hcju/isabelleps huggingface dataset:
strict_monoD: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "order.strict_mono less f"   and "less x y" shows "f x < f y"
fac_acc_body_final_def: shows "fac_acc_body_final \<equiv> \<Lambda> r n acc. if n = 0 then up\<cdot>acc else r\<cdot>(n -\<^sub># 1)\<cdot>(n *\<^sub># acc)"
assms_2: shows "h \<in> R \<rightarrow>\<^sub>M state_measure V \<Gamma>"
assms: shows "homeomorphic_maps (nsphere p) (nsphere p) f g"
irreducible_def: fixes p :: "'a" shows "irreducible p = (p \<noteq> (0::'a) \<and> \<not> is_unit p \<and> (\<forall>a b. p = a * b \<longrightarrow> is_unit a \<or> is_unit b))"
map_add_def: fixes m1 :: "'c \<Rightarrow> 'd option"   and m2 :: "'c \<Rightarrow> 'd option" shows "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b * (a * c) = a * (b * c)"
IDirProd_eq_subgroup_mult: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G" shows "IDirProd G H J = H <#> J"
cases: fixes x :: "'b ltlr" obtains \<mu> :: "'b ltlr" and \<psi> :: "'b ltlr" where "x = \<mu>__ and\<^sub>r \<psi>__"   | \<mu> :: "'b ltlr" and \<psi> :: "'b ltlr" where "x = \<mu>'__ U\<^sub>r \<psi>'__"   | \<mu> :: "'b ltlr" and \<psi> :: "'b ltlr" where "x = \<mu>''__ R\<^sub>r \<psi>''__"   | \<mu> :: "'b ltlr" and \<psi> :: "'b ltlr" where "x = \<mu> or\<^sub>r \<psi>"   | "x = true\<^sub>r"   | "x = false\<^sub>r"   | v :: "'b" where "x = prop\<^sub>r(v__)"   | v :: "'b" where "x = nprop\<^sub>r(va__)"   | v :: "'b ltlr" where "x = X\<^sub>r v"
T5: shows "B Out C0 C \<and> Cong B C0 B A"
leI: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" assumes "\<not> lt_of_key_order ko x y" shows "le_of_key_order ko y x"
induct: fixes x1 :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and x2 :: "'e"   and x3 :: "'f \<Rightarrow> 'g"   and x4 :: "'e"   and x5 :: "'f \<Rightarrow> 'g"   and P :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint) \<Rightarrow> 'e \<Rightarrow> ('f \<Rightarrow> 'g) \<Rightarrow> 'e \<Rightarrow> ('f \<Rightarrow> 'g) \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<rightarrow>* \<langle>x4, x5\<rangle>"   and "\<And>A l u. P A l u l u"   and "\<And>A l u l' u' l'' u''. \<lbrakk>A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>; A \<turnstile> \<langle>l', u'\<rangle> \<rightarrow>* \<langle>l'', u''\<rangle>; P A l' u' l'' u''\<rbrakk> \<Longrightarrow> P A l u l'' u''" shows "P x1 x2 x3 x4 x5"
silent_moves_preds_transfers_2: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s')" shows "transfers (map f as) s = s'"
simps_2: fixes arity_sym :: "'a \<Rightarrow> enat"   and arity_var :: "'b \<Rightarrow> enat"   and f :: "'a" shows "arity.arity_hd arity_sym arity_var (Sym f) = arity_sym f"
nb: fixes z :: "complex" assumes "bounded (path_component_set (- S) z)" shows "z \<in> S"
is_rewritable_spp_def: fixes bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and u :: "'t" shows "is_rewritable_spp bs p u = (\<exists>b\<in>set bs. fst b adds\<^sub>t u \<and> rword_strict p b)"
induct: fixes P :: "iface \<Rightarrow> (iface \<Rightarrow> ('a word \<times> nat) list option) \<Rightarrow> 'a common_primitive rule list \<Rightarrow> 'a word set \<Rightarrow> 'a word set \<Rightarrow> bool"   and a0 :: "iface"   and a1 :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and a2 :: "'a common_primitive rule list"   and a3 :: "'a word set"   and a4 :: "'a word set" assumes "\<And>iface ipassmt allowed denied1. P iface ipassmt [] allowed denied1"   and "\<And>iface ipassmt m rs allowed denied1. P iface ipassmt rs (allowed \<union> get_exists_matching_src_ips iface m) denied1 \<Longrightarrow> P iface ipassmt (Rule m Accept # rs) allowed denied1"   and "\<And>iface ipassmt m rs allowed denied1. P iface ipassmt rs allowed (denied1 \<union> (get_all_matching_src_ips iface m - allowed)) \<Longrightarrow> P iface ipassmt (Rule m Drop # rs) allowed denied1"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Log # va) ux_ uy_"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Reject # va) ux_ uy_"   and "\<And>uu_ uv_ vb vd va ux_ uy_. P uu_ uv_ (Rule vb (Call vd) # va) ux_ uy_"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Return # va) ux_ uy_"   and "\<And>uu_ uv_ vb vd va ux_ uy_. P uu_ uv_ (Rule vb (Goto vd) # va) ux_ uy_"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Empty # va) ux_ uy_"   and "\<And>uu_ uv_ vb va ux_ uy_. P uu_ uv_ (Rule vb Unknown # va) ux_ uy_" shows "P a0 a1 a2 a3 a4"
termi: shows "\<Gamma>\<turnstile>cs,css \<Down> s"
eval_gext: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and w :: "vals"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (w, s')" shows "snd s\<le>|snd s'"
product_over_sum_left: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y \<and> O x z" shows "P (x \<otimes> (y \<oplus> z)) (x \<otimes> y \<oplus> x \<otimes> z)"
zip_null_gr: fixes xs :: "'b list"   and k :: "nat"   and ys :: "'c list" assumes "length xs = k"   and "length ys = k"   and "zip xs ys \<noteq> []" shows "0 < k"
l9_9: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TS Bet P Q A B" shows "\<not> Tarski_neutral_dimensionless.OS Bet P Q A B"
in_image_msetE: fixes x :: "'a"   and f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" assumes "x \<in># image_mset f M" obtains y :: "'b" where "y \<in># M" and "x = f y"
ring_mat_simps_3: fixes ty :: "'b itself"   and n :: "nat"   and b :: "'c" shows "\<one>\<^bsub>ring_mat ty n b\<^esub> = 1\<^sub>m n"
setdist_eq_0_compact_closed: fixes S :: "'b set"   and T :: "'b set" assumes "compact S"   and "closed T" shows "(setdist S T = 0) = (S = {} \<or> T = {} \<or> S \<inter> T \<noteq> {})"
lex_convex2': fixes q :: "real \<times> real"   and p :: "real \<times> real"   and r :: "real \<times> real"   and u :: "real" assumes "Counterclockwise_2D_Arbitrary.lex q p"   and "Counterclockwise_2D_Arbitrary.lex r p"   and "0 \<le> u"   and "u \<le> 1" shows "Counterclockwise_2D_Arbitrary.lex (u *\<^sub>R q + (1 - u) *\<^sub>R r) p"
sp_fundefs_add: fixes F :: "'fenv_ubx"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and f :: "'fun" assumes "sp_fundefs (Fubx_get F)"   and "sp_fundef (Fubx_get F) fd (body fd) = Ok [None]"   and "all_same_arities (Fubx_get F) (Fubx_get (Fubx_add F f fd))" shows "sp_fundefs (Fubx_get (Fubx_add F f fd))"
st_hrabs: fixes x :: "real star" assumes "x \<in> HFinite" shows "\<bar>st x\<bar> = st \<bar>x\<bar>"
s1_def: shows "s1 = (if conv_abscissa f = - \<infinity> then s \<bullet> (1::'a) - 2 else s \<bullet> (1::'a) * 1 / 3 + real_of_ereal (conv_abscissa f) * 2 / 3)"
simps: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and a1 :: "('b, 'a) tm"   and a2 :: "('b, 'a) tm" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf" shows "epo.gt ground_heads_var gt_sym extf a1 a2 = ((\<exists>t s. a1 = t \<and> a2 = s \<and> is_App t \<and> (epo.gt ground_heads_var gt_sym extf (chop t) s \<or> chop t = s)) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> ground_heads.gt_hd gt_sym ground_heads_var (head t) (head s) \<and> is_Sym (head s) \<and> epo.chkchop (epo.gt ground_heads_var gt_sym extf) t s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> head t = head s \<and> epo.chkchop_same (epo.gt ground_heads_var gt_sym extf) t s \<and> (\<forall>f\<in>ground_heads.ground_heads ground_heads_var (head t). extf f (epo.gt ground_heads_var gt_sym extf) (args t) (args s))))"
ccpo_Sup_singleton: fixes Sup :: "'d set \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.ccpo Sup less_eq less" shows "Sup {x} = x"
fds_deriv_zeta': shows "fds_deriv fds_zeta = - fds (\<lambda>x. of_real (ln (real x)))"
list_all2: shows "list_all2 (\<lambda>x y. triple_eq_pair x y (\<lambda>tp'. filter_a1_map pos tp' a1_map) (\<lambda>ts' tp'. filter_a2_map I ts' tp' a2_map)) (drop (length done) auxlist) (zip (linearize tss) [tp - len..<tp])"
impl_prop: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(\<lceil>P\<rceil> \<subseteq> \<lceil>Q\<rceil>) = (\<forall>s. P s \<longrightarrow> Q s)"
simps_2: fixes i\<^sub>1 :: "int"   and i\<^sub>2 :: "int" shows "binop (Add, Intg i\<^sub>1, Intg i\<^sub>2) = \<lfloor>Intg (i\<^sub>1 + i\<^sub>2)\<rfloor>"
compactin_euclidean_iff: fixes S :: "'a set" shows "compactin euclidean S = compact S"
summands_valid: fixes A :: "'c set"   and a :: "'c \<Rightarrow> 'a" assumes "A \<subseteq> carrier V"   and "a \<in> A \<rightarrow> carrier K" shows "\<forall>v\<in>A. a v \<odot>\<^bsub>V\<^esub> v \<in> carrier V"
qAFresh_qSwap_exchange1: fixes zs :: "'d"   and z2 :: "'e"   and z1 :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "qAFresh zs z2 (X #[[z1 \<and> z2]]_zs) = qAFresh zs z1 X"
OUTfromVCorrect2SYSTEM_def: shows "OUTfromVCorrect2SYSTEM \<equiv> \<forall>x. OUTfromVCorrect2 x"
is_extensional: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "natural_transformation A B F G \<tau>"   and "\<not> partial_magma.arr A f" shows "\<tau> f = partial_magma.null B"
s\<^sub>1_r: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV T)) e1 e2])"
simple_tycon_compare: shows "OFCLASS(simple_tycon, compare_class)"
Dirichlet_kernel_def: shows "Dirichlet_kernel \<equiv> \<lambda>n x. if x = 0 then real n + 1 / 2 else sin ((real n + 1 / 2) * x) / (2 * sin (x / 2))"
irreducible\<^sub>dD_2: fixes p :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" assumes "irreducible\<^sub>d p"   and "degree q < degree p"   and "degree r < degree p" shows "p \<noteq> q * r"
content_ball_conv_unit_ball: fixes r :: "real"   and c :: "'a" assumes "0 \<le> r" shows "content (ball c r) = r ^ DIM('a) * content (ball (0::'a) 1)"
map_IInf_def: fixes F :: "'a upset \<Rightarrow> 'b upset" shows "map_IInf F = IInf \<circ> F \<circ> us"
ipresIGOp_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'c, 'e, 'h) model_scheme"   and MOD' :: "('index, 'bindex, 'i, 'j, 'opSym, 'k, 'd, 'f, 'l) model_scheme" shows "ipresIGOp h hA MOD MOD' \<equiv> \<forall>delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> h (igOp MOD delta inp binp) = igOp MOD' delta (lift h inp) (lift hA binp)"
comp_red_basis_lt: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> set (comp_red_basis xs)" obtains q :: "'t \<Rightarrow>\<^sub>0 'c" where "q \<in> set xs" and "q \<noteq> 0" and "lt q = lt p"
ucl'_def: shows "ucl' = map getGMUserCom (tl (purgeIdle sl'))"
oexp_Limit: fixes \<beta> :: "V"   and \<alpha> :: "V" assumes "Limit \<beta>" shows "\<alpha> \<up> \<beta> = (if \<alpha> = 0 then 0 else \<Squnion> ((\<up>) \<alpha> ` elts \<beta>))"
choice_Basis_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and P :: "'a \<Rightarrow> real \<Rightarrow> bool" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(\<forall>i\<in>Basis. \<exists>x. P i x) = (\<exists>x. \<forall>i\<in>Basis. P i (inner x i))"
eq_refl: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
cdf_distribution_def: fixes M :: "real measure" shows "cdf_distribution M \<equiv> real_distribution M"
i_nrows_iarray: shows "i < nrows_iarray (matrix_to_iarray A)"
det_identical_rows: fixes A :: "'b mat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "i \<noteq> j"   and "i < n"   and "j < n"   and "row A i = row A j" shows "det A = (0::'b)"
assms_4: shows "\<GG>' = \<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH>"
pfinite_sim: shows "pfinite_converter \<I>_real \<I>_ideal sim"
coprime_0_1: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "comm_monoid_mult.coprime times one p zero = dvd.dvd times p one"
Tsb: shows "T \<subseteq> span ((+) (- a) ` S)"
unit_simps_3: fixes a :: "'a" assumes "S.obj a" shows "S.trg \<i>[a] = a"
cod_closed: fixes f :: "'a \<times> 'a" assumes "B.arr (fst f) \<and> B.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (snd f)" shows "B.arr (fst (B.VxV.cod f)) \<and> B.arr (snd (B.VxV.cod f)) \<and> src\<^sub>B (fst (B.VxV.cod f)) = trg\<^sub>B (snd (B.VxV.cod f))"
axioms_2: fixes s :: "(real \<times> real) set"   and i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain_typeI :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and two_chain_typeII :: "(real \<times> real \<Rightarrow> real \<times> real) set" assumes "green_typeI_typeII_chain s i j F two_chain_typeI two_chain_typeII" shows "green_typeI_chain i j F two_chain_typeI s"
t_twice_append_pre: fixes ires :: "cell list"   and rs :: "nat"   and n :: "nat"   and stp :: "nat"   and ln :: "nat"   and rn :: "nat" assumes "steps0 (Suc 0, Bk # Bk # ires, Oc \<up> Suc rs @ Bk \<up> n) t_twice stp = (Suc t_twice_len, Bk \<up> ln @ Bk # Bk # ires, Oc \<up> Suc (2 * rs) @ Bk \<up> rn)" shows "steps0 (Suc 0 + length t_wcode_main_first_part div 2, Bk # Bk # ires, Oc \<up> Suc rs @ Bk \<up> n) (t_wcode_main_first_part @ shift t_twice (length t_wcode_main_first_part div 2) @ [(L, 1), (L, 1)] @ shift t_fourtimes (t_twice_len + 13) @ [(L, 1), (L, 1)]) stp = (Suc t_twice_len + length t_wcode_main_first_part div 2, Bk \<up> ln @ Bk # Bk # ires, Oc \<up> Suc (2 * rs) @ Bk \<up> rn)"
prob_space_distr: fixes f :: "'b \<times> 'b \<Rightarrow> 'c"   and M' :: "'c measure" assumes "P.random_variable M' f" shows "prob_space (distr P M' f)"
zero_pp: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat_pp :: "'a \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'a"   and lex_comp' :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and deg' :: "'a \<Rightarrow> nat" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'" shows "rep_nat_pp zero = 0"
downset_set_ext: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "id \<le> ord.downset_set less_eq"
sorted_add_spvec_helper: fixes arr :: "(nat \<times> 'a) list"   and brr :: "(nat \<times> 'a) list"   and ab :: "nat"   and bb :: "'a"   and list :: "(nat \<times> 'a) list" assumes "add_spvec arr brr = (ab, bb) # list" shows "arr \<noteq> [] \<and> ab = fst (hd arr) \<or> brr \<noteq> [] \<and> ab = fst (hd brr)"
assms_3: shows "\<forall>z\<in>V. \<forall>zs\<in>fZs z. qFresh zs z X"
expand\<^sub>T_def: fixes n_ns :: "'a node \<times> 'a node set" shows "expand\<^sub>T n_ns \<equiv> REC\<^sub>T (\<lambda>expand (n, ns). if new n = {} then if \<exists>n'\<in>ns. old n' = old n \<and> next n' = next n then RETURN (name n, upd_incoming n ns) else expand (\<lparr>name = expand_new_name (name n), incoming = {name n}, new = next n, old = {}, next = {}\<rparr>, {n} \<union> ns) else SPEC (\<lambda>x. x \<in> new n) \<bind> (\<lambda>\<phi>. let n = n\<lparr>new := new n - {\<phi>}\<rparr> in if \<exists>q. \<phi> = prop\<^sub>r(q) \<or> \<phi> = nprop\<^sub>r(q) then if not\<^sub>r \<phi> \<in> old n then RETURN (name n, ns) else expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = true\<^sub>r then expand (n\<lparr>old := {\<phi>} \<union> old n\<rparr>, ns) else if \<phi> = false\<^sub>r then RETURN (name n, ns) else if \<exists>\<nu> \<mu>. \<phi> = \<nu> and\<^sub>r \<mu> \<or> \<phi> = X\<^sub>r \<nu> then expand (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) else expand (n\<lparr>new := new1 \<phi> \<union> new n, old := {\<phi>} \<union> old n, next := next1 \<phi> \<union> next n\<rparr>, ns) \<bind> (\<lambda>(nm, nds). expand (n\<lparr>name := nm, new := new2 \<phi> \<union> new n, old := {\<phi>} \<union> old n\<rparr>, nds)))) n_ns"
rat_to_normalized_int_poly_of_int_2: fixes p :: "int poly"   and c :: "rat"   and q :: "int poly" assumes "rat_to_normalized_int_poly (of_int_poly p) = (c, q)"   and "p \<noteq> 0" shows "c = rat_of_int (content p) \<and> q = primitive_part p"
d': shows "D' = {#L \<in># D. L \<noteq> Pos A#}"
box_empty_imp: fixes a :: "'a"   and b :: "'a" assumes "box a b = {}" shows "content (cbox a b) = 0"
nat_to_rat_surj_def: fixes n :: "nat" shows "nat_to_rat_surj n = (let (a, b) = prod_decode n in Fract (int_decode a) (int_decode b))"
Un_insert_left: fixes a :: "'a"   and B :: "'a set"   and C :: "'a set" shows "{a} \<union> B \<union> C = {a} \<union> (B \<union> C)"
preserves_cod: fixes f :: "'a" assumes "arr f" shows "cod (src f) = src (cod f)"
lookup_tensor_from_lookup: fixes "is" :: "nat list"   and ds :: "nat list"   and e :: "nat list \<Rightarrow> 'b" assumes "is \<lhd> ds" shows "lookup (tensor_from_lookup ds e) is = e is"
init_ps_def: fixes am :: "nat \<Rightarrow> nat list" shows "init_ps am \<equiv> assert (is_adj_map am) \<bind> (\<lambda>_. return am)"
inj_on_xa_A: shows "inj_on (\<lambda>x. x * a) A"
prv_imp_cnjL: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp (cnj \<phi> \<chi>) \<phi>)"
conjugation_action_def: fixes p :: "nat" shows "conjugation_action p = (\<lambda>g\<in>carrier G. \<lambda>P\<in>subgroups_of_size p. g <# (P #> inv g))"
mod_0_less_div_conv: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n mod m = 0" shows "(k * m < n) = (k < n div m)"
finite_dom_map_of: fixes l :: "('a \<times> 'b) list" shows "finite (dom (map_of l))"
fg: shows "\<forall>\<^sub>F w in F. f w = y \<longrightarrow> g y = z"
simps_2: shows "stock_measure INTEG = count_space (range IntVal)"
refl: fixes a :: "('c, 'd) Term.term" shows "a \<sqsubseteq> a"
only_fork_introduces_rids: fixes r :: "'r"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "revision_step r s s'"   and "\<And>\<sigma> \<tau> \<E> e. s r \<noteq> Some (\<sigma>, \<tau>, \<E> [Rfork e])" shows "RID\<^sub>G s' \<subseteq> RID\<^sub>G s"
cases: fixes a :: "'form"   and t :: "('form, 'rule, 'subst, 'a) itnode rose_tree"   and "is" :: "nat list" assumes "a \<in> hyps_along t is" obtains is' :: "nat list" and i :: "nat" and h :: "('form, 'var) out_port" where "a = subst (iSubst (tree_at t is')) (freshen (iAnnot (tree_at t is')) (labelsOut (iNodeOf (tree_at t is')) h))" and "prefix (is' @ [i]) is" and "i < length (inPorts' (iNodeOf (tree_at t is')))" and "hyps (iNodeOf (tree_at t is')) h = Some (inPorts' (iNodeOf (tree_at t is')) ! i)"
triangular_to_def: fixes k :: "nat"   and A :: "'a mat" shows "local.triangular_to k A \<equiv> \<forall>j<k. local.triangular_column j A"
plus_1_less: fixes x :: "'b word" shows "(x + 1 \<le> x) = (x = - 1)"
StepReachable: fixes cfg :: "('p, 'v, 's) configuration"   and cfg' :: "('p, 'v, 's) configuration"   and msg :: "('p, 'v) message" assumes "reachable cfg cfg'"   and "enabled cfg msg"   and "\<not> enabled cfg' msg" shows "stepReachable cfg msg cfg'"
prod_cases3: fixes y :: "'a \<times> 'b \<times> 'c" obtains a :: "'a" and b :: "'b" and c :: "'c" where "y = (a, b, c)"
simps_1: fixes x :: "'b" shows "splay x \<langle>\<rangle> = \<langle>\<rangle>"
l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_axioms: shows "l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t type_wf"
fn: fixes A :: "'a trm"   and A' :: "'a trm"   and x :: "'a"   and T :: "PreSimplyTyped.type" assumes "A \<rightarrow>\<beta> A'" shows "Fn x T A \<rightarrow>\<beta> Fn x T A'"
d_def: shows "d = poly_deg p - poly_deg (homogenize x (dehomogenize x p))"
\<psi>_char_4: shows "\<exists>!\<psi>. \<guillemotleft>\<psi> : D'.unity \<rightarrow>\<^sub>D F C'.unity\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<iota>\<^sub>D = F'.\<iota>\<^sub>1 \<cdot>\<^sub>D D'.T (\<psi>, \<psi>)"
polymul_1_2: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "(1)\<^sub>p *\<^sub>p p = p"
acc_downward: fixes b :: "'c"   and r :: "('c \<times> 'c) set"   and a :: "'c" assumes "b \<in> Wellfounded.acc r"   and "(a, b) \<in> r" shows "a \<in> Wellfounded.acc r"
nop_v11_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "nop_v11 A n f \<equiv> v11 f \<and> nop_v11_axioms A n f"
bags_in_V: fixes t :: "'c" assumes "t \<in> V\<^bsub>T\<^esub>" shows "bag t \<subseteq> V"
stream_eq_Stream_iff: fixes s :: "'c stream"   and x :: "'c"   and t :: "'c stream" shows "(s = x ## t) = (shd s = x \<and> stl s = t)"
STL2: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> F"
Suc_1: fixes j :: "nat"   and i :: "nat" assumes "x = j - i"   and "i \<le> j" shows "subgraph (S i) (S j)"
topspace_def: fixes T :: "'c topology" shows "topspace T = \<Union> {S. openin T S}"
visit_subset_dfs: fixes S :: "'b"   and xs :: "'a list"   and y :: "'a" assumes "invariant S"   and "list_all isNode xs"   and "isNode y"   and "memb y S" shows "memb y (dfs S xs)"
merge_restrict: fixes t1 :: "'a \<Rightarrow> 'b option"   and S :: "'a set"   and t :: "'a \<Rightarrow> 'b option"   and t2 :: "'a \<Rightarrow> 'b option" assumes "t1 |` S = t |` S"   and "t2 |` S = t |` S" shows "merge t1 t2 |` S = t |` S"
Q_invar_def: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and Q :: "nat list" shows "Q_invar f Q \<equiv> distinct Q \<and> set Q = {v \<in> V - {s, t}. excess f v \<noteq> (0::'capacity)}"
connected_append_edge: fixes u :: "nat"   and v :: "nat"   and w :: "nat" assumes "connected u v"   and "(v, w) \<in> E" shows "connected u w"