hcju/isabelleps huggingface dataset:
vector_derivative_cong_eq: fixes A :: "real set"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and x :: "real"   and y :: "real"   and B :: "real set" assumes "\<forall>\<^sub>F x in nhds x. x \<in> A \<longrightarrow> f x = g x"   and "x = y"   and "A = B"   and "x \<in> A" shows "vector_derivative f (at x within A) = vector_derivative g (at y within B)"
assms_3: fixes e1 :: "pexpr"   and e2 :: "pexpr" assumes "e = PExpr1 (PAdd e1 e2)" shows "P"
independent_injective_image: fixes s :: "'b set" assumes "vs1.independent s"   and "inj_on f (vs1.span s)" shows "vs2.independent (f ` s)"
prems_7: shows "fmrel_on_fset (frees (name $$ ts)) vrelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2"
pos_decomp_append: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "(ps @ qs)\<^sub>+ = ps\<^sub>+ @ qs\<^sub>+"
r_cancel_one': fixes x :: "'c \<Rightarrow> 'a"   and a :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "a \<in> carrier F" shows "(x = a \<oplus>\<^bsub>F\<^esub> x) = (a = \<zero>\<^bsub>F\<^esub>)"
nat_ex_greater_finite_Max_conv: fixes A :: "nat set"   and n :: "nat" assumes "A \<noteq> {}" shows "(\<exists>x\<in>A. n < x) = (finite A \<longrightarrow> n < Max A)"
all_final_except_def: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and Ts :: "'b set" shows "all_final_except s Ts \<equiv> \<forall>t. not_final_thread s t \<longrightarrow> t \<in> Ts"
to_nat_from_nat_id: fixes x :: "nat" assumes "x < CARD('b)" shows "to_nat (from_nat x) = x"
return_ok: fixes h :: "'a"   and x :: "'c" shows "h \<turnstile> ok return x"
induct: fixes P :: "('a, 'b) Combinators \<Rightarrow> bool"   and Combinators :: "('a, 'b) Combinators" assumes "P DenyAll"   and "\<And>x1 x2. P (DenyAllFromTo x1 x2)"   and "\<And>x1 x2 x3. P (AllowPortFromTo x1 x2 x3)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 \<oplus> x2)" shows "P Combinators"
is_bintree_list_merge: fixes l :: "nat"   and t1 :: "('c, 'd) bintree"   and t2 :: "('c, 'd) bintree" assumes "is_bintree_list l (children t1)"   and "is_bintree_list l (children t2)" shows "is_bintree_list (Suc l) (children (merge t1 t2))"
mon_c_add_mset_unconc: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and x :: "'b list"   and b :: "'b list multiset" shows "mon_c fg (add_mset x b) = mon_s fg x \<union> mon_c fg b"
psubst_subst: fixes c :: "name"   and \<theta> :: "(name \<times> lam) list"   and s :: "lam"   and t :: "lam" assumes "c \<sharp> \<theta>" shows "\<theta><t>[c::=s] = ((c, s) # \<theta>)<t>"
prefix_fininf_eq_inf_transp: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>2 =\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
summedBidVectorEquivalence: fixes x :: "'c \<times> 'd set"   and N :: "'c set"   and \<Omega> :: "'d set"   and a :: "('c \<times> 'd set) set" assumes "x \<in> N \<times> (Pow \<Omega> - {{}})" shows "summedBidVector (pseudoAllocation a <| (N \<times> finestpart \<Omega>)) N \<Omega> x = summedBidVectorAlg (maxbidAlg a N \<Omega>) N \<Omega> x"
inv1_inv1: shows "inv1 \<circ> inv1 = id"
ideal_npow_closed: fixes I :: "'a set"   and x :: "'a"   and n :: "nat" assumes "ideal R I"   and "x \<in> I"   and "0 < n" shows "x^\<^bsup>R n\<^esup> \<in> I"
bij_betw_ide_Obj_3: fixes A :: "bool" assumes "A \<in> {False, True}" shows "Dom (MkIde A) = A"
MM3: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and B :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and C :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>\<not> A \<longrightarrow> \<box>(A \<and> B \<longrightarrow> C)"
finer_than_def: fixes T1 :: "'c topology"   and T2 :: "'c topology" shows "(T1 finer_than T2) = continuous_map T1 T2 (\<lambda>x. x)"
right_zero: fixes a :: "'a set" assumes "a \<in> additive.Partition" shows "a [\<cdot>] additive.Class \<zero> = additive.Class \<zero>"
lunit_closed: fixes f :: "'a" assumes "Arr f"   and "B.ide f" shows "Arr (B.\<ll> f)"
Cons_eq: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "\<not> x < y"   and "\<not> y < x"   and "ord_class.lexordp xs ys" shows "ord_class.lexordp (x # xs) (y # ys)"
incrIndexList_less_eq: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat"   and n :: "nat" assumes "incrIndexList ls m nmax"   and "Suc n < |ls|" shows "ls ! n \<le> ls ! Suc n"
mult_vv_scale_sv1: fixes k :: "complex"   and v1 :: "complex \<times> complex"   and v2 :: "complex \<times> complex" shows "k *\<^sub>s\<^sub>v v1 *\<^sub>v\<^sub>v v2 = k * v1 *\<^sub>v\<^sub>v v2"
RSpan_lincomb_closed: fixes rs :: "'r list"   and ms :: "'m list"   and ns :: "'m list" assumes "set rs \<subseteq> R"   and "set ms \<subseteq> M"   and "set ns \<subseteq> RSpan ms" shows "rs \<bullet>\<cdot> ns \<in> RSpan ms"
transitive_top_closed: shows "top \<sqinter> top \<le> top"
map_\<I>_id: fixes \<I> :: "('c, 'd) \<I>" shows "map_\<I> id id \<I> = \<I>"
C_eq_RS1n: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" shows "Cp (list2FWpolicy (removeShadowRules1_alternative p)) = Cp (list2FWpolicy p)"
dual_iff: fixes x :: "'b"   and y :: "'b" shows "(- x = y) = (x = - y)"
the_elem_simp: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and x :: "'e" assumes "ring_hom_ring R S h"   and "x \<in> carrier R" shows "the_elem (h ` (a_kernel R S h +>\<^bsub>R\<^esub> x)) = h x"
has_partial_deriv: shows "\<forall>p\<in>D. has_partial_vector_derivative F i (F_i p) p"
pair_fin_digraph_mk_symmetric: fixes a :: "'a \<times> nat \<times> 'a" shows "pair_fin_digraph (mk_symmetric (add_arc a))"
E28: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>F \<and> \<box>\<diamond>G \<longrightarrow> \<box>\<diamond>(F \<and> G)"
DerivMono: fixes X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and B :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "(X, c, B) \<in> Deriv" shows "\<exists>A R S \<phi>. B = Sec A R S (\<phi> (FIX \<phi>)) \<and> Monotone \<phi>"
F_def: shows "F = {g. current \<Gamma> g \<and> wave \<Gamma> g \<and> g \<le> f \<and> \<E> (TER g) = \<E> (TER f)}"
prod_smult: fixes c :: "'b \<Rightarrow> 'a"   and p :: "'b \<Rightarrow> 'a poly"   and A :: "'b set" shows "(\<Prod>x\<in>A. Polynomial.smult (c x) (p x)) = Polynomial.smult (prod c A) (prod p A)"
assms_2: shows "Q \<sqsubseteq> P ;; Q"
elist_var_inj_term: fixes t :: "expr list"   and w :: "Term.var" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
continuous_on_closed: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on S f = (\<forall>T. closedin (top_of_set (f ` S)) T \<longrightarrow> closedin (top_of_set S) (S \<inter> f -` T))"
naive_algorithm_def: fixes s :: "'b list"   and t :: "'b list" shows "naive_algorithm s t \<equiv> let i = 0; j = 0; pos = None in WHILE\<^sub>T\<^bsup>I_out_na s t\<^esup> (\<lambda>(i, uu_, pos). i + length s \<le> length t \<and> pos = None) (\<lambda>(i, j, pos). WHILE\<^sub>T\<^bsup>I_in_na s t i\<^esup> (\<lambda>(j, pos). t ! (i + j) = s ! j \<and> pos = None) (\<lambda>(j, uu_). let j = j + 1 in if j = length s then RETURN (j, Some i) else RETURN (j, None)) (j, pos) \<bind> (\<lambda>(uu_, pos). if pos = None then let i = i + 1; j = 0 in RETURN (i, j, None) else RETURN (i, j, Some i))) (i, j, pos) \<bind> (\<lambda>(uu_, uu_, pos). RETURN pos)"
eval_cases_8: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and V :: "char list"   and T :: "ty"   and e\<^sub>1 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>{V:T; e\<^sub>1},s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" and l\<^sub>0 :: "char list \<Rightarrow> val option" and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" and l\<^sub>1 :: "char list \<Rightarrow> val option" where "s = (h\<^sub>0, l\<^sub>0)" and "s' = (h\<^sub>1, l\<^sub>1(V := l\<^sub>0 V))" and "P \<turnstile> \<langle>e\<^sub>1,(h\<^sub>0, l\<^sub>0(V := None))\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>1, l\<^sub>1)\<rangle>"
simple_distributed_def: fixes M :: "'d measure"   and X :: "'d \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> real" shows "simple_distributed M X f = ((\<forall>x. 0 \<le> f x) \<and> distributed M (count_space (X ` space M)) X (\<lambda>x. ennreal (f x)) \<and> finite (X ` space M))"
semigroup_add_ow_Ball_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semigroup_add_ow S pls = ((\<forall>a\<in>S. \<forall>b\<in>S. \<forall>c\<in>S. pls (pls a b) c = pls a (pls b c)) \<and> (\<forall>a\<in>S. \<forall>b\<in>S. pls a b \<in> S))"
rbt_greater_prop: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and t :: "('b, 'c) RBT_Impl.rbt" shows "ord.rbt_greater less k t = (\<forall>x\<in>set (RBT_Impl.keys t). less k x)"
fmimage_dom: fixes m :: "('d, 'c) fmap" shows "fmimage m (fmdom m) = fmran m"
dg_Dom_app_in_Obj: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<AA> i\<lparr>Arr\<rparr>" shows "\<AA> i\<lparr>Dom\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Obj\<rparr>"
reset_trap_val_def: fixes state :: "'b sparc_state" shows "reset_trap_val state \<equiv> get_reset_trap (state_var state)"
distinct_15: fixes x2 :: "nat"   and x4 :: "nat" shows "Number x2 \<noteq> Key x4"
h_def: shows "h \<equiv> \<lambda>x. if x = 0 then 0 else 1"
differentiable_at_polynomial_function: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and S :: "'a set" assumes "polynomial_function f" shows "f differentiable at a within S"
Ass: fixes v :: "Term.var"   and Env :: "env"   and B :: "lname set"   and V :: "assigned"   and e :: "expr"   and A :: "assigned" assumes "\<forall>vn. v \<noteq> LVar vn"   and "Env\<turnstile> B \<guillemotright>\<langle>v\<rangle>\<guillemotright> V"   and "Env\<turnstile> nrm V \<guillemotright>\<langle>e\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>v:=e\<rangle>\<guillemotright> A"
member_class_to_carrier: fixes x :: "'a \<times> 'a"   and r :: "'a"   and s :: "'a"   and y :: "'a \<times> 'a"   and r' :: "'a"   and s' :: "'a" assumes "x \<in> (r |\<^bsub>rel\<^esub> s)"   and "y \<in> (r' |\<^bsub>rel\<^esub> s')" shows "(fst x \<otimes> fst y, snd x \<otimes> snd y) \<in> carrier rel"
sup_commute: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type) = sup y x"
xor_one_left: fixes x :: "'a" shows "\<one> \<oplus> x = \<sim> x"
bounded_lattice_ow_def: fixes U :: "'c set"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ls :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" shows "bounded_lattice_ow U inf le ls sup bot top \<equiv> lattice_ow U inf le ls sup \<and> order_bot_ow U bot le ls \<and> order_top_ow U le ls top"
l2_inv7_init: shows "init l2 \<subseteq> l2_inv7"
os__coplanar: fixes A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "A B OS C D" shows "Coplanar A B C D"
i_set_mult_mod_0_div_i_set: fixes I :: "nat set"   and k :: "nat"   and d :: "nat" assumes "I \<in> i_set_mult k"   and "k mod d = 0" shows "I \<oslash> d \<in> i_set"
\<alpha>_gen_lookup: fixes A :: "'s"   and bound :: "natural"   and n :: "nat"   and hs :: "natural \<times> natural \<times> natural" assumes "invar A" shows "local.gen_lookup bound A n hs = Benchmark_Comparison.gen_lookup bound (\<alpha> A) n hs"
inter_guards_Seq: fixes a1 :: "('d, 'e, 'f) com"   and a2 :: "('d, 'e, 'f) com"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((Seq a1 a2 \<inter>\<^sub>g c2) = Some c) = (\<exists>b1 b2 d1 d2. c2 = Seq b1 b2 \<and> (a1 \<inter>\<^sub>g b1) = Some d1 \<and> (a2 \<inter>\<^sub>g b2) = Some d2 \<and> c = Seq d1 d2)"
CallRec': fixes p :: "'d"   and Procs :: "'d set"   and \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and r :: "(('e \<times> 'd) \<times> 'e \<times> 'd) set"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'d \<Rightarrow> 'g \<Rightarrow> 'e set"   and Q :: "'d \<Rightarrow> 'g \<Rightarrow> 'e set"   and A :: "'d \<Rightarrow> 'g \<Rightarrow> 'e set"   and F :: "'f set"   and Z :: "'g" assumes "p \<in> Procs"   and "Procs \<subseteq> dom \<Gamma>"   and "wf r"   and "\<forall>p\<in>Procs. \<forall>\<tau> Z. \<Gamma>,\<Theta> \<union> (\<Union>q\<in>Procs. \<Union>Z. {(P q Z \<inter> {s. ((s, q), \<tau>, p) \<in> r}, q, Q q Z, A q Z)})\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({\<tau>} \<inter> P p Z) the (\<Gamma> p) (Q p Z),(A p Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P p Z) Call p (Q p Z),(A p Z)"
dims_Ts': fixes T :: "real tensor" assumes "T \<in> set (map (\<lambda>j. A $$ (i, j) \<cdot> Ts $ j) [0..<dim_vec Ts])" shows "dims T = ds"
\<chi>\<sigma>z': shows "subst \<chi>\<sigma> (Var z') inp = exi zz (cnj (subst (subst \<sigma> (Var zz) out) (Var z') inp) (subst \<chi> (Var zz) inp))"
multcol_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and k :: "nat"   and a :: "'b" assumes "A \<in> carrier_mat nr n" shows "multcol k a A = A * multrow_mat n k a"
rel_frontier_def: fixes S :: "'c set" shows "rel_frontier S = closure S - rel_interior S"
collision_excluded: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and T :: "real set"   and S :: "real set" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"   and "\<And>t. t \<in> T \<Longrightarrow> movement.s a\<^sub>e v\<^sub>e s\<^sub>e t \<noteq> movement.s a\<^sub>o v\<^sub>o s\<^sub>o t" shows "safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o S = safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o (S - T)"
sum_upto_altdef: fixes f :: "nat \<Rightarrow> 'b"   and x :: "real" shows "sum_upto f x = sum f {0<..nat \<lfloor>x\<rfloor>}"
ctx_equiv_def: fixes e :: "exp"   and e' :: "exp" shows "e \<simeq> e' \<equiv> \<forall>C ob. FV (plug C e) = {} \<and> FV (plug C e') = {} \<longrightarrow> plug C e \<Down> ob = plug C e' \<Down> ob"
lookup_empty_empty: fixes t :: "('b, 'c) mapping_rbt" assumes "ID ccompare \<noteq> None" shows "(RBT_Mapping2.lookup t = Map.empty) = (t = RBT_Mapping2.empty)"
cases: fixes x :: "tm" obtains z :: "name" and r :: "tm" and z' :: "name" and x :: "name" and y :: "name" and x' :: "name" and y' :: "name" where "atom z \<sharp> (r, z', x, y, x', y')" and "atom z' \<sharp> (r, x, y, x', y')" and "atom x \<sharp> (r, y, x', y')" and "atom y \<sharp> (r, x', y')" and "atom x' \<sharp> (r, y')" and "atom y' \<sharp> r" and "xa__ = r"
dist_commute_lessI: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and y :: "'d"   and x :: "'d"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist y x < e" shows "dist x y < e"
naturality: fixes f :: "'d" assumes "D.arr f" shows "\<tau>'.map (D.cod f) \<cdot>\<^sub>C F f = F' f \<cdot>\<^sub>C \<tau>'.map (D.dom f)"
hom_minus: fixes x :: "int"   and y :: "int" shows "of_int (x - y) = of_int x - of_int y"
swapEnv_sym: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" shows "rho &[x \<and> y]_zs = rho &[y \<and> x]_zs"
sum_not_inf_dest: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "a + b \<noteq> \<infinity>" shows "a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
approx_slp_outer_lengthD: fixes p :: "nat"   and d :: "nat"   and slp :: "floatarith list"   and a :: "(real \<times> real pdevs) list"   and xs :: "(real \<times> real pdevs) list" assumes "approx_slp_outer p d slp a = Some xs" shows "length xs = min d (length slp + length a)"
simps_14: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case T_i64 of T_i32 \<Rightarrow> f1 | T_i64 \<Rightarrow> f2 | T_f32 \<Rightarrow> f3 | T_f64 \<Rightarrow> f4) = f2"
three_neq_5: assumes "three TYPE('d)" shows "three.three\<^sub>3 \<noteq> three.three\<^sub>1"
2: shows "{s. timpls_transformable_to' TI t s} \<subseteq> {s. timpls_transformable_to_pred (the_Abs ` {f \<in> funs_term t. is_Abs f} \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) t s}"
3: shows "(v \<squnion> e\<^sup>T * top)\<^sup>T \<le> r\<^sup>T * (t \<squnion> e)\<^sup>\<star>"
fps_to_fls_uminus: fixes f :: "'a fps" shows "fps_to_fls (- f) = - fps_to_fls f"
cont: fixes i :: "'b" assumes "i \<in> I" shows "continuous_map (subtopology X (T i)) Y (f i)"
gram_schmidt0_result_4: fixes ws :: "complex vec list" assumes "set ws \<subseteq> carrier_vec d" shows "local.span (set ws) = local.span (set (gram_schmidt0 d ws))"
hyps: fixes n :: "int"   and m :: "int" assumes "length\<cdot>xs = MkI\<cdot>n"   and "n \<le> m" shows "(xs ++ ys) !! MkI\<cdot>m = ys !! MkI\<cdot>(m - n)"
eqvt: fixes x1 :: "(name \<times> ty) list"   and x2 :: "trm"   and x3 :: "ty"   and pi :: "(name \<times> name) list" assumes "x1 \<turnstile> x2 : x3" shows "pi \<bullet> x1 \<turnstile> pi \<bullet> x2 : pi \<bullet> x3"
enumerator_increase: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat"   and as :: "nat list"   and bs :: "nat list" assumes "ls \<in> set (enumerator m nmax)"   and "as @ bs = ls" shows "\<forall>x\<in>set as. \<forall>y\<in>set bs. x \<le> y"
base_enabled: fixes vs :: "state \<Rightarrow> 'b"   and c :: "'b"   and F :: "(nat \<Rightarrow> state) \<Rightarrow> bool"   and s :: "nat \<Rightarrow> state" assumes "basevars vs"   and "\<And>u. vs (first u) = c \<Longrightarrow> F (first s ## u)" shows "(Enabled F) s"
Unit_eq_dividesone: fixes u :: "'a" assumes "u \<in> carrier R" shows "(u \<in> Units R) = u divides \<one>"
add_mset_lt_lt_lt: fixes a :: "'b"   and b :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" assumes "a < b"   and "A < B" shows "add_mset a A < add_mset b B"
inj_on_f'': fixes H :: "'a set"   and a :: "'a" assumes "H \<subseteq> carrier G"   and "a \<in> carrier G" shows "inj_on ((\<otimes>) (inv a)) (a <# H)"
abs_eq: fixes x :: "nat" shows "inc_Arity (to_Arity x) = to_Arity (Suc x)"
zero_less_hSuc: fixes n :: "nat star" shows "0 < hSuc n"
assms_5: fixes a :: "'a"   and xs :: "'a fset" shows "P (insert_fset a (insert_fset a xs)) (insert_fset a xs)"
ss_rewrite2: fixes h :: "'a \<times> 'a"   and a :: "'a"   and b :: "'a"   and e :: "nat"   and z :: "nat"   and e' :: "nat"   and z' :: "nat" assumes "fst h \<in> carrier \<G>"   and "snd h \<in> carrier \<G>"   and "a \<in> carrier \<G>"   and "b \<in> carrier \<G>"   and "e < order \<G>"   and "a \<otimes> fst h [^] e = \<^bold>g [^] z"   and "b \<otimes> snd h [^] e = g' [^] z"   and "e' < e"   and "a \<otimes> fst h [^] e' = \<^bold>g [^] z'"   and "b \<otimes> snd h [^] e' = g' [^] z'" shows "snd h = g' [^] ((int z - int z') * Number_Theory_Aux.inverse (e - e') (order \<G>) mod int (order \<G>))"
var_def: shows "var = inv_image (brk_rel (finite_psupset (E\<^sup>* `` U0) <*lex*> measure (card \<circ> dom))) (\<lambda>(res, V, ws). (res \<noteq> None, V, ws))"
progress_sat_cong: fixes \<pi> :: "'a prefix"   and \<sigma> :: "'a trace"   and \<sigma>' :: "'a trace"   and i :: "nat"   and v :: "'b list" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi> \<sigma>'"   and "i < progress \<pi>" shows "sat \<sigma> v i = sat \<sigma>' v i"
coreflexive_conv_closed: fixes x :: "'a" assumes "x \<le> top" shows "id x \<le> top"
filterlim_If: fixes f :: "'a \<Rightarrow> 'b"   and G :: "'b filter"   and F :: "'a filter"   and P :: "'a \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'b" assumes "filterlim f G (inf F (principal {x. P x}))"   and "filterlim g G (inf F (principal {x. \<not> P x}))" shows "LIM x F. if P x then f x else g x :> G"
uint_nonnegative: fixes w :: "'b word" shows "0 \<le> uint w"
deterministic_def: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set" shows "deterministic I = ((\<forall>s t x ta' x' m' ta'' x'' m''. s \<in> I \<longrightarrow> thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<longrightarrow> t \<turnstile> \<langle>x, shr s\<rangle> -ta'\<rightarrow> \<langle>x', m'\<rangle> \<longrightarrow> t \<turnstile> \<langle>x, shr s\<rangle> -ta''\<rightarrow> \<langle>x'', m''\<rangle> \<longrightarrow> actions_ok s t ta' \<longrightarrow> actions_ok s t ta'' \<longrightarrow> ta' = ta'' \<and> x' = x'' \<and> m' = m'') \<and> invariant3p redT I)"
arr_cod_iff_arr: fixes f :: "'b" shows "\<phi>.B.arr (\<phi>.B.cod f) = \<phi>.B.arr f"
v_neq_0_imp: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and cfg :: "'a cfg" assumes "Reachability_Problem K S S1 S2"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S"   and "Reachability_Problem.v K S S1 S2 cfg \<noteq> 0" shows "state cfg \<in> Reachability_Problem.S\<^sub>r K S S1 S2 \<union> S2"
abs_complex_def: shows "abs = complex_of_real \<circ> cmod"
obtain_qFreshEnv: fixes V :: "'var set"   and XS :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and Rho :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm option) set"   and xs :: "'varSort" assumes "|V| <o |UNIV| \<or> finite V"   and "|XS| <o |UNIV| \<or> finite XS"   and "Ball XS qGood"   and "|Rho| <o |UNIV| \<or> finite Rho"   and "Ball Rho qGoodEnv" shows "\<exists>z. z \<notin> V \<and> (\<forall>X\<in>XS. qFresh xs z X) \<and> (\<forall>rho\<in>Rho. qFreshEnv xs z rho)"
scale_scale_spmf: fixes r :: "real"   and r' :: "real"   and p :: "'c spmf" shows "scale_spmf r (scale_spmf r' p) = scale_spmf (r * max 0 (min (inverse (weight_spmf p)) r')) p"
rtl_inter_closed: fixes L1 :: "'a tree set"   and A :: "'a \<Rightarrow> nat option"   and L2 :: "'a tree set" assumes "L1 \<in> regular_languages A"   and "L2 \<in> regular_languages A" shows "L1 \<inter> L2 \<in> regular_languages A"
fw_upd'_def: fixes m :: "nat \<times> nat \<Rightarrow> 'b"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "fw_upd' m k i j = RETURN (op_mtx_set m (i, j) (min (op_mtx_get m (i, j)) (op_mtx_get m (i, k) + op_mtx_get m (k, j))))"
satisfies_tableau_def: fixes v :: "nat \<Rightarrow> 'a"   and t :: "(nat \<times> linear_poly) list" shows "v \<Turnstile>\<^sub>t t \<equiv> \<forall>e\<in>set t. v \<Turnstile>\<^sub>e e"
less_card: shows "mod_type_class.to_nat a + 1 < CARD('a)"