hcju/isabelleps huggingface dataset:
lr_of_tran_s1_def: fixes rt :: "('b, 'c) routing_rule_scheme list" shows "lr_of_tran_s1 rt = map (\<lambda>r. (route2match r, routing_oiface r)) rt"
assms_6: shows "is_Groebner_basis (fst ` set gs)"
diff_add_assoc2: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b" shows "b + c - a = b - a + c"
create_gba__fin: shows "create_gba \<phi> \<le> SPEC (\<lambda>\<A>. finite (g_V \<A>))"
AA: shows "swaprows I (Suc I) A \<in> carrier_mat n n"
rightmost_element_property: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'a list" and x' :: "'a" and zs :: "'a list" where "xs = ys @ x' # zs" and "P x'" and "\<forall>z\<in>set zs. \<not> P z"
max: shows "\<forall>r. Well_order r \<and> downset_on (Field r) p \<and> extension_on (Field r) r p \<and> (m, r) \<in> I \<longrightarrow> r = m"
restriction_add_res: fixes v :: "view"   and u :: "view"   and w :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=u--w \<longrightarrow> |restrict v (res ts) c| = |restrict u (res ts) c| + |restrict w (res ts) c|"
fr_eq: fixes p :: "fm"   and vs :: "'b list"   and bs :: "'b list"   and x :: "'b" assumes "islin p" shows "(\<exists>x. Ifm vs (x # bs) p) = (Ifm vs (x # bs) (minusinf p) \<or> Ifm vs (x # bs) (plusinf p) \<or> (\<exists>(n, t)\<in>set (uset p). \<exists>(m, s)\<in>set (uset p). Ifm vs ((- Itm vs (x # bs) t / \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (2::'b) # bs) p))"
sep_conj_sep_impl: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and h :: "'a"   and Q :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "P h"   and "\<And>h. sep_algebra.sep_conj plus sep_disj P Q h \<Longrightarrow> R h" shows "sep_algebra.sep_impl plus sep_disj Q R h"
exe_check_proof_iff_check_proof: fixes e :: "exetheory"   and P :: "proofterm"   and res :: "Core.term" shows "exe_check_proof e P res = check_proof (translate_theory e) P res"
finfun_ext: fixes f :: "'c \<Rightarrow>f 'd"   and g :: "'c \<Rightarrow>f 'd" assumes "\<And>a. f $ a = g $ a" shows "f = g"
cancels_to_1_preserves_generators: fixes l :: "(bool \<times> 'c) list"   and l' :: "(bool \<times> 'c) list"   and gens :: "'c set" assumes "cancels_to_1 l l'"   and "l \<in> lists (UNIV \<times> gens)" shows "l' \<in> lists (UNIV \<times> gens)"
order_iff_strict: fixes ko :: "'a key_order"   and b :: "'a"   and a :: "'a" shows "ko.le ko b a = (ko.lt ko b a \<or> a = b)"
INF_image: fixes Inf :: "'b set \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "Inf (g ` f ` A) = Inf ((g \<circ> f) ` A)"
basic_subtype_asym: fixes \<tau> :: "'a basic_type"   and \<sigma> :: "'a basic_type" assumes "\<tau> \<sqsubset>\<^sub>B \<sigma>"   and "\<sigma> \<sqsubset>\<^sub>B \<tau>" shows "False"
conf_Boolean: fixes G :: "prog"   and s :: "st"   and v :: "val" assumes "G,s\<turnstile>v\<Colon>\<preceq>PrimT Boolean" shows "\<exists>b. v = Bool b"
insert_Diff1: fixes x :: "'d"   and B :: "'d set"   and A :: "'d set" assumes "x \<in> B" shows "insert x A - B = A - B"
borel_compact: fixes A :: "'a set" assumes "compact A" shows "A \<in> sets borel"
wivs_mask_def: shows "wivs_mask = 2 ^ dflt_size - 1"
normalize_guards: fixes gs :: "('f \<times> 'd set) list"   and c :: "('d, 'e, 'f) com" shows "Language.normalize (guards gs c) = guards gs (Language.normalize c)"
differing_vars_def: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "differing_vars mem\<^sub>1 mem\<^sub>2 = {x. mem\<^sub>1 x \<noteq> mem\<^sub>2 x}"
simps_3: fixes sh :: "sheap"   and v :: "val" shows "iconf sh (Val v) = True"
prv_all_congW: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<in> var"   and "bprv (imp \<phi> \<chi>)"   and "bprv (all x \<phi>)" shows "bprv (all x \<chi>)"
approx_tse_form_def: fixes prec :: "nat"   and t :: "nat"   and s :: "nat"   and f :: "form" shows "approx_tse_form prec t s f = (case f of Bound x a b f \<Rightarrow> x = Var 0 \<and> (case (approx prec a [None], approx prec b [None]) of (None, b) \<Rightarrow> False | (Some ivl, None) \<Rightarrow> False | (Some ivl, Some ivl') \<Rightarrow> approx_tse_concl prec t f s ivl ivl') | _ \<Rightarrow> False)"
convex_bound_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and a :: "'b"   and y :: "'b"   and u :: "'b"   and v :: "'b" assumes "class.linordered_semiring_1 plus minus less_eq less zero times one"   and "less_eq x a"   and "less_eq y a"   and "less_eq zero u"   and "less_eq zero v"   and "plus u v = one" shows "less_eq (plus (times u x) (times v y)) a"
measure_Diff: fixes M :: "'b measure"   and A :: "'b set"   and B :: "'b set" assumes "emeasure M A \<noteq> \<infinity>"   and "A \<in> sets M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M B"
uint_of_uint16_def: shows "uint_of_uint16 \<equiv> map_fun Rep_uint16 Abs_uint ucast"
interleave_assoc2: fixes a :: "'b list"   and ys :: "'b list"   and zs :: "'b list"   and b :: "'b list"   and xs :: "'b list" assumes "a \<in> ys \<otimes> zs"   and "b \<in> xs \<otimes> a" shows "\<exists>c. c \<in> xs \<otimes> ys \<and> b \<in> c \<otimes> zs"
ipurge_tr_rev_t_inv_2_def: fixes P :: "'c process"   and I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and X :: "('c, 'd) ipurge_rec" shows "ipurge_tr_rev_t_inv_2 P I D xs X \<equiv> secure P I D \<longrightarrow> xs \<in> traces P \<longrightarrow> Out X @ In X \<in> traces P"
is_gcd_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and gcd' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.GCD_ring times one plus zero minus uminus" shows "GCD_ring.is_gcd times gcd' = (\<forall>a b. dvd.dvd times (gcd' a b) a \<and> dvd.dvd times (gcd' a b) b \<and> (\<forall>d'. dvd.dvd times d' a \<and> dvd.dvd times d' b \<longrightarrow> dvd.dvd times d' (gcd' a b)))"
assms: fixes s :: "'a"   and x :: "'b" shows "RETURN (f s x) \<le> F s x"
Kelly_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a set"   and A :: "'a set" shows "Kelly R B A = (\<forall>a\<in>A. \<forall>b\<in>B. R b a)"
measurable_dens_ctxt_measure_component: fixes x :: "nat"   and \<rho> :: "nat \<Rightarrow> val" assumes "x \<in> V" shows "(\<lambda>\<sigma>. \<sigma> x) \<in> dens_ctxt_measure \<Y> \<rho> \<rightarrow>\<^sub>M stock_measure (\<Gamma> x)"
eventually_sup: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and F' :: "'a filter" shows "eventually P (sup F F') = (eventually P F \<and> eventually P F')"
env_restr_join: fixes S :: "'a set"   and m1 :: "'a \<Rightarrow> 'b"   and m2 :: "'a \<Rightarrow> 'b" shows "(m1 \<squnion> m2) f|` S = m1 f|` S \<squnion> m2 f|` S"
length_univ_list_card: shows "length univ_list = CARD('b)"
UP_smult_assoc1: fixes a :: "'a"   and b :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "p \<in> carrier P" shows "a \<otimes> b \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
prev_in_basics_1: fixes u :: "'a" assumes "u \<in> {a<..b}" shows "prev_in A u \<in> A"
s2: shows "(0::'a) \<in> {x. f x = (0::'b)}"
ofilter_def: fixes A :: "'a set" shows "local.ofilter A \<equiv> A \<subseteq> Field r \<and> (\<forall>a\<in>A. local.under a \<subseteq> A)"
l7_21_R1: 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"   and P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C"   and "B \<noteq> D"   and "Cong A B C D"   and "Cong B C D A"   and "Tarski_neutral_dimensionless.Col Bet A P C"   and "Tarski_neutral_dimensionless.Col Bet B P D" shows "Tarski_neutral_dimensionless.Midpoint Bet Cong P A C"
not_red_map_in_map_not_red: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G :: "'b inference set"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and Prec_F_g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and N :: "'a set" assumes "tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F_g" shows "\<Union> (\<G>_F ` N) - Red_F_G (\<Union> (\<G>_F ` N)) \<subseteq> \<Union> (\<G>_F ` (N - tiebreaker_lifting.Red_F_\<G> Red_F_G \<G>_F Prec_F_g N))"
snd_diff: fixes x :: "'b \<times> 'a"   and y :: "'b \<times> 'a" shows "snd (x - y) = snd x - snd y"
Entry_noteq_Exit: assumes "(_Entry_) = (_Exit_)" shows "False"
key_order_of_nat_term_order_inv_def: fixes to :: "'b nat_term_order" shows "key_order_of_nat_term_order_inv to = Abs_key_order (nat_term_compare_inv to)"
fin: shows "finite ((\<lambda>x. {x}) ` (topspace X \<inter> S))"
retraction_def: fixes S :: "'a set"   and T :: "'a set"   and r :: "'a \<Rightarrow> 'a" shows "retraction S T r = (T \<subseteq> S \<and> continuous_on S r \<and> r ` S \<subseteq> T \<and> (\<forall>x\<in>T. r x = x))"
f2: shows "rk {a, c, \<beta>} = 2"
distinct_99: fixes x4 :: "val"   and x15 :: "expr" shows "Val x4 \<noteq> throw x15"
eq_\<I>_attach_on': fixes \<I>' :: "('y, 'z) \<I>"   and res :: "('y, 'z) resource"   and \<I> :: "('aa, 'ab) \<I>"   and conv :: "('aa, 'ab, 'y, 'z) converter"   and conv' :: "('aa, 'ab, 'y, 'z) converter"   and A :: "'aa set" assumes "\<I>' \<turnstile>res res \<surd>"   and "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"   and "A \<subseteq> outs_\<I> \<I>" shows "A \<turnstile>\<^sub>R conv \<rhd> res \<sim> conv' \<rhd> res"
IH: shows "(Fun f (S @ t # T), Fun g (S @ u # T)) \<in> (timpl_closure'_step c)\<^sup>*"
range_subsetD: fixes f :: "'e \<Rightarrow> 'd"   and B :: "'d set"   and i :: "'e" assumes "range f \<subseteq> B" shows "f i \<in> B"
minim_Under: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "r.minim B \<in> r.Under B"
equiv_appR: fixes t :: "dB"   and t' :: "dB"   and s :: "dB" assumes "t \<leftrightarrow> t'" shows "s \<degree> t \<leftrightarrow> s \<degree> t'"
cod_def: fixes f :: "(bool, 'arr list) concrete_category.arr" shows "cod f = (if codomains f \<noteq> {} then SOME b. b \<in> codomains f else null)"
pdevs_val_nlex_pdevs: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and x :: "(real \<times> real) pdevs" assumes "e \<in> UNIV \<rightarrow> I"   and "uminus ` I = I" obtains e' :: "nat \<Rightarrow> real" where "e' \<in> UNIV \<rightarrow> I" and "pdevs_val e x = pdevs_val e' (nlex_pdevs x)"
norm1_def: fixes p :: "'b poly" shows "norm1 p = sum_list (map abs (coeffs p))"
Quadratic_Reciprocity: fixes p :: "nat"   and q :: "nat" assumes "prime p"   and "2 < p"   and "prime q"   and "2 < q"   and "p \<noteq> q" shows "Legendre (int p) (int q) * Legendre (int q) (int p) = (- 1) ^ ((p - 1) div 2 * ((q - 1) div 2))"
assms_2: shows "full_functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F"
renaming: fixes xvec :: "name list"   and Tvec :: "'a list"   and p :: "(name \<times> name) list"   and T :: "'c" assumes "length xvec = length Tvec"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "(p \<bullet> xvec) \<sharp>* T" shows "substCond T xvec Tvec = substCond (p \<bullet> T) (p \<bullet> xvec) Tvec"
set_spmf_sample_uniform: fixes n :: "nat" assumes "0 < n" shows "set_spmf (sample_uniform n) = {..<n}"
word_succ_rbl: fixes w :: "'b word"   and bl :: "bool list" assumes "to_bl w = bl" shows "to_bl (word_succ w) = rev (rbl_succ (rev bl))"
bounded_closure: fixes S :: "'a set" assumes "bounded S" shows "bounded (closure S)"
rcs_eqTr1: fixes H :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G" shows "(a \<in> H \<bullet> b) = (a \<cdot> \<rho> b \<in> H)"
adj_sub_geq2: fixes \<sigma> :: "('b, 'c, 'a) subst"   and x2 :: "('b, 'a) trm"   and x1 :: "('b, 'a) trm" shows "(\<Union>i\<in>SIGT x2. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x) \<subseteq> \<Union> (SFV \<sigma> ` (SDom \<sigma> \<inter> SIGF (Geq x1 x2)))"
preserves_comp_2: fixes f' :: "'a"   and f :: "'a" assumes "S.seq f' f" shows "\<ll> (f' \<cdot>\<^sub>S f) = \<ll> f' \<cdot>\<^sub>S L f"
that: fixes \<pi> :: "'a \<Rightarrow> 'a" assumes "\<pi> permutes A"   and "R' = map_relation A \<pi> R" shows "thesis"
diff_cancel2: fixes m :: "nat"   and k :: "nat"   and n :: "nat" shows "m + k - (n + k) = m - n"
bexE: fixes A :: "'e set"   and P :: "'e \<Rightarrow> bool" assumes "\<exists>x\<in>A. P x" obtains x :: "'e" where "x \<in> A" and "P x"
cr_Gromov_completion_def: shows "cr_Gromov_completion \<equiv> \<lambda>x y. Gromov_completion_rel x x \<and> abs_Gromov_completion x = y"
assms: shows "\<exists>r o\<^sub>1 o\<^sub>2. Some r = d\<^sub>2 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> (o\<^sub>1, o\<^sub>2) \<in> ex2 r v"
image2p_def: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "image2p f g R = (\<lambda>x y. \<exists>x' y'. R x' y' \<and> f x' = x \<and> g y' = y)"
comm_ring_ow: shows "class.comm_ring = comm_ring_ow UNIV"
gets_the_conv: fixes B :: "'c \<Rightarrow> 'b option"   and s :: "'c" shows "gets_the B s = (case B s of None \<Rightarrow> ({}, True) | Some r' \<Rightarrow> ({(r', s)}, False))"
lists_infinite_bij_betw: fixes A :: "'a set" assumes "infinite A" shows "\<exists>f. bij_betw f (lists A) A"
continuous_at_imp_continuous_within: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'e"   and s :: "'d set" assumes "isCont f x" shows "continuous (at x within s) f"
exhaust: fixes y :: "'b dual" obtains x :: "'b" where "y = dual x"
nnP: fixes s :: "'s"   and a :: "'a" shows "0 \<le> P s a"
distinct_18: fixes x5 :: "char list" shows "Class x5 \<noteq> Integer"
NotActualNotD: fixes v :: "i"   and \<phi> :: "\<o>" assumes "\<not> [\<^bold>\<A>\<^bold>\<not>\<phi> in v]" shows "\<not> [\<^bold>\<not>\<^bold>\<A>\<phi> in v]"
closed: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set" assumes "semilattice_set f"   and "finite A"   and "A \<noteq> {}"   and "\<And>x y. f x y \<in> {x, y}" shows "semilattice_set.F f A \<in> A"
extend_quasinorm: fixes F :: "'c set"   and C :: "real"   and N :: "'c \<Rightarrow> ennreal" assumes "quasinorm_on F C N" shows "quasinorm_on UNIV C (\<lambda>x. if x \<in> F then N x else \<infinity>)"
If_transT_invert: fixes b :: "'test"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and phi :: "bool" assumes "(if b then c1 else c2, s) \<rightarrow>t s'" shows "phi"
case_prod_Pair_iden: fixes p :: "'d \<times> 'e" shows "(case p of (x, y) \<Rightarrow> (x, y)) = p"
simps: fixes x :: "'a extC"   and y :: "'a extC" shows "x + y = extC.Val const"
is_prefix_length: fixes a :: "'b list"   and b :: "'b list" assumes "is_prefix a b" shows "length a \<le> length b"
Iod_carrier: fixes T :: "'a set" assumes "T \<subseteq> carrier D" shows "carrier (Iod D T) = T"
p_imp_p: fixes P :: "'b upred" shows "(P \<Rightarrow> P) = true"
Gcd_int_def: fixes K :: "int set" shows "Gcd K = int (Gcd ((nat \<circ> abs) ` K))"
conforms_xgext: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and x' :: "abopt"   and s' :: "st" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "(x', s')\<Colon>\<preceq>(G, L)"   and "s'\<le>|s"   and "dom (locals s') \<subseteq> dom (locals s)" shows "(x', s)\<Colon>\<preceq>(G, L)"
close_noneq_nonempty: fixes A :: "'b list"   and x :: "'b"   and a :: "'b" assumes "List.member A x"   and "x \<noteq> a" shows "close A a \<noteq> []"
zero_less_mult_iff_hmset: fixes m :: "hmultiset"   and n :: "hmultiset" shows "(0 < m * n) = (0 < m \<and> 0 < n)"
singleton_or_lem: fixes x1 :: "'a attree"   and x :: "'a set \<times> 'a set" assumes "\<turnstile>[x1] \<oplus>\<^sub>\<or>\<^bsup>x\<^esup>" shows "fst x \<subseteq> fst (attack x1)"
min_gallery_adj: fixes x :: "'a set"   and y :: "'a set" assumes "chamber x"   and "chamber y"   and "x \<sim> y"   and "x \<noteq> y" shows "min_gallery [x, y]"
nths_vars1': shows "nths d.dims0 d.vars1' = d.dims1"
equivalence_of_partition: shows "partition.Equivalence e2.Partition = {(x, y). stk2.rel x y}"
iIN_inext_if: fixes t :: "nat"   and n :: "nat"   and d :: "nat" shows "inext t [n\<dots>,d] = (if n \<le> t \<and> t < n + d then Suc t else t)"
simps: fixes a1 :: "'a list"   and a2 :: "('b, 'c) sequent" shows "((a1, a2) \<in> modRules2) = (\<exists>ps c M Ms. a1 = ps \<and> a2 = c \<and> ps \<noteq> [] \<and> ModalSequents.mset c = \<LM> Modal M Ms  \<RM>)"
Reception: fixes evsr :: "event list"   and A :: "agent"   and B :: "agent"   and X :: "msg" assumes "evsr \<in> orb"   and "Says A B X \<in> set evsr" shows "Gets B X # evsr \<in> orb"
simps: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c" shows "Domainp r a = (\<exists>aa b. a = aa \<and> r aa b)"
put_get: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and v :: "'b" assumes "weak_lens x" shows "get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v"
simps: fixes A :: "'b set"   and a :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "af A a = ((\<exists>P. a = P \<and> (\<forall>x xa. x \<in> A \<longrightarrow> xa \<in> A \<longrightarrow> P x xa)) \<or> (\<exists>P. a = P \<and> (\<forall>x. x \<in> A \<longrightarrow> af A (\<lambda>y z. P y z \<or> P x y))))"
poly_f_roots_prop_1: fixes qs :: "real poly list" assumes "check_all_const_deg qs = False" shows "\<forall>x1 x2. x1 < x2 \<and> (\<exists>q1\<in>set qs. q1 \<noteq> 0 \<and> poly q1 x1 = 0) \<and> (\<exists>q2\<in>set qs. q2 \<noteq> 0 \<and> poly q2 x2 = 0) \<longrightarrow> (\<exists>q>x1. q < x2 \<and> poly (poly_f qs) q = 0)"
leq_head_def: fixes t :: "('a, 'b) Term.term"   and u :: "('a, 'b) Term.term" shows "leq_head t u = (case (root t, root u) of (None, xa) \<Rightarrow> True | (Some a, None) \<Rightarrow> False | (Some a, Some g) \<Rightarrow> a \<le> g)"
map_disc_iff: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c list" shows "(map f a = []) = (a = [])"
strictify_cancel: fixes f :: "'c \<rightarrow> 'b" assumes "f\<cdot>\<bottom> = \<bottom>" shows "strictify\<cdot>f = f"
ord_le_eq_trans: fixes x :: "'o"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "le x a b"   and "b = c" shows "le x a c"
worklists_vacant_to_def: fixes c :: "('loc, 't) configuration"   and t :: "'t" shows "worklists_vacant_to c t = (\<forall>loc1 loc2 s t'. s \<in>\<^sub>A path_summary loc1 loc2 \<and> t' \<in>#\<^sub>z c_work c loc1 \<longrightarrow> \<not> results_in t' s \<le> t)"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "P \<langle>\<rangle>"   and "\<And>l a r. P (merge l r) \<Longrightarrow> P \<langle>l, a, r\<rangle>" shows "P a0"
not_le_real: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less a zero \<or> a = zero \<or> less zero a"   and "less b zero \<or> b = zero \<or> less zero b" shows "(\<not> less_eq b a) = less a b"
poly_zero: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and p :: "'b list" assumes "class.idom_char_0 one times plus zero minus uminus" shows "(semiring_0.poly plus zero times p = semiring_0.poly plus zero times []) = (\<forall>c\<in>set p. c = zero)"
comp_null_2: fixes f :: "'a" shows "f \<cdot>\<^sub>A A.null = A.null"
simps: fixes p :: "'node"   and S :: "'node set"   and a :: "'node" shows "reachable_through p S a = (a = p \<or> (\<exists>p' Sl p''. a = p'' \<and> reachable_through p S p' \<and> p' \<in> W \<and> Sl \<in> slices p' \<and> Sl \<subseteq> S \<and> p'' \<in> Sl))"
incseq_Suc_iff: fixes f :: "nat \<Rightarrow> 'a" shows "incseq f = (\<forall>n. f n \<le> f (Suc n))"
simps_2: fixes x :: "'a"   and n :: "nat" shows "power_sq x (Suc n) = 1\<^sub>\<sigma> + x \<cdot> power_sq x n"
extend_sum: fixes S :: "'e set"   and A :: "'e set"   and a :: "'e \<Rightarrow> 'c" assumes "finite S"   and "A \<subseteq> S"   and "a \<in> A \<rightarrow> carrier M" shows "(\<Oplus>\<^bsub>M\<^esub>x\<in>S. if x \<in> A then a x else \<zero>\<^bsub>M\<^esub>) = finsum M a A"
except_Int_keys: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and U :: "'a set" shows "except p (U \<inter> keys p) = except p U"
keys_dehomogenizeE: 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 (dehomogenize x p)" obtains s :: "'b \<Rightarrow>\<^sub>0 nat" where "s \<in> keys p" and "t = except s {x}"
n\<^sub>2_def: shows "n\<^sub>2 = size (mset_heap ts\<^sub>2)"
gallery_obtain_pgallery: fixes x :: "'a set"   and y :: "'a set"   and xs :: "'a set list" assumes "x \<noteq> y"   and "gallery (x # xs @ [y])" shows "\<exists>ys. order.greater_eq (set xs) (set ys) \<and> order.greater_eq (length xs) (length ys) \<and> pgallery (x # ys @ [y])"
orbit_cong0: fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "x \<in> A"   and "f \<in> A \<rightarrow> A"   and "\<And>y. y \<in> A \<Longrightarrow> f y = g y" shows "orbit f x = orbit g x"
borel_measurable_vimage_halfspace_component_lt: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>a i. i \<in> Basis \<longrightarrow> {x \<in> S. f x \<bullet> i < a} \<in> sets (lebesgue_on S))"
dgrad_p_set_closed_mult_scalar: fixes d :: "'a \<Rightarrow> nat"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and m :: "nat"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "p \<in> punit.dgrad_p_set d m"   and "r \<in> dgrad_p_set d m" shows "p \<odot> r \<in> dgrad_p_set d m"
less_add_eq_less: fixes k :: "nat"   and l :: "nat"   and m :: "nat"   and n :: "nat" assumes "k < l"   and "m + l = k + n" shows "m < n"
F_def: shows "F \<equiv> RR (Var yy) \<langle>\<phi>R\<rangle>"
hyps_2: fixes Vs :: "char list list"   and ls :: "val list" assumes "fv e \<subseteq> set Vs"   and "l \<subseteq>\<^sub>m [Vs [\<mapsto>] ls]"   and "length Vs + max_vars e \<le> length ls" shows "\<exists>ls'. compP\<^sub>1 P \<turnstile>\<^sub>1 \<langle>compE\<^sub>1 Vs e,(h, ls)\<rangle> \<Rightarrow> \<langle>fin\<^sub>1 (addr a),(h\<^sub>1, ls')\<rangle> \<and> l\<^sub>1 \<subseteq>\<^sub>m [Vs [\<mapsto>] ls']"
\<Gamma>_prefix_conv: fixes p :: "'a Trace.prefix"   and s :: "'a Trace.trace"   and s' :: "'a Trace.trace"   and i :: "nat" assumes "prefix_of p s"   and "prefix_of p s'"   and "i < plen p" shows "\<Gamma> s i = \<Gamma> s' i"