spec_position
stringlengths 41
140
| minilang
stringlengths 4
2.76M
⌀ | proof_position
stringlengths 43
144
| isar
stringlengths 1
726k
⌀ |
|---|---|---|---|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:192
|
RULE subset_antisym subsetI
(*goal: "\<I> (\<V> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) \<subseteq> \<surd>ideal F" and "\<surd>ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subseteq> \<I> (\<V> F)" *)
RULE subset_antisym subsetI
(*goal: "\<And>x::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b. x \<in> \<I> (\<V> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) \<Longrightarrow> x \<in> \<surd>ideal F" *)
INTRO VARS f
(*fixes variables x :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" *)
(*introduces facts assm0: "(f__::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<in> \<I> (\<V> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set))" *)
NEXT WITH Nullstellensatz assms assm0
(*goal: "\<surd>ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subseteq> \<I> (\<V> F)" *)
END WITH ideal_ofI variety_ofD variety_of_radical_ideal
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:195:1
|
proof (intro subset_antisym subsetI)
fix f
assume "f \<in> \<I> (\<V> F)"
with assms show "f \<in> \<surd>ideal F"
by (rule Nullstellensatz)
qed (metis ideal_ofI variety_ofD variety_of_radical_ideal)
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:201
|
HAVE fact0: "\<I> (\<V> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) = \<surd>ideal F"
END WITH strong_Nullstellensatz assms(1,2)
(*introduces facts local.fact0: "\<I> (\<V> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) = \<surd>ideal F" *)
END WITH radical_ideal_eq_UNIV_iff[symmetric] assms(3) fact0 WITHOUT radical_ideal_eq_UNIV_iff
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:204:1
|
proof -
from assms(1, 2) have "\<I> (\<V> F) = \<surd>ideal F"
by (rule strong_Nullstellensatz)
thus ?thesis
by (simp add: assms(3) flip: radical_ideal_eq_UNIV_iff)
qed
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:264
|
END WITH generates_max_ideal_def assms
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:267:3
|
using assms
by (simp add: generates_max_ideal_def)
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:269
|
RULE generates_max_idealI WITH assms(1)
(*goal: "\<And>F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set. \<lbrakk>F' \<subseteq> P[X::'a set]; ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subset> ideal F'\<rbrakk> \<Longrightarrow> ideal F' = UNIV" *)
INTRO VARS F'
(*fixes variables F' :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" *)
(*introduces facts assm0: "(F'__::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subseteq> P[X::'a set]" and assm1: "ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subset> ideal (F'__::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)" *)
HAVE fact1: "\<not> (F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subseteq> ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)"
END WITH ideal.span_subset_spanI assm1
(*introduces facts local.fact1: "\<not> (F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subseteq> ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)" *)
CONSIDER p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where fact2: "p \<in> F'" and fact3: "p \<notin> ideal F"
(*goal: "\<exists>p::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b. p \<in> (F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<and> p \<notin> ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)" *)
END WITH fact1
(*fixes variables p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" *)
(*introduces facts local.fact2: "(p::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<in> (F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)" and local.fact3: "(p::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<notin> ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)" *)
HAVE fact4: "(p::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<in> P[X::'a set]"
END WITH fact2 assm0
(*introduces facts local.fact4: "(p::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<in> P[X::'a set]" *)
HAVE fact5: "(1::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<in> ideal (insert (p::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set))"
END WITH fact4 fact3 assms(2)
(*introduces facts local.fact5: "(1::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<in> ideal (insert (p::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set))" *)
HAVE fact6: "ideal (insert (p::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) \<subseteq> ideal ((F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<union> F)"
END WITH ideal.span_mono \<open>p \<in> F'\<close>
(*introduces facts local.fact6: "ideal (insert (p::('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) \<subseteq> ideal ((F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<union> F)" *)
HAVE fact7: "ideal ((F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<union> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) = ideal (ideal F' \<union> ideal F)"
END WITH ideal.span_Un ideal.span_span
(*introduces facts local.fact7: "ideal ((F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<union> (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set)) = ideal (ideal F' \<union> ideal F)" *)
HAVE fact8: "ideal (F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<union> ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) = ideal F'"
END WITH assm1
(*introduces facts local.fact8: "ideal (F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<union> ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) = ideal F'" *)
END WITH subsetD ideal_eq_UNIV_iff_contains_one ideal.span_span fact5 fact6 fact7 fact8
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:272:3
|
using assms(1)
proof (rule generates_max_idealI)
fix F'
assume "F' \<subseteq> P[X]" and sub: "ideal F \<subset> ideal F'"
from this(2) ideal.span_subset_spanI have "\<not> F' \<subseteq> ideal F"
by blast
then obtain p where "p \<in> F'" and "p \<notin> ideal F"
by blast
from this(1) \<open>F' \<subseteq> P[X]\<close> have "p \<in> P[X]" ..
hence "1 \<in> ideal (insert p F)"
using \<open>p \<notin> _\<close>
by (rule assms(2))
also have "\<dots> \<subseteq> ideal (F' \<union> F)"
by (rule ideal.span_mono) (simp add: \<open>p \<in> F'\<close>)
also have "\<dots> = ideal (ideal F' \<union> ideal F)"
by (simp add: ideal.span_Un ideal.span_span)
also from sub have "ideal F' \<union> ideal F = ideal F'"
by blast
finally show "ideal F' = UNIV"
by (simp only: ideal_eq_UNIV_iff_contains_one ideal.span_span)
qed
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:286
|
NEXT WITH generates_max_ideal_def assms
(*goal: "\<lbrakk>(F'::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subseteq> P[X::'a set]; ideal (F::(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set) \<subset> ideal F'\<rbrakk> \<Longrightarrow> ideal F' = UNIV" *)
END WITH generates_max_ideal_def assms
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:289:3
|
using assms
by (simp_all add: generates_max_ideal_def)
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:291
|
END WITH generates_max_ideal_def assms
|
./contrib/afp-2025-02-12/thys/Nullstellensatz/Nullstellensatz_Field.thy:294:3
|
using assms
by (auto simp: generates_max_ideal_def)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneGraphIso.thy:10
|
END WITH image_iff
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneGraphIso.thy:11:1
|
by (auto simp: image_iff)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:10
|
END
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:12:1
|
by simp
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListAux.thy:14
|
END
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListAux.thy:15:1
|
by auto
|
./contrib/afp-2025-02-12/thys/Formal_SSA/While_Combinator_Exts.thy:11
|
APPLY (insert assms)
(*goal: "\<lbrakk>while_option (b::'a \<Rightarrow> bool) (c::'a \<Rightarrow> 'a) (s::'a) = None; wf (r::('a \<times> 'a) set); (I::'a \<Rightarrow> bool) s; \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s); \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> r\<rbrakk> \<Longrightarrow> False" *)
APPLY (drule wf_rel_while_option_Some [of r I b c])
(*goal: "\<And>sa::'a. \<lbrakk>while_option (b::'a \<Rightarrow> bool) (c::'a \<Rightarrow> 'a) (s::'a) = None; (I::'a \<Rightarrow> bool) s; \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s); \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> (r::('a \<times> 'a) set); I sa \<and> b sa\<rbrakk> \<Longrightarrow> (c sa, sa) \<in> r" and "\<And>sa::'a. \<lbrakk>while_option (b::'a \<Rightarrow> bool) (c::'a \<Rightarrow> 'a) (s::'a) = None; (I::'a \<Rightarrow> bool) s; \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s); \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> (r::('a \<times> 'a) set); I sa \<and> b sa\<rbrakk> \<Longrightarrow> I (c sa)" and "\<lbrakk>while_option (b::'a \<Rightarrow> bool) (c::'a \<Rightarrow> 'a) (s::'a) = None; (I::'a \<Rightarrow> bool) s; \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s); \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> (r::('a \<times> 'a) set)\<rbrakk> \<Longrightarrow> I (?s7::'a)" and "\<lbrakk>while_option (b::'a \<Rightarrow> bool) (c::'a \<Rightarrow> 'a) (s::'a) = None; (I::'a \<Rightarrow> bool) s; \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s); \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> (r::('a \<times> 'a) set); \<exists>t::'a. while_option b c (?s7::'a) = Some t\<rbrakk> \<Longrightarrow> False" *)
NEXT
(*goal: "\<And>sa::'a. \<lbrakk>while_option (b::'a \<Rightarrow> bool) (c::'a \<Rightarrow> 'a) (s::'a) = None; (I::'a \<Rightarrow> bool) s; \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s); \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> (r::('a \<times> 'a) set); I sa \<and> b sa\<rbrakk> \<Longrightarrow> I (c sa)" *)
NEXT
(*goal: "\<lbrakk>while_option (b::'a \<Rightarrow> bool) (c::'a \<Rightarrow> 'a) (s::'a) = None; (I::'a \<Rightarrow> bool) s; \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s); \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> (r::('a \<times> 'a) set)\<rbrakk> \<Longrightarrow> I (?s7::'a)" *)
NEXT
(*goal: "\<lbrakk>while_option (b::'a \<Rightarrow> bool) (c::'a \<Rightarrow> 'a) (s::'a) = None; (I::'a \<Rightarrow> bool) s; \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> I (c s); \<And>s::'a. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> (r::('a \<times> 'a) set); \<exists>t::'a. while_option b c s = Some t\<rbrakk> \<Longrightarrow> False" *)
END
|
./contrib/afp-2025-02-12/thys/Formal_SSA/While_Combinator_Exts.thy:16:3
|
using assms
by -(drule wf_rel_while_option_Some [of r I b c], auto)
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:14
|
END WITH three_phase_def
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:15:1
|
by (simp add: three_phase_def)
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:12
|
END WITH two_phase_def
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:13:1
|
by (simp add: two_phase_def)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/SimplConv.thy:31
|
NEXT
(*goal: "switch (v::'e \<Rightarrow> 'h) ((a::'h set, b::('e, 'f, 'g) com) # (vs::('h set \<times> ('e, 'f, 'g) com) list)) \<equiv> IF \<acute>v \<in> a THEN
b
ELSE
switch v vs
FI" *)
END
|
./contrib/afp-2025-02-12/thys/AutoCorres2/SimplConv.thy:34:3
|
by auto
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:14
|
END
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:15:1
|
by simp
|
./contrib/afp-2025-02-12/thys/Dependent_SIFUM_Refinement/CompositionalRefinement.thy:10
|
END WITH card_inj_on_le
|
./contrib/afp-2025-02-12/thys/Dependent_SIFUM_Refinement/CompositionalRefinement.thy:12:3
|
by (blast intro: card_inj_on_le)
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Consensus_Misc.thy:17
|
RULE option.expand
(*goal: "\<lbrakk>((option::'a option) = None) = ((option'::'a option) = None); \<And>(x::'a) y::'a. \<lbrakk>option = Some x; option' = Some y\<rbrakk> \<Longrightarrow> x = y\<rbrakk> \<Longrightarrow> (option = None) = (option' = None)" and "\<lbrakk>((option::'a option) = None) = ((option'::'a option) = None); \<And>(x::'a) y::'a. \<lbrakk>option = Some x; option' = Some y\<rbrakk> \<Longrightarrow> x = y; option \<noteq> None; option' \<noteq> None\<rbrakk> \<Longrightarrow> the option = the option'" *)
NEXT
(*goal: "\<lbrakk>((option::'a option) = None) = ((option'::'a option) = None); \<And>(x::'a) y::'a. \<lbrakk>option = Some x; option' = Some y\<rbrakk> \<Longrightarrow> x = y; option \<noteq> None; option' \<noteq> None\<rbrakk> \<Longrightarrow> the option = the option'" *)
END
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Consensus_Misc.thy:20:3
|
by (rule option.expand, auto)
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Trace.thy:14
|
APPLY (coinduction arbitrary: n)
(*goal: "\<And>n::'a. \<forall>n::'a. n \<le> (f::'a \<Rightarrow> 'a) n \<Longrightarrow> \<exists>s::'a stream. siterate f n = s \<and> shd s \<le> shd (stl s) \<and> ((\<exists>n::'a. stl s = siterate f n \<and> (\<forall>n::'a. n \<le> f n)) \<or> ssorted (stl s))" *)
END
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Trace.thy:15:3
|
by (coinduction arbitrary: n) auto
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Quasi_Order.thy:22
|
END WITH subseteq_qle_def in_qle_def
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Quasi_Order.thy:23:1
|
by (auto simp add: subseteq_qle_def in_qle_def)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ScoreProps.thy:10
|
END WITH deleteAround_def
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ScoreProps.thy:11:1
|
by (simp add: deleteAround_def)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:24
|
END WITH ogets_def spec_monad_eqI runs_to_iff(1-31)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:25:3
|
apply (rule spec_monad_eqI)
apply (auto simp add: runs_to_iff ogets_def)
done
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:13
|
INDUCT n arbitrary: x
(*fixes variables x :: nat *)
(*goal: "\<And>x::nat. tabulate (f::nat \<Rightarrow> 'a) x (0::nat) = map f [x..<x + (0::nat)]" and "\<And>(n::nat) x::nat. (\<And>x::nat. tabulate (f::nat \<Rightarrow> 'a) x n = map f [x..<x + n]) \<Longrightarrow> tabulate f x (Suc n) = map f [x..<x + Suc n]" *)
NEXT WITH upt_rec not_le Suc_le_eq
(*fixes variables n :: nat and x :: nat *)
(*introduces facts Suc.IH: "tabulate (f::nat \<Rightarrow> 'a) (?x::nat) (na__::nat) = map f [?x..<?x + na__]" *)
(*goal: "tabulate (f::nat \<Rightarrow> 'a) (x::nat) (Suc (na__::nat)) = map f [x..<x + Suc na__]" *)
END WITH upt_rec not_le Suc_le_eq Suc
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:14:3
|
by (induct n arbitrary: x) (auto simp: not_le Suc_le_eq upt_rec)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Computation/ArchComp.thy:11
|
APPLY (eval)
(*no goal remains!*)
END
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Computation/ArchComp.thy:12:1
|
by eval
|
./contrib/afp-2025-02-12/thys/AutoCorres2/LocalVarExtract.thy:18
|
END WITH set_eqI
|
./contrib/afp-2025-02-12/thys/AutoCorres2/LocalVarExtract.thy:20:3
|
by (fastforce intro: set_eqI)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListSum.thy:18
|
INDUCT V
(*goal: "(\<Sum>\<^bsub>v::'a\<in>[]\<^esub> (0::nat)) = (0::nat)" and "\<And>(a::'a) V::'a list. (\<Sum>\<^bsub>v::'a\<in>V\<^esub> (0::nat)) = (0::nat) \<Longrightarrow> (\<Sum>\<^bsub>v::'a\<in>a # V\<^esub> (0::nat)) = (0::nat)" *)
NEXT
(*fixes variables a :: 'a and V :: "'a list" *)
(*introduces facts Cons.IH: "(\<Sum>\<^bsub>v::'a\<in>Va__::'a list\<^esub> (0::nat)) = (0::nat)" *)
(*goal: "(\<Sum>\<^bsub>v::'a\<in>(a__::'a) # (Va__::'a list)\<^esub> (0::nat)) = (0::nat)" *)
END WITH Cons
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListSum.thy:18:64
|
by (induct V) simp_all
|
./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:5
|
END WITH AB AP BP
|
./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:10:1
|
proof -
from AB AP BP show ?thesis
by blast
qed
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/EnumeratorProps.thy:9
|
INDUCT xs
(*fixes variables x :: 'a *)
(*goal: "\<And>x::'a. |hideDupsRec x []| = |[]|" and "\<And>(a::'a) (xs::'a list) x::'a. (\<And>x::'a. |hideDupsRec x xs| = |xs|) \<Longrightarrow> |hideDupsRec x (a # xs)| = |a # xs|" *)
NEXT
(*fixes variables a :: 'a and xs :: "'a list" and x :: 'a *)
(*introduces facts Cons.IH: "|hideDupsRec (?x::'a) (xsa__::'a list)| = |xsa__|" *)
(*goal: "|hideDupsRec (x::'a) ((a__::'a) # (xsa__::'a list))| = |a__ # xsa__|" *)
END WITH Cons
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/EnumeratorProps.thy:10:1
|
by (induct xs) auto
|
./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/Memcpy.thy:18
|
END WITH intvl_Suc ptr_aligned_def c_null_guard_def c_guard_def
|
./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/Memcpy.thy:20:3
|
unfolding c_guard_def c_null_guard_def ptr_aligned_def
by (clarsimp simp: intvl_Suc)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/L1Peephole.thy:17
|
END WITH L1_seq_def bind_assoc
|
./contrib/afp-2025-02-12/thys/AutoCorres2/L1Peephole.thy:18:3
|
apply (clarsimp simp: L1_seq_def bind_assoc)
done
|
./contrib/afp-2025-02-12/thys/Formal_SSA/RBT_Mapping_Exts.thy:12
|
APPLY (transfer)
(*goal: "\<And>(f::'a \<Rightarrow> 'b) r::('a, unit) RBT.rbt. (Some \<circ> f) |` RBT_Set.Set r = RBT.lookup (RBT.map (\<lambda>(a::'a) _::unit. f a) r)" *)
END WITH restrict_map_def RBT_Set.Set_def
|
./contrib/afp-2025-02-12/thys/Formal_SSA/RBT_Mapping_Exts.thy:14:3
|
by transfer (auto simp: RBT_Set.Set_def restrict_map_def)
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:17
|
END WITH three_step_def
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:18:1
|
by (simp add: three_step_def)
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:15
|
END WITH two_step_def
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:16:1
|
by (simp add: two_step_def)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneGraphIso.thy:36
|
END WITH Iso_def
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneGraphIso.thy:37:1
|
by (simp add:Iso_def)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Worklist.thy:20
|
END WITH worklist_aux_def while_option_unfold
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Worklist.thy:21:1
|
by (simp add: worklist_aux_def while_option_unfold)
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Nat_Mod_Helper.thy:7
|
END WITH add_diff_cancel_left' mod_if mod_mult_div_eq mod_mult_self1_is_0
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Nat_Mod_Helper.thy:9:3
|
by (metis add_diff_cancel_left' mod_if mod_mult_div_eq mod_mult_self1_is_0)
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Trace.thy:17
|
INDUCT i arbitrary: s
(*fixes variables s :: "'a stream" *)
(*introduces facts 0.prems: "ssorted (sa__::'a stream)" *)
(*goal: "\<And>s::'a stream. ssorted s \<Longrightarrow> s !! (0::nat) \<le> stl s !! (0::nat)" and "\<And>(i::nat) s::'a stream. \<lbrakk>\<And>s::'a stream. ssorted s \<Longrightarrow> s !! i \<le> stl s !! i; ssorted s\<rbrakk> \<Longrightarrow> s !! Suc i \<le> stl s !! Suc i" *)
NEXT WITH ssorted.cases "0"
(*fixes variables i :: nat and s :: "'a stream" *)
(*introduces facts Suc.IH: "ssorted (?s::'a stream) \<Longrightarrow> ?s !! (ia__::nat) \<le> stl ?s !! ia__" and Suc.prems: "ssorted (s::'a stream)" *)
(*goal: "(s::'a stream) !! Suc (ia__::nat) \<le> stl s !! Suc ia__" *)
END WITH ssorted.cases Suc
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Trace.thy:18:3
|
by (induct i arbitrary: s) (auto elim: ssorted.cases)
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:16
|
INDUCT n arbitrary: x
(*fixes variables x :: nat *)
(*goal: "\<And>x::nat. length (tabulate (f::nat \<Rightarrow> 'a) x (0::nat)) = (0::nat)" and "\<And>(n::nat) x::nat. (\<And>x::nat. length (tabulate (f::nat \<Rightarrow> 'a) x n) = n) \<Longrightarrow> length (tabulate f x (Suc n)) = Suc n" *)
NEXT
(*fixes variables n :: nat and x :: nat *)
(*introduces facts Suc.IH: "length (tabulate (f::nat \<Rightarrow> 'a) (?x::nat) (na__::nat)) = na__" *)
(*goal: "length (tabulate (f::nat \<Rightarrow> 'a) (x::nat) (Suc (na__::nat))) = Suc na__" *)
END WITH Suc
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:17:3
|
by (induction n arbitrary: x) simp_all
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneProps.thy:11
|
HAVE fact0: "(f::face) \<in> \<F> (g::graph)"
END WITH inv_mgp minGraphProps assms(1,3,4)
(*introduces facts local.fact0: "(f::face) \<in> \<F> (g::graph)" *)
END WITH finalGraph_face assms(2) fact0
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneProps.thy:13:1
|
proof -
from assms(1,3,4) have "f \<in> \<F> g"
by (blast intro: minGraphProps inv_mgp)
with assms(2) show ?thesis
by (rule finalGraph_face)
qed
|
./contrib/afp-2025-02-12/thys/Green/Derivs.thy:5
|
END WITH has_real_derivative_iff_has_vector_derivative
|
./contrib/afp-2025-02-12/thys/Green/Derivs.thy:7:3
|
by (simp add: has_real_derivative_iff_has_vector_derivative)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/LocalVarExtract.thy:22
|
END WITH set_eqI
|
./contrib/afp-2025-02-12/thys/AutoCorres2/LocalVarExtract.thy:24:3
|
by (fastforce intro: set_eqI)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/ExceptionRewrite.thy:16
|
END WITH top_post_state_def always_fail_def
|
./contrib/afp-2025-02-12/thys/AutoCorres2/ExceptionRewrite.thy:17:3
|
by (clarsimp simp: always_fail_def top_post_state_def)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:29
|
END WITH ogets_def spec_monad_eqI runs_to_iff(1-31)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:30:3
|
apply (rule spec_monad_eqI)
apply (auto simp add: runs_to_iff ogets_def)
done
|
./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:14
|
END WITH AB AP BP CP
|
./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:20:1
|
proof -
from AB AP BP CP show ?thesis
by blast
qed
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:17
|
END
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:19:1
|
by simp
|
./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/FactorialTest.thy:23
|
END WITH fact.simps
|
./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/FactorialTest.thy:24:3
|
by (subst fact.simps) simp
|
./contrib/afp-2025-02-12/thys/AutoCorres2/SimplConv.thy:36
|
APPLY (subst signed.less_le)
(*goal: "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word)\<rbrakk> \<Longrightarrow> a \<le>s n \<and> a \<noteq> n" *)
APPLY (safe)
(*goal: "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word)\<rbrakk> \<Longrightarrow> a \<le>s n" *)
APPLY (subst word_sle_msb_le)
(*goal: "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word)\<rbrakk> \<Longrightarrow> (msb n \<longrightarrow> msb a) \<and> (msb a \<and> \<not> msb n \<or> a \<le> n)" *)
APPLY (safe)
(*goal: "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word); msb n\<rbrakk> \<Longrightarrow> msb a" and "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word); \<not> a \<le> n\<rbrakk> \<Longrightarrow> msb a" and "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word); \<not> a \<le> n; msb n\<rbrakk> \<Longrightarrow> False" *)
NEXT WITH not_msb_from_less
(*goal: "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word); \<not> a \<le> n\<rbrakk> \<Longrightarrow> msb a" *)
NEXT
(*goal: "\<lbrakk>(a::'a word) < (n::'a word); n \<le> (2::'a word) ^ (LENGTH('a) - (1::nat)) - (1::'a word); \<not> a \<le> n; msb n\<rbrakk> \<Longrightarrow> False" *)
END
|
./contrib/afp-2025-02-12/thys/AutoCorres2/SimplConv.thy:38:3
|
apply (subst signed.less_le)
apply safe
apply (subst word_sle_msb_le)
apply safe
apply (force simp: not_msb_from_less)
apply simp
apply simp
done
|
./contrib/afp-2025-02-12/thys/Ordered_Resolution_Prover/Lazy_List_Liminf.thy:24
|
END
|
./contrib/afp-2025-02-12/thys/Ordered_Resolution_Prover/Lazy_List_Liminf.thy:25:3
|
by simp
|
./contrib/afp-2025-02-12/thys/LocalLexing/TheoremD10.thy:7
|
END WITH admissible_wellformed_tokens "\<P>_are_admissible"
|
./contrib/afp-2025-02-12/thys/LocalLexing/TheoremD10.thy:8:1
|
using \<P>_are_admissible admissible_wellformed_tokens
by blast
|
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_VCG.thy:8
|
END WITH HT'_def BB_PROTECT_def assms[unfolded "BB_PROTECT_def" "HT'_def"]
|
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_VCG.thy:11:5
|
using assms
unfolding BB_PROTECT_def HT'_def
by auto
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/counting/Count_Util.thy:12
|
END WITH gr0I count_list_0_iff
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/counting/Count_Util.thy:14:3
|
by (meson count_list_0_iff gr0I)
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Rotated_Substring.thy:21
|
NEXT WITH inc_one_bounded_def
(*goal: "\<lbrakk>inc_one_bounded (n::nat) (xs::nat list); (i::nat) < length xs\<rbrakk> \<Longrightarrow> xs ! i < n" *)
END WITH inc_one_bounded_def
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Rotated_Substring.thy:24:3
|
using inc_one_bounded_def
by blast+
|
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_Var_Postprocessor.thy:8
|
END WITH RENAMING_def
|
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_Var_Postprocessor.thy:8:59
|
unfolding RENAMING_def
by auto
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/EnumeratorProps.thy:12
|
CASE_SPLIT xs
(*introduces facts Nil.prem0: "(xs::'a list) = []" *)
(*goal: "(xs::'a list) = [] \<Longrightarrow> |hideDups xs| = |xs|" and "\<And>(a::'a) list::'a list. (xs::'a list) = a # list \<Longrightarrow> |hideDups xs| = |xs|" *)
NEXT WITH Nil
(*fixes variables a :: 'a and list :: "'a list" *)
(*introduces facts Cons.prem0: "(xs::'a list) = (a__::'a) # (list__::'a list)" *)
(*goal: "|hideDups (xs::'a list)| = |xs|" *)
END WITH Cons
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/EnumeratorProps.thy:13:1
|
by (cases xs) simp_all
|
./contrib/afp-2025-02-12/thys/AutoCorres2/L1Peephole.thy:21
|
NEXT WITH L1_skip_def L1_seq_def
(*goal: "L1_seq L1_skip (A::(unit option, unit, 'a) spec_monad) = A" *)
END WITH L1_skip_def L1_seq_def
|
./contrib/afp-2025-02-12/thys/AutoCorres2/L1Peephole.thy:24:3
|
apply (clarsimp simp: L1_seq_def L1_skip_def)+
done
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:20
|
END WITH three_step_def three_phase_def
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/MRU/Three_Steps.thy:21:3
|
by (auto simp add: three_phase_def three_step_def)
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:18
|
END WITH two_step_def two_phase_def
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Two_Steps.thy:19:3
|
by (auto simp add: two_phase_def two_step_def)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/Split_Heap.thy:20
|
END WITH array_fields_def
|
./contrib/afp-2025-02-12/thys/AutoCorres2/Split_Heap.thy:21:3
|
by (auto simp add: array_fields_def)
|
./contrib/afp-2025-02-12/thys/MDP-Rewards/Blinfun_Util.thy:11
|
NEXT WITH blinfun_eqI
(*goal: "(f::'a \<Rightarrow>\<^sub>L 'b) o\<^sub>L id_blinfun = f" *)
END WITH blinfun_eqI
|
./contrib/afp-2025-02-12/thys/MDP-Rewards/Blinfun_Util.thy:14:3
|
by (auto intro!: blinfun_eqI)
|
./contrib/afp-2025-02-12/thys/Bernoulli/Bernoulli.thy:14
|
END WITH power_Suc numeral_eq_Suc
|
./contrib/afp-2025-02-12/thys/Bernoulli/Bernoulli.thy:15:3
|
by (simp only: numeral_eq_Suc power_Suc)
|
./contrib/afp-2025-02-12/thys/Ordered_Resolution_Prover/Map2.thy:22
|
END WITH zip_Nil zip_Cons_Cons zip.simps list.simps Nil_is_map_conv
|
./contrib/afp-2025-02-12/thys/Ordered_Resolution_Prover/Map2.thy:23:3
|
by (metis Nil_is_map_conv list.exhaust list.simps(3) zip.simps(1) zip_Cons_Cons zip_Nil)
|
./contrib/afp-2025-02-12/thys/MDP-Rewards/MDP_reward_Util.thy:8
|
END WITH mult.commute assms
|
./contrib/afp-2025-02-12/thys/MDP-Rewards/MDP_reward_Util.thy:12:3
|
using assms
by (auto simp: mult.commute)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/AbstractArrays.thy:26
|
CASE_SPLIT n
(*introduces facts 0.prem0: "(n::nat) = (0::nat)" *)
(*goal: "(n::nat) = (0::nat) \<Longrightarrow> ((x::'a ptr) \<in> set (array_addrs x n)) = ((0::nat) < n)" and "\<And>nat::nat. (n::nat) = Suc nat \<Longrightarrow> ((x::'a ptr) \<in> set (array_addrs x n)) = ((0::nat) < n)" *)
NEXT WITH array_addrs.simps(2) "0"
(*fixes variables nat :: nat *)
(*introduces facts Suc.prem0: "(n::nat) = Suc (nat__::nat)" *)
(*goal: "((x::'a ptr) \<in> set (array_addrs x (n::nat))) = ((0::nat) < n)" *)
END WITH array_addrs.simps(2) Suc
|
./contrib/afp-2025-02-12/thys/AutoCorres2/AbstractArrays.thy:28:3
|
by (cases n, auto simp: array_addrs.simps(2))
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/FaceDivisionProps.thy:13
|
INDUCT g
(*fixes variables x1a :: "face list" and x2a :: nat and x3a :: "face list list" and x4a :: "nat list" *)
(*goal: "\<And>(x1a::face list) (x2a::nat) (x3a::face list list) x4a::nat list. vertices (makeFaceFinal (f::face) (Graph x1a x2a x3a x4a)) = vertices (Graph x1a x2a x3a x4a)" *)
END WITH makeFaceFinal_def vertices_graph_def
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/FaceDivisionProps.thy:14:1
|
by (induct g)(simp add:vertices_graph_def makeFaceFinal_def)
|
./contrib/afp-2025-02-12/thys/Amicable_Numbers/Amicable_Numbers.thy:18
|
END WITH assms
|
./contrib/afp-2025-02-12/thys/Amicable_Numbers/Amicable_Numbers.thy:22:3
|
using assms
by linarith
|
./contrib/afp-2025-02-12/thys/HOL-CSP/Stop.thy:56
|
END WITH FAILURES_def Failures.rep_eq STOP.rep_eq
|
./contrib/afp-2025-02-12/thys/HOL-CSP/Stop.thy:57:3
|
by (simp add: FAILURES_def Failures.rep_eq STOP.rep_eq)
|
./contrib/afp-2025-02-12/thys/Formal_SSA/Serial_Rel.thy:21
|
INDUCT m
(*goal: "iterated_serial_on (A::'a set) (r::('a \<times> 'a) set) (x::'a) ((n::nat) + (0::nat)) = iterated_serial_on A r (iterated_serial_on A r x n) (0::nat)" and "\<And>m::nat. iterated_serial_on (A::'a set) (r::('a \<times> 'a) set) (x::'a) ((n::nat) + m) = iterated_serial_on A r (iterated_serial_on A r x n) m \<Longrightarrow> iterated_serial_on A r x (n + Suc m) = iterated_serial_on A r (iterated_serial_on A r x n) (Suc m)" *)
NEXT
(*fixes variables m :: nat *)
(*introduces facts Suc.IH: "iterated_serial_on (A::'a set) (r::('a \<times> 'a) set) (x::'a) ((n::nat) + (ma__::nat)) = iterated_serial_on A r (iterated_serial_on A r x n) ma__" *)
(*goal: "iterated_serial_on (A::'a set) (r::('a \<times> 'a) set) (x::'a) ((n::nat) + Suc (ma__::nat)) = iterated_serial_on A r (iterated_serial_on A r x n) (Suc ma__)" *)
END WITH Suc
|
./contrib/afp-2025-02-12/thys/Formal_SSA/Serial_Rel.thy:22:1
|
by (induction m) auto
|
./contrib/afp-2025-02-12/thys/Combinatorics_Words_Graph_Lemma/Glued_Codes.thy:16
|
END WITH last_append if_split
|
./contrib/afp-2025-02-12/thys/Combinatorics_Words_Graph_Lemma/Glued_Codes.thy:18:3
|
by (auto simp only: last_append split: if_split)
|
./contrib/afp-2025-02-12/thys/Euler_Partition/Euler_Partition.thy:17
|
END WITH assms nonzero_mult_div_cancel_right le_add_diff_inverse2 less_not_refl2 power_add power_not_zero
|
./contrib/afp-2025-02-12/thys/Euler_Partition/Euler_Partition.thy:21:1
|
by (metis assms nonzero_mult_div_cancel_right le_add_diff_inverse2 less_not_refl2 power_add power_not_zero)
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Nat_Mod_Helper.thy:11
|
END
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/util/Nat_Mod_Helper.thy:13:3
|
using order_le_less_trans
by blast
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:19
|
INDUCT n arbitrary: x
(*fixes variables x :: nat *)
(*goal: "\<And>x::nat. map (f::'b \<Rightarrow> 'a) (tabulate (g::nat \<Rightarrow> 'b) x (0::nat)) = tabulate (\<lambda>x::nat. f (g x)) x (0::nat)" and "\<And>(n::nat) x::nat. (\<And>x::nat. map (f::'b \<Rightarrow> 'a) (tabulate (g::nat \<Rightarrow> 'b) x n) = tabulate (\<lambda>x::nat. f (g x)) x n) \<Longrightarrow> map f (tabulate g x (Suc n)) = tabulate (\<lambda>x::nat. f (g x)) x (Suc n)" *)
NEXT
(*fixes variables n :: nat and x :: nat *)
(*introduces facts Suc.IH: "map (f::'b \<Rightarrow> 'a) (tabulate (g::nat \<Rightarrow> 'b) (?x::nat) (na__::nat)) = tabulate (\<lambda>x::nat. f (g x)) ?x na__" *)
(*goal: "map (f::'b \<Rightarrow> 'a) (tabulate (g::nat \<Rightarrow> 'b) (x::nat) (Suc (na__::nat))) = tabulate (\<lambda>x::nat. f (g x)) x (Suc na__)" *)
END WITH Suc
|
./contrib/afp-2025-02-12/thys/MFOTL_Monitor/Table.thy:20:3
|
by (induction n arbitrary: x) simp_all
|
./contrib/afp-2025-02-12/thys/Green/DiamExample.thy:7
|
END WITH dual_order.order_iff_strict abs_if
|
./contrib/afp-2025-02-12/thys/Green/DiamExample.thy:10:3
|
by (simp add: abs_if dual_order.order_iff_strict)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/Memcpy.thy:24
|
CASE_SPLIT x
(*fixes variables x :: "32 word" *)
(*introduces facts Ptr.prem0: "(x::'b ptr) = PTR('b) (xa__::32 word)" *)
(*goal: "\<And>xa::32 word. (x::'b ptr) = PTR('b) xa \<Longrightarrow> ptr_val (PTR_COERCE('b \<rightarrow> 'a) x +\<^sub>p (a::int)) = ptr_val x + word_of_int a * of_nat_addr (size_of TYPE('a))" *)
END WITH CTypesDefs.ptr_add_def Ptr
|
./contrib/afp-2025-02-12/thys/AutoCorres2/tests/examples/Memcpy.thy:25:3
|
apply (case_tac x)
apply (clarsimp simp: CTypesDefs.ptr_add_def)
done
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Worklist.thy:23
|
END WITH worklist_aux_def while_option_unfold
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Worklist.thy:25:1
|
by (simp add: worklist_aux_def while_option_unfold)
|
./contrib/afp-2025-02-12/thys/CoreC++/Auxiliary.thy:19
|
END
|
./contrib/afp-2025-02-12/thys/CoreC++/Auxiliary.thy:21:2
|
by arith
|
./contrib/afp-2025-02-12/thys/AutoCorres2/L2Peephole.thy:17
|
END
|
./contrib/afp-2025-02-12/thys/AutoCorres2/L2Peephole.thy:18:3
|
by simp
|
./contrib/afp-2025-02-12/thys/Formal_SSA/SSA_Transfer_Rules.thy:16
|
END WITH right_totalE rel_funE
|
./contrib/afp-2025-02-12/thys/Formal_SSA/SSA_Transfer_Rules.thy:17:3
|
by (auto 4 4 elim: right_totalE rel_funE)
|
./contrib/afp-2025-02-12/thys/CoreC++/WWellForm.thy:16
|
END WITH wwf_mdecl_def
|
./contrib/afp-2025-02-12/thys/CoreC++/WWellForm.thy:19:1
|
by (simp add:wwf_mdecl_def)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListSum.thy:20
|
INDUCT xs
(*goal: "ListSum (filter (\<lambda>x::'a. \<not> (P::'a \<Rightarrow> bool) x) []) (f::'a \<Rightarrow> nat) + ListSum (filter P []) f = ListSum [] f" and "\<And>(a::'a) xs::'a list. ListSum (filter (\<lambda>x::'a. \<not> (P::'a \<Rightarrow> bool) x) xs) (f::'a \<Rightarrow> nat) + ListSum (filter P xs) f = ListSum xs f \<Longrightarrow> ListSum (filter (\<lambda>x::'a. \<not> P x) (a # xs)) f + ListSum (filter P (a # xs)) f = ListSum (a # xs) f" *)
NEXT
(*fixes variables a :: 'a and xs :: "'a list" *)
(*introduces facts Cons.IH: "ListSum (filter (\<lambda>x::'a. \<not> (P::'a \<Rightarrow> bool) x) (xsa__::'a list)) (f::'a \<Rightarrow> nat) + ListSum (filter P xsa__) f = ListSum xsa__ f" *)
(*goal: "ListSum (filter (\<lambda>x::'a. \<not> (P::'a \<Rightarrow> bool) x) ((a__::'a) # (xsa__::'a list))) (f::'a \<Rightarrow> nat) + ListSum (filter P (a__ # xsa__)) f = ListSum (a__ # xsa__) f" *)
END WITH Cons
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/ListSum.thy:22:2
|
by (induct xs) simp_all
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Voting/OneThirdRule_Proofs.thy:13
|
HAVE fact0: "((HOL.eq::nat \<Rightarrow> nat \<Rightarrow> bool) ((Groups.plus_class.plus::nat \<Rightarrow> nat \<Rightarrow> nat) ((Finite_Set.card::'a set \<Rightarrow> nat) (S::'a set)) ((Finite_Set.card::'a set \<Rightarrow> nat) ((Groups.uminus_class.uminus::'a set \<Rightarrow> 'a set) (S::'a set)))) ((Finite_Set.card::'a set \<Rightarrow> nat) (Orderings.top_class.top::'a set)))"
RULE card_Un_disjoint[of S "-S", simplified Compl_partition, symmetric]
(*goal: "finite (S::'a set)" and "finite (- (S::'a set))" and "(S::'a set) \<inter> - S = {}" *)
NEXT
(*goal: "finite (- (S::'a set))" *)
NEXT
(*goal: "(S::'a set) \<inter> - S = {}" *)
END
(*introduces facts local.fact0: "card (S::'a set) + card (- S) = card UNIV" *)
END WITH fact0
|
./contrib/afp-2025-02-12/thys/Consensus_Refined/Voting/OneThirdRule_Proofs.thy:16:1
|
proof-
have "card S + card (-S) = card (UNIV :: 'a set)"
by (rule card_Un_disjoint[of S "-S", simplified Compl_partition, symmetric]) (auto)
thus ?thesis
by simp
qed
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneProps.thy:18
|
END WITH finalVertex_def nonFinals_def filter_empty_conv plane_final_facesAt
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/PlaneProps.thy:20:1
|
by (auto simp add: finalVertex_def nonFinals_def filter_empty_conv plane_final_facesAt)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/c-parser/CLanguage.thy:18
|
END WITH addr_card_def card_word
|
./contrib/afp-2025-02-12/thys/AutoCorres2/c-parser/CLanguage.thy:19:3
|
by (simp add: addr_card_def card_word)
|
./contrib/afp-2025-02-12/thys/IMP2/IMP2.thy:14
|
END
|
./contrib/afp-2025-02-12/thys/IMP2/IMP2.thy:14:63
|
by auto
|
./contrib/afp-2025-02-12/thys/Case_Labeling/Examples/Hoare/Labeled_Hoare_Examples.thy:12
|
APPLY (vcg_simp)
(*no goal remains!*)
END
|
./contrib/afp-2025-02-12/thys/Case_Labeling/Examples/Hoare/Labeled_Hoare_Examples.thy:19:1
|
by vcg_simp
|
./contrib/afp-2025-02-12/thys/AutoCorres2/lib/Mutual_CCPO_Recursion.thy:18
|
END WITH image_def fun_lub_def fun_eq_iff Inf_apply
|
./contrib/afp-2025-02-12/thys/AutoCorres2/lib/Mutual_CCPO_Recursion.thy:19:3
|
unfolding fun_lub_def fun_eq_iff Inf_apply image_def
by simp
|
./contrib/afp-2025-02-12/thys/AutoCorres2/AutoCorres.thy:33
|
END WITH o_def
|
./contrib/afp-2025-02-12/thys/AutoCorres2/AutoCorres.thy:34:3
|
by (simp add: o_def)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Computation/ArchComp.thy:14
|
APPLY (eval)
(*no goal remains!*)
END
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/Computation/ArchComp.thy:15:1
|
by eval
|
./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:24
|
END WITH AB AP BP CP DP
|
./contrib/afp-2025-02-12/thys/LocalLexing/InductRules.thy:31:1
|
proof -
from AB AP BP CP DP show ?thesis
by blast
qed
|
./contrib/afp-2025-02-12/thys/ROBDD/BDD_Code.thy:8
|
END WITH blit'_def
|
./contrib/afp-2025-02-12/thys/ROBDD/BDD_Code.thy:11:33
|
by (simp add: blit'_def)
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/TameProps.thy:10
|
INDUCT xs
(*introduces facts Nil.prems: "\<forall>x::'a\<in>set []. \<not> ((P::'a \<Rightarrow> bool) x \<and> (Q::'a \<Rightarrow> bool) x)" *)
(*goal: "\<forall>x::'a\<in>set []. \<not> ((P::'a \<Rightarrow> bool) x \<and> (Q::'a \<Rightarrow> bool) x) \<Longrightarrow> |filter P []| + |filter Q []| \<le> |[]|" and "\<And>(a::'a) xs::'a list. \<lbrakk>\<forall>x::'a\<in>set xs. \<not> ((P::'a \<Rightarrow> bool) x \<and> (Q::'a \<Rightarrow> bool) x) \<Longrightarrow> |filter P xs| + |filter Q xs| \<le> |xs|; \<forall>x::'a\<in>set (a # xs). \<not> (P x \<and> Q x)\<rbrakk> \<Longrightarrow> |filter P (a # xs)| + |filter Q (a # xs)| \<le> |a # xs|" *)
NEXT WITH Nil
(*fixes variables a :: 'a and xs :: "'a list" *)
(*introduces facts Cons.IH: "\<forall>x::'a\<in>set (xsa__::'a list). \<not> ((P::'a \<Rightarrow> bool) x \<and> (Q::'a \<Rightarrow> bool) x) \<Longrightarrow> |filter P xsa__| + |filter Q xsa__| \<le> |xsa__|" and Cons.prems: "\<forall>x::'a\<in>set ((a__::'a) # (xsa__::'a list)). \<not> ((P::'a \<Rightarrow> bool) x \<and> (Q::'a \<Rightarrow> bool) x)" *)
(*goal: "|filter (P::'a \<Rightarrow> bool) ((a__::'a) # (xsa__::'a list))| + |filter (Q::'a \<Rightarrow> bool) (a__ # xsa__)| \<le> |a__ # xsa__|" *)
END WITH Cons
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/TameProps.thy:12:1
|
by (induct xs) auto
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/counting/Rank_Select.thy:15
|
END WITH rank_def count_list_eq_count
|
./contrib/afp-2025-02-12/thys/BurrowsWheeler/counting/Rank_Select.thy:17:3
|
by (simp add: count_list_eq_count rank_def)
|
./contrib/afp-2025-02-12/thys/Directed_Sets/Well_Order_Connection.thy:11
|
END WITH refl_on_def reflexive_def relation_of_def
|
./contrib/afp-2025-02-12/thys/Directed_Sets/Well_Order_Connection.thy:12:3
|
by (auto simp: refl_on_def reflexive_def relation_of_def)
|
./contrib/afp-2025-02-12/thys/LocalLexing/Limit.thy:9
|
INDUCT n
(*introduces facts 0.prems: "setmonotone (f::'a set \<Rightarrow> 'a set)" *)
(*goal: "setmonotone (f::'a set \<Rightarrow> 'a set) \<Longrightarrow> setmonotone (funpower f (0::nat))" and "\<And>n::nat. \<lbrakk>setmonotone (f::'a set \<Rightarrow> 'a set) \<Longrightarrow> setmonotone (funpower f n); setmonotone f\<rbrakk> \<Longrightarrow> setmonotone (funpower f (Suc n))" *)
NEXT WITH setmonotone_def "0"
(*fixes variables n :: nat *)
(*introduces facts Suc.IH: "setmonotone (f::'a set \<Rightarrow> 'a set) \<Longrightarrow> setmonotone (funpower f (na__::nat))" and Suc.prems: "setmonotone (f::'a set \<Rightarrow> 'a set)" *)
(*goal: "setmonotone (funpower (f::'a set \<Rightarrow> 'a set) (Suc (na__::nat)))" *)
END WITH setmonotone_def Suc
|
./contrib/afp-2025-02-12/thys/LocalLexing/Limit.thy:10:3
|
by (induct n, auto simp add: setmonotone_def)
|
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_VCG.thy:14
|
END WITH HT'_partial_def BB_PROTECT_def assms[unfolded "BB_PROTECT_def" "HT'_partial_def"]
|
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_VCG.thy:17:5
|
using assms
unfolding BB_PROTECT_def HT'_partial_def
by auto
|
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_Var_Postprocessor.thy:9
|
END WITH assms[unfolded "RENAMING_def"]
|
./contrib/afp-2025-02-12/thys/IMP2/automation/IMP2_Var_Postprocessor.thy:11:15
|
using assms
unfolding RENAMING_def
by simp
|
./contrib/afp-2025-02-12/thys/MonoBoolTranAlgebra/Mono_Bool_Tran.thy:28
|
END WITH image_comp fun_eq_iff
|
./contrib/afp-2025-02-12/thys/MonoBoolTranAlgebra/Mono_Bool_Tran.thy:30:3
|
by (simp add: fun_eq_iff image_comp)
|
./contrib/afp-2025-02-12/thys/CoreC++/CWellForm.thy:20
|
END WITH wf_C_mdecl_def
|
./contrib/afp-2025-02-12/thys/CoreC++/CWellForm.thy:27:1
|
by (simp add:wf_C_mdecl_def)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/c-parser/umm_heap/SepTactic.thy:21
|
END WITH sep_conj_extract_def
|
./contrib/afp-2025-02-12/thys/AutoCorres2/c-parser/umm_heap/SepTactic.thy:23:3
|
by (simp add: sep_conj_extract_def)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:34
|
END WITH ogets_def spec_monad_eqI runs_to_iff(1-31)
|
./contrib/afp-2025-02-12/thys/AutoCorres2/TypeStrengthen.thy:35:3
|
apply (simp add: gets_the_def assert_opt_def[abs_def] ogets_def gets_def)
apply (rule spec_monad_eqI)
apply (auto simp add: runs_to_iff)
done
|
./contrib/afp-2025-02-12/thys/AutoCorres2/ExceptionRewrite.thy:19
|
END WITH always_fail_def run_bind
|
./contrib/afp-2025-02-12/thys/AutoCorres2/ExceptionRewrite.thy:20:3
|
apply (clarsimp simp: always_fail_def run_bind)
done
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:21
|
END
|
./contrib/afp-2025-02-12/thys/Flyspeck-Tame/LowerBound.thy:22:1
|
by simp
|
./contrib/afp-2025-02-12/thys/Affine_Arithmetic/Counterclockwise_Vector.thy:13
|
END WITH ac_simps(1-48)
|
./contrib/afp-2025-02-12/thys/Affine_Arithmetic/Counterclockwise_Vector.thy:15:3
|
by (auto simp: ac_simps)
|
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