hcju/isabelleps huggingface dataset:
parts_crypt_prikey_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and K :: "key"   and A :: "nat" assumes "s \<turnstile> s'"   and "Crypt K (PriKey A) \<in> parts (used s')"   and "Crypt K (PriKey A) \<notin> parts (used s)" shows "(\<exists>n. K = Auth_ShaKey n \<and> (Asset n, Crypt (Auth_ShaKey n) (PriKey A)) \<in> s') \<or> {PriKey A, Key K} \<subseteq> spied s'"
Cons: fixes x :: "'b"   and xs' :: "'b list" assumes "xs = x # xs'"   and "R x y"   and "list_all2 R xs' ys'" shows "phi"
intTF_def: shows "M.intTF \<equiv> InfModel.intTF"
cmod_eq_Im: fixes z :: "complex" assumes "is_imag z" shows "cmod z = \<bar>Im z\<bar>"
coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b \<le> c" shows "min a b \<le> c"
epsclo_def: fixes Q :: "hf set" shows "epsclo Q \<equiv> nfa.states M \<inter> (\<Union>q\<in>Q. {q'. (q, q') \<in> (eps M)\<^sup>*})"
restrict_in: fixes x :: "'d"   and A :: "'d set"   and m :: "'d \<Rightarrow> 'e option" assumes "x \<in> A" shows "(m|\<^bsub>A\<^esub>) x = m x"
eq: shows "C =m \<Prod>\<^sub># (factors_of_factor_tree (Factor_Node f l r))"
unit_interval_convex_hull: shows "cbox (0::'b) One = convex hull {x. \<forall>i\<in>Basis. x \<bullet> i = 0 \<or> x \<bullet> i = 1}"
next_InvTempUpright: fixes s :: "('c, 'd) configuration stream" assumes "holds InvTempUpright s"   and "next s" shows "nxt (holds InvTempUpright) s"
assms: shows "rel_option (R \<upharpoonleft> P \<otimes> Q) x y"
measure_le_0_iff: fixes M :: "'d measure"   and X :: "'d set" shows "(Sigma_Algebra.measure M X \<le> 0) = (Sigma_Algebra.measure M X = 0)"
compI_1: fixes CC :: "('b, 'c, 'd) category_scheme"   and f :: "'c"   and g :: "'c" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>"   and "g \<in> Ar\<^bsub>CC\<^esub>"   and "Cod\<^bsub>CC\<^esub> f = Dom\<^bsub>CC\<^esub> g" shows "g \<bullet>\<^bsub>CC\<^esub> f \<in> Ar\<^bsub>CC\<^esub>"
assms: shows "safe_distance_1r < s\<^sub>o - s\<^sub>e"
roundup_nat_ge_8_help: fixes M :: "bit list"   and emBits :: "nat" assumes "length (sha1 M) + sLen + 16 \<le> emBits" shows "8 \<le> roundup emBits 8 * 8 - (length (sha1 M) + 8)"
Assign_blah: fixes t :: "char list \<Rightarrow> int"   and x :: "char list"   and e :: "aexp"   and v :: "int" assumes "t x = aval e (t(x := v))" shows "(x ::= e, t(x := v)) \<Rightarrow> t"
hyps: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat" assumes "outstanding_refs (Not \<circ> is_volatile_Read\<^sub>s\<^sub>b) xs \<inter> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "reads_consistent pending_write \<O> m xs" shows "reads_consistent pending_write \<O> (flush sb m) xs"
wellformed_all_edges: fixes G :: "nat set \<times> nat set set" assumes "uwellformed G" shows "uedges G \<subseteq> all_edges (uverts G)"
inf_drop_def: fixes i :: "nat"   and s :: "nat \<Rightarrow> 'a" shows "inf_drop i s \<equiv> \<lambda>j. s (i + j)"
FEval_cong: fixes A :: "form"   and MI :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)" shows "\<forall>e1 e2. (\<forall>x. x \<in> set (fv A) \<longrightarrow> e1 x = e2 x) \<longrightarrow> FEval MI e1 A = FEval MI e2 A"
cod_simp: fixes f :: "'b \<times> 'b" assumes "B.VV.arr f" shows "B.VV.cod f = B.VxV.cod f"
dghm_HomCod: shows "\<FF>\<lparr>HomCod\<rparr> = \<BB>"
ordered_nats: shows "0 \<le> i \<and> i < j \<and> j < l \<and> l < card X"
qAFreshAll_qFreshAll_preserves_alphaAll: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and z :: "'var"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "(qGood X \<longrightarrow> qAFresh zs z X \<longrightarrow> (\<forall>X'. X #= X' \<longrightarrow> qFresh zs z X')) \<and> (qGoodAbs A \<longrightarrow> qAFreshAbs zs z A \<longrightarrow> (\<forall>A'. A $= A' \<longrightarrow> qFreshAbs zs z A'))"
all_info2: shows "set (characterize_consistent_signs_at_roots p qs2) \<subseteq> set signs2"
inducts: fixes x1 :: "(C, BD) sequent list"   and x2 :: "(C, BD) sequent"   and P :: "(C, BD) sequent list \<Rightarrow> (C, BD) sequent \<Rightarrow> bool" assumes "(x1, x2) \<in> g3mod1"   and "\<And>A. P [ \<Empt> \<Rightarrow>* \<LM> A  \<RM>] ( \<Empt> \<Rightarrow>* \<LM> \<box> A  \<RM>)"   and "\<And>A. P [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>] ( \<LM> \<diamond> A  \<RM> \<Rightarrow>* \<Empt>)" shows "P x1 x2"
ET_Source_Target: fixes SA :: "('d, 'e, 'f) seqauto"   and ST :: "('d, 'e, 'f) status"   and TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" assumes "SA \<in> SAs (HA ST)"   and "TS \<subseteq> ET ST"   and "States SA \<inter> Source TS = {}" shows "States SA \<inter> Target TS = {}"
stern_brocot_iterate_def: shows "stern_brocot_iterate \<equiv> map_tree mediant (stern_brocot_iterate_aux I)"
filter_False: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> P x" shows "filter P xs = []"
step_2_def: fixes A :: "'b mat" shows "step_2 A = step_2_main (dim_row A) 0 A"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_on_left_moving_2_B ires rs (l, r) = (\<exists>ml mr rn. l = Bk \<up> ml @ Oc # Bk # Oc # ires \<and> r = Bk \<up> mr @ Oc \<up> Suc rs @ Bk \<up> rn \<and> Suc 0 < ml + mr \<and> 0 < mr)"
dghm_up_components_2: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<CC> :: "V"   and \<phi> :: "V \<Rightarrow> V" shows "dghm_up I \<AA> \<CC> \<phi>\<lparr>ArrMap\<rparr> = (\<lambda>f\<in>\<^sub>\<circ>\<CC>\<lparr>Arr\<rparr>. \<lambda>i\<in>\<^sub>\<circ>I. \<phi> i\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>)"
not_pred_eq: fixes P :: "unit Predicate.pred" shows "Predicate.not_pred P = (if pred.eval P () then \<bottom> else Predicate.single ())"
c_def: shows "c__ = unit_factor (gcd G1 G2)"
HStartBallot_blocksOf_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p"   and "p \<noteq> q" shows "blocksOf s' q \<subseteq> blocksOf s q"
ipurge_ref_aux_less_def: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_ref_aux_less I D U xs X \<equiv> {x \<in> X. \<forall>v\<in>sinks_aux_less I D U xs. (v, D x) \<notin> I}"
power2_le_imp_le: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq (power.power one times x 2) (power.power one times y 2)"   and "less_eq zero y" shows "less_eq x y"
next_plane0_incr_finV: fixes g' :: "graph"   and p :: "nat"   and g :: "graph" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "minGraphProps g" shows "\<forall>v\<in>\<V> g. v \<in> \<V> g' \<and> ((\<forall>f\<in>\<F> g. v \<in> \<V> f \<longrightarrow> final f) \<longrightarrow> (\<forall>f\<in>\<F> g'. v \<in> \<V> f \<longrightarrow> f \<in> \<F> g))"
ipath_def: fixes E :: "('a \<times> 'a) set"   and r :: "nat \<Rightarrow> 'a" shows "ipath E r \<equiv> \<forall>i. (r i, r (Suc i)) \<in> E"
head_fun: fixes s :: "('a, 'b) tm" shows "head (fun s) = head s"
cross_basis_5: shows "cross3 (axis 3 1) (axis 1 1) = axis 2 1"
degree_div_less: fixes p :: "'a poly"   and r :: "'a poly" assumes "p \<noteq> 0"   and "r dvd p"   and "degree r \<noteq> 0" shows "degree (p div r) < degree p"
intro: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "ergodic_qmpt M T"   and "conservative_mpt M T" shows "ergodic_conservative_mpt M T"
intro: fixes T :: "real set"   and X :: "'b set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b" assumes "local_lipschitz T X f"   and "\<And>x. x \<in> X \<Longrightarrow> continuous_on T (\<lambda>t. f t x)"   and "open T"   and "open X" shows "ll_on_open_axioms T f X"
terminalE: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "category C"   and "category.terminal C b"   and "partial_magma.ide C a" obtains f :: "'a" where "partial_magma.in_hom C f a b" and "\<And>f'. partial_magma.in_hom C f' a b \<Longrightarrow> f' = f"
intro: fixes c :: "nat \<times> nat \<Rightarrow> int"   and N :: "nat" assumes "Graph.V c \<subseteq> {0..<N}" shows "Edka_Impl_axioms c N"
inf_bot_right: fixes x :: "'a multiset" shows "x \<inter># \<Empt> = \<Empt>"
UnionI: fixes X :: "'a set"   and C :: "'a set set"   and A :: "'a" assumes "X \<in> C"   and "A \<in> X" shows "A \<in> \<Union> C"
deg_nzero_nzero: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "deg Zp p \<noteq> 0" shows "p \<noteq> \<zero>\<^bsub>UP Zp\<^esub>"
float_divl: fixes prec :: "nat"   and x :: "float"   and y :: "float" shows "real_of_float (float_divl prec x y) \<le> real_of_float x / real_of_float y"
asymmetric_inf: fixes x :: "'a" shows "asymmetric x = irreflexive_inf (x * x)"
slice_replace_prefix: fixes R :: "'c list set"   and \<phi> :: "'c MFOTL.formula"   and \<pi> :: "(char list \<times> 'c list) prefix"   and \<sigma> :: "(char list \<times> 'c list) trace" assumes "prefix_of (pmap_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> R) \<pi>) \<sigma>" shows "map_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> R) (replace_prefix \<pi> \<sigma>) = map_\<Gamma> (\<lambda>D. D \<inter> relevant_events \<phi> R) \<sigma>"
dim_subset: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T" shows "local.dim S \<le> local.dim T"
kop_mono_iff: fixes f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'e \<Rightarrow> 'd set" shows "(op\<^sub>K \<circ> f = op\<^sub>K \<circ> g) = (f = g)"
permutation_of_cycle: fixes cs :: "'a list" shows "permutation (cycle_of_list cs)"
UP_subring_compose_monom: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "a \<in> S" shows "Cring_Poly.compose R (monom (UP R) a n) g = Cring_Poly.compose (R\<lparr>carrier := S\<rparr>) (monom (UP (R\<lparr>carrier := S\<rparr>)) a n) g"
euler_liar_mult': fixes n :: "nat"   and y :: "int"   and x :: "int" assumes "1 < n"   and "coprime y (int n)"   and "euler_liar y n"   and "euler_witness x n" shows "euler_witness (x * y) n"
acquired_append_Prog\<^sub>s\<^sub>b: fixes pending_write :: "bool"   and sb :: "'b memref list"   and p\<^sub>1 :: "'b"   and p\<^sub>2 :: "'b"   and mis :: "instr list"   and S :: "nat set" shows "acquired pending_write (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ [Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis])) S = acquired pending_write (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S"
assms: shows "aEvalUni (LessUni (a, b, c)) x \<noteq> aEvalUni (LessUni (a, b, c)) y \<and> x < y"
pathStarts: fixes subs :: "'b \<Rightarrow> 'b set"   and gamma :: "'b"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool" shows "path subs gamma P 0 = gamma"
step_3: fixes gpv :: "('b, 'a, 'c) gpv" shows "interaction_bound' (gpv_stop gpv) = interaction_bound'' gpv"
stop_type: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" shows "\<turnstile> \<Gamma>,\<S>,P {Stop} \<Gamma>,\<S>,P"
lcm_proj2_iff_nat: fixes m :: "nat"   and n :: "nat" shows "(lcm m n = n) = (m dvd n)"
\<sigma>\<^sub>O\<^sub>T\<^sub>O\<^sub>Y_def: shows "\<sigma>\<^sub>O\<^sub>T\<^sub>O\<^sub>Y \<equiv> {(toy_init, \<Gamma>\<^sub>T\<^sub>O\<^sub>Y PToy)}"
const_ring_hom: fixes R :: "('c, 'd) ring_scheme" assumes "UP_ring R" shows "(\<lambda>a. monom (UP R) a 0) \<in> ring_hom R (UP R)"
Polygamma_approx_1_real: fixes x :: "real"   and m :: "nat" assumes "0 < x" shows "Polygamma_approx (Suc 0) m x = ln x - 1 / (2 * x) + stirling_sum (Suc 0) m x"
inv_closed: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> x \<in> carrier P"
ale_minus: fixes x :: "ant"   and y :: "ant" assumes "x \<le> y" shows "- y \<le> - x"
star_trans_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.B2_algebra plus times one zero less_eq less star" shows "times (star x) (star x) = star x"
cod_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "CCC.cod f = (if arr (fst f) \<and> CC.arr (snd f) then (cod (fst f), CC.cod (snd f)) else (null, CC.null))"
bounded_clinear_zero: shows "bounded_clinear (\<lambda>x. 0::'e)"
prod_mOp_mem_i: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and X :: "'d \<Rightarrow> 'e"   and i :: "'d" assumes "\<forall>j\<in>I. aGroup (A j)"   and "X \<in> carr_prodag I A"   and "i \<in> I" shows "prod_mOp I A X i = -\<^sub>a\<^bsub>A i\<^esub> X i"
jkbpCn_length: fixes t :: "'s Trace"   and n :: "nat" assumes "t \<in> jkbpC\<^bsub>n\<^esub>" shows "tLength t = n"
simps: fixes s1 :: "'e"   and s2 :: "('a, 'f, 'g, 'h) SLin_state_scheme" shows "P20 (s1, s2) = Ball (abortVals s2) ((\<preceq>) (dstate s2))"
secrecy_lemma: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and NA :: "msg"   and K :: "nat"   and NB :: "msg" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> otway" shows "Says Server B \<lbrace>NA, Crypt (shrK A) \<lbrace>NA, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs \<longrightarrow> Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs \<longrightarrow> Key K \<notin> analz (knows Spy evs)"
assms_3: shows "Prj\<^sub>1\<^sub>1 \<cdot> h = Prj\<^sub>1\<^sub>1 \<cdot> h'"
cong_add_lcancel: fixes a :: "'a"   and x :: "'a"   and y :: "'a"   and n :: "'a" shows "[a + x = a + y] (mod n) = [x = y] (mod n)"
image_compactin: fixes X :: "'d topology"   and S :: "'d set"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e" assumes "compactin X S"   and "continuous_map X Y f" shows "compactin Y (f ` S)"
card_M_le_sum_card_Mi: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and L :: "'b \<Rightarrow> nat" assumes "matching V E M"   and "OSC L E" shows "card M \<le> (\<Sum>i\<in>L ` V. card (matching_i i V E M L))"
u: shows "(s \<cdot> r) \<^sup>@ Suc i = u"
code_lemmas_4: fixes cs :: "QDelta ns_constraint list"   and asi :: "('a, ('a \<times> QDelta atom) list) mapping"   and tv :: "((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list"   and s :: "('a, QDelta) state" shows "solution_simplex' (cs, (asi, tv), s) = \<langle>from_ns (solution_nsc \<V> ((asi, tv), s)) cs\<rangle>"
assms: fixes Pd :: "'d \<Rightarrow> ('x \<times> 'x) set"   and Ch :: "'h \<Rightarrow> 'x set \<Rightarrow> 'x set"   and d :: "'d"   and Pd' :: "('x \<times> 'x) set"   and y :: "'x" assumes "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch"   and "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (Pd(d := Pd')) Ch"   and "d \<in> ds"   and "y \<in> \<phi> (Pd(d := Pd')) Ch ds"   and "y \<in> Field (Pd d)"   and "\<forall>x\<in>dX (\<phi> Pd Ch ds) d. x \<noteq> y \<and> (x, y) \<in> Pd d" shows "False"
simps: fixes p :: "'a \<times> 'b"   and a :: "'a" shows "fstsp p a = (a = fst p)"
seq_imp_conv: fixes s :: "'a"   and ss :: "('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(s, ss) \<in> seq ars" shows "(s, map (Pair True) ss) \<in> conv ars \<and> lst_conv (s, map (Pair True) ss) = lst (s, ss) \<and> labels (s, ss) = labels_conv (s, map (Pair True) ss)"
elims: fixes x :: "'c"   and xa :: "('c \<times> 'd) list"   and y :: "('c \<times> 'd) list" assumes "delete_aux x xa = y" obtains k :: "'c" where "x = ka__" and "xa = []" and "y = []"   | k :: "'c" and k' :: "'c" and v :: "'d" and xs :: "('c \<times> 'd) list" where "x = k" and "xa = (k', v) # xs" and "y = (if k = k' then xs else (k', v) # delete_aux k xs)"
set_iterator_rev_linord_def: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'a set" shows "set_iterator_rev_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>x y. y \<le> x)"
uQ\<phi>: shows "unitary (Q \<circ> \<phi>)"
eventually_at_right_field: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "eventually P (at_right x) = (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
star_inductr_var_equiv: fixes y :: "'a"   and x :: "'a" shows "(y ; x \<le> y) = (y ; x\<^sup>\<star> \<le> y)"
tr_s_mcallI: fixes var :: "char list"   and T :: "x \<Rightarrow> x option"   and var' :: "char list"   and x :: "x"   and x' :: "x"   and y_y'_list :: "(x \<times> x) list"   and meth :: "char list" assumes "(case T (x_var var) of None \<Rightarrow> var | Some (x_var var') \<Rightarrow> var' | Some x_this \<Rightarrow> var) = var'"   and "(case T x of None \<Rightarrow> x | Some x' \<Rightarrow> x') = x'"   and "list_all (\<lambda>f. f) (map (\<lambda>(y_XXX, y_'). (case T y_XXX of None \<Rightarrow> y_XXX | Some x' \<Rightarrow> x') = y_') y_y'_list)" shows "tr_s T (s_call var x meth (map (\<lambda>(y_XXX, y_'). y_XXX) y_y'_list)) (s_call var' x' meth (map (\<lambda>(y_XXX, y_'). y_') y_y'_list))"
rel_fundef_body_nth: fixes fd1 :: "('dyn, 'a, 'b, 'c, 'opinl) Inca.instr fundef"   and fd2 :: "('dyn, 'a, 'b, 'c, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat" assumes "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"   and "pc < length (body fd1)" shows "body fd1 ! pc = norm_instr (body fd2 ! pc)"
strict_mono_less_oLimit: fixes f :: "nat \<Rightarrow> ordinal"   and n :: "nat" assumes "OrdinalInduct.strict_mono f" shows "f n < oLimit f"
wf_sort_def: fixes sub :: "(String.literal \<times> String.literal) set"   and S :: "String.literal set" shows "wf_sort sub S = (normalized_sort sub S \<and> sort_ex sub S)"
strict_mono_sets_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b set" shows "strict_mono_sets A f \<equiv> \<forall>x\<in>A. \<forall>y\<in>A. x < y \<longrightarrow> f x \<lless> f y"
lnth_0_conv_lhd: fixes xs :: "'c llist" assumes "\<not> lnull xs" shows "lnth xs 0 = lhd xs"
A8_DAcc_level0: shows "DAcc level0 sA8 = {sA9}"
Sup_downset_id: shows "Sup \<circ> \<down> = id"
image_subset_iff: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'b set" shows "(\<P> f A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
intro: fixes get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>h new_shadow_root_ptr h' r ptr'. \<lbrakk>h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr; h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'; r \<in> get_shadow_root_locs ptr'\<rbrakk> \<Longrightarrow> r h h'" shows "l_new_shadow_root_get_shadow_root_axioms get_shadow_root_locs"
map_bset_eqvt: fixes p :: "perm"   and f :: "'c \<Rightarrow> 'a"   and B :: "'c set['b]" shows "p \<bullet> map_bset f B = map_bset (p \<bullet> f) (p \<bullet> B)"
assms_2: shows "transition stateA stateB F0 decisionVars"
not\<^sub>r_semantics: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr" shows "w \<Turnstile>\<^sub>r not\<^sub>r \<phi> = (\<not> w \<Turnstile>\<^sub>r \<phi>)"
prod_dfs_inj: fixes x :: "nat \<times> nat"   and i1 :: "nat"   and i2 :: "nat"   and j1 :: "nat"   and j2 :: "nat"   and k :: "nat" assumes "prod_is_node A B x"   and "dfa_is_node A i1"   and "dfa_is_node B i2"   and "dfa_is_node A j1"   and "dfa_is_node B j2"   and "fst (prod_dfs A B x) ! i1 ! i2 = Some k"   and "fst (prod_dfs A B x) ! j1 ! j2 = Some k" shows "(i1, i2) = (j1, j2)"
alt_effective_matrix_tensor_elements: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and i :: "nat"   and M2 :: "'b list list"   and M3 :: "'b list list"   and j :: "nat" assumes "Matrix_Tensor.mult id f"   and "(i < mult.row_length M2 * mult.row_length M3 \<and> j < length M2 * length M3) \<and> mat (mult.row_length M2) (length M2) M2 \<and> mat (mult.row_length M3) (length M3) M3" shows "mult.Tensor f M2 M3 ! j ! i = f (M2 ! (j div length M3) ! (i div mult.row_length M3)) (M3 ! (j mod length M3) ! (i mod mult.row_length M3))"
implies_dist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<leadsto> y \<sqinter> z = (x \<leadsto> y) \<sqinter> (x \<leadsto> z)"
t_C_p_2: fixes t :: "nat" assumes "t_C \<le> t"   and "enat t < llength Sts" shows "C \<in> P_of_wstate (lnth Sts t)"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t_mono: fixes M :: "('a \<times> ('b, 'c) stateful_strand_step) list"   and N :: "('a \<times> ('b, 'c) stateful_strand_step) list" assumes "set M \<subseteq> set N" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t M \<subseteq> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t N"
assms: shows "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"
Mp_coeff: fixes f :: "int poly"   and i :: "nat" shows "coeff (Mp f) i = M (coeff f i)"
CastRedsAddr: fixes 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)"   and a :: "nat"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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 D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>addr a,s'\<rangle>"   and "hp s' a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "P \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>addr a,s'\<rangle>"
r_tl_def: shows "r_tl \<equiv> let r = Cn 1 r_pdec1 [r_dec] in Cn 1 r_ifz [recf.Id 1 0, Z, Cn 1 r_ifz [r, Z, Cn 1 S [Cn 1 r_prod_encode [Cn 1 r_dec [r], Cn 1 r_pdec22 [r_dec]]]]]"
minGraphProps9': fixes g :: "graph"   and f :: "face"   and v :: "nat" assumes "minGraphProps g"   and "f \<in> \<F> g"   and "v \<in> \<V> f" shows "v < countVertices g"
simps_8: fixes uv :: "Unboxed.type"   and x :: "'a" shows "norm_instr (IStoreUbx uv x) = Inca.instr.IStore x"
Prover9Argument: assumes "\<not> E (\<^bold>\<tau>x. \<Phi> x) \<longrightarrow> (\<^bold>\<exists>x. G x (\<^bold>\<tau>x. \<Phi> x) \<and> C x)"   and "\<And>x. atomic (G x)" shows "E g"
bb_0: fixes ord :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "1 < card X"   and "finite F"   and "F \<subseteq> P[X]"   and "f \<in> F"   and "\<And>f'. f' \<in> F \<Longrightarrow> homogeneous f'"   and "\<And>f'. f' \<in> F \<Longrightarrow> poly_deg f' \<le> poly_deg f"   and "0 < poly_deg f"   and "ideal {f} \<noteq> ideal F" shows "pm_powerprod.bb ord X F 0 \<le> max (pm_powerprod.aa X f F 1) (pm_powerprod.bb ord X F 1)"
bigo_zero: fixes g :: "'a \<Rightarrow> 'b" shows "0 \<in> O(g)"
completeness: fixes z :: "(nat, nat) form list"   and p :: "(nat, nat) form" assumes "\<forall>e f g. e,f,g,z \<Turnstile> p" shows "z \<turnstile> p"
assms_4: fixes n :: "nat" assumes "R n" shows "Q (Suc n)"
is_processT5_S1: fixes s :: "'a event list"   and X :: "'a event set"   and P :: "'a process"   and Y :: "'a event set" assumes "(s, X) \<in> F P"   and "(s, X \<union> Y) \<notin> F P" shows "\<exists>c. c \<in> Y \<and> (s @ [c], {}) \<in> F P"