hcju/isabelleps huggingface dataset:
l_get_ancestors_wf_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog" shows "l_get_ancestors_wf heap_is_wellformed parent_child_rel known_ptr known_ptrs type_wf get_ancestors get_ancestors_locs get_child_nodes get_parent \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_get_ancestors_wf_axioms heap_is_wellformed parent_child_rel known_ptrs type_wf get_ancestors get_ancestors_locs get_child_nodes get_parent"
USUP_as_Inf_collect': fixes f :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Squnion> P \<bullet> f P) = \<Squnion>range f"
foldli_foldl: fixes xs :: "'h list"   and f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and \<sigma> :: "'g" shows "foldli xs (\<lambda>_. True) f \<sigma> = foldl (\<lambda>\<sigma> x. f x \<sigma>) \<sigma> xs"
abs_eq: fixes x :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool" shows "Semantics.d\<^sub>1 (make\<Pi>\<^sub>1 x) = Some x"
transfer: shows "pcr_pfun (=) (=) Map.empty {}\<^sub>p"
simps: fixes a :: "('sequent \<times> 'rule) tree" shows "local.wf a = (\<exists>t. a = t \<and> snd (tree.root t) \<in> R \<and> effStep (tree.root t) ((fst \<circ> tree.root) |`| cont t) \<and> (\<forall>x. x |\<in>| cont t \<longrightarrow> local.wf x))"
redundancy_and_semantics: fixes C :: "'at Literal set"   and S :: "'at Literal set set"   and I :: "'at set" assumes "redundant C S"   and "I \<Turnstile> S" shows "I \<Turnstile> C"
select_convs_5: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'a" shows "cTemperatureF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cTemperatureF"
C_eq_None: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "p \<noteq> []"   and "applied_rule_rev C x p = \<bottom>" shows "C (list2FWpolicy p) x = \<bottom>"
raw_match'_extend_map_preserve: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs'' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and p :: "variable \<times> String.literal set" assumes "raw_match' T U subs = Some subs'"   and "subs' \<subseteq>\<^sub>m subs''"   and "p \<in> tvsT T" shows "subs'' p = subs' p"
both_empty2: fixes t :: "'b ttree" shows "t \<otimes>\<otimes> TTree.empty = t"
bij_betw_same_card: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "card A = card B"
apex_def: shows "Dom\<mu>_Dom\<nu>.apex \<equiv> C.dom Dom\<mu>_Dom\<nu>.leg0"
distinct_85: fixes x8 :: "'a fm"   and x91 :: "nat"   and x92 :: "'a fm" shows "AllQ x8 \<noteq> ExN x91 x92"
signs_smash_property: fixes signs1 :: "'a list list"   and signs2 :: "'a list list" shows "\<forall>elem. elem \<in> set (signs_smash signs1 signs2) \<longrightarrow> (\<exists>n m. elem = signs1 ! n @ signs2 ! m)"
isMaxim_def: fixes A :: "('b + 'c) set"   and b :: "'b + 'c" shows "st.isMaxim A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (a, b) \<in> s +o t)"
ineq_model_def: fixes \<I> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "ineq_model \<I> X F \<equiv> \<forall>\<delta>. subst_domain \<delta> = set X \<and> ground (subst_range \<delta>) \<longrightarrow> list_ex (\<lambda>f. fst f \<cdot> \<delta> \<circ>\<^sub>s \<I> \<noteq> snd f \<cdot> \<delta> \<circ>\<^sub>s \<I>) F"
path_image_compose: fixes f :: "'c \<Rightarrow> 'b"   and p :: "real \<Rightarrow> 'c" shows "path_image (f \<circ> p) = f ` path_image p"
degqf: shows "degree (Polynomial.smult (of_nat CARD('q)) (#f)) = degree (#f)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) C.map C.map C.map"
mset_ssort': fixes empty :: "'a"   and is_empty :: "'a \<Rightarrow> bool"   and of_list :: "'b list \<Rightarrow> 'a"   and multiset :: "'a \<Rightarrow> 'b multiset"   and remove_max :: "'a \<Rightarrow> 'b \<times> 'a"   and inv :: "'a \<Rightarrow> bool"   and l :: "'a"   and sl :: "'b list" assumes "RemoveMax empty is_empty of_list multiset remove_max inv"   and "inv l" shows "mset (RemoveMax.ssort' is_empty remove_max l sl) = multiset l + mset sl"
resComm: fixes x :: "name"   and y :: "name"   and Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "\<And>Q. (\<lparr>\<nu>x\<rparr>\<lparr>\<nu>y\<rparr>Q, \<lparr>\<nu>y\<rparr>\<lparr>\<nu>x\<rparr>Q) \<in> Rel" shows "\<lparr>\<nu>x\<rparr>\<lparr>\<nu>y\<rparr>P \<leadsto>[Rel] \<lparr>\<nu>y\<rparr>\<lparr>\<nu>x\<rparr>P"
r: fixes z :: "nat" shows "eval (Cn 2 r_result1 [Cn 2 r_pdec2 [recf.Id 2 0], Cn 2 r_s [Cn 2 r_append_zeros [recf.Id 2 1, Cn 2 r_pdec1 [recf.Id 2 0]]], Cn 2 r_add [Cn 2 r_length [recf.Id 2 1], Cn 2 r_pdec1 [recf.Id 2 0]]]) [z, b] = eval r_result1 [pdec2 z, the (s (e_append_zeros b (pdec1 z))), e_length b + pdec1 z]"
IH_1: shows "2 ^ min_height l \<le> size1 l"
widen_Class_Class_eq: fixes G :: "prog"   and C :: "qtname"   and cm :: "qtname" shows "G\<turnstile>Class C\<preceq>Class cm = G\<turnstile>C\<preceq>\<^sub>C cm"
elims: fixes x :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and xa :: "'a list"   and xb :: "'b list"   and y :: "assn" assumes "list_assn x xa xb = y" obtains P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" where "x = P__" and "xa = []" and "xb = []" and "y = emp"   | P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" and a :: "'a" and as :: "'a list" and c :: "'b" and cs :: "'b list" where "x = P" and "xa = a # as" and "xb = c # cs" and "y = P a c * list_assn P as cs"   | uu_ :: "'a \<Rightarrow> 'b \<Rightarrow> assn" and v :: "'a" and va :: "'a list" where "x = uu___" and "xa = v__ # va__" and "xb = []" and "y = false"   | uu_ :: "'a \<Rightarrow> 'b \<Rightarrow> assn" and v :: "'b" and va :: "'b list" where "x = uu_" and "xa = []" and "xb = v # va" and "y = false"
le_neq_trans: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b"   and "a \<noteq> b" shows "a < b"
hitting_paths_def: shows "hitting_paths \<equiv> \<lambda>x. x \<noteq> v0 \<and> ((\<exists>xs\<in>paths. x \<in> set xs) \<or> x = v1)"
kbo_strongly_normalizing: fixes s :: "('f, 'b) Term.term" shows "SN_on {(s, t). S s t} {s}"
eq\<^sub>C: shows "equal_class.equal \<equiv> local.ltln\<^sub>C_eq_test"
F: shows "\<forall>y. O y (\<sigma>v. F v) = (\<exists>x. F x \<and> O y x)"
psubset_card_mono: fixes B :: "'c set"   and A :: "'c set" assumes "finite B"   and "A \<subset> B" shows "card A < card B"
d_OUT_alt_def2: fixes g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "d_OUT g x = (\<Sum>\<^sup>+ y\<in>{y. (x, y) \<in> support_flow g}. g (x, y))"
fields_table_SomeD: fixes G :: "prog"   and C :: "qtname"   and P :: "fspec \<Rightarrow> Decl.field \<Rightarrow> bool"   and fn :: "fspec"   and T :: "ty" assumes "fields_table G C P fn = Some T"   and "unique (DeclConcepts.fields G C)" shows "\<exists>f. table_of (DeclConcepts.fields G C) fn = Some f \<and> type f = T"
bitor_one: fixes y :: "bit" shows "(\<one> \<or>\<^sub>b y) = \<one>"
19: shows "padic_mult p f g (Suc (nat (padic_val p f + padic_val p g))) mod p ^ Suc (nat (padic_val p f + padic_val p g)) mod p ^ (nat (padic_val p f) + nat (padic_val p g)) = i * j * p ^ (nat (padic_val p f) + nat (padic_val p g)) mod p ^ (nat (padic_val p f) + nat (padic_val p g))"
comp_apply: fixes f :: "'l \<Rightarrow> 'k"   and g :: "'m \<Rightarrow> 'l"   and x :: "'m" shows "(f \<circ> g) x = f (g x)"
combined_size_bound_integer_log: fixes x :: "int" assumes "x \<in> {fs.\<mu>' i j |i j. j \<le> i \<and> i < length fs_init} \<union> {fs.\<sigma>s l i j |i j l. i < length fs_init \<and> j \<le> i \<and> l < j}"   and "length fs_init \<noteq> 0"   and "x \<noteq> 0" shows "log 2 \<bar>real_of_int x\<bar> \<le> log 2 (real (length fs_init)) + real (3 + 3 * length fs_init) * log 2 (real_of_rat fs.gs.N)"
tagged_partial_division_ofD_2: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set"   and x :: "'a"   and K :: "'a set" assumes "s tagged_partial_division_of i"   and "(x, K) \<in> s" shows "x \<in> K"
opeS: shows "openin (top_of_set (S \<union> T)) S"
assms_1: shows "ins n (d + 1) x r = Unbal r'"
poly_offset: fixes p :: "'a poly"   and a :: "'a" shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
repr_inj: fixes x :: "'a"   and y :: "'a" shows "(repr isom x = repr isom y) = (x = y)"
lnull_cons'_def: shows "local.lnull_cons' = lnull_cons (lgenerator g)"
eval_stmt_eq: fixes G :: "prog"   and s :: "state"   and t :: "stmt"   and w :: "vals"   and s' :: "state" shows "G\<turnstile>s \<midarrow>In1r t\<succ>\<rightarrow> (w, s') = (w = \<diamondsuit> \<and> G\<turnstile>s \<midarrow>t\<rightarrow> s')"
powr_non_neg: fixes a :: "real"   and x :: "real" shows "\<not> a powr x < 0"
j: shows "has_bochner_integral (lebesgue_on {c..b}) f (integral\<^sup>L (lebesgue_on {c..b}) f)"
terminates_dynCall: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and s :: "'e"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile>call init (p s) return c \<down> Normal s" shows "\<Gamma>\<turnstile>dynCall init p return c \<down> Normal s"
is_prefix_def: fixes xs :: "'d list"   and ys :: "'d list" shows "is_prefix xs ys \<equiv> CATCH (ebind (CHECK (length xs \<le> length ys) False) (\<lambda>_. ebind (EWHILEIT (\<lambda>i. i \<le> length xs \<and> take i xs = take i ys) (\<lambda>i. i < length xs) (\<lambda>i. ebind (EASSERT (i < length xs \<and> i < length ys)) (\<lambda>_. ebind (CHECK (xs ! i = ys ! i) False) (\<lambda>_. ERETURN (i + 1)))) 0) (\<lambda>_. THROW True))) ERETURN"
p1_def: shows "p1 = p \<circ>\<^sub>p [:lb, Complex (Re ub - Re lb) 0:]"
linear_with_def: fixes plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f" shows "linear_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 f \<equiv> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 \<and> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<and> module_hom_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 f"
subseq_append_iff: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "subseq xs (ys @ zs) = (\<exists>xs1 xs2. xs = xs1 @ xs2 \<and> subseq xs1 ys \<and> subseq xs2 zs)"
Domain_fst: fixes r :: "('c \<times> 'd) set" shows "Domain r = fst ` r"
lift_poly_def: shows "lift_poly = (\<lambda>p. p \<circ>\<^sub>p [:0, fpxs_X_power r:]) \<circ> map_poly ((\<lambda>x. fpxs_compose_power x (1 / rat_of_nat b)) \<circ> fpxs_of_fls \<circ> fps_to_fls)"
elims: fixes x :: "'b symb"   and xa :: "'b exp list"   and y :: "'b exp" assumes "mk x xa = y" obtains xb :: "'b" where "x = v xb" and "xa = []" and "y = V xb"   | n :: "nat" where "x = c n" and "xa = []" and "y = C n"   | a :: "'b exp" and b :: "'b exp" where "x = p" and "xa = [a__, b]" and "y = exp.Plus a__ b"   | v :: "nat" and va :: "'b exp" and vb :: "'b exp list" where "x = c v__" and "xa = va__ # vb__" and "y = undefined"   | "x = p" and "xa = []" and "y = undefined"   | v :: "'b exp" where "x = p" and "xa = [vc__]" and "y = undefined"   | v :: "'b exp" and vb :: "'b exp" and vd :: "'b exp" and ve :: "'b exp list" where "x = p" and "xa = vd__ # vba__ # vda__ # ve__" and "y = undefined"   | vb :: "'b" and va :: "'b exp" and vaa :: "'b exp list" where "x = v vbb__" and "xa = va # vaa" and "y = undefined"   | a :: "'b symb" and v :: "'b exp" where "x = aa__" and "xa = [vf__]" and "y = undefined"   | a :: "'b symb" and v :: "'b exp" and vb :: "'b exp" and vd :: "'b exp" and ve :: "'b exp list" where "x = a" and "xa = v # vb # vd # ve" and "y = undefined"
Mp_0: fixes m :: "int" shows "poly_mod.Mp m 0 = 0"
mult_mult_of: fixes R :: "('c, 'd) ring_scheme" shows "(\<otimes>\<^bsub>mult_of R\<^esub>) = (\<otimes>\<^bsub>R\<^esub>)"
Inf_set_def: fixes A :: "'a set set" shows "\<Inter> A = {x. \<Sqinter>B\<in>A. s2p B x}"
poly_subst_sum: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "'e \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and A :: "'e set" shows "poly_subst f (sum p A) = (\<Sum>a\<in>A. poly_subst f (p a))"
sigma_algebra_iff: fixes \<Omega> :: "'a set"   and M :: "'a set set" shows "sigma_algebra \<Omega> M = (algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> \<Union> (range A) \<in> M))"
11: shows "\<bar>c' - c\<bar> = \<bar>c - c'\<bar>"
expand_superset: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "{a. g a \<noteq> \<^bold>1} \<subseteq> A" shows "G g = F.F g A"
poly2_x_y: fixes x :: "'a"   and y :: "'a" shows "poly2 x_y x y = x - y"
not_final_thread_conv: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x1 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "r1.not_final_thread s t = (thr s t \<noteq> None \<and> \<not> r1.final_thread s t)"
finsum_over_indexes_to_vectors: fixes vs :: "'a vec list"   and l :: "nat"   and g :: "nat \<Rightarrow> int" assumes "set vs \<subseteq> carrier_vec n"   and "l = length vs" shows "\<exists>c. (\<Oplus>\<^bsub>V\<^esub>x\<in>{0..<l}. of_int (g x) \<cdot>\<^sub>v vs ! x) = (\<Oplus>\<^bsub>V\<^esub>v\<in>set vs. of_int (c v) \<cdot>\<^sub>v v)"
cd_diff_r: fixes m :: "nat"   and n :: "nat"   and x :: "nat" assumes "m \<le> n" shows "cd x m n = cd x m (n - m)"
Domain_unfold: fixes r :: "('a \<times> 'b) set" shows "Domain r = {x. \<exists>y. (x, y) \<in> r}"
ipassmt_inj_k: fixes k :: "iface"   and ips' :: "('a word \<times> nat) list" assumes "ipassmt k = Some ips'"   and "p_dst p \<in> ipcidr_union_set (set ips')" shows "k = ifce"
sq_mtx_one_ith_simps_2: fixes i :: "'a"   and j :: "'a" assumes "i \<noteq> j" shows "1 $$ i $ j = 0"
finite_kmax: fixes f :: "nat \<Rightarrow> real" shows "finite (kmax f PR)"
sv01_def: fixes e :: "nat" shows "s\<^bsub>01\<^esub> e \<equiv> eval r_sv01 [e]"
exhaust: fixes y :: "simple_action" obtains "y = Accept"   | "y = Drop"
hyps_3: shows "\<forall>x. var_asm_not_written mds\<^sub>C' x \<longrightarrow> mem\<^sub>C' x = mem\<^sub>C'' x \<and> dma\<^sub>C mem\<^sub>C' x = dma\<^sub>C mem\<^sub>C'' x"
TOTAL_wrt_phi_eq_TOTAL: shows "TOTAL_wrt \<phi> = TOTAL"
0: fixes \<delta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and x :: "('fun, 'atom) Term.term \<times> nat"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)"   and "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var x) = \<Gamma> (Var y)"   and "t \<sqsubseteq> \<delta> x"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>" shows "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var y) = \<Gamma> t"
of_real_poly_map_smult: fixes s :: "real"   and p :: "real poly" shows "map_poly of_real (smult s p) = smult (of_real s) (map_poly of_real p)"
comp_fun_commute_insort: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "comp_fun_commute (linorder.insort_key less_eq (\<lambda>x. x))"
Collect_case_prod_in_rel_leE: fixes X :: "('a \<times> 'b) set"   and Y :: "('a \<times> 'b) set" assumes "X \<subseteq> {(x, y). in_rel Y x y}" obtains "X \<subseteq> Y"
simps_1: fixes A :: "'at" shows "negative (Pos A) = False"
return_writer_def: fixes return :: "'h \<times> 'g list \<Rightarrow> 'i"   and x :: "'h" shows "return_writer return x = WriterT (return (x, []))"
Shift_def: fixes Kl :: "'d list set"   and k :: "'d" shows "BNF_Greatest_Fixpoint.Shift Kl k = {kl. k # kl \<in> Kl}"
wt_ge_\<delta>: fixes s :: "('s, 'v) tm" shows "\<delta> \<le> wt s"
markov_inequality: fixes X :: "'a \<Rightarrow> real"   and t :: "real" assumes "\<And>a. 0 \<le> X a"   and "integrable M X"   and "0 < t" shows "prob {a \<in> space M. t \<le> X a} \<le> expectation X / t"
q_def: fixes y :: "nat"   and t :: "nat" shows "q y t = (if y = p (c_snd t) then pA (c_snd t) (c_fst t) else 1)"
not_le_imp_less: fixes y :: "'a"   and x :: "'a" assumes "\<not> y \<le> x" shows "x < y"
deformation_retraction_imp_homotopy_equivalent_space: fixes X :: "'b topology"   and s :: "'c \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'c"   and Y :: "'c topology" assumes "homotopic_with (\<lambda>x. True) X X (s \<circ> r) id"   and "retraction_maps X Y r s" shows "X homotopy_equivalent_space Y"
explicit_sctn_set_def: fixes po :: "'c \<Rightarrow> 'd set"   and X0 :: "'d set" shows "explicit_sctn_set po X0 = isEmpty_spec X0 \<bind> (\<lambda>e. WHILE\<^bsup>\<lambda>(e, X, Y). (e \<longrightarrow> X = {}) \<and> X0 = X \<union> (\<Union>(sctn, IS)\<in>Y. IS) \<and> (\<forall>(sctn, IS)\<in>Y. IS \<subseteq> po sctn)\<^esup> (\<lambda>(e, X, Y). \<not> e) (\<lambda>(e, X, Y). split_with_invar po X \<bind> (\<lambda>((S, sctn), X'). isEmpty_spec X' \<bind> (\<lambda>e. RETURN (e, X', insert (sctn, S) Y)))) (e, X0, {}) \<bind> (\<lambda>(uu_, uu_, Xis). RETURN Xis))"
tensor_preserves_can: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.can f"   and "\<F>C.can g" shows "\<F>C.can (f \<otimes> g)"
not_add_less1: fixes i :: "nat"   and j :: "nat" shows "\<not> order.greater i (i + j)"
blues_seen_same: fixes n :: "nat"   and p :: "'person"   and w :: "'person \<Rightarrow> 'color"   and w' :: "'person \<Rightarrow> 'color" assumes "local.possible n p w w'" shows "local.blues_seen w' p = local.blues_seen w p"
v11_vcomp_vconverse: fixes r :: "V" assumes "v11 r" shows "r\<inverse>\<^sub>\<circ> \<circ>\<^sub>\<circ> r = vid_on (\<D>\<^sub>\<circ> r)"
hcomp_ide_dom: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_transformation A B F G \<tau>" shows "\<tau> \<circ> identity_functor.map A = \<tau>"
impl_blue_dfs_def: shows "impl_blue_dfs \<equiv> \<lambda>x. NDFS_SI_Statistics.start_nres \<bind> (\<lambda>xa. LIST_FOREACH' (RETURN (id (gi_V0 x))) (\<lambda>(xb, xc, xd). xd = NO_CYC) (\<lambda>xb (xc, xd, xe). if \<not> map2set_memb (\<lambda>k t. rbt_lookup t k) xb xc then REC\<^sub>T (\<lambda>xf (xg, xh, xi, xj). let y' = map2set_insert rbt_insert xj xg; y'a = map2set_insert rbt_insert xj xi; y'b = bgi_F x xj in NDFS_SI_Statistics.vis_blue_nres \<bind> (\<lambda>xk. LIST_FOREACH' (RETURN (id (gi_E x xj))) (\<lambda>(xl, xm, xn, xo). xo = NO_CYC) (\<lambda>xl (xm, xn, xo, xp). if map2set_memb (\<lambda>k t. rbt_lookup t k) xl xo \<and> (y'b \<or> bgi_F x xl) then RETURN (xm, xn, xo, init_wit_blue_early xj xl) else if \<not> map2set_memb (\<lambda>k t. rbt_lookup t k) xl xm then xf (xm, xn, xo, xl) \<bind> (\<lambda>(xq, xr, xs, xt). RETURN (xq, xr, xs, prep_wit_blue xj xt)) else NDFS_SI_Statistics.match_blue_nres \<bind> (\<lambda>xq. RETURN (xm, xn, xo, xp))) (y', xh, y'a, NO_CYC) \<bind> (\<lambda>(xl, xm, xn, xo). (if xo = NO_CYC \<and> y'b then impl_red_dfs (gi_E x) xn xm xj \<bind> (\<lambda>(xp, xq). RETURN (xp, init_wit_blue xj xq)) else RETURN (xm, xo)) \<bind> (\<lambda>(xp, xq). let y'c = ord.rbt_delete (<) xj xn in RETURN (xl, xp, y'c, xq))))) (xc, xd, rbt.Empty, xb) \<bind> (\<lambda>(xf, xg, xh, xi). RETURN (xf, xg, xi)) else RETURN (xc, xd, NO_CYC)) (rbt.Empty, rbt.Empty, NO_CYC) \<bind> (\<lambda>(xb, xc, xd). NDFS_SI_Statistics.stop_nres \<bind> (\<lambda>xe. RETURN (extract_res xd))))"
less_bool_rel_iff: fixes a :: "bool"   and b :: "bool" shows "((a, b) \<in> less_bool_rel) = (\<not> a \<and> b)"
ltrm_eq_imp_deg_drop: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "monom P (p (deg R p)) (deg R p) = monom P (q (deg R q)) (deg R q)"   and "0 < deg R p" shows "deg R (p \<ominus>\<^bsub>P\<^esub> q) < deg R p"
dual_equivalence: shows "internal_equivalence g f (local.inv \<epsilon>) (local.inv \<eta>)"
nv2T: fixes x :: "Preliminaries.var" assumes "\<not> infTp (tpOfV x)"   and "\<exists>T\<in>set Tl. x \<in> nv2T T" shows "\<exists>l. wtL l \<and> \<not> I.satL \<xi> l \<and> isGuard x l"
singular_chain_of: fixes p :: "nat"   and X :: "'c topology"   and c :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" shows "singular_chain p X (frag_of c) = singular_simplex p X c"
finite_vars_term: fixes t :: "('d, 'c) Term.term" shows "finite (fv t)"
5: shows "range (\<lambda>s. trees_cyl (map_tree (\<lambda>_. space M) s)) \<subseteq> trees_cyl ` trees (sets M)"
assms: shows "m1 \<in> carrier_mat d1 d1"
ideal_axioms: shows "ideal additive.Ker R (+) (\<cdot>) \<zero> \<one>"
normal_inv_iff: fixes N :: "'a set" shows "N \<lhd> add_monoid R = (subgroup N (add_monoid R) \<and> (\<forall>x\<in>carrier R. \<forall>h\<in>N. x \<oplus> h \<oplus> \<ominus> x \<in> N))"
AA19: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and A :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and v :: "'d \<Rightarrow> 'e" shows "\<turnstile> \<box>P \<and> \<diamond>\<langle>A\<rangle>_v \<longrightarrow> \<diamond>\<langle>P \<and> A\<rangle>_v"
AlgorithmForAgent_axioms: shows "AlgorithmForAgent jkbp envInit envAction envTrans envVal jview envObs jviewInit jviewIncr simf simRels simVal simAbs simObs simInit simTrans simAction aOps tOps"
Order_comp_eq: fixes E :: "'c Order"   and F :: "'c Order" assumes "carrier E = carrier F"   and "rel E = rel F" shows "E = F"