hcju/isabelleps huggingface dataset:
Union: fixes a :: "nat \<Rightarrow> 'd set"   and sp :: "'d set"   and A :: "'d set set" assumes "\<And>i. a i \<in> sigma_sets sp A" shows "\<Union> (range a) \<in> sigma_sets sp A"
thin: fixes i :: "nat" shows "thin_set (f -` {i} \<inter> \<F>)"
nth_append: fixes xs :: "'a list"   and ys :: "'a list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < |xs| then xs ! n else ys ! (n - |xs|))"
sees_field_def: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" shows "P \<turnstile> C sees F,b:T in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of (map (\<lambda>((F, D), b, T). (F, D, b, T)) FDTs) F = \<lfloor>(D, b, T)\<rfloor>"
continuous_on_map_topology: fixes T :: "'c topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map T (map_topology f T) f"
inj: shows "inj ((*) (1 / pi) \<circ> (\<lambda>x. if - 1 \<le> x \<and> x \<le> 1 then arccos x else if x < - 1 then - x + pi else 1 - x) \<circ> (\<lambda>t. 2 * x_coord (1 - t)))"
add_le_cancel_left: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and c :: "'g"   and a :: "'g"   and b :: "'g" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus c a) (plus c b) = less_eq a b"
append_natpermute_less_eq_1: fixes xs :: "nat list"   and ys :: "nat list"   and n :: "nat"   and k :: "nat" assumes "xs @ ys \<in> natpermute n k" shows "sum_list xs \<le> n"
"3_3": shows "C (C (C (m # ms) # ms') # ns) \<in> hbase_ext (Suc c)"
simps_2: fixes prec :: "nat"   and f :: "form"   and s :: "nat"   and n :: "nat"   and ivl :: "float interval"   and bs :: "float interval option list"   and ss :: "nat list" shows "approx_form' prec f (Suc s) n ivl bs ss = (let (ivl1, ivl2) = split_float_interval ivl in if approx_form' prec f s n ivl1 bs ss then approx_form' prec f s n ivl2 bs ss else False)"
next_plane0_len_filter_eq: fixes g' :: "graph"   and p :: "nat"   and g :: "graph"   and v :: "nat"   and P :: "face \<Rightarrow> bool" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "Invariants.inv g"   and "v \<in> \<V> g"   and "finalVertex g v" shows "|filter P (facesAt g' v)| = |filter P (facesAt g v)|"
sinpoly_def: fixes n :: "nat" shows "sinpoly n = (\<lambda>x. \<Sum>k<n. sin_coeff k * x ^ k)"
IdNatTransCompDefCod: fixes f :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" assumes "NatTrans f" shows "f \<approx>>\<bullet> IdNatTrans (NTCod f)"
eq_\<I>_gpv_2: shows "\<I> \<turnstile>g gpv \<surd>"
char_to_digit_simps_3: shows "char_to_digit CHR ''2'' = 2"
AnalBinds_not_there: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and exp :: "exp \<Rightarrow> 'c \<rightarrow> 'b"   and ae :: "Vars.var \<Rightarrow> 'c\<^sub>\<bottom>" assumes "x \<notin> domA \<Gamma>" shows "(ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae) x = \<bottom>"
nfa_is_node_def: fixes A :: "bool list bdd list \<times> bool list" shows "nfa_is_node A = (\<lambda>qs. length qs = length (fst A))"
odlist_all_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c odlist" shows "odlist_all P xs = (\<forall>x\<in>toSet xs. P x)"
comp: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "map f \<circ> map g = map (f \<circ> g)"
common_summand_in_product: fixes x :: "i"   and y :: "i"   and z :: "i" shows "P x ((x \<oplus> y) \<otimes> (x \<oplus> z))"
that: fixes t :: "real" assumes "\<bar>t\<bar> < 1 / 2"   and "(1 + x) powr p = 1 + p * x + p * (p - 1) * (1 + t) powr (p - 2) / 2 * x\<^sup>2" shows "thesis"
gfp_fixpoint: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "f (gfp f) = gfp f"
list_all_length: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list" shows "list_all P xs = (\<forall>n<length xs. P (xs ! n))"
range_enum: fixes N :: "nat set" assumes "infinite N" shows "range (enum N) = N"
borders_compare: fixes x :: "'b list"   and w :: "'b list"   and x' :: "'b list" assumes "x \<le>b w"   and "x' \<le>b w"   and "\<^bold>|x'\<^bold>| < \<^bold>|x\<^bold>|" shows "x' \<le>b x"
wt_elim_cases_3: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and i :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In2 (e.[i])\<Colon>T" obtains T :: "ty" where "Ta__ = Inl T" and "E,dt\<Turnstile>e\<Colon>-T.[]" and "E,dt\<Turnstile>i\<Colon>-PrimT Integer"
convex_sum: fixes S :: "'c set"   and C :: "'d set"   and a :: "'c \<Rightarrow> real"   and y :: "'c \<Rightarrow> 'd" assumes "finite S"   and "convex C"   and "sum a S = 1"   and "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i"   and "\<And>i. i \<in> S \<Longrightarrow> y i \<in> C" shows "(\<Sum>j\<in>S. a j *\<^sub>R y j) \<in> C"
less_asym': fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and P :: "bool" assumes "a \<prec> b"   and "b \<prec> a" shows "P"
shows_list_literal_def: fixes ss :: "String.literal list" shows "shows_list ss = shows_string (concat (map literal.explode ss))"
sqrt_approx_def: fixes \<epsilon> :: "'a"   and x :: "'a" shows "sqrt_approx \<epsilon> x \<equiv> if (0::'a) < \<epsilon> then if x = (0::'a) then 0::'a else let xpos = \<bar>x\<bar> in sqrt_approx_main_impl \<epsilon> xpos (xpos + (1::'a)) else (0::'a)"
HTsetI: assumes "wf (inv_image R (f \<circ> snd))"   and "\<And>Pa ca Qa s\<^sub>0. \<lbrakk>HTset_r (\<lambda>c' s'. ((c', s'), ca, s\<^sub>0) \<in> inv_image R (f \<circ> snd)) \<pi> {(P, c, Q)}; (Pa, ca, Qa) \<in> {(P, c, Q)}; Pa s\<^sub>0\<rbrakk> \<Longrightarrow> wp \<pi> ca (Qa s\<^sub>0) s\<^sub>0" shows "HTset \<pi> {(P, c, Q)}"
integral_join: fixes f :: "'c \<Rightarrow> real"   and N :: "'c measure"   and B :: "real"   and M :: "'c measure measure"   and B' :: "real" assumes "f \<in> borel_measurable N"   and "\<And>x. x \<in> space N \<Longrightarrow> \<bar>f x\<bar> \<le> B"   and "sets M = sets (subprob_algebra N)"   and "finite_measure M"   and "AE M' in M. emeasure M' (space M') \<le> ennreal B'" shows "integral\<^sup>L (join M) f = LINT M'|M. integral\<^sup>L M' f"
simps: fixes a1 :: "'a"   and a2 :: "'a"   and r :: "('a \<times> 'a) set" shows "((a1, a2) \<in> r\<^sup>+) = ((\<exists>a b. a1 = a \<and> a2 = b \<and> (a, b) \<in> r) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>+ \<and> (b, c) \<in> r))"
S_reachable: shows "S (A (Q' Q \<pi> u) (\<pi>' Q \<pi> u)) \<subseteq> nodes rg"
measure_diff_le_measure_setdiff: fixes S :: "'a set"   and M :: "'a measure"   and T :: "'a set" assumes "S \<in> fmeasurable M"   and "T \<in> fmeasurable M" shows "Sigma_Algebra.measure M S - Sigma_Algebra.measure M T \<le> Sigma_Algebra.measure M (S - T)"
BodyN: fixes P :: "'b assn"   and pn :: "pname"   and Q :: "'b assn"   and G :: "'b triple set" assumes "insert ({P}. BODY pn .{Q}) G|-{P}. the (body pn) .{Q}" shows "G|-{P}. BODY pn .{Q}"
g: shows "g = fpxs_compose_power (fpxs_of_fls g') (1 / rat_of_nat s)"
Un_commute: fixes A :: "'x set"   and B :: "'x set" shows "A \<union> B = B \<union> A"
assms_2: shows "\<forall>\<^sub>F x in at_top. f x \<le> u x"
dual_category_def: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag" shows "dual_category C \<equiv> category C"
primepow_prime_power: fixes p :: "'a"   and n :: "nat" assumes "prime p" shows "primepow (p ^ n) = (0 < n)"
Union_of_finite_supp_sets: fixes S :: "'e set" assumes "finite S" shows "finite (\<Union> (supp ` S))"
div_right_ineq: fixes x :: "nat"   and y :: "nat"   and z :: "nat" assumes "x < y * z" shows "x div y < z"
simps_1: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d"   and q :: "'d" shows "app_pair f (Inl (p, q)) = Inl (f p, f q)"
group_sizes_def: shows "group_sizes \<equiv> {card G |G. G \<in> \<G>}"
indecomposable_ordertype_eq: fixes \<alpha> :: "V"   and A :: "V set"   and B :: "V set" assumes "indecomposable \<alpha>"   and "tp A = \<alpha>"   and "B \<subseteq> A"   and "small A" shows "tp B = \<alpha> \<or> tp (A - B) = \<alpha>"
inputBoundOutputTrans: fixes a :: "name"   and x :: "name"   and P :: "pi"   and b :: "name"   and y :: "name"   and P' :: "pi" assumes "a<x>.P \<longmapsto> b<\<nu>y> \<prec> P'" shows "False"
qGood_qTerm_induct: fixes phi :: "('c, 'd, 'varSort, 'var, 'e) qTerm \<Rightarrow> bool"   and phiAbs :: "('c, 'd, 'varSort, 'var, 'e) qAbs \<Rightarrow> bool"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and A :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>|{i. inp i \<noteq> None}| <o |UNIV|; |{i. binp i \<noteq> None}| <o |UNIV|; liftAll (\<lambda>X. qGood X \<and> phi X) inp; liftAll (\<lambda>A. qGoodAbs A \<and> phiAbs A) binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. \<lbrakk>qGood X; \<And>Y. qGood Y \<and> qSkel Y = qSkel X \<Longrightarrow> phi Y; \<And>Y. (X, Y) \<in> qSwapped \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (qAbs xs x X)" shows "(qGood X \<longrightarrow> phi X) \<and> (qGoodAbs A \<longrightarrow> phiAbs A)"
is_path_def: fixes \<pi> :: "nat \<Rightarrow> 'n" shows "is_path \<pi> = (\<forall>n. (\<pi> n, \<pi> (Suc n)) \<in> local.edges)"
InitBlockThrow: fixes V :: "char list"   and T :: "ty"   and v :: "val"   and a :: "nat"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; Throw a},s\<rangle> \<rightarrow> \<langle>Throw a,s\<rangle>"
has_integral_integral_lebesgue: fixes f :: "'a \<Rightarrow> 'b" assumes "integrable lebesgue f" shows "(f has_integral integral\<^sup>L lebesgue f) UNIV"
is_span: shows "C.span \<mu>.leg0 \<mu>.leg1"
AssmBigAnd: fixes G :: "'b formula list"   and F :: "'b formula" shows "(set G \<turnstile> F) = ({} \<turnstile> \<^bold>\<And>G \<^bold>\<rightarrow> F)"
max_coeff_bound: fixes xs :: "nat list"   and ys :: "nat list" assumes "(xs, ys) \<in> local.Minimal_Solutions" shows "(\<forall>x\<in>set xs. x \<le> maxne0 ys b) \<and> (\<forall>y\<in>set ys. y \<le> maxne0 xs a)"
IH_2: shows "KD_Tree.size_kdt r \<le> 2 ^ height r"
that: fixes k :: "nat" assumes "f k \<le> n"   and "n < f (Suc k)" shows "thesis"
simps_7: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "Bag \<tau> \<squnion> \<sigma> = (case \<sigma> of Collection \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Set \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | OrderedSet \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | Bag \<rho> \<Rightarrow> Bag (\<tau> \<squnion> \<rho>) | Sequence \<rho> \<Rightarrow> Collection (\<tau> \<squnion> \<rho>) | _ \<Rightarrow> OclSuper)"
induct: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b"   and a1 :: "'a list"   and a2 :: "'b list" assumes "\<And>f accs. P f [] accs"   and "\<And>f a as accs. P f as (f a # accs) \<Longrightarrow> P f (a # as) accs" shows "P a0 a1 a2"
countable_UN': fixes X :: "'d set"   and A :: "'d \<Rightarrow> 'e set"   and M :: "'e measure" assumes "countable X"   and "A ` X \<subseteq> sets M" shows "\<Union> (A ` X) \<in> sets M"
lm023: fixes pair :: "'e \<times> 'f set"   and bids :: "'e \<times> 'f set \<Rightarrow> 'd" shows "snd (summedBid bids pair) = sum bids (omega pair)"
l11_41: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C"   and "Bet B A D"   and "A \<noteq> D" shows "Tarski_neutral_dimensionless.LtA Bet Cong A C B C A D \<and> Tarski_neutral_dimensionless.LtA Bet Cong A B C C A D"
error_free_set_lvars: fixes s :: "state"   and l :: "lname \<Rightarrow> val option" assumes "error_free s" shows "error_free (set_lvars l s)"
ball_eq_ball_iff: fixes x :: "'c"   and d :: "real"   and y :: "'c"   and e :: "real" shows "(ball x d = ball y e) = (d \<le> 0 \<and> e \<le> 0 \<or> x = y \<and> d = e)"
inf_float_def: fixes a :: "float"   and b :: "float" shows "inf a b = min a b"
fconverse_fcomp: fixes r :: "V"   and s :: "V" shows "(r \<circ>\<^sub>\<bullet> s)\<inverse>\<^sub>\<bullet> = s\<inverse>\<^sub>\<bullet> \<circ>\<^sub>\<bullet> r\<inverse>\<^sub>\<bullet>"
assms: shows "map_poly to_fract p = smult c (map_poly to_fract q)"
IH: fixes \<Gamma> :: "name list" shows "wellformed' (length \<Gamma>) (nterm_to_term \<Gamma> t)"
OclIf_false: fixes B\<^sub>1 :: "'a state \<times> 'a state \<Rightarrow> 'b"   and B\<^sub>2 :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(if false then B\<^sub>1 else B\<^sub>2 endif) = B\<^sub>2"
addrow_mat_carrier: fixes n :: "nat"   and a :: "'a"   and k :: "nat"   and l :: "nat" shows "addrow_mat n a k l \<in> carrier_mat n n"
unlabel_mono: fixes A :: "('a \<times> 'b) list"   and B :: "('a \<times> 'b) list" assumes "set A \<subseteq> set B" shows "set (unlabel A) \<subseteq> set (unlabel B)"
coinduct_strong: fixes R :: "'c llist \<Rightarrow> 'c llist \<Rightarrow> bool"   and llist :: "'c llist"   and llist' :: "'c llist" assumes "R llist llist'"   and "\<And>llist llist'. R llist llist' \<Longrightarrow> lnull llist = lnull llist' \<and> (\<not> lnull llist \<longrightarrow> \<not> lnull llist' \<longrightarrow> lhd llist = lhd llist' \<and> (R (ltl llist) (ltl llist') \<or> ltl llist = ltl llist'))" shows "llist = llist'"
scale_left_distrib: fixes a :: "'a \<Rightarrow>\<^sub>0 'c"   and b :: "'a \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c" shows "(a + b) \<odot> x = a \<odot> x + b \<odot> x"
bij_betw_iff_has_inverse: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "map \<alpha> S T" shows "bij_betw \<alpha> S T = (\<exists>\<beta>\<in>T \<rightarrow>\<^sub>E S. compose S \<beta> \<alpha> = identity S \<and> compose T \<alpha> \<beta> = identity T)"
assms_1: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and H :: "'a set" assumes "geodesic_segment_between G x y"   and "geodesic_segment_between H x y"   and "G \<inter> H = {x, y}" shows "x = y"
WhileFalse: fixes tst :: "'test"   and s :: "'state"   and c :: "('test, 'atom) com" assumes "\<not> tval tst s" shows "(while tst do c, s) \<rightarrow>t s"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "ord x y"   and "ord y z" shows "ord x z"
Red_F_model: fixes Inf :: "'a inference set"   and Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and M :: "'a set"   and N :: "'a set" assumes "calculus_with_standard_redundancy Inf Bot entails"   and "entails M (N - calculus_with_standard_redundancy.Red_F entails N)" shows "entails M N"
minus_mult_minus: fixes a :: "'a"   and b :: "'a" shows "- a * - b = a * b"
fmlookup_add: fixes m :: "('d, 'c) fmap"   and n :: "('d, 'c) fmap"   and x :: "'d" shows "(m ++\<^sub>f n) $$ x = (if x |\<in>| fmdom n then n $$ x else m $$ x)"
ahm_invar_impl_of: fixes hm :: "('a, 'b) hashmap" shows "ArrayHashMap_Impl.ahm_invar (impl_of hm)"
degree_trunc_pdevs_le: fixes X :: "'a pdevs"   and n :: "nat"   and p :: "nat" assumes "degree X \<le> n" shows "degree (trunc_pdevs p X) \<le> n"
inclusion: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "arr (fst f) \<and> VV.arr (snd f) \<and> src (fst f) = trg (fst (snd f))" shows "VxVxV.arr f"
distinct_group_sort: fixes f :: "'a \<Rightarrow> 'b"   and merge :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a list \<Rightarrow> 'c"   and xs :: "'a list" assumes "groupsort f merge g" shows "distinct (map f (groupsort.group_sort f merge xs))"
ENext_def: fixes p :: "PrIds" shows "ENext p = (\<exists>l. #l \<in> #MemLoc \<and> Call memCh p #(read l))"
return_node_call_of_return_node: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.return_node targetnode valid_edge get_return_edges n" shows "\<exists>!n'. CFG.call_of_return_node sourcenode targetnode valid_edge get_return_edges n n'"
assms_3: shows "\<forall>x\<in>S. \<bar>f x\<bar> \<le> b"
fs': shows "fs' = fs[i := fs ! i - c \<cdot>\<^sub>v fs ! j]"
interior_mono: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T" shows "interior S \<subseteq> interior T"
prod_invariant_def: fixes A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list" shows "prod_invariant A B = (\<lambda>(tab, ps). length tab = length (fst A) \<and> (\<forall>tab'\<in>set tab. length tab' = length (fst B)))"
prognosis_not_called: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and x :: "Vars.var"   and as :: "Arity list"   and a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "ae x = \<bottom>" shows "prognosis ae as a (\<Gamma>, e, S) \<sqsubseteq> prognosis ae as a (delete x \<Gamma>, e, S)"
prod_cong: fixes A :: "'a set"   and f :: "'a \<Rightarrow> int" assumes "finite A" shows "(\<Otimes>i\<in>A. f i mod int p) = prod f A mod int p"
LID\<^sub>SI_1: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and l :: "'b"   and v :: "('a, 'b, 'c) val" assumes "\<sigma> l = Some v" shows "l \<in> LID\<^sub>S \<sigma>"
\<gamma>_st_def: fixes \<gamma> :: "'c \<Rightarrow> 'b set"   and F :: "'c Abs_State.st" shows "\<gamma>_st \<gamma> F = {f. \<forall>x. f x \<in> \<gamma> (lookup F x)}"
Fun: fixes x2a :: "('f, 'a) Term.term" assumes "x2a \<in> set ss" shows "NS x2a x2a"
awlast_of_awalk: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v"   and "NOMATCH (awlast u p) v" shows "awlast u p = v"
llength_lzip: fixes xs :: "'b llist"   and ys :: "'c llist" shows "llength (xs \<bar>\<bar> ys) = min (llength xs) (llength ys)"
x_pow_n: fixes n :: "nat" shows "Polynomial.monom (1::'a) 1 ^ n = Polynomial.monom (1::'a) n"
fr: fixes j :: "'a"   and x :: "'b" assumes "j \<in> I"   and "a \<le> j"   and "j < b" shows "(f(i := x) \<longlongrightarrow> u j) (at_right j)"
pfinite_gpv_mono: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and \<I>' :: "('h, 'i) \<I>" assumes "pfinite_gpv \<I> gpv"   and "\<I> \<le> \<I>'"   and "\<I> \<turnstile>g gpv \<surd>" shows "pfinite_gpv \<I>' gpv"
leadsETo_weaken: fixes F :: "'d program"   and A :: "'d set"   and CC' :: "'d set set"   and A' :: "'d set"   and B :: "'d set"   and B' :: "'d set"   and CC :: "'d set set" assumes "F \<in> A leadsTo[CC'] A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'"   and "CC' \<subseteq> CC" shows "F \<in> B leadsTo[CC] B'"
rtrancl_beta_sred: fixes r :: "dB"   and r' :: "dB" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* r'" shows "r \<rightarrow>\<^sub>s r'"
range_None: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "MaxR r"   and "MaxR.MaxR_opt r A = None" shows "A \<inter> Field r = {}"
xs_eq: shows "[0..<min (dim_row A) (dim_col A) - 1] = [0..<min (nrows A') (ncols A') - 1]"
Seq_restr: fixes s :: "hf"   and k :: "hf" assumes "Seq s (succ k)" shows "Seq (hrestrict s k) k"
inverse_matrix_iarray_def: fixes A :: "'b iarray iarray" shows "inverse_matrix_iarray A = (if invertible_iarray A then Some (fst (Gauss_Jordan_iarrays_PA A)) else None)"
IH_1: fixes w :: "nat \<Rightarrow> 'a set" assumes "\<mu>_stable \<phi>1 w"   and "w \<Turnstile>\<^sub>n \<phi>1" shows "\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi>1 (prefix i w)[{\<psi>. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)} \<inter> restricted_subformulas \<phi>1]\<^sub>\<nu>"
iso_iff_mono_and_retraction: fixes f :: "'a" shows "C.iso f = (C.mono f \<and> C.retraction f)"
continuous_map_cases_function: fixes X :: "'a topology"   and Z :: "'b topology"   and p :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and Y :: "'c topology"   and f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "continuous_map X Z p"   and "continuous_map (subtopology X {x \<in> topspace X. p x \<in> Z closure_of U}) Y f"   and "continuous_map (subtopology X {x \<in> topspace X. p x \<in> Z closure_of (topspace Z - U)}) Y g"   and "\<And>x. \<lbrakk>x \<in> topspace X; p x \<in> Z frontier_of U\<rbrakk> \<Longrightarrow> f x = g x" shows "continuous_map X Y (\<lambda>x. if p x \<in> U then f x else g x)"
empty_iff: fixes c :: "'x" shows "(c \<in> {}) = False"
assms_1: shows "f piecewise_differentiable_on {a..c}"
height_bound': fixes u :: "nat" assumes "u \<in> V" shows "l u \<le> 2 * card V - 1"
lift_is_hom: fixes f :: "'a \<Rightarrow> int"   and gens :: "'a set" assumes "f \<in> gens \<rightarrow> carrier int_group" shows "int.lift f \<in> hom \<F>\<^bsub>gens\<^esub> int_group"
hypsy: shows "y \<in> \<rat> \<and> y ^ 3 = 2"
IH: fixes k :: "nat"   and I :: "'a \<Rightarrow> 'b option" assumes "k < length (trace_parallel_plan_sas_plus I \<psi>)" shows "trace_parallel_plan_sas_plus I \<psi> ! k = execute_parallel_plan_sas_plus I (take k \<psi>)"
p_eq_0_imp: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg"   and "p (state cfg) = 0" shows "v cfg = 0"
ih_1: assumes "\<not> getConflictFlag state'"   and "getQ state' \<noteq> []" shows "exhaustiveUnitPropagate_dom (applyUnitPropagate state')"
monadic_WHILEIT_def: fixes I :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool nres"   and f :: "'b \<Rightarrow> 'b nres"   and s :: "'b" shows "monadic_WHILEIT I b f s \<equiv> REC\<^sub>T (\<lambda>D s. ASSERT (I s) \<bind> (\<lambda>_. b s \<bind> (\<lambda>bv. if bv then f s \<bind> D else RETURN s))) s"
ge: fixes n :: "nat" shows "ennreal (1 - e n) \<le> emeasure M (U n)"
gen_analz_insert_eq: fixes X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "X \<in> analz G"   and "G \<subseteq> H" shows "analz (insert X H) = analz H"
xmem: fixes x :: "bool stream set" assumes "x \<in> {pseudo_proj_True n -` B \<inter> space M |B. B \<in> events}" shows "x \<in> sets (nat_filtration (Suc n))"
q_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "ego.q x \<le> ego.q y"
split_paired_all: fixes P :: "'b \<times> 'c \<Rightarrow> prop" shows "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"
list_length_hint4: fixes x :: "'d list" assumes "length x \<le> Suc 0"   and "x \<noteq> []" shows "length x = Suc 0"
rrule_sound: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and j :: "nat"   and L :: "rrule" assumes "rrule_ok SG C i j L"   and "i < length SG"   and "j < length (snd (SG ! i))"   and "sound (SG, C)" shows "sound (close (SG @ Rrule_result L j (SG ! i)) (SG ! i), C)"
not_zero_suc_i_suc_k: shows "\<not> is_zero_row_upt_k (i + (1::'rows)) (Suc k) B"
append_dropped_prefix: fixes u :: "'a list"   and v :: "'a list"   and w :: "'a list" assumes "is_prefix u v"   and "drop (length u) v = w" shows "u @ w = v"
isoI: fixes f :: "'a1"   and g :: "'a1" assumes "C1.inverse_arrows f g" shows "C1.iso f"
prems: shows "Q t\<^sub>1 t\<^sub>2"
lemma_2_5_17: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a l\<rightarrow> b \<le> (b l\<rightarrow> c) r\<rightarrow> a l\<rightarrow> c"
pac_idealI3: fixes p :: "int mpoly"   and A :: "int mpoly set"   and q :: "int mpoly" assumes "p \<in> More_Modules.ideal A" shows "p * q \<in> pac_ideal A"
isomorphic_categories_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "isomorphic_categories A B \<equiv> category A \<and> category B \<and> isomorphic_categories_axioms A B"