hcju/isabelleps huggingface dataset:
nat_to_set_lub: fixes x :: "nat"   and u :: "nat" assumes "x \<in> nat_to_set u" shows "x \<le> log2 u"
evalRR_simps_2: fixes p :: "nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> int"   and x :: "int" assumes "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "x \<in> carrier R" shows "eval R R id x (p \<ominus>\<^bsub>UP R\<^esub> q) = eval R R id x p \<ominus>\<^bsub>R\<^esub> eval R R id x q"
tan_periodic_int: fixes x :: "real"   and i :: "int" shows "tan (x + real_of_int i * pi) = tan x"
chine_simps_2: shows "C.dom trg_f.chine = trg_f.dom.apex"
equivI: fixes A :: "'g set"   and r :: "('g \<times> 'g) set" assumes "refl_on A r"   and "sym r"   and "trans r" shows "equiv A r"
between_sucs: fixes x :: "nat"   and n :: "nat" assumes "x < Suc n"   and "\<not> x < n" shows "x = n"
matchrel_sound: fixes P :: "(('d, 'e) Term.term \<times> ('d, 'f) Term.term) multiset"   and \<sigma> :: "'e \<Rightarrow> ('d, 'f) Term.term option"   and d :: "'e \<Rightarrow> ('d, 'f) Term.term" assumes "((P, Map.empty), {#}, \<sigma>) \<in> matchrel\<^sup>*" shows "subst_of_map d \<sigma> \<in> matchers (set_mset P)"
simps_9: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Dom \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = (Dom t \<^bold>\<star> Dom u) \<^bold>\<star> Dom v"
zle_zless_trans: fixes i :: "int"   and j :: "int"   and k :: "int" assumes "i \<le> j"   and "j < k" shows "i < k"
simps_3: fixes uu :: "const" shows "plus_const uu Any = Any"
ttree_join_is_either: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "t \<squnion> t' = t \<oplus>\<oplus> t'"
orthogonal_rvsum: fixes s :: "'a set"   and x :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "finite s"   and "\<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)" shows "orthogonal x (sum f s)"
arr_dom_iff_arr: fixes f :: "'a \<times> 'a" shows "VoV.arr (VoV.dom f) = VoV.arr f"
Ide_implies_Arr: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Ide V src trg t" shows "bicategorical_language.Arr V src trg t"
ran_map_of: fixes al :: "('d \<times> 'c) list" shows "ran (map_of al) = snd ` set (AList.clearjunk al)"
objE: fixes a :: "'a" assumes "B.obj a" obtains "B.ide a" and "src\<^sub>B a = a" and "trg\<^sub>B a = a"
col_rw: shows "col (submatrix B UNIV J) j = col B (pick J j)"
lambda_system_Int: fixes x :: "'a set"   and f :: "'a set \<Rightarrow> ennreal"   and y :: "'a set" assumes "x \<in> lambda_system \<Omega> M f"   and "y \<in> lambda_system \<Omega> M f" shows "x \<inter> y \<in> lambda_system \<Omega> M f"
conv_tail1_2: fixes s :: "'a"   and d :: "bool"   and \<alpha> :: "'b"   and t :: "'a"   and xs :: "(bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, (d, \<alpha>, t) # xs) \<in> conv ars"   and "d" shows "(s, \<alpha>, t) \<in> ars"
preserves_dom: fixes f :: "'c" assumes "C.arr f" shows "D.dom (G f) = G (C.dom f)"
extensible_def: fixes R :: "('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph"   and G :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set" shows "extensible R G f \<equiv> \<exists>g. graph_homomorphism (snd R) G g \<and> agree_on (fst R) f g"
Lt_le_LeI: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "Lt a \<le> Le b"
red_single_monom_mult: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a"   and c :: "'c"   and s :: "'a" assumes "red_single p q f t"   and "c \<noteq> (0::'c)" shows "red_single (monom_mult c s p) (monom_mult c s q) f (s + t)"
finite_tvsT: fixes T :: "typ" shows "finite (tvsT T)"
decreasing_tendsto: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and l :: "'b"   and f :: "'c \<Rightarrow> 'b"   and F :: "'c filter" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in F. less_eq l (f n)"   and "\<And>x. less l x \<Longrightarrow> \<forall>\<^sub>F n in F. less (f n) x" shows "filterlim f (topological_space.nhds open l) F"
cycle_def: fixes p :: "'b list" shows "cycle p \<equiv> \<exists>u. awalk u p u \<and> distinct (tl (awalk_verts u p)) \<and> p \<noteq> []"
restrict_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else SOME y. True)"
sgn_mult: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "sgn (times a b) = times (sgn a) (sgn b)"
assms_1: fixes c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>c (c', s')" shows "phi"
le_env_def: fixes A :: "'a list"   and B :: "'a list" shows "(A \<le> B) = (length B = length A \<and> (\<forall>i<length A. A ! i \<le> B ! i))"
path_connected_space_def: fixes X :: "'b topology" shows "path_connected_space X \<equiv> \<forall>x\<in>topspace X. \<forall>y\<in>topspace X. \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
idem: fixes j :: "nat" assumes "j < max_Var_floatarith fa" shows "vs ! s j = vs ! j"
distinct_1: fixes x11 :: "nat"   and x12 :: "'a"   and x21 :: "nat"   and x22 :: "'a" shows "Leq x11 x12 \<noteq> Geq x21 x22"
Not_Zero: shows "Coinductive_Language.Not Zero = \<Sigma>\<^sup>*"
cobounded2: fixes b :: "'b"   and a :: "'b" shows "b \<le> a \<sqinter> b"
measurable_case_nat': fixes g :: "'a \<Rightarrow> val"   and R :: "'a measure"   and t :: "pdf_type"   and h :: "'a \<Rightarrow> nat \<Rightarrow> val"   and V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" assumes "g \<in> R \<rightarrow>\<^sub>M stock_measure t"   and "h \<in> R \<rightarrow>\<^sub>M state_measure V \<Gamma>" shows "(\<lambda>x. case_nat (g x) (h x)) \<in> R \<rightarrow>\<^sub>M state_measure (shift_var_set V) (case_nat t \<Gamma>)"
replace_primitive_matchexpr: fixes m :: "'b match_expr"   and disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and C :: "'c \<Rightarrow> 'b"   and \<gamma> :: "('b \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and replace_f :: "'c negation_type \<Rightarrow> 'b match_expr"   and a :: "action"   and p :: "'d" assumes "normalized_nnf_match m"   and "wf_disc_sel disc_sel C"   and "\<forall>pt. matches \<gamma> (replace_f pt) a p = matches \<gamma> (negation_type_to_match_expr_f C pt) a p" shows "matches \<gamma> (replace_primitive_matchexpr disc_sel replace_f m) a p = matches \<gamma> m a p"
cases: fixes x :: "('a, 'b, 'c) formula list \<times> ('a, 'b, 'c) formula list" obtains ante :: "('a, 'b, 'c) formula list" and succ :: "('a, 'b, 'c) formula list" where "x = (ante, succ)"
enum_eq_p: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "n \<le> j" shows "enum i j = p"
h1: shows "a \<le> c \<or> c \<le> a"
mod_mult_self1: fixes a :: "'b"   and c :: "'b"   and b :: "'b" shows "(a + c * b) mod b = a mod b"
valid_conj_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(conj_PL p q \<in> Cn {}) = (p \<in> Cn {} \<and> q \<in> Cn {})"
cases: fixes a1 :: "'b com option \<times> 'b"   and a2 :: "'b com option \<times> 'b" assumes "a1 -c\<rightarrow> a2" obtains f :: "'b \<Rightarrow> 'b" and s :: "'b" where "a1 = (Some (Basic f), sa__)" and "a2 = (None, f sa__)"   | P0 :: "'b com" and s :: "'b" and t :: "'b" and P1 :: "'b com" where "a1 = (Some (Seq P0__ P1__), sb__)" and "a2 = (Some P1__, t__)" and "(Some P0__, sb__) -c\<rightarrow> (None, t__)"   | P0 :: "'b com" and s :: "'b" and P2 :: "'b com" and t :: "'b" and P1 :: "'b com" where "a1 = (Some (Seq P0 P1a__), sc__)" and "a2 = (Some (Seq P2__ P1a__), ta__)" and "(Some P0, sc__) -c\<rightarrow> (Some P2__, ta__)"   | s :: "'b" and b :: "'b set" and P1 :: "'b com" and P2 :: "'b com" where "a1 = (Some (Cond b__ P1b__ P2a__), sd__)" and "a2 = (Some P1b__, sd__)" and "sd__ \<in> b__"   | s :: "'b" and b :: "'b set" and P1 :: "'b com" and P2 :: "'b com" where "a1 = (Some (Cond ba__ P1 P2), se__)" and "a2 = (Some P2, se__)" and "se__ \<notin> ba__"   | s :: "'b" and b :: "'b set" and P :: "'b com" where "a1 = (Some (While bb__ P__), sf__)" and "a2 = (None, sf__)" and "sf__ \<notin> bb__"   | s :: "'b" and b :: "'b set" and P :: "'b com" where "a1 = (Some (While bc__ Pa__), sg__)" and "a2 = (Some (Seq Pa__ (While bc__ Pa__)), sg__)" and "sg__ \<in> bc__"   | s :: "'b" and b :: "'b set" and P :: "'b com" and t :: "'b" where "a1 = (Some (Await b P), s)" and "a2 = (None, t)" and "s \<in> b" and "(Some P, s) -c*\<rightarrow> (None, t)"
thenOrdering_def: shows "thenOrdering = (\<Lambda> x y. case x of LT \<Rightarrow> LT | EQ \<Rightarrow> y | GT \<Rightarrow> GT)"
normalizer_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "normalizer G H = stabilizer G (\<lambda>g. \<lambda>H\<in>{H. H \<subseteq> carrier G}. g <#\<^bsub>G\<^esub> H #>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> g) H"
2: shows "fmlookup (fmrestrict_set {v \<in> fmdom' s2. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s2 v} s2) \<subseteq>\<^sub>m fmlookup (fmrestrict_set {v \<in> fmdom' s1. v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup s1 v} s1)"
simps_15: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = insert x21 (set x22)"
order_iff_strict: fixes ko :: "'a key_order"   and a :: "'a"   and b :: "'a" shows "ko.le ko a b = (ko.lt ko a b \<or> a = b)"
OrdP_I: fixes y :: "name"   and x :: "tm"   and H :: "fm set"   and z :: "name" assumes "insert (Var y IN x) H \<turnstile> Var y SUBS x"   and "insert (Var z IN Var y) (insert (Var y IN x) H) \<turnstile> Var z SUBS Var y"   and "atom y \<sharp> (x, z)"   and "\<forall>B\<in>H. atom y \<sharp> B"   and "atom z \<sharp> x"   and "\<forall>B\<in>H. atom z \<sharp> B" shows "H \<turnstile> OrdP x"
a_def: shows "a \<equiv> SOME x. x \<in> X"
evalid_def: fixes P :: "assn"   and e :: "expr"   and Q :: "vassn" shows "\<Turnstile>\<^sub>e {P} e {Q} \<equiv> \<forall>s v t. P s \<longrightarrow> (\<exists>n. s -e\<succ>v-n\<rightarrow> t) \<longrightarrow> Q v t"
image_Un: fixes f :: "'h \<Rightarrow> 'g"   and A :: "'h set"   and B :: "'h set" shows "f ` (A \<union> B) = f ` A \<union> f ` B"
simps_2: fixes "is" :: "'b set"   and t :: "'b" shows "redT_updI is (ClearInterrupt t) = is - {t}"
indT: fixes P :: "state \<Rightarrow> bool"   and F :: "behavior \<Rightarrow> bool" shows "\<turnstile> \<box>(Init P \<and> \<not> \<box>F \<longrightarrow> Init P$ \<and> F) \<longrightarrow> Init P \<longrightarrow> \<box>F"
assms_1: shows "path \<sigma> l = path \<sigma>' l'"
set_intros_2: fixes y :: "'b"   and x22 :: "'b list"   and x21 :: "'b" assumes "y \<in>. x22" shows "y \<in>. x21 # x22"
fc: shows "cod\<^bsub>CanonicalCat' T\<^esub> f = Y"
inj_iff_eq_0: shows "inj f = (\<forall>x. f x = (0::'c) \<longrightarrow> x = (0::'b))"
h4: shows "\<forall>csa. List.member (find_sgas (map real_of_rat_poly (fst (factorize_polys qs)))) csa = (csa \<in> consistent_sign_vectors (fst (factorize_polys qs)) UNIV \<and> has_no_zeros csa \<or> csa \<in> consistent_sign_vectors (fst (factorize_polys qs)) UNIV \<and> \<not> has_no_zeros csa)"
Radj_cancel: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and x :: "'a"   and y :: "'a"   and \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.Radj x y (ids.Radj x y \<nu>) = \<nu>"
cring: fixes R :: "('e, 'f) ring_scheme" assumes "cring R" shows "cring R"
pirreducible_degree: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)"   and "pirreducible\<^bsub>poly_ring R\<^esub> K p" shows "1 \<le> degree p"
l5_5_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<exists>X. Bet A B X \<and> Cong A X C D" shows "A B Le C D"
nn: shows "nprv (insert (subst \<phi> (Var y) x) F) \<psi>"
scaleC_right_mono_neg: fixes b :: "complex"   and a :: "complex"   and c :: "'b" assumes "b \<le> a"   and "c \<le> (0::'b)" shows "a *\<^sub>C c \<le> b *\<^sub>C c"
pullback_topology_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b topology" shows "pullback_topology A f T = topology (\<lambda>S. \<exists>U. openin T U \<and> S = f -` U \<inter> A)"
I: shows "A \<otimes> B \<notin> parts (used u)"
simps_1: fixes x :: "'a"   and y :: "'a"   and p :: "'b"   and z :: "('a, 'b) Combinators list" shows "removeShadowRules2 (AllowPortFromTo x y p # z) = (if DenyAllFromTo x y \<in> set z then removeShadowRules2 z else AllowPortFromTo x y p # removeShadowRules2 z)"
inf_commute: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y = y \<cdot> x"
append_take_drop_id: fixes n :: "nat"   and xs :: "'c list" shows "take n xs @ drop n xs = xs"
elims_3: fixes x :: "'a list" assumes "\<not> Lyndon x" obtains w :: "'a list" where "x = w" and "\<not> (w \<noteq> \<epsilon> \<and> (\<forall>n. 0 < n \<and> n < \<^bold>|w\<^bold>| \<longrightarrow> w <lex rotate n w))"
simps: fixes x :: "'c nonzero"   and a :: "'c nonzero" shows "proj_rel x a = (\<exists>c. a = c *\<^sub>R x \<and> c \<noteq> 0)"
path_from_toI: fixes xs :: "'c list"   and v :: "'c"   and w :: "'c" assumes "T.path xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "v \<leadsto>xs\<leadsto>\<^bsub>T\<^esub> w"
WN_on_def: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" shows "WN_on r A = (\<forall>a\<in>A. \<exists>b. (a, b) \<in> r\<^sup>!)"
inductive_invariant_def: fixes P :: "('r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option) \<Rightarrow> bool" shows "inductive_invariant P \<equiv> (\<forall>s. (\<exists>e r. s = \<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e)) \<and> program_expr e) \<longrightarrow> P s) \<and> (\<forall>s s'. s \<leadsto> s' \<longrightarrow> P s \<longrightarrow> P s')"
cong_sym_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b c a = unique_euclidean_semiring.cong modulo c b a"
degree_map_poly: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'a poly" assumes "\<And>x. x \<noteq> (0::'a) \<Longrightarrow> f x \<noteq> (0::'b)" shows "degree (map_poly f p) = degree p"
map_bits_rev_to_bl: fixes x :: "'a word" shows "map ((!!) x) [0..<size x] = rev (to_bl x)"
predicate2I: fixes P :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Q :: "'n \<Rightarrow> 'o \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
weight_spmf_of_set: fixes A :: "'c set" shows "weight_spmf (spmf_of_set A) = (if finite' A then 1 else 0)"
foldSetD_closed: fixes B :: "'e set"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and A :: "'e set"   and z :: "'f"   and e :: "'f" assumes "LCD B D f"   and "(A, z) \<in> foldSetD D f e" shows "z \<in> D"
dualPO: fixes cl :: "'b potype" assumes "PO cl" shows "dual cl \<in> PartialOrder"
simps_2: fixes a :: "'b"   and as :: "'b list" shows "annotate_rlen (a # as) = (length as, a) # annotate_rlen as"
SAMEQUOT_def: fixes a :: "'c"   and A :: "'c list set" shows "SAMEQUOT a A = {fin_cut_same a x @ replicate m a |x m. x \<in> A}"
dgrad_set_le_subset: fixes S :: "'a set"   and T :: "'a set"   and d :: "'a \<Rightarrow> nat" assumes "S \<subseteq> T" shows "dgrad_set_le d S T"
constant_rules_3: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph"   and c :: "'b" assumes "standard' C G"   and "c \<in> C" shows "maintained (const_prop c) G"
a1: shows "MPoly (\<Sum>m | m \<in> monomials p \<and> lookup m var = x. monomial (MPoly_Type.coeff p m) (Poly_Mapping.update var 0 m)) = q"
KBasic_1: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<phi>) in v]"
steps_height: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat"   and up :: "nat" assumes "finite A" shows "steps A f 0 up 0 = up * Max (f ` A \<union> {0})"
strict_monoidal_functor_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>C :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>C :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>C :: "'d"   and D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and T\<^sub>D :: "'e \<times> 'e \<Rightarrow> 'e"   and \<alpha>\<^sub>D :: "'e \<times> 'e \<times> 'e \<Rightarrow> 'e"   and \<iota>\<^sub>D :: "'e"   and F :: "'d \<Rightarrow> 'e" shows "strict_monoidal_functor C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D F \<equiv> (monoidal_category C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C \<and> monoidal_category D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D) \<and> functor C D F \<and> strict_monoidal_functor_axioms C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D F"
translation_map: fixes a :: "'a" assumes "a \<in> G" shows "(a)\<^sub>L \<in> G \<rightarrow>\<^sub>E G"
well_order_on_Field: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "well_order_on A r" shows "A = Field r"
finite_assignment_rhs\<^sub>s\<^sub>t: fixes S :: "('a, 'b) strand_step list" shows "finite (assignment_rhs\<^sub>s\<^sub>t S)"
lv_def: fixes p :: "(nat \<times> int) list"   and d :: "nat" shows "lv p d = fst (p ! d)"
similar_mat_witD_6: fixes n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "P \<in> carrier_mat n n"
Seq: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>0 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e\<^sub>0,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>0;; e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>2,s\<^sub>2\<rangle>"
ContingentlyTrue_def: shows "ContingentlyTrue \<equiv> \<lambda>p. p \<^bold>& \<^bold>\<diamond>\<^bold>\<not>p"
Un_range_binary: fixes a :: "'c set"   and b :: "'c set" shows "a \<union> b = \<Union> (range (binary a b))"
shift_eq_scons: fixes u :: "'a list"   and v :: "'a stream"   and a :: "'a"   and w :: "'a stream" shows "(u @- v = a ## w) = (u = [] \<and> v = a ## w \<or> (\<exists>u'. u = a # u' \<and> u' @- v = w))"
add_0_l: fixes a :: "'b"   and p :: "'b point" shows "ell_field_class.add a Infinity p = p"
l_inv_ex: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<exists>y\<in>carrier P. y \<oplus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>"
transition_eqvt': fixes P :: "'state"   and \<alpha> :: "'act"   and Q :: "'state"   and p :: "perm" assumes "P \<rightarrow> \<langle>\<alpha>,Q\<rangle>" shows "p \<bullet> P \<rightarrow> \<langle>p \<bullet> \<alpha>,p \<bullet> Q\<rangle>"
betapply_Abs_fv: fixes t :: "Core.term"   and T :: "typ"   and v :: "variable" assumes "is_closed t" shows "Abs_fv v T t \<bullet> Fv v T = t"
normalized_nnf_match_normalize_match: fixes m :: "'e match_expr" shows "\<forall>m'\<in>set (normalize_match m). normalized_nnf_match m'"
mpj_mHom: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "submodule R M H" shows "mpj M H \<in> mHom R M (M /\<^sub>m H)"
valid_index_dimsE: fixes "is" :: "nat list"   and A :: "'b tensor" assumes "is \<lhd> dims A" obtains "isb__ = []" and "dims A = []"   | "is" :: "nat list" and ds :: "nat list" and i :: "nat" and d :: "nat" where "isb__ = i # is" and "dims A = d # ds" and "is \<lhd> ds" and "i < d"
nonzeros_One_pdevs_raw: shows "{i. One_pdevs_raw i \<noteq> (0::'b)} = - {length Basis_list..}"
to_int_mod_ring_times: fixes x :: "'a mod_ring"   and y :: "'a mod_ring" shows "to_int_mod_ring (x * y) = M (to_int_mod_ring x * to_int_mod_ring y)"
order_prop: fixes x :: "('a \<times> 'a set) set"   and y :: "('a \<times> 'a set) set" shows "(x \<subseteq> y) = (\<exists>z. x \<union> z = y)"
guaranteeStripPair_def: fixes f :: "'c"   and g :: "'d set" shows "guaranteeStripPair f g = (f, g)"
well_def_wp_healthy: fixes prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "well_def prog" shows "healthy (wp prog)"
scalar_prod_uminus_right: fixes v :: "'a vec"   and w :: "'a vec" assumes "dim_vec v = dim_vec w" shows "v \<bullet> - w = - (v \<bullet> w)"
constrains_Int_distrib: fixes C :: "'c set"   and A :: "'c set"   and B :: "'c set" shows "C co A \<inter> B = (C co A) \<inter> (C co B)"
z1: shows "lincomb c W = 0\<^sub>v n"
insert_remdups: fixes x :: "'a"   and xs :: "'a list" shows "List.insert x (remdups xs) = remdups (List.insert x xs)"
psimps: fixes s :: "char list \<Rightarrow> int"   and b :: "bexp"   and C :: "acom" assumes "times_dom (s, b, C)" shows "Nielson_VCG.times s b C = (if bval b s then Suc (Nielson_VCG.times (postQ C s) b C) else 0)"
preprm_apply_unequal: fixes x :: "'b"   and y :: "'b"   and \<pi> :: "('b \<times> 'b) list" assumes "x \<noteq> y" shows "preprm_apply \<pi> x \<noteq> preprm_apply \<pi> y"
simps: fixes l :: "rat"   and r :: "rat"   and x :: "real" shows "in_interval (l, r) x = (real_of_rat l \<le> x \<and> x \<le> real_of_rat r)"
\<phi>J_def: fixes \<psi> :: "'fmla" shows "\<phi>J \<psi> \<equiv> instInpP \<psi> 0 (\<tau>J \<psi>)"
subgroup_is_comm_group: fixes H :: "('c \<Rightarrow> 'a) set" assumes "subgroup H (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)" shows "comm_group (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<lparr>carrier := H\<rparr>)"
atleast_free_weaken: fixes m :: "'a \<Rightarrow> 'b option"   and n :: "nat" assumes "atleast_free m (Suc n)" shows "atleast_free m n"
vreal_inverse_closed: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>" shows "x\<inverse>\<^sub>\<real> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"
IA2: shows "Iagree I J {Inl x |x. x \<in> SIGT t2}"
vc: shows "(ftv T1 - ftv \<Gamma>) \<sharp>* \<theta>"