hcju/isabelleps huggingface dataset:
is_regular_sequenceD: fixes fs :: "('c \<Rightarrow>\<^sub>0 'd) list"   and j :: "nat"   and q :: "'c \<Rightarrow>\<^sub>0 'd" assumes "is_regular_sequence fs"   and "j < length fs"   and "q * fs ! j \<in> ideal (set (take j fs))" shows "q \<in> ideal (set (take j fs))"
sum_list_distinct_conv_sum_set: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "distinct xs" shows "sum_list (map f xs) = sum f (set xs)"
subst_lit_is_pos: fixes L :: "'a literal"   and \<sigma> :: "'s" shows "is_pos (L \<cdot>l \<sigma>) = is_pos L"
verticesFrom_is_nextElem_last: fixes v' :: "nat"   and f :: "face"   and v :: "nat" assumes "v' \<in> \<V> f"   and "distinct (vertices f)"   and "is_nextElem (verticesFrom f v') (last (verticesFrom f v')) v" shows "v = v'"
closedin_contains_derived_set: fixes X :: "'a topology"   and S :: "'a set" shows "closedin X S = (X derived_set_of S \<subseteq> S \<and> S \<subseteq> topspace X)"
lcs_plus_right: fixes s :: "'a"   and u :: "'a"   and t :: "'a" shows "lcs (s + u) (t + u) = lcs s t + u"
nrows_le: fixes A :: "'b matrix"   and n :: "nat" shows "(nrows A \<le> n) = (\<forall>j i. n \<le> j \<longrightarrow> Rep_matrix A j i = (0::'b))"
circuit_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "circuit_in \<E> X" shows "X \<subseteq> \<E>"
iso_verts_iso_inv: fixes hom :: "('a, 'a \<times> nat \<times> 'a, 'c, 'd) digraph_isomorphism"   and u :: "'c" assumes "H.digraph_isomorphism hom"   and "u \<in> verts (app_iso hom (app_iso iso_f G))" shows "iso_verts hom (iso_verts (H.inv_iso hom) u) = u"
to_st_trms_tfr\<^sub>s\<^sub>e\<^sub>t: fixes A :: "('fun, 'var) extstrand_step list" assumes "tfr\<^sub>s\<^sub>e\<^sub>t (\<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A))" shows "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t (to_st A))"
S_def: shows "S \<equiv> \<lambda>n. \<Sum>i = 1..n. of_nat i powr - s"
cos_conv_pre_cos: fixes x :: "real" shows "cos x = powser (msllist_of_msstream cos_series_stream) (x\<^sup>2)"
multiplicative_function'_axioms: shows "multiplicative_function' g g_prime_power g_prime"
middle_def: fixes x :: "'a"   and y :: "'a" shows "middle x y = (SOME z. dist x z = dist x y / 2 \<and> dist z y = dist x y / 2)"
div_mult_self1: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and b :: "'e"   and a :: "'e"   and c :: "'e" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times c b)) b = plus c (divide a b)"
irreducible_connect: fixes f :: "int poly" shows "irreducible\<^sub>d_m f = irreducible_m f"
simps: fixes f :: "int poly"   and us :: "int poly list" shows "LLL_many_reconstruction f us = (let d = degree f; d2 = d div 2 in Let (find_map_filter (\<lambda>u. gcd f (LLL_short_polynomial pl (Suc d2) u)) (\<lambda>f2. let deg = degree f2 in 0 < deg \<and> deg < d) (filter (\<lambda>u. degree u \<le> d2) us)) (case_option [f] (\<lambda>f2. let f1 = f div f2; (us1, us2) = List.partition (\<lambda>gi. p.dvdm gi f1) us in LLL_many_reconstruction f1 us1 @ LLL_many_reconstruction f2 us2)))"
narrow_type: fixes \<Delta> :: "binding list"   and Q :: "POPLmark.type"   and \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type"   and P :: "POPLmark.type" assumes "\<Delta> @ TVarB Q \<Colon> \<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> P <: Q" shows "\<Delta> @ TVarB P \<Colon> \<Gamma> \<turnstile> t : T"
take_rows_def: fixes A :: "'b mat"   and inds :: "nat list" shows "take_rows A inds = mat_of_rows (dim_col A) (map ((!) (rows A)) (filter (\<lambda>y. y < dim_row A) inds))"
valid_intv_distinct: fixes c :: "nat"   and I :: "intv"   and I' :: "intv"   and x :: "'b"   and u :: "'b \<Rightarrow> real" assumes "valid_intv c I"   and "valid_intv c I'"   and "intv_elem x u I"   and "intv_elem x u I'" shows "I = I'"
\<tau>exec_moves_def: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap" shows "\<tau>exec_moves ci P t es h = (\<lambda>(stk, loc, pc, xcp) s'. exec_moves ci P t es h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s' \<and> \<tau>moves2 P h stk es pc xcp)"
AssignAddr: fixes a :: "expr"   and \<theta> :: "nat \<Rightarrow> nat option"   and volatile :: "bool"   and e :: "expr"   and A :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and L :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and R :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and W :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and t :: "nat" assumes "\<forall>sop. a \<noteq> Tmp sop" shows "\<theta>\<turnstile> (Assign volatile a e A L R W, t) \<rightarrow>\<^sub>p ((Assign volatile (Tmp (eval_expr t a)) e A L R W, t + used_tmps a), issue_expr t a)"
distinct_insert: fixes x :: "'a"   and xs :: "'a list" shows "distinct (List.insert x xs) = distinct xs"
pmf_of_set_singleton: fixes x :: "'a" shows "pmf_of_set {x} = return_pmf x"
P: shows "P \<in> Units (ring_mat TYPE('a) nr b)"
leg1_simps_4: shows "trg p\<^sub>1 = trg (r\<^sub>0\<^sup>* \<star> s\<^sub>1)"
generate_is_subgroup: fixes G :: "('f, 'g) monoid_scheme"   and H :: "'f set" assumes "Group.group G"   and "H \<subseteq> carrier G" shows "subgroup (generate G H) G"
ord_eq_le_trans: fixes a :: "('x, nat) pp"   and b :: "('x, nat) pp"   and c :: "('x, nat) pp" assumes "a = b"   and "ord b c" shows "ord a c"
continuous: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> R" shows "continuous_on S f"
distinct_update_with_aux: fixes v :: "'b"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and ps :: "('a \<times> 'b) list" shows "distinct (map fst (AList.update_with_aux v k f ps)) = distinct (map fst ps)"
simps_2: fixes w :: "nat \<Rightarrow> 'b"   and p :: "'b \<Rightarrow> bool" shows "w \<Turnstile>\<^sub>p atom\<^sub>p(p) = p (w 0)"
mat_def: fixes nr :: "nat"   and nc :: "nat"   and m :: "'a list list" shows "mat nr nc m = (length m = nc \<and> Ball (set m) (vec nr))"
red: shows "t \<turnstile> \<langle>x, m\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"
simps: fixes a :: "(name \<times> ty) list" shows "valid a = (a = [] \<or> (\<exists>\<Gamma> x T. a = (x, T) # \<Gamma> \<and> valid \<Gamma> \<and> x \<sharp> \<Gamma>))"
assms_1: shows "Domain a \<inter> X \<noteq> {}"
path_join_imp: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "path g1"   and "path g2"   and "pathfinish g1 = pathstart g2" shows "path (g1 +++ g2)"
transitive: fixes M :: "'a trm"   and M' :: "'a trm"   and M'' :: "'a trm" assumes "M \<rightarrow>\<beta>\<^sup>* M'"   and "M' \<rightarrow>\<beta> M''" shows "M \<rightarrow>\<beta>\<^sup>* M''"
in_var_def: fixes x :: "'b \<Longrightarrow> 'c" shows "$x = x ;\<^sub>L fst\<^sub>L"
bexI: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and A :: "'d set" assumes "P x"   and "x \<in> A" shows "\<exists>x\<in>A. P x"
vec_matrix: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f" shows "vec_to_iarray (A $ i) = matrix_to_iarray A !! mod_type_class.to_nat i"
is_inf_def: fixes x :: "nat"   and y :: "nat"   and i :: "nat" shows "nat_dvd.is_inf x y i = (i dvd x \<and> i dvd y \<and> (\<forall>z. z dvd x \<and> z dvd y \<longrightarrow> z dvd i))"
arcs_blocks_ends: fixes x :: "'a"   and y :: "'a" assumes "x \<rightarrow> y" shows "{x, y} \<in># arcs_blocks"
lookup_Mapping: fixes t :: "('d, 'e) RBT.rbt" shows "Mapping.lookup (RBT_Mapping.Mapping t) = RBT.lookup t"
vsubstAbs_ident: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and zs :: "'varSort"   and z :: "'var" assumes "goodAbs A" shows "A $[z // z]_zs = A"
unit_r_inv: fixes x :: "'a" assumes "unit x" shows "x ** local.inv x = one"
interp_def: fixes C :: "'a literal multiset" shows "interp C = \<Union> (production ` {D. D < C})"
vec_nth_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> UNIV" obtains x :: "('b, 'a) Finite_Cartesian_Product.vec" where "y = ($h) x"
assms_2: fixes j :: "'j" assumes "J.arr j" shows "\<guillemotleft>F j : b \<rightarrow> D j\<guillemotright>"
path_defs_1: fixes g :: "real \<Rightarrow> 'c" shows "path g = continuous_on {0..1} g"
chainmap_preserves_length: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and \<Pi> :: "('c, 'd) chain" shows "chainlen (chainmap f g \<Pi>) = chainlen \<Pi>"
bounded_subset_ballI: fixes S :: "'a set"   and x :: "'a"   and r :: "real" assumes "S \<subseteq> ball x r" shows "bounded S"
trcl_rev_cons: fixes c :: "'a"   and w :: "'b list"   and ch :: "'a"   and T :: "('a \<times> 'b \<times> 'a) set"   and e :: "'b"   and c' :: "'a" assumes "(c, w, ch) \<in> trcl T"   and "(ch, e, c') \<in> T" shows "(c, w @ [e], c') \<in> trcl T"
dom_trms_subset: fixes C :: "'a literal set"   and E :: "'a trm set" shows "dom_trms C E \<subseteq> E"
prefix_Cons_elim: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "prefix (x # xs) ys" obtains ys' :: "'b list" where "ys = x # ys'" and "prefix xs ys'"
rupture_def: fixes R :: "('d, 'e) ring_scheme"   and K :: "'d set"   and p :: "'d list" shows "Rupt\<^bsub>R\<^esub> K p = K [X]\<^bsub>R\<^esub> Quot PIdl\<^bsub>K [X]\<^bsub>R\<^esub>\<^esub> p"
aHom_func: fixes f :: "'c \<Rightarrow> 'd"   and F :: "('c, 'e) aGroup_scheme"   and G :: "('d, 'f) aGroup_scheme" assumes "f \<in> aHom F G" shows "f \<in> carrier F \<rightarrow> carrier G"
insert_not_empty: fixes a :: "'b"   and A :: "'b set" shows "insert a A \<noteq> {}"
simps_2: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and uu :: "variable"   and S :: "String.literal set" shows "typ_ok_sig \<Sigma> (Tv uu S) = wf_sort (subclass (osig \<Sigma>)) S"
simps_2: fixes x :: "'a"   and X :: "'a set" shows "linear_cf (insert x X) = (if x \<in> Field r then if linear_cf X = {} then {x} else {local.MaxR.maxR x y |y. y \<in> linear_cf X} else linear_cf X)"
priority_fst: fixes xp :: "'c \<times> 'd"   and q :: "('c, 'd) pq" assumes "xp \<in> set (alist_of q)" shows "priority q (fst xp) = Some (snd xp)"
sympE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "symp r"   and "r b a" obtains "r a b"
unitary11_gen_def: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary11_gen M = (\<exists>k. k \<noteq> 0 \<and> congruence M (1, 0, 0, - 1) = k *\<^sub>s\<^sub>m (1, 0, 0, - 1))"
assms_2: shows "strict_monoidal_functor C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D G"
orbit_eq_singleton_iff: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "(orbit f x = {x}) = (f x = x)"
errMOD_iwlsFSbSwTR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.iwlsFSb Delta MOD" shows "FixSyn.iwlsFSbSwTR Delta (FixSyn.errMOD Delta MOD)"
poly_roots_index_lemma: fixes p :: "'a list"   and x :: "'a"   and n :: "nat" assumes "poly p x \<noteq> poly [] x"   and "length p = n" shows "\<exists>i. \<forall>x. poly p x = (0::'a) \<longrightarrow> (\<exists>m\<le>n. x = i m)"
equivpartpCI: fixes x :: "'c"   and y :: "'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "x \<noteq> y \<Longrightarrow> sympartp r x y" shows "equivpartp r x y"
exec': shows "exec_move ci P t body h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h (stk', loc', pc', xcp')"
list_ops_insertions: fixes ops :: "('c \<times> ('c, 'd) list_op) list" assumes "list_ops ops" shows "insert_ops (insertions ops)"
i0_lb: fixes n :: "enat" shows "0 \<le> n"
calc_atm_2: fixes x :: "name" shows "[] \<bullet> x = x"
poly_sum: fixes p :: "'c \<Rightarrow> 'b poly"   and A :: "'c set"   and x :: "'b" shows "poly (sum p A) x = (\<Sum>k\<in>A. poly (p k) x)"
iso_set_sym: fixes h :: "'c \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> Group.iso H H" shows "inv_into (carrier H) h \<in> Group.iso H H"
dgrad_set_leI: fixes S :: "'c set"   and T :: "'c set"   and d :: "'c \<Rightarrow> nat" assumes "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. d s \<le> d t" shows "dgrad_set_le d S T"
SubstTermP_Var_same: fixes v :: "tm"   and t :: "tm" shows "{VarP v, TermP t} \<turnstile> SubstTermP v t v t"
Gromov_converging_at_boundary_converges': fixes u :: "nat \<Rightarrow> 'c" assumes "Gromov_converging_at_boundary u" shows "convergent (\<lambda>n. to_Gromov_completion (u n))"
col_dim: fixes A :: "'a mat"   and i :: "nat" shows "col A i \<in> carrier_vec (dim_row A)"
Partition_sum_Partition_diff: fixes k :: "nat"   and m :: "nat" assumes "k \<le> m" shows "Partition m k = sum (Partition (m - k)) {..k}"
vector_ssub_ldistrib: fixes c :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec"   and y :: "('a, 'b) Finite_Cartesian_Product.vec" shows "c *s (x - y) = c *s x - c *s y"
VectorSpaceIso_axioms_def: fixes V :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and W :: "'b set" shows "VectorSpaceIso_axioms V T W \<equiv> bij_betw T V W"
lnths_LNil: fixes A :: "nat set" shows "lselect A <> = <>"
intro: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" assumes "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>" shows "func f S T"
subfilt_filt_equiv: fixes F :: "'b \<Rightarrow> 'c measure"   and M :: "'c measure"   and N :: "'c measure"   and G :: "'b \<Rightarrow> 'c measure" assumes "filt_equiv F M N"   and "\<forall>t. subalgebra (F t) (G t)"   and "\<forall>s t. s \<le> t \<longrightarrow> subalgebra (G t) (G s)" shows "filt_equiv G M N"
fresh_star_prod_list: fixes xs :: "'d list"   and a :: "'e"   and b :: "'f" shows "xs \<sharp>* (a, b) = (xs \<sharp>* a \<and> xs \<sharp>* b)"
g'_def: shows "g' \<equiv> \<lambda>x. if x \<in> cbox w z then g x else x"
coerce_sfun: shows "COERCE('e \<rightarrow>! 'f, 'g \<rightarrow>! 'h) = sfun_map\<cdot>COERCE('g, 'e)\<cdot>COERCE('f, 'h)"
check_instr_pc: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and ins :: "'b instr"   and P :: "(nat \<times> nat \<times> 'b instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'a"   and stk :: "'b val list"   and loc :: "'b val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'b frame list"   and pc' :: "nat" assumes "JVM_heap_base.check_instr typeof_addr ins P h stk loc C M pc frs"   and "ins_jump_ok ins pc'" shows "JVM_heap_base.check_instr typeof_addr ins P h stk loc C M pc' frs"
assms_2: fixes i :: "nat" assumes "(f i, f (Suc i)) \<notin> r \<or> (f i, a) \<notin> r" shows "P"
truncate_up_pos: fixes x :: "real"   and p :: "nat" assumes "0 < x" shows "0 < truncate_up p x"
total_pseudo_ordered_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "total_pseudo_ordered_set A less_eq \<equiv> connex A less_eq \<and> antisymmetric A less_eq"
aboveS_AboveS_singl: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "aboveS r a = AboveS r {a}"
RG_proj_in_RG: fixes A :: "'b set"   and x :: "('a, 'b) aezfun" shows "aezfun_setspan_proj A x \<in> aezfun_setspan A"
streams_shd: fixes s :: "'a stream"   and A :: "'a set" assumes "s \<in> streams A" shows "shd s \<in> A"
r_def: shows "r = {(\<P>1, \<P>2). \<P>1 \<subseteq> \<P>2 \<and> \<P>1 \<in> Field_r \<and> \<P>2 \<in> Field_r}"
sum_ereal_right_distrib: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ereal"   and r :: "ereal" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i" shows "r * sum f A = (\<Sum>n\<in>A. r * f n)"
IH: shows "p \<bullet> P' \<Rightarrow> p \<bullet> P''"
wf_fut_derivs: fixes k :: "nat" shows "local.wf idx (fut_derivs k \<psi>)"
fpos: fixes x :: "'a" assumes "x \<in> S" shows "0 \<le> f x"
summarize_threshold_def: fixes p :: "nat"   and t :: "real"   and x :: "'a pdevs"   and y :: "'b" shows "summarize_threshold p t x y = (t * infnorm (eucl_truncate_up p (tdev' p x)) \<le> infnorm y)"
part_implies_overlap: fixes x :: "i"   and y :: "i" assumes "PM"   and "P x y" shows "O x y"
interval_lebesgue_integrable_def: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a" shows "interval_lebesgue_integrable M a b f = (if a \<le> b then set_integrable M (einterval a b) f else set_integrable M (einterval b a) f)"
ample_subset: fixes q :: "'state"   and A :: "'action set" assumes "ample_set q A" shows "A \<subseteq> {a. en a q}"
hyps: fixes sa :: "nat"   and lm' :: "nat list" assumes "abc_steps_l (0, lm) B n = (sa, lm')" shows "abc_steps_l (length A, lm) (A [+] B) n = (sa + length A, lm')"
upd_rev_cs_Cons_intra: fixes a :: "'edge"   and cs :: "'edge list"   and as :: "'edge list" assumes "intra_kind (kind a)" shows "upd_rev_cs cs (a # as) = upd_rev_cs cs as"
ys_def: shows "ys = coeffs q @ replicate (n - length (coeffs q)) (0::'b)"
ex1_iff_ex_Uniq: fixes P :: "'c \<Rightarrow> bool" shows "(\<exists>!x. P x) = ((\<exists>x. P x) \<and> (\<exists>\<^sub>\<le>\<^sub>1 x. P x))"
mono_def: fixes f :: "'a \<Rightarrow> 'c" shows "mono f = (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
preserves_seq: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" shows "B.seq (B.map g) (B.map f)"
reds: shows "P,E(Vs [\<mapsto>] Ts) \<turnstile> \<langle>e,(h\<^sub>0, l\<^sub>0(Vs [\<mapsto>] vs'))\<rangle> \<rightarrow>* \<langle>e',(h\<^sub>1, l\<^sub>1)\<rangle>"
rAlpha: fixes \<alpha> :: "'a action"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and C :: "'d"   and p :: "(name \<times> name) list"   and P' :: "('a, 'b, 'c) psi" assumes "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "bn \<alpha> \<sharp>* bn (p \<bullet> \<alpha>)"   and "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))"   and "distinctPerm p"   and "bn (p \<bullet> \<alpha>) \<sharp>* \<alpha>"   and "bn (p \<bullet> \<alpha>) \<sharp>* P'"   and "Prop C \<Psi> P \<alpha> P'" shows "Prop C \<Psi> P (p \<bullet> \<alpha>) (p \<bullet> P')"
ring_iso_imp_img_cring: fixes h :: "'c \<Rightarrow> 'e"   and S :: "('e, 'f) ring_scheme" assumes "h \<in> ring_iso (S\<lparr>zero := h \<zero>\<rparr>) S" shows "cring (S\<lparr>zero := h \<zero>\<^bsub>S\<lparr>zero := h \<zero>\<rparr>\<^esub>\<rparr>)"
span_base: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> S" shows "a \<in> cspan S"
insertCI: fixes a :: "'l"   and B :: "'l set"   and b :: "'l" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
1: shows "pmf (bv n) as = (1 / 2) ^ n"
theNFT_def: fixes cl :: "('a, 'b, 'c) cmd list" shows "theNFT cl \<equiv> {n. n < length cl \<and> \<not> finished (cl ! n)}"
\<alpha>_\<beta>: shows "LeftDerivation \<alpha> D \<beta>"
Inf_counterex_reducing: fixes N :: "'a literal multiset set"   and D :: "'a literal multiset" assumes "N \<inter> {{#}} = {}"   and "D \<in> N"   and "\<not> I_of N \<TTurnstile>e {D}"   and "\<And>C. \<lbrakk>C \<in> N; \<not> I_of N \<TTurnstile>e {C}\<rbrakk> \<Longrightarrow> D \<le> C" shows "\<exists>\<iota>\<in>Inf. inference.prems_of \<iota> \<noteq> [] \<and> Standard_Redundancy_Criterion.main_prem_of \<iota> = D \<and> set (Standard_Redundancy_Criterion.side_prems_of \<iota>) \<subseteq> N \<and> I_of N \<TTurnstile>e set (Standard_Redundancy_Criterion.side_prems_of \<iota>) \<and> \<not> I_of N \<TTurnstile>e {Calculus.inference.concl_of \<iota>} \<and> Calculus.inference.concl_of \<iota> < D"
WTrtCallNT: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and es :: "char list exp list"   and Ts :: "ty list"   and M :: "char list"   and T :: "ty" assumes "P,E,h \<turnstile> e : NT"   and "P,E,h \<turnstile> es [:] Ts" shows "P,E,h \<turnstile> e\<bullet>M(es) : T"
simps_2: fixes x :: "'m"   and f :: "'m \<Rightarrow> ('k, 'l) result" shows "Ok x \<bind> f = f x"
MKT_2: shows "avl (MKT n l (MKT rn rl rr rh) h)"
get_ancestors_ok: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ptr |\<in>| object_ptr_kinds h"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h" shows "h \<turnstile> ok Shadow_DOM.get_ancestors ptr"
sigma_sets_mono'': fixes A :: "'d set"   and C :: "'d set"   and D :: "'d set set"   and B :: "'d set set" assumes "A \<in> sigma_sets C D"   and "B \<subseteq> D"   and "D \<subseteq> Pow C" shows "sigma_sets A B \<subseteq> sigma_sets C D"
simps_2: fixes x21 :: "'c"   and x22 :: "'c" shows "Infinity \<noteq> Point x21 x22"
ref_succeed_def: fixes m :: "'b nres"   and m' :: "'b nres" shows "ref_succeed m m' = (m \<le> m' \<and> (m = SUCCEED \<longrightarrow> m' = SUCCEED))"
rel_cases: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a le_constraint"   and b :: "'b le_constraint" assumes "rel_le_constraint R a b" obtains x1a :: "le_rel" and x2a :: "linear_poly" and x3a :: "'a" and y1a :: "le_rel" and y2a :: "linear_poly" and y3a :: "'b" where "a = Le_Constraint x1a x2a x3a" and "b = Le_Constraint y1a y2a y3a" and "x1a = y1a" and "x2a = y2a" and "R x3a y3a"
Rep_downset_inject: fixes x :: "'a downset"   and y :: "'a downset" shows "(Rep_downset x = Rep_downset y) = (x = y)"
Gets_imp_knows_analz: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> bankerb_gets" shows "X \<in> analz (knows B evs)"
circle_left_edge_def: shows "circle_left_edge = (- 1, \<lambda>y. (- (d / 2), 0::'b))"
prefix_elem_to_carriers: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat"   and x :: "('a \<times> ('a, 'b) operation) event" assumes "xs prefix of i"   and "x \<in> set xs" shows "x \<in> set (history i)"
v_in_preconditions_of_op: shows "v \<in> set (precondition_of op)"
a_closed: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "x \<oplus>\<^bsub>K [X]\<^esub> y \<in> carrier (K [X])"
constr_rep_2: fixes x2 :: "act"   and x1 :: "ccs" shows "Rep_ccs (x2.x1) = Action_Rep x2 (Rep_ccs x1)"
mult_mat_vec_def: fixes A :: "'b mat"   and v :: "'b Matrix.vec" shows "A *\<^sub>v v \<equiv> Matrix.vec (dim_row A) (\<lambda>i. Matrix.row A i \<bullet> v)"
induct: fixes P :: "'c list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'c list"   and a1 :: "nat" assumes "\<And>x uu_. P (x # uu_) 0"   and "\<And>uv_ xs n. P xs n \<Longrightarrow> P (uv_ # xs) (Suc n)"   and "\<And>ux_. P [] ux_"   and "\<And>v. P [] (Suc v)" shows "P a0 a1"
pos_minus_divideR_le_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "real"   and b :: "'a"   and a :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less_eq (uminus (scaleR (inverse c) b)) a = less_eq (uminus b) (scaleR c a)"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b OR a OR c = a OR b OR c"
HEndPhase0_HInv4c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4c s q" shows "HInv4c s' q"
not_ordLess_ordLeq: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r <o r'" shows "(r', r) \<notin> ordLeq"