hcju/isabelleps huggingface dataset:
spoly_red_zero_imp_crit_pair_cbelow_on: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "p \<in> dgrad_p_set d m"   and "q \<in> dgrad_p_set d m"   and "p \<noteq> 0"   and "q \<noteq> 0"   and "(red F)\<^sup>*\<^sup>* (spoly p q) 0" shows "crit_pair_cbelow_on d m F p q"
WT_converter_id: fixes \<I> :: "('h, 'i) \<I>" shows "\<I>, \<I> \<turnstile>\<^sub>C 1\<^sub>C \<surd>"
chambersubcomplex_image: shows "folding_g.ChamberSubcomplex (f \<turnstile> X)"
exhaust: fixes y :: "'a local.term" obtains x1 :: "'a" where "y = \<^bold>\<langle>x1\<^bold>\<rangle>"   | "y = \<^bold>\<I>"   | x31 :: "'a local.term" and x32 :: "'a local.term" where "y = x31 \<^bold>\<otimes> x32"   | x41 :: "'a local.term" and x42 :: "'a local.term" where "y = x41 \<^bold>\<cdot> x42"   | x5 :: "'a local.term" where "y = \<^bold>\<l>\<^bold>[x5\<^bold>]"   | x6 :: "'a local.term" where "y = \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[x6\<^bold>]"   | x7 :: "'a local.term" where "y = \<^bold>\<r>\<^bold>[x7\<^bold>]"   | x8 :: "'a local.term" where "y = \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[x8\<^bold>]"   | x91 :: "'a local.term" and x92 :: "'a local.term" and x93 :: "'a local.term" where "y = \<^bold>\<a>\<^bold>[x91, x92, x93\<^bold>]"   | x101 :: "'a local.term" and x102 :: "'a local.term" and x103 :: "'a local.term" where "y = \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[x101, x102, x103\<^bold>]"
replicate_add: fixes n :: "nat"   and m :: "nat"   and x :: "'a" shows "replicate (n + m) x = replicate n x @ replicate m x"
slp_of_fa_un_slp_of_fa_rev_un: fixes Binop :: "floatarith \<Rightarrow> floatarith"   and a :: "floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and M2 :: "(floatarith, nat) mapping"   and slp2 :: "floatarith list" shows "slp_of_fa_rev_un Binop a M slp (length slp) M2 slp2 (length slp2) = (let (M, slp') = slp_of_fa_un Binop a M (rev slp) M2 (rev slp2) in (M, rev slp', length slp'))"
differentiable_vec: fixes S :: "'a set" shows "vec differentiable_on S"
distinct_5: fixes x11 :: "CAttId"   and x12 :: "nat"   and x41 :: "Arraytype"   and x42 :: "nat"   and x43 :: "nat" shows "objLoc x11 x12 \<noteq> arrLoc x41 x42 x43"
lebS': fixes T :: "(real, 'n) vec set" assumes "T \<subseteq> g ` S'"   and "T \<in> sets lebesgue" shows "{x \<in> S'. g x \<in> T} \<in> sets lebesgue"
List_app: fixes h :: "nat \<Rightarrow> nat option"   and x :: "nat"   and as :: "nat list"   and bs :: "nat list" shows "List h x (as @ bs) = (\<exists>y. Path h x as y \<and> List h y bs)"
lift_preserves_snd_I: fixes F :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and i :: "nat" assumes "F \<in> preserves snd" shows "lift i F \<in> preserves snd"
diamond_x_def: shows "diamond_x \<equiv> \<lambda>t. (t - 1 / 2) * d"
powsum_split_var4: fixes x :: "'a"   and m :: "nat"   and n :: "nat" shows "x\<^bsub>0\<^esub>\<^bsup>m + n\<^esup> + x\<^bsub>m\<^esub>\<^bsup>n\<^esup> = x\<^bsub>0\<^esub>\<^bsup>m + n\<^esup>"
3: shows "map ((!) (as @ bs)) (snd xs) = map ((!) as) (snd xs)"
MGU_def: fixes \<sigma> :: "('b \<times> 'b trm) list"   and t :: "'b trm"   and u :: "'b trm" shows "MGU \<sigma> t u = (Unifier \<sigma> t u \<and> (\<forall>\<theta>. Unifier \<theta> t u \<longrightarrow> (\<exists>\<gamma>. \<theta> \<doteq> \<sigma> \<lozenge> \<gamma>)))"
summable_deriv_ln_Gamma: fixes z :: "'a" assumes "z \<noteq> (0::'a)" shows "summable (\<lambda>n. inverse (of_nat (Suc n)) - inverse (z + of_nat (Suc n)))"
pexpr_cases: fixes e :: "pexpr" obtains c :: "int" where "e = PExpr1 (PCnst c)"   | n :: "nat" where "e = PExpr1 (PVar n__)"   | e1 :: "pexpr" and e2 :: "pexpr" where "e = PExpr1 (PAdd e1__ e2__)"   | e1 :: "pexpr" and e2 :: "pexpr" where "e = PExpr1 (PSub e1a__ e2a__)"   | e' :: "pexpr" where "e = PExpr1 (PNeg e'__)"   | e1 :: "pexpr" and e2 :: "pexpr" where "e = PExpr2 (PMul e1 e2)"   | e' :: "pexpr" and n :: "nat" where "e = PExpr2 (PPow e' n)"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "subseqs (x # xs) = (let xss = subseqs xs in map ((#) x) xss @ xss)"
le_M: fixes t :: "'s"   and t' :: "'s" assumes "(s, t) \<in> acc_on (- H)"   and "t \<notin> H"   and "t' \<in> set_pmf (K t)" shows "\<rho> t + \<iota> t t' \<le> M"
Step1_2_5: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> ImpNext p \<and> \<not> unchanged (e p, c p, r p, m p, rmhist!p) \<and> $S5 rmhist p \<longrightarrow> S6 rmhist p$ \<and> RPCReply crCh rmCh rst p \<and> unchanged (e p, c p, m p) \<or> S6 rmhist p$ \<and> RPCFail crCh rmCh rst p \<and> unchanged (e p, c p, m p)"
assert_def: fixes p :: "'c Assertion" shows "{\<cdot> p } = Rep_Assertion p"
lemVecPlusMinus: fixes u :: "'a Vector"   and v :: "'a Vector" shows "u \<oplus> ~ v = u \<ominus> v"
ws_idecl_def: fixes G :: "prog"   and I :: "qtname"   and si :: "qtname list" shows "ws_idecl G I si = (\<forall>J\<in>set si. is_iface G J \<and> (J, I) \<notin> (subint1 G)\<^sup>+)"
elem_set: fixes x :: "'a"   and xo :: "\<langle>'a\<rangle>\<^sub>\<bottom>" shows "(x \<in> set_option xo) = (xo = \<lfloor>x\<rfloor>)"
gd_powerprod_axioms: fixes cmp_term :: "'b nat_term_order" shows "gd_powerprod (\<lambda>s t. le_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min))) (\<lambda>s t. lt_of_nat_term_order cmp_term (fst (s, the_min)) (fst (t, the_min)))"
listrel1p_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "listrel1p r xs ys = ((xs, ys) \<in> listrel1 {(x, y). r x y})"
norm_cvec_square: fixes z :: "complex \<times> complex" shows "\<langle>z\<rangle>\<^sup>2 = Re \<langle>z,z\<rangle>"
path_image_rectpath_inter_box: fixes a :: "complex"   and b :: "complex" assumes "Re a \<le> Re b"   and "Im a \<le> Im b" shows "path_image (rectpath a b) \<inter> box a b = {}"
hoare1_def: fixes p :: "'b Assertion"   and S :: "'b"   and q :: "'b Assertion" shows "hoare1 p S q = ([\<cdot> p ] * S * [\<cdot> - q ] = \<top>)"
from_nat_eq_imp_eq: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and x :: "nat"   and xa :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.from_nat Abs x = mod_type.from_nat Abs xa"   and "x < CARD('a)"   and "xa < CARD('a)" shows "x = xa"
point_simps_1: fixes n :: "nat" shows "left (point n) = n"
permutation_permutes: fixes p :: "'d \<Rightarrow> 'd" shows "permutation p = (\<exists>S. finite S \<and> p permutes S)"
config'_append2: fixes step :: "'b \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'b"   and A :: "('b \<Rightarrow> 'c) \<times> ('b \<times> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'c)"   and s :: "'b \<times> 'c"   and xs :: "'d list"   and ys :: "'d list" shows "On_Off.config' step A s (xs @ ys) = On_Off.config' step A (On_Off.config' step A s xs) ys"
cases: fixes x :: "nat \<times> ('ekey \<times> 'dkey \<times> 'cipher list) option \<times> 'cipher" obtains \<eta> :: "nat" and cipher :: "'cipher" where "x = (\<eta>'__, None, ciphera__)"   | \<eta> :: "nat" and ekey :: "'ekey" and dkey :: "'dkey" and cstars :: "'cipher list" and cipher :: "'cipher" where "x = (\<eta>, Some (ekey, dkey, cstars), cipher)"
hyps: fixes v :: "nat" assumes "xs !. v = Some x" shows "mapi f xs !. v = Some (f v x)"
red_simps_2: fixes t :: "'a local.term" assumes "Ide t" shows "Trg (t\<^bold>\<down>) = Trg t"
add_right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> E"   and "y \<in> E"   and "z \<in> E" shows "(y + x = z + x) = (y = z)"
directedLength_def: fixes f :: "face"   and a :: "nat"   and b :: "nat" shows "directedLength f a b \<equiv> if a = b then 0 else |between (vertices f) a b| + 1"
abs_eq: shows "{}\<^sub>f = ffun.Abs_pfun {}\<^sub>p"
mds_eq: shows "conc.low_mds_eq mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C"
bij': shows "bij_betw (inv_into (uverts G\<^sub>1) f) (uverts G\<^sub>2) (uverts G\<^sub>1)"
compl_bot_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus bot = top"
assms_2: shows "\<guillemotleft>w : src u \<rightarrow> src f\<guillemotright>"
holf: shows "f holomorphic_on ball 0 1 - {0}"
nop_v11_axioms_def: fixes A :: "V"   and n :: "V"   and f :: "V" shows "nop_v11_axioms A n f \<equiv> n \<in>\<^sub>\<circ> \<omega> \<and> \<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n \<and> \<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
distr_cong_AE: fixes M :: "'a measure"   and K :: "'a measure"   and N :: "'b measure"   and L :: "'b measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = K"   and "sets N = sets L"   and "AE x in M. f x = g x"   and "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> K \<rightarrow>\<^sub>M L" shows "distr M N f = distr K L g"
effect_shiftr: fixes h :: "heap"   and i :: "nat ref"   and j :: "nat"   and a :: "'b array"   and key :: "'b"   and h' :: "heap"   and u :: "unit" assumes "Ref.get h i = j"   and "j < Array.length h a"   and "sorted (take j (Array.get h a))"   and "effect (while (shiftr_p a key i) (shiftr_f a i)) h h' u" shows "Array.length h a = Array.length h' a \<and> Ref.get h' i \<le> j \<and> mset ((Array.get h a)[j := key]) = mset ((Array.get h' a)[Ref.get h' i := key]) \<and> ini h a j = both h' a j i \<and> sorted (both h' a j i) \<and> (\<forall>x\<in>set (right h' a j i). key < x)"
reachNT_stateO_aux: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and s :: "'a" assumes "BD_Security.reachNT istate step T s" shows "s = istate \<or> (\<exists>sh a ou. IO_Automaton.reach istate step sh \<and> step sh a = (ou, s) \<and> \<not> T (Trans sh a ou s))"
dp_map2: fixes x :: "'a"   and y :: "'a" assumes "is_direct_product x y" shows "is_map y"
not_eq_extremum: fixes a :: "'b list" shows "(a \<noteq> \<epsilon>) = (\<epsilon> <p a)"
butterfly_scaleC_right: fixes \<psi> :: "'b"   and c :: "complex"   and \<phi> :: "'a" shows "butterfly \<psi> (c *\<^sub>C \<phi>) = cnj c *\<^sub>C butterfly \<psi> \<phi>"
yun_erel_1_eq: fixes F :: "int poly"   and f :: "rat poly" assumes "yun_erel F f" shows "(F = 1) = (f = 1)"
plusplus_closed: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c list"   and y :: "'c" assumes "Semilat A r f"   and "set x \<subseteq> A"   and "y \<in> A" shows "x \<Squnion>\<^bsub>f\<^esub> y \<in> A"
new_element_is_element_ptr: fixes h :: "(_) heap"   and new_element_ptr :: "(_) element_ptr" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr" shows "is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_element_ptr"
i: shows "subset.chain {B. \<not> B \<turnstile> \<phi>} (f ` A')"
rref_Gauss_Jordan: fixes A :: "(('b, 'c) vec, 'd) vec" shows "reduced_row_echelon_form (Gauss_Jordan A)"
real_less_lsqrt: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y"   and "x < y\<^sup>2" shows "sqrt x < y"
a_n: shows "mod_type_class.to_nat a \<le> n"
cases: fixes x :: "'a Literal" obtains A :: "'a" where "x = Pos Aa__"   | A :: "'a" where "x = Neg A"
some_child_sm: fixes t :: "('b \<times> 'c) list"   and y :: "'c"   and xs :: "('b \<times> 'c) list"   and ls :: "('b \<times> 'c) list"   and sub :: "'b"   and sep :: "'c"   and rs :: "('b \<times> 'c) list" assumes "linear_split_help t y xs = (ls, (sub, sep) # rs)" shows "y \<le> sep"
expand: fixes tllist :: "('f, 'g) tllist"   and tllist' :: "('f, 'g) tllist" assumes "is_TNil tllist = is_TNil tllist'"   and "\<lbrakk>is_TNil tllist; is_TNil tllist'\<rbrakk> \<Longrightarrow> terminal tllist = terminal tllist'"   and "\<lbrakk>\<not> is_TNil tllist; \<not> is_TNil tllist'\<rbrakk> \<Longrightarrow> thd tllist = thd tllist' \<and> ttl tllist = ttl tllist'" shows "tllist = tllist'"
borel_measurable_vimage_open: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>T. open T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
card_of_diff: fixes A :: "'a set"   and B :: "'a set" shows "|A - B| \<le>o |A|"
filter_greater: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "B \<subseteq> carrier"   and "A \<in> F" shows "B \<in> F"
simps_2: fixes y :: "'a"   and v :: "nat \<times> 'a"   and va :: "(nat \<times> 'a) list" shows "addmult_spvec y [] (v # va) = smult_spvec y (v # va)"
1: shows "continuous_on (path_image (circlepath z r)) (\<lambda>x. 1 / (2 * complex_of_real pi * \<i>) * g x)"
lunstream_def: fixes g :: "'c \<Rightarrow> ('d, 'c) step" shows "lunstream g \<equiv> llist.fixp_fun (\<lambda>lunstream s. case g s of Done \<Rightarrow> LNil | Skip s' \<Rightarrow> lunstream s' | Yield x s' \<Rightarrow> LCons x (lunstream s'))"
finfun_apply_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> finfun" obtains x :: "('a, 'b) finfun" where "y = finfun_apply x"
subseteq_guards_Basic: fixes c :: "('d, 'e, 'f) com"   and f :: "'d \<Rightarrow> 'd" assumes "c \<subseteq>\<^sub>g Basic f" shows "c = Basic f"
measure_eqI_countable: fixes M :: "'c measure"   and A :: "'c set"   and N :: "'c measure" assumes "sets M = Pow A"   and "sets N = Pow A"   and "countable A"   and "\<And>a. a \<in> A \<Longrightarrow> emeasure M {a} = emeasure N {a}" shows "M = N"
assoc_oo: fixes f :: "'d \<rightarrow> 'c"   and g :: "'e \<rightarrow> 'd"   and h :: "'b \<rightarrow> 'e" shows "f oo g oo h = (f oo g) oo h"
subset: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "subgroup H G" shows "H \<subseteq> carrier G"
ideal_points_proportional: fixes H :: "circline"   and i1 :: "complex_homo"   and i2 :: "complex_homo"   and k :: "real" assumes "is_poincare_line H"   and "ideal_points H = {i1, i2}"   and "to_complex i1 = cor k * to_complex i2" shows "0\<^sub>h \<in> circline_set H"
divide_inverse: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a b = times a (inverse b)"
insertCI: fixes a :: "'c"   and B :: "'c set"   and b :: "'c" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
Progress: fixes P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and s :: "heap \<times> sfields \<times> sheap"   and e :: "char list exp"   and T :: "ty"   and b :: "bool" assumes "wf_J_prog P"   and "P,E,s \<turnstile> e : T \<surd>"   and "\<D> e \<lfloor>dom (lcl s)\<rfloor>"   and "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>"   and "\<not> final e" shows "\<exists>e' s' b'. P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>"
strict_coboundedI1: fixes c :: "'c"   and a :: "'c"   and b :: "'c" assumes "c < a" shows "c < max a b"
last_in_set: fixes as :: "'g list" assumes "as \<noteq> []" shows "last as \<in> set as"
2546: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<squnion> (- - x \<sqinter> - y \<squnion> z) = - x \<squnion> (- y \<squnion> z)"
less_eq_cblinfun_def_heterogenous: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(A \<le> B) = (if heterogenous_same_type_cblinfun TYPE('a) TYPE('b) then \<forall>\<psi>. 0 \<le> \<psi> \<bullet>\<^sub>C ((B - A) *\<^sub>V heterogenous_cblinfun_id *\<^sub>V \<psi>) else A = B)"
echelon_form_of_column_k_efficient_def: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A' :: "(('d, 'e) vec, 'f) vec \<times> nat"   and k :: "nat" shows "echelon_form_of_column_k_efficient bezout A' k = (let (A, i) = A'; from_nat_k = mod_type_class.from_nat k; from_nat_i = mod_type_class.from_nat i; all_zero_below_i = \<forall>m>from_nat_i. A $ m $ from_nat_k = (0::'d) in if i = nrows A \<or> A $ from_nat_i $ from_nat_k = (0::'d) \<and> all_zero_below_i then (A, i) else if all_zero_below_i then (A, i + 1) else let n = LEAST n. A $ n $ from_nat_k \<noteq> (0::'d) \<and> from_nat_i \<le> n; interchange_A = interchange_rows A from_nat_i n in (bezout_iterate interchange_A (nrows A - 1) from_nat_i from_nat_k bezout, i + 1))"
exhaust: fixes y :: "finite_1" obtains "y = a\<^sub>1"
rec_less_def: shows "rec_less = CN rec_sign [rec_swap rec_minus]"
bezout_ext_def: fixes a :: "'a"   and b :: "'a" shows "bezout_ext a b = (SOME (p, q, u, v, d). p * a + q * b = d \<and> d dvd a \<and> d dvd b \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd d) \<and> d * u = - b \<and> d * v = a)"
coincidence_formula: fixes \<phi> :: "('a, 'b, 'c) formula"   and I :: "('a, 'b, 'c) interp"   and J :: "('a, 'b, 'c) interp"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec"   and \<nu>' :: "(real, 'c) vec \<times> (real, 'c) vec" assumes "fsafe \<phi>"   and "Iagree I J (SIGF \<phi>)"   and "Vagree \<nu> \<nu>' (FVF \<phi>)" shows "(\<nu> \<in> fml_sem I \<phi>) = (\<nu>' \<in> fml_sem J \<phi>)"
length: shows "length (map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))) = length (xs ! i)"
sel_1: fixes x1 :: "'e" shows "the_Var (Var x1) = x1"
CFG_SSA_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option" shows "CFG_SSA \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis \<equiv> CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<and> CFG_SSA_axioms \<alpha>n inEdges' defs phis"
flowpipe_exivl: fixes X0 :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set"   and hl :: "real"   and hu :: "real"   and CX :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set"   and X1 :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set"   and h :: "real"   and x0 :: "'c"   and d0 :: "'c \<Rightarrow>\<^sub>L 'c" assumes "flowpipe X0 hl hu CX X1"   and "hl \<le> h"   and "h \<le> hu"   and "(x0, d0) \<in> X0" shows "h \<in> existence_ivl0 x0"
tm_collapse_apps: fixes s :: "('c, 'd) tm" shows "apps (Hd (head s)) (args s) = s"
infinite_Complement_set_less_eq: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less"   and "finite A"   and "finite B"   and "infinite UNIV" shows "\<not> ord.set_less_eq less_eq (- A) B"
scale_bind_spmf: fixes r :: "real"   and p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" assumes "r \<le> 1" shows "scale_spmf r (p \<bind> f) = p \<bind> (\<lambda>x. scale_spmf r (f x))"
dzero': fixes i :: "nat" assumes "i \<le> m" shows "d fs' i \<noteq> 0"
ring_hom_a_inv: fixes S :: "('e, 'f) ring_scheme"   and h :: "'a \<Rightarrow> 'e"   and b :: "'a" assumes "ring S"   and "h \<in> ring_hom R S"   and "b \<in> carrier R" shows "h (\<ominus> b) = \<ominus>\<^bsub>S\<^esub> h b"
simps_2: fixes m :: "'a SecurityInvariant"   and Ms :: "'a SecurityInvariant list"   and G :: "'a list_graph" shows "TopoS_Composition_Theory_impl.generate_valid_topology_some (m # Ms) G = (if implc_sinvar m G then TopoS_Composition_Theory_impl.generate_valid_topology_some Ms G else FiniteListGraph.delete_edges (TopoS_Composition_Theory_impl.generate_valid_topology_some Ms G) (TopoS_Interface_impl.minimalize_offending_overapprox (implc_sinvar m) (edgesL G) [] G))"
atLeastAtMost_shift_0: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "sum g {m..n} = sum (g \<circ> (+) m) {0..n - m}"
prebicategory_axioms_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "prebicategory_axioms V H \<equiv> (\<forall>\<mu>. partial_magma.arr V \<mu> \<longrightarrow> weak_composition.sources V H \<mu> \<noteq> {}) \<and> (\<forall>\<mu>. partial_magma.arr V \<mu> \<longrightarrow> weak_composition.targets V H \<mu> \<noteq> {})"
takeWhile_sing_pow: fixes a :: "'b"   and w :: "'b list" shows "(takeWhile (\<lambda>x. x = a) w = w) = (w = [a] \<^sup>@ \<^bold>|w\<^bold>|)"
replacement_monotonic: fixes v :: "'a trm"   and \<sigma> :: "('a \<times> 'a trm) list"   and u :: "'a trm"   and t :: "'a trm"   and p :: "indices list"   and s :: "'a trm" assumes "(v \<lhd> \<sigma>, u \<lhd> \<sigma>) \<in> trm_ord"   and "subterm t p u"   and "replace_subterm t p v s" shows "(s \<lhd> \<sigma>, t \<lhd> \<sigma>) \<in> trm_ord"
ptensor_mat_trace: fixes m1 :: "'a mat"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2" shows "trace (ptensor_mat m1 m2) = trace m1 * trace m2"
l\<^sub>1\<^sub>2: shows "instance_of\<^sub>l l\<^sub>1 l\<^sub>2"
dvd_mod_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.semidom_modulo divide plus minus zero times one modulo"   and "dvd.dvd times c b" shows "dvd.dvd times c (modulo a b) = dvd.dvd times c a"
deg_add_eq: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "deg R q \<noteq> deg R p" shows "deg R (p \<oplus>\<^bsub>UP R\<^esub> q) = max (deg R p) (deg R q)"
snoc: shows "concat (map2 (\<lambda>x i. replicate (scf i) x) (bef @ i # aft) [0..<length (bef @ i # aft)]) = concat (map2 (\<lambda>x i. replicate (scf i) x) bef [0..<length bef]) @ replicate (scf (length bef)) i @ concat (map2 (\<lambda>x i. replicate (scf (Suc (length bef + i))) x) aft [0..<length aft])"
dom_char: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "A_B.dom f = (if A_B.arr f then A_B.MkIde (A_B.Dom f) else A_B.null)"
sum_nonzero_representation_eq: fixes basis :: "'b set"   and v :: "'b" assumes "vs1.independent basis"   and "v \<in> vs1.span basis" shows "(\<Sum>b | vs1.representation basis v b \<noteq> (0::'a). vs1.representation basis v b *a b) = v"
semiring_normalization_rules_2: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and m :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "plus (times a m) m = times (plus a one) m"
object_empty: fixes x :: "(_) Object" shows "\<lparr>RObject.nothing = (), \<dots> = RObject.more x\<rparr> = x"