hcju/isabelleps huggingface dataset:
conga_diff56: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C CongA A' B' C'" shows "C' \<noteq> B'"
CLAMP_def: fixes f :: "nat \<Rightarrow> real"   and x :: "nat" shows "CLAMP f x = (if x < 3 then 0 else f x)"
eq: shows "\<bar>real (count_derangements n) - fact n / exp 1\<bar> = exp t / real (n + 1)"
eq: shows "\<Union> (Collect ((\<lambda>x. x \<in> \<B>) relative_to U)) = U"
try_gpv_cong: fixes gpv :: "('d, 'e, 'f) gpv"   and gpv'' :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv"   and gpv''' :: "('d, 'e, 'f) gpv" assumes "gpv = gpv''"   and "\<not> colossless_gpv \<I>_full gpv'' \<Longrightarrow> gpv' = gpv'''" shows "TRY gpv ELSE gpv' = TRY gpv'' ELSE gpv'''"
Methd: fixes Env :: "env"   and D :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd"   and B :: "lname set"   and A :: "assigned" assumes "methd (prg Env) D sig = Some m"   and "Env\<turnstile> B \<guillemotright>\<langle>Body (declclass m) (stmt (mbody (mthd m)))\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>Methd D sig\<rangle>\<guillemotright> A"
assms_3: fixes b :: "'a" assumes "b dvd a" shows "a dvd b \<or> b dvd (1::'a)"
collect_clki_def: fixes I :: "'g \<Rightarrow> ('e, 'f) cconstraint" shows "collect_clki I = \<Union> {collect_clock_pairs (I x) |x. True}"
closed_closure_eq: fixes T :: "'b set set"   and a :: "'b set" assumes "a closed\<^bsub>T\<^esub>" shows "closure\<^bsub>T\<^esub> a = a"
power_Suc0_right: fixes a :: "'b list" shows "a \<^sup>@ Suc 0 = a"
GSMP_disjointE: fixes n :: "'b"   and A :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and m :: "'b"   and Sec :: "('fun, 'var) Term.term set" assumes "GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj n A)) (trms\<^sub>l\<^sub>s\<^sub>t (proj m A)) Sec" shows "GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n A)) \<inter> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj m A)) \<subseteq> Sec \<union> {m. {} \<turnstile>\<^sub>c m}"
rel_fun_pos_distr: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and uu :: "('e \<times> 'f \<times> 'g) itself"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B' :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "rel_fun_pos_distr_cond A A' uu" shows "(A ===> B) OO (A' ===> B') \<le> A OO A' ===> B OO B'"
trms\<^sub>s\<^sub>s\<^sub>t_subst_cons: fixes a :: "('a, 'b) stateful_strand_step"   and A :: "('a, 'b) stateful_strand_step list"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "trms\<^sub>s\<^sub>s\<^sub>t (a # A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>) = trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) \<union> trms\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
retraction_def: fixes S :: "'b set"   and T :: "'b set"   and r :: "'b \<Rightarrow> 'b" 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))"
measure_Union: fixes M :: "'c measure"   and A :: "'c set"   and B :: "'c set" assumes "emeasure M A \<noteq> \<infinity>"   and "emeasure M B \<noteq> \<infinity>"   and "A \<in> sets M"   and "B \<in> sets M"   and "A \<inter> B = {}" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M B"
Least_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool" shows "ord.Least less_eq P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq x y))"
rg_def: fixes g :: "'a ugraph"   and r :: "'a" shows "Prim.rg g r \<equiv> component_of g r"
suf_prod_longer: fixes u :: "'b list"   and w :: "'b list"   and z :: "'b list"   and v :: "'b list" assumes "u \<le>s w \<cdot> z"   and "v \<le>s w"   and "\<^bold>|v \<cdot> z\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "v \<cdot> z \<le>s u"
resNilLeft_1: fixes Rel :: "(pi \<times> pi) set"   and x :: "name"   and y :: "name"   and P :: "pi" shows "\<zero> \<leadsto>[Rel] <\<nu>x>x<y>.P"
inEdges_def: fixes inEdges' :: "'c \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'b) list"   and g :: "'c"   and n :: "'a" shows "graph_path_base.inEdges inEdges' g n \<equiv> map (\<lambda>(f, d). (f, d, n)) (inEdges' g n)"
bezout_imp_all_fin_gen_ideals_are_principal_unsatisfactory: assumes "class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus" shows "\<forall>I. finitely_generated_ideal I \<longrightarrow> principal_ideal I"
sign_changes_def: fixes ps :: "real poly list"   and x :: "real" shows "sign_changes ps x = length (remdups_adj (list_neq (map (\<lambda>p. sgn (poly p x)) ps) 0)) - 1"
base: fixes C :: "sequent"   and R :: "(sequent list \<times> sequent) set" assumes "([], C) \<in> R" shows "(C, 0) \<in> derivable R"
m_closed: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "x \<otimes> y \<in> carrier Zp"
orderI: fixes a :: "'g"   and b :: "'g" assumes "a = min a b" shows "a \<le> b"
f: fixes l :: "nat list" shows "f l + sum_list l < ack kf (sum_list l)"
mset_lt_single_iff: fixes x :: "'a"   and y :: "'a" shows "({#x#} < {#y#}) = (x < y)"
that: fixes N :: "nat set"   and \<psi> :: "nat \<Rightarrow> ('a, 'b) chart" assumes "\<And>i. i \<in> N \<Longrightarrow> \<psi> i \<in> atlas"   and "(domain \<circ> \<psi>) ` N refines \<X> ` I"   and "regular_cover N \<psi>" shows "thesis"
small_bounded_rank_le: fixes a :: "V" shows "small {x. rank x \<le> a}"
IH: fixes ei :: "expr"   and si :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and T :: "ty" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ei,si\<rangle>"   and "P,E \<turnstile> e :: T"   and "P,E \<turnstile> s\<^sub>0 \<surd>" shows "throw e' = ei \<and> s\<^sub>1 = si"
elem_index_inj: fixes x :: "'a"   and y :: "'a" assumes "elem_index x = elem_index y" shows "x = y"
Sigma_max_point: fixes ma :: "nat"   and f :: "nat list \<Rightarrow> nat"   and xs :: "nat list"   and w :: "nat" assumes "\<forall>k<ma. f (xs @ [k]) = 1"   and "\<forall>k\<ge>ma. f (xs @ [k]) = 0"   and "ma \<le> w" shows "UF.Sigma f (xs @ [w]) = ma"
veq_structure_eq_left: fixes t :: "value"   and u :: "value" assumes "veq_structure t u"   and "vno_abs t" shows "t = u"
tdghm_dghm_comp_NTMap_vdomain: fixes \<NN> :: "V" shows "\<D>\<^sub>\<circ> ((\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>)\<lparr>NTMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
h2: assumes "\<exists>(a', b', c')\<in>set les. a' = 0 \<and> b' \<noteq> 0 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>- (c' / b'). \<forall>x\<in>{- (c' / b')<..y'}. d * x\<^sup>2 + e * x + f < 0)" shows "\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0"
\<theta>_double_less: fixes n :: "nat" assumes "0 < n" shows "\<theta> (2 * real n) - \<theta> (real n) < real n * ln 4"
frechet_derivative_fst: fixes f :: "'b \<Rightarrow> 'c \<times> 'd"   and x :: "'b" assumes "f differentiable at x" shows "frechet_derivative (\<lambda>x. fst (f x)) (at x) = (\<lambda>xa. fst (frechet_derivative f (at x) xa))"
semiring_hom_axioms: shows "semiring_hom poly_y_x"
maxR_idem: fixes x :: "'a" shows "maxR x x = x"
lsu_lax: fixes x :: "'b"   and y :: "'b" shows "\<nu>\<^sup>\<natural> x \<cdot> \<nu>\<^sup>\<natural> y \<le> \<nu>\<^sup>\<natural> (x \<cdot> y)"
linear_conv_bounded_linear: fixes f :: "'d \<Rightarrow> 'e" shows "linear f = bounded_linear f"
abs_eq: fixes xb :: "'b"   and xa :: "'c \<Rightarrow> 'c"   and x :: "'b \<Rightarrow> 'c option" shows "map_entry xb xa (Mapping x) = Mapping (case x xb of None \<Rightarrow> x | Some v \<Rightarrow> x(xb \<mapsto> xa v))"
zero_matrix_def: shows "0 = Abs_matrix (\<lambda>j i. 0::'b)"
monom_one_comm: fixes k :: "nat" shows "monom P \<one> k \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 = monom P \<one> 1 \<otimes>\<^bsub>P\<^esub> monom P \<one> k"
imp: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla" shows "imp \<phi>1 \<phi>2 \<in> fmla"
to_hma\<^sub>v_def: fixes v :: "'d Matrix.vec" shows "HMA_Connect.to_hma\<^sub>v v = (\<chi>i. v $v Bij_Nat.to_nat i)"
Smith_normal_form_diagonal_to_Smith: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'c) vec, 'd) vec" assumes "is_bezout_ext bezout"   and "isDiagonal A" shows "Smith_normal_form (diagonal_to_Smith A bezout)"
orderE: fixes a :: "'e"   and b :: "'e" assumes "a \<le> b" obtains "a = inf a b"
times_fun_prod: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and g :: "'c \<times> 'd \<Rightarrow> 'e" shows "f \<oplus> g = (\<lambda>(x, y). \<Squnion>{f (x, y1) \<oplus> g (x, y2) |y1 y2. D y1 y2 \<and> y = y1 \<oplus> y2})"
comparator_nat_term_compare: fixes to :: "'a nat_term_order" shows "comparator (nat_term_compare to)"
LIM_cos_div_sin: shows "(\<lambda>x. cos x / sin x) \<midarrow>pi / 2\<rightarrow> 0"
forest_components_idempotent: fixes x :: "'a" assumes "injective x" shows "forest_components (forest_components x) = forest_components x"
sinks_aux_member_ipurge_tr_aux: fixes u :: "'b"   and I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and U :: "'b set"   and V :: "'b set"   and xs :: "'c list"   and w :: "'b" assumes "u \<in> sinks_aux I D (U \<union> V) xs"   and "(u, w) \<in> I"   and "\<not> (\<exists>v\<in>sinks_aux I D V xs. (v, w) \<in> I)" shows "u \<in> sinks_aux I D U (ipurge_tr_aux I D V xs)"
elims: fixes x :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort"   and y :: "'v" assumes "edge_begin x = y" obtains v1 :: "'v" and p1 :: "'outPort" and v2 :: "'v" and p2 :: "'inPort" where "x = ((v1, p1), v2, p2)" and "y = v1"
psi_v_1: shows "psi_v \<in> \<R>\<^sup>2"
Cidm: fixes C :: "('d, 'e, 'f) Category_scheme"   and X :: "'d" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X maps\<^bsub>C\<^esub> X to X"
all_zero2: shows "(\<forall>m>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'a)) = vector_all_zero_from_index (i + 1, column_iarray k (matrix_to_iarray A))"
map_le_def: fixes m\<^sub>1 :: "'a \<Rightarrow> 'b option"   and m\<^sub>2 :: "'a \<Rightarrow> 'b option" shows "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) = (m\<^sub>1 = m\<^sub>2 on dom m\<^sub>1)"
reflect_poly_def: fixes p :: "'b poly" shows "reflect_poly p = Poly (rev (coeffs p))"
repeat_mset_right: fixes a :: "nat"   and b :: "nat"   and A :: "'k multiset" shows "repeat_mset a (repeat_mset b A) = repeat_mset (a * b) A"
fresh_asTerm_qFresh: fixes qX :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "qGood qX" shows "fresh xs x (asTerm qX) = qFresh xs x qX"
dist_s_A: fixes n :: "nat" shows "distinct (s_A n)"
equals_minim_Under: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> B"   and "a \<in> r.Under B" shows "a = r.minim B"
semiring_norm_45: shows "\<bar>1::'a\<bar> = (1::'a)"
rel_FGco_def: fixes L1 :: "'o \<Rightarrow> 'ab \<Rightarrow> bool"   and Co1 :: "'p \<Rightarrow> 'ac \<Rightarrow> bool"   and Co2 :: "'r \<Rightarrow> 'ae \<Rightarrow> bool"   and Co3 :: "'s \<Rightarrow> 'af \<Rightarrow> bool"   and Co4 :: "'t \<Rightarrow> 'ag \<Rightarrow> bool"   and Co5 :: "'q \<Rightarrow> 'ad \<Rightarrow> bool"   and Co6 :: "'x \<Rightarrow> 'aj \<Rightarrow> bool"   and Contra1 :: "'u \<Rightarrow> 'ah \<Rightarrow> bool"   and Contra2 :: "'v \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra3 :: "'y \<Rightarrow> 'ak \<Rightarrow> bool"   and Contra4 :: "'z \<Rightarrow> 'al \<Rightarrow> bool" shows "rel_FGco L1 Co1 Co2 Co3 Co4 Co5 Co6 Contra1 Contra2 Contra3 Contra4 = rel_F L1 Co1 Co5 (rel_G Co1 Co2 Co3 Co4 Contra1 Contra2) Co3 Co6 Contra1 Contra3 Contra4"
fact_binomial: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "fact k * of_nat (n choose k) = fact n / fact (n - k)"
comp_in_homI: fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and g :: "'c"   and c :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>C c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>C f : a \<rightarrow>\<^sub>C c\<guillemotright>"
s_Inj1_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Inj1 e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Inj1 e'a" and "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"
simps: fixes s :: "nat"   and lm :: "nat list"   and a :: "abc_inst option" shows "abc_step_l (s, lm) a = (case a of None \<Rightarrow> (s, lm) | Some (Inc n) \<Rightarrow> let nv = abc_lm_v lm n in (s + 1, abc_lm_s lm n (nv + 1)) | Some (Dec n e) \<Rightarrow> let nv = abc_lm_v lm n in if nv = 0 then (e, abc_lm_s lm n 0) else (s + 1, abc_lm_s lm n (nv - 1)) | Some (Goto n) \<Rightarrow> (n, lm))"
h1: shows "0 < poly (qs ! a) x \<and> 0 < poly (prod_list (retrieve_polys qs xa)) x \<longrightarrow> prod_list (map ((!) sign) (a # xa)) = 1"
gamma1_def: shows "gamma1 \<equiv> $pc1 = #g \<and> pc1$ = #a \<and> sem$ = Suc<$sem> \<and> Unchanged (x, y, pc2)"
homogeneous_set_Polys: fixes X :: "'b set" shows "homogeneous_set P[X]"
flush_reads_program: fixes sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" assumes "\<forall>r\<in>set sb. is_Read\<^sub>s\<^sub>b r \<or> is_Prog\<^sub>s\<^sub>b r \<or> is_Ghost\<^sub>s\<^sub>b r" shows "\<exists>\<O>' \<R>' \<S>''. (m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f\<^sup>* (m, [], \<O>', \<R>', \<S>'')"
qe: fixes as :: "'a list" assumes "\<forall>a\<in>set as. depends\<^sub>0 a" shows "is_dnf_qe qe as"
set_of_seq_def: shows "set_of_seq = set_of_pred \<circ> pred_of_seq"
Quotient_rep_equiv1: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "Quotient R Abs Rep T"   and "T a b" shows "R a (Rep b)"
orderI: fixes a :: "'b"   and b :: "'b" assumes "a = a \<squnion> b" shows "b \<le> a"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e blindable\<^sub>h \<Rightarrow> 'c"   and blindable\<^sub>m :: "('d, 'e) blindable\<^sub>m" shows "P (case blindable\<^sub>m of Unblinded x \<Rightarrow> f1 x | Blinded x \<Rightarrow> f2 x) = ((\<forall>x1. blindable\<^sub>m = Unblinded x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. blindable\<^sub>m = Blinded x2 \<longrightarrow> P (f2 x2)))"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "(symbol \<times> character list) set" assumes "\<Z> x xa = y" obtains k :: "nat" where "x = ka__" and "xa = 0" and "y = {}"   | k :: "nat" and u :: "nat" where "x = k" and "xa = Suc u" and "y = \<Y> (\<Z> k u) (\<P> k u) k"
point_replication_number_def: fixes B :: "'c set multiset"   and x :: "'c" shows "B rep x \<equiv> int (size (filter_mset ((\<in>) x) B))"
coinduct: fixes R :: "'b tree \<Rightarrow> 'b tree \<Rightarrow> bool"   and tree :: "'b tree"   and tree' :: "'b tree" assumes "R tree tree'"   and "\<And>tree tree'. R tree tree' \<Longrightarrow> root tree = root tree' \<and> R (left tree) (left tree') \<and> R (right tree) (right tree')" shows "tree = tree'"
iMODb_div_right_inj_on: fixes k :: "nat"   and m :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < k"   and "k \<le> m" shows "inj_on (\<lambda>x. x div k) [ r, mod m, c ]"
Rep_model_cases: fixes y :: "object set \<times> (predicate \<Rightarrow> object list \<Rightarrow> bool)" assumes "y \<in> model" obtains x :: "model" where "y = Rep_model x"
wr_com_loop: fixes c :: "command" shows "wr_com (Loop c) = wr_com c"
neg_le_iff_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus b) (uminus a) = less_eq a b"
kSup_distr: fixes F :: "('b \<Rightarrow> 'd set) set"   and g :: "'d \<Rightarrow> 'c set" shows "\<Squnion> F \<circ>\<^sub>K g = (\<Squnion>f\<in>F. f \<circ>\<^sub>K g)"
class_add_sees_methods: fixes P :: "jvm_prog"   and D :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> jvm_method) \<times> char list) option"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> D sees_methods Mm"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> D sees_methods Mm"
g: fixes x :: "'a" assumes "x \<in> X" shows "inj_on (g x) (B x)"
finprod_mono_neutral_cong_left: fixes B :: "'d set"   and A :: "'d set"   and h :: "'d \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "finite B"   and "A \<subseteq> B"   and "\<And>i. i \<in> B - A \<Longrightarrow> h i = \<one>"   and "\<And>x. x \<in> A \<Longrightarrow> g x = h x"   and "h \<in> B \<rightarrow> carrier G" shows "finprod G g A = finprod G h B"
between_endpoint_in_S: fixes a :: "proj2"   and b :: "proj2"   and c :: "proj2" assumes "a \<noteq> b"   and "b \<noteq> c"   and "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S"   and "c \<in> hyp2 \<union> S"   and "B\<^sub>\<real> cart2_pt a cart2_pt b cart2_pt c" shows "endpoint_in_S a b = endpoint_in_S b c"
assms_1: shows "(u \<longlongrightarrow> ereal l) F"
simps: fixes x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> bval) list"   and v1 :: "val"   and g1 :: "bval set"   and v2 :: "val"   and g2 :: "bval set"   and r :: "bool" shows "good_entry x e \<rho> ((v1, g1), v2, g2) r = ((\<forall>v\<in>g1. \<exists>v'. (x, v) # \<rho> \<turnstile> e \<Down> v' \<and> v' \<in> g2) \<and> r)"
partList4_disjoint: fixes s :: "'b set"   and ts :: "'b set list" assumes "s \<subseteq> \<Union> (set ts)"   and "disjoint_list_rec ts" shows "disjoint_list_rec (partList4 s ts)"
gcdI: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times c a"   and "dvd.dvd times c b"   and "\<And>d. \<lbrakk>dvd.dvd times d a; dvd.dvd times d b\<rbrakk> \<Longrightarrow> dvd.dvd times d c"   and "normalize c = c" shows "c = gcd a b"
lmeasurable_cbox: fixes a :: "'a"   and b :: "'a" shows "cbox a b \<in> lmeasurable"
l7_21_R1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and P :: "'p" assumes "\<not> Col A B C"   and "B \<noteq> D"   and "Cong A B C D"   and "Cong B C D A"   and "Col A P C"   and "Col B P D" shows "P Midpoint A C"
affine_Inter: fixes f :: "'a set set" assumes "\<And>s. s \<in> f \<Longrightarrow> affine s" shows "affine (\<Inter> f)"
simps_90: fixes x1 :: "'b" shows "atoms (Atom (x1::'b)) = {x1}"
in_set_insert: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "List.insert x xs = xs"
strict_implies_not_eq: fixes b :: "'b"   and a :: "'b" assumes "b < a" shows "a \<noteq> b"
1: assumes "Obj t" shows "t \<in> HHom t t \<and> t \<in> VHom t t"
mod_ring_inverse: fixes p :: "int" assumes "prime_field TYPE('a) p" shows "(mod_ring_locale.mod_ring_rel ===> mod_ring_locale.mod_ring_rel) (inverse_p p) inverse"
mc_lookup: fixes xs :: "'a option list"   and tstp :: "nat + nat" assumes "Mapping.lookup mc xs = Some tstp" shows "tstp_lt tstp (cur - (left I - 1)) tp \<and> isl tstp = (0 < left I)"
k_def: shows "k \<equiv> \<lfloor>(r - Arg z + pi) / (2 * pi)\<rfloor>"
split_max: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and i :: "'a"   and j :: "'a" assumes "class.linorder less_eq less" shows "P (ord.max less_eq i j) = ((less_eq i j \<longrightarrow> P j) \<and> (\<not> less_eq i j \<longrightarrow> P i))"
eq1: shows "punit.lt (rep_list m) = punit.lt (rep_list p)"
zero_mat_def: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<equiv> Matrix.mat nr nc (\<lambda>ij. 0::'b)"
Image_Id_on: fixes A :: "'a set"   and B :: "'a set" shows "Id_on A `` B = A \<inter> B"
continuous_within_Arctan: fixes z :: "complex"   and s :: "complex set" assumes "Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1" shows "continuous (at z within s) Arctan"
option2list_def: fixes n :: "'a option" shows "option2list n \<equiv> case n of None \<Rightarrow> [] | Some s \<Rightarrow> [s]"
that: fixes c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>c (c', s')"   and "s' \<approx> t'"   and "c' \<approx>s d'" shows "thesis"
nat_power_of_nat: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" assumes "class.nat_power one times plus zero nat_power"   and "0 < m" shows "nat_power m (semiring_1.of_nat one plus zero n) = semiring_1.of_nat one plus zero (m ^ n)"
rconj_relator_freeword: fixes s :: "'w"   and t :: "'w"   and xs :: "('w \<times> bool) list" assumes "s \<in> S"   and "t \<in> S"   and "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "rconjby (Abs_freeword xs) (pair_relator_freeword s t) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
monoid_vec_simps_2: fixes ty :: "'a itself"   and n :: "nat" shows "carrier (monoid_vec ty n) = carrier_vec n"
finite_imp_closedin: fixes S :: "'a set"   and T :: "'a set" assumes "finite S"   and "S \<subseteq> T" shows "closedin (top_of_set T) S"
simps_1: fixes P :: "char list"   and ts :: "'a list" shows "Pos P ts\<^sup>c = Neg P ts"
terminates_Normal_elim_cases_2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and g :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>Guard f g c \<down> Normal s" obtains "s \<in> g" and "\<Gamma>\<turnstile>c \<down> Normal s"   | "s \<notin> g"
bind_eq_OkE: fixes x :: "('k, 'm) result"   and f :: "'m \<Rightarrow> ('k, 'l) result"   and z :: "'l" assumes "x \<bind> f = Ok z" obtains xa :: "'m" where "x = Ok xa \<and> f xa = Ok z"
diff_add_zero: fixes a :: "'b"   and b :: "'b" shows "a - (a + b) = (0::'b)"
isomorphic_trans: fixes smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set"   and smult'' :: "('b, 'a) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and X :: "'e set" assumes "isomorphic smult' W"   and "\<exists>T. FGModuleIso G smult' W smult'' T X" shows "isomorphic smult'' X"
antimono_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less" shows "order.antimono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f y \<le> f x)"
light_year_def: shows "light_year = QCOERCE[L] (\<^bold>c \<^bold>\<cdot> julian_year)"