hcju/isabelleps huggingface dataset:
size_list_simp_2: fixes xs :: "'d list" shows "length xs = (if xs = [] then 0 else Suc (length (tl xs)))"
None_eq_map_option_iff: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c option" shows "(None = map_option f x) = (x = None)"
interior_of_topspace: fixes X :: "'c topology" shows "X interior_of topspace X = topspace X"
9: shows "M \<in> {G. proper_filter G \<and> F \<subseteq> G \<or> G = {}} \<and> (\<forall>X\<in>{G. proper_filter G \<and> F \<subseteq> G \<or> G = {}}. M \<subseteq> X \<longrightarrow> X = M)"
simps_2: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and f :: "'c \<times> 'c"   and fs :: "('c \<times> 'c) list"   and keep :: "('c \<times> 'c) list"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" shows "SecurityInvariant_withOffendingFlows.minimalize_offending_overapprox sinvar (f # fs) keep G nP = (if sinvar (delete_edges_list G (fs @ keep)) nP then SecurityInvariant_withOffendingFlows.minimalize_offending_overapprox sinvar fs keep G nP else SecurityInvariant_withOffendingFlows.minimalize_offending_overapprox sinvar fs (f # keep) G nP)"
xor_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<oplus> b) \<oplus> c = a \<oplus> b \<oplus> c"
le_Liminf_iff: fixes C :: "'c"   and F :: "'d filter"   and X :: "'d \<Rightarrow> 'c" shows "(C \<le> Liminf F X) = (\<forall>y<C. \<forall>\<^sub>F x in F. y < X x)"
simps_215: fixes x131 :: "expr"   and x132 :: "expr"   and x133 :: "expr"   and x121 :: "expr"   and x122 :: "expr" shows "if (x131) x132 else x133 \<noteq> x121;; x122"
simps_6: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cAmountF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cAmountF"
pref_profile_unique_favorites_axioms_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "pref_profile_unique_favorites_axioms R \<equiv> has_unique_favorites R"
pd_intro: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and l :: "'a"   and k :: "'a"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "IFC prob"   and "IFC_def.is_pd prob l k"   and "IFC_def.is_path prob \<pi>"   and "\<pi> 0 = k"   and "\<pi> n = IFC_def.return prob" obtains i :: "nat" where "i \<le> n" and "\<pi> i = l"
span_superset: fixes S :: "'b set" shows "S \<subseteq> vs1.span S"
ran_restrict_map_def: fixes f :: "'c \<Rightarrow> 'd option"   and B :: "'d set" shows "f\<upharpoonleft>\<^bsub>B\<^esub> = (\<lambda>x. f x \<bind> (\<lambda>v. if v \<in> B then Some v else None))"
boundI: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d" assumes "\<And>x. x \<in> X \<Longrightarrow> r x b" shows "bound X r b"
strongly_strategyproof: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R"   and "i \<in> agents"   and "total_preorder_on alts Ri'" shows "strongly_strategyproof_profile R i Ri'"
K_parcontractE: fixes r :: "comb" assumes "K \<Rrightarrow>\<^sup>1 r" obtains "r = K"
commuteUnionFinestpart: fixes XX :: "'c set set" shows "\<Union> (finestpart ` XX) = finestpart (\<Union> XX)"
step: shows "\<forall>ts. order.greater (length ss) (length ts) \<longrightarrow> ts \<in> lists S \<longrightarrow> sum_list ts = (0::'w) \<longrightarrow> Abs_freelist ts \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
INF_Inf: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type"   and X :: "'c::type set" assumes "class.refinement_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Inf::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` (X::'c::type set)) = Inf {f x |x::'c::type. x \<in> X}"
single_valued_relcomp: fixes r :: "('b \<times> 'c) set"   and s :: "('c \<times> 'd) set" assumes "single_valued r"   and "single_valued s" shows "single_valued (r O s)"
space_in_prod_algebra: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
y_1: shows "y \<in> B\<^sub>2 - (B\<^sub>1 - {x})"
VInter_iff: fixes A :: "V"   and a :: "V" assumes "A \<noteq> 0" shows "(a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> A) = (\<forall>x\<in>elts A. a \<in>\<^sub>\<circ> x)"
progress_binop_relop_i: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and cs :: "e list"   and t :: "t"   and e :: "b_e"   and iop :: "binop_i"   and irop :: "relop_i"   and s :: "s"   and vs :: "v list"   and i :: "nat" assumes "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [t, t]"   and "is_int_t t"   and "const_list cs"   and "e = Binop_i t iop \<or> e = Relop_i t irop" shows "\<exists>a s' vs' es'. \<lparr>s;vs;cs @ [$e]\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
monic_0_iff: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "(punit.monic p = 0) = (p = 0)"
ann_matches_imp_assertionsR': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a :: "('e, 'd, 'f) ann"   and c :: "('e, 'd, 'f) com"   and a' :: "'e set"   and Q :: "'e set"   and A :: "'e set" assumes "ann_matches \<Gamma> \<Theta> a c"   and "a' \<in> pre_set a" shows "assertionsR \<Gamma> \<Theta> Q A a c a'"
pnormal_id: 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" shows "semiring_0.pnormal zero p = (0 < length p \<and> last p \<noteq> zero)"
abs_minus_cancel: fixes a :: "'a" shows "\<bar>- a\<bar> = \<bar>a\<bar>"
IH: fixes x :: "complex" assumes "0 < Re x" shows "Polygamma_approx (Suc (Suc j)) m x = stirling_sum' (Suc j) m x"
length_tabulate: fixes f :: "nat \<Rightarrow> 'b"   and x :: "nat"   and n :: "nat" shows "length (tabulate f x n) = n"
UN_extend_simps_10: fixes B :: "'c \<Rightarrow> 'a set"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "(\<Union>a\<in>A. B (f a)) = \<Union> (B ` f ` A)"
sum_cls_val_process0: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and pf2 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market" assumes "portfolio pf1"   and "portfolio pf2" shows "\<forall>w. cls_val_process Mkt (qty_sum pf1 pf2) 0 w = cls_val_process Mkt pf1 0 w + cls_val_process Mkt pf2 0 w"
inj_on_\<pp>_G: shows "inj_on \<pp> G"
elims_1: fixes x :: "'v"   and xa :: "('v \<times> 'w \<times> 'v) list"   and xb :: "'v"   and y :: "bool" assumes "is_path x xa xb = y" obtains v :: "'v" and v' :: "'v" where "x = vb__" and "xa = []" and "xb = v'a__" and "y = (vb__ = v'a__ \<and> v'a__ \<in> V)"   | v :: "'v" and v1 :: "'v" and w :: "'w" and v2 :: "'v" and p :: "('v \<times> 'w \<times> 'v) list" and v' :: "'v" where "x = v" and "xa = (v1, w, v2) # p" and "xb = v'" and "y = (v = v1 \<and> (v1, w, v2) \<in> E \<and> is_path v2 p v')"
l2_inv4'_def: shows "l2_inv4' \<equiv> {s. \<forall>Rb A B gnx. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<longrightarrow> in_progress (progress s Rb) xsk \<longrightarrow> guessed_frame Rb xgnx = Some gnx \<longrightarrow> guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny)))}"
l8_7: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Per A B C"   and "Per A C B" shows "B = C"
inversion_sym: shows "inversion = reciprocal \<circ> conjugate"
uint32_divmod_def: fixes x :: "uint32"   and y :: "uint32" shows "uint32_divmod x y = (if y = 0 then (undefined (div) x 0, undefined (mod) x 0) else (x div y, x mod y))"
bounded_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "path g" shows "bounded (path_image g)"
induct: fixes P :: "real poly list \<Rightarrow> real \<Rightarrow> bool"   and a0 :: "real poly list"   and a1 :: "real" assumes "\<And>p x. P [p] x"   and "\<And>p q x. P [p, q] x"   and "\<And>p q r ps x. \<lbrakk>poly p x \<noteq> 0 \<and> poly q x = 0 \<Longrightarrow> P (r # ps) x; \<not> (poly p x \<noteq> 0 \<and> poly q x = 0) \<Longrightarrow> P (q # r # ps) x\<rbrakk> \<Longrightarrow> P (p # q # r # ps) x"   and "\<And>b. P [] b" shows "P a0 a1"
Suc_1: fixes j :: "nat" assumes "k = j - i"   and "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "\<forall>x\<in>rvars (\<T> (l ! j)). \<langle>\<V> (l ! j)\<rangle> x = \<langle>\<V> (l ! i)\<rangle> x \<or> \<langle>\<V> (l ! j)\<rangle> x = the (\<B>\<^sub>l (l ! i) x) \<or> \<langle>\<V> (l ! j)\<rangle> x = the (\<B>\<^sub>u (l ! i) x)"
pelims: fixes x :: "'e \<Rightarrow> 'd"   and xa :: "'e \<times> 'e"   and y :: "'d \<times> 'd" assumes "pairself x xa = y"   and "pairself_dom (x, xa)" obtains f :: "'e \<Rightarrow> 'd" and a :: "'e" and b :: "'e" where "x = f" and "xa = (a, b)" and "y = (f a, f b)" and "pairself_dom (f, a, b)"
intro: fixes f :: "'a \<Rightarrow> 'b" assumes "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" shows "bounded_linear_axioms f"
assms_2: fixes ys :: "bool list" assumes "xs = True # ys" shows "P"
semiring_normalization_rules_24: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "plus a c = plus c a"
Abs_upset_inverse: fixes y :: "'a set" assumes "y \<in> range \<up>" shows "Rep_upset (Abs_upset y) = y"
lhs: shows "id_func AA \<^bsub>\<a>\<^esub> (Id A) = Id A"
assms_1: shows "isFDERIV_aform prec N xs fas AS"
match_raw_ternary: fixes \<beta> :: "'b \<Rightarrow> 'c \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'c \<Rightarrow> bool"   and expr :: "'b"   and a :: "action"   and p :: "'c" shows "matches (\<beta>, \<alpha>) (Match expr) a p = (case \<beta> expr p of TernaryTrue \<Rightarrow> True | TernaryFalse \<Rightarrow> False | TernaryUnknown \<Rightarrow> \<alpha> a p)"
prefix_drop: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "drop (length xs) ys = zs"   and "prefix xs ys" shows "ys = xs @ zs"
A_card_eq: shows "card A = nat ((int p - 1) div 2)"
alist_of: fixes x :: "('c, 'd) pq" shows "alist_of x \<in> {xs. distinct (map fst xs) \<and> sorted (map snd xs)}"
simps_1: fixes b :: "'a"   and x :: "'a" shows "nextElem [] b x = b"
LLL_many_factorization: fixes f :: "int poly"   and gs :: "int poly list" assumes "LLL_many_factorization f = gs"   and "square_free f"   and "degree f \<noteq> 0" shows "f = prod_list gs \<and> (\<forall>g\<in>set gs. irreducible\<^sub>d g)"
prime_elem_linear_poly: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)"   and "coprime a b" shows "prime_elem [:a, b:]"
progress_prefix_conv: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<sigma>' :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi> \<sigma>'" shows "Monitor.progress \<sigma> P \<phi> (plen \<pi>) = Monitor.progress \<sigma>' P \<phi> (plen \<pi>)"
non_empty_backward_terminating_path_root_iff': fixes x :: "'a" shows "backward_finite_path_root (start_points x) x = (backward_terminating_path x \<and> x \<noteq> 0)"
target_Target: fixes t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and T :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" assumes "t \<in> T" shows "target t \<in> Target T"
shunting: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (inf x y) z = less_eq x (sup z (uminus y))"
Domain_Un_eq: fixes A :: "('a \<times> 'b) set"   and B :: "('a \<times> 'b) set" shows "Domain (A \<union> B) = Domain A \<union> Domain B"
gauss_int_norm_uminus: fixes z :: "gauss_int" shows "gauss_int_norm (- z) = gauss_int_norm z"
psiFreshVec_5: fixes xvec :: "name list"   and x :: "name"   and P :: "('d, 'e, 'f) psi" shows "xvec \<sharp>* \<lparr>\<nu>x\<rparr>P = xvec \<sharp>* [x].P"
lub_eqI: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b set"   and l :: "'b" assumes "class.po below"   and "po.is_lub below M l" shows "po.lub below M = l"
C: shows "\<forall>f g K. theLe \<circ> K = id \<longrightarrow> finite (g ` {d |d. f M (v c) = K d})"
rel_compp: fixes R1 :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and S1 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and S2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_prod (R1 OO S1) (R2 OO S2) = rel_prod R1 R2 OO rel_prod S1 S2"
nat_mult_zero: fixes x :: "nat \<Rightarrow> int" shows "[0] \<cdot> x = \<zero>"
matrix_compose_gen: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec"   and g :: "('b, 'd) vec \<Rightarrow> ('b, 'e) vec" assumes "linear (*s) (*s) f"   and "linear (*s) (*s) g" shows "matrix (g \<circ> f) = matrix g ** matrix f"
\<tau>rtrancl3p_trans: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g"   and tls :: "'h list"   and s' :: "'g"   and tls' :: "'h list"   and s'' :: "'g" assumes "\<tau>trsys.\<tau>rtrancl3p trsys \<tau>move s tls s'"   and "\<tau>trsys.\<tau>rtrancl3p trsys \<tau>move s' tls' s''" shows "\<tau>trsys.\<tau>rtrancl3p trsys \<tau>move s (tls @ tls') s''"
trivial_limit_def: fixes F :: "'c filter" shows "(F = bot) = (\<forall>\<^sub>F x in F. False)"
subst_fml_valid: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<phi> :: "('sf, 'sc, 'sz) formula" assumes "Fadmit \<sigma> \<phi>"   and "fsafe \<phi>"   and "ssafe \<sigma>"   and "valid \<phi>" shows "valid (Fsubst \<phi> \<sigma>)"
absorb_iff1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> U"   and "b \<in> U" shows "(a \<le>\<^sub>o\<^sub>w b) = (local.min a b = a)"
mem_simps_6: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
f_range: fixes X :: "'a set" shows "f_range_on X linear_cf"
key_sets_def: fixes X :: "msg"   and H :: "msg set" shows "key_sets X H \<equiv> {InvKey ` set KS |KS. foldr Crypt KS X \<in> H}"
fs_int_d_pos: fixes n :: "nat"   and fs :: "int vec list"   and k :: "nat" assumes "fs_int_indpt n fs"   and "k \<le> length fs" shows "0 < fs_int.d n fs k"
execute_parallel_operator_effect_3: fixes s :: "'b \<Rightarrow> bool option"   and ops :: "'b strips_operator list"   and v :: "'b" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops" shows "(\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)) \<longrightarrow> execute_parallel_operator s ops v = s v"
list_tail_coinc: fixes n1 :: "'b"   and r1 :: "'b list"   and n2 :: "'b"   and r2 :: "'b list" assumes "n1 # r1 = n2 # r2" shows "n1 = n2 \<and> r1 = r2"
strict_partial_order_subset: shows "strict_partial_order {(x, y). x \<subset> y}"
plus_gt_left_mono: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<succ> y" shows "x + z \<succ> y + z"
T4: shows "serial (\<lambda>w v. w r v \<and> (\<exists>x. \<forall>xa. \<P> xa v = (\<^bold>\<box>xa x) v)) \<longrightarrow> universal (\<lambda>w v. w r v \<longrightarrow> (\<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa v = (\<^bold>\<box>xa x) v)))"
UPFDefs_8: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" shows "m2 \<Oplus> m1 = (\<lambda>x. case m2 x of \<bottom> \<Rightarrow> m1 x | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>y\<rfloor>)"
psubset_card_mono: fixes B :: "'e set"   and A :: "'e set" assumes "finite B"   and "A \<subset> B" shows "card A < card B"
compatible_iff_seq: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" shows "(A1xA2.domains g \<inter> A1xA2.codomains f \<noteq> {}) = A1xA2.seq g f"
necessitation_2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "\<And>s. P s" shows "(\<diamond>P) \<sigma>"
liftAll2_True: fixes inp :: "'f \<Rightarrow> 'd option"   and inp' :: "'f \<Rightarrow> 'e option" shows "liftAll2 (\<lambda>v v'. True) inp inp'"
Bernstein_reciprocal_translate: fixes p :: "nat"   and i :: "nat" shows "reciprocal_poly p (Bernstein_Poly_01 i p) \<circ>\<^sub>p [:1, 1:] = monom (real (p choose i)) (p - i)"
emb_step_chop: fixes t :: "('c, 'd) tm" assumes "is_App t" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b chop t"
memoization_correct_2: fixes dp\<^sub>T :: "'a \<Rightarrow> ('b, 'c) state"   and x :: "'a"   and v :: "'c"   and m :: "'b" assumes "consistentDP dp\<^sub>T"   and "run_state (dp\<^sub>T x) empty = (v, m)" shows "cmem m"
take_Cons: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
conjuncts_qf: fixes p :: "fm" assumes "qfree p" shows "\<forall>q\<in>set (conjuncts p). qfree q"
ileq_interleave_alt: fixes l' :: "'d list"   and l :: "'d list" shows "l' \<preceq> l \<equiv> \<exists>lb. l \<in> lb \<otimes> l'"
swapEnv_psubstEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "goodEnv rho"   and "goodEnv rho'" shows "rho &[rho'] &[z1 \<and> z2]_zs = rho &[z1 \<and> z2]_zs &[rho' &[z1 \<and> z2]_zs]"
lemParallelNotMeet: fixes lineA :: "'a Line"   and lineB :: "'a Line"   and x :: "'a Point" assumes "lineA \<parallel>  lineB"   and "direction lineA \<noteq> 0"   and "direction lineB \<noteq> 0"   and "inLine x lineA"   and "\<not> inLine x lineB" shows "\<not> meets lineA lineB"
braun_height_l_le: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>" shows "height l \<le> Suc (height r)"
is_lubD: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and u :: "'b" assumes "is_lub r x y u" shows "is_ub r x y u \<and> (\<forall>z. is_ub r x y z \<longrightarrow> (u, z) \<in> r)"
nres_inequalities_3: fixes x :: "'a" shows "FAIL \<noteq> RETURN x"
binop_app_in_vrange: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"   and "b \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "a +\<^sub>\<rat> b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_plus"
init_eq: fixes k :: "nat" shows "f' k \<triangleright> a k = f k \<triangleright> a k"
index_smult_mat_2: fixes a :: "'c"   and A :: "'c mat" shows "dim_row (a \<cdot>\<^sub>m A) = dim_row A"
freshEnv_swapEnv_id: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "goodEnv rho"   and "freshEnv xs x1 rho"   and "freshEnv xs x2 rho" shows "rho &[x1 \<and> x2]_xs = rho"
Suffixes_conc: fixes A :: "'a list set"   and B :: "'a list set" assumes "A \<noteq> {}" shows "Suffixes (A @@ B) = Suffixes B \<union> Suffixes A @@ B"
irrefl_def: fixes r :: "('c \<times> 'c) set" shows "irrefl r = (\<forall>a. (a, a) \<notin> r)"
parPres: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<sim>\<^sub>e Q" shows "P \<parallel> R \<sim>\<^sub>e Q \<parallel> R"
unit_in_vhom: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>\<i>\<^sub>C[a] : a \<star>\<^sub>C a \<Rightarrow>\<^sub>C a\<guillemotright>"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs" shows "l_new_document_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_tag_name get_tag_name_locs"
compact_space: fixes X :: "'c topology" shows "compact_space X = (\<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> \<Union> \<U> = topspace X \<longrightarrow> (\<exists>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<and> \<Union> \<F> = topspace X))"
Int: fixes S :: "'c set set"   and a :: "'c set"   and b :: "'c set" assumes "generate_topology S a"   and "generate_topology S b" shows "generate_topology S (a \<inter> b)"
ids_def: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and t :: "'a" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.ids frees consts t = frees t |\<union>| consts t"
integral_density: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> g x" shows "integral\<^sup>L (density M (\<lambda>x. ennreal (g x))) f = LINT x|M. g x *\<^sub>R f x"
finite_polytope_faces: fixes S :: "'a set" assumes "polytope S" shows "finite {F. F face_of S}"
inner_scaleR_right: fixes x :: "'b"   and r :: "'a"   and y :: "'b" shows "inner x (scale r y) = cnj_class.cnj r * inner x y"
new_keys_not_analzd: fixes evs :: "event list"   and K :: "nat" assumes "evs \<in> kerbIV_gets"   and "K \<in> symKeys"   and "Key K \<notin> used evs" shows "K \<notin> keysFor (analz (knows Spy evs))"
right_neutral: fixes a :: "'b::monoid_add" shows "(a::'b::monoid_add) + (0::'b::monoid_add) = a"
simps_1: fixes r :: "recf" shows "r_prepend [] r = r"
Compl: fixes a :: "'d set"   and sp :: "'d set"   and A :: "'d set set" assumes "a \<in> sigma_sets sp A" shows "sp - a \<in> sigma_sets sp A"
exhaust: fixes y :: "('e, 'f) atm" obtains x11 :: "'e trm" and x12 :: "'e trm" where "y = Eq x11 x12"   | x21 :: "'f" and x22 :: "'e trm list" where "y = Pr x21 x22"