hcju/isabelleps huggingface dataset:
undirected: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Graph G" shows "v \<rightarrow>\<^bsub>G\<^esub> w = w \<rightarrow>\<^bsub>G\<^esub> v"
rGamma_series_LIMSEQ: fixes z :: "'a" shows "rGamma_series z \<longlonglongrightarrow> rGamma z"
rell_FGl_def: fixes L1 :: "'q \<Rightarrow> 'ad \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 'ae \<Rightarrow> bool"   and L3 :: "'x \<Rightarrow> 'af \<Rightarrow> bool" shows "rell_FGl L1 L2 L3 = rel_FGl L1 L2 L3 (=) (=) (=) (=) (=) (=) (=) (=)"
join_def: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = (if \<exists>z. {x, y} <<| z then lub {x, y} else x)"
"2_1": fixes xa :: "(real \<times> real) list \<times> (real \<times> real) list"   and xb :: "(real \<times> real) list"   and y :: "(real \<times> real) list"   and xaa :: "real \<times> real" assumes "xa = partition (coll 0 z) zs"   and "(xb, y) = xa"   and "xaa = z + sum_list xb"   and "x \<in> set (independent_pdevs y)"   and "y \<in> set (independent_pdevs y)"   and "\<And>x. x \<in> set y \<Longrightarrow> x \<noteq> 0"   and "x \<noteq> y" shows "det3 0 x y \<noteq> 0"
ix: shows "ix < length (snd r) \<and> snd r ! ix = \<gamma> ! j"
finite_keys: fixes f :: "'c \<Rightarrow>\<^sub>0 'd" shows "finite (keys f)"
emsapss_decode_help3_def: fixes mHash :: "bit list"   and EM :: "bit list"   and emBits :: "nat" shows "emsapss_decode_help3 mHash EM emBits = emsapss_decode_help4 mHash (generate_maskedDB EM emBits (length mHash)) (generate_H EM emBits (length mHash)) emBits"
Quo_rec_quo: fixes y :: "nat"   and x :: "nat" shows "rec_eval rec_quo [y, x] = Quo x y"
Obj_Trg: fixes t :: "'a E.term" assumes "E.Arr t" shows "E.Obj (E.Trg t)"
is_processT3_ST: fixes s :: "'c event list"   and t :: "'c event list"   and P :: "'c process" assumes "s @ t \<in> T P" shows "s \<in> T P"
uminus_minus_left_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ab_group_add plus zero minus uminus" shows "minus (uminus a) (plus b c) = minus (uminus b) (plus a c)"
lift_minus: fixes f :: "nat \<Rightarrow> real"   and z :: "nat"   and c :: "real mpoly"   and C :: "real" assumes "insertion f (liftPoly 0 z (c - Const C)) = 0" shows "insertion f (liftPoly 0 z c) = C"
ipathI: fixes r :: "nat \<Rightarrow> 'h"   and E :: "('h \<times> 'h) set" assumes "\<And>i. (r i, r (Suc i)) \<in> E" shows "ipath E r"
filtermap_def: fixes f :: "'c \<Rightarrow> 'b"   and F :: "'c filter" shows "filtermap f F = Abs_filter (\<lambda>P. \<forall>\<^sub>F x in F. P (f x))"
sigma_finite: obtains A :: "nat \<Rightarrow> 'a set" where "range A \<subseteq> sets M" and "\<Union> (range A) = space M" and "\<And>i. emeasure M (A i) \<noteq> \<infinity>"
eq: shows "(\<Sum>X | X \<subseteq> {..<n} \<and> root' X \<noteq> 0. exp (root' X)) = - of_nat A"
iffd_def: shows "fd\<^sup>-\<^sub>\<F> = op\<^sub>K \<circ> bd\<^sup>-\<^sub>\<F>"
simps_2: fixes \<rho> :: "'b run"   and K :: "clock"   and n :: "nat" shows "#\<^sub>\<le> \<rho> K Suc n = (if hamlet (Rep_run \<rho> (Suc n) K) then 1 + #\<^sub>\<le> \<rho> K n else #\<^sub>\<le> \<rho> K n)"
gt_unary: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wt t = wt s"   and "\<not> head t \<le>\<ge>\<^sub>h\<^sub>d head s"   and "num_args t = 1"   and "\<exists>f\<in>local.ground_heads (head t). arity_sym f = 1 \<and> wt_sym f = 0"   and "arg t \<ge>\<^sub>t s" shows "t >\<^sub>t s"
norm_matrix_bound: fixes x :: "('b, 'c) vec"   and A :: "(('b, 'c) vec, 'd) vec" assumes "\<parallel>x\<parallel> = 1" shows "\<parallel>A *v x\<parallel> \<le> \<parallel>(\<chi>i j. \<parallel>A $ i $ j\<parallel>) *v 1\<parallel>"
per_exp: fixes u :: "'b list"   and r :: "'b list"   and k :: "nat" assumes "u \<le>p r\<^sup>\<omega>" shows "u \<le>p r \<^sup>@ k \<cdot> u"
CFG_SSA_def: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g"   and defs :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set"   and "uses" :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set"   and phis :: "'f \<Rightarrow> 'g \<times> 'i \<Rightarrow> 'i list option" shows "CFG_SSA \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<equiv> CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<and> CFG_SSA_axioms \<alpha>n inEdges' defs phis"
simps_3: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and N :: "'b ltln set" shows "\<phi> R\<^sub>n \<psi>[N]\<^sub>\<Pi>\<^sub>2 = \<phi>[N]\<^sub>\<Pi>\<^sub>2 R\<^sub>n \<psi>[N]\<^sub>\<Pi>\<^sub>2"
HaddP_inv2: fixes x :: "tm"   and y :: "tm"   and z :: "tm"   and y' :: "tm" shows "{HaddP x y z, HaddP x y' z, OrdP x} \<turnstile> y' EQ y"
measure_preserving_def: fixes M :: "'b measure"   and N :: "'c measure" shows "measure_preserving M N = {f \<in> M \<rightarrow>\<^sub>M N. \<forall>A\<in>sets N. emeasure M (f -` A \<inter> space M) = emeasure N A}"
exec_mbindFSave_success: fixes ioprog :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and a :: "'f"   and \<sigma> :: "'e"   and b :: "'d"   and \<sigma>' :: "'e"   and S :: "'f list"   and M :: "'d list \<Rightarrow> 'e \<Rightarrow> (bool \<times> 'e) option" assumes "ioprog a \<sigma> = Some (b, \<sigma>')" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e (a # S) ioprog) M) = (\<sigma>' \<Turnstile> _bind_SE s (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e S ioprog) (M (b # s)))"
sum_list_mono: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> g x" shows "sum_list (map f xs) \<le> sum_list (map g xs)"
last_message_replicate: fixes n :: "nat"   and m :: "'b message_af" assumes "0 < n" shows "last_message m\<^bsup>n\<^esup> = m"
R: shows "v \<oplus>\<^bsub>V\<^esub> w \<in> span U"
succeed_t_inclusion: fixes x :: "nat"   and n :: "nat"   and i :: "nat" assumes "rank x n = Some i"   and "token_run x n \<notin> F - {q\<^sub>0}"   and "token_run x (Suc n) \<in> F" shows "n \<in> succeed_t i"
generate_subgroup_incl: fixes K :: "'c set"   and H :: "'c set" assumes "K \<subseteq> H"   and "subgroup H H" shows "generate H K \<subseteq> H"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> a2 = ((\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel))"
nilpotent_def: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c" shows "nilpotent R a = (\<exists>n. a^\<^bsup>R n\<^esup> = \<zero>\<^bsub>R\<^esub>)"
NatTransCompNatTrans': fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "\<eta>1 \<approx>>\<bullet> \<eta>2" shows "NatTransP (\<eta>1 \<bullet>1 \<eta>2)"
cong_reflexivity: fixes A :: "'p"   and B :: "'p" shows "Cong A B A B"
max_0_1_2: 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" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq one zero = one"
prime_elem_power_mult_dvdI: fixes p :: "'a"   and n :: "nat"   and c :: "'a"   and b :: "'a" assumes "prime_elem p"   and "p ^ n dvd c"   and "b dvd c"   and "\<not> p dvd b" shows "p ^ n * b dvd c"
changes_alt_itv_smods_def: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" shows "changes_alt_itv_smods a b p q = (let ps = smods p q in changes_alt_poly_at ps a - changes_alt_poly_at ps b)"
map_fst_zip: fixes xs :: "'d list"   and ys :: "'e list" assumes "length xs = length ys" shows "map fst (xs || ys) = xs"
mset_set_set: fixes xs :: "'k list" assumes "distinct xs" shows "mset_set (set xs) = mset xs"
lm023: fixes X :: "'a set"   and mx :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "finite X"   and "mx \<in> X"   and "\<forall>x\<in>X. x \<noteq> mx \<longrightarrow> f x < f mx" shows "argmax f X = {mx}"
ZeroHeightReachableElim: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and r :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, 0, r) \<in> Reachable" shows "r = s"
degree_polysub: fixes p :: "'b poly"   and n0 :: "nat"   and q :: "'b poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "Polynomial_Expression.degree (p -\<^sub>p q) \<le> max (Polynomial_Expression.degree p) (Polynomial_Expression.degree q)"
pdevs_of_list_dense_list_of_pdevs: fixes x :: "'b pdevs" shows "pdevs_of_list (dense_list_of_pdevs x) = x"
of_int_complex_interval_tendsto: fixes n :: "int" shows "(\<lambda>i. of_int_complex_interval n) \<longlonglongrightarrow>\<^sub>c of_int n"
tauStepChainResPres: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and x :: "name" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"   and "x \<sharp> \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<Longrightarrow>\<^sub>\<tau> \<lparr>\<nu>x\<rparr>P'"
phl_test_while: fixes p :: "'a"   and b :: "'a"   and x :: "'a" assumes "test p"   and "test b"   and "\<lparr>p \<cdot> b\<rparr>x\<lparr>x \<cdot> p\<rparr>" shows "\<lparr>p\<rparr>while b do x od\<lparr>(while b do x od) \<cdot> (p \<cdot> !b)\<rparr>"
hom_image_conc: fixes f :: "'b list \<Rightarrow> 'c list"   and A :: "'b list set"   and B :: "'b list set" assumes "\<And>xs ys. f (xs @ ys) = f xs @ f ys" shows "f ` (A @@ B) = f ` A @@ f ` B"
weakPsiCongruence_def: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "env.weakPsiCongruence substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P Q \<equiv> (\<Psi>, P, Q) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<and> env.weakCongSimulation substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq') Q \<and> env.weakCongSimulation substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> Q (env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq') P"
simps: fixes "consider" :: "'b \<Rightarrow> bool"   and gpv :: "('d, 'b, 'e) gpv" shows "interaction_bound consider gpv = (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound consider (c input)) else \<Squnion>input. interaction_bound consider (c input))"
mono2mono_ltakeWhile: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd llist"   and P :: "'d \<Rightarrow> bool" assumes "monotone orda (\<sqsubseteq>) t" shows "monotone orda (\<sqsubseteq>) (\<lambda>x. ltakeWhile P (t x))"
is_category: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "concrete_category Obj Hom Id Comp" shows "category (concrete_category.COMP Obj Hom Comp)"
termMOD_igSubstAbsIPresIGWlsAbs: shows "igSubstAbsIPresIGWlsAbs termMOD"
indexlt_rel_def: shows "indexlt_rel = less_than <*lex*> less_than"
M2_def: shows "M2 \<equiv> y$ = Suc<$y> \<and> x$ = $x"
tendsto_enn2ereal_iff: fixes f :: "'a \<Rightarrow> ennreal"   and x :: "ennreal"   and F :: "'a filter" shows "((\<lambda>i. enn2ereal (f i)) \<longlongrightarrow> enn2ereal x) F = (f \<longlongrightarrow> x) F"
inv_comp_left_2: fixes g :: "'b"   and f :: "'b" assumes "D.iso (g \<cdot>\<^sub>D f)"   and "D.iso g" shows "D.iso f"
CTermP_graph_aux_def: shows "CTermP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>CTermP_aux k s t vf. x1 = (vf, t) \<and> x2 = SyntaxN.Ex s (SyntaxN.Ex k (SeqCTermP vf (Var s) (Var k) t)) \<and> atom k \<sharp> (s, t) \<and> atom s \<sharp> t)"
finite_UN: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "\<Union> (A ` I) \<in> sets M"
L_def: shows "L \<equiv> Sup {\<parallel>A t\<parallel>\<^sub>o\<^sub>p |t. t \<in> T}"
ZObis_def: shows "ZObis \<equiv> bis ZOretr"
conversep_le_swap: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "(r \<le> s\<inverse>\<inverse>) = (r\<inverse>\<inverse> \<le> s)"
simps_3: fixes v :: "nat"   and va :: "real mpoly"   and L :: "atom list" shows "find_lucky_eq v (Less va # L) = find_lucky_eq v L"
SPEC_nofail: fixes X :: "'b nres"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "X \<le> SPEC \<Phi>" shows "nofail X"
approx_trans2: fixes r :: "'a star"   and x :: "'a star"   and s :: "'a star" assumes "r \<approx> x"   and "s \<approx> x" shows "r \<approx> s"
right_idem: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "semilattice f" shows "f (f a b) b = f a b"
db_defs_1: fixes inp :: "state \<Rightarrow> 'b"   and mid :: "state \<Rightarrow> 'b"   and out :: "state \<Rightarrow> 'b"   and q1 :: "state \<Rightarrow> 'b list"   and q2 :: "state \<Rightarrow> 'b list"   and ic :: "state \<Rightarrow> 'c"   and q :: "state \<Rightarrow> 'c list"   and oc :: "state \<Rightarrow> 'c" assumes "DBuffer inp mid out q1 q2" shows "BInit ic q oc \<equiv> $q = #[] \<and> $ic = $oc"
l0: shows "e * c / (f2 * (min e c - 1)) \<le> real l0"
that: fixes g :: "complex \<Rightarrow> complex" assumes "g holomorphic_on S"   and "\<And>z. z \<in> S \<Longrightarrow> f z = cos (g z)" shows "thesis"
minus_flow_simps: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and g :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" shows "(f \<ominus> g) (x, y) = (if edge \<Delta> x y then f (x, y) - g (x, y) else if edge \<Delta> y x then g (y, x) - f (y, x) else 0)"
set_times_elim: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<in> A * B" obtains a :: "'b" and b :: "'b" where "x = a * b" and "a \<in> A" and "b \<in> B"
mrw_before: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and r :: "nat"   and w :: "nat" assumes "E \<in> \<E>"   and "P,E \<turnstile> r \<leadsto>mrw w"   and "non_speculative P (\<lambda>_. {}) (ltake (enat r) (lmap snd E))" shows "w < r"
List_explode_def: fixes xss :: "'b list list" shows "List_explode xss = concat (map (\<lambda>xs. if xs = [] then [None] else map Some xs) xss)"
formulaFalseAppend: fixes formula :: "Literal list list"   and valuation :: "Literal list"   and formula' :: "Literal list list" assumes "formulaFalse formula valuation" shows "formulaFalse (formula @ formula') valuation"
nres_rel_comp: fixes A :: "('a \<times> 'c) set"   and B :: "('c \<times> 'b) set" shows "\<langle>A\<rangle>nres_rel O \<langle>B\<rangle>nres_rel = \<langle>A O B\<rangle>nres_rel"
inf_Sup: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and a :: "'b"   and B :: "'b set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf a (Sup B) = Sup (inf a ` B)"
permute_perm_def: fixes p :: "perm"   and q :: "perm" shows "p \<bullet> q = p + q - p"
SubspaceI: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and N :: "'b set" assumes "VectorSpace smult V"   and "Group N \<and> N \<subseteq> V"   and "\<And>r n. \<lbrakk>r \<in> UNIV; n \<in> N\<rbrakk> \<Longrightarrow> smult r n \<in> N" shows "RModule UNIV smult N \<and> N \<subseteq> V"
io_snd: shows "inj_on snd {(u, p). gen_iapath V u p v}"
aLeq: shows "2 ^ 31 \<le> 9223372039002259455"
in_hhomE: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and a :: "'b"   and b :: "'b" assumes "horizontal_homs V src trg"   and "horizontal_homs.in_hhom V src trg \<mu> a b" obtains "partial_magma.arr V \<mu>" and "horizontal_homs.obj V src a" and "horizontal_homs.obj V src b" and "src \<mu> = a" and "trg \<mu> = b"
distinct_remdups: fixes xs :: "'e list" shows "distinct (remdups xs)"
conjugate_square_eq_0_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "(v \<bullet>c v = (0::'a)) = (v = 0\<^sub>v n)"
cases: fixes x :: "ereal" obtains r :: "real" where "x = ereal r"   | "x = PInfty"   | "x = MInfty"
is_ground_atm_list_def: fixes As :: "'a list" shows "is_ground_atm_list As = Ball (set As) is_ground_atm"
pair_graph_def: fixes G :: "'b pair_pre_digraph" shows "pair_graph G \<equiv> pair_digraph G \<and> pair_pseudo_graph G"
name_abs_eq: fixes a :: "name"   and x :: "'b"   and b :: "name"   and y :: "'b" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> x = [(a, b)] \<bullet> y \<and> a \<sharp> y)"
list_all2': shows "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ A_x # A_xs) @ A_ws @ A_ys) ((zs @ join x False y # xs) @ ws @ ys)"
signed_take_bit_eq_iff_take_bit_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c"   and b :: "'c" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "(ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n b) = (take_bit (Suc n) a = take_bit (Suc n) b)"
intros: fixes ts :: "('c \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c memref list \<times> 'd \<times> 'e \<times> 'f) list"   and ts\<^sub>d :: "('c \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> 'g \<times> 'h) list" assumes "length ts = length ts\<^sub>d"   and "\<forall>i<length ts. \<exists>\<O>' \<D>' \<R>'. let (p, is, \<theta>, sb, \<D>, \<O>, \<R>) = ts\<^sub>d ! i in ts ! i = (p, is, \<theta>, [], \<D>', \<O>', \<R>')" shows "ts \<sim> ts\<^sub>d "
ct_list_eq_take: fixes xs :: "ct list"   and ys :: "ct list"   and n :: "nat" assumes "ct_list_eq xs ys" shows "ct_list_eq (take n xs) (take n ys)"
singleton_conv: fixes a :: "'a" shows "{x. x = a} = \<eta> a"
AAcc_\<tau>red1r_xt1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr expr1"   and xs :: "'addr val list"   and a' :: "'addr expr1"   and xs' :: "'addr val list"   and i :: "'addr expr1" assumes "\<tau>red1gr uf P t h (a, xs) (a', xs')" shows "\<tau>red1gr uf P t h (a\<lfloor>i\<rceil>, xs) (a'\<lfloor>i\<rceil>, xs')"
prems_1: shows "s (a # t)\<Longrightarrow>\<^bsub>SES\<^esub> s'"
assms_3: fixes \<chi> :: "'a ltl" assumes "\<chi> \<in> dom \<pi>\<^sub>\<A>" shows "\<pi>\<^sub>\<A> \<chi> = mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G (Abs (theG \<chi>)) w {q. dom \<pi>\<^sub>\<A> \<up>\<Turnstile>\<^sub>P q}"
hcomp_can: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term"   and w :: "'a local.term" assumes "Ide t"   and "Ide u"   and "Ide v"   and "Ide w"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>"   and "\<^bold>\<lfloor>v\<^bold>\<rfloor> = \<^bold>\<lfloor>w\<^bold>\<rfloor>"   and "Src t = Trg v"   and "Src u = Trg w" shows "can u t \<star> can w v = can (u \<^bold>\<star> w) (t \<^bold>\<star> v)"
induct: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a list" assumes "\<And>x. P x []"   and "\<And>x y xs. P x xs \<Longrightarrow> P x (y # xs)" shows "P a0 a1"
orecvmsg_def: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and a :: "'e seq_action" shows "orecvmsg P \<sigma> a \<equiv> case a of receive m \<Rightarrow> P \<sigma> m | _ \<Rightarrow> True"
lookup_single_not_eq: fixes k :: "'a"   and k' :: "'a"   and v :: "'b" assumes "k \<noteq> k'" shows "lookup (Poly_Mapping.single k v) k' = (0::'b)"
on_triple_def: fixes R :: "('d \<times> 'e) set" shows "on_triple R \<equiv> {((l, s, t), l', s', t'). l = l' \<and> (s, s') \<in> R \<and> (t, t') \<in> R}"
prv_ball_NumI: fixes x :: "'var"   and \<phi> :: "'fmla"   and n :: "nat" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<And>i. i \<le> n \<Longrightarrow> prv (subst \<phi> (Num i) x)" shows "prv (ball x (Num n) \<phi>)"
reduct_matrix: fixes a :: "'a list"   and M :: "'a list list" assumes "mat (row_length (a # M)) (length (a # M)) (a # M)" shows "mat (row_length M) (length M) M"
surj_permute: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "perm" assumes "class.pt permute" shows "surj (permute p)"
min_extend_neg: shows "sint 0xFFFFFFFF80000001 < 0"
row_rank_eq_col_rank: fixes A :: "(('a, 'b) vec, 'c) vec" shows "row_rank A = col_rank A"
left_neutral: fixes a :: "'c" shows "\<top> \<sqinter> a = a"
rev_lang: fixes r :: "'a rexp" shows "rev ` lang r = lang (Rev r)"
top_finite_def: shows "\<top> = \<Squnion> UNIV"
squanderFace_def: fixes n :: "nat" shows "\<d> n \<equiv> if n = 3 then 0 else if n = 4 then 2058 else if n = 5 then 4819 else if n = 6 then 7120 else squanderTarget"
sum_less_suminf2: fixes n :: "nat"   and i :: "nat" assumes "summable f"   and "\<And>m. n \<le> m \<Longrightarrow> (0::'a) \<le> f m"   and "n \<le> i"   and "(0::'a) < f i" shows "sum f {..<n} < suminf f"
simps_1: fixes xs :: "('a, 'b) Combinators list" shows "insertDeny (DenyAll # xs) = DenyAll # xs"
elims_1: fixes x :: "real_alg_2"   and y :: "bool" assumes "invariant_2 x = y" obtains n :: "nat" and rai :: "int poly \<times> rat \<times> rat" where "x = Irrational n rai" and "y = (invariant_1_2 rai \<and> n = card {y. y \<le> real_of_1 rai \<and> poly (real_of_int_poly (poly_real_alg_1 rai)) y = 0})"   | r :: "rat" where "x = Rational r" and "y"
tarjan_fin_nofail: fixes u :: "'v"   and s' :: "'v tarjan_state" assumes "pre_on_finish u s'" shows "nofail (tarjan_fin u s')"
coboundedI2: fixes c :: "'a multiset"   and b :: "'a multiset"   and a :: "'a multiset" assumes "c \<subseteq># b" shows "c \<subseteq># a \<union># b"