hcju/isabelleps huggingface dataset:
bind_lunit: fixes x :: "'e"   and f :: "'e \<Rightarrow> 'd option" shows "Some x \<bind> f = f x"
length_rbtreeify_g: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "0 < n"   and "n \<le> Suc (length kvs)" shows "length (snd (rbtreeify_g n kvs)) = Suc (length kvs) - n"
nop_onto_axioms: shows "nop_onto A (1\<^sub>\<nat>) f"
parSym: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q \<sim> Q \<parallel> P"
a_very_costrict': fixes x :: "'a" shows "(ad x = (1::'a)) = (x = (0::'a))"
UNIV_I: fixes x :: "'q" shows "x \<in> UNIV"
poly_poly_of_focus_monomial: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "poly (poly_of_focus x p) (monomial (1::'d) (monomial 1 x)) = p"
subo_subcls_trans: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and C' :: "char list"   and Cs' :: "char list list" assumes "Subobjs P C (Cs @ C' # Cs')" shows "\<forall>C''\<in>set Cs. (C'', C') \<in> (subcls1 P)\<^sup>+"
INF_mono: fixes B :: "'a::type set"   and A :: "'b::type set"   and f :: "'b::type \<Rightarrow> 'c::complete_lattice"   and g :: "'a::type \<Rightarrow> 'c::complete_lattice" assumes "\<And>m::'a::type. m \<in> (B::'a::type set) \<Longrightarrow> \<exists>n::'b::type\<in>A::'b::type set. (f::'b::type \<Rightarrow> 'c::complete_lattice) n \<le> (g::'a::type \<Rightarrow> 'c::complete_lattice) m" shows "\<Sqinter> ((f::'b::type \<Rightarrow> 'c::complete_lattice) ` (A::'b::type set)) \<le> \<Sqinter> ((g::'a::type \<Rightarrow> 'c::complete_lattice) ` (B::'a::type set))"
eq: shows "{0..<length dims0} = ind_in_set vars0 ` ({0..<length dims} \<inter> vars0)"
tri_next3_id: fixes f :: "face"   and v :: "nat" assumes "triangle f"   and "distinct (vertices f)"   and "v \<in> \<V> f" shows "f \<bullet> (f \<bullet> (f \<bullet> v)) = v"
connectedin_def: fixes X :: "'a topology"   and S :: "'a set" shows "connectedin X S \<equiv> S \<subseteq> topspace X \<and> connected_space (subtopology X S)"
assms_2: shows "w\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>3"
ActualActualE: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<A>\<^bold>\<A>\<phi> in v]" shows "[\<^bold>\<A>\<phi> in v]"
fps_deriv_nth: fixes f :: "'a fps"   and n :: "nat" shows "fps_deriv f $ n = of_nat (n + 1) * f $ (n + 1)"
trunc_bl2bin_len: fixes bl :: "bool list" shows "take_bit (length bl) (bl_to_bin bl) = bl_to_bin bl"
induct: fixes P :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and x1 :: "('a \<times> real) msllist"   and x2 :: "real \<Rightarrow> real"   and x3 :: "(real \<Rightarrow> real) list"   and Pa :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool" assumes "ms_closure P x1 x2 x3"   and "\<And>xs f basis. P xs f basis \<Longrightarrow> Pa xs f basis"   and "\<And>xs f basis g. \<lbrakk>ms_closure P xs f basis; Pa xs f basis; \<forall>\<^sub>F x in at_top. f x = g x\<rbrakk> \<Longrightarrow> Pa xs g basis"   and "\<And>xs f C b e basis. \<lbrakk>ms_closure P xs (\<lambda>x. f x - eval C x * b x powr e) (b # basis); Pa xs (\<lambda>x. f x - eval C x * b x powr e) (b # basis); is_expansion C basis; \<And>e'. e < e' \<Longrightarrow> f \<in> o(\<lambda>x. b x powr e'); ms_exp_gt e (ms_aux_hd_exp xs)\<rbrakk> \<Longrightarrow> Pa (MSLCons (C, e) xs) f (b # basis)"   and "\<And>xs f basis ys g. \<lbrakk>ms_closure P xs f basis; Pa xs f basis; ms_closure P ys g basis; Pa ys g basis\<rbrakk> \<Longrightarrow> Pa (plus_ms_aux xs ys) (\<lambda>x. f x + g x) basis"   and "\<And>xs f basis ys g. \<lbrakk>ms_closure P xs f basis; Pa xs f basis; ms_closure P ys g basis; Pa ys g basis\<rbrakk> \<Longrightarrow> Pa (times_ms_aux xs ys) (\<lambda>x. f x * g x) basis"   and "\<And>xs f b basis C e. \<lbrakk>ms_closure P xs f (b # basis); Pa xs f (b # basis); is_expansion C basis\<rbrakk> \<Longrightarrow> Pa (scale_shift_ms_aux (C, e) xs) (\<lambda>x. eval C x * b x powr e * f x) (b # basis)"   and "\<And>xs f basis. is_expansion_aux xs f basis \<Longrightarrow> Pa xs f basis" shows "Pa x1 x2 x3"
canonical_proj_ker: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "a_kernel R (RDirProd (R Quot I) (R Quot J)) (canonical_proj I J) = I \<inter> J"
qml_2: fixes \<phi> :: "\<o>" shows "Axioms.axiom (\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<phi>)"
lc_obj: fixes f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type" shows "lc (Obj f T) = (\<forall>l\<in>dom f. body (the (f l)))"
llist_all2_lnthD2: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'e llist"   and ys :: "'f llist"   and n :: "nat" assumes "llist_all2 P xs ys"   and "enat n < llength ys" shows "P (lnth xs n) (lnth ys n)"
1: shows "\<guillemotleft>\<l>[a] : \<I> \<otimes> a \<rightarrow> a\<guillemotright> \<and> \<I> \<otimes> \<l>[a] = (\<iota> \<otimes> a) \<cdot> local.inv \<a>[\<I>, \<I>, a]"
qSwapAbs_preserves_qGoodAbs1: fixes A :: "('f, 'g, 'e, 'd, 'h) qAbs"   and zs :: "'e"   and x :: "'d"   and y :: "'d" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGoodAbs A" shows "FixVars.qGoodAbs (A $[[x \<and> y]]_zs)"
lastMarkedIsMarkedElement: fixes M :: "('b \<times> bool) list" assumes "markedElements M \<noteq> []" shows "lastMarked M \<in> set (markedElements M)"
epi_compose_rev: fixes f :: "'g \<Rightarrow> 'h"   and G :: "('g, 'i) monoid_scheme"   and H :: "('h, 'j) monoid_scheme"   and g :: "'h \<Rightarrow> 'k"   and K :: "('k, 'l) monoid_scheme" assumes "f \<in> hom G H"   and "g \<in> hom H K"   and "g \<circ> f \<in> epi G K" shows "g \<in> epi H K"
intros_2: fixes c :: "('c, 'd) lit list"   and \<sigma> :: "'tp" assumes "\<forall>x\<in>nvC c. tpOfV x \<noteq> \<sigma>" shows "\<sigma> \<turnstile> c"
triangle_points_closer: fixes x :: "complex"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and y :: "complex" assumes "x \<in> convex hull {a, b, c}"   and "y \<in> convex hull {a, b, c}" shows "cmod (x - y) \<le> cmod (a - b) \<or> cmod (x - y) \<le> cmod (b - c) \<or> cmod (x - y) \<le> cmod (c - a)"
clock_simAction': fixes t :: "'s Trace"   and aec :: "'s odlist"   and a :: "'a"   and cec :: "'s odlist" assumes "t \<in> Clock.jkbpC"   and "toSet aec = agent_abs a t"   and "toSet cec = common_abs t" shows "set (local.clock_simAction a (cec, aec)) = set (jAction Clock.MC t a)"
map_fst_zip: fixes xs :: "'b list"   and ys :: "'c list" assumes "length xs = length ys" shows "map fst (zip xs ys) = xs"
mapi_branch_add_oob: fixes branch :: "(('b, 'c) fm list \<times> 'c) list"   and v :: "nat"   and v' :: "nat"   and xs :: "(nat \<times> nat) set"   and f :: "('b, 'c) fm \<Rightarrow> ('b, 'c) fm" assumes "length branch \<le> v" shows "mapi_branch (mapper f ({(v, v')} \<union> xs)) branch = mapi_branch (mapper f xs) branch"
seq: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and t :: "state"   and rs\<^sub>2 :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1 @ rs\<^sub>2, Undecided\<rangle> \<Rightarrow> t'"
r: shows "rhs t \<noteq> trm_rep (rhs t) S \<longrightarrow> (trm_rep (rhs t) S, rhs t) \<in> trm_ord"
cring_simprules_5: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
S\<^sub>i\<^sub>i: fixes x :: "i"   and y :: "i" shows "(E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E x \<^bold>\<leftarrow> E (dom x)) \<^bold>\<and> (E y \<^bold>\<leftarrow> E (cod y))"
filter_comm: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "filter (\<lambda>x. P x \<and> Q x) xs = filter (\<lambda>x. Q x \<and> P x) xs"
not_distinct_conv_prefix: fixes as :: "'a list" shows "(\<not> distinct as) = (\<exists>xs y ys. y \<in> set xs \<and> distinct xs \<and> as = xs @ y # ys)"
ed_ou_witness_union: fixes X :: "ed_ou_witness set set" assumes "\<forall>x\<in>X. open x" shows "open (\<Union> X)"
Vinv2_invariant: shows "reach v_TS \<subseteq> Vinv2"
eq_iff: fixes a :: "'a list"   and b :: "'a list" shows "(a = b) = (subseq b a \<and> subseq a b)"
y: shows "\<langle>k, y\<rangle> \<^bold>\<in> s"
that: fixes as :: "'edge list"   and n' :: "'node SDG_node" assumes "parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n'"   and "n' \<in> HRB_slice S"   and "V \<in> Use\<^bsub>SDG\<^esub> n'"   and "\<forall>n''. valid_SDG_node n'' \<and> parent_node n'' \<in> set (sourcenodes as) \<longrightarrow> V \<notin> Def\<^bsub>SDG\<^esub> n''" shows "thesis"
sq_norm_vec_vCons: fixes a :: "'b"   and v :: "'b vec" shows "\<parallel>vCons a v\<parallel>\<^sup>2 = sq_norm_conjugate a + \<parallel>v\<parallel>\<^sup>2"
sexp_CONS_D: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" assumes "CONS M N \<in> sexp" shows "M \<in> sexp \<and> N \<in> sexp"
\<ll>_ide_simp: fixes f :: "'a" assumes "ide f" shows "\<ll> f = \<l>[f]"
hRe_add: fixes x :: "complex star"   and y :: "complex star" shows "hRe (x + y) = hRe x + hRe y"
3: shows "\<guillemotleft>trnr\<^sub>\<epsilon> (g \<star> f\<^sup>*) \<rho> : g \<star> f\<^sup>* \<Rightarrow> g \<star> f\<^sup>*\<guillemotright> \<and> local.iso (trnr\<^sub>\<epsilon> (g \<star> f\<^sup>*) \<rho>)"
extended_def: fixes S :: "'b formula set" shows "extended S = \<Union> {extender n S |n. True}"
onorm_zero: shows "onorm (\<lambda>x. 0::'e) = 0"
n_max: fixes A :: "nat set"   and n :: "nat" assumes "A \<subseteq> {i. i \<le> n}"   and "A \<noteq> {}" shows "n_max A \<in> A \<and> (\<forall>x\<in>A. x \<le> n_max A)"
Quo2: fixes b :: "nat"   and a :: "nat" shows "b * Quo b a + a mod b = a"
HNextPart_allInput_or_NotAnInput: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HNextPart s s'"   and "Inv2a_innermost s p (dblock s' p)" shows "inp (dblock s' p) \<in> allInput s' \<union> {NotAnInput}"
contour_integral_lmul: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and c :: "complex" assumes "f contour_integrable_on g" shows "contour_integral g (\<lambda>x. c * f x) = c * contour_integral g f"
gt0_iff_gem1: fixes x :: "'b word" shows "(0 < x) = (x - 1 < x)"
map_rel_def: fixes K :: "('c \<times> 'e) set"   and V :: "('d \<times> 'f) set" shows "\<langle>K, V\<rangle>map_rel \<equiv> (K \<rightarrow> \<langle>V\<rangle>option_rel) \<inter> {(mi, m). dom mi \<subseteq> Domain K \<and> dom m \<subseteq> Range K}"
P: shows "P \<equiv> P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr,Up\<rbrakk>"
N_fact_le: fixes m :: "nat" shows "fb.N (fact m) \<le> 1 / (2 - 1) * 1 / 2 ^ m"
cexpr_sem_Add: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr"   and \<sigma> :: "nat \<Rightarrow> val"   and V :: "nat set" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : REAL"   and "\<sigma> \<in> space (state_measure V \<Gamma>)"   and "free_vars e1 \<subseteq> V"   and "free_vars e2 \<subseteq> V" shows "extract_real (cexpr_sem \<sigma> (e1 +\<^sub>c e2)) = extract_real (cexpr_sem \<sigma> e1) + extract_real (cexpr_sem \<sigma> e2)"
non_empty_max: fixes i :: "nat_int" shows "i \<noteq> bot \<longrightarrow> (\<exists>m. maximum i = m)"
natToVertexList_nths1: fixes f :: "face"   and v :: "nat"   and vs :: "nat list"   and es :: "nat list"   and n :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "vs = verticesFrom f v"   and "incrIndexList es |es| |vs|"   and "n \<le> |es|" shows "nths (take (Suc (es ! (n - 1))) vs) (set (take n es)) = removeNones (take n (natToVertexList v f es))"
crypts_mpair_1: fixes X :: "msg"   and Y :: "msg" shows "crypts {\<lbrace>X, Y\<rbrace>} \<subseteq> insert \<lbrace>X, Y\<rbrace> (crypts {X} \<union> crypts {Y})"
map_converter_id14: fixes b :: "'e \<Rightarrow> 'b"   and out :: "'f \<Rightarrow> 'c"   and res :: "('a, 'e, 'f, 'd) converter" shows "map_converter id b out id res = map_converter' b out res"
simps: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('c, 'b) justifying_execution" shows "(P \<turnstile> (E, ws) weakly_justified_by J) = (is_commit_sequence E J \<and> justification_well_formed P J \<and> committed_subset_actions J \<and> happens_before_committed_weak P E ws J \<and> value_written_committed P E J \<and> write_seen_committed ws J \<and> uncommitted_reads_see_hb P J \<and> committed_reads_see_committed_writes_weak ws J \<and> wf_action_translations E J)"
ide_cod: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "CC.ide (CC.cod f)"
simps_3: fixes t :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" shows "dual\<^sub>s\<^sub>t (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S) = receive\<langle>t\<rangle>\<^sub>s\<^sub>t # dual\<^sub>s\<^sub>t S"
elementLevel_def: fixes e :: "'b"   and t :: "('b \<times> bool) list" shows "elementLevel e t = length (markedElementsTo e t)"
image_cong: fixes M :: "'e set"   and N :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
cnf_of_encode_interfering_operator_exclusion_is: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_interfering_operator_exclusion \<Pi> t) = \<Union> (\<Union>(op\<^sub>1, op\<^sub>2)\<in>{(op\<^sub>1, op\<^sub>2). (op\<^sub>1, op\<^sub>2) \<in> set (\<Pi>\<^sub>\<O>) \<times> set (\<Pi>\<^sub>\<O>) \<and> index (\<Pi>\<^sub>\<O>) op\<^sub>1 \<noteq> index (\<Pi>\<^sub>\<O>) op\<^sub>2 \<and> are_operators_interfering op\<^sub>1 op\<^sub>2}. {{{(Operator k (index (\<Pi>\<^sub>\<O>) op\<^sub>1))\<inverse>, (Operator k (index (\<Pi>\<^sub>\<O>) op\<^sub>2))\<inverse>}} |k. k \<in> {0..<t}})"
RelStay: fixes P :: "pi"   and Q :: "pi"   and a :: "name"   and b :: "name" assumes "(P, Q) \<in> Rel"   and "a \<noteq> b" shows "([a\<noteq>b]P, Q) \<in> Rel"
cfbi: shows "f O b\<inverse> \<subseteq> b\<inverse>"
exp_of_nat_mult: fixes n :: "nat"   and x :: "'c" shows "exp (of_nat n * x) = exp x ^ n"
prj_char_4: shows "SPN_fgh.Prj\<^sub>1 = \<lbrakk>\<lbrakk>TfTgh.p\<^sub>1\<rbrakk>\<rbrakk>"
chamberD_simplex: fixes x :: "'a set" assumes "folding_g.chamber x" shows "x \<in> X"
elims: fixes x :: "'a literal"   and y :: "'a trm multiset" assumes "mset_lit x = y" obtains t :: "'a trm" and s :: "'a trm" where "x = Pos (Eq ta__ sa__)" and "y = {#ta__, sa__#}"   | t :: "'a trm" and s :: "'a trm" where "x = Neg (Eq t s)" and "y = {#t, t, s, s#}"
rComm1: fixes \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and xvec :: "name list"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "yvec \<sharp>* M"   and "yvec \<sharp>* K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* C" shows "Prop (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
new_tv_Suc_list: fixes n :: "nat"   and A :: "typ list" shows "new_tv n A \<longrightarrow> new_tv (Suc n) (TVar n # A)"
res_ts': shows "res ts' c = res ts c \<squnion> clm ts c"
CD_on_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.CD_on Pd ds X = (\<Union>d\<in>ds. Contracts.Cd Pd d X)"
tensor_vec_def: fixes v1 :: "'b vec"   and v2 :: "'b vec" shows "tensor_vec v1 v2 = Matrix.vec d (\<lambda>i. v1 $ encode1 i * v2 $ encode2 i)"
one_interval_def: shows "1 \<equiv> Interval (1::'b, 1::'b)"
key_sets_msg_eq: fixes X :: "msg" shows "key_sets_msg X X = {{}}"
simps_2: fixes x :: "('b, 'c) Combinators"   and v :: "('b, 'c) Combinators"   and va :: "('b, 'c) Combinators list" shows "list2policyR (x # v # va) = list2policyR (v # va) \<oplus> x"
filterlim_tendsto_add_at_top_iff: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F" shows "(LIM x F. f x + g x :> at_top) = filterlim g at_top F"
defval_conf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and T :: "ty"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "is_type P T" shows "P,h \<turnstile> default_val T :\<le> T"
split_minD: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "Tree_Set.split_min t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "x # inorder t' = inorder t"
add_point_sub_des: fixes point_set :: "'c set"   and block_collection :: "'c set multiset"   and p :: "'c" assumes "design point_set block_collection" shows "sub_design point_set block_collection (incidence_system.add_point point_set p) block_collection"
seqI: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c" assumes "C.VV.arr f"   and "C.VV.arr g"   and "C.VV.dom g = C.VV.cod f" shows "C.VV.seq g f"
while_inv_def: fixes p :: "'a"   and i :: "'a"   and x :: "'a" shows "while p inv i do x od = while p do x od"
intros_20: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> =n\<Rightarrow> t"   and "\<not> isAbr t" shows "\<Gamma>\<turnstile> \<langle>Catch c\<^sub>1 c\<^sub>2,Normal s\<rangle> =n\<Rightarrow> t"
sub_upd_fresh: fixes i :: "'b"   and p :: "('c, 'b) fm"   and f :: "'b \<Rightarrow> 'd"   and j :: "'d" assumes "i \<notin> nominals p" shows "sub (f(i := j)) p = sub f p"
invar_merge: fixes q1 :: "'a tree"   and q2 :: "'a tree" assumes "heap q1"   and "heap q2" shows "heap (merge q1 q2)"
extendID: fixes S :: "sequent"   and i :: "nat"   and xs :: "NominalSequents.var list"   and \<Gamma> :: "form multiset"   and \<Delta> :: "form multiset" assumes "extend S ( \<LM> At i xs  \<RM> \<Rightarrow>* \<LM> At i xs  \<RM>) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "At i xs \<in># \<Gamma> \<and> At i xs \<in># \<Delta>"
abs_eq: fixes x :: "'c" shows "diag x = of_fun (\<lambda>i j. if i = j then x else (0::'c))"
before_in_index: fixes l :: "'c list"   and x :: "'c"   and y :: "'c" assumes "set l = {x, y}"   and "length l = 2"   and "x \<noteq> y" shows "x < y in l = (index l x = 0)"
member_def: fixes xs :: "'b list"   and x :: "'b" shows "List.member xs x = (x \<in> set xs)"
\<oo>_Star: fixes r :: "'a language" shows "\<oo> (Star r)"
asIMOD_igSwap: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c) semDom"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'c) \<Rightarrow> 'c" shows "igSwap (asIMOD SEM) zs z1 z2 X = (\<lambda>val. if sWlsVal SEM val then X (val ^[z1 \<and> z2]_zs) else undefined)"
inj_on_imp_bij_betw: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" assumes "inj_on f A" shows "bij_betw f A (f ` A)"
dg_is_arrD_5: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" shows "\<AA>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
insertI: fixes A :: "'q set"   and a :: "'q" assumes "finite A" shows "finite (insert a A)"
is_serial_solution_for_problem_def: fixes \<Pi> :: "('b, 'c) strips_problem_scheme"   and \<pi> :: "'b strips_operator list" shows "is_serial_solution_for_problem \<Pi> \<pi> \<equiv> \<Pi>\<^sub>G \<subseteq>\<^sub>m execute_serial_plan (\<Pi>\<^sub>I) \<pi> \<and> list_all (\<lambda>op. ListMem op (\<Pi>\<^sub>\<O>)) \<pi>"
const_rv: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and c :: "real" assumes "measure_space M" shows "(\<lambda>x. c) \<in> rv M"
while_proof': fixes Q :: "('Q, 'L tree) HashMap.hashmap \<times> ('Q list \<times> 'Q list) \<times> (('Q, 'L) ta_rule, nat) HashMap.hashmap \<times> 'Q option \<Rightarrow> bool" assumes "\<forall>s. s \<in> wa_invar (det_wa_wa (brec_det_algo (build_rqrm \<delta>) Qi \<delta>)) \<and> s \<notin> wa_cond (det_wa_wa (brec_det_algo (build_rqrm \<delta>) Qi \<delta>)) \<longrightarrow> Q s" shows "Q loop"
BCh'_simps_4: fixes h :: "'c"   and X :: "'a set" assumes "h \<notin> {Xh d1h1, Xh d1h2, Xh d1h3}" shows "BCh' h X = {}"
next_symbol_predicts: fixes x :: "item"   and N :: "symbol"   and a :: "symbol list"   and k :: "nat" assumes "next_symbol x = Some N"   and "(N, a) \<in> \<RR>"   and "k = item_end x" shows "init_item (N, a) k \<in> Predict k {x}"
R23sD: fixes s :: "l2_state"   and t :: "l3_state" assumes "(s, t) \<in> R23s" shows "ik s = ik t \<inter> payload \<and> secret s = secret t \<and> progress s = progress t \<and> l1_state.signals s = l1_state.signals t \<and> chan s = local.abs (ik t) \<and> l2_state.bad s = l3_state.bad t"
f'_le: fixes x :: "real" assumes "x \<in> {real_of_float l..real_of_float u}" shows "f' x \<le> max (f' (real_of_float l)) (f' (real_of_float u))"
Cd_range': fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and d :: "'e"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch"   and "x \<in> Contracts.Cd Pd d X" shows "x \<in> X \<inter> Field (Pd d)"
2: fixes v :: "nat \<times> 'b" assumes "v \<in> gunodes A w" shows "snd v \<in> dom (fst ((p ## r) !! fst v))"
mat_multcol_def: fixes k :: "nat"   and a :: "'b"   and A :: "'b mat" shows "multcol k a A = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = j then a * A $$ (i, j) else A $$ (i, j))"
support_on_cong: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>x. x \<in> S \<Longrightarrow> (f x = (0::'b)) = (g x = (0::'c))" shows "support_on S f = support_on S g"
hom_one: fixes a :: "'c" shows "poly 1 a = (1::'c)"
disc_17: fixes x21 :: "variable"   and x22 :: "typ" shows "\<not> is_Abs (Fv x21 x22)"
ACI_norm_alt_Plus: fixes r :: "'c rexp"   and s :: "'c rexp"   and t :: "'c rexp" assumes "ACI_norm_alt r = Plus s t" shows "\<exists>s t. r = Plus s t"