hcju/isabelleps huggingface dataset:
norm_1_non_empty: shows "{\<parallel>f *\<^sub>v x\<parallel> |x. \<parallel>x\<parallel> = 1} \<noteq> {}"
map_nth: fixes xs :: "'f list" shows "map ((!) xs) [0..<length xs] = xs"
invc_baldR: fixes a :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and b :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c \<times> 'd" assumes "invc a"   and "invc2 b"   and "color a = Black" shows "invc (baldR a x b)"
leq: shows "(Cs, Ds) \<in> (leq_path1 P C)\<^sup>+"
le_infE: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x (inf a b)" obtains "less_eq x a" and "less_eq x b"
N_def: fixes n :: "int" shows "N n = (\<Sum>k. if k = 0 \<or> int k dvd n then 0 else 1 / q ^ k)"
results: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>" shows "results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV"
simps_9: fixes va :: "nat"   and vb :: "pol"   and v :: "nat"   and Q :: "pol" shows "isnorm (PX (Pinj va vb) (Suc v) Q) = (isnorm (Pinj va vb) \<and> isnorm Q)"
to_Gromov_completion_homeomorphism: shows "homeomorphism_on UNIV to_Gromov_completion"
combine_with_key_def: shows "combine_with_key \<equiv> map_fun id (map_fun rep (map_fun rep Mapping)) (\<lambda>f m1 m2 x. combine_options (f x) (m1 x) (m2 x))"
ffold_finsert_idem2: fixes z :: "'m"   and x :: "'m"   and A :: "'m fset" shows "ffold funionM z (finsert x A) = ffold funionM (funionM x z) A"
a: shows "\<^bold>g [^] fst z \<otimes> g' [^] snd z = a \<otimes> h [^] e"
non_const: shows "\<exists>w\<in>s. f w \<noteq> 0"
assms_4: shows "b c \<congruent>\<^sub>K b d"
mismatchId: fixes a :: "name"   and b :: "name"   and P :: "pi" assumes "a \<noteq> b" shows "[a\<noteq>b]P \<approx> P"
a2_map''_def: shows "a2_map'' = Mapping.delete (tp - len) a2_map'"
simps_8: fixes \<A> :: "'a ltl set"   and v :: "'a ltl" shows "\<A> \<Turnstile>\<^sub>P G v = (G v \<in> \<A>)"
elementary_cartesian_category_axioms: shows "elementary_cartesian_category (\<cdot>) pr1 pr0 \<I> \<tau>"
that: fixes g :: "'a \<Rightarrow>\<^sub>C 'b" assumes "f = apply_bcontfun g" shows "thesis"
zip_replicate: fixes i :: "nat"   and x :: "'g"   and j :: "nat"   and y :: "'h" shows "zip (replicate i x) (replicate j y) = replicate (min i j) (x, y)"
iso_imp_homomorphism: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" assumes "h \<in> Group.iso G H" shows "h \<in> hom G H"
MakeCat_def: fixes C :: "('g, 'h, 'i) Category_scheme" shows "MakeCat C \<equiv> \<lparr>Obj = obj\<^bsub>C\<^esub>, Mor = mor\<^bsub>C\<^esub>, Dom = restrict (Dom C) (mor\<^bsub>C\<^esub>), Cod = restrict (Cod C) (mor\<^bsub>C\<^esub>), Id = restrict (Category.Id C) (obj\<^bsub>C\<^esub>), Comp = \<lambda>f g. (\<lambda>(x, y)\<in>{(f, g) |f g. f \<approx>>\<^bsub>C\<^esub> g}. x ;;\<^bsub>C\<^esub> y) (f, g), \<dots> = Category.more C\<rparr>"
foldli_cong: fixes l :: "'a list"   and l' :: "'a list"   and \<sigma> :: "'b"   and \<sigma>' :: "'b"   and c :: "'b \<Rightarrow> bool"   and c' :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "l = l'"   and "\<sigma> = \<sigma>'"   and "c = c'"   and "\<And>\<sigma> x. \<lbrakk>x \<in> set l'; c' \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> = f' x \<sigma>" shows "foldli l c f \<sigma> = foldli l' c' f' \<sigma>'"
cdf_nondecreasing: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "C x \<le> C y"
lcos_module_imp: fixes H :: "'d set"   and G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and x' :: "'d" assumes "subgroup H G"   and "Group.group G"   and "x \<in> carrier G"   and "x' \<in> x <#\<^bsub>G\<^esub> H" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x' \<in> H"
th3_1: fixes a :: "(real, 'b) vec"   and b :: "(real, 'b) vec" shows "B\<^sub>\<real> a b b"
assms: shows "freshEnv ys y (rho [x \<leftarrow> X]_xs)"
basis_subspace_exists: fixes S :: "'b set" assumes "subspace S" obtains B :: "'b set" where "finite B" and "B \<subseteq> S" and "independent B" and "span B = S" and "card B = dim S"
min_set_Union: fixes X :: "'c fset set set" assumes "finite X" shows "min_set (\<Union> (min_set ` X)) = min_set (\<Union> X)"
map_eq_imp_length_eq: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list"   and g :: "'f \<Rightarrow> 'd"   and ys :: "'f list" assumes "map f xs = map g ys" shows "length xs = length ys"
module_hom_id: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "module scale" shows "module_hom scale scale id"
SUP_bot_conv_1: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and B :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Sup (B ` A) = bot) = (\<forall>x\<in>A. B x = bot)"
le_funI: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h" assumes "\<And>x. f x \<le> g x" shows "f \<le> g"
igFreshAbs_termMOD: shows "igFreshAbs termMOD = freshAbs"
that: fixes s :: "'c \<Rightarrow>\<^sub>0 nat" assumes "s \<in> keys p"   and "t = (\<Sum>x\<in>keys s. monomial (lookup s x) (f x))" shows "thesis"
meet_assoc: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "lower_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> y \<sqinter>\<^bsub>L\<^esub> z = x \<sqinter>\<^bsub>L\<^esub> (y \<sqinter>\<^bsub>L\<^esub> z)"
VcompNml_Nml_Ide: fixes t :: "'a local.term"   and a :: "'a local.term" assumes "Nml t"   and "Ide a"   and "Dom t = Cod a" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> a = t"
WTrt_elim_cases_5: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp"   and T :: "ty" assumes "P,E,h \<turnstile> try e\<^sub>1 catch(C V) e\<^sub>2 : T" obtains T\<^sub>1 :: "ty" where "P,E,h \<turnstile> e\<^sub>1 : T\<^sub>1" and "P,E(V \<mapsto> Class C),h \<turnstile> e\<^sub>2 : T" and "P \<turnstile> T\<^sub>1 \<le> T"
inv_comp_left_1: fixes g :: "'b"   and f :: "'b" assumes "D.iso (g \<cdot>\<^sub>D f)"   and "D.iso g" shows "D.inv (g \<cdot>\<^sub>D f) = D.inv f \<cdot>\<^sub>D D.inv g"
ifmi_notouch_dcl: fixes v :: "nat"   and ni1 :: "nat"   and ni2 :: "nat"   and s :: "bdd"   and ni :: "nat"   and s' :: "bdd" assumes "ifmi v ni1 ni2 s = (ni, s')" shows "dcl s' = dcl s"
not_infI: fixes cfg :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option" assumes "\<And>f. \<lbrakk>f 0 = cfg; \<And>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)\<rbrakk> \<Longrightarrow> False" shows "\<not> \<Gamma>\<turnstile> cfg \<rightarrow> \<dots>(\<infinity>)"
M_unqiue: fixes s :: "'a"   and t :: "'a" assumes "s \<in> M"   and "t \<in> M" shows "\<Delta> s = \<Delta> t"
conjugate_square_positive: fixes conjugate :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.conjugatable_ordered_ring conjugate plus zero minus uminus times less_eq less" shows "less_eq zero (times a (conjugate a))"
addrow_mat_inv: fixes k :: "nat"   and n :: "nat"   and l :: "nat"   and a :: "'c" assumes "k < n"   and "l < n"   and "k \<noteq> l" shows "addrow_mat n a k l * addrow_mat n (- a) k l = 1\<^sub>m n"
DC: fixes K :: "nat"   and X :: "freemsg" shows "DECRYPT K (CRYPT K X) \<sim> X"
is_category: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool" assumes "subcategory C Arr" shows "category (subcategory.comp C Arr)"
g_def: fixes z :: "'c" shows "g z \<equiv> snd (inv h z)"
pivot_funD_4: fixes A :: "'c mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr"   and "f i < nc" shows "A $$ (i, f i) = (1::'c)"
gis_done_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat" shows "gis_done_code node_eq_impl node_hash_impl \<equiv> \<lambda>x xa. case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl x (case snd xa of (xb, xc, xd, xe) \<Rightarrow> xd) of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then False else True"
AbstFormP_Ex: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm" shows "{AbstFormP v (SUCC i) x x'} \<turnstile> AbstFormP v i (Q_Ex x) (Q_Ex x')"
SUP_absorb: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and 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 bot :: "'d"   and top :: "'d"   and k :: "'e"   and I :: "'e set"   and A :: "'e \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "k \<in> I" shows "sup (A k) (Sup (A ` I)) = Sup (A ` I)"
infinite_not_subset_finite: fixes A :: "'a set"   and B :: "'a set" assumes "infinite A"   and "finite B" shows "\<not> A \<subseteq> B"
min_extend_val: shows "- 0x7FFFFFFF = 0xFFFFFFFF80000001"
step_step': fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set"   and S :: "(char list \<Rightarrow> 'a) option"   and C :: "(char list \<Rightarrow> 'a) option acom" assumes "Abs_Int_fun num' plus' \<gamma>" shows "step (\<gamma>_option (\<gamma>_fun \<gamma>) S) (ACom.map_acom (\<gamma>_option (\<gamma>_fun \<gamma>)) C) \<le> ACom.map_acom (\<gamma>_option (\<gamma>_fun \<gamma>)) (Abs_Int_fun.step' num' plus' S C)"
multlist_ee_cong: fixes fs :: "'a list"   and fs' :: "'a list" assumes "essentially_equal G fs fs'"   and "set fs \<subseteq> carrier G"   and "set fs' \<subseteq> carrier G" shows "foldr (\<otimes>) fs \<one> \<sim> foldr (\<otimes>) fs' \<one>"
in_strand_fv_subset: fixes x :: "('a, 'b) strand_step"   and S :: "('a, 'b) strand_step list" assumes "x \<in> set S" shows "vars\<^sub>s\<^sub>t\<^sub>p x \<subseteq> vars\<^sub>s\<^sub>t S"
Fubini_integral: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "integrable (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). f x y)" shows "LINT y|M2. LINT x|M1. f x y = LINT x|M1. integral\<^sup>L M2 (f x)"
pickBinp_def: fixes binp :: "'d \<Rightarrow> 'e set option" shows "pickBinp binp \<equiv> lift pick binp"
fiter_def: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "finite_iteration.fiter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (c::'b::refinement_lattice) \<equiv> gfp (\<lambda>x::'b::refinement_lattice. nil \<sqinter> seq c x)"
elims_1: fixes x :: "'v"   and xa :: "'v list"   and y :: "bool" assumes "adj_path x xa = y" obtains v :: "'v" where "x = vb__" and "xa = []" and "y = (vb__ \<in> V)"   | v :: "'v" and u :: "'v" and us :: "'v list" where "x = v" and "xa = u # us" and "y = (adjacent v u \<and> adj_path u us)"
Bang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> Rs"   and "guarded P" shows "\<Psi> \<rhd> !P \<longmapsto> Rs"
pred_intros_logic_5: fixes Q :: "'a \<Rightarrow> bool"   and M :: "'a measure"   and P :: "'a \<Rightarrow> bool" assumes "Measurable.pred M Q"   and "Measurable.pred M P" shows "Measurable.pred M (\<lambda>x. Q x \<or> P x)"
Free: fixes \<alpha> :: "freeRes"   and Q' :: "pi" assumes "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
gcd_smult_left: fixes c :: "'a"   and f :: "'a poly"   and g :: "'a poly" assumes "c \<noteq> (0::'a)" shows "gcd (Polynomial.smult c f) g = gcd f g"
3: shows "\<forall>i\<in>{1..m}. sum t ((A(x := A x - {Suc j})) i) = (T(x := T x - t (Suc j))) i"
of_int_nonneg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "0 \<le> z" shows "less_eq zero (ring_1.of_int one plus zero minus z)"
k_lte_length_\<pi>_plus_one: shows "k < length \<pi> + 1"
eval_in_hom: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "\<guillemotleft>hfsetcat.eval b c : HF'.prod (hfsetcat.exp b c) b \<rightarrow> c\<guillemotright>"
singleton': fixes a :: "nat" shows "\<not> eventually ((=) a) \<U>"
map_id_ran: fixes xs :: "'c set" shows "ran (map_id_on xs) = xs"
SMP_union: fixes A :: "('fun, 'var) Term.term set"   and B :: "('fun, 'var) Term.term set" shows "SMP (A \<union> B) = SMP A \<union> SMP B"
below: fixes x :: "'a" assumes "x \<in> A" shows "i \<sqsubseteq> x"
return_3: shows "Lfilled j lholed (vs @ [$Return]) es"
SD_preorderI: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'b pmf" assumes "Order_Predicates.preorder_on carrier le"   and "p \<in> lotteries_on carrier"   and "q \<in> lotteries_on carrier"   and "\<And>x. x \<in> carrier \<Longrightarrow> measure_pmf.prob q (preferred_alts le x) \<le> measure_pmf.prob p (preferred_alts le x)" shows "SD le q p"
s_AuthP: fixes v :: "Syntax.term"   and \<pi> :: "Syntax.term list" assumes "closed \<lparr>v\<rparr>"   and "value v" shows "\<lless>\<pi>, Auth v\<ggreater> P\<rightarrow> \<lless>\<pi>, Hashed (hash \<lparr>v\<rparr>) v\<ggreater>"
map_cong: fixes x :: "('a, 'b) blindable\<^sub>m"   and ya :: "('a, 'b) blindable\<^sub>m"   and f1 :: "'a \<Rightarrow> 'c"   and g1 :: "'a \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'd"   and g2 :: "'b \<Rightarrow> 'd" assumes "x = ya"   and "\<And>z1. z1 \<in> set1_blindable\<^sub>m ya \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> set2_blindable\<^sub>m ya \<Longrightarrow> f2 z2 = g2 z2" shows "map_blindable\<^sub>m f1 f2 x = map_blindable\<^sub>m g1 g2 ya"
2: shows "mkArr A B F \<in> hom (mkIde A) (mkIde B) \<and> Fun (mkArr A B F) = restrict F (local.set (mkIde A))"
unitary_unitary_gen: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary M" shows "unitary_gen M"
weave_def: fixes A :: "nat set"   and xs :: "'m list"   and ys :: "'m list" shows "weave A xs ys = map (\<lambda>i. if i \<in> A then xs ! card {a \<in> A. a < i} else ys ! card {a \<in> - A. a < i}) [0..<length xs + length ys]"
LIMSEQ_root: shows "(\<lambda>n. root n (real n)) \<longlonglongrightarrow> 1"
INFM_mono: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<exists>\<^sub>\<infinity>x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "\<exists>\<^sub>\<infinity>x. Q x"
runit_char_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "prebicategory_with_homs_and_units V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "H (prebicategory_with_homs_and_units.runit V H \<a> \<i> src f) (src f) = V (H f (\<i> (src f))) (\<a> f (src f) (src f))"
lift_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'m" shows "lift f x = bind x (\<lambda>x. return (f x))"
inner_simps_1: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x + y) \<bullet> z = x \<bullet> z + y \<bullet> z"
lmeasurable_frontier: fixes S :: "'c set" assumes "bounded S" shows "frontier S \<in> lmeasurable"
op_set_empty_def: shows "op_set_empty \<equiv> {}"
out1_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool"   and \<sigma> :: "bool" shows "out1 A B \<sigma> = (let (a1, a2) = A; (b1, b2) = B in return_spmf (\<sigma>, \<sigma> \<oplus> (a1 \<oplus> b1 \<and> a2 \<oplus> b2)))"
record_simps_2: fixes f :: "(char list, char list, 'b) namespace \<Rightarrow> (char list, char list, 'a) namespace"   and x :: "'b sem_env" shows "c (update_v f x) = c x"
null: fixes G :: "prog"   and R :: "ref_ty" shows "G\<turnstile>NT\<preceq>RefT R"
list_singleton_induct: fixes P :: "'b list \<Rightarrow> bool"   and xs :: "'b list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y xs. P (y # xs) \<Longrightarrow> P (x # y # xs)" shows "P xs"
intros: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t"   and "wt s < wt t" shows "gt_wt t s"
red_preserves_iconf: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "iconf sh e" shows "iconf sh' e'"
ltrm_coeff: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "coeff (UP R) (monom (UP R) (f (deg R f)) (deg R f)) n = (if n = deg R f then f (deg R f) else \<zero>\<^bsub>R\<^esub>)"
basis_enum_of_vec_mult: fixes v :: "complex vec"   and c :: "complex" assumes "dim_vec v = length canonical_basis" shows "basis_enum_of_vec (c \<cdot>\<^sub>v v) = c *\<^sub>C basis_enum_of_vec v"
Basis_pair_def: shows "local.Basis_pair = CBasis \<times> {0::'b} \<union> {0::'a} \<times> CBasis"
C: shows "diag_block_mat (map fst Ms) \<in> carrier_mat m m"
him_hv: fixes x :: "(real, 7) vec" shows "HIm (Hv x) = x"
iwlsFSbSwTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "iwlsFSbSwTR MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWlsSTR MOD \<and> igSubstAllIPresIGWlsAllSTR MOD \<and> igFreshClsSTR MOD \<and> igSubstClsSTR MOD \<and> igAbsRenSTR MOD"
minus_remove1_mset_if: fixes A :: "'a multiset"   and B :: "'a multiset"   and b :: "'a" shows "A - remove1_mset b B = (if b \<in># B \<and> b \<in># A \<and> count B b \<le> count A b then {#b#} + (A - B) else A - B)"
normf_def: fixes R :: "('c \<Rightarrow> real) set"   and S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "function_ring_on R S" shows "function_ring_on.normf S f \<equiv> SUP x\<in>S. \<bar>f x\<bar>"
degree_f_id: shows "degree (\<Prod>i\<in>UNIV. to_fun (\<^bold>X - \<^bold>C A) i i) = CARD('n)"
pr_sesk_card_user: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and m :: "nat"   and n :: "nat"   and run :: "nat"   and s :: "rat"   and a :: "rat"   and b :: "rat"   and c :: "rat"   and s' :: "rat"   and b' :: "rat"   and a' :: "rat"   and d :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "User m \<noteq> Spy"   and "NonceS (S (User m, n, run)) = Some s"   and "IntMapK (S (User m, n, run)) = Some a"   and "ExtMapK (S (User m, n, run)) = Some b"   and "IntAgrK (S (User m, n, run)) = Some c"   and "NonceS (S (Card n, n, run)) = Some s'"   and "IntMapK (S (Card n, n, run)) = Some b'"   and "ExtMapK (S (Card n, n, run)) = Some a'"   and "IntAgrK (S (Card n, n, run)) = Some d"   and "ExtAgrK (S (Card n, n, run)) = Some (c * (s + a * b))"   and "s' + a' * b' = s + a * b" shows "Key (sesK (c * d * (s + a * b))) \<notin> A"
distinct_19: fixes x2 :: "('e, 'f) Term.term"   and x61 :: "poscheckvariant"   and x62 :: "('e, 'f) Term.term"   and x63 :: "('e, 'f) Term.term" shows "receive\<langle>x2\<rangle> \<noteq> \<langle>x61: x62 \<in> x63\<rangle>"
hIm_diff: fixes x :: "complex star"   and y :: "complex star" shows "hIm (x - y) = hIm x - hIm y"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and s :: "'c stream" shows "(x # xs) @- s = x ## xs @- s"
mult_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero b"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
nat_term_compare_LEX: shows "nat_term_compare Term_Order.LEX = lex_comp"
rtranclp_r_diff_Id: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(inf r (\<noteq>))\<^sup>*\<^sup>* = r\<^sup>*\<^sup>*"
mset_merge_pairs: fixes hs :: "'b hp list" shows "mset_heap (merge_pairs hs) = \<Sum>\<^sub># (image_mset mset_hp (mset hs))"
cross_back_edge: fixes s :: "('v, 'es) state_scheme"   and u :: "'v"   and v :: "'v"   and s' :: "('v, 'es) state_scheme" assumes "DFS_invar G param s"   and "I s"   and "cond s"   and "\<not> is_break param s"   and "stack s \<noteq> []"   and "(u, v) \<in> pending s"   and "u = hd (stack s)"   and "v \<in> dom (discovered s)"   and "discovered s' = discovered s"   and "finished s' = finished s"   and "stack s' = stack s"   and "tree_edges s' = tree_edges s"   and "counter s' = counter s"   and "pending s' = pending s - {(u, v)}"   and "cross_edges s' \<union> back_edges s' = cross_edges s \<union> back_edges s \<union> {(u, v)}"   and "state.more s' = state.more s" shows "on_cross_edge param u v s' \<le>\<^sub>n SPEC (\<lambda>x. DFS_invar G param (s'\<lparr>state.more := x\<rparr>) \<longrightarrow> I (s'\<lparr>state.more := x\<rparr>))"
set_mset_filter: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a multiset" shows "set_mset (filter_mset P M) = {a. a \<in># M \<and> P a}"
liberalE: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and i :: "'c" assumes "liberal scf A Is"   and "i \<in> Is" shows "\<exists>x\<in>A. \<exists>y\<in>A. x \<noteq> y \<and> decisive scf A Is {i} x y \<and> decisive scf A Is {i} y x"
\<epsilon>_props_2: shows "\<epsilon> < 1 / real k"
da_def: shows "da = max (0::'a) (min wa (d - wa))"
spliter: shows "s_TS init h ((as @ [x] @ bs @ [x]) @ cs1) (length (as @ [x] @ bs @ [x] @ cs1)) = s_TS init h ((as @ [x] @ bs @ [x]) @ cs) (length (as @ [x] @ bs @ [x]) + k + 1)"
unification: fixes L :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "finite L"   and "unifier\<^sub>l\<^sub>s \<sigma> L" shows "\<exists>\<theta>. mgu\<^sub>l\<^sub>s \<theta> L"