hcju/isabelleps huggingface dataset:
swap_set_not_in: fixes a :: "atom"   and S :: "atom set"   and b :: "atom" assumes "a \<notin> S"   and "b \<notin> S" shows "(a \<rightleftharpoons> b) \<bullet> S = S"
norm_triangle_sub: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm x \<le> norm y + norm (minus x y)"
float_sqrt_def: fixes a :: "('c, 'd) IEEE.float" shows "float_sqrt a = fsqrt To_nearest a"
partial_summation_strong: fixes X :: "real set"   and y :: "real"   and x :: "real"   and f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and a :: "nat \<Rightarrow> 'b" assumes "finite X"   and "0 \<le> y"   and "y < x"   and "\<And>z. z \<in> {y..x} - X \<Longrightarrow> (f has_vector_derivative f' z) (at z)"   and "continuous_on {y..x} f" shows "((\<lambda>t. sum_upto a t * f' t) has_integral sum_upto a x * f x - sum_upto a y * f y - (\<Sum>n\<in>real -` {y<..x}. a n * f (real n))) {y..x}"
cc_restr_lub: fixes S :: "Vars.var set"   and X :: "CoCalls set" shows "lub X G|` S = (\<Squnion>G\<in>X. G G|` S)"
IH: fixes bs :: "'b list" shows "\<forall>i<length (fst (filter2 P as bs)). P (fst (filter2 P as bs) ! i) (snd (filter2 P as bs) ! i)"
sum_eq_def: fixes eql :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and eqr :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and s1 :: "'c + 'e"   and s2 :: "'d + 'f" shows "sum_eq eql eqr s1 s2 \<equiv> case (s1, s2) of (Inl x1, Inl x2) \<Rightarrow> eql x1 x2 | (Inl x1, Inr ba) \<Rightarrow> False | (Inr x1, Inl a) \<Rightarrow> False | (Inr x1, Inr x2) \<Rightarrow> eqr x1 x2"
new_character_data_get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'k" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"   and "ptr \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
insert_Diff_single: fixes a :: "'b"   and A :: "'b set" shows "Set.insert a (A - {a}) = Set.insert a A"
integrable_on_subdivision: fixes \<D> :: "'c set set"   and i :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "\<D> division_of i"   and "f integrable_on S"   and "i \<subseteq> S" shows "f integrable_on i"
acc_induct: fixes a :: "'c"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "a \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>x \<in> Wellfounded.acc r; \<forall>y. (y, x) \<in> r \<longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
simps_2: fixes p :: "'b list \<Rightarrow> bool"   and a :: "'b"   and S :: "'b list" shows "N p (a # S) = p S"
assms_2: shows "\<guillemotleft>\<nu> : b \<rightarrow> c\<guillemotright>"
opt: fixes F :: "atom fm"   and L :: "real list" shows "eval ((push_forall \<circ> nnf \<circ> unpower 0 \<circ> groupQuantifiers \<circ> clearQuantifiers) F) L = eval F L"
holomorphic_convex_primitive: fixes S :: "complex set"   and K :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "convex S"   and "finite K"   and "continuous_on S f"   and "\<And>x. x \<in> interior S - K \<Longrightarrow> f field_differentiable at x" obtains g :: "complex \<Rightarrow> complex" where "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative f x) (at x within S)"
subset_imp_LeadsTo: fixes A :: "'b set"   and B :: "'b set"   and F :: "'b program" assumes "A \<subseteq> B" shows "F \<in> A \<longmapsto>w B"
zero_interval_def: shows "0 \<equiv> Interval (0::'a, 0::'a)"
monotone_on_id: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "monotone_on X r r id"
comp_of_local_ring_morphisms: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and addA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multA :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zeroA :: "'d"   and oneA :: "'d"   and B :: "'e set"   and addB :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multB :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zeroB :: "'e"   and oneB :: "'e"   and g :: "'e \<Rightarrow> 'f"   and C :: "'f set"   and addC :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and multC :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zeroC :: "'f"   and oneC :: "'f" assumes "local_ring_morphism f A addA multA zeroA oneA B addB multB zeroB oneB"   and "local_ring_morphism g B addB multB zeroB oneB C addC multC zeroC oneC" shows "local_ring_morphism (g \<circ> f \<down> A) A addA multA zeroA oneA C addC multC zeroC oneC"
absolutely_integrable_component_ubound: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f integrable_on A"   and "g absolutely_integrable_on A"   and "\<And>x b. \<lbrakk>x \<in> A; b \<in> Basis\<rbrakk> \<Longrightarrow> f x \<bullet> b \<le> g x \<bullet> b" shows "f absolutely_integrable_on A"
simps_1: fixes l :: "'b tree"   and v :: "'b"   and r :: "'b tree" shows "left \<langle>l, v, r\<rangle> = l"
arr_char: fixes f :: "'a \<times> 'a" shows "C.VV.arr f = (C.arr (fst f) \<and> C.arr (snd f) \<and> src\<^sub>C (fst f) = trg\<^sub>C (snd f))"
kernel_axioms: shows "Matrix_Kernel.kernel nr nc (gauss_jordan_single A)"
in_hd_or_tl_conv: fixes l :: "'a list"   and x :: "'a" assumes "l \<noteq> []" shows "(x = hd l \<or> x \<in> set (tl l)) = (x \<in> set l)"
bounded_clinear_const_scaleC: fixes g :: "'a \<Rightarrow> 'b"   and c :: "complex" assumes "bounded_clinear g" shows "bounded_clinear (\<lambda>x. c *\<^sub>C g x)"
\<eta>_def: shows "F'G'.\<eta> \<equiv> transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map F'G'.\<eta>o"
vsv_vconverse: shows "vsv (s\<inverse>\<^sub>\<circ>)"
list_update_beyond: fixes xs :: "'g list"   and i :: "nat"   and x :: "'g" assumes "length xs \<le> i" shows "xs[i := x] = xs"
perpendicular_to_x_axis_intersects_x_axis: fixes H :: "circline" assumes "is_poincare_line H"   and "perpendicular_to_x_axis H" shows "intersects_x_axis H"
comp_simp: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "g \<cdot> f = C g f"
folding_ex: fixes C :: "'a set" assumes "folding_g.chamber C"   and "C \<in> f \<turnstile> X" shows "\<exists>x. folding_g.chamber x \<and> x \<notin> f \<turnstile> X \<and> f ` x = C"
proj_def: fixes v :: "'a"   and u :: "'a" shows "proj v u = (v \<bullet> u / (u \<bullet> u)) *\<^sub>R u"
synthetic_divmod_pCons: fixes a :: "'b"   and p :: "'b poly"   and c :: "'b" shows "synthetic_divmod (pCons a p) c = (case synthetic_divmod p c of (q, r) \<Rightarrow> (pCons r q, a + c * r))"
tcg_add: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<tau> x = \<tau> y" shows "\<tau> (x + z) = \<tau> (y + z)"
sum_eq: shows "SWA.sum as i j = SWA.sum (butlast as) i j"
rel_witness_gpv_def: fixes A :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'l \<Rightarrow> bool"   and R :: "'j \<Rightarrow> 'n \<Rightarrow> bool"   and R' :: "'n \<Rightarrow> 'm \<Rightarrow> bool" shows "rel_witness_gpv A C R R' = corec_gpv (map_spmf (map_generat id id (\<lambda>(rpv, rpv'). Inr \<circ> rel_witness_fun R R' (rpv, rpv')) \<circ> rel_witness_generat) \<circ> rel_witness_spmf (rel_generat A C (R OO R' ===> rel_gpv'' A C (R OO R'))) \<circ> map_prod the_gpv the_gpv)"
reds_final_same: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   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" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"   and "final e" shows "e = e' \<and> s = s' \<and> b = b'"
emeasure_real: fixes A :: "bool stream set" shows "\<exists>r\<ge>0. emeasure M A = ennreal r"
iMOD_cut_le: fixes r :: "nat"   and m :: "nat"   and t :: "nat" shows "[ r, mod m ] \<down>\<le> t = (if t < r then {} else [ r, mod m, (t - r) div m ])"
H2: shows "ExcessNotAtRec ps g = ListSum (ExcessNotAtRecList ps g) E"
ex: shows "{0..t} \<subseteq> existence_ivl0 x"
sifum_example2_axioms_def: shows "sifum_example2_axioms \<equiv> \<forall>lc lc' lc''. eval_abv\<^sub>C lc lc' \<longrightarrow> eval_abv\<^sub>C lc lc'' \<longrightarrow> lc' = lc''"
the_coext_def: fixes x' :: "'d"   and f :: "'c" shows "\<epsilon>x.the_coext x' f = (THE g. \<epsilon>x.is_coext x' f g)"
vcons_eqD_1: fixes x\<^sub>1 :: "V"   and x\<^sub>2 :: "V" assumes "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2" shows "xs\<^sub>1 = xs\<^sub>2"
Abs_quot_inverse: fixes y :: "'b set" assumes "y \<in> quot" shows "Rep_quot (Abs_quot y) = y"
updownI': fixes d :: "nat"   and a :: "rat array"   and v :: "(nat \<times> int) list \<Rightarrow> real" assumes "d \<le> dm" shows "<gridI a v> updownI' d a <\<lambda>r. gridI a (updown' dm lm d v)>\<^sub>t"
map_def: fixes f :: "'a" shows "I.map f = (if arr f then MC.unity else null)"
op_map_pick_remove_alt: fixes m :: "'c \<Rightarrow> 'd option"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'b nres" shows "op_map_pick_remove m \<bind> (\<lambda>((k, v), m). f k v m) = SPEC (\<lambda>(k, v). m k = Some v) \<bind> (\<lambda>(k, v). Let (m |` (- {k})) (f k v))"
top_empty_eq: shows "\<top> = (\<lambda>x. x \<in> UNIV)"
wf_imp_nf_ex: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "wfP r\<inverse>\<inverse>" obtains b :: "'b" where "r\<^sup>*\<^sup>* a b" and "relation.is_final r b"
of_int_eq_1_iff: fixes 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 z :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus z = one) = (z = 1)"
simps_3: fixes x :: "real"   and y :: "real" shows "(real_min_top.R x \<le> real_min_top.R y) = (x \<le> y)"
star_pratt2_def: fixes x :: "pratt2" shows "x\<^sup>\<star> \<equiv> if x = \<bottom>\<^sub>2 \<or> x = [0]\<^sub>2 then [0]\<^sub>2 else \<top>\<^sub>2"
homeomorphic_space_def: fixes X :: "'d topology"   and Y :: "'e topology" shows "X homeomorphic_space Y \<equiv> \<exists>f g. homeomorphic_maps X Y f g"
\<psi>\<^sub>1\<^sub>0_tensor: fixes n :: "nat" assumes "1 \<le> n" shows "Matrix.mat (2 ^ 1) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ 1)) \<Otimes> Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n)) = Matrix.mat (2 ^ Suc n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ Suc n))"
\<tau>1: shows "typ_of (Abs \<tau>' s) = Some (\<tau>' \<rightarrow> \<tau>)"
lossless_response_OR: fixes R :: "'i \<times> 'e \<times> 'f + 'c \<times> 'e \<times> 'k"   and W :: "'h + 'b"   and s :: "'e" shows "lossless_spmf (response_OR R W s)"
ground_aux_def: fixes t :: "tm"   and S :: "atom set" shows "ground_aux t S \<equiv> supp t \<subseteq> S"
hom_sum: fixes f :: "'c \<Rightarrow> 'b poly"   and X :: "'c set"   and a :: "'b" shows "poly (sum f X) a = (\<Sum>x\<in>X. poly (f x) a)"
SD_iff_expected_utilities_le: fixes p :: "'alt pmf"   and q :: "'alt pmf" assumes "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts" shows "SD (R i) p q = (\<forall>u. is_vnm_utility u \<longrightarrow> measure_pmf.expectation p u \<le> measure_pmf.expectation q u)"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'b"   and v :: "'d pmf" shows "map_pmf g (map_pmf f v) = map_pmf (g \<circ> f) v"
ereal_mult_strict_left_mono: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a < b"   and "0 < c"   and "c < \<infinity>" shows "c * a < c * b"
simps_1: fixes uu :: "String.literal"   and T :: "typ" shows "Types (Ct uu T) = {T}"
assms: shows "Var x \<rightarrow>\<^sub>\<beta>\<^sup>* t"
c: shows "divisor_set (m * n) = {m, n, m * n, 1}"
assms_2: shows "f \<circ> Inr = g \<circ> Inr"
vec_of_list_def: shows "vec_of_list \<equiv> map_fun id Abs_vec (\<lambda>v. (length v, mk_vec (length v) ((!) v)))"
Cauchy_theorem_disc_simple: fixes f :: "complex \<Rightarrow> complex"   and a :: "complex"   and e :: "real"   and g :: "real \<Rightarrow> complex" assumes "f holomorphic_on ball a e"   and "valid_path g"   and "path_image g \<subseteq> ball a e"   and "pathfinish g = pathstart g" shows "(f has_contour_integral 0) g"
realizable_Append_ics_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "realizable n ns n''"   and "n'' ics-ns'\<rightarrow>\<^sub>d* n'" obtains ns'' :: "'node SDG_node list" where "realizable n (ns @ ns'') n'"
simps_14: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b"   and va :: "value list" shows "GExp_Lexorder.less_gexp_aux (Gt e1 e2) (In v va) = True"
disc_27: fixes x61 :: "'a OCL_Types.type"   and x62 :: "String.literal"   and x63 :: "'a expr list" shows "\<not> is_If (StaticOperationCall x61 x62 x63)"
Trans: shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> M\<lparr>N\<rparr> \<prec> R"
eq_numeral_iff_iszero_6: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and y :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "(one = numeral.numeral one plus y) = ring_1.iszero zero (neg_numeral.sub minus plus one num.One y)"
F_def2: fixes K :: "'b set" assumes "K \<in> normal" shows "F K = {a. - Rep_G a + u \<in> K}"
pos_assign_def: fixes \<alpha> :: "'c \<Rightarrow> 'd" shows "pos_assign \<alpha> = (\<forall>x. (0::'d) \<le> \<alpha> x)"
elims_2: fixes x :: "'a ext0"   and xa :: "'a ext0" assumes "x < xa" obtains x :: "'a ext0" and y :: "'a ext0" where "xa__ = x" and "xa = y" and "x \<le> y \<and> \<not> y \<le> x"
rel_interior_closure_convex_segment: fixes S :: "'c set"   and a :: "'c"   and b :: "'c" assumes "convex S"   and "a \<in> rel_interior S"   and "b \<in> closure S" shows "open_segment a b \<subseteq> rel_interior S"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) A'.map A'.map A'.map"
State_Idle_def: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and state :: "'e" shows "State_Idle localState output_fun trans_fun state \<equiv> \<forall>c. localState c = state \<longrightarrow> localState (trans_fun NoMsg c) = state \<and> output_fun (trans_fun NoMsg c) = NoMsg"
assms_1: shows "P \<Longrightarrow>\<^sup>^\<tau> \<prec> P'"
safe: shows "safe (s\<lparr>isin := (isin s)(r1 := isin s r1 \<union> {g1}), roomk := (roomk s)(r1 := k1), safe := (safe s)(r1 := owns s r1 = \<lfloor>g1\<rfloor> \<and> isin s r1 = {} \<and> k1 = currk s r1 \<or> safe s r1)\<rparr>) r"
distinct_7: fixes x21 :: "nat"   and x22 :: "'b list"   and x23 :: "'b list"   and x24 :: "'b list"   and x25 :: "'b list"   and x31 :: "nat"   and x32 :: "'b list"   and x33 :: "'b list" shows "Rev x21 x22 x23 x24 x25 \<noteq> App x31 x32 x33"
assms_1: shows "A \<ge>\<^sub>m 0\<^sub>m n n"
lossless_D: shows "\<forall>view. lossless_spmf (D view)"
document_ptr_no_node_ptr_cast: fixes node_ptr :: "(_) node_ptr" shows "\<not> is_document_ptr_kind (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)"
connected_eq_connected_components_eq: fixes s :: "'a set" shows "connected s = (\<forall>c\<in>components s. \<forall>c'\<in>components s. c = c')"
extended_Gromov_distance_commute: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_Gromov_distance x y = extended_Gromov_distance y x"
diff_strict_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and d :: "'b"   and c :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less a b"   and "less d c" shows "less (minus a c) (minus b d)"
s'_def: fixes t :: "real" shows "s' t = (if t \<le> t_stop then p' t else 0)"
chain_def: fixes A :: "'b set set"   and C :: "'b set set" shows "chain_on (\<subset>) C A = (C \<subseteq> A \<and> (\<forall>x\<in>C. \<forall>y\<in>C. (\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x))"
surjective: fixes r :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" shows "r = \<lparr>CatDom = CatDom r, CatCod = CatCod r, MapM = (##) r, \<dots> = Functor.more r\<rparr>"
normalized_single: fixes t :: "('c, 'd) bintree" shows "normalized [Some t]"
len_lt_eq: shows "length (pivot_positions (gauss_jordan_single (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))\<^sup>T)) \<le> dim_col (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets)))"
nact_active: fixes id :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n'' :: "nat" assumes "\<parallel>id\<parallel>\<^bsub>t n\<^esub>"   and "n \<le> n''"   and "\<not> (\<exists>n'\<ge>n. n' < n'' \<and> \<parallel>id\<parallel>\<^bsub>t n'\<^esub>)" shows "n = n''"
ring_in_Units_imp_not_zero: fixes a :: "'a" assumes "\<one> \<noteq> \<zero>"   and "a \<in> Units R" shows "a \<noteq> \<zero>"
transitionI_3: fixes P :: "pi"   and P''' :: "pi"   and a :: "name"   and x :: "name"   and P'' :: "pi"   and u :: "name"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P'''"   and "P''' \<longmapsto> a<x> \<prec> P''"   and "P''[x::=u] \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'"
image_preserves_per: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and A' :: "'a"   and X :: "'a"   and Y :: "'a"   and B :: "'a"   and B' :: "'a"   and C :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Reflect Bet Cong A A' X Y"   and "Tarski_neutral_dimensionless.Reflect Bet Cong B B' X Y"   and "Tarski_neutral_dimensionless.Reflect Bet Cong C C' X Y"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C" shows "Tarski_neutral_dimensionless.Per Bet Cong A' B' C'"
last_index_drop: fixes xs :: "'b list"   and x :: "'b"   and i :: "nat" assumes "last_index xs x < i" shows "x \<notin> set (drop i xs)"
simps_1: fixes F :: "'a filter" shows "landau_dominating_chain' F [] = True"
true_lit_simps_1: fixes I :: "'a set"   and A :: "'a" shows "(I \<Turnstile>l Pos A) = (A \<in> I)"
hyp: fixes L :: "'a language"   and K :: "'a language" assumes "R L K" shows "\<oo> L \<le> \<oo> K \<and> (\<forall>x. Plus_Times_pre_cong R (\<dd> L x) (\<dd> K x))"
domD: fixes a :: "'d"   and m :: "'d \<Rightarrow> 'e option" assumes "a \<in> dom m" shows "\<exists>b. m a = \<lfloor>b\<rfloor>"
empty_def: shows "Graph.empty \<equiv> \<lparr>nodes = {}, edges = {}\<rparr>"
cEnum_unit_def: shows "CENUM(unit) = Some (enum_class.enum, enum_class.enum_all, enum_class.enum_ex)"
consistent_one_step_axioms: shows "consistent_one_step t0 (t0 + e1') solution (euler_increment f) 1 (euler_C TYPE('a)) (r - b) B'"
finite_set_plus: fixes s :: "'a set"   and t :: "'a set" assumes "finite s"   and "finite t" shows "finite (s + t)"
sappend_scons: fixes x :: "'a"   and xs :: "[:'a:]"   and ys :: "[:'a:]" shows "(x :# xs) :@ ys = x :# xs :@ ys"
weakBisimWeakCoinductAux: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi"   and X :: "('e \<times> ('d, 'e, 'f) psi \<times> ('d, 'e, 'f) psi) set" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> env.weakStatImp substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P X Q \<and> env.weakSimulation substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P X Q \<and> (\<forall>\<Psi>'. (SCompose' \<Psi> \<Psi>', P, Q) \<in> X) \<and> (\<Psi>, Q, P) \<in> X" shows "(\<Psi>, P, Q) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
iT_Div_Un: fixes A :: "nat set"   and B :: "nat set"   and k :: "nat" shows "A \<union> B \<oslash> k = (A \<oslash> k) \<union> (B \<oslash> k)"
assms_3: fixes J :: "'a set" assumes "J \<in> I" shows "A \<in> Pi\<^sub>F {J} M \<rightarrow>\<^sub>M N"
pext: fixes tcp_flags2 :: "tcp_flag set"   and payload2 :: "char list"   and aux :: "'c" shows "simple_matches m \<lparr>p_iiface = pc_iiface c, p_oiface = pc_oiface c, p_src = s, p_dst = d, p_proto = pc_proto c, p_sport = pc_sport c, p_dport = pc_dport c, p_tcp_flags = tcp_flags2, p_payload = payload2, \<dots> = aux\<rparr> = simple_matches m \<lparr>p_iiface = pc_iiface c, p_oiface = pc_oiface c, p_src = s, p_dst = d, p_proto = pc_proto c, p_sport = pc_sport c, p_dport = pc_dport c, p_tcp_flags = {TCP_SYN}, p_payload = []\<rparr>"
non_null_2: fixes i :: "'b" assumes "i \<in> S" shows "0 < d i"
sup_ge2_classes1: fixes d :: "classes1"   and c :: "classes1" shows "d \<le> c \<squnion> d"
homeomorphic_maps_subtopologies: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c"   and S :: "'c set"   and T :: "'d set" assumes "homeomorphic_maps X Y f g"   and "f ` (topspace X \<inter> S) = topspace Y \<inter> T" shows "homeomorphic_maps (subtopology X S) (subtopology Y T) f g"
notsym_Elem: fixes a :: "ZF"   and b :: "ZF" shows "\<not> (Elem a b \<and> Elem b a)"
compact_Un: fixes s :: "'b set"   and t :: "'b set" assumes "compact s"   and "compact t" shows "compact (s \<union> t)"
wf_envs_mk_db: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat" shows "wf_envs \<sigma> j Map.empty Map.empty Map.empty (mk_db (\<Gamma> \<sigma> j))"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and f :: "'fsym"   and Tl :: "'fsym trm list" shows "Ik.int \<xi> (Fn f Tl) = intF f (map (Ik.int \<xi>) Tl)"
fresh_atm_1: fixes a :: "tyvrs"   and b :: "tyvrs" shows "a \<sharp> b = (a \<noteq> b)"
m_max: shows "\<forall>n\<in>N. n < m"
canonical_fun_def: fixes U :: "'a set"   and x :: "'b" shows "stk1.canonical_fun U x = stk1.class_of U x"