hcju/isabelleps huggingface dataset:
asymp_equiv_mult: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<sim>[F] g1"   and "f2 \<sim>[F] g2" shows "(\<lambda>x. f1 x * f2 x) \<sim>[F] (\<lambda>x. g1 x * g2 x)"
rel_conversep: fixes R :: "'d \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_H R\<inverse>\<inverse> = (rel_H R)\<inverse>\<inverse>"
distinct_6: fixes x31 :: "expr"   and x32 :: "expr"   and x2 :: "int" shows "Add x31 x32 \<noteq> Lit x2"
monomial_decomp_split_2: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite X"   and "U \<subseteq> X"   and "finite S" shows "monomial_decomp (snd (split t U S))"
trans_NS_point: fixes x :: "('f, 'v) Term.term"   and y :: "('f, 'v) Term.term"   and z :: "('f, 'v) Term.term" assumes "(x, y) \<in> NS"   and "(y, z) \<in> NS" shows "(x, z) \<in> NS"
intros: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Bs :: "base list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and D :: "char list" assumes "class P C = \<lfloor>(Bs, rest)\<rfloor>"   and "D \<in> baseClasses Bs" shows "P \<turnstile> C \<prec>\<^sup>1 D"
subset_diff: fixes B :: "'b set"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'a measure" assumes "B \<subseteq> A"   and "finite A" shows "sup_measure.F g A = sup (sup_measure.F g (A - B)) (sup_measure.F g B)"
assms_1: shows "C\<^sub>1 \<subseteq> C\<^sub>2"
bis_imp_5: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "c1 \<approx>wT c2" shows "c1 \<approx>w c2"
b: shows "r2 = (ps, c) \<and> r2 \<in> upRules"
A'_eq: shows "A' = (fst A', fst (snd A'), snd (snd A'))"
simps: fixes g :: "igraph"   and d :: "igraph \<Rightarrow> location \<Rightarrow> ((actor \<Rightarrow> bool) \<times> action set) set" shows "graphI (Infrastructure g d) = g"
ct_suffix_extend_any1: fixes xs :: "ct list"   and ys :: "ct list" assumes "ct_suffix xs ys"   and "length xs < length ys" shows "ct_suffix (TAny # xs) ys"
general_U_ordinal_Utility: shows "ordinal_utility local.\<P> \<R> local.general_U"
id_stutter_sampler: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "stutter_sampler id \<sigma>"
D_compE\<^sub>1': fixes e :: "char list exp"   and V :: "char list"   and Vs :: "char list list" assumes "\<D> e \<lfloor>set (V # Vs)\<rfloor>"   and "fv e \<subseteq> set (V # Vs)"   and "distinct (V # Vs)" shows "\<D> (compE\<^sub>1 (V # Vs) e) \<lfloor>{..length Vs}\<rfloor>"
length_drop: fixes n :: "nat"   and xs :: "'b list" shows "length (xs \<up> n) = length xs - n"
le: fixes y :: "nat"   and x :: "nat" assumes "0 < y" shows "2 * (x mod y) \<le> x + y"
pinduct: fixes a0 :: "int"   and a1 :: "int"   and a2 :: "int"   and P :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool" assumes "tak_dom (a0, a1, a2)"   and "\<And>x y z. \<lbrakk>tak_dom (x, y, z); \<not> x \<le> y \<Longrightarrow> P (x - 1) y z; \<not> x \<le> y \<Longrightarrow> P (y - 1) z x; \<not> x \<le> y \<Longrightarrow> P (z - 1) x y; \<not> x \<le> y \<Longrightarrow> P (tak (x - 1) y z) (tak (y - 1) z x) (tak (z - 1) x y)\<rbrakk> \<Longrightarrow> P x y z" shows "P a0 a1 a2"
restricted_advice_sets_def: fixes \<phi> :: "'b ltln" shows "restricted_advice_sets \<phi> = List.product (subseqs (filter (\<lambda>x. x \<in> restricted_subformulas \<phi>) (subformulas\<^sub>\<mu>_list \<phi>))) (subseqs (filter (\<lambda>x. x \<in> restricted_subformulas \<phi>) (subformulas\<^sub>\<nu>_list \<phi>)))"
Suc: shows "prv (eql (pls (Num m) (Num n)) (Num (m + n)))"
height_up\<^sub>i_merge: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and l :: "'c btree"   and a :: "'c"   and r :: "'c btree"   and t :: "'c btree"   and ls :: "('c btree \<times> 'c) list"   and x :: "'c"   and rs :: "('c btree \<times> 'c) list"   and tt :: "'c btree" assumes "split split"   and "height_up\<^sub>i (Up\<^sub>i l a r) = height t" shows "height (Node (ls @ (t, x) # rs) tt) = height (Node (ls @ (l, a) # (r, x) # rs) tt)"
unit_factor_nat_def: fixes n :: "nat" shows "unit_factor n = (if n = 0 then 0 else 1)"
LocalSecretsComposition_exprChannel_s: fixes x :: "chanID"   and Secrets :: "Secrets"   and P :: "specID"   and Q :: "specID" assumes "exprChannel x (sE Secrets)"   and "\<not> ine P (sE Secrets)"   and "\<not> ine Q (sE Secrets)"   and "\<not> (x \<notin> ins P \<and> x \<notin> ins Q)" shows "False"
uncurry_apply: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and a :: "'f"   and b :: "'g" shows "uncurry f (a, b) = f a b"
class_of_1_eq: fixes U :: "'a set"   and U' :: "'a set" assumes "U \<in> {U. is_open U \<and> x \<in> U}"   and "U' \<in> {U. is_open U \<and> x \<in> U}" shows "stx.class_of U \<one>\<^bsub>U\<^esub> = stx.class_of U' \<one>\<^bsub>U'\<^esub>"
rdvd_left_0_eq: fixes t :: "real" shows "(0 rdvd t) = (t = 0)"
DomainI: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "r a b" shows "Domainp r a"
fls_deriv_X_inv: shows "fls_deriv fls_X_inv = - fls_X_inv\<^sup>2"
ntrs_modify_context_s: fixes s :: "'n list"   and c :: "'n list multiset"   and ee :: "('p, 'ba) label list"   and s' :: "'n list"   and c' :: "'n list multiset"   and cn :: "'n list multiset" assumes "((s, c), ee, s', c') \<in> ntrs fg"   and "mon_w fg ee \<inter> mon_c fg cn = {}" shows "\<exists>csp. c' = csp + c \<and> mon_c fg csp = {} \<and> ((s, cn), ee, s', csp + cn) \<in> ntrs fg"
sum_def: fixes U :: "'b set"   and V :: "'b set" shows "U + V = {u + v |u v. u \<in> U \<and> v \<in> V}"
rev_take: fixes i :: "nat"   and xs :: "'b list" shows "rev (take i xs) = drop (length xs - i) (rev xs)"
face: fixes C1 :: "'a set"   and C2 :: "'a set" assumes "C1 \<in> \<M>"   and "C2 \<in> \<M>" shows "C1 \<inter> C2 face_of C1"
real_cond_exp_nested_subalg: fixes M :: "'c measure"   and F :: "'c measure"   and G :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "subalgebra M G"   and "subalgebra G F"   and "integrable M f" shows "AE x in M. real_cond_exp M F (real_cond_exp M G f) x = real_cond_exp M F f x"
mcont: fixes f :: "'b \<Rightarrow> 'a" assumes "mcont \<Or> (\<sqsubseteq>) Sup (\<le>) f" shows "mcont \<Or> (\<sqsubseteq>) Sup (\<le>) (F f)"
insort: fixes l :: "nat"   and s :: "'a list"   and v :: "'a" assumes "Suc l < length s"   and "s ! l < v"   and "v < s ! (l + 1)"   and "sorted_wrt (<) s" shows "sorted_wrt (<) (take (Suc l) s @ v # drop (Suc l) s)"
length_rev: fixes xs :: "'g list" shows "length (rev xs) = length xs"
eval_fm_All2_Iff_elts: fixes t :: "tm"   and e :: "(name, hf) finfun"   and i :: "name"   and A :: "fm" assumes "ground t" shows "eval_fm e (All2 i t A) = (\<forall>u\<in>elts t. eval_fm e (A(i::=u)))"
lens_equiv_pres_indep': fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and Z :: "'g \<Longrightarrow> 'e" assumes "X \<approx>\<^sub>L Y"   and "Z \<bowtie> Y" shows "Z \<bowtie> X"
cosafety_lr: fixes nfv :: "nat"   and fv :: "nat set"   and sat :: "'c trace \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool"   and \<sigma> :: "'c trace"   and v :: "'d list"   and i :: "nat" assumes "cosafety_fo_spec nfv fv sat"   and "sat \<sigma> v i" shows "\<exists>\<pi>. prefix_of \<pi> \<sigma> \<and> (\<forall>\<sigma>'. prefix_of \<pi> \<sigma>' \<longrightarrow> sat \<sigma>' v i)"
drop_Cons': fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
bind_gpv_sel': fixes r :: "('d, 'b, 'c) gpv"   and f :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "the_gpv (r \<bind> f) = the_gpv r \<bind> (\<lambda>generat. if is_Pure generat then the_gpv (f (generat.result generat)) else return_spmf (IO (generat.output generat) (\<lambda>input. continuation generat input \<bind> f)))"
union_commute: fixes M :: "'d multiset"   and N :: "'d multiset" shows "M + N = N + M"
add_minus_id: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "local.add a (local.add a p\<^sub>1 p\<^sub>2) (opp p\<^sub>2) = p\<^sub>1"
insertion_lowerPoly1: fixes i :: "nat"   and p :: "'a mpoly"   and prfx :: "'a list"   and x :: "'a"   and L :: "'a list" assumes "i \<notin> vars p"   and "length prfx = i" shows "insertion (nth_default (0::'a) (prfx @ x # L)) p = insertion (nth_default (0::'a) (prfx @ L)) (lowerPoly i 1 p)"
c1_term: shows "(c1, s) \<Rightarrow> p \<Down> t"
preserves_arr: fixes f :: "'c" assumes "Cop.arr f" shows "S.arr (Y a' f)"
update_convs_2: fixes one' :: "'c \<Rightarrow> 'c"   and carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "one_update one' \<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr> = \<lparr>carrier = carrier, mult = mult, one = one' one, \<dots> = more\<rparr>"
existence_ivl_empty1: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<notin> T" shows "existence_ivl t0 x0 = {}"
y_2: shows "range (flow0 y) \<subseteq> \<omega>_limit_set x"
packet_independent_unknown_match_tacs_1: shows "packet_independent_\<alpha> in_doubt_allow"
quantale_closed_unital_maps_def: shows "quantale_closed_unital_maps = {f. (\<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y)) \<and> (1::'d) \<le> f (1::'c)}"
partIps_length: fixes s :: "'a wordinterval"   and ts :: "'a wordinterval list" shows "length (partIps s ts) \<le> length ts * 2"
list_ex_simps_2: fixes P :: "'d \<Rightarrow> bool" shows "list_ex P [] = False"
check_extension_l_no_new_var_err_impl_def: fixes p :: "'a" shows "check_extension_l_no_new_var_err_impl p = ''No new variable could be found in polynomial '' @ show p"
coreflexive_comp_inf_complement: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> top" shows "x \<sqinter> y \<sqinter> - z = x \<sqinter> y \<sqinter> - (x \<sqinter> z)"
gc_complete_Liminf: fixes Ns :: "('a literal multiset \<times> label) set llist"   and B :: "'a literal multiset" assumes "chain (\<leadsto>GC) Ns"   and "FL.active_subset (lhd Ns) = {}"   and "FL.passive_subset (Liminf_llist Ns) = {}"   and "B \<in> {{#}}"   and "fst ` lhd Ns \<TTurnstile>\<G>e {B}" shows "\<exists>BL\<in>{{#}} \<times> UNIV. BL \<in> Liminf_llist Ns"
setVar'_def: fixes gl :: "bool"   and v :: "String.literal"   and idx :: "integer option"   and val :: "integer"   and g :: "'a gState_scheme"   and p :: "pState" shows "setVar' gl v idx val g p = (if gl then if v = STR ''_'' then (g, p) else case lm.lookup v (gState.vars g) of None \<Rightarrow> abortv STR ''Unknown global variable: '' v (\<lambda>_. (g, p)) | Some x \<Rightarrow> (g\<lparr>gState.vars := lm.update v (editVar x idx val) (gState.vars g)\<rparr>, p) else case lm.lookup v (pState.vars p) of None \<Rightarrow> abortv STR ''Unknown proc variable: '' v (\<lambda>_. (g, p)) | Some x \<Rightarrow> (g, p\<lparr>pState.vars := lm.update v (editVar x idx val) (pState.vars p)\<rparr>))"
inres_SPEC: fixes M :: "'b nres"   and x :: "'b"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "inres M x"   and "M \<le> SPEC \<Phi>" shows "\<Phi> x"
deg'_pp_def: fixes x :: "('a, 'b) pp" shows "deg' x = rep_nat (deg_pp x)"
comm_monoid_axioms: shows "comm_monoid (AND) (- (1::'a))"
substitute_mono2: fixes t :: "'b ttree"   and t' :: "'b ttree"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set" assumes "paths t \<subseteq> paths t'" shows "paths (substitute f T t) \<subseteq> paths (substitute f T t')"
axioms: fixes A :: "V"   and f :: "V" assumes "unop_v11 A f" shows "nop_v11 A (1\<^sub>\<nat>) f"
on_triple_dest: fixes R :: "('f \<times> 'e) set"   and E :: "('d \<times> 'f \<times> 'f) set"   and G :: "('d \<times> 'e \<times> 'e) set"   and l :: "'d"   and x :: "'f"   and y :: "'f"   and xx :: "'e"   and yy :: "'e" assumes "on_triple R `` E \<subseteq> G"   and "(l, x, y) \<in> E"   and "(x, xx) \<in> R"   and "(y, yy) \<in> R" shows "(l, xx, yy) \<in> G"
bilinear_continuous_on_compose: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "continuous_on S f"   and "continuous_on S g"   and "bilinear h" shows "continuous_on S (\<lambda>x. h (f x) (g x))"
eq: fixes p :: "tm"   and vs :: "'b list"   and bs :: "'b list" assumes "allpolys isnpoly p" shows "Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
prob_Max_IID_geometric_atMost: fixes p :: "real"   and n :: "nat"   and i :: "nat" assumes "p \<in> {0..1}" shows "measure_pmf.prob (random_skip_list.H\<^sub>N p n) {..i} = measure_pmf.prob (geometric_pmf p) {..i} ^ n"
assms_2: assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "thesis"
w_defs_2: shows "w2 = fst (bv_chop (snd (bv_chop w (Suc l))) (Suc j))"
lve_Entry_Exit_edge: fixes e :: "'g LDCFG_node \<times> ('h, 'i, 'j, 'k) edge_kind \<times> 'g LDCFG_node"   and valid_edge :: "'l \<Rightarrow> bool"   and src :: "'l \<Rightarrow> 'g"   and trg :: "'l \<Rightarrow> 'g"   and knd :: "'l \<Rightarrow> ('h, 'i, 'j, 'k) edge_kind"   and E :: "'g"   and X :: "'g" assumes "e = (NewEntry, (\<lambda>s. False)\<^sub>\<surd>, NewExit)" shows "lift_valid_edge valid_edge src trg knd E X e"
permutes_univ: fixes p :: "'a \<Rightarrow> 'a" shows "(p permutes UNIV) = (\<forall>y. \<exists>!x. p x = y)"
TS_c': fixes x :: "nat"   and y :: "nat"   and h :: "nat list"   and qs :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "h = [] \<or> (\<exists>hs. h = [x, x] @ hs)"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])" shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs \<le> 2 * T\<^sub>p [x, y] qs (OPT2 qs [x, y]) \<and> TS_inv' (Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs) (last qs) [x, y]"
AA25: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and w :: "'b \<Rightarrow> 'd" assumes "|~ \<langle>P\<rangle>_v \<longrightarrow> \<langle>Q\<rangle>_w" shows "\<turnstile> \<diamond>\<langle>P\<rangle>_v \<longrightarrow> \<diamond>\<langle>Q\<rangle>_w"
degree_linear_factors: fixes f :: "'c \<Rightarrow> 'b"   and as :: "'c list" shows "degree (\<Prod>a\<leftarrow>as. [:f a, 1::'b:]) = length as"
nths_atLeastLessThan_0_take: fixes xs :: "'b list"   and m :: "nat" shows "nths xs {0..<m} = take m xs"
V: fixes i :: "'a" shows "V i \<le> Uinv *\<^sub>S \<top>"
DF_def: shows "DF \<equiv> rec_nat (\<lambda>i\<in>{..<0}. {}, {}, \<lambda>p. {}, N) F"
assms: shows "f \<in> mor\<^bsub>CatDom F\<^esub>"
in_keys_lookup_in_range: fixes k :: "'a"   and f :: "'a \<Rightarrow>\<^sub>0 'b" assumes "k \<in> keys f" shows "lookup f k \<in> Poly_Mapping.range f"
per_supset_rel_dom: fixes p1 :: "('a \<times> 'a) set"   and p2 :: "('a \<times> 'a) set" assumes "(p1, p2) \<in> local.per_supset_rel" shows "Domain p2 \<subseteq> Domain p1"
downset_prop: fixes x :: "'b" shows "\<down> x = {y. y \<le> x}"
intro: fixes remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>ptr shadow_root_ptr w element_ptr h h' r. \<lbrakk>ptr \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr; w \<in> remove_shadow_root_locs element_ptr shadow_root_ptr; h \<turnstile> w \<rightarrow>\<^sub>h h'; r \<in> get_child_nodes_locs ptr\<rbrakk> \<Longrightarrow> r h h'" shows "l_remove_shadow_root_get_child_nodes get_child_nodes_locs remove_shadow_root_locs"
eqpoll_iff_card: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "(A \<approx> B) = (card A = card B)"
pequiv_antisym: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<sqsubseteq> b"   and "b \<sqsubseteq> a" shows "a \<simeq> b"
WbisT_prefix: shows "WbisT \<subseteq> WretrT WbisT"
adjacentchain_obtain_proper: fixes x :: "'b set"   and y :: "'b set"   and xs :: "'b set list" assumes "x \<noteq> y"   and "adjacentchain (x # xs @ [y])" shows "\<exists>zs. order.greater_eq (set xs) (set zs) \<and> order.greater_eq (length xs) (length zs) \<and> padjacentchain (x # zs @ [y])"
lt: fixes numnoabs :: "Ferrack.num \<Rightarrow> bool"   and t :: "Ferrack.num"   and bs :: "real list" assumes "numnoabs t" shows "Ifm bs (case rsplit0 t of (x, xa) \<Rightarrow> lt x xa) = Ifm bs (Lt t) \<and> isrlfm (case rsplit0 t of (x, xa) \<Rightarrow> lt x xa)"
perm_bij: fixes pi :: "(name \<times> name) list"   and x :: "'b"   and y :: "'b" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
valid_refs_invD_4: fixes x :: "'a"   and m :: "'b"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state"   and y :: "'a" assumes "x \<in> mut_m.tso_store_refs m s"   and "(x reaches y) s"   and "valid_refs_inv s" shows "\<exists>obj. sys_heap s y = Some obj"
intros: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'BExp" shows "\<Gamma> \<turnstile>\<^sub>b e \<in> max_dom (to_total \<Gamma> ` bexp_vars e) "
le_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
enum_eq: fixes j :: "nat" assumes "0 < j"   and "j \<le> n" shows "s.enum j = t.enum j"
unstream_Cons_trans_False: fixes x :: "'e"   and g :: "('e, 'f) generator"   and s :: "'f" shows "unstream (Cons_trans x g) (False, s) = unstream g s"
keys_hom_componentD_2: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" assumes "t \<in> keys (hom_component p n)" shows "deg_pm t = n"
neq_le_trans: fixes a :: "'c list"   and b :: "'c list" assumes "a \<noteq> b"   and "suffix a b" shows "strict_suffix a b"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c"   and order :: "order" shows "P (case order of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = (\<not> (order = Eq \<and> \<not> P f1 \<or> order = Lt \<and> \<not> P f2 \<or> order = Gt \<and> \<not> P f3))"
tfr_subset_3: fixes B :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set"   and A :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term set" assumes "tfr\<^sub>s\<^sub>e\<^sub>t B"   and "SMP A \<subseteq> SMP B" shows "tfr\<^sub>s\<^sub>e\<^sub>t A"
CallRec: fixes P :: "'e set"   and p :: "'f"   and Q :: "'e set"   and A :: "'e set"   and Specs :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and \<Gamma> :: "'f \<Rightarrow> ('e, 'f, 'g) com option"   and \<Theta> :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and F :: "'g set" assumes "(P, p, Q, A) \<in> Specs"   and "\<forall>(P, p, Q, A)\<in>Specs. p \<in> dom \<Gamma> \<and> \<Gamma>,\<Theta> \<union> Specs\<turnstile>\<^bsub>/F \<^esub>P the (\<Gamma> p) Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Call p Q,A"
type_less_x_OrderedSet: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> < OrderedSet \<sigma>" obtains \<rho> :: "'a OCL_Types.type" where "\<tau> = OrderedSet \<rho>" and "\<rho> < \<sigma>"
A_diff_ge: fixes \<alpha> :: "real"   and x :: "real" assumes "\<alpha> \<in> {0<..<1}"   and "1 / \<alpha> \<le> x" shows "- ln \<alpha> - 2 * M \<le> A x - A (\<alpha> * x)"
prj_chine_assoc'_3: shows "fgh.Prj\<^sub>0 \<cdot> fgh.chine_assoc' = fgh.Prj\<^sub>0\<^sub>0"
submonoid_axioms: shows "submonoid G M (\<cdot>) \<one>"
assms_2: shows "\<forall>\<^sub>F x in F. 0 \<le> f x"
h2: shows "b \<le> c \<or> c \<le> b"
simps_2: fixes uu :: "'a" shows "incr [uu] = True"
removeLeaf_val_val: fixes t :: "'c Tree" assumes "snd (removeLeaf t) \<noteq> E"   and "t \<noteq> E" shows "val t = val (snd (removeLeaf t))"
xcpt_eff_def: fixes i :: "instr"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and et :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list" shows "xcpt_eff i P pc \<tau> et = (let (ST, LT) = \<tau> in map (\<lambda>(f, t, C, h, d). (h, \<lfloor>(Class C # drop (length ST - d) ST, LT)\<rfloor>)) (relevant_entries P i pc et))"
max_reg_V_R: fixes n :: "nat" shows "max_reg (V (R n)) = Some n"
wp_under_wlp_Bind: fixes a :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>s. wp_under_wlp (a (f s))" shows "wp_under_wlp (Bind f a)"
X1: fixes a :: "'a"   and ps :: "('a \<times> 'b) list"   and m :: "'a \<Rightarrow> 'b option"   and b :: "'b" assumes "a \<notin> fst ` set ps" shows "fold (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m(a \<mapsto> b) = fold (\<lambda>(k, v) m. m(k \<mapsto> v)) ps (m(a \<mapsto> b))"
generates_max_idealI: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and X :: "'c set" assumes "ideal F \<noteq> UNIV"   and "\<And>F'. \<lbrakk>F' \<subseteq> P[X]; ideal F \<subset> ideal F'\<rbrakk> \<Longrightarrow> ideal F' = UNIV" shows "generates_max_ideal X F"
normalize_field: fixes x :: "'a" shows "normalize x = (if x = (0::'a) then 0::'a else (1::'a))"
prv_eql_subst_trm: fixes x :: "'var"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "bprv (imp (eql t1 t2) (imp (subst \<phi> t1 x) (subst \<phi> t2 x)))"
unit_is_iso: shows "local.iso (local.inv \<epsilon>\<^sub>1)"
vec_is_zero: fixes v :: "'a vec"   and n :: "nat" assumes "dim_vec v = n" shows "vec_is_zero n v = (v = 0\<^sub>v n)"
get_heap_returns_heap: fixes h :: "'a"   and f :: "'a \<Rightarrow> ('a, 'b, 'c) prog"   and h'' :: "'a" shows "h \<turnstile> get_heap \<bind> f \<rightarrow>\<^sub>h h'' = h \<turnstile> f h \<rightarrow>\<^sub>h h''"
c_drop_def: shows "c_drop = PrimRecOp (\<lambda>x. x) (\<lambda>x y z. c_tl y)"
mset_drop_upto: fixes a :: "nat"   and N :: "'c list" shows "mset (drop a N) = (!) N `# mset_set {a..<length N}"
execn_normalize_iff_execn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile> \<langle>Language.normalize c,s\<rangle> =n\<Rightarrow> t = \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
hyps_1: shows "\<Gamma>\<turnstile> (c\<^sub>1, Abrupt s) \<rightarrow>\<^sup>* (Skip, Abrupt s)"
inv_naturality: fixes f :: "'c" assumes "C.arr f" shows "(F \<circ> G) f \<cdot>\<^sub>C C.inv (\<epsilon> (C.dom f)) = C.inv (\<epsilon> (C.cod f)) \<cdot>\<^sub>C C.map f"
representsE_full: fixes p :: "int poly"   and x :: "'b" assumes "p represents x" obtains "p \<noteq> 0" and "ipoly p x = (0::'b)" and "degree p \<noteq> 0"
f: shows "\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
induct: fixes P :: "lam \<Rightarrow> bool"   and lam :: "lam" assumes "\<And>name. P (Var name)"   and "\<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (lam1 \<degree> lam2)"   and "\<And>name lam. P lam \<Longrightarrow> P (Lam [name].lam)" shows "P lam"
add_eq_conv_diff: fixes a :: "'k"   and M :: "'k multiset"   and b :: "'k"   and N :: "'k multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> M = add_mset b (N - {#a#}) \<and> N = add_mset a (M - {#b#}))"
gamma2_differentiable: shows "\<forall>x\<in>{0..1}. \<gamma>2 differentiable at x"