hcju/isabelleps huggingface dataset:
elims_2: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and m :: "('a, 'b) insert_message"   and x :: "'a extended"   and xa :: "'a extended"   and xb :: "('a, 'b option) insert_message" assumes "integrate_insert_commute_insert M a s m"   and "integrate_insert_commute_insert.is_concurrent a s x xa xb" obtains pm :: "'a extended" and sm :: "'a extended" and xc :: "('a, 'b option) insert_message" where "x = pm" and "xa = sm" and "xb = xc" and "xc \<in> set s \<and> Psi.subset (a pm, a sm) (a (P xc), a (S xc)) \<and> elem (a \<lbrakk>I xc\<rbrakk>) (a pm, a sm)"
lub: shows "(\<forall>y. z = y \<longrightarrow> P y (\<sigma>x. z = x)) \<and> (\<forall>v. (\<forall>y. z = y \<longrightarrow> P y v) \<longrightarrow> P (\<sigma>x. z = x) v)"
cp\<^sub>P\<^sub>O\<^sub>S\<^sub>T': fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and x :: "'b"   and res :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'res"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "POST self x res \<tau> = POST (\<lambda>_. self \<tau>) (f_lam x \<tau>) (\<lambda>_. res \<tau>) \<tau>"
next_plane0_incr_squander_lb: fixes g' :: "graph"   and p :: "nat"   and g :: "graph" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "Invariants.inv g" shows "squanderLowerBound g \<le> squanderLowerBound g'"
steps_def: shows "steps \<equiv> \<lambda>uu uua. rec_message (\<lambda>p v cfg1 cfg2. (\<forall>s. (s = p \<longrightarrow> states cfg2 p = trans p (states cfg1 p) (Bool v)) \<and> (s \<noteq> p \<longrightarrow> states cfg2 s = states cfg1 s)) \<and> enabled cfg1 <p, inM v> \<and> msgs cfg2 = sends p (states cfg1 p) (Bool v) \<union># msgs cfg1 -# <p, inM v>) (\<lambda>v cfg1 cfg2. False) (\<lambda>p v cfg1 cfg2. (\<forall>s. (s = p \<longrightarrow> states cfg2 p = trans p (states cfg1 p) (Value v)) \<and> (s \<noteq> p \<longrightarrow> states cfg2 s = states cfg1 s)) \<and> enabled cfg1 <p, v> \<and> msgs cfg2 = sends p (states cfg1 p) (Value v) \<union># msgs cfg1 -# <p, v>) uua uu"
ipresIGFreshAll_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('g, 'h, 'i, 'j, 'k, 'l, 'c, 'e, 'm) model_scheme"   and MOD' :: "('n, 'o, 'i, 'p, 'q, 'l, 'd, 'f, 'r) model_scheme" shows "ipresIGFreshAll h hA MOD MOD' \<equiv> ipresIGFresh h MOD MOD' \<and> ipresIGFreshAbs hA MOD MOD'"
H\<^sub>L_preserved_along_iso: fixes a :: "'a"   and a' :: "'a" assumes "weak_unit a"   and "a \<cong> a'" shows "endofunctor (local.Left a) (H\<^sub>L a')"
assms_2: shows "\<forall>w. 0 \<le> f w"
inf_last_ti_nonempty: fixes s :: "nat \<Rightarrow> 'a list"   and t :: "nat"   and k :: "nat" assumes "s t \<noteq> []" shows "inf_last_ti s (t + k) \<noteq> []"
intro: fixes invar :: "'c \<Rightarrow> bool"   and iteratei :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e set" assumes "\<And>S. invar S \<Longrightarrow> set_iterator (iteratei S) (\<alpha> S)" shows "set_iteratei_axioms \<alpha> invar iteratei"
real_nonneg_mat_def: fixes A :: "complex mat" shows "real_nonneg_mat A \<equiv> \<forall>a\<in>elements_mat A. a \<in> \<real> \<and> 0 \<le> Re a"
zero_less_power: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero a" shows "less zero (power.power one times a n)"
stationary_distribution_alt_def: fixes N :: "'i pmf" shows "stationary_distribution N = (N = pmf_of_st_vec (A *st st_vec_of_pmf N))"
divide_diff_eq_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "z \<noteq> zero" shows "minus (divide x z) y = divide (minus x (times y z)) z"
signof_def: fixes p :: "nat \<Rightarrow> nat" shows "signof p = (if sign p = 1 then 1::'c else - (1::'c))"
from_nat_to_nat: fixes j :: "nat"   and k :: "'a" assumes "mod_type_class.from_nat j = k"   and "j < CARD('a)" shows "mod_type_class.to_nat k = j"
borel_measurable_vimage_borel: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
preserves_comp: fixes g :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr" assumes "S.seq g f" shows "S.src (g \<cdot>\<^sub>S f) = S.src g \<cdot>\<^sub>S S.src f"
kruskal_forest_components_inf: fixes f :: "'a"   and w :: "'a" assumes "f \<le> w \<squnion> w\<^sup>T"   and "injective w" shows "forest_components f \<le> forest_components (forest_components f \<sqinter> w)"
prob_space_axioms: fixes i :: "'i" shows "prob_space (distr M borel (Y i))"
nnegI2: fixes P :: "'c \<Rightarrow> 'd" assumes "(\<lambda>s. 0::'d) \<le> P" shows "nneg P"
path_unique_def: fixes 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 C :: "char list"   and D :: "char list" shows "P \<turnstile> Path C to D unique \<equiv> \<exists>!Cs. Subobjs P C Cs \<and> last Cs = D"
dense_list_of_pdevs_pdevs_of_list: fixes xs :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> (0::'a)" shows "dense_list_of_pdevs (pdevs_of_list xs) = xs"
\<theta>_asymptotics: shows "\<theta> \<sim>[at_top] (\<lambda>x. x)"
finite_ordLess_infinite2: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "infinite B" shows "|A| <o |B|"
map_indets_times: fixes f :: "'d \<Rightarrow> 'b"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "map_indets f (p * q) = map_indets f p * map_indets f q"
ptensor_mat_def: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "stm2.ptensor_mat m1 m2 = tensor_mat stm2.dims0 stm2.vars1' m1 m2"
stutter_equivI: fixes f :: "nat \<Rightarrow> nat"   and \<sigma> :: "nat \<Rightarrow> 'f"   and g :: "nat \<Rightarrow> nat"   and \<tau> :: "nat \<Rightarrow> 'f" assumes "stutter_sampler f \<sigma>"   and "stutter_sampler g \<tau>"   and "\<sigma> \<circ> f = \<tau> \<circ> g" shows "\<sigma> \<approx> \<tau>"
euclidean_size_numeral: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and k :: "num" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size (numeral.numeral one plus k) = numeral k"
switch_triangle: fixes v :: "view"   and c :: "cars"   and u :: "view"   and d :: "cars"   and w :: "view" shows "v = c > u \<and> v = d > w \<longrightarrow> u = d > w"
eq_fst_iff: fixes a :: "'a"   and p :: "'a \<times> 'c" shows "(a = fst p) = (\<exists>b. p = (a, b))"
ext_irrefl_trans_strong_axioms_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_irrefl_trans_strong_axioms ext \<equiv> \<forall>zs ys xs gt. (\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x) \<longrightarrow> ext gt zs ys \<longrightarrow> ext gt ys xs \<longrightarrow> ext gt zs xs"
add: fixes f :: "'i \<Rightarrow> 'j"   and x :: "'i"   and y :: "'i" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
normalize_in_carrier: fixes p :: "'a list" assumes "set p \<subseteq> carrier R" shows "set (local.normalize p) \<subseteq> carrier R"
fminusI: fixes c :: "'b"   and A :: "'b fset"   and B :: "'b fset" assumes "c |\<in>| A"   and "c |\<notin>| B" shows "c |\<in>| A |-| B"
mono1: fixes y :: "'b" assumes "y \<in> Y" shows "monotone (\<sqsubseteq>) (\<le>) (\<lambda>x. f x y)"
uint64_of_nat_conv_def: fixes x :: "nat" shows "uint64_of_nat_conv x = x"
pair_complete_def: shows "pair_complete \<equiv> {{x, y} |. x y}-complete"
disjI1_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p" shows "A \<turnstile> p .\<or>. q"
id: shows "inverse (2 * \<i>) * (x - cnj x) = complex_of_real (Im x)"
expl_cond_expect_disc_fct: fixes Y :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and X :: "'c \<Rightarrow> real" assumes "disc_fct Y" shows "disc_fct (expl_cond_expect M Y X)"
inf_sup_aci_6: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.lattice inf less_eq less sup" shows "sup (sup x y) z = sup x (sup y z)"
hyps: fixes v :: "symbol list"   and w :: "symbol list"   and u :: "symbol list" assumes "LeftDerivation v D w"   and "is_word u" shows "LeftDerivation (u @ v) (derivation_shift D 0 (length u)) (u @ w)"
vec_of_basis_enum_scaleC: fixes c :: "complex"   and b :: "'a" shows "vec_of_basis_enum (c *\<^sub>C b) = c \<cdot>\<^sub>v vec_of_basis_enum b"
gfp_ordinal_induct: fixes f :: "'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" assumes "mono f"   and "\<And>S. \<lbrakk>P S; gfp f \<le> S\<rbrakk> \<Longrightarrow> P (f S)"   and "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (\<Sqinter>M)" shows "P (gfp f)"
f_sup: fixes x :: "'a node \<times> 'a node set" shows "f x \<le> expand x"
m_closed: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<otimes>\<^bsub>P\<^esub> y \<in> carrier P"
sumlist_append: fixes xs :: "'a vec list"   and ys :: "'a vec list" assumes "set xs \<subseteq> carrier_vec n"   and "set ys \<subseteq> carrier_vec n" shows "M.sumlist (xs @ ys) = M.sumlist xs + M.sumlist ys"
p2: shows "\<S>\<^sub>2 = \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2"
neg_fun_distr1: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and S' :: "'i \<Rightarrow> 'h \<Rightarrow> bool" assumes "left_unique R"   and "right_total R"   and "right_unique R'"   and "left_total R'" shows "rel_fun (R OO R') (S OO S') \<le> rel_fun R S OO rel_fun R' S'"
Impl_Succ_axioms: shows "Impl_Succ absG succ isG succ_impl"
alternating_order2_cancel_1left: fixes s :: "'a"   and n :: "nat"   and t :: "'a" assumes "s + s = (0::'a)" shows "sum_list (s # alternating_list (Suc n) s t) = sum_list (alternating_list n t s)"
hyps_4: shows "mds\<^sub>2'''__ \<sqsubseteq> mds\<^sub>1'"
non_comm_hds_neq: fixes u :: "'b list"   and v :: "'b list" assumes "u \<cdot> v \<noteq> v \<cdot> u" shows "hd (Dec \<BB>\<^sub>F {u, v} u) \<noteq> hd (Dec \<BB>\<^sub>F {u, v} v)"
T3: shows "Col A B C \<and> \<not> Col A B P \<longrightarrow> (\<exists>Q. A B Perp Q C \<and> Coplanar A B P Q)"
distinct_31: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> While x61 x62"
Units_l_inv_ex: fixes x :: "'a mat" assumes "x \<in> Units (ring_mat TYPE('a) n b)" shows "\<exists>y\<in>carrier (ring_mat TYPE('a) n b). y \<otimes>\<^bsub>ring_mat TYPE('a) n b\<^esub> x = \<one>\<^bsub>ring_mat TYPE('a) n b\<^esub>"
step_foldL_step_cong: fixes pri :: "'a"   and pri' :: "'a"   and onxt :: "edgeIndex option"   and onxt' :: "edgeIndex option"   and s :: "'b list"   and s' :: "'b list"   and d :: "nat \<times> edgeIndex \<times> (String.literal, nat) assoc_list \<times> 'c list \<times> 'c list"   and d' :: "nat \<times> edgeIndex \<times> (String.literal, nat) assoc_list \<times> 'c list \<times> 'c list"   and g :: "'b \<Rightarrow> (String.literal, nat) assoc_list \<times> 'a \<times> nat \<times> edgeIndex \<times> edgeIndex option \<times> bool \<Rightarrow> 'c list \<times> edgeIndex \<times> (String.literal, nat) assoc_list"   and g' :: "'b \<Rightarrow> (String.literal, nat) assoc_list \<times> 'a \<times> nat \<times> edgeIndex \<times> edgeIndex option \<times> bool \<Rightarrow> 'c list \<times> edgeIndex \<times> (String.literal, nat) assoc_list" assumes "pri = pri'"   and "onxt = onxt'"   and "s = s'"   and "d = d'"   and "\<And>x d. x \<in> set s \<Longrightarrow> g x d = g' x d" shows "step_foldL_step g pri onxt s d = step_foldL_step g' pri' onxt' s' d'"
nth_derivative_differentiable: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and n :: "nat"   and x :: "'a"   and v :: "'a" assumes "higher_differentiable_on S f (Suc n)"   and "x \<in> S" shows "(\<lambda>x. nth_derivative n f x v) differentiable at x"
T_F: fixes s :: "'a event list"   and P :: "'a process" assumes "s \<in> T P" shows "(s, {}) \<in> F P"
arev_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(arev x = arev y) = (x = y)"
intvl: shows "{..<n - m} = insert 0 (Suc ` {..<n - Suc m})"
integrable_distr_eq: fixes g :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "g \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> borel_measurable N" shows "integrable (distr M N g) f = integrable M (\<lambda>x. f (g x))"
2: shows "bijective (i * top) \<and> bijective (e[[top]])"
deg_in: shows "in_degree G u + 1 = out_degree G u"
lfp_exp_greatest: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real" assumes "\<And>P. \<lbrakk>t P \<tturnstile> P; sound P\<rbrakk> \<Longrightarrow> Q \<tturnstile> P"   and "sound Q"   and "t R \<tturnstile> R"   and "sound R" shows "Q \<tturnstile> lfp_exp t"
wn_cs_butlast: fixes \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "butlast (cs\<^bsup>\<pi>\<^esup> m) = butlast (cs\<^bsup>\<pi>\<^esup> n)"   and "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"   and "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> n"   and "m < n" shows "i < j"
simps_3: fixes K :: "nat"   and X :: "freemsg" shows "freediscrim (CRYPT K X) = freediscrim X + 2"
intensional_simps_27: fixes y :: "'b \<Rightarrow> bool" shows "y \<or> y \<equiv> y"
r_prefixes: fixes i :: "nat"   and n :: "nat" shows "eval r_prefixes [i, n] \<down>= Partial_Recursive.list_encode (prefixes i n)"
theI': fixes P :: "'b \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "P (THE x. P x)"
older_seniors_cases_subset: fixes x :: "nat"   and n :: "nat"   and y :: "nat" obtains "older_seniors x n \<subset> older_seniors y n"   | "older_seniors x n = older_seniors y n"   | "older_seniors y n \<subset> older_seniors x n"
normalize_positive_ports_step_nnf: fixes m :: "'b common_primitive match_expr"   and disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and m' :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<not> has_disc_negated disc False m"   and "m' \<in> set (normalize_positive_ports_step (disc, sel) C m)" shows "normalized_nnf_match m'"
poly_linear_linear_factor: fixes b :: "'a"   and as :: "'a list" assumes "[:b, 1::'a:] dvd (\<Prod>a\<leftarrow>as. [:a, 1::'a:])" shows "b \<in> set as"
subspace_on_def: fixes T :: "'b set" shows "local.subspace T = ((0::'b) \<in> T \<and> (\<forall>x\<in>T. \<forall>y\<in>T. x + y \<in> T) \<and> (\<forall>c. \<forall>x\<in>T. c *s x \<in> T))"
exhaust: fixes y :: "freeRes" obtains x11 :: "name" and x12 :: "name" where "y = x11[x12]"   | "y = \<tau>"
simps: fixes a1 :: "'d DBMEntry"   and a2 :: "'d DBMEntry" shows "(a1 \<prec> a2) = ((\<exists>uu_. a1 = Lt uu_ \<and> a2 = \<infinity>) \<or> (\<exists>uv_. a1 = Le uv_ \<and> a2 = \<infinity>) \<or> (\<exists>a b. a1 = Le a \<and> a2 = Le b \<and> a < b) \<or> (\<exists>a b. a1 = Le a \<and> a2 = Lt b \<and> a < b) \<or> (\<exists>a b. a1 = Lt a \<and> a2 = Le b \<and> a \<le> b) \<or> (\<exists>a b. a1 = Lt a \<and> a2 = Lt b \<and> a < b))"
assms_2: fixes A :: "'a set" assumes "A \<in> sets M"   and "0 < emeasure M A" shows "\<exists>n>0. (T ^^ n) -` A \<inter> A \<noteq> {}"
idealprod_whole_l: fixes I :: "'a set" assumes "ideal R I" shows "carrier R \<diamondsuit>\<^sub>r I = I"
G: shows "0 < offs_num (length ns) (y # xs) index key mi ma i"
assms_3: fixes zs :: "('a \<times> 'b) list \<times> 'o" assumes "oalist_inv zs" shows "g zs = zs"
beta_equiv_eq_1_3: fixes \<phi> :: "\<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<o>"   and \<psi> :: "\<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<o>"   and v :: "i"   and x :: "\<nu>"   and y :: "\<nu>"   and z :: "\<nu>" assumes "IsProperInXYZ \<phi>"   and "IsProperInXYZ \<psi>"   and "\<And>x y z. [\<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) in v]" shows "[\<lparr>\<^bold>\<lambda>\<^sup>3 (\<lambda>x y z. \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)),x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>\<^bold>\<lambda>\<^sup>3 (\<lambda>x y z. \<psi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P)),x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr> in v]"
B_def: shows "B = (\<Sum>\<^sub>ax\<in>T. norm (\<mu> x))"
norm_minus_commute: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus a b) = norm (minus b a)"
collapse_6: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "is_InSet stateful_strand_step" shows "\<langle>the_check stateful_strand_step: the_elem_term stateful_strand_step \<in> the_set_term stateful_strand_step\<rangle> = stateful_strand_step"
polymul_commute: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "p *\<^sub>p q = q *\<^sub>p p"
prj0_simps_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "partial_magma.arr C (prj0 f g)"
induct: fixes P :: "'b wordinterval \<Rightarrow> 'b wordinterval \<Rightarrow> bool"   and a0 :: "'b wordinterval"   and a1 :: "'b wordinterval" assumes "\<And>s e ms me. P (WordInterval s e) (WordInterval ms me)"   and "\<And>r1 r2 t. \<lbrakk>P r1 t; P r2 t\<rbrakk> \<Longrightarrow> P (RangeUnion r1 r2) t"   and "\<And>v va r1 r2. \<lbrakk>P (WordInterval v va) r1; P (wordinterval_setminus' (WordInterval v va) r1) r2\<rbrakk> \<Longrightarrow> P (WordInterval v va) (RangeUnion r1 r2)" shows "P a0 a1"
expands_to_exp_0_pull_out2: fixes f :: "real \<Rightarrow> real"   and c :: "real"   and b :: "real \<Rightarrow> real"   and xs :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "((\<lambda>x. exp (f x - c * ln (b x))) expands_to MS xs g) (b # basis)"   and "basis_wf (b # basis)" shows "((\<lambda>x. exp (f x)) expands_to MS (shift_ms_aux c xs) (\<lambda>x. b x powr c * g x)) (b # basis)"
add_instr_sub1_privilege: fixes s' :: "'b sparc_state"   and instr_name :: "sparc_operation"   and result :: "word32"   and rs1_val :: "word32"   and operand2 :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (add_instr_sub1 instr_name result rs1_val operand2 s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
dd: fixes x :: "'a" shows "x ^ d = x ^ (d + 1 - 1)"
inverse_arrows_sym: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" shows "VoV.inverse_arrows f g = VoV.inverse_arrows g f"
g': fixes x :: "'a" assumes "x \<in> topspace X" shows "g' x = g x"
card_of_ordLeqI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "inj_on f A"   and "\<And>a. a \<in> A \<Longrightarrow> f a \<in> B" shows "|A| \<le>o |B|"
smc_SemiCAT_composable_arrs_dg_SemiCAT: fixes \<alpha> :: "V" shows "composable_arrs (dg_SemiCAT \<alpha>) = composable_arrs (smc_SemiCAT \<alpha>)"
LEQ: shows "\<forall>p\<in>P. \<exists>r\<in>R. p \<le>o r"
is_pure_conv: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> assn" shows "is_pure R = (\<exists>R'. R = pure R')"
PLam_transient: fixes i :: "nat"   and I :: "nat set"   and F :: "nat \<Rightarrow> ('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and A :: "((nat \<Rightarrow> 'c) \<times> 'd) set" assumes "i \<in> I" shows "(PLam I F \<in> transient A) = (\<exists>i\<in>I. lift i (F i) \<in> transient A)"
PowI: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "A \<in> Pow B"
X: fixes i :: "'a" assumes "i \<in> I" shows "X i \<in> M.events i"
2: fixes k :: "nat" shows "reduced_run q (stake k v) (sdrop k v) (l k) (w k) (w\<^sub>1 k) (w\<^sub>2 k) (u k)"
close_def: fixes g :: "graph"   and u :: "nat"   and v :: "nat" shows "close g u v \<equiv> \<exists>f\<in>set (facesAt g u). if |vertices f| = 4 then v = f \<bullet> u \<or> v = f \<bullet> (f \<bullet> u) else v = f \<bullet> u"
aforms_relD: fixes a :: "('a \<times> 'a pdevs) list"   and b :: "'a list set" assumes "(a, b) \<in> aforms_rel" shows "b = Joints a"
pure_tree_simps_2: fixes x :: "'b" shows "left (pure x) = pure x"
hcomp_simps_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.hseq \<nu> \<mu>" shows "trg\<^sub>B (\<nu> \<star>\<^sub>B \<mu>) = trg\<^sub>B \<nu>"
restrict_map_id: fixes m :: "'c \<Rightarrow> 'd option" shows "m |` dom m = m"
ordered_sel_resolvent_def: fixes atom_ordering :: "('a \<times> 'a) set"   and Sel :: "'a set"   and P1 :: "'a Literal set"   and P2 :: "'a Literal set"   and C :: "'a Literal set" assumes "propositional_atoms atom_ordering" shows "propositional_atoms.ordered_sel_resolvent atom_ordering Sel P1 P2 C \<equiv> \<exists>A. C = P1 - {Pos A} \<union> (P2 - {Neg A}) \<and> propositional_atoms.strictly_maximal_literal atom_ordering P1 (Pos A) \<and> selected_part Sel P1 = {} \<and> (propositional_atoms.strictly_maximal_literal atom_ordering P2 (Neg A) \<and> selected_part Sel P2 = {} \<or> propositional_atoms.strictly_maximal_literal atom_ordering (selected_part Sel P2) (Neg A))"
pathfinish_reversepath: fixes g :: "real \<Rightarrow> 'b" shows "pathfinish (reversepath g) = pathstart g"
lt_less: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "p \<noteq> 0"   and "\<And>u. v \<preceq>\<^sub>t u \<Longrightarrow> lookup p u = (0::'c)" shows "lt p \<prec>\<^sub>t v"
Suc: shows "0 < emeasure M {w \<in> space M. stake n w = stake n x}"
pair_prob_space_axioms: shows "pair_prob_space M1 M2"
prefixE: fixes f :: "nat \<Rightarrow> nat option"   and vs :: "nat list"   and x :: "nat" assumes "prefix f (length vs - 1) = vs"   and "f \<in> \<R>"   and "0 < length vs"   and "x < length vs" shows "f x \<down>= vs ! x"