hcju/isabelleps huggingface dataset:
MGFn_lemma: fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and t :: "Term.term" assumes "\<And>n. \<forall>C sig. G,A\<turnstile>{=:n} In1l (Methd C sig)\<succ> {G\<rightarrow>}"   and "wf_prog G" shows "G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>}"
fst_comp_apsnd: fixes f :: "'b \<Rightarrow> 'c" shows "(fst \<circ>\<circ> apsnd) f = fst"
SL\<^sub>N_def: fixes n :: "nat" shows "SL\<^sub>N n = SL {..<n}"
inject: fixes x1 :: "'f"   and x2 :: "'g"   and y1 :: "'f"   and y2 :: "'g" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
the_ext_prop_1: fixes y' :: "'c"   and g :: "'d" assumes "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y y' g" shows "\<guillemotleft>y\<eta>.the_ext y' g : Fo y \<rightarrow>\<^sub>C y'\<guillemotright>"
False: shows "\<not> (\<forall>k. f \<bullet> ns k \<noteq> \<zero>)"
complex_roots_complex_def: fixes p :: "complex poly" shows "complex_roots_complex p = (SOME as. Polynomial.smult (lead_coeff p) (\<Prod>a\<leftarrow>as. [:- a, 1:]) = p \<and> length as = degree p)"
dvd_mult_left: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times (times a b) c" shows "dvd.dvd times a c"
disc2_noProt: shows "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated is_Dst_Ports False m"
assms_4: shows "X \<subseteq> (\<Oplus>G\<leftarrow>Gs. G)"
croot_unique: fixes n :: "nat"   and y :: "complex"   and x :: "complex" assumes "n \<noteq> 0"   and "y ^ n = x"   and "\<And>z. z ^ n = x \<Longrightarrow> Re z < Re y \<or> Re z = Re y \<and> Im z \<le> Im y" shows "croot n x = y"
is_group: shows "Group.group (G\<lparr>carrier := generate G {a}\<rparr>)"
rel_map_2: fixes Sa :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and g :: "'e \<Rightarrow> 'd"   and y :: "'e option" shows "rel_option Sa x (map_option g y) = rel_option (\<lambda>x y. Sa x (g y)) x y"
p: shows "prv (neg (PP \<langle>fls\<rangle>))"
in_codomains_char: fixes b :: "('o, 'a) arr"   and f :: "('o, 'a) arr" shows "(b \<in> codomains f) = ((f \<noteq> Null \<and> Dom f \<in> Obj \<and> Cod f \<in> Obj \<and> Map f \<in> Hom (Dom f) (Cod f)) \<and> b = MkIde (Cod f))"
14: shows "w \<sqinter> top * e * - F \<le> F * (w \<sqinter> top * e * - F) * w\<^sup>\<star>"
social_decision_scheme_axioms: shows "social_decision_scheme agents alts sds"
disc_9: fixes x4 :: "'a" shows "\<not> isSend (Snapshot x4)"
unique_r: fixes x :: "'a" assumes "x \<in> \<Union> K" shows "\<exists>y\<in>\<Union> K'. T x y"
Child: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and d :: "nat"   and dir :: "dir" assumes "p \<in> grid b ds"   and "d \<in> ds" shows "child p dir d \<in> grid b ds"
numeral_plus_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.numeral plus" shows "plus (numeral.numeral one plus m) (numeral.numeral one plus n) = numeral.numeral one plus (m + n)"
restrict_Pi_cancel: fixes x :: "'b \<Rightarrow> 'c"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" shows "(restrict x I \<in> Pi I A) = (x \<in> Pi I A)"
comp_arr_dom: fixes f :: "'c"   and a :: "'c" assumes "Cop.arr f"   and "Cop.dom f = a" shows "f \<cdot>\<^sup>o\<^sup>p a = f"
in_prod_fst_sndI: fixes x :: "'a \<times> 'b"   and A :: "'a set"   and B :: "'b set" assumes "fst x \<in> A"   and "snd x \<in> B" shows "x \<in> A \<times> B"
SUP_const: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and f :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Sup ((\<lambda>i. f) ` A) = f"
inject: fixes x3 :: "'a"   and y3 :: "'a" shows "(Match x3 = Match y3) = (x3 = y3)"
IsoIsMor: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "ciso\<^bsub>C\<^esub> f" shows "f \<in> mor\<^bsub>C\<^esub>"
components_are_iso: fixes a :: "'a" assumes "S.ide a" shows "S.iso (R a)"
disj: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (Ferrack.disj p q) = Ifm bs (Or p q)"
directed_complete_def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "directed_complete A r \<equiv> {X. directed X r \<and> X \<noteq> {}}-complete A r"
mod_aux: fixes i :: "nat"   and n :: "nat"   and a :: "nat"   and j :: "nat" assumes "i < n"   and "a mod n = i"   and "j < a + n"   and "j mod n = i"   and "a < j" shows "False"
powerstar: fixes x :: "'a"   and n :: "nat" shows "x\<^sup>\<star> = (x\<^bsup>n + 1\<^esup>)\<^sup>\<star> \<cdot> x\<^bsub>0\<^esub>\<^bsup>n\<^esup>"
complement_proper_design: assumes "\<And>bl. bl \<in># \<B> \<Longrightarrow> incomplete_block bl" shows "proper_design \<V> \<B>\<^sup>C"
ring_simprules_9: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<ominus> x \<oplus> x = \<zero>"
apply_guards_subset_append: fixes G :: "vname gexp list"   and G' :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" assumes "set G \<subseteq> set G'" shows "apply_guards (G @ G') s = apply_guards G' s"
elims_2: fixes x :: "'b trm" assumes "is_a_variable x" obtains xa :: "'b" where "x = Var xa"
h4a: shows "\<exists>x. \<forall>y. insertion (nth_default 0 (L[var := y])) A = x"
real_euclid_B_def: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec"   and c :: "(real, 'a) vec" shows "B\<^sub>\<real> a b c \<equiv> \<exists>l\<ge>0. l \<le> 1 \<and> b - a = l *\<^sub>R (c - a)"
timpl_closure_set_mono: fixes M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and M' :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set"   and TI' :: "('c set \<times> 'c set) set" assumes "M \<subseteq> M'"   and "TI \<subseteq> TI'" shows "timpl_closure_set M TI \<subseteq> timpl_closure_set M' TI'"
homotopic_constant_maps: fixes X :: "'b topology"   and X' :: "'c topology"   and a :: "'c"   and b :: "'c" shows "homotopic_with (\<lambda>x. True) X X' (\<lambda>x. a) (\<lambda>x. b) = (topspace X = {} \<or> path_component_of X' a b)"
ereal_zero_less_0_iff: fixes a :: "ereal"   and b :: "ereal" shows "(0 < a * b) = (0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0)"
is_lasso_prpl_pre_conv: fixes m :: "'a"   and prpl :: "'Q list \<times> 'Q list" shows "graph.is_lasso_prpl_pre (to_gbg_ext m) prpl = (snd prpl \<noteq> [] \<and> graph.is_lasso_pre (to_gbg_ext m) (lasso_of_prpl prpl))"
AE_distr_iff: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "{x \<in> space N. P x} \<in> sets N" shows "almost_everywhere (distr M N f) P = (AE x in M. P (f x))"
ordinal_plus_absorb: fixes x :: "ordinal"   and y :: "ordinal" assumes "x * \<omega> \<le> y" shows "x + y = y"
cINF_greatest: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b set"   and m :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq m (f x)" shows "less_eq m (Inf (f ` A))"
prod2: shows "mat_of_rows n gs = of_int_hom.mat_hom U2 * mat_of_rows n fs_init"
content_cbox: fixes a :: "'a"   and b :: "'a" assumes "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i" shows "content (cbox a b) = (\<Prod>i\<in>Basis. b \<bullet> i - a \<bullet> i)"
stabilizer_def: fixes m :: "'a set" shows "conjP.stabilizer m = {g \<in> carrier (G\<lparr>carrier := P\<rparr>). conjugation_action p g m = m}"
matches_case_ternaryvalue_tuple: fixes \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and a :: "action"   and p :: "'d" shows "Matching_Ternary.matches (\<beta>, \<alpha>) m a p = (case ternary_ternary_eval (map_match_tac \<beta> p m) of TernaryTrue \<Rightarrow> True | TernaryFalse \<Rightarrow> False | TernaryUnknown \<Rightarrow> \<alpha> a p)"
VInter_vunion: fixes A :: "V"   and B :: "V" assumes "A \<noteq> 0"   and "B \<noteq> 0" shows "\<Inter>\<^sub>\<circ> (A \<union>\<^sub>\<circ> B) = \<Inter>\<^sub>\<circ> A \<inter>\<^sub>\<circ> \<Inter>\<^sub>\<circ> B"
prefixes_snoc: fixes xs :: "'a list"   and x :: "'a" shows "prefixes (xs @ [x]) = prefixes xs @ [xs @ [x]]"
disc_15: fixes x2 :: "'g" shows "is_Set (Set x2)"
pt_bij: fixes pi :: "('b \<times> 'b) list"   and x :: "'a"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
inject_2: fixes x31 :: "'a"   and x32 :: "'a"   and x33 :: "'b"   and y31 :: "'a"   and y32 :: "'a"   and y33 :: "'b" shows "(AllowPortFromTo x31 x32 x33 = AllowPortFromTo y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
S_call: fixes u :: "'a"   and q :: "'b"   and v :: "'a"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and M :: "'d set"   and P :: "'b multiset"   and k :: "nat"   and Ms :: "'d set"   and Ps :: "'b multiset"   and P' :: "'b multiset" assumes "(u, Call q, v) \<in> edges fg"   and "(u, M, P) \<in> S_cs fg k"   and "(return fg q, Ms, Ps) \<in> S_cs fg k"   and "P' \<subseteq># P + Ps"   and "size P' \<le> k" shows "(v, mon fg q \<union> M \<union> Ms, P') \<in> S_cs fg k"
assign_ge_0: 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 A :: "'a pvar \<Rightarrow> zhmultiset"   and x :: "'a pvar" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.legal_zpassign ground_heads_var \<delta> arity_sym wt_sym A" shows "0 \<le> A x"
expected_value_mix_pmf_distrib: fixes p :: "'b pmf"   and q :: "'b pmf"   and a :: "real"   and f :: "'b \<Rightarrow> real" assumes "finite (set_pmf p)"   and "finite (set_pmf q)"   and "a \<in> {0<..<1}" shows "measure_pmf.expectation (mix_pmf a p q) f = a * measure_pmf.expectation p f + (1 - a) * measure_pmf.expectation q f"
disc_3: fixes x3 :: "'a" shows "is_Inner \<langle>x3\<rangle>"
vars_def: shows "vars \<equiv> map_fun Rep_linear_poly id fun_vars"
bigI: fixes \<sigma>' :: "'state"   and P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat" assumes "\<sigma>' \<in> small_nstep P \<sigma> n"   and "\<sigma>' \<in> endset" shows "\<sigma>' \<in> big P \<sigma>"
1: shows "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>A. norm (f x n) \<le> g n"
hurwitz_zeta_real: fixes a :: "real"   and x :: "real" assumes "0 < a" shows "hurwitz_zeta a (complex_of_real x) \<in> \<real>"
extract_var_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and v :: "nat" shows "extract_var (monom m a) v = monom (remove_key v m) (monom (Poly_Mapping.single v (lookup m v)) a)"
connected_component_intermediate_subset: fixes U :: "'c set"   and a :: "'c"   and T :: "'c set" assumes "connected_component_set U a \<subseteq> T"   and "T \<subseteq> U" shows "connected_component_set T a = connected_component_set U a"
list_rel_append2: fixes l :: "'a list"   and as :: "'b list"   and bs :: "'b list"   and R :: "('a \<times> 'b) set" shows "((l, as @ bs) \<in> \<langle>R\<rangle>list_rel) = (\<exists>cs ds. l = cs @ ds \<and> (cs, as) \<in> \<langle>R\<rangle>list_rel \<and> (ds, bs) \<in> \<langle>R\<rangle>list_rel)"
l2_defs_5: shows "l2 \<equiv> \<lparr>init = l2_init, trans = l2_trans, obs = id\<rparr>"
i_le_m: shows "mod_type_class.from_nat i \<le> m"
dense_interval: fixes L :: "real set"   and U :: "real set"   and l :: "real"   and u :: "real"   and x :: "real"   and P :: "real \<Rightarrow> bool" assumes "finite L"   and "finite U"   and "l \<in> L"   and "u \<in> U"   and "l < x"   and "x < u"   and "P x"   and "\<And>y l u. \<lbrakk>\<forall>y\<in>{l<..<x}. y \<notin> L; \<forall>y\<in>{x<..<u}. y \<notin> U; l < x; x < u; l < y; y < u\<rbrakk> \<Longrightarrow> P y" shows "\<exists>l\<in>L. \<exists>u\<in>U. l < u \<and> (\<forall>y. l < y \<and> y < u \<longrightarrow> P y)"
sp_one: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and e :: "('c \<times> nat) edge"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.subpath g v1 [e] v2 subs = (sub_rel_of g subs \<and> (v1 = src e \<or> (v1, src e) \<in> subs\<^sup>+) \<and> e \<in> edges g \<and> (tgt e = v2 \<or> (tgt e, v2) \<in> subs\<^sup>+))"
strong_cases: fixes a1 :: "ccs"   and a2 :: "act"   and a3 :: "ccs"   and x :: "name" assumes "a1 \<longmapsto>a2 \<prec> a3" obtains \<alpha> :: "act" and P :: "ccs" where "a1 = \<alpha>__.P__" and "a2 = \<alpha>__" and "a3 = P__"   | P :: "ccs" and \<alpha> :: "act" and P' :: "ccs" and Q :: "ccs" where "a1 = Pa__ \<oplus> Q__" and "a2 = \<alpha>'__" and "a3 = P'__" and "Pa__ \<longmapsto>\<alpha>'__ \<prec> P'__"   | Q :: "ccs" and \<alpha> :: "act" and Q' :: "ccs" and P :: "ccs" where "a1 = Pb__ \<oplus> Qa__" and "a2 = \<alpha>''__" and "a3 = Q'__" and "Qa__ \<longmapsto>\<alpha>''__ \<prec> Q'__"   | P :: "ccs" and \<alpha> :: "act" and P' :: "ccs" and Q :: "ccs" where "a1 = Pc__ \<parallel> Qb__" and "a2 = \<alpha>'''__" and "a3 = P'a__ \<parallel> Qb__" and "Pc__ \<longmapsto>\<alpha>'''__ \<prec> P'a__"   | Q :: "ccs" and \<alpha> :: "act" and Q' :: "ccs" and P :: "ccs" where "a1 = Pd__ \<parallel> Qc__" and "a2 = \<alpha>''''__" and "a3 = Pd__ \<parallel> Q'a__" and "Qc__ \<longmapsto>\<alpha>''''__ \<prec> Q'a__"   | P :: "ccs" and a :: "act" and P' :: "ccs" and Q :: "ccs" and Q' :: "ccs" where "a1 = Pe__ \<parallel> Q" and "a2 = \<tau>" and "a3 = P'b__ \<parallel> Q'" and "Pe__ \<longmapsto>a \<prec> P'b__" and "Q \<longmapsto>coAction a \<prec> Q'" and "a \<noteq> \<tau>"   | P :: "ccs" and \<alpha> :: "act" and P' :: "ccs" where "\<lbrakk>x \<sharp> a1; x \<sharp> a2; x \<sharp> a3\<rbrakk> \<Longrightarrow> a1 = \<lparr>\<nu>x\<rparr>Pf__ \<and> a2 = \<alpha>'''''__ \<and> a3 = \<lparr>\<nu>x\<rparr>P'c__ \<and> Pf__ \<longmapsto>\<alpha>'''''__ \<prec> P'c__ \<and> x \<sharp> \<alpha>'''''__"   | P :: "ccs" and \<alpha> :: "act" and P' :: "ccs" where "a1 = !P" and "a2 = \<alpha>" and "a3 = P'" and "P \<parallel> !P \<longmapsto>\<alpha> \<prec> P'"
rotate11: fixes A11 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and A8 :: "fm"   and A9 :: "fm"   and A10 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A11 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 H)))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 H)))))))))) \<turnstile> B"
supp_fresh_iff: fixes x :: "'b" shows "supp x = {a. \<not> a \<sharp> x}"
Suc: shows "g\<cdot>(iterate i\<cdot>f\<cdot>\<bottom>) = iterate i\<cdot>h\<cdot>(g\<cdot>\<bottom>)"
ipset_prefix_match_distinct: fixes rpm :: "'b word set \<times> 'b word set"   and pfx :: "'b prefix_match"   and rg :: "'b word set" assumes "rpm = ipset_prefix_match pfx rg" shows "fst rpm \<inter> snd rpm = {}"
intro: fixes charts :: "('c, 'd) chart set"   and k :: "enat" assumes "\<And>c1 c2. \<lbrakk>c1 \<in> charts; c2 \<in> charts\<rbrakk> \<Longrightarrow> k-smooth_compat c1 c2" shows "c_manifold charts k"
universal: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "\<exists>!l. \<p>\<^sub>1[f, g] \<cdot> l = h \<and> \<p>\<^sub>0[f, g] \<cdot> l = k"
intros: fixes association_classes :: "'a \<rightharpoonup>\<^sub>f 'b"   and associations :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f 'd"   and \<A> :: "'a"   and role :: "'c"   and end' :: "'d"   and "end" :: "'d" assumes "association_class_end' association_classes associations \<A> role end'"   and "end \<noteq> end'" shows "association_class_end_not_unique association_classes associations \<A> role end"
f'_1: shows "f' \<in> borel_measurable borel"
add_left_imp_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.cancel_semigroup_add plus"   and "plus a b = plus a c" shows "b = c"
While': fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile> {\<lambda>s. P s \<and> bval b s} c {P}"   and "\<forall>s. P s \<and> \<not> bval b s \<longrightarrow> Q s" shows "\<turnstile> {P} WHILE b DO c {Q}"
per_cong_mid_R2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and C :: "'a"   and A :: "'a"   and H :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "B \<noteq> C"   and "Bet A B C"   and "Cong A H C H"   and "Tarski_neutral_dimensionless.Per Bet Cong H B C" shows "Tarski_neutral_dimensionless.Midpoint Bet Cong B A C"
subtype_implies_fresh: fixes \<Gamma> :: "binding list"   and S :: "ty"   and T :: "ty"   and X :: "tyvrs" assumes "\<Gamma>\<turnstile>S<:T"   and "X \<sharp> \<Gamma>" shows "X \<sharp> S \<and> X \<sharp> T"
extf_singleton_nil_if_\<delta>\<^sub>h_eq_\<epsilon>\<^sub>h: 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 f :: "'b"   and gt :: "('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool"   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 "of_nat \<delta> = of_nat \<epsilon>" shows "extf f gt [s] []"
alpha_preserves_qGood1: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "X #= X'"   and "FixVars.qGood X" shows "FixVars.qGood X'"
sync_order_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and a :: "nat"   and a' :: "nat" shows "(P,E \<turnstile> a \<le>so a') = (a \<in> sactions P E \<and> a' \<in> sactions P E \<and> E \<turnstile> a \<le>a a')"
stop_rel: fixes mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "sifum_refinement_eg1_eg2" shows "(\<langle>Stop, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>C\<rangle>\<^sub>A, \<langle>Stop, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
val_ty_observe: fixes A :: "val"   and v :: "exp"   and v' :: "exp"   and ob :: "obs" assumes "A \<in> E v []"   and "A \<in> E v' []"   and "observe v ob"   and "isval v'"   and "isval v" shows "observe v' ob"
rot_same_dir_2: fixes x1 :: "'a"   and a :: "'a"   and b :: "'a"   and x2 :: "'a"   and y :: "'a" assumes "x1 \<in> {a<--<b}"   and "x2 \<in> {x1<--<b}" shows "(y \<bullet> rot (a - b) < 0) = (y \<bullet> rot (x1 - x2) < 0)"
cancel_comp2: fixes l\<^sub>1 :: "'a literal"   and l\<^sub>2 :: "'a literal" assumes "l\<^sub>1\<^sup>c = l\<^sub>2\<^sup>c" shows "l\<^sub>1 = l\<^sub>2"
select_convs_4: fixes set_dom :: "'c set"   and set_func :: "'c \<Rightarrow> 'c"   and set_cod :: "'c set"   and more :: "'b" shows "set_arrow.more \<lparr>set_dom = set_dom, set_func = set_func, set_cod = set_cod, \<dots> = more\<rparr> = more"
IH: fixes gpv :: "('a, 'b, 'c) gpv" shows "expectation_gpv' gpv \<le> expectation_gpv'' gpv"
assms: shows "e \<in> set_spmf (sample_coprime N)"
ex_map_conv: fixes ys :: "'d list"   and f :: "'c \<Rightarrow> 'd" shows "(\<exists>xs. ys = map f xs) = (\<forall>y\<in>set ys. \<exists>x. y = f x)"
cqr: shows "lookup (q + r) (t \<oplus> lt f) \<noteq> (0::'b)"
cptn_iff_cptn_mod: fixes c :: "('b com option \<times> 'b) list" shows "(c \<in> cptn) = (c \<in> cptn_mod)"
circ_separate_1: fixes y :: "'a"   and x :: "'a" assumes "y * x \<le> x * y" shows "(x \<squnion> y)\<^sup>\<star> = x\<^sup>\<star> * y\<^sup>\<star>"
iMOD_iMODb_neq: fixes m :: "nat"   and r :: "nat"   and r' :: "nat"   and m' :: "nat"   and c' :: "nat" assumes "0 < m" shows "[ r, mod m ] \<noteq> [ r', mod m', c' ]"
le_cases: fixes x :: "'c"   and y :: "'c" obtains "x \<le> y"   | "y \<le> x"
antisym_conv3: fixes y :: "'b"   and x :: "'b" assumes "\<not> y < x" shows "(\<not> x < y) = (x = y)"
hrealpow_two_le: fixes r :: "real star" shows "0 \<le> r ^ Suc (Suc 0)"
mid_le_1: fixes i :: "float interval" shows "lower i \<le> mid i"
assms_1: shows "wcode_erase1 ires rs (b, Bk # ba)"
not_S_Var: fixes x :: "'b"   and t :: "('f, 'b) Term.term" shows "\<not> S (Var x) t"
WHILEI_def: shows "cWHILEI \<equiv> \<lambda>I b f. REC (WHILEI_body cbind creturn I b f)"
set_removeAll: fixes x :: "'b"   and xs :: "'b list" shows "set (removeAll x xs) = set xs - {x}"
mset_neg_sum_set: fixes M :: "'a set"   and f :: "'a \<Rightarrow> 'b zmultiset" assumes "\<And>m. m \<in> M \<Longrightarrow> mset_neg (f m) = {#}" shows "mset_neg (sum f M) = {#}"
consume_def: fixes \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and trn :: "('a, 'b, 'c) trans"   and vl :: "'d list"   and vl' :: "'d list" shows "BD_Security.consume \<phi> f trn vl vl' \<equiv> if \<phi> trn then vl \<noteq> [] \<and> f trn = hd vl \<and> vl' = tl vl else vl' = vl"
sum_power2_eq_zero_iff: 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 x :: "'b"   and y :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(plus (power.power one times x 2) (power.power one times y 2) = zero) = (x = zero \<and> y = zero)"
eq_is_eq: fixes x :: "'a"   and y :: "'a" shows "Predicate.eq x y \<equiv> x = y"
prod_list_induct3_snoc_Conssnoc_Cons_pairwise: fixes Q :: "'d list \<times> 'b list \<times> 'c list \<Rightarrow> bool"   and t :: "'d list \<times> 'b list \<times> 'c list" assumes "\<And>ys zs. Q ([], ys, zs)"   and "\<And>xs zs. Q (xs, [], zs)"   and "\<And>xs ys. Q (xs, ys, [])"   and "\<And>xs x y z zs. Q (xs @ [x], [y], z # zs)"   and "\<And>xs x y ys w z zs. \<lbrakk>Q (xs, ys, zs); Q (xs, ys @ [w], z # zs); Q (xs @ [x], y # ys, zs)\<rbrakk> \<Longrightarrow> Q (xs @ [x], y # ys @ [w], z # zs)" shows "Q t"
map_comp: fixes g1 :: "'c \<Rightarrow> 'a"   and g2 :: "'d \<Rightarrow> 'b"   and f1 :: "'e \<Rightarrow> 'c"   and f2 :: "'f \<Rightarrow> 'd"   and v :: "'e \<times> 'f" shows "map_prod g1 g2 (map_prod f1 f2 v) = map_prod (g1 \<circ> f1) (g2 \<circ> f2) v"