hcju/isabelleps huggingface dataset:
ih: fixes tp' :: "cell list \<times> cell list" assumes "steps0 (1, tp) A n = (0, tp')" shows "\<exists>n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')"
hyps: fixes i :: "nat"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis # sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p, is, \<theta>, sb, \<D>', \<O>, \<R>')]) m = flush_all_until_volatile_write ts m"
sel_3: fixes x21 :: "'k"   and x22 :: "'k list" shows "tl (x21 # x22) = x22"
fin_atom: fixes a :: "'b"   and q :: "bool list" shows "fin (atom a) q = (q = [False])"
\<F>_semantics': fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F> \<phi> w = subformulas\<^sub>\<mu> \<phi> \<inter> {\<psi>. w \<Turnstile>\<^sub>n F\<^sub>n \<psi>}"
phis'_code_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and "next" :: "'node \<times> 'val" shows "uninst_code.phis'_code p g next \<equiv> map_values (\<lambda>(n, v) vs. if v = snd next then None else Some (map (uninst_code.substNext_code p g next) vs)) (p g)"
step: shows "\<Gamma>\<turnstile> cfg \<rightarrow> cfg'"
leI: fixes x :: "'a word"   and y :: "'a word" assumes "\<not> x <s y" shows "y \<le>s x"
validF: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
keys_homogenizeE: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "t \<in> keys (homogenize x p)" obtains t' :: "'b \<Rightarrow>\<^sub>0 nat" where "t' \<in> keys p" and "t = monomial (poly_deg p - deg_pm t') x + t'"
induced_automorphism_respects_labels: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and B :: "'c set"   and \<phi> :: "'b \<Rightarrow> 'c"   and v :: "'b" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "ChamberComplex.label_wrt X B \<phi>"   and "v \<in> \<Union> X" shows "\<phi> (OpposedThinChamberComplexFoldings.induced_automorphism X f g v) = \<phi> v"
distinct_155: shows "GreaterOrEqual \<noteq> BinXor"
reindex_bij_witness: fixes S :: "'c set"   and i :: "'d \<Rightarrow> 'c"   and j :: "'c \<Rightarrow> 'd"   and T :: "'d set"   and h :: "'d \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'e" assumes "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"   and "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"   and "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"   and "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"   and "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" shows "sum g S = sum h T"
simps_6: fixes \<phi> :: "atom fm" shows "clearQuantifiers (Neg \<phi>) = neg (clearQuantifiers \<phi>)"
decseq_emeasure: fixes B :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range B \<subseteq> sets M"   and "decseq B" shows "decseq (\<lambda>i. emeasure M (B i))"
If2: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>1 :: "com"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com"   and e' :: "(char list \<Rightarrow> int) \<Rightarrow> nat" assumes "\<turnstile>\<^sub>1 {\<lambda>l s. P l s \<and> bval b s} c\<^sub>1 { e \<Down> Q}"   and "\<turnstile>\<^sub>1 {\<lambda>l s. P l s \<and> \<not> bval b s} c\<^sub>2 { e \<Down> Q}"   and "\<And>l s. P l s \<Longrightarrow> e s + 1 = e' s" shows "\<turnstile>\<^sub>1 {P} IF b THEN c\<^sub>1 ELSE c\<^sub>2 { e' \<Down> Q}"
permBottom: fixes p :: "(name \<times> name) list" shows "p \<bullet> \<one> = \<one>"
vpowv_inv_gr_z: shows "0 < 1 / real (card (uverts H) ^ card (uverts H))"
cfg_approx_def: fixes prog :: "'b program_scheme" shows "cfg_approx prog \<equiv> \<Union> {approx_reachable (body p) |p. p \<in> set (program.processes prog)}"
lift_allowed_redex: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<theta> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and u :: "'b trm"   and C :: "'b eclause" assumes "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>"   and "allowed_redex u C \<sigma>" shows "allowed_redex u C \<theta>"
Pd_range': fixes x :: "'x"   and d :: "'d" assumes "x \<in> Field (Pd d)" shows "Xd x = d"
lt_setI: fixes f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "f \<in> F"   and "f \<noteq> 0" shows "lpp f \<in> punit.lt_set F"
k: shows "set p \<subseteq> insert (Upair x x) F"
suffix1: fixes i :: "nat" assumes "i < n" shows "suffix k1 w \<Turnstile>\<^sub>n af (formula i) (w [index i \<rightarrow> k1])[X]\<^sub>\<nu>"
IH: shows "integer_hull (Q + B) + cone C = polyhedron A' b'"
is_unit_iff_degree: fixes p :: "'a poly" assumes "p \<noteq> 0" shows "is_unit p = (degree p = 0)"
P2: shows "Bet Z A Z' \<and> Cong A Z' Z A"
final_state: shows "passive_subset (Liminf_llist Ns) = {}"
Sup_to_Inf_var: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and dual :: "'b \<Rightarrow> 'b"   and X :: "'b set" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Sup X = dual (Inf (dual ` X))"
wt_cont_eff_Par_1: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'a) cmd list"   and n :: "nat"   and i :: "nat"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "PL cval"   and "properL cl"   and "n < length cl"   and "i < PL.brn (cl ! n)" shows "PL.wt aval tval cval (Par cl) s (PL.brnL cl n + i) = 1 / real (length cl) * PL.wt aval tval cval (cl ! n) s i"
le_ccInf_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and b :: "'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A" shows "less_eq b (Inf A) = (\<forall>a\<in>A. less_eq b a)"
remdups_adj_set: fixes xs :: "'g list" shows "set (remdups_adj xs) = set xs"
negGorbit_list_def: fixes gs :: "'g list"   and T :: "'a \<Rightarrow> ('f, 'g) aezfun \<Rightarrow> 'v"   and as :: "'a list" shows "negHorbit_list gs T as \<equiv> map (\<lambda>g. map ((*\<currency>) (- g) \<circ> T) as) gs"
openin_discrete_topology: fixes U :: "'b set"   and S :: "'b set" shows "openin (discrete_topology U) S = (S \<subseteq> U)"
lex_take_index: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" assumes "(xs, ys) \<in> lex r" obtains i :: "nat" where "i < length xs" and "i < length ys" and "take i xs = take i ys" and "(xs ! i, ys ! i) \<in> r"
g: fixes s :: "complex" assumes "s0 < ereal (Re s)" shows "exp (g s) = eval_fds f s"
at_ds9: fixes pi2 :: "('a \<times> 'a) list"   and pi1 :: "('a \<times> 'a) list" assumes "at TYPE('a)" shows " (rev pi2 @ rev pi1) \<triangleq> (rev pi1 @ rev (pi1 \<bullet> pi2)) "
flatten_pi_1_length: fixes \<phi> :: "'b ltln"   and M :: "'b ltln set" shows "size (\<phi>[M]\<^sub>\<Pi>\<^sub>1) \<le> size \<phi>"
uf: shows "unique_factorization_m (smult d f) (c, fs)"
pair_measure_eq_bind: shows "M2 \<Otimes>\<^sub>M M1 = M2 \<bind> (\<lambda>x. M1 \<bind> (\<lambda>y. return (M2 \<Otimes>\<^sub>M M1) (x, y)))"
IH: assumes "local.path g (ns @ [n])" shows "local.path g (ns @ [n, m])"
ereal_ennreal_cases: fixes a :: "ereal" obtains b :: "ennreal" where "0 \<le> a" and "a = enn2ereal b"   | "a < 0"
Gamma_def: fixes z :: "'a" shows "Gamma z = inverse (rGamma z)"
semilat_le_err_OK1: fixes x :: "'b"   and A :: "'b set"   and y :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b err \<Rightarrow> 'b err \<Rightarrow> 'b err"   and z :: "'b" assumes "x \<in> A"   and "y \<in> A"   and "semilat (err A, le r, f)"   and "OK x \<squnion>\<^bsub>f\<^esub> OK y = OK z" shows "x \<sqsubseteq>\<^bsub>r\<^esub> z"
ts_tuple_rel_dest: fixes t :: "nat"   and as :: "'a option list"   and ys :: "(nat \<times> 'a option list set) set" assumes "(t, as) \<in> ts_tuple_rel ys" shows "\<exists>X. (t, X) \<in> ys \<and> as \<in> X"
Nil_is_map_conv: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "([] = map f xs) = (xs = [])"
simps_1: fixes f :: "'a" shows "Dom \<^bold>\<langle>f\<^bold>\<rangle> = \<^bold>\<langle>C.dom f\<^bold>\<rangle>"
getNeg_Neg_empty: fixes X :: "'b \<Rightarrow> 'a"   and src :: "'b list" shows "getPos (NegPos_map X (map Neg src)) = []"
inf_to_sup: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = - (- x \<squnion> - y)"
f_def: shows "\<forall>x\<in>{1..j}. x \<in> A (f x) \<and> f x \<in> {1..m}"
t1: shows "t1 = \<lfloor>t\<rfloor>"
l9_4_2_aux: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a"   and C :: "'a"   and R :: "'a"   and A :: "'a"   and P :: "'a"   and Q :: "'a"   and U :: "'a"   and V :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong S C R A"   and "Tarski_neutral_dimensionless.TS Bet P Q A C"   and "Tarski_neutral_dimensionless.Col Bet R P Q"   and "Tarski_neutral_dimensionless.Perp Bet Cong P Q A R"   and "Tarski_neutral_dimensionless.Col Bet S P Q"   and "Tarski_neutral_dimensionless.Perp Bet Cong P Q C S"   and "Tarski_neutral_dimensionless.Out Bet R U A"   and "Tarski_neutral_dimensionless.Out Bet S V C" shows "Tarski_neutral_dimensionless.TS Bet P Q U V"
aless: fixes i :: "nat"   and x :: "real" assumes "l i \<le> x" shows "a < ereal x"
assms_2: shows "y \<in> set (remdups_wrt f xs)"
id_act: fixes b :: "proj2" shows "apply_cltn2 b \<one>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> = b"
Dag_upd_same_l: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag" shows "Dag p (l(p := p)) r t = (p = Null \<and> t = Tip)"
tl_append2: fixes xs :: "'d list"   and ys :: "'d list" assumes "xs \<noteq> []" shows "tl (xs @ ys) = tl xs @ ys"
inf_mono: fixes a :: "'g multiset"   and c :: "'g multiset"   and b :: "'g multiset"   and d :: "'g multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<inter># b \<subseteq># c \<inter># d"
optimize_matches_def: fixes f :: "'b match_expr \<Rightarrow> 'b match_expr"   and rs :: "'b rule list" shows "optimize_matches f rs = optimize_matches_option (\<lambda>m. if matcheq_matchNone (f m) then None else Some (f m)) rs"
merge_coeffs_alt_def: fixes p :: "(char list list \<times> int) list" shows "(RETURN \<circ> merge_coeffs) p = REC\<^sub>T (\<lambda>f p. case p of [] \<Rightarrow> RETURN [] | [(aa, b)] \<Rightarrow> RETURN p | (aa, b) # (ys, m) # p \<Rightarrow> if aa = ys then if b + m \<noteq> 0 then f ((aa, b + m) # p) else f p else f ((ys, m) # p) \<bind> (\<lambda>p. RETURN ((aa, b) # p))) p"
interp_factorization_def: shows "interp_factorization = (\<lambda>(a, cs). smult a (\<Prod>(c, n)\<leftarrow>cs. [:- c, 1::'b:] ^ Suc n))"
isint_add: fixes a :: "MIR.num"   and bs :: "real list"   and b :: "MIR.num" assumes "isint a bs"   and "isint b bs" shows "isint (Add a b) bs"
derivableI: fixes X :: "'a set"   and x :: "'a" assumes "derivation X"   and "x \<in> X" shows "derivable x"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and p :: "'psym"   and Tl :: "'fsym trm list" shows "I.satA \<xi> (Pr p Tl) = intP p (map (I.int \<xi>) Tl)"
comm_monoid_add_ow_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "comm_monoid_add_ow S pls z \<equiv> ab_semigroup_add_ow S pls \<and> comm_monoid_add_ow_axioms S pls z"
assign_eval\<^sub>w: fixes x :: "'Var"   and e :: "'AExp"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>x \<leftarrow> e, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>Stop, mds, mem(x := eval\<^sub>A mem e)\<rangle>\<^sub>w"
add_inf_distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semilattice_inf_ab_group_add plus zero minus uminus less_eq less inf" shows "plus a (inf b c) = inf (plus a b) (plus a c)"
hcomp_in_hhom': fixes \<mu> :: "'b"   and \<nu> :: "'b"   and a :: "'b"   and c :: "'b" assumes "D.arr \<mu>"   and "D.arr \<nu>"   and "src\<^sub>D \<mu> = a"   and "trg\<^sub>D \<nu> = c"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : a \<rightarrow>\<^sub>D c\<guillemotright>"
g''_def: shows "g'' \<equiv> of_int_poly (Mp g)"
1: shows "submatrix A I UNIV \<in> carrier_mat (card {i. i < n \<and> i \<in> I}) nc"
H_cons: fixes p :: "'a"   and p' :: "'a"   and q' :: "'a"   and q :: "'a"   and x :: "'a" assumes "\<tt>\<tt> p \<le> \<tt>\<tt> p'"   and "\<tt>\<tt> q' \<le> \<tt>\<tt> q"   and "H p' x q'" shows "H p x q"
Sum1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> Rs"   and "guarded P" shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<top> Q \<longmapsto> Rs"
chine_assoc'_def: shows "chine_assoc' \<equiv> \<langle>\<langle>Prj\<^sub>1 \<lbrakk>\<mu>.leg0, \<nu>.leg1\<rbrakk> Prj\<^sub>1\<^sub>0\<rangle> \<lbrakk>\<nu>.leg0 \<cdot> \<mu>\<nu>.prj\<^sub>0, \<pi>.leg1\<rbrakk> Prj\<^sub>0\<^sub>0\<rangle>"
appR: fixes s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "u $ s \<rightarrow>\<^sub>\<eta> u $ t"
cexpr_typing_subst: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type"   and e' :: "cexpr"   and t' :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c e : t"   and "case_nat t \<Gamma> \<turnstile>\<^sub>c e' : t'" shows "\<Gamma> \<turnstile>\<^sub>c cexpr_subst 0 e e' : t'"
quot_to_fract_quot_of_fract: fixes x :: "'b fract" shows "quot_to_fract (quot_of_fract x) = x"
euler_theorem: fixes m :: "int"   and a :: "int" assumes "residues m"   and "coprime a m" shows "[a ^ totient (nat m) = 1] (mod m)"
r: shows "\<forall>n. N < r n \<and> r n < r (Suc n) \<and> (\<forall>i\<in>{N<..r (Suc n)}. u i \<le> u (r (Suc n)))"
ISpec_BInv: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "\<turnstile> ISpec ic q oc \<longrightarrow> \<box>BInv ic q oc"
scope_valid: fixes v :: "'form \<times> nat list"   and i :: "('form, 'var) antecedent"   and v' :: "'form \<times> nat list" assumes "scope' v i v'" shows "v' |\<in>| vertices"
IH: assumes "rga_ops (pre @ xs @ [(oid, ref)])"   and "\<And>i r. (i, Some r) \<in> set xs \<Longrightarrow> r \<noteq> oid"   and "\<And>r. ref = Some r \<Longrightarrow> r \<notin> fst ` set xs" shows "interp_rga (pre @ (oid, ref) # xs) = interp_rga (pre @ xs @ [(oid, ref)])"
Gets_imp_knows_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> kerbIV_gets" shows "X \<in> knows Spy evs"
relH_D: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap" assumes "relH as h h'" shows "in_range (h, as) \<and> in_range (h', as)"
T: shows "\<forall>y\<in>T. g y \<in> S \<and> f (g y) = y"
coinduct_strong: fixes R :: "('a, 'b, 'c) gpv \<Rightarrow> ('a, 'b, 'c) gpv \<Rightarrow> bool"   and gpv :: "('a, 'b, 'c) gpv"   and gpv' :: "('a, 'b, 'c) gpv" assumes "R gpv gpv'"   and "\<And>gpv gpv'. R gpv gpv' \<Longrightarrow> rel_spmf (rel_generat (=) (=) (rel_fun (=) (\<lambda>gpv gpv'. R gpv gpv' \<or> gpv = gpv'))) (the_gpv gpv) (the_gpv gpv')" shows "gpv = gpv'"
square_free_MP_Rel: shows "(MP_Rel ===> (=)) square_free_m square_free"
inject: fixes x1 :: "'b assn"   and x2 :: "Term.term"   and x3 :: "'b assn"   and y1 :: "'b assn"   and y2 :: "Term.term"   and y3 :: "'b assn" shows "({x1} x2\<succ> {x3} = {y1} y2\<succ> {y3}) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
integrable_Dirichlet_kernel_mul_expand: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and n :: "nat" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue" shows "integrable (lebesgue_on S) (\<lambda>x. Dirichlet_kernel n x * f x) = integrable (lebesgue_on S) (\<lambda>x. sin ((real n + 1 / 2) * x) * f x / (2 * sin (x / 2)))"
line_integral_distrib_2: fixes f :: "'a \<Rightarrow> 'a"   and basis :: "'a set"   and g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "line_integral_exists f basis g1"   and "line_integral_exists f basis g2"   and "valid_path g1"   and "valid_path g2" shows "line_integral_exists f basis (g1 +++ g2)"
concat_map_concat: fixes xs :: "'c list list list" shows "concat (map concat xs) = concat (concat xs)"
prj_in_hom_1: shows "\<guillemotleft>\<nu>\<pi>\<rho>.Prj\<^sub>1\<^sub>1 : HH\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<nu>.apex\<guillemotright>"
redT_normal: fixes t :: "'t"   and x :: "'x2"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x' :: "'x2"   and m' :: "'m2"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "r2.r_syntax t x (shr s) ta x' m'"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "r2.actions_ok s t ta"   and "redT_upd s t ta x' m' s'" shows "s -2-t\<triangleright>ta\<rightarrow> s'"
prime_elem_gauss_cnj: fixes z :: "gauss_int" assumes "prime_elem z" shows "prime_elem (gauss_cnj z)"
induct: fixes x :: "('b, 'c) ipurge_rec"   and Y :: "('b, 'c) ipurge_rec"   and P :: "('b, 'c) ipurge_rec \<Rightarrow> bool" assumes "x \<in> ipurge_fail_aux_t_set Y"   and "P Y"   and "\<And>I D U xs X. \<lbrakk>\<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = None, Set = X\<rparr> \<in> ipurge_fail_aux_t_set Y; P \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = None, Set = X\<rparr>\<rbrakk> \<Longrightarrow> P \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some [], Set = ipurge_ref_aux I D U [] X\<rparr>"   and "\<And>I D U x xs ys X. \<lbrakk>\<lparr>Pol = I, Map = D, Doms = U, List = x # xs, ListOp = Some ys, Set = X\<rparr> \<in> ipurge_fail_aux_t_set Y; P \<lparr>Pol = I, Map = D, Doms = U, List = x # xs, ListOp = Some ys, Set = X\<rparr>; \<exists>u\<in>U. (u, D x) \<in> I\<rbrakk> \<Longrightarrow> P \<lparr>Pol = I, Map = D, Doms = U, List = ipurge_tr I D (D x) xs, ListOp = Some ys, Set = ipurge_ref I D (D x) xs X\<rparr>"   and "\<And>I D U x xs ys X. \<lbrakk>\<lparr>Pol = I, Map = D, Doms = U, List = x # xs, ListOp = Some ys, Set = X\<rparr> \<in> ipurge_fail_aux_t_set Y; P \<lparr>Pol = I, Map = D, Doms = U, List = x # xs, ListOp = Some ys, Set = X\<rparr>; \<not> (\<exists>u\<in>U. (u, D x) \<in> I)\<rbrakk> \<Longrightarrow> P \<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some (ys @ [x]), Set = X\<rparr>" shows "P x"
wall_crossings_subset_walls_betw: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "min_gallery (C # Cs @ [D])" shows "order.greater_eq (walls_betw C D) (set (wall_crossings (C # Cs @ [D])))"
inv_eq_1_iff: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "(inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>) = (x = \<one>\<^bsub>G\<^esub>)"
id3_5: shows "xsj = x (Suc (l + j))"
pp_pos: fixes n :: "nat" shows "0 < pp (n + 1)"
eq_bottom_iff: fixes x :: "'c" shows "(x = \<bottom>) = (x \<sqsubseteq> \<bottom>)"
Max_wrt_among_nonempty: fixes B :: "'alt set" assumes "B \<inter> alts \<noteq> {}"   and "finite (B \<inter> alts)" shows "Max_wrt_among (Pareto R) B \<noteq> {}"
gr_implies_not_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "'c"   and n :: "'c" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less m n" shows "n \<noteq> zero"
in_red_subsets: shows "List.member (reduction_subsets p qs signs subsets (M_mat signs subsets)) x"
None_not_eq: fixes x :: "'c option" shows "(None \<noteq> x) = (\<exists>y. x = Some y)"
basis_in_subset_eq: fixes \<E> :: "'a set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set" assumes "\<E> \<subseteq> carrier"   and "basis_in \<E> B\<^sub>1"   and "basis_in \<E> B\<^sub>2"   and "B\<^sub>1 \<subseteq> B\<^sub>2" shows "B\<^sub>1 = B\<^sub>2"
simps_3: fixes OPERATION :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b"   and x :: "'b"   and v :: "'b"   and va :: "'b list" shows "rexp_of_list OPERATION N (x # v # va) = OPERATION x (rexp_of_list OPERATION N (v # va))"
assms_4: shows "\<theta>\<^sub>r = (s \<star> \<theta>) \<cdot> (s \<star> \<a>\<^sup>-\<^sup>1[s\<^sub>0, p\<^sub>0, w]) \<cdot> \<a>[s, s\<^sub>0, p\<^sub>0 \<star> w] \<cdot> (\<sigma> \<star> p\<^sub>0 \<star> w) \<cdot> \<a>[s\<^sub>1, p\<^sub>0, w] \<cdot> (r\<^sub>0s\<^sub>1.\<phi> \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[r\<^sub>0, p\<^sub>1, w]"
whisker_left: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "horizontal_composition V H src trg"   and "partial_magma.ide V f"   and "partial_magma.arr V (V \<nu> \<mu>)" shows "H f (V \<nu> \<mu>) = V (H f \<nu>) (H f \<mu>)"
set_img: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f" shows "set_category_data.set S (set_category.some_img S \<AA>) (set_category.img S \<AA> f) = set_category_given_img.Fun S (set_category.some_img S \<AA>) f ` set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f)"
step_1_most_general: fixes x :: "'a pmf"   and y :: "'a pmf"   and \<alpha> :: "real"   and \<beta> :: "real" assumes "x \<succeq>[\<R>] y \<and> (y, x) \<notin> \<R>"   and "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "\<beta> < \<alpha>" shows "mix_pmf \<alpha> x y \<succeq>[\<R>] mix_pmf \<beta> x y \<and> (mix_pmf \<beta> x y, mix_pmf \<alpha> x y) \<notin> \<R>"
minus_of_int: fixes na :: "nat"   and x :: "real"   and n :: "int" shows "pbernpoly na (x - real_of_int n) = pbernpoly na x"
c_0: fixes s_id :: "'b"   and s_prod :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c_id :: "'b"   and c_prod :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and U :: "'b"   and meet :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and nc :: "'b"   and x :: "'b" assumes "class.c_lattice s_id s_prod c_id c_prod plus less_eq less zero U meet nc" shows "meet x c_id = s_prod x zero"
left_adjoint_determines_right_up_to_iso: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.adjoint_pair f g"   and "Span.adjoint_pair f g'" shows "Span.isomorphic g g'"