hcju/isabelleps huggingface dataset:
integral_dis: shows "integral s (\<lambda>x. partial_vector_derivative (\<lambda>x. F x \<bullet> j) i x - partial_vector_derivative (\<lambda>x. F x \<bullet> i) j x) = integral s (partial_vector_derivative (\<lambda>x. F x \<bullet> j) i) + integral s (\<lambda>x. - partial_vector_derivative (\<lambda>x. F x \<bullet> i) j x)"
poly_smult_zero_iff: fixes a :: "'c"   and p :: "'c poly"   and x :: "'c" shows "(poly (smult a p) x = (0::'c)) = (a = (0::'c) \<or> poly p x = (0::'c))"
unique_connecting_path_decomp_2: fixes v :: "'a"   and w :: "'a"   and ps :: "'a list"   and u :: "'a"   and ps' :: "'a list" assumes "v \<in> V"   and "w \<in> V"   and "v \<leadsto> w = ps @ u # ps'" shows "u # ps' = u \<leadsto> w"
map_upd_Some_unfold: fixes m :: "'c \<Rightarrow> 'b option"   and a :: "'c"   and b :: "'b"   and x :: "'c"   and y :: "'b" shows "((m(a \<mapsto> b)) x = \<lfloor>y\<rfloor>) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = \<lfloor>y\<rfloor>)"
linear: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "linear f"
fcompose_mult: fixes p1 :: "'a poly"   and p2 :: "'a poly"   and q1 :: "'a poly"   and q2 :: "'a poly" shows "fcompose (p1 * p2) q1 q2 = fcompose p1 q1 q2 * fcompose p2 q1 q2"
Id_onE: fixes c :: "'d \<times> 'd"   and A :: "'d set" assumes "c \<in> Id_on A" obtains x :: "'d" where "x \<in> A" and "c = (x, x)"
pair_loopfree_digraph_subdivide: fixes G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b"   and w :: "'b" assumes "pair_loopfree_digraph G"   and "e \<in> parcs G"   and "w \<notin> pverts G" shows "pair_loopfree_digraph (subdivide G e w)"
ras_get_refine: fixes i' :: "nat"   and l :: "'c list"   and a :: "'d array \<times> nat"   and R :: "('d \<times> 'c) set"   and i :: "nat" assumes "i' < length l"   and "(a, l) \<in> \<langle>R\<rangle>ras_rel"   and "(i, i') \<in> nat_rel" shows "(ras_get a i, l ! i') \<in> R"
omega_exp: fixes a :: "ordinal" assumes "additive_principal a" shows "\<exists>x. a = \<omega> ** x"
distinct_length_le: fixes s :: "'b set"   and xs :: "'b list" assumes "finite s"   and "set xs \<subseteq> s"   and "distinct xs" shows "length xs \<le> card s"
AbstrLevels_S6_A41: fixes i :: "AbstrLevelsID" assumes "sS6 \<in> AbstrLevel i" shows "sA41 \<notin> AbstrLevel i"
bounded_quasi_semilattice_set_axioms: shows "bounded_quasi_semilattice_set (\<^bold>*) \<^bold>\<top> \<^bold>\<bottom> normalize"
Compl_Diff_eq: fixes A :: "'b set"   and B :: "'b set" shows "- (A - B) = - A \<union> B"
intro: fixes Basis :: "'a set"   and scale :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" assumes "finite Basis"   and "\<not> module.dependent scale Basis"   and "module.span scale Basis = UNIV" shows "finite_dimensional_vector_space_axioms scale Basis"
ah: fixes C :: "'a set" assumes "C \<in> components (U - S)"   and "C \<inter> K \<noteq> {}" shows "a C \<in> C \<and> a C \<in> L \<and> continuous_on (S \<union> (C - {a C})) (h C) \<and> h C ` (S \<union> (C - {a C})) \<subseteq> T \<and> (\<forall>x\<in>S. h C x = f x)"
Abs_finfun_inject: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> finfun"   and "y \<in> finfun" shows "(Abs_finfun x = Abs_finfun y) = (x = y)"
trace_same_ext: fixes A :: "('e, 'd, 'f) ioa_scheme"   and B :: "('g, 'd, 'h) ioa_scheme"   and e :: "'i \<times> ('d \<times> 'j) list" assumes "externals (ioa.asig A) = externals (ioa.asig B)" shows "trace (ioa.asig A) e = trace (ioa.asig B) e"
l_known_ptrs_axioms: shows "l_known_ptrs ShadowRootClass.known_ptr ShadowRootClass.known_ptrs"
sumZero: fixes P :: "pi" shows "P \<oplus> \<zero> \<sim>\<^sup>s P"
geodesic_segment_disconnection: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "geodesic_segment_between G x y"   and "z \<in> G" shows "connected (G - {z}) = (z = x \<or> z = y)"
cnorm_le: fixes x :: "'a"   and y :: "'b" shows "(norm x \<le> norm y) = (x \<bullet>\<^sub>C x \<le> y \<bullet>\<^sub>C y)"
fold_congs_6: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "'c"   and f :: "'c \<Rightarrow> 'c"   and f' :: "'c \<Rightarrow> 'c" assumes "r = r'"   and "more r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "more_update f r = more_update f' r'"
put_det: fixes \<sigma> :: "'b"   and v :: "'a"   and \<rho> :: "'b" shows "put \<sigma> v = put \<rho> v"
implies_galois_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> z) = (y \<le> x \<leadsto> z)"
fix_eq: shows "P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>P n,Up\<^sub>P\<rbrakk> = (\<Lambda> X. (\<lambda>ps. Mprefix (Tr\<^sub>P n ps) (\<lambda>e. X (Up\<^sub>P ps e))))\<cdot>(P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>P n,Up\<^sub>P\<rbrakk>)"
least_consequence_graph_def: fixes t :: "'b itself"   and Rs :: "(('c, 'd) labeled_graph \<times> ('c, 'd) labeled_graph) set"   and S :: "('c, 'e) labeled_graph"   and G :: "('c, 'e) labeled_graph" shows "least_consequence_graph t Rs S G \<equiv> consequence_graph Rs G \<and> least t Rs S G"
to_poly_closed: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "to_polynomial Zp a \<in> carrier (UP Zp)"
IH: fixes gpv :: "('d, 'b, 'c) gpv"   and s :: "'a" shows "ord_spmf (=) (exec_gpv1 gpv s) (exec_gpv2 gpv s)"
subseqs_Cons: fixes x :: "'c"   and xs :: "'c list" shows "subseqs (x # xs) = map ((#) x) (subseqs xs) @ subseqs xs"
neutr_eq_iff: fixes a :: "nat"   and b :: "nat" shows "(0 = gcd a b) = (a = 0 \<and> b = 0)"
free_generator_def: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and H :: "'g set" shows "free_generator R M H = (generator R M H \<and> (\<forall>n s f. s \<in> {j. j \<le> n} \<rightarrow> carrier R \<and> f \<in> {j. j \<le> n} \<rightarrow> H \<and> inj_on f {j. j \<le> n} \<and> l_comb R M n s f = \<zero>\<^bsub>M\<^esub> \<longrightarrow> s \<in> {j. j \<le> n} \<rightarrow> {\<zero>\<^bsub>R\<^esub>}))"
reflects_iso: fixes \<mu> :: "'a" assumes "arr \<mu>"   and "S.iso (S.UP \<mu>)" shows "local.iso \<mu>"
block_def: fixes init :: "'d \<Rightarrow> 'd"   and bdy :: "('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" shows "block init bdy return c = DynCom (\<lambda>s. Seq (Catch (Seq (Basic init) bdy) (Seq (Basic (return s)) Throw)) (DynCom (\<lambda>t. Seq (Basic (return s)) (c s t))))"
subst_with_ax1: fixes M :: "trm"   and x :: "name"   and a :: "coname"   and y :: "name" shows "M{x:=<a>.Ax y a} \<longrightarrow>\<^sub>a* M[x\<turnstile>n>y]"
extend_indets_def: shows "extend_indets = poly_subst extend_indets_subst"
sum_of_powers_nat_aux: fixes a :: "nat"   and b :: "real"   and c :: "real"   and b' :: "nat"   and c' :: "nat" assumes "real a = b / c"   and "real b' = b"   and "real c' = c" shows "a = b' div c'"
post_mult_testNk: fixes k :: "nat" assumes "f k" shows "post * testN k = testN k"
intS: shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S"
uniform_limit_interval_integral_right: fixes a :: "ereal"   and b :: "ereal"   and g :: "real \<Rightarrow> real"   and Y :: "'a set"   and f :: "'a \<Rightarrow> real \<Rightarrow> 'b" assumes "interval_lebesgue_integrable lborel a b g"   and "\<And>y. y \<in> Y \<Longrightarrow> set_borel_measurable borel (einterval a b) (f y)"   and "\<And>y. y \<in> Y \<Longrightarrow> AE x\<in>einterval a b in lborel. norm (f y x) \<le> g x"   and "a < b" shows "uniform_limit Y (\<lambda>b' y. interval_lebesgue_integral lborel a b' (f y)) (\<lambda>y. interval_lebesgue_integral lborel a b (f y)) (at_left b)"
GE_Guard_not_unprot_protCl: fixes f :: "'fsym"   and al :: "univ list" assumes "wtFsym f"   and "\<not> unprot (resOf f)"   and "\<not> protCl (resOf f)"   and "list_all2 eintT (arOf f) al" shows "eintP (Guard (resOf f)) [intF f al]"
constrainsI: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set" assumes "\<And>act s s'. \<lbrakk>act \<in> Acts F; (s, s') \<in> act; s \<in> A\<rbrakk> \<Longrightarrow> s' \<in> A'" shows "F \<in> A co A'"
prime_factorization_prime_power: fixes p :: "'a"   and n :: "nat" assumes "prime p" shows "prime_factorization (p ^ n) = replicate_mset n p"
absc_def: shows "absc \<equiv> \<lambda>a. Fun (Abs a) []"
set_map: fixes f :: "'f \<Rightarrow> 'd"   and v :: "('e, 'f) fmap" shows "fmran' (fmmap f v) = f ` fmran' v"
fixed_def: fixes a :: "'a" shows "fixed a = (\<forall>s. s \<noteq> \<dd> \<longrightarrow> s \<circ> a = a)"
sq_mtx_vec_mult_sum_cols: fixes A :: "'b sq_mtx"   and x :: "(real, 'b) vec" shows "A *\<^sub>V x = (\<Sum>i\<in>UNIV. x $ i *\<^sub>R \<c>\<o>\<l> i A)"
fmfilter_cong': fixes m :: "('d, 'e) fmap"   and n :: "('d, 'e) fmap"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "m = n"   and "\<And>x. x \<in> fmdom' m \<Longrightarrow> P x = Q x" shows "fmfilter P m = fmfilter Q n"
add_casew: fixes a :: "'b"   and b :: "'b"   and p :: "'b point"   and q :: "'b point"   and P :: "'b point \<Rightarrow> 'b point \<Rightarrow> 'b point \<Rightarrow> bool" assumes "on_curve a b p"   and "on_curve a b q"   and "\<And>p. P Infinity p p"   and "\<And>p. P p Infinity p"   and "\<And>p. on_curve a b p \<Longrightarrow> P p (opp p) Infinity"   and "\<And>p\<^sub>1 x\<^sub>1 y\<^sub>1 p\<^sub>2 x\<^sub>2 y\<^sub>2 p\<^sub>3 x\<^sub>3 y\<^sub>3 l. \<lbrakk>p\<^sub>1 = Point x\<^sub>1 y\<^sub>1; p\<^sub>2 = Point x\<^sub>2 y\<^sub>2; p\<^sub>3 = Point x\<^sub>3 y\<^sub>3; p\<^sub>3 = ell_field_class.add a p\<^sub>1 p\<^sub>2; p\<^sub>1 \<noteq> opp p\<^sub>2; x\<^sub>1 = x\<^sub>2 \<and> y\<^sub>1 = y\<^sub>2 \<and> l = ((3::'b) * x\<^sub>1\<^sup>2 + a) / ((2::'b) * y\<^sub>1) \<or> x\<^sub>1 \<noteq> x\<^sub>2 \<and> l = (y\<^sub>2 - y\<^sub>1) / (x\<^sub>2 - x\<^sub>1); x\<^sub>3 = l\<^sup>2 - x\<^sub>1 - x\<^sub>2; y\<^sub>3 = - y\<^sub>1 - l * (x\<^sub>3 - x\<^sub>1)\<rbrakk> \<Longrightarrow> P p\<^sub>1 p\<^sub>2 p\<^sub>3" shows "P p q (ell_field_class.add a p q)"
exhaust: fixes y :: "Value" obtains "y = \<bottom>"   | cfun :: "Value \<rightarrow> Value" where "y = Fn\<cdot>cfun"   | discr :: "bool discr" where "y = B\<cdot>discr"
reversepath_joinpaths: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "pathfinish g1 = pathstart g2" shows "reversepath (g1 +++ g2) = reversepath g2 +++ reversepath g1"
fold_congs_2: fixes r :: "'a control_state_scheme"   and r' :: "'a control_state_scheme"   and v' :: "bool"   and f :: "bool \<Rightarrow> bool"   and f' :: "bool \<Rightarrow> bool" assumes "r = r'"   and "return_status r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "return_status_update f r = return_status_update f' r'"
Body: fixes Env :: "env"   and B :: "lname set"   and c :: "stmt"   and C :: "assigned"   and A :: "assigned"   and D :: "qtname" assumes "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C"   and "jumpNestingOkS {Ret} c"   and "Result \<in> nrm C"   and "nrm A = B"   and "brk A = (\<lambda>l. UNIV)" shows "Env\<turnstile> B \<guillemotright>\<langle>Body D c\<rangle>\<guillemotright> A"
shows_prec_xmldoc_def: fixes d :: "nat"   and doc :: "xmldoc" shows "shows_prec d doc = shows_xmldoc doc"
card_2_iff: fixes S :: "'a set" shows "(card S = 2) = (\<exists>x y. S = {x, y} \<and> x \<noteq> y)"
cd_disj: shows "h\<^sub>0' \<bottom> h\<^sub>1'"
extf_mono: fixes gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and gt' :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and f :: "'s" assumes "gt \<le> gt'" shows "extf f gt \<le> extf f gt'"
mantissa_neg_iff: fixes x :: "float" shows "(mantissa x < 0) = (x < 0)"
disc_rfr_proc_borel_measurable: fixes r :: "real"   and n :: "nat"   and M :: "'b measure" shows "disc_rfr_proc r n \<in> borel_measurable M"
gt_iff_wt_unary_diff_same: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "(t >\<^sub>t s) = (gt_wt t s \<or> gt_unary t s \<or> gt_diff t s \<or> gt_same t s)"
induct: fixes x :: "'a literal multiset"   and CC :: "'a literal multiset set"   and P :: "'a literal multiset \<Rightarrow> bool" assumes "x \<in> saturate CC"   and "\<And>C. C \<in> CC \<Longrightarrow> P C"   and "\<And>CC' D E. \<lbrakk>Infer CC' D E \<in> \<Gamma>; \<And>C'. C' \<in># CC' \<Longrightarrow> C' \<in> saturate CC; \<And>C'. C' \<in># CC' \<Longrightarrow> P C'; D \<in> saturate CC; P D\<rbrakk> \<Longrightarrow> P E" shows "P x"
prefix_suffix_subsequence: fixes j :: "nat"   and w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "prefix i (suffix j w) = w [j \<rightarrow> i + j]"
scaleR_cont: fixes f :: "'c \<Rightarrow> 'd"   and a :: "real" assumes "f \<in> bcontfun" shows "(\<lambda>x. a *\<^sub>R f x) \<in> bcontfun"
induct: fixes x1 :: "com"   and x2 :: "state"   and x3 :: "nat"   and x4 :: "state"   and P :: "com \<Rightarrow> state \<Rightarrow> nat assn" assumes "<x1,x2> -x3-> x4"   and "\<And>s n. P SKIP s n s"   and "\<And>X a s n. P (X:==a) s n (s[X::=a s])"   and "\<And>c s0 Y a n s1. \<lbrakk><c,s0[Loc Y::=a s0]> -n-> s1; P c (s0[Loc Y::=a s0]) n s1\<rbrakk> \<Longrightarrow> P (LOCAL Y:=a IN c) s0 n (s1[Loc Y::=s0<Y>])"   and "\<And>c0 s0 n s1 c1 s2. \<lbrakk><c0,s0> -n-> s1; P c0 s0 n s1; <c1,s1> -n-> s2; P c1 s1 n s2\<rbrakk> \<Longrightarrow> P (c0;; c1) s0 n s2"   and "\<And>b s c0 n s1 c1. \<lbrakk>b s; <c0,s> -n-> s1; P c0 s n s1\<rbrakk> \<Longrightarrow> P (IF b THEN c0 ELSE c1) s n s1"   and "\<And>b s c1 n s1 c0. \<lbrakk>\<not> b s; <c1,s> -n-> s1; P c1 s n s1\<rbrakk> \<Longrightarrow> P (IF b THEN c0 ELSE c1) s n s1"   and "\<And>b s c n. \<not> b s \<Longrightarrow> P (WHILE b DO c) s n s"   and "\<And>b s0 c n s1 s2. \<lbrakk>b s0; <c,s0> -n-> s1; P c s0 n s1; <WHILE b DO c,s1> -n-> s2; P (WHILE b DO c) s1 n s2\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s0 n s2"   and "\<And>pn s0 n s1. \<lbrakk><the (body pn),s0> -n-> s1; P (the (body pn)) s0 n s1\<rbrakk> \<Longrightarrow> P (BODY pn) s0 (Suc n) s1"   and "\<And>pn s0 a n s1 X. \<lbrakk><BODY pn,setlocs s0 newlocs[Loc Arg::=a s0]> -n-> s1; P (BODY pn) (setlocs s0 newlocs[Loc Arg::=a s0]) n s1\<rbrakk> \<Longrightarrow> P (X:=CALL pn(a)) s0 n (setlocs s1 (loc s0)[X::=s1<Res>])" shows "P x1 x2 x3 x4"
assms_6: shows "((\<lambda>x. set_lebesgue_integral lborel (X x) g) \<longlongrightarrow> set_lebesgue_integral lborel X' g) F"
first_step: fixes C :: "('b \<times> 'b) set"   and A :: "('b \<times> 'b) set"   and B :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "C = A \<union> B"   and "(x, y) \<in> C\<^sup>*"   and "(y, z) \<in> B" shows "\<exists>y. (x, y) \<in> A\<^sup>* O B"
check_induct': fixes s :: "('i, 'a) state"   and P :: "('i, 'a) state \<Rightarrow> bool" assumes "\<nabla> s"   and "\<triangle> (\<T> s)"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<And>s x\<^sub>i dir I. \<lbrakk>dir = Positive \<or> dir = Negative; \<not> \<U> s; min_lvar_not_in_bounds s = Some x\<^sub>i; \<lhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x\<^sub>i) (LB dir s x\<^sub>i); min_rvar_incdec dir s x\<^sub>i = Inl I; P s\<rbrakk> \<Longrightarrow> P (set_unsat I s)"   and "\<And>s x\<^sub>i x\<^sub>j l\<^sub>i. \<lbrakk>\<triangle> (\<T> s); \<nabla> s; x\<^sub>i \<in> lvars (\<T> s); x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i); P s\<rbrakk> \<Longrightarrow> P (pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s)"   and "P s" shows "P (check s)"
bij: shows "card (carrier (G Mod stabilizer m)) = card (orbit m)"
assms_3: shows "f21 = fst (snd (splitFace g v u f ws))"
to_d_repr: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "state = (f, mu, ds)" shows "d_repr ds fs"
simps_3: fixes lev :: "nat"   and t :: "Core.term"   and u :: "Core.term" shows "decr lev (t $ u) = decr lev t $ decr lev u"
ReadInner_enabled: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and l :: "Locs"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals" assumes "basevars (rtrner ch!p, rs!p)" shows "\<turnstile> Calling ch p \<and> arg<ch!p> = #(read l) \<longrightarrow> Enabled ReadInner ch mm rs p l"
valid_dbm_non_empty_diag: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry" assumes "valid_dbm M"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}" shows "\<forall>k\<le>n. \<one> \<le> M k k"
sappend_assoc: fixes xs :: "[:'a:]"   and ys :: "[:'a:]"   and zs :: "[:'a:]" shows "(xs :@ ys) :@ zs = xs :@ ys :@ zs"
elims: fixes x :: "nat"   and y :: "nat" assumes "fib x = y" obtains "x = 0" and "y = 0"   | "x = Suc 0" and "y = 1"   | n :: "nat" where "x = Suc (Suc n)" and "y = fib (Suc n) + fib n"
Abinds_env_restr_cong: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "ae f|` domA \<Delta> = ae' f|` domA \<Delta>" shows "ArityAnalysis.ABinds Aexp \<Delta>\<cdot>ae = ArityAnalysis.ABinds Aexp \<Delta>\<cdot>ae'"
subspace_span: fixes S :: "'b set" assumes "S \<subseteq> U\<^sub>V\<^sub>S" shows "local.subspace (local.span S)"
assms: shows "\<exists>o\<^sub>1 o\<^sub>2. Some (\<omega>\<nu> o\<^sub>1) = d\<^sub>\<kappa> x \<and> Some (\<omega>\<nu> o\<^sub>2) = d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2"
product_category_def: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "product_category C1 C2 \<equiv> category C1 \<and> category C2"
assms_1: shows "bijective (a * top) \<and> bijective (a[[top]])"
all_acquired_unshared_acquired: fixes a :: "nat"   and sb :: "'b memref list"   and \<O> :: "nat set" assumes "a \<in> all_acquired sb"   and "a \<notin> all_shared sb" shows "a \<in> acquired True sb \<O>"
ideal_ideal_generated: fixes S :: "'a set" shows "ideal (ideal_generated S)"
alw_ks: shows "alw (\<lambda>ks. (shd (stl ks), shd ks) \<in> Ord) ks"
DynCastRedsRef: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   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 D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and C :: "char list"   and Cs' :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs),s'\<rangle>"   and "hp s' a = \<lfloor>(D, S)\<rfloor>"   and "P \<turnstile> Path D to C via Cs' "   and "P \<turnstile> Path D to C unique" shows "P,E \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs'),s'\<rangle>"
bdd_below_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.preorder less_eq less" shows "preorder.bdd_below less_eq A = (\<exists>m. \<forall>x\<in>A. less_eq m x)"
mask_lower_twice2: fixes a :: "'b word"   and n :: "nat"   and m :: "nat" shows "(a AND NOT (mask n)) AND NOT (mask m) = a AND NOT (mask (max n m))"
yC: shows "lesvars upds \<inter> varacom C = {}"
shows_prec_xml_def: fixes d :: "nat"   and xml :: "xml" shows "shows_prec d xml = shows_XML_indent [] 2 xml"
minus_mult_distrib_mat_vec: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat"   and v :: "'c vec" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc" shows "(A - B) *\<^sub>v v = A *\<^sub>v v - B *\<^sub>v v"
binaryset_def: fixes A :: "'c set"   and B :: "'c set" shows "binaryset A B = (\<lambda>x. {})(0 := A, Suc 0 := B)"
exhaust: fixes y :: "'b sem_env" obtains x1 :: "(char list, char list, 'b) namespace" and x2 :: "(char list, char list, nat \<times> tid_or_exn) namespace" where "y = make_sem_env x1 x2"
hyps: shows "bound (a * of_nat n ^ deg) \<le> bound a * of_nat n ^ deg"
elims_2: fixes x :: "('a, 'b) rbt" assumes "bd_inv x" obtains "x = Leaf"   | l :: "('a, 'b) rbt" and c :: "color" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l c k v r" and "bd_inv l \<and> bd_inv r \<and> black_depth l = black_depth r"
d_2: shows "root_newton_int_main x n = (if x < 0 \<or> n < 0 then (0, False) else if x ^ p \<le> n then (x, x ^ p = n) else root_newton_int_main ((n div x ^ pm + x * int pm) div int p) n)"
find_shortest_augmenting_spec_cf_refine: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'a" assumes "RGraph c s t cf" shows "Network.find_shortest_augmenting_spec_cf c s t cf \<le> Network.find_shortest_augmenting_spec c s t (Network.flow_of_cf c cf)"
lincomb_insert: fixes S :: "'a Matrix.vec set"   and v :: "'a Matrix.vec"   and a :: "'a Matrix.vec \<Rightarrow> 'a" assumes "finite S"   and "S \<subseteq> carrier_vec n"   and "True"   and "v \<notin> S"   and "v \<in> carrier_vec n" shows "NC.lincomb a (S \<union> {v}) = a v \<cdot>\<^sub>v v + NC.lincomb a S"
foundation24: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> 'e"   and Y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> not (X \<triangleq> Y)) = (X \<tau> \<noteq> Y \<tau>)"
plus_case: fixes x1a :: "'b extended"   and x2a :: "'b extended" shows "x1a + x2a = (case x1a of Fin xda \<Rightarrow> case x2a of Fin x2da \<Rightarrow> Fin (xda + x2da) | \<infinity> \<Rightarrow> \<infinity> | -\<infinity> \<Rightarrow> -\<infinity> | \<infinity> \<Rightarrow> case x2a of Fin xba \<Rightarrow> \<infinity> | _ \<Rightarrow> \<infinity> | -\<infinity> \<Rightarrow> case x2a of \<infinity> \<Rightarrow> \<infinity> | _ \<Rightarrow> -\<infinity>)"
measure_times: fixes A :: "'i \<Rightarrow> 'a set" assumes "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)" shows "emeasure (Pi\<^sub>M J M) (Pi\<^sub>E J A) = (\<Prod>i\<in>J. emeasure (M i) (A i))"
strict_assoc: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.ide \<a>\<^sub>D[f, g, h]"
entailst_def: fixes A :: "assn"   and B :: "assn" shows "A \<Longrightarrow>\<^sub>t B \<equiv> A \<Longrightarrow>\<^sub>A B * true"
agents_eq: shows "agents = {i. R2 i \<noteq> (\<lambda>_ _. False)}"
timpl_closure_is_timpl_closure': fixes s :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and TI :: "('g set \<times> 'g set) set" shows "(s \<in> timpl_closure t TI) = ((t, s) \<in> timpl_closure' TI)"
intro: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>C :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>C :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d" assumes "monoidal_category C T\<^sub>C \<alpha>\<^sub>C \<iota>" shows "opposite_monoidal_category C T\<^sub>C \<alpha>\<^sub>C \<iota>"