hcju/isabelleps huggingface dataset:
tts_rel_set_transfer: fixes S :: "'b set"   and A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "S \<subseteq> Collect (Domainp A)" shows "\<exists>S'. rel_set A S S'"
ivl_pi_def: shows "compute_bound_fa prec Pi [] = Some ivl_pi"
push_forall_negation_free_all: fixes \<phi> :: "atom fm" assumes "negation_free \<phi>" shows "negation_free (push_forall (AllQ \<phi>))"
col_convergence: fixes X :: "nat \<Rightarrow> 'a sq_mtx"   and L :: "((real, 'a) vec, 'a) vec" assumes "\<forall>i. (\<lambda>n. \<c>\<o>\<l> i (X n)) \<longlonglongrightarrow> L $ i" shows "X \<longlonglongrightarrow> to_mtx (Finite_Cartesian_Product.transpose L)"
bi_total_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "bi_total R = ((\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y))"
simps: fixes \<Gamma>l :: "String.literal \<Rightarrow> unit option"   and \<Gamma>g :: "String.literal \<Rightarrow> unit option"   and ls :: "local_state"   and gs :: "global_state" shows "ty_fs (\<Gamma>l, \<Gamma>g) (ls, gs) = (ty_local_state \<Gamma>l ls \<and> ty_global_state \<Gamma>g gs)"
that: fixes h :: "nat \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int" assumes "\<And>p. h p 0 = 0"   and "\<And>p c1 c2. h p (c1 - c2) = h p c1 - h p c2"   and "\<And>p X c. singular_chain p X c \<Longrightarrow> singular_chain (Suc p) X (h p c)"   and "\<And>p X c. singular_chain p X c \<Longrightarrow> chain_boundary (Suc p) (h p c) + h (p - Suc 0) (chain_boundary p c) = (singular_subdivision p ^^ n) c - c" shows "thesis"
assms_4: shows "b \<in>\<^sub>A frontier (c_imp c loc2)"
cball_translation: fixes a :: "'c"   and c :: "'c"   and r :: "real" shows "cball (a + c) r = (+) a ` cball c r"
DERIV_nonpos_imp_nonincreasing: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. (f has_real_derivative y) (at x) \<and> y \<le> 0" shows "f b \<le> f a"
prems_6: shows "oldf \<in> set (F ! k) \<longrightarrow> set newfs \<inter> set (F ! k) \<subseteq> {oldf}"
arr_tensor: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g" shows "arr (f \<otimes>\<^sub>1 g)"
Abs_nebset_inject: fixes x :: "'d set"   and y :: "'d set" assumes "x \<in> {A. A \<noteq> {} \<and> |A| <o natLeq +c |UNIV|}"   and "y \<in> {A. A \<noteq> {} \<and> |A| <o natLeq +c |UNIV|}" shows "(Abs_nebset x = Abs_nebset y) = (x = y)"
usubstappt_differential_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<theta> :: "trm" assumes "usubstappt \<sigma> U (Differential \<theta>) \<noteq> undeft" shows "usubstappt \<sigma> Denotational_Semantics.allvars \<theta> \<noteq> undeft"
hom_swaprows: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_row A" shows "swaprows i j (mat\<^sub>h A) = mat\<^sub>h (swaprows i j A)"
mopup_jump_over1_2_aft_erase_a: fixes n :: "nat"   and l :: "cell list"   and xs :: "cell list"   and lm :: "nat list"   and ires :: "cell list" assumes "mopup_jump_over1 (Suc (2 * n), l, Bk # xs) lm n ires" shows "mopup_aft_erase_a (Suc (Suc (2 * n)), Bk # l, xs) lm n ires"
fset_mset_def: fixes N :: "'c multiset" shows "fset_mset N \<equiv> Abs_fset (set_mset N)"
X_def_2: shows "f 0 = Q\<^sub>x"
elims_2: fixes x :: "'c nres"   and xa :: "'c nres" assumes "x \<le> xa" obtains uu_ :: "'c nres" where "x = uu_" and "xa = FAILi"   | a :: "'c set" and b :: "'c set" where "x = RES a" and "xa = RES b" and "a \<subseteq> b"
hm_isEmpty_rule: fixes m :: "'a \<Rightarrow> 'b option"   and ht :: "('a, 'b) hashtable" shows "<is_hashmap m ht> hm_isEmpty ht <\<lambda>r. is_hashmap m ht * \<up> (r = (m = Map.empty))>"
unit_eq_div1: 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 "dvd.dvd times b one" shows "(divide a b = c) = (a = times c b)"
f_eq_g: shows "\<forall>x\<in>set_of_vector X. g x = f x"
always_lft_rel_comp_a: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "nat \<Rightarrow> 'd"   and r' :: "'e \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and v :: "nat \<Rightarrow> 'e" shows "(\<box> lft_rel_st r) u OO (\<box> lft_rel_st r') v = (\<box> lft_rel_st (\<lambda>(u, v) (u', v'). r u u' OO r' v v')) (u || v)"
norm_memb_eq_rep_ARR: fixes f :: "'c local.term set"   and t :: "'c local.term" assumes "ARR f"   and "t \<in> f" shows "\<^bold>\<parallel>t\<^bold>\<parallel> = rep f"
le_length_mono: fixes s :: "'a list"   and t :: "'a list" assumes "s \<le> t" shows "length s \<le> length t"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "\<And>x. hom x dvd (1::'b) \<Longrightarrow> x dvd (1::'a)" shows "unit_preserving_hom_axioms hom"
set_removeAll: fixes x :: "'c"   and xs :: "'c list" shows "set (removeAll x xs) = set xs - {x}"
IfFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" assumes "\<not> bval b s" shows "(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<rightarrow> (c\<^sub>2, s)"
cart_product_add_1_filter: fixes P :: "'b \<times> 'c \<Rightarrow> bool"   and a :: "'b"   and M :: "'b multiset"   and N :: "'c multiset" shows "filter_mset P (add_mset a M \<times># N) = filter_mset P (M \<times># N) + filter_mset P ({#a#} \<times># N)"
pair_measure_def: fixes A :: "'a measure"   and B :: "'b measure" shows "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B) {a \<times> b |a b. a \<in> sets A \<and> b \<in> sets B} (\<lambda>X. \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>B \<partial>A)"
of_bits_int_def: fixes bs :: "bool list" shows "of_bits bs = (if bs = [] \<or> \<not> last bs then horner_sum of_bool 2 bs else horner_sum of_bool 2 bs - 2 ^ length bs)"
WTrts_iffs_3: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and es :: "'addr expr list"   and Ts :: "ty list" shows "(P,E,h \<turnstile> e # es [:] Ts) = (\<exists>T Ts. Ts = T # Ts \<and> P,E,h \<turnstile> e : T \<and> P,E,h \<turnstile> es [:] Ts)"
dim_subset_UNIV: fixes S :: "('a, 'b) vec set" shows "vec.dim S \<le> finite_dimensional_vector_space.dimension cart_basis"
simps_1: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b" shows "[] \<Squnion>\<^bsub>f\<^esub> y = y"
rank_le_ncols: fixes A :: "(('a, 'b) vec, 'c) vec" shows "rank A \<le> ncols A"
wfp_def: fixes p :: "prog" shows "wfp p \<equiv> distinct (para p) \<and> wfe (ctxt p) (body p)"
one_star_app_congR: fixes t1 :: "lam"   and t2 :: "lam"   and s :: "lam" assumes " t1 \<longrightarrow>\<^sub>1\<^sup>* t2" shows " App s t1 \<longrightarrow>\<^sub>1\<^sup>* App s t2"
sum_pow2_upto: fixes x :: "nat" shows "sum ((^) 2) {0..<x} = 2 ^ x - 1"
l2wi_negation_type_intersect: fixes l :: "('a word \<times> 'a word) negation_type list" shows "wordinterval_to_set (l2wi_negation_type_intersect l) = (\<Inter>(i, j)\<in>set (getPos l). {i..j}) - (\<Union>(i, j)\<in>set (getNeg l). {i..j})"
list_all2_map1: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'i \<Rightarrow> 'g"   and as :: "'i list"   and bs :: "'h list" shows "list_all2 P (map f as) bs = list_all2 (\<lambda>x. P (f x)) as bs"
quasi_inverses_def: fixes f :: "'c"   and g :: "'c" shows "C.quasi_inverses f g \<equiv> \<exists>\<phi> \<psi>. C.internal_equivalence f g \<phi> \<psi>"
imirror_bounds_elem_conv: fixes I :: "nat set"   and n :: "nat"   and l :: "nat"   and r :: "nat" assumes "finite I"   and "n \<le> l + r"   and "Max I \<le> l + r" shows "(nat_mirror n l r \<in> imirror_bounds I l r) = (n \<in> I)"
bind_pure_I: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x"   and g :: "'z \<Rightarrow> ('x, 'y, 'aa) prog" assumes "pure f h"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> pure (g x) h" shows "pure (Heap_Error_Monad.bind f g) h"
fact_mod: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "fact n mod fact m = (0::'c)"
all_info1: shows "set (characterize_consistent_signs_at_roots p qs1) \<subseteq> set signs1"
l2_inv8_trans: shows "{l2_inv8 \<inter> l2_inv1 \<inter> l2_inv2 \<inter> l2_inv3 \<inter> l2_inv4 \<inter> l2_inv4'} TS.trans l2 {> l2_inv8}"
minus_fract_cancel: fixes a :: "'a"   and b :: "'a" shows "Fraction_Field.Fract (- a) (- b) = Fraction_Field.Fract a b"
neqE: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "x \<noteq> y" obtains "x \<sqsubset> y"   | "y \<sqsubset> x"
max_fun_diff_le_eq: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and x :: "'e" assumes "wo_rel2 r s"   and "(wo_rel.max_fun_diff s f g, x) \<in> s"   and "f \<noteq> g"   and "f \<in> FinFunc r s"   and "g \<in> FinFunc r s"   and "x \<noteq> wo_rel.max_fun_diff s f g" shows "f x = g x"
env_restr_fun_upd_other: fixes x :: "'a"   and S :: "'a set"   and m1 :: "'a \<Rightarrow> 'b"   and v :: "'b" assumes "x \<notin> S" shows "m1(x := v) f|` S = m1 f|` S"
wprepare_loop_start_in_middle_Oc: fixes m :: "nat"   and lm :: "nat list"   and b :: "cell list"   and list :: "cell list" assumes "wprepare_loop_start_in_middle m lm (b, Oc # list)" shows "wprepare_loop_start_in_middle m lm (Oc # b, list)"
add_left_cancel: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.cancel_semigroup_add plus" shows "(plus a b = plus a c) = (b = c)"
d_cospoly_sinpoly: fixes n :: "nat"   and x :: "real" shows "(cospoly (Suc n) has_real_derivative - sinpoly n x) (at x)"
pset_pnPlus: fixes r :: "'a rexp"   and s :: "'a rexp" shows "pset (pnPlus r s) = pset (Plus r s)"
cases: fixes a :: "trace"   and p :: "prog" assumes "a \<in> traces p" obtains \<sigma> :: "char list \<Rightarrow> nat option" and \<sigma>' :: "char list \<Rightarrow> nat option" where "a = \<lparr>initial = \<sigma>, final = \<sigma>'\<rparr>" and "\<sigma> \<in> states p" and "body p \<rhd> \<sigma> \<leadsto> Some \<sigma>'"
g_1: fixes y :: "'b" assumes "y \<in> T" shows "f (g y) = y"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>0 'b"   and xa :: "'a \<Rightarrow>\<^sub>0 'b" shows "(x \<le> xa) = (less_fun (lookup x) (lookup xa) \<or> lookup x = lookup xa)"
hext_objD: fixes h :: "'heap"   and h' :: "'heap"   and a :: "'addr"   and C :: "String.literal" assumes "h \<unlhd> h'"   and "typeof_addr h a = \<lfloor>Class_type C\<rfloor>" shows "typeof_addr h' a = \<lfloor>Class_type C\<rfloor>"
acomplete_balance_list: fixes xs :: "'a list" shows "acomplete (balance_list xs)"
rtrancl_restrictI: fixes u :: "'a"   and v :: "'a"   and E :: "('a \<times> 'a) set"   and R :: "'a set" assumes "(u, v) \<in> (E - UNIV \<times> R)\<^sup>*"   and "u \<notin> R" shows "(u, v) \<in> (rel_restrict E R)\<^sup>*"
prems_1: shows "(tab(x # xs [\<mapsto>] ys)) vn = Some el"
et_var: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and x :: "nat" shows "\<Gamma> \<turnstile> Var x : \<Gamma> x"
prems_2: shows "(l # ts) ! i = (p, is, \<theta>, Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis # sb, \<D>, \<O>, \<R>)"
g1_scale_j_contin: shows "continuous_on (cbox a b) (\<lambda>x. (0::'a, g1 x))"
swap_ident: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "X #[x \<and> x]_xs = X"
filtermap_def: fixes f :: "'d \<Rightarrow> 'c"   and F :: "'d filter" shows "filtermap f F = Abs_filter (\<lambda>P. \<forall>\<^sub>F x in F. P (f x))"
map_of_inject_set: fixes xs :: "('e \<times> 'f) list"   and ys :: "('e \<times> 'f) list" assumes "distinct (map fst xs)"   and "distinct (map fst ys)" shows "(map_of xs = map_of ys) = (set xs = set ys)"
wf_Int1: fixes r :: "('g \<times> 'g) set"   and r' :: "('g \<times> 'g) set" assumes "wf r" shows "wf (r \<inter> r')"
cols_nth: fixes i :: "nat"   and A :: "'a mat" assumes "i < dim_col A" shows "cols A ! i = col A i"
Aexp_pre_simps_1: fixes x :: "Vars.var"   and a :: "Arity" shows "Aexp_syn' a (Var x) = esing x\<cdot>(up\<cdot>a)"
atomic_step_precondition_def: fixes s :: "state_t"   and tid :: "thread_id_t"   and ipt :: "int_point_t" shows "atomic_step_precondition s tid ipt \<equiv> case ipt of SK_IPC dir PREP partner page \<Rightarrow> True | SK_IPC dir WAIT partner page \<Rightarrow> ipc_precondition tid dir partner page s | SK_IPC dir (BUF page') partner page \<Rightarrow> ipc_precondition tid dir partner page s \<and> ipc_precondition partner (opposite_ipc_direction dir) tid page' s | SK_EV_SIGNAL EV_SIGNAL_PREP partner \<Rightarrow> True | SK_EV_SIGNAL EV_SIGNAL_FINISH partner \<Rightarrow> ev_signal_precondition tid partner s | _ \<Rightarrow> True"
iterate_to_list_correct: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list"   and S0 :: "'a set" assumes "set_iterator it S0" shows "set (iterate_to_list it) = S0 \<and> distinct (iterate_to_list it)"
invc_baliL: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invc2 l"   and "invc r" shows "invc (baliL l a r)"
DERIV_cong: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b"   and F :: "'b filter"   and Y :: "'b" assumes "(f has_field_derivative X) F"   and "X = Y" shows "(f has_field_derivative Y) F"
ide_dom: fixes f :: "'c" assumes "Fop.Bop.arr f" shows "Fop.Bop.ide (Fop.Bop.dom f)"
index_eq_iff: fixes xs :: "'c list"   and x :: "'c"   and i :: "nat" shows "(index xs x = i) = (i \<le> length xs \<and> (\<forall>j<i. xs ! j \<noteq> x) \<and> (i < length xs \<longrightarrow> xs ! i = x))"
int_iffD2: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" assumes "\<turnstile> F = G" shows "\<turnstile> G \<longrightarrow> F"
prems: shows "\<Gamma> \<turnstile>\<^sub>L expr # exprs : \<xi>"
clinearI: fixes f :: "'c \<Rightarrow> 'd" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>C b) = r *\<^sub>C f b" shows "clinear f"
map\<^sub>0_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>D (\<tau>\<^sub>0 a) = map\<^sub>0 a"
HPhase1or2Write_HInv4a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2Write s s' p d"   and "HInv4a s q" shows "HInv4a s' q"
LLL_measure_approx_fs_init: fixes upw :: "bool"   and i :: "nat" assumes "LLL_invariant upw i fs_init"   and "4 / 3 < \<alpha>"   and "m \<noteq> 0" shows "real (LLL_measure i fs_init) \<le> real m + real (2 * m * m) * log base (real N)"
psi_def: shows "psi = (\<lambda>(x, t). dist (fx (x, t)) (fx (x0, t)))"
transition_def: fixes stateA :: "State"   and stateB :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "transition stateA stateB F0 decisionVars \<equiv> appliedDecide stateA stateB decisionVars \<or> appliedUnitPropagate stateA stateB F0 \<or> appliedBacktrack stateA stateB F0"
content_ff_0_iff: fixes p :: "'b fract poly" shows "(content_ff p = 0) = (p = 0)"
cdf_at_left: fixes a :: "real" shows "(cdf M \<longlongrightarrow> Sigma_Algebra.measure M {..<a}) (at_left a)"
diff_delta_arctan_upper_12_def: shows "diff_delta_arctan_upper_12 \<equiv> \<lambda>x. - 4 * x ^ 4 / ((x\<^sup>2 + 3)\<^sup>2 * (1 + x\<^sup>2))"
RN_deriv_unique: fixes f :: "'a \<Rightarrow> ennreal"   and N :: "'a measure" assumes "f \<in> borel_measurable M"   and "density M f = N" shows "AE x in M. f x = RN_deriv M N x"
prv_neg_excl: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "bprv \<phi>"   and "bprv (neg \<phi>)" shows "False"
evaluation_map_axioms: shows "Coherence.evaluation_map (\<cdot>\<^sub>B) src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if B.arr \<mu> then \<mu> else B.null)"
foldD_def: fixes D :: "'b set"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and e :: "'b"   and A :: "'c set" shows "foldD D f e A = (THE x. (A, x) \<in> foldSetD D f e)"
dagger_prod_unfold: fixes x :: "'a"   and y :: "'a" shows "1' + x ; (y ; x)\<^sup>\<star> ; y = (x ; y)\<^sup>\<star>"
ru: shows "has_lock (redT_updLs ls t las $ L) T"
comp_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "L'.C.in_hom f a b"   and "L'.C.in_hom g b c" shows "L'.C.in_hom (local.Left a' g f) a c"
rep_eq: fixes x :: "'b ltln\<^sub>P"   and xa :: "'b ltln\<^sub>P" shows "ltln\<^sub>P_eq_test x xa = rep_ltln\<^sub>P x \<sim>\<^sub>P rep_ltln\<^sub>P xa"
gb_schema_dummy_simp: fixes data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "struct_spec sel ap ab compl" shows "gb_schema_dummy data D bs ps = (if ps = [] then (gs @ bs, D) else let sps = sel gs bs ps (snd data); ps0 = ps -- sps; aux = compl gs bs ps0 sps (snd data); remcomps = fst data - count_const_lt_components (fst aux) in if remcomps = 0 then (full_gb (gs @ bs), D) else let (hs, data') = add_indices aux (snd data) in gb_schema_dummy (remcomps, data') (D \<union> (set hs \<times> (set gs \<union> set bs \<union> set hs) \<union> set (ps -- sps) -\<^sub>p set (ap gs bs ps0 hs data'))) (ab gs bs hs data') (ap gs bs ps0 hs data'))"
not_eq_order_implies_strict: fixes a :: "'a list"   and b :: "'a list" assumes "a \<noteq> b"   and "Sublist.prefix a b" shows "strict_prefix a b"
usubstr_cons_undef: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and A :: "fml"   and B :: "fml list"   and C :: "fml" shows "(usubstr \<sigma> (A # B, C) \<noteq> None) = (usubstr \<sigma> ([A], C) \<noteq> None \<and> usubstr \<sigma> (B, C) \<noteq> None)"
beta_l_def: fixes l :: "nat" shows "beta_l l = complex_of_real (sin ((real l + 1 / 2) * \<theta>))"
poincare_line_non_homogenous: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<noteq> \<infinity>\<^sub>h"   and "v \<noteq> \<infinity>\<^sub>h"   and "u \<noteq> v"   and "u \<noteq> inversion v" shows "let u' = to_complex u; v' = to_complex v; A = \<i> * (u' * cnj v' - v' * cnj u'); B = \<i> * (v' * cor ((cmod u')\<^sup>2 + 1) - u' * cor ((cmod v')\<^sup>2 + 1)) in poincare_line u v = mk_circline A B (cnj B) A"
cri_less_eq_def: shows "cri_less_eq = (\<lambda>(loc1, t1) (loc2, t2). \<exists>s. s \<in>\<^sub>A path_summary loc1 loc2 \<and> results_in t1 s \<le> t2)"
semiring_normalization_rules_2: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and m :: "'c" assumes "class.comm_semiring_1 times one plus zero" shows "plus (times a m) m = times (plus a one) m"
isoE: fixes f :: "'a" assumes "C.iso f" obtains g :: "'a" where "C.inverse_arrows f g"
mono: fixes x :: "('a, 'b, 'c) gpv"   and f :: "'a \<Rightarrow> ennreal"   and \<I> :: "('b, 'c) \<I>"   and fail :: "ennreal" shows "lfp.mono_body (\<lambda>expectation_gpv. \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. expectation_gpv (c r)) \<partial>measure_spmf (the_gpv x) + fail * ennreal (pmf (the_gpv x) None))"
marg_dens2_integral: fixes x :: "nat"   and y :: "nat"   and X :: "val set"   and \<rho> :: "nat \<Rightarrow> val" assumes "x \<in> V"   and "y \<in> V"   and "x \<noteq> y"   and "X \<in> sets (stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y)))"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "set_nn_integral (stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y))) X (marg_dens2 \<Y> x y \<rho>) = \<integral>\<^sup>+\<sigma>\<in>(\<lambda>\<sigma>. <|\<sigma> x, \<sigma> y|>) -` X \<inter> space (state_measure V \<Gamma>). \<delta> (merge V V' (\<sigma>, \<rho>))\<partial>state_measure V \<Gamma>"
get_elements_by_tag_name_result_in_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and name :: "char list"   and results :: "(_) element_ptr list"   and to :: "(_) object_ptr list"   and element_ptr :: "(_) element_ptr" assumes "h \<turnstile> get_elements_by_tag_name ptr name \<rightarrow>\<^sub>r results"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "element_ptr \<in> set results" shows "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr \<in> set to"
elims_2: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and x :: "(char list, char list, v) namespace" assumes "cakeml_static_env static_cenv"   and "cakeml_static_env.is_cupcake_ns static_cenv x" obtains v0 :: "(char list \<times> v) list" where "x = Bind v0 []" and "list_all (cakeml_static_env.is_cupcake_value static_cenv \<circ> snd) v0"
improve_lottery_support_subset: fixes p :: "'alt pmf"   and q :: "'alt pmf" assumes "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts"   and "p \<prec>[Pareto (SD \<circ> R)] q"   and "set_pmf p = set_pmf q" obtains r :: "'alt pmf" where "r \<in> lotteries_on alts" and "Pareto (SD \<circ> R) q r" and "set_pmf r \<subset> set_pmf p"
collapse: fixes SkewBinomialTree :: "('c, 'd) SkewBinomialTree" shows "Node (val SkewBinomialTree) (prio SkewBinomialTree) (rank SkewBinomialTree) (children SkewBinomialTree) = SkewBinomialTree"
instInpP_fmla: fixes \<phi> :: "'fmla"   and \<tau> :: "'fmla"   and n :: "nat" assumes "\<phi> \<in> fmla"   and "\<tau> \<in> fmla" shows "instInpP \<phi> n \<tau> \<in> fmla"
lift_pI: fixes P :: "'a \<Rightarrow> bool"   and f :: "('a, 'b) state" assumes "\<And>heap x heap'. \<lbrakk>P heap; run_state f heap = (x, heap')\<rbrakk> \<Longrightarrow> P heap'" shows "lift_p P f"
eq_fold: fixes g :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "sum g A = Finite_Set.fold ((+) \<circ> g) (0::'c) A"
SATp_SAT_eq: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "SATp \<Gamma> f = (\<lambda>x. x \<in> SAT \<Gamma> f)"
relcomp_distrib2: fixes S :: "('e \<times> 'g) set"   and T :: "('e \<times> 'g) set"   and R :: "('g \<times> 'f) set" shows "(S \<union> T) O R = S O R \<union> T O R"
Nfunc_LeastTr2_4: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set"   and i :: "nat" assumes "\<forall>l\<le>n. G \<guillemotright> f l "   and "\<forall>l\<le>n. G \<guillemotright> g l "   and "i \<le> n"   and "f i \<noteq> g i" shows "(LEAST k. f k \<noteq> g k) \<le> n"
pre_adj_if_postL: fixes l :: "('c \<times> nat) tree"   and a :: "'c"   and lv :: "nat"   and r :: "('c \<times> nat) tree"   and l' :: "('c \<times> nat) tree"   and b :: "'c" assumes "invar \<langle>l, (a, lv), r\<rangle>"   and "post_del l l'" shows "pre_adjust \<langle>l', (b, lv), r\<rangle>"
coboundedI2: fixes c :: "'c \<times> 'd"   and b :: "'c \<times> 'd"   and a :: "'c \<times> 'd" assumes "less_eq_prod c b" shows "less_eq_prod c (sup_prod a b)"
to_poly_closed: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R" shows "to_polynomial R a \<in> carrier (UP R)"
filter_ASSUME_def: fixes c :: "'a \<Rightarrow> bool"   and m :: "'a nres" shows "filter_ASSUME c m \<equiv> m \<bind> (\<lambda>x. ASSUME (c x) \<bind> (\<lambda>_. RETURN x))"