hcju/isabelleps huggingface dataset:
finite_cyclic_subgroup_int: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "finite (carrier (subgroup_generated (relative_homology_group (int n) (nsphere n) (equator n)) {x})) = (\<exists>i. i \<noteq> 0 \<and> x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> i = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub>)"
suminf_of_real: fixes X :: "nat \<Rightarrow> real" assumes "summable X" shows "of_real (\<Sum>n. X n) = (\<Sum>n. of_real (X n))"
abelian_groupE_6: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "abelian_group R"   and "x \<in> carrier R" shows "\<exists>y\<in>carrier R. y \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
unlift_ap: shows "unlift_vars 0 (x \<diamondop> y) = unlift_vars 0 x \<degree> unlift_vars 0 y"
locally_lipschitz_imp_lipschitz_aux: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and M :: "real" assumes "a \<le> b"   and "continuous_on {a..b} f"   and "\<And>x. x \<in> {a..<b} \<Longrightarrow> \<exists>y\<in>{x<..b}. dist (f y) (f x) \<le> M * (y - x)" shows "dist (f b) (f a) \<le> M * (b - a)"
coprime_1_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a one"
Lambert_W'_gtI: fixes x :: "real"   and y :: "real" assumes "x < y * exp y"   and "y \<le> - 1"   and "- exp (- 1) \<le> x" shows "y \<le> Lambert_W' x"
translation_unit_eq: shows "identity G = (\<one>)\<^sub>L"
indiff_imp_same_utility_value: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and \<alpha> :: "real"   and \<beta> :: "real" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "some_best outcomes \<R> \<succeq>[\<R>] some_worst outcomes \<R> \<and> (some_worst outcomes \<R>, some_best outcomes \<R>) \<notin> \<R>"   and "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "mix_pmf \<beta> (some_best outcomes \<R>) (some_worst outcomes \<R>) \<approx>[\<R>] mix_pmf \<alpha> (some_best outcomes \<R>) (some_worst outcomes \<R>)" shows "\<beta> = \<alpha>"
h_1: shows "path_image h \<subseteq> path_image (f +++ g)"
drop_bit_half: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n (divide a (numeral.numeral one plus (num.Bit0 num.One))) = divide (drop_bit n a) (numeral.numeral one plus (num.Bit0 num.One))"
path_component_maximal: fixes x :: "'a"   and T :: "'a set"   and S :: "'a set" assumes "x \<in> T"   and "path_connected T"   and "T \<subseteq> S" shows "T \<subseteq> path_component_set S x"
StepSeqWhile: fixes b :: "expr"   and c' :: "cmd"   and l :: "nat"   and s :: "char list \<Rightarrow> val option" assumes "labels (while (b) c') l (Skip;; while (b) c')" shows "while (b) c' \<turnstile> \<langle>Skip;; while (b) c',s,l\<rangle> \<leadsto> \<langle>while (b) c',s,0\<rangle>"
exec: shows "steps0 (st, l, r) ap stp = (st', l', r')"
Shuffle_ZeroR: fixes r :: "'a language" shows "Shuffle r Zero = Zero"
finite_set_intervals2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and l :: "'a"   and u :: "'a"   and S :: "'a set" assumes "class.linorder less_eq less"   and "P x"   and "less_eq l x"   and "less_eq x u"   and "l \<in> S"   and "u \<in> S"   and "finite S"   and "\<forall>x\<in>S. less_eq l x"   and "\<forall>x\<in>S. less_eq x u" shows "(\<exists>s\<in>S. P s) \<or> (\<exists>a\<in>S. \<exists>b\<in>S. (\<forall>y. less a y \<and> less y b \<longrightarrow> y \<notin> S) \<and> less a x \<and> less x b \<and> P x)"
subst_Var: fixes i :: "nat"   and s :: "dB"   and k :: "nat" shows "dB.Var i[s/k] = (if k < i then dB.Var (i - 1) else if i = k then s else dB.Var i)"
exe_trap_wr_pc_def: fixes uu :: "unit" shows "exe_trap_wr_pc uu \<equiv> do psr_val \<leftarrow> gets (cpu_reg_val PSR); new_psr_val \<leftarrow> gets (\<lambda>s. update_S 1 psr_val); write_cpu new_psr_val PSR; reset_trap \<leftarrow> gets reset_trap_val; tbr_val \<leftarrow> gets (cpu_reg_val TBR); if reset_trap = False then do write_cpu tbr_val PC; write_cpu (tbr_val + 4) nPC; return () od else do write_cpu 0 PC; write_cpu 4 nPC; set_reset_trap False; return () od od"
T_meas: shows "T \<in> M \<rightarrow>\<^sub>M M"
da_elim_cases_38: fixes Env :: "env"   and B :: "lname set"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1l Super\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>" and "This \<in> B"
assms: shows "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"
lang_flatten_PLUS: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and project :: "'c \<Rightarrow> 'c"   and X :: "'d rexp set"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "project \<Sigma> project"   and "finite X" shows "project.lang \<Sigma> project lookup n (PLUS (sorted_list_of_set X)) = \<Union> (project.lang \<Sigma> project lookup n ` X)"
prv_eqv_imp_transi: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv (eqv \<phi>1 \<phi>2)"   and "bprv (imp \<phi>2 \<chi>)" shows "bprv (imp \<phi>1 \<chi>)"
ent_conjE2: fixes B :: "assn"   and C :: "assn"   and A :: "assn" assumes "B \<Longrightarrow>\<^sub>A C" shows "A \<and>\<^sub>A B \<Longrightarrow>\<^sub>A C"
bounded_clinear_const_scaleC: fixes g :: "'d \<Rightarrow> 'e"   and c :: "complex" assumes "bounded_clinear g" shows "bounded_clinear (\<lambda>x. c *\<^sub>C g x)"
drop_res_def: fixes k :: "int"   and f :: "nat \<Rightarrow> int"   and n :: "nat" shows "drop_res k f n = (if f n = k then 0 else f n)"
simps_1: fixes S :: "'b set" shows "sstart S [] = streams S"
reaches_def: fixes x :: "'d"   and y :: "'d" shows "x reaches y = (\<lambda>s. (\<lambda>x y. (x points_to y) s)\<^sup>*\<^sup>* x y)"
qtm_conc: fixes q :: "('c, 'd) SkewBinomialTree list"   and q' :: "('c, 'd) SkewBinomialTree list" shows "queue_to_multiset (q @ q') = queue_to_multiset q + queue_to_multiset q'"
sub_wrap: fixes x :: "'b word"   and z :: "'b word" shows "(x \<le> x - z) = (z = 0 \<or> x < z)"
reduced_GB_lc: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "punit.is_reduced_GB G"   and "g \<in> G" shows "lcf g = (1::'b)"
is_ang_distinct: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "A B C Ang a" shows "A \<noteq> B \<and> C \<noteq> B"
ereal_less_ereal_Ex: fixes x :: "ereal"   and r :: "real" shows "(x < ereal r) = (x = - \<infinity> \<or> (\<exists>p<r. x = ereal p))"
restrict_def: fixes ord :: "WS1S_Prelim.order"   and P :: "'a fset" shows "restrict ord P = (case ord of FO \<Rightarrow> P \<noteq> {||} | SO \<Rightarrow> True)"
max_Var_floatariths_fold_const_fa: fixes xs :: "floatarith list" shows "max_Var_floatariths (map fold_const_fa xs) \<le> max_Var_floatariths xs"
pref_antisym: fixes a :: "'b list"   and b :: "'b list" assumes "a \<le>p b"   and "b \<le>p a" shows "a = b"
set_intros_2: fixes y :: "'h"   and x22 :: "'h list"   and x21 :: "'h" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
piecewise_C1_differentiable_on_mult: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "f piecewise_C1_differentiable_on S"   and "g piecewise_C1_differentiable_on S" shows "(\<lambda>x. f x * g x) piecewise_C1_differentiable_on S"
parZeroLeft: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "\<And>Q. (Q \<parallel> \<zero>, Q) \<in> Rel" shows "P \<parallel> \<zero> \<leadsto>[Rel] P"
triangle': fixes f :: "'d"   and g :: "'d" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "f \<star>\<^sub>D \<l>\<^sub>D[g] = (\<r>\<^sub>D[f] \<star>\<^sub>D g) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, src\<^sub>D f, g]"
r_into_trancl: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "r a b" shows "r\<^sup>+\<^sup>+ a b"
Pi_cong: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c set" assumes "\<And>w. w \<in> A \<Longrightarrow> f w = g w" shows "(f \<in> Pi A B) = (g \<in> Pi A B)"
winning_regionI: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "v \<in> V\<^bsub>G\<^esub>"   and "ParityGame.strategy G p \<sigma>"   and "ParityGame.winning_strategy G p \<sigma> v" shows "v \<in> ParityGame.winning_region G p"
L_V: fixes l :: "'s"   and u :: "'c \<Rightarrow> real" assumes "(l, u) \<in> S" shows "l \<in> L"
a: shows "\<Delta> @ [(x, U)] @ \<Gamma> \<turnstile> t : T"
split_zip: fixes P :: "('c \<times> 'd) list \<Rightarrow> prop" shows "(\<And>x. PROP P x) \<equiv> (\<And>y z. length y = length z \<Longrightarrow> PROP P (y || z))"
FSwImorph_igWlsInp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'e option"   and h :: "'e \<Rightarrow> 'h"   and hA :: "'f \<Rightarrow> 'i"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'h, 'i, 'j) model_scheme" assumes "igWlsInp MOD delta inp"   and "FSwImorph h hA MOD MOD'" shows "igWlsInp MOD' delta (lift h inp)"
oreach_def: fixes S :: "('b, 'a) spec" shows "oreach S \<equiv> obs S ` reach S"
prod_le_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq zero (f x) \<and> less_eq (f x) one" shows "less_eq (comm_monoid_mult.prod times one f A) one"
af: shows "a \<in>\<^sub>\<circ> f (0\<^sub>\<nat>)"
cong_gcd_eq_1_nat: fixes m :: "nat"   and a :: "nat"   and b :: "nat"   and n :: "nat" assumes "0 < m"   and "[a ^ m = 1] (mod b)"   and "[a ^ n = 1] (mod b)" shows "[a ^ gcd m n = 1] (mod b)"
integral_cong: fixes M :: "'d measure"   and N :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L N g"
disjoint_second_summands: fixes v :: "i"   and x :: "i"   and y :: "i"   and z :: "i" assumes "CEM"   and "P v (x \<oplus> y) \<and> P v (x \<oplus> z)"   and "\<not> O y z" shows "P v x"
monom_mult_in_pmdl: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and b :: "'a \<Rightarrow>\<^sub>0 'd"   and B :: "('a \<Rightarrow>\<^sub>0 'd) set"   and c :: "'d"   and t :: "'b" assumes "term_powerprod pair_of_term term_of_pair"   and "b \<in> B" shows "term_powerprod.monom_mult pair_of_term term_of_pair c t b \<in> module.span (term_powerprod.mult_scalar pair_of_term term_of_pair) B"
intro: fixes point_set :: "'b set" assumes "finite point_set" shows "finite_incidence_system_axioms point_set"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>uu_. P [] uu_"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>v va. P (v # va) []" shows "P a0 a1"
no_trailing_map: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "no_trailing P (map f xs) = no_trailing (P \<circ> f) xs"
simps: fixes a1 :: "('f \<times> 'l) set"   and a2 :: "('f \<times> 'l) set" shows "(a1 \<rhd>L a2) = (\<exists>M N. a1 = M \<and> a2 = N \<and> M - N \<subseteq> Red_F_\<G> N)"
the_\<theta>_props_2: shows "local.iso \<chi>.the_\<theta>"
measurable_id: fixes M :: "'c measure" shows "(\<lambda>x. x) \<in> M \<rightarrow>\<^sub>M M"
Confidentiality_Serv_A: fixes A :: "agent"   and authK :: "nat"   and Ta :: "nat"   and evs :: "event list"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat" assumes "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredAK Ta evs"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbV" shows "Key servK \<notin> analz (knows Spy evs)"
reachable_imp_\<S>\<^sub>G: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "reachable s" shows "\<S>\<^sub>G s"
hfref_def: fixes P :: "'c \<Rightarrow> bool"   and RS :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and T :: "'d \<Rightarrow> 'b \<Rightarrow> assn" shows "[P]\<^sub>a RS \<rightarrow> T \<equiv> {(f, g). \<forall>c a. P a \<longrightarrow> hn_refine (fst RS a c) (f c) (snd RS a c) T (g a)}"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_same a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "head t = head s" and "chksubs (>\<^sub>t) t s" and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)"
Inv_def: fixes ES :: "('a list set \<times> 'a trm set) set" shows "Inv ES = (invariant ES \<and> (\<exists>rhs. (X, rhs) \<in> ES))"
reduced_word_for_def: fixes A :: "'a set"   and a :: "'a"   and as :: "'a list" shows "reduced_word_for A a as \<equiv> is_arg_min length (word_for A a) as"
dube_eq_0: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F" shows "(\<lambda>z. z + int (card X) - 1 gchoose (card X - 1)) = (\<lambda>z. z - int (poly_deg f) + int (card X) - 1 gchoose (card X - 1) + pm_powerprod.Hilbert_poly X (pm_powerprod.aa X f F) z + pm_powerprod.Hilbert_poly X (pm_powerprod.bb ord X F) z)"
bigtheta_def: fixes F :: "'c filter"   and g :: "'c \<Rightarrow> 'd" shows "\<Theta>[F](g) = O[F](g) \<inter> \<Omega>[F](g)"
lexLessAppend: fixes b :: "('a \<times> bool) list"   and a :: "('a \<times> bool) list" assumes "b \<noteq> []" shows "(a @ b, a) \<in> lexLess"
curry_preserves_transformations: fixes F :: "'a2 \<times> 'a1 \<Rightarrow> 'b"   and G :: "'a2 \<times> 'a1 \<Rightarrow> 'b"   and \<tau> :: "'a2 \<times> 'a1 \<Rightarrow> 'b" assumes "natural_transformation A2xA1.comp (\<cdot>\<^sub>B) F G \<tau>" shows "natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>[\<^sub>A\<^sub>1,\<^sub>B\<^sub>]) (local.curry F F F) (local.curry G G G) (local.curry F G \<tau>)"
sup_assoc: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
deterministic: fixes lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc'' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "lc \<leadsto> lc'"   and "lc \<leadsto> lc''" shows "lc' = lc''"
\<alpha>_ide_simp: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "MC.assoc a b c = \<a>[a, b, c]"
admissible_rel_spmf: shows "ccpo.admissible (prod_lub lub_spmf lub_spmf) (rel_prod (ord_spmf (=)) (ord_spmf (=))) (\<lambda>(x, y). rel_spmf R x y)"
image_image: fixes f :: "'b::type \<Rightarrow> 'a::type"   and g :: "'c::type \<Rightarrow> 'b::type"   and A :: "'c::type set" shows "(f::'b::type \<Rightarrow> 'a::type) ` (g::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set) = (\<lambda>x::'c::type. f (g x)) ` A"
list_combE_2: fixes f :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "\<not> shadows_consts (list_comb f xs)"   and "x \<in> set xs" shows "\<not> shadows_consts x"
out546_suma__conga: 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 E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.Out Bet E D F" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C G H I"
igFreshCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b, 'c, 'd) model_scheme" shows "igFreshCls MOD \<equiv> igFreshIGVar MOD \<and> igFreshIGAbs1 MOD \<and> igFreshIGAbs2 MOD \<and> igFreshIGOp MOD"
simps_1: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and p :: "char"   and \<theta> :: "trm" shows "usubstappf \<sigma> U (Pred p \<theta>) = (case usubstappt \<sigma> U \<theta> of None \<Rightarrow> undeff | Some \<sigma>\<theta> \<Rightarrow> case SPreds \<sigma> p of None \<Rightarrow> Afml (Pred p \<sigma>\<theta>) | Some r \<Rightarrow> if FVF r \<inter> U = {} then usubsta (dotsubstt \<sigma>\<theta>) r else undeff)"
Seq: fixes c\<^sub>1 :: "com"   and s\<^sub>1 :: "char list \<Rightarrow> int"   and s\<^sub>2 :: "char list \<Rightarrow> int"   and c\<^sub>2 :: "com"   and s\<^sub>3 :: "char list \<Rightarrow> int" assumes "(c\<^sub>1, s\<^sub>1) \<Rightarrow> s\<^sub>2"   and "(c\<^sub>2, s\<^sub>2) \<Rightarrow> s\<^sub>3" shows "(c\<^sub>1;; c\<^sub>2, s\<^sub>1) \<Rightarrow> s\<^sub>3"
in_ivp_sols_ivl: fixes t :: "real"   and s :: "'a" assumes "t \<in> T"   and "s \<in> S" shows "(\<lambda>t. \<phi> t s) \<in> Sols (\<lambda>t. f) (\<lambda>s. {0--t}) S 0 s"
is_le_relE: fixes rel :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "ordered_powerprod_lin.is_le_rel rel" obtains "rel = (=)"   | "rel = (\<preceq>)"   | "rel = (\<prec>)"
run_of_lasso_of_prpl: fixes pl :: "'e list"   and pr :: "'e list" assumes "pl \<noteq> []" shows "run_of_lasso (lasso_of_prpl (pr, pl)) = pr \<frown> pl\<^sup>\<omega>"
inv_solve_left: fixes a :: "'a \<Rightarrow> complex"   and b :: "'a \<Rightarrow> complex"   and c :: "'a \<Rightarrow> complex" assumes "a \<in> carrier (Characters G)"   and "b \<in> carrier (Characters G)"   and "c \<in> carrier (Characters G)" shows "(a = inv\<^bsub>Characters G\<^esub> b \<otimes>\<^bsub>Characters G\<^esub> c) = (c = b \<otimes>\<^bsub>Characters G\<^esub> a)"
d_3: fixes F :: "'a formula"   and G :: "'a formula"   and H :: "'a formula" assumes "Con F G H"   and "F \<in> pcp_lim C S" shows "G \<triangleright> H \<triangleright> pcp_lim C S \<in> C"
ds_dom_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y" shows "d x \<sqsubseteq> d y"
large: shows "\<forall>v n. v dvd Polynomial.smult lu u \<longrightarrow> degree v \<le> degree_bound vs \<longrightarrow> 2 * \<bar>coeff v n\<bar> < m"
set_eq_pairI: fixes S :: "('c, 'd) eq_object_scheme"   and x :: "'c"   and x' :: "'c"   and y :: "'c" assumes "equivalence S"   and "x .=\<^bsub>S\<^esub> x'"   and "x \<in> carrier S"   and "x' \<in> carrier S"   and "y \<in> carrier S" shows "{x, y} {.=}\<^bsub>S\<^esub> {x', y}"
r_one: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.monoid G"   and "x \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x"
loxodromic_repelling_fixed_point_3: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "0 < additive_strength f (repelling_fixed_point f)"
face_cycles_dv: shows "H.face_cycle_sets = face_cycle_sets"
simps_3: fixes bs :: "real list"   and a :: "Ferrack.num" shows "Ifm bs (Lt a) = (Inum bs a < 0)"
preserves_arr: fixes f :: "'j" assumes "J.arr f" shows "Cop_S.arr ((local.map \<circ> D) f)"
sorted_wrt2: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and zs :: "'b list" assumes "transp P" shows "sorted_wrt P (x # y # zs) = (P x y \<and> sorted_wrt P (y # zs))"
analytic_pre_zeta_aux: fixes a :: "real"   and n :: "nat" assumes "0 < a" shows "pre_zeta_aux n a analytic_on {s. - 2 * real n < Re s}"
tau_add: fixes x :: "'a"   and y :: "'a" shows "\<tau> (x + y) = \<tau> x + \<tau> y"
D_2: fixes n :: "nat" shows "\<exists>N. C n \<inter> {N..} \<subseteq> D"
R2_def: fixes M :: "'a \<times> 'a"   and v :: "bool" shows "R2 M v = (let (m0, m1) = M in sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. let c\<^sub>v = a * b mod order \<G> in sample_uniform (order \<G>) \<bind> (\<lambda>c\<^sub>v'. sample_uniform_units (order \<G>) \<bind> (\<lambda>r0. sample_uniform_units (order \<G>) \<bind> (\<lambda>s0. let w0 = (\<^bold>g [^] a) [^] s0 \<otimes> \<^bold>g [^] r0; z = (\<^bold>g [^] c\<^sub>v') [^] s0 \<otimes> (\<^bold>g [^] b) [^] r0 in sample_uniform_units (order \<G>) \<bind> (\<lambda>r1. sample_uniform_units (order \<G>) \<bind> (\<lambda>s1. let w1 = (\<^bold>g [^] a) [^] s1 \<otimes> \<^bold>g [^] r1; z' = (\<^bold>g [^] c\<^sub>v) [^] s1 \<otimes> (\<^bold>g [^] b) [^] r1; enc_m = z \<otimes> (if v then m0 else m1); enc_m' = z' \<otimes> (if v then m1 else m0) in return_spmf (v, \<^bold>g [^] a, \<^bold>g [^] b, \<^bold>g [^] c\<^sub>v, w0, enc_m, w1, enc_m')))))))))"
continuous_on_imp_closedin: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "continuous_on S f"   and "closedin (top_of_set (f ` S)) T" shows "closedin (top_of_set S) (S \<inter> f -` T)"
subfield_iff_2: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set" assumes "subfield K L" shows "field (L\<lparr>carrier := K\<rparr>)"
no_icd_pd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "\<forall>l\<ge>n. \<not> IFC_def.is_icdi prob k \<pi> l"   and "n \<le> k" shows "IFC_def.is_pd prob (\<pi> k) (\<pi> n)"
indecomposable_def: fixes \<alpha> :: "V" shows "indecomposable \<alpha> \<equiv> Ord \<alpha> \<and> (\<forall>\<beta>\<in>elts \<alpha>. \<forall>\<gamma>\<in>elts \<alpha>. \<beta> + \<gamma> \<in> elts \<alpha>)"