hcju/isabelleps huggingface dataset:
mm_equiv_sym: fixes c\<^sub>1 :: "'Com"   and mds\<^sub>1 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "'Com"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds\<^sub>1, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" shows "\<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle> \<approx> \<langle>c\<^sub>1, mds\<^sub>1, mem\<^sub>1\<rangle>"
weight_spmf'_def: shows "weight_spmf' = weight_spmf"
assms_2: fixes x :: "'a" assumes "consider' (Inl x)" shows "consider x"
assms_5: shows "s2' = s2(V := \<lbrakk>e\<rbrakk>s2)"
ide_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C" shows "partial_magma.ide C a = (partial_magma.arr C a \<and> partial_magma.dom C a = a \<and> partial_magma.cod C a = a)"
sig_red_zeroE: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red_zero sing_reg F r" obtains s :: "'t \<Rightarrow>\<^sub>0 'b" where "(sig_red sing_reg (\<preceq>) F)\<^sup>*\<^sup>* r s" and "rep_list s = 0"
pdevs_apply_One_pdevs: fixes i :: "nat" shows "pdevs_apply One_pdevs i = (if i < length Basis_list then Basis_list ! i else (0::'b))"
addrow_carrier: fixes ad :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and k :: "nat"   and l :: "nat"   and A :: "'c mat"   and n :: "nat"   and nc :: "nat" shows "(mat_addrow_gen ad mul a k l A \<in> carrier_mat n nc) = (A \<in> carrier_mat n nc)"
insert_fset_left_comm: fixes x :: "'b"   and y :: "'b"   and S :: "'b fset" shows "insert_fset x (insert_fset y S) = insert_fset y (insert_fset x S)"
inf_greatest: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y"   and "less_eq x z" shows "less_eq x (inf y z)"
eval_prefs_from_table_2: fixes i :: "'d"   and y :: "'c set list"   and xs :: "('d \<times> 'c set list) list" shows "prefs_from_table ((i, y) # xs) i = of_weak_ranking y"
prime_imp_sigma: fixes p :: "nat" assumes "prime p" shows "sigma p = p + 1"
eq_fract_1: fixes b :: "'b"   and d :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<noteq> (0::'b)"   and "d \<noteq> (0::'b)" shows "(Fraction_Field.Fract a b = Fraction_Field.Fract c d) = (a * d = c * b)"
simps_18: fixes x21 :: "('d, 'c) tm"   and x22 :: "('d, 'c) tm" shows "vars (App x21 x22) = vars x21 \<union> vars x22"
hide_rules_1: fixes c :: "'b \<Rightarrow> 'c"   and B :: "'c set"   and P :: "'b \<Rightarrow> 'c process" assumes "\<forall>y. c y \<notin> B" shows "read c UNIV P \ B = (c`?`x \<rightarrow> P x \ B)"
spmf_eqI: fixes p :: "'c spmf"   and q :: "'c spmf" assumes "\<And>i. spmf p i = spmf q i" shows "p = q"
h_case: fixes M :: "(action \<times> nat) list"   and ns :: "nat list" assumes "halts M ns" shows "{\<lambda>tp. tp = ([Bk], <(code M, ns)>)} H {\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0>)}"
less_char_def: fixes c :: "char"   and d :: "char" shows "less_char c d = (of_char c < of_char d)"
pquvd: shows "(case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> p, case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> q, case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> u, case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> v, case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j)"
assms_1: shows "homotopic_with (\<lambda>x. True) X X id f"
supp_def: fixes x :: "'a" shows "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
eval_map_ms: fixes f :: "'b \<Rightarrow> 'a"   and F :: "'b ms" shows "eval (map_ms f F) = eval F"
absolutely_integrable_imp_borel_measurable: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "f absolutely_integrable_on S"   and "S \<in> sets lebesgue" shows "f \<in> borel_measurable (lebesgue_on S)"
elims: fixes x :: "'b kdt"   and y :: "nat" assumes "min_height x = y" obtains uu_ :: "(real, 'b) vec" where "x = Leaf uu_" and "y = 0"   | uv_ :: "'b" and uw_ :: "real" and l :: "'b kdt" and r :: "'b kdt" where "x = Node uv_ uw_ l r" and "y = min (min_height l) (min_height r) + 1"
RS2_NMT: fixes p :: "('b, 'c) Combinators list" assumes "p \<noteq> []" shows "removeShadowRules2 p \<noteq> []"
\<gamma>': fixes x :: "real" shows "(\<gamma> has_vector_derivative \<gamma>' x) (at x)"
permute_flip_cancel: fixes a :: "'f"   and b :: "'f"   and x :: "'e" shows "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"
farkas_lemma_matrix: fixes A :: "rat mat"   and nr :: "nat"   and nc :: "nat"   and b :: "rat vec" assumes "A \<in> carrier_mat nr nc"   and "b \<in> carrier_vec nr" shows "(\<exists>x\<in>carrier_vec nc. A *\<^sub>v x \<le> b) = (\<forall>y\<ge>0\<^sub>v nr. mat_of_row y * A = 0\<^sub>m 1 nc \<longrightarrow> 0 \<le> y \<bullet> b)"
converse_rtrancl_induct_red: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and R :: "char list exp \<Rightarrow> heap \<Rightarrow> sfields \<Rightarrow> sheap \<Rightarrow> bool \<Rightarrow> char list exp \<Rightarrow> heap \<Rightarrow> sfields \<Rightarrow> sheap \<Rightarrow> bool \<Rightarrow> bool" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l', sh'),b'\<rangle>"   and "\<And>e h l sh b. R e h l sh b e h l sh b"   and "\<And>e\<^sub>0 h\<^sub>0 l\<^sub>0 sh\<^sub>0 b\<^sub>0 e\<^sub>1 h\<^sub>1 l\<^sub>1 sh\<^sub>1 b\<^sub>1 e' h' l' sh' b'. \<lbrakk>P \<turnstile> \<langle>e\<^sub>0,(h\<^sub>0, l\<^sub>0, sh\<^sub>0),b\<^sub>0\<rangle> \<rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),b\<^sub>1\<rangle>; R e\<^sub>1 h\<^sub>1 l\<^sub>1 sh\<^sub>1 b\<^sub>1 e' h' l' sh' b'\<rbrakk> \<Longrightarrow> R e\<^sub>0 h\<^sub>0 l\<^sub>0 sh\<^sub>0 b\<^sub>0 e' h' l' sh' b'" shows "R e h l sh b e' h' l' sh' b'"
deriv_in_subterms_3: fixes p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "p1 \<oplus> p2 \<in> subterms p" shows "p1 \<in> subterms p"
is_prebicategory: shows "prebicategory (\<cdot>) (\<star>) \<a>"
fract_poly_eq_0_iff: fixes p :: "'b poly" shows "(map_poly to_fract p = 0) = (p = 0)"
assms: shows "range_vars \<delta> \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t S = {}"
rel_spmf_mono: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a spmf"   and g :: "'b spmf"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_spmf B f g"
cnf2bhv_suc: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "\<langle>c \<and> t\<rangle> \<le> n" shows "\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>Suc n = Suc (\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n)"
Dom_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VxV.seq g f" shows "(if B.arr (fst (B.VxV.comp g f)) \<and> B.arr (snd (B.VxV.comp g f)) then (B.dom (fst (B.VxV.comp g f)), B.dom (snd (B.VxV.comp g f))) else (B.null, B.null)) = (if B.arr (fst f) \<and> B.arr (snd f) then (B.dom (fst f), B.dom (snd f)) else (B.null, B.null))"
nrows_diagonal_step: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and j :: "nat"   and c :: "'a"   and v :: "'a" shows "nrows (diagonal_step A i j c v) = nrows A"
prod_mset_prime_factorization: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.factorial_semiring_multiplicative divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "comm_monoid_mult.prod_mset times one (factorial_semiring.prime_factorization zero times one normalize x) = normalize x"
HS_If: fixes G :: "Var \<Rightarrow> L"   and b :: "BExpr"   and t :: "L"   and p :: "L"   and c :: "IMP"   and H :: "Var \<Rightarrow> L"   and d :: "IMP" assumes "(G, b, t) \<in> HS_B"   and "(LUB p t, G, c, H) \<in> HS"   and "(LUB p t, G, d, H) \<in> HS" shows "(p, G, Iff b c d, H) \<in> HS"
supp_1: fixes x1 :: "name" shows "supp (Var x1) = supp x1"
step_invariantD: fixes A :: "('a, 'b) automaton"   and I :: "'b \<Rightarrow> bool"   and P :: "'a \<times> 'b \<times> 'a \<Rightarrow> bool"   and s :: "'a"   and a :: "'b"   and s' :: "'a" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) P"   and "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "P (s, a, s')"
lqI: fixes u :: "'b list"   and z :: "'b list"   and v :: "'b list" assumes "u \<cdot> z = v" shows "u\<inverse>\<^sup>>v = z"
N_def: shows "N \<equiv> distr M (stock_measure REAL) (lift_RealIntVal2 f' f)"
SumZero: fixes P :: "pi" shows "P \<oplus> \<zero> \<equiv>\<^sub>e P"
mset_insert: fixes q :: "'a tree"   and x :: "'a" assumes "heap q" shows "mset_tree (insert x q) = {#x#} + mset_tree q"
max_vars_inline_call: fixes e' :: "('a, 'b, 'c) exp"   and e :: "('a, 'b, 'c) exp" shows "max_vars (inline_call e' e) \<le> max_vars e + max_vars e'"
con_rews_3: fixes a :: "'r"   and tree1 :: "'r tree"   and tree2 :: "'r tree" shows "(Node\<cdot>a\<cdot>tree1\<cdot>tree2 = \<bottom>) = (a = \<bottom>)"
row_upd_row_If: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and x :: "('c, 'd) vec"   and j :: "'d" shows "Square_Matrix.row (upd_row M i x) j = (if i = j then x else Square_Matrix.row M j)"
opeT: shows "openin (top_of_set (U - S)) T"
integral_cong: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "integral s f = integral s g"
assms: shows "\<forall>f\<in>set L. freeIn var f"
homotopic_with_imp_subset1: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a set"   and Y :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "homotopic_with_canon P X Y f g" shows "f ` X \<subseteq> Y"
end_point_no_successor: fixes x :: "'a" shows "x\<^sup>T ; end_points x = 0"
assms: shows "v \<Turnstile>\<^sub>e (x\<^sub>j, lp')"
vector_scaleR_commute: fixes A :: "(('a, 'c) vec, 'b) vec"   and c :: "real"   and x :: "('a, 'c) vec" shows "A *v c *\<^sub>R x = c *\<^sub>R (A *v x)"
semantics_ltln_ite_simps_1: fixes w :: "nat \<Rightarrow> 'a set"   and P :: "bool" shows "w \<Turnstile>\<^sub>n (if P then true\<^sub>n else false\<^sub>n) = P"
Suc_6: shows "x \<in> rvars (\<T> (l ! i))"
a_not_b: shows "to_nat a \<noteq> to_nat b"
iso_char: fixes \<phi> :: "'a" assumes "Right_a.arr \<phi>" shows "Right_a.iso \<phi> = local.iso \<phi>"
Y: fixes xs :: "'a llist"   and ys :: "'b llist" assumes "(xs, ys) \<in> Y" shows "llist_all2 P xs ys"
subset_ball: fixes d :: "real"   and e :: "real"   and x :: "'b" assumes "d \<le> e" shows "ball x d \<subseteq> ball x e"
S_selects_neg_lits: fixes L :: "'a literal"   and C :: "'a literal multiset" assumes "L \<in># S C" shows "is_neg L"
elims_1: fixes x :: "arctic"   and xa :: "arctic"   and y :: "bool" assumes "(x \<le> xa) = y" obtains xb :: "arctic" where "x = MinInfty" and "xa = xb__" and "y"   | uu_ :: "int" where "x = Num_arc uu_" and "xa = MinInfty" and "\<not> y"   | ya :: "int" and xb :: "int" where "x = Num_arc ya" and "xa = Num_arc xb" and "y = (ya \<le> xb)"
A_def: shows "A = card {X \<in> Pow {..<n}. root' X = 0}"
G_def: shows "G = {x \<in> space M. (\<lambda>n. u n x / real n) \<longlonglongrightarrow> 0 \<and> (\<lambda>n. birkhoff_sum (\<lambda>x. \<bar>u s x / real s\<bar>) n x / real n) \<longlonglongrightarrow> real_cond_exp M Invariants (\<lambda>x. \<bar>u s x / real s\<bar>) x \<and> (\<lambda>n. birkhoff_sum F n x / real n) \<longlonglongrightarrow> real_cond_exp M Invariants F x \<and> real_cond_exp M Invariants F x + real_cond_exp M Invariants (\<lambda>x. \<bar>u s x / real s\<bar>) x = real_cond_exp M Invariants F2 x}"
f'_vals_closed: assumes "a \<in> carrier Zp" shows "f' \<bullet> a \<in> carrier Zp"
step_eq_fetch: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and as :: "nat"   and ins :: "abc_inst"   and s :: "nat"   and b :: "cell"   and ac :: "action"   and ns :: "nat" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "abc_fetch as ap = Some ins"   and "fetch (ci ly (start_of ly as) ins) (Suc s - start_of ly as) b = (ac, ns)"   and "ns \<noteq> 0" shows "fetch tp s b = (ac, ns)"
box_diamond_galois_2: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "d p = p"   and "d q = q" shows "( |x\<rangle> p \<le> q) = (p \<le> [x| q)"
def_x: shows "\<tau> \<Turnstile> \<delta> x"
basis_finmap_eq_empty: assumes "local.basis_proj = {}" shows "local.basis_finmap = {Pi' {} undefined}"
mem_CD_on_Cd: fixes x :: "'a"   and ds :: "'b set"   and X :: "'a set" shows "(x \<in> Singleton_for_d.CD_on ds X) = (x \<in> Singleton_for_d.Cd (Xd x) X \<and> Xd x \<in> ds)"
add_closed: fixes V :: "'c set"   and x :: "'c"   and y :: "'c" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V" shows "x + y \<in> V"
3: shows "alternate f g (k + n) \<le> alternate f g k"
that_2: assumes "x \<notin> A"   and "x \<in> B" shows "thesis"
jle: shows "j < length (list_of (ltake (enat (Suc (max i j))) lst))"
mem_imp_elem: fixes S :: "('d, 'e) eq_object_scheme"   and x :: "'d"   and A :: "'d set" assumes "equivalence S"   and "x \<in> carrier S"   and "x \<in> A" shows "x .\<in>\<^bsub>S\<^esub> A"
dom: fixes f :: "'b \<Rightarrow> 'b" assumes "f \<in> F" shows "has_dom f (arcs G)"
indep_in_iff_rank_of: fixes X :: "'a set" assumes "X \<subseteq> \<E>" shows "indep_in \<E> X = (rank_of X = card X)"
Limsup_mono: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x \<le> g x" shows "Limsup F f \<le> Limsup F g"
T_def: fixes s :: "'s" shows "T s = distr (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x)))) (stream_space (count_space UNIV)) (walk s)"
lm154: fixes X :: "'c set"   and Y :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "graph (X \<inter> Y) f = graph X f || Y"
fold_keys_def: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('c, 'd) RBT.rbt"   and A :: "'b" shows "fold_keys f t A = RBT.fold (\<lambda>k _. f k) t A"
assms: shows "rel_gpv (R \<upharpoonleft> P \<otimes> Q) (R' \<upharpoonleft> P' \<otimes> Q') x y"
Diff: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> M"   and "b \<in> M" shows "a - b \<in> M"
sum4sq_nat_def: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" shows "sum4sq_nat a b c d \<equiv> a\<^sup>2 + b\<^sup>2 + c\<^sup>2 + d\<^sup>2"
extensional_arb: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c" assumes "f \<in> extensional A"   and "x \<notin> A" shows "f x = undefined"
A: shows "WtNFT cl = 1 - WtFT cl"
af_letter_nested_prop_atoms: fixes \<phi> :: "'b ltln"   and \<nu> :: "'b set" shows "nested_prop_atoms (af_letter \<phi> \<nu>) \<subseteq> nested_prop_atoms \<phi>"
wellfounded: shows "wf {(nxt x, x) |x. cnt x}"
r_psi_def: shows "r_psi \<equiv> Cn 2 r_last [r_prefixes]"
size_eq_Suc_imp_eq_union: fixes M :: "'b multiset"   and n :: "nat" assumes "size M = Suc n" shows "\<exists>a N. M = a, N"
nth_mapWithIndex: fixes i :: "nat"   and xs :: "'a list"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "i < length xs" shows "mapWithIndex f xs ! i = f i (xs ! i)"
sel_2: fixes x :: "gauss_int"   and y :: "gauss_int" shows "ImZ (x + y) = ImZ x + ImZ y"
Int_pairs_def: fixes A :: "('e \<times> 'e) set"   and B :: "('e \<times> 'e) set" shows "A \<inter>\<^sub>p B = A \<inter> (B \<union> prod.swap ` B)"
abs_abs: fixes a :: "'a" shows "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
H'_lim: fixes x :: "'s"   and y :: "'s" shows "(\<lambda>n. T.prob x (H' y n)) \<longlonglongrightarrow> H x y"
of_bool_eq_1: fixes one :: "'b"   and zero :: "'b" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero False = zero"
l: assumes "set (y # map snd txs) \<subseteq> var"   and "set (s # map fst txs) \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla" shows "length (getFrN (y # map snd txs) (s # map fst txs) [\<phi>] (length txs)) = length txs"
hyp2_rep_abs_examples_5: shows "hyp2_rep (hyp2_abs (vector [1 / 2, 1 / 2])) = vector [1 / 2, 1 / 2]"
distinct_70: fixes pi' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "!pi' \<noteq> [name1\<frown>name2]pi"
bisim: shows "strong_reduction_simulation Rel (STCal Source Target) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> (P', Q') \<in> Rel))"
unique_fields: fixes G :: "prog"   and C :: "qtname" assumes "is_class G C"   and "wf_prog G" shows "unique (DeclConcepts.fields G C)"
njmiq: shows "occurs_on (t ! (j - 1)) \<noteq> occurs_on (t ! j)"
i_set0_ind_empty: shows "{} \<in> i_set0_ind"
empty_eq_map_add_iff: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "(Map.empty = f ++ g) = (f = Map.empty \<and> g = Map.empty)"
fiter_induct: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and x :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"   and "(x::'b::refinement_lattice) \<sqsubseteq> (d::'b::refinement_lattice) \<sqinter> (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) x" shows "(x::'b::refinement_lattice) \<sqsubseteq> (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (finite_iteration.fiter seq (nil::'b::refinement_lattice) (c::'b::refinement_lattice)) (d::'b::refinement_lattice)"
pr1_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "pr1 a b = (if ide a \<and> ide b then mkArr (local.set (local.prod a b)) (local.set a) (\<lambda>x. UP (hfst (DOWN x))) else null)"
mult_scaleC_left: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "complex"   and x :: "'b"   and y :: "'b" assumes "class.complex_algebra scaleR scaleC plus zero minus uminus times" shows "times (scaleC a x) y = scaleC a (times x y)"
binop_app_in_vrange: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"   and "b \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>" shows "a +\<^sub>\<real> b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_plus"
Sideals: fixes I :: "'a set" assumes "I \<in> S" shows "ideal I R"
closure_complement: fixes A :: "'b list set" assumes "regular A" shows "regular (- A)"
merges_property1: fixes qs :: "(nat \<times> nat list) list"   and ss :: "nat list list" assumes "\<forall>(q, \<tau>)\<in>set qs. q < length ss \<and> \<tau> \<in> A"   and "set ss \<subseteq> A"   and "length ss = n"   and "distinct (map fst qs)" shows "\<forall>(q, \<tau>)\<in>set qs. merges f qs ss ! q = \<tau> \<squnion>\<^bsub>f\<^esub> ss ! q"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "Oldf x1 \<noteq> Wit x2"
intros: fixes is' :: "nat list"   and i :: "nat"   and "is" :: "nat list"   and t :: "('form, 'rule, 'subst, 'a) itnode rose_tree"   and h :: "('form, 'var) out_port" assumes "prefix (is' @ [i]) is"   and "i < length (inPorts' (iNodeOf (tree_at t is')))"   and "hyps (iNodeOf (tree_at t is')) h = Some (inPorts' (iNodeOf (tree_at t is')) ! i)" shows "subst (iSubst (tree_at t is')) (freshen (iAnnot (tree_at t is')) (labelsOut (iNodeOf (tree_at t is')) h)) \<in> hyps_along t is"
ltl_llist_inf: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "xs \<noteq> LNil"   and "ys \<noteq> LNil"   and "lhd xs = lhd ys" shows "ltl (xs \<sqinter> ys) = ltl xs \<sqinter> ltl ys"
set_n_lists: fixes n :: "nat"   and xs :: "'a list" shows "set (List.n_lists n xs) = {ys. length ys = n \<and> set ys \<subseteq> set xs}"
IHAPP_2: shows "atU U ({#s'#} + c')"
all_clauses_def: fixes S :: "'b literal set set" shows "all_clauses S = {s. s \<subseteq> {k\<^sup>+ |k. k \<in> atoms_of_cnf S} \<union> {k\<inverse> |k. k \<in> atoms_of_cnf S}}"
oppositeIsDifferentFromLiteral: fixes literal :: "Literal" shows "opposite literal \<noteq> literal"
ctrm_zcf: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "zcf f = \<zero>" shows "monom P (f 0) 0 = \<zero>\<^bsub>P\<^esub>"
det_nonzero_congruence: fixes A :: "'c mat"   and M :: "'c mat"   and B :: "'c mat"   and n :: "nat" assumes "A * M = B * M"   and "det M \<noteq> (0::'c)"   and "M \<in> carrier_mat n n"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "A = B"
ChamberSubcomplexD_sub: fixes X :: "'b set set"   and Y :: "'b set set" assumes "ChamberComplex X"   and "ChamberComplex.ChamberSubcomplex X Y" shows "order.greater_eq X Y"
essentialI: fixes G :: "('a, 'b) graph_scheme"   and x :: "'a"   and p :: "'a list"   and y :: "'a"   and B :: "'a set"   and S :: "'a set" assumes "path G x p y"   and "y \<in> B"   and "\<And>z. \<lbrakk>x \<noteq> y; z \<in> set p\<rbrakk> \<Longrightarrow> z = x \<or> z \<notin> S" shows "essential G B S x"
d2_ex0: fixes y :: "'a"   and t :: "real" assumes "(y, t) \<in> cball (x, 0) d2"   and "0 < t" shows "y \<in> X"
r_le_def: shows "r_le \<equiv> Cn 2 r_ifle [recf.Id 2 0, recf.Id 2 1, r_constn 1 0, r_constn 1 1]"
lOfFax_def: fixes f :: "'fsym" shows "lOfFax f = Fn (Tag (resOf f)) [rOfFax f]"
simps_1: fixes a :: "Arity"   and e :: "exp"   and x :: "Vars.var" shows "\<T>\<^bsub>a\<^esub> (App e x) = App (\<T>\<^bsub>inc\<cdot>a\<^esub> e) x"
f11: shows "t \<otimes> (snd x \<otimes> r) = t \<otimes> (s \<otimes> fst x)"
cod_char: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" shows "J_C.cod f = (if J_C.arr f then J_C.MkIde (J_C.Cod f) else J_C.null)"
ideal_deterministic: fixes e :: "Syntax.term" assumes "\<lless>[], t\<ggreater> I\<rightarrow>i \<lless>[], e\<ggreater>" shows "e = Inj1 Unit"
CompDiag_Ide_Diag: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and a :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t"   and "monoidal_language.Ide C a"   and "monoidal_language.Dom C a = monoidal_language.Cod C t" shows "monoidal_language.CompDiag C a t = t"
naturality: fixes f :: "'a" assumes "B.arr f" shows "B.\<rr> (B.cod f) \<cdot>\<^sub>B B.R f = B.map f \<cdot>\<^sub>B B.\<rr> (B.dom f)"
type_less_x_Tuple': fixes \<tau> :: "'a OCL_Types.type"   and \<xi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" assumes "\<tau> < Tuple \<xi>"   and "acyclicP_on (fmran' \<xi>) (\<sqsubset>)" obtains \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" where "\<tau> = Tuple \<pi>" and "strict_subtuple (\<le>) \<pi> \<xi>"
s_prod_isol: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y" shows "z \<cdot> x \<le> z \<cdot> y"
assms: shows "Derivation \<alpha> (D @ E) \<gamma>"