hcju/isabelleps huggingface dataset:
game3_def: shows "game3 = exec_gpv rf_encrypt \<A>1 Map.empty \<bind> (\<lambda>(((p0, p1), \<sigma>), s1). if valid_plain p0 \<and> valid_plain p1 then coin_spmf \<bind> (\<lambda>b. let pb = if b then p0 else p1 in spmf_of_set (nlists UNIV len) \<bind> (\<lambda>r. spmf_of_set (nlists UNIV len) \<bind> (\<lambda>pad. let cipher = (r, pb [\<oplus>] pad) in exec_gpv rf_encrypt (\<A>2 cipher \<sigma>) s1 \<bind> (\<lambda>(b', s2). return_spmf (b' = b))))) else coin_spmf)"
Ay: shows "A $ x $ (LEAST k. A $ y $ k \<noteq> (0::'a)) = (0::'a)"
RedNew: fixes h' :: "'heap"   and a :: "'addr"   and h :: "'heap"   and C :: "String.literal"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "(h', a) \<in> allocate h (Class_type C)" shows "extTA,P,t \<turnstile> \<langle>new C,(h, l)\<rangle> -\<lbrace>NewHeapElem a (Class_type C)\<rbrace>\<rightarrow> \<langle>addr a,(h', l)\<rangle>"
simps_6: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "not\<^sub>n (\<phi> or\<^sub>n \<psi>) = not\<^sub>n \<phi> and\<^sub>n not\<^sub>n \<psi>"
adjoint_def: fixes f :: "'b \<Rightarrow> 'a" shows "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
that: fixes r :: "real" assumes "x = of_real r"   and "0 \<le> r" shows "thesis"
4: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
a1: shows "d y \<le> d z \<cdot>  |x] y"
r_neq: fixes x :: "nat"   and y :: "nat" shows "eval r_neq [x, y] \<down>= (if x = y then 1 else 0)"
max_ideal_shape_aux: fixes a :: "'b \<Rightarrow> 'c"   and X :: "'b set" shows "(\<lambda>x. monomial (1::'c) (monomial 1 x) - monomial (a x) 0) ` X \<subseteq> P[X]"
face_of_subset_rel_frontier: fixes T :: "'a set"   and S :: "'a set" assumes "T face_of S"   and "T \<noteq> S" shows "T \<subseteq> rel_frontier S"
compositionality_BSI: fixes ES1 :: "'b ES_rec"   and ES2 :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>1 :: "'b V_rec"   and \<V>2 :: "'b V_rec" assumes "Compositionality ES1 ES2 \<V> \<V>1 \<V>2"   and "BSD \<V>1 Tr\<^bsub>ES1\<^esub>"   and "BSD \<V>2 Tr\<^bsub>ES2\<^esub>"   and "BSI \<V>1 Tr\<^bsub>ES1\<^esub>"   and "BSI \<V>2 Tr\<^bsub>ES2\<^esub>" shows "BSI \<V> Tr\<^bsub>ES1 \<parallel> ES2\<^esub>"
equiv_eq_right_lang: shows "equiv (states Collapse_dfa) Collapse.eq_right_lang"
filter_M_reads: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> ('b, 'c, bool) prog"   and h :: "'b"   and h' :: "'b"   and S :: "('b \<Rightarrow> 'b \<Rightarrow> bool) set" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h'"   and "\<And>x. x \<in> set xs \<Longrightarrow> reads S (P x) h h'"   and "\<forall>P\<in>S. reflp P \<and> transp P" shows "reads S (filter_M P xs) h h'"
row_echelon_form_def: fixes A :: "'c mat" shows "row_echelon_form A \<equiv> \<exists>f. pivot_fun A f (dim_col A)"
cont_compose2: fixes c :: "'c \<Rightarrow> 'b \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'b" assumes "\<And>y. cont (\<lambda>x. c x y)"   and "\<And>x. cont (c x)"   and "cont f"   and "cont g" shows "cont (\<lambda>x. c (f x) (g x))"
monad_fail_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b" assumes "monad_fail return bind fail" shows "monad_fail (return_env return) (bind_env bind) (fail_env fail)"
hyps1: shows "card {M. set_mset M \<subseteq> A \<and> size M = Suc k} = card A + Suc k - 1 choose Suc k"
dual_Sup: fixes A :: "'b set"   and inf :: "'b" shows "is_Sup (dual ` A) (dual inf) = is_Inf A inf"
unit_imp_dvd: fixes b :: "'b"   and a :: "'b" assumes "is_unit b" shows "b dvd a"
elims_3: fixes x :: "'a"   and xa :: "'b"   and xb :: "('c, 'd, 'a, 'b, 'e) qAbs" assumes "\<not> qFreshAbs x xa xb" obtains xs :: "'a" and xc :: "'b" and ys :: "'a" and y :: "'b" and X :: "('c, 'd, 'a, 'b, 'e) qTerm" where "x = xs" and "xa = xc" and "xb = qAbs ys y X" and "\<not> (xs = ys \<and> xc = y \<or> qFresh xs xc X)"
1: shows "\<forall>x\<in>MPLS_s PROB s. x \<le> 2 ^ card (prob_dom PROB) - 1"
try_spmf_def: fixes p :: "'c spmf"   and q :: "'c spmf" shows "TRY p ELSE q = p \<bind> (\<lambda>x. case x of None \<Rightarrow> q | Some y \<Rightarrow> return_spmf y)"
axioms_3: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "currying A1 A2 B" shows "category B"
d_of: shows "\<forall>E\<in>EE. DD_of E \<subseteq> CC \<union> {{#- L#} |L. L \<in># E} \<and> finite (DD_of E) \<and> \<not> satisfiable (DD_of E)"
nuc_Inf_closed: fixes f :: "'b \<Rightarrow> 'b" assumes "nucleus f" shows "Inf_closed_set (Fix f)"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> R"   and "y \<in> R" shows "\<eta> (x \<cdot> y) = \<eta> x \<cdot>' \<eta> y"
p_2: shows "\<nexists>a. (\<^bold>\<not> p\<langle>\<^bold>\<dagger> a []/0\<rangle>) \<in> Extend S f"
intros: fixes s :: "'b + 'c"   and x :: "'c" assumes "s = Inr x" shows "x \<in> Basic_BNFs.setr s"
cases: fixes x :: "nat \<times> nat \<times> nat" obtains n :: "nat" and p :: "nat" and q :: "nat" where "x = (Suc (Suc n), pa__, q__)"   | p :: "nat" and q :: "nat" where "x = (0, pb__, qa__)"   | p :: "nat" and q :: "nat" where "x = (Suc 0, p, q)"
bind_tick: fixes m :: "'m"   and f :: "unit \<Rightarrow> 'm" shows "bind (tick m) f = tick (bind m f)"
add_im_sheaf_def: shows "add_im_sheaf \<equiv> \<lambda>V. add_str\<^sub>X (f \<^sup>\<inverse> X V)"
A: fixes w :: "'a" assumes "w \<in> {y--z}" shows "infdist w (\<Union>x\<in>X. cball x r) \<le> C + 8 * deltaG TYPE('a)"
minus_suc: fixes n :: "nat" assumes "0 < n" shows "n - Suc 0 \<noteq> n"
cong_sum_mono_neutral_right: fixes T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and n :: "'b" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. [g i = 0::'b] (mod n)" shows "[sum g T = sum g S] (mod n)"
pderiv_char_poly_erase_mat: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "monom (1::'a) 1 * pderiv (charpoly A) = (\<Sum>i\<in>UNIV. charpoly (erase_mat A i i))"
simps_1: fixes xs :: "'b list" shows "braun_list \<langle>\<rangle> xs = (xs = [])"
rtrancl_list_conv: fixes s :: "'a"   and t :: "'a"   and R :: "('a \<times> 'a) set" shows "((s, t) \<in> R\<^sup>*) = (\<exists>ts. last (s # ts) = t \<and> (\<forall>i<length ts. ((s # ts) ! i, (s # ts) ! Suc i) \<in> R))"
simps_14: fixes v :: "fm"   and va :: "fm" shows "disjuncts (Imp v va) = [Imp v va]"
idem: fixes a :: "'a" shows "a \<cdot> a = a"
dom_comp: fixes g :: "'s"   and f :: "'s" assumes "S.seq g f" shows "S.dom (g \<cdot>\<^sub>S f) = S.dom f"
AbGroup_inner_dirsum: fixes Gs :: "'a set list" assumes "\<forall>G\<in>set Gs. AbGroup G" shows "AbGroup (\<Oplus>G\<leftarrow>Gs. G)"
cbilinear_add_left: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a"   and b :: "'a"   and c :: "'b" assumes "cbilinear f" shows "f (a + b) c = f a c + f b c"
lunit_in_hom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "'a \<times> 'a \<Rightarrow> 'a"   and \<alpha> :: "'a \<times> 'a \<times> 'a \<Rightarrow> 'a"   and \<iota> :: "'a"   and a :: "'a" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.in_hom C (monoidal_category.lunit C T \<alpha> \<iota> a) (T (monoidal_category.unity C \<iota>, a)) a"
h1: shows "sum f {0..<Suc n} = sum f {0..<n} + f n"
bij_lassocr_swap_sum: shows "bij lassocr_swap_sum"
Nats_def: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_1 one times plus zero" shows "semiring_1.Nats one plus zero = range (semiring_1.of_nat one plus zero)"
except_id_iff: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and S :: "'c set" shows "(except p S = p) = (keys p \<inter> S = {})"
f_fin: fixes i :: "nat" shows "f (A i - \<Inter> (range A)) \<noteq> \<infinity>"
p_props_2: shows "\<not> distinct (awalk_verts u p)"
lookup_zip_idx_eq: fixes params :: "'a list"   and args :: "'b list"   and i :: "nat"   and k :: "'a" assumes "length params = length args"   and "i < length args"   and "distinct params"   and "k = params ! i" shows "map_of (zip params args) k = Some (args ! i)"
Abs_inverse: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and y :: "'e" assumes "type_definition Rep Abs A"   and "y \<in> A" shows "Rep (Abs y) = y"
not_finite_existsD: fixes P :: "'a \<Rightarrow> bool" assumes "infinite {a. P a}" shows "\<exists>a. P a"
mayDiverge_If_R: fixes tst :: "'test"   and s :: "'state"   and c2 :: "('test, 'atom) com"   and c1 :: "('test, 'atom) com" assumes "\<not> tval tst s"   and "mayDiverge c2 s" shows "mayDiverge (if tst then c1 else c2) s"
subprob_measurableD_1: fixes N :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and S :: "'d measure"   and x :: "'c" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S"   and "x \<in> space M" shows "space (N x) = space S"
c_of_initial_no_mon: fixes csp :: "'n list multiset" assumes "\<And>s. s \<in># csp \<Longrightarrow> \<exists>p. s = [entry fg p] \<and> initialproc fg p" shows "mon_c fg csp = {}"
aux: fixes n :: "nat" assumes "n \<le> length T" shows "(Fun f T, Fun g (take n S @ drop n T)) \<in> timpl_closure' c"
contour_integrable_add: fixes f1 :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and f2 :: "complex \<Rightarrow> complex" assumes "f1 contour_integrable_on g"   and "f2 contour_integrable_on g" shows "(\<lambda>x. f1 x + f2 x) contour_integrable_on g"
cring_simprules_11: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z)"
fcompress_def: shows "fcompress \<equiv> map_fun fset Abs_fset compress"
lex_fun_plus_monotone: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c"   and u :: "'a \<Rightarrow> 'c" assumes "rlex_fun t s" shows "rlex_fun (t + u) (s + u)"
bit_mask_iff: fixes m :: "nat"   and n :: "nat" shows "bit (mask m) n = ((2::'b) ^ n \<noteq> (0::'b) \<and> n < m)"
mapl_F_cong: fixes x :: "('n, 'o, 'p, 'q, 'r, 's, 't, 'u, 'v, 'w) F"   and l1 :: "'n \<Rightarrow> 'x"   and l1' :: "'n \<Rightarrow> 'x"   and l2 :: "'o \<Rightarrow> 'y"   and l2' :: "'o \<Rightarrow> 'y"   and l3 :: "'p \<Rightarrow> 'z"   and l3' :: "'p \<Rightarrow> 'z" assumes "\<And>z. z \<in> set1_F x \<Longrightarrow> l1 z = l1' z"   and "\<And>z. z \<in> set2_F x \<Longrightarrow> l2 z = l2' z"   and "\<And>z. z \<in> set3_F x \<Longrightarrow> l3 z = l3' z" shows "mapl_F l1 l2 l3 x = mapl_F l1' l2' l3' x"
lfp_gfp: fixes f :: "'b set \<Rightarrow> 'b set" shows "lfp f = - gfp (\<lambda>s. - f (- s))"
Bell_Stirling_eq: fixes m :: "nat" shows "Bell m = sum (Stirling m) {..m}"
tranclpD: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "R\<^sup>+\<^sup>+ x y" shows "\<exists>z. R x z \<and> R\<^sup>*\<^sup>* z y"
f: fixes i :: "nat" shows "f i (n i) = f (Suc i) 0"
unitor_coincidence_1: fixes a :: "'b" assumes "obj a" shows "\<l>[a] = \<i>[a]"
simps_3: fixes t :: "dB"   and s :: "dB"   and k :: "nat" shows "dB.Abs t[s/k] = dB.Abs (t[lift s 0/k + 1])"
evenperm_swap: fixes a :: "'c"   and b :: "'c" shows "evenperm (Fun.swap a b id) = (a = b)"
parAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<parallel> Q) \<parallel> R \<sim>\<^sup>s P \<parallel> Q \<parallel> R"
If_THE_True: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c1 :: "com"   and c2 :: "com" assumes "bval b s"   and "\<down> (c1, s)" shows "Suc (\<down>\<^sub>t (c1, s)) = \<down>\<^sub>t (IF b THEN c1 ELSE c2, s)"
distinct_split1: fixes xs :: "'c list"   and y :: "'c list"   and r :: "'c"   and z :: "'c list" assumes "distinct xs"   and "xs = y @ [r] @ z" shows "r \<notin> set y"
assms_2: shows "b = aa \<and> Oc # list = ba"
calculus_axioms: shows "calculus Bot_F Inf_F (\<lambda>N1 N2. \<G>_Fset N1 \<Turnstile>G \<G>_Fset N2) Red_I_\<G> local.Red_F_\<G>"
drlex_pp_refl: fixes s :: "('a, 'b) pp" shows "drlex_pp s s"
order_eq_iff: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (x \<le> y \<and> y \<le> x)"
D_def: shows "D = c * Beta (- (real_of_int (- int n) / 2) - 1 / 2) (1 / 2) / 2"
intT_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP" shows "ProblemIkTpart_TEModel.intT prot protFw eintT eintF \<sigma> a \<equiv> if ProblemIkTpart.unprot prot protFw \<sigma> then eintT \<sigma> a else if protFw \<sigma> then eintT \<sigma> a \<and> eintF (Tag \<sigma>) [a] = a else eintT \<sigma> a \<and> a \<in> (eintF (Tag \<sigma>) \<circ> singl) ` {b. eintT \<sigma> b}"
induced_automorphism_on_chamber_vertices_f: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and v :: "'b" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "SimplicialComplex.maxsimp X C"   and "v \<in> C" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g v = (if C \<in> f \<turnstile> ChamberComplex.chamber_system X then g v else f v)"
simps_2: shows "X4d X4.Xd1' = D1"
represents_minus: fixes A :: "'b set"   and r :: "'b \<Rightarrow>\<^sub>0 'a"   and x :: "'b"   and B :: "'b set"   and s :: "'b \<Rightarrow>\<^sub>0 'a"   and y :: "'b" assumes "represents A r x"   and "represents B s y" shows "represents (A \<union> B) (r - s) (x - y)"
weakCongSimClosed_1: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "eqvt Rel"   and "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<leadsto>\<guillemotleft>Rel\<guillemotright> p \<bullet> Q"
child_level_gt: fixes p :: "(nat \<times> int) list"   and dir :: "dir"   and d :: "nat" shows "level p \<le> level (child p dir d)"
rR: shows "r \<in> {hs i |i. i < len}"
hyps: fixes xs :: "'a list"   and ys :: "'a list" shows "(replicate n x = xs @ ys) = (\<exists>m\<le>n. xs = replicate m x \<and> ys = replicate (n - m) x)"
mono2_Step: fixes f :: "char list \<Rightarrow> aexp \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "bexp \<Rightarrow> 'b \<Rightarrow> 'b"   and C1 :: "'b acom"   and C2 :: "'b acom"   and S1 :: "'b"   and S2 :: "'b" assumes "\<And>x e S1 S2. S1 \<le> S2 \<Longrightarrow> f x e S1 \<le> f x e S2"   and "\<And>b S1 S2. S1 \<le> S2 \<Longrightarrow> g b S1 \<le> g b S2"   and "C1 \<le> C2"   and "S1 \<le> S2" shows "Step f g S1 C1 \<le> Step f g S2 C2"
map_nth_drop: fixes i :: "nat"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "i < length xs" shows "map f (nth_drop i xs) = nth_drop i (map f xs)"
empt_invariant: shows "{} \<subseteq> reach (set \<Sigma>) \<delta> q\<^sub>0"
scale_left_commute: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> S" shows "a *\<^sub>R b *\<^sub>R x = b *\<^sub>R a *\<^sub>R x"
Y_def: shows "Y \<equiv> X \<union> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t T1"
coprime_gcd: fixes h :: "'b mod_ring poly"   and c1 :: "'b mod_ring"   and c2 :: "'b mod_ring"   and f :: "'b mod_ring poly" assumes "algebraic_semidom_class.coprime (h - [:c1:]) (h - [:c2:])" shows "algebraic_semidom_class.coprime (gcd f (h - [:c1:])) (gcd f (h - [:c2:]))"
keys_for_def: fixes H :: "chmsg set" shows "keys_for H \<equiv> {K. \<exists>c M. DynCh c K M \<in> H}"
DIAnd_result_correct: shows "proof_result (proof_take 61 DIAndProof) = DIAnd"
Lab: fixes G :: "prog"   and s0 :: "st"   and c :: "stmt"   and s1 :: "state"   and l :: "jump" assumes "G\<turnstile>Norm s0 \<midarrow>c\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>l\<bullet> c\<rightarrow> abupd (absorb l) s1"
meet_mono: fixes x :: "'c"   and z :: "'c"   and y :: "'c"   and w :: "'c" assumes "x \<sqsubseteq> z"   and "y \<sqsubseteq> w" shows "x \<sqinter> y \<sqsubseteq> z \<sqinter> w"
is_regular_uminus_sheaf_spec: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set" assumes "is_regular s U"   and "U \<subseteq> Spec" shows "is_regular (uminus_sheaf_spec U s) U"
derive_pos_mono: fixes x :: "'b sort_key extended"   and y :: "'b sort_key extended"   and h :: "'b \<times> sort_dir" assumes "x \<le> y" shows "derive_pos h x \<le> derive_pos h y"
superposition_preserves_finiteness: fixes P1 :: "'a eclause"   and P2 :: "'a eclause"   and C :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "finite (cl_ecl P1)"   and "finite (cl_ecl P2)"   and "superposition P1 P2 C \<sigma> k C'" shows "finite (cl_ecl C) \<and> finite C'"
even_count_lconjseq_rev_relator: fixes s :: "'w"   and t :: "'w"   and x :: "'w" assumes "s \<in> S"   and "t \<in> S" shows "gcd_nat.greater_eq (count_list (lconjseq (rev (pair_relator_list s t))) x) 2"
seq_extract_def: fixes A :: "nat set"   and xs :: "'b list" shows "A \<upharpoonleft>\<^sub>l xs = nths xs A"
incl_subgroup: fixes J :: "('c \<Rightarrow> 'a) set"   and I :: "('c \<Rightarrow> 'a) set" assumes "subgroup J (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"   and "subgroup I (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<lparr>carrier := J\<rparr>)" shows "subgroup I (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"
lim_g: shows "(sum g \<longlongrightarrow> infsetsum' g A) (finite_subsets_at_top A)"
IH: shows "local.coeff (p @ q) = (\<lambda>i. if i < length q then local.coeff q i else local.coeff p (i - length q))"
ABsubset: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "cbox (A n) (B n) \<subseteq> cbox (A m) (B m)"
ls_delete_map_of: fixes l :: "('a \<times> 'b) list"   and k :: "'a"   and x :: "'a" assumes "distinct (map fst l)" shows "map_of (fst (ls_delete k l)) x = (map_of l |` (- {k})) x"
c_ipurge_aux_singleton: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and u :: "'f"   and xs :: "'e list" shows "c_ipurge_aux I D {u} xs = c_ipurge I D u xs"
split_conv: fixes f :: "'l \<Rightarrow> 'm \<Rightarrow> 'k"   and a :: "'l"   and b :: "'m" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
fall2_intro: fixes \<pi>l :: "('state \<times> 'aprop set) stream list"   and F :: "('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<And>\<pi> \<pi>'. \<lbrakk>wfp AP' \<pi>; wfp AP' \<pi>'; \<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l); \<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0; stateOf \<pi>' = stateOf \<pi>\<rbrakk> \<Longrightarrow> F \<pi> \<pi>' \<pi>l" shows "fall2 F \<pi>l"
strip_context_liftn: fixes n :: "nat"   and m :: "nat"   and t :: "dB"   and k :: "nat" shows "strip_context n (liftn (m + n) t k) k = liftn m t k"
iso_sym: fixes H :: "('c, 'd) monoid_scheme" assumes "G \<cong> H" shows "H \<cong> G"
normal_inv_iff: fixes N :: "'a set" shows "N \<lhd> G\<lparr>carrier := S\<rparr> = (subgroup N (G\<lparr>carrier := S\<rparr>) \<and> (\<forall>x\<in>carrier (G\<lparr>carrier := S\<rparr>). \<forall>h\<in>N. x \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> h \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x \<in> N))"
HC_intros_5: fixes F :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H F \<^bold>\<rightarrow> F \<^bold>\<or> G"
set_take_subset: fixes n :: "nat"   and xs :: "'a list" shows "set (take n xs) \<subseteq> set xs"
inf_exists: fixes A :: "'a set" assumes "A \<subseteq> carrier (L\<lparr>carrier := f ` carrier L\<rparr>)" shows "\<exists>i. is_glb (L\<lparr>carrier := f ` carrier L\<rparr>) i A"
uniformity_quat_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
that: fixes r :: "V" assumes "x = \<langle>r, \<F>\<^sub>\<circ> r\<rangle>"   and "r \<in>\<^sub>\<circ> A" shows "thesis"
f: shows "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
that: assumes "h' = snd (f h)"   and "r = fst (f h)" shows "thesis"
arr_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VxV.arr f = (arr (fst f) \<and> arr (snd f))"
not_eq_order_implies_strict: fixes a :: "'a list"   and b :: "'a list" assumes "a \<noteq> b"   and "prefix a b" shows "strict_prefix a b"
bheight_baliR: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "bheight l = bheight r" shows "bheight (baliR l a r) = Suc (bheight l)"
less_option_None_Some_code: fixes x :: "'a" shows "(None < Some x) = True"
distinguishing_weak_formula_supp_aux: fixes P :: "'state"   and Q :: "'state" assumes "\<not> P \<equiv>\<cdot> Q" shows "supp (Abs_bset {- p \<bullet> (SOME x. weak_formula x \<and> supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) |p. True}) \<subseteq> supp P"
well_def_subsets1: shows "all_list_constr_R subsets1 (length qs1)"
a: assumes "x = y" shows "filter P (merge_wrt rel xs ys) = filter P xs"
is_closed_eq_rev: fixes S :: "('a, 'b) eq_object_scheme"   and A :: "'a set"   and x' :: "'a"   and x :: "'a" assumes "equivalence S"   and "is_closed\<^bsub>S\<^esub> A"   and "x' \<in> A"   and "x .=\<^bsub>S\<^esub> x'"   and "x \<in> carrier S" shows "x \<in> A"
refl_rtrancl: fixes r :: "('g \<times> 'g) set" shows "refl (r\<^sup>*)"
drop_half_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "drop_half (map f xs) = map f (drop_half xs)"
inv_loop_nonempty_7: fixes x :: "nat"   and xs :: "cell list" shows "inv_loop6 x ([], Bk # xs) = False"
dom_MkArr: fixes A :: "'c \<Rightarrow> 's"   and B :: "'c \<Rightarrow> 's"   and F :: "'c \<Rightarrow> 's" assumes "Cop_S.arr (Cop_S.MkArr A B F)" shows "Cop_S.dom (Cop_S.MkArr A B F) = Cop_S.MkIde A"
simps_2: fixes xs :: "'a list" shows "local.merge xs [] = xs"
horizontal_composition_axioms: shows "horizontal_composition (\<cdot>) (\<star>) src trg"
divide_real_def: fixes x :: "real"   and y :: "real" shows "x / y = x * inverse y"
isArrV_def: fixes v :: "Value" shows "isArrV v = (case v of arrV T a \<Rightarrow> True | _ \<Rightarrow> False)"
chain': fixes x :: "'b" assumes "x \<in> Y" shows "Complete_Partial_Order.chain (\<le>) (f x ` Y)"
A7_NSources: shows "\<forall>C\<in>AbstrLevel level0. C \<noteq> sA9 \<and> C \<noteq> sA8 \<longrightarrow> sA7 \<notin> Sources level0 C"
less_top_ennreal: fixes x :: "ennreal" shows "(x < top) = (\<exists>r\<ge>0. x = ennreal r)"
filter_nth: fixes i :: "nat"   and P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "i < length (filter P xs)" shows "P (filter P xs ! i)"
atLeastLessThan_iff: fixes i :: "'d Word.word"   and l :: "'d Word.word"   and u :: "'d Word.word" shows "(i \<in> signed.atLeastLessThan l u) = (l \<le>s i \<and> i <s u)"
intro: fixes R :: "'a set"   and M :: "'b set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "R_scalar_mult R"   and "Group M"   and "RModule_axioms R smult M" shows "RModule R smult M"
count_of_multiset: fixes xs :: "('a \<times> nat) list" shows "count_of xs \<in> multiset"