hcju/isabelleps huggingface dataset:
rtranclp_idemp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "r\<^sup>*\<^sup>*\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
oprod_zero_2: fixes r :: "('d \<times> 'd) set" shows "r *o {} = {}"
uf_graph_relsndD: fixes a :: "('a \<times> 'a) set"   and b :: "'edge set"   and c :: "'edge set" assumes "((a, b), c) \<in> uf_graph_rel" shows "b = c"
Entry_Exit_edge: shows "\<exists>a. valid_edge a \<and> sourcenode a = (_Entry_) \<and> targetnode a = (_Exit_) \<and> kind a = (\<lambda>s. False)\<^sub>\<surd>"
results'_gpv_Fail: shows "results'_gpv Fail = {}"
convergent_subseq_convergent: fixes X :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> nat" assumes "convergent X"   and "strict_mono f" shows "convergent (X \<circ> f)"
distinct_14: fixes x51 :: "'b gexp"   and x52 :: "'b gexp"   and x21 :: "'b aexp"   and x22 :: "'b aexp" shows "Nor x51 x52 \<noteq> Eq x21 x22"
pairwise_compat: fixes c1 :: "('a, 'b) chart"   and c2 :: "('a, 'b) chart" assumes "c1 \<in> charts1"   and "c2 \<in> charts1" shows "k-smooth_compat c1 c2"
secure_def: shows "secure \<equiv> \<forall>tr vl vl1. validFrom istate tr \<and> never T tr \<and> B vl vl1 \<and> V tr = vl \<longrightarrow> (\<exists>tr1. validFrom istate tr1 \<and> O tr1 = O tr \<and> V tr1 = vl1)"
Z_eq_Arg_plus_def: fixes n :: "nat" shows "(Z=Arg+n) = (\<lambda>Z s. Z = s<Arg> + n)"
vals1: fixes prim :: "prim"   and p :: "lambda" assumes "P prim \<in> vals p" shows "prim \<in> prims p"
bisim_inv1: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and ta1 :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s1' :: "'x1 \<times> 'm1" assumes "t \<turnstile> s1 \<approx> s2"   and "t \<turnstile> s1 -1-ta1\<rightarrow> s1'" obtains s2' :: "'x2 \<times> 'm2" where "t \<turnstile> s1' \<approx> s2'"
strip_while_idem: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "no_trailing P xs" shows "strip_while P xs = xs"
hom_mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "act (x \<otimes>\<^bsub>G\<^esub> y) = act x \<otimes>\<^bsub>BijGroup X\<^esub> act y"
B: fixes self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "(\<lambda>_. Set{\<lambda>_. self \<tau>.salary@pre} \<tau>) = (\<lambda>_. Set{self.salary@pre} \<tau>)"
less_multiset_ext\<^sub>D\<^sub>M_imp_mult: fixes N :: "'b multiset"   and A :: "'b set"   and M :: "'b multiset"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "set_mset N \<subseteq> A"   and "set_mset M \<subseteq> A"   and "less_multiset_ext\<^sub>D\<^sub>M R M N" shows "(M, N) \<in> mult {(x, y). x \<in> A \<and> y \<in> A \<and> R x y}"
pmf_of_alist_support: fixes \<mu> :: "('a \<times> real) list" assumes "\<forall>(uu_, p)\<in>set \<mu>. 0 \<le> p"   and "sum_list (map snd \<mu>) = 1"   and "distinct (map fst \<mu>)" shows "set_pmf (pmf_of_alist \<mu>) \<subseteq> fst ` set \<mu>"
pmf_of_set: fixes S :: "'c set"   and x :: "'c" assumes "S \<noteq> {}"   and "finite S" shows "pmf (local.pmf_of_set S) x = indicat_real S x / real (card S)"
add_zero: fixes P :: "'a mpoly" shows "P + Const (0::'a) = P"
y1_prop: shows "q < y1 \<and> (\<forall>a\<in>set leq. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y1}. d * x\<^sup>2 + e * x + f \<le> 0)"
ii: shows "divisor_set 75 = {1, 3, 5, 15, 25, 75}"
\<pi>_bigo: shows "\<pi> \<in> O(\<lambda>x. x / ln x)"
rightreg_scalar_multD1: fixes x :: "'r"   and r :: "'r"   and f :: "'r \<Rightarrow> 'a" assumes "x \<in> R" shows "(r \<currency> f) x = f (x * r)"
approx_def: fixes x :: "'c star"   and y :: "'c star" shows "(x \<approx> y) = (x - y \<in> Infinitesimal)"
isTable_eq: fixes E :: "'a \<Rightarrow> 'b"   and vs :: "'a list"   and a :: "'a"   and b :: "'b"   and ps :: "('a \<times> 'b) list" assumes "isTable E vs ((a, b) # ps)" shows "b = E a"
A_def: shows "A \<equiv> sum_upto (\<lambda>n. a n / real n)"
finprod_0': fixes f :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "f \<in> {..n} \<rightarrow> carrier (ideals_set R)" shows "f 0 \<otimes>\<^bsub>ideals_set R\<^esub> finprod (ideals_set R) f {Suc 0..n} = finprod (ideals_set R) f {..n}"
has_integral_diff: fixes f :: "'a \<Rightarrow> 'b"   and k :: "'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "(f has_integral k) S"   and "(g has_integral l) S" shows "((\<lambda>x. f x - g x) has_integral k - l) S"
dist': fixes P :: "face \<Rightarrow> bool" shows "distinct (filter P (facesAt g' v))"
range_vars_def: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "range_vars \<sigma> = \<Union> (fv ` subst_range \<sigma>)"
sorted_append: fixes xs :: "'a list list"   and ys :: "'a list list" shows "linorder.sorted (\<le>lex) (xs \<cdot> ys) = (linorder.sorted (\<le>lex) xs \<and> linorder.sorted (\<le>lex) ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. x \<le>lex y))"
comm_mult_strict_left_mono: fixes a :: "'e"   and b :: "'e"   and c :: "'e" assumes "a < b"   and "(0::'e) < c" shows "c * a < c * b"
map_fun2_id: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'e"   and x :: "'f \<Rightarrow> 'g" shows "map_fun f g x = g \<circ> map_fun f id x"
cos_add: fixes x :: "'a"   and y :: "'a" shows "cos (x + y) = cos x * cos y - sin x * sin y"
hurwitz_zeta_one_half_left: fixes s :: "complex" assumes "s \<noteq> 1" shows "hurwitz_zeta (1 / 2) s = (2 powr s - 1) * zeta s"
lcs_adds: fixes s :: "'a"   and u :: "'a"   and t :: "'a" assumes "s adds u"   and "t adds u" shows "lcs s t adds u"
complete2: fixes A :: "'a set"   and B :: "'a set"   and M :: "'a measure" assumes "A \<subseteq> B"   and "B \<in> null_sets (completion M)" shows "A \<in> null_sets (completion M)"
finsum_vec_def: fixes ty :: "'c itself"   and n :: "nat" shows "finsum_vec ty n = finprod (monoid_vec ty n)"
simps: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "nfoldli l c f s = (case l of [] \<Rightarrow> RETURN s | x # ls \<Rightarrow> if c s then f x s \<bind> nfoldli ls c f else RETURN s)"
adjacent_append_atomics_reusables: fixes w :: "nat \<times> nat"   and t :: "'a tree" assumes "0 < fst w"   and "fst w \<le> snd w"   and "local.valid t"   and "l t \<le> fst w"   and "r t \<le> snd w" shows "adjacent w (local.atomics (max (fst w) (Suc (r t))) (snd w) @ reusables t w)"
GB_alt_3_finite: fixes G :: "(('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite G" shows "pp_pm.extended_ord.punit.is_Groebner_basis G = (\<forall>f\<in>MPoly_Type_Class.punit.pmdl G. f \<noteq> 0 \<longrightarrow> (\<exists>g\<in>G. g \<noteq> 0 \<and> MPoly_Type_Class.punit.adds_term (pp_pm.extended_ord.lpp g) (pp_pm.extended_ord.lpp f)))"
fromInteger_Integer_def: shows "fromInteger = (\<Lambda> x. x)"
M: fixes x :: "'a" assumes "x \<in> S" shows "norm (f x) \<le> M"
face_of_aff_dim_lt: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "T face_of S"   and "T \<noteq> S" shows "aff_dim T < aff_dim S"
cSup_greaterThanLessThan: fixes y :: "'a"   and x :: "'a" assumes "y < x" shows "Sup {y<..<x} = x"
H_1: shows "Pi\<^sub>E UNIV X \<subseteq> U"
ListMem_iff: fixes x :: "'d"   and xs :: "'d list" shows "ListMem x xs = (x \<in> set xs)"
wp_skip_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wp \<pi> SKIP Q s = Q s"
purge_def: fixes execs :: "'dom_t \<Rightarrow> 'action_t list list"   and u :: "'dom_t" shows "purge execs u \<equiv> \<lambda>d. if unrelated d u then SOME alpha. realistic_execution alpha else execs d"
comp_monoid_morphisms: fixes \<eta> :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and multA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and oneA :: "'d"   and B :: "'e set"   and multB :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and oneB :: "'e"   and \<theta> :: "'e \<Rightarrow> 'f"   and C :: "'f set"   and multC :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and oneC :: "'f" assumes "monoid_homomorphism \<eta> A multA oneA B multB oneB"   and "monoid_homomorphism \<theta> B multB oneB C multC oneC" shows "monoid_homomorphism (\<theta> \<circ> \<eta> \<down> A) A multA oneA C multC oneC"
spec\<^sub>1_def: fixes p :: "prog" shows "spec\<^sub>1 p \<equiv> wfe \<Gamma>\<^sub>x\<^sub>y (body p) \<and> para p = [''x'', ''y''] \<and> (\<forall>x y. eval (\<E>\<^sub>x\<^sub>y x y) (body p) = Some (f x y)) \<and> coste (body p) \<le> 3"
S4ReadInner: fixes p :: "PrIds"   and l :: "Locs"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> ReadInner rmCh mm ires p l \<and> $S4 rmhist p \<and> unchanged (e p, c p, r p) \<and> HNext rmhist p \<and> $MemInv mm l \<longrightarrow> S4 rmhist p$ \<and> unchanged (rmhist!p)"
case_prodI2: fixes p :: "'h \<times> 'i"   and c :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
cone_cone_hull: fixes S :: "'a set" shows "cone (cone hull S)"
ipurge_ref_empty: fixes v :: "'c"   and u :: "'c"   and I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and X :: "'d set" assumes "v = u \<or> v \<in> sinks I D u xs"   and "\<And>x. x \<in> X \<Longrightarrow> (v, D x) \<in> I" shows "ipurge_ref I D u xs X = {}"
summable_ratio_test: fixes c :: "real"   and N :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "c < 1"   and "\<And>n. N \<le> n \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)" shows "summable f"
safe_safe: fixes s :: "event list"   and r :: "room" assumes "safe s r" shows "safe\<^sub>0 s r"
a_Hcarr: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme"   and h :: "'a" assumes "additive_subgroup H G"   and "h \<in> H" shows "h \<in> carrier G"
gt_trans: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and u :: "('b, 'a) tm"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "arity.wary arity_sym arity_var u"   and "arity.wary arity_sym arity_var t"   and "arity.wary arity_sym arity_var s"   and "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym u t"   and "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym t s" shows "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym u s"
pid_system_automaton_simps_1: shows "pid.V = UNIV"
cases: fixes a :: "'b sequent list \<times> 'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set" assumes "a \<in> R*" obtains r :: "'b sequent list \<times> 'b sequent" and seq :: "'b sequent" where "a = extendRule seq r" and "r \<in> R"
sym_def: fixes r :: "('e \<times> 'e) set" shows "sym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
strict_trans1: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "(b::'b) \<le> (a::'b)"   and "(c::'b) < (b::'b)" shows "(c::'b) < (a::'b)"
flopped_half_apartments_fg: shows "folding_f.Y = g \<turnstile> X"
Nil: fixes x :: "'a vec" assumes "x \<in> set []" shows "x \<in> carrier_vec n"
simps: fixes m :: "'d \<Rightarrow> nat"   and f :: "'d \<Rightarrow> 'd option"   and x :: "'d" shows "fixpoint m f x = (case f x of None \<Rightarrow> None | Some x' \<Rightarrow> if m x' < m x then fixpoint m f x' else Some x')"
Limsup_mono: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. f x \<le> g x" shows "Limsup F f \<le> Limsup F g"
A: fixes i :: "'i" assumes "i \<in> I \<union> J" shows "A i \<in> sets (M i)"
assms_1: shows "u \<in> \<langle>\<C>\<rangle>"
fags_ippTr: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int" assumes "fags_cond f z i a" shows "fags_ipp f z i a (n\<Odot>a\<^bsub>f,i,z\<^esub>) = (- n)\<Odot>a\<^bsub>f,i,z\<^esub>"
linear_simps_1: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c" assumes "bounded_linear f" shows "f (a + b) = f a + f b"
expand_def: fixes f :: "'c \<Rightarrow> 'd set" shows "expand f = {g. \<forall>x. g x \<in> f x}"
linear_componentwise: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and j :: "'e" assumes "linear f" shows "f x \<bullet> j = (\<Sum>i\<in>Basis. x \<bullet> i * (f i \<bullet> j))"
measurable_envelopeD_4: fixes M :: "'b measure"   and A :: "'b set"   and E :: "'b set" assumes "measurable_envelope M A E" shows "A \<subseteq> space M"
sc_0_m: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and m :: "'e" assumes "R module M"   and "m \<in> carrier M" shows "\<zero>\<^bsub>R\<^esub> \<cdot>\<^sub>s\<^bsub>M\<^esub> m = \<zero>\<^bsub>M\<^esub>"
rel_transp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transp R" shows "transp (list_all2 R)"
nonzero_coeffs_diff: fixes A :: "('a \<times> 'b) multiset"   and B :: "('a \<times> 'b) multiset" assumes "nonzero_coeffs A" shows "nonzero_coeffs (A - B)"
path_components_of_eq_empty: fixes X :: "'b topology" shows "(path_components_of X = {}) = (topspace X = {})"
binomial_Suc: fixes i :: "nat" assumes "i \<in> {1..n}" shows "Suc n choose i = n choose (i - 1) + (n choose i)"
simps_3: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and uw :: "'a"   and ux :: "'a list" shows "lex_ext_unbounded f [] (uw # ux) = (False, False)"
Shuffle_comm: fixes r :: "'a language"   and s :: "'a language" shows "Shuffle r s = Shuffle s r"
antisymE: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> r"   and "(b, a) \<in> r" shows "a = b"
sep_conjD: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P Q h" shows "\<exists>x y. sep_disj x y \<and> h = plus x y \<and> P x \<and> Q y"
None_lcl_casts_values: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and V :: "char list"   and T :: "ty"   and v' :: "val" assumes "wf_prog wf_md P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "l V = None"   and "E V = \<lfloor>T\<rfloor>"   and "l' V = \<lfloor>v'\<rfloor>" shows "P \<turnstile> T casts v' to v' "
assms_4: shows "fmrestrict_set vsa fm = fmrestrict_set (vsa \<inter> vs) fma"
fps_mult_nth_outside_subdegrees_2: fixes n :: "nat"   and g :: "'a fps"   and f :: "'a fps" assumes "n < subdegree g" shows "fps_nth (f * g) n = (0::'a)"
connected_append_edge: fixes c :: "nat \<times> nat \<Rightarrow> 'c"   and u :: "nat"   and v :: "nat"   and w :: "nat" assumes "Graph.connected c u v"   and "(v, w) \<in> Graph.E c" shows "Graph.connected c u w"
elims: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b list"   and xb :: "'a list"   and y :: "'a list" assumes "map_tailrec_rev x xa xb = y" obtains f :: "'b \<Rightarrow> 'a" and bs :: "'a list" where "x = fa__" and "xa = []" and "xb = bs__" and "y = bs__"   | f :: "'b \<Rightarrow> 'a" and a :: "'b" and as :: "'b list" and bs :: "'a list" where "x = f" and "xa = a # as" and "xb = bs" and "y = map_tailrec_rev f as (f a # bs)"
cls_val_process_def: fixes Mkt :: "('b, 'c) discrete_market"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real" shows "cls_val_process Mkt p = (if \<not> portfolio p then \<lambda>n w. 0 else up_cl_proc Mkt p)"
ThrowReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>throw e,s,b\<rangle> \<rightarrow>* \<langle>throw e',s',b'\<rangle>"
HIm_def: fixes q :: "octo" shows "HIm q \<equiv> vector [Im1 q, Im2 q, Im3 q, Im4 q, Im5 q, Im6 q, Im7 q]"
simps: fixes c :: "nat" shows "left c = lo c (Pi 0)"
simps_15: fixes x2 :: "'e" shows "set_option (Some x2) = {x2}"
inj_on_image_mem_iff: fixes f :: "'d \<Rightarrow> 'e"   and B :: "'d set"   and a :: "'d"   and A :: "'d set" assumes "inj_on f B"   and "a \<in> B"   and "A \<subseteq> B" shows "(f a \<in> f ` A) = (a \<in> A)"
f1: fixes f :: "'fmla"   and fa :: "'fmla"   and fb :: "'fmla" shows "f \<notin> fmla \<or> \<not> prv (imp fa fb) \<or> fb \<notin> fmla \<or> fa \<notin> fmla \<or> prv (imp fa (dsj fb f))"
f_carrier: fixes n :: "nat"   and fs :: "'b vec list"   and i :: "nat" assumes "gram_schmidt_fs_Rn n fs"   and "i < length fs" shows "fs ! i \<in> carrier_vec n"
Un_UNIV_left: fixes B :: "'b set" shows "UNIV \<union> B = UNIV"
multiplicative_function'_axioms: shows "multiplicative_function' (ind squarefree) (\<lambda>p k. if 1 < k then 0::'a else (1::'a)) (\<lambda>_. 1::'a)"
upper_bound_def: fixes D :: "('h, 'i) Order_scheme"   and S :: "'h set"   and b :: "'h" shows "ub\<^bsub>D\<^esub> S b = (b \<in> carrier D \<and> (\<forall>s\<in>S. s \<preceq>\<^bsub>D\<^esub> b))"
preorder_on_restrict_subset: fixes A :: "'a set" assumes "A \<subseteq> carrier" shows "Order_Predicates.preorder_on A (restrict_relation A le)"
upper_inv_eq: fixes y :: "'a" assumes "y \<in> carrier \<Y>\<^bsub>F\<^esub>" shows "\<pi>\<^sup>*\<^bsub>F\<^esub> (\<pi>\<^sub>*\<^bsub>F\<^esub> y) = y"
A_1: shows "p div d = sturm_squarefree' p ! 0"
app_iso_inv_subgraph: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and H :: "('a, 'b) pre_digraph" assumes "digraph_isomorphism hom"   and "subgraph H G" shows "app_iso (inv_iso hom) (app_iso hom H) = H"
that: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "homeomorphism T T f g"   and "f a = b"   and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"   and "bounded {x. \<not> (f x = x \<and> g x = x)}" shows "thesis"
sup_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> B = Collect (sup (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
metric_Cauchy_iff2: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity X = (\<forall>j. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < inverse (real (Suc j)))"
ineffectual_const_get: fixes x :: "'b \<Longrightarrow> 'c" assumes "ief_lens x" shows "\<exists>v. \<forall>\<sigma>\<in>\<S>\<^bsub>x\<^esub>. get\<^bsub>x\<^esub> \<sigma> = v"
fs_int_mu_d_Z: fixes n :: "nat"   and fs :: "int vec list"   and j :: "nat"   and ii :: "nat" assumes "fs_int_indpt n fs"   and "j \<le> ii"   and "ii < length fs" shows "rat_of_int (fs_int.d n fs (Suc j)) * gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) ii j \<in> \<int>"
R: shows "(\<le>) = (\<le>) \<or> (\<le>) = (\<lambda>x y. y \<le> x)"
fmsubset_restrict_set_mono: fixes m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap"   and A :: "'a set" assumes "m \<subseteq>\<^sub>f n" shows "fmrestrict_set A m \<subseteq>\<^sub>f fmrestrict_set A n"
row_add_iterate_eq_row_add: fixes a :: "'a"   and i :: "'a"   and n :: "nat"   and A :: "(('b, 'c) vec, 'a) vec"   and j :: "'c"   and b :: "'c" assumes "a \<noteq> i"   and "n < nrows A"   and "mod_type_class.to_nat a \<le> n" shows "row_add_iterate A n i j $ a $ b = row_add A a i (- A $ a $ j) $ a $ b"
cblinfun_image_0: fixes U :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "U *\<^sub>S 0 = 0"
compact_space_alt: fixes X :: "'c topology" shows "compact_space X = (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> topspace X \<subseteq> \<Union> \<U> \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> topspace X \<subseteq> \<Union> \<F>))"
mHom_l_zero: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "tOp_mHom R M N (mzeromap M N) f = f"
q_defs_1: fixes i :: "name" shows "q_Var i \<equiv> succ (ord_of (nat_of_name i))"
transfer: shows "pcr_sq_matrix (=) (=) (\<lambda>i j. 0::'d) 0"
l2wi: fixes l :: "('a word \<times> 'a word) list" shows "wordinterval_to_set (l2wi l) = (\<Union>(i, j)\<in>set l. {i..j})"
ivl_splice_Int: fixes n :: "nat" shows "(*) 2 ` {0..<n} \<inter> (\<lambda>i. Suc (2 * i)) ` {0..<n} = {}"
hyp: shows "(\<forall>s. s \<noteq> \<dd> \<longrightarrow> s \<circ> a = a) \<and> (\<forall>s. s \<noteq> \<dd> \<longrightarrow> s \<circ> b = b)"
meval_sched_snocD: fixes ns :: "nat list"   and n :: "nat"   and cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>1' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns @ [n]\<^esub> (cms\<^sub>1', mem\<^sub>1')" shows "\<exists>cms\<^sub>1'''' mem\<^sub>1''''. (cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns\<^esub> (cms\<^sub>1'''', mem\<^sub>1'''') \<and> (cms\<^sub>1'''', mem\<^sub>1'''') \<leadsto>\<^bsub>n\<^esub> (cms\<^sub>1', mem\<^sub>1')"
is_open_from_ind_is_open: fixes U :: "'a set" assumes "is_open U"   and "it.ind_is_open U" shows "is_open U"
Int_UNIV_right: fixes A :: "'c set" shows "A \<inter> UNIV = A"
get_shadow_root_new_character_data: fixes h :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) element_ptr" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "r \<in> get_shadow_root_locs ptr'" shows "r h h'"
bs_clos: fixes \<rho> :: "(nat \<times> bval) list"   and \<rho>' :: "(nat \<times> exp) list"   and x :: "nat"   and e :: "exp" assumes "bs_env \<rho> \<rho>'"   and "FV (ELam x (psubst ((x, EVar x) # \<rho>') e)) = {}" shows "bs_val (BClos x e \<rho>) (ELam x (psubst ((x, EVar x) # \<rho>') e))"