hcju/isabelleps huggingface dataset:
ccBinds_Cons: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and x :: "Vars.var"   and e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "CoCallAnalysis.ccBinds ccExp ((x, e) # \<Gamma>) = CoCallAnalysis.ccBind ccExp x e \<squnion> CoCallAnalysis.ccBinds ccExp (delete x \<Gamma>)"
argument_principle_poly: fixes p :: "complex poly"   and g :: "real \<Rightarrow> complex" assumes "p \<noteq> 0"   and "valid_path g"   and "pathfinish g = pathstart g"   and "path_image g \<subseteq> - proots p" shows "contour_integral g (\<lambda>x. deriv (poly p) x / poly p x) = 2 * complex_of_real pi * \<i> * (\<Sum>x\<in>proots p. winding_number g x * of_nat (order x p))"
assms: shows "limsup (\<lambda>x. ereal (u x)) < \<infinity>"
avoid_def: shows "avoid \<equiv> \<lambda>p e. \<forall>w\<in>cball p e. w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pz)"
code: fixes z :: "octo" shows "Octonions.cnj z = Octo (Ree z) (- Im1 z) (- Im2 z) (- Im3 z) (- Im4 z) (- Im5 z) (- Im6 z) (- Im7 z)"
terminals_def: fixes ts :: "(symbol \<times> character list) list" shows "terminals ts = map terminal_of_token ts"
le_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less" shows "less_eq x y = (less x y \<or> x = y)"
UN_finite_bound: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a set" assumes "finite A"   and "A \<subseteq> \<Union> (range f)" shows "\<exists>m. A \<subseteq> \<Union> (f ` {..m})"
homeomorphic_imp_homotopy_eqv: fixes S :: "'b set"   and T :: "'c set" assumes "S homeomorphic T" shows "S homotopy_eqv T"
measurable_restrict_space3: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> A \<rightarrow> B" shows "f \<in> restrict_space M A \<rightarrow>\<^sub>M restrict_space N B"
nonneg_indicator: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and V :: "nat set" assumes "\<Gamma> \<turnstile>\<^sub>c e : BOOL"   and "free_vars e \<subseteq> V" shows "nonneg_cexpr V \<Gamma> (\<langle>e\<rangle>\<^sub>c)"
locally_essentially_surjective: fixes a :: "'b"   and b :: "'b"   and g :: "'c" assumes "C.obj a"   and "C.obj b"   and "\<guillemotleft>g : G.map\<^sub>0 a \<rightarrow>\<^sub>D G.map\<^sub>0 b\<guillemotright>"   and "D.ide g" shows "\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<and> C.ide f \<and> D.isomorphic (G f) g"
is_commit_sequence_def: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and J :: "nat \<Rightarrow> ('b, 'a) justifying_execution" shows "is_commit_sequence E J = (committed (J 0) = {} \<and> (\<forall>n. action_translation (J n) ` committed (J n) \<subseteq> action_translation (J (Suc n)) ` committed (J (Suc n))) \<and> actions E = (\<Union>n. action_translation (J n) ` committed (J n)))"
add_closed: fixes g :: "'g"   and h :: "'g" assumes "g \<in> G"   and "h \<in> G" shows "g + h \<in> G"
convex_onD_Icc: fixes x :: "'c"   and y :: "'c"   and f :: "'c \<Rightarrow> real"   and t :: "real" assumes "convex_on {x..y} f"   and "x \<le> y"   and "0 \<le> t"   and "t \<le> 1" shows "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
induct: fixes x1 :: "('a, 'b, 'c) subst"   and x2 :: "('a, 'c) trm"   and P :: "('a, 'b, 'c) subst \<Rightarrow> ('a, 'c) trm \<Rightarrow> bool" assumes "Tadmit x1 x2"   and "\<And>\<sigma> \<theta>. \<lbrakk>Tadmit \<sigma> \<theta>; P \<sigma> \<theta>; TUadmit \<sigma> \<theta> UNIV\<rbrakk> \<Longrightarrow> P \<sigma> (Differential \<theta>)"   and "\<And>\<sigma> args f f'. \<lbrakk>\<And>i. Tadmit \<sigma> (args i); \<And>i. P \<sigma> (args i); SFunctions \<sigma> f = Some f'; TadmitFO (\<lambda>i. Tsubst (args i) \<sigma>) f'\<rbrakk> \<Longrightarrow> P \<sigma> ($f f args)"   and "\<And>\<sigma> args f. \<lbrakk>\<And>i. Tadmit \<sigma> (args i); \<And>i. P \<sigma> (args i); SFunctions \<sigma> f = None\<rbrakk> \<Longrightarrow> P \<sigma> ($f f args)"   and "\<And>\<sigma> \<theta>1 \<theta>2. \<lbrakk>Tadmit \<sigma> \<theta>1; P \<sigma> \<theta>1; Tadmit \<sigma> \<theta>2; P \<sigma> \<theta>2\<rbrakk> \<Longrightarrow> P \<sigma> (Plus \<theta>1 \<theta>2)"   and "\<And>\<sigma> \<theta>1 \<theta>2. \<lbrakk>Tadmit \<sigma> \<theta>1; P \<sigma> \<theta>1; Tadmit \<sigma> \<theta>2; P \<sigma> \<theta>2\<rbrakk> \<Longrightarrow> P \<sigma> (Times \<theta>1 \<theta>2)"   and "\<And>\<sigma> x. P \<sigma> ($' x)"   and "\<And>\<sigma> x. P \<sigma> (Var x)"   and "\<And>\<sigma> r. P \<sigma> (Const r)" shows "P x1 x2"
valid_lists_mono1: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and G :: "CoCalls" assumes "S \<subseteq> S'" shows "valid_lists S G \<subseteq> valid_lists S' G"
finsert_ident: fixes x :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "x |\<notin>| A"   and "x |\<notin>| B" shows "(finsert x A = finsert x B) = (A = B)"
rq_suf_suf: fixes r :: "'b list"   and s :: "'b list" assumes "r \<le>s s" shows "s\<^sup><\<inverse>r  \<le>p s"
induced_modulus_def: fixes d :: "nat" shows "principal.induced_modulus d = (d dvd n \<and> (\<forall>a. coprime a n \<and> [a = 1] (mod d) \<longrightarrow> principal_dchar n a = 1))"
LAssRedsThrow: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 E :: "char list \<Rightarrow> ty option"   and V :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>V:=e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
HC_intros_13: fixes F :: "'b formula" shows "AX10 \<turnstile>\<^sub>H (\<^bold>\<not> F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> F"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "inf b (inf a c) = inf a (inf b c)"
row_space_mult: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nr" shows "row_space (B * A) \<subseteq> row_space A"
inj2_def: fixes M1 :: "('g, 'f) module"   and M2 :: "('g, 'e) module" shows "inj2 M1 M2 = Pair \<zero>\<^bsub>M1\<^esub>"
comp_fun_idem_sample_var: fixes X :: "'v \<Rightarrow> int pmf" shows "comp_fun_idem (\<lambda>x m. bind (memo (sample_var X) x) (\<lambda>_. m))"
finite_measure_axioms: fixes s :: "'a" shows "finite_measure (K1.T s)"
\<rho>': shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r' \<rho>' f g"
id: shows "coeffs (smult a p) = map ((*) a) (coeffs p)"
hyps_1: fixes H :: "'a set" assumes "n = card H"   and "finite H"   and "H \<subseteq> carrier G"   and "J \<subseteq> carrier G" shows "card (H <#> J) \<le> card H * card J"
sum_representation_eq: fixes basis :: "'d set"   and v :: "'d"   and B :: "'d set" assumes "cindependent basis"   and "v \<in> cspan basis"   and "finite B"   and "basis \<subseteq> B" shows "(\<Sum>b\<in>B. crepresentation basis v b *\<^sub>C b) = v"
FOREACHoci_emp: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<Phi> :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma> :: "'c" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,\<Phi>\<^esup> {} c f \<sigma> = ASSERT (\<Phi> {} \<sigma>) \<bind> (\<lambda>_. RETURN \<sigma>)"
nres_of_simps_3: fixes x :: "'a" shows "nres_of (dRETURN x) = RETURN x"
assms_2: shows "S \<subseteq> U\<^sub>M\<^sub>_\<^sub>1"
RBT_HM_inverse: fixes y :: "(uint32, ('a, 'b) assoc_list) RBT.rbt" assumes "y \<in> {hm. HashMap_Impl.invar hm}" shows "impl_of_RBT_HM (RBT_HM y) = y"
assoc_inv: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "inverse_arrows (MC.assoc a b c) (MC.assoc' a b c)"
strict_sorted_imp_sorted: fixes xs :: "'a list" assumes "local.strict_sorted xs" shows "local.sorted xs"
forest_bot: shows "bot \<sqinter> id bot \<le> top \<and> bot \<sqinter> top \<le> - top"
lower_interval_of: fixes a :: "'a" shows "lower (interval_of a) = a"
children_def: fixes BinomialTree :: "('c, 'd) BinomialTree" shows "children BinomialTree = (case BinomialTree of Node x1 x2 x3 x4 \<Rightarrow> x4)"
LeftDerivationFix_implies_ex_ladder: fixes a :: "symbol list"   and i :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationFix a i D j \<gamma>" shows "\<exists>L. LeftDerivationLadder a D L \<gamma> \<and> ladder_last_j L = j \<and> ladder_last_n L = length D"
vsv_axioms: shows "vsv (\<BB>\<lparr>Dom\<rparr>)"
mult1_def: fixes r :: "('d \<times> 'd) set" shows "mult1 r = {(N, M). \<exists>a M0 K. M = add_mset a M0 \<and> N = M0 + K \<and> (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"
desc_nec_encode: fixes v :: "i"   and \<phi> :: "\<Pi>\<^sub>1 \<Rightarrow> \<o>"   and G :: "\<Pi>\<^sub>1" shows "[\<lbrace>\<^bold>\<iota>x. \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& (\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<phi> F),G\<rbrace> \<^bold>\<equiv> \<^bold>\<A>\<phi> G in v]"
le_trans: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "weak_partial_order L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "y \<sqsubseteq>\<^bsub>L\<^esub> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> z"
hcomplex_approx_iff: fixes x :: "complex star"   and y :: "complex star" shows "(x \<approx> y) = (hRe x \<approx> hRe y \<and> hIm x \<approx> hIm y)"
denom_dvd_ints: fixes r :: "rat"   and z :: "int"   and n :: "int"   and i :: "int" assumes "quotient_of r = (z, n)"   and "rat_of_int i * r \<in> \<int>" shows "n dvd i"
WT1: shows "\<I>, \<I>' \<turnstile>\<^sub>C conv1 \<surd>"
set_intros_2: fixes y :: "'d"   and x22 :: "'d list"   and x21 :: "'d" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
len_match: shows "length (rows_to_keep (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))) = dim_col (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))"
QQ7_a_def: shows "QQ7_a \<equiv> [: p, t, left, right, atom, mrk \<leadsto> p', t', left', right', atom', mrk' . p \<noteq> nil \<and> \<not> gg mrk atom left p \<and> \<not> gg mrk atom right p \<and> t = nil \<and> p' = nil \<and> t' = t \<and> mrk' = mrk \<and> left' = left \<and> right' = right \<and> atom' = atom :]"
poly_map_poly: fixes p :: "'a poly poly"   and x :: "'a poly" shows "poly (map_poly (map_poly hom) p) (map_poly hom x) = map_poly hom (poly p x)"
assms_2: assumes "rt1 \<sqsubseteq>\<^bsub>i\<^esub> rt2"   and "\<not> rt2 \<sqsubseteq>\<^bsub>i\<^esub> rt1" shows "P rt1 rt2 i"
flowsto_congI2: fixes A :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set"   and B :: "real set"   and C :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set"   and D :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set" assumes "c1_on_open_euclidean.flowsto g f' Y A B C D" shows "flowsto A B C D"
typ_of1_fun_typ: fixes Ts :: "typ list"   and t :: "Core.term"   and u :: "Core.term"   and R :: "typ"   and U :: "typ" assumes "typ_of1 Ts (t $ u) = Some R"   and "typ_of1 Ts t = Some (U \<rightarrow> R)" shows "typ_of1 Ts u = Some U"
sdescending_scons: fixes a :: "'b"   and w :: "'b stream" assumes "sdescending (a ## w)" shows "sdescending w"
ih: fixes y :: "name"   and \<theta> :: "(name \<times> trm) list"   and s :: "trm" assumes "y \<sharp> \<theta>" shows "\<theta><t[y::=s]> = \<theta><t>[y::=\<theta>
]"
exhaust: fixes y :: "enat" obtains nat :: "nat" where "y = enat nat"   | "y = \<infinity>"
var: fixes x :: "'b" shows "PVar x \<approx> PVar x"
simps: fixes B :: "ty"   and X :: "ntrm set" shows "NOTRIGHT (NOT B) X = {<a>:NotR x.M a |a x M. fic (NotR x.M a) a \<and> x:M \<in> X}"
comp_arr_dom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.dom C f = a" shows "C f a = f"
char_matrix_def: fixes A :: "'a mat"   and e :: "'a" shows "char_matrix A e = A + - e \<cdot>\<^sub>m 1\<^sub>m (dim_row A)"
simps_2: fixes f :: "char list"   and ts :: "fterm list"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "Fun f ts \<cdot>\<^sub>t \<sigma> = Fun f (ts \<cdot>\<^sub>t\<^sub>s \<sigma>)"
point_mult_mult: fixes a :: "int"   and b :: "int"   and p :: "int point"   and m :: "nat"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p"   and "nonsingular a b" shows "point_mult a (m * n) p = point_mult a n (point_mult a m p)"
Z: shows "esqrt 2 = eexp (ereal (ln 2 / 2))"
terminates_drop_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step" assumes "terminates g" shows "terminates (drop_raw g)"
eventually_uniformity_metric: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and P :: "'d \<times> 'd \<Rightarrow> bool" assumes "class.uniformity_dist dist uniformity" shows "eventually P uniformity = (\<exists>e>0. \<forall>x y. dist x y < e \<longrightarrow> P (x, y))"
fin: shows "emeasure MX {x \<in> space MX. Px x \<noteq> 0} \<noteq> top"
wf_disc_sel': shows "wf_disc_sel (disc, sel) C"
cmod_sin_le_exp: fixes u :: "real"   and z :: "complex" assumes "0 \<le> u"   and "u \<le> 1" shows "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
DINING_is_DINING\<^sub>n\<^sub>o\<^sub>r\<^sub>m: shows "DINING = DINING\<^sub>n\<^sub>o\<^sub>r\<^sub>m N (replicate N 0, replicate N 0)"
rel_inject_4: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x4 :: "'a \<times> nat"   and y4 :: "'b \<times> nat" shows "rel_SDG_node R (Actual_in x4) (Actual_in y4) = rel_prod R (=) x4 y4"
restrict_inj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and A1 :: "'c set" assumes "inj_on f A"   and "A1 \<subseteq> A" shows "inj_on f A1"
Abs_aezfun_inject: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> aezfun_set"   and "y \<in> aezfun_set" shows "(Abs_aezfun x = Abs_aezfun y) = (x = y)"
nat_approx_posE: fixes e :: "'b" assumes "(0::'b) < e" obtains n :: "nat" where "(1::'b) / of_nat (Suc n) < e"
suminf_split_head: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f" shows "(\<Sum>n. f (Suc n)) = suminf f - f 0"
induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "star r x1 x2"   and "\<And>x. P x x"   and "\<And>x y z. \<lbrakk>r x y; star r y z; P y z\<rbrakk> \<Longrightarrow> P x z" shows "P x1 x2"
thin_triangles: fixes Gxy :: "'a set"   and x :: "'a"   and y :: "'a"   and Gxz :: "'a set"   and z :: "'a"   and Gyz :: "'a set"   and w :: "'a" assumes "geodesic_segment_between Gxy x y"   and "geodesic_segment_between Gxz x z"   and "geodesic_segment_between Gyz y z"   and "w \<in> Gyz" shows "infdist w (Gxy \<union> Gxz) \<le> 4 * deltaG TYPE('a)"
perm_append_single: fixes a :: "'a"   and xs :: "'a list" shows "a # xs <~~> xs @ [a]"
iso_\<phi>: fixes f :: "'c" assumes "C.ide f" shows "D.iso (\<tau>.\<phi> f)"
simps_8: fixes V :: "'c set"   and t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and S :: "('d, 'c) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (select\<langle>t,s\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t (V \<union> fv t \<union> fv s) S"
type_definition_conjugate_space: shows "type_definition from_conjugate_space to_conjugate_space UNIV"
pdivides_imp_degree_le: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subring K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "q \<noteq> []"   and "p pdivides\<^bsub>poly_ring R\<^esub> q" shows "degree p \<le> degree q"
plus_u_def: fixes x :: "'a\<^sub>\<bottom>"   and y :: "'a\<^sub>\<bottom>" shows "x + y = (case x of up\<cdot>a \<Rightarrow> \<Lambda> (up\<cdot>b). up\<cdot>(a + b))\<cdot>y"
reflexive: fixes p :: "'c"   and A :: "('d, 'c) nba" shows "p \<in> NBA.reachable A p"
not_less_if_bot: fixes b :: "alt"   and p :: "alt \<Rightarrow> real"   and c :: "alt" assumes "b \<cdot>< p" shows "\<not> p c < p b"
perm_refl: fixes l :: "'b list" shows "l <~~> l"
space_extend_measure: fixes G :: "'c \<Rightarrow> 'b set"   and I :: "'c set"   and \<Omega> :: "'b set"   and \<mu> :: "'c \<Rightarrow> ennreal" assumes "G ` I \<subseteq> Pow \<Omega>" shows "space (extend_measure \<Omega> I G \<mu>) = \<Omega>"
minus_u_def: fixes x :: "'b\<^sub>\<bottom>"   and y :: "'b\<^sub>\<bottom>" shows "x - y = (case x of up\<cdot>a \<Rightarrow> \<Lambda> (up\<cdot>b). up\<cdot>(a - b))\<cdot>y"
ord_le_eq_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<sqsubseteq> b"   and "b = c" shows "a \<sqsubseteq> c"
eqS: shows "\<Union> (range S) = (\<Union>n. \<Union> (S ` {..n}))"
VarNonOccForm_imp_dbfm_fresh: fixes v :: "hf"   and x :: "hf"   and e :: "(name, hf) finfun" assumes "VarNonOccForm v x" shows "\<exists>A. wf_dbfm A \<and> x = \<lbrakk>quot_dbfm A\<rbrakk>e \<and> atom (decode_Var v) \<sharp> A"
decode_plan_step_distinct: fixes k :: "nat"   and t :: "nat"   and \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "k < t" shows "distinct ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"
read_def: fixes c :: "'e \<Rightarrow> 'c"   and x :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)"   and P :: "('c, 'd) action" shows "read c x P \<equiv> iPrefix (\<lambda>A. c (select x A)) (\<lambda>(s, s'). \<exists>a. s' = increase x a s) (Denotational_Semantics.Let x) (\<lambda>_. range c) P"
star_of_numeral: fixes k :: "num" shows "star_of (numeral k) = numeral k"
A2: shows "0 \<le> padic_val p (a \<oplus>\<^bsub>padic_int p\<^esub> b)"
execn_elim_cases_3: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Abrupt s\<rangle> =n\<Rightarrow> t" obtains "t = Abrupt s"
accepts_atom: fixes a :: "'b"   and w :: "'b list" shows "NAe.accepts (atom a) w = (w = [a])"
integral_one_over_x_ln_x_squared_3: fixes a :: "real" assumes "1 < a" shows "((\<lambda>t. 1 / (t * (ln t)\<^sup>2)) has_integral 1 / ln a) {a<..}"
mult_of_def: fixes R :: "('e, 'f) ring_scheme" shows "mult_of R \<equiv> \<lparr>carrier = carrier R - {\<zero>\<^bsub>R\<^esub>}, mult = (\<otimes>\<^bsub>R\<^esub>), one = \<one>\<^bsub>R\<^esub>\<rparr>"
transfer_prover_start: fixes x :: "'a"   and x' :: "'a"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b" assumes "x = x'"   and "Transfer.Rel R x' y" shows "Transfer.Rel R x y"
empty_imp_negligible: fixes S :: "'c set" assumes "S = {}" shows "negligible S"
hom_zero: fixes hom :: "'a \<Rightarrow> 'b" assumes "zero_hom hom" shows "hom (0::'a) = (0::'b)"
fU: fixes U :: "'a set" assumes "U \<subseteq> S" shows "f ` U = T \<inter> g -` U"
poly_mult: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p1 :: "'b list"   and p2 :: "'b list"   and x :: "'b" assumes "class.comm_semiring_0 plus zero times" shows "semiring_0.poly plus zero times (semiring_0.pmult plus zero times p1 p2) x = times (semiring_0.poly plus zero times p1 x) (semiring_0.poly plus zero times p2 x)"
simps_3: fixes e :: "'c match_expr"   and v :: "'c match_expr"   and va :: "'c match_expr list" shows "MatchAnd_foldr (e # v # va) = MatchAnd e (MatchAnd_foldr (v # va))"
5: shows "d\<^sup>T * top \<le> - ((w \<sqinter> - d)\<^sup>T\<^sup>\<star> * e\<^sup>T * top)"
simps_2: fixes p :: "perm"   and x :: "'b fs_set \<times> 'a" shows "p \<bullet> EF x = EF (p \<bullet> x)"
prems_4: shows "path A (w # ws || r1 # r1s) p1 \<and> path B (w # ws || r2 # r2s) p2 \<or> target (w # ws || r1 # r1s || r2 # r2s) (p1, p2) = FAIL \<and> 0 < length (w # ws) \<and> path A (butlast (w # ws || r1 # r1s)) p1 \<and> path B (butlast (w # ws || r2 # r2s)) p2 \<and> succ A (last (w # ws)) (target (butlast (w # ws || r1 # r1s)) p1) = {} \<and> succ B (last (w # ws)) (target (butlast (w # ws || r2 # r2s)) p2) \<noteq> {}"
OUT_f_outside: fixes x :: "'v"   and n :: "nat" assumes "x \<notin> \<^bold>V" shows "d_OUT (f n) \<langle>x\<rangle> = 0"
eval': shows "(cms, subst (to_partial mem |` X) mem) \<leadsto> (cms', subst (g (to_partial mem |` X)) mem')"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and xa :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (x - xa) = (\<lambda>k. lookup x k - lookup xa k)"
convex_hull_empty_1: fixes n :: "nat" shows "gram_schmidt.convex_hull n {} = {}"
multiple_1_same: shows "sys1.multiple_blocks 1 = \<B>"
exec_fdom_empty_prob: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" assumes "prob_dom PROB = {}"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "exec_plan s as = fmempty"
a3: fixes x :: "'a" assumes "x \<in> G" shows "A x = B x"
a_distrib: shows "SIDE_PRECOND (\<forall>x y. a (x \<union> y) = a x \<inter> a y)"
prod_ord_rawD: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ys :: "('a \<times> 'f) list \<times> 'o"   and P :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and k :: "'a" assumes "oalist_inv xs"   and "oalist_inv ys"   and "prod_ord_raw P xs ys"   and "k \<in> fst ` set (fst xs) \<union> fst ` set (fst ys)" shows "P k (lookup_raw xs k) (lookup_raw ys k)"
main_eq: shows "map vlist_of_vfsequence (vlist_of_vfsequence xs) = ys"
\<delta>: shows "item_\<beta> x = t # \<delta>"
Inf_pres_Inf_closed: fixes f :: "'b \<Rightarrow> 'c" assumes "Inf_pres f" shows "Inf_closed_set (range f)"
card_Diff1_le: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "card (A - {x}) \<le> card A"
prod_ins_def: shows "prod_ins = (\<lambda>(i, j) (tab, ps). (tab[i := (tab ! i)[j := Some (length ps)]], ps @ [(i, j)]))"
in_sccsI: fixes c :: "('a, 'b) pre_digraph" assumes "induced_subgraph c G"   and "strongly_connected c"   and "\<nexists>c'. induced_subgraph c' G \<and> strongly_connected c' \<and> verts c \<subset> verts c'" shows "c \<in> sccs"
iterative_HSem: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> Value" assumes "x \<notin> domA \<Gamma>" shows "\<lbrace>(x, e) # \<Gamma>\<rbrace>\<rho> = (\<mu> xa. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<lbrace>\<Gamma>\<rbrace>xa))(x := \<lbrakk> e \<rbrakk>\<^bsub>xa\<^esub>))"
ImpL: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and G :: "'b formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n"   and "G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F \<^bold>\<rightarrow> G, \<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
skip: fixes p :: "assertion" shows "prov_triple (p, Skip, p)"
WTrtBinOpEq: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E,h,sh \<turnstile> e\<^sub>1 : T\<^sub>1"   and "P,E,h,sh \<turnstile> e\<^sub>2 : T\<^sub>2" shows "P,E,h,sh \<turnstile> e\<^sub>1 \<guillemotleft>Eq\<guillemotright> e\<^sub>2 : Boolean"
line_integral_ex: shows "line_integral_exists F {b} \<gamma>2"
tarski: fixes x :: "'a" assumes "regular x"   and "x \<noteq> bot" shows "times_top_class.total (top * x)"