hcju/isabelleps huggingface dataset:
char_one: shows "\<chi> \<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> = 1"
nontriv_solution_of_nat_of_nat': fixes z :: "nat \<times> nat" shows "nontriv_solution (case z of (a, b) \<Rightarrow> (of_nat a, of_nat b)) = nontriv_solution z"
p\<^sub>2: shows "on_curve a b p\<^sub>2"
simps_4: fixes x2 :: "'a"   and x1 :: "'a" shows "Neg x2 \<noteq> Pos x1"
cong: fixes x :: "'a" assumes "x \<in> S" shows "f x = g x"
build_ip_partition_complete: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "\<Union> (wordinterval_to_set ` set (build_ip_partition c rs)) = UNIV"
NSLIM_I: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd"   and L :: "'d" assumes "\<And>x. \<lbrakk>x \<noteq> star_of a; x \<approx> star_of a\<rbrakk> \<Longrightarrow> (*f* f) x \<approx> star_of L" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"
unit_simps_5: fixes a :: "'a" assumes "S.obj a" shows "S.cod \<i>[a] = a"
TryCatch: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list"   and e\<^sub>2 :: "char list exp"   and V :: "char list"   and e\<^sub>2' :: "char list exp"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"   and "h\<^sub>1 a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C"   and "P \<turnstile> \<langle>e\<^sub>2,(h\<^sub>1, l\<^sub>1(V \<mapsto> Addr a), sh\<^sub>1)\<rangle> \<Rightarrow> \<langle>e\<^sub>2',(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>" shows "P \<turnstile> \<langle>try e\<^sub>1 catch(C V) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>2',(h\<^sub>2, l\<^sub>2(V := l\<^sub>1 V), sh\<^sub>2)\<rangle>"
inv_ordsetmap_strict: fixes a :: "'b"   and b :: "'b" assumes "a \<in> f ` P"   and "b \<in> f ` P"   and "codomain.greater b a" shows "the_inv_into P f b \<succ> the_inv_into P f a"
Smith_1x2_eucl_def: fixes A :: "(('b, 2) Finite_Cartesian_Product.vec, 1) Finite_Cartesian_Product.vec" shows "Smith_1x2_eucl A = (if A $h 0 $h 0 = (0::'b) \<and> A $h 0 $h 1 \<noteq> (0::'b) then let Q = interchange_columns (Finite_Cartesian_Product.mat (1::'b)) 0 1; A' = interchange_columns A 0 1 in (A', Q) else if A $h 0 $h 0 \<noteq> (0::'b) \<and> A $h 0 $h 1 \<noteq> (0::'b) then let bezout_matrix_right = Finite_Cartesian_Product.transpose (bezout_matrix (Finite_Cartesian_Product.transpose A) 0 1 0 euclid_ext2) in (A ** bezout_matrix_right, bezout_matrix_right) else (A, Finite_Cartesian_Product.mat (1::'b)))"
nsqn\<^sub>r_inv: fixes dsn :: "nat"   and dsk :: "k"   and hops :: "nat"   and nhip :: "nat" shows "nsqn\<^sub>r (dsn, dsk, Aodv_Basic.inv, hops, nhip) = dsn - 1"
of_int_le_0_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus z) zero = (z \<le> 0)"
LBot: shows "([],  \<LM> ff  \<RM> \<Rightarrow>* \<Empt>) \<in> Ax"
nz: fixes x :: "'b" assumes "x \<in> A" shows "f x \<noteq> (0::'a)"
map_pmf_comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and M :: "'d pmf" shows "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
generate_m_inv_closed: fixes H :: "'a set"   and h :: "'a" assumes "H \<subseteq> carrier G"   and "h \<in> generate G H" shows "inv h \<in> generate G H"
assms_1: fixes p :: "perm" shows "(p \<bullet> x = x') = (p \<bullet> y = y')"
section_retraction_of_iso_2: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.iso f"   and "VV.ide (VV.comp f g)" shows "VV.inverse_arrows f g"
wI_2: shows "wf_interp_for_formula (w, I) (FAnd \<phi> \<psi>)"
fix_argument_promote_fun: fixes g :: "'a \<Rightarrow> 'b \<rightarrow> 'b" shows "(\<lambda>x. fix\<cdot>(g x)) = (\<mu> x. (\<lambda>xa. g xa\<cdot>(x xa)))"
antisym: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "(a, b) \<in> rel D"   and "(b, a) \<in> rel D" shows "a = b"
CatchMiss: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "'b"   and t :: "('b, 'c) xstate"   and c\<^sub>2 :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> t"   and "\<not> isAbr t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
append_vec_le: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec"   and v' :: "'a vec"   and w' :: "'a vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "(v @\<^sub>v v' \<le> w @\<^sub>v w') = (v \<le> w \<and> v' \<le> w')"
non_zero_components: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and n :: "nat"   and y :: "bool stream" assumes "val_process Mkt pf n y \<noteq> 0"   and "stock_portfolio Mkt pf" shows "pf stk (Suc n) y \<noteq> 0 \<or> pf risk_free_asset (Suc n) y \<noteq> 0"
assms: shows "x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> y"
collapse: fixes option :: "\<langle>'d\<rangle>\<^sub>\<bottom>" assumes "option \<noteq> None" shows "\<lfloor>the option\<rfloor> = option"
trm_dom_subst: fixes \<Gamma> :: "binding list"   and X :: "tyvrs"   and T :: "ty" shows "trm_dom (\<Gamma>[X \<mapsto> T]\<^sub>e) = trm_dom \<Gamma>"
ideal_iff_lideal: fixes I :: "'a set" shows "ideal I R (+) (\<cdot>) \<zero> \<one> = lideal I R (+) (\<cdot>) \<zero> \<one>"
span_Int: fixes A :: "'b set"   and B :: "'b set" shows "local.span (local.span A \<inter> local.span B) = local.span A \<inter> local.span B"
F: fixes x :: "'b" shows "f -` {x} \<inter> space M \<in> sets (completion M)"
MkIde_Dom: fixes a :: "('a, 'a set) arr" assumes "arr a" shows "MkIde (local.Dom a) = local.dom a"
elims_2: fixes x :: "nat"   and xa :: "'person"   and xb :: "'person \<Rightarrow> 'color" assumes "local.leaves x xa xb" obtains n :: "nat" and p :: "'person" and w :: "'person \<Rightarrow> 'color" where "x = n" and "xa = p" and "xb = w" and "\<forall>w'. local.possible n p w w' \<longrightarrow> w' p = w p"
Chn_LHS_eq: shows "Chn LHS = \<lbrakk>\<lbrakk>HHfgh_HfHgh.chine\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>THfgh_HHfgh.chine\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>TTfgh_THfgh.chine\<rbrakk>\<rbrakk>"
fresh_def: fixes a :: "'b"   and x :: "'c" shows "a \<sharp> x = (a \<notin> supp x)"
simps_1: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex"   and var :: "'a"   and val :: "bool" shows "restrict (IF v t e) var val = (let rt = restrict t var val; re = restrict e var val in if v = var then if val then rt else re else IF v rt re)"
weaken_left: fixes G :: "'b pl set"   and H :: "'b pl set"   and p :: "'b pl" assumes "G \<subseteq> H"   and "G |- p" shows "H |- p"
eq_\<I>_attach_on: fixes \<I>' :: "('i, 'j) \<I>"   and res :: "('i, 'j) resource"   and A :: "'k set"   and conv :: "('k, 'l, 'i, 'j) converter"   and conv' :: "('k, 'l, 'i, 'j) converter" assumes "\<I>' \<turnstile>res res \<surd>"   and "\<I>_uniform A UNIV,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'" shows "A \<turnstile>\<^sub>R conv \<rhd> res \<sim> conv' \<rhd> res"
upper_round_ivl: fixes p :: "nat"   and x :: "float interval" shows "upper (round_interval p x) = float_round_up p (upper x)"
reach_def: fixes tr :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and p :: "'a"   and as :: "'b list"   and q :: "'a" shows "reach tr p as q = (q = foldl tr p as)"
imp_eqvt: fixes p :: "perm"   and A :: "bool"   and B :: "bool" shows "p \<bullet> (A \<longrightarrow> B) = (p \<bullet> A \<longrightarrow> p \<bullet> B)"
H: fixes n :: "nat"   and m :: "nat" assumes "n < m"   and "0 < n" shows "\<exists>q. MPoly_Type.monom (monomial (Suc 0) x') 1 ^ n = MPoly_Type.monom (monomial (Suc 0) x') 1 + q * ((MPoly_Type.monom (monomial (Suc 0) x') 1)\<^sup>2 - MPoly_Type.monom (monomial (Suc 0) x') 1)"
assms: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "r a b = r' a b"
postListpostSet: fixes xs :: "('a \<times> ('b \<Rightarrow> 'c)) list"   and l :: "'a \<Rightarrow> 'c"   and s :: "'b" assumes "postSet (set xs) l s" shows "postList xs l s"
f: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A"   and "x \<noteq> 0" shows "f\<lparr>x\<rparr> \<in>\<^sub>\<circ> x"
f3: fixes r :: "real"   and ra :: "real"   and rb :: "real" shows "r - ra + rb + ra - r = rb"
\<tau>multithreaded_wf_init_fin: shows "\<tau>multithreaded_wf r2.init_fin_final r2.init_fin r2.init_fin_\<tau>move"
IH_6: assumes "\<langle>c: t \<in> s\<rangle> \<in> set S" shows "fv t \<union> fv s \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
Basis_nonneg: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'a" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "less_eq zero i"
assms_3: fixes dir :: "('i, 'a) Direction"   and x :: "nat"   and c :: "'a" assumes "dir = Positive \<or> dir = Negative"   and "a = LE dir x c"   and "x \<notin> lvars (\<T> s)"   and "lt dir c (\<langle>\<V> s\<rangle> x)"   and "\<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)"   and "\<not> \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)" shows "P' (lt dir) (UBI dir) (LBI dir) (UB dir) (LB dir) (UBI_upd dir) (UI dir) (LI dir) (LE dir) (GE dir) (update x c (update\<B>\<I> (UBI_upd dir) i x c s))"
subset_eq: fixes A :: "'k set"   and B :: "'k set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
comp_quasi_inverse_1: fixes f :: "'b" assumes "D.equivalence_map f" shows "f\<^sup>~\<^sup>D \<star>\<^sub>D f \<cong>\<^sub>D src\<^sub>D f"
G_af_simp: fixes \<phi> :: "'b ltl"   and w :: "'b set list" shows "\<^bold>G (af \<phi> w) = \<^bold>G \<phi>"
monom_times_poly_shift: fixes n :: "nat"   and p :: "'b poly"   and c :: "'b" assumes "n \<le> poly_subdegree p" shows "monom c n * poly_shift n p = smult c p"
signed_take_bit_take_bit: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and "and" :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and or :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and xor :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and mask :: "nat \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and not :: "'k \<Rightarrow> 'k"   and m :: "nat"   and n :: "nat"   and a :: "'k" 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 m (take_bit n a) = (if n \<le> m then take_bit n else ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m) a"
inheritedI: fixes P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and subs :: "'b \<Rightarrow> 'b set" assumes "\<And>A B. (P A \<and> P B) = P (A \<union> B)"   and "\<And>A. P A = P (incLevel ` A)"   and "\<And>n Gamma A. \<not> terminal subs Gamma \<Longrightarrow> P A = P (insert (n, Gamma) A)"   and "P {}" shows "inherited subs P"
add_mono_thms_linordered_semiring_1: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i \<le> j \<and> k \<le> l" shows "i + k \<le> j + l"
2: shows "(local.inv \<eta> \<star> g) \<cdot> (g \<star> local.inv \<epsilon>) = g"
incrs: shows "incrs \<equiv> \<lambda>h t x. incr h t x + s h t x"
pcr_sq_matrix_def: fixes T :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and Ta :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "pcr_sq_matrix T Ta \<equiv> rel_fun T (rel_fun T Ta) OO cr_sq_matrix"
assms_5: shows "ideal F \<div> monomial (1::'a) t = ideal (monomial (1::'a) ` S)"
mem_aug_minf: fixes a :: "ant" assumes "a \<in> Z\<^sub>-\<^sub>\<infinity>" shows "a = - \<infinity> \<or> (\<exists>z. a = ant z)"
hom_restrict: fixes h :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c"   and H :: "('c, 'd) monoid_scheme"   and t :: "('b \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c" assumes "h \<in> hom (free_Abelian_group T) H"   and "\<And>g. g \<in> carrier (free_Abelian_group T) \<Longrightarrow> h g = t g" shows "t \<in> hom (free_Abelian_group T) H"
continuous_on_generate_topology: fixes X :: "'a set set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "open = generate_topology X"   and "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A" shows "continuous_on A f"
assms_1: fixes n :: "nat" shows "0 \<le> a n"
invrules_2: shows "\<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2) = U - V"
principal_eq_bot_iff: fixes X :: "'b set" shows "(principal X = bot) = (X = {})"
normalize_mult_unit_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a one" shows "normalize (times a b) = normalize b"
liftAll_mono: fixes phi :: "'b \<Rightarrow> bool"   and chi :: "'b \<Rightarrow> bool"   and inp :: "'c \<Rightarrow> 'b option" assumes "\<And>v. phi v \<Longrightarrow> chi v"   and "liftAll phi inp" shows "liftAll chi inp"
lossless_sample_uniform: fixes n :: "nat" shows "lossless_spmf (sample_uniform n) = (0 < n)"
assms_1: fixes i :: "nat" assumes "i \<le> n" shows "\<exists>j. suffix (i + j) w \<Turnstile>\<^sub>n af \<phi> (w [i \<rightarrow> j + i])[X]\<^sub>\<nu>"
padic_int_simps_3: fixes p :: "int" shows "(\<oplus>\<^bsub>padic_int p\<^esub>) = padic_add p"
comp_preserves_can: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.can f"   and "\<F>C.can g"   and "\<F>C.dom f = \<F>C.cod g" shows "\<F>C.can (f \<cdot> g)"
totient_primorial'_asymp_equiv: shows "(\<lambda>x. real (totient (primorial' x))) \<sim>[sequentially] (\<lambda>k. third_mertens_const * real (primorial' k) / ln (real k))"
2: shows "\<p>\<^sub>0[\<nu>.cod.leg0, \<mu>.cod.leg1] \<cdot> chine_hcomp \<nu> \<mu> = \<mu>.chine \<cdot> \<p>\<^sub>0[\<nu>.dom.leg0, \<mu>.dom.leg1]"
mset_union_subset: fixes A :: "'c multiset"   and B :: "'c multiset"   and C :: "'c multiset" assumes "A + B \<subseteq># C" shows "A \<subseteq># C \<and> B \<subseteq># C"
singleton_def: fixes default :: "unit \<Rightarrow> 'a"   and A :: "'a Predicate.pred" shows "Predicate.singleton default A = (if \<exists>!x. pred.eval A x then THE x. pred.eval A x else default ())"
cont_prodD2: fixes luba :: "'b set \<Rightarrow> 'b"   and lubb :: "'c set \<Rightarrow> 'c"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubc :: "'d set \<Rightarrow> 'd"   and ordc :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<times> 'c \<Rightarrow> 'd"   and y :: "'c" assumes "cont (prod_lub luba lubb) (rel_prod orda ordb) lubc ordc f"   and "class.preorder ordb (mk_less ordb)"   and "lub_singleton lubb" shows "cont luba orda lubc ordc (\<lambda>x. f (x, y))"
f_alt_def: fixes x :: "nat \<Rightarrow>\<^sub>0 nat" shows "f x = MPoly_Type.monom x (MPoly_Type.coeff p x)"
succeed_t_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat" shows "mojmir_def.succeed_t \<Sigma> \<delta> q\<^sub>0 w F i = {n. \<exists>q. semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = Some i \<and> q \<notin> F - {q\<^sub>0} \<and> \<delta> q (w n) \<in> F}"
mult_le_cancel_right1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq c (times b c) = ((less zero c \<longrightarrow> less_eq one b) \<and> (less c zero \<longrightarrow> less_eq b one))"
append_Cons_eq_iff: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list"   and xs' :: "'a list"   and ys' :: "'a list" assumes "x \<notin> set xs"   and "x \<notin> set ys" shows "(xs @ x # ys = xs' @ x # ys') = (xs = xs' \<and> ys = ys')"
preserves_dom: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "functor A B F"   and "partial_magma.arr A f" shows "partial_magma.dom B (F f) = F (partial_magma.dom A f)"
is_bound_def: fixes Q\<^sub>b :: "'a"   and Q :: "'a set" shows "is_bound Q\<^sub>b Q \<equiv> \<exists>f. is_bound_f Q\<^sub>b Q f"
wf_clinit_def: fixes ms :: "'b mdecl list" shows "wf_clinit ms = (\<exists>m. (clinit, Static, [], Void, m) \<in> set ms)"
distr_pair_swap: fixes M1 :: "'a measure"   and M2 :: "'b measure" assumes "pair_sigma_finite M1 M2" shows "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))"
alternating_list_SucSuc_ConsCons: fixes k :: "nat"   and s :: "'b"   and t :: "'b" shows "alternating_list (Suc (Suc k)) s t = s # t # alternating_list k s t"
slide_mem: fixes i :: "nat"   and j :: "nat"   and l :: "nat" assumes "i \<le> j"   and "l \<in> {k. k \<le> j - i}" shows "slide i l \<in> nset i j"
slide_def: fixes i :: "nat"   and j :: "nat" shows "slide i j \<equiv> i + j"
axioms_1: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'a" assumes "GroupEnd G T" shows "GroupHom G T"
seqE: fixes g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "seq g f" obtains "arr f" and "arr g" and "local.dom g = cod f"
CB_project_not_bot: fixes scrut :: "CValue"   and v\<^sub>1 :: "CValue"   and v\<^sub>2 :: "CValue" shows "(CB_project\<cdot>scrut\<cdot>v\<^sub>1\<cdot>v\<^sub>2 \<noteq> \<bottom>) = (\<exists>b. scrut = CB\<cdot>(Discr b) \<and> (if b then v\<^sub>1 else v\<^sub>2) \<noteq> \<bottom>)"
subspace_def: fixes S :: "'d set" shows "csubspace S = ((0::'d) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *\<^sub>C x \<in> S))"
PfP_quot_contra: fixes \<alpha> :: "fm" assumes "ground_fm \<alpha>" shows "{} \<turnstile> PfP \<guillemotleft>\<alpha>\<guillemotright> IMP PfP \<guillemotleft>Neg \<alpha>\<guillemotright> IMP PfP \<guillemotleft>Fls\<guillemotright>"
assms_3: shows "mat (rat_poly.row_length m3) nc m3"
mem_Times_iff: fixes x :: "'c \<times> 'd"   and A :: "'c set"   and B :: "'d set" shows "(x \<in> A \<times> B) = (fst x \<in> A \<and> snd x \<in> B)"
poly_rel_normalize: shows "(poly_rel ===> poly_rel) (normalize_poly_i ops) normalization_semidom_class.normalize"
simps_4: fixes v :: "nat"   and ts :: "nat" shows "max_tstp (Inr v) (Inl ts) = Inl ts"
funcset_compose: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "compose A g f \<in> A \<rightarrow> C"
tendsto_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b"   and g :: "'c \<Rightarrow> 'b"   and y :: "'b" assumes "class.linorder_topology less_eq less open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open x) F"   and "filterlim g (topological_space.nhds open y) F"   and "\<forall>\<^sub>F x in F. less_eq (g x) (f x)" shows "less_eq y x"
project_inject_cancel: fixes x :: "'a" shows "project (inject x) = x"
wf_merge: fixes n :: "nat"   and I :: "nat set"   and t1 :: "'a option list"   and J :: "nat set"   and t2 :: "'a option list"   and V :: "nat set"   and t :: "'a option list" assumes "wf_tuple n I t1"   and "wf_tuple n J t2"   and "V = I \<union> J"   and "t = merge t1 t2" shows "wf_tuple n V t"
supset_circuit_imp_dep: fixes C :: "'a set"   and X :: "'a set" assumes "\<E>.circuit C \<and> C \<subseteq> X" shows "\<not> indep_in \<E> X"
that: fixes ys1 :: "'a list"   and ys2 :: "'a list" assumes "ys1 \<in> listset xs1"   and "ys2 \<in> listset xs2"   and "ys = ys1 @ ys2" shows "thesis"
divide_right_mono_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less_eq c zero" shows "less_eq (divide b c) (divide a c)"
less_term: fixes x :: "'a"   and y :: "'a" assumes "rep_nat_term x < rep_nat_term y" shows "x < y"
emb_step_fun: fixes t :: "('a, 'b) tm" assumes "is_App t" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b fun t"
S_o_alpha_typ: fixes S :: "nat \<Rightarrow> typ"   and alpha :: "nat \<Rightarrow> nat"   and t :: "typ" shows "$ (S \<circ> alpha) t = $ S ($ (\<lambda>x. TVar (alpha x)) t)"
Rel_eq_refl: fixes x :: "'e" shows "Transfer.Rel (=) x x"
map\<^sub>1_simps_4: fixes f :: "'a" assumes "C.ide f" shows "D.dom (\<l>\<^sub>D\<^sup>-\<^sup>1[G f] \<cdot>\<^sub>D \<r>\<^sub>D[G f]) = G f \<star>\<^sub>D G.map\<^sub>0 (src\<^sub>C f)"
size_5: fixes x2 :: "'b" shows "size (In2 x2) = Suc 0"
crdt_ops_spec_ops_exist: fixes xs :: "('a \<times> 'b) list"   and deps :: "'b \<Rightarrow> 'a set" assumes "crdt_ops xs deps" shows "\<exists>ys. set xs = set ys \<and> spec_ops ys deps"