hcju/isabelleps huggingface dataset:
mkAlgAuto_def: fixes aOps :: "('f, 'e, 'd list) MapOps"   and tOps :: "('g, 'e \<times> 'c, 'e) MapOps"   and simObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'c"   and simInit :: "'b \<Rightarrow> 'c \<Rightarrow> 'e"   and simTrans :: "'b \<Rightarrow> 'e \<Rightarrow> 'e list"   and simAction :: "'b \<Rightarrow> 'e \<Rightarrow> 'd list"   and frontier :: "'b \<Rightarrow> 'e list" shows "mkAlgAuto aOps tOps simObs simInit simTrans simAction frontier \<equiv> \<lambda>a. let auto = alg_dfs aOps tOps (simObs a) (simTrans a) (simAction a) (frontier a) in \<lparr>pInit = simInit a, pTrans = \<lambda>obs ec. the (MapOps.lookup tOps (aTrans auto) (ec, obs)), pAct = \<lambda>ec. the (MapOps.lookup aOps (aActs auto) ec)\<rparr>"
ordered_semiring_record_simps_12: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'b" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
taut_test: fixes b :: "'b" shows "taut_test ifex_of b = (\<forall>s. val b s)"
head_in_verts: fixes e :: "'b \<times> 'b" assumes "e \<in> arcs (with_proj (list_digraph G_list))" shows "head (with_proj (list_digraph G_list)) e \<in> verts (with_proj (list_digraph G_list))"
useful: fixes \<phi> :: "'a \<Rightarrow> bool"   and \<psi> :: "bool" assumes "\<forall>x. \<phi> x \<longrightarrow> \<psi>" shows "(\<exists>x. \<phi> x) \<longrightarrow> \<psi>"
TYREL_def: fixes R :: "('c \<times> 'd) set" shows "TYREL R \<equiv> True"
modhyp: shows "\<not> module.lin_dep class_ring (module_vec TYPE('a) nc) S"
assms_2: shows "(\<sigma>', k') \<cc> (\<sigma>, k)"
that: fixes as :: "'edge list" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'"   and "as \<noteq> []" shows "thesis"
distinct_12: fixes x3 :: "'a formula" shows "\<^bold>\<not> x3 \<noteq> \<bottom>"
prod_prime_factors: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "(\<Prod>p\<in>prime_factors x. p ^ multiplicity p x) = normalize x"
Inv_Ide: fixes a :: "'a local.term" assumes "Ide a" shows "Inv a = a"
r_one: fixes x :: "'a \<Rightarrow> 'b" assumes "x \<in> carrier (DirProds G I)" shows "x \<otimes>\<^bsub>DirProds G I\<^esub> \<one>\<^bsub>DirProds G I\<^esub> = x"
fin_cut_same_Nil: fixes x :: "'c" shows "fin_cut_same x [] = []"
convergent_ereal: fixes X :: "nat \<Rightarrow> 'b" shows "convergent X = (limsup X = liminf X)"
ED: fixes x :: "'s"   and y :: "'s" assumes "x \<in> X"   and "y \<in> set_pmf (K x)" shows "y \<in> X"
mtx_vec_scaleR_commute: fixes A :: "'a sq_mtx"   and c :: "real"   and x :: "(real, 'a) vec" shows "A *\<^sub>V (c *\<^sub>R x) = c *\<^sub>R A *\<^sub>V x"
1: fixes n :: "nat" shows "X_poly_plus R a n \<in> S"
axioms_2: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "total_ordered_set A less_eq" shows "antisymmetric A less_eq"
dvdm_factor: fixes f :: "int poly"   and g :: "int poly"   and h :: "int poly" assumes "pl.dvdm f g" shows "pl.dvdm f (g * h)"
of_nat_zero_less_power_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less zero (power.power one times (semiring_1.of_nat one plus zero x) n) = (0 < x \<or> n = 0)"
rep_eq: fixes x :: "'b blinop"   and xa :: "'b blinop" shows "Rep_blinop (x - xa) = Rep_blinop x - Rep_blinop xa"
that_1: fixes i :: "'a" assumes "i \<in> Basis"   and "y = i" shows "thesis"
invoke_ptr_in_heap: fixes h :: "(_) heap"   and xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> 'd \<Rightarrow> ((_) heap, exception, 'c) prog)) list"   and ptr :: "(_) object_ptr"   and args :: "'d" assumes "h \<turnstile> ok invoke xs ptr args" shows "ptr |\<in>| object_ptr_kinds h"
one_im_sheaf_def: shows "one_im_sheaf \<equiv> \<lambda>V. one_str (f \<^sup>\<inverse> X V)"
subst_pp_plus: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and s :: "'d \<Rightarrow>\<^sub>0 nat"   and t :: "'d \<Rightarrow>\<^sub>0 nat" shows "subst_pp f (s + t) = subst_pp f s * subst_pp f t"
of_hypreal_eq_0_iff: fixes x :: "real star" shows "(of_hypreal x = 0) = (x = 0)"
asm_2: shows "\<forall>n\<in>set ns. v \<notin> defs g n"
field_inverse_zero: shows "inverse (0::'b) = (0::'b)"
currentD_finite_IN: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "current \<Gamma> f" shows "d_IN f x \<noteq> \<top>"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) Cod_f.map Cod_f.map Cod_f.map"
ideal_subset: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and h :: "'e" assumes "Ring R"   and "ideal R I"   and "h \<in> I" shows "h \<in> carrier R"
HInfinite_gt_zero_gt_one: fixes x :: "real star" assumes "x \<in> HInfinite"   and "0 < x" shows "1 < x"
nonzero_imp_inverse_nonzero: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<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 a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse a \<noteq> zero"
base_in_grid: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and ds :: "nat set" assumes "p \<in> sparsegrid' dm" shows "base ds p \<in> grid (start dm) {0..<dm}"
semiring_simprules_5: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"
supp_asToHeap: fixes as :: "assn" shows "supp (asToHeap as) = supp as"
2: shows "is_left_adjoint (k \<star> p\<^sub>1)"
vdiff_vempty_left: fixes A :: "V" shows "0 -\<^sub>\<circ> A = 0"
seq_comp_traces: fixes P :: "'c option process"   and Q :: "'c option process" assumes "weakly_sequential P" shows "traces (P ; Q) = Domain (seq_comp_failures P Q)"
transp_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "transp r = (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
prime_power_eq_one_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "(power.power one times p n = one) = (n = 0)"
upset_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.upset less_eq = ord.upset_set less_eq \<circ> (\<lambda>x. {x})"
indep_vars_compose2: fixes M :: "'a measure"   and M' :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> 'a \<Rightarrow> 'c"   and I :: "'b set"   and Y :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and N :: "'b \<Rightarrow> 'd measure" assumes "prob_space M"   and "prob_space.indep_vars M M' X I"   and "\<And>i. i \<in> I \<Longrightarrow> Y i \<in> M' i \<rightarrow>\<^sub>M N i" shows "prob_space.indep_vars M N (\<lambda>i x. Y i (X i x)) I"
tail_arev: fixes a :: "'b" assumes "a \<in> arcs H" shows "tail H (rev_H a) = head H a"
properL_def: fixes cl :: "('a, 'b, 'c) cmd list" shows "properL cl \<equiv> cl \<noteq> [] \<and> (\<forall>c\<in>set cl. proper c)"
cases: fixes a1 :: "nat"   and a2 :: "intv" assumes "valid_intv a1 a2" obtains d :: "nat" and c :: "nat" where "a1 = c__" and "a2 = Const d__" and "0 \<le> d__" and "d__ \<le> c__"   | d :: "nat" and c :: "nat" where "a1 = ca__" and "a2 = Intv d" and "0 \<le> d" and "d < ca__"   | c :: "nat" where "a1 = c" and "a2 = Greater c"
fsmultD: fixes smult :: "('b, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'b"   and v :: "'a" shows "aezfun_scalar_mult.fsmult smult a v = smult (a \<delta>\<delta> (0::'c)) v"
FormP_quot_dbfm: fixes A :: "dbfm" assumes "wf_dbfm A" shows "{} \<turnstile> FormP (quot_dbfm A)"
gmean_def: fixes s :: "real list" shows "gmean s = root (length s) (prod_list s)"
ord_strict_p_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "(p \<prec>\<^sub>p q) = (\<exists>v. lookup p v = (0::'c) \<and> lookup q v \<noteq> (0::'c) \<and> (\<forall>u. v \<prec>\<^sub>t u \<longrightarrow> lookup p u = lookup q u))"
sorted_wrt_filter: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "sorted_wrt f xs" shows "sorted_wrt f (filter P xs)"
rmigeq: fixes ni1 :: "nat"   and n1 :: "nat ifex"   and s :: "bdd"   and ni2 :: "nat"   and n2 :: "nat ifex" assumes "Rmi_g ni1 n1 s"   and "Rmi_g ni2 n2 s"   and "ni1 = ni2" shows "n1 = n2"
Uniq_I: fixes P :: "'g \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> y = x" shows "Uniq P"
minus_in_float_intervalI: fixes a :: "real"   and A :: "float interval"   and b :: "real"   and B :: "float interval" assumes "a \<in>\<^sub>r A"   and "b \<in>\<^sub>r B" shows "a - b \<in>\<^sub>r A - B"
vsubstEnv_preserves_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and z :: "'var"   and ys :: "'varSort"   and y :: "'var"   and y1 :: "'var" assumes "goodEnv rho"   and "freshEnv zs z rho"   and "zs \<noteq> ys \<or> z \<notin> {y, y1}" shows "freshEnv zs z (rho &[y1 // y]_ys)"
seq_mono: fixes c\<^sub>1 :: "'a::refinement_lattice"   and d\<^sub>1 :: "'a::refinement_lattice"   and c\<^sub>2 :: "'a::refinement_lattice"   and d\<^sub>2 :: "'a::refinement_lattice" assumes "(c\<^sub>1::'a::refinement_lattice) \<sqsubseteq> (d\<^sub>1::'a::refinement_lattice)"   and "(c\<^sub>2::'a::refinement_lattice) \<sqsubseteq> (d\<^sub>2::'a::refinement_lattice)" shows "(c\<^sub>1::'a::refinement_lattice) ; (c\<^sub>2::'a::refinement_lattice) \<sqsubseteq> (d\<^sub>1::'a::refinement_lattice) ; (d\<^sub>2::'a::refinement_lattice)"
simps_1: fixes x1 :: "('c, 'd) Term.term"   and y1 :: "('c, 'd) Term.term" shows "(send\<langle>x1\<rangle>\<^sub>s\<^sub>t = send\<langle>y1\<rangle>\<^sub>s\<^sub>t) = (x1 = y1)"
take_greater_eqI: fixes c :: "nat"   and xs :: "'a list"   and ys :: "'a list"   and a :: "nat" assumes "take c xs = take c ys"   and "a \<le> c" shows "take a xs = take a ys"
bhv2cnf_greater_lActive: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1 < n" shows "\<langle>c \<and> t\<rangle> < \<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n"
prime_dvd_power_int_iff: fixes p :: "int"   and n :: "nat"   and x :: "int" assumes "prime p"   and "0 < n" shows "(p dvd x ^ n) = (p dvd x)"
simp1: shows "\<forall>y. a * (- b / a * y + - c / a) = a * (- b / a) * y + a * (- c / a)"
le_B0: fixes w :: "complex" assumes "cmod (w - z) \<le> r" shows "cmod (f w - y) \<le> B0"
sqn_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat" shows "sqn rt dip \<equiv> case rt dip of None \<Rightarrow> 0 | Some r \<Rightarrow> \<pi>\<^sub>2 r"
amin_le_l: fixes x :: "ant"   and y :: "ant" shows "amin x y \<le> x"
e_1: fixes i :: "nat" shows "partial_state.encode2 dims vars2 i = partial_state.encode1 dims vars1 i"
nz_b_least2_impl_less: fixes z :: "nat"   and y :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" assumes "z < y"   and "f x z \<noteq> 0" shows "b_least2 f x y < y"
comm_semiring_0_cancel_ow_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "comm_semiring_0_cancel_ow U plus minus zero times \<equiv> comm_semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero"
map_comp0: fixes g1 :: "'g \<Rightarrow> 'e"   and f1 :: "'c \<Rightarrow> 'g"   and g2 :: "'h \<Rightarrow> 'f"   and f2 :: "'d \<Rightarrow> 'h" shows "map_coF0 (g1 \<circ> f1) (g2 \<circ> f2) = map_coF0 g1 g2 \<circ> map_coF0 f1 f2"
less_irrefl: fixes x :: "'a list" shows "\<not> strict_prefix x x"
conversep_conversep: fixes r :: "'n \<Rightarrow> 'o \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<inverse>\<inverse> = r"
fiter_unfold: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'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)" shows "finite_iteration.fiter (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice) (c::'b::refinement_lattice) = nil \<sqinter> seq c (finite_iteration.fiter seq nil c)"
cases: fixes a :: "dag"   and nodes :: "ref set"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref" assumes "a \<in> Dags nodes low high" obtains t :: "dag" and p :: "ref" where "a = t" and "set_of t \<subseteq> nodes" and "Dag p low high t" and "t \<noteq> Tip"
mcont_sup1: fixes x :: "'a" shows "mcont Sup (\<le>) Sup (\<le>) ((\<squnion>) x)"
sums_mult: fixes f :: "nat \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c" assumes "f sums a" shows "(\<lambda>n. c * f n) sums (c * a)"
assms_2: shows "g ` A \<subseteq> {s. conv_abscissa f < ereal (s \<bullet> (1::'a))}"
derives_count_terminals_leq: fixes a :: "symbol list"   and b :: "symbol list" assumes "derives a b" shows "count_terminals a \<le> count_terminals b"
matrix_vector_mult_def: fixes m :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and x :: "('b, 'd) Finite_Cartesian_Product.vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. m $h i $h j * x $h j"
l3_inv2_init: shows "init l3 \<subseteq> l3_inv2"
finite_csucc: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<omega>" shows "csucc a = ZFC_in_HOL.succ a"
46: fixes x :: "'a" shows "- bot \<squnion> - - x = - bot"
intersect'_language: fixes A :: "('d, 'e) dbta"   and B :: "('d, 'f) dbta" shows "DGBTA.language (intersect' A B) = DBTA.language A \<inter> DBTA.language B"
sup_idem: fixes x :: "'d multiset" shows "x \<union># x = x"
hd: shows "dens_ctxt_\<alpha> ([], vs @ vs', \<Gamma>, CReal 1) \<turnstile>\<^sub>d e \<Rightarrow> (\<lambda>x xa. ennreal (eval_cexpr f x xa))"
reach_l: shows "s\<turnstile> l reachable_from x"
integral_combine_tagged_division_bottomup: fixes p :: "('a \<times> 'a set) set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "p tagged_division_of cbox a b"   and "\<And>x k. (x, k) \<in> p \<Longrightarrow> f integrable_on k" shows "integral (cbox a b) f = (\<Sum>(x, k)\<in>p. integral k f)"
simps_5: fixes uv :: "typ"   and v :: "typ"   and va :: "Core.term" shows "eta_reducible (Abs uv (Abs v va)) = eta_reducible (Abs v va)"
deterministic: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and lc :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and lc' :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and lc'' :: "('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)" assumes "sifum_security_init dma \<C>_vars eval"   and "(lc, lc') \<in> eval"   and "(lc, lc'') \<in> eval" shows "lc' = lc''"
p_def: shows "p = Cn 1 r_pdec1 [Cn 1 r_parallel [at0, at1, c]]"
execute_while_unfolds_2: fixes p :: "bool Heap"   and h :: "heap"   and f :: "'b Heap" assumes "ro_cond p"   and "success p h"   and "\<not> cond p h" shows "execute (while p f) h = execute (return ()) h"
star_induct: fixes w :: "'a list"   and A :: "'a list set"   and P :: "'a list \<Rightarrow> bool" assumes "w \<in> star A"   and "P []"   and "\<And>u v. \<lbrakk>u \<in> A; v \<in> star A; P v\<rbrakk> \<Longrightarrow> P (u @ v)" shows "P w"
abs_0: fixes abs :: "'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs zero = zero"
order_root: fixes p :: "'b poly"   and a :: "'b" shows "(poly p a = (0::'b)) = (p = 0 \<or> order a p \<noteq> 0)"
assms: shows "exec_step_input P C M pc ics = StepT Cs a"
cond_antimono: fixes x :: "nat"   and xs :: "nat list"   and s :: "nat"   and x' :: "nat"   and s' :: "nat" assumes "C (x # xs) s"   and "x' \<le> x"   and "s' \<le> s" shows "C (x' # xs) s'"
permutation_mat_both: fixes A :: "'a mat"   and n :: "nat"   and p :: "nat \<Rightarrow> nat" assumes "A \<in> carrier_mat n n"   and "p permutes {..<n}" shows "permutation_mat n p * Matrix.mat n n (\<lambda>(i, j). A $$ (p i, p j)) * permutation_mat n (inv p) = A"
memFresh: fixes x :: "name"   and l :: "'a list"   and p :: "'a" assumes "x \<sharp> l"   and "p mem l" shows "x \<sharp> p"
convergent_powser'_eventually: fixes cs :: "real msllist"   and f :: "real \<Rightarrow> real" assumes "convergent_powser' cs f" shows "\<forall>\<^sub>F x in nhds 0. powser cs x = f x"
plus_prod_def: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "x + y = (fst x + fst y, snd x + snd y)"
lessThan_insert: fixes n :: "'a" shows "insert n {..<n} = {..n}"
success_not_cond_effect: fixes p :: "bool Heap"   and h :: "heap" assumes "ro_cond p"   and "success p h"   and "\<not> cond p h" shows "effect p h h False"
sum_not_inf_dest: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" assumes "a + b \<noteq> \<infinity>" shows "a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
openin_connected_component_locally_connected: fixes S :: "'a set"   and x :: "'a" assumes "locally connected S" shows "openin (top_of_set S) (connected_component_set S x)"
cap_simps_1: fixes y :: "'v" shows "cap (None, \<langle>y\<rangle>) = (if y \<in> \<^bold>V then u y else 0)"
foo5b: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and c :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp b)"   and "\<And>r. r \<in> set c \<Longrightarrow> x \<notin> dom (Cp r)" shows "applied_rule_rev Cp x (b # c) = \<lfloor>b\<rfloor>"
mult_le_cancel_left: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times c a) (times c b) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
degree_lcoeff_sum: fixes f :: "'c \<Rightarrow> 'b poly"   and q :: "'c"   and n :: "nat"   and S :: "'c set"   and c :: "'b" assumes "degree (f q) = n"   and "finite S"   and "q \<in> S"   and "\<And>p. p \<in> S - {q} \<Longrightarrow> degree (f p) < n"   and "coeff (f q) n = c" shows "degree (sum f S) = n \<and> coeff (sum f S) n = c"
char_to_digit_simps_2: shows "char_to_digit CHR ''1'' = 1"
union_mset_add_mset_right: fixes A :: "'c multiset"   and a :: "'c"   and B :: "'c multiset" shows "A + add_mset a B = add_mset a (A + B)"
has_integral_f: fixes k :: "nat" shows "(f k has_integral F k) {0..c}"
eval_fds_zeta_nonzero: assumes "1 < s \<bullet> 1" shows "eval_fds fds_zeta s \<noteq> 0"
replace4: fixes f :: "'c"   and ls :: "'c list"   and oldF :: "'c"   and fs :: "'c list" assumes "f \<in> set ls"   and "oldF \<noteq> f" shows "f \<in> set (replace oldF fs ls)"
eFreshInp_simp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'a option"   and ys :: "'varSort"   and y :: "'var" assumes "igWlsInp MOD delta inp" shows "eFreshInp MOD ys y (OKI inp) = igFreshInp MOD ys y inp"
ht_hash_def: fixes l :: "('a \<times> 'b) list list" shows "ht_hash l \<equiv> \<forall>i<length l. \<forall>x\<in>set (l ! i). bounded_hashcode_nat (length l) (fst x) = i"
add2_def: fixes x :: "'b"   and y :: "'b" shows "Sep_Eq.add2 x y \<equiv> if x ## y then Some (x + y) else None"
finite_set_avoid: fixes S :: "'c set"   and a :: "'c" assumes "finite S" shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
assms: shows "ref = None \<or> ref = Some r \<and> r \<in> set xs"
countable_image_inj_Int_vimage: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and A :: "'d set" assumes "inj_on f S"   and "countable A" shows "countable (S \<inter> f -` A)"
set_state_iff: fixes x :: "'b"   and m :: "('c, 'b) state" shows "(x \<in> set_state m) = (\<exists>s s'. run_state m s = (x, s'))"
d3_var: fixes x :: "'a"   and y :: "'a" shows "d (x \<sqinter> d y) \<le> d (x \<sqinter> y)"
assms: shows "((\<sigma>, SubnetS s t), a, \<sigma>', st') \<in> opnet_sos (automaton.trans (opnet np p1)) (automaton.trans (opnet np p2))"
Some: shows "\<tau> = \<lfloor>\<tau>\<^sub>1\<rfloor>"
face_of_face: fixes T :: "'b set"   and S :: "'b set"   and f :: "'b set" assumes "T face_of S" shows "(f face_of T) = (f face_of S \<and> f \<subseteq> T)"
Rcd_type1: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type"   and fs :: "(char list \<times> trm) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list" assumes "\<Gamma> \<turnstile> t : T"   and "t = Rcd fs"   and "\<Gamma> \<turnstile> T <: RcdT fTs" shows "\<forall>(l, U)\<in>set fTs. \<exists>u. fs\<langle>l\<rangle>\<^sub>? = \<lfloor>u\<rfloor> \<and> \<Gamma> \<turnstile> u : U"
d_IN_restrict_current_outside: fixes x :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "x \<in> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "d_IN (g \<upharpoonleft> \<Gamma> / f) x = 0"
subst_cls_lists_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and Cs :: "'b literal multiset list"   and \<sigma>s :: "'c list" shows "substitution_ops.subst_cls_lists subst_atm Cs \<sigma>s = Map2.map2 (substitution_ops.subst_cls subst_atm) Cs \<sigma>s"
map_lists_surjective: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" assumes "f ` A = B" shows "map f ` lists A = lists B"
fst_propa: shows "fst (propa f (step (hd w) (ss ! hd w)) ss (tl w)) = merges f (step (hd w) (ss ! hd w)) ss"
reachable_del_vertD: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> w" shows "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"
assms_2: shows "Q \<in> carrier_mat (dim_col A) (dim_col A)"
loc_srj_2: shows "nat_to_sch (c_pair 3 0) = Base_snd"
distinct_3: fixes x1 :: "'a" shows "Action x1 \<noteq> Undefined"
enat_0_iff_2: fixes x :: "nat" shows "(0 = enat x) = (x = 0)"