hcju/isabelleps huggingface dataset:
id_upt_k_def: fixes A :: "(('c, 'd) vec, 'd) vec"   and k :: "nat" shows "id_upt_k A k = (\<forall>i j. mod_type_class.to_nat i < k \<and> mod_type_class.to_nat j < k \<longrightarrow> (i = j \<longrightarrow> A $ i $ j = (1::'c)) \<and> (i \<noteq> j \<longrightarrow> A $ i $ j = (0::'c)))"
aless_natless: fixes n :: "nat"   and m :: "nat" shows "(an n < an m) = (n < m)"
B_2: fixes z :: "'a" assumes "z \<in> B" shows "s0 \<bullet> (1::'a) < z \<bullet> (1::'a)"
insert_commute: fixes x :: "'c"   and y :: "'c"   and A :: "'c set" shows "insert x (insert y A) = insert y (insert x A)"
inf_le2: 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" 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 "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type)) y"
is_ub_def: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and u :: "'b" shows "is_ub r x y u = ((x, u) \<in> r \<and> (y, u) \<in> r)"
dim_eq_card_independent: fixes B :: "'b poly set" assumes "poly_vs.independent B" shows "poly_vs.dim B = card B"
well_order_on_Well_order: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "well_order_on A r" shows "A = Field r \<and> Well_order r"
encode_problem_with_operator_interference_exclusion_has_model_then_also_partial_encodings_2: fixes \<A> :: "sat_plan_variable \<Rightarrow> bool"   and \<Pi> :: "'b strips_problem"   and t :: "nat" assumes "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t" shows "\<A> \<Turnstile> encode_operators \<Pi> t"
IH: fixes \<rho>' :: "(nat \<times> exp) list" assumes "bs_env \<rho> \<rho>'" shows "closed_env \<rho>'"
that: fixes n' :: "'node" assumes "Suc (shortestPath g n') = shortestPath g n"   and "n' \<in> set (predecessors g n)" shows "thesis"
preservation_of_binary_predicates_and_closures_1: fixes Rel :: "('c \<times> 'c) set"   and Pred :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_preserves_binary_pred Rel Pred" shows "rel_preserves_binary_pred (Rel\<^sup>=) Pred"
induct: fixes x1 :: "nat list"   and x2 :: "nat list"   and P :: "nat list \<Rightarrow> nat list \<Rightarrow> bool" assumes "x1 \<lhd> x2"   and "P [] []"   and "\<And>is ds i d. \<lbrakk>is \<lhd> ds; P is ds; i < d\<rbrakk> \<Longrightarrow> P (i # is) (d # ds)" shows "P x1 x2"
alw_holds2: fixes P :: "'b \<Rightarrow> bool"   and ss :: "'b stream" shows "alw (holds P) ss = (P (shd ss) \<and> alw (holds P) (stl ss))"
sg_sg: fixes K :: "'a set"   and H :: "'a set" assumes "G \<guillemotright> K "   and "G \<guillemotright> H "   and "H \<subseteq> K" shows "\<natural>K \<guillemotright> H "
osum_wf_Id: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Total r"   and "Total r'"   and "wf (r - Id)"   and "wf (r' - Id)" shows "wf (r +o r' - Id)"
component_of_term_of_pair: fixes t :: "'a"   and k :: "'k" shows "component_of_term (term_of_pair (t, k)) = k"
set_lebesgue_integral_eq_integral_2: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on S" shows "set_lebesgue_integral lebesgue S f = integral S f"
sorted_fst_def: fixes ps :: "(int \<times> int) list" shows "sorted_fst ps = sorted_wrt (\<lambda>p\<^sub>0 p\<^sub>1. fst p\<^sub>0 \<le> fst p\<^sub>1) ps"
M_Cons: fixes fs :: "(char list \<times> trm) list"   and l :: "char list"   and t :: "trm"   and p :: "pat"   and ts :: "trm list"   and fps :: "(char list \<times> pat) list"   and us :: "trm list" assumes "fs\<langle>l\<rangle>\<^sub>? = \<lfloor>t\<rfloor>"   and "\<turnstile> p \<rhd> t \<Rightarrow> ts"   and "\<turnstile> fps [\<rhd>] fs \<Rightarrow> us" shows "\<turnstile> (l, p) \<Colon> fps [\<rhd>] fs \<Rightarrow> ts @ us"
rec_np_def: shows "rec_np = (let Rr = Cn 2 rec_conj [Cn 2 rec_less [recf.id 2 0, recf.id 2 1], Cn 2 rec_prime [recf.id 2 1]] in Cn 1 (rec_Minr Rr) [recf.id 1 0, Cn 1 s [rec_fac]])"
image_Union: fixes f :: "'f \<Rightarrow> 'e"   and S :: "'f set set" shows "f ` \<Union> S = \<Union> ((`) f ` S)"
star_star_closure: fixes x :: "'a"   and z :: "'a" assumes "x\<^sup>\<star> \<le> z\<^sup>\<star>" shows "(x\<^sup>\<star>)\<^sup>\<star> \<le> z\<^sup>\<star>"
assms_5: shows "\<forall>A\<in>\<Gamma>. typ_of A = Some propT"
intros_2: fixes F :: "'b formula"   and G :: "'b formula" shows "(F::'b formula) \<^bold>\<rightarrow> F \<^bold>\<or> (G::'b formula) \<in> AX10"
while_default_rule: fixes P :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and Q :: "'a \<Rightarrow> bool"   and s :: "'a"   and d :: "'a" assumes "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "\<And>s. \<lbrakk>P s; \<not> b s\<rbrakk> \<Longrightarrow> Q s"   and "P s"   and "Q d" shows "Q (while_default d b c s)"
degree_bound_def: fixes vs :: "'a poly list" shows "degree_bound vs = max_factor_degree (map degree vs)"
INF_image: fixes Inf :: "'a set \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "Inf (g ` f ` A) = Inf ((g \<circ> f) ` A)"
not_square: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> (\<not> [A]_v) = <\<not> A>_v"
oseqp_sos_subterms: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp"   and \<sigma> :: "nat \<Rightarrow> 'f"   and a :: "'g seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'f"   and p' :: "('f, 'g, 'e, 'h) seqp"   and i :: "nat" assumes "wellformed \<Gamma>"   and "\<exists>pn. p \<in> subterms (\<Gamma> pn)"   and "((\<sigma>, p), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i" shows "\<exists>pn. p' \<in> subterms (\<Gamma> pn)"
fst_A': shows "matrix_to_iarray (fst A') = fst B"
set_of_multisets_eq: fixes x :: "'b"   and A :: "'b set"   and k :: "nat" assumes "x \<notin> A" shows "{M. set_mset M \<subseteq> insert x A \<and> size M = Suc k} = {M. set_mset M \<subseteq> A \<and> size M = Suc k} \<union> (\<lambda>M. M + {#x#}) ` {M. set_mset M \<subseteq> insert x A \<and> size M = k}"
r_minus: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
vsv_vimageI2': fixes b :: "V"   and a :: "V" assumes "b = \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>)"
disj_conj_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<^bold>\<sqinter> z \<^bold>\<squnion> x = (y \<^bold>\<squnion> x) \<^bold>\<sqinter> (z \<^bold>\<squnion> x)"
postdominate_def: fixes n' :: "'node"   and n :: "'node" shows "n' postdominates n \<equiv> valid_node n \<and> valid_node n' \<and> (\<forall>as. n -as\<rightarrow>* (_Exit_) \<longrightarrow> n' \<in> set (sourcenodes as))"
is_acc_def: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat" shows "b_graph.is_acc (degeneralize_ext m) r \<equiv> \<exists>\<^sub>\<infinity>i. r i \<in> bg_F (degeneralize_ext m)"
nested_union_vs: fixes I :: "'e set"   and N :: "'e \<Rightarrow> 'c set"   and N' :: "'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> LinearCombinations.submodule R (N i) M"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> \<exists>k. k \<in> I \<and> N i \<subseteq> N k \<and> N j \<subseteq> N k"   and "N' = \<Union> (N ` I)"   and "I \<noteq> {}" shows "LinearCombinations.submodule R N' M"
sup_apply: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "sup f g x = sup (f x) (g x)"
sdrop_simps_1: fixes n :: "nat"   and s :: "'b stream" shows "shd (sdrop n s) = s !! n"
S2S3S4S5S6LeadstoS6: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds"   and sigma :: "behavior" assumes "sigma \<Turnstile> S2 rmhist p \<leadsto> S3 rmhist p"   and "sigma \<Turnstile> S3 rmhist p \<leadsto> S4 rmhist p \<or> S6 rmhist p"   and "sigma \<Turnstile> S4 rmhist p \<and> ires!p = #NotAResult \<leadsto> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<or> S5 rmhist p"   and "sigma \<Turnstile> S4 rmhist p \<and> ires!p \<noteq> #NotAResult \<leadsto> S5 rmhist p"   and "sigma \<Turnstile> S5 rmhist p \<leadsto> S6 rmhist p" shows "sigma \<Turnstile> S2 rmhist p \<or> S3 rmhist p \<or> S4 rmhist p \<or> S5 rmhist p \<or> S6 rmhist p \<leadsto> S6 rmhist p"
g: shows "group_isomorphisms (relative_homology_group (int n) (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0}) (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) (hom_boundary (int n) (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0}) gp"
subspace_imp_affine: fixes s :: "'b set" assumes "subspace s" shows "affine s"
add_ideals: fixes R :: "('c, 'd) ring_scheme"   and I :: "'c set"   and J :: "'c set" assumes "ring R"   and "ideal I R"   and "ideal J R" shows "ideal (I <+>\<^bsub>R\<^esub> J) R"
igWlsInp_def: fixes MOD :: "('h, 'i, 'j, 'sort, 'k, 'l, 'm, 'n, 'o) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'm option" shows "igWlsInp MOD delta inp \<equiv> wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 (igWls MOD) (arOf delta) inp"
assign: fixes x_update :: "('c \<Rightarrow> 'd) \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b \<Rightarrow> 'd"   and P :: "'b set" shows "\<turnstile> \<lbrace>\<acute>(x_update (\<lambda>_. \<acute>a)) \<in> P\<rbrace> Basic (\<lambda>s. x_update (\<lambda>_. a s) s) P"
MP: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and H' :: "fm set" assumes "H \<turnstile> A IMP B"   and "H' \<turnstile> A" shows "H \<union> H' \<turnstile> B"
remove_nth_sound_l: fixes p :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "p < n" shows "remove_nth n xs ! p = xs ! p"
Diff_eq: fixes A :: "'e set"   and B :: "'e set" shows "A - B = A \<inter> - B"
measure_eq: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "emeasure (distr (density lborel f) lborel h) {a..b} = emeasure (density lborel g) {a..b}"
composition_conseq2l: fixes l :: "fterm literal"   and \<sigma>\<^sub>1 :: "char list \<Rightarrow> fterm"   and \<sigma>\<^sub>2 :: "char list \<Rightarrow> fterm" shows "l \<cdot>\<^sub>l \<sigma>\<^sub>1 \<cdot>\<^sub>l \<sigma>\<^sub>2 = l \<cdot>\<^sub>l (\<sigma>\<^sub>1 \<cdot> \<sigma>\<^sub>2)"
box_casedist: fixes x :: "'a Box" obtains "x = \<bottom>"   | u :: "'a" where "x = box\<cdot>u"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>a as b bs. \<lbrakk>a = b \<Longrightarrow> P as (b # bs); \<lbrakk>a \<noteq> b; b < a\<rbrakk> \<Longrightarrow> P (a # as) bs\<rbrakk> \<Longrightarrow> P (a # as) (b # bs)"   and "\<And>uu_. P [] uu_"   and "\<And>a uv_. P (a # uv_) []" shows "P a0 a1"
assms_3: shows "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"
MkIde_Dom: fixes a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr a" shows "J_C.MkIde (J_C.Dom a) = J_C.dom a"
simps: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" shows "checked_simplex cs J (Simplex_State s) = checked_simplex' cs J s"
singletonE: fixes b :: "'a"   and a :: "'a" assumes "b \<in> {a}" obtains "b = a"
strict_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b"   and "b < c" shows "a < c"
overlap_chain: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c]]"   and "[[b c d]]" shows "ch {a, b, c, d}"
Assume: fixes A :: "fm"   and H :: "fm set" shows "insert A H \<turnstile> A"
mk_tree_Suc_debug: fixes n :: "nat" shows "mk_tree (Suc n) = (let _ = Debug.flush STR ''tick''; t = mk_tree n in \<triangle> \<^bold>l: t \<^bold>r: t)"
tendsto_of_real_0_I: fixes f :: "'a \<Rightarrow> real"   and G :: "'a filter" assumes "(f \<longlongrightarrow> 0) G" shows "((\<lambda>x. of_real (f x)) \<longlongrightarrow> (0::'b)) G"
get_root_node_def: shows "Shadow_DOM.get_root_node \<equiv> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node Shadow_DOM.get_parent"
of_real_1: shows "of_real 1 = (1::'a)"
elims_2: fixes x :: "'at set"   and xa :: "'at Literal set" assumes "x \<Turnstile> xa" obtains I :: "'at set" and C :: "'at Literal set" where "x = I" and "xa = C" and "\<exists>L. L \<in> C \<and> I \<Turnstile> L"
Rep_pfun_inject: fixes x :: "('a, 'b) pfun"   and y :: "('a, 'b) pfun" shows "(Rep_pfun x = Rep_pfun y) = (x = y)"
bernoulli'_fps_def: shows "bernoulli'_fps = fps_X / (1 - fps_exp (- (1::'a)))"
FGModule_Im: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "FGModuleHom G smult V smult' T" shows "FGModule G smult' (T ` V)"
zip_map_fst_snd: fixes zs :: "('b \<times> 'c) list" shows "zip (map fst zs) (map snd zs) = zs"
sync_commute: shows "(P \<lbrakk>C\<rbrakk> Q) = (Q \<lbrakk>C\<rbrakk> P)"
isomorphic_group_homology_group_prod_retract: fixes S :: "'a set"   and X :: "'a topology"   and p :: "int" assumes "S = {} \<or> S retract_of_space X" shows "homology_group p X \<cong> homology_group p (subtopology X S) \<times>\<times> relative_homology_group p X S"
LeadsTo_Un: fixes F :: "'c program"   and A :: "'c set"   and C :: "'c set"   and B :: "'c set" assumes "F \<in> A \<longmapsto>w C"   and "F \<in> B \<longmapsto>w C" shows "F \<in> A \<union> B \<longmapsto>w C"
is_equivalence: shows "internal_equivalence (h \<star> f) (g \<star> k) \<eta> \<epsilon>"
measurable_finmap_compose: fixes J :: "'f set"   and f :: "'f \<Rightarrow> 'd"   and M :: "'e measure" shows "(\<lambda>m. compose J m f) \<in> Pi\<^sub>M (f ` J) (\<lambda>_. M) \<rightarrow>\<^sub>M Pi\<^sub>M J (\<lambda>_. M)"
eq: shows "fst ` (set gs \<union> set (ab gs bs hs data')) = fst ` (set gs \<union> set bs) \<union> fst ` set hs"
assms_1: shows "verts3 (with_proj G) \<subseteq> V"
dagger_unfoldr_distr: fixes y :: "'a"   and x :: "'a" shows "y + x\<^sup>\<dagger> \<cdot> x \<cdot> y = x\<^sup>\<dagger> \<cdot> y"
assms_1: shows "\<rho> \<in> carrier_mat d d"
xsnb_in_pret: shows "\<forall>x\<in>set_of nort. var x < nb \<and> x \<in> set_of pret"
matcheq_matchNone_not_matches: fixes m :: "'b match_expr"   and \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a :: "action"   and p :: "'c" assumes "matcheq_matchNone m" shows "\<not> matches \<gamma> m a p"
orbit_refl: fixes x :: "'c" assumes "x \<in> E" shows "x \<in> orbit G \<phi> x"
character_axioms_def: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" shows "character_axioms G \<chi> \<equiv> \<chi> \<one>\<^bsub>G\<^esub> \<noteq> 0 \<and> (\<forall>a. a \<notin> carrier G \<longrightarrow> \<chi> a = 0) \<and> (\<forall>a b. a \<in> carrier G \<longrightarrow> b \<in> carrier G \<longrightarrow> \<chi> (a \<otimes>\<^bsub>G\<^esub> b) = \<chi> a * \<chi> b)"
prj_simps_5: shows "C.dom f_src.prj\<^sub>0 = f.leg0 \<down>\<down> src.leg1"
simps: fixes xs :: "'d"   and x :: "'e"   and ys :: "'d"   and y :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "qFreshAbs xs x (qAbs ys y X) = (xs = ys \<and> x = y \<or> qFresh xs x X)"
prems_2: shows "\<forall>a\<in>set vs. case a of (x, \<tau>) \<Rightarrow> (x, \<tau>) \<notin> fv (Abs T s) \<union> FV \<Gamma>"
one_time_pad: fixes x :: "'e" assumes "x \<in> carrier L" shows "map_spmf ((\<oplus>) x) (spmf_of_set (carrier L)) = spmf_of_set (carrier L)"
induct: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "'c"   and a2 :: "('b \<times> 'c) tree" assumes "\<And>k p. P k p \<langle>\<rangle>"   and "\<And>k p l k1 p1 r. \<lbrakk>k < k1 \<Longrightarrow> P k p l; \<lbrakk>\<not> k < k1; k1 < k\<rbrakk> \<Longrightarrow> P k p r\<rbrakk> \<Longrightarrow> P k p \<langle>l, (k1, p1), r\<rangle>" shows "P a0 a1 a2"
forall_spec: fixes P :: "'h \<Rightarrow> bool"   and Q :: "'h \<Rightarrow> bool"   and a :: "'h" assumes "\<forall>b. P b \<longrightarrow> Q b"   and "P a" shows "Q a"
update_arg_wf_def: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" shows "update_arg_wf r \<equiv> \<pi>\<^sub>4 r = val \<and> (\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk) \<and> (\<pi>\<^sub>3 r = unk \<longrightarrow> \<pi>\<^sub>5 r = 1)"
simps: fixes x :: "'a extN"   and y :: "'a extN" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
is_Ghost\<^sub>s\<^sub>b_def: fixes r :: "'b memref" shows "is_Ghost\<^sub>s\<^sub>b r = (case r of Ghost\<^sub>s\<^sub>b x xa xb xc \<Rightarrow> True | _ \<Rightarrow> False)"
valid_SE_def: fixes \<sigma> :: "'a"   and m :: "'a \<Rightarrow> (bool \<times> 'a) option" shows "(\<sigma> \<Turnstile> m) = (m \<sigma> \<noteq> None \<and> fst (the (m \<sigma>)))"
real_cond_exp_intg_1: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M (\<lambda>x. f x * g x)"   and "f \<in> borel_measurable F"   and "g \<in> borel_measurable M" shows "integrable M (\<lambda>x. f x * real_cond_exp M F g x)"
cont_apply: fixes t :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "cont t"   and "\<And>x. cont (f x)"   and "\<And>y. cont (\<lambda>x. f x y)" shows "cont (\<lambda>x. f x (t x))"
p: shows "p = [:a0, a1, a2, a3, a4:]"
subset_insertI2: fixes A :: "'d set"   and B :: "'d set"   and b :: "'d" assumes "A \<subseteq> B" shows "A \<subseteq> insert b B"
unique_connecting_path_rev: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "v \<leadsto>\<^bsub>G\<^esub> w = rev (w \<leadsto>\<^bsub>G\<^esub> v)"
cases: fixes a1 :: "value"   and a2 :: "value" assumes "\<turnstile>\<^sub>v a1 \<approx> a2" obtains ts :: "value list" and us :: "value list" and name :: "name" where "a1 = Vconstr namea__ ts" and "a2 = Vconstr namea__ us" and "list_all2 vrelated ts us"   | cs :: "(Term_Class.term \<times> sterm) list" and \<Gamma>\<^sub>1 :: "(name, value) fmap" and \<Gamma>\<^sub>2 :: "(name, value) fmap" where "a1 = Vabs cs \<Gamma>\<^sub>1__" and "a2 = Vabs cs \<Gamma>\<^sub>2__" and "fmrel_on_fset (frees (Sabs cs)) vrelated \<Gamma>\<^sub>1__ \<Gamma>\<^sub>2__" and "fmrel_on_fset (consts (Sabs cs)) vrelated (fmap_of_list rs) \<Gamma>\<^sub>2__"   | \<Gamma>\<^sub>1 :: "(name, value) fmap" and \<Gamma>\<^sub>2 :: "(name, value) fmap" and css :: "(name, (Term_Class.term \<times> sterm) list) fmap" and name :: "name" where "a1 = Vrecabs css name \<Gamma>\<^sub>1" and "a2 = Vrecabs css name \<Gamma>\<^sub>2" and "pred_fmap (\<lambda>cs. fmrel_on_fset (frees (Sabs cs)) vrelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2 \<and> fmrel_on_fset (consts (Sabs cs)) vrelated (fmap_of_list rs) (\<Gamma>\<^sub>2 ++\<^sub>f mk_rec_env css \<Gamma>\<^sub>2)) css" and "name |\<in>| fmdom css"
subspace_diff: fixes Sa :: "('a \<Rightarrow> real) set"   and x :: "'a \<Rightarrow> real"   and y :: "'a \<Rightarrow> real" assumes "Sa \<subseteq> diff_fun_space"   and "diff_fun_space.m1.subspace Sa"   and "x \<in> Sa"   and "y \<in> Sa" shows "x - y \<in> Sa"
eq: shows "monom a b = [:a:] \<otimes>\<^bsub>R\<^esub> monom 1 b"
assms_3: shows "x \<in> circline_set l \<inter> circline_set x_axis"
sum_character: shows "sum (principal_char (Characters G)) (carrier (Characters G)) = (if principal_char (Characters G) = principal_char (Characters G) then of_nat (order (Characters G)) else 0)"
abs_eq: fixes y :: "8 word" shows "Abs_uint8' y = Abs_uint8 y"
eq_finfun_All_ext: fixes f :: "('a, 'b) finfun"   and g :: "('a, 'b) finfun" shows "(f = g) = finfun_All (finfun_comp (\<lambda>(x, y). x = y) (finfun_Diag f g))"
rtos_hom0: fixes r :: "nat"   and s :: "nat"   and i :: "nat" assumes "0 < r"   and "0 < s"   and "i \<le> r * s - Suc 0" shows "i div s < r"
purgeIdle_Nil_iff: fixes sl :: "('St, 'U, 'C) state list" shows "(purgeIdle sl = []) = (\<forall>s\<in>set sl. \<not> isState s)"
INTF_OF_REL_def: fixes R :: "('e \<times> 'f) set"   and I :: "'g itself" shows "INTF_OF_REL R I \<equiv> True"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_plus" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vint_plus" and "vint_plus\<lparr>a\<rparr> = b"
not_on_closed_segmentI: fixes z :: "'a"   and a :: "'a"   and b :: "'a" assumes "norm (z - a) *\<^sub>R (b - z) \<noteq> norm (b - z) *\<^sub>R (z - a)" shows "z \<notin> closed_segment a b"
bst_join: fixes l :: "('b \<times> color) tree"   and a :: "'b"   and n :: "color"   and r :: "('b \<times> color) tree" assumes "Tree2.bst \<langle>l, (a, n), r\<rangle>" shows "Tree2.bst (Set2_Join_RBT.join l a r)"
inf_mono: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "a \<le> c"   and "b \<le> d" shows "inf a b \<le> inf c d"
mat_extension_hermitian: fixes m :: "'a mat"   and dims :: "nat list"   and vars :: "nat set" assumes "m \<in> carrier_mat (partial_state.d1 dims vars) (partial_state.d1 dims vars)"   and "hermitian m" shows "hermitian (mat_extension dims vars m)"
assms_1: fixes s :: "char list"   and t :: "char list" shows "l s @ t = l (s @ t)"
ElemLambdaFun: fixes u :: "ZF"   and f :: "ZF \<Rightarrow> ZF"   and U :: "ZF" assumes "\<And>x. Elem x u \<Longrightarrow> Elem (f x) U" shows "Elem (Lambda u f) (Fun u U)"
False: shows "\<not> (Field r = {} \<or> Field s = {})"
simps_2: fixes l :: "dag"   and a :: "ref"   and r :: "dag"   and t :: "dag" shows "subdag (Node l a r) t = (t = l \<or> t = r \<or> subdag l t \<or> subdag r t)"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (\<tau> f) = A.arr f"
assms_1: fixes x :: "real" assumes "real_of_int a \<le> x" shows "f x = g x"
im: fixes a :: "'n"   and b :: "'n" shows "Sigma_Algebra.measure lebesgue (f ` cbox a b) = m * Sigma_Algebra.measure lebesgue (cbox a b)"
update_def: fixes k :: "'b"   and v :: "'c" shows "Assoc_List.update k v = Assoc_List.update_with v k (\<lambda>_. v)"
reflp_refl_eq: fixes r :: "('b \<times> 'b) set" shows "reflp (\<lambda>x y. (x, y) \<in> r) = refl r"
pcr_cr_eq: shows "pcr_ccsubspace (=) = cr_ccsubspace"
word_power_less_1: fixes sz :: "nat" assumes "sz < LENGTH('b)" shows "2 ^ sz - 1 < 2 ^ sz"
embed_measure_prod: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and M :: "'a measure"   and N :: "'c measure" assumes "inj f"   and "inj g"   and "sigma_finite_measure M"   and "sigma_finite_measure N" shows "embed_measure M f \<Otimes>\<^sub>M embed_measure N g = embed_measure (M \<Otimes>\<^sub>M N) (\<lambda>(x, y). (f x, g y))"
cval_add_def: fixes u :: "'c \<Rightarrow> 'd"   and d :: "'d" shows "u \<oplus> d = (\<lambda>x. u x + d)"
get_document_ptr_simp1: fixes document_ptr :: "(_) document_ptr"   and document :: "('m, 'n, 'o, 'p option) RDocument_scheme"   and h :: "(_) heap" shows "get document_ptr (put document_ptr document h) = Some document"
poly_mult_zero_2: fixes p :: "'a list list" assumes "set p \<subseteq> carrier (poly_ring R)" shows "UP.poly_mult p [] = []"
assms_6: shows "\<forall>m\<ge>(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows). A $ m $ mod_type_class.from_nat k = (0::'a)"
ortho_1: fixes w :: "'a vec" assumes "w \<in> W" shows "w \<bullet> normal_vector W = (0::'a)"
ms_nil: fixes e :: "exp" shows "e \<longrightarrow>* e"
that: fixes a :: "V"   and b :: "V" assumes "x = \<langle>a, b\<rangle>"   and "b \<in>\<^sub>\<circ> A"   and "r\<lparr>a\<rparr> = b" shows "thesis"
real_power_up_fl: fixes p :: "nat"   and x :: "float"   and n :: "nat" shows "real_of_float (power_up_fl p x n) = power_up p (real_of_float x) n"