hcju/isabelleps huggingface dataset:
add_commute: fixes a :: "'b hrel"   and b :: "'b hrel" shows "a \<sqinter> b = b \<sqinter> a"
result_def: fixes generat :: "('a, 'b, 'c) generat" shows "generat.result generat = (case generat of Pure x1 \<Rightarrow> x1)"
pSubst: shows "presSubstAll (rec MOD) (recAbs MOD) MOD"
fmlookup_default_def: fixes d :: "'a"   and m :: "('b, 'a) fmap"   and x :: "'b" shows "fmlookup_default d m x = (case fmlookup m x of None \<Rightarrow> d | Some v \<Rightarrow> v)"
mat_row_eq: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list" assumes "mat nr nc m1"   and "mat nr nc m2" shows "(m1 = m2) = (\<forall>i<nr. row m1 i = row m2 i)"
poly_of_list_impl: fixes as :: "'b list" shows "coeffs (poly_of_list as) = strip_while ((=) (0::'b)) as"
Bseq_isLub: fixes X :: "nat \<Rightarrow> real" assumes "Bseq X" shows "\<exists>U. isLub UNIV {x. \<exists>n. X n = x} U"
pos_decomp_map: fixes f :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<times> 'x set) list" shows "(map (apfst f) ps)\<^sub>+ = map (apfst f) (ps\<^sub>+)"
reindex_cong: fixes l :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'c" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "prod g A = prod h B"
append_replicate_commute: fixes n :: "nat"   and x :: "'a"   and k :: "nat" shows "replicate n x @ replicate k x = replicate k x @ replicate n x"
add_idem': fixes x :: "('a \<times> 'a set) set" shows "x \<union> x = x"
edges_ins_edge: fixes e :: "'a \<times> 'a"   and g :: "'a ugraph" shows "edges (ins_edge e g) = (if fst e = snd e then edges g else {e, prod.swap e} \<union> edges g)"
min_def: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k"   and b :: "'k" shows "ord.min less_eq a b = (if less_eq a b then a else b)"
lfilter_LCons_found: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a llist" assumes "P x" shows "lfilter P (x #\<^sub>l xs) = x #\<^sub>l lfilter P xs"
rel_generat_mono: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and C :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "('g, 'i, 'k) generat"   and y :: "('h, 'j, 'l) generat"   and A' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and C' :: "'k \<Rightarrow> 'l \<Rightarrow> bool" assumes "rel_generat A B C x y"   and "\<And>x y. A x y \<Longrightarrow> A' x y"   and "\<And>x y. B x y \<Longrightarrow> B' x y"   and "\<And>x y. C x y \<Longrightarrow> C' x y" shows "rel_generat A' B' C' x y"
mult: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "f \<in> R"   and "g \<in> R" shows "(\<lambda>x. f x * g x) \<in> R"
fixpoint: shows "(f \<circ> g \<circ> h) I = I"
list_all2_induct: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and R :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "list_all2 P xs ys"   and "R [] []"   and "\<And>x xs y ys. \<lbrakk>P x y; list_all2 P xs ys; R xs ys\<rbrakk> \<Longrightarrow> R (x # xs) (y # ys)" shows "R xs ys"
Join_ensures: fixes F :: "'c program"   and G :: "'c program"   and A :: "'c set"   and B :: "'c set" shows "(F \<squnion> G \<in> A ensures B) = (F \<in> A - B co A \<union> B \<and> G \<in> A - B co A \<union> B \<and> (F \<in> transient (A - B) \<or> G \<in> transient (A - B)))"
diff_fun_between_chartsD: fixes c :: "('a, 'b) chart"   and x :: "'a" assumes "c \<in> atlas"   and "x \<in> domain c" shows "k-smooth_on (codomain c) (b \<circ> inv_chart c)"
C_restr_below_cong: fixes r :: "C"   and f :: "C \<rightarrow> 'a"   and g :: "C \<rightarrow> 'a" assumes "\<And>r'. r' \<sqsubseteq> r \<Longrightarrow> f\<cdot>r' \<sqsubseteq> g\<cdot>r'" shows "f|\<^bsub>r\<^esub> \<sqsubseteq> g|\<^bsub>r\<^esub>"
vector_mul_rcancel: fixes a :: "'b"   and x :: "('b, 'c) vec"   and b :: "'b" shows "(a *s x = b *s x) = (a = b \<or> x = 0)"
chain_consE: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b llist" assumes "chain R (LCons x xs)" obtains "xs = LNil"   | "chain R xs" and "R x (lhd xs)"
assms_1: shows "(g has_derivative g' y) (at y within X)"
Delta_CDMode_CTRL: shows "Delta CDMode_CTRL = CDMode_CTRL_Delta"
connected_frontier_simple: fixes S :: "'a set" assumes "connected S"   and "connected (- S)" shows "connected (frontier S)"
6: shows "span ((\<lambda>x. x /\<^sub>R norm x) ` B) = S"
dist_les_6: shows "GT \<notsqsubseteq> EQ"
simps_2: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "tfr\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t = ((\<forall>x\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X. \<exists>a. \<Gamma> (Var x) = Var a) \<or> (\<forall>f T. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<longrightarrow> T = [] \<or> (\<exists>s\<in>set T. s \<notin> Var ` set X)))"
abs_eq: fixes x :: "real" shows "ennreal x = e2ennreal (sup 0 (ereal x))"
TS_step_d_def: fixes s :: "'b list \<times> 'b list"   and q :: "'b" shows "TS_step_d s q = ((let li = index (snd s) q in if li = length (snd s) then 0 else let sincelast = take li (snd s); S = {x. x < q in fst s \<and> count_list sincelast x \<le> 1} in if S = {} then 0 else index (fst s) q - Min (index (fst s) ` S), []), q # snd s)"
subspace_Polys: fixes X :: "'d set" shows "phull.subspace P[X]"
stake_strict_1: shows "stake\<cdot>\<bottom> = \<bottom>"
netgmap_subnet_split1: fixes sr :: "'e \<Rightarrow> 'c \<times> 'd"   and s1 :: "'e net_state"   and s2 :: "'e net_state"   and n1 :: "net_tree"   and n2 :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'c"   and \<zeta> :: "'d net_state" assumes "netgmap sr (SubnetS s1 s2) = netmask (net_tree_ips (n1 \<parallel> n2)) (\<sigma>, \<zeta>)"   and "net_tree_ips n1 \<inter> net_tree_ips n2 = {}"   and "net_ips s1 = net_tree_ips n1"   and "net_ips s2 = net_tree_ips n2" shows "netgmap sr s1 = netmask (net_tree_ips n1) (\<sigma>, snd (netgmap sr s1))"
Psize: shows "P \<in> subgroups_of_size (p ^ a)"
assms_3: fixes \<sigma> :: "'\<sigma>"   and k :: "'u"   and it :: "'u set"   and v :: "'v" assumes "c \<sigma>"   and "k \<in> dom (\<alpha> m) - it"   and "\<alpha> m k = Some v"   and "it \<subseteq> dom (\<alpha> m)"   and "I it \<sigma>" shows "I (insert k it) (f (k, v) \<sigma>)"
mergeable_envs_def: fixes n :: "nat"   and S :: "'b list set" shows "mergeable_envs n S = (\<forall>v1\<in>S. \<forall>v2\<in>S. \<forall>A B f. (\<forall>x\<in>A. x < n \<and> v1 ! x = f x) \<and> (\<forall>x\<in>B. x < n \<and> v2 ! x = f x) \<longrightarrow> (\<exists>v\<in>S. \<forall>x\<in>A \<union> B. v ! x = f x))"
le_iff_inf: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<cdot> y = x)"
assms_6: shows "prv (eqv \<chi>1 \<chi>2)"
g_Zorn_lemma1: fixes C :: "'a set" assumes "inductive_set D"   and "Chain D C" shows "\<exists>m. maximal m \<and> m \<in> upper_bounds D C"
prodM_sprod_def: fixes R :: "('h, 'k) Ring_scheme"   and I :: "'i set"   and A :: "'i \<Rightarrow> ('j, 'h, 'l) Module_scheme" shows "prodM_sprod R I A = (\<lambda>a\<in>carrier R. \<lambda>g\<in>carr_prodag I A. \<lambda>j\<in>I. a \<cdot>\<^sub>s\<^bsub>A j\<^esub> g j)"
\<I>\<^sub>\<tau>_3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>\<^sub>\<tau>)"
chain_with_def: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and X :: "'a set" shows "[[.. x .. y .. z ..]X] \<equiv> [[x y z]] \<and> x \<in> X \<and> y \<in> X \<and> z \<in> X \<and> (\<exists>f. TernaryOrdering.ordering f betw X)"
inf_measure_def: fixes a :: "'a measure"   and b :: "'a measure" shows "inf a b = Inf {a, b}"
simps_3: fixes x :: "int"   and y :: "int"   and xs :: "int list" shows "take_incr (x # y # xs) = (if x < y then x # take_incr (y # xs) else [x])"
card_partition_on: fixes A :: "'c set"   and k :: "nat" assumes "finite A" shows "card {P. partition_on A P \<and> card P = k} = Stirling (card A) k"
HMM2_axioms: shows "HMM2 \<O>\<^sub>s \<K> \<O> \<S>"
simps: fixes f\<^sub>1 :: "'c val \<Rightarrow> 'b val"   and f\<^sub>2 :: "'d \<Rightarrow> 'b val"   and v :: "'d" shows "F_Times1 f\<^sub>1 f\<^sub>2 v = Seq (f\<^sub>1 Void) (f\<^sub>2 v)"
Skip2: fixes l :: "nat" shows "\<turnstile> SKIP : l"
assms: shows "get_delayed_pool s = [] \<and> get_trap_set s = {} \<and> snd (execute_instruction () s) = False \<and> s' = snd (fst (execute_instruction () s)) \<and> get_S (cpu_reg_val PSR s) = 0"
iso_DirProds_IDirProds: fixes S :: "'d \<Rightarrow> 'a set"   and I :: "'d set" assumes "is_idirprod (carrier G) S I"   and "finite I" shows "(\<lambda>x. finprod G x I) \<in> Group.iso (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) G"
intros: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set" assumes "kuhn_simplex p n base upd s" shows "ksimplex p n s"
distinct_selsort: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "distinct (linorder_list0.selsort le xs) = distinct xs"
chainE: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "class.po below"   and "po.chain below Y" shows "below (Y i) (Y (Suc i))"
mpoly_base_conv_2: fixes c :: "'b"   and x :: "'b" shows "c = poly [:c:] x"
assms_2: fixes c' :: "real" assumes "c' \<in> {c<..<u}" shows "\<forall>\<^sub>F x in F. f x \<le> c' * g x"
sdiv_poly_def: fixes p :: "'b poly"   and a :: "'b" shows "sdiv_poly p a = map_poly (\<lambda>c. c div a) p"
iT_Minus_empty: fixes k :: "nat" shows "k \<ominus> {} = {}"
subtuple_eq_fmrel_fmrestrict_fset: fixes ym :: "'d \<rightharpoonup>\<^sub>f 'f"   and r :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xm :: "'d \<rightharpoonup>\<^sub>f 'e" shows "subtuple r xm ym = fmrel r (fmrestrict_fset (fmdom ym) xm) ym"
common_abs_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agents :: "'b odlist"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and t :: "'e Trace" assumes "FiniteLinorderEnvironment jkbp agents" shows "FiniteLinorderEnvironment.common_abs jkbp envInit envAction envTrans envVal envObs t \<equiv> tLast ` PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal (Environment.clock_jview envObs) (tLength t)"
cod_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "C.CCC.cod f = (if C.arr (fst f) \<and> C.CC.arr (snd f) then (C.cod (fst f), C.CC.cod (snd f)) else (C.null, C.CC.null))"
eventually_at_bot_linorder: fixes P :: "'c \<Rightarrow> bool" shows "eventually P at_bot = (\<exists>N. \<forall>n\<le>N. P n)"
suf_len': fixes u :: "'b list"   and z :: "'b list" shows "\<^bold>|u\<^bold>| \<le> \<^bold>|z \<cdot> u\<^bold>|"
eucl_rel_poly_pCons: fixes x :: "'b poly"   and y :: "'b poly"   and q :: "'b poly"   and r :: "'b poly"   and b :: "'b"   and a :: "'b" assumes "eucl_rel_poly x y (q, r)"   and "y \<noteq> 0"   and "b = coeff (pCons a r) (degree y) / lead_coeff y" shows "eucl_rel_poly (pCons a x) y (pCons b q, pCons a r - smult b y)"
intros_2: fixes u :: "'b list"   and C :: "'b list set"   and b :: "'b list" assumes "u \<in> no_head_gen C b" shows "u \<cdot> b \<in> no_head_gen C b"
filtermap_ln_at_top: shows "filtermap ln at_top = at_top"
Ipoly_CInt: fixes bs :: "'b list"   and i :: "int" shows "\<lparr>(i)\<^sub>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = of_int i"
maxsimp_im_max: fixes x :: "'a set"   and w :: "'a set" assumes "maxsimp x"   and "w \<in> X"   and "order.greater_eq (f ` w) (f ` x)" shows "f ` w = f ` x"
induct: fixes x1 :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and x2 :: "(nat \<times> ('sf, 'sc, 'sz) step) list"   and P :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list \<Rightarrow> (nat \<times> ('sf, 'sc, 'sz) step) list \<Rightarrow> bool" assumes "deriv_ok x1 x2"   and "\<And>R. P R []"   and "\<And>R i S SS. \<lbrakk>step_ok R i S; 0 \<le> i; i < length (fst R); deriv_ok (step_result R (i, S)) SS; P (step_result R (i, S)) SS\<rbrakk> \<Longrightarrow> P R ((i, S) # SS)" shows "P x1 x2"
rcs_fixed1: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and a :: "'e"   and h :: "'e" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G"   and "h \<in> H" shows "H \<bullet>\<^bsub>G\<^esub> a = H \<bullet>\<^bsub>G\<^esub> (h \<cdot>\<^bsub>G\<^esub> a)"
PDGBackwardSliceCorrect: fixes prog :: "cmd" shows "BackwardSlice sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (WStandardControlDependence.PDG_BS_s prog)"
check_eqv_sound: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" assumes "slow.check_eqvRE \<Sigma> n r s"   and "WS1S_Equivalence_Checking.wf \<Sigma> n r"   and "WS1S_Equivalence_Checking.wf \<Sigma> n s" shows "lang \<Sigma> n r = lang \<Sigma> n s"
orthogonal_unit_vec: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n"   and "j < n"   and "i \<noteq> j" shows "\<langle>unit_vec n i|unit_vec n j\<rangle> = 0"
gnz: fixes w :: "complex" assumes "w \<in> ball \<xi> r" shows "g w \<noteq> 0"
assms_2: shows "subs' \<subseteq>\<^sub>m subs''"
linear_sum: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and S :: "'c set" assumes "linear f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
s\<^sub>2s\<^sub>3: shows "(s\<^sub>2, s\<^sub>3) \<in> [\<leadsto>]\<^sup>="
cond_unit_T: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<triangleleft> true \<triangleright>  Q) = P"
alwaysActive: fixes active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and k :: "cnf" assumes "singleton active" shows "\<exists>id. active id k"
my_vidx_inj: shows "inj_on vidx (fset vertices)"
diff_fun_space_extensionalD: fixes g :: "'a \<Rightarrow> real" assumes "g \<in> diff_fun_space" shows "extensional0 carrier g"
disjoint_family_disjointed: fixes A :: "nat \<Rightarrow> 'c set" shows "disjoint_family (disjointed A)"
Re_Reals_divide: fixes r :: "complex"   and z :: "complex" assumes "r \<in> \<real>" shows "Re (r / z) = Re r * Re z / (cmod z)\<^sup>2"
arithmetic_simps_78: fixes a :: "'a" shows "(1::'a) * a = a"
wfp_on_restrict_to_tranclp': fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wfp_on (restrict_to P A)\<^sup>+\<^sup>+ A" shows "wfp_on P A"
Id_in_Acts: fixes F :: "'c program" shows "Id \<in> Acts F"
ipath_conc_conv: fixes E :: "('e \<times> 'e) set"   and u :: "'e list"   and v :: "nat \<Rightarrow> 'e" shows "ipath E (u \<frown> v) = (\<exists>a. path E a u (v 0) \<and> ipath E v)"
inj_term_sym_simps_4: fixes es :: "expr list" shows "In3 es = \<langle>es\<rangle>"
coprime_quot_of_fract: fixes x :: "'a fract" shows "coprime (fst (quot_of_fract x)) (snd (quot_of_fract x))"
assms_1: shows "i \<in> kD (rt \<xi>)"
fresh_right: fixes pi :: "(name \<times> name) list"   and a :: "name"   and x :: "'a" shows "pi \<bullet> a \<sharp> x = a \<sharp> rev pi \<bullet> x"
assms: fixes B :: "'a set"   and x :: "'a"   and y :: "'a" assumes "B \<subseteq> A"   and "(x, y) \<in> rwp_on A f"   and "x \<in> f B"   and "y \<in> B" shows "y \<in> f B"
length_rk2_fas: fixes ode_ops :: "ode_ops"   and rkp :: "floatarith"   and x0 :: "floatarith list"   and h :: "floatarith"   and cx :: "floatarith list"   and s2 :: "floatarith" shows "length (ode_operations.rk2_fas ode_ops rkp x0 h cx s2) = 2 * length x0"
simps: fixes D :: "ast_domain"   and n :: "char list"   and params :: "(variable \<times> PDDL_STRIPS_Semantics.type) list"   and pre :: "PDDL_STRIPS_Semantics.term atom formula"   and eff :: "PDDL_STRIPS_Semantics.term ast_effect" shows "ast_domain.wf_action_schema D (Action_Schema n params pre eff) = (let tyt = ty_term (map_of params) (ast_domain.constT D) in distinct (map fst params) \<and> ast_domain.wf_fmla D tyt pre \<and> ast_domain.wf_effect D tyt eff)"
toCard_def: fixes A :: "'c set"   and r :: "('d \<times> 'd) set" shows "toCard A r \<equiv> SOME f. toCard_pred A r f"
PNT4_imp_PNT5: assumes "\<theta> \<sim>[at_top] (\<lambda>x. x)" shows "\<psi> \<sim>[at_top] (\<lambda>x. x)"
antimono_pow: shows "decseq (\<lambda>i. (f ^^ i) top)"
permutation_inverse_compose: fixes p :: "'a \<Rightarrow> 'a"   and q :: "'a \<Rightarrow> 'a" assumes "permutation p"   and "permutation q" shows "inv (p \<circ> q) = inv q \<circ> inv p"
cdist_def: fixes z1 :: "complex"   and z2 :: "complex" shows "cdist z1 z2 \<equiv> cmod (z2 - z1)"
F_def: shows "F \<equiv> \<lambda>lset xs. case xs of LNil \<Rightarrow> {} | LCons x xs \<Rightarrow> insert x (lset xs)"
assms_2: fixes u :: "nat \<Rightarrow> 'a" assumes "\<forall>n. u n \<in> space\<^sub>N N"   and "\<forall>n. Norm N (u n) \<le> c n" shows "\<exists>x\<in>space\<^sub>N N. tendsto_in\<^sub>N N (\<lambda>n. sum u {0..<n}) x"
vinsert_vdiff_nin: fixes a :: "V"   and B :: "V"   and A :: "V" assumes "a \<notin>\<^sub>\<circ> B" shows "vinsert a (A -\<^sub>\<circ> B) = vinsert a A -\<^sub>\<circ> B"
same: fixes a :: "'a\<^sub>m" assumes "a \<in> A" shows "m a a = Some a"
isTrue_neg: fixes \<phi> :: "'c" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "isTrue \<phi> \<or> isTrue (B.neg \<phi>)"
x_half: fixes x :: "real" shows "2 * cos (x / 2) * cos (x / 2) - 1 = cos x"
ide_hcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.ide \<nu>"   and "B.ide \<mu>"   and "src\<^sub>B \<nu> = trg\<^sub>B \<mu>" shows "B.ide (\<nu> \<star>\<^sub>B \<mu>)"
vec_to_iarray_column: fixes i :: "'f"   and A :: "(('d, 'f) vec, 'e) vec" shows "vec_to_iarray (column i A) = column_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)"
hfref_weaken_pre_nofail: fixes f :: "'a \<Rightarrow> 'b Heap"   and g :: "'c \<Rightarrow> 'd nres"   and P :: "'c \<Rightarrow> bool"   and R :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and S :: "'d \<Rightarrow> 'b \<Rightarrow> assn" assumes "(f, g) \<in> [P]\<^sub>a R \<rightarrow> S" shows "(f, g) \<in> [\<lambda>x. nofail (g x) \<longrightarrow> P x]\<^sub>a R \<rightarrow> S"
preserves_left_adjoint: fixes f :: "'a" assumes "C.is_left_adjoint f" shows "D.is_left_adjoint (F f)"
prj1_in_hom: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.cospan f g"   and "a = f \<down>\<down> g"   and "b = C.dom f" shows "\<guillemotleft>\<p>\<^sub>1[f, g] : a \<rightarrow> b\<guillemotright>"
restrict_apply: fixes f :: "'ah \<Rightarrow> 'ag"   and A :: "'ah set"   and x :: "'ah" shows "restrict f A x = (if x \<in> A then f x else undefined)"
unitor_coincidence_1: shows "\<l>[\<I>] = \<iota>"
not_prv_prv_neg_PPf: fixes \<phi> :: "'fmla"   and n :: "'trm" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "\<not> prv \<phi>"   and "n \<in> num" shows "bprv (neg (PPf n \<langle>\<phi>\<rangle>))"
y: shows "y = mat_to_polys (Keys_to_list ps) (row_echelon (Macaulay_mat ps)) ! i2"
dghm_cn_comp_def: fixes \<GG> :: "V"   and \<FF> :: "V" shows "\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF> = [\<GG>\<lparr>ObjMap\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ObjMap\<rparr>, \<GG>\<lparr>ArrMap\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr>, op_dg (\<FF>\<lparr>HomDom\<rparr>), \<GG>\<lparr>HomCod\<rparr>]\<^sub>\<circ>"
rep_eq: fixes x :: "(('b, 'a) vec, 'a) vec" shows "to_fun (from_vec x) = (\<lambda>i. ($) (x $ i))"
update_convs_4: fixes more' :: "'m \<Rightarrow> 'm"   and NTDom :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NTCod :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and NatTransMap :: "'g \<Rightarrow> 'j"   and more :: "'m" shows "NatTrans.more_update more' \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more' more\<rparr>"
thetaPar_sym: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "sym (PL_Indis.thetaPar tval aval indis)"
mult_vm_cancel_r: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and v :: "complex \<times> complex"   and v' :: "complex \<times> complex" assumes "mat_det A \<noteq> 0"   and "v *\<^sub>v\<^sub>m A = v' *\<^sub>v\<^sub>m A" shows "v = v'"
disj_comb: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "heap_sep_algebra.tern_rel plus sep_disj x y z"   and "sep_disj x w"   and "sep_disj y w" shows "sep_disj z w"
gen_inv: fixes x :: "'d"   and G :: "('d, 'e) monoid_scheme"   and gens :: "'d set" assumes "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
list_slice_concat: fixes xs :: "'b list"   and k :: "nat" shows "concat (list_slice xs k) = xs \<down> (length xs div k * k)"
lift_CFGExit_wf: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and H :: "'i set"   and L :: "'i set" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "CFGExit_wf src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_Def Def Entry Exit H L) (lift_Use Use Entry Exit H L) state_val NewExit"
int_eq_sint: fixes x :: "nat" assumes "x < 2 ^ (LENGTH('b) - 1)" shows "sint (word_of_nat x) = int x"
that: fixes y :: "'a" assumes "y \<in> C"   and "x < y" shows "thesis"
comp_char: fixes g :: "'b"   and f :: "'b" shows "hhom\<^sub>D.comp g f = (if hhom\<^sub>D.arr f \<and> hhom\<^sub>D.arr g \<and> D.seq g f then g \<cdot>\<^sub>D f else D.null)"
select_convs_3: fixes V :: "'b set"   and N :: "'b set"   and C :: "'b set"   and more :: "'c" shows "C \<lparr>V = V, N = N, C = C, \<dots> = more\<rparr> = C"
case_prodE2: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'd \<Rightarrow> 'a"   and z :: "'b \<times> 'd" assumes "Q (case z of (a, b) \<Rightarrow> P a b)" obtains x :: "'b" and y :: "'d" where "z = (x, y)" and "Q (P x y)"
dim_eq_full: fixes S :: "'c set" shows "(vs2.dim S = vs2.dimension) = (vs2.span S = UNIV)"
RID\<^sub>SI: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and l :: "'b"   and v :: "('a, 'b, 'c) val"   and r :: "'a" assumes "\<sigma> l = Some v"   and "r \<in> RID\<^sub>V v" shows "r \<in> RID\<^sub>S \<sigma>"
not_eq_last_diff_paths_ii: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" assumes "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "\<not> FactoredSystem.inj last (state_set (state_list s as)) (valid_states PROB)" shows "\<exists>l1 l2. l1 \<in> state_set (state_list s as) \<and> l2 \<in> state_set (state_list s as) \<and> last l1 = last l2 \<and> l1 \<noteq> l2"
ANR_imp_locally_connected: fixes S :: "'a set" assumes "ANR S" shows "locally connected S"
euler_fas'_def: fixes D :: "nat" shows "euler_fas' D = map fold_const_fa (euler_fas (map floatarith.Var [0..<D]) (floatarith.Var (2 * D)) (map floatarith.Var [D..<2 * D]))"
assms_1: shows "chain_reparam_chain one_chain1 one_chain2"