hcju/isabelleps huggingface dataset:
vsv_vimage_eqI: fixes r :: "V"   and a :: "V"   and b :: "V"   and A :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> r `\<^sub>\<circ> A"
subset_code_1: fixes xs :: "'m list"   and B :: "'m set" shows "(set xs \<subseteq> B) = (\<forall>x\<in>set xs. x \<in> B)"
poly_divide_tendsto_aux: fixes p :: "'a poly" shows "((\<lambda>x. poly p x / x ^ degree p) \<longlongrightarrow> lead_coeff p) at_infinity"
distinct_edgesI: fixes p :: "nat list" assumes "distinct p" shows "distinct (uwalk_edges p)"
fw_upd_out_of_bounds1: fixes i :: "nat"   and i' :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and j :: "nat"   and j' :: "nat" assumes "i < i'" shows "fw_upd M k i j i' j' = M i' j'"
AR_ball: fixes a :: "'a"   and r :: "real" shows "AR (ball a r) = (0 < r)"
rep_eq: fixes x :: "real_alg_3" shows "get_itvl_3 x = get_itvl_2 (rep_real_alg_3 x)"
fnE: fixes x :: "'b"   and T :: "PreSimplyTyped.type"   and A :: "'b ptrm"   and Y :: "'b ptrm" assumes "PFn x T A \<approx> Y" obtains B :: "'b ptrm" where "Y = PFn x T Ba__" and "A \<approx> Ba__"   | b :: "'b" and B :: "'b ptrm" where "Y = PFn b T B" and "x \<noteq> b" and "A \<approx> [x \<leftrightarrow> b] \<bullet> B" and "x \<notin> ptrm_fvs B"
is_parallel_solution_for_problem_operator_set: fixes \<Pi> :: "'a strips_problem"   and \<pi> :: "'a strips_operator list list"   and ops :: "'a strips_operator list"   and op :: "'a strips_operator" assumes "is_parallel_solution_for_problem \<Pi> \<pi>"   and "ops \<in> set \<pi>"   and "op \<in> set ops" shows "op \<in> set (\<Pi>\<^sub>\<O>)"
orderE: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" obtains "a = inf a b"
fold_def_alt: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "('b, 'c) RBT.rbt" shows "RBT.fold f t = List.fold (\<lambda>(x, y). f x y) (entries t)"
simps_1: fixes env :: "v sem_env"   and e :: "'b error_result" shows "combine_dec_result env (Rerr e) = Rerr e"
represents_mult_rat: fixes r :: "rat"   and p :: "int poly"   and x :: "'a" assumes "r \<noteq> 0"   and "p represents x" shows "poly_mult_rat r p represents of_rat r * x"
crel_vs1_state_of: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f Heap" assumes "crel_vs1 R a b" shows "state_dp_consistency.crel_vs R a (state_of b)"
min_0_1_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.min less_eq one zero = zero"
b_least2_at_zero: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" shows "b_least2 f x 0 = 0"
tak_m3_def: shows "tak_m3 = (\<lambda>(x, y, z). nat (x - Min {x, y, z}))"
ord_less_eq_subst: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "a < b"   and "f b = c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "f a < c"
constant_functionE: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "constant_function (carrier R) a b = a"
card_a: fixes k :: "nat" shows "card (a k) = \<Sqinter> (d k)"
mem_rel_interior: fixes x :: "'b"   and S :: "'b set" shows "(x \<in> rel_interior S) = (\<exists>T. open T \<and> x \<in> T \<inter> S \<and> T \<inter> affine hull S \<subseteq> S)"
PPs_singleton: fixes x :: "'b" shows ".[{x}] = range (Poly_Mapping.single x)"
satPB_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "U.satPB \<xi> \<Phi> = F.satPB \<xi> \<Phi>"
has_label: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>" shows "\<exists>l. l \<in> labels \<Gamma> p"
associatedD2: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a" assumes "monoid_cancel G"   and "a \<sim>\<^bsub>G\<^esub> b"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "\<exists>u\<in>Units G. a = b \<otimes>\<^bsub>G\<^esub> u"
diff_numeral_special_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus one (numeral.numeral one plus n) = neg_numeral.sub minus plus one num.One n"
intro: fixes \<Delta> :: "('a, 'b) network_scheme" assumes "countable_network \<Delta>"   and "flow_attainability_axioms \<Delta>" shows "flow_attainability \<Delta>"
wf_tuple_length: fixes n :: "nat"   and V :: "nat set"   and x :: "'c option list" assumes "wf_tuple n V x" shows "length x = n"
IH_1: fixes ps :: "(real, 'a) vec list" assumes "invar l"   and "sorted_wrt_dist p ps"   and "set ps \<inter> set_kdt l = {}"   and "distinct ps"   and "0 < n" shows "\<forall>q\<in>set_kdt l \<union> set ps - set (nearest_nbors n ps p l). dist (last (nearest_nbors n ps p l)) p \<le> dist q p"
integral_normal_moment_even: fixes \<sigma> :: "real"   and \<mu> :: "real"   and k :: "nat" assumes "0 < \<sigma>" shows "LBINT x. normal_density \<mu> \<sigma> x * (x - \<mu>) ^ (2 * k) = fact (2 * k) / ((2 / \<sigma>\<^sup>2) ^ k * fact k)"
FAssert_def: fixes x1 :: "'d" shows "\<langle>\<epsilon>, x1\<rangle> \<equiv> Abs_frame (FAssert_Rep x1)"
ovr_overshadow_left: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "overrider F" shows "F (F x y) z = F x z"
cSup_eq_maximum: fixes z :: "'c multiset"   and X :: "'c multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq># z" shows "Sup X = z"
induction_axioms_hold: fixes A :: "fm"   and e :: "(name, hf) finfun" assumes "A \<in> induction_axioms" shows "eval_fm e A"
bnd: shows "\<forall>g k. g dvd Polynomial.smult (lead_coeff f) f \<longrightarrow> degree g \<le> degree_bound hs \<longrightarrow> 2 * \<bar>coeff g k\<bar> < p ^ n"
piecewise_continuous_on_AE_boundedE: fixes a :: "real"   and I :: "real \<Rightarrow> real set"   and f :: "real \<Rightarrow> complex" assumes "\<And>k. piecewise_continuous_on a k (I k) f" obtains B :: "real \<Rightarrow> real" where "\<And>k. AE x\<in>{a..k} in lebesgue. cmod (f x) \<le> B k"
face_face_op_Seed: fixes p :: "nat" shows "face_face_op Seed\<^bsub>p\<^esub>"
index_add_vec_1: fixes i :: "nat"   and v\<^sub>2 :: "'b Matrix.vec"   and v\<^sub>1 :: "'b Matrix.vec" assumes "i < dim_vec v\<^sub>2" shows "(v\<^sub>1 + v\<^sub>2) $v i = v\<^sub>1 $v i + v\<^sub>2 $v i"
fsafe_Exists_simps: fixes x :: "'c"   and p :: "('a, 'b, 'c) formula" shows "fsafe (Exists x p) = fsafe p"
h: fixes i :: "'a" assumes "i \<in> I" shows "homologous_rel p X S (f i) (g i)"
strictly_exact_def: fixes P :: "'a \<Rightarrow> bool" shows "strictly_exact P \<equiv> \<forall>h h'. P h \<and> P h' \<longrightarrow> h = h'"
evalS_append: fixes M :: "model"   and phi :: "vbl \<Rightarrow> object"   and Gamma :: "formula list"   and Delta :: "formula list" shows "evalS M phi (Gamma @ Delta) = (evalS M phi Gamma \<or> evalS M phi Delta)"
normal_poly_of_roots: fixes p :: "real poly" assumes "\<And>z. poly (map_poly complex_of_real p) z = 0 \<Longrightarrow> Re z \<le> 0 \<and> (cmod z)\<^sup>2 \<le> 4 * (Re z)\<^sup>2"   and "monic p" shows "normal_poly p"
inside_subset: fixes U :: "'a set"   and T :: "'a set"   and S :: "'a set" assumes "connected U"   and "\<not> bounded U"   and "T \<union> U = - S" shows "inside S \<subseteq> T"
isGlb_isLb: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isGlb R S x" shows "isLb R S x"
ideal_condition1: fixes I :: "'a set" assumes "I \<subseteq> carrier R"   and "I \<noteq> {}"   and "\<forall>x\<in>I. \<forall>y\<in>I. x \<plusminus> y \<in> I"   and "\<forall>r\<in>carrier R. \<forall>x\<in>I. r \<cdot>\<^sub>r x \<in> I" shows "ideal R I"
word_eq_iff_unsigned: fixes v :: "'c word"   and w :: "'c word" shows "(v = w) = (unsigned v = unsigned w)"
valid_countings_eq_zero: fixes a :: "nat"   and b :: "nat" assumes "a \<le> b"   and "0 < b" shows "valid_countings a b = 0"
lcm_D: fixes a :: "'a" shows "a l\<rightarrow> a = (1::'a)"
uprod_def: fixes A :: "('a, 'b) node set set"   and B :: "('a, 'b) node set set" shows "uprod A B \<equiv> \<Union>x\<in>A. \<Union>y\<in>B. {Scons x y}"
loxodromic_attracting_fixed_point_attracts: fixes f :: "'b \<Rightarrow> 'b"   and xi :: "'b Gromov_completion" assumes "loxodromic_isometry f"   and "xi \<noteq> repelling_fixed_point f" shows "(\<lambda>n. (Gromov_extension f ^^ n) xi) \<longlonglongrightarrow> attracting_fixed_point f"
ind_all: shows "\<forall>i<r. terminate f (xs @ [i]) \<and> (\<forall>x xa xb. rec_ci f = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (xb - Suc xa) @ xc})) \<and> 0 < rec_exec f (xs @ [i])"
aligned_add_mask_lessD: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word" assumes "x + mask n < y"   and "is_aligned x n" shows "x < y"
dBox_assigns: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and \<Phi> :: "'b upred" shows "\<^bold>[\<langle>\<sigma>\<rangle>\<^sub>a\<^bold>]\<Phi> = \<sigma> \<dagger> \<Phi>"
g: shows "Seed\<^bsub>p\<^esub> [next_plane0\<^bsub>p\<^esub>]\<rightarrow>* g"
t: fixes s' :: "'s \<times> ('c \<Rightarrow> real)" assumes "s' \<in> set_pmf (rept s (action cfg))" shows "t (cont cfg (abss s')) = s'"
mono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'b" shows "incseq f = (\<forall>n. f n \<le> f (Suc n))"
in_existence_between_zeroI: fixes T :: "real set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and t :: "real"   and t0 :: "real"   and x0 :: "'a"   and s :: "real" assumes "ll_on_open_it T f X"   and "t \<in> ll_on_open.existence_ivl T f X t0 x0"   and "s \<in> {t0--t}" shows "s \<in> ll_on_open.existence_ivl T f X t0 x0"
distinct_1: fixes x11 :: "'c option"   and x12 :: "'d"   and x2 :: "'c" shows "Insert x11 x12 \<noteq> Delete x2"
cosafety_lr: fixes \<sigma> :: "'a trace"   and v :: "'b list"   and i :: "nat" assumes "sat \<sigma> v i" shows "\<exists>\<pi>. prefix_of \<pi> \<sigma> \<and> (\<forall>\<sigma>'. prefix_of \<pi> \<sigma>' \<longrightarrow> sat \<sigma>' v i)"
MapsToId: fixes C :: "('d, 'e, 'f) Category_scheme"   and X :: "'d" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X \<approx>>\<^bsub>C\<^esub> id\<^bsub>C\<^esub> X"
cases: fixes a1 :: "ccs"   and a2 :: "ccs" assumes "bangPred a1 a2" obtains P :: "ccs" where "a1 = P__" and "a2 = !P__"   | P :: "ccs" where "a1 = P" and "a2 = P \<parallel> !P"
nsets_eq_empty_iff: fixes A :: "'b set"   and r :: "nat" shows "([A]\<^bsup>r\<^esup> = {}) = (finite A \<and> card A < r)"
distinct_11: fixes x1 :: "('e, 'f) Term.term"   and x71 :: "'f list"   and x72 :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list"   and x73 :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list" shows "send\<langle>x1\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
inject_2: fixes x2 :: "('e \<times> 'f) set['g]"   and y2 :: "('e \<times> 'f) set['g]" shows "(BPS x2 = BPS y2) = (x2 = y2)"
CSHO_trans_alt_def: fixes snd_f :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f"   and nxt_st :: "nat \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> ('d \<Rightarrow> 'f option) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and HOs :: "nat \<Rightarrow> 'd \<Rightarrow> 'd set"   and SHOs :: "nat \<Rightarrow> 'd \<Rightarrow> 'd set"   and coords :: "nat \<Rightarrow> 'd \<Rightarrow> 'd" shows "CSHO_trans_alt snd_f nxt_st HOs SHOs coords \<equiv> \<Union>r \<mu>. {((r, cfg), Suc r, cfg') |cfg cfg'. \<forall>p. \<mu> p \<in> get_msgs (snd_f r) cfg (HOs r) (SHOs r) p \<and> (\<forall>p. nxt_st r p (cfg p) (\<mu> p) (coords r p) (cfg' p))}"
subsequence_def: fixes w :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" shows "w [i \<rightarrow> j] \<equiv> map w [i..<j]"
adds_termI: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "'a"   and t :: "'b"   and u :: "'a" assumes "term_powerprod pair_of_term term_of_pair"   and "v = term_powerprod.splus pair_of_term term_of_pair t u" shows "term_powerprod.adds_term pair_of_term u v"
p_in_reachable: fixes x :: "'a"   and y :: "'a"   and K :: "'a \<Rightarrow> 'a pmf"   and n :: "nat" assumes "(x, y) \<notin> (SIGMA x:UNIV. set_pmf (K x))\<^sup>*" shows "MC_syntax.p K x y n = 0"
intros_3: fixes k :: "nat"   and xs :: "nat list"   and ys :: "nat list"   and zs :: "nat list" assumes "Form_Body (Suc k) k xs ys zs"   and "0 < k" shows "Form (2 * k) {xs, ys}"
Subgraph_def: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and E' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "Subgraph E E' \<equiv> \<forall>a b. E' a b \<longrightarrow> E a b"
serialize_of_matches_def: fixes pids :: "char list \<Rightarrow> char list" shows "serialize_of_matches pids \<equiv> (@) ''hard_timeout=0,idle_timeout=0,'' \<circ> intersperse CHR '','' \<circ> map (serialize_of_match pids) \<circ> sorted_list_of_set"
subseteq_Liminf_state_eventually_always: fixes CC :: "'a literal multiset set" assumes "finite CC"   and "CC \<noteq> {}"   and "CC \<subseteq> Q_of_state (Liminf_state Sts)" shows "\<exists>j. enat j < llength Sts \<and> (\<forall>j'. enat j \<le> enat j' \<longrightarrow> enat j' < llength Sts \<longrightarrow> CC \<subseteq> Q_of_state (lnth Sts j'))"
real_inner_1_left: fixes x :: "real" shows "1 \<bullet> x = x"
translate_plus3: fixes a :: "'a"   and x :: "'a"   and b :: "'a"   and c :: "'a" shows "ccw (a - x) (b - x) c = ccw a b (c + x)"
find_first_some_strong: fixes x :: "'a"   and n :: "nat"   and xs :: "'a list" assumes "x \<in> set (take n xs)"   and "n \<le> length xs" obtains i :: "nat" where "find_first x xs = Some i" and "i < n"
config_rand: fixes A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and init :: "'b list"   and qs :: "'b list" shows "\<forall>x\<in>set_pmf (config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). set (fst x) = set init \<and> distinct (fst x) = distinct init \<and> length (fst x) = length init"
pw_le_iff: fixes S :: "'a nres"   and S' :: "'a nres" shows "(S \<le> S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x))"
map_sel_2: fixes a :: "'b tree"   and f :: "'b \<Rightarrow> 'c" assumes "a \<noteq> Leaf" shows "r (map_tree f a) = r a"
sum_list_us_gt0: fixes y :: "nat list"   and i :: "nat" assumes "sum_list y \<noteq> 0" shows "0 < sum_list (\<^bold>u y i)"
Lt_cases: 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 "Tarski_neutral_dimensionless.Lt Bet Cong A B C D \<or> Tarski_neutral_dimensionless.Lt Bet Cong B A C D \<or> Tarski_neutral_dimensionless.Lt Bet Cong A B D C \<or> Tarski_neutral_dimensionless.Lt Bet Cong B A D C" shows "Tarski_neutral_dimensionless.Lt Bet Cong A B C D"
domain_non_negative: shows "\<forall>x. 0 \<le> domain x"
theI': fixes P :: "'d \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "P (THE x. P x)"
hom_compose: fixes f :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme"   and g :: "'c \<Rightarrow> 'f"   and I :: "('f, 'g) monoid_scheme" assumes "f \<in> hom G H"   and "g \<in> hom H I" shows "g \<circ> f \<in> hom G I"
Subobjs_isSubobj: 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 Cs :: "char list list" assumes "Subobjs P C Cs" shows "is_subobj P (C, Cs)"
mset_left_cancel_union: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" assumes "a \<in># A + B"   and "a \<notin># A" shows "a \<in># B"
vle_antisym: fixes x :: "V"   and y :: "V" assumes "x \<unlhd> y"   and "y \<unlhd> x" shows "x = y"
mr_not_DA: fixes s :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "(int \<times> int \<times> protocol) set set"   and ab :: "(int \<times> int \<times> protocol) set set" assumes "wellformed_policy1_strong s"   and "applied_rule_rev Cp x p = \<lfloor>DenyAllFromTo a ab\<rfloor>"   and "set p = set s" shows "applied_rule_rev Cp x s \<noteq> \<lfloor>DenyAll\<rfloor>"
minimal_unsat_core_def: fixes I :: "'a set"   and ics :: "('a \<times> constraint) list" shows "minimal_unsat_core I ics = (I \<subseteq> fst ` set ics \<and> (\<nexists>v. (I, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set ics) \<and> (distinct_indices ics \<longrightarrow> (\<forall>J\<subset>I. \<exists>v. (J, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set ics)))"
subst_not_dom_fixed: fixes v :: "'a"   and t :: "('b, 'a) Term.term"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<in> fv t"   and "v \<notin> subst_domain s" shows "v \<in> fv (t \<cdot> s)"
mtf2_forward_effect2: fixes q :: "'a"   and xs :: "'a list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q = i" shows "index (mtf2 n q xs) (xs ! i) = index xs q - n \<and> index xs q - n = index (mtf2 n q xs) (xs ! i)"
mf_fm: fixes J :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and f' :: "'e \<Rightarrow> 'd"   and x :: "'d \<Rightarrow> 'f"   and M :: "'f measure" assumes "function_to_finmap J f f'"   and "x \<in> space (Pi\<^sub>M J (\<lambda>_. M))" shows "function_to_finmap.mf J f (function_to_finmap.fm J f f' x) = x"
tagrel_sub_inv: fixes sub :: "'b run"   and r :: "'b run"   and c\<^sub>1 :: "clock"   and c\<^sub>2 :: "clock"   and R :: "'b tag_const \<times> 'b tag_const \<Rightarrow> bool" assumes "sub \<lless> r"   and "r \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "sub \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
vunion_vsingleton_iff: fixes A :: "V"   and B :: "V"   and x :: "V" shows "(A \<union>\<^sub>\<circ> B = set {x}) = (A = 0 \<and> B = set {x} \<or> A = set {x} \<and> B = 0 \<or> A = set {x} \<and> B = set {x})"
Image_runiq_eq_eval: fixes x :: "'a"   and R :: "('a \<times> 'b) set" assumes "x \<in> Domain R"   and "runiq R" shows "R `` {x} = {R ,, x}"
stable_rank_at_def: fixes x :: "nat"   and n :: "nat" shows "stable_rank_at x n \<equiv> \<exists>i. \<forall>m\<ge>n. rank x m = Some i"
distinct_remdups: fixes xs :: "'g list" shows "distinct (remdups xs)"
hrestrict_iff: fixes z :: "hf"   and r :: "hf"   and A :: "hf" shows "(z \<^bold>\<in> hrestrict r A) = (z \<^bold>\<in> r \<and> (\<exists>x y. z = \<langle>x, y\<rangle> \<and> x \<^bold>\<in> A))"
left_inverse_linear: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'd) vec"   and g :: "('c, 'd) vec \<Rightarrow> ('c, 'd) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "g \<circ> f = id" shows "Vector_Spaces.linear (*s) (*s) g"
catch_gpv_bind_gpv: fixes gpv :: "('g, 'e, 'f) gpv"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "catch_gpv (gpv \<bind> f) = catch_gpv gpv \<bind> (\<lambda>x. case x of None \<Rightarrow> Generative_Probabilistic_Value.Done None | Some x' \<Rightarrow> catch_gpv (f x'))"
singleton_not_empty: fixes a :: "'a" shows "{a} \<noteq> {}"
zero_flow: fixes \<Delta> :: "('a, 'b) network_scheme" assumes "countable_network \<Delta>" shows "flow \<Delta> (\<lambda>_. 0)"
cnf_of_literal_formula: fixes f :: "'b formula" assumes "is_literal_formula f" shows "cnf f = {{literal_formula_to_literal f}}"
POT_def: fixes to :: "'a nat_term_order" shows "POT to = Abs_nat_term_order (pot_comp (nat_term_compare to))"
partition_between_poly_def: shows "partition_between_poly = partition_between_ref (\<le>) fst"
exec_move_Call1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and M :: "String.literal"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e\<bullet>M(es)) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
count_add_mset: fixes b :: "'c"   and A :: "'c multiset"   and a :: "'c" shows "count (add_mset b A) a = (if b = a then Suc (count A a) else count A a)"
1: fixes S :: "'b set"   and a :: "'b" assumes "card S \<le> 0" shows "card (a \<triangleright> S) \<le> 1"
rename_JN: fixes h :: "'c \<Rightarrow> 'd"   and I :: "'e set"   and F :: "'e \<Rightarrow> 'c program" assumes "bij h" shows "rename h (JOIN I F) = (\<Squnion>i\<in>I. rename h (F i))"
d\<^sub>2_inject: fixes x :: "\<Pi>\<^sub>2"   and y :: "\<Pi>\<^sub>2" assumes "d\<^sub>2 x = d\<^sub>2 y" shows "x = y"
assms_5: shows "b \<otimes> deriv f a = to_fun f a"
stalk_is_ring: fixes V :: "'a set" assumes "is_open V"   and "x \<in> V" shows "ring stk1.carrier_stalk stk1.add_stalk stk1.mult_stalk (stk1.zero_stalk V) (stk1.one_stalk V)"
sup_pres_disj: fixes x :: "'b"   and y :: "'b" assumes "x \<in> disjunctive"   and "y \<in> disjunctive" shows "x \<squnion> y \<in> disjunctive"
Seq_succ_insf: fixes s :: "hf"   and k :: "hf" assumes "Seq s (succ k)" shows "\<exists>y. s = insf s k y"
history_order_def: fixes x :: "'a event"   and i :: "nat"   and z :: "'a event" shows "x \<sqsubset>\<^sup>i z \<equiv> \<exists>xs ys zs. xs @ x # ys @ z # zs = history i"
abrupt_if_def: fixes c :: "bool"   and x' :: "abopt"   and x :: "abopt" shows "abrupt_if c x' x = (if c \<and> x = None then x' else x)"
predict_imp_adapt: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and N :: "'b measure" assumes "predict_stoch_proc F X N" shows "adapt_stoch_proc F X N"
h_q1_len: shows "1 < length qs \<longrightarrow> 0 < length (take (length qs div 2) qs)"
error_free_FVar_lemma: fixes s :: "state"   and stat :: "bool"   and a :: "val" assumes "error_free s" shows "error_free (abupd (if stat then id else np a) s)"
scaleR_right: fixes a :: "'c \<Rightarrow>\<^sub>L 'b"   and r :: "real"   and b :: "'c" shows "blinfun_apply a (r *\<^sub>R b) = r *\<^sub>R blinfun_apply a b"
alw_holds: fixes P :: "'b \<Rightarrow> bool"   and h :: "'b"   and t :: "'b stream" shows "alw (holds P) (h ## t) = (P h \<and> alw (holds P) t)"
apex_def: shows "Cod_rf.apex \<equiv> C.dom Cod_rf.leg0"
bind_returns_heap_E: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and h'' :: "'a" assumes "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>h h''" obtains x :: "'d" and h' :: "'a" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''"
mC_ZOC_def: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_ZOC theta c d s t I0 P F \<equiv> mC_ZOC_part c d s t I0 P F \<and> inj_on F P \<and> mC_ZOC_wt c d s t I0 P F \<and> mC_ZOC_eff_cont0 theta c d s t I0 F \<and> mC_ZOC_eff_cont theta c d s t I0 P F"
DleqC': shows "P \<turnstile> D \<preceq>\<^sup>* C'"