hcju/isabelleps huggingface dataset:
decr_qf: fixes p :: "fm" assumes "bound0 p" shows "qfree (decr p)"
Image_eq_UN: fixes r :: "('b \<times> 'a) set"   and B :: "'b set" shows "r `` B = \<mu> (\<P> (\<lambda>y. r `` \<eta> y) B)"
d: shows "the (delay i x) = Suc (GREATEST j. j < e_length x \<and> 0 < the (eval r_result1 [e_length x, i, e_take (Suc j) x]))"
finite_some_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "finite (some_GB F)"
ssorted_sdrop: fixes s :: "'b stream"   and i :: "nat" assumes "ssorted s" shows "ssorted (sdrop i s)"
sum_of_Zp_cauchy_is_Zp_cauchy: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and t :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_cauchy s"   and "is_Zp_cauchy t" shows "is_Zp_cauchy (s \<oplus>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> t)"
form_of_cnf_def: fixes F :: "'a literal list list" shows "form_of_cnf F \<equiv> \<^bold>\<And>map disj_of_clause F"
Signature_axioms: shows "Signature TYPE('tp) wtFsym wtPsym"
hyps: fixes a :: "real" assumes "0 < a" shows "poly p a \<noteq> 0"
plus_of_int: fixes x :: "'a"   and n :: "int" shows "sin (x + of_int n * ((2::'a) * of_real pi)) = sin x"
simps_2: fixes f :: "'a \<Rightarrow> 'a"   and n :: "nat"   and p :: "'a"   and d :: "'a" shows "iter f (Suc n) p d = (if f p = p then p else iter f n (f p) d)"
analytic_on_ident: fixes S :: "complex set" shows "(\<lambda>x. x) analytic_on S"
assms: shows "WHILE\<^sub>T b f s0 \<noteq> FAIL"
length_take: fixes n :: "nat"   and xs :: "'a list" shows "|take n xs| = min |xs| n"
size_index_conv: fixes xs :: "'a list"   and n :: "nat"   and x :: "'a" assumes "length xs = n" shows "(n = index xs x) = (x \<notin> set xs)"
counit_is_iso: shows "local.iso \<epsilon>\<^sub>1"
inj_on_insert': fixes A :: "'b set set"   and x :: "'b" assumes "\<And>B. B \<in> A \<Longrightarrow> x \<notin> B" shows "inj_on (insert x) A"
type_definition_bcontfun: shows "type_definition apply_bcontfun Bcontfun bcontfun"
ext_compat_list_strong_def: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" shows "ext_compat_list_strong ext \<equiv> ext_compat_cons ext \<and> ext_compat_snoc ext \<and> ext_singleton ext"
stut_not: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "STUTINV F" shows "STUTINV \<not> F"
path_connected_space_def: fixes X :: "'c topology" shows "path_connected_space X \<equiv> \<forall>x\<in>topspace X. \<forall>y\<in>topspace X. \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
permutation_Ex_bij: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs <~~> ys" shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! f i)"
assms_3: shows "fst ` (set gs \<union> set bs) \<subseteq> dgrad_p_set d m"
init_induct: fixes dummy :: "'d"   and phi1 :: "('d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) \<Rightarrow> bool"   and phi2 :: "('d IIT \<Rightarrow> ((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) \<Rightarrow> bool" assumes "closed dummy phi1 phi2" shows "(\<forall>x\<in>car_init1 dummy. phi1 x) \<and> (\<forall>x\<in>car_init2 dummy. phi2 x)"
CondFalse: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "s \<notin> b" shows "\<Gamma>\<turnstile> (Cond b c\<^sub>1 c\<^sub>2 # cs, css, Normal s) \<rightarrow> (c\<^sub>2 # cs, css, Normal s)"
measurable_envelopeD_2: fixes M :: "'b measure"   and A :: "'b set"   and E :: "'b set" assumes "measurable_envelope M A E" shows "E \<in> sets M"
simps_5: fixes F :: "'b formula"   and G :: "'b formula" shows "depth (F \<^bold>\<or> G) = Suc (max (depth F) (depth G))"
content_dvd_1: fixes f :: "'a poly"   and g :: "'a poly" assumes "content f = (1::'a)"   and "g dvd f" shows "content g = (1::'a)"
flip_in_conv: fixes tup :: "'a \<times> 'b"   and R :: "('a \<times> 'b) set" assumes "tup \<in> R" shows "flip tup \<in> R\<inverse>"
rtrancl_path_nth: fixes R :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and xs :: "'k list"   and y :: "'k"   and i :: "nat" assumes "rtrancl_path R x xs y"   and "i < length xs" shows "R ((x # xs) ! i) (xs ! i)"
abpc: shows "a \<otimes> b = p \<otimes> c"
subset_iff: fixes A :: "'b set"   and B :: "'b set" shows "((A::'b set) \<subseteq> (B::'b set)) = (\<forall>t::'b. t \<in> A \<longrightarrow> t \<in> B)"
x_prop: shows "(\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0)"
M_elem_map_carrier: fixes M2 :: "'a set" assumes "M2 \<in> M" shows "(SOME x. x \<in> carrier G \<and> M1 #> x = M2) \<in> carrier G"
simps_18: fixes V :: "'a"   and a :: "'b"   and e :: "('a, 'a, 'b) exp" shows "syncvars (insync\<^bsub>V\<^esub> (a) e) = (syncvars e \<and> V \<notin> fv e)"
inv_is_inverse: fixes f :: "'s" assumes "S.iso f" shows "S.inverse_arrows f (S.inv f)"
nf_inres_def: fixes m :: "'b nres"   and x :: "'b" shows "nf_inres m x \<equiv> nofail m \<and> inres m x"
arcs_distinct1: fixes i :: "nat"   and xs :: "nat list"   and j :: "nat"   and a :: "nat"   and b :: "nat" assumes "i \<notin> set xs"   and "j \<notin> set xs"   and "distinct xs"   and "xs \<noteq> []"   and "(a, b) \<in> set (arcs i j xs)" shows "a \<noteq> b"
binomial_n_n: fixes n :: "nat" shows "n choose n = 1"
uminus_eq_vec: fixes v :: "'b vec"   and w :: "'b vec" shows "(- v = - w) = (v = w)"
le_Suc_ex: fixes k :: "nat"   and l :: "nat" assumes "k \<le> l" shows "\<exists>n. l = k + n"
\<R>\<^sub>1_mem_eq: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" shows "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2"
affine_hull_translation: fixes a :: "'b"   and S :: "'b set" shows "affine hull (+) a ` S = (+) a ` (affine hull S)"
cases: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and a :: "'a stream" assumes "(\<phi> until \<psi>) a" obtains xs :: "'a stream" where "a = xs__" and "\<psi> xs__"   | xs :: "'a stream" where "a = xs" and "\<phi> xs" and "(\<phi> until \<psi>) (stl xs)"
first_countable_topology_ow_axioms_def: fixes U :: "'c set"   and \<tau> :: "'c set \<Rightarrow> bool" shows "first_countable_topology_ow_axioms U \<tau> \<equiv> \<forall>x\<in>U. \<exists>B. (\<forall>i. B i \<subseteq> U \<and> x \<in> B i \<and> \<tau> (B i)) \<and> (\<forall>S. S \<subseteq> U \<and> \<tau> S \<and> x \<in> S \<longrightarrow> (\<exists>i. B i \<subseteq> S))"
obs_intrap_obs_intra_eq: fixes n :: "'node"   and S :: "'node set" shows "obs_intrap n (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> obs_intra n S)"
Limsup_lessD: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'c"   and y :: "'c" assumes "Limsup F f < y" shows "\<forall>\<^sub>F x in F. f x < y"
scale_left_imp_eq: fixes a :: "complex"   and x :: "'a"   and y :: "'a" assumes "a \<noteq> 0"   and "a *\<^sub>C x = a *\<^sub>C y" shows "x = y"
cring_simprules_6: shows "(1::'b) \<in> UNIV"
encode_problem_with_operator_interference_exclusion_has_model_then_also_partial_encodings_1: 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> \<Phi>\<^sub>I \<Pi>"
relpow_imp_rtrancl: fixes p :: "'c \<times> 'c"   and n :: "nat"   and R :: "('c \<times> 'c) set" assumes "p \<in> R ^^ n" shows "p \<in> R\<^sup>*"
emeasure_subadditive: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M (A \<union> B) \<le> emeasure M A + emeasure M B"
B_A_step: fixes a' :: "'b"   and b' :: "'b"   and a :: "'a" assumes "B a' b'"   and "a \<sim> a'"   and "PA a"   and "PB a'" shows "\<exists>b. A a b \<and> b \<sim> b'"
lenh: shows "0 < length (fst (factorize_polys qs))"
NegN: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and p :: "('c, 'b) fm" assumes "Hintikka A H"   and "(\<^bold>\<not> \<^bold>\<not> p) at i in' H" shows "p at i in' H"
vector_component_2: fixes x :: "('b, 'c) Finite_Cartesian_Product.vec"   and y :: "('b, 'c) Finite_Cartesian_Product.vec"   and i :: "'c" shows "(x + y) $h i = x $h i + y $h i"
bounded_generate_check_axioms_def: fixes C\<^sub>2 :: "nat list \<Rightarrow> nat \<Rightarrow> bool"   and C\<^sub>1 :: "nat list \<Rightarrow> nat list \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> bool"   and B\<^sub>1 :: "nat list \<Rightarrow> nat" shows "bounded_generate_check_axioms C\<^sub>2 C\<^sub>1 B\<^sub>1 \<equiv> \<forall>b ys. ys \<in> fst ` set (bounded_gen_check.gen_check C\<^sub>2 b) \<longrightarrow> bounded_gen_check (C\<^sub>1 b ys) (B\<^sub>1 b)"
it_to_it_fold: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list"   and it' :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b" assumes "proper_it it it'" shows "it_to_it it = it'"
leof_fun_conv_le: fixes f :: "'c \<Rightarrow> 'b nres"   and x :: "'c"   and M :: "'c \<Rightarrow> 'b nres" shows "(f x \<le>\<^sub>n M x) = (f x \<le> (if nofail (f x) then M x else FAIL))"
gcd_eq_1_imp_coprime: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "gcd a b = one" shows "algebraic_semidom.coprime times one a b"
RS3_NMT: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "DenyAll \<in> set p" shows "rm_MT_rules Cp p \<noteq> []"
case_prod_eta: fixes f :: "'a \<times> 'b \<Rightarrow> 'c" shows "(\<lambda>(x, y). f (x, y)) = f"
cauchy_filter_def: fixes F :: "'a filter" shows "cauchy_filter F = (F \<times>\<^sub>F F \<le> uniformity)"
prod_set_simps_1: fixes x :: "'c"   and y :: "'d" shows "Basic_BNFs.fsts (x, y) = {x}"
semiring_normalization_rules_4: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "plus m m = times (plus one one) m"
AbGroup_inner_dirsum_el_decomp_ex1: fixes Gs :: "'a set list" assumes "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs" shows "\<forall>x\<in>\<Oplus>G\<leftarrow>Gs. G. \<exists>!gs. gs \<in> listset Gs \<and> x = sum_list gs"
simps_1: fixes nodesL :: "'b list"   and edgesL :: "('b \<times> 'b) list"   and more :: "'c" shows "nodesL \<lparr>nodesL = nodesL, edgesL = edgesL, \<dots> = more\<rparr> = nodesL"
tr_par_vars_disj: fixes A' :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and D :: "('lbl strand_label \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "A' \<in> set (tr\<^sub>p\<^sub>c A D)"   and "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel D) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"   and "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}" shows "fv\<^sub>l\<^sub>s\<^sub>t A' \<inter> bvars\<^sub>l\<^sub>s\<^sub>t A' = {}"
simps_16: fixes V :: "'a"   and T :: "ty"   and vo :: "'c val option"   and e :: "('a, 'b, 'c) exp" shows "max_vars {V:T=vo; e} = max_vars e + 1"
CollectE: fixes a :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "a \<in> {x. P x}" obtains "P a"
uint_nat: fixes w :: "'a word" shows "uint w = int (unat w)"
col_dim: fixes A :: "'b mat"   and i :: "nat" shows "col A i \<in> carrier_vec (dim_row A)"
eq: shows "(\<Sum>i = 0..<n. monomial k x) = monomial (k * n) x"
zmod_eq_diff_mod_0_conv: fixes a :: "int"   and m :: "int"   and b :: "int" shows "(a mod m = b mod m) = ((b - a) mod m = 0)"
extr_empty_chan: fixes bad :: "agent set"   and IK :: "msg set" shows "extr bad IK {} = IK"
exact_seq_solvable_recip: fixes G3 :: "'b monoid"   and G2 :: "'b monoid"   and G1 :: "'b monoid"   and g1 :: "'b \<Rightarrow> 'b"   and g2 :: "'b \<Rightarrow> 'b" assumes "exact_seq ([G1], [])  \<longlongrightarrow>\<^bsub>g1\<^esub> G2  \<longlongrightarrow>\<^bsub>g2\<^esub> G3"   and "inj_on g1 (carrier G1)"   and "g2 ` carrier G2 = carrier G3"   and "solvable G1 \<and> solvable G3" shows "solvable G2"
assms_5: fixes x :: "'a" assumes "x \<in> A" shows "f x = f' x"
k_invariantTD: fixes ec :: "'rep"   and A :: "('ma, 'mt) AlgState"   and obs :: "'obs" assumes "k_isNode ec"   and "k_memb ec A"   and "obs \<in> simObs a ` set (k_succs ec)"   and "k_invariant A" shows "\<exists>ec'. lookup tOps (aTrans A) (ec, obs) = Some ec' \<and> simAbs ec' \<in> simAbs ` set (k_succs ec) \<and> simObs a ec' = obs"
wary_\<rho>: shows "wary_subst \<rho>"
inv_BijGroup: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set" assumes "f \<in> Bij S" shows "inv\<^bsub>BijGroup S\<^esub> f = restrict (inv_into S f) S"
insert_body_insert': fixes xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool" shows "insert' xs e None = Some (insert_body xs e)"
uminus_perm_def: fixes p :: "'b perm" shows "- p = Abs_perm (inv (Rep_perm p))"
DSourcesA2_L0: shows "DSources level0 sA2 = {sA1, sA4}"
lessE: fixes a :: "'k"   and b :: "'k" assumes "a < b" obtains c :: "'k" where "b = a + c" and "c \<noteq> (0::'k)"
axioms_1: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<lhd> G" shows "subgroup H G"
BoundOutputS_def: fixes x1 :: "name" shows "BoundOutputS x1 \<equiv> Abs_subject (BoundOutputS_Rep x1)"
zip_same: fixes a :: "'b"   and b :: "'b"   and xs :: "'b list" shows "((a, b) \<in> set (zip xs xs)) = (a \<in> set xs \<and> a = b)"
nat_term_compare_inv_def: fixes to :: "'a nat_term_order" shows "nat_term_compare_inv to = (\<lambda>x y. nat_term_compare to y x)"
zero_neq_one: fixes one :: "'g"   and zero :: "'g" assumes "class.zero_neq_one one zero" shows "zero \<noteq> one"
connected_Ioo: fixes a :: "'a"   and b :: "'a" shows "connected {a<..<b}"
map_le_empty2: fixes m :: "'c \<Rightarrow> 'd option" shows "(m \<subseteq>\<^sub>m Map.empty) = (m = Map.empty)"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "B.seq (G g) (G f)"
FOREACHcd_def: fixes S :: "'d set"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma> :: "'c" shows "FOREACHcd S c f \<sigma> \<equiv> ASSERT (finite S) \<bind> (\<lambda>_. SPEC (\<lambda>l. set l = S) \<bind> (\<lambda>l. nfoldli l c f \<sigma>))"
spr_ii_simVal_def: shows "spr_ii_simVal \<equiv> envVal \<circ> snd"
ma_normalize_def: fixes x :: "rat \<times> rat \<times> nat" shows "ma_normalize x \<equiv> case x of (a, b, c) \<Rightarrow> if b = 0 then (a, 0, 0) else (a, b, c)"
mat_col_first_carrier: fixes A :: "'b mat"   and n :: "nat" shows "mat_col_first A n \<in> carrier_mat (dim_row A) n"
elims: fixes x :: "'a tree23s"   and y :: "nat" assumes "len x = y" obtains uu_ :: "'a tree23" where "x = T uu_" and "y = 1"   | uv_ :: "'a tree23" and uw_ :: "'a" and ts :: "'a tree23s" where "x = TTs uv_ uw_ ts" and "y = len ts + 1"
size_list_def: shows "Context_Free_Grammar_Example.size_list = length"
TransBound_subst: fixes a :: "'c"   and e :: "exp"   and x :: "Vars.var"   and y :: "Vars.var" shows "(trans a e)[x::=y] = trans a e[x::=y]"
irrefl: fixes a :: "nat list" shows "\<not> a <\<^sub>v a"
\<epsilon>_map_simp: fixes x :: "'c" assumes "C.ide x" shows "transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) \<eta>\<epsilon>.FG.map \<epsilon>o x = \<psi> x (G x)"
x1_def: shows "x1 \<equiv> (- (v\<^sub>o - v\<^sub>e) + sqrt D2) / (2 * ((a\<^sub>o - a\<^sub>e) / 2))"
mono_D_syn: fixes S :: "'c process"   and A :: "'c set"   and Q :: "'c process" shows "D (S \<lbrakk>A\<rbrakk> Q) = D (Q \<lbrakk>A\<rbrakk> S)"
antipar_2: shows "S.src (S.UP g) = S.trg (S.UP f)"
in_measure: fixes x :: "'e"   and y :: "'e"   and f :: "'e \<Rightarrow> nat" shows "((x, y) \<in> measure f) = (f x < f y)"
assms_2: shows "vcard \<SS> = 2\<^sub>\<nat>"
tj_to_tSi: shows "can_occur (t ! j) (S t (Suc i))"
MGF_asm: fixes G :: "prog"   and A :: "state triple set"   and t :: "Term.term" assumes "\<forall>C sig. is_methd G C sig \<longrightarrow> G,A\<turnstile>{\<doteq>} In1l (Methd C sig)\<succ> {G\<rightarrow>}"   and "wf_prog G" shows "G,A\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}"
least_A_le_least_H: shows "(LEAST n. A $ ia $ n \<noteq> (0::'a)) \<le> (LEAST n. H $ ia $ n \<noteq> (0::'a))"
dither_halts_aux: fixes m :: "nat" shows "steps0 (1, Bk \<up> m, [Oc, Oc]) dither 2 = (0, Bk \<up> m, [Oc, Oc])"
pran_comp: fixes g :: "('c, 'a) pfun"   and f :: "('b, 'c) pfun" shows "pran (g \<circ>\<^sub>p f) = pran (pran f \<lhd>\<^sub>p g)"
InitNone\<^sub>1: fixes sh :: "sheap"   and C :: "char list"   and P :: "nat exp prog"   and C' :: "char list"   and Cs :: "char list list"   and e :: "nat exp"   and h :: "heap"   and l :: "val list"   and e' :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap" assumes "sh C = None"   and "P \<turnstile>\<^sub>1 \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh(C \<mapsto> (sblank P C, Prepared)))\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
diff_le_mono: fixes m :: "nat"   and n :: "nat"   and l :: "nat" assumes "m \<le> n" shows "m - l \<le> n - l"
Rep: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and x :: "'d" assumes "type_definition Rep Abs A" shows "Rep x \<in> A"
Pow_insert: fixes a :: "'a"   and A :: "'a set" shows "Pow (insert a A) = Pow A \<union> insert a ` Pow A"
wt_s\<^sub>1: shows "wtc c pc (wtl (take pc ins) c 0 s\<^sub>0) \<noteq> \<top>"
is_prefix_length: fixes a :: "'a list"   and b :: "'a list" assumes "is_prefix a b" shows "length a \<le> length b"
Suc: shows "ffact n (x + y) = (\<Sum>k = 0..n. of_nat (n choose k) * ffact k x * ffact (n - k) y)"
extensional_proj: fixes P :: "'d \<Rightarrow>\<^sub>F 'e" shows "(P)\<^sub>F \<in> extensional (domain P)"
has_integral_Beta3: fixes a :: "real"   and b :: "real" assumes "a < - 1 / 2"   and "0 < b" shows "((\<lambda>x. (b + x\<^sup>2) powr a) has_integral Beta (- a - 1 / 2) (1 / 2) / 2 * b powr (a + 1 / 2)) {0<..}"
NSCauchy_Cauchy_iff: fixes X :: "nat \<Rightarrow> 'a" shows "NSCauchy X = Cauchy X"
that: fixes t :: "'a list" assumes "x \<cdot> t = u"   and "t \<cdot> v = y" shows "thesis"
necessity_D': assumes "\<forall>A. admits_diagonal_reduction A" shows "\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})"
map_pmf_rep_eq: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d pmf" shows "measure_pmf (map_pmf f M) = distr (measure_pmf M) (count_space UNIV) f"
subset_Diff_insert: fixes A :: "'e set"   and B :: "'e set"   and x :: "'e"   and C :: "'e set" shows "(A \<subseteq> B - insert x C) = (A \<subseteq> B - C \<and> x \<notin> A)"
elims_2: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'b list" assumes "binrelchain x xa" obtains P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" where "x = P__" and "xa = []"   | P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xb :: "'b" where "x = Pa__" and "xa = [xb__]"   | P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and xb :: "'b" and y :: "'b" and xs :: "'b list" where "x = P" and "xa = xb # y # xs" and "P xb y \<and> binrelchain P (y # xs)"
simps_8: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'a" shows "(case \<turnstile> of \<turnstile> \<Rightarrow> f1 | \<lbrakk>x\<rbrakk> \<Rightarrow> f2 x | \<stileturn> \<Rightarrow> f3) = f1"
odd_pow: fixes n :: "nat"   and P :: "pol" assumes "odd n" shows "Commutative_Ring.pow n P = P \<langle>*\<rangle> Commutative_Ring.pow (n div 2) (sqr P)"
char_mult: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and a :: "'c"   and b :: "'c" assumes "character G \<chi>"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "\<chi> (a \<otimes>\<^bsub>G\<^esub> b) = \<chi> a * \<chi> b"
limit_is_idempotent: fixes f :: "'a set \<Rightarrow> 'a set"   and I :: "'a set" assumes "regular f" shows "limit f (limit f I) = limit f I"
sup_ge2: fixes y :: "'a multiset"   and x :: "'a multiset" shows "y \<subseteq># x \<union># y"
effective_redundancy_criterion_def: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" shows "effective_redundancy_criterion \<Gamma> Rf Ri \<equiv> redundancy_criterion \<Gamma> Rf Ri \<and> effective_redundancy_criterion_axioms \<Gamma> Rf Ri"
b': shows "\<forall>r'\<in>R. leftPrincipal r' (Compound F Fs) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* Em) \<in> set (fst r')"
emptiness_preservance_steps: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and \<R> :: "('e \<Rightarrow> real) set set"   and l :: "'d"   and R :: "('e \<Rightarrow> real) set"   and l' :: "'d"   and R' :: "('e \<Rightarrow> real) set" assumes "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle>"   and "R = {}" shows "R' = {}"
alw_cong: fixes P :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream"   and Q1 :: "'b stream \<Rightarrow> bool"   and Q2 :: "'b stream \<Rightarrow> bool" assumes "alw P \<omega>"   and "\<And>\<omega>. P \<omega> \<Longrightarrow> Q1 \<omega> = Q2 \<omega>" shows "alw Q1 \<omega> = alw Q2 \<omega>"
ResComm: fixes x :: "name"   and y :: "name"   and P :: "pi" shows "<\<nu>x><\<nu>y>P \<equiv>\<^sub>e <\<nu>y><\<nu>x>P"
undeclared_in_def: fixes G :: "prog"   and m :: "memberid"   and C :: "qtname" shows "G\<turnstile> m undeclared_in C = (case m of fid fn \<Rightarrow> cdeclaredfield G C fn = None | mid sig \<Rightarrow> cdeclaredmethd G C sig = None)"
alpha1: shows "(qbinp, qbinp') \<in> alphaBinp"
subgroupE_1: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := S\<rparr>)" shows "H \<subseteq> carrier (G\<lparr>carrier := S\<rparr>)"
Lim_transform_eventually: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "(f \<longlongrightarrow> l) F"   and "\<forall>\<^sub>F x in F. f x = g x" shows "(g \<longlongrightarrow> l) F"
a1: shows "s' = snd (fst (write_cpu (update_PSR_icc n_val z_val v_val c_val (cpu_reg_val PSR s)) PSR s)) \<and> get_S (cpu_reg_val PSR s) = 0"