hcju/isabelleps huggingface dataset:
mUnionM_add: fixes A :: "'m multiset"   and B :: "'m multiset" shows "mUnionM (A + B) = munionM (mUnionM A) (mUnionM B)"
map_int_pair_REAL: fixes f :: "int \<Rightarrow> int \<Rightarrow> 'a"   and g :: "val \<Rightarrow> 'a"   and i :: "real"   and j :: "real" shows "map_int_pair f g <|RealVal i, RealVal j|> = g <|RealVal i, RealVal j|>"
eq_FF_2: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and xs :: "'a"   and ys :: "'a" assumes "class.Eq_def eq" shows "(eq\<cdot>xs\<cdot>ys = FF) = (xs \<noteq> \<bottom> \<and> ys \<noteq> \<bottom> \<and> xs \<noteq> ys)"
mono_def: fixes f :: "'a \<Rightarrow> 'a" shows "preord_class.mono f = (\<forall>x y. x \<sqsubseteq> y \<longrightarrow> f x \<sqsubseteq> f y)"
hyps_1: fixes y :: "nat" assumes "y < i" shows "local.bad (local.worst_chain (Suc y))"
assms_3: fixes x :: "'a" assumes "x \<in> cbox a b" shows "norm (f x) \<le> B"
map_upds_apply_eq_Some: fixes m :: "'b \<Rightarrow> 'a option"   and xs :: "'b list"   and ys :: "'a list"   and x :: "'b"   and y :: "'a" shows "((m(xs [\<mapsto>] ys)) x = \<lfloor>y\<rfloor>) = (let xs' = take (length ys) xs in if x \<in> set xs' then ys ! last_index xs' x = y else m x = \<lfloor>y\<rfloor>)"
assms_1: shows "well_com (Measure x1 x2a x3a)"
unidx_idx: fixes x :: "'a" assumes "x \<in> agents" shows "unidx (idx x) = x"
list_rel_pres_length: fixes x' :: "'a list"   and x :: "'b list"   and A :: "('a \<times> 'b) set" assumes "(x', x) \<in> \<langle>A\<rangle>list_rel" shows "length x' = length x"
FAss_\<tau>red1t_xt1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and F :: "String.literal"   and D :: "String.literal"   and e2 :: "'addr expr1" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (e\<bullet>F{D} := e2, xs) (e'\<bullet>F{D} := e2, xs')"
abs_inner_Basis: fixes u :: "'a"   and v :: "'a" assumes "\<bar>u\<bar> \<in> Basis"   and "\<bar>v\<bar> \<in> Basis" shows "u \<bullet> v = (if u = v then 1 else if u = - v then - 1 else 0)"
is_natural_1: fixes f :: "'a" assumes "A.arr f" shows "H f \<cdot>\<^sub>B \<tau> (A.dom f) = \<tau> f"
Ax_canonical: fixes k :: "'b"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" shows "Atom k, \<Gamma> \<Rightarrow> Atom k, \<Delta>"
of_int_dvd_imp_dvd_gauss_cnj: fixes n :: "int"   and z :: "gauss_int" assumes "of_int n dvd z" shows "of_int n dvd gauss_cnj z"
Keys_to_list_eq_pps_to_list: fixes ps :: "(('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.extended_ord.punit.Keys_to_list ps = pp_pm.extended_ord.punit.pps_to_list (Keys (set ps))"
istopology_subbase: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set" shows "istopology (arbitrary union_of (finite intersection_of P relative_to S))"
assms_2: fixes K :: "nat" assumes "K \<le> k"   and "P K"   and "\<And>k. k < K \<Longrightarrow> \<not> P k" shows "Q"
ideD_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.arr C a"
inj_on_minus: fixes y :: "nat"   and q :: "nat" shows "inj_on (\<lambda>b. (y + (q - b)) mod q) {..<q}"
has_domain_iff_arr: fixes f :: "'a \<times> 'a \<times> 'a" shows "(B.VVV.domains f \<noteq> {}) = B.VVV.arr f"
SigmaD2: fixes a :: "'g"   and b :: "'h"   and A :: "'g set"   and B :: "'g \<Rightarrow> 'h set" assumes "(a, b) \<in> Sigma A B" shows "b \<in> B a"
fBall_alt_def: fixes S :: "'a fset"   and P :: "'a \<Rightarrow> bool" shows "fBall S P = (\<forall>x. x |\<in>| S \<longrightarrow> P x)"
less_oSuc0: fixes x :: "ordinal" shows "(x < oSuc 0) = (x = 0)"
semiring_simprules_15: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "semiring R"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
less_eq_rexp_def: fixes r :: "'a rexp"   and s :: "'a rexp" shows "r \<le> s \<equiv> le_rexp r s"
act2: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and p :: "'d" assumes "unital_quantale_module act" shows "act (1::'c) p = p"
periodic_arg_plus_0: fixes x :: "'a" shows "g x (0::'a) = x"
assms: shows "\<exists>o\<^sub>1 y. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<omega>\<nu> y"
comp_ide_arr: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and f :: "'a" assumes "partial_magma C"   and "partial_magma.ide C b"   and "partial_magma.arr C (C b f)" shows "C b f = f"
simps_2: fixes x :: "'h"   and xs :: "'h list"   and c :: "'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and \<sigma> :: "'g" shows "foldli (x # xs) c f \<sigma> = (if c \<sigma> then foldli xs c f (f x \<sigma>) else \<sigma>)"
circ_decompose_9: shows "top = bounded_distrib_lattice_class.comp_inf.tc (bounded_distrib_lattice_class.comp_inf.tc top)"
Input: fixes a :: "name"   and x :: "name"   and P :: "pi"   and u :: "name" shows "a<x>.P \<Longrightarrow>\<^sub>lu in P\<rightarrow>a<x> \<prec> P[x::=u]"
add_ac_1: fixes a :: "'k"   and b :: "'k"   and c :: "'k" shows "a + b + c = a + (b + c)"
Upper_memI: fixes A :: "'a set"   and L :: "('a, 'b) gorder_scheme"   and x :: "'a" assumes "\<And>y. y \<in> A \<Longrightarrow> y \<sqsubseteq>\<^bsub>L\<^esub> x"   and "x \<in> carrier L" shows "x \<in> Upper L A"
finite_Int: fixes F :: "'c set"   and G :: "'c set" assumes "finite F \<or> finite G" shows "finite (F \<inter> G)"
Sup_le_iff: fixes A :: "V set"   and a :: "V" assumes "small A" shows "(\<Squnion> A \<le> a) = (\<forall>x\<in>A. x \<le> a)"
const_type_inv_wf: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and c :: "'a"   and X :: "('a, 'b) Term.term list"   and a :: "'c" assumes "typed_model arity public Ana \<Gamma>"   and "\<Gamma> (Fun c X) = Var a"   and "intruder_model.wf\<^sub>t\<^sub>r\<^sub>m arity (Fun c X)" shows "X = []"
tag_name_eq2_h: fixes doc_ptr :: "(_) element_ptr" shows "|h \<turnstile> get_tag_name doc_ptr|\<^sub>r = |h2 \<turnstile> get_tag_name doc_ptr|\<^sub>r"
T_BIT_absch: fixes n :: "nat" assumes "n \<le> length qs" shows "T_BIT n \<le> 7 / 4 * real_of_int (T_A' n) - 3 / 4 * real n"
h2'_eq: fixes z :: "complex" assumes "0 < Re z"   and "Re z < 1" shows "h2' (z - 1) = h2' z"
hyps: assumes "\<exists>g\<in>set G. gval g s = invalid" shows "gval (fold gAnd G (Bc True)) s = invalid"
monad_alt_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_alt return bind alt" shows "monad_alt (return_env return) (bind_env bind) (alt_env alt)"
atUV_union: fixes U :: "'f set"   and V :: "'f set"   and c1 :: "'f list multiset"   and c2 :: "'f list multiset" shows "atUV U V (c1 + c2) = (atUV U V c1 \<or> atUV U V c2 \<or> atU U c1 \<and> atU V c2 \<or> atU V c1 \<and> atU U c2)"
resChainI: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi"   and lst :: "name list" assumes "eqvt Rel"   and "\<And>R S y. (R, S) \<in> Rel \<Longrightarrow> (<\<nu>y>R, <\<nu>y>S) \<in> Rel"   and "P \<leadsto><Rel> Q" shows "resChain lst P \<leadsto><Rel> resChain lst Q"
get_attribute_locs_impl: shows "get_attribute_locs = a_get_attribute_locs"
inner_node_def: fixes n :: "'b LDCFG_node" shows "local.CFGExit.inner_node n \<equiv> CFGExit.valid_node n \<and> n \<noteq> NewEntry \<and> n \<noteq> NewExit"
elem_mset_set: fixes A :: "'c set"   and x :: "'c" assumes "finite A" shows "(x \<in># mset_set A) = (x \<in> A)"
U_def: shows "Multirelations.U \<equiv> {(a, A) |a A. a \<in> UNIV \<and> A \<subseteq> UNIV}"
map_ident: fixes t :: "'g list" shows "map (\<lambda>x. x) t = t"
exhaust: fixes y :: "Comparator.order" obtains "y = Eq"   | "y = Lt"   | "y = Gt"
finite_ordering_img: fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "X \<noteq> {}"   and "finite X"   and "TernaryOrdering.ordering f ord X" shows "f ` {0..card X - 1} = X"
homeomorphic_map_swap: fixes X :: "'c topology"   and Y :: "'d topology" shows "homeomorphic_map (prod_topology X Y) (prod_topology Y X) (\<lambda>(x, y). (y, x))"
bounded_iff: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
partition_on_insert: fixes A :: "'b set"   and P :: "'b set set"   and X :: "'b set"   and A' :: "'b set" assumes "partition_on A P"   and "disjnt A X"   and "X \<noteq> {}"   and "A \<union> X = A'" shows "partition_on A' (insert X P)"
upred_lattice_def: shows "\<P> = \<lparr>carrier = UNIV, eq = (=), le = (\<sqsubseteq>)\<rparr>"
satisfies_gen_restrict: fixes r :: "'k \<Rightarrow> 'v \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula"   and V :: "nat list"   and k :: "'k" shows "(satisfies_gen r \<AA> \<phi> \<and> (\<forall>x\<in>set V. restrict k (x\<^bsup>\<AA>\<^esup>k))) = satisfies_gen r \<AA> (foldr (\<lambda>x. FAnd (Restrict k x)) V \<phi>)"
aux1: fixes P :: "('a, 'b, 'c) psi" shows "bangPred P (!P)"
not_SN_on_rel_succ: fixes E :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and s :: "'a" assumes "\<not> SN_on (E\<^sup>* O R O E\<^sup>*) {s}" shows "\<exists>t u. (s, t) \<in> E\<^sup>* \<and> (t, u) \<in> R \<and> \<not> SN_on (E\<^sup>* O R O E\<^sup>*) {u}"
regular_def: fixes e :: "cars"   and ts :: "traffic"   and c :: "cars" shows "regular e ts c \<equiv> if e = c then physical_size ts c + braking_distance ts c else physical_size ts c"
higher_differentiable_on_scaleR: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real"   and n :: "nat"   and g :: "'b \<Rightarrow> 'c" assumes "higher_differentiable_on S f n"   and "higher_differentiable_on S g n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. f x *\<^sub>R g x) n"
eval_inv_character: fixes \<chi> :: "'c \<Rightarrow> complex"   and j :: "'c" shows "inv_character \<chi> j = cnj (\<chi> j)"
chine_assoc_props_1: shows "\<guillemotleft>cods.chine_assoc : Chn ((cod \<mu> \<star> cod \<nu>) \<star> cod \<pi>) \<rightarrow>\<^sub>C Chn (cod \<mu> \<star> cod \<nu> \<star> cod \<pi>)\<guillemotright>"
Ord_LeastI: fixes k :: "V"   and P :: "V \<Rightarrow> bool" assumes "Ord k"   and "P k" shows "P (LEAST i. Ord i \<and> P i)"
steps_appendD3: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "steps (xs @ [x, y])" shows "steps (xs @ [x]) \<and> E x y"
delete_conv: fixes k :: "'e"   and al :: "('e \<times> 'd) list"   and k' :: "'e" shows "map_of (delete k al) k' = ((map_of al)(k := None)) k'"
USpec_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan" shows "USpec ch \<equiv> \<exists>\<exists> mm rs. IUSpec ch mm rs"
lt_monom_mult: fixes c :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "c \<noteq> (0::'c)"   and "p \<noteq> 0" shows "punit.lt (local.punit.monom_mult c t p) = local.punit.splus t (punit.lt p)"
COMB_no_paid: fixes s :: "nat list"   and "is" :: "bool list \<times> nat list + nat list"   and q :: "nat" shows "\<forall>((free, paid), t)\<in>set_pmf (snd (COMB []) (s, is) q). paid = []"
assms_2: shows "ineq_model (\<delta> \<circ>\<^sub>s \<theta>) X F"
allocationInverseRangeDomainProperty: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> allAllocations N G" shows "a\<inverse> \<in> injections (Range a) N \<and> Range a partitions G \<and> Domain a \<subseteq> N"
bb: shows "drop (Suc l) (snd (Strat ! x)) = drop l (snd ((take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat) ! x))"
tip_in_cone: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" shows "h \<in> cone (h, U)"
3: shows "\<Union> {if u \<in> Fvars \<chi> then FvarsT t else {} |t u. (t, u) \<in> set (zip (map fst txs) us)} = \<Union> {if x \<in> Fvars \<phi> then FvarsT t else {} |t x. (t, x) \<in> set txs}"
homotopic_on_emptyI: fixes X :: "'b topology"   and P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and X' :: "'c topology" assumes "topspace X = {}"   and "P f"   and "P g" shows "homotopic_with P X X' f g"
fimage_finsert: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and B :: "'c fset" shows "f |`| finsert a B = finsert (f a) (f |`| B)"
set_tag_name_defs: fixes ptr :: "(_) element_ptr"   and tag :: "char list" shows "local.a_set_tag_name ptr tag = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr tag_name_update tag)"
bind_nonempty'': fixes f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "space M \<noteq> {}" shows "M \<bind> f = join (distr M (subprob_algebra N) f)"
rel_pmf_return_pmf1: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and M :: "'c pmf" shows "rel_pmf R (return_pmf x) M = (\<forall>a\<in>set_pmf M. R x a)"
ord_Zp_ultrametric: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp"   and "x \<oplus> y \<in> nonzero Zp" shows "min (ord_Zp x) (ord_Zp y) \<le> ord_Zp (x \<oplus> y)"
4: shows "(f \<sqinter> q) * forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e) \<le> forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
list_all2_alt_def: shows "list_all2_alt \<equiv> \<lambda>uu uua. rec_list (\<lambda>P ys. case ys of [] \<Rightarrow> True | a # list \<Rightarrow> False) (\<lambda>x xs xsa P ys. case ys of [] \<Rightarrow> False | y # ys \<Rightarrow> P x y \<and> xsa P ys) uua uu"
\<alpha>0: fixes F :: "nat \<Rightarrow> 'a fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a fract"   and G1 :: "'a poly"   and G2 :: "'a poly"   and i :: "nat" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2"   and "1 < i" shows "(subresultant_prs_locale.\<alpha> \<delta> f i = 0) = (k < i - 1)"
simps_2: fixes l :: "'a bdd"   and r :: "'a bdd"   and b :: "bool"   and bs :: "bool list" shows "bdd_lookup (Branch l r) (b # bs) = bdd_lookup (if b then r else l) bs"
proj_specific_secrets_anti_mono: fixes l :: "'a"   and t :: "('fun, 'var) Term.term"   and \<A> :: "('a strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and Sec' :: "('fun, 'var) Term.term set" assumes "proj_specific l t \<A> Sec"   and "Sec' \<subseteq> Sec" shows "proj_specific l t \<A> Sec'"
times_OrdSum_def: fixes a :: "('b, 'c) OrdSum"   and b :: "('b, 'c) OrdSum" shows "a * b \<equiv> if second a = (1::'c) \<and> second b = (1::'c) then Abs_OrdSum (first a * first b, 1::'c) else if first a = (1::'b) \<and> first b = (1::'b) then Abs_OrdSum (1::'b, second a * second b) else if first a = (1::'b) \<and> second b = (1::'c) then b else a"
assms_5: shows "i *\<^sub>R p + j *\<^sub>R q + k *\<^sub>R r = 0"
is_category_with_pullbacks: shows "category_with_pullbacks (\<cdot>)"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r" shows "b = c"
mat_multrow_def: fixes mul :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat"   and a :: "'a"   and A :: "'a mat" shows "mat_multrow_gen mul k a A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = i then mul a (A $$ (i, j)) else A $$ (i, j))"
Max_wrt_preorder: shows "Max_wrt (Pareto R) = {x \<in> alts. \<forall>y\<in>alts. Pareto R x y \<longrightarrow> Pareto R y x}"
inverse_numeral_1: shows "inverse Numeral1 = Numeral1"
is_monomial_monomial: fixes p :: "'b \<Rightarrow>\<^sub>0 'c" assumes "is_monomial p" obtains c :: "'c" and t :: "'b" where "c \<noteq> (0::'c)" and "p = monomial c t"
inj_iff_eq_0: fixes A :: "(('b, 'c) vec, 'd) vec" shows "inj ((*v) A) = (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
valofn_def: fixes x :: "('c, 'd) IEEE.float" shows "valofn x = 2 ^ IEEE.exponent x / 2 ^ bias TYPE(('c, 'd) IEEE.float) * (1 + real (fraction x) / 2 ^ LENGTH('d))"
iso_dghm_HomDom_is_arr_conv: fixes f :: "V"   and a :: "V"   and b :: "V" assumes "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>" shows "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b"
pheap_merge_pairs: fixes hs :: "'b heap list" assumes "\<forall>h\<in>set hs. pheap h" shows "pheap (merge_pairs hs)"
Image_eq_UN: fixes r :: "('e \<times> 'd) set"   and B :: "'e set" shows "r `` B = (\<Union>y\<in>B. r `` {y})"
\<phi>_props_2: shows "\<phi> \<in> allVars g"
strict_trans1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "ordering less_eq less"   and "less_eq a b"   and "less b c" shows "less a c"
matchs_some_eq_length: fixes xs :: "term list"   and ys :: "'a list"   and env :: "(name, 'a) fmap" assumes "matchs xs ys = Some env" shows "length xs = length ys"
dvd_diff_commute: 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 uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_ring divide plus minus zero times one modulo euclidean_size uminus" shows "dvd.dvd times a (minus c b) = dvd.dvd times a (minus b c)"
substitutes_on_antimono: fixes B :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set" assumes "substitutes_on B f"   and "A \<subseteq> B" shows "substitutes_on A f"
ccExp_IfThenElse: fixes scrut :: "exp"   and e1 :: "exp"   and a :: "Arity"   and e2 :: "exp" shows "\<G>\<^bsub>0\<^esub> scrut \<squnion> (\<G>\<^bsub>a\<^esub> e1 \<squnion> \<G>\<^bsub>a\<^esub> e2) \<squnion> edom (\<A>\<^bsub>0\<^esub> scrut) G\<times> (edom (\<A>\<^bsub>a\<^esub> e1) \<union> edom (\<A>\<^bsub>a\<^esub> e2)) \<sqsubseteq> \<G>\<^bsub>a\<^esub> (scrut ? e1 : e2)"
distinct_7: fixes x2 :: "nat" shows "[x2]\<^sub>1 \<noteq> \<infinity>\<^sub>1"
Derivs_pderivs: fixes s :: "'b list"   and r :: "'b rexp" shows "Derivs s (lang r) = \<Union> (lang ` pderivs s r)"
elementary_cartesian_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and trm :: "'b \<Rightarrow> 'b" shows "elementary_cartesian_category C pr0 pr1 one trm \<equiv> elementary_category_with_binary_products C pr0 pr1 \<and> elementary_category_with_terminal_object C one trm"
simps_1: fixes S :: "'b formula \<Rightarrow> 'b formula"   and k :: "'b" shows "tseytin_tran1 S (Atom k) = [Atom k \<^bold>\<leftrightarrow> S (Atom k)]"
nth_take_eq: fixes n :: "nat"   and ls :: "'a list"   and ls' :: "'a list"   and i :: "nat" assumes "take n ls = take n ls'"   and "i < n" shows "ls ! i = ls' ! i"
rev_swap: fixes xs :: "'a list"   and ys :: "'a list" shows "(rev xs = ys) = (xs = rev ys)"
constrains_Un_distrib: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<union> B co C = (A co C) \<inter> (B co C)"
simps_1: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "l_sort_aux (x # xs, ys, []) = l_sort_aux (xs, [], ys @ [x])"
sum_upto_plus1: assumes "0 \<le> x - 1" shows "sum_upto f (x - 1 + 1) = sum_upto f (x - 1) + f (Suc (nat \<lfloor>x - 1\<rfloor>))"
r2f_mono: shows "\<forall>f g. \<F> \<circ> f = \<F> \<circ> g \<longrightarrow> f = g"
elims: fixes x :: "lambda"   and y :: "prim set" assumes "prims x = y" obtains uu_ :: "nat" and vs :: "(nat \<times> char list) list" and c :: "call" where "x = Lambda uu_ vs c" and "y = primsC c"
even_def: shows "HOL_Light_Maps.even = semiring_parity_class.even"
Block_def: shows "Class = (\<lambda>a\<in>S. THE A. a \<in> A \<and> A \<in> Partition)"
BinOp: fixes G :: "prog"   and s0 :: "st"   and e1 :: "expr"   and v1 :: "val"   and s1 :: "state"   and binop :: "binop"   and e2 :: "expr"   and v2 :: "val"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e1-\<succ>v1\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>(if need_second_arg binop v1 then In1l e2 else In1r Skip)\<succ>\<rightarrow> (In1 v2, s2)" shows "G\<turnstile>Norm s0 \<midarrow>BinOp binop e1 e2-\<succ>eval_binop binop v1 v2\<rightarrow> s2"
exhaust: fixes y :: "ltkey" obtains x1 :: "agent" where "y = sharK x1"   | x2 :: "agent" where "y = publK x2"   | x3 :: "agent" where "y = privK x3"
assms: fixes i :: "'a" assumes "i \<in> Basis" shows "(\<lambda>x. f x \<bullet> i) C1_differentiable_on s"
simps_3: fixes x2 :: "real" shows "real_max_bot.R x2 \<noteq> MInfty"
137: fixes x :: "'a"   and y :: "'a" shows "(- x \<squnion> y) \<sqinter> - - x = y \<sqinter> - - x"
XBnd: shows "X \<subseteq> Bounded_vec Bnd"
exhaust: fixes y :: "'d \<times> 'e" obtains x1 :: "'d" and x2 :: "'e" where "y = (x1, x2)"
poly_rat_3: fixes x :: "rat" shows "poly_rat x \<noteq> 0"
hom_group_mul_eq: shows "((\<lambda>(x, y). x \<otimes> y) \<in> hom (subgroup_generated G A \<times>\<times> subgroup_generated G B) G) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<otimes> y = y \<otimes> x)"
relInvImage_Gr: fixes R :: "('d \<times> 'd) set"   and B :: "'d set"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "R \<subseteq> B \<times> B" shows "BNF_Greatest_Fixpoint.relInvImage A R f = BNF_Def.Gr A f O R O (BNF_Def.Gr A f)\<inverse>"
rec_halt_def: shows "rec_halt = Mn (Suc (Suc 0)) rec_nonstop"
assms_2: shows "v \<rightarrow>\<^sub>v v'"
assms_3: shows "a [^] n \<noteq> \<zero>"
sel_1: fixes x :: "gauss_int" shows "ReZ (- x) = - ReZ x"
filter_concat: fixes p :: "'a \<Rightarrow> bool"   and xs :: "'a list list" shows "filter p (concat xs) = concat (map (filter p) xs)"
a_star_CutL: fixes M :: "trm"   and M' :: "trm"   and a :: "coname"   and x :: "name"   and N :: "trm" assumes "M \<longrightarrow>\<^sub>a* M'" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>a* Cut <a>.M' x.N"
is_closed_eq_rev: fixes A :: "'a set"   and x' :: "'a"   and x :: "'a" assumes "is_closed A"   and "x' \<in> A"   and "x .= x'"   and "x \<in> carrier L" shows "x \<in> A"
proj_unl_cons_3: fixes l :: "'b"   and a :: "'a"   and A :: "('b strand_label \<times> 'a) list" shows "proj_unl l ((\<star>, a) # A) = a # proj_unl l A"
simps_2: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" shows "height \<langle>l, a, r\<rangle> = max (height l) (height r) + 1"
empty_not_insert: fixes a :: "'d"   and A :: "'d set" shows "\<not> {} = insert a A"
sinh_real_le_iff: fixes x :: "real"   and y :: "real" shows "(sinh x \<le> sinh y) = (x \<le> y)"
closed_halfspace_Im_eq: fixes b :: "real" shows "closed {z. Im z = b}"
gt_zero: fixes V :: "'b set"   and norm :: "'b \<Rightarrow> real"   and x :: "'b" assumes "normed_vectorspace V norm"   and "x \<in> V"   and "x \<noteq> (0::'b)" shows "0 < norm x"