hcju/isabelleps huggingface dataset:
twiddleHeap_inverse: fixes \<beta> :: "(Location \<times> Location) set"   and s :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "twiddleHeap \<beta> s t" shows "twiddleHeap (Pbij_inverse \<beta>) t s"
derivation_sound_refute: fixes Cs :: "fterm literal set set"   and Cs' :: "fterm literal set set"   and F :: "char list \<Rightarrow> 'b list \<Rightarrow> 'b"   and G :: "char list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "resolution_deriv Cs Cs' \<and> {} \<in> Cs'" shows "\<not> eval\<^sub>c\<^sub>s F G Cs"
sub: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c" assumes "submonoid N M composition unit"   and "a \<in> N" shows "a \<in> M"
ord_spmf_return_spmf: fixes x :: "'c"   and p :: "'c spmf" shows "ord_spmf (=) (return_spmf x) p = (p = return_spmf x)"
eligible: shows "eligible (S_M S M) \<sigma> As (D + negs (mset As))"
in_open_segment_rotD: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> {a<--<b}" shows "(x - a) \<bullet> rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a \<bullet> (b - a)<..<b \<bullet> (b - a)}"
index_le_size: fixes xs :: "'b list"   and x :: "'b" shows "index xs x \<le> length xs"
l2_step3_partners: fixes A :: "agent"   and B :: "agent"   and Rb :: "fid_t" assumes "guessed_runs test = \<lparr>role = Init, owner = A, partner = B\<rparr>"   and "guessed_frame test xsk = Some (NonceF (Rb $ sk))"   and "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "guessed_frame Rb xpkE = Some (epubKF (test $ kE))" shows "Rb \<in> partners"
Osum_Total: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "Total r"   and "Total r'" shows "Total (r \<union>o r')"
\<beta>'_\<gamma>': shows "LeftDerivationFix \<beta>' (i - length \<delta> + ix) (derivation_shift D (length \<delta>) 0) (j - length \<delta>) \<gamma>'"
ring_record_simps_6: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'c" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
emb_cfun_def: shows "EMB('e \<rightarrow> 'f) = EMB('e\<^sub>\<bottom> \<rightarrow>! 'f) oo encode_cfun"
hasRead_allBlks: fixes s :: "state"   and p :: "Proc" assumes "Inv2c_inner s p"   and "phase s p = 0" shows "\<forall>d\<in>{d. hasRead s p d p}. disk s d p \<in> allBlocksRead s p"
linear_surjective_right_inverse: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "linear (*s) (*s) f"   and "surj f" shows "\<exists>g. linear (*s) (*s) g \<and> f \<circ> g = id"
bind_assoc: fixes x :: "'m"   and f :: "'a \<times> 'w list \<Rightarrow> 'm"   and g :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
atU_add_mset: fixes U :: "'f set"   and c :: "'f list"   and c2 :: "'f list multiset" shows "atU U (add_mset c c2) = (atU_s U c \<or> atU U c2)"
range_constant: fixes x :: "'c" shows "range (\<lambda>_. x) = {x}"
rbt_lookup_from_in_tree: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt"   and k :: "'a" assumes "rbt_sorted t1"   and "rbt_sorted t2"   and "\<And>v. ((k, v) \<in> set (entries t1)) = ((k, v) \<in> set (entries t2))" shows "rbt_lookup t1 k = rbt_lookup t2 k"
p: shows "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>)"
arrE: fixes f :: "'a \<times> 'a" assumes "H.VV.arr f" shows "V.arr (fst f) \<and> V.arr (snd f) \<and> local.dom (fst f) = cod (snd f)"
F3: shows "(0::'a) \<le> \<bar>inverse c\<bar>"
all_f_coeffs: shows "\<forall>n. n \<noteq> 0 \<and> n \<noteq> p \<longrightarrow> (1 + fps_X) ^ p $ n mod int p = int (0 mod p)"
icd_uniq: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and m :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat" assumes "IFC_def.is_icdi prob m \<pi> n"   and "IFC_def.is_icdi prob m \<pi> n'" shows "n = n'"
intros_3: fixes x :: "real" assumes "x \<in> radical_sqrt"   and "x \<noteq> 0" shows "1 / x \<in> radical_sqrt"
assms: fixes \<sigma> :: "(name list \<times> 'a list) list"   and \<Psi> :: "'b" assumes "wellFormedSubst \<sigma>" shows "\<Psi> \<rhd> P[<\<sigma>>] \<doteq> Q[<\<sigma>>]"
mono_mprefix_FD: fixes A :: "'d set"   and P :: "'d \<Rightarrow> 'd process"   and P' :: "'d \<Rightarrow> 'd process" assumes "\<forall>x\<in>A. P x \<le> P' x" shows "Mprefix A P \<le> Mprefix A P'"
K: fixes x :: "nat" shows "norm (f (g x)) \<le> K"
Cons_5: shows "xs = as @ concat (map ((#) i) (zs # xss)) @ xs'"
poly_decompose_lemma: fixes p :: "'a poly" assumes "\<not> (\<forall>z. z \<noteq> (0::'a) \<longrightarrow> poly p z = (0::'a))" shows "\<exists>k a q. a \<noteq> (0::'a) \<and> Suc (psize q + k) = psize p \<and> (\<forall>z. poly p z = z ^ k * poly (pCons a q) z)"
IH_2: assumes "braun_indices r = {1..size r}" shows "braun r"
is_minimal_basisD2: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_minimal_basis B"   and "p \<in> B"   and "q \<in> B"   and "p \<noteq> q" shows "\<not> lt p adds\<^sub>t lt q"
xor_one_left: fixes x :: "'a" shows "- (1::'a) XOR x = NOT x"
C: shows "\<not> ((v, w) \<in> I \<or> (\<exists>v'\<in>sinks I D v xs. (v', w) \<in> I))"
Lit: fixes env :: "v sem_env"   and s :: "'a state"   and l :: "lit" shows "Evaluate_Single.evaluate env s (Lit l) = (s, Rval (Litv l))"
get_empty: fixes x :: "'fun" shows "Finca_get Finca_empty x = None"
assms_4: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "U \<subseteq> X"   and "finite S"   and "0 \<in> S" shows "P t U S ([(monomial (1::'a) t, U)], [])"
noDAsep: fixes p :: "('b, 'c) Combinators list" assumes "noDenyAll p" shows "noDenyAll (separate p)"
TNil_Lazy_tllist: fixes b :: "'b" shows "TNil b = Lazy_tllist (\<lambda>_. Inr b)"
invh_baldR_invc: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invh l"   and "invh r"   and "bheight l = bheight r + 1"   and "invc l" shows "invh (baldR l a r) \<and> bheight (baldR l a r) = bheight l"
ltln_weak_strong_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = w \<Turnstile>\<^sub>n G\<^sub>n \<psi> or\<^sub>n (\<phi> M\<^sub>n \<psi>)"
WTf: fixes x :: "'c" assumes "x \<in> results_gpv \<I> gpv" shows "\<I> \<turnstile>g f x \<surd>"
until_false: fixes \<phi> :: "'a stream \<Rightarrow> bool" shows "\<phi> until (\<lambda>xs. False) = alw \<phi>"
Join_transient: fixes F :: "'c program"   and G :: "'c program"   and A :: "'c set" shows "(F \<squnion> G \<in> transient A) = (F \<in> transient A \<or> G \<in> transient A)"
set_def: fixes a :: "'t" shows "S'.set a = S'.some_img ` S'.hom S'.unity a"
is_operator_applicable_in_def: fixes s :: "'a \<Rightarrow> 'b option"   and op :: "('a, 'b) sas_plus_operator" shows "is_operator_applicable_in s op \<equiv> map_of (precondition_of op) \<subseteq>\<^sub>m s"
simps_3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and k :: "nat"   and i :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "linordered_ab_monoid_add.fw plus less_eq m n k (Suc i) 0 = linordered_ab_monoid_add.fw_upd plus less_eq (linordered_ab_monoid_add.fw plus less_eq m n k i n) k (Suc i) 0"
pid_system_automaton_simps_2: shows "pid.E = lih'.pid_step"
simps_4: fixes x :: "quat" shows "quat_proj x (Suc (Suc (Suc 0))) = Im3 x"
winning_strategy_on_deadends: fixes v :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v \<in> VV p"   and "deadend v"   and "strategy p \<sigma>" shows "winning_strategy p** \<sigma> v"
x1: shows "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2"
ldc_def: fixes S :: "char list \<times> char list list" shows "ldc S = last (snd S)"
f2: shows "Deliver (i, Append i e) \<notin> set (history k)"
olen_add_eqv: fixes y :: "'b word"   and x :: "'b word" shows "(y \<le> y + x) = (x \<le> y + x)"
poly_decompose_by_proots: fixes p :: "'a poly"   and P :: "'a \<Rightarrow> bool" assumes "p \<noteq> 0" shows "\<exists>q r. p = q * r \<and> lead_coeff q = (1::'a) \<and> (\<forall>x\<in>proots q. P x) \<and> (\<forall>x\<in>proots r. \<not> P x)"
set_insert_nth: fixes i :: "nat"   and x :: "'b"   and xs :: "'b list" shows "set (insert_nth i x xs) = insert x (set xs)"
is_acc_def: fixes m :: "'b"   and r :: "nat \<Rightarrow> 'Q" shows "gb_graph.is_acc (to_gbg_ext m) r \<equiv> \<forall>A\<in>gbg_F (to_gbg_ext m). \<exists>\<^sub>\<infinity>i. r i \<in> A"
demand_contravariant: fixes f :: "C \<rightarrow> 'a"   and g :: "C \<rightarrow> 'a" assumes "f \<sqsubseteq> g" shows "demand g \<sqsubseteq> demand f"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.cod (g \<cdot> f) = C.cod g"
next_execs_subset: shows "\<forall>s execs u. actions_in_execution (int.next_execs s execs u) \<subseteq> actions_in_execution (execs u)"
u_in_isolated_verts: shows "u \<in> isolated_verts"
int: shows "integrable MX (\<lambda>x. - Px x * log b (Px x))"
rep_eq: fixes x :: "'b fls" shows "fpxs_nth (fpxs_of_fls x) = (\<lambda>r. if r \<in> \<int> then fls_nth x \<lfloor>r\<rfloor> else (0::'b))"
\<omega>_limit_set_in_compact_nonempty: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "trapped_forward x K" shows "\<omega>_limit_set x \<noteq> {}"
UN_extend_simps_3: fixes A :: "'e set"   and B :: "'f \<Rightarrow> 'e set"   and C :: "'f set" shows "A \<union> \<Union> (B ` C) = (if C = {} then A else \<Union>x\<in>C. A \<union> B x)"
dghm_proj_ArrMap_vdomain: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" shows "\<D>\<^sub>\<circ> (\<pi>\<^sub>D\<^sub>G I \<AA> i\<lparr>ArrMap\<rparr>) = dg_prod I \<AA>\<lparr>Arr\<rparr>"
emeasure_eq_AE: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure" assumes "AE x in M. (x \<in> A) = (x \<in> B)"   and "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M A = emeasure M B"
not_empty_iff_iEx_True: fixes I :: "nat set" shows "(I \<noteq> {}) = (\<diamond> t I. True)"
assms_2: fixes v :: "'a" assumes "v \<in> verts G" shows "out_degree G v = \<r>"
failingSub_def: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and gamma :: "'b" shows "failingSub subs P gamma = (SOME sigma. sigma \<in> subs gamma \<and> \<not> P (tree subs sigma))"
closed_Un': fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set"   and opena :: "'a set \<Rightarrow> \<bool>" assumes "U \<noteq> {}"   and "\<forall>x\<in>S. x \<in> U"   and "\<forall>x\<in>T. x \<in> U"   and "topological_space_ow U opena"   and "closed_ow U opena S"   and "closed_ow U opena T" shows "closed_ow U opena (S \<union> T)"
it_to_sorted_list_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d set" shows "it_to_sorted_list R s \<equiv> SPEC (\<lambda>l. distinct l \<and> s = set l \<and> sorted_wrt R l)"
list_all2_refl: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "\<And>x. P x x" shows "list_all2 P xs xs"
Cd_preferred: fixes x :: "'x"   and y :: "'x"   and d :: "'d"   and X :: "'x set" assumes "(x, y) \<in> Pd d"   and "x \<in> Cd d X"   and "y \<in> X" shows "y = x"
tpair: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and A :: "'a trm"   and \<tau> :: "PreSimplyTyped.type"   and B :: "'a trm"   and \<sigma> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> A : \<tau>"   and "\<Gamma> \<turnstile> B : \<sigma>" shows "\<Gamma> \<turnstile> SimplyTyped.Pair A B : TPair \<tau> \<sigma>"
m_closed: fixes x :: "nat"   and y :: "nat" assumes "x \<in> H"   and "y \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> H"
oalist_inv_raw_singleton: fixes x :: "'o"   and k :: "'a"   and v :: "'d" shows "oalist_inv_raw x [(k, v)] = (v \<noteq> (0::'d))"
decomp_rm\<^sub>e\<^sub>s\<^sub>t_single_1: fixes t :: "('fun, 'var) Term.term" shows "decomp_rm\<^sub>e\<^sub>s\<^sub>t [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)] = [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)]"
diff_right_commute: fixes M :: "'b multiset"   and N :: "'b multiset"   and Q :: "'b multiset" shows "M - N - Q = M - Q - N"
weakSimulation_def: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" shows "P \<leadsto><Rel> Q \<equiv> (\<forall>a x Q'. Q \<longmapsto> a<\<nu>x> \<prec> Q' \<and> x \<sharp> P \<longrightarrow> (\<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel)) \<and> (\<forall>\<alpha> Q'. Q \<longmapsto> \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
insert_def: fixes e :: "'c"   and a :: "'d"   and q :: "('c, 'd) BinomialHeap" shows "local.insert e a q \<equiv> Abs_BinomialHeap (BinomialHeapStruc.insert e a (Rep_BinomialHeap q))"
\<eta>_p: shows "C\<^sub>1 \<cdot>\<^sub>l\<^sub>s \<eta> = C\<^sub>1' \<and> C\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<eta> = C\<^sub>2'"
Rep_p_point: fixes x :: "p_point" shows "Rep_p_point x \<in> {z. z \<in> unit_disc}"
of_nat_tendsto_top_ennreal: shows "of_nat \<longlonglongrightarrow> \<top>"
Agent: fixes agt :: "agent"   and H :: "msg set" shows "Agent agt \<in> synth H"
assert_cs_ok: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state"   and j :: "'a"   and s' :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" assumes "invariant_simplex' cs J s"   and "assert_simplex' j s = Inr s'" shows "invariant_simplex' cs (insert j J) s'"
ANDLEFT2_eqvt_name: fixes pi :: "(name \<times> name) list"   and A :: "ty"   and B :: "ty"   and X :: "ntrm set" shows "pi \<bullet> ANDLEFT2 (A AND B) X = ANDLEFT2 (A AND B) (pi \<bullet> X)"
simps: fixes a :: "'b fm" shows "(\<turnstile> a) = ((\<exists>p. a = p \<and> tautology p) \<or> (\<exists>i p q. a = (K i p \<^bold>\<and> K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i q)) \<or> (\<exists>p q. a = q \<and> \<turnstile> p \<and> \<turnstile> (p \<^bold>\<longrightarrow> q)) \<or> (\<exists>p i. a = K i p \<and> \<turnstile> p))"
rel_fun_inf: fixes Q :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_fun Q R \<sqinter> rel_fun Q R' = rel_fun Q (R \<sqinter> R')"
wf_append_exec: fixes V :: "'d set"   and S :: "('e, 'd) strand_step list"   and S' :: "('e, 'd) strand_step list" assumes "wf\<^sub>s\<^sub>t V (S @ S')" shows "wf\<^sub>s\<^sub>t (V \<union> wfvarsoccs\<^sub>s\<^sub>t S) S'"
Stable_def: fixes A :: "'b set" shows "Stable A \<equiv> A Co A"
int_gS': shows "f integrable_on g ` S' \<and> integral (g ` S') f \<le> integral S' (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x))"
eq: fixes s :: "'b \<Rightarrow>\<^sub>0 nat" shows "subst_pp (\<lambda>x. if x \<in> X then monomial (a x) 0 else monomial (1::'a) (monomial 1 x)) s = monomial (\<Prod>x\<in>keys s \<inter> X. a x ^ lookup s x) (except s X)"
fcard_fminus_fsingleton: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<in>| A" shows "fcard (A |-| {|x|}) = fcard A - 1"
g_sort: fixes xs :: "'a list" shows "g (local.sort xs) = g xs"
numsubst0_numbound0: fixes t :: "MIR.num"   and a :: "MIR.num" assumes "numbound0 t" shows "numbound0 (numsubst0 t a)"
simps_2: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'c" shows "\<one>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
ide_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "partial_magma C" shows "partial_magma.ide C a \<equiv> C a a \<noteq> partial_magma.null C \<and> (\<forall>f. (C f a \<noteq> partial_magma.null C \<longrightarrow> C f a = f) \<and> (C a f \<noteq> partial_magma.null C \<longrightarrow> C a f = f))"
Integer_arith_simps_3: fixes a :: "int"   and b :: "int" shows "MkI\<cdot>a - MkI\<cdot>b = MkI\<cdot>(a - b)"
distinct_5: fixes x2 :: "'b"   and x3 :: "'c" shows "OpUbx1 x2 \<noteq> OpUbx2 x3"
onorm_def: fixes f :: "'a \<Rightarrow> 'b" shows "onorm f = (SUP x. (\<parallel>f x\<parallel>) / (\<parallel>x\<parallel>))"
rbt_union_with_def: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "rbt_union_with f = rbt_union_with_key (\<lambda>_. f)"
ideal_generated_subset2: fixes B :: "'b set"   and A :: "'b set" assumes "\<forall>b\<in>B. b \<in> ideal_generated A" shows "ideal_generated B \<subseteq> ideal_generated A"
is_natural_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b"   and f :: "'a" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (G f) (\<tau> (partial_magma.dom A f)) = \<tau> f"
fw_impl'_correct: shows "(fw_impl', fw_spec) \<in> nat_rel \<rightarrow> br curry (\<lambda>_. True) \<rightarrow> \<langle>br curry (\<lambda>_. True)\<rangle>nres_rel"
inverse_less_imp_less_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less (inverse a) (inverse b)"   and "less b zero" shows "less b a"
indexed_var_not_index_free: fixes i :: "'c" shows "\<not> index_free \<X>\<^bsub>i\<^esub> i"
\<L>\<^sub>P_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.\<L>\<^sub>P \<NN> \<TT> \<RR> \<SS> = {uu_. \<exists>u v. uu_ = u \<and> CFG.is_word \<TT> u \<and> CFG.is_derivation \<NN> \<TT> \<RR> \<SS> (u @ v)}"
closed_map_fst: fixes T :: "'a set"   and S :: "'b set"   and c :: "('b \<times> 'a) set" assumes "compact T"   and "closedin (top_of_set (S \<times> T)) c" shows "closedin (top_of_set S) (fst ` c)"
subspace_Span_is_findim: fixes vs :: "'b list"   and W :: "'b set" assumes "set vs \<subseteq> V"   and "Subspace W"   and "W \<subseteq> RSpan vs" shows "\<exists>ms. set ms \<subseteq> W \<and> RSpan ms = W"
rel_fun_comm: fixes \<rho> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "class.rel_ab_semigroup \<rho>" shows "rel_magma.times_rel_fun \<rho> f g = rel_magma.times_rel_fun \<rho> g f"
matrix_equality: shows "mat_of_rows n (map of_int_hom.vec_hom fs') = gs2.M (length (map of_int_hom.vec_hom fs')) * mat_of_rows n (map gs2.gso [0..<length (map of_int_hom.vec_hom fs')])"
intro: fixes f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "injective f"
insert: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<notin> A" shows "sorted_list_of_set (insert x A) = insort x (sorted_list_of_set A)"
ResComm: fixes c :: "name"   and d :: "name"   and Q :: "pi" shows "(<\<nu>c><\<nu>d>Q, <\<nu>d><\<nu>c>Q) \<in> Rel"
right_cancel: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)"   and "z \<in> carrier (poly_ring R)" shows "(y \<oplus>\<^bsub>poly_ring R\<^esub> x = z \<oplus>\<^bsub>poly_ring R\<^esub> x) = (y = z)"
bintrunc_Suc_numeral_3: fixes n :: "nat"   and w :: "num" shows "take_bit (Suc n) (numeral (num.Bit0 w)) = 2 * take_bit n (numeral w)"
separating_of_bipartite_aux: fixes \<Gamma> :: "('a, 'b) web_scheme"   and S :: "('a + 'a) set"   and x :: "'a"   and p :: "'a list"   and y :: "'a" assumes "web \<Gamma>"   and "separating (bipartite_web_of \<Gamma>) S"   and "path \<Gamma> x p y"   and "y \<in> B \<Gamma>"   and "x \<in> A \<Gamma> \<or> Inr x \<in> S" shows "(\<exists>z\<in>set p. z \<in> web.separating_of_bipartite \<Gamma> S) \<or> x \<in> web.separating_of_bipartite \<Gamma> S"
checkI_def: fixes einp :: "'c \<Rightarrow> 'd withERR option" shows "checkI einp \<equiv> lift check einp"
cimage_cinsert: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and B :: "'d cset" shows "f `\<^sub>c cinsert a B = cinsert (f a) (f `\<^sub>c B)"
sup_assoc: fixes x :: "'e multiset"   and y :: "'e multiset"   and z :: "'e multiset" shows "x \<union># y \<union># z = x \<union># (y \<union># z)"
lambda_zero: shows "(\<lambda>h. 0::'a) = (*) (0::'a)"
lcm_left_idem: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "lcm a (lcm a b) = lcm a b"
L_post: fixes s :: "'s" assumes "wf_state s" shows "L (post s) = L s"
Cignorabimus_step: fixes S :: "variable set set"   and \<alpha> :: "game"   and V :: "variable set" assumes "\<And>M. M \<in> S \<Longrightarrow> M \<in> Cignorabimus \<alpha> V" shows "\<Union> S \<in> Cignorabimus \<alpha> V"
apath_Cons_iff: fixes u :: "'a"   and e :: "'b"   and es :: "'b list"   and w :: "'a" shows "G.apath u (e # es) w = (e \<in> arcs G \<and> tail G e = u \<and> G.apath (head G e) es w \<and> tail G e \<notin> set (G.awalk_verts (head G e) es))"
pmf_cond: fixes x :: "'a" shows "pmf local.cond_pmf x = (if x \<in> s then pmf p x / measure_pmf.prob p s else 0)"
compact_path_image: fixes g :: "real \<Rightarrow> 'c" assumes "path g" shows "compact (path_image g)"
Abs_inject: fixes x :: "atom"   and y :: "atom" assumes "x \<in> {a. sort_of a \<in> range sort_fun}"   and "y \<in> {a. sort_of a \<in> range sort_fun}" shows "(Abs x = Abs y) = (x = y)"
reciprocal_0_iff: fixes P :: "'a poly"   and p :: "nat" assumes "degree P \<le> p" shows "(reciprocal_poly p P = 0) = (P = 0)"
aform_scaleR_def: fixes x :: "real \<times> real pdevs"   and y :: "'a" shows "aform_scaleR x y = (fst x *\<^sub>R y, pdevs_scaleR (snd x) y)"
subgroupE_3: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := S\<rparr>)"   and "a \<in> H" shows "inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> a \<in> H"
Prim2I: fixes r :: "recf"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "prim_recfn 2 r"   and "\<And>x y. f x y = eval r [x, y]" shows "f \<in> Prim2"
simps: fixes a :: "nat"   and x :: "nat"   and xs :: "nat list"   and s :: "nat" shows "incs a x (xs, s) = (let t = s + a * x in if C (x # xs) t then (x # xs, t) # incs a (Suc x) (xs, s) else [])"
linear_def: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" shows "linear s1 s2 f \<equiv> vector_space s1 \<and> vector_space s2 \<and> module_hom s1 s2 f"