x) = nth_default (0::'b) x"
id_eq_1: fixes v :: "i"   and x :: "'b" shows "[x \<^bold>= x in v]"
distrib: fixes g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "(\<Sum>x\<in>A. g x + h x) = sum g A + sum h A"
in_minus_Un_2: fixes i :: "'a"   and I :: "'a set" assumes "i \<in> I" shows "{i} \<union> (I - {i}) = I"
closed_segment_subset_domain: fixes t0 :: "real"   and x0 :: "'a"   and t :: "real" assumes "t0 \<in> existence_ivl0 x0"   and "t \<in> existence_ivl0 x0" shows "{t0--t} \<subseteq> existence_ivl0 x0"
dist_self: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.metric_space dist uniformity open" shows "dist x x = 0"
simps_14: fixes e1 :: "('f, 'g, 'h) exp"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "('f, 'g, 'h) exp"   and e3 :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3)) A = (\<D> e1 A \<and> \<D> e2 (A \<squnion> \<A> e1) \<and> \<D> e3 (A \<squnion> \<A> e1 \<squnion> \<A> e2))"
inverse_equality: fixes u :: "'a set"   and v :: "'a set" assumes "u [\<cdot>] v = Class \<one>"   and "v [\<cdot>] u = Class \<one>"   and "u \<in> Partition"   and "v \<in> Partition" shows "quotient.inverse u = v"
neg: fixes x :: "'a" assumes "vectorspace V"   and "x \<in> V" shows "f (- x) = - f x"
gbinomial_pochhammer': fixes a :: "'b"   and k :: "nat" shows "a gchoose k = pochhammer (a - of_nat k + (1::'b)) k / fact k"
from_nat_into_surj: fixes A :: "'a set"   and a :: "'a" assumes "countable A"   and "a \<in> A" shows "\<exists>n. from_nat_into A n = a"
cartesian_product: fixes g :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and B :: "'f set"   and A :: "'e set" shows "(\<Sum>x\<in>A. sum (g x) B) = (\<Sum>(x, y)\<in>A \<times> B. g x y)"
ring_primeE_3: fixes p :: "'a" assumes "p \<in> carrier R"   and "ring_prime p" shows "prime R p"
edom_perm_rev: fixes \<pi> :: "perm"   and f :: "'c \<Rightarrow> 'd" shows "\<pi> \<bullet> edom f = edom (\<pi> \<bullet> f)"
enumerator_hd: fixes ls :: "nat list"   and m :: "nat"   and n :: "nat" assumes "ls \<in> set (enumerator m n)" shows "hd ls = 0"
rpr: fixes R :: "('b \<times> 'b) set" assumes "R \<subseteq> Id" shows "p2r (r2p R) = R"
sum_list_replicate: fixes n :: "nat"   and c :: "'a" shows "sum_list (replicate n c) = of_nat n * c"
stable_rank_equiv_token_squats: fixes x :: "nat" shows "token_squats x = (\<exists>i. stable_rank x i)"
\<alpha>_fin: fixes G :: "nat set \<times> nat set set" assumes "finite (uverts G)" shows "\<alpha> G \<noteq> \<infinity>"
assms_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
extended_Gromov_product_at_topological: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_Gromov_product_at e x y = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e (u n) (v n))) |u v. (\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> x \<and> (\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> y}"
l1_inv1_trans: shows "{l1_inv1} TS.trans l1 {> l1_inv1}"
3: shows "(negatex \<circ> sqprojection \<circ> (\<lambda>w. (f \<circ> (\<lambda>x. x $ 1)) w - (g \<circ> (\<lambda>x. x $ 2)) w)) ` cbox (- 1) 1 \<subseteq> cbox (- 1) 1"
trans: fixes b :: "'h multiset"   and a :: "'h multiset"   and c :: "'h multiset" assumes "b \<subseteq># a"   and "c \<subseteq># b" shows "c \<subseteq># a"
unfold2_def: fixes s1 :: "'e \<Rightarrow> ('d, 'e, 'f) F1"   and s2 :: "'f \<Rightarrow> ('d, 'e, 'f) F2"   and x :: "'f" shows "unfold2 s1 s2 x = Abs_JF2 ((Equiv_Relations.proj (lsbis2 carT1 carT2 strT1 strT2) \<circ> beh2 s1 s2) x)"
noProt: shows "\<forall>a. \<not> disc (Prot a)"
rcis_def: fixes r :: "real"   and a :: "real" shows "rcis r a = complex_of_real r * cis a"
Dube_def: fixes n :: "nat"   and d :: "nat" shows "Dube n d = (if n \<le> 1 \<or> d = 0 then d else Dube_aux n d 1)"
mirlfr_def: shows "mirlfr = DJ redlove \<circ> ferrack01 \<circ> simpfm \<circ> exsplit"
supports_def: fixes S :: "atom set"   and x :: "'b" shows "S supports x \<equiv> \<forall>a b. a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
derived_quot_is_group: shows "Group.group (G Mod derived G (carrier G))"
frechet_derivative: fixes x :: "'e \<Rightarrow> 'a"   and a :: "'e"   and y :: "'e \<Rightarrow> 'b" assumes "x differentiable at a"   and "y differentiable at a" shows "frechet_derivative (\<lambda>a. x a ** y a) (at a) = (\<lambda>h. x a ** frechet_derivative y (at a) h + frechet_derivative x (at a) h ** y a)"
budan_fourier_real_3: fixes p :: "real poly"   and b :: "real" assumes "p \<noteq> 0"   and "all_roots_real p" shows "int (proots_count p {x. b < x}) = changes_gt_der b p"
lproject_prefix: fixes u :: "'b llist"   and v :: "'b llist"   and A :: "'b set" assumes "u \<le> v" shows "lproject A u \<le> lproject A v"
plus_monom_zero: fixes p :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "p + monom m (0::'a) = p"
end_smult_distrib_right: fixes T :: "'v \<Rightarrow> 'v"   and a :: "'f"   and b :: "'f" assumes "range T \<subseteq> V" shows "(a + b) \<cdot>\<cdot> T = a \<cdot>\<cdot> T + b \<cdot>\<cdot> T"
AE_1_1: shows "AE x in M. liminf (\<lambda>xa. ereal (- u xa x / real xa)) = - subcocycle_lim_ereal u x"
fraction_scaleC_times: fixes u :: "num"   and v :: "num"   and w :: "num"   and a :: "'b" shows "(numeral u / numeral v) *\<^sub>C (numeral w * a) = (numeral u * numeral w / numeral v) *\<^sub>C a"
sorted_single: fixes cmp :: "'b comparator"   and x :: "'b" shows "Sorting_Algorithms.sorted cmp [x] = True"
strictly_preserves_runit: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "F (\<F>C.runit a) = \<r>\<^sub>D[F a]"
comp_arr_dom: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c" assumes "CopxC.arr f"   and "CopxC.dom f = a" shows "f \<odot> a = f"
s_fin: shows "\<integral>\<^sup>+ x. ennreal (norm (s i x)) \<partial>M < \<infinity>"
Ssegment_def: fixes D :: "'c Order"   and a :: "'c" shows "Ssegment D a = (if a \<notin> carrier D then carrier D else {x. x \<prec>\<^bsub>D\<^esub> a \<and> x \<in> carrier D})"
dfs_inductive: fixes g :: "(node \<times> node) list"   and x :: "node"   and xs :: "node list"   and ys :: "node list" shows "dfs g (x # xs) ys = (if List.member ys x then dfs g xs ys else dfs g (nexts g x @ xs) (x # ys))"