hcju/isabelleps huggingface dataset:
geq': fixes x :: "real" assumes "x \<in> {0..t}" shows "0 \<le> f' x s"
borel_measurable_iff_halfspace_greater: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a < f w \<bullet> i} \<in> sets M)"
induct: fixes x :: "('d, 'e) ODE"   and P :: "('d, 'e) ODE \<Rightarrow> bool" assumes "osafe x"   and "\<And>c. P (OVar c)"   and "\<And>\<theta> x. dfree \<theta> \<Longrightarrow> P (OSing x \<theta>)"   and "\<And>ODE1 ODE2. \<lbrakk>osafe ODE1; P ODE1; osafe ODE2; P ODE2; ODE_dom ODE1 \<inter> ODE_dom ODE2 = {}\<rbrakk> \<Longrightarrow> P (OProd ODE1 ODE2)" shows "P x"
relpow_imp_rtrancl: fixes p :: "'a \<times> 'a"   and n :: "nat"   and R :: "('a \<times> 'a) set" assumes "p \<in> R ^^ n" shows "p \<in> R\<^sup>*"
path_connected_space_iff_components_eq: fixes X :: "'a topology" shows "path_connected_space X = (\<forall>C\<in>path_components_of X. \<forall>C'\<in>path_components_of X. C = C')"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "H.VV.seq g f = (H.VV.arr f \<and> H.VV.arr g \<and> H.VxV.seq g f)"
sum: fixes X :: "'c set"   and a :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "finite X"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> ccw (0::'a) a (f x)" shows "ccw (0::'a) a (sum f X)"
comp_simps_6: fixes i :: "'h" shows "local.start i \<equiv> {s. \<forall>p. status s p = (if (0::'h) < i then Sleep else Ready) \<and> dstate s = \<bottom> \<and> (if (0::'h) < i then \<not> initialized s else initialized s) \<and> initVals s = {} \<and> abortVals s = {}}"
fun_eq_iff: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" shows "(f = g) = (\<forall>x. f x = g x)"
nofail_imp_RES_UNIV: fixes s :: "'a nres" assumes "nofail s" shows "s \<le> RES UNIV"
IH: assumes "degree p2 < degree p1" shows "fcompose (p1 + p2) q1 q2 = fcompose p1 q1 q2 + q2 ^ (degree p1 - degree p2) * fcompose p2 q1 q2"
closure_sound: fixes r :: "'b rexp"   and s :: "'b rexp"   and N :: "('s \<times> 's) set" assumes "closure ([(init r, init s)], {(post (init r), post (init s))}) = Some ([], N)"   and "local.wf n r"   and "local.wf n s" shows "lang n r = lang n s"
prob_le_1: fixes M :: "'d measure"   and A :: "'d set" assumes "prob_space M" shows "Sigma_Algebra.measure M A \<le> 1"
assms_1: shows "\<exists>\<^sub>\<infinity>i. 0 < n i"
dX_range: fixes X :: "'x set" shows "\<forall>d. dX X d \<subseteq> {x. Xd x = d}"
map_eq_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and g :: "'b \<Rightarrow> 'a" shows "(map f xs = map g xs) = (\<forall>x\<in>list.set xs. f x = g x)"
execute_of_list: fixes xs :: "'b list"   and h :: "heap" shows "execute (Array.of_list xs) h = Some (Array.alloc xs h)"
monofun_por: fixes y :: "ValD" shows "monofun (\<lambda>x. x por y)"
simps_2: fixes c :: "int \<times> int"   and c' :: "poly"   and n' :: "nat"   and p' :: "poly" shows "C c +\<^sub>p CN c' n' p' = CN (C c +\<^sub>p c') n' p'"
new_a_odd: shows "odd (nat \<bar>a\<bar>)"
R_def: shows "R = {(V', V''). \<exists>c c' W W'. V' = c # W \<and> V'' = c' # W' \<and> W \<noteq> [] \<and> W' \<noteq> [] \<and> ([c], [c']) \<in> Rc \<and> (W, W') \<in> RV}"
\<Gamma>_sat_preserving: fixes N :: "'a literal multiset set" assumes "satisfiable N" shows "satisfiable (N \<union> concls_of (inferences_from N))"
Eq4: shows "VC q (te q (i + 1)) = IC q (i + 1) (te q (i + 1))"
induct: fixes P :: "'c rule list \<Rightarrow> bool"   and a0 :: "'c rule list" assumes "P []"   and "P [Rule MatchAny Accept]"   and "P [Rule MatchAny Drop]"   and "P [Rule MatchAny Reject]"   and "\<And>uu_ uv_ rs. P (Rule uu_ (Goto uv_) # rs)"   and "\<And>uw_ ux_ rs. P (Rule uw_ (Call ux_) # rs)"   and "\<And>uy_ rs. P (Rule uy_ Return # rs)"   and "\<And>uz_ rs. P (Rule uz_ Unknown # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (Match vc) Accept # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (Match vc) Drop # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (Match vc) Log # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (Match vc) Reject # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (Match vc) Empty # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (MatchNot vc) Accept # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (MatchNot vc) Drop # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (MatchNot vc) Log # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (MatchNot vc) Reject # rs)"   and "\<And>vc rs. P rs \<Longrightarrow> P (Rule (MatchNot vc) Empty # rs)"   and "\<And>vc vd rs. P rs \<Longrightarrow> P (Rule (MatchAnd vc vd) Accept # rs)"   and "\<And>vc vd rs. P rs \<Longrightarrow> P (Rule (MatchAnd vc vd) Drop # rs)"   and "\<And>vc vd rs. P rs \<Longrightarrow> P (Rule (MatchAnd vc vd) Log # rs)"   and "\<And>vc vd rs. P rs \<Longrightarrow> P (Rule (MatchAnd vc vd) Reject # rs)"   and "\<And>vc vd rs. P rs \<Longrightarrow> P (Rule (MatchAnd vc vd) Empty # rs)"   and "\<And>va rs. P rs \<Longrightarrow> P (Rule (Match va) Drop # rs)"   and "\<And>va rs. P rs \<Longrightarrow> P (Rule (MatchNot va) Drop # rs)"   and "\<And>va vb rs. P rs \<Longrightarrow> P (Rule (MatchAnd va vb) Drop # rs)"   and "\<And>v va vb. P (va # vb) \<Longrightarrow> P (Rule v Drop # va # vb)"   and "\<And>v rs. P rs \<Longrightarrow> P (Rule v Log # rs)"   and "\<And>va rs. P rs \<Longrightarrow> P (Rule (Match va) Reject # rs)"   and "\<And>va rs. P rs \<Longrightarrow> P (Rule (MatchNot va) Reject # rs)"   and "\<And>va vb rs. P rs \<Longrightarrow> P (Rule (MatchAnd va vb) Reject # rs)"   and "\<And>v va vb. P (va # vb) \<Longrightarrow> P (Rule v Reject # va # vb)"   and "\<And>v rs. P rs \<Longrightarrow> P (Rule v Empty # rs)"   and "\<And>vc v vb. P (v # vb) \<Longrightarrow> P (Rule vc Accept # v # vb)"   and "\<And>vc v vb. P (v # vb) \<Longrightarrow> P (Rule vc Drop # v # vb)"   and "\<And>vc v vb. P (v # vb) \<Longrightarrow> P (Rule vc Log # v # vb)"   and "\<And>vc v vb. P (v # vb) \<Longrightarrow> P (Rule vc Reject # v # vb)"   and "\<And>vc v vb. P (v # vb) \<Longrightarrow> P (Rule vc Empty # v # vb)" shows "P a0"
comp_ide_self: fixes a :: "'a" assumes "S.ide a" shows "a \<cdot>\<^sub>S a = a"
Join_lower: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "a \<sqsubseteq> \<Squnion>A"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G"
Step1_5_2a: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> Init (ImpInit p \<and> HInit rmhist p) \<longrightarrow> Init PInit (resbar rmhist) p"
case_6: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f5 :: "('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f6 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f7 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x61 :: "poscheckvariant"   and x62 :: "('d, 'e) Term.term"   and x63 :: "('d, 'e) Term.term" shows "(case \<langle>x61: x62 \<in> x63\<rangle> of send\<langle>x\<rangle> \<Rightarrow> f1 x | receive\<langle>x\<rangle> \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle> \<Rightarrow> f3 x xa xb | insert\<langle>x,xa\<rangle> \<Rightarrow> f4 x xa | delete\<langle>x,xa\<rangle> \<Rightarrow> f5 x xa | \<langle>x: xa \<in> xb\<rangle> \<Rightarrow> f6 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa \<or>\<notin>: xb\<rangle> \<Rightarrow> f7 x xa xb) = f6 x61 x62 x63"
Ide_Nmlize_Ide: fixes t :: "'a local.term" assumes "Ide t" shows "Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>"
less_eq_list_nil_inv: fixes xs :: "'a list" assumes "local.less_eq_list xs []" shows "xs = []"
simps: fixes a1 :: "('rule \<times> 'form) NatRule"   and a2 :: "'form fset \<times> 'form"   and a3 :: "('form fset \<times> 'form) fset" shows "eff a1 a2 a3 = ((\<exists>con \<Gamma>. a1 = Axiom \<and> a2 = (\<Gamma> \<turnstile> con) \<and> a3 = {||} \<and> con |\<in>| \<Gamma>) \<or> (\<exists>rule c ants \<Gamma> a s. a1 = NatRule rule \<and> a2 = (\<Gamma> \<turnstile> subst s (freshen a c)) \<and> a3 = (\<lambda>ant. (\<lambda>p. subst s (freshen a p)) |`| a_hyps ant |\<union>| \<Gamma> \<turnstile> subst s (freshen a (a_conc ant))) |`| ants \<and> natEff_Inst rule c ants \<and> (\<forall>x xa. x |\<in>| ants \<longrightarrow> xa |\<in>| \<Gamma> \<longrightarrow> freshenLC a ` a_fresh x \<inter> lconsts xa = {}) \<and> (\<forall>x. x |\<in>| ants \<longrightarrow> freshenLC a ` a_fresh x \<inter> subst_lconsts s = {})) \<or> (\<exists>\<Gamma> c'. a1 = Cut \<and> a2 = (\<Gamma> \<turnstile> c') \<and> a3 = {|\<Gamma> \<turnstile> c'|}))"
R_def: shows "R = (\<lambda>n. A.birkhoff_sum phiA n x)"
remove_elt_pairs_sorted: fixes ys :: "('a \<times> 'b) list"   and x :: "'a" assumes "Lists_Ex.strict_sorted (map fst ys)" shows "Lists_Ex.strict_sorted (map fst (remove_elt_pairs x ys))"
mult_inverse: fixes a :: "'a"   and v :: "'c"   and w :: "'c" assumes "a \<in> carrier K"   and "v \<in> carrier V"   and "a \<odot>\<^bsub>V\<^esub> v = w"   and "a \<noteq> \<zero>\<^bsub>K\<^esub>" shows "v = inv\<^bsub>K\<^esub> a \<odot>\<^bsub>V\<^esub> w"
rotate8: fixes A8 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and A5 :: "fmla"   and A6 :: "fmla"   and A7 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A8 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 H))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 H))))))) \<turnstile> B"
yields_isomorphic_representation_2: shows "local.iso (trnr\<^sub>\<epsilon> (g \<star> f\<^sup>*) \<rho>)"
pop_def: fixes stk :: "nat list"   and v :: "int" shows "local.pop stk v \<equiv> dropWhile (\<lambda>j. v \<le> xs ! j) stk"
a_get_attribute_def: fixes ptr :: "(_) element_ptr"   and k :: "char list" shows "l_get_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_attribute ptr k = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. return (fmlookup m k))"
fPs: fixes Q :: "'a set" assumes "Q \<subseteq> P" shows "norm (sum f Q) \<le> e"
fdisjnt_union_left: fixes A :: "'b fset"   and M :: "'b fset"   and B :: "'b fset" assumes "fdisjnt A M"   and "fdisjnt B M" shows "fdisjnt (A |\<union>| B) M"
abs_prod: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "\<bar>prod f A\<bar> = (\<Prod>x\<in>A. \<bar>f x\<bar>)"
drop_bit_of_nat: fixes n :: "nat"   and m :: "nat" shows "drop_bit n (of_nat m) = of_nat (drop_bit n m)"
Inl_def: fixes a :: "V" shows "ZFC_Cardinals.Inl a \<equiv> \<langle>0, a\<rangle>"
absolutely_integrable_sin_product: fixes f :: "real \<Rightarrow> real"   and k :: "real" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<lambda>x. sin (k * x) * f x) absolutely_integrable_on {- pi..pi}"
nat_term_order_eq_def: fixes to1 :: "'a nat_term_order"   and to2 :: "'a nat_term_order"   and dg :: "bool"   and ps :: "bool" shows "nat_term_order_eq to1 to2 dg ps = (\<forall>u v. (dg \<longrightarrow> deg_pp (fst (rep_nat_term u)) = deg_pp (fst (rep_nat_term v))) \<longrightarrow> (ps \<longrightarrow> snd (rep_nat_term u) = snd (rep_nat_term v)) \<longrightarrow> nat_term_compare to1 u v = nat_term_compare to2 u v)"
id_oracle_def: fixes s :: "'b"   and x :: "'c" shows "id_oracle s x = Generative_Probabilistic_Value.Pause x (\<lambda>x. Generative_Probabilistic_Value.Done (x, s))"
nodes_balanced_ordered: shows "\<forall>no\<in>set (prx @ node # sfx). no \<noteq> Null \<and> (low no = Null) = (high no = Null) \<and> low no \<notin> set (prx @ node # sfx) \<and> high no \<notin> set (prx @ node # sfx) \<and> isLeaf_pt no low high = (var no \<le> 1) \<and> (low no \<noteq> Null \<longrightarrow> rep (low no) \<noteq> Null) \<and> (rep \<propto> low) no \<notin> set (prx @ node # sfx)"
Lim_null_comparison: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> real"   and net :: "'a filter" assumes "\<forall>\<^sub>F x in net. norm (f x) \<le> g x"   and "(g \<longlongrightarrow> 0) net" shows "(f \<longlongrightarrow> (0::'b)) net"
disjoint_family_onD: fixes A :: "'b \<Rightarrow> 'c set"   and I :: "'b set"   and i :: "'b"   and j :: "'b" assumes "disjoint_family_on A I"   and "i \<in> I"   and "j \<in> I"   and "i \<noteq> j" shows "A i \<inter> A j = {}"
exhaust: fixes y :: "kon_node" obtains "y = a"   | "y = b"   | "y = c"   | "y = d"
wf_tuple_upd_None: fixes n :: "nat"   and A :: "nat set"   and xs :: "'a option list"   and i :: "nat"   and B :: "nat set" assumes "wf_tuple n A xs"   and "A - {i} = B" shows "wf_tuple n B (xs[i := None])"
recurrent_subset_incl_2: fixes A :: "'a set" shows "recurrent_subset_infty A \<subseteq> A"
min_element: fixes P :: "nat \<Rightarrow> bool" assumes "\<exists>m. P m" shows "\<exists>mm. P mm \<and> (\<forall>m'<mm. \<not> P m')"
weakenStatImpMonotonic: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and A :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and B :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<A> Q"   and "A \<subseteq> B" shows "\<Psi> \<rhd> P \<lessapprox>\<^sub>w<B> Q"
simps_2: fixes a :: "'f"   and S :: "'f list"   and iostep :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e" shows "mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e (a # S) iostep \<sigma> = (case iostep a \<sigma> of None \<Rightarrow> mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e S iostep \<sigma> | Some (out, \<sigma>') \<Rightarrow> case mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e S iostep \<sigma>' of None \<Rightarrow> None | Some (outs, \<sigma>'') \<Rightarrow> Some (out # outs, \<sigma>''))"
return_env_def: fixes x :: "'a" shows "local.return_env x = EnvT (\<lambda>_. return x)"
less_1: fixes p :: "'a list" assumes "degree p < degree pa__"   and "p \<in> carrier (poly_ring R)"   and "\<And>q. \<lbrakk>q \<in> carrier (poly_ring R); pirreducible (carrier R) q; q pdivides p\<rbrakk> \<Longrightarrow> degree q \<le> 1" shows "splitted p"
cSUP_upper: fixes x :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset" assumes "x \<in> A"   and "subset_mset.bdd_above (f ` A)" shows "f x \<subseteq># \<Squnion> (f ` A)"
universal_property_def: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and N :: "('i, 'e, 'h) Module_scheme"   and MN :: "('j, 'e, 'h) Module_scheme"   and f :: "'g \<times> 'i \<Rightarrow> 'j" shows "universal_property R M N MN f = (bilinear_map f R M N MN \<and> (\<forall>Z g. R module Z \<and> bilinear_map g R M N Z \<longrightarrow> (\<exists>!h. h \<in> mHom R MN Z \<and> compose (M \<times>\<^sub>c N) h f = g)))"
inj: fixes v :: "'a vec"   and v' :: "'a vec" assumes "v \<in> gen"   and "v' \<in> gen"   and "B *\<^sub>v v = B *\<^sub>v v'" shows "v = v'"
subset_insertI: fixes B :: "'b set"   and a :: "'b" shows "order.greater_eq (insert a B) B"
F: fixes x' :: "'a"   and x :: "'b"   and \<sigma>' :: "'c"   and \<sigma> :: "'d" assumes "(x', x) \<in> S"   and "(\<sigma>', \<sigma>) \<in> R" shows "f' x' \<sigma>' \<le> \<Down> R (f x \<sigma>)"
assms_3: shows "q \<in> dgrad_p_set d m"
assms_1: fixes x :: "'a" assumes "x \<in> S" shows "f x \<le> g x"
simps_2: fixes t :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list" shows "ik\<^sub>s\<^sub>t (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S) = insert t (ik\<^sub>s\<^sub>t S)"
divmod_numeral_simps_34: fixes m :: "num"   and n :: "num" shows "(num.Bit0 m = num.Bit1 n) = False"
seqE: fixes g :: "'b"   and f :: "'b" assumes "J0.seq g f" obtains "J0.arr f" and "J0.arr g" and "J0.dom g = J0.cod f"
eqvt_at_apply': fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and p :: "perm" assumes "eqvt_at f x" shows "p \<bullet> f x = f (p \<bullet> x)"
is_Ghost\<^sub>s\<^sub>b_simps_3: fixes p :: "'b"   and p' :: "'b"   and "is" :: "instr list" shows "is_Ghost\<^sub>s\<^sub>b (Prog\<^sub>s\<^sub>b p p' is) = False"
obs_def: fixes \<O>\<^sub>s :: "'a set" shows "HMM_defs.obs \<O>\<^sub>s \<equiv> SOME x. x \<in> \<O>\<^sub>s"
elims_1: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list"   and y :: "bool" assumes "successively x xa = y" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "xa__ = P__" and "xa = []" and "ya__"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and x :: "'a" where "xa__ = Pa__" and "xa = [xb__]" and "ya__"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and x :: "'a" and y :: "'a" and xs :: "'a list" where "xa__ = P" and "xa = x # y # xs" and "ya__ = (P x y \<and> successively P (y # xs))"
\<section>: assumes "a \<noteq> midpoint a b"   and "b - midpoint a b \<noteq> - 1 *\<^sub>R (a - midpoint a b)" shows "b = midpoint a b"
Id_in_Hom: fixes A :: "'o" assumes "A \<in> Obj" shows "Id A \<in> Hom A A"
inf2I: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "A x y"   and "B x y" shows "inf A B x y"
that: assumes "\<not> prefix xs ys \<and> \<not> prefix ys xs" shows "thesis"
M_def: shows "M = Pi\<^sub>M UNIV (\<lambda>i. measure_pmf (bernoulli_pmf p))"
assms_4: shows "\<exists>ch. ch \<in> ins P \<and> exprChannel ch E \<and> ch \<notin> loc PQ"
Quotient3_tllist_all2_tmap_tmapI: fixes R1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs1 :: "'g \<Rightarrow> 'h"   and Rep1 :: "'h \<Rightarrow> 'g"   and R2 :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and Abs2 :: "'i \<Rightarrow> 'j"   and Rep2 :: "'j \<Rightarrow> 'i"   and ts :: "('h, 'j) tllist" assumes "Quotient3 R1 Abs1 Rep1"   and "Quotient3 R2 Abs2 Rep2" shows "tllist_all2 R1 R2 (tmap Rep1 Rep2 ts) (tmap Rep1 Rep2 ts)"
def_hashmap_size: fixes def_hashmap_size :: "'a itself \<Rightarrow> nat" assumes "class.hashable def_hashmap_size" shows "1 < def_hashmap_size TYPE('a)"
confs_widens: fixes P :: "'a prog"   and h :: "heap"   and vs :: "val list"   and Ts :: "ty list"   and Ts' :: "ty list" assumes "P,h \<turnstile> vs [:\<le>] Ts"   and "P \<turnstile> Ts [\<le>] Ts'" shows "P,h \<turnstile> vs [:\<le>] Ts'"
almost_full_on_trees: fixes P :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool" assumes "almost_full_on P F" shows "almost_full_on (emb P) trees"
zero_simplex_sing: fixes a :: "'b" shows "0 simplex {a}"
tfin: shows "t \<in> UNIV\<^sup>\<star>"
append_is_Nil_conv: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs \<cdot> ys = []) = (xs = [] \<and> ys = [])"
singleton_Un_iff: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "({x} = A \<union> B) = (A = {} \<and> B = {x} \<or> A = {x} \<and> B = {} \<or> A = {x} \<and> B = {x})"
F2set3_JF2set_incl_JF2set: fixes a' :: "'d JF2"   and a :: "'d JF2" assumes "a' \<in> F2set3 (dtor2 a)" shows "JF2set a' \<subseteq> JF2set a"
DDH_advantage_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "DDH_advantage \<A> = \<bar>spmf (DDH0 \<A>) True - spmf (DDH1 \<A>) True\<bar>"
2: shows "pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` B)"
dg_prod_2_ObjI: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr>"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "R.in_hom f a b" obtains "R.arr f" and "R.dom f = a" and "R.cod f = b"
le0: shows "- f 0 \<le>\<^sub>L - f 0"
resolvingBidEquivalence: fixes x :: "'c \<times> 'd set"   and N :: "'c set"   and \<Omega> :: "'d list"   and b :: "'c \<times> 'd set \<Rightarrow> 'e"   and r :: "integer" assumes "x \<in> N \<times> (Pow (set \<Omega>) - {{}})"   and "0 < card N"   and "distinct \<Omega>" shows "tiebids' (randomEl (takeAll (\<lambda>x. winningAllocationRel N (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg N \<Omega>)) r) N (set \<Omega>) x = resolvingBidAlg N \<Omega> b r x"
11: shows "e * prim_P w v e\<^sup>T\<^sup>\<star> * (w \<sqinter> - prim_EP w v e)\<^sup>\<star> \<le> v * - v\<^sup>T"
supteq_trans: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term"   and u :: "('d, 'e) Term.term" assumes "s \<unrhd> t"   and "t \<unrhd> u" shows "s \<unrhd> u"
inducts: fixes x :: "('fun, 'var) Term.term"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool" assumes "public_ground_wf_term x"   and "\<And>f T. \<lbrakk>public f; arity f = length T; \<And>t. t \<in> set T \<Longrightarrow> public_ground_wf_term t; \<And>t. t \<in> set T \<Longrightarrow> P t\<rbrakk> \<Longrightarrow> P (Fun f T)" shows "P x"
to_Gromov_completion_lim: fixes u :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter" shows "((\<lambda>n. to_Gromov_completion (u n)) \<longlongrightarrow> to_Gromov_completion a) F = (u \<longlongrightarrow> a) F"
assms: fixes x :: "'a" assumes "x \<in> A"   and "x \<in> B" shows "False"
Cons_3: shows "C \<in> f \<turnstile> folding_g.\<C>"
IH: fixes A :: "'b set"   and x :: "'a"   and y :: "'a" assumes "card A < n"   and "A \<subseteq> B"   and "(A, x) \<in> foldSetD D (\<cdot>) e"   and "(A, y) \<in> foldSetD D (\<cdot>) e" shows "y = x"
assms_3: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U :: "'x set"   and h' :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U' :: "'x set" assumes "(h, U) \<in> set (ps\<^sub>+)"   and "(h', U') \<in> set (ps\<^sub>+)"   and "poly_deg h = poly_deg h'" shows "(h, U) = (h', U')"
that: fixes i :: "nat"   and N :: "nat set" assumes "i < 2"   and "infinite N"   and "N \<subseteq> M"   and "h ` {l \<in> A. list.set l \<subseteq> N} \<subseteq> {i}" shows "thesis"
sub_sem: fixes I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I" shows "\<nu> \<in> fml_sem I (Fsubst \<phi> \<sigma>)"
dvd_PM_iff: fixes p :: "'b poly"   and q :: "'b poly" shows "(p dvd q) = (divides_ff (content_ff (map_poly to_fract p)) (content_ff (map_poly to_fract q)) \<and> map_poly to_fract p dvd map_poly to_fract q)"
bra_bra_vec: fixes v :: "complex vec" shows "\<langle>|v\<rangle>| = bra_vec v"
assms_3: shows "D.cones_map f \<chi> = \<chi>'"
no_unacquired_write_to_read_only_reads: fixes \<S> :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and \<O> :: "nat set"   and a :: "nat" assumes "no_write_to_read_only_memory \<S> sb"   and "non_volatile_owned_or_read_only True \<S> \<O> sb"   and "sharing_consistent \<S> \<O> sb"   and "a \<in> read_only_reads \<O> sb"   and "a \<notin> \<O> \<union> all_acquired sb" shows "a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b sb"
subst_cnj: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "subst (cnj \<phi> \<chi>) t x = cnj (subst \<phi> t x) (subst \<chi> t x)"
simple_bochner_integral_linear: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'e measure"   and g :: "'e \<Rightarrow> 'c" assumes "linear f"   and "simple_bochner_integrable M g" shows "simple_bochner_integral M (\<lambda>x. f (g x)) = f (simple_bochner_integral M g)"
pure_def: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x" shows "pure f h = (h \<turnstile> ok f \<longrightarrow> h \<turnstile> f \<rightarrow>\<^sub>h h)"
limr: shows "limit_mat (mat_add_seq (M0 * \<rho> * adjoint M0) (matrix_sum d (\<lambda>k. D0 k (DS (M1 * \<rho> * adjoint M1))))) (M0 * \<rho> * adjoint M0 + DW (DS (M1 * \<rho> * adjoint M1))) d"
nth_append_length_plus: fixes xs :: "'a list"   and ys :: "'a list"   and n :: "nat" shows "(xs @ ys) ! (length xs + n) = ys ! n"
powr_eventually_exp_ln: fixes p :: "real" shows "\<forall>\<^sub>F x in at_top. x powr p = exp (p * ln x)"