hcju/isabelleps huggingface dataset:
lookup_operator_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list" shows "ast_problem.lookup_operator problem name \<equiv> find (\<lambda>(n, uu_, uu_, uu_). n = name) (ast_problem.ast\<delta> problem)"
bind_spmf_assoc: fixes x :: "'t spmf"   and f :: "'t \<Rightarrow> 's spmf"   and g :: "'s \<Rightarrow> 'r spmf" shows "x \<bind> f \<bind> g = x \<bind> (\<lambda>y. f y \<bind> g)"
is_false_def: fixes d :: "dynamic" shows "is_false d = (d = DBool False)"
intro: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "VectorSpace smult V"   and "FinDimVectorSpace_axioms smult V" shows "FinDimVectorSpace smult V"
I_1: shows "NominalSequents.mset C \<equiv> \<LM> Cpd0 F Fs  \<RM>"
topspace_Euclidean_space: fixes n :: "nat" shows "topspace (Euclidean_space n) = {x. \<forall>i\<ge>n. x i = 0}"
restriction_is_subrel: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "P || X \<subseteq> P"
add_mset_remove_trivial_eq: fixes N :: "'k multiset"   and a :: "'k" shows "(N = add_mset a (N - {#a#})) = (a \<in># N)"
u_def: shows "u \<equiv> \<lambda>X\<in>carrier_direct_lim. let x = SOME x. x \<in> X in \<psi> (fst x) (snd x)"
matchT_TMT_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.matchT_TMT tval aval indis c d \<equiv> \<forall>s t s'. PL.mustT tval aval c s \<and> PL.mustT tval aval d t \<and> (s, t) \<in> indis \<and> PL.transT tval aval (c, s) s' \<longrightarrow> (\<exists>t'. PL.MtransT tval aval (d, t) t' \<and> (s', t') \<in> indis)"
ZOretr_sym: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and theta :: "(('c, 'd, 'a) cmd \<times> ('c, 'd, 'a) cmd) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "sym theta" shows "sym (PL_Indis.ZOretr aval tval cval indis theta)"
ball_trivial: fixes x :: "'c" shows "ball x 0 = {}"
not_le_imp_less: fixes y :: "'d word"   and x :: "'d word" assumes "\<not> y \<le>s x" shows "x <s y"
strongBisimSubstWeakBisimSubst: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and Q :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "(\<Psi>, P, Q) \<in> substPsi.closeSubst substTerm substAssert substCond (env.bisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq')" shows "(\<Psi>, P, Q) \<in> substPsi.closeSubst substTerm substAssert substCond (env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq')"
ins_ins2_3: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree"   and n' :: "nat"   and h' :: "nat" assumes "ins2 n d x t = Unbal2 t' n' h'" shows "ins n d x t = Unbal t' \<and> n' = size t' \<and> h' = height t'"
inline1_def: fixes callee :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd, 'e, 'f) gpv" shows "inline1 callee \<equiv> curry (spmf.fixp_fun (\<lambda>inline1 (gpv, s). the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). curry inline1 (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))))"
cases: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'e" assumes "(r OO s) a1 a2" obtains a :: "'c" and b :: "'d" and c :: "'e" where "a1 = a" and "a2 = c" and "r a b" and "s b c"
path_concatenation_cycle_free: fixes x :: "'a"   and y :: "'a" assumes "forward_terminating_path x"   and "backward_terminating_path y"   and "end_points x = start_points y"   and "x ; 1 \<cdot> y\<^sup>T ; 1 = 0" shows "path (x + y)"
Inv_preserves_Can_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Dom C (monoidal_language.Inv C t) = monoidal_language.Cod C t"
P16: shows "Z Midpoint Q Q'' \<and> Cong Y Q Y Q'"
ub2: shows "\<forall>x\<ge>ub2. sgn (poly p x) = sgn_pos_inf p"
sub_is_proper: assumes "\<U> \<noteq> \<V>" shows "proper_sub_set_system \<U> \<A> \<V> \<B>"
rcos_const: fixes h :: "nat" assumes "Group.group G"   and "h \<in> H" shows "H #>\<^bsub>G\<^esub> h = H"
integral_bounded_linear: fixes T :: "'c \<Rightarrow> 'd"   and M :: "'e measure"   and f :: "'e \<Rightarrow> 'c" assumes "bounded_linear T"   and "integrable M f" shows "LINT x|M. T (f x) = T (integral\<^sup>L M f)"
decomps\<^sub>e\<^sub>s\<^sub>t_decomp_rm\<^sub>e\<^sub>s\<^sub>t_empty: fixes D :: "('fun, 'var) extstrand_step list"   and M :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t M N \<I>" shows "decomp_rm\<^sub>e\<^sub>s\<^sub>t D = []"
exhaust_sel: fixes list :: "'c list" obtains "list = []"   | "list = lead_coeff list # tl list"
module_ow_axioms: shows "module_ow U\<^sub>M\<^sub>_\<^sub>1 (+) (0::'b) (-) uminus (*s\<^sub>1)"
lemma_nat2Nat: fixes n :: "nat"   and m :: "nat" assumes "nat2Nat n = nat2Nat (n + m)"   and "0 < m" shows "False"
apply_inject: fixes x :: "'a perm"   and y :: "'a perm" shows "((\<langle>$\<rangle>) x = (\<langle>$\<rangle>) y) = (x = y)"
norm_minus_commute: fixes a :: "'a"   and b :: "'a" shows "norm (a - b) = norm (b - a)"
P2: shows "Bet B C C' \<and> Cong C C' E F"
sum_defs_3: fixes b :: "V" shows "ZFC_Cardinals.Inr b \<equiv> \<langle>1, b\<rangle>"
elims_2: fixes x :: "'b kdt" assumes "KD_Tree.complete x" obtains uu_ :: "(real, 'b) vec" where "x = Leaf uu_"   | uv_ :: "'b" and uw_ :: "real" and l :: "'b kdt" and r :: "'b kdt" where "x = Node uv_ uw_ l r" and "KD_Tree.complete l \<and> KD_Tree.complete r \<and> height l = height r"
errMOD_igSwapIGOpSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igWlsAbsIsInBar MOD"   and "igOpIPresIGWls MOD"   and "igSwapIPresIGWls MOD"   and "igSwapAbsIPresIGWlsAbs MOD"   and "igWlsDisj MOD"   and "igWlsAbsDisj MOD"   and "igSwapIGOp MOD" shows "igSwapIGOpSTR (errMOD MOD)"
vwalk_to_vpath_subset: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set (vwalk_to_vpath xs)" shows "x \<in> set xs"
length_mergesort: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "length (mergesort f xs) = length xs"
map\<^sub>0_simps_1: fixes a :: "'c" assumes "C.obj a" shows "D.obj (map\<^sub>0 a)"
w_simps_4: shows "trg \<omega>.chine = src t\<^sub>0"
existence_ivl_trans: fixes s :: "real"   and x0 :: "'a"   and t :: "real" assumes "s \<in> existence_ivl0 x0"   and "t \<in> existence_ivl0 (flow0 x0 s)" shows "s + t \<in> existence_ivl0 x0"
h3: shows "Inf \<circ> (`) (\<lambda>x. x \<circ> f) = (\<lambda>x. x \<circ> f) \<circ> Inf"
p_prime: fixes p :: "nat" assumes "mersenne_prime p" shows "prime p"
Ex1_Normal_form_part1: fixes f :: "int \<Rightarrow> int" assumes "f \<in> CofGroups.Ex1" shows "\<exists>k. \<forall>n. f n = n + k"
whiletrue: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<iota> :: "nat"   and c :: "('exp, 'id) MWLsCom" assumes "BMap (E b m) = True" shows "\<langle>while\<^bsub>\<iota>\<^esub> b do c od,m\<rangle> \<rightarrow>\<lhd>[]\<rhd> \<langle>Some (c;while\<^bsub>\<iota>\<^esub> b do c od),m\<rangle>"
not_alw_not: fixes \<phi> :: "'c stream \<Rightarrow> bool" shows "(\<lambda>xs. \<not> alw (\<lambda>xs. \<not> \<phi> xs) xs) = ev \<phi>"
fb0: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "a \<le> 1"   and "0 \<le> b"   and "b \<le> 1" shows "0 \<le> (1 - a) * f b + a * b"
app_insf_LstSeq_if: fixes s :: "hf"   and k :: "hf"   and z :: "hf"   and y :: "hf"   and k' :: "hf" assumes "LstSeq s k z" shows "app (insf s (succ k) y) k' = (if k' = succ k then y else app s k')"
2: shows "F * e\<^sup>T * e \<le> w\<^sup>T\<^sup>\<star>"
head_\<omega>_times: fixes m :: "hmultiset"   and n :: "hmultiset" shows "head_\<omega> (m * n) = head_\<omega> m * head_\<omega> n"
add_mset_delete_index: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "add_mset (xs ! i) (mset (delete_index i xs)) = mset xs"
square_free_multD_1: fixes f :: "'a poly"   and g :: "'a poly"   and h :: "'a poly" assumes "square_free (f * g)"   and "h dvd f"   and "h dvd g" shows "degree h = 0"
nat_power_functions: fixes R :: "('a, 'b) ring_scheme"   and n :: "nat" assumes "U_function_ring R" shows "U_function_ring.nat_power R n \<in> carrier (function_ring (carrier R) R)"
oprod_Total: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Total r"   and "Total r'" shows "Total (r *o r')"
subgame_\<omega>: fixes V' :: "'a set" shows "\<omega>\<^bsub>G'.subgame V'\<^esub> = \<omega>\<^bsub>G'\<^esub>"
ord_of_nonzero_2: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "0 \<le> ord_Zp x" shows "x \<in> nonzero Zp"
prob_space_C: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure"   and x :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" assumes "Ionescu_Tulcea P M"   and "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "prob_space (Ionescu_Tulcea.C P M n m x)"
simps_2: fixes f\<^sub>1 :: "'b val \<Rightarrow> 'b val"   and f\<^sub>2 :: "'b val \<Rightarrow> 'b val"   and v :: "'b val" shows "F_Plus f\<^sub>1 f\<^sub>2 (val.Left v) = val.Left (f\<^sub>1 v)"
distinct_card: fixes xs :: "'c list" assumes "distinct xs" shows "card (set xs) = List.length xs"
nat_pow_eone: fixes x :: "'a list" assumes "x \<in> carrier (poly_ring R)" shows "x [^]\<^bsub>poly_ring R\<^esub> 1 = x"
long_pref: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>p v"   and "\<^bold>|v\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "u = v"
intro: fixes m_ops :: "('a, 'b, 'c, 'd) map_ops_scheme"   and s_ops :: "('e, 'b, 'f) set_ops_scheme" assumes "StdMap m_ops"   and "StdSet s_ops" shows "index_loc m_ops s_ops"
ordered_sel_resolvent_upon_is_resolvent: fixes Sel :: "'at set"   and P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set"   and A :: "'at" assumes "ordered_sel_resolvent_upon Sel P1 P2 C A" shows "ordered_sel_resolvent Sel P1 P2 C"
in_notinI: fixes r :: "('c \<times> 'c) set"   and j :: "'c"   and i :: "'c" assumes "wo_rel r"   and "(j, i) \<notin> r \<or> j = i"   and "i \<in> Field r"   and "j \<in> Field r" shows "(i, j) \<in> r"
simps: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" shows "node\<^sub>i k ts t = (if length ts \<le> 2 * k then T\<^sub>i (Node ts t) else case split_half ts of (ls, (sub, sep) # rs) \<Rightarrow> Up\<^sub>i (Node ls sub) sep (Node rs t))"
wt_elim_cases_5: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and T' :: "ty"   and i :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (New T'[i])\<Colon>T" obtains "T = Inl (T'.[])" and "is_acc_type (prg E) (pkg E) T'" and "E,dt\<Turnstile>i\<Colon>-PrimT Integer"
ucast_down_minus: fixes a :: "'b word"   and b :: "'b word" assumes "is_down UCAST('b \<rightarrow> 'c)" shows "UCAST('b \<rightarrow> 'c) (a - b) = UCAST('b \<rightarrow> 'c) a - UCAST('b \<rightarrow> 'c) b"
le_infI1: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<le> x" shows "a \<cdot> b \<le> x"
obtain_fresh_variable': fixes \<Gamma> :: "Core.term set"   and \<tau> :: "typ"   and t :: "Core.term"   and u :: "Core.term" assumes "finite \<Gamma>" obtains x :: "variable" where "(x, \<tau>) \<notin> fv t \<union> fv u \<union> FV \<Gamma>"
augment_cf_def: fixes edges :: "(nat \<times> nat) set"   and cap :: "'capacity" shows "cf.augment_cf edges cap \<equiv> \<lambda>e. if e \<in> edges then cf e - cap else if prod.swap e \<in> edges then cf e + cap else cf e"
div_dvd_iff_mult: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "b \<noteq> zero"   and "dvd.dvd times b a" shows "dvd.dvd times (divide a b) c = dvd.dvd times a (times c b)"
bdd_below_mono: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set"   and A :: "'c set" assumes "class.preorder less_eq less"   and "preorder.bdd_below less_eq B"   and "A \<subseteq> B" shows "preorder.bdd_below less_eq A"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "y \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "x \<otimes>\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> y \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)"
subset_chain_def: fixes \<A> :: "'c set set"   and \<C> :: "'c set set" shows "subset.chain \<A> \<C> = (\<C> \<subseteq> \<A> \<and> (\<forall>X\<in>\<C>. \<forall>Y\<in>\<C>. X \<subseteq> Y \<or> Y \<subseteq> X))"
project_stable: fixes F :: "'c program"   and A :: "'a set" shows "(project h UNIV F \<in> stable A) = (F \<in> stable (extend_set h A))"
Nil_is_rev_conv: fixes xs :: "'g list" shows "([] = rev xs) = (xs = [])"
inject: fixes x21 :: "dag"   and x22 :: "ref"   and x23 :: "dag"   and y21 :: "dag"   and y22 :: "ref"   and y23 :: "dag" shows "(Node x21 x22 x23 = Node y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
chamber_image_subset: fixes D :: "'a set" assumes "D \<in> folding.\<C> - f \<turnstile> folding.\<C>" shows "order.greater_eq {B \<in> folding.\<C>. order.greater (chamber_distance B D) (chamber_distance B (f ` D))} (f \<turnstile> folding.\<C>)"
test_comp_mult2: fixes p :: "'a" assumes "test p" shows "p \<cdot> !p = (0::'a)"
dense_repr_monom_closed: fixes K :: "'a list set"   and p :: "'a list list"   and t :: "'a list \<times> nat" assumes "subring K (K [X])"   and "set p \<subseteq> K"   and "t \<in> set (UP.dense_repr p)" shows "UP.monom (fst t) (snd t) \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"
r_altdef: shows "r = monom (1::'a) 1 + [:- a / of_nat n:]"
rGamma_series_aux: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and rGamma :: "'a \<Rightarrow> 'a"   and z :: "'a" assumes "class.Gamma dist uniformity open divide inverse times one plus zero minus uminus scaleR norm sgn rGamma"   and "\<And>n. z \<noteq> uminus (semiring_1.of_nat one plus zero n)" shows "let fact' = \<lambda>n. comm_monoid_mult.prod times one (semiring_1.of_nat one plus zero) {1..n}; exp = \<lambda>x. THE e. LIM n sequentially. comm_monoid_add.sum plus zero (\<lambda>k. scaleR (inverse (fact k)) (power.power one times x k)) {..<n} :> topological_space.nhds open e; pochhammer' = \<lambda>a n. comm_monoid_mult.prod times one (\<lambda>n. plus a (semiring_1.of_nat one plus zero n)) {0..n} in LIM n sequentially. divide (pochhammer' z n) (times (fact' n) (exp (times z (scaleR (ln (real n)) one)))) :> topological_space.nhds open (rGamma z)"
\<tau>: shows "pseudonatural_equivalence V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
in_front: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o" shows "s\<^sub>e < s\<^sub>o"
size_def: fixes m :: "('d, 'e) mapping" shows "Mapping.size m = (if finite (keys m) then card (keys m) else 0)"
fold_mset_union: fixes s :: "'m"   and M :: "'m multiset"   and N :: "'m multiset" shows "fold_mset munionM s (M + N) = fold_mset munionM (fold_mset munionM s M) N"
scale_left_imp_eq: fixes a :: "'b"   and x :: "'b"   and y :: "'b" assumes "a \<noteq> (0::'b)"   and "a * x = a * y" shows "x = y"
assms_2: shows "c = l_child l p \<or> c = r_child l p"
moebius_pt_comp_inv_image_right: fixes M :: "moebius"   and A :: "complex_homo set" shows "moebius_pt M ` moebius_pt (moebius_inv M) ` A = A"
A_SET_INV_defs_1: fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "a_set_inv\<^bsub>G\<^esub> H = set_inv\<^bsub>add_monoid G\<^esub> H"
removeAll_id: fixes x :: "'c"   and xs :: "'c list" assumes "x \<notin> set xs" shows "removeAll x xs = xs"
the_coext_prop_2: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' y f" shows "f = e \<cdot>\<^sub>C F (the_coext x' f)"
assms_1: shows "\<not> Vbl \<subseteq> vars (elements (getM state))"
zip_Cons_Cons: fixes x :: "'b"   and xs :: "'b list"   and y :: "'c"   and ys :: "'c list" shows "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
init_syzygy_list_def: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "init_syzygy_list bs = map_idx (lift_poly_syz (length bs)) bs 0"
almost_diff_var: shows "valid (Equals (Differential (Var vid1)) ($' vid1))"
weight_finite: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a" assumes "web \<Gamma>" shows "weight \<Gamma> x \<noteq> \<top>"
two_comparisons_into_case_order_1: fixes x :: "'b"   and y :: "'b"   and P :: "'d"   and Q :: "'d"   and R :: "'d" shows "(if c1.le x y then if x = y then P else Q else R) = (case c1 x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
FRadmit_box_simps: fixes a :: "('sf, 'sc, 'sz) hp"   and f :: "('sf, 'sc, 'sz) formula" shows "FRadmit ([[a]]f) = ((\<exists>\<theta>1 \<theta>2. ([[a]]f) = Geq \<theta>1 \<theta>2) \<or> (\<exists>p args. ([[a]]f) = $\<phi> p args) \<or> (\<exists>\<phi>. ([[a]]f) = ! \<phi> \<and> FRadmit \<phi>) \<or> (\<exists>\<phi> \<psi>. ([[a]]f) = (\<phi> && \<psi>) \<and> FRadmit \<phi> \<and> FRadmit \<psi>) \<or> (\<exists>\<phi>. (\<exists>x. ([[a]]f) = Exists x \<phi>) \<and> FRadmit \<phi>) \<or> (\<exists>\<alpha> \<phi>. ([[a]]f) = (\<langle> \<alpha> \<rangle> \<phi>) \<and> PRadmit \<alpha> \<and> FRadmit \<phi>))"
simps_2: fixes x :: "'b"   and hs :: "'b hp list" shows "del_min (Some (Hp x hs)) = pass\<^sub>2 (pass\<^sub>1 hs)"
Chains_FieldD: fixes M :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b" assumes "M \<in> Chains r"   and "x \<in> M" shows "x \<in> Field r"
set_dom_cod_object_subset_3: fixes f :: "'b set_arrow"   and U :: "'b set" assumes "f \<in> Ar\<^bsub>set_cat U\<^esub>" shows "set_cod f \<subseteq> U"
prove_SubstFormP: fixes i :: "name"   and A :: "fm" shows "{} \<turnstile> SubstFormP \<guillemotleft>Var i\<guillemotright> \<guillemotleft>\<guillemotleft>A\<guillemotright>\<guillemotright> \<guillemotleft>A\<guillemotright> \<guillemotleft>A(i::=\<guillemotleft>A\<guillemotright>)\<guillemotright>"
C_slide: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y)\<^sup>\<star> \<cdot> x = x \<cdot> (y \<cdot> x)\<^sup>\<star>"
ab_def_1: fixes n :: "nat" shows "AB n = (f ^^ n) (a, b)"
finite_sum_univ_singleton: fixes g :: "'e \<Rightarrow> 'd"   and i :: "'e" shows "sum g UNIV = sum g {i} + sum g (UNIV - {i})"
lemma_2_10_24: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r" shows "less_eq a (imp_r (imp_l a b) b)"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [v, $Tee_local i] : ts _> ts'"
inject_2: fixes x1 :: "name"   and x2 :: "'d frame"   and y1 :: "name"   and y2 :: "'d frame" shows "(\<lparr>\<nu>x1\<rparr>x2 = \<lparr>\<nu>y1\<rparr>y2) = ([x1].x2 = [y1].y2)"
sum_nonpos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) zero" shows "less_eq (comm_monoid_add.sum plus zero f A) zero"
CardI: fixes i :: "V" assumes "Ord i"   and "\<And>j. \<lbrakk>j < i; Ord j\<rbrakk> \<Longrightarrow> \<not> elts j \<approx> elts i" shows "Card i"
homE_1: fixes CC :: "('d, 'e, 'f) category_scheme"   and A :: "'d"   and B :: "'d"   and f :: "'e" assumes "category CC"   and "A \<in> Ob\<^bsub>CC\<^esub>"   and "B \<in> Ob\<^bsub>CC\<^esub>"   and "f \<in> Hom\<^bsub>CC\<^esub> A B" shows "Dom\<^bsub>CC\<^esub> f = A"
summable_norm_cancel: assumes "summable (\<lambda>n. norm (f n))" shows "summable f"
minus_gt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat" shows "less a b = less zero (minus b a)"
cnf_semantics: fixes F :: "'b formula"   and \<A> :: "'b \<Rightarrow> bool" assumes "is_nnf F" shows "cnf_semantics \<A> (cnf F) = \<A> \<Turnstile> F"
has_codomain_iff_arr: fixes f :: "'a" shows "(C.codomains f \<noteq> {}) = C.arr f"
assms_2: fixes S :: "'a set"   and T :: "'a set" assumes "P S"   and "P T"   and "interior S \<inter> interior T = {}" shows "P (S \<union> T)"
witness_prop_def: fixes \<delta> :: "('a, 'b) ta_rule set"   and m :: "'a \<Rightarrow> 'b tree option" shows "witness_prop \<delta> m \<equiv> \<forall>q t. m q = Some t \<longrightarrow> accs \<delta> t q"
prems_3: shows "\<forall>a\<in>set (map fst (ys @ [y])). oid < a"
conjugate_circline_set': fixes H :: "circline" shows "conjugate ` circline_set H \<subseteq> circline_set (conjugate_circline H)"
In1_def: fixes M :: "('a, 'b) node set" shows "In1 M \<equiv> Scons (Numb 1) M"
one_mat_def: fixes n :: "nat" shows "1\<^sub>m n \<equiv> mat n n (\<lambda>(i, j). if i = j then 1::'c else (0::'c))"
cos_minus_pi: fixes x :: "real" shows "cos (x - pi) = - cos x"
pick_addr_def: fixes HA :: "('a \<times> 'b) set" shows "pick_addr HA = (SOME ha. ha \<in> HA)"
x_def: shows "x \<equiv> THE x. abss x = s \<and> x \<in> set_pmf (action cfg)"
Close2: fixes P :: "pi"   and a :: "name"   and y :: "name"   and P' :: "pi"   and Q :: "pi"   and Q'' :: "pi"   and x :: "name"   and Q' :: "pi" assumes "P \<Longrightarrow>\<^sub>l a<\<nu>y> \<prec> P'"   and "Q \<Longrightarrow>\<^sub>ly in Q''\<rightarrow>a<x> \<prec> Q'"   and "y \<sharp> P"   and "y \<sharp> Q" shows "P \<parallel> Q \<Longrightarrow>\<^sub>l \<tau> \<prec> <\<nu>y>(P' \<parallel> Q')"