hcju/isabelleps huggingface dataset:
T4: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b"   and w :: "'a \<Rightarrow> 'c" shows "\<turnstile> \<box>[P]_v \<longrightarrow> \<box>[_actrans P v]_w"
uminus_scene_twice: fixes X :: "'a scene" shows "- (- X) = X"
Nonce_LtK: fixes X :: "nonce_t"   and X' :: "ltkey" shows "Nonce X \<noteq> LtK X'"
valid_add_new_ts_msaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "'msaux"   and auxlist :: "(nat \<times> event_data option list set) list"   and nt :: "nat" assumes "valid_msaux args cur aux auxlist"   and "cur \<le> nt" shows "valid_msaux args nt (add_new_ts_msaux args nt aux) (filter (\<lambda>(t, rel). enat (nt - t) \<le> right (args_ivl args)) auxlist)"
intro: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah" assumes "category A"   and "category B" shows "functor_category A B"
card_mono_strict_subset: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" assumes "finite A"   and "finite B"   and "finite C"   and "A \<inter> B \<noteq> {}"   and "C = A - B" shows "card C < card A"
prems_3: shows "m = factor_sq * factor_sq * factor_pr * n"
real_affinity_le: fixes m :: "'c"   and x :: "'c"   and c :: "'c"   and y :: "'c" assumes "(0::'c) < m" shows "(m * x + c \<le> y) = (x \<le> inverse m * y + - (c / m))"
is_red_reduced_GB_monomial_lt_GB_Polys: fixes X :: "'x set"   and G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "G \<subseteq> P[X]"   and "0 \<notin> G" shows "punit.is_red (punit.reduced_GB (monomial (1::'b) ` lpp ` G)) = punit.is_red G"
det_addrow: fixes l :: "nat"   and n :: "nat"   and k :: "nat"   and A :: "'a mat"   and a :: "'a" assumes "l < n"   and "k \<noteq> l"   and "A \<in> carrier_mat n n" shows "det (addrow a k l A) = det A"
12639: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (- y \<sqinter> x) = - x"
UNTIL_def: shows "(until) \<equiv> \<lambda>\<phi> \<psi>. gfp (\<lambda>p x. (\<exists>xs. x = xs \<and> \<psi> xs) \<or> (\<exists>xs. x = xs \<and> \<phi> xs \<and> p (stl xs)))"
pullback_pseudominimal_in: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "ordering.has_bottom less_eq P"   and "x \<in> P"   and "ordering.pseudominimal_in less_eq' less' (f ` P) (f x)" shows "ordering.pseudominimal_in less_eq less P x"
Suc_6: shows "infs (degen cs) (w ||| k ## sscan (count cs) w k)"
assms_2: fixes w1 :: "'a" assumes "w1 \<in> verts G" shows "P w1 [] w1"
h'_g: fixes y :: "'v" assumes "y \<in> E' `` X" shows "(\<Sum>\<^sup>+ x\<in>X. h' x y) = g y"
neg_of_aezfun_is_aezfun: fixes f :: "'c \<Rightarrow> 'd" assumes "f \<in> aezfun_set" shows "- f \<in> aezfun_set"
Rep_inject: fixes x :: "'b word"   and y :: "'b word" shows "(uint x = uint y) = (x = y)"
has_derivative_zero_constant: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "convex s"   and "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative (\<lambda>h. 0::'b)) (at x within s)" shows "\<exists>c. \<forall>x\<in>s. f x = c"
elims: fixes idx0 :: "'a bound \<Rightarrow> 'a \<Rightarrow> nat"   and idx1 :: "'b bound \<Rightarrow> 'b \<Rightarrow> nat"   and size1 :: "'b bound \<Rightarrow> nat"   and x :: "('a \<times> 'b) bound"   and xa :: "'a \<times> 'b"   and y :: "nat" assumes "prod_index_def.idx idx0 idx1 size1 x xa = y" obtains l0 :: "'a" and r0 :: "'b" and l1 :: "'a" and r1 :: "'b" and a :: "'a" and b :: "'b" where "x = Bound (l0, r0) (l1, r1)" and "xa = (a, b)" and "y = idx0 (Bound l0 l1) a * size1 (Bound r0 r1) + idx1 (Bound r0 r1) b"
polymap_def: fixes p :: "'f poly" shows "polymap p \<equiv> coeffs p \<bullet>\<cdot>\<cdot> map endpow [0..<Suc (degree p)]"
invalidate_changes: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "invalidate rt dests dip = Some (dsn, dsk, flag, hops, nhip, pre)" shows "dsn = (case dests dip of None \<Rightarrow> \<pi>\<^sub>2 (the (rt dip)) | Some rsn \<Rightarrow> rsn) \<and> dsk = \<pi>\<^sub>3 (the (rt dip)) \<and> flag = (if dests dip = None then \<pi>\<^sub>4 (the (rt dip)) else Aodv_Basic.inv) \<and> hops = \<pi>\<^sub>5 (the (rt dip)) \<and> nhip = \<pi>\<^sub>6 (the (rt dip)) \<and> pre = \<pi>\<^sub>7 (the (rt dip))"
finite_Domain: fixes r :: "('b \<times> 'c) set" assumes "finite r" shows "finite (Domain r)"
intro: fixes \<I> :: "('e, 'f) \<I>"   and I :: "'d \<Rightarrow> bool"   and \<I>' :: "('g, 'h) \<I>"   and callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd, 'g, 'h) gpv" assumes "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> {s. I s}"   and "\<And>x s. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> \<I>' \<turnstile>g callee s x \<surd>" shows "raw_converter_invariant \<I> \<I>' callee I"
that: fixes a :: "'a multiset" assumes "a \<in># A"   and "x \<in># a" shows "thesis"
tree2_induct: fixes P :: "('c \<times> 'd) tree \<Rightarrow> bool"   and treea :: "('c \<times> 'd) tree" assumes "P \<langle>\<rangle>"   and "\<And>x1 a b x3. \<lbrakk>P x1; P x3\<rbrakk> \<Longrightarrow> P \<langle>x1, (a, b), x3\<rangle>" shows "P treea"
relabel_pres_Labeling: fixes u :: "nat" assumes "relabel_precond f (relabel_effect f l u) u" shows "Labeling c s t f (relabel_effect f (relabel_effect f l u) u)"
Int_absorb2: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subseteq> B" shows "A \<inter> B = A"
neq_imp_neq_div_or_mod: fixes i :: "nat"   and j :: "nat"   and n :: "nat" assumes "i \<noteq> j" shows "i div 2 ^ n \<noteq> j div 2 ^ n \<or> i mod 2 ^ n \<noteq> j mod 2 ^ n"
dgrad_p_set_closed_monom_mult_zero: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and c :: "'c" assumes "p \<in> dgrad_p_set d m" shows "monom_mult c (0::'a) p \<in> dgrad_p_set d m"
RPCFwd_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCFwd send rcv rst p \<equiv> $Calling send p \<and> $(rst!p) = #rpcA \<and> IsLegalRcvArg<arg<$(send!p)>> \<and> Call rcv p RPCRelayArg<arg<send!p>> \<and> (rst!p)$ = #rpcB \<and> unchanged (rtrner send!p)"
angelic_bottom: fixes R :: "'c \<Rightarrow> 'd" shows "{: R :} \<bottom> = {}"
get_var_coeff_def: fixes lp :: "(nat, rat) fmap"   and v :: "nat" shows "get_var_coeff lp v \<equiv> case fmlookup lp v of None \<Rightarrow> 0 | Some c \<Rightarrow> c"
basis_orthogonal': fixes xs :: "('a, 'c) vec list" shows "length (Gram_Schmidt xs) = length xs \<and> span (set (Gram_Schmidt xs)) = span (set xs) \<and> pairwise orthogonal (set (Gram_Schmidt xs))"
oalist_inv_raw_sort_oalist_aux: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "oalist_inv_raw ko (sort_oalist_aux ko xs)"
pnormal_last_nonzero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b list" assumes "class.semiring_0 plus zero times"   and "semiring_0.pnormal zero p" shows "last p \<noteq> zero"
neqE: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" obtains "x \<prec> y"   | "y \<prec> x"
absolutely_integrable_spike_set_eq: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "negligible {x \<in> S - T. f x \<noteq> (0::'b)}"   and "negligible {x \<in> T - S. f x \<noteq> (0::'b)}" shows "(f absolutely_integrable_on S) = (f absolutely_integrable_on T)"
rep_eq: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and x :: "'c word" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero x = semiring_1.of_nat one plus zero (nat (take_bit LENGTH('c) (Word.rep x)))"
neg_one_odd_power: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "odd n" shows "power.power one times (uminus one) n = uminus one"
assoc: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd"   and c :: "'c \<times> 'd" shows "sup_prod (sup_prod a b) c = sup_prod a (sup_prod b c)"
distinct_43: fixes x41 :: "poly"   and x42 :: "poly"   and x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly" shows "Sub x41 x42 \<noteq> CN x81 x82 x83"
rank_card_indpt: fixes A :: "'a mat"   and nc :: "nat"   and S :: "'a vec set" assumes "A \<in> carrier_mat n nc"   and "maximal S (\<lambda>T. T \<subseteq> set (cols A) \<and> lin_indpt T)" shows "rank A = card S"
eq: fixes P :: "real \<Rightarrow> real \<Rightarrow> bool" shows "((\<lambda>(x, y). P x y) \<in> POW \<Otimes>\<^sub>M COCOUNT \<rightarrow>\<^sub>M BOOL) = (P \<in> POW \<rightarrow>\<^sub>M EXP)"
single_eq_add_mset: fixes a :: "'b"   and b :: "'b"   and M :: "'b multiset" shows "(a, {#} = b, M) = (b = a \<and> M = {#})"
F: fixes x :: "nat" shows "(x \<in> fail_t) = (r x \<in> fail\<^sub>R)"
iso_inclE: fixes A :: "'c ring"   and B :: "'c ring" assumes "ring A"   and "ring B"   and "A \<lesssim> B" shows "ring_hom_ring A B id"
ok_Join_iff2: fixes G :: "'c program"   and H :: "'c program"   and F :: "'c program" shows "G \<squnion> H ok F = (G ok F \<and> H ok F)"
taut_conj: fixes A :: "'b upred"   and B :: "'b upred" shows "`A \<and> B` = (`A` \<and> `B`)"
unitor_coincidence_2: fixes a :: "'b" assumes "obj a" shows "\<r>[a] = \<i>[a]"
T: fixes f :: "'s stream \<Rightarrow> ennreal"   and s :: "'s" assumes "f \<in> borel_measurable (stream_space (count_space UNIV))" shows "integral\<^sup>N (T s) f = \<integral>\<^sup>+ \<omega>. f (walk s \<omega>) \<partial>stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x)))"
l_neg: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "\<ominus>\<^bsub>K [X]\<^esub> x \<oplus>\<^bsub>K [X]\<^esub> x = \<zero>\<^bsub>K [X]\<^esub>"
elims: fixes x :: "dir"   and y :: "dir" assumes "Grid_Point.inv x = y" obtains "x = left" and "y = right"   | "x = right" and "y = left"
ras_push_def: fixes x :: "'c"   and s :: "'c array \<times> nat" shows "ras_push x s \<equiv> let (a, n) = s; a = if n = array_length a then array_grow a (max 4 (2 * n)) x else a; a = array_set a n x in (a, n + 1)"
brI: fixes a :: "'g"   and \<alpha> :: "'h \<Rightarrow> 'g"   and c :: "'h"   and I :: "'h \<Rightarrow> bool" assumes "a = \<alpha> c"   and "I c" shows "(c, a) \<in> br \<alpha> I"
IH: assumes "wfrestrictedvars\<^sub>s\<^sub>t S \<subseteq> V" shows "wf\<^sub>s\<^sub>t V S"
zero_left: fixes b :: "'b" shows "(0::'a) ** b = (0::'c)"
lemma_2_5_13_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "b r\<rightarrow> c \<le> a r\<rightarrow> c"
rbt_greater_simps_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and c :: "color"   and lt :: "('a, 'b) rbt"   and kt :: "'a"   and v :: "'b"   and rt :: "('a, 'b) rbt" shows "ord.rbt_greater less k (Branch c lt kt v rt) = (less k kt \<and> ord.rbt_greater less k lt \<and> ord.rbt_greater less k rt)"
map_cong: fixes x :: "('d, 'e) optionT"   and ya :: "('d, 'e) optionT"   and f1 :: "'d \<Rightarrow> 'f"   and g1 :: "'d \<Rightarrow> 'f"   and f2 :: "'e \<Rightarrow> 'g"   and g2 :: "'e \<Rightarrow> 'g" assumes "x = ya"   and "\<And>z1. z1 \<in> phantom_optionT ya \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> set_optionT ya \<Longrightarrow> f2 z2 = g2 z2" shows "map_optionT' f1 f2 x = map_optionT' g1 g2 ya"
snd_image_times: fixes A :: "'c set"   and B :: "'b set" shows "snd ` (A \<times> B) = (if A = {} then {} else B)"
Mp_smult_p_removal: fixes k :: "nat"   and f :: "int poly" assumes "poly_mod.Mp (p * p ^ k) (smult p f) = 0" shows "poly_mod.Mp (p ^ k) f = 0"
inf_Int_eq: fixes R :: "'e set"   and S :: "'e set" shows "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
app_mono_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and app :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and A :: "'b set" shows "app_mono r app n A = (\<forall>s p t. s \<in> A \<and> p < n \<and> s \<sqsubseteq>\<^bsub>r\<^esub> t \<longrightarrow> app p t \<longrightarrow> app p s)"
transfer: shows "rel_fun pcr_real (rel_fun pcr_real pcr_real) (\<lambda>X Y n. X n * Y n) (*)"
interaction_bound_fixp_induct: fixes P :: "(('d, 'out, 'e) gpv \<Rightarrow> enat) \<Rightarrow> bool" assumes "lfp.admissible P"   and "P (\<lambda>_. 0)"   and "\<And>interaction_bound'. \<lbrakk>P interaction_bound'; \<And>gpv. interaction_bound' gpv \<le> local.interaction_bound gpv; \<And>gpv. interaction_bound' gpv \<le> (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))\<rbrakk> \<Longrightarrow> P (\<lambda>gpv. \<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))" shows "P local.interaction_bound"
asset_seskey_other: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset n, SesKey SK) \<in> s" shows "\<exists>A B C D. fst (snd SK) = {A, B} \<and> snd (snd SK) = {C, D} \<and> (Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s \<and> (Asset n, \<lbrace>Num 4, PubKey D\<rbrace>) \<in> s \<and> (Asset n, Token n (Auth_PriK n) B C SK) \<in> s"
PiE_restrict: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and B :: "'f \<Rightarrow> 'g set" assumes "f \<in> Pi\<^sub>E A B" shows "restrict f A = f"
act1: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, uu_). castmsg (I \<sigma>) a)"
lotteries_on_def: fixes Oc :: "'b set" shows "lotteries_on Oc = {p. set_pmf p \<subseteq> Oc}"
dual_dual: fixes x :: "'a" shows "(x ^ o) ^ o = x"
update_kno_dsn_greater_zero: fixes dsn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and hops :: "nat"   and ip :: "nat"   and npre :: "nat set" assumes "1 \<le> dsn" shows "1 \<le> sqn (update rt dip (dsn, kno, val, hops, ip, npre)) dip"
observable_moves_equal_length_2: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) =as\<Rightarrow> (ms',s')" shows "length ms' = length s'"
tensor_prod_def: fixes A :: "'a tensor"   and B :: "'a tensor" shows "local.tensor_prod_otimes A B = tensor_from_vec (dims A @ dims B) (concat (map (\<lambda>a. vec_smult a (vec B)) (vec A)))"
if_True: fixes x :: "'a"   and y :: "'a" shows "(if True \<Rightarrow> x | otherwise \<Rightarrow> y) = x"
f_is_measure: shows "\<forall>\<sigma>. \<not> exec_stop \<sigma> \<and> cond \<sigma> \<and> P \<sigma> \<longrightarrow> M \<sigma> \<noteq> None \<and> f (snd (the (M \<sigma>))) < f \<sigma>"
var_pow_closed: fixes K :: "'a set"   and n :: "nat" assumes "subring K R" shows "X [^]\<^bsub>K [X]\<^esub> n \<in> carrier (K [X])"
simps_2: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
cases: fixes M :: "'a literal multiset set"   and a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'a literal multiset" assumes "gr.ord_resolve M a1 a2 a3 a4 a5" obtains CAs :: "'a literal multiset list" and n :: "nat" and Cs :: "'a literal multiset list" and AAs :: "'a multiset list" and As :: "'a list" and D :: "'a literal multiset" where "a1 = CAs" and "a2 = D + negs (mset As)" and "a3 = AAs" and "a4 = As" and "a5 = \<Sum>\<^sub># (mset Cs) + D" and "length CAs = n" and "length Cs = n" and "length AAs = n" and "length As = n" and "n \<noteq> 0" and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)" and "\<forall>i<n. AAs ! i \<noteq> {#}" and "\<forall>i<n. \<forall>A\<in>#AAs ! i. A = As ! i" and "gr.eligible M As (D + negs (mset As))" and "\<forall>i<n. gr.strictly_maximal_wrt (As ! i) (Cs ! i)" and "\<forall>i<n. S_M S M (CAs ! i) = {#}"
fold_bind_random_permutation_foldr: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b pmf"   and x :: "'c" assumes "finite A" shows "fold_bind_random_permutation f g x A = pmf_of_set (permutations_of_set A) \<bind> (\<lambda>xs. g (foldr f xs x))"
hom_power: fixes x :: "'a poly"   and n :: "nat"   and p :: "'a poly" shows "x ^ n \<circ>\<^sub>p p = (x \<circ>\<^sub>p p) ^ n"
that: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> 0 = x"   and "\<pi> n = local.return"   and "\<forall>i>0. \<pi> i \<noteq> x" shows "thesis"
d: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" shows "(p permutes S) = ((\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y))"
ext_inject: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'a"   and break_status' :: "bool"   and return_status' :: "bool"   and more' :: "'a" shows "(\<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = \<lparr>break_status = break_status', return_status = return_status', \<dots> = more'\<rparr>) = (break_status = break_status' \<and> return_status = return_status' \<and> more = more')"
rep_eq: fixes x :: "'d \<Rightarrow>\<^sub>L 'e"   and xa :: "'d \<Rightarrow>\<^sub>L 'e" shows "blinfun_apply (x - xa) = (\<lambda>xb. blinfun_apply x xb - blinfun_apply xa xb)"
S_2: shows "\<not> sat S (fst (tree.root t))"
snd_quot_of_fract_to_fract: fixes x :: "'a" shows "snd (quot_of_fract (to_fract x)) = (1::'a)"
pred_intros_conj2': fixes K :: "bool"   and P :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "K \<Longrightarrow> pred M P" shows "pred M (\<lambda>x. P x \<and> K)"
len_faces_sum: fixes g :: "graph" shows "|faces g| = |finals g| + |nonFinals g|"
fps_lr_inverse_starting0_1: fixes f :: "'a fps" shows "fps_left_inverse f (0::'a) = 0"
size_map_tree: fixes f :: "'d \<Rightarrow> 'c"   and t :: "'d tree" shows "size (map_tree f t) = size t"
in_br_conv: fixes c :: "'b"   and a :: "'c"   and \<alpha> :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" shows "((c, a) \<in> br \<alpha> I) = (a = \<alpha> c \<and> I c)"
diff_left: fixes a :: "'a"   and a' :: "'a"   and b :: "'b" shows "(a - a') ** b = a ** b - a' ** b"
wm_to_state_works_for_wf_wm: fixes M :: "object atom formula set"   and fmla :: "object atom formula" assumes "wf_world_model M"   and "fmla \<in> M" shows "close_eq (wm_to_state M) \<Turnstile> fmla"
h: shows "- (c / b) = (- aa - sqrt (aa\<^sup>2 - 4 * a * ba)) / (2 * a)"
foldli_cong: fixes l :: "'c list"   and l' :: "'c list"   and \<sigma> :: "'d"   and \<sigma>' :: "'d"   and c :: "'d \<Rightarrow> bool"   and c' :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd \<Rightarrow> 'd" assumes "l = l'"   and "\<sigma> = \<sigma>'"   and "c = c'"   and "\<And>\<sigma> x. \<lbrakk>x \<in> set l'; c' \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> = f' x \<sigma>" shows "foldli l c f \<sigma> = foldli l' c' f' \<sigma>'"
big_step_post_step: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int"   and C :: "(char list \<Rightarrow> int) set acom"   and S :: "(char list \<Rightarrow> int) set" assumes "(c, s) \<Rightarrow> t"   and "strip C = c"   and "s \<in> S"   and "step S C \<le> C" shows "t \<in> post C"
EQ_SS_DOM: fixes ss :: "('d, 'e) fmap set"   and vs1 :: "'d set"   and vs2 :: "'d set" assumes "ss \<noteq> {}"   and "stateSpace ss vs1"   and "stateSpace ss vs2" shows "vs1 = vs2"
PowI: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "A \<in> Pow B"
measurable_equality_set: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{x \<in> space M. f x = g x} \<in> sets M"
cases: fixes a :: "('test, 'atom, 'choice) cmd" assumes "siso a" obtains c :: "('test, 'atom, 'choice) cmd" where "a = c" and "\<And>s i. i < brn c \<Longrightarrow> siso (local.cont c s i)" and "\<And>s t i. i < brn c \<and> s \<approx> t \<Longrightarrow> eff c s i \<approx> eff c t i \<and> wt c s i = wt c t i \<and> local.cont c s i = local.cont c t i"
atm_of_eq_atm_of: fixes L :: "'a literal"   and L' :: "'a literal" shows "(atm_of L = atm_of L') = (L = L' \<or> L = - L')"
matrix_vector_mul_diag_mat: fixes f :: "'b \<Rightarrow> 'a"   and s :: "('a, 'b) vec" shows "diag_mat f *v s = (\<chi>i. f i * s $ i)"
generate_check_def: fixes aa :: "nat list"   and ba :: "nat list"   and a :: "nat list"   and b :: "nat list" shows "c12_generate_check aa ba a b = concat (map (\<lambda>ys. map (\<lambda>xs. (xs, ys)) (bounded_gen_check.gen_check (cond1 b (fst ys)) a)) (c2_gen_check aa ba b))"
product_integral_fold: fixes I :: "'i set"   and J :: "'i set"   and f :: "('i \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "I \<inter> J = {}"   and "finite I"   and "finite J"   and "integrable (Pi\<^sub>M (I \<union> J) M) f" shows "integral\<^sup>L (Pi\<^sub>M (I \<union> J) M) f = LINT x|Pi\<^sub>M I M. LINT y|Pi\<^sub>M J M. f (merge I J (x, y))"
LCons_lprefix_conv: fixes x :: "'c"   and xs :: "'c llist"   and ys :: "'c llist" shows "LCons x xs \<sqsubseteq> ys = (\<exists>ys'. ys = LCons x ys' \<and> xs \<sqsubseteq> ys')"
trg_def: fixes \<mu> :: "'a arrow_of_spans_data" shows "trg \<mu> \<equiv> if arr \<mu> then mkObj (C.cod (Leg1 (Dom \<mu>))) else null"
cr_prod2_simps: fixes x :: "'h"   and A :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and a :: "'i"   and b :: "'h"   and c :: "'j" shows "cr_prod2 x A a (b, c) = (A a c \<and> x = b)"
prems: shows "is_tangle_diagram (basic b) \<and> is_tangle_diagram (a * as)"
get_min_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and t :: "'b trm"   and E :: "('b trm \<times> 'b trm) set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.get_min trm_ord t E = (if basic_superposition.min_trms trm_ord E = {} then t else SOME x. x \<in> basic_superposition.min_trms trm_ord E)"
surjective: fixes r :: "('b, 'c) state_scheme" shows "r = \<lparr>nu = nu r, m = m r, l = l r, hare = hare r, tortoise = tortoise r, \<dots> = more r\<rparr>"
is_processT6: fixes P :: "'b process" shows "\<forall>s X. (s @ [tick], {}) \<in> F P \<longrightarrow> (s, X - {tick}) \<in> F P"
linear_simps_5: fixes f :: "'a \<Rightarrow> 'b"   and s :: "real"   and v :: "'a" assumes "bounded_linear f" shows "f (s *\<^sub>R v) = s *\<^sub>R f v"
all_args_def: fixes f :: "'aa \<Rightarrow> ('x, 'y, 'z) prog" shows "all_args f = (\<Union>arg. {f arg})"