hcju/isabelleps huggingface dataset:
lexext_Cons: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and ys :: "'b list"   and xs :: "'b list" assumes "gt y x" shows "lexext gt (y # ys) (x # xs)"
verticesFrom_Def_1: fixes f :: "face" shows "verticesFrom f \<equiv> rotate_to (vertices f)"
HF: shows "(H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. nil \<sqinter> (c::'a::refinement_lattice) ; (c\<^sup>\<star> \<sqinter> x))"
nth_akiyama_tanigawa_step_aux: fixes n :: "nat"   and xs :: "real list"   and m :: "nat" assumes "n < length xs - 1" shows "akiyama_tanigawa_step_aux m xs ! n = real (m + n) * (xs ! n - xs ! Suc n)"
simps_3: fixes x :: "octo"   and y :: "octo" shows "Im2 (x - y) = Im2 x - Im2 y"
essential_eq_leI: fixes \<Gamma> :: "('a, 'b) web_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> g"   and "wave \<Gamma> g"   and "\<And>e. f e \<le> g e"   and "\<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g) \<subseteq> TER\<^bsub>\<Gamma>\<^esub> f" shows "\<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) = \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> g)"
M': fixes s :: "'a"   and t :: "'a" assumes "M' s"   and "(s, t) \<in> R' \<union> Rw'" shows "M' t"
Q_min_is_light: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option"   and u :: "'a"   and d :: "nat" assumes "Prim_Invar2_ctd_loc w g r Q \<pi>"   and "Q u = enat d"   and "\<forall>v. enat d \<le> Q v" obtains v :: "'a" where "\<pi> u = Some v" and "Prim.light_edge w g r (Prim.S r (A Q \<pi>)) v u"
jordan_block_def: fixes n :: "nat"   and a :: "'a" shows "jordan_block n a = Matrix.mat n n (\<lambda>(i, j). if i = j then a else if Suc i = j then 1::'a else (0::'a))"
mtf2_forward_effect4: fixes q :: "'a"   and xs :: "'a list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "i < index xs q - n" shows "index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < index xs q - n"
cases: fixes a1 :: "com list"   and a2 :: "state"   and a3 :: "com list"   and a4 :: "state" assumes "(a1, a2) \<rightarrow> (a3, a4)" obtains t :: "state" and f :: "state \<Rightarrow> state set" and s :: "state" and cs :: "com list" where "a1 = Do f__ # cs__" and "a2 = s__" and "a3 = cs__" and "a4 = t" and "t \<in> f__ s__"   | c1 :: "com" and c2 :: "com" and cs :: "com list" and s :: "state" where "a1 = (c1__; c2__) # csa__" and "a2 = sa__" and "a3 = c1__ # c2__ # csa__" and "a4 = sa__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and c2 :: "com" and cs :: "com list" where "a1 = (IF b__ THEN c1a__ ELSE c2a__) # csb__" and "a2 = sb__" and "a3 = c1a__ # csb__" and "a4 = sb__" and "b__ sb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and c2 :: "com" and cs :: "com list" where "a1 = (IF ba__ THEN c1 ELSE c2) # csc__" and "a2 = sc__" and "a3 = c2 # csc__" and "a4 = sc__" and "\<not> ba__ sc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and cs :: "com list" where "a1 = (WHILE bb__ DO c__) # csd__" and "a2 = sd__" and "a3 = csd__" and "a4 = sd__" and "\<not> bb__ sd__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and cs :: "com list" where "a1 = (WHILE b DO ca__) # cse__" and "a2 = se__" and "a3 = ca__ # (WHILE b DO ca__) # cse__" and "a4 = se__" and "b se__"   | cs :: "com list" and s :: "state" where "a1 = CALL # csf__" and "a2 = sf__" and "a3 = body # csf__" and "a4 = sf__"   | f :: "state \<Rightarrow> state" and c :: "com" and g :: "state \<Rightarrow> state \<Rightarrow> state" and cs :: "com list" and s :: "state" where "a1 = (LOCAL f; c; g) # cs" and "a2 = s" and "a3 = c # Do (\<lambda>t. {g s t}) # cs" and "a4 = f s"
rename_gens_canceled: fixes g :: "'d \<Rightarrow> 'e"   and l :: "(bool \<times> 'd) list"   and f :: "bool \<Rightarrow> bool" assumes "inj_on g (snd ` set l)"   and "canceled l" shows "canceled (map (map_prod f g) l)"
map_map: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'i \<Rightarrow> 'h"   and xs :: "'i list" shows "map f (map g xs) = map (f \<circ> g) xs"
matrix_vector_mult_matrix_matrix_mult_compose: fixes A :: "(('d, 'g) vec, 'f) vec"   and B :: "(('d, 'e) vec, 'g) vec" shows "(*v) (A ** B) = (*v) A \<circ> (*v) B"
rep_eq: shows "key_compare key_order_of_le = comparator_of"
suffix_Nil: fixes xs :: "'b list" shows "xs \<le>s \<epsilon> = (xs = \<epsilon>)"
mult_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_mult [x, y] = x * y"
h_le_lim_h_plus: fixes e :: "'v \<times> 'v" shows "h e \<le> (\<Squnion>n. h_plus n e)"
ccField_ccProd: fixes S :: "Vars.var set"   and S' :: "Vars.var set" shows "ccField (S G\<times> S') = (if S = {} then {} else if S' = {} then {} else S \<union> S')"
rtrancl_eq_or_trancl: fixes x :: "'e"   and y :: "'e"   and R :: "('e \<times> 'e) set" shows "((x, y) \<in> R\<^sup>*) = (x = y \<or> x \<noteq> y \<and> (x, y) \<in> R\<^sup>+)"
snoc: fixes m :: "'a"   and t :: "'a list" shows "fst (foldl f (m, t) xs) = Max (SelectionSort_Functional.set (m # xs))"
assms: fixes theta :: "('a \<times> 'a) set" assumes "theta \<in> Theta" shows "sym theta"
R1: fixes xps :: "('e \<times> 'f) list"   and uu :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and s :: "'g"   and x :: "'e" assumes "(xps, uu) \<in> c_failures step out s\<^sub>0"   and "s = foldl step s\<^sub>0 (map fst xps)" shows "(xps @ [(x, out s x)], {(y, p). p \<noteq> out (step s x) y}) \<in> c_failures step out s\<^sub>0"
IMP_PRED: fixes P1 :: "'e \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "IMP_PRE P1 P2"   and "P1 x" shows "P2 x"
fmap_of_list_simps_2: fixes k :: "'c"   and v :: "'d"   and kvs :: "('c \<times> 'd) list" shows "fmap_of_list ((k, v) # kvs) = fmap_of_list kvs(k $$:= v)"
subst0_nb: fixes t :: "tm"   and p :: "fm" assumes "tmbound0 t"   and "qfree p" shows "bound0 (subst0 t p)"
diff_zero_right: fixes x :: "'a" assumes "x \<in> E" shows "(0::'a) - x = - x"
right_neutral: fixes a :: "'d" shows "a + (0::'d) = a"
BS3: fixes f :: "'a"   and f' :: "'a"   and \<mu> :: "'a"   and \<mu>' :: "'a" assumes "is_left_adjoint f"   and "is_left_adjoint f'"   and "\<guillemotleft>\<mu> : f \<Rightarrow> f'\<guillemotright>"   and "\<guillemotleft>\<mu>' : f \<Rightarrow> f'\<guillemotright>" shows "local.iso \<mu> \<and> local.iso \<mu>' \<and> \<mu> = \<mu>'"
left_idem: fixes a :: "'a"   and b :: "'a" shows "inf a (inf a b) = inf a b"
to_language_Neg: fixes A :: "'b list set" shows "to_language (- A) = Coinductive_Language.Not (to_language A)"
abs_eq_iff: fixes x :: "('a, 'b, 'c) var0"   and y :: "('a, 'b, 'c) var0" shows "(abs_var x = abs_var y) = x \<sim> y"
supp_conv_fresh: fixes x :: "'e" shows "supp x = {a. \<not> a \<sharp> x}"
FAssThrow2: fixes v :: "val"   and F :: "char list"   and D :: "char list"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>Val v\<bullet>F{D} := throw e,s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
inverse: fixes h :: "'a \<Rightarrow> real" assumes "h \<in> G" shows "(\<lambda>x. inverse (h x)) \<in> G"
Re_divide: fixes x :: "complex"   and y :: "complex" shows "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
less_def: shows "less = (\<lambda>(a, b, c) (a', b', c'). a < a' \<or> a = a' \<and> (b < b' \<or> b = b' \<and> c < c'))"
a_3: fixes uu :: "('a \<times> 'c) list"   and x :: "'b" assumes "distinct (map fst (rev uu))"   and "x \<notin> range f" shows "None = fold (\<lambda>(x, v) m. m(f x \<mapsto> v)) uu Map.empty x"
derivablep_derivable_eq: fixes R :: "('a sequent list \<times> 'a sequent) set" shows "derivablep (\<lambda>x. x \<in> R) = (\<lambda>x xa. (x, xa) \<in> derivable R)"
rep_eq: fixes x :: "'b pmf" shows "set_pmf x = {xa. measure_pmf.prob x {xa} \<noteq> 0}"
map_def: shows "local.map \<equiv> local.curry Hom.sym Hom.sym Hom.sym"
global_lipschitz_axioms: shows "global_lipschitz T X' f B'"
inducts: fixes x :: "nat set"   and k :: "nat"   and P :: "nat set \<Rightarrow> bool" assumes "x \<in> i_set_mult k"   and "\<And>n. P [n\<dots>]"   and "\<And>n. P [\<dots>n]"   and "\<And>n d. P [n\<dots>,d]"   and "\<And>r m. P [ r, mod m * k ]"   and "\<And>r m c. P [ r, mod m * k, c ]" shows "P x"
rep_eq: fixes x :: "'c program"   and xa :: "'c program" shows "(x \<le> xa) = rep_program x \<sqsubseteq> rep_program xa"
octo_unfold: fixes q :: "octo" shows "q = Ree q *\<^sub>R e0 + Im1 q *\<^sub>R e1 + Im2 q *\<^sub>R e2 + Im3 q *\<^sub>R e3 + Im4 q *\<^sub>R e4 + Im5 q *\<^sub>R e5 + Im6 q *\<^sub>R e6 + Im7 q *\<^sub>R e7"
ord_eq_le_trans: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a = b"   and "b \<subseteq># c" shows "a \<subseteq># c"
rw: shows "S * (k \<cdot>\<^sub>m 1\<^sub>m n) = Matrix.mat m n (\<lambda>(i, j). S $$ (i, j) * k)"
assms: shows "(u, w) \<in> (insert (v1, v2) E)\<^sup>+"
set_new_pairs_naive: fixes gs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "set (punit.new_pairs_naive gs bs hs data) = Pair True ` (set hs \<times> (set gs \<union> set bs)) \<union> set (pairs (punit.add_pairs_single_naive data) False hs)"
affine_independent_1: fixes a :: "'a" shows "\<not> affine_dependent {a}"
eq_Min_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and m :: "'a" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(m = linorder.Min less_eq A) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq m a))"
w: fixes w :: "'v" assumes "v \<noteq> w" shows "f w = g w"
ct_list_eq_length: fixes xs :: "ct list"   and ys :: "ct list" assumes "ct_list_eq xs ys" shows "length xs = length ys"
ln_x_over_x_tendsto_0: shows "((\<lambda>x. ln x / x) \<longlongrightarrow> 0) at_top"
update_type_mismatch: fixes x :: "Value"   and l :: "Location"   and s :: "Store" assumes "\<not> \<tau> x \<le> ltype l" shows "s\<langle>l := x\<rangle> = s"
good_subst: fixes i :: "'a" shows "dsafe (\<sigma> i)"
trace_callee_eq_def: fixes callee1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and A :: "'e set"   and p :: "'d spmf"   and q :: "'g spmf" shows "trace_callee_eq callee1 callee2 A p q = (\<forall>xs. set xs \<subseteq> A \<times> UNIV \<longrightarrow> (\<forall>x\<in>A. trace_callee callee1 p xs x = trace_callee callee2 q xs x))"
set_of_mkt_bal_r: fixes l :: "'b tree"   and r :: "'b tree"   and n :: "'b" assumes "avl l"   and "avl r" shows "set_of (mkt_bal_r n l r) = Set.insert n (set_of l \<union> set_of r)"
prems_4: shows "\<forall>k. i \<le> k \<and> k < j \<longrightarrow> occurs_on (t ! k) \<noteq> p"
prime_factors_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "(0::'b) \<notin> f ` A" shows "prime_factors (prod f A) = \<Union> ((prime_factors \<circ> f) ` A)"
floor_divide_of_nat_eq: fixes m :: "nat"   and n :: "nat" shows "\<lfloor>of_nat m / of_nat n\<rfloor> = int (m div n)"
IH: assumes "bounded_WHILE n cond step m0 = FAIL"   and "m0 \<noteq> FAIL" shows "\<exists>n'<n. \<exists>x X. bounded_WHILE n' cond step m0 = RES X \<and> x \<in> X \<and> cond x \<and> step x = FAIL"
rem_unPath_edges: fixes ps :: "('a \<times> 'b \<times> 'a) list"   and G :: "('a, 'b) graph" shows "edges (rem_unPath ps G) = edges G - (set ps \<union> set (rev_path ps))"
d\<^sub>2_def: shows "Semantics.d\<^sub>2 \<equiv> map_fun eval\<Pi>\<^sub>2 id Some"
has_real_derivative_iff: fixes f :: "real \<Rightarrow> real"   and F :: "real filter" shows "(\<exists>c. (f has_real_derivative c) F) = (\<exists>D. (f has_derivative D) F)"
nat_power_nonzero: fixes n :: "nat"   and z :: "'a" assumes "0 < n" shows "nat_power n z \<noteq> (0::'a)"
unfold: fixes g :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c \<times> 'b)"   and s :: "'b" assumes "fst g s"   and "snd (snd g s) \<in> terminates_on g" shows "s \<in> terminates_on g"
5: shows "x \<in> T (v (\<Squnion>i. Y i))"
amalgamate: fixes i :: "nat"   and j :: "nat" shows "\<phi> (amalgamate i j) = amalgamation i j"
trace_eq_simcl_bind1: fixes p :: "'a spmf"   and S :: "'b spmf \<Rightarrow> 'c spmf \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and q :: "'c spmf" assumes "\<forall>x\<in>set_spmf p. trace_eq_simcl S (f x) q"   and "lossless_spmf p" shows "trace_eq_simcl S (p \<bind> f) q"
binaryset_sums: fixes f :: "'c set \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" assumes "f {} = (0::'b)" shows "(\<lambda>n. f (binaryset A B n)) sums (f A + f B)"
fls_base_factor_mult: fixes f :: "'b fls"   and g :: "'b fls" shows "fls_base_factor (f * g) = fls_base_factor f * fls_base_factor g"
lex_pm_strict_alt: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "lex_pm_strict s t = (\<not> lex_pm t s)"
infsetsum'_converges_cong: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "infsetsum'_converges f A = infsetsum'_converges g A"
Irnat_C: fixes i :: "nat"   and "is" :: "int list"   and ls :: "int list list"   and vs :: "nat list" shows "Irnat (NC i) is ls vs = i"
triangle_reindex: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" shows "(\<Sum>(i, j)\<in>{(i, j). i + j < n}. g i j) = (\<Sum>k<n. \<Sum>i\<le>k. g i (k - i))"
oLimit_def: fixes f :: "nat \<Rightarrow> ordinal" shows "oLimit f = (LEAST k. \<forall>n. f n \<le> k)"
tensor_compose_condn: fixes nr :: "nat"   and nc :: "nat"   and A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list" assumes "mat nr nc (A1 \<circ> A2 \<otimes> B1 \<circ> B2)"   and "mat nr nc ((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2))"   and "\<forall>j<nc. \<forall>i<nr. (A1 \<circ> A2 \<otimes> B1 \<circ> B2) ! j ! i = ((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2)) ! j ! i" shows "A1 \<circ> A2 \<otimes> B1 \<circ> B2 = (A1 \<otimes> B1) \<circ> (A2 \<otimes> B2)"
qSwap_twice_preserves_qGood: fixes zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" shows "qGood (X #[[x \<and> y]]_zs #[[x' \<and> y']]_zs') = qGood X"
orderE: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" obtains "a = inf a b"
mix_pmf_comp_with_dif_equiv: fixes \<alpha> :: "real"   and \<beta> :: "real"   and p :: "'a pmf"   and q :: "'a pmf" assumes "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "\<beta> < \<alpha>" shows "mix_pmf (\<beta> / \<alpha>) (mix_pmf \<alpha> p q) q = mix_pmf \<beta> p q"
M0: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>F \<longrightarrow> \<box>[F \<longrightarrow> \<circle>F]_v"
\<I>_trivial_def: fixes \<I> :: "('d, 'e) \<I>" shows "\<I>_trivial \<I> = (outs_\<I> \<I> = UNIV)"
brw_invar_def: fixes \<delta> :: "('a, 'b) ta_rule set" shows "brw_invar \<delta> \<equiv> brw_invar_add \<delta> \<inter> {s. brw_\<alpha> s \<in> br'_invar \<delta>}"
cases: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "char list exp list"   and a3 :: "ty list" assumes "P,a1,h,sh \<turnstile> a2 [:] a3" obtains E :: "char list \<Rightarrow> ty option" where "a1 = Ea__" and "a2 = []" and "a3 = []"   | E :: "char list \<Rightarrow> ty option" and e :: "char list exp" and T :: "ty" and es :: "char list exp list" and Ts :: "ty list" where "a1 = E" and "a2 = e # es" and "a3 = T # Ts" and "P,E,h,sh \<turnstile> e : T" and "P,E,h,sh \<turnstile> es [:] Ts"
d_Suc: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat" assumes "gram_schmidt_fs_lin_indpt n fs" shows "gram_schmidt.Gramian_determinant n fs (Suc i) = gram_schmidt_fs_lin_indpt.\<mu>' n fs i i"
butlast_drop: fixes n :: "nat"   and xs :: "'g list" shows "butlast (drop n xs) = drop n (butlast xs)"
code: shows "\<Sigma>\<^sup>* = Lang True (\<lambda>_. \<Sigma>\<^sup>*)"
pderiv_deg_0: fixes f :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP INTEG)"   and "deg INTEG f = 0" shows "INTEG.pderiv f = \<zero>\<^bsub>UP INTEG\<^esub>"
p'_def: shows "p' = (\<lambda>a. if a \<in> A then p (f a) a else a)"
m_diff_image_weak: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "S \<in> lmeasurable"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S" shows "f ` S \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
j_1: fixes i :: "nat" assumes "i < length YS" shows "ys ! i = xs ! j i"
store_typing_imp_glob_agree_1: fixes s :: "s"   and \<S> :: "s_context"   and i :: "nat"   and j :: "nat" assumes "store_typing s \<S>"   and "i < length (s_inst \<S>)"   and "j < length (global (s_inst \<S> ! i))" shows "sglob_ind s i j < length (s_globs \<S>)"
mapping_eq_iff: fixes m1 :: "('a, 'b) mapping"   and m2 :: "('a, 'b) mapping" shows "(m1 = m2) = (\<forall>k. Mapping.lookup m1 k = Mapping.lookup m2 k)"
monom_eq_const_iff: fixes c :: "'c"   and n :: "nat"   and d :: "'c" shows "(monom c n = [:d:]) = (c = d \<and> (c = (0::'c) \<or> n = 0))"
cos_2pi_minus: fixes x :: "real" shows "cos (2 * pi - x) = cos x"
uniqueness_of_induced_hom': fixes T :: "'a freeword set permutation \<Rightarrow> 'b" assumes "GroupHom (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) T"   and "\<forall>s\<in>S. T (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {Abs_freeletter s})) = f s" shows "T \<circ> (\<lambda>a. if a \<in> FreeGroup S then BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {a}) else 0) = F \<circ> (\<lambda>a. if a \<in> FreeGroup S then BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {a}) else 0)"
res_zero_eq: fixes m :: "int" assumes "residues m" shows "\<zero>\<^bsub>residue_ring m\<^esub> = 0"
L4_2__aux: fixes \<phi> :: "'b ltlr"   and qs :: "'b node set"   and \<sigma> :: "nat \<Rightarrow> 'b node"   and \<mu> :: "'b ltlr"   and \<eta> :: "'b ltlr" assumes "create_gba_from_nodes_precond \<phi> qs"   and "ipath (g_E (create_gba_from_nodes \<phi> qs)) \<sigma>"   and "\<mu> U\<^sub>r \<eta> \<in> old (\<sigma> 0)"   and "\<forall>j. (\<forall>i<j. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i)) \<longrightarrow> \<eta> \<notin> old (\<sigma> j)" shows "\<forall>i. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i) \<and> \<eta> \<notin> old (\<sigma> i)"
le_minus_one_simps_3: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and uminus :: "'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sgn :: "'g \<Rightarrow> 'g"   and abs :: "'g \<Rightarrow> 'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and one :: "'g" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq zero (uminus one)"
obs_consistent_conj1: fixes R :: "('a \<times> 'b) set"   and pi :: "'c \<Rightarrow> 'd"   and Sa :: "('a, 'd) spec"   and Sc :: "('b, 'c) spec"   and R' :: "('a \<times> 'b) set" assumes "obs_consistent R pi Sa Sc" shows "obs_consistent (R \<inter> R') pi Sa Sc"
distinct_1: fixes x2 :: "name" shows "DBZero \<noteq> DBVar x2"
elims: fixes x :: "('d, 'c) Term.term"   and xa :: "('d, 'c) Term.term"   and y :: "('c \<Rightarrow> ('d, 'c) Term.term) option" assumes "mgu x xa = y" obtains s :: "('d, 'c) Term.term" and t :: "('d, 'c) Term.term" where "x = s" and "xa = t" and "y = (case unify [(s, t)] [] of None \<Rightarrow> None | Some res \<Rightarrow> Some (subst_of res))"
assms: shows "\<angle> z1 z2 \<le> 0"
substEnv_idEnv: fixes xs :: "'varSort"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and x :: "'var" shows "idEnv &[X / x]_xs = idEnv [x \<leftarrow> X]_xs"
prems: shows "set (Vs @ [x]) \<subseteq> carrier_vec n"
wlp_close: fixes dims :: "nat list"   and S :: "com"   and P :: "complex mat" assumes "state_sig.well_com dims S"   and "state_sig.is_quantum_predicate dims P" shows "state_sig.is_quantum_predicate dims (state_sig.wlp dims S P)"
lm133: fixes f :: "('a \<times> 'b) set" shows "- Domain f \<subseteq> (f Else (0::'b)) -` {0::'b}"
left_translations_of_subgroup_are_transformation_group: shows "transformation_group (left.translation ` H) G"
f_nonneg: fixes i :: "nat"   and x :: "'a" shows "0 \<le> f i x"
div_Suc: fixes m :: "nat"   and n :: "nat" shows "Suc m div n = (if Suc m mod n = 0 then Suc (m div n) else m div n)"
d1: shows "dist u v / 2 \<le> dist (flow0 x su1) v"
elims_3: fixes x :: "const"   and xa :: "const" assumes "\<not> x \<sqsubseteq> xa" obtains n :: "int" and m :: "int" where "x = Const n" and "xa = Const m" and "n \<noteq> m"   | uv_ :: "int" where "x = Any" and "xa = Const uv_"
zero_le_power: fixes a :: "'a"   and n :: "nat" assumes "(0::'a) \<le> a" shows "(0::'a) \<le> a ^ n"
assms_1: shows "\<forall>x\<in>fv a. (\<exists>f. \<theta> x = Fun f []) \<or> (\<exists>y. \<theta> x = Var y)"
iso_cancel_left: fixes f :: "'b"   and g :: "'b"   and g' :: "'b" assumes "\<phi>.B.iso f"   and "B f g = B f g'"   and "\<phi>.B.seq f g" shows "g = g'"
codomain_smult_zero: fixes r :: "'r" assumes "r \<in> R" shows "r \<star> (0::'n) = (0::'n)"
linorders_on_def: fixes A :: "'b set" shows "linorders_on A = {R. linorder_on A R}"
permutes_inj_on: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and A :: "'c set" assumes "f permutes S" shows "inj_on f A"
code_blue_dfs_ahs_correct: fixes G :: "'a b_graph_rec"   and Gi :: "('a \<Rightarrow> bool, 'a \<Rightarrow> 'a list, 'a list, 'a \<Rightarrow> bool) gen_bg_impl" assumes "b_graph G"   and "finite ((g_E G)\<^sup>* `` g_V0 G)"   and "(Gi, G) \<in> bg_impl_rel_ext unit_rel Id" shows "RETURN (code_blue_dfs_ahs Gi) \<le> SPEC (\<lambda>r. case r of None \<Rightarrow> \<forall>prpl. \<not> b_graph.is_lasso_prpl G prpl | Some L \<Rightarrow> b_graph.is_lasso_prpl G L)"
op_ias_empty_sz_def: fixes N :: "nat" shows "op_ias_empty_sz N \<equiv> op_set_empty"
const_unconst: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and u :: "'c"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "unconst u = Some name" shows "u = const name"
den'_eq_chop_num': shows "den' = tree_chop num'"