hcju/isabelleps huggingface dataset:
digraph_iso_def: fixes G :: "('e, 'f) pre_digraph"   and H :: "('g, 'h) pre_digraph" shows "digraph_iso G H \<equiv> \<exists>f. pre_digraph.digraph_isomorphism G f \<and> H = app_iso f G"
simps_13: fixes V :: "char list"   and T :: "ty"   and e :: "char list exp" shows "fv {V:T; e} = fv e - {V}"
openin_open_Int': fixes S :: "'c set"   and U :: "'c set" assumes "open S" shows "openin (top_of_set U) (S \<inter> U)"
elims_2: fixes x :: "('a, 'b) Term.term"   and xa :: "('a, 'b) Term.term" assumes "subterm_of x xa" obtains t :: "('a, 'b) Term.term" and y :: "'b" where "x = t__" and "xa = Var y" and "t__ = Var y"   | t :: "('a, 'b) Term.term" and f :: "'a" and T :: "('a, 'b) Term.term list" where "x = t" and "xa = Fun f T" and "t = Fun f T \<or> list_ex (subterm_of t) T"
pred_entailment_un: fixes P :: "'BExp set"   and P' :: "'BExp set"   and P'' :: "'BExp set" assumes "P \<turnstile> P'"   and "P \<turnstile> P''" shows "P \<turnstile> (P' \<union> P'')"
Infinitesimal_HFinite_mult: fixes x :: "'b star"   and y :: "'b star" assumes "x \<in> Infinitesimal"   and "y \<in> HFinite" shows "x * y \<in> Infinitesimal"
K_elem_abs_inj: fixes \<mu> :: "('s \<times> ('c \<Rightarrow> real)) pmf"   and l :: "'s"   and u :: "'c \<Rightarrow> real" assumes "\<mu> \<in> K (l, u)" shows "inj_on (\<lambda>(l, u). (l, [u]\<^sub>\<R>)) (set_pmf \<mu>)"
raise_if_SomeD: fixes c :: "bool"   and x :: "xname"   and y :: "abopt"   and z :: "abrupt" assumes "raise_if c x y = Some z" shows "c \<and> z = Xcpt (Std x) \<and> y = None \<or> y = Some z"
set_child_nodes_locs_def: shows "Core_DOM_Functions.set_child_nodes_locs \<equiv> l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_child_nodes_locs"
in_radicals_smaller_order: fixes s :: "expr"   and r :: "expr" assumes "s \<in> radicals r" shows "order s < order r"
either_above_arg1: fixes t :: "'a ttree"   and t' :: "'a ttree" shows "t \<sqsubseteq> t \<oplus>\<oplus> t'"
le_trans_equiv_trans: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "le_trans t u"   and "equiv_trans u v" shows "le_trans t v"
substAll_terminates: fixes g :: "'g" shows "while_option (uninst_code.cond (u_g g s1_uses) (p_g g s1_phis) g) (uninst.step ((usesOf \<circ>\<circ>\<circ> u_g) g s1_uses) (\<lambda>ga. Mapping.lookup (p_g g s1_phis ga)) g) ((usesOf \<circ>\<circ>\<circ> u_g) g s1_uses g, Mapping.lookup (p_g g s1_phis g)) \<noteq> None"
is_shadow_root_ptr_kind_obtains: fixes ptr :: "(_) object_ptr" assumes "is_shadow_root_ptr_kind ptr" obtains shadow_root_ptr :: "(_) shadow_root_ptr" where "ptr = cast shadow_root_ptr"
inject_3: fixes x2 :: "name"   and x1 :: "ctx"   and y2 :: "name"   and y1 :: "ctx" shows "(CLam [x2].x1 = CLam [y2].y1) = (x2 = y2 \<and> x1 = y1)"
xor_assoc: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "boolean_algebra.xor conj disj compl (boolean_algebra.xor conj disj compl a b) c = boolean_algebra.xor conj disj compl a (boolean_algebra.xor conj disj compl b c)"
emb_prepend_coinduct: fixes X :: "'b llist \<Rightarrow> 'b llist \<Rightarrow> bool"   and x1 :: "'b llist"   and x2 :: "'b llist" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> (\<exists>xs. x1 = LNil \<and> x2 = xs \<and> lfinite xs) \<or> (\<exists>xs ys x zs. x1 = LCons x xs \<and> x2 = prepend zs (LCons x ys) \<and> (prepend_cong1 (X xs) ys \<or> emb xs ys))" shows "emb x1 x2"
multi_union_self_other_eq: fixes A :: "'a multiset"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "A + X = A + Y" shows "X = Y"
mult_pCons_right: fixes p :: "'a poly"   and a :: "'a"   and q :: "'a poly" shows "p * pCons a q = Polynomial.smult a p + pCons (0::'a) (p * q)"
aux3: fixes ys :: "('c \<times> 'd) list"   and a :: "'c" assumes "distinct (map fst ys)"   and "a \<notin> fst ` set ys" shows "map_of ys a = None"
distinct_5: fixes x1 :: "'a" shows "Normal x1 \<noteq> Stuck"
cscc_invar_def: shows "cscc_invar \<equiv> \<lambda>v0 D0 (l, p, D, pE). cscc_invar_loc G v0 D0 l p D pE"
simps_4: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b msllist \<Rightarrow> 'a" shows "(case MSLNil of MSLNil \<Rightarrow> f1 | MSLCons x xa \<Rightarrow> f2 x xa) = f1"
Zero_fract_def: shows "0 = Fract (0::'b) (1::'b)"
Px: shows "x mod n < n \<and> [a * (x mod n) = b] (mod n)"
assms: shows "nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> (\<exists>s'. (x, s') \<in> R \<and> inres S' s'))"
uminus_ratfps_code: fixes x :: "'a ratfps" shows "local.quot_of_ratfps (- x) = (let (a, b) = local.quot_of_ratfps x in (- a, b))"
power2_abs: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "power.power one times (abs a) 2 = power.power one times a 2"
or_dfa_accepts: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and N :: "nat bdd list \<times> bool list"   and bs :: "bool list list" assumes "wf_dfa M n"   and "wf_dfa N n"   and "list_all (is_alph n) bs" shows "dfa_accepts (or_dfa M N) bs = (dfa_accepts M bs \<or> dfa_accepts N bs)"
map_upds_cong_ext: fixes tab :: "'b \<Rightarrow> 'a option"   and z :: "'b"   and tab' :: "'b \<Rightarrow> 'a option"   and xs :: "'b list"   and ys :: "'a list" assumes "tab z = tab' z" shows "(tab(xs [\<mapsto>] ys)) z = (tab'(xs [\<mapsto>] ys)) z"
below_trans: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.po below"   and "below x y"   and "below y z" shows "below x z"
a_consistent_app\<^sub>2: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, Lam [y]. e, Arg x # S)" shows "ArityAnalysisLetSafe.a_consistent Aexp (ae, pred\<cdot>a, as) (\<Gamma>, e[y::=x], S)"
e: fixes z :: "'a" assumes "z \<in> X" shows "dist x z \<le> e"
is_extensional: fixes f :: "'a1" assumes "\<not> A1.arr f" shows "\<tau> f = A2_B.null"
elims: fixes x :: "('b \<Rightarrow> 'c) \<times> ('b \<times> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'c)"   and y :: "('b \<Rightarrow> 'c pmf) \<times> ('b \<times> 'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) pmf)" assumes "On_Off.embed x = y" obtains A :: "('b \<Rightarrow> 'c) \<times> ('b \<times> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'c)" where "x = A" and "y = (\<lambda>s. return_pmf (fst A s), \<lambda>s r. return_pmf (snd A s r))"
dvd_gcdD1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd gcd b c" shows "a dvd b"
rep_eq: fixes x :: "\<o>" shows "eval\<o> (\<^bold>\<A>x) = (\<lambda>s w. eval\<o> x s dw)"
power_is_relpow: fixes X :: "('a \<times> 'a) set"   and i :: "nat" shows "rel_d.power X i = X ^^ i"
partner_runs_def: fixes R :: "fid_t"   and R' :: "fid_t" shows "partner_runs R R' \<equiv> role (guessed_runs R) = role_comp (role (guessed_runs R')) \<and> owner (guessed_runs R) = partner (guessed_runs R') \<and> owner (guessed_runs R') = partner (guessed_runs R) \<and> matching (sklvl1.guessed_frame R) (sklvl1.guessed_frame R')"
infinite_rev_existence_ivl0_rewrites_2: fixes x :: "'a" shows "({..0} \<subseteq> existence_ivl0 x) = ({0..} \<subseteq> rev.existence_ivl0 x)"
interpret_Floatreal: fixes f :: "float"   and vs :: "real list" shows "interpret_floatarith (Num f) vs = real_of_float f"
distinct_37: shows "AtOp \<noteq> string_binop.LessOp"
intro: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set" assumes "\<forall>m. Preorder (Pm m) \<and> Total (Pm m)"   and "\<forall>w. Preorder (Pw w) \<and> Total (Pw w)" shows "StableMarriage Pm Pw"
partition_onD4: fixes A :: "'a set"   and P :: "'a set set"   and p :: "'a set"   and q :: "'a set"   and x :: "'a" assumes "partition_on A P"   and "p \<in> P"   and "q \<in> P"   and "x \<in> p"   and "x \<in> q" shows "p = q"
distinct_vertices: shows "distinct [vertex_1, vertex_2, vertex_3]"
unique_f: shows "transforms_to_unique_keys (m1.\<alpha> m) f"
SUP_le_iff: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and u :: "'b" shows "(\<Squnion> (f ` A) \<le> u) = (\<forall>i\<in>A. f i \<le> u)"
cases: fixes a1 :: "'b"   and a2 :: "('a, 'b, 'c) psi"   and a3 :: "('a, 'b, 'c) psi" assumes "a1 \<rhd> a2 \<sim> a3" obtains P :: "('a, 'b, 'c) psi" and \<Psi> :: "'b" and Q :: "('a, 'b, 'c) psi" where "a1 = \<Psi>" and "a2 = P" and "a3 = Q" and "insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi>" and "\<Psi> \<rhd> P \<leadsto>[bisim] Q" and "\<forall>\<Psi>''. \<Psi> \<otimes> \<Psi>'' \<rhd> P \<sim> Q" and "\<Psi> \<rhd> Q \<sim> P"
sum_pos_lt_pair: fixes f :: "nat \<Rightarrow> real"   and k :: "nat" assumes "summable f"   and "\<And>d. 0 < f (k + Suc (Suc 0) * d) + f (k + (Suc (Suc 0) * d + 1))" shows "sum f {..<k} < suminf f"
functor_axioms_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" shows "functor_axioms A B F \<equiv> ((\<forall>f. \<not> partial_magma.arr A f \<longrightarrow> F f = partial_magma.null B) \<and> (\<forall>f. partial_magma.arr A f \<longrightarrow> partial_magma.arr B (F f))) \<and> (\<forall>f. partial_magma.arr A f \<longrightarrow> partial_magma.dom B (F f) = F (partial_magma.dom A f)) \<and> (\<forall>f. partial_magma.arr A f \<longrightarrow> partial_magma.cod B (F f) = F (partial_magma.cod A f)) \<and> (\<forall>g f. partial_magma.arr A (A g f) \<longrightarrow> F (A g f) = B (F g) (F f))"
start_of_less: fixes e :: "nat"   and as :: "nat"   and ly :: "nat list" assumes "e < as" shows "start_of ly e \<le> start_of ly as"
convex_lincomb_def: fixes c :: "'a vec \<Rightarrow> 'a"   and Vs :: "'a vec set"   and b :: "'a vec" shows "convex_lincomb c Vs b = (nonneg_lincomb c Vs b \<and> sum c Vs = (1::'a))"
ennexp_mono2: fixes x :: "ereal"   and y :: "ereal" assumes "x \<le> y" shows "ennexp x \<le> ennexp y"
inj_ring_hom_axioms: shows "inj_ring_hom (map_poly hom)"
prems: fixes x :: "'a" assumes "x \<in> insert x__ A" shows "0 \<le> f x"
generate_valid_stateful_policy_IFSACS_select_simps_3: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and edgesList :: "('b \<times> 'b) list" shows "flows_state (generate_valid_stateful_policy_IFSACS G M edgesList) \<subseteq> set edgesList"
consis_m: shows "reads_consistent pending_write \<O> m (r # sb)"
b: fixes b :: "'a" assumes "b \<in> CBasis" shows "\<langle>x, b\<rangle> = \<langle>y, b\<rangle>"
fps_deriv_mult_const_left: fixes c :: "'a"   and f :: "'a fps" shows "fps_deriv (fps_const c * f) = fps_const c * fps_deriv f"
retraction_map_fst: fixes X :: "'b topology"   and Y :: "'c topology" shows "retraction_map (prod_topology X Y) X fst = (topspace Y = {} \<longrightarrow> topspace X = {})"
ssubst_preserves_lc: fixes t :: "sterm"   and t' :: "sterm"   and x :: "char list" assumes "lc t"   and "lc t'" shows "lc ([x \<rightarrow> t'] t)"
mod_ident_iff: fixes m :: "int"   and x :: "int" assumes "0 < m" shows "(x mod m = x) = (x \<in> {0..<m})"
WHILE_Its: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat" assumes "(WHILE b DO c, s) \<Rightarrow> t"   and "Q l t" shows "\<exists>n. wpw b c n Q l s"
instantiate_var_same_typ': fixes a :: "Core.term"   and \<tau> :: "typ"   and B :: "Core.term"   and x :: "variable" assumes "typ_of a = Some \<tau>"   and "is_closed B" shows "subst_bv a (bind_fv (x, \<tau>) B) = subst_term [((x, \<tau>), a)] B"
totalize_constrains_iff: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set" shows "(totalize F \<in> A co B) = (F \<in> A co B)"
map_in_list_rel_conv: fixes l :: "'b list"   and l' :: "'c list"   and \<alpha> :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" shows "((l, l') \<in> \<langle>br \<alpha> I\<rangle>list_rel) = ((\<forall>x\<in>set l. I x) \<and> l' = map \<alpha> l)"
Collect_mem_eq: fixes A :: "'a set" shows "{x. s2p A x} = A"
algebra_axioms: fixes M :: "'b measure" shows "algebra (space M) (sets M)"
cScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<longmapsto> \<alpha> \<prec> \<lparr>\<nu>x\<rparr>P'"
ti_GSMP: shows "t\<^sub>i \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
that: fixes Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'"   and "insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi>"   and "(\<Psi>, P, Q') \<in> Rel" shows "thesis"
dagger_unfoldr: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<dagger> \<cdot> x \<le> x\<^sup>\<dagger>"
wt0_cong: fixes \<zeta> :: "('s, 'v) hd"   and A :: "'v pvar \<Rightarrow> zhmultiset"   and B :: "'v pvar \<Rightarrow> zhmultiset" assumes "\<forall>x\<in>vars_hd \<zeta>. A (PWt x) = B (PWt x)" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt0 \<zeta>)) = eval_tpoly B (map_tpoly (\<lambda>x. x) zhmset_of (wt0 \<zeta>))"
nths_vars2': shows "nths d.dims0 (- d.vars1') = d.dims2"
intro: fixes R :: "('c, 'd) ring_scheme"   and \<phi> :: "'c \<Rightarrow> nat" assumes "domain R"   and "euclidean_domain_axioms R \<phi>" shows "euclidean_domain R \<phi>"
is_set_to_sorted_list_def: fixes ordR :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Rk :: "('g \<times> 'f) set"   and Rs :: "('g \<times> 'f) set \<Rightarrow> ('h \<times> 'f set) set"   and tsl :: "'h \<Rightarrow> 'g list" shows "is_set_to_sorted_list ordR Rk Rs tsl \<equiv> \<forall>s s'. (s, s') \<in> \<langle>Rk\<rangle>Rs \<longrightarrow> (\<exists>l'. (tsl s, l') \<in> \<langle>Rk\<rangle>list_rel \<and> RETURN l' \<le> it_to_sorted_list ordR s')"
renaming_preserves_satisfiability: fixes S :: "'at Literal set set"   and Sel :: "'at set" assumes "satisfiable S" shows "satisfiable (rename_formula Sel S)"
diff_union_swap: fixes a :: "'a"   and b :: "'a"   and M :: "'a multiset" assumes "a \<noteq> b" shows "add_mset b (M - {#a#}) = add_mset b M - {#a#}"
"0_3": assumes "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> set TI"
Continuous_def: fixes H :: "'d upred \<Rightarrow> 'd upred" shows "Continuous H = (\<forall>A. \<not> A = {} \<longrightarrow> H (\<Or>A) = \<Or>(H ` A))"
int_pow_int: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and n :: "nat" shows "x [^]\<^bsub>G\<^esub> int n = x [^]\<^bsub>G\<^esub> n"
in_abs_intervalI1: fixes lx :: "'b"   and ux :: "'b"   and xa :: "'b" assumes "lx < (0::'b)"   and "(0::'b) < ux"   and "(0::'b) \<le> xa"   and "xa \<le> max (- lx) ux" shows "xa \<in> abs ` {lx..ux}"
induced_vector_def: fixes v :: "'v" shows "induced_vector v \<equiv> if v \<in> V then \<lambda>y. if y \<in> FH then FG_proj y \<cdot> v else (0::'v) else 0"
block_closed: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A"   and "A \<in> Partition" shows "a \<in> S"
permutation_Ex_bij: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs <~~> ys" shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! f i)"
Un: fixes a :: "'c set"   and M :: "'c measure"   and b :: "'c set" assumes "a \<in> fmeasurable M"   and "b \<in> fmeasurable M" shows "a \<union> b \<in> fmeasurable M"
inf_chain_RS: shows "\<forall>i. (f (Suc i), f i) \<in> R \<union> S"
assms_2: shows "a \<in> Units (R\<lparr>carrier := K\<rparr>)"
exec_to_exec_normalize: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Language.normalize c,s\<rangle> \<Rightarrow> t"
Nle0_iff: fixes x :: "int \<times> int" assumes "isnormNum x" shows "(INum x \<le> (0::'a)) = 0\<ge>\<^sub>N x"
gbinomial_eq_0_int: fixes n :: "nat"   and k :: "nat" assumes "n < k" shows "int n gchoose k = 0"
prime_multiplicity_prime: fixes p :: "'a"   and q :: "'a" assumes "prime p"   and "prime q" shows "multiplicity p q = (if p = q then 1 else 0)"
distinct_107: fixes x3 :: "val"   and x201 :: "char list"   and x202 :: "'a exp"   and x203 :: "char list list"   and x204 :: "'a exp" shows "Val x3 \<noteq> RI (x201,x202) ; x203 \<leftarrow> x204"
assoc_simps_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "arr \<a>[f, g, h]"
not_fresh_inner_floatariths: fixes xs :: "floatarith list"   and ys :: "floatarith list"   and i :: "nat" assumes "length xs = length ys" shows "(\<not> fresh_floatarith (inner_floatariths xs ys) i) = (\<not> fresh_floatariths xs i \<or> \<not> fresh_floatariths ys i)"
remove_all_mset_minus: fixes ys :: "'b list"   and xs :: "'b list" assumes "mset ys \<subseteq># mset xs" shows "mset (remove_all xs ys) = mset xs - mset ys"
closed_f: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "x \<squnion>\<^bsub>f\<^esub> y \<in> A"
graph: shows "graph_of_mat a = {(Bij_Nat.to_nat i, Bij_Nat.to_nat j) |i j. A $h i $h j \<noteq> (0::'a)}"
lens_override_id: fixes S\<^sub>1 :: "'d"   and S\<^sub>2 :: "'d" shows "S\<^sub>1 \<triangleleft>\<^bsub>1\<^sub>L\<^esub> S\<^sub>2 = S\<^sub>2"
remap_dta: fixes f :: "'Q \<Rightarrow> 'b" assumes "inj_on f Q" shows "det_tree_automaton (ta_remap f TA) A"
valid_before: shows "valid_mmuaux' args cur dt (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) auxlist"
disc_2: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lnull xs \<or> \<not> lnull ys" shows "\<not> lnull (lappend xs ys)"
validity_implies_entailment: fixes S1 :: "'at Literal set set"   and S2 :: "'at Literal set set" assumes "\<forall>I. I \<Turnstile> S1 \<longrightarrow> I \<Turnstile> S2" shows "entails_formula S1 S2"
product_listset_def: fixes xs :: "real list set"   and ys :: "real list set" shows "product_listset xs ys = (\<lambda>(x, y). x @ y) ` (xs \<times> ys)"
inf_commute: fixes x :: "'d upred"   and y :: "'d upred" shows "(x \<and> y) = (y \<and> x)"
divisor_count_asymptotics_aux: shows "(\<lambda>x. sum_upto (\<lambda>n. sum_upto (\<lambda>_. 1) (x / real n)) (sqrt x) - (x * ln x / 2 + euler_mascheroni * x)) \<in> O(sqrt)"
cobounded1: fixes a :: "'d multiset"   and b :: "'d multiset" shows "a \<inter># b \<subseteq># a"
assms_5: fixes I :: "'a set"   and a :: "'a"   and b :: "'a"   and J :: "'a set"   and c :: "'a"   and d :: "'a" assumes "Q I a b"   and "Q J c d"   and "I \<subseteq> A"   and "J \<subseteq> A" shows "(a = b \<and> b = c \<and> c = d \<longrightarrow> P I J) \<and> (a = b \<and> b \<noteq> c \<and> c = d \<longrightarrow> P I J) \<and> (a = b \<and> b = c \<and> c \<noteq> d \<longrightarrow> P I J) \<and> (a = b \<and> b \<noteq> c \<and> c \<noteq> d \<and> a \<noteq> d \<longrightarrow> P I J) \<and> (a \<noteq> b \<and> b = c \<and> c \<noteq> d \<and> a = d \<longrightarrow> P I J) \<and> ([[a b c]] \<and> a = d \<longrightarrow> P I J) \<and> ([[b a c]] \<and> a = d \<longrightarrow> P I J)"
Var\<^sub>0_power: fixes v :: "'a"   and n :: "nat" shows "Var\<^sub>0 v ^ n = monomial (1::'b) (monomial n v)"
Rep_process: fixes x :: "'b process" shows "Rep_process x \<in> process_set"
pairwise_rel_prime: shows "pairwise_coprime_list (fst (factorize_polys qs))"
uminus_in_Nats_iff: fixes x :: "'a" shows "(- x \<in> \<nat>) = (x \<in> \<int>\<^sub>\<le>\<^sub>0)"
subset_trans: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "A \<subseteq> C"
l12_6: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ParStrict Bet A B C D" shows "Tarski_neutral_dimensionless.OS Bet A B C D"
poly_uminus_irreducible: fixes p :: "int poly" assumes "irreducible p"   and "degree p \<noteq> 0" shows "irreducible (poly_uminus p)"
swap_disjunct_emb_step_at: fixes p :: "dir list"   and q :: "dir list"   and d2 :: "dir"   and d1 :: "dir"   and t :: "('a, 'b) tm" assumes "length p \<le> length q \<Longrightarrow> take (length p) q \<noteq> p"   and "length q \<le> length p \<Longrightarrow> take (length q) p \<noteq> q" shows "emb_step_at q d2 (emb_step_at p d1 t) = emb_step_at p d1 (emb_step_at q d2 t)"
config_n_init2: fixes init :: "'b list"   and qs :: "'b list" shows "\<forall>(uu_, uu_, x)\<in>set_pmf (config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs). x = init"
true_iff: fixes P :: "'d upred" shows "(P \<Leftrightarrow> true) = P"
f: shows "\<forall>x. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> \<not> P (cbox (fst (f x)) (snd (f x))) \<and> (\<forall>i\<in>Basis. fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and> fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and> snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and> 2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)"
prems_1: shows "set (u # us) \<subseteq> carrier_vec n"
simps_1: fixes x :: "'n"   and uu :: "('m, 'n) result" shows "Ok x <|> uu = Ok x"
conc_Diff_if_Nil2: fixes B :: "'a list set"   and A :: "'a list set" assumes "[] \<in> B" shows "A @@ B = A @@ (B - {[]}) \<union> A"
compact_sums: fixes s :: "'a set"   and t :: "'a set" assumes "compact s"   and "compact t" shows "compact {x + y |x y. x \<in> s \<and> y \<in> t}"
match_ex_table_append: fixes P :: "'a prog"   and C :: "String.literal"   and pc :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and xt' :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" shows "match_ex_table P C pc (xt @ xt') = (case match_ex_table P C pc xt of None \<Rightarrow> match_ex_table P C pc xt' | \<lfloor>pcd\<rfloor> \<Rightarrow> \<lfloor>pcd\<rfloor>)"
whr_inv_auto_3: fixes x :: "name"   and t12 :: "trm"   and t2 :: "trm"   and t :: "trm" assumes "App (Lam [x].t12) t2 \<leadsto> t" obtains xa :: "name" and t\<^sub>1 :: "trm" where "[x].t12 = [xa].t\<^sub>1" and "t = t\<^sub>1[xa::=t2]"   | t\<^sub>1' :: "trm" where "t = App t\<^sub>1' t2" and "Lam [x].t12 \<leadsto> t\<^sub>1'"