hcju/isabelleps huggingface dataset:
red_monomial_keys: fixes r :: "'t \<Rightarrow>\<^sub>0 'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_monomial r"   and "red {r} p q" shows "card (keys p) = Suc (card (keys q))"
Borsuk_separation_theorem: fixes S :: "'a set" assumes "compact S"   and "2 \<le> DIM('a)" shows "connected (- S) = (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere (0::'a) 1 \<longrightarrow> (\<exists>c. homotopic_with_canon (\<lambda>x. True) S (sphere (0::'a) 1) f (\<lambda>x. c)))"
rivest_bind: fixes \<eta> :: "nat"   and \<A> :: "nat \<times> nat \<Rightarrow> (nat \<times> nat \<times> (nat \<times> nat) \<times> nat \<times> nat \<times> nat) spmf" shows "rivest_commit.bind_advantage \<eta> \<A> \<le> 1 / real (q \<eta>)"
outer_transitivity_between: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Bet A B C"   and "Bet B C D"   and "B \<noteq> C" shows "Bet A B D"
mult2_alt_s_implies_mult2_alt_ns: fixes X :: "'b multiset"   and Y :: "'b multiset"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "(X, Y) \<in> mult2_alt_s ns s" shows "(X, Y) \<in> mult2_alt_ns ns s"
block_comp_elem_alt_left: fixes x :: "'a"   and bl :: "'a set"   and ps :: "'a set" assumes "x \<in> bl"   and "ps \<subseteq> bl\<^sup>c" shows "x \<notin> ps"
integrable_on_empty: fixes f :: "'a \<Rightarrow> 'b" shows "f integrable_on {}"
continuous_at_extension_sequentially': fixes b :: "'c"   and T :: "'c set"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "b \<in> T"   and "\<And>u b. \<lbrakk>\<forall>n. u n \<in> S; b \<in> T; u \<longlonglongrightarrow> b\<rbrakk> \<Longrightarrow> convergent (\<lambda>n. f (u n))"   and "\<And>b. b \<in> T \<Longrightarrow> \<exists>u. (\<forall>n. u n \<in> S) \<and> u \<longlonglongrightarrow> b \<and> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f b" shows "continuous (at b within S \<union> T) f"
inject_1: fixes x1 :: "('e, 'f) Term.term"   and y1 :: "('e, 'f) Term.term" shows "(send\<langle>x1\<rangle> = send\<langle>y1\<rangle>) = (x1 = y1)"
ide_dom: fixes f :: "'a" assumes "hom'.arr f" shows "hom'.ide (hom'.dom f)"
sup_aci_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> y \<squnion> z = x \<squnion> (y \<squnion> z)"
ord_equiv_reflex: fixes D :: "('c, 'd) Order_scheme" assumes "Order D" shows "ord_equiv D D"
fix_is_fix_of_limit: fixes f :: "'b set \<Rightarrow> 'b set"   and I :: "'b set" assumes "f I = I" shows "limit f I = I"
ParComp: fixes i :: "nat"   and Ps :: "'b com option list"   and s :: "'b"   and r :: "'b com option"   and t :: "'b" assumes "i < length Ps"   and "(Ps ! i, s) -c\<rightarrow> (r, t)" shows "(Ps, s) -pc\<rightarrow> (Ps[i := r], t)"
right: shows "ix p' d \<le> (ix p d + 1) * 2 ^ (lv p' d - lv p d)"
l_one: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "\<one> \<otimes> x = x"
localp: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and \<zeta> :: "'a \<Rightarrow> 'b"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and \<zeta>' :: "'a \<Rightarrow> 'b"   and s :: "'c"   and a :: "'d"   and s' :: "'c" assumes "\<forall>j\<in>J. \<sigma> j = \<zeta> j"   and "\<forall>j\<in>J. \<sigma>' j = \<zeta>' j"   and "P ((\<sigma>, s), a, \<sigma>', s')" shows "P ((\<zeta>, s), a, \<zeta>', s')"
linked_paths_def: fixes atends :: "bool"   and g :: "real \<Rightarrow> 'a"   and h :: "real \<Rightarrow> 'a" shows "linked_paths atends g h \<equiv> if atends then pathstart h = pathstart g \<and> pathfinish h = pathfinish g else pathfinish g = pathstart g \<and> pathfinish h = pathstart h"
cases: fixes x :: "rat + int poly \<times> nat" obtains q :: "rat" where "x = Inl q"   | f :: "int poly" and n :: "nat" where "x = Inr (f, n)"
foldmatrix_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "foldmatrix f g A m n \<equiv> foldseq_transposed g (\<lambda>j. foldseq f (A j) n) m"
simps_1: fixes f1 :: "'e \<Rightarrow> 'c"   and f2 :: "'f \<Rightarrow> 'd"   and a :: "'e" shows "map_sum f1 f2 (Inl a) = Inl (f1 a)"
not_free_2: shows "\<forall>i\<in>frees t. n \<le> i"
prob_space_lim_stream: fixes x :: "'a" assumes "x \<in> space M" shows "prob_space (lim_stream x)"
homeomorphic_punctured_sphere_affine: fixes r :: "real"   and b :: "'c"   and a :: "'c"   and T :: "'d set" assumes "0 < r"   and "b \<in> sphere a r"   and "affine T"   and "aff_dim T + 1 = int DIM('c)" shows "sphere a r - {b} homeomorphic T"
vec_cases: fixes v :: "'a vec" obtains "v = vNil"   | a :: "'a" and w :: "'a vec" where "v = vCons a w"
invertible_iff_is_unit: fixes A :: "(('a, 'b) vec, 'b) vec" shows "invertible A = (det A dvd (1::'a))"
inv_inv_eq: fixes f :: "'d \<Rightarrow> 'e" assumes "bij f" shows "inv (inv f) = f"
assms: shows "\<bar>b\<bar> < \<bar>a\<bar>"
emb_step_at_if_position: fixes t :: "('a, 'b) tm"   and p :: "dir list"   and d :: "dir" assumes "position_of t (p @ [d])" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b emb_step_at p d t"
s0_defs_1: shows "s0 \<equiv> \<lparr>init = s0_init, trans = s0_trans, obs = id\<rparr>"
sams2_suma2__conga456: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "SAMS A B C D E F"   and "SAMS A B C D' E' F'"   and "A B C D E F SumA G H I"   and "A B C D' E' F' SumA G H I" shows "D E F CongA D' E' F'"
ENC_def: fixes n :: "nat"   and V :: "nat set" shows "ENC n V = INTERSECT (sorted_list_of_set (valid_ENC n ` V))"
full_elims_1: fixes n :: "nat" assumes "full n Empty" obtains "n = 0"
sees_methods_fun: fixes P :: "'a prog"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option"   and Mm' :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" assumes "P \<turnstile> C sees_methods Mm"   and "P \<turnstile> C sees_methods Mm'" shows "Mm' = Mm"
\<epsilon>_leg_simps_3: shows "\<epsilon>.dom.leg0 = fg.leg0"
scon_def: fixes \<phi>s :: "((('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list \<Rightarrow> bool) set"   and \<pi>l :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list" shows "scon \<phi>s \<pi>l \<equiv> \<forall>\<phi>\<in>\<phi>s. \<phi> \<pi>l"
fdia_star_unfoldr: fixes y :: "'a"   and x :: "'a" shows " |(1::'a)\<rangle> y +  |x\<^sup>\<star>\<rangle>  |x\<rangle> y =  |x\<^sup>\<star>\<rangle> y"
fail_def: shows "fail = altc cempty (\<lambda>_. undefined)"
FAssThrow2: fixes E :: "char list \<Rightarrow> ty option"   and v :: "val"   and F :: "char list"   and Cs :: "char list list"   and r :: "nat \<times> char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>Val v\<bullet>F{Cs} := Throw r,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
fbox_star_induct_eq: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "d z \<cdot>  |x] y = d y" shows "d y \<le>  |x\<^sup>\<star>] z"
ClassicInit_def: shows "ClassicInit = {(p, t, left, right, atom, mrk). atom = atom0 \<and> left = left0 \<and> right = right0 \<and> finite (- mrk) \<and> mrk = {}}"
reds_doesnt_forget: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z" shows "domA \<Gamma> \<subseteq> domA \<Delta>"
return_writer_def: fixes x :: "'a" shows "local.return_writer x = WriterT (return (x, []))"
backward_simulation_axioms: shows "backward_simulation Sstd.step Sinca.step Sstd.final Sinca.final (\<lambda>_ _. False) (\<lambda>_. (\<sim>))"
complement_overlap: fixes x :: "i"   and y :: "i" assumes "\<not> O x y" shows "O x (\<midarrow> y)"
i_set0_def: shows "i_set0 \<equiv> insert {} i_set"
independence_symmetric: fixes a :: "'action"   and b :: "'action" assumes "ind a b" shows "ind b a"
in_idealE_Polys: fixes B :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and X :: "'a set"   and p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "B \<subseteq> P[X]"   and "p \<in> P[X]"   and "p \<in> ideal B" obtains A :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" and q :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where "finite A" and "A \<subseteq> B" and "\<And>b. q b \<in> P[X]" and "p = (\<Sum>b\<in>A. q b * b)"
foo: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and q :: "nat \<Rightarrow> nat" assumes "i < Suc n"   and "j < Suc n"   and "q permutes {0..<n}" shows "{(i', permutation_insert i j q i') |i'. i' \<in> {0..<Suc n} - {i}} = {(insert_index i i'', insert_index j (q i'')) |i''. i'' < n}"
that: fixes \<psi> :: "'a" assumes "\<guillemotleft>\<psi> : g \<star> f\<^sup>* \<Rightarrow> r\<guillemotright>"   and "local.iso \<psi>" shows "thesis"
assms_2: assumes "\<And>s r x. p s = Inr (x, r) \<Longrightarrow> length r \<le> length s" shows "P"
steps_c\<^sub>1: shows "\<Gamma>\<turnstile> (c\<^sub>1, Fault f) \<rightarrow>\<^sup>* (Skip, Fault f)"
h2: shows "\<forall>elem1 elem2. elem1 \<in> set subsets1 \<and> elem2 \<in> set subsets2 \<longrightarrow> list_constr (elem1 @ map ((+) (length qs1)) elem2) (length (qs1 @ qs2))"
IH: assumes "valid_reqs M"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>" shows "all_security_requirements_fulfilled M (generate_valid_topology_SOME M \<lparr>nodes = V, edges = E\<rparr>)"
s': shows "s' \<in> pmdl.syzygy_module (set bs)"
0: shows "finite (the_Abs ` {f \<in> funs_term t. is_Abs f} \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+)"
dgrad_p_set_subset: fixes m :: "nat"   and n :: "nat"   and d :: "'a \<Rightarrow> nat" assumes "m \<le> n" shows "dgrad_p_set d m \<subseteq> dgrad_p_set d n"
B_def: fixes V :: "'a set"   and f :: "'a \<Rightarrow> real" shows "B V f \<equiv> {0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel> |x. x \<noteq> (0::'a) \<and> x \<in> V}"
radical_def: 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 divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.radical zero times one normalize x = (if x = zero then zero else comm_monoid_mult.prod times one (\<lambda>x. x) (set_mset (factorial_semiring.prime_factorization zero times one normalize x)))"
qo_on_LEXEQ: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "transp_on P A" shows "qo_on (LEXEQ P) (SEQ A)"
pochhammer_of_nat_eq_0_lemma': fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "pochhammer (- of_nat n) k \<noteq> (0::'b)"
update_simps_5: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set" assumes "ip \<in> kD rt"   and "\<pi>\<^sub>3 r = unk"   and "(\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk)" shows "update rt ip r = rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 (addpre r (\<pi>\<^sub>7 (the (rt ip)))), \<pi>\<^sub>4 (addpre r (\<pi>\<^sub>7 (the (rt ip)))), \<pi>\<^sub>5 (addpre r (\<pi>\<^sub>7 (the (rt ip)))), \<pi>\<^sub>6 (addpre r (\<pi>\<^sub>7 (the (rt ip)))), \<pi>\<^sub>7 (addpre r (\<pi>\<^sub>7 (the (rt ip))))))"
degree_to_poly: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R" shows "deg R (to_polynomial R a) = 0"
Lim_bounded2: fixes f :: "nat \<Rightarrow> 'c"   and l :: "'c"   and N :: "nat"   and C :: "'c" assumes "f \<longlonglongrightarrow> l"   and "\<forall>n\<ge>N. C \<le> f n" shows "C \<le> l"
none: fixes \<F>' :: "'a set set" assumes "finite \<F>'"   and "\<F>' \<subseteq> \<F>" shows "\<Inter> \<F>' \<noteq> {}"
relation_rules_3: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "P x \<Longrightarrow> Q x" shows "(P \<longrightarrow> Q) x"
assms_3: fixes \<sigma> :: "'\<sigma>"   and k :: "'u"   and it :: "'u set"   and v :: "'v" assumes "c \<sigma>"   and "k \<in> it"   and "\<forall>j\<in>it. j \<le> k"   and "\<forall>j\<in>dom (\<alpha> m) - it. k \<le> j"   and "\<alpha> m k = Some v"   and "it \<subseteq> dom (\<alpha> m)"   and "I it \<sigma>" shows "I (it - {k}) (f (k, v) \<sigma>)"
QSPS: fixes \<sigma> :: "'a"   and \<sigma>' :: "'a"   and a :: "'c" assumes "QS \<sigma> \<sigma>' a" shows "PS \<sigma> \<sigma>' a"
integral_map_pmf: fixes g :: "'c \<Rightarrow> 'b"   and p :: "'c pmf"   and f :: "'b \<Rightarrow> 'a" shows "measure_pmf.expectation (map_pmf g p) f = measure_pmf.expectation p (\<lambda>x. f (g x))"
tr_rev_step_4: shows "\<forall>p. ps c p \<noteq> None"
pair_graphI_complete_bipartite: fixes m :: "nat"   and n :: "nat"   and G :: "'b pair_pre_digraph" assumes "K\<^bsub>m,n\<^esub> (with_proj G)" shows "pair_graph G"
p_residue_range: fixes m :: "nat"   and n :: "int" shows "p_residue m n \<in> {0..<p ^ m}"
swap_P2: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "f \<in> \<P>\<^sup>2" shows "(\<lambda>x y. f y x) \<in> \<P>\<^sup>2"
basic_cqe_conv1_1: fixes p :: "'a poly" shows "(\<exists>x. poly p x = (0::'a) \<and> poly 0 x \<noteq> (0::'a)) = False"
bmod_assoc_weak: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'c"   and x :: "'a" shows "\<otimes> \<rho> (\<otimes> \<rho> f g) h x = \<otimes> \<rho> f (\<otimes> \<rho> g h) x"
mem_convex_alt: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and u :: "real"   and v :: "real" assumes "convex S"   and "x \<in> S"   and "y \<in> S"   and "0 \<le> u"   and "0 \<le> v"   and "0 < u + v" shows "(u / (u + v)) *\<^sub>R x + (v / (u + v)) *\<^sub>R y \<in> S"
continuous_on_joinpaths: fixes g1 :: "real \<Rightarrow> 'b"   and g2 :: "real \<Rightarrow> 'b" assumes "continuous_on {0..1} g1"   and "continuous_on {0..1} g2"   and "pathfinish g1 = pathstart g2" shows "continuous_on {0..1} (g1 +++ g2)"
lemma2_6_2_a: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset" assumes "trans r"   and "M \<subseteq># N" shows "(M, N) \<in> mul_eq r"
assms: shows "adm (\<lambda>p. f (fst p) (snd p))"
upper_asymptotic_density_finite_Union: fixes I :: "'a set"   and A :: "'a \<Rightarrow> nat set" assumes "finite I" shows "upper_asymptotic_density (\<Union> (A ` I)) \<le> (\<Sum>i\<in>I. upper_asymptotic_density (A i))"
singular_chain_imp_relboundary: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p (subtopology X S) c" shows "singular_relboundary p X S c"
var_fresh_for_tyS: fixes x :: "W.var"   and S :: "tyS" shows "x \<sharp> S"
hoare_choice: fixes p :: "'b Assertion"   and x :: "'b"   and y :: "'b"   and q :: "'b Assertion" shows "hoare p (x \<sqinter> y) q = (hoare p x q \<and> hoare p y q)"
mat_default_ge_0: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and default :: "'b"   and n :: "nat" assumes "order_pair gt default" shows "default\<^sub>m default n \<ge>\<^sub>m 0\<^sub>m n n"
rep_eq: fixes x :: "('b, unit) RBT.rbt" shows "r_min x = rbt_min (rbt.impl_of x)"
smulextp_cong: fixes xs1 :: "'a multiset"   and ys1 :: "'a multiset"   and xs2 :: "'a multiset"   and ys2 :: "'a multiset"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool" assumes "xs1 = ys1"   and "xs2 = ys2"   and "\<And>x x'. \<lbrakk>x \<in># ys1; x' \<in># ys2\<rbrakk> \<Longrightarrow> f x x' = g x x'" shows "smulextp f xs1 xs2 = smulextp g ys1 ys2"
permutation_insert_row_step: fixes i :: "nat"   and j :: "nat"   and p :: "nat \<Rightarrow> nat" shows "permutation_insert (Suc i) j p \<circ> Fun.swap i (Suc i) id = permutation_insert i j p"
simps_4: fixes h1 :: "'b tree"   and h2 :: "'b tree" shows "U Merge [h1, h2] = log 2 (real (size h1 + size h2 + 1)) + 2"
eq: shows "mbuf2_take f (mbuf2_add xs ys buf) = (zs, buf')"
strict_order_def: fixes a :: "'a G"   and b :: "'a G" shows "a < b \<equiv> a \<le> b \<and> \<not> b \<le> a"
elims: fixes x :: "'b extended"   and xa :: "'b extended"   and y :: "'b extended" assumes "x + xa = y" obtains xb :: "'b" and ya :: "'b" where "x = Fin xb__" and "xa = Fin ya__" and "y = Fin (xb__ + ya__)"   | xb :: "'b" where "x = Fin xba__" and "xa = \<infinity>" and "y = \<infinity>"   | xb :: "'b" where "x = \<infinity>" and "xa = Fin xbb__" and "y = \<infinity>"   | "x = \<infinity>" and "xa = \<infinity>" and "y = \<infinity>"   | ya :: "'b" where "x = -\<infinity>" and "xa = Fin ya" and "y = -\<infinity>"   | xb :: "'b" where "x = Fin xb" and "xa = -\<infinity>" and "y = -\<infinity>"   | "x = -\<infinity>" and "xa = -\<infinity>" and "y = -\<infinity>"   | "x = -\<infinity>" and "xa = \<infinity>" and "y = \<infinity>"   | "x = \<infinity>" and "xa = -\<infinity>" and "y = \<infinity>"
width_spec_def: fixes X :: "'a" shows "width_spec X = SPEC top"
map: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "nat"   and x2 :: "(nat \<times> linear_poly) list"   and x3 :: "('b \<times> QDelta atom) list"   and x4 :: "linear_poly \<Rightarrow> nat option"   and x5 :: "'b list" shows "map_istate f (IState x1 x2 x3 x4 x5) = IState x1 x2 (map (map_prod f id) x3) x4 (map f x5)"
curryI: fixes f :: "'h \<times> 'i \<Rightarrow> bool"   and a :: "'h"   and b :: "'i" assumes "f (a, b)" shows "curry f a b"
sum_list_less_diff_Ex: fixes u :: "nat list"   and y :: "nat list" assumes "u \<le>\<^sub>v y"   and "sum_list u < sum_list y" shows "\<exists>i<length y. u ! i < y ! i"
eqc_listassoc_cong: fixes as :: "'a list"   and bs :: "'a list" assumes "as [\<sim>] bs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "map (assocs G) as = map (assocs G) bs"
r_pdec2_prim: shows "prim_recfn 1 r_pdec2"
is_nan_plus: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x \<or> IEEE.is_nan y" shows "IEEE.is_nan (x + y)"
Subject_reduction: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and T :: "ty" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"   and "P,E,s \<turnstile> e : T \<surd>" shows "P,E,hp s' \<turnstile> e' :\<^bsub>NT\<^esub> T"
f3: shows "\<forall>mo. Deliver (i, Store e2 mo i) \<notin> set (history k)"
mono_Always_o: fixes h :: "'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'b" assumes "mono h" shows "Always {s. f s \<le> g s} \<subseteq> Always {s. h (f s) \<le> h (g s)}"
InvariantGetReasonIsReasonAfterApplyBackjump: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "getConflictFlag state"   and "InvariantUniqC (getC state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)"   and "0 < currentLevel (getM state)"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"   and "0 < getBackjumpLevel state \<longrightarrow> getF state \<noteq> [] \<and> last (getF state) = getC state" shows "let state' = applyBackjump state in InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state'))"
1: shows "\<nu> \<star> \<tau> \<noteq> null"
restrict_star_right_induct: fixes ms :: "'c list"   and g :: "'c \<times> 'c \<Rightarrow> 'd"   and l :: "'c"   and f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "distinct ms"   and "ms \<langle>g\<rangle> [l] \<odot> [l] \<langle>f\<rangle> [l] \<preceq> ms \<langle>g\<rangle> [l]" shows "ms \<langle>g\<rangle> [l] \<odot> [l] \<langle>star \<circ> f\<rangle> [l] \<preceq> ms \<langle>g\<rangle> [l]"
psubset_imp_ex_mem: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subset> B" shows "\<exists>b. b \<in> B - A"
p_id_term: fixes x :: "'a" assumes "x \<cdot> 1\<^sub>\<pi> = x" shows "x \<le> 1\<^sub>\<pi>"
is_hom_GB_boundI: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and b :: "nat" assumes "\<And>g. \<lbrakk>\<And>f. f \<in> F \<Longrightarrow> homogeneous f; g \<in> punit.reduced_GB F\<rbrakk> \<Longrightarrow> poly_deg g \<le> b" shows "is_hom_GB_bound F b"
analyzed_in_def: fixes t :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term set" shows "analyzed_in t M \<equiv> \<forall>K R. Ana t = (K, R) \<and> (\<forall>k\<in>set K. M \<turnstile>\<^sub>c k) \<longrightarrow> (\<forall>r\<in>set R. M \<turnstile>\<^sub>c r)"
le_bottom: fixes x :: "'a" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> \<bottom>) = (x = \<bottom>)"
infinite_imp_nonempty: fixes S :: "'d set" assumes "infinite S" shows "S \<noteq> {}"
simps: fixes a :: "'a list"   and n :: "nat" shows "array_get (Array a) n = a ! n"
resumption_ord_chain_resume: fixes R :: "('f, 'g, 'h) Resumption.resumption set"   and inp :: "'h" assumes "Complete_Partial_Order.chain resumption_ord R" shows "Complete_Partial_Order.chain resumption_ord ((\<lambda>c. c inp) ` Resumption.resumption.resume ` (R \<inter> {r. \<not> Resumption.resumption.is_Done r}))"
adjoint_def: fixes f :: "'a set \<Rightarrow> 'a set" shows "adjoint f \<equiv> \<lambda>p. - f (- p)"
two_chain_integral_eq_integral_divisable: fixes twoChain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and F :: "real \<times> real \<Rightarrow> real"   and s :: "(real \<times> real) set" assumes "\<forall>twoCube\<in>twoChain. F integrable_on cubeImage twoCube"   and "gen_division s (cubeImage ` twoChain)"   and "valid_two_chain twoChain" shows "integral s F = two_chain_integral twoChain F"
dvd_mult_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a * b dvd c" shows "a dvd c"
assms_5: shows "h \<turnstile> get_elements_by_tag_name ptr idd \<rightarrow>\<^sub>h h'"
IH_2: fixes zs :: "'a list"   and n :: "nat"   and ys :: "'a list" assumes "(zs, n) \<in> set_pmf (rquicksort R ys)"   and "length ys < length xs"   and "set ys \<subseteq> set xs" shows "mset zs = mset ys"