hcju/isabelleps huggingface dataset:
sum_of_two_squares': fixes n :: "nat" shows "(\<exists>a b. n = a\<^sup>2 + b\<^sup>2) = (\<forall>p. prime p \<and> [p = 3] (mod 4) \<longrightarrow> even (multiplicity p n))"
quicksort_acc_conv_sort: fixes ac :: "'a list"   and xs :: "'a list" shows "quicksort_acc ac xs = sort xs @ ac"
wide_subdg_Obj: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
invim_def: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" shows "invim f A B = {x \<in> A. f x \<in> B}"
image_mset_def: fixes f :: "'c \<Rightarrow> 'd" shows "image_mset f = fold_mset (add_mset \<circ> f) {#}"
minus_interval_tendsto: fixes X :: "nat \<Rightarrow> 'a interval"   and x :: "'a"   and Y :: "nat \<Rightarrow> 'a interval"   and y :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>i x"   and "Y \<longlonglongrightarrow>\<^sub>i y" shows "(\<lambda>i. X i - Y i) \<longlonglongrightarrow>\<^sub>i x - y"
igFreshIGAbs1STR_imp_igFreshIGAbs1: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "igFreshIGAbs1STR MOD" shows "FixSyn.igFreshIGAbs1 Delta MOD"
zdvd_zdiffD: fixes k :: "int"   and m :: "int"   and n :: "int" assumes "k dvd m - n"   and "k dvd n" shows "k dvd m"
conv_dist_comp: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<odot> y)\<^sup>t = y\<^sup>t \<odot> x\<^sup>t"
preserves_hom: fixes f :: "'a arrow_of_spans_data"   and a :: "'a arrow_of_spans_data"   and b :: "'a arrow_of_spans_data" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>L f : L a \<Rightarrow> L b\<guillemotright>"
solves_ode_def: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set" shows "(y solves_ode f) T X = ( D y = (\<lambda>t. f t (y t)) on T \<and> y \<in> T \<rightarrow> X)"
cdf_diff_eq: fixes M :: "real measure"   and x :: "real"   and y :: "real" assumes "finite_borel_measure M"   and "x < y" shows "cdf M y - cdf M x = Sigma_Algebra.measure M {x<..y}"
equivalent_objects_transitive: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.equivalent_objects V H \<a> \<i> src trg a b"   and "bicategory.equivalent_objects V H \<a> \<i> src trg b c" shows "bicategory.equivalent_objects V H \<a> \<i> src trg a c"
preorder_on_def: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Order_Predicates.preorder_on carrier le \<equiv> ((\<forall>x y. le x y \<longrightarrow> x \<in> carrier) \<and> (\<forall>x y. le x y \<longrightarrow> y \<in> carrier)) \<and> (\<forall>x. x \<in> carrier \<longrightarrow> le x x) \<and> (\<forall>x y z. le x y \<longrightarrow> le y z \<longrightarrow> le x z)"
intros: fixes c :: "('test, 'atom) com" assumes "\<And>s c' s'. (c, s) \<rightarrow>c (c', s') \<Longrightarrow> siso c'"   and "\<And>s t c' s'. \<lbrakk>s \<approx> t; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>c (c', t')"   and "\<And>s t s'. \<lbrakk>s \<approx> t; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>t t'" shows "siso c"
f_def: shows "f \<equiv> \<lambda>n. real_of_int (b n) / real_of_int (prod a {..n})"
Ssub: shows "S \<subseteq> \<Union> \<G>"
Proc_CFG_CondThen: fixes c\<^sub>1 :: "cmd"   and n :: "label"   and et :: "p_edge_kind"   and n' :: "label"   and b :: "expr"   and c\<^sub>2 :: "cmd" assumes "c\<^sub>1 \<turnstile> n -et\<rightarrow>\<^sub>p n'"   and "n \<noteq> Entry" shows "if (b) c\<^sub>1 else c\<^sub>2 \<turnstile> n \<oplus> 1 -et\<rightarrow>\<^sub>p n' \<oplus> 1"
coeff_diff: fixes p :: "'a poly"   and q :: "'a poly"   and n :: "nat" shows "coeff (p - q) n = coeff p n - coeff q n"
onorm_scaleR_left: fixes r :: "'b \<Rightarrow> real"   and f :: "'c" assumes "bounded_linear r" shows "onorm (\<lambda>x. r x *\<^sub>R f) = onorm r * norm f"
Pi_split_insert_domain: fixes x :: "'b \<Rightarrow> 'c"   and i :: "'b"   and I :: "'b set"   and X :: "'b \<Rightarrow> 'c set" shows "(x \<in> Pi (insert i I) X) = (x \<in> Pi I X \<and> x i \<in> X i)"
quotient_imp_surjective_map: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "quotient_map X X' f" shows "f ` topspace X = topspace X'"
negligible_frontier_interval: fixes a :: "'a"   and b :: "'a" shows "negligible (cbox a b - box a b)"
runProc_cl_inv: fixes prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState"   and name :: "String.literal"   and args :: "expr list" assumes "program_inv prog"   and "gState_inv prog g"   and "pState_inv prog p"   and "cl_inv (g, p)" shows "cl_inv (runProc name args prog g p)"
a2: shows "\<forall>pkt. (pkt \<inter> fmask1 = c1) = (pkt \<inter> fmask2 = c2)"
cut_add: fixes n :: "nat"   and m :: "nat"   and P :: "'v" shows "cut n (add m P) = (if n \<le> m then cut n P else add m (cut n P))"
refl: fixes S :: "('c, 'd) eq_object_scheme"   and x :: "'c" assumes "equivalence S"   and "x \<in> carrier S" shows "x .=\<^bsub>S\<^esub> x"
A: shows "{msg. msg \<in># msgs init \<and> (\<exists>p v. <p, v> = msg)} = {}"
dickson_less_pD3: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'d \<Rightarrow> nat"   and m :: "nat"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "gd_term.dickson_less_p pair_of_term ord_term_strict d m p q" shows "ordered_term.ord_strict_p ord_term_strict p q"
\<Phi>_def: shows "\<Phi> \<equiv> \<lambda>NL N. N \<subseteq> hd NL \<and> Inf (hd NL) < Inf N \<and> infinite N \<and> decides_all (set (map Inf NL)) N"
finals_def: fixes A :: "'b list set" shows "finals A \<equiv> {\<approx>A `` {s} |s. s \<in> A}"
finite_digraph_axioms: shows "finite_digraph \<alpha> \<CC>"
set_subst_ident: fixes M :: "('b, 'a) Term.term set"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv\<^sub>s\<^sub>e\<^sub>t M \<inter> subst_domain \<theta> = {}" shows "M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta> = M"
ideD_3: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.cod C a = a"
a: shows "\<theta> / 2 = arccos (sqrt (real (N - M) / real N))"
indep_sets_cong: fixes I :: "'d set"   and J :: "'d set"   and F :: "'d \<Rightarrow> 'a set set"   and G :: "'d \<Rightarrow> 'a set set" assumes "I = J"   and "\<And>i. i \<in> I \<Longrightarrow> F i = G i" shows "indep_sets F I = indep_sets G J"
is_float_inverse_aform': fixes p :: "nat"   and X :: "real \<times> real pdevs" shows "is_float (fst (fst (inverse_aform' p X)))"
assms_1: shows "\<phi> (\<sigma>\<^bsub>c\<^esub>t n)"
perm_restrict_union: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "perm_restrict f A permutes A"   and "perm_restrict f B permutes B"   and "A \<inter> B = {}" shows "perm_restrict f A \<circ> perm_restrict f B = perm_restrict f (A \<union> B)"
M_def: shows "M \<equiv> \<lambda>A. trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t (unlabel A)"
berlekamp_basis_def: fixes u :: "'a mod_ring poly" shows "local.berlekamp_basis u = map (Poly \<circ> list_of_vec) (find_base_vectors (local.berlekamp_resulting_mat u))"
represents_cnj: fixes p :: "int poly"   and x :: "complex" assumes "p represents x" shows "p represents cnj x"
comp_functor_identity: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd" assumes "functor A B F" shows "F \<circ> identity_functor.map A = F"
add_cancel_right_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus a b) = (b = zero)"
convolution_of_aezfun_is_aezfun: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> aezfun_set"   and "g \<in> aezfun_set" shows "convolution f g \<in> aezfun_set"
assms_4: shows "subst_domain \<theta>\<^sub>1 \<inter> vars\<^sub>s\<^sub>t S\<^sub>1 = {}"
nin_P: fixes s :: "real" assumes "0 < s"   and "s < h" shows "flow0 x0 s \<notin> P"
select_index_def: fixes x :: "'a" shows "select_index x \<equiv> SOME i. i \<in> index \<and> x \<in> cover i"
fps_to_fls_eq_zero_iff: fixes f :: "'b fps" shows "(fps_to_fls f = 0) = (f = 0)"
linear_injective_isomorphism: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f"   and "vec.dim UNIV = vec.dim UNIV" shows "\<exists>f'. Vector_Spaces.linear (*s) (*s) f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
arcs_ends_symmetric: fixes G :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "symmetric G"   and "u \<rightarrow>\<^bsub>G\<^esub> v" shows "v \<rightarrow>\<^bsub>G\<^esub> u"
Zero_in_aug_inf: shows "0 \<in> Z\<^sub>\<infinity>"
star_rtc1_eq: fixes x :: "'a" shows "(1::'a) + x + x\<^sup>\<star> \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
foundation11: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> x"   and "\<tau> \<Turnstile> \<delta> y" shows "(\<tau> \<Turnstile> x or y) = (\<tau> \<Turnstile> x \<or> \<tau> \<Turnstile> y)"
assms_2: fixes a' :: "'a" assumes "a \<in> carrier S"   and "a' \<in> A"   and "a .= a'" shows "P"
mono: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "le_utrans t u"   and "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "\<And>P. unitary P \<Longrightarrow> unitary (u P)" shows "le_utrans (T t) (T u)"
FiniteSingleAgentEnvironment_axioms_def: fixes agent :: "'a" shows "FiniteSingleAgentEnvironment_axioms agent \<equiv> \<forall>a. a = agent"
xs_eq: shows "[i + 1..<min (dim_row A) (dim_col A)] = [i + 1..<min (nrows A') (ncols A')]"
exec_n_split_full: fixes P :: "instr list"   and P' :: "instr list"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and k :: "nat"   and j :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list" assumes "P @ P' \<turnstile> (0, s, stk) \<rightarrow>^k (j, s', stk')"   and "size P \<le> j"   and "closed P"   and "exits P' \<subseteq> {0..}" shows "\<exists>k1 k2 s'' stk''. P \<turnstile> (0, s, stk) \<rightarrow>^k1 (size P, s'', stk'') \<and> P' \<turnstile> (0, s'', stk'') \<rightarrow>^k2 (j - size P, s', stk')"
simps_2: fixes s :: "('a, 'b option) insert_message list"   and m :: "'a delete_message" shows "integrate s (Delete m) = integrate_delete m s"
rrep_1_update_changes: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l = PRrep-:1 \<longrightarrow> rt \<xi> \<noteq> update (rt \<xi>) (dip \<xi>) (dsn \<xi>, kno, val, hops \<xi> + 1, sip \<xi>))"
induct: fixes P :: "('c, 'd) strand_step list \<Rightarrow> bool"   and a0 :: "('c, 'd) strand_step list" assumes "P []"   and "\<And>t S. P S \<Longrightarrow> P (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>t S. P S \<Longrightarrow> P (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v va vb S. P S \<Longrightarrow> P (\<langle>v: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v va S. P S \<Longrightarrow> P (\<forall>v\<langle>\<or>\<noteq>: va\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0"
inf_dual_def: fixes x :: "'a dual"   and y :: "'a dual" shows "x \<sqinter> y = \<partial> (\<partial>\<^sup>- x \<squnion> \<partial>\<^sup>- y)"
pp_is_le_rel_canc_right: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a"   and u :: "'a"   and t :: "'a" assumes "ordered_powerprod_lin.is_le_rel rel" shows "rel (s + u) (t + u) = rel s t"
map_iterator_dom_def: fixes it :: "('g \<Rightarrow> bool) \<Rightarrow> ('h \<times> 'i \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g" shows "map_iterator_dom it = set_iterator_image fst it"
iwlsFSbSw_iterAll_termFSwSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSbSw MOD" shows "termFSwSbImorph (iter MOD) (iterAbs MOD) MOD"
intro: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd" assumes "network history msg_id"   and "causal_network_axioms history" shows "causal_network history msg_id"
map: fixes z :: "nat"   and L :: "real list" shows "eval (liftmap (\<lambda>x A. fm.Atom (linear_substitution (var + x) (liftPoly 0 x b) (liftPoly 0 x c) A)) \<phi> (z + x1)) L = eval (liftmap (\<lambda>x A. fm.Atom (linear_substitution (var + x1 + x) (liftPoly 0 (x + x1) b) (liftPoly 0 (x + x1) c) A)) \<phi> z) L"
extract_def: shows "extract \<equiv> \<lambda>a ts. if a \<in> ts then 1 else 0"
simps_18: fixes x :: "float" shows "fold_const_fa (Num x) = Num x"
measurable_sets_Collect: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "{x \<in> space N. P x} \<in> sets N" shows "{x \<in> space M. P (f x)} \<in> sets M"
arev_arev_raw: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "bidirected_digraph G arev"   and "a \<in> arcs G" shows "arev (arev a) = a"
assn2_lupd: fixes x :: "char list"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat"   and n :: "nat" assumes "x \<notin> support Q" shows "Q (l(x := n)) = Q l"
simps_1: fixes ops :: "'b arith_ops_record"   and f :: "'b"   and x :: "'b"   and xs :: "'b list" shows "pderiv_main_i ops f (x # xs) = cCons_i ops (arith_ops_record.times ops f x) (pderiv_main_i ops (arith_ops_record.plus ops f (arith_ops_record.one ops)) xs)"
distinct_9: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and x25 :: "nat"   and x31 :: "nat \<Rightarrow> nat option"   and x32 :: "nat" shows "Rrep x21 x22 x23 x24 x25 \<noteq> Rerr x31 x32"
aux_lemma2: fixes l :: "'a list"   and q :: "'a list"   and n :: "nat" assumes "set l \<subseteq> K"   and "q \<in> carrier (K [X])" shows "UP.eval (map poly_of_const l) q = UP.eval (map poly_of_const (replicate n \<zero> @ l)) q"
SUP_mono': fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'d \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` A))"
cosh_minus: fixes x :: "'a" shows "cosh (- x) = cosh x"
fresh_image_atom: fixes A :: "'a set"   and i :: "atom" assumes "finite A" shows "i \<sharp> atom ` A = i \<sharp> A"
consistent_length_tcsigs_def: fixes tcs :: "'a \<Rightarrow> ('b \<Rightarrow> 'c list option) option" shows "consistent_length_tcsigs tcs \<equiv> \<forall>ars\<in>ran tcs. \<forall>ss\<^sub>1\<in>ran ars. \<forall>ss\<^sub>2\<in>ran ars. length ss\<^sub>1 = length ss\<^sub>2"
iprog_trans: fixes i :: "init_state"   and i' :: "init_state"   and i'' :: "init_state" assumes "i \<le>\<^sub>i i'"   and "i' \<le>\<^sub>i i''" shows "i \<le>\<^sub>i i''"
bind_map_spmf: fixes f :: "'f \<Rightarrow> 'e"   and p :: "'f spmf"   and g :: "'e \<Rightarrow> 'd spmf" shows "map_spmf f p \<bind> g = p \<bind> g \<circ> f"
valid_c_offending_flows: fixes m :: "'b SecurityInvariant_configured"   and G :: "'b graph" assumes "configured_SecurityInvariant m" shows "c_offending_flows m G = {F. F \<subseteq> edges G \<and> \<not> c_sinvar m G \<and> c_sinvar m (delete_edges G F) \<and> (\<forall>(e1, e2)\<in>F. \<not> c_sinvar m (add_edge e1 e2 (delete_edges G F)))}"
LIST_FOREACH'_refine: fixes tsl' :: "'d list nres"   and c' :: "'c \<Rightarrow> bool"   and f' :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma>' :: "'c"   and \<Phi> :: "'d set \<Rightarrow> 'c \<Rightarrow> bool" shows "LIST_FOREACH' tsl' c' f' \<sigma>' \<le> LIST_FOREACH \<Phi> tsl' c' f' \<sigma>'"
encoding_wrt_barbs_axioms: shows "encoding_wrt_barbs Source Target SWB TWB"
inf_left_commute: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (inf (y::'b::type) (z::'b::type)) = inf y (inf x z)"
HNextPart_Inv2c_chosen: fixes s :: "state"   and s' :: "state" assumes "HNextPart s s'"   and "Inv2c s"   and "\<forall>p. outpt s' p = (if phase s' p = 3 then inp (dblock s' p) else NotAnInput)"   and "\<forall>p. inp (dblock s' p) \<in> allInput s' \<union> {NotAnInput}" shows "chosen s' \<in> allInput s' \<union> {NotAnInput}"
phis'_aux_def: fixes g :: "'g"   and v :: "'var"   and ns :: "'node set"   and phis :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" shows "phis'_aux g v ns phis \<equiv> Mapping (\<lambda>(m, v\<^sub>2). if v\<^sub>2 = v \<and> m \<in> \<Union> (phiDefNodes_aux g v (filter (\<lambda>n. (n, v) \<notin> Mapping.keys phis) (\<alpha>n g)) ` ns) \<and> v \<in> vars g then Some (map (\<lambda>m. lookupDef g m v) (predecessors g m)) else Mapping.lookup phis (m, v\<^sub>2))"
PQ: fixes t :: "('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b" assumes "P t" shows "Q t"
distinct_10: fixes x6 :: "Arraytype" shows "ArrT x6 \<noteq> BoolT"
top_greatest: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c"   and a :: "'c" assumes "class.order_top less_eq less top" shows "less_eq a top"
length_rule: fixes a :: "'b array"   and xs :: "'b list" shows "<a \<mapsto>\<^sub>a xs> Array.len a <\<lambda>r. a \<mapsto>\<^sub>a xs * \<up> (r = length xs)>"
prems_3: shows "\<forall>f\<in>funs_term (Fun f T). is_Abs f \<longrightarrow> the_Abs f \<in> A"
counit_in_hom_2: fixes a :: "'d" assumes "D.obj a" shows "\<guillemotleft>\<eta>'.counit a : \<eta>\<^sub>0 a \<star>\<^sub>D \<eta>'.map\<^sub>0 a \<Rightarrow>\<^sub>D GF.map\<^sub>0 a\<guillemotright>"
l_inv_gHom: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "gbij\<^bsub>F,G\<^esub> f" shows "(f\<inverse>\<^bsub>carrier G,carrier F\<^esub>) \<circ>\<^bsub>F\<^esub> f = I\<^bsub>F\<^esub>"
borel_open: fixes A :: "'a set" assumes "open A" shows "A \<in> sets borel"
subgroups_of_size_def: fixes p :: "nat" shows "groupP.subgroups_of_size p = {H. subgroup H (G\<lparr>carrier := P\<rparr>) \<and> card H = p}"
rewriteR_comp_comp2: fixes g :: "'d \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'd"   and r1 :: "'e \<Rightarrow> 'c"   and r2 :: "'b \<Rightarrow> 'e"   and f :: "'c \<Rightarrow> 'f"   and l :: "'e \<Rightarrow> 'f" assumes "g \<circ> h = r1 \<circ> r2"   and "f \<circ> r1 = l" shows "f \<circ> g \<circ> h = l \<circ> r2"
word_of_int_conv_set_bits: fixes i :: "int" shows "word_of_int i = (BITS n. i !! n)"
Eq5: shows "IC q (i + 1) (te q (i + 1)) = cfn q (\<theta> q (i + 1))"
parallel_reduction_sndE: fixes P :: "'a trm"   and X :: "'a trm" assumes "Snd P >> X" shows "(\<exists>P'. P >> P' \<and> X = Snd P') \<or> (\<exists>A B B'. P = SimplyTyped.Pair A B \<and> B >> B' \<and> X = B')"
reach_obs_upd: fixes x :: "'a \<Rightarrow> 'b"   and S :: "('a, 'b, 'c) spec_scheme" shows "reach (S\<lparr>obs := x\<rparr>) = reach S"
doesnt_modify_def: fixes c :: "'Com\<^sub>A"   and x :: "'Var\<^sub>A" shows "abs.doesnt_modify c x = (\<forall>mds mem c' mds' mem'. \<langle>c, mds, mem\<rangle>\<^sub>A \<leadsto>\<^sub>A \<langle>c', mds', mem'\<rangle>\<^sub>A \<longrightarrow> mem x = mem' x \<and> dma\<^sub>A mem x = dma\<^sub>A mem' x)"
ceq_expr'_def: shows "CEQ('a expr') = (case ID CEQ('a) of None \<Rightarrow> None | Some eq \<Rightarrow> Some (expr'_rel eq))"
ap_specD2: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) set"   and m :: "nat"   and h :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and g :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "ap_spec ap"   and "dickson_grading d"   and "set gs \<union> set bs \<union> set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "(h, g) \<in>\<^sub>p set hs \<times> (set gs \<union> set bs \<union> set hs)"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "h \<noteq> g"   and "fst h \<noteq> 0"   and "fst g \<noteq> 0"   and "\<And>a b. \<lbrakk>(a, b) \<in>\<^sub>p set (ap gs bs ps hs data); fst a \<noteq> 0; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)"   and "\<And>a b. \<lbrakk>a \<in> set gs \<union> set bs; b \<in> set gs \<union> set bs; fst a \<noteq> 0; fst b \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)" shows "crit_pair_cbelow_on d m (fst ` B) (fst h) (fst g)"
Sup1: fixes x :: "'a Assertion"   and A :: "'a Assertion set" assumes "x \<in> A" shows "x \<le> \<Squnion>A"
compact_filter: fixes U :: "'b set" shows "compact U = (\<forall>F. F \<noteq> bot \<longrightarrow> (\<forall>\<^sub>F x in F. x \<in> U) \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case EV_WAIT of EV_PREP \<Rightarrow> f1 | EV_WAIT \<Rightarrow> f2 | EV_FINISH \<Rightarrow> f3) = f2"