hcju/isabelleps huggingface dataset:
card_order_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "card_order_on A r \<equiv> well_order_on A r \<and> (\<forall>r'. well_order_on A r' \<longrightarrow> r \<le>o r')"
local_lipschitz_affine: fixes T :: "real set"   and S :: "('b, 'c) vec set"   and A :: "real \<Rightarrow> (('b, 'c) vec, 'd) vec"   and B :: "real \<Rightarrow> ('b, 'd) vec" assumes "open T"   and "open S"   and "\<And>\<tau> \<epsilon>. \<lbrakk>0 < \<epsilon>; \<tau> \<in> T; cball \<tau> \<epsilon> \<subseteq> T\<rbrakk> \<Longrightarrow> bdd_above {\<parallel>A t\<parallel>\<^sub>o\<^sub>p |t. t \<in> cball \<tau> \<epsilon>}" shows "local_lipschitz T S (\<lambda>t s. A t *v s + B t)"
tends: fixes x :: "'a" assumes "x \<notin> N" shows "(\<lambda>n. g n x) \<longlonglongrightarrow> (if x \<in> S then h x else (0::'b))"
the_\<nu>_simps_2: shows "src \<omega>.the_\<nu> = src r\<^sub>1"
intros_8: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "'e"   and s' :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,s'\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Seq c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
arith_extra_simps_19: fixes a :: "'c" shows "a * (0::'c) = (0::'c)"
flincomb_closed: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and rs :: "'b list"   and ms :: "'c list" assumes "FGModule G smult V"   and "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "scalar_mult.lincomb (aezfun_scalar_mult.fsmult smult) rs ms \<in> V"
dube_eq_1: 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 "(\<lambda>z. z + int (card X) - 1 gchoose (card X - 1)) = (\<lambda>z. z - int (poly_deg f) + int (card X) - 1 gchoose (card X - 1) + (z - int (poly_deg f) + int (card X) gchoose card X) + (z + int (card X) gchoose card X) - 2 - (\<Sum>i = 1..card X. z - int (pm_powerprod.aa X f F i) + int i - 1 gchoose i + (z - int (pm_powerprod.bb ord X F i) + int i - 1 gchoose i)))"
sel_1: fixes x1 :: "exesignature"   and x2 :: "Core.term list" shows "exesig (ExeTheory x1 x2) = x1"
strict_coboundedI2: fixes b :: "'a multiset"   and c :: "'a multiset"   and a :: "'a multiset" assumes "b \<subset># c" shows "a \<inter># b \<subset># c"
s1_def: shows "s1 = the_inv_into (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) fundantivertex (\<phi> v1)"
fixed_point_imp_closed_orbit_period_zero_1: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "x \<in> X"   and "f x = (0::'b)" shows "auto_ll_on_open.closed_orbit f X x"
hyps: fixes ys :: "('t \<Rightarrow>\<^sub>0 'b) list" shows "set ys \<subseteq> set (comp_red_basis_aux xs ys)"
fresh_3: fixes a :: "name"   and x2 :: "ty"   and x1 :: "ty" shows "a \<sharp> x2 \<otimes> x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
size_del_max: fixes t :: "('b \<times> nat) tree" assumes "t \<noteq> \<langle>\<rangle>" shows "size t = Suc (size (snd (del_max t)))"
has_derivative_at: fixes f :: "'a \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "(f has_derivative D) (at x) = (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) \<midarrow>0::'a\<rightarrow> 0)"
TS_EventSet: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<subseteq> ET ST" shows "\<Union> (Actevent (Label TS)) \<subseteq> HAEvents (HA ST)"
shows_attrs_def: fixes as :: "(char list \<times> char list) list" shows "shows_attrs as = foldr (\<lambda>a. shows_string '' '' \<circ> shows_attr a) as"
ind_eq_1_iff: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" shows "(ind P n = (1::'b)) = (0 < n \<and> P n)"
simps_4: fixes s :: "'state_t"   and d :: "'dom_t"   and a :: "'action_t"   and as :: "'action_t list"   and execs' :: "'action_t list list" shows "CISK_control s d ((a # as) # execs') = (if aborting s d a then (None, execs', set_error_code s a) else if waiting s d a then (Some a, (a # as) # execs', s) else (Some a, as # execs', s))"
finite_comm_group_axioms: shows "finite_comm_group (G\<lparr>carrier := H'\<rparr>)"
arrI: fixes f :: "('o, 'a) arr" assumes "f \<noteq> Null"   and "Dom f \<in> Obj"   and "Cod f \<in> Obj"   and "Map f \<in> Hom (Dom f) (Cod f)" shows "arr f"
inj_imp_inv_eq: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "inj f"   and "\<forall>x. f (g x) = x" shows "inv f = g"
x_in_MPLS_if: fixes x :: "nat"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" assumes "x \<in> MPLS PROB" shows "\<exists>s as. s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<and> x = Inf (PLS s as)"
point_indices_balanced: shows "point_indices \<t> = {\<Lambda>\<^sub>t}"
2: fixes i :: "nat" assumes "i < length (t # T)" shows "term_variants_pred P ((t # T) ! i) ((s # S) ! i)"
fsubset_not_fsubset_eq: fixes A :: "'b fset"   and B :: "'b fset" shows "(A |\<subset>| B) = (A |\<subseteq>| B \<and> \<not> B |\<subseteq>| A)"
less_imp_le: fixes x :: "'t"   and y :: "'t" assumes "x \<prec>\<^sub>t y" shows "x \<preceq>\<^sub>t y"
foundation23: fixes \<tau> :: "'a state \<times> 'a state"   and P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> P) = (\<tau> \<Turnstile> \<lambda>_. P \<tau>)"
quotient_web_trimming: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "wave \<Gamma> f"   and "trimming \<Gamma> f g" shows "quotient_web \<Gamma> f = quotient_web \<Gamma> g"
finite_imp_closed: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.t1_space open"   and "finite S" shows "topological_space.closed open S"
permute_profile_def: fixes \<sigma> :: "'d \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'c" shows "permute_profile \<sigma> R = (\<lambda>i x y. R i (inv \<sigma> x) (inv \<sigma> y))"
is_set_to_sorted_listE: fixes ordR :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and Rk :: "('g \<times> 'f) set"   and Rs :: "('g \<times> 'f) set \<Rightarrow> ('h \<times> 'f set) set"   and tsl :: "'h \<Rightarrow> 'g list"   and s :: "'h"   and s' :: "'f set" assumes "is_set_to_sorted_list ordR Rk Rs tsl"   and "(s, s') \<in> \<langle>Rk\<rangle>Rs" obtains l' :: "'f list" where "(tsl s, l') \<in> \<langle>Rk\<rangle>list_rel" and "RETURN l' \<le> it_to_sorted_list ordR s'"
Scons_inject1: fixes M :: "('b, 'c) Old_Datatype.node set"   and N :: "('b, 'c) Old_Datatype.node set"   and M' :: "('b, 'c) Old_Datatype.node set"   and N' :: "('b, 'c) Old_Datatype.node set" assumes "Scons M N = Scons M' N'" shows "M = M'"
that: fixes p' :: "'t \<Rightarrow>\<^sub>0 'b"   and q' :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(red F)\<^sup>*\<^sup>* p p'"   and "(red F)\<^sup>*\<^sup>* q q'"   and "r = p' - q'" shows "thesis"
unique_session_keys: fixes A :: "agent"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and Tk' :: "nat"   and B' :: "agent"   and X' :: "msg" assumes "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "Says Server A' (Crypt (shrK A') \<lbrace>Number Tk', Agent B', Key K, X'\<rbrace>) \<in> set evs"   and "evs \<in> bankerberos" shows "A = A' \<and> Tk = Tk' \<and> B = B' \<and> X = X'"
in_set_insert: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set xs" shows "List.insert x xs = xs"
is_stmt_rews_4: fixes x :: "stmt" shows "is_stmt (In1r x)"
coclop_alt: fixes f :: "'b \<Rightarrow> 'b" shows "coclop f = (\<forall>x y. (f x \<le> y) = (f x \<le> f y))"
Sup_lexord1: fixes A :: "'b set"   and k :: "'b \<Rightarrow> 'c set"   and P :: "'b \<Rightarrow> bool"   and c :: "'b set \<Rightarrow> 'b"   and s :: "'b set \<Rightarrow> 'b" assumes "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> k a = \<Union> (k ` A)"   and "P (c A)" shows "P (Sup_lexord k c s A)"
Teq: shows "T = \<Union> (range C) \<union> K"
xor_list_commute: fixes xs :: "'b list"   and ys :: "'b list" shows "xs [\<oplus>] ys = ys [\<oplus>] xs"
composite_simps_2: shows "trg ((\<nu> \<star> \<pi>) \<star> \<rho>) = trg (\<nu> \<star> \<pi>)"
exponent10_def: fixes float10 :: "float10" shows "exponent10 float10 = (case float10 of x1 \<e> x2 \<Rightarrow> x2)"
inj: shows "inj ((\<langle>$\<rangle>) f)"
igSubstIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" shows "igSubstIGOp MOD \<equiv> \<forall>ys y Y delta inp binp. igWls MOD (asSort ys) Y \<and> igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSubst MOD ys Y y (igOp MOD delta inp binp) = igOp MOD delta (igSubstInp MOD ys Y y inp) (igSubstBinp MOD ys Y y binp)"
product_category_axioms: shows "product_category (\<cdot>\<^sub>1) (\<cdot>\<^sub>2)"
splus_right_canc: fixes t :: "'a"   and v :: "'a"   and s :: "'a" shows "(local.punit.splus t v = local.punit.splus s v) = (t = s)"
dcd_pdg: fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and v :: "'var"   and m :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n"   and m' :: "nat" assumes "n dcd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m via \<pi>' m'" obtains l :: "'n" where "(\<pi> m, l) \<in> pdg" and "(l, \<pi> n) \<in> pdg"
lconjby_bij: fixes x :: "'a" shows "bij (lconjby x)"
simps_2: fixes a :: "'d \<times> ('e, 'f) fmap"   and as :: "('d \<times> ('e, 'f) fmap) list" shows "rem_effectless_act (a # as) = (if fmdom' (snd a) \<noteq> {} then a # rem_effectless_act as else rem_effectless_act as)"
HS_B_eval: fixes G :: "Var \<Rightarrow> L"   and b :: "BExpr"   and t :: "L"   and pp :: "L"   and r :: "Var \<Rightarrow> Val"   and s :: "Var \<Rightarrow> Val" assumes "(G, b, t) \<in> HS_B"   and "EQ G pp r s"   and "LEQ t pp" shows "evalB b r = evalB b s"
finite_dimensional_vector_space_pair_1_on_def: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and basis1 :: "'d set" shows "finite_dimensional_vector_space_pair_1_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 basis1 \<equiv> finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 basis1 \<and> vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2"
ID_ccompare_neq_None': shows "ID ccompare \<noteq> None"
reads_consistent_unbuffered_snoc: fixes sb :: "'b memref list"   and a :: "nat"   and m :: "nat \<Rightarrow> nat"   and v :: "nat"   and pending_write :: "bool"   and \<O> :: "nat set"   and volatile :: "bool"   and t :: "nat" assumes "buffered_val sb a = None"   and "m a = v"   and "reads_consistent pending_write \<O> m sb"   and "volatile \<longrightarrow> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "reads_consistent pending_write \<O> m (sb @ [Read\<^sub>s\<^sub>b volatile a t v])"
terminatesD: fixes g :: "'e \<Rightarrow> ('f, 'e) step"   and s :: "'e" assumes "terminates g" shows "s \<in> terminates_on g"
w2: shows "(OF_priority_match OF_match_fields_safe oft p = Action []) = (simple_linux_router_nol12 rt fw p = None)"
TR4_sub_carr: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'h) Module_scheme"   and P :: "('g \<times> 'i, 'a, 'h) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "TR4 R M N P \<subseteq> carrier P"
inverse_ereal_tendsto_pos: fixes x :: "ereal" assumes "0 < x" shows "inverse \<midarrow>x\<rightarrow> inverse x"
map_closed: fixes \<alpha> :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and a :: "'b" assumes "map \<alpha> S T"   and "a \<in> S" shows "\<alpha> a \<in> T"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'c tree234 \<Rightarrow> 'b"   and f2 :: "'c tree234 \<Rightarrow> 'c \<Rightarrow> 'c tree234 \<Rightarrow> 'b"   and up\<^sub>i :: "'c up\<^sub>i" shows "P (case up\<^sub>i of T\<^sub>i x \<Rightarrow> f1 x | Up\<^sub>i x xa xb \<Rightarrow> f2 x xa xb) = ((\<forall>x1. up\<^sub>i = T\<^sub>i x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22 x23. up\<^sub>i = Up\<^sub>i x21 x22 x23 \<longrightarrow> P (f2 x21 x22 x23)))"
zero_divides: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "ring R" shows "\<zero>\<^bsub>R\<^esub> divides\<^bsub>R\<^esub> a = (a = \<zero>\<^bsub>R\<^esub>)"
homeomorphism_on_def: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "homeomorphism_on S f = (\<exists>g. homeomorphism S (f ` S) f g)"
pure_filter_parametric: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun A (rel_filter A) pure_filter pure_filter"
standard_identity_rules_2: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph" assumes "standard' C G" shows "maintained (transitive_rule S_Idt) G"
scale_zero_left: fixes x :: "'c" shows "(0::'a) *b x = (0::'c)"
AE_ball_countable: fixes X :: "'b set"   and P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'c measure" assumes "countable X" shows "(AE x in M. \<forall>y\<in>X. P x y) = (\<forall>y\<in>X. AE x in M. P x y)"
loop_stepA: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"   and "B\<^sub>1 = {}"   and "L = {}"   and "u \<in> S" shows "inv\<^sub>3 P\<^sub>1 P\<^sub>2 {u} B\<^sub>2 (S - {u}) L"
strict_trans1: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "lex_pm a b"   and "lex_pm_strict b c" shows "lex_pm_strict a c"
eval_simps'_5: fixes f :: "'c local.term" assumes "Arr f" shows "cod \<lbrace>f\<rbrace> = \<lbrace>Cod f\<rbrace>"
ordinal_utility_expected_value: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "continuous_vnm (\<P> outcomes) \<R>" shows "ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x (ocU outcomes \<R>))"
Sup_upto_llist_enat_0: fixes Xs :: "'b set llist" shows "Sup_upto_llist Xs (enat 0) = (if lnull Xs then {} else lhd Xs)"
wt_sat_if_simple: fixes S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and \<I>' :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "simple S"   and "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S \<theta>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S)"   and "\<forall>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<longrightarrow> ineq_model \<I>' X F"   and "ground (subst_range \<I>')"   and "subst_domain \<I>' = {x \<in> vars\<^sub>s\<^sub>t S. \<exists>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<and> x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X}"   and "list_all tfr\<^sub>s\<^sub>t\<^sub>p S" shows "\<exists>\<I>. (subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)) \<and> \<I> \<Turnstile>\<^sub>c \<langle>S,\<theta>\<rangle> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<I>)"
mult_smult_right: fixes p :: "'b poly"   and a :: "'b"   and q :: "'b poly" shows "p * smult a q = smult a (p * q)"
distinct_88: fixes pi' :: "pi"   and pi1 :: "pi"   and pi2 :: "pi" shows "!pi' \<noteq> pi1 \<parallel> pi2"
reversepath_simps_4: fixes g :: "real \<Rightarrow> 'c" shows "pathfinish (reversepath g) = pathstart g"
K: fixes \<phi> :: "i \<Rightarrow> bool"   and \<psi> :: "i \<Rightarrow> bool" shows "\<lfloor>\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>\<rightarrow> \<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi>\<rfloor>"
image_subgroup: fixes G :: "'a set" assumes "subgroup G G (\<cdot>) \<one>" shows "subgroup (left.translation ` G) (left.translation ` G) (compose G) (identity G)"
l8_14_2_1b_bis: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and X :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D"   and "Tarski_neutral_dimensionless.Col Bet X A B"   and "Tarski_neutral_dimensionless.Col Bet X C D" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D"
f4: shows "0 \<le> sqrt (real 2) + - 1 * sqrt ((Ratreal p)\<^sup>2)"
pi_dummy_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_dummy_pi [x, y] = Pi y"
assms: shows "rat_to_normalized_int_poly p = (d, q)"
sorted_distinct: fixes xs :: "'b list" assumes "sorted xs"   and "distinct xs" shows "\<forall>i<length xs - 1. xs ! i < xs ! (i + 1)"
induct: fixes x1 :: "collection_ternop"   and x2 :: "'a OCL_Types.type"   and x3 :: "'b OCL_Types.type"   and x4 :: "'a OCL_Types.type"   and x5 :: "'a OCL_Types.type"   and P :: "collection_ternop \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'b OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "collection_ternop_type x1 x2 x3 x4 x5"   and "\<And>\<sigma> \<rho> \<tau>. \<lbrakk>\<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1]; \<rho> \<le> \<tau>\<rbrakk> \<Longrightarrow> P InsertAtOp (OrderedSet \<tau>) \<sigma> \<rho> (OrderedSet \<tau>)"   and "\<And>\<sigma> \<rho> \<tau>. \<lbrakk>\<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1]; \<rho> \<le> \<tau>\<rbrakk> \<Longrightarrow> P InsertAtOp (Sequence \<tau>) \<sigma> \<rho> (Sequence \<tau>)"   and "\<And>\<sigma> \<rho> \<tau>. \<lbrakk>\<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1]; \<rho> = UnlimitedNatural[1]\<midarrow>Integer[1]\<rbrakk> \<Longrightarrow> P SubOrderedSetOp (OrderedSet \<tau>) \<sigma> \<rho> (OrderedSet \<tau>)"   and "\<And>\<sigma> \<rho> \<tau>. \<lbrakk>\<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1]; \<rho> = UnlimitedNatural[1]\<midarrow>Integer[1]\<rbrakk> \<Longrightarrow> P SubSequenceOp (Sequence \<tau>) \<sigma> \<rho> (Sequence \<tau>)" shows "P x1 x2 x3 x4 x5"
functionOnFirstEqualsSecond: fixes R :: "('c \<times> 'd) set"   and z :: "'c \<times> 'd" assumes "runiq R"   and "z \<in> R" shows "R ,, fst z = snd z"
triangle_G: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<eta> :: "'d \<Rightarrow> 'd"   and \<epsilon> :: "'c \<Rightarrow> 'c" assumes "unit_counit_adjunction C D F G \<eta> \<epsilon>" shows "vertical_composite.map C D (\<eta> \<circ> G) (G \<circ> \<epsilon>) = G"
bres_curry: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "prod.bres x (prod.bres y z) = prod.bres (times_prod' y x) z"
bij_betwI: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'd" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> A"   and "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x"   and "\<And>y. y \<in> B \<Longrightarrow> f (g y) = y" shows "bij_betw f A B"
ax_methods_spec: fixes G :: "prog"   and A :: "'b triple set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b triple"   and ms :: "('c \<times> 'd) set"   and C :: "'c"   and sig :: "'d" assumes "G,A|\<turnstile>(\<lambda>(x, y). f x y) ` ms"   and "(C, sig) \<in> ms" shows "G,A\<turnstile>f C sig"
null_char: shows "partial_magma.null cospan_shape.comp = cospan_shape.Null"
minus_minus: fixes x :: "'c" assumes "x \<in> carrier H" shows "\<ominus>\<^bsub>H\<^esub> (\<ominus>\<^bsub>H\<^esub> x) = x"
mset_merge_pairs: fixes hs :: "'b heap list" shows "mset_heap (merge_pairs hs) = \<Sum>\<^sub># (image_mset mset_heap (mset hs))"
D: shows "y * \<top> \<sqinter> y ^ o = y"
a_zero: shows "ad (zero_class.zero::'a) = (1::'a)"
intros_2: fixes u :: "'b list"   and C :: "'b list set"   and b :: "'b list" assumes "u \<in> C"   and "u \<noteq> b" shows "u \<in> no_head C b"
tuple_encode_decode': fixes n :: "nat"   and i :: "nat" shows "tuple_encode n (tuple_decode n i) = i"
linepath_def: fixes a :: "'b"   and b :: "'b" shows "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
ladder_i_eq_last_j: fixes L :: "(nat \<times> nat \<times> nat) list"   and L' :: "(nat \<times> nat \<times> nat) list" assumes "L \<noteq> []" shows "ladder_i (L @ L') (length L) = ladder_last_j L"
bl2nat_exp_zero: fixes y :: "nat" shows "bl2nat (Oc \<up> y) 0 = 2 ^ y - Suc 0"
abc_SucTr0: fixes a :: "nat"   and c :: "nat"   and b :: "nat" assumes "0 < a"   and "c \<le> b"   and "a - Suc 0 = b - c" shows "a = Suc b - c"
inverse_aform_err_float_2: fixes p :: "nat"   and x :: "(real \<times> real pdevs) \<times> real"   and a :: "real"   and b :: "real pdevs"   and ba :: "real" assumes "inverse_aform_err p x = Some ((a, b), ba)" shows "ba \<in> float"
bind_pure_returns_result_I: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a"   and x :: "'c"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog"   and y :: "'d" assumes "pure f h"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
k2: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "2 \<le> k"
fold_set_fold_remdups: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c"   and xs :: "'b list" assumes "comp_fun_commute f" shows "Finite_Set.fold f y (set xs) = fold f (remdups xs) y"
constant_transformation_axioms: shows "constant_transformation (\<cdot>\<^sub>J) (\<cdot>) a'"
deg_zero_ltrm_smult_cring: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "p \<in> carrier P"   and "a \<in> carrier R"   and "deg R p = 0" shows "monom P ((a \<odot>\<^bsub>P\<^esub> p) (deg R (a \<odot>\<^bsub>P\<^esub> p))) (deg R (a \<odot>\<^bsub>P\<^esub> p)) = a \<odot>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)"
dims2_def: shows "st2.dims2 = nths dims {Suc k}"