hcju/isabelleps huggingface dataset:
main_part_null_part_Int: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets (completion M)" shows "main_part M S \<inter> null_part M S = {}"
case_x_not_in_range: shows "\<forall>x. x \<notin> cbox a b \<longrightarrow> integral UNIV (\<lambda>y. f (x, y)) = (0::'c)"
v: shows "v \<in> L - {0\<^sub>v n}"
ceiling_one: shows "\<lceil>1::'a\<rceil> = 1"
ANDSepaux2: fixes x :: "('b set set, 'c) Combinators"   and y :: "('b set set, 'c) Combinators"   and z :: "('b set set, 'c) Combinators list" assumes "allNetsDistinct (x # y # z)"   and "allNetsDistinct (separate (y # z))" shows "allNetsDistinct (x # separate (y # z))"
assms: fixes p :: "('b, 'c, 'a, 'd) seqp"   and pn :: "'a" assumes "p \<in> subterms (\<Gamma> pn)" shows "\<exists>!l. labels \<Gamma> p = {l}"
join_empty_left: fixes pos :: "bool"   and t :: "'b option list set" shows "join {} pos t = {}"
K_def: shows "K = backdoor (\<lambda>x. \<^bold>\<exists>F. \<lbrace>x,F\<rbrace> \<^bold>& \<^bold>\<not>\<lparr>F,x\<rparr>)"
s2': shows "s2' \<in> pmdl.syzygy_module (set bs)"
P6: shows "A \<noteq> B \<and> C \<noteq> B \<and> D \<noteq> E \<and> F \<noteq> E"
augment_cf_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and edges :: "(nat \<times> nat) set"   and cap :: "'a" shows "Graph.augment_cf c edges cap \<equiv> \<lambda>e. if e \<in> edges then c e - cap else if prod.swap e \<in> edges then c e + cap else c e"
proots_upper_cindex_eq: fixes p :: "complex poly" assumes "lead_coeff p = 1"   and "\<forall>x\<in>proots p. Im x \<noteq> 0" shows "real (proots_upper p) = real_of_int (int (degree p) - cindex_poly_ubd (map_poly Im p) (map_poly Re p)) / 2"
inject_1: fixes x1 :: "nat"   and y1 :: "nat" shows "(Var x1 = Var y1) = (x1 = y1)"
perm_bij: fixes pi :: "(name \<times> name) list"   and x :: "'a"   and y :: "'a" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
remap_polys_l2_def: fixes spec :: "(char list list \<times> int) list" shows "remap_polys_l2 spec = (\<lambda>\<V> A. upper_bound_on_dom A \<bind> (\<lambda>n. RETURN (2 ^ 64 \<le> n) \<bind> (\<lambda>b. if b then remap_polys_l_dom_err \<bind> (\<lambda>c. RETURN (error_msg 0 c, \<V>, fmempty)) else nfoldli [0..<n] (\<lambda>_. True) (\<lambda>i (b, \<V>, A'). if i \<in># dom_m A then ASSERT (fmlookup A i \<noteq> None) \<bind> (\<lambda>_. full_normalize_poly (the (fmlookup A i)) \<bind> (\<lambda>p. weak_equality_l p spec \<bind> (\<lambda>eq. RETURN (\<V> \<union> vars_llist (the (fmlookup A i))) \<bind> (\<lambda>\<V>. RETURN (b \<or> eq, \<V>, fmupd i p A'))))) else RETURN (b, \<V>, A')) (False, \<V>, fmempty) \<bind> (\<lambda>(b, \<V>, A). RETURN (if b then CFOUND else CSUCCESS, \<V>, A)))))"
exhaust_3: fixes x :: "3" shows "x = 1 \<or> x = 2 \<or> x = 3"
intro: fixes G :: "'b pair_pre_digraph" assumes "pair_digraph G"   and "pair_pseudo_graph G" shows "pair_graph G"
conv: shows "weak_conv_m (\<lambda>n. distr P borel (\<lambda>x. (g x + birkhoff_sum f n x) / B n)) Z"
typeof_addr_hext_mono: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'d prog"   and h :: "'c"   and h' :: "'c"   and a :: "'a"   and hT :: "htype" assumes "heap addr2thread_id thread_id2addr allocate typeof_addr heap_write P"   and "heap_base.hext typeof_addr h h'"   and "typeof_addr h a = \<lfloor>hT\<rfloor>" shows "typeof_addr h' a = \<lfloor>hT\<rfloor>"
assms_2: fixes x :: "'a" assumes "x \<in> space N" shows "f' (f x) = x"
wt_getTvars: fixes \<sigma>l :: "'tp list" shows "list_all TE.wt (TE.getTvars \<sigma>l)"
\<omega>_dec_of_Cantor: fixes \<alpha>s :: "V list"   and ms :: "nat list" assumes "sorted_wrt (\<lambda>x y. y < x) \<alpha>s"   and "length \<alpha>s = length ms" shows "sorted_wrt (\<lambda>x y. y \<le> x) (\<omega>_of_Cantor \<alpha>s ms)"
put_writer_def: fixes s :: "'s"   and m :: "('w, 'a, 'm) writerT" shows "local.put_writer s m = WriterT (put s (run_writer m))"
rel_gpv_lift_spmf: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and p :: "'d spmf"   and q :: "'e spmf" shows "rel_gpv A C (lift_spmf p) (lift_spmf q) = rel_spmf A p q"
intro: fixes M :: "'c measure" assumes "\<exists>A. countable A \<and> A \<subseteq> sets M \<and> \<Union> A = space M \<and> (\<forall>a\<in>A. emeasure M a \<noteq> \<infinity>)" shows "sigma_finite_measure M"
us: shows "us = lzip (ltake (llength us) xs) (ltake (llength us) ys)"
simps_1: fixes P :: "'a" shows "strip (SKIP {P}) = com.SKIP"
assms: fixes \<psi> :: "'a" shows "\<psi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>) = \<psi> \<bullet>\<^sub>C (B *\<^sub>V \<psi>)"
orthogonalD_out: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "orthogonal \<Delta> f S"   and "edge \<Delta> x y"   and "x \<in> S"   and "y \<notin> S" shows "f (x, y) = capacity \<Delta> (x, y)"
None_add2: fixes x :: "'b option" shows "total_partial.add2 None x = None"
fmlookup_inject: fixes x :: "('a, 'b) fmap"   and y :: "('a, 'b) fmap" shows "(fmlookup x = fmlookup y) = (x = y)"
card_eq_0_iff: fixes A :: "'e set" shows "(card A = 0) = (A = {} \<or> infinite A)"
rel_FGl_def: fixes L1 :: "'n \<Rightarrow> 'aa \<Rightarrow> bool"   and L2 :: "'o \<Rightarrow> 'ab \<Rightarrow> bool"   and L3 :: "'u \<Rightarrow> 'ag \<Rightarrow> bool"   and Co1 :: "'p \<Rightarrow> 'ac \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'ad \<Rightarrow> bool"   and Co3 :: "'v \<Rightarrow> 'ah \<Rightarrow> bool"   and Co4 :: "'w \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra1 :: "'r \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra2 :: "'s \<Rightarrow> 'af \<Rightarrow> bool"   and Contra3 :: "'x \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra4 :: "'y \<Rightarrow> 'ak \<Rightarrow> bool" shows "rel_FGl L1 L2 L3 Co1 Co2 Co3 Co4 Contra1 Contra2 Contra3 Contra4 = rel_F (rel_G L1 L2 Co1 Co2 Contra1 Contra2) L1 L3 Co1 Co3 Co4 Contra1 Contra3 Contra4"
elims: fixes x :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem"   and y :: "('a \<times> 'b) multiset" assumes "BsSkewBinomialHeapStruc.elem_to_mset x = y" obtains e :: "'a" and a :: "'b" and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" where "x = BsSkewBinomialHeapStruc.Element e a q" and "y = {#(e, a)#} + \<Sum>\<^sub># (image_mset BsSkewBinomialHeapStruc.elem_to_mset (image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q))))"
proper_it_def: fixes it :: "('b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and it' :: "('d \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "proper_it it it' \<equiv> \<exists>l. it = foldli l \<and> it' = foldli l"
roots: shows "{z. poly q' z = 0} = set (map fst (snd fctrs))"
steps_Fault: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and f :: "'c" shows "\<Gamma> \<turnstile> (c, Fault f) \<rightarrow>\<^sup>* (Skip, Fault f)"
S_transform_weakly_bisimilar_is_weak_bisimulation: shows "is_weak_bisimulation (\<approx>\<cdot>\<^sub>S)"
Aheap_heap3: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" assumes "x \<in> thunks \<Gamma>" shows "(Aheap \<Gamma> e\<cdot>a) x = up\<cdot>0"
orth2: shows "corthogonal ((adjuster n w us + w) # us)"
power_eq_if: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and m :: "nat" shows "power.power one times p m = (if m = 0 then one else times p (power.power one times p (m - 1)))"
rel_mset_Plus: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and M :: "'a multiset"   and N :: "'b multiset" assumes "R a b"   and "rel_mset R M N" shows "rel_mset R (add_mset a M) (add_mset b N)"
vimage_singleton_eq: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'd"   and b :: "'d" shows "(a \<in> f -` {b}) = (f a = b)"
find_mu_def: shows "find_mu \<equiv> (\<lambda>s. s\<lparr>m := 0, tortoise := x0, hare := seq (l s)\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) (\<lambda>s. s\<lparr>tortoise := f (tortoise s), hare := f (hare s), m := m s + 1\<rparr>)"
editVar_variable_inv: fixes v :: "variable"   and idx :: "integer option"   and val :: "integer" assumes "variable_inv v" shows "variable_inv (editVar v idx val)"
mat_adjoint_def': fixes M :: "'a mat" shows "mat_adjoint M = (map_mat conjugate M)\<^sup>T"
residualFreshChainSimp_1: fixes xvec :: "name list"   and M :: "'d"   and N :: "'d"   and P :: "('d, 'e, 'f) psi" shows "xvec \<sharp>* (M\<lparr>N\<rparr> \<prec> P) = (xvec \<sharp>* M \<and> xvec \<sharp>* N \<and> xvec \<sharp>* P)"
fuse_converter_def: shows "fuse_converter = CNV (stateless_callee fuse_callee) ()"
ms_of_greek_cons: fixes x :: "accent \<times> 'a"   and s :: "(accent \<times> 'a) list" shows "ms_of_greek (x # s) = {#adj_msog [] s (x, [])#} + image_mset (adj_msog [x] []) (ms_of_greek s)"
map_eq: fixes x :: "nat" shows "map (\<lambda>xa. if xa = s x then Num 1 else Num 0) xs = map (\<lambda>xa. if xa = x then Num 1 else Num 0) xs"
inj_on_inverseI: fixes A :: "'c set"   and g :: "'d \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x" shows "inj_on f A"
greaterThan_iff: fixes i :: "'b"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b" shows "(i \<in> ord.greaterThan less k) = less k i"
elims: fixes x :: "term"   and y :: "name option" assumes "unfree x = y" obtains name :: "name" where "x = Free name" and "y = Some name"   | v :: "name" where "x = Const v__" and "y = None"   | v :: "term" where "x = \<Lambda> va__" and "y = None"   | v :: "nat" where "x = Bound vb__" and "y = None"   | v :: "term" and va :: "term" where "x = v $ va" and "y = None"
inject_5: fixes x61 :: "'b formula"   and x62 :: "'b formula"   and y61 :: "'b formula"   and y62 :: "'b formula" shows "(x61 \<^bold>\<rightarrow> x62 = y61 \<^bold>\<rightarrow> y62) = (x61 = y61 \<and> x62 = y62)"
fpairs_vsingleton: fixes a :: "V"   and b :: "V" shows "fpairs (set {[a, b]\<^sub>\<circ>}) = set {[a, b]\<^sub>\<circ>}"
insert_def: fixes x :: "'b"   and t :: "'b tree23" shows "insert x t = treeI (ins x t)"
simps: fixes a1 :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)"   and a2 :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset \<times> ('e \<Rightarrow> ('d, 'f) Term.term option)" shows "MATCH1 a1 a2 = ((\<exists>\<sigma> x t P. a1 = (P + {#(Var x, t)#}, \<sigma>) \<and> a2 = (P, \<sigma>(x \<mapsto> t)) \<and> (\<sigma> x = None \<or> \<sigma> x = Some t)) \<or> (\<exists>ss ts P f \<sigma>. a1 = (P + {#(Fun f ss, Fun f ts)#}, \<sigma>) \<and> a2 = (P + mset (zip ss ts), \<sigma>) \<and> length ss = length ts))"
assms_1: fixes j :: "nat" assumes "j < D * E" shows "isDERIV i (A ! j) xs"
disjuncts_NFOR: fixes X :: "('a, 'b) aformula set" assumes "finite X"   and "X \<noteq> {}"   and "\<forall>x\<in>X. \<not> is_FOr x" shows "disjuncts (NFOR X) = X"
i_less: fixes a :: "real"   and b :: "real pdevs"   and i :: "nat" assumes "(a, b) \<in> set as"   and "i < degree b" shows "i < min d da"
T_vrestr_meas_2: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> sets M" shows "(T ^^ n) --` A \<in> sets M"
connected_frontier_disjoint: fixes S :: "'a set"   and T :: "'a set" assumes "connected S"   and "connected T"   and "disjnt S T"   and "frontier S \<subseteq> frontier T" shows "connected (frontier S)"
WtFT_def: fixes cl :: "('d, 'e, 'f) cmd list" shows "WtFT cl \<equiv> \<Sum>n\<in>theFT cl. 1 / real (length cl)"
mult_float_mono2: fixes a :: "real"   and b :: "real"   and ab :: "real"   and bb :: "real"   and aa :: "real"   and ba :: "real"   and ac :: "real"   and bc :: "real"   and prec :: "nat" assumes "a \<le> b"   and "ab \<le> bb"   and "aa \<le> a"   and "b \<le> ba"   and "ac \<le> ab"   and "bb \<le> bc" shows "plus_up prec (pprt b * pprt bb) (plus_up prec (pprt a * nprt bb) (plus_up prec (nprt b * pprt ab) (nprt a * nprt ab))) \<le> plus_up prec (pprt ba * pprt bc) (plus_up prec (pprt aa * nprt bc) (plus_up prec (nprt ba * pprt ac) (nprt aa * nprt ac)))"
prCons_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> ('varSort, 'b) sAbs"   and SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b, 'c) semDom_scheme" shows "prCons g gA SEM \<equiv> prVar g SEM \<and> prAbs g gA SEM \<and> prOp g gA SEM"
assoc'_naturality: fixes f0 :: "'d"   and f1 :: "'d"   and f2 :: "'d" assumes "D.arr f0"   and "D.arr f1"   and "D.arr f2" shows "((f0 \<otimes>\<^sub>D f1) \<otimes>\<^sub>D f2) \<cdot>\<^sub>D D.assoc' (D.dom f0) (D.dom f1) (D.dom f2) = D.assoc' (D.cod f0) (D.cod f1) (D.cod f2) \<cdot>\<^sub>D (f0 \<otimes>\<^sub>D f1 \<otimes>\<^sub>D f2)"
params_subterms: fixes X :: "('c, 'd) Term.term list"   and f :: "'c" shows "set X \<subseteq> subterms (Fun f X)"
ux_frees: shows "\<forall>i\<in>frees (unlift_vars n x'). n \<le> i"
ne: shows "OPT2 (a # rest1) [x, y] \<noteq> []"
get_tag_name_new_element: fixes ptr' :: "(_) element_ptr"   and new_element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "r \<in> get_tag_name_locs ptr'" shows "r h h'"
While_transT_invert: fixes tst :: "'test"   and c1 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" assumes "(while tst do c1, s) \<rightarrow>t s'" obtains "\<not> tval tst s" and "s' = s"
filter_insort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map f xs)"   and "P x" shows "filter P (linorder.insort_key less_eq f x xs) = linorder.insort_key less_eq f x (filter P xs)"
assms_1: shows "\<forall>n w. prices Mkt asset (Suc n) w \<noteq> 0"
T1: fixes u :: "'b"   and \<omega> :: "'b" assumes "D.ide u"   and "\<guillemotleft>\<omega> : D.dom \<omega> \<Rightarrow>\<^sub>D F r \<star>\<^sub>D u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. D.ide w \<and> \<guillemotleft>\<theta> : F f \<star>\<^sub>D w \<Rightarrow>\<^sub>D u\<guillemotright> \<and> \<guillemotleft>\<nu> : D.dom \<omega> \<Rightarrow>\<^sub>D F g \<star>\<^sub>D w\<guillemotright> \<and> D.iso \<nu> \<and> ((F r \<star>\<^sub>D \<theta>) \<cdot>\<^sub>D \<a>\<^sub>D[F r, F f, w] \<cdot>\<^sub>D (D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho> \<star>\<^sub>D w)) \<cdot>\<^sub>D \<nu> = \<omega>"
S: fixes x :: "'a" assumes "x \<in> topspace X"   and "f x \<in> topspace Y" shows "(f x \<in> T) = (x \<in> S)"
complex_rotations_fix: fixes r :: "Vertex \<Rightarrow> Vertex"   and a :: "Vertex"   and b :: "Vertex" assumes "r \<in> complex_rotations"   and "r a = a"   and "r b = b"   and "a \<noteq> b" shows "r = id"
simps_6: fixes sh :: "sheap"   and V :: "'a"   and e :: "'a exp" shows "iconf sh (V:=e) = iconf sh e"
rsum_append: fixes D1 :: "(real \<times> real \<times> real) list"   and D2 :: "(real \<times> real \<times> real) list"   and f :: "real \<Rightarrow> real" shows "rsum (D1 @ D2) f = rsum D1 f + rsum D2 f"
snoc_sublist_snoc: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list"   and y :: "'c" shows "sublist (xs @ [x]) (ys @ [y]) = (x = y \<and> suffix xs ys \<or> sublist (xs @ [x]) ys)"
in_set: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list"   and I :: "nat list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec_copr qs x"   and "list_constr I (length qs)" shows "sign \<in> set (characterize_consistent_signs_at_roots_copr p qs)"
LTL_G_one_step_unfolding: fixes w :: "nat \<Rightarrow> 'c set"   and \<phi> :: "'c ltl" shows "w \<Turnstile> G \<phi> = (w \<Turnstile> \<phi> \<and> w \<Turnstile> X (G \<phi>))"
fresh_23: fixes a :: "name"   and x4 :: "coname"   and x5 :: "trm"   and x2 :: "name"   and x3 :: "trm"   and x1 :: "name" shows "a \<sharp> ImpL <x4>.x5 x2.x3 x1 = (a \<sharp> [x4].x5 \<and> a \<sharp> [x2].x3 \<and> a \<sharp> x1)"
fup_Aexp_edom: fixes e :: "exp"   and a :: "Arity\<^sub>\<bottom>" shows "edom (Aexp_bot_syn a e) \<subseteq> fv e"
prems_1: shows "\<I> \<turnstile>g gpv \<surd>"
case_2: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and f3 :: "'g \<Rightarrow> 'f \<Rightarrow> 'e"   and x2 :: "'f" shows "(case Skip x2 of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = f2 x2"
rel_inject_3: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x3 :: "'a"   and y3 :: "'b" shows "rel_behaviour R (Goes_wrong x3) (Goes_wrong y3) = R x3 y3"
meta_eq_to_obj_eq: fixes A :: "'a"   and B :: "'a" assumes "A \<equiv> B" shows "A = B"
Fubini_integral: fixes f :: "'a \<Rightarrow> ('i \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "integrable (M i \<Otimes>\<^sub>M Pi\<^sub>M I M) (\<lambda>(x, y). f x y)" shows "LINT y|Pi\<^sub>M I M. LINT x|M i. f x y = LINT x|M i. integral\<^sup>L (Pi\<^sub>M I M) (f x)"
intro: fixes e :: "real"   and c :: "real"   and f2 :: "real"   and l0 :: "nat" assumes "1 < e"   and "1 < c"   and "0 < f2"   and "e * c / (f2 * (min e c - 1)) \<le> real l0" shows "Optimal f2 c e l0"
cases: fixes x :: "'b word \<times> nat" obtains pre :: "'b word" and len :: "nat" where "x = (pre, len)"
unif: shows "\<forall>i<n. \<forall>A\<in>#AAs ! i. A \<cdot>a \<sigma> = As ! i \<cdot>a \<sigma>"
containsE: fixes A :: "'c set"   and x :: "'c" assumes "Predicate_Compile.contains A x" obtains A' :: "'c set" and x' :: "'c" where "A = A'" and "x = x'" and "x \<in> A"
integral_cmul: fixes S :: "'b set"   and c :: "real"   and f :: "'b \<Rightarrow> 'a" shows "integral S (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral S f"
framed_linkrel_def: fixes x :: "wall"   and y :: "wall" shows "framed_linkrel x y = (framed_uncross x y \<or> pull x y \<or> straighten x y \<or> swing x y \<or> Tangle_Moves.rotate x y \<or> compress x y \<or> slide x y)"
subalbegra_incl_imp_finite_dimension: fixes K :: "'a set"   and E :: "'a set"   and V :: "'a set" assumes "subfield K R"   and "finite_dimension K E"   and "subalgebra K V R"   and "V \<subseteq> E" shows "finite_dimension K V"
elims_1: fixes x :: "'b trm"   and xa :: "'b trm"   and y :: "bool" assumes "x \<prec> xa = y" obtains u :: "'b trm" and v :: "'b" where "x = u__" and "xa = Var v" and "\<not> y"   | u :: "'b trm" and c :: "'b" where "x = ua__" and "xa = Const c" and "\<not> y"   | u :: "'b trm" and M :: "'b trm" and N :: "'b trm" where "x = u" and "xa = M \<cdot> N" and "y = (u = M \<or> u = N \<or> u \<prec> M \<or> u \<prec> N)"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "stails\<cdot>(x :# xs) = (x :# xs) :# stails\<cdot>xs"
upRuleCharacterise: fixes Ps :: "'a list"   and C :: "('b, 'c) sequent" assumes "(Ps, C) \<in> upRules" shows "\<exists>F Fs. C = ( \<Empt> \<Rightarrow>* \<LM> Compound F Fs  \<RM>) \<or> C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* \<Empt>)"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "U.satL \<xi> (Pos a) = U.satA \<xi> a"
orbit_unfold_image: fixes f :: "'b perm"   and a :: "'b" shows "orbit f a = (\<lambda>n. (f ^ n) \<langle>$\<rangle> a) ` {..<order f a}"
del_unEdge_def: fixes v :: "'d"   and e :: "'e"   and v' :: "'d"   and g :: "('d, 'e, 'f) graph_scheme" shows "del_unEdge v e v' g \<equiv> \<lparr>nodes = nodes g, edges = edges g - {(v, e, v'), (v', e, v)}\<rparr>"
add_le_same_cancel1: fixes b :: "'b zmultiset"   and a :: "'b zmultiset" shows "(b + a \<subseteq>#\<^sub>z b) = (a \<subseteq>#\<^sub>z {#}\<^sub>z)"
analz_insert_cong: fixes H :: "msg set"   and H' :: "msg set"   and X :: "msg" assumes "analz H = analz H'" shows "analz (insert X H) = analz (insert X H')"
InputPres: fixes P :: "pi"   and Q :: "pi"   and x :: "name"   and a :: "name" assumes "\<forall>y\<in>supp (P, Q, x). P[x::=y] \<equiv>\<^sub>e Q[x::=y]" shows "a<x>.P \<equiv>\<^sub>e a<x>.Q"
symrun_interp_right_idem: fixes \<Gamma>\<^sub>1 :: "'a constr list"   and \<Gamma>\<^sub>2 :: "'a constr list" shows "\<lbrakk>\<lbrakk> (\<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2) @ \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
Red_F_Bot_F: fixes B :: "'f"   and N :: "'f set" assumes "B \<in> Bot_F"   and "\<G>_Fset N \<Turnstile>G \<G>_Fset {B}" shows "\<G>_Fset (N - local.Red_F_\<G> N) \<Turnstile>G \<G>_Fset {B}"
relcomp_Image: fixes X :: "('b \<times> 'c) set"   and Y :: "('c \<times> 'a) set"   and Z :: "'b set" shows "(X ; Y) `` Z = Y `` X `` Z"
not_nofail_iff: fixes S :: "'c nres" shows "(\<not> nofail S) = (S = FAIL)"
take_bit_length_eq: fixes w :: "'a word" shows "take_bit LENGTH('a) w = w"
fp_cop_F_induct: fixes P :: "'x set \<Rightarrow> bool"   and ds :: "'d set" assumes "P {}"   and "\<And>fp. P fp \<Longrightarrow> P (cop_F ds fp)" shows "P (fp_cop_F ds)"
SDom_def: fixes \<sigma> :: "('d, 'e, 'f) subst" shows "SDom \<sigma> = {Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}"
assms: shows "\<lbrace>\<alpha> x \<cdot> \<tau> y\<rbrace>z\<lbrace>w\<rbrace>"
pCons_as_add: fixes a :: "'a"   and p :: "'a poly" shows "pCons a p = [:a:] + monom (1::'a) 1 * p"
unop_ontoI: fixes A :: "V"   and f :: "V" assumes "nop_onto A (1\<^sub>\<nat>) f" shows "unop_onto A f"
refine: fixes D :: "nat"   and Xi :: "(real list \<times> real list) list"   and X :: "'c set"   and Yi :: "(real list \<times> real list) list"   and Y :: "'c set"   and Mi :: "nat"   and M :: "nat" assumes "DIM_precond TYPE('c) D"   and "(Xi, X) \<in> clw_rel lvivl_rel"   and "(Yi, Y) \<in> clw_rel lvivl_rel"   and "(Mi, M) \<in> nat_rel" shows "(nres_of (subset_spec_ivls_clw_impl D Mi Xi Yi), subset_spec_ivls_clw M X Y) \<in> \<langle>bool_rel\<rangle>nres_rel"
extended_superset: fixes S :: "'b formula set" shows "S \<subseteq> extended S"
nfv_def: fixes \<phi> :: "formula" shows "nfv \<phi> = Max (insert 0 (Suc ` fv \<phi>))"
flrestriction_on_vempty: fixes r :: "V" shows "r \<restriction>\<^sup>l\<^sub>\<bullet> []\<^sub>\<circ> = []\<^sub>\<circ>"
jumpF_pathstart_def: fixes g :: "real \<Rightarrow> complex"   and z :: "complex" shows "jumpF_pathstart g z = jumpF (\<lambda>t. Im (g t - z) / Re (g t - z)) (at_right 0)"
linear_simps_3: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "f (0::'d) = (0::'e)"
not\<^sub>n_semantics: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n not\<^sub>n \<phi> = (\<not> w \<Turnstile>\<^sub>n \<phi>)"
get_parent_child_dual: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some ptr" obtains children :: "(_) node_ptr list" where "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children" and "child \<in> set children"
absolutely_integrable_mult_Dirichlet_kernel: fixes f :: "real \<Rightarrow> real"   and n :: "nat" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<lambda>x. Dirichlet_kernel n x * f x) absolutely_integrable_on {- pi..pi}"
abs_eq: fixes xa :: "cmsg"   and x :: "cmsg" shows "Exp (Ab xa) (Ab x) = Ab (cExp xa x)"